{"group":{"id":1,"name":"Community","lockable":false,"created_at":"2012-01-18T18:02:15.000Z","updated_at":"2026-04-16T00:12:35.000Z","description":"Problems submitted by members of the MATLAB Central community.","is_default":true,"created_by":161519,"badge_id":null,"featured":false,"trending":false,"solution_count_in_trending_period":0,"trending_last_calculated":"2026-04-16T00:00:00.000Z","image_id":null,"published":true,"community_created":false,"status_id":2,"is_default_group_for_player":false,"deleted_by":null,"deleted_at":null,"restored_by":null,"restored_at":null,"description_opc":null,"description_html":null,"published_at":null},"problems":[{"id":58478,"title":"Optimal saving in Solow's classical growth model","description":"Let us consider a simplified version of Solow's classical growth model. Let , , ,   and  denote production, the capital stock, labor, (gross) investment, savings and consumption at time  respectively (all variables are in real rather than nominal terms), and assume that output is produced using a neoclassical production function using capital and labor as inputs, , satisfying the following conditions:\r\nThe marginal product of capital and labor is positive: , and .\r\nThe marginal product of capital and labor is diminishing: , and .\r\nProduction exhibits constant returns to scale:  is homogenous of degree one, i.e.  for all .\r\n satisfies the Inada conditions: , and .\r\nCapital in the economy accumulates according to the law of motion , where  is the rate of depreciation; investment equals savings, which are assumed to be a constant fraction of output,  for all , for some . Output that is not saved is consumed (in other words, we assume a closed economy with no government activity), so that  for all .\r\nAssume that the population and hence the labor force is constant (this kind of defeats the purpose of a growth model, but we are considering a simplified version only). It is helpful to recast the model in per-capita (technically, per-laborer) terms by dividing by  throughout and taking advantage of the fact that  is homogenous of degree one. We use lower-case letters for per-capita terms:  is the capital intensity,  is output per capita, and so on. We also write ;  is the intensive form of the production function .\r\nThe model economy is in its steady state when the per-capita variables do not change; denote the steady-state capital intensity by . An expression implicitly characterizing  can be derived from the law of motion for capital by moving to per-capita variables and replacing  and  with  throughout.\r\nSince in the steady state,  is constant, so is output per capita  and hence consumption per capita .  depends on three things: the depreciation rate , the savings rate , and the macroeconomic production function  (equivalently, ). A social planner seeking to maximize steady-state per-capita consumption may not be able to change  or , but can maximize  by influencing . We will call the savings rate that maximizes per-capita consumption the golden rule savings rate and denote it ; similarly, we will denote steady-state values for ,  etc. implied by  as ,  and so forth.\r\nTo find , we proceed as follows:\r\nfind an expression for  by using the relationship , moving to per-capita terms, and using the expression characterizing  to replace the term  with ;\r\ntake the derivative w.r.t. , keeping in mind that  depends on ;\r\nset the resulting expression to zero, obtaining an equality identifying  with the marginal product of capital, in per-capita terms, when the economy follows the golden rule;\r\nsubstitute this expression back into the expression characterizing  and solving for .\r\n can then be found by again considering the relationship  in per-capita terms in the steady state, with .\r\nYour task is now simple (in principle): assume that macroeconomic production follows a Cobb-Douglas relationship, ,  (you may verify that this satisfies the conditions listed above). For given values of the (constant) technology parameter , the capital elasticity of output  and the depreciation rate , please compute the golden rule savings rate , and the resulting steady-state capital intensity , per-capita output  and per-capita consumption .","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 965.688px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 482.844px; transform-origin: 407px 482.844px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 84.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42.25px; text-align: left; transform-origin: 384px 42.25px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eLet us consider a simplified version of Solow's classical growth model. Let \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14.5\" height=\"20\" style=\"width: 14.5px; height: 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margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg 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style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e denote production, the capital stock, labor, (gross) investment, savings and consumption at time \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003et\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e respectively (all variables are in real rather than nominal terms), and assume that output is produced using a neoclassical production function using capital and labor as inputs, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"81.5\" height=\"19\" style=\"width: 81.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, satisfying the following conditions:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003col style=\"block-size: 143.438px; font-family: Helvetica, Arial, sans-serif; list-style-type: decimal; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 391px 71.7188px; transform-origin: 391px 71.7188px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 35px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 17.5px; text-align: left; transform-origin: 363px 17.5px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eThe marginal product of capital and labor is positive: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-15px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"83\" height=\"35\" style=\"width: 83px; height: 35px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-15px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"81\" height=\"35\" style=\"width: 81px; height: 35px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 37px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 18.5px; text-align: left; transform-origin: 363px 18.5px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eThe marginal product of capital and labor is diminishing: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-16px\"\u003e\u003cimg 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width=\"95\" height=\"37\" style=\"width: 95px; height: 37px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-16px\"\u003e\u003cimg 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width=\"91.5\" height=\"37\" style=\"width: 91.5px; height: 37px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 40.9375px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 20.4688px; text-align: left; transform-origin: 363px 20.4688px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eProduction exhibits constant returns to scale: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eF\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e is homogenous of degree one, i.e. \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"141\" height=\"19\" style=\"width: 141px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e for all \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEgAAAAkCAYAAAAq23xmAAAAAXNSR0IArs4c6QAAA6NJREFUaEPtmGmoTVEYhp9rTEjIDwo/JEmh/KBMUaYUKTLLlCF+GCJCMmZWpsiUKWNJKRFKhpIMZS4lFEkIGRN669u3fdbd5949nHtuae26dc89a6/1rWd963vf75bwfz91gHXATGAKsAf4G9pyX+AUcAiYD3x1cZT833wYbJt/BIwEnjv7bQjsBYYAY4AjVQloBNAc2Ae8L8LB1AW2AeOBXcBs4LuzrhJkEbACOA1MAj6GxxQzg0YDh4EPwGY7uTeVCKq9XZ/WwBxbM2q5QcAZ4AswALheVYBqAL2AJUB3A7Ub2AG8rARQwcY1tbL3eJ41ugLX7Ltplm2lQ+NkUDWgIzAUaAE8s89vgZXAq4Sbc0Hp5HQFthYY1EJgtcXWzc2MUMxtDF4Hy7IFwK/g+4oAqWYIQitgFnDbVKAPcAE4CUx1721MYFGgDliQKqZhtYk5Zc7BLwX0oycuoDK1qjxAnYDttsAE4LH9XhNYBcwzYEpfZVXaR6B62nz9rBZIetcDT1KCkryrzunwkgA6D6hWlopIPkBtgf12pSR/l53dC9444BygSbOcdjC1C0p/PwisSQGqlr0n5UoCSAczGfhU3hULm6stZqB+pk2PFO9FgToKbADuAX9izikxWJ4QUKwr1gM4BtQzA3UpZkCFHtYM2AQMt4lf2GGdiLmQrtfOhICkqHOBH/kyKGycbgCjAAVWzKeJ1Q5dj0bAU1M4tQOfEwQSV+a7mODUj/JLbg3SIBXmsWbRZ1jRTBBX6qFNzckGYO4DG83slemRYqwSlu+4RlEtR86NcQE1tn5EapLPnseILdEQF8wtU7CzEa1BkonDrYYULcffhCYK/NJFE54cd+8CamAdr0yhUjpfBskfqWVIc7JBbFFglgEKtFCiEHTr8m+S79cO4fB+5fMkSjmK7AKqbuZKCiDfMwx4GJpU4/sDMooakwaQC+YqsLbAYIKQw4oc1a0HAHUoKurv3BSN8kGBB+oMSF4XW6FuCUw3Gy5vohYhyaPiq1NSv6PiK/8kM3gF+J1kooRjta7ibWce54G9r/2o3qqtUN2LFKN8RlESOxEYCKjK3wV00pL/mwm8SHgvQTevfytIvqWScT1NQiZlhte2RlkxfLPapkyWMSy31lXUi2UNLPx+b5PpO0UEkzn+YgLKHGxVTOABVUDdA/KAsl1Mn0E+g3wGZSPgMygbP1+DfAb5DMpGwGdQNn6+BvkM8hmUjYDPoGz8fA2qgN8/iFPVJd5S/rIAAAAASUVORK5CYII=\" width=\"36\" height=\"18\" style=\"width: 36px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 30.5px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 15.25px; text-align: left; transform-origin: 363px 15.25px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eF\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e satisfies the Inada conditions: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-15px\"\u003e\u003cimg 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margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eCapital in the economy accumulates according to the law of motion \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg 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width=\"63\" height=\"18\" style=\"width: 63px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e is the rate of depreciation; investment equals savings, which are assumed to be a constant fraction of output, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" 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style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"61\" height=\"18\" style=\"width: 61px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e. Output that is not saved is consumed (in other words, we assume a closed economy with no government activity), so that \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"71\" height=\"20\" style=\"width: 71px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e for all \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003et\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 124.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 62.25px; text-align: left; transform-origin: 384px 62.25px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eAssume that the population and hence the labor force is constant (this kind of defeats the purpose of a growth model, but we \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eare\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e considering a simplified version only). It is helpful to recast the model in per-capita (technically, per-laborer) terms by dividing by \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14.5\" height=\"20\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e throughout and taking advantage of the fact that \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eF\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e is homogenous of degree one. We use lower-case letters for per-capita terms: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"44.5\" height=\"40\" style=\"width: 44.5px; height: 40px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e is the capital intensity, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"43\" height=\"40\" style=\"width: 43px; height: 40px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e is output per capita, and so on. We also write \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"99\" height=\"20\" style=\"width: 99px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e; \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ef\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e is the intensive form of the production function \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eF\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.75px; text-align: left; transform-origin: 384px 31.75px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eThe model economy is in its steady state when the per-capita variables do not change; denote the steady-state capital intensity by \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15\" height=\"19\" style=\"width: 15px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e. An expression implicitly characterizing \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15\" height=\"19\" style=\"width: 15px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e can be derived from the law of motion for capital by moving to per-capita variables and replacing \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"25.5\" height=\"20\" style=\"width: 25.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"13\" height=\"20\" style=\"width: 13px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e with \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15\" height=\"19\" style=\"width: 15px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e throughout.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 105.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 52.75px; text-align: left; transform-origin: 384px 52.75px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eSince in the steady state, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15\" height=\"19\" style=\"width: 15px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e is constant, so is output per capita \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"16\" height=\"19\" style=\"width: 16px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and hence consumption per capita \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15\" height=\"19\" style=\"width: 15px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAB4AAAAmCAYAAADTGStiAAAAAXNSR0IArs4c6QAAAf1JREFUWEft1UuoTWEYBuDnEKWIYqQcuUyUgTJwGaBToiiEklBM5FYkJo4oREZSBkoZuBaJkbuSKAMMkIkBmTBwSxLl0l/frm3lnPMva532ZO9atVd9//t+3/t/77s6tOjX0SJe/UE8AJPxLp5/ztYfxBNxHlewrydF6yIegUW4i484gYu4jHl4j/vNTdRFPCWm/IQtQXYLs3EIe3AAPxvkdREnvMFIDSzGTLzCpZj0A36XnTgtSwJchk68jPe0PPvxJgAbxEsxA89wDfdC/r+uu6+JxwT4BGzFo+h8Lm7gAtZjOE5jFDZiDm6iC3uxHUfwK0fqqTgWhWvxIv4PivvaEY2sCNssxGO8jeVKi5Wknh8qPc2RehJOhrSrcKdgi9TUGlzF9cL9jcfZsNPBMnYagsPYjKPYie8lEi5d3zh8DhtlB8issMZQLMHtEqTZpcXlSu+7InEeYCVeZ6OVKCwSD4uFWo1T2IQvJfCyS4vEI3Emkuc4tuFbNlqJwiJx8mPK2RQWvU2c/J3S6GsJrl4DZGDk6u7w7XI8bzqRGk2+TAGSamojThwND0/DOXTHgo3FBvyI4K909z1F5miswwJMx5PI3PSdfdgcfXVJ/b84pc/19ZEoDZh7oE2cq1TlurbUlSXMBWhLnatU5bq21JUlzAVoS52rVOW6lkn9B7+JXifHusByAAAAAElFTkSuQmCC\" width=\"15\" height=\"19\" style=\"width: 15px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e depends on three things: the depreciation rate \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eδ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, the savings rate \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003es\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, and the macroeconomic production function \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eF\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e (equivalently, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ef\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e). A social planner seeking to maximize steady-state per-capita consumption may not be able to change \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eδ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e or \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eF\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, but can maximize \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAB4AAAAmCAYAAADTGStiAAAAAXNSR0IArs4c6QAAAf1JREFUWEft1UuoTWEYBuDnEKWIYqQcuUyUgTJwGaBToiiEklBM5FYkJo4oREZSBkoZuBaJkbuSKAMMkIkBmTBwSxLl0l/frm3lnPMva532ZO9atVd9//t+3/t/77s6tOjX0SJe/UE8AJPxLp5/ztYfxBNxHlewrydF6yIegUW4i484gYu4jHl4j/vNTdRFPCWm/IQtQXYLs3EIe3AAPxvkdREnvMFIDSzGTLzCpZj0A36XnTgtSwJchk68jPe0PPvxJgAbxEsxA89wDfdC/r+uu6+JxwT4BGzFo+h8Lm7gAtZjOE5jFDZiDm6iC3uxHUfwK0fqqTgWhWvxIv4PivvaEY2sCNssxGO8jeVKi5Wknh8qPc2RehJOhrSrcKdgi9TUGlzF9cL9jcfZsNPBMnYagsPYjKPYie8lEi5d3zh8DhtlB8issMZQLMHtEqTZpcXlSu+7InEeYCVeZ6OVKCwSD4uFWo1T2IQvJfCyS4vEI3Emkuc4tuFbNlqJwiJx8mPK2RQWvU2c/J3S6GsJrl4DZGDk6u7w7XI8bzqRGk2+TAGSamojThwND0/DOXTHgo3FBvyI4K909z1F5miswwJMx5PI3PSdfdgcfXVJ/b84pc/19ZEoDZh7oE2cq1TlurbUlSXMBWhLnatU5bq21JUlzAVoS52rVOW6lkn9B7+JXifHusByAAAAAElFTkSuQmCC\" width=\"15\" height=\"19\" style=\"width: 15px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e by influencing \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003es\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e. We will call the savings rate that maximizes per-capita consumption the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-style: italic; \"\u003egolden rule\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e savings rate and denote it \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14\" height=\"20\" style=\"width: 14px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e; similarly, we will denote steady-state values for \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ek\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ec\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e etc. implied by \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"38.5\" height=\"20\" style=\"width: 38.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e as \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14.5\" height=\"20\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14.5\" height=\"20\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and so forth.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.75px; text-align: left; transform-origin: 384px 10.75px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eTo find \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14\" height=\"20\" style=\"width: 14px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, we proceed as follows:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003col style=\"block-size: 123.75px; font-family: Helvetica, Arial, sans-serif; list-style-type: decimal; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 391px 61.875px; transform-origin: 391px 61.875px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 42px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 21px; text-align: left; transform-origin: 363px 21px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003efind an expression for \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15\" height=\"19\" style=\"width: 15px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e by using the relationship \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"71\" height=\"20\" style=\"width: 71px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, moving to per-capita terms, and using the expression characterizing \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15\" height=\"19\" style=\"width: 15px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e to replace the term \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"40\" height=\"20\" style=\"width: 40px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e with \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"23\" height=\"19\" style=\"width: 23px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4375px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2188px; text-align: left; transform-origin: 363px 10.2188px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003etake the derivative w.r.t. \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003es\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, keeping in mind that \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAB4AAAAmCAYAAADTGStiAAAAAXNSR0IArs4c6QAAAn1JREFUWEft1lvIjWkUB/Df55jTxbiSQ4wUaqJcuHAoakxJY2aijGOEkCuH5JArprhgIhqKqBnDmBkzZHIsSuRiHKK5UmqaGlPKjFyIwrRqfbXb7W+/77tv7ad2+93PXuv5r/Vfa/2ft0Pj1RMr8XF+BuAWFuDPLnwqbXcUWPfFPqzAt1iLF5UQujAuAv4gAWdhPb4uCdoPY/EHXjbyKQIeg9MYgZm4WRJ4Hk5hSlc+RcCzcRbXsRB/NwGODCfgV0TAUaKleI5g7HKtfzPg7tiBLTiMdV3RlsGswe7MdC9W45vcm4q5uNYZeDPg2vrGIQFetMInQD7HqKT5FzysD7oZ8Dj8hP74FHcQe0F5fHfDvbTpbKJO4MhuKG7gN9zH69qomwEHwHe4mvP7NB0/xNGk8We8zf3PcgIuYheW4Qi+wkf4EreLqO6VzlHXGKHNGfFAbEAA3q3jPaidmBnG84Fsrpj76biAZ0XAg3EC07AonwdhFY6XUK85OIZParMsQ/XkjPCf7MY3Sfee2qibdFpvjMRjvKoiIJHZIVzCOWzC/qT9XVFrl/m/UXP1SYAAr13RZEvwpMzBRTaNgIfje0zCVjzAScQNFbU7U3Romf8bAc9IeXuU9f0rx+cLhBgsx79lDm9mUw8cv7elVIZ4xHUYWhtgMZMxGiEOobshqTEmv+O/qoHUAwedB7EY23P4o5mGpDgE0I85VsMyiBCIhp1bJePR+AHjcwavpHMEOD87PYI7n4q1EVGSyqs+485rsNFrTo98HdqZGh0jVq9epQMouo9LH1TVsA1clbGW7dtUt0xdVcc21VUZa9m+TXXL1FV1bFNdlbGW7d8/qv8Hq2N9JwVuavQAAAAASUVORK5CYII=\" width=\"15\" height=\"19\" style=\"width: 15px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e depends on \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003es\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 40.875px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 20.4375px; text-align: left; transform-origin: 363px 20.4375px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eset the resulting expression to zero, obtaining an equality identifying \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eδ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e with the marginal product of capital, in per-capita terms, when the economy follows the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-style: italic; \"\u003egolden rule\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4375px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2188px; text-align: left; transform-origin: 363px 10.2188px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003esubstitute this expression back into the expression characterizing \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15\" height=\"19\" style=\"width: 15px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and solving for \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003es\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e\u003cdiv style=\"block-size: 21.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.75px; text-align: left; transform-origin: 384px 10.75px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14.5\" height=\"20\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e can then be found by again considering the relationship \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"71\" height=\"20\" style=\"width: 71px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e in per-capita terms in the steady state, with \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"38.5\" height=\"20\" style=\"width: 38.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 87px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 43.5px; text-align: left; transform-origin: 384px 43.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eYour task is now simple (in principle): assume that macroeconomic production follows a Cobb-Douglas relationship, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-8px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"88.5\" height=\"22\" style=\"width: 88.5px; height: 22px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"63.5\" height=\"18\" style=\"width: 63.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e (you may verify that this satisfies the conditions listed above). For given values of the (constant) technology parameter \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eA\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, the capital elasticity of output \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eα\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and the depreciation rate \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eδ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, please compute the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-style: italic; \"\u003egolden rule\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e savings rate \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14\" height=\"20\" style=\"width: 14px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, and the resulting steady-state capital intensity \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14.5\" height=\"20\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, per-capita output \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14.5\" height=\"20\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and per-capita consumption \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14.5\" height=\"20\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [sg, kg, yg, cg] = Solow(A, alpha, delta)\r\n    \r\nend","test_suite":"%%\r\n[s, k, y, c] = Solow(1, 0.3, 0.05);\r\nassert(max(abs([s k y c] - [0.3 12.931373133239163 2.155228855539861 1.508660198877902])) \u003c 1e-12)\r\n\r\n%%\r\n[s, k, y, c] = Solow(2, 0.4, 0.025);\r\nassert(max(abs([s k y c] - [0.4 322.5397887730876 20.158736798317975 12.095242078990784])) \u003c 1e-12)\r\n\r\n%%\r\n[s, k, y, c] = Solow(3, 0.5, 0.1);\r\nassert(max(abs([s k y c] - [0.5 225 45 22.5])) \u003c 1e-12)\r\n","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":332395,"edited_by":332395,"edited_at":"2023-07-02T18:27:24.000Z","deleted_by":null,"deleted_at":null,"solvers_count":3,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2023-07-01T21:55:49.000Z","updated_at":"2026-04-18T04:33:17.000Z","published_at":"2023-07-01T21:55:49.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eLet us consider a simplified version of Solow's classical growth model. Let \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eY_t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eK_t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eL_t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eI_t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eS_t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eC_t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e denote production, the capital stock, labor, (gross) investment, savings and consumption at time \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003et\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e respectively (all variables are in real rather than nominal terms), and assume that output is produced using a neoclassical production function using capital and labor as inputs, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eY = F(K, L)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, satisfying the following conditions:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe marginal product of capital and labor is positive: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eF_K = \\\\frac{\\\\partial F}{\\\\partial K} \u0026gt; 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eF_L = \\\\frac{ \\\\partial F}{ \\\\partial L} \u0026gt; 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe marginal product of capital and labor is diminishing: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eF_{KK} = \\\\frac{ \\\\partial^2 F }{ \\\\partial K^2 } \u0026lt; 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eF_{LL} = \\\\frac{ \\\\partial^2 F }{ \\\\partial L^2 } \u0026lt; 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eProduction exhibits constant returns to scale: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eF\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is homogenous of degree one, i.e. \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eF(cK, cL) = c F(K, L)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e for all \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ec \u0026gt; 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eF\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e satisfies the Inada conditions: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\lim_{K\\\\to 0} F_K = \\\\lim_{L\\\\to 0} F_L = \\\\infty\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\lim_{K\\\\to\\\\infty} = \\\\lim_{L\\\\to\\\\infty} F_L = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eCapital in the economy accumulates according to the law of motion \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eK_{t+1} = (1 - \\\\delta) K_t + I_t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, where \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e0 \u0026lt; \\\\delta \u0026lt; 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the rate of depreciation; investment equals savings, which are assumed to be a constant fraction of output, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eS_t = sY_t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e for all \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003et\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, for some \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e0 \u0026lt; s \u0026lt; 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. Output that is not saved is consumed (in other words, we assume a closed economy with no government activity), so that \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eY_t = C_t + I_t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e for all \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003et\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAssume that the population and hence the labor force is constant (this kind of defeats the purpose of a growth model, but we \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eare\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e considering a simplified version only). It is helpful to recast the model in per-capita (technically, per-laborer) terms by dividing by \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eL_t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e throughout and taking advantage of the fact that \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eF\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is homogenous of degree one. We use lower-case letters for per-capita terms: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek_t = \\\\frac{ K_t }{ L_t }\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the capital intensity, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey_t = \\\\frac{ Y_t }{ L_t }\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is output per capita, and so on. We also write \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef(k_t) = F(k_t, 1)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e; \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the intensive form of the production function \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eF\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe model economy is in its steady state when the per-capita variables do not change; denote the steady-state capital intensity by \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. An expression implicitly characterizing \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e can be derived from the law of motion for capital by moving to per-capita variables and replacing \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek_{t+1}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek_t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e throughout.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSince in the steady state, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is constant, so is output per capita \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and hence consumption per capita \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ec^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ec^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e depends on three things: the depreciation rate \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\delta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, the savings rate \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, and the macroeconomic production function \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eF\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e (equivalently, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e). A social planner seeking to maximize steady-state per-capita consumption may not be able to change \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\delta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e or \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eF\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, but can maximize \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ec^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e by influencing \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. We will call the savings rate that maximizes per-capita consumption the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003egolden rule\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e savings rate and denote it \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es_g\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e; similarly, we will denote steady-state values for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ec\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e etc. implied by \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es = s_g\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e as \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek_g\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ec_g\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and so forth.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eTo find \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es_g\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, we proceed as follows:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003efind an expression for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ec^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e by using the relationship \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eY_t = C_t + I_t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, moving to per-capita terms, and using the expression characterizing \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e to replace the term \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003esf(k^*)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\delta k^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003etake the derivative w.r.t. \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, keeping in mind that \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e depends on \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eset the resulting expression to zero, obtaining an equality identifying \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\delta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e with the marginal product of capital, in per-capita terms, when the economy follows the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003egolden rule\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003esubstitute this expression back into the expression characterizing \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and solving for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ec_g\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e can then be found by again considering the relationship \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eY_t = C_t + I_t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e in per-capita terms in the steady state, with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es = s_g\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eYour task is now simple (in principle): assume that macroeconomic production follows a Cobb-Douglas relationship, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eY_t = AK_t^\\\\alpha L_t^{1-\\\\alpha}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e0 \u0026lt; \\\\alpha \u0026lt; 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e (you may verify that this satisfies the conditions listed above). For given values of the (constant) technology parameter \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eA\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, the capital elasticity of output \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\alpha\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and the depreciation rate \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\delta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, please compute the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003egolden rule\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e savings rate \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es_g\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, and the resulting steady-state capital intensity \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek_g\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, per-capita output \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey_g\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and per-capita consumption \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ec_g\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"}],"problem_search":{"errors":[],"problems":[{"id":58478,"title":"Optimal saving in Solow's classical growth model","description":"Let us consider a simplified version of Solow's classical growth model. Let , , ,   and  denote production, the capital stock, labor, (gross) investment, savings and consumption at time  respectively (all variables are in real rather than nominal terms), and assume that output is produced using a neoclassical production function using capital and labor as inputs, , satisfying the following conditions:\r\nThe marginal product of capital and labor is positive: , and .\r\nThe marginal product of capital and labor is diminishing: , and .\r\nProduction exhibits constant returns to scale:  is homogenous of degree one, i.e.  for all .\r\n satisfies the Inada conditions: , and .\r\nCapital in the economy accumulates according to the law of motion , where  is the rate of depreciation; investment equals savings, which are assumed to be a constant fraction of output,  for all , for some . Output that is not saved is consumed (in other words, we assume a closed economy with no government activity), so that  for all .\r\nAssume that the population and hence the labor force is constant (this kind of defeats the purpose of a growth model, but we are considering a simplified version only). It is helpful to recast the model in per-capita (technically, per-laborer) terms by dividing by  throughout and taking advantage of the fact that  is homogenous of degree one. We use lower-case letters for per-capita terms:  is the capital intensity,  is output per capita, and so on. We also write ;  is the intensive form of the production function .\r\nThe model economy is in its steady state when the per-capita variables do not change; denote the steady-state capital intensity by . An expression implicitly characterizing  can be derived from the law of motion for capital by moving to per-capita variables and replacing  and  with  throughout.\r\nSince in the steady state,  is constant, so is output per capita  and hence consumption per capita .  depends on three things: the depreciation rate , the savings rate , and the macroeconomic production function  (equivalently, ). A social planner seeking to maximize steady-state per-capita consumption may not be able to change  or , but can maximize  by influencing . We will call the savings rate that maximizes per-capita consumption the golden rule savings rate and denote it ; similarly, we will denote steady-state values for ,  etc. implied by  as ,  and so forth.\r\nTo find , we proceed as follows:\r\nfind an expression for  by using the relationship , moving to per-capita terms, and using the expression characterizing  to replace the term  with ;\r\ntake the derivative w.r.t. , keeping in mind that  depends on ;\r\nset the resulting expression to zero, obtaining an equality identifying  with the marginal product of capital, in per-capita terms, when the economy follows the golden rule;\r\nsubstitute this expression back into the expression characterizing  and solving for .\r\n can then be found by again considering the relationship  in per-capita terms in the steady state, with .\r\nYour task is now simple (in principle): assume that macroeconomic production follows a Cobb-Douglas relationship, ,  (you may verify that this satisfies the conditions listed above). For given values of the (constant) technology parameter , the capital elasticity of output  and the depreciation rate , please compute the golden rule savings rate , and the resulting steady-state capital intensity , per-capita output  and per-capita consumption .","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 965.688px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 482.844px; transform-origin: 407px 482.844px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 84.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42.25px; text-align: left; transform-origin: 384px 42.25px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eLet us consider a simplified version of Solow's classical growth model. Let \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14.5\" height=\"20\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"16\" height=\"20\" style=\"width: 16px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAB0AAAAoCAYAAAACJPERAAAAAXNSR0IArs4c6QAAAfNJREFUWEft1kuozVEUx/HPTVEkJSFzycCjZGRiRBQjIY+ZARkZiEKSSAakGIgYeERJHkUyMZKJiZkwMEApSYooj5bW/3bu6Z67T+ffzuTs2pP/Wet8/+u31v7t/4jymooz2JGh23CtnNY7YqSP5HkJWYlX2IAXfeT1DOkHugIPMR23suIvtaF7cCohh3AMf2pCo7pz2J6QVXjcBhi5JXkX4CaW4Cm24G1t6HrcTch5hNTfa0JDhcO5g7MTAW69JpJ3Bi7mEfmK6Oez1sRCTxfnEZmPJ9iK97WhAbmakNPYj581oZNxIgcnOK2tr/Nle/V0EOubhnV4hM8TKdILOoj1xaCtyTb8GATaaX1HEHsi65ubk34Jt0t9H6/Sbusr9TOAJxHTvgkvO6BzcBDH8aF5Ph40km9gYQZdwFG8w+98FoM2G6uxF2GX3Y41M4fxG/ZhVPJOaLxxXNQbsagk0Ti/N4pMSY/eheV4kPt6M2Alwx+A/S8l/vdAwjd3X/q1oI2F/sJufOrnnA5aYZPXWOjlHLKAj65alTZX4tr81BlTRA3opJz2AHYfodGGt5WyO38WruBjDlOc1ee15V2G+zibXxn38KY2dCnu4HX6cFQ5xkJr9LTYriG0KFGbgKG8bdQr5g7lLUrUJmAobxv1irn/Rd6/MrFcKdq8SKgAAAAASUVORK5CYII=\" width=\"14.5\" height=\"20\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"11\" height=\"20\" style=\"width: 11px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"13.5\" height=\"20\" style=\"width: 13.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"16\" height=\"20\" style=\"width: 16px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e denote production, the capital stock, labor, (gross) investment, savings and consumption at time \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003et\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e respectively (all variables are in real rather than nominal terms), and assume that output is produced using a neoclassical production function using capital and labor as inputs, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"81.5\" height=\"19\" style=\"width: 81.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, satisfying the following conditions:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003col style=\"block-size: 143.438px; font-family: Helvetica, Arial, sans-serif; list-style-type: decimal; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 391px 71.7188px; transform-origin: 391px 71.7188px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 35px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 17.5px; text-align: left; transform-origin: 363px 17.5px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eThe marginal product of capital and labor is positive: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-15px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"83\" height=\"35\" style=\"width: 83px; height: 35px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-15px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAKIAAABGCAYAAABPJ0+PAAAAAXNSR0IArs4c6QAACstJREFUeF7tnQWsLEsRhr+HS4AQHBIkOAkWEiB4cHf34O6uAYK7O8FdgntwCxIkBIfg7u7ku6l+9M6ZOdMzu3PO7E538nLvfdvTU931b3fVX9W1R1HbplbgdMArYrCbAT/uGPjmwMvis+cD9wT+2tH32sDtgJsAv96UoHMc56g5CrXFMt0eeB5wG+DFLfM4JvAo4IHx2R0AwdjVLgrcGLg38LctXpde0SsQe5doUAeB83HgOR3gOXnsmlcEfgpcDfj8Pm84K3A24J2DpNjCzhWIm1Xa2YHXAT/rOE4vALwdOA3wHsAj/FebFWE7R6tA3KzeEhD/HEfq9xvDp6Pb//1Y4KHAv7M+pwQeBDwa+OVmRZv3aBWI4/RzekCn4yrxuGD6MPAR4NnAX4AbAN/Ihj8+8FRAMNpuCrwq+1xdaFueKoD433GibedTFYjD9HbcAMsjA3T3B74N6IQ8IBwRR/xSCxDPALwauAjwrdgxvxevP3H0v1sc1x8cJtb2965ALNfhscIBeRzw1tjZfp49fh7gjYAOxgcCaPnxejngfT2vc1eVqvlJuVi70bMCsUyPrtONgpr5bvz9a41Hk3143gCkx+zvsz5SNo+Jf2sbagd6/Dq24H0G8IUWu7FMwi3vVYFYpsAzAq8EpGd0Jp7QcDIcJfeIm47IicJ21Eu2XR54f4v9+C7gbWUi7VavPiCeBZDdvypw8YFT7yNrBw53qN09LgWiu+D1gK+2SHP1OLL9qOmI5LvlJ1s8aoGq3fk0oOlpH+rER7xcO/rKwDWAHwZVJc5eDnwM+E/bmH1ATM8kotZ//zEMa3mw3LM7DnCOMNo9xi4JfHTEROb2iPPSLjQUZwjvzrEGuZx5xKQNrDlI28J62p9nBr4D/GtuCzBAHh0ymQHXzHVKXyodtBcBbwnzRHprpZUC8RLhJfpwHxF7QeDpwC0b9MWA+cyq60kDgFI1D8s841xICWq/8ZcNW+9+wN+jw9Cw3qwmP0CYU0S40jW4bsMxS9TUC2KjenLzC1cKxD4iNpfXY+ghgFTEbwdMZK5dTxZ83xVajtwks+bLmyJsd8PGSZCH9TxNrgR8Yq6THSmXOFLfmhbuerdu0f2ZgNcAcrDNNTrisfW1JhHr2b+fQX3q4MreAfyjb/At+Dx3NJq2n+I7X48dd0y5xOa3PXdiDoKeUV8PD3tMGintzFMudX4i5IxA/s4cRzIE+alRBMSciG0jaqec4BzG9suqfSjA7hU2UJIr5xZfGED8TUPo/DTRfhKsU35Bc4V/Fngi4KbQlWq2iTXOOdL9NirX8SlthH/Jjpi/pM1Yd+J6fO4KeUhrExOcyxiJvvldRFZMajAacnfgLjF3HRqP3rxpX+qc6GnbHhH/TR2+UzaPR+VzI/lKfJEk3Pc4ChtY5AQw5y819emOMRP74McrFFYfEP38wZmB3rbtOuC1YreY8lu3gfVaawjBeNdgAwSkzaP2DcA3GwyCFIaxZnMJtS1Tc7e0v6li/jn1sdkEpBvF44HXbxCQxwuQ36kjtJkves6+rJwufUA8SXzb9YJs2kGfAo4BuBO6yPeN7b8tEXQtzXc87JfBHXidtjQTow2Qzww24A/rLCRwgmBJjCT1rWsOxBUGog+Iefy0S14D+AL1y2tOqPTxCsTSldrbT0Aa3XFnl92Q5xOQL13jKkLOKgwB4gqf2gdEj5fXxnyaRKyekrFTebI2knf8ctUnp16BJiA1GbRxXzICkPmpOQSIK47bfkAsIWLdnf7U8CSnXsQ6/uZWoA2QAkQzy6sMJS330ocAsfho7rtfIVDlq0yJ2u/eRclkap/DWwE3IyNnzwXOGWJ8KNLcNLv6Wr5hDQGiySMmhxxp++2IffcrfNakiB9NzFH1LUT9fNwKqD99AJN7zQ2wvTccTzPNh8S8S+mb3NRb4Rv3A+KQsN64pRj3VHVWxq1bekrG43zAfTIAGpaTaDYzqDU7pueVesPvBoxClRDaexJDuoDYd79iyFLIqZkIcceYuKSnXJZXJCVa88tDJeNWIJas0t4+bQA09vsk4IsjAZjekof4mtGn1CfnG/XSDQQcTa53AbF5v2IT9EyqcGCWivTBuvzVOHUs76k2AKoDveSvN4j4sauTZ9d0pcolTJ07KCR9i6NbFxDzsJ5hoWba+1CB85w+S2gYDpw6zDVUxl3r3wSgJ5G6NPa8KQDma5bSwC7ckl2TA/VZkfCwEoVrA6JHqfcpLHNhEzTGLL0iObadNlKprA/jztgVixw7fn1udXM5f6TiGXoVgNbakZbx1uCUG4B3b7Q1NbfkllP9Hx1fr9nKrmhaNRNDVrxm3XBLYdwiC9I7PSdi9oZxVWOUKc46RPnJmJUWuC3wiyEP176DViDZ9yZamBFk+ZMfDBphvc4nDIdFp0UuUnkMA2oXdnrjfZGV9UT6/9PJvTcSI/c4hBrYlAxLGUen4Jqh9FJS+tDX5iCAmCeWmsksVVBbXYGVFTgIIKYbbL64WYZjW9Vx7DBhvN2oTa3tawKBdIjeYFfOXzJ/vEx0nUg8SGuQUsQs0uS9oKG01rau5RG5DwKI6QbbLtE2rpu2j56fxr8eqnawmSzmJkpP5VUg2kCSF+wUxNrm2lCLbFMDMY9D3gPQdW9+0w28e2fachz/nJkWvOTjLjeXZixY6mXn2tRATIkT7hZSCc3iQr7f9HF3lzlyixWIBwT5qYGYEidMFWtGZzzOvFopryTJfVCJtVMvbQpBCmKLdra1vsymqWWc3fhTAdFxDeVIhN8qZm25Ceu9eINNT/picf9jl2xHp5ouCHVdxrdPX2bT7IAytUBTAXFquec8fgLifldH55rZdGjrWoE4buk1Ky4Uu71/pp+esKSG5YeteND10xVDCxaMk3DLnqpAHK4wKzsYizeEZgKBdX7MJMorPjhqFxCXXrCgdcUrEIcB0frWAsw4alt5kfzI7bpMnxe06kqZGibVDvSuQCxXosF8C3R6kdxkAuPnzQhKXsmgK0E0xd1988q9jXJRdq9nBWK5Tt0F3cGkovZUs4phcpA1q8Lapa9ybLk0O9azArFMobmDsSfNPYbIk3+7brP1VY4tk2YHe1Uglik1dzC6SjKn5N9LRe7dyp2MeE1f5dgyaXawVwVimVJNfzcWnoj4tkKbl4kUN/uYWfPmxtAlBQua0qgfK7BafcyLZjvbKhDLVJuKB3UduaaC6chYNdUipm1Z6GPCeqaXectOpyb9OFCZxFvWqwKxTGGpGFXbT5s5wqXjVwcMX3alc+VhvbYfBGpKYhFQKSIvJa1UVy0Tebt6VSCW6cuCm9aD8Yhu/rRt4hYFo16zF5XarkLkHGPfD4a7w5oIInHuffD8N/vKJN6yXhWI5QpL9I1HsHeCzau05IrFAs4Vt+a0C9sqJZhzadKsybC2z0Rh089lZYwNGwp4qzCYu2n2d18tmXLpZ96zArFcQQLF345xh7IyvnyihLZ3hf3FqBRvzkeU0rl+XJEQWENb3845dLzZ9q9AnK1qliVYBeKy9D3b2VYgzlY1yxKsAnFZ+p7tbCsQZ6uaZQlWgbgsfc92thWIs1XNsgSrQFyWvmc72wrE2apmWYJVIC5L37OdbQXibFWzLMEqEJel79nOtgJxtqpZlmAViMvS92xnW4E4W9UsS7AKxGXpe7azrUCcrWqWJVgF4rL0PdvZ/g97Nmhl+Z+OPAAAAABJRU5ErkJggg==\" width=\"81\" height=\"35\" style=\"width: 81px; height: 35px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 37px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 18.5px; text-align: left; transform-origin: 363px 18.5px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eThe marginal product of capital and labor is diminishing: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-16px\"\u003e\u003cimg 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width=\"95\" height=\"37\" style=\"width: 95px; height: 37px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-16px\"\u003e\u003cimg 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width=\"91.5\" height=\"37\" style=\"width: 91.5px; height: 37px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 40.9375px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 20.4688px; text-align: left; transform-origin: 363px 20.4688px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eProduction exhibits constant returns to scale: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eF\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e is homogenous of degree one, i.e. \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"141\" height=\"19\" style=\"width: 141px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e for all \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"36\" height=\"18\" style=\"width: 36px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 30.5px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 15.25px; text-align: left; transform-origin: 363px 15.25px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eF\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e satisfies the Inada conditions: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-15px\"\u003e\u003cimg 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margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eCapital in the economy accumulates according to the law of motion \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg 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width=\"63\" height=\"18\" style=\"width: 63px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e is the rate of depreciation; investment equals savings, which are assumed to be a constant fraction of output, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" 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style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHoAAAAkCAYAAABG3S5jAAAAAXNSR0IArs4c6QAABHFJREFUeF7tmlmoVVUYx39XQzNJUZNIpBCCFCkFC8RMBcMBNMscsBwSrUTsIRscQohEREQlHFBUsFJMMbXBx8CgSUXEJxND7SFDMX1QsSQH/vAt2Bz3Ofesvdc+h+5eGy5czll7re///dc3nxbSn+eBXcBZYAbwZ2JZF2A9MByYBfxQZY/4cXgN9AemGx/jgVHASWAqcLrWcS0pX3YDtgCTgfeAdcDdinUTga+Ab4A3gUvhMcUdUzTQAbhln88EPstD9EjggG02Fvgp5cCngD3AAOBVYH8DaGkHDATmAp+2doMbIE+WI0JieB3YmZXo9sByYAnwM/Aa8EcKoq7ANmCSWf+7wM0syOt4xynnfWBavcDq2LeRS4rAkIvoJIGHLD5fTdFIJ3PpbwOHAR16IbDmpJwhwELgFeCauSqFknMp4STw8UG2KxJDLqKfBL4EBtVhqcuAT4C/ACUGx4OoBh6wRO8DYHRggnsC7wAKT7qgutiSX7HvioWEEDCKxODky0V0X2Av8LQH0Tp4aJVY7qO0opXjkkzJ+jJwFJDFKZncDEyw5NJH5sq1RWNInpeLaJVVP9puyrxrxV5n0VqulF/lWJanI/AisAh4IbAFJ+WRl/oWuAxMAX6zL3tZUrMU+DULAKBRGIIR7ZTxmKdFZ7EGxflxgFz0cwUS7JSTxPaWJZMqGyXHx8BW4HdPohuNIRjRybLJx6J9XHc15awBznsq2me5XPd2S+yOALOBUz4bJNY2C0MwouXG5IJHeFi0smF1aOpxew9Z+aa4+LBJrf+/AP7NqHSf1+R5dJbO/hpQ1XDRZwOg2RiCJGP1llcPArLA+a3U22k6VDeuj8V/tVCl9GPAKuC7gglXsrTYLptk2wB8mKEH0EwMQYgWgI9MEfU2THYAC4Abnpah5Y/bZZFVd28Q4Y+at5J1yxupRv8+g+zulWZg0Nm5sm5t4Fqg12vUx8l6Wy1Jxb48T5HKGgz8DZxJCDjM+gVKOlcCqiBu5wHQhEubm2g3nVLTfJ7d/kodvGQxTn1wlVahkigpfo65dWfhq82lZ22xikS1cT9PgFC42Gidv002vPknJ9Hu9SIwpImWm2ht6saUGoFVTqfcRZDLU5miTlropwfwhnWxnjCXnoVwl0sogUqGl2QLN4RHSsMfCkM13QYhWrFa3SO5NY3C1lqSpGRGCZQbfChD/y80y4n9dKk0D1fbUqWfb9LmLFdYJPdB65H3A3abJ8qSeftAzosh7SzxsMISSbVwXaevqlxp8+jkYsVOufBn7EcIva0ckdtr5GChs03KNB+/U8+g3UCot+2yeZGrbF9Dmmet1St3niWJ9CHarc2KIXnWIzYWVqNJf+6RAewDfrHW7n2lamtEZwFU5DtyuWOAEwHzgiLlTdu7KRj+b0Q3mpQ2c14kus1QWRtIJDoSXRINlARmtOhIdEk0UBKY0aIj0SXRQElgRouORJdEAyWBGS06El0SDZQEZrToSHRJNFASmNGiI9El0UBJYEaLLgnR9wBoqlY0eQscxQAAAABJRU5ErkJggg==\" width=\"61\" height=\"18\" style=\"width: 61px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e. Output that is not saved is consumed (in other words, we assume a closed economy with no government activity), so that \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"71\" height=\"20\" style=\"width: 71px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e for all \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003et\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 124.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 62.25px; text-align: left; transform-origin: 384px 62.25px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eAssume that the population and hence the labor force is constant (this kind of defeats the purpose of a growth model, but we \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eare\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e considering a simplified version only). It is helpful to recast the model in per-capita (technically, per-laborer) terms by dividing by \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14.5\" height=\"20\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e throughout and taking advantage of the fact that \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eF\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e is homogenous of degree one. We use lower-case letters for per-capita terms: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"44.5\" height=\"40\" style=\"width: 44.5px; height: 40px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e is the capital intensity, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"43\" height=\"40\" style=\"width: 43px; height: 40px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e is output per capita, and so on. We also write \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"99\" height=\"20\" style=\"width: 99px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e; \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ef\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e is the intensive form of the production function \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eF\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.75px; text-align: left; transform-origin: 384px 31.75px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eThe model economy is in its steady state when the per-capita variables do not change; denote the steady-state capital intensity by \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15\" height=\"19\" style=\"width: 15px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e. An expression implicitly characterizing \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15\" height=\"19\" style=\"width: 15px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e can be derived from the law of motion for capital by moving to per-capita variables and replacing \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"25.5\" height=\"20\" style=\"width: 25.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"13\" height=\"20\" style=\"width: 13px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e with \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15\" height=\"19\" style=\"width: 15px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e throughout.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 105.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 52.75px; text-align: left; transform-origin: 384px 52.75px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eSince in the steady state, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15\" height=\"19\" style=\"width: 15px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e is constant, so is output per capita \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACAAAAAmCAYAAAClI5npAAAAAXNSR0IArs4c6QAAAl5JREFUWEft1kuojWEUBuDnRO6FiQgZEAMTl4EolEsxkEsGihmSFEkuJSkSSpgoyqWIDISMJCmFmZIJJTJwzT255taqtU//Oe2jf5+zd2eyv9nerf9b73rXu971tejm09LN+TUSQNw9Gn/wpKNCGwlgMM7gGTbiWzUQ9QbQG4vxAA+xF2+wH9MxAFfxuwKm3gBG4hTGYC2G4CP64yjOJxtfGgUg7u2JscnEDHzCFdzAq9REazc6YmAUJmEyJmI8duNEoY89sAFbcQCH8KsAYAlm4Wkmv47X+FvUQkcA+qIfxuEgpuASVuJDXtArQW3BXSzDOxzDbKzDQLzFoAR4EtvwvZYWLEw1xzfzcbtQQfR2M2ZieQKYl1TfSxG+TxFGzA/cKrJQRoTDcBpzsAN72tEYfV6Ulf0sgIsxDEE+x6Zi1WVaUIyJXu/M5BewKoUVMVFAzHj0+WKVOR+e/wWIqqcMA/HhXFzDIyzF/bwt2AlxBjMvO2PrZQHEVJzDNKzA2UwWfe/TbjpqwlEWQExFjNmaFFZUPDSr397Z6is9LIs4eh0jGf6+HqvxosBG2XvaxJVlID4Ktd/EnTSeBanuii80HED4e3h52GxUHqZU9ISGAwhXO55TEBrYl9bbqcS1OGEltiLEEekFsVi6fGrRQDjbkfT60EJdTlkAERfK/5ptaLPRuoKkLIBYJLGUovetj4muJP6fBqLXE/Ip9TjfBbHbQ3Sf65G0eEc1BmK8Qu2xRi9n0l35u975qz7Lp6bQItnhdLrY4w05ZTXQkOS17oKGgGgy0GSgyUCTgSYD/wD0fHInPgHIPwAAAABJRU5ErkJggg==\" width=\"16\" height=\"19\" style=\"width: 16px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and hence consumption per capita \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15\" height=\"19\" style=\"width: 15px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15\" height=\"19\" style=\"width: 15px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e depends on three things: the depreciation rate \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eδ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, the savings rate \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003es\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, and the macroeconomic production function \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eF\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e (equivalently, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ef\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e). A social planner seeking to maximize steady-state per-capita consumption may not be able to change \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eδ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e or \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eF\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, but can maximize \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15\" height=\"19\" style=\"width: 15px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e by influencing \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003es\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e. We will call the savings rate that maximizes per-capita consumption the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-style: italic; \"\u003egolden rule\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e savings rate and denote it \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14\" height=\"20\" style=\"width: 14px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e; similarly, we will denote steady-state values for \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ek\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ec\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e etc. implied by \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"38.5\" height=\"20\" style=\"width: 38.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e as \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14.5\" height=\"20\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14.5\" height=\"20\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and so forth.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.75px; text-align: left; transform-origin: 384px 10.75px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eTo find \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14\" height=\"20\" style=\"width: 14px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, we proceed as follows:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003col style=\"block-size: 123.75px; font-family: Helvetica, Arial, sans-serif; list-style-type: decimal; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 391px 61.875px; transform-origin: 391px 61.875px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 42px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 21px; text-align: left; transform-origin: 363px 21px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003efind an expression for \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15\" height=\"19\" style=\"width: 15px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e by using the relationship \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"71\" height=\"20\" style=\"width: 71px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, moving to per-capita terms, and using the expression characterizing \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15\" height=\"19\" style=\"width: 15px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e to replace the term \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"40\" height=\"20\" style=\"width: 40px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e with \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"23\" height=\"19\" style=\"width: 23px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4375px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2188px; text-align: left; transform-origin: 363px 10.2188px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003etake the derivative w.r.t. \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003es\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, keeping in mind that \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15\" height=\"19\" style=\"width: 15px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e depends on \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003es\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 40.875px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 20.4375px; text-align: left; transform-origin: 363px 20.4375px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eset the resulting expression to zero, obtaining an equality identifying \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eδ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e with the marginal product of capital, in per-capita terms, when the economy follows the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-style: italic; \"\u003egolden rule\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4375px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2188px; text-align: left; transform-origin: 363px 10.2188px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003esubstitute this expression back into the expression characterizing \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAB4AAAAmCAYAAADTGStiAAAAAXNSR0IArs4c6QAAAn1JREFUWEft1lvIjWkUB/Df55jTxbiSQ4wUaqJcuHAoakxJY2aijGOEkCuH5JArprhgIhqKqBnDmBkzZHIsSuRiHKK5UmqaGlPKjFyIwrRqfbXb7W+/77tv7ad2+93PXuv5r/Vfa/2ft0Pj1RMr8XF+BuAWFuDPLnwqbXcUWPfFPqzAt1iLF5UQujAuAv4gAWdhPb4uCdoPY/EHXjbyKQIeg9MYgZm4WRJ4Hk5hSlc+RcCzcRbXsRB/NwGODCfgV0TAUaKleI5g7HKtfzPg7tiBLTiMdV3RlsGswe7MdC9W45vcm4q5uNYZeDPg2vrGIQFetMInQD7HqKT5FzysD7oZ8Dj8hP74FHcQe0F5fHfDvbTpbKJO4MhuKG7gN9zH69qomwEHwHe4mvP7NB0/xNGk8We8zf3PcgIuYheW4Qi+wkf4EreLqO6VzlHXGKHNGfFAbEAA3q3jPaidmBnG84Fsrpj76biAZ0XAg3EC07AonwdhFY6XUK85OIZParMsQ/XkjPCf7MY3Sfee2qibdFpvjMRjvKoiIJHZIVzCOWzC/qT9XVFrl/m/UXP1SYAAr13RZEvwpMzBRTaNgIfje0zCVjzAScQNFbU7U3Romf8bAc9IeXuU9f0rx+cLhBgsx79lDm9mUw8cv7elVIZ4xHUYWhtgMZMxGiEOobshqTEmv+O/qoHUAwedB7EY23P4o5mGpDgE0I85VsMyiBCIhp1bJePR+AHjcwavpHMEOD87PYI7n4q1EVGSyqs+485rsNFrTo98HdqZGh0jVq9epQMouo9LH1TVsA1clbGW7dtUt0xdVcc21VUZa9m+TXXL1FV1bFNdlbGW7d8/qv8Hq2N9JwVuavQAAAAASUVORK5CYII=\" width=\"15\" height=\"19\" style=\"width: 15px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and solving for \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003es\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e\u003cdiv style=\"block-size: 21.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.75px; text-align: left; transform-origin: 384px 10.75px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14.5\" height=\"20\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e can then be found by again considering the relationship \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"71\" height=\"20\" style=\"width: 71px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e in per-capita terms in the steady state, with \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"38.5\" height=\"20\" style=\"width: 38.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 87px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 43.5px; text-align: left; transform-origin: 384px 43.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eYour task is now simple (in principle): assume that macroeconomic production follows a Cobb-Douglas relationship, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-8px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"88.5\" height=\"22\" style=\"width: 88.5px; height: 22px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"63.5\" height=\"18\" style=\"width: 63.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e (you may verify that this satisfies the conditions listed above). For given values of the (constant) technology parameter \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eA\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, the capital elasticity of output \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eα\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and the depreciation rate \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eδ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, please compute the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-style: italic; \"\u003egolden rule\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e savings rate \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14\" height=\"20\" style=\"width: 14px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, and the resulting steady-state capital intensity \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAB0AAAAoCAYAAAACJPERAAAAAXNSR0IArs4c6QAAAqpJREFUWEft10uszVcUx/HP1YYIHZSJaLwGDSYMJRhURUNEQ1oRigmiYeSRhjYmaGvSEInQpkTjFW8aIS0DEY9BvULEgAEhSKRUDETjlSXryL2He879u/8zu3tyXuu/v2utvdZvr9Ok9voYWzAeC7G6jn2bfm6qYzUIu9Af43CqTbvWMaoH/RIHcRzf4E6joR9gBZbiVyzAk0ZDm5/ntwkug6lWeodgD7pjAs4hvos0x2snXEibK0WyUAsam2/FMUzD/QxzADZiPfbiRdHwW4N2xqo8x2iTJfgfPbAoYeeLwir2rUF7Yxs+w/R83wtzsRk33xcYz7UGHYEjuIev8TxT/AsetAdYCxoRbcBf+BPfYW0q0stGQLvm5gFuvqKgZuJuI6D9sB3D8T0uYQc+wlfY1wjoGPyNa3met7JFJmE/ZuFhe8DVhRSff0j5C2GYjUcJ+h2P05FwKmRyFM7ivyJOVEMjheswA8vwI6JwPskrLiC7s3X6pANh87Q90IHYiaH4Akdzs3BualZ0OHYolWhxHkMR5lt9WrnKTmdfNheBDzEHK1Nzo42qVSn0eBjGZvR9M3OXm3tV7z4tEkGX7OeAzsftrI1ovQginLxaSxyKwCq2o7O656VsRi1EXUR9hKpF671eZUZaAUzJ9FeKMkadEJUbjYDGVbg8BeQiPsemTPEfeNYIaLecGCOyM1n9AQv5bNFSZaY32m1N9vbJWkVRJjQGuJ/wD37OCfKdclkWNNolNDlS+ymikmPFNBmTx5vzLKt6e+ZocwCHUzYDHlGPxMRqESkj0lCxmImrh/H4dxCjTYjCibIVKVL5GyZXRRTaHfdvnHWLEaeMSEOTQ3Ei0rgIrmNwzlXxP+jf6kouA1pYMjughVNW5IGO9BbJVmHbjvQWTlmRB14B+0qIKXSh+E4AAAAASUVORK5CYII=\" width=\"14.5\" height=\"20\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, per-capita output \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14.5\" height=\"20\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and per-capita consumption \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14.5\" height=\"20\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [sg, kg, yg, cg] = Solow(A, alpha, delta)\r\n    \r\nend","test_suite":"%%\r\n[s, k, y, c] = Solow(1, 0.3, 0.05);\r\nassert(max(abs([s k y c] - [0.3 12.931373133239163 2.155228855539861 1.508660198877902])) \u003c 1e-12)\r\n\r\n%%\r\n[s, k, y, c] = Solow(2, 0.4, 0.025);\r\nassert(max(abs([s k y c] - [0.4 322.5397887730876 20.158736798317975 12.095242078990784])) \u003c 1e-12)\r\n\r\n%%\r\n[s, k, y, c] = Solow(3, 0.5, 0.1);\r\nassert(max(abs([s k y c] - [0.5 225 45 22.5])) \u003c 1e-12)\r\n","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":332395,"edited_by":332395,"edited_at":"2023-07-02T18:27:24.000Z","deleted_by":null,"deleted_at":null,"solvers_count":3,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2023-07-01T21:55:49.000Z","updated_at":"2026-04-18T04:33:17.000Z","published_at":"2023-07-01T21:55:49.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eLet us consider a simplified version of Solow's classical growth model. Let \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eY_t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eK_t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eL_t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eI_t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eS_t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eC_t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e denote production, the capital stock, labor, (gross) investment, savings and consumption at time \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003et\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e respectively (all variables are in real rather than nominal terms), and assume that output is produced using a neoclassical production function using capital and labor as inputs, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eY = F(K, L)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, satisfying the following conditions:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe marginal product of capital and labor is positive: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eF_K = \\\\frac{\\\\partial F}{\\\\partial K} \u0026gt; 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eF_L = \\\\frac{ \\\\partial F}{ \\\\partial L} \u0026gt; 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe marginal product of capital and labor is diminishing: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eF_{KK} = \\\\frac{ \\\\partial^2 F }{ \\\\partial K^2 } \u0026lt; 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eF_{LL} = \\\\frac{ \\\\partial^2 F }{ \\\\partial L^2 } \u0026lt; 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eProduction exhibits constant returns to scale: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eF\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is homogenous of degree one, i.e. \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eF(cK, cL) = c F(K, L)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e for all \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ec \u0026gt; 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eF\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e satisfies the Inada conditions: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\lim_{K\\\\to 0} F_K = \\\\lim_{L\\\\to 0} F_L = \\\\infty\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\lim_{K\\\\to\\\\infty} = \\\\lim_{L\\\\to\\\\infty} F_L = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eCapital in the economy accumulates according to the law of motion \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eK_{t+1} = (1 - \\\\delta) K_t + I_t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, where \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e0 \u0026lt; \\\\delta \u0026lt; 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the rate of depreciation; investment equals savings, which are assumed to be a constant fraction of output, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eS_t = sY_t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e for all \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003et\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, for some \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e0 \u0026lt; s \u0026lt; 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. Output that is not saved is consumed (in other words, we assume a closed economy with no government activity), so that \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eY_t = C_t + I_t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e for all \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003et\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAssume that the population and hence the labor force is constant (this kind of defeats the purpose of a growth model, but we \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eare\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e considering a simplified version only). It is helpful to recast the model in per-capita (technically, per-laborer) terms by dividing by \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eL_t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e throughout and taking advantage of the fact that \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eF\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is homogenous of degree one. We use lower-case letters for per-capita terms: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek_t = \\\\frac{ K_t }{ L_t }\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the capital intensity, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey_t = \\\\frac{ Y_t }{ L_t }\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is output per capita, and so on. We also write \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef(k_t) = F(k_t, 1)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e; \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the intensive form of the production function \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eF\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe model economy is in its steady state when the per-capita variables do not change; denote the steady-state capital intensity by \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. An expression implicitly characterizing \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e can be derived from the law of motion for capital by moving to per-capita variables and replacing \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek_{t+1}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek_t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e throughout.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSince in the steady state, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is constant, so is output per capita \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and hence consumption per capita \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ec^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ec^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e depends on three things: the depreciation rate \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\delta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, the savings rate \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, and the macroeconomic production function \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eF\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e (equivalently, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e). A social planner seeking to maximize steady-state per-capita consumption may not be able to change \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\delta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e or \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eF\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, but can maximize \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ec^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e by influencing \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. We will call the savings rate that maximizes per-capita consumption the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003egolden rule\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e savings rate and denote it \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es_g\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e; similarly, we will denote steady-state values for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ec\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e etc. implied by \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es = s_g\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e as \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek_g\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ec_g\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and so forth.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eTo find \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es_g\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, we proceed as follows:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003efind an expression for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ec^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e by using the relationship \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eY_t = C_t + I_t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, moving to per-capita terms, and using the expression characterizing \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e to replace the term \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003esf(k^*)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\delta k^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003etake the derivative w.r.t. \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, keeping in mind that \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e depends on \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eset the resulting expression to zero, obtaining an equality identifying \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\delta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e with the marginal product of capital, in per-capita terms, when the economy follows the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003egolden rule\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003esubstitute this expression back into the expression characterizing \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and solving for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ec_g\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e can then be found by again considering the relationship \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eY_t = C_t + I_t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e in per-capita terms in the steady state, with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es = s_g\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eYour task is now simple (in principle): assume that macroeconomic production follows a Cobb-Douglas relationship, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eY_t = AK_t^\\\\alpha L_t^{1-\\\\alpha}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e0 \u0026lt; \\\\alpha \u0026lt; 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e (you may verify that this satisfies the conditions listed above). For given values of the (constant) technology parameter \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eA\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, the capital elasticity of output \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\alpha\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and the depreciation rate \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\delta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, please compute the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003egolden rule\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e savings rate \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es_g\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, and the resulting steady-state capital intensity \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek_g\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, per-capita output \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey_g\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and per-capita consumption \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ec_g\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"}],"term":"tag:\"solow\"","current_player_id":null,"fields":[{"name":"page","type":"integer","callback":null,"default":1,"directive":null,"facet":null,"facet_method":"and","operator":null,"param":null,"static":null,"prepend":true},{"name":"per_page","type":"integer","callback":null,"default":50,"directive":null,"facet":null,"facet_method":"and","operator":null,"param":null,"static":null,"prepend":true},{"name":"sort","type":"string","callback":null,"default":null,"directive":null,"facet":null,"facet_method":"and","operator":null,"param":null,"static":null,"prepend":true},{"name":"body","type":"text","callback":null,"default":"*:*","directive":null,"facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":false},{"name":"group","type":"string","callback":null,"default":null,"directive":"group","facet":true,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"difficulty_rating_bin","type":"string","callback":null,"default":null,"directive":"difficulty_rating_bin","facet":true,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"id","type":"integer","callback":null,"default":null,"directive":"id","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"tag","type":"string","callback":null,"default":null,"directive":"tag","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"product","type":"string","callback":null,"default":null,"directive":"product","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"created_at","type":"timeframe","callback":{},"default":null,"directive":"created_at","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"profile_id","type":"integer","callback":null,"default":null,"directive":"author_id","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"created_by","type":"string","callback":null,"default":null,"directive":"author","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"player_id","type":"integer","callback":null,"default":null,"directive":"solver_id","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"player","type":"string","callback":null,"default":null,"directive":"solver","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"solvers_count","type":"integer","callback":null,"default":null,"directive":"solvers_count","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"comments_count","type":"integer","callback":null,"default":null,"directive":"comments_count","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"likes_count","type":"integer","callback":null,"default":null,"directive":"likes_count","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"leader_id","type":"integer","callback":null,"default":null,"directive":"leader_id","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"leading_solution","type":"integer","callback":null,"default":null,"directive":"leading_solution","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true}],"filters":[{"name":"asset_type","type":"string","callback":null,"default":null,"directive":null,"facet":null,"facet_method":"and","operator":null,"param":null,"static":"\"cody:problem\"","prepend":true},{"name":"profile_id","type":"integer","callback":{},"default":null,"directive":null,"facet":null,"facet_method":"and","operator":null,"param":"author_id","static":null,"prepend":true}],"query":{"params":{"per_page":50,"term":"tag:\"solow\"","current_player":null,"sort":"map(difficulty_value,0,0,999) 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