Problem 58478. Optimal saving in Solow's classical growth model
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 t 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:
- The marginal product of capital and labor is positive: , and .
- The marginal product of capital and labor is diminishing: , and .
- Production exhibits constant returns to scale: F is homogenous of degree one, i.e. for all .
- F satisfies the Inada conditions: , and .
Capital 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 t, 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 t.
Assume 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 F 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 ; f is the intensive form of the production function F.
The 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.
Since in the steady state, is constant, so is output per capita and hence consumption per capita . depends on three things: the depreciation rate δ, the savings rate s, and the macroeconomic production function F (equivalently, f). A social planner seeking to maximize steady-state per-capita consumption may not be able to change δ or F, but can maximize by influencing s. 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 k, c etc. implied by as , and so forth.
To find , we proceed as follows:
- find an expression for by using the relationship , moving to per-capita terms, and using the expression characterizing to replace the term with ;
- take the derivative w.r.t. s, keeping in mind that depends on s;
- set the resulting expression to zero, obtaining an equality identifying δ with the marginal product of capital, in per-capita terms, when the economy follows the golden rule;
- substitute this expression back into the expression characterizing and solving for s.
can then be found by again considering the relationship in per-capita terms in the steady state, with .
Your 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 A, the capital elasticity of output α and the depreciation rate δ, please compute the golden rule savings rate , and the resulting steady-state capital intensity , per-capita output and per-capita consumption .
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