Question

Question 2. In this problem, we will consider how the rental price of capital Rt and the wage rate we are determined under the assumptions of the Solow growth model. Suppose there exists a representative firm in this economy with Cobb-Douglas production function given by Y = K L-, and that the price of its output P has been normalized to 1. a) Write out the firm's profit function. (Hint: think about what total revenues and total costs are if the firm uses Kunits of capital and Lt units of labor in production.) b) Solve the firm's profit maximization problem. That is, find the optimal captal-labor ratio ky := (Hint: take two first-order conditions of the profit function with respect to the inputs of production, take the ratio of these equations, and solve for kt.) c) Find output per capita y = and consumption per capita q = as a function of the capital-labor ratio k. (Hint: recall how total consumption is determined by the exogenous saving rate s and output Yt in the Solow growth model.) d) Let f(kt) := denote output per capita (your solution to part c). Using the two first-order conditions you wrote down in part b, show that the rental price of capital Rt and the wage rate we can be written in terms of the function f(kt). (Hint: look for where the capital-labor ratio appears in your derivatives of the production function.) e) Under the assumptions of the Solow growth model, find the law of motion for the capital-labor ratio. Then find the steady state capital-labor ratio k. (Hint: start with the definition of the capital-labor ratio, log-differentiate this, and make the necessary substitutions given the assumptions we make in the model.) f) Write out the steady state rental rate of capital and wage rate in terms of ks. What happens to R and w as the exogenous saving rate s increases? (Hint: you should write the rental price and wage rate as a function of s and differentiate this.)

Answer 1

a) Profit function of the firm is

Π = Total revenue – Total cost

    = KtαLt1-α   - wtLt -RtKt    (since Price P=1)

Where wt is the wage rate and Rt is the rental rat of capital.

b) Profit maximization condition:

Max Π = KtαLt1-α   - wtLt -RtKt

FoCs: dΠ/dKt =   α Ktα-1Lt1-α   -Rt = 0   => Rt = α Ktα-1Lt1-α   ………………..(1)

          : dΠ/dLt = (1-α) KtαLt-α   -wt = 0 => wt = (1-α) KtαLt-α …………………..(2)

Now kt = Kt/Lt

Divided by (1) / (2) => => Rt / wt = α Ktα-1Lt1-α / (1-α) KtαLt-α

= > Lt / Kt = Rt (1-α)/ α wt

             = >κt = (K/L)t = α wt/ (1-α) Rt  

This is optimal capital- Labor ratio.

c) yt = Yt/Lt = KtαLt1-α   / Lt

= > yt = (Kt/Lt)α = (κt)α

Consumption function in Solow model : Ct = (1-s)Yt

             = > (C/L)t   = (1-s)Yt /Lt

            => ct = (1-s) yt = (1-s)(κt)α

d) f(κt) = Yt/Lt

From equation (1) in part b,

=> Rt = α Ktα-1Lt1-α   = α Ktα Lt1-α/ K = αYt/Kt

=> Rt / L = αYt/KL = αyt/K

=> Rt = αytL/K

= > Rt = αf(κ) /κ   (as K/L = κ)

From equation (2) in part b,

wt = (1-α) KtαLt-α = (1-α) Ktα/ Ltα

      = (1-α) f(κ)            (as f(κ) = yt= Ktα/ Ltα = κtα)