2. Consider a small island nation. Assume the economy is following the Solow Gro

Question

2. Consider a small island nation. Assume the economy is following the Solow Growth Model, Let K = $100 Billion dollars and L =100 million persons. The production function is Y -K3/10 L710. Let savings rate-: 10% and depreciation rate 5%. a. Is the production function a Cobb-Douglas function? Explain how you know. (short answer) b. Does the production function have diminishing marginal returns to capital? Explain how you know. (short answer) c. What is the production function for output per capita, y? (show your work) d. What is the steady-state value of capital per capita, k*? (show your work) e. What is steady-state value of output per capita, y*? (show your work) f. What is steady-state value of investment per capita, ? (show your work) g. What is the steady-state value of consumption per capita, c*? (Show your work)

Explanation / Answer

a) We know that it is a C-D function because the labor share and capital share in come are added to 1. This is seen from 0.3 + 0.7 = 1 from the exponents of capital and labor

b) Yes. Find the MPK = 0.5*3/10*(L/K)^7/10 and see that MPK is a decling function. Here dMPK/dK is 0.15*-7/10*K^-17/10L^7/10 = -0.105K^-17/10L^7/10 which is < 0. There are diminishing marginal returns to capital when increase in capital by 1 unit reduces MPK.

c) Output per capita is Y/K

Y/K = (K/L)^3/10 (L/L)^7/10

y = k^0.3

d) At the steady state, k/y = s/d

k/k^0.3 = 10%/5%

k* = 2^(1/0.3) = 10.08

e) y* = (2^(1/0.3)^0.3 = 2

f) i* = sy = 10%*2 = 0.2

g) c* = (1 - s)*y = 0.9*2 = 1.8

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