Problem 5. (15 points) Suppose that in the Solow model, instead of the productio

Question

Problem 5. (15 points) Suppose that in the Solow model, instead of the production function Y = F(K, N), we instead have Y-F(K, hN) where h is the level of human capital per person in the economy. For simplicity, h grows exogenously at the rate g, which means that = h(1 g). The function F is still constant returns to scale in its two inputs K and hN. where hN is interpreted as "effective" labor. Using the constant returns to scale property of F, we are going to define "per effective labor" production function f as K hN where kX We would like to derive the equation that characterizes the steady state k of this altered model. To do so, let's start with the law of capital accumulation K'=(1-d) K +1 Substitute the asset market clearing condition S = 1 to the equation above to obtain K'-(1-d) K +S Then, use the rule of thumb for savings S- sY to obtain F(K, hN) to obtain K' = sF(K.AN) + (1-d) K Then, use the production function Y Problem. (a) (5 points) Divide the last equation by hN and derive the equation that describes k as a function of k. You must show all the work for full credit. Hint: In the original solow model, this equation was k, = What is its equivalent in this altered model? Problem. b) (5 points) Use your answer from Part (a), along with the steady state condi- tion k' = k to derive the equation that determines the steady state k. of this altered model. Hint: In the original solow model his equation was (n dk szf(k). What is its equivalent in this altered model?] Problent. (c) (5 points) Just like the original Solow model, this altered model will also have that k is constant at the steady state, which means y is also constant at the steady state. Then, what is the rate at which the aggregate capital K and the aggregate output Y is growing at the steady state?

Explanation / Answer

A) The production function is given as Y = KN1-

Multiply all the inputs by A

Y = A(AK)(AN)1-

Y = A*AKN1-

Y = A*Y

Constant returns to scale is shown by a production function when increasing the inputs in the given proportion increases the output in same proportion. This implies that the output should be equal to the proportionate increase in the inputs. Here output increases by A when inputs are increased by A so there are constant returns to scale

ii) Find the Marginal product of labor

MPN = d(KN1-)/dN = (1-)(K/N)

When N is increased, MPN falls. Hence there are diminshing returns to labor

iii) Find the Marginal product of capital

MPK = d(K-1N)/dK = (N/K)1-

When K is increased, MPK falls. Hence there are diminshing returns to capital

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