2. Assume output (real GDP) is produced with capital and labor with the Cobb-Dou
ID: 1115851 • Letter: 2
Question
2. Assume output (real GDP) is produced with capital and labor with the Cobb-Douglas production function: Y = AK" N", . An individual's flow of utility, as a function of consumption, C, and hours worked, N, is given by: U(C, N)-Ln(C)-N ) Express the marginal product of labor of firms in terms of output and hours. Calculate the marginal rate of substitution (ratio of marginal disutility of hours worked to marginal utility of consumption b) Assuming firms equate marginal product to the real wage and workers equate marginal rate of substitution to the real wage, solve for equilibrium hours worked as a function of Y/C c) Suppose the economy has no government spending nor net exports, so Y= C+1 . Suppose the ratio I/Y goes from 0.2 to 0.25. Explain by what percent hours worked will change, given the assumptions in part (b).Explanation / Answer
Constant returns to scale implies that output increases proportionately to increase in inputs. Let both K and L increase by x. So the new production function Y' becomes:
Y'= A (xK)0.36 (xL)1-0.36 ....substitute for alpha
Y'= A (xK)0.36 (xL)0.64
Y'= A (x0.36K0.36) (x0.64L0.64) ....opening the brackets
Y'= A (x0.36 x0.64 )(K0.36) (L0.64) ..........taking x components together
Y'= A (x0.36 +0.64 )(K0.36) (L0.64) .... (rule of ab ac = ab+c)
Y'= A (x1 )(K0.36) (L0.64)
Y'= x A(K0.36) (L0.64)
Y'= x Y
Hence ouput increased in proportion to increase in inputs.
b) For this part, suppose r= rent/price of capital and w= wage /price for labour.
In order to find the factor share we need to maximise profits we need to set up
profit = production - cost
p = A (K)0.36 (L)0.64 - wL - rK
now profit maximising equation has dp/dk =0 and dp/dL=0
dp/dk = A (0.36K)0.36-1 (L)0.64 - rK =0
therefore MPK = r (MPK= marginal product of capital)
share of capital = r.K/Y = (MPK. K )/ Y
similarly for labour, MPL = w
share of labour = w.L/Y = (MPL.L) / Y
c) When labour increases by 20% now L' = (1+0.2)L = 1.2L. so production function becomes:
Y'= A (K)0.36 (L')0.64
Y'= A (K)0.36 (1.2L)0.64
Y'= (1.20.64 [A (K)0.36 (L)0.64]
Y'= 1.20.64 Y
from part b) retal price of capital = r = MPK
solve for dY'/dk= d(1.20.64 [A (K)0.36 (L)0.64]/ dk = 1.20.64 [A (0.36K)0.36-1 (L)0.64]
similarly,
retal price of labour= w = MPL
solve for dY'/dL= d(1.20.64 [A (K)0.36 (L)0.64]/ dk = 1.20.64 [A (K)0.36 (0.64L)0.64-1]
d) goes the same way as part c) but here K' = (1+0.2)K = 1.2K
substitute in the production function to get the rest of the part.
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