Macroeconomics Intermediate Homework 2 Jongwook Park

Homework 2

Econ 4002.01 Intermediate Macroeconomic Theory

Autumn 2015

*** The due date is September 18 at the start of class.

*** Each question is worth 10 points (30 points for bonus).

*** Show all your work for the full credit.

1. (Returns to scale) Example on Slide 7 in LN3.

Determine whether each of these production functions has constant, decreasing, or increasing returns to scale:

(a) F(K,L)=K+L

F(K,L) = K + L

F (zK, zL) =zK + zL

=z(K+L)

=zF(K,L)

There are constant returns to scale for any z > 0

(b) FK,L=KαL1-α

(Cobb-Douglas production function)

F (K, L) = A Kα L1- α

F(zK,zL) = A(zK)α (zL)1- α

F(zK,zL) = Az α Kαz1- αL1- α

F(zK,zL) = Az α z1- α KαL1- α

F(zK,zL) = Az α+1- α KαL1- α

F(zK,zL) = Az KαL1- α = zA KαL1- α = zF(K,L) = zY

Cobb-Douglas production function has constant returns to scale.

(b) F(K,L)=K1/2+L1/2

(d) F(K,L)= K2+L2

(e) F(K,L)= K2L

F(zK,zL)= (zK)^2 / (zL)

=(z^2 * K^2)/zL

=z ( k^2 / L)

=zF (K,L)

There are constant returns to scale for any z > 0

2. (MPL and demand for labor) Example on Slide 21 in LN3.

1. If L=3 then the firm should hire more labor. The benefit of hiring the fourth laborer, MPL=7, is greater than the cost, W/P=6, so it is beneficial.

2. If L=7, then no, the firm should not hire more laborers. The 8th worker adds only 3 units MPL but costs W/P 6.

3. (Cobb-Douglas production function) Problem 2 on Slide 29-34 in LN3.

Suppose the production function in medieval Europe is Y=K0.5L0.5, where K is the amount of land and L is the amount of labor. The economy begins with 100 units of land and 100 units of labor.

a. How much output does the economy produce?

Y=k^.05 * L ^ .05

Y=100^.05 * 100^.05

=100

b. What are the (real) wage and the (real) rental price of land?

(MPL=0.5K0.5L-0.5, MPK=0.5K-0.5L0.5)

MPL=(1-a)A K^a L^-a;

MPK=a A K^(a-1) L^(1-a)

a=.5

A=1

MPL=.5

MPK=.5

c. What share of output does labor receive?

MPL *L...