Question

1. For each of the following production functions (a-c) answer the following questions (i-v):

i) Calculate the marginal products MPL and MPK.

ii) Calculate MRTSL;K and determine if this is diminishing as good L increases.

iii) If Q0 = 100, w = 8 and r = 2, determine long-run cost minimizing combination of labor and capital and the associated total cost.

iv) If Q0 = 100, w = 8 and r = 2, determine short-run cost minimizing combination of labor and capital and the associated total cost if the firm is stuck with K = 4 unit of capital in the short run.

v) Do we have too much or too little capital in the short run, and how much money is the firm losing by being stuck with K = 4 unit of capital in the short run?

(a) Q(L;K) = 10L^1/2K^1/2.

(b) Q(L;K) = 4L^2 + 4K^2.

(c) Q(L;K) = 4LK + 6K.

Answer 1

We will be using the following to answer these questions.

a)

Q(L,K)=10L^1/2K^1/2

i)

MPL= 10(1/2)L^-1/2K^1/2

MPL= 5L^-1/2K^1/2

MPK= 10(1/2)L^1/2K^-1/2

MPL= 5L^1/2K^-1/2

ii)

As L increases, K decreases.

iii)

Q=100

w=8

r=2

Therefore the cost function can be written as C=wL+rK

C = 8L+2K

and 100=10L^1/2K^1/2

Therefore we create the Lagrangian, minimize C=8L+2K subject to the constraint 100=10L^1/2K^1/2 which gives us

Differentiating the Lagrangian wrt L

Differentiating the Lagrangian wrt K

Equating both the equations for Lamda we get,

Using this relationship in the constraint equation we get

Total cost = 8L+2K

Therefore, TC=8(5)+2(20)

TC=80

iv)

In the short run assuming that K=4

C=8L+8

subject to the constraint 100=10L^1/2(2)

100=20L^1/2

L^1/2=5

Therefore L=25

Calculating Total cost for this level L and K-

TC=8*25+2*4

TC=208

v)

In the short run the firm has too little capital as the equilibrium level K=20 while in the short run we have only 4 units of K available. As a result of this firm loses profits due to increasing total costs. The difference in nthe TC being 208-80=128 is the loss of profits.