Question

A firm uses only two inputs L and K, which have prices PLand PK. Its production function is q = 4L3K2

a. (3) Derive themarginal product of K and L (you can leave it unsimplified).

b. (3) Derive the firm’s marginal rate of substitution including the formula you use (you can leave it unsimplified)..

c. (3) Assume now instead that q = LK. Using this, solve for K(as a function of L and q).

d. (8) Assume PL= 2, Pk= 1, and q = 2. Substitute these values into the cost expression and derive using calculus how much L it employs to minimize costs. Points for (explaining)each stepand you may leave your answer as a fraction. From this, find how much capital is used andthe cost of producing 2 units.

Please do all parts

Answer #1

The question seems not posted properly as there are some
unreadable signs in the production function. Assuming that the
prodcution function is q=4L^{3}K^{2}.

**A.** The marginal prodict of K would be

MPK=dq/dK=8L^{3}K^{1}.

Marginal product of L would be

MPL=dq/dL=12L^{2}K^{2}.

**B.** We know that

MRS=MPL/MPK. So, using MPK and MPL from above, we get

MRS=12L^{2}K^{2}/8L^{3}K^{1}.

MRS=3K/2L.

**C.** The new production function is given as
q=LK.

Solving for K, in terms of q and L, we get

K=q/L

**D.** We know that

Cost function= P_{L}L+P_{K}K

We also established, in part C, K=q/L. Putting this and the values given above, we get

Cost function=2L+2/L.

The cost would be minimum where the differentiation of the cost function would be zero. In other words, where

2-2/L^{2}=0

**L=1 will minimize costs**

Since K=q/L and q=2,

**K=2/1=2.**

So, the cost of producing 2 units is

2*1+1*2**=4**

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