Question

In: Economics

Consider the same firm from above but this time assume the firm is operating in the...

Consider the same firm from above but this time assume the firm is operating in the long-run, i.e. it can vary K as well. As before, the firm’s production function is q = 4K0.5L 0.5 and the wage rate is w = $8 and the rental rate of capital is r = $2.

a) What is the optimal long-run relationship between K and L?

b) If the firm is producing 24 units, how much labor will it employ?

c) How much capital will it employ?

d) What is the (long-run) cost of producing 24 units?

e) If the total cost function is linear then, using your answer from (d), what is the slope of this line and the long-run total cost function?

f) Comparing your answers to 3(f) and 2(c), why is the average total cost of producing 8 units the same in the short- and long-run? Why is the long-run ATC of producing a different quantity smaller than the short run ATC?

Thank you for your help, I appreciate it!

Expert Solution

a) Optimal long-run relationship between K and L is determined at the point where MRTS = w/r
So, MRTS = MPL/MPK
MPL =

MPK = $\frac{\partial q}{\partial K} = 4(0.5)K^{0.5-1}L^{0.5} = 2K^{-0.5}L^{0.5}$

So, MRTS =
So, K/L = w/r = 8/2 = 4
So, K = 4L

b) q = 4K0.5L 0.5 = 24
So, q = 4(4L)0.5L 0.5 = 24
So, 4*2L^0.5+0.5 = 24
So, 8L = 24
So, L = 24/8 = 3

L = 3

c) K = 4L = 4(3) = 12
So, K = 12

d) Cost = wL + rK = 8(3) + 2(12) = 24 + 24 = 48

e) Slope = -dK/dL = -(dC/dL)/(dC/dK) = -w/r = -8/2 = -4

f) Parts 3f and 2c are not given. So this can not be answered.

Rahul Sunny answered 2 years ago

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