Problem 6 . Let Q = KL½ where Q is the output, K is capital and L is labor. a. W
ID: 1124847 • Letter: P
Question
Problem 6 . Let Q = KL½ where Q is the output, K is capital and L is labor. a. Write the equation of a sample isoquant. b. Determine whether the production exhibits constant/decreasing/increasing returns to scale. Explairn c. Let w16 and 4 where w is the wage rate and v is the rental rate of capital. Solve for the optimal your answer. input combination to produce 100 units of output. What is the minimized total cost of producing 100 units of output? d. For arbitrary values of w, v and Q solve for the long run input demand curves. e. Suppose that the firms capital is fixed in the short run at K = 100, Find the firm's short run demand function for labor. Find the amount of labor the firm must employ to produce 100 units of output in the short run f. How much money is the firm sacrificing by not having the ability to choose its level of capital optimally in the short run?Explanation / Answer
a) Say, y = 100, the equation of a sample isoquant is 100 = (KL)^0.5
b) Since sum of share of labor and capital in output, (indices or power) is 1, there are constant returns. See that we increase inputs by 'a'
y = (aKaL)^0.5
y = (a^2KL)^0.5
y = a x (KL)^0.5
New y = a x old y
Since output also increases by a, there are CRS
c) Isocost line has an equation C = 16L + 4K. From the optimum rule we have MRTS = w/v or K/L = 16/4. This gives K = 4L and so we have
y = (4L x L)^0.5
100 = 2L
L* = 50 and so K* = 4*50 = 200 units
Minimized cost is C = 50*16 + 200*4 = $1600
d) In the long run we have y = (4L x L)^0.5 or L* = y/2 and K* = 2y. This is the demand functions for inputs. This implies cost function is C = wy/2 + 2yv. This is the long run cost functions
e) Capital is 100. This implies we have y = (100L)^0.5. Demand function for labor is therefore L* = y^2/100. For y = 100. labor units are L* = 100 x 100/100 = 100 units
f) This costs C = 16*100 + 4*100 = $2000. It costs $400 more.