Question 1 Suppose a firm has the Cobb-Douglas production function Q = f(K, L) =

Question

Question 1

Suppose a firm has the Cobb-Douglas production function Q = f(K, L) = AK^alpha L^beta, where alpha beta > 0, alpha + beta = 1; A is a positive parameter that represents technology, technology, K is capital, and L is labor. Using this function, show the following. Does this production function exhibit increasing, constant, or decreasing returns to scale? Why? Please draw the set of three isoquants that support your answer in part (1). What are the characteristics of these isoquants? Please find MP_L and MP_K. Please show diminishing marginal productivities for both K and L. Given the total cost outlay such as C = wL + rK where w is the wage of labor and r is the rental price of capital. Please determine the amount of labor (L*) and capital (K.*) that the firm should use in order to maximize output. Suppose that Q = 5 Squareroot KL, w = 5, r = 20, C = 1000. Please find the optimal quantities of labor and capital that this firm should hire. How much output can the firm produce?

Explanation / Answer

1) This function exhibits constant return to scale because a+B = 1.

3) MPL = dQ/dL = A BKaLB-1

and MPK = dQ/dK = AaKa-1LB

4) MPK = AaKa-1LB

let K = nK where n is a positive integer.

New MPK = Aana-1Ka-1LB = na-1(AaKa-1LB ) = na-1(old MPK)

ans as we know a+b = 1 that is a<1 which means n>na-1 therfore by increasing K with n we get return in MPK where we dont get 1 unit f n fully. Hence it is diminishing.

Similarly,

MPL = ABKaLB-1

let L = nL where n is a positive integer.

New MPL = ABKanB-1LB-1 = nB-1(ABKaLB-1 ) = nB-1(old MPL)

ans as we know a+b = 1 that is b<1 which means n>nb-1 therfore by increasing L with n we get return in MPL where we dont get 1 unit f n fully. Hence it is diminishing. also.

5) At the optimal allocation to maximize profit, MPK/MPL = r/w

AaKa-1LB / ABKaLB-1 = r/w

aL/BK = r/w

L* = BKr/aw

and K* = aLw/Br

6) Optimal unit of L = BKr/aw = 0.5K20/0.5*5 = 4K

and from cost function we get, K = (C-wL)r = (1000-5L)/20 = 50-0.25L

Thus L = 4K or L = 4(50-0.25L) or L = 100 labors

Similarly, Optimal unit of K =  aLw/Br = 0.5L5/0.5*20 = 0.25L

again from cost function, L = (C-rk)/w = (1000-20K)/5 = 200-4K

and thus K = 0.25L or K = 0.25(200-4K) or K = 25 units.

Output = 5(100*25)0.5 = 250 units.

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