The firm Avanti produces custom-made sports cars that appeal to a certain client

Question

The firm Avanti produces custom-made sports cars that appeal to a certain clientele because of their high price and thus exclusivity. Avanti has the following production function: q = f(k, l) = k 2/3 l 1/3(K to the power of 2 over 3, times L to the power of 1 over e) , where k and l are the capital and labor input respectively.

(a) What additional output does one extra marginal unit of capital produce when the capital-labor ratio is (k/l) = 8? (2)

(b) If you employ one marginal worker less, how many additional units of capital do you need to produce the same output? Again, assume that the capital-labor ratio is (k/l) = 8. [Do not assume this any further.] (2)

(c) How many units of labor does the firm need to produce q units of output in the short run when the capital stock is fixed at a level of k = 5? (2)

Explanation / Answer

K/L = 8

a) additional output by one extra marginal unit of capital = MPK

MPK = dq/dK = 2/3 x (L/K)1/3 = 2/3 x 1/2 = 1/3

b) MPL = dq/dL = 1/3 x (K/L)2/3 = 1/3 x 4 = 4/3

Additional units of capital needed = MPL/MPK = 4/3 x 3/1 = 4

c) q = 52/3 x L1/3

L = q3/25 (L = labour required)

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