Question

Suppose the production function for Alfredo Barbua, a maker of fine violins, is given by Q = 0.5K^0.5L^0.5.

A) If Alfredo is currently using 1 unit of capital, how many violins will he produce if his workers work for 4 hours?

B) If the wage rate is $20 and rental rate is $15, what is Alfredo’s cost minimizing ratio of labor to capital?

C) Alfredo needs to supply an orchestra with as many violins as he can, but his costs cannot exceed $12,000, what’s the maximum number of violins he could make for them?

D) If the orchestra pays him $500 per violin, what are his potential profits?

Answer 1

a) Q = 0.5 k^0.5 L^0.5

K = 1

L = 4

Q = 0.5 (1)^0.5 (4)^0.5

Q = 1

b) w = 20

r = 15

MPL = dQ/dL = d [0.5 k^0.5 L^0.5] /dL

MPL = 0.25 k^0.5 L^-0.5

MPK = dQ/dK = d [0.5 k^0.5 L^0.5] /dK

MPL = 0.25 k^-0.5 L^0.5

(MPL/w) = (MPK/r)

(0.25 k^0.5 L^-0.5)/ 20 = (0.25 L^0.5 K^-0.5)/ 15

( k^0.5 L^-0.5)/ (L^0.5 K^-0.5) = (20/15)

(K/L) = (4/3)

c) Using rental rate = 15$

wage rate = 20

C = L w + K r

Q = 0.5 L^0.5 K^0.5

Q/0.5= (LK)^0.5

2Q = (LK)^0.5

SOBS

4Q² = LK

4Q²/K = L

K = 1 from ques A

L = 4Q²

C = L w + K r

12000 = 4Q² x 20 + 1x15

12000 = 80Q² + 15

Q² = 11985

Q = 109.47 app 110 violins