Suppose the production function for Alfredo Barbua, a maker of fine violins, is given by Q = 0.5K0.5L0.5.
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?
Suppose the production function for Alfredo Barbua, a maker of fine violins, is given by Q = 0.5K0.5L0.5.
A) K = 1 L = 4, Q = 0.5*(1^0.5)*(4^0.5) = 1 violin
B) It will be the ratio that satisfies |MRTS| = w/r
0.5*(K/L)^0.5 / 0.5*(L/K)^0.5 = 20/15
K/L = 4/3
This 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?
C = 12000
wL + rK = 12000
20L + 15K = 12000
20L + 15*(4/3L) = 12000
20L + 20L = 12000
L* = 300 and K* = 400.
Maximum number of violines Q = 0.5*(400^0.5)*(300^0.5) = 173 violins.
D) If the orchestra pays him $500 per violin, what are his potential profits?
173*500 - 12000 = 74500.
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