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A birdfeeder manufacturer uses capital (K) and labor (L) and has the production function Q = 5K12L, where is the number of birdfeeders the firm makes per hour
A birdfeeder manufacturer uses capital (K) and labor (L) and has the production function Q = 5K12L, where is the number of birdfeeders the firm makes per hour. Suppose the hourly cost of a unit of K is $60 and the hourly wage rate for L is $20. In the short run, the firm must use 64 units of K. a. What is the marginal product of labor (MP) in the short run? b. What does it cost the firm to make 2,400 birdfeeders per hour in the short run? c. What is the firm's short-run marginal cost of making a birdfeeder? d. Are returns to scale in birdfeeder manufacturing increasing, decreasing or constant?
Expert Solution
16.
Q= 5K1/2 L
Cost of K= r= 60
Wage= w= 20
If K=64, then:
Q= 5(64)1/2 L
Q= 5(8)L
Q= 40L Short run production function
L= Q/40 Equation 1
a.
MPL= dQ/dL= 40
b.
Total cost= Kr+Lw= 60(64)+20L= 3840+20L
From equation 1:
Total cost= 3840+20(Q/40)
Total cost= 3840+(Q/2) Equation 2
If Q= 2400, then:
Total cost= 3840+(2400/2)= 3840+1200= 5040
c.
From equation 2:
Total cost(TC)= 3840+(Q/2)
Marginal cost= d(TC)/dQ= 1/2= 0.5
d.
Q= 5K1/2 L
If both K and L increases by 2 times, then:
Q'= 5(2K)1/2 (2L)= 5(2)1/2 K1/2 (2)(L)= 2.83(5K1/2 L)= 2.83Q
As Quantity increases by 2.83 times when both variables increases by 2 times this condition implies that production function exhibits increasing returns to scale.
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