Suppose a firm's production function is given by the equation Q = 12L.5K.5 . This firm operates in the short run where capital (K) is fixed at a quantity of 16. If the price per unit of the good is $1.8 and labor costs $10 per unit. Then the profit maximizing amount of labor in the short run is ___?
Answer
Total Cost = wL + vK where w = wage rate = 10 , v = rent(and is constant)
=> Total Cost = 10L + 16v
Total revenue = Price *Quantity = 1.8Q = 1.8*12L.5K.5 = 1.8*12L.516.5 = 86.4L.5
Profit(Pr) = Total revenue - Total Cost
=> Profit(Pr) = 86.4L.5 - 10L + 16v
First order condition;
d(Pr)/dL = 0 => 0.5*86.4/L.5 - 10 = 0
=> L = (0.5*86.4/10)2
=> L = 18.66
Thus, the profit maximizing amount of labor in the short run is 18.66 or 19 (approx) units
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