Firm A’s production function and cost line are given by:
Q=Q(L,K)=2 L^(1/2) K^(1/2) (The production function)
?L+?=30000 (The cost line of firm A):
?L is the amount of labor hired.
?K is the amount of capital hired.
??p_L or the price of labor is 1 dollar per unit.
??p_K or the price of capital is 1 dollars per unit.
C or cost (think of it as the firm’s budget) is 30000 dollars.
How much labor and capital should this firm optimally hire?
Ans. Production function, Q = 2*L^(0.5) * K^(0.5)
Marginal product of labour, MPL = dQ/dL = (K/L)^0.5
Margial product of capital, MPK = dQ/dK = (L/K)^0.5
=> Marginal rate of technical substitution, MRTS = dK/dL = MPL/MPK = K/L
At optimal level,
MRTS = pL/pK
=> K/L = 1/1
=> K = L ---> Eq1
Substituting Eq1 in the cost line, L+ K = 30000, we get,
K + K = 30000
=> K = L = 15000 units
Thus, optimal level of labour hired 15000 units and capital hired is 15000 units.
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