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Homework answers / question archive / A competitive firm produces output y using two inputs, labour L and capital K, The production function is given by F(K,L)=K1/31/3L1/31/3
A competitive firm produces output y using two inputs, labour L and capital K, The production function is given by F(K,L)=K1/31/3L1/31/3.The firm takes the input and output prices as given and they are: output price p=12, price of labour w=2 and price of capital r=2. Find the value of L that maximises the firm's profit?
The production function is given by F(K,L)=K1/31/3L1/31/3
Price of output is , p=12
Price of Labour, w=2
Price of Capital, r=2
Profit=Revenue - Cost
Revenue= p* K1/31/3L1/31/3
Cost=w*L+r*K
Differentiating profit w.r.t K
d(Profit)/dK=d(Revenue)/dK-d(Cost)/dK
=4K^(-2/3)L^(1/3)-2
Using F.O.C
4K^(-2/3)L^(1/3)=2 --------------------------------------------(i)
Similarly, differentiating profit w.r.t L
4L^(-2/3)K^(1/3)=2 --------------------------------------------(ii)
(i)/(ii)
K/L=1
K=L
Putting the value in equation (i)
L=8
K=8
The profit is maximum when labour and capital are 8 units each.
The profit is 16 units
