Tricky Production Function Optimisation

[paddy1]

Suppose that a firm in perfect competitive markets has the following production function:
1 1
Q = f (K,L) = K^(1/3)*L^(1/3) ,
where Q,K and L denote the production level, capital inputs and labour inputs.

The firm's cost function is given by
TC = r*K + w*L ,
where r is the costs of using each unit of capital and w is the costs of using each unit of labour.

Suppose that capital costs are $2 per unit, that is r=$2 , labour costs are $1 per unit, that is w=$1 and the product sells at $3 per unit.

Solve for the optimal value of K and L that achieves the maximum level of
profit.

 

[Aaron Bex]

Let P be the profit.
P = TR - TC
= 3Q - (2K + L)
= 3(K^(1/3)*L^(1/3)) - (2K + L)
= 3(K^(1/3)*L^(1/3)) - 2K - L

We want to maximize P, so we take the partial derivatives of P with respect to K and L and set them to zero.

dP/dK = 0
=> 3*(1/3)*K^(-2/3)*L^(1/3) - 2 = 0
=> K^(-2/3)*L^(1/3) = 2/3
=> K = (2/3)^3 * L

dP/dL = 0
=> 3*(1/3)*K^(1/3)*L^(-2/3) - 1 = 0
=> K^(1/3)*L^(-2/3) = 1/3
=> K = (1/3)^3 * L

Comparing these two results, we see that K = (2/3)^3 * L = (1/3)^3 * L.

Therefore, the optimal value of K and L that achieves the maximum level of profit is:
K =