Answer to Question #218092 in Microeconomics for Yves10-Official

Question

A firm's unit capital and labor costs are $1 and $2 respectively. If the production function is given by Q = 4LK + L^2. Find the maximum output and the level of K and L at which it is achieved when the total in put cost is fixed at $105. Verify that the ratio of marginal product to price is the same for both inputs at the optimum

Answer 1

The maximum output of the production function can be determined by finding the marginal product of labor.

The marginal product of labor refers to the change in the level of output that arises when the producer adds additional units of labor inputs.

Total cost refers to the fixed cost and variable cost. It refers to the overall cost of input that is used to produce output.

The marginal revenue product of labor refers to the change in the marginal revenue when there is a change in the level of labor input used.

The output can be maximized when the marginal product is maximized.

$Q=4LK+L^2\\Marginal\space product=\frac{dQ}{dL}\\=4K+2L\\4K+2L=0\\2L=-4K\\L=-2K$

$\frac{d\space Marginal\space Product}{dL}=2>0,$ Hence the output cannot be maximized

$Q=4(-2K)K+(-2K)^2\\=-8K^2+4K^2\\=-4K^2$

The output is maximized only when the second-order differential equation of the production function (i.e., first-order differentiation of marginal product) is less than 0. If the second-order differential equation of the production function is greater than 0, the output is minimized.