Question

Consider a competitive firm.

a) Sketch the cost function as a function of the price of good 1, assuming that the production function is Leontiff.

b) Sketch the cost function as a function of the price of good 2, assuming the production function is linear in good 2.

Answer 1

a) The following is an example of a Leontief production function:

The firm's cost function is given by:

From the given production function, the firm treats labor and capital as complements and always hires equal amounts of both at equilibrium. Hence, L = K = Q. The cost function now becomes:

In order to maximize profit, a competitive firm sets price equal to its marginal cost.

Hence,

Plotting the firm's cost with price on the vertical axis and quantity on the horizontal axis.

b) An example of a linear production function is:

The firm's cost function is given by:

Since the firm treats the two goods as perfect substitutes, it will only hire the factor which is cheaper. Hence, the firm's cost function is given by:

Following the same process as the previous part, a competitive firm sets price equal to its marginal cost to maximize profit. Cost function is:

Plotting the firm's cost with price on the vertical axis and quantity on the horizontal axis.