A competitive, cost-minimizing firm has the production function f (x, y) = x + 2y and uses positive amounts of both inputs. If the price of x doubles and the price of y triples, then the cost of production will more than double.

Question

Quizplus · Accepted Answer

The cost of production will not more than double. In fact, we can calculate the new cost of production:

New cost = (2p_x)x + (3p_y)y

Original cost = px + py

If we assume that the original cost is equal to 1 (for simplicity), then the new cost is:

New cost = 2x + 3y

Using the production function, we can substitute x + 2y for f(x,y) in the new cost equation:

New cost = 2x + 3y = 2(f(x,y) - 2y) + 3y = 2f(x,y) - y

So, the new cost is twice the original cost minus some amount of y. Since the firm is cost-minimizing, it will choose the input combination (x,y) that minimizes the cost of production. This means that the firm will adjust its input usage in response to the price changes to minimize the new cost. Specifically, the firm will use less y and more x than before, since the price of y has increased more than the price of x. Therefore, the new cost will be lower than it would be if the firm used the same input combination as before. Hence, the statement "the cost of production will more than double" is false.

A Competitive, Cost-Minimizing Firm Has the Production Function F (X