Question

A perfectly competitive firm's total cost function is tc = q² - 25q + 144. a) If the market price is $75, what is the profit-maximizing output (q*)? b) Calculate the break-even point.

Answer 1

To find the profit-maximizing output, set the marginal cost equal to the market price. The profit-maximizing output is 50 units. The break-even point is the quantity of output at which total cost equals total revenue, and it is approximately 9.56 units.

Step-by-step explanation:

A perfectly competitive firm determines its profit-maximizing output by equating marginal revenue (which is equal to the market price) with marginal cost. To find the profit-maximizing output, we need to find the quantity of output (q) where marginal cost equals the market price.

a) To find the profit-maximizing output:

1. Calculate the marginal cost by taking the derivative of the total cost function: MC = d(tc)/dq = 2q - 25.
2. Set the marginal cost equal to the market price: 2q - 25 = 75.
3. Solve for q: 2q = 100, q = 50.

The profit-maximizing output (q*) is 50 units.

b) To calculate the break-even point:

The break-even point is the quantity of output at which total cost equals total revenue.

1. Calculate the average cost by dividing the total cost by the quantity of output: AC = tc/q = (q^2 - 25q + 144)/q = q - 25 + 144/q.
2. Set the average cost equal to the market price: q - 25 + 144/q = 75.
3. Solve for q using algebra or a numerical method: q = 9.56 (rounded to 2 decimal places).

The break-even point is approximately 9.56 units.