Question

A firm produces a product in a competitive industry and has a total cost function \(C=50+4 q+2 q^{2}\) and a marginal cost function \(\mathrm{MC}=4+4 q\). At the given market price of \(\$ 20,\) the firm is producing 5 units of output. Is the firm maximizing its profit? What quantity of output should the firm produce in the long run?

Short answer

No, the firm is not maximizing its profit by producing 5 units of output as the marginal cost (24) is more than the market price (20). The firm should produce 4 units of output in the long run to maximize its profit.

Step-by-step solution

Calculate the Marginal Cost for current production

The given marginal cost function is MC=4+4q. Substitute q=5 (current production level) into the MC function: MC = 4 + 4*5 = 24.

Check Profit Maximization

Compare the marginal cost with the market price. Here, MC (24) > Price (20), which means that the firm is not maximizing its profit. The firm is producing more than the profit maximizing quantity as the cost of producing an additional unit is more than what it can be sold for.

Determine the Optimal Output Level

Set the marginal cost equal to the market price to find the profit maximizing output level: 4 + 4q = 20. Solve this equation to find the value of q. From this, we get q = 4. So, the firm should produce 4 units in the long run to maximize its profit.