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. The firm should produce 4 units of output in the long run to maximize profit.

Step-by-step solution

Determine current profit situation

Compute the Marginal Revenue (MR) at the given conditions. In a competitive market, MR is equal to the price of the output commodity, which is $20. Compare this with the Marginal Cost (MC) at the current quantity of 5 units, using the given MC function, \(\mathrm{MC}=4+4 q\). Substituting \(q = 5\) into the equation gives \(\mathrm{MC} = 24\). The comparison shows that the MR is less than MC at the current level of production, indicating that the firm is not maximizing its profit.

Determine optimal quantity of output

The firm maximizes its profit when MC equals MR. Set the MC function equal to the price since in a competitive market, price equals MR. Solve the equation \(4+4q = 20\) to get the optimal quantity \(q = 4\). So, in long run, the firm should produce 4 units of output.

Key concepts

Understanding Marginal Cost in Profit Maximization

Marginal cost is a crucial concept in economics, particularly when it comes to maximizing profits. It refers to the additional cost incurred by producing one more unit of a product. The formula for marginal cost is often derived from the total cost function. In the provided exercise, the marginal cost function is given as \( \mathrm{MC} = 4 + 4q \). This function tells us how the cost of production changes with each additional unit produced.

In a competitive market environment, firms constantly compare their marginal cost with the market price to decide how much to produce. If the marginal cost of producing an extra unit is less than the market price, the firm can increase profit by producing more. Conversely, if the marginal cost is greater, the firm should reduce production. In this exercise, at 5 units, the marginal cost was 24, which exceeded the market price of \( \$20 \), indicating a need to adjust output.

Key Points to Consider:

• The marginal cost rises as more units are produced due to factors like increased use of inputs.
• Understanding the marginal cost helps in identifying the optimal level of production where profits can be maximized.

Characteristics of a Competitive Market

A competitive market is characterized by many firms, none of which can influence the market price. This means each firm acts as a price taker, basing decisions on the prevailing market price. In such markets, the marginal revenue (MR) of producing an additional unit equals the market price, simplifying profit maximization calculations.

In the exercise scenario, the relevant market price for the firm was given as \( \\(20 \). This constant price is crucial because:

• It ensures firms adjust production based on changes in marginal cost rather than price strategies.
• In the long run, it leads to firms meeting their marginal cost at the market price level.This balances output to the point where no economic profits are earned, only normal profits.

For students, understanding how price-taking behavior influences a firm's production decisions is essential. It clarifies why, at 5 units, the firm's marginal cost was higher than \( \\)20 \), showing a non-optimal level of production for profit maximization.

Reaching the Long-run Output

In the long run, a firm in a competitive market adjusts its output to optimize profits. Long-run output is reached when the marginal cost (MC) equals the marginal revenue (MR), which is also the market price in this context. For the firm in the exercise above, this point is found by solving the equation \(4 + 4q = 20\), leading to the conclusion that it should produce 4 units of output.

Key considerations for long-run output:

• The firm aims for a level where its profits are maximized, meaning no opportunity is left for earning additional profit by changing production levels.
• Over time, firms can adjust all inputs to ensure they operate efficiently at this equilibrium point.Reaching this balance implies the firm is earning just enough to cover all costs, including opportunity costs, ensuring sustainability.

The importance of understanding long-run output is that it helps in planning strategic decisions for future growth. It ensures firms remain competitive and efficient within the market landscape.