Question

A firm in a perfectly competitive industry has patented a new process for making widgets. The new process lowers the firm's average cost, meaning that this firm alone (although still a price taker) can earn real economic profits in the long run. a. If the market price is \(\$ 20\) per widget and the firm's marginal cost is given by \(M C=0.4 q\), where \(q\) is the daily widget production for the firm, how many widgets will the firm produce? b. Suppose a government study has found that the firm's new process is polluting the air and estimates the social marginal cost of widget production by this firm to be \(S M C=0.5 q .\) If the market price is still \(\$ 20,\) what is the socially optimal level of production for the firm? What should be the rate of a government-imposed excise tax to bring about this optimal level of production? c. Graph your results.

Short answer

Answer: The socially optimal output level for widget production, considering the pollution costs, is 40 widgets. The government should impose an excise tax of $4 per widget to achieve this level.

Step-by-step solution

Determine the profit-maximization condition for the firm

To maximize profit, the firm should produce output at the level where the marginal cost (MC) equals the market price (P). In this case, the market price is \(P = \$20\), and the marginal cost function is \(MC = 0.4q\).

Set the marginal cost equal to the market price and solve for production level q

To find the profit-maximizing output level, set the marginal cost (MC) equal to the market price (P) and solve for q: \(0.4q = \$20\) Now, solve for q: \(q = \frac{\$20}{0.4}\) \(q = 50\) So, the firm will produce 50 widgets to maximize its profit. b. Finding the socially optimal level of production and the excise tax

Determine the socially optimal output level

The firm's socially optimal output level is where the social marginal cost (SMC) equals the market price (P). The social marginal cost is given by \(SMC = 0.5q\). Set SMC equal to the market price and solve for q: \(0.5q = \$20\) Now, solve for q: \(q = \frac{\$20}{0.5}\) \(q = 40\) The socially optimal output level is 40 widgets.

Calculate the rate of the excise tax

To find the excise tax rate, we need to find the difference between the social marginal cost (SMC) and the original marginal cost (MC) at the socially optimal output level: \(SMC - MC = 0.5(40) - 0.4(40)\) \(20 - 16 = \$4\) The government should impose an excise tax of \( \$4\) per widget to achieve the socially optimal level of output. c. Graphing the results

Plot the graphs

Plot separate graphs for the marginal cost (MC) and social marginal cost (SMC) curves. On each graph, the x-axis represents the output level (number of widgets) and the y-axis represents the price and cost. - On the MC graph, plot a line with the equation \(MC = 0.4q\). - On the SMC graph, plot a line with the equation \(SMC = 0.5q\).

Add the market price line

On both graphs, draw a horizontal line at the market price of \(\$ 20\).

Find the firm's output levels and indicate them on the graphs

On each graph, mark the points where the MC or SMC curve intersects with the market price: - MC graph: The optimal output level with no external costs is 50 widgets. - SMC graph: The socially optimal output level, considering pollution costs, is 40 widgets.

Add the excise tax rate

On the SMC graph, draw an arrow on the y-axis representing the excise tax rate of \(\$ 4\) from the original MC curve to the SMC curve at the socially optimal output level of 40 widgets.

Key concepts

Economic Profit

In a perfectly competitive market, economic profit is the financial gain a firm makes after accounting for both its explicit and implicit costs. It's a measure of profitability that indicates whether the company is generating a return that exceeds the opportunity costs of its resources. Unlike accounting profit, which only considers explicit costs, economic profit evaluates the true financial performance by including the cost of opportunities foregone.

For example, a firm that innovates a more efficient production process, thereby lowering average costs, has the potential to earn an economic profit. This is because they can sell their products at market price while incurring lower costs than competitors, which leads to a return above the normal profit level (break-even point). However, in perfectly competitive markets over the long run, the entry of new firms attracted by the prospect of profit will drive prices down, eventually eroding economic profits.

Social Marginal Cost

The term social marginal cost (SMC) represents the total cost to society of producing one additional unit of a good or service. This includes not just the private costs incurred by the producer, such as materials and labor, but also external costs, like environmental pollution, that are not reflected in the market price.

An understanding of SMC is crucial for recognizing the true cost of an economic activity. In the case of a firm that causes pollution, the SMC would be higher than the private marginal cost because it reflects the harm done to the environment. When firms only consider their private costs, they may produce more than the socially optimal output level, leading to negative externalities.

Excise Tax

An excise tax is a targeted tax charged on specific goods, such as tobacco, alcohol, and, in the context of our exercise, polluting widgets. This type of tax is usually leveraged to correct for externalities, with the aim of reducing the consumption or production of goods that have additional social costs.

In our example, if a firm's widget production pollutes the environment, an excise tax can be set to encourage the firm to reduce production to the socially optimal level. The tax rate is ideally set equal to the difference between the social marginal cost and the private marginal cost. This encourages firms to internalize the external costs, aligning private incentives with social welfare.

Profit Maximization

The goal of profit maximization is at the heart of most enterprises and refers to the process of determining the optimal price and quantity of goods to sell in order to maximize earnings. In a perfectly competitive market, a firm maximizes profit by producing goods up until the point where the additional cost of producing one more unit – the marginal cost (MC) – is equal to the market price (P) of the good.

This condition ensures that the firm is not spending more to produce a unit than what it can sell it for. However, this principle does not account for social costs, and that's where the concept of social marginal cost (SMC) becomes important for evaluating the broader impact on society.

Externalities

The concept of externalities is key to understanding the impact of economic activities that extend beyond private transactions. Externalities are costs or benefits arising from a production or consumption activity that affect third parties who are not involved in the transaction.

Negative externalities, such as pollution, result in social costs that exceed the private costs of production. These can lead to overproduction relative to the socially optimal level as the marketplace does not naturally charge for these additional costs. Conversely, positive externalities, such as the benefits from vaccinations, can lead to under-consumption as the marketplace does not naturally reward the additional benefits. Governments often intervene to correct these externalities, using instruments such as taxes, subsidies, or regulations.