Question

A firm that produces steel is polluting the air. Assume that the marginal cost of producing steel is constant at \(£ 4\). The inverse market demand for steel is \(P=44-\) \(2 \mathrm{Q}\), where \(P\) is the price of steel and \(Q\) is the 1 quantity of steel. The air pollution associated with steel production is creating an externality given by \(£ 2 \mathrm{Q}\). Assuming that the market for steel is competitive, what is the profit- maximizing level of steel when only marginal private costs are taken into account? The marginal social costs are given by the sum of the marginal private costs plus the externality. What is the social level of steel output? Show your solution graphically. What is the social loss associated with the externality? How can we solve this externality problem using taxation?

Short answer

Profit-maximizing output is 20 units; socially optimal output is 10 units. Social loss is £100. Taxation of £2Q can resolve the externality.

Step-by-step solution

Find Profit-Maximizing Quantity Considering Marginal Private Costs

In a competitive market, firms maximize profit where marginal cost (MC) equals marginal revenue (MR). Given the inverse market demand \(P = 44 - 2Q\), the marginal revenue \(MR\) for a competitive firm is equal to the price \(P\). Since the marginal cost (MC) is constant at \(£4\), set \(MC = MR\):\[4 = 44 - 2Q\]Solving for \(Q\), we have:\[2Q = 40\] \[Q = 20\]So, the profit-maximizing level of steel output is 20 units.

Find Socially Optimal Quantity Including Externality

The marginal social cost (MSC) is the sum of the marginal private cost (MPC) and the externality. Given \(MPC = £4\) and the externality as \(£2Q\), the MSC is:\[MSC = 4 + 2Q\]Set the MSC equal to the current market price \(P\) for social optimization:\[4 + 2Q = 44 - 2Q\]Solving for \(Q\), we get:\[4Q = 40\] \[Q = 10\]The socially optimal level of steel output is 10 units.

Calculate Social Loss Due to Externality

The social loss is the difference in cost due to the externality for the profit-maximizing vs. socially optimal output. Calculate the loss as the area of the triangle formed between MSC and MPC from \(Q = 10\) to \(Q = 20\): the externality (\(£2Q\)) adds costs between \(Q = 10\) and \(Q = 20\), with a cost range from \(20\) to \(40\):\[\text{Loss} = \frac{1}{2} \times (20 - 10) \times (40 - 20) = 100\]The social loss is £100.

Solve Externality Problem using Taxation

To correct for the externality, impose a tax equal to the marginal external cost at the socially optimal level of output, which is \(£2Q\). At \(Q = 10\), this is \(£20\), ensuring MSC equals MPC. This tax will shift the MPC curve upward by \(£2Q\), aligning MPC with MSC at the socially optimal \(Q = 10\). By doing so, the market participants internalize the externality costs, reducing output to the socially optimal level.

Key concepts

Marginal Cost

Marginal cost is an essential concept in economics. It represents the cost to produce one more unit of a good or service. In the case of the steel-producing firm, the marginal cost is constant at £4 per unit. This means that for every additional ton of steel produced, the firm incurs an additional cost of £4.

Understanding marginal cost helps businesses determine the optimal level of production. By analyzing their costs in this way, firms can decide the quantity of goods to produce that will maximize profits. In our exercise, the firm operates in a competitive market where it balances marginal cost with the marginal revenue (price). This means they produce up to the point where the cost of the last unit produced equals the revenue it generates.

• Marginal Cost = Additional cost of producing one more unit.
• In the example, Constant Marginal Cost = £4.
• Profit Maximization occurs where Marginal Cost = Marginal Revenue.

It is crucial for businesses to know this so they can avoid producing more than what the market can bear without losing money.

Social Loss

Social loss arises when a firm's activities create a negative externality that is not considered in its decision-making process. Externalities occur when the production or consumption of a product affects third parties who are not directly involved in the transaction. In our exercise, the pollution from steel production impacts the environment and the community, without the firm accounting for this in its cost structure.

The social loss is measured by the difference between the social cost of producing the steel and the private cost that the firm considers. In this scenario, we calculated the social loss as the area between the marginal social cost and marginal private cost curves for the quantity range of 10 to 20 units. The social loss amounts to £100, reflecting the overall negative effect on society that occurs because the firm only maximizes its private profit.

• Social Loss = Extra cost to society from externality.
• In the example above, Social Loss = £100.
• Essential for understanding true cost of negative externalities.

Recognizing social loss helps policymakers draft better regulations to mitigate these negative impacts.

Taxation

Taxation can be a powerful tool to address the problem of externalities. By imposing a tax on each unit of the good that generates the externality, the government can encourage firms to take into account the social costs caused by their production activities. This method is known as a Pigovian tax, named after economist Arthur Pigou.

In the steel example, a tax equivalent to the marginal external cost, which is £2 per unit at the socially optimal output, is imposed. This effectively raises the firm's marginal cost to include the external damage cost. As a result, the marginal private cost curve shifts upward, aligning with the marginal social cost. By doing so, the firm's decision-making now reflects the true cost of production, including the social damage.

• Tax = Additional charge to correct externalities.
• Encourages firms to internalize societal costs.
• Pigovian Tax ensures alignment of MPC and MSC.

This leads to socially optimal levels of output where the external damages are minimized, promoting overall societal welfare.

Marginal Social Cost

Marginal social cost represents the total cost to society of producing one additional unit of a good. It includes both the private cost borne by the producer and the externality cost imposed on society.

In the given exercise, the marginal social cost of steel is the sum of the marginal private cost (which is constant at £4) and the external cost due to pollution (given by £2Q). Therefore, the formula for marginal social cost becomes:\[MSC = MPC + External Cost = 4 + 2Q\]

Calculating marginal social cost is essential for determining the socially optimal level of output. This is the level at which the marginal social cost equals the price consumers are willing to pay (P). In our example, this occurs at 10 units of steel, which minimizes societal harm while maintaining economic activity.

• Marginal Social Cost = Sum of private and external costs.
• Reflects true cost of production from society's perspective.
• Essential for determining socially optimal production levels.

Understanding and calculating marginal social cost aids in making informed policy decisions that can enhance societal welfare.