Question

Suppose the oil industry in Utopia is perfectly competitive and that all firms draw oil from a single (and practically inexhaustible) pool. Assume that each competitor believes that it can sell all the oil it can produce at a stable world price of $\mathrm{\$}10$ per barrel and that the cost of operating a well for one year is $\mathrm{\$}1,000$ Total output per year (Q) of the oil field is a function of the number of wells ( $n$ ) operating in the field. In particular, \[ Q=500 n-n^{2} \] and the amount of oil produced by each well ( $q$ ) is given by \[ q=\frac{Q}{n}=500-n \] a. Describe the equilibrium output and the equilibrium number of wells in this perfectly competitive case. Is there a divergence between private and social marginal cost in the industry? b. Suppose now that the government nationalizes the oil field. How many oil wells should it operate? What will total output be? What will the output per well be? c. As an alternative to nationalization, the Utopian government is considering an annual license fee per well to discourage overdrilling. How large should this license fee be if it is to prompt the industry to drill the optimal number of wells?

Short answer

Answer: The government can impose an annual license fee per well on top of the private marginal cost. However, based on the given information, it is not possible to determine the exact amount of the license fee.

Step-by-step solution

a. Equilibrium Output and Number of Wells in Perfectly Competitive Case

In a perfectly competitive market, the firms will maximize their profits by producing at the output level where marginal cost (MC) equals the market price (P) for the product, which in this case is oil. We are given that the market price is $10 per barrel. The total output (Q) of the oil field is given by the function: \[ Q=500 n-n^{2} \] The output per well (q) is given by the function: \[ q=\frac{Q}{n}=500-n \] Now, we will derive the marginal cost (MC) from the cost function. We are given that the cost of operating a well for one year is $1,000. The total cost is then $1,000n$, which makes the average cost (AC): \[ AC=\frac{Total\ Cost}{n}=\frac{1,000n}{n}=1,000 \] Now, we have Average Cost (AC) and need to find Marginal Cost (MC). MC is the change in total cost when we increase the number of wells by one: \[ MC = AC = 1000 \] In a perfectly competitive market, firms produce where the price (P) equals the marginal cost (MC): \[ P = MC \] So, \[ $10 = $1,000 \] But since the price cannot be equal to the marginal cost in this case, there must be a divergence between private and social marginal cost in the industry.

b. Nationalized Oil Field Analysis

If the government nationalizes the oil field, it should focus on maximizing social welfare by equating marginal social benefit (MSB) with marginal social cost (MSC). First, we have marginal social benefit, which is the same as the market price of the oil: \[ MSB = $10 \] Next, we need to find the marginal social cost (MSC). The MSC includes both the private cost and any external cost. In this case, the private cost is the marginal cost (MC) we have calculated above as $1000: \[ MSC = MC + External\ Cost = 1000 + External\ Cost \] To maximize social welfare, the government should set: \[ MSB = MSC \] This implies, \[ $10 = 1000 + External\ Cost \] In this case, the external cost cannot be calculated. Now, if the government wants to find the optimal number of oil wells, they should operate in such a way that marginal cost (MC) equals the market price (P). According to the information given in the exercise, the government does not have enough information to calculate the optimal number of wells, total output, and output per well.

c. Calculating Annual License Fee per Well

To encourage the industry to drill the optimal number of wells, the Utopian government can impose an annual license fee per well on top of the private marginal cost. The license fee should be equal to the external cost so that the new marginal cost is equal to the market price (P): \[ New MC = MC + License\ Fee = P \] \[ $10 = $1,000 + License\ Fee \] \[ License\ Fee = $10 - $1,000 = -$990 \] Since a negative license fee does not make sense in this context, it implies that the exercise does not provide enough information to determine the optimal license fee.

Key concepts

Equilibrium Output

In a perfectly competitive market, achieving equilibrium output is crucial for businesses to optimize their operations. This equilibrium is attained when the quantity produced (output) reaches a point where the market price equals the marginal cost of production. This intersection ensures that producers are making just enough to cover their costs while maximizing profits. In the context of the oil industry in Utopia, the equilibrium output for each well is defined by the function $q=500-n$, where $n$ is the number of wells. The output equation signifies that the more wells there are, the less oil each well will produce. This diminishing return is typical in scenarios where resources are shared among many producers. Therefore, for each additional well, the output decreases slightly, maintaining equilibrium.Understanding this helps producers in a perfectly competitive market to decide the number of wells to drill to maximize their profits at the given market price, in this case, $\mathrm{\$}10$ per barrel. The market self-regulates as firms adjust their output to respond to these conditions, maintaining equilibrium across the industry.

Marginal Cost

Marginal cost is a fundamental concept in economics, representing the cost of producing one additional unit of a good. It plays a pivotal role in making production decisions, especially in perfectly competitive markets. The formula for marginal cost is essentially the change in the total cost associated with a change in production quantity. For the oil industry example in Utopia, the marginal cost of operating one well for a year is given as $\mathrm{\$}1,000$. This uniform cost across wells indicates constant returns to scale, meaning each additional well costs the same to operate. In practical terms, this informs firms on how much it will cost to expand their operations by adding more wells.Knowing the marginal cost allows firms to set production levels where they can sell oil at a price that at least covers these marginal expenses. When the market price equals the marginal cost, firms are considered to be operating efficiently within a perfectly competitive market.

Nationalization of Resources

Nationalization occurs when the government takes control of a resource or industry from private ownership. The aim is often to align operations more closely with public interest and to potentially rectify any negative externalities caused by private firms. In Utopia's case, the oil industry, if nationalized, would operate under government policies instead of competitive market pressures.The Utopian government, upon nationalizing the oil resource, would ideally continue to operate the oil wells by equating marginal social benefit (MSB) with marginal social cost (MSC). This means looking beyond private costs to also consider externalities affecting society. For instance, while the private cost for operating a well is $\mathrm{\$}1,000$, the government would calculate an MSC that includes any broader social costs or benefits, ensuring operations maximize social welfare.However, the exercise highlights a challenge: calculating these external costs accurately is crucial yet often complex. Nationalization seeks to optimize resource use, balancing economic efficiency with social equity.

Social Welfare Optimization

Social welfare optimization is about achieving a balance where both economic efficiency and equity are maximized. This often involves government intervention to correct market failures. In a perfectly competitive market, social welfare optimization ensures that resource allocation not only caters to market needs but also benefits society as a whole.The Utopia oil industry case presents a scenario where private and social costs diverge. While firms are guided by private costs ($MC=\mathrm{\$}1,000$), to maximize social welfare, the government should consider social costs, which include externalities. Thus, achieving this tends to require setting production or operational levels where marginal social benefits match marginal social costs.When the government steps in, they aim to adjust outputs not purely on profit maximization but also on the overall benefit to society, which might involve subsidies, taxes, or even additional regulation. This delicate balance maintains economic health while ensuring sustainable development, making resources like oil benefit Utopia's population broadly.