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 he or she can sell all the oil he or she can produce at a stable world price of \(\$ 10\) per barrel and that the cost of operating a well for one year is \(\$ 1,000\). 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. \(\quad\) 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: In the equilibrium, firms produce as much oil as possible with no additional costs. The optimal number of wells for maximizing social welfare is 250, with a total output of 62,500 barrels and output per well of 250 barrels. To achieve the optimal number of wells, the government should impose a license fee of $10 per well.

Step-by-step solution

a. Equilibrium Output and Equilibrium Number of Wells

To find the equilibrium output and the equilibrium number of wells, we need to find where marginal cost equals marginal revenue. In a perfectly competitive market, every firm faces the same price, so the marginal revenue is simply the world price of \(\$ 10\) per barrel. To determine the marginal cost per well, we first need to find the average cost per well, which is given by: $$AC = \frac{Total \, cost}{quantity} = \frac{1,000}{q}$$ Next, we find the marginal cost: $$MC = \frac{d(Total \, cost)}{dq} = \frac{d(AC \times q)}{dq} = \frac{d(\frac{1,000}{q} \times q)}{dq} = \frac{d(1,000)}{dq} = 0$$ Given that the marginal cost is zero and the marginal revenue is \(\$ 10\), we reach the equilibrium at the point where the firms produce as much as they can without the marginal cost exceeding the marginal revenue. In other words, the firms will keep producing more oil as long as the price remains at \(\$ 10\) without incurring additional costs. Calculate equilibrium quantity and the number of wells using: $$q=500-N$$ $$N_{equilibrium}=500-q_{equilibrium}$$ For the equilibrium output, the q is not bound by N, so the companies will tend to produce as much as possible given the constraints. The social marginal cost in this case is the same as private as there is no external cost, so there is no divergence.

b. Optimal Number of Wells, Total Output, and Output per Well

If the government nationalizes the oil field, they will maximize social welfare rather than profit. Since there are no external costs, the government will also aim to produce where MC=MR. Under this condition, the optimal number of wells (N) can be determined by setting q to maximize total output: $$Q=500 N-N^2$$ Then, we can find the maximum value of Q by taking the derivative and setting it to zero: $$\frac{d(Q)}{d(N)}=500-2N=0$$ Solve for N: $$N_{optimal}=250$$ Now, we can determine total output and output per well: $$Q_{optimal}=500 N_{optimal}-N_{optimal}^2$$ $$Q_{optimal}=500(250)-(250^2)=62500$$ And: $$q_{optimal}=\frac{Q_{optimal}}{N_{optimal}}$$ $$q_{optimal}=\frac{62,500}{250}=250$$

c. License Fee for Optimal Number of Wells

To encourage the optimal number of wells, the government has to make it costly for firms to operate more wells than the socially optimal number (250). Therefore, we need to find the license fee that makes the marginal cost of drilling additional wells equal to the marginal benefit from producing more oil. Let's call the license fee L, then the new average cost of operating a well is: $$AC = \frac{1,000+L}{q}$$ The new marginal cost is: $$MC_L = \frac{d(AC \times q)_L}{dq} = \frac{d(\frac{1,000+L}{q} \times q)}{dq} = \frac{d(1,000+L)}{dq} = 0$$ Now, we need to find L such that MC_L = MR at the optimal output level q_optimal: $$MC_L = 10$$ $$0 + L = 10$$ $$L=\$10$$ The Utopian government should impose a license fee of \(\$10\) per well to encourage the optimal number of wells.