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Chapter 23 / Externalities

23.5 Tragedy of the Commons

Consider a town of 35 people. Each person can choose to fish or to hunt in the nearby forest. If $L$ people choose to fish, the total number of fish caught on a lake is given by $f(L) = 40L - L^2$.

Suppose that fish sell for $€10$ each, and each person who hunts always gets $€100$ of meat.

Pareto Efficiency

What is the optimal number of fish to catch? Total revenue, as a function of the number of fisherpeople $L$, is given by \(TR(L) = \overbrace{40L - L^2}^\text{total fish} \times \overbrace{10}^\text{price per fish} + \overbrace{(35 - L)}^\text{number of hunters} \times \overbrace{100}^\text{each hunter's payoff}\) or \(TR(L) = \overbrace{400L - 10L^2}^\text{total revenue from fish} + \overbrace{3,500 - 100L}^\text{total revenue from hunting}\) The change in profit when society devotes another person to fishing rather than hunting is \(MR(L) = \overbrace{400 - 20L}^\text{marginal revenue from fish} - \overbrace{100}^\text{opportunity cost from one less hunter}\) Setting this equal to zero yields $L = 15$. Under this “social planner’s optimum,” the total number of fish caught will be \(f(15)=40L-L^2 = 600 - 225 = 375\) and total revenue from the lake will be $375 \times 10 = 3,750$. Since 15 people will fish, the other 20 will choose to hunt and earn 100 each, so total revenue for the town will be 5,750.

Competitive Equilibrium

Suppose each fisherperson who chooses to fish on the lake receives the average amount of fish caught; so fishing has a payoff to each fisherperson of \(\pi_F(L) = AR(L) = \frac{(40L - L^2)\text{ total fish}\times €10\text{ per fish}}{L\text{ fisherpeople}} = 400 - 10L\) It’s easy to see that the more people fish on the lake, the fewer fish are caught by each person. If people can freely choose whether to fish or hunt, the equilibrium payoff to fishing ($\pi_F$, above) and hunting ($\pi_L = 100$) must be the same – otherwise, people would switch from the less-profitable task to the more-profitable one. In this “free entry” equilibrium, therefore, we would have \(\begin{aligned}\pi_F &= \pi_H\\\\ 400 - 10L &= 100\\\\ L &= 30\end{aligned}\) Hence the total number of fish caught will be \(f(30)=40L-L^2 = 1200 - 900 = 300\) and total revenue from the lake will be $300 \times 10 = 3,000$. Since 30 people will choose to fish, the other 5 will choose to hunt and earn 100 each, so total revenue for the town will be 3,500 – much less than before!

Analysis of the Problem

What’s happening here? Each person who fishes exerts a negative externality on the others who are already fishing, by decreasing the average catch for everyone by 10. When there are 15 people fishing, therefore, each fisherperson gets a total benefit of $400 - 10\times 15 = 250$, but the marginal fisherperson generates a negative externality of $€150$; so the total benefit of 250 is balanced by the opportunity cost of 100 (what they could have gotten by hunting) and the marginal external cost of 150. By contrast, the 30th person to choose fishing earns a private benefit of 100, which just offsets their opportunity cost of hunting; but also generates a whopping $€300$ negative externality on their fellow fisherpeople; so the 30th fisherperson actually decreases the town’s overall revenue by $€300$!

Potential Solutions

One solution for this problem is to have fishing permits. Suppose the village issued fifteen fishing permits and sold them for price $c$. In this case, the benefit to fishing would be \(\pi_F(L|c) = 400 - 10L - c\) As we found before, when $L = 15$ each fisherperson would earn a private benefit of $€250$ from fishing; so they would be willing to pay at most $\overline c = 150$ for the permit.

An alternative solution would be to have a sort of pigouvian tax. In this case, the market price of fish would be $p - t$, so if fish sold for $€10$ each, we would have \(\pi_F(L|t) = (40 - L)\times(10 - t) = 400 - 10L - (40-L)t\) Again, at the optimal number of $L = 15$, this would be equal to $250 - 25t$; equating this to the opportunity cost of hunting, we would therefore need to set a tax of $t = 6$.

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Copyright (c) Christopher Makler /