Unit 10 Market successes and failures: The societal effects of private decisions

Extension 10.2 The external effects of pollution

We used Figure 10.2, reproduced here as Figure E10.1, to show the marginal private and social costs of banana production. When the plantations choose their output, taking into account only their private costs, the equilibrium is at point A, which is not Pareto efficient. Point B, where the marginal social cost of bananas is equal to the world price, would be a Pareto-efficient outcome.

This is an example of a method for analysing external effects that is very widely used, particularly for understanding environmental problems like pollution. But it relies critically on the assumption that the firms or consumers for whom the level of pollution matters have quasi-linear preferences.

We discuss quasi-linear preferences in some detail in Unit 5 (in particular Extension 5.4). As explained there, quasi-linear preferences are a useful simplification because they allow us to measure utility (benefits and costs) in terms of money income. The general form of a quasi-linear utility function is:

\[u(m, x)=m+f(x)\]

where x is a good (or a ‘bad’, if its effect on utility is negative) and m is the individual’s other income (spent on other goods). This function is linear in m, and has the important property that the marginal utility of the good x is independent of income:

\[\frac{\partial u}{\partial x}=f'(x)\]

In other words (since the marginal utility of income is 1), the marginal rate of substitution (MRS) between income and x depends only on the level of x.

So if x represents the level of air pollution in a town, for example, and all citizens have the same quasi-linear utility function, then the marginal utility of a unit of pollution will be negative, but a citizen’s MRS between pollution and income won’t depend on whether they are rich or poor. This is a questionable assumption: we might expect that rich people would be more prepared to give up income for an improvement in air quality.

Preferences, profits, and Pareto efficiency in the Weevokil model

In our Weevokil model, we have effectively adopted quasi-linear preferences by assuming that both plantation owners and fisherman only care about Weevokil because it affects the profits from growing bananas and catching fish. For both groups, their utility depends only on their net monetary pay-offs. So we can write their pay-offs in quasi-linear form.

Think first about what matters to the fishermen. Their livelihoods depend on the profits from fishing, but they are affected by the pollution resulting from banana production. We will write the fishermen’s utility as the sum of two terms, one that depends on banana output, Q, and one that does not:

\[u(m_f, Q)=m_f -C_e(Q)\]

Here, C_e(Q) represents the costs imposed on the fisherman by the plantations: the external costs of banana production. And m_f represents ‘other income’—it includes all net income that is not a function of Q. We assume that \(C^{\prime}_e(Q)>0\), which means that the fishermen’s utility decreases with Q; in other words, bananas are a ‘bad’ for them.

To simplify the explanation, we will suppose that there is a single plantation owner on the island, who chooses the level of Q. Remember that bananas are sold at the world market price, P^W, and so the owner is a price-taker, not a monopolist. Hence this assumption makes no difference to the outcome—it just means that we don’t have to find the profit-maximizing output of each plantation and then add them up. We write C_p(Q) for the private costs of banana-growing for the plantation. And as above, it is helpful to include explicitly a term, m_p, for any other net income the owner receives. Then we can write the plantation owner’s pay-off as:

\[y(m_p, Q)= m_p + (P^W Q-C_p(Q))\]

We can now match the elements of the model with the situation illustrated in Figure E10.1. The world price of bananas, P^W, is $400, and the marginal private cost for the plantations of growing bananas is:

\[\text{MPC} = C^{\prime}_p(Q)\]

In the figure, MPC increases with Q, corresponding to the property, \(C_p''(Q) > 0\). The marginal external cost of bananas is the marginal cost experienced by the fishermen, which is:

\[\text{MEC} = C^{\prime}_e(Q)\]

and the marginal social cost is the sum of the two costs:

\[\text{MSC}=\text{MPC}+\text{MEC} = C^{\prime}_p(Q) + C^{\prime}_e(Q)\]

Therefore, the line representing MSC lies above the MPC line. We assumed that MEC also increases with Q (\(C_e''(Q)>0\)) so MSC slopes upward, more steeply than MPC.

The private choice of banana output is not Pareto efficient

The plantation owner chooses Q to maximize profit. Differentiating the expression above for the owner’s pay-off and setting it to zero, we find that the owner’s profit-maximizing choice, Q_p, satisfies the first-order condition:

\[\begin{align*} \frac{\partial y}{\partial Q} = P^W-C^{\prime}_p(Q)&=0\\ \Rightarrow P^W&=C^{\prime}_p(Q_p) \end{align*}\]

This is point A in Figure E10.1, where the MPC is equal to P^W, the marginal social benefit of banana-growing. Checking the second-order condition, we can confirm that since \(-C_p''(Q)<0\), A is a maximum point.

We argued in the main part of this section that the outcome at A, \(Q=Q_p\), is not Pareto efficient, because it is possible to achieve a Pareto improvement. For a mathematical version of the argument, suppose that output were to be increased by a small (infinitesimal) amount. Then, the corresponding change in profit is:

\[\frac{\partial y}{\partial Q} = P - C^{\prime}_p(Q_p)\]

which is equal to zero (for an infinitesimal change). But the effect on the fishermen is:

\[\frac{\partial u}{\partial Q} = - C^{\prime}_e(Q_p) < 0\]

The effects of an infinitesimal decrease in Q would have the opposite sign. So the fishermen would be better off by \(C^{\prime}_e(Q_p)>0\), and it would have no effect on the plantation owner. And if the fisherman were to make a payment, \(\tau\) (an income transfer), to the plantation owner in return for a small reduction in Q, where \(0<\tau< C^{\prime}_e(Q_p)\), both would be better off:

• Gain for plantation = \(\tau\)
• Gain for fishermen = \(C^{\prime}_e(Q_p)- \tau\)

If you apply the same analysis to the outcome at point B, where the marginal social cost is equal to the price (\(C^{\prime}_p(Q) +C^{\prime}_e(Q) = P^W\)), you will find that an infinitesimal change in Q will lead to a gain for one group that is exactly equal to the loss for the other. So in this case, the gainer cannot make a transfer payment to compensate the loser and remain better off: this outcome is Pareto efficient.

This analysis and the diagrammatic equivalent in the main part of the section require quasi-linearity. Without it, the MEC suffered by the fisherman would change if they made or received a transfer payment. The diagram would no longer help us to locate Pareto-efficient allocations, because the change in the MEC would move the MSC line.

Finding Pareto-efficient allocations

An allocation is Pareto efficient if at least one individual can be made better off without making anyone else worse off. A general method for finding Pareto-efficient allocations is to consider whether the allocation of goods could be rearranged to increase one individual’s pay-off, for given levels of the pay-offs for everyone else. The allocations that maximize the individual’s pay-off are Pareto efficient.

In our model where everyone cares about two things—the amount of a particular good or bad, and other income—rearranging the allocation means redistributing these two things between those affected.

Specifically, we consider how the pay-off of one group (fishermen or banana producers) could be maximized by changing the quantity of bananas produced and at the same time transferring income between the groups, without changing the pay-off for the other group. We can answer this question by solving the following constrained choice problem, in which \(\tau\) is defined as a monetary transfer from the fishermen to the plantation owner. Note, however, that the value of \(\tau\) could turn out to be negative, representing a payment by the plantation owner to the fishermen.

Solving this problem means finding the possible values of \(\tau\) and Q that maximize the fisherman’s pay-off, u, given that the plantation owner receives a particular pay-off level, y₀. We have written \(m^0_f\) for the fishermen’s income in the situation when both Q and \(\tau\) are zero—their income if bananas were not produced—and assumed that the plantation owner has no income in this situation (for convenience; this assumption does not affect our conclusions).

Finding the solution for all possible levels of y₀ will give us all the Pareto-efficient allocations. It is straightforward to solve this problem using the substitution method. The constraint can be rearranged to obtain:

\[\tau =y_0 - P^W Q + C_p(Q)\]

Substituting this into the fishermen’s pay-off gives us the objective function (the function we want to maximize) in terms of Q only:

\[u=m_f^0 - y_0 +P^WQ - C_p(Q) - C_e(Q)\]

Differentiating gives us the first-order condition for Q:

\[\begin{align*} \frac{du}{dQ} = P^W - C^{\prime}_p(Q) - C^{\prime}_e(Q) &= 0 \\ \Rightarrow C^{\prime}_p(Q) + C^{\prime}_e(Q)& = P^W \end{align*}\]

The first-order condition tells us that the level of Q at any Pareto-efficient allocation satisfies the condition:

\[\text{MPC} + \text{MEC} = P^W\]

That is, the marginal social cost is equal to the marginal social benefit. You can check that under our assumptions about the cost functions, the second-order condition is satisfied, and that there can only be one Pareto-efficient value of Q: it is the value at point B in Figure E10.1, which we will call Q^*.

What about the transfer payment, \(\tau\)? Substituting the Pareto-efficient Q back into the expression above gives us:

\[\tau^* =y_0 - P^WQ^* + C_p(Q^*)\text{ or equivalently } y_0 =\tau^* + P^W Q^* - C_p(Q^*)\]

Therefore, the transfer payment required is the payment that will ensure that, together with the plantation profits at \(Q=Q*\), the owner’s total pay-off will be y₀. The transfer payment depends on y₀, and could be positive or negative.

In summary, for Pareto efficiency in this model the level of banana output must be \(Q=Q^*\), irrespective of the levels of other income. And \(Q^*\) does not depend on the value we specify for the plantation’s pay-off, y₀. Any particular value of y₀ can be achieved through a transfer between the fisherman and the plantation owner.

In other words, income does not matter for Pareto efficiency of banana production (although it matters very much to everyone concerned). This property is a direct result of the assumption of quasi-linearity. Everyone simply wants to maximize their total monetary pay-off, and the marginal costs of banana production do not vary with income.

You may have wondered why we expressed the Pareto efficiency problem as maximizing the fisherman’s utility given a particular pay-off for the plantation owner. Why not the other way around? The answer is that we could equally well have maximized the plantation owner’s pay-off given a particular level of utility for the fisherman. We would have reached the same conclusions.

Pareto improvements

We can use this approach to work out how, given any particular initial allocation, it is possible to reach a Pareto-efficient allocation that makes everyone better off.

Suppose we start from the allocation in which the level of output is chosen by the plantation owner: that is, point A in Figure E10.1, where \(Q=Q_p\). The owner’s pay-off at this point is:

\[y_0= P^W Q_p - C_p(Q_p)\]

Solving the constrained choice problem for this value of y₀ tells us we can make the fisherman as well off as possible without reducing the owner’s pay-off if banana output is reduced to \(Q^*\) and the fishermen exactly compensate the owner by paying a transfer equal to the reduction in banana profits—a Pareto-efficient outcome and a Pareto improvement.

Alternatively, we could consider a situation in which no bananas could be grown without Weevokil, and Weevokil could not be used unless the fisherman agreed to it. In the absence of an agreement, the owner’s pay-off is zero. Solving the problem above for the case, \(y_0=0\) gives us \(Q=Q^*\) and \(\tau^*= - (P^W Q^*-C_p(Q^*))\). This represents an agreement in which the plantation owner produces Q^* and transfers all of the profits to the fisherman. Again, this is a Pareto-efficient outcome and a Pareto improvement.

In these two examples the plantation owner’s pay-off doesn’t change. But by choosing a higher level of y₀, we could find Pareto-efficient outcomes that made both sides strictly better off.

What if preferences are not quasi-linear?

Although the result that there is a unique Pareto-efficient level of output depends on the assumption of quasi-linearity, it is true in general that the plantation’s private choice of Q will not be Pareto efficient. For a more general model, suppose the fishermen’s utility is given by:

\[u(m_f, Q) \text{ where } \frac{\partial u}{\partial m_f}>0 \text{ and } \frac{\partial u}{\partial Q} < 0\]

As before, the marginal external costs of banana production are given by \(-\frac{\partial u}{\partial Q}\); this is the marginal reduction in utility caused by a marginal increase in Q. But in general, this will be a function of both m_f and Q:

\[\text{MEC} = -\frac{\partial u}{\partial Q}(m_f, Q)\]

In other words, the marginal external costs may depend on whether the fishermen’s other income is high or low.

How does this affect our analysis? The method for finding Pareto-efficient allocations described above is quite general—we can use it even if preferences are not quasi-linear. The constrained choice problem becomes:

As before, we can solve it using the substitution method, but we need to be careful with the differentiation step. Substituting \(\tau =y_0 - P^W Q + C_p(Q)\) into the fishermen’s utility, we can write the objective function as:

\[u(m_f, Q) \text{ where } m_f= m_f^0 - y_0 +P^W Q - C_p(Q)\]

Note that both the arguments of the function u depend on Q. Differentiating (and using the chain rule for the first argument), we obtain the first-order condition:

\[\begin{align*} \frac{du}{dQ} =\frac{\partial u}{\partial m_f}\frac{dm_f}{dQ}+ \frac{\partial u}{\partial Q}&= 0 \\ \Rightarrow \frac{\partial u}{\partial m_f}(P^W - C^{\prime}_p(Q)) + \frac{\partial u}{\partial Q}&= 0 \end{align*}\]

As before, Pareto-efficient allocations \((Q,\tau)\) satisfy this equation, and \(\tau =y_0 - P^W Q + C_p(Q)\). But remember that the partial derivatives in the first-order condition depend on m_f as well as Q. So in this case the Pareto-efficient levels of both Q and \(\tau\) depend on the value we choose for y₀.

We conclude that there is a set of Pareto efficient allocations \((Q,\tau)\) corresponding to different values of y₀.

We can use the equations above to demonstrate that—even when there are many Pareto-efficient allocations—the level of Q chosen by the plantations cannot be Pareto efficient. At \(Q=Q_p\):

\[P^W=C^{\prime}_p(Q) \Rightarrow \frac{dm_f}{dQ}=0\]

Hence the derivative of the objective function is:

\[\frac{du}{dQ} =\frac{\partial u}{\partial Q} <0\]

So the first-order condition for Pareto efficiency cannot be satisfied at \(q=Q_p\). The plantations care only about their own profit, and do not take into account the negative effect on fishermen.

By evaluating the second derivative, we can deduce something more: the plantation’s choice, Q_p, is always too high. You can check that (provided MEC increases with Q as in the quasi-linear model) \(\frac{d^2u}{dQ^2}<0\) at this point. Since \(\frac{du}{dQ}\) is also negative there, output must be reduced to satisfy the first-order condition for Pareto efficiency.