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

10.2 The external effects of pollution: Private and social costs and benefits

external cost, negative externality, external diseconomy
A negative external effect: that is, a negative effect of an economic decision on other people, that is not taken into account by the decision-maker. It may be described as an external cost, or a negative externality, or an external diseconomy. See also: external effect.
marginal private cost, MPC
The cost for the producer of producing an additional unit of output. It is called the marginal private cost, or MPC (rather than simply the marginal cost) when we want to emphasise that it doesn’t include any external costs that production imposes on others. See also: marginal external cost, marginal social cost.
marginal social cost, MSC
The marginal social cost (MSC) is the cost of producing an additional unit of output, including both the cost for the producer (marginal private cost) and the costs imposed on others (the MEC). MSC = MPC + MEC.
marginal external cost, MEC
The marginal external cost (MEC) is the cost of an additional unit of output that is incurred by someone other than the producer (or the sum of these costs if several others are affected). The marginal social cost is the sum of the MEC and the marginal private cost to the producer: MSC = MEC + MPC.

We begin with the diagnosis and treatment of a case like the pesticides in Martinique and Guadeloupe, in which the production of a good creates an external cost: pollution. We will model the case of an imaginary Caribbean island where a fictional pesticide called Weevokil is used to grow bananas, but pollutes coastal waters and kills fish. In our model there is just one external cost, to the livelihoods of fishermen—Weevokil does not affect the health of the population directly.

external effect, externality
An external effect occurs when a person’s action confers a benefit or imposes a cost on others and this cost or benefit is not taken into account by the individual taking the action. External effects are also called externalities.

This is called an external effect (or sometimes an externality), because the pollution has no effect on the plantation owners who decide how much pesticide to use. It affects fishermen who play no part in their decisions. Imagine for a minute that the same company owned the banana plantations and fisheries, thereby hiring fishermen and selling what they catch for profit. The company owners would choose the level of pesticide to use, taking account of its downstream effects. They would trade off the profits from the banana part of their business against the losses from the fisheries. Joint ownership of the plantations and fisheries would internalize this effect. The externality arises when they are under separate ownership.

Figure 10.1 shows the marginal costs of growing bananas using Weevokil. Remember, the marginal cost is the additional cost of producing one more unit: in this case, one more ton of bananas.

The marginal cost for the growers is labelled as the marginal private cost (MPC). It slopes upward because the cost of an additional ton of bananas increases as the land is more intensively used, thereby requiring more Weevokil. Use the analysis in Figure 10.1 to compare the MPC with the marginal social cost (MSC), which includes the costs borne by fishermen whose waters are contaminated by Weevokil.

In this diagram, the horizontal axis shows the quantity of bananas Q in tons per year, and ranges from 0 to 100000. The vertical axis shows the costs in dollars, and ranges from 0 to 900. Coordinates are (quantity, costs). An upward-sloping straight line passes through points (0, 200) and (80000, 400) and is labelled marginal private cost. Another convex, upward-sloping curve passes through points (0, 50) and (80000, 275), is labelled marginal external cost and is below the marginal private cost line at all points. Another convex, upward-sloping curve passes through points (0, 250) and (80000, 675), is labelled marginal social cost and is above the marginal private cost line at all points.
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Figure 10.1 Marginal costs of banana production using Weevokil.

The marginal private cost: In this diagram, the horizontal axis shows the quantity of bananas Q in tons per year, and ranges from 0 to 100000. The vertical axis shows the costs in dollars, and ranges from 0 to 900. Coordinates are (quantity, costs). An upward-sloping straight line passes through points (0, 200) and (80000, 400) and is labelled marginal private cost.
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The marginal private cost

The upward-sloping line is the marginal cost for the growers: the marginal private cost (MPC) of banana production. The MPC rises as the land is more intensively used.

The marginal external cost: In this diagram, the horizontal axis shows the quantity of bananas Q in tons per year, and ranges from 0 to 100000. The vertical axis shows the costs in dollars, and ranges from 0 to 900. Coordinates are (quantity, costs). An upward-sloping straight line passes through points (0, 200) and (80000, 400) and is labelled marginal private cost. Another convex, upward-sloping curve passes through points (0, 50) and (80000, 275), is labelled marginal external cost and is below the marginal private cost line at all points.
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The marginal external cost

The lower curve shows the marginal cost imposed by the plantations on fishermen: the marginal external cost (MEC). It is the cost of the reduction in fish numbers caused by each additional ton of bananas.

The marginal social cost: In this diagram, the horizontal axis shows the quantity of bananas Q in tons per year, and ranges from 0 to 100000. The vertical axis shows the costs in dollars, and ranges from 0 to 900. Coordinates are (quantity, costs). An upward-sloping straight line passes through points (0, 200) and (80000, 400) and is labelled marginaltexttonn private cost. Another convex, upward-sloping curve passes through points (0, 50) and (80000, 275), is labelled marginal external cost and is below the marginal private cost line at all points. Another convex, upward-sloping curve passes through points (0, 250) and (80000, 675), is labelled marginal social cost and is above the marginal private cost line at all points.
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The marginal social cost

Adding together the MPC and the MEC, we get the full marginal cost of banana production: the marginal social cost (MSC). This is the top curve in the diagram.

The total external cost: In this diagram, the horizontal axis shows the quantity of bananas Q in tons per year, and ranges from 0 to 100000. The vertical axis shows the costs in dollars, and ranges from 0 to 900. Coordinates are (quantity, costs). An upward-sloping straight line passes through points (0, 200) and (80000, 400) and is labelled marginal private cost. Another convex, upward-sloping curve passes through points (0, 50) and (80000, 275), is labelled marginal external cost and is below the marginal private cost line at all points. Another convex, upward-sloping curve passes through points (0, 250) and (80000, 675), is labelled marginal social cost and is above the marginal private cost line at all points. A vertical line passes through point (70000, 0). The area between the vertical line, the marginal social cost curve and the marginal private cost line is the external cost of producing 100000 tons of bananas.
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The total external cost

For each ton of bananas produced, MEC = MSC – MPC. The total cost imposed on the fishermen is the sum of the MECs for every ton produced. The shaded area in the figure shows the total external cost of producing 100,000 tons of bananas.

Since banana production has a negative external effect—a marginal external cost—the marginal social cost of banana production is higher than the marginal private cost. When production is low and little Weevokil is used, the external harm is small, but it grows rapidly as production rises.

For more explanation about why firms in a competitive market produce where the marginal cost is equal to the market price, read Section 8.4.

To focus on the essentials, we will assume that most bananas are exported, the world market for bananas is competitive, and the market price is $400 per ton. To maximize their profit, the plantation owners will choose their output so that their marginal cost—that is, the marginal private cost—is equal to $400.

Figure 10.2 shows that their total output will be 80,000 tons of bananas (point A). The shaded area shows the total cost imposed on the fishing industry. Although 80,000 tons maximizes profits for banana producers, it does not allow for this external cost, so it is not a Pareto-efficient outcome.

In this diagram, the horizontal axis shows the quantity of bananas Q in tons per year, and ranges from 0 to 100000. The vertical axis shows the costs in dollars, and ranges from 0 to 900. Coordinates are (quantity, costs). An upward-sloping straight line passes through points (0, 200) and A (80000, 400) and is labelled marginal private cost. Another convex, upward-sloping curve passes through points (0, 250) and (80000, 675), is labelled marginal social cost and is above the marginal private cost line at all points. A vertical line passes through point (70000, 0). The area between the vertical line, the marginal social cost curve and the marginal private cost line is the external cost of producing 70000 tons of bananas. A horizontal line passes through points A and B (38000, 400) and is labelled price.
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Figure 10.2 The plantations’ choice of banana output.

To understand this idea, think about what would happen if the plantations were to produce one ton less. The fishermen would gain $275—they would no longer suffer the loss of fishing revenue caused by the production of the 80,000th ton of bananas. The plantations would lose hardly anything. Their revenues would fall by $400, but their costs would fall by almost as much, because at 80,000 tons the marginal private cost is also $400.

So both groups could be better off with 79,999 tons of bananas, if they could somehow arrange for the fishermen to pay some part of their $275 gain to the plantations.

A similar argument tells us that they could do better still by reducing further. If the output of bananas is reduced by 1 ton, the gain for the fishermen is the MEC of that ton, and the loss to the plantations is the surplus they would have made from it, which is P – MPC. Whenever MEC > P – MPC, it would be possible for both groups to be better off—if the fishermen shared the benefits of the reduction by making a payment to the plantations.

How much should production be reduced? Applying the argument above tells us:

\[\text{Reduce output if } P < \text{MEC + MPC}\]

Since MEC + MPC is the marginal social cost of banana production, this is equivalent to:

\[\text{Reduce output if } P < \text{MSC}\]

The Pareto-efficient level of output—where it is not possible to make one group better off without making the other worse off—is 38,000 tons at point B in Figure 10.2, where:

\[\text{The price of bananas, } P = \text{the marginal social cost of production, MSC}\]

At this point, although fishing revenues are still affected by pollution, the maximum payment they would be willing to make would not be enough to induce the plantations to cut production further. Below 38,000 tons, the marginal social cost of bananas is less than their price (P > MSC) so it would be better to raise output.

To summarize:

  • The plantations choose to produce 80,000 tons of bananas, where P = MPC.
  • The Pareto-efficient level of output is 38,000 tons of bananas, where P = MSC.
marginal social benefit, MSB
The marginal social benefit (MSB) is the benefit of the production or consumption of an additional unit of a good, including both the benefit for the producer or consumer (marginal private benefit) and the benefits conferred on others. MSB = MPB + MEB.

The price of bananas represents what people are willing to pay for them on the world market. We can think of this as the marginal social benefit (MSB) of banana production, so the condition for Pareto efficiency is MSB = MSC.

The condition for Pareto efficiency

The level of production and consumption of a good is Pareto efficient if and only if the marginal social cost is equal to the marginal social benefit.

At this level, producing another unit of the good would cost more to some members of society than it would be valued by others. Similarly, producing one unit less would reduce the benefits to some by more than the cost savings for others.

If a single company owned both the plantations and fisheries, its private costs would include the effect of the pesticide on fish stocks. This company would choose to produce 38,000 tons of bananas, where its MPC would be $400.

The negative external effects of pollutants like Weevokil are sometimes called environmental spillovers. The pollutants bring private benefits to those who decide to use them, but by damaging the environment they impose external costs on other firms or on households that rely on environmental resources. For society as a whole, this is a market failure: compared with the Pareto-efficient allocation, the pollutant is overused, and too much of the associated good (bananas, in our example) is produced.

Question 10.1 Choose the correct answer(s)

A factory is situated next to a dormitory for nurses who work night shifts. The factory produces 120 robots a day. The production process is rather noisy, and the nurses often complain that their sleep is disturbed. Based on this information, read the following statements and choose the correct option(s).

  • The marginal private cost is the factory’s total cost of producing 120 robots a day.
  • The marginal social cost is the noise cost incurred by the nurses from production of an additional robot.
  • The marginal external cost is the cost to the factory, plus the noise cost incurred by the nurses, when an additional robot is produced.
  • The total external cost is the total costs per day imposed on the nurses by the factory’s production.
  • The marginal private cost (MPC) is the factory’s cost of producing one additional robot, not the total cost.
  • The statement describes the marginal external cost (MEC). The marginal social cost (MSC) is the sum of the costs to the factory and the nurses of producing the additional robot: MSC = MPC + MEC.
  • The MEC is the noise cost incurred by the nurses when an additional robot is produced.
  • The total external cost is the sum of the marginal external costs of production of all the robots. This is the total cost imposed on the nurses.

Extension 10.2 The external effects of pollution

This extension explains the underlying model for the main section. Using calculus and assuming quasi-linearity, we derive the marginal costs from plantations’ profits and fishermen’s preferences, and show that the plantations’ private choice of output is not Pareto efficient.

The second part generalizes these results (and is more challenging mathematically). With quasi-linearity, there is only one Pareto-efficient output level; without it there are many, depending on the incomes of the two parties. Either way, the private choice is never Pareto efficient, and both parties can be better off through a change in output and a monetary transfer.

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.

The plantations’ choice of banana output.
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Figure E10.1 The plantations’ choice of banana output.

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, Ce(Q) represents the costs imposed on the fisherman by the plantations: the external costs of banana production. And mf 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, PW, 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 Cp(Q) for the private costs of banana-growing for the plantation. And as above, it is helpful to include explicitly a term, mp, 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, PW, 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, Qp, 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 PW, 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, \(τ\) (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 = \(τ\)
  • Gain for fishermen = \(C^{\prime}_e(Q_p)- τ\)

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.

Exercise E10.1 Comparing the private and Pareto-efficient outcomes

A premium toy manufacturer has the cost function, \(C(Q) = 2Q^2 + 2Q + 5\), where Q is the number of toys produced. Producing toys results in environmental pollution costs of \(\frac{1}{6}Q^3 + \frac{1}{2}Q^2\). The world price for premium toys is $50 per unit.

  1. How many units would the manufacturer choose to produce? Calculate the manufacturer’s profit and the environment pollution costs under this scenario.
  2. Draw a diagram to show the marginal private, external, and social costs of toy production.
  3. Which output would be Pareto efficient?
  4. Suppose that a philanthropist concerned about pollution offers to pay the manufacturer to reduce output. How much would she need to pay to compensate the manufacturer for reducing output to the Pareto-efficient level? What would be the reduction in pollution costs?

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.

Pareto efficiency problem: Quasi-linear case

Choose \(\tau\) and Q to maximize \(m_f^0 - \tau - C_e(Q)\) subject to the constraint, \(\tau + P^W Q-C_p(Q)=y_0\).

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 mf 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:

Pareto efficiency problem: General case

Choose \(\tau\) and Q to maximize \(u(m_f^0 - \tau, Q)\) subject to the constraint \(\tau + P^W Q-C_p(Q)=y_0\).

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 mf 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.

A similar result arises in the model of the interaction between Angela and Bruno in Unit 5. When Angela has quasi-linear preferences between grain and free time, there is only one Pareto-efficient level of free time (satisfying the condition MRS = MRT)—irrespective of how much grain she receives. But in Extension 5.9, we show that if Angela’s preferences are not quasi-linear, there is a set of Pareto-efficient allocations with different combinations of grain and free time.

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, Qp, 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.