# 23.3 Consumption Externalities in the Edgeworth Box

Let’s start out where we’ve been for a while — in the Edgeworth Box. Up until now, we’ve assumed \(u^A (x_1^A,x_2^A,x_1^B,x_2^B) = u^A (x_1^A,x_2^A)\) \(u^B (x_1^A,x_2^A,x_1^B,x_2^B) = u^B (x_1^B,x_2^B)\) That is, we’ve thought of utility being a function solely of one’s own consumption. Under this assumption (and other maintained assumptions, like smooth utility functions) we’ve shown that the set of Pareto efficient allocations is the one such that \(MRS^A (x_1^A,x_2^A) = MRS^B (x_1^B,x_2^B)\) What happens, though, if your consumption affects my enjoyment? It turns out we can use the Edgeworth Box to analyze situations like this as well.

## Problem: Smoking Roommates

Suppose Chris and Ken are roommates. Chris is a non-smoker and doesn’t like smoke. Ken is a smoker and enjoys smoking quite a bit.

Let’s think of each of their utility as being a function of two things: the amount Ken smokes (good 1) and the amount of money they have (good 2). Let $s$ be the number of hours that Ken spends smoking, and let’s assume that Ken could spend up to 10 hours a day smoking: that is, $10 − s$ is the number of hours each day he *doesn’t* spend smoking. In a way, we can think of $x_1^K = s$ and $x_1^C = 10 − s$ — that is, Chris’s “consumption good” is the number of smoke free hours, and Ken’s “consumption good” is the number of hours spent smoking. (Assume that Ken would like to smoke for the full 10 hours.) Finally, let’s assume that Chris and Ken each have $€100$.

Just as we used an Edgeworth Box to analyze allocations of goods among people, we can use one to analyze possible outcomes in this model. The height of the box is the total amount of money Chris and Ken have ($€200$). The width of the box is the 10 hours in which Ken could smoke or not smoke:

Let’s suppose that both Chris and Ken have *quasilinear* preferences over smoking and money: in particular, Ken’s preferences are given by
\(u^K(x_1^K,x_2^K) = k \ln x_1^K + x_2^K\)
and
\(u^C(x_1^C,x_2^C) = c \ln x_1^C + x_2^C\)
Since money is “good 2” and smoke is “good 1,” this means that each person’s MRS is their willingness to pay for an hour of smoke (in Ken’s case) or an hour less of smoke (in Chris’s case):
\(MRS^K = {k \over x_1^K}\)
\(MRS^C = {c \over x_1^C}\)
Now, we have that the amount of “good 1” that Ken has is the amount he smokes ($s$), and the amount of “good 1” that Chris has is the amount Ken doesn’t smoke ($10 - s$), so we along the contract curve, when these are equated, Ken’s marginal utility from the last hour he spends smoking is exactly equal to Chris’s marginal disutility from that hour. Furthermore, note that because we used quasilinear utility functions, the efficient amount of smoke doesn’t depend on money at all: it just depends on the amount of smoke! In particular, when these are equal, we have
\(\begin{aligned}
{k \over s} &= {c \over 10-s}\\
k(10-s) &= cs\\
10k &= (k + c)s\\
s &= 10 \times {k \over k + c}
\end{aligned}\)
Note that $k$ and $c$ represent the intensity with which Ken and Chris think about smoking. If $k = c$, then the efficient quantity of smoke is 5; if $k > c$, it’s efficient for Ken to smoke more than 5 hours; and if $k < c$, it’s efficient for Ken to smoke less than 5 hours:

So, there is a single efficient quantity of smoke…how do we get there?

## Property Rights and the Coase Theorem

In the Edgeworth Box, we generally start out with the notion of an “endowment.” As far as an endowment goes, the money part is easy: we’ve assumed each of them starts with €100, so $e_2^C=e_2^K = 100$. This means that we’re starting off with an endowment somewhere along the horizontal line in the diagram below:

But how can we think of an “endowment” of smoking time? One way of thinking of it is in terms of *property rights*. Suppose this is Ken’s house, and Chris is only paying rent; we might then assume that Ken has the right to smoke as much as he likes in his house. This would correspond to the point at the far right-hand end of the line in the box above. Likewise, if it’s Chris’s house, we might assume that he has the right to demand a smoke-free living environment; this would correspond to the point at the far left-hand end of the line. You could also imagine various different institutional arrangements which would give Ken the right to smoke some number $s$ hours per day. You can drag the point in the graph above left and right to assign different property rights.

Now, in general, most endowments won’t be Pareto efficient, unless they *happen* to lie directly on the contract curve. What then? Well, if we allow Chris and Ken to trade, everything we know from the arguments in the last chapter go through: they’ll keep trading until they end up on the contract curve, and we will end up at a Pareto efficient point. In fact, if we allow for there to be a *market for the externality itself* — that is, a market in which people can trade the right to smoke — then the competitive price in that market will converge to the price which lands you on the contract curve:

The Coase Theorem, named after Ronald Coase and based on his 1960 paper “The Problem of Social Costs,” essentially stipulates that as long as the costs of negotiation are sufficiently low, the problem of the inefficient externality can be solved by assigning property rights to the externality itself and allowing the interested parties to bargain with one another; and furthermore, that as long as everyone’s preferences are quasilinear, the efficient outcome (in this case, the quantity of smoke) will be the result *regardless of the initial allocation of property rights*. Of course, the property rights are themselves valuable: Ken end up better off if he starts out with the right to smoke, and Chris ends up better off if he starts out with the rights to clean air. But either way, creating a market for smoking means that in equilibrium, the amount of smoke is the Pareto efficient amount.

In addition to quasilinear preferences, the Coase theorem relies on some pretty strong assumptions about how the agents in the model come together. The last diagram shows a competitive equilibrium; but if we’re just talking about two roommates, the assumptions of competitive equilibrium (many buyers and sellers, etc.) won’t apply; so bargaining power can start to play a large role. All parties may not recognize the property rights of the others, or agree on a third party to impose those rights. And especially if the stakes are high, bargaining may not be costless; once the lawyers get involved, things can get expensive fast. Oh, and there could be asymmetric information, incentives to overstate the harm one feels from the externality, and so on. But the central Coasean idea is a powerful one: that the efficiency of markets can be brought to bear on things, like smoking, for which there is no existing market.