# Externalities, the Coase Theorem and the Edgeworth Box

We use the Edgeworth Box to present the Coase Theorem and analyse its consequences. Suppose there are two parties: a factory (F) and a group of residents (R). There are two commodities in this economy: the factory's output, Q, and another good that can be transferred costlessly between the parties, which we shall call money.

Suppose that the factory produces output of Q > 0. The utility to the factory is: where MF is the amount of money that the factory 'consumes'.1 We assume that the marginal utility of Q is positive but declining with the level of activity.2 We also assume that the factory prefers more money to less, but that the marginal utility of money is declining.3 The factory also starts out with some amount of money, Mf. A typical indifference curve for the factory is plotted in Figure 3.2.1.

The residents also start out with some amount of money, Mr . We assume that the production of Q by the factory generates uncompensated disutility for the residents. In other words, the production of Q by the factory generates a negative externality. The residents' utility function is therefore: where the marginal utility of Q is now negative. The marginal disutility (or, effectively, the marginal cost) is also assumed to increase with the Figure 3.2.1 An indifference curve for the factory Figure 3.2.2 An indifference curve for the residents

level of activity.4 Again, we assume that the residents prefer more money to less, but that the marginal utility of money is declining.5 A typical indifference curve for the residents is plotted in Figure 3.2.2.

To see why the residents' indifference curve is shaped like this, consider a move from point A to point B in Figure 3.2.2. At point B, the residents are consuming the same amount of money as at point A, but output Q is higher. Since the residents dislike Q, this move makes them worse off. The only way that indifference can be restored is if they consume more money (say, point C). Thus, the indifference curve must be upward sloping.

To understand why the indifference curve has a concave shape, note that the marginal disutility of Q increases with Q. This means that every additional unit of Q which is produced by the factory requires ever greater amounts of compensation in terms of money in order for the residents to remain indifferent. This means that the typical indifference curve must be increasing and concave as shown in Figure 3.2.2. Finally, suppose that there is some maximum amount of Q that the factory can actually produce (because, for example, of capacity constraints). Let this maximum amount be given by Q .

Instead of the residents having preferences over Q and money, with Q entering the utility function as a 'bad', define another variable, Q - Q, to be the absence of Q. The variable Q - Q then enters the residents' utility or benefit function as a good in the standard fashion.

In other words, let us define variables in such a way that:  Figure 3.2.3 Preferences for the absence of Q

with: and We can then plot indifference curves for money and the absence of Q in the usual way. This is done in Figure 3.2.3.