# When does the invariance version of the Coase Theorem hold? The special case of quasi-linear preferences

In the previous section we investigated the efficiency version of the Coase Theorem and saw that in the absence of transaction costs, parties would bargain over the level of *Q* until there were no further gains from trade left to exploit. However, that analysis showed that even in the absence of transaction costs, the invariance version of the Theorem does not usually hold.

Let us now investigate the invariance version in a little more detail. It turns out that the invariance version of the Coase Theorem holds if the preferences of the parties take the quasi-linear form. Consider again the utility function of the factory. Suppose that its utility function is:

where *u'(Q*) > 0 and *u"(Q*) < 0. These are called quasi-linear preferences because they are partially linear in one of the goods - in this case money. The marginal rate of substitution of money for *Q* is:

which is independent of the consumption of *M _{F}.* This means that if we horizontally shift the consumer's indifference curve, for any fixed level of Q, their marginal rate of substitution does not change. Effectively, this means that, for a given set of relative prices or rate of exchange between production of

*Q*and money, changes in money do not change the factory's demand for

*Q*, as shown in Figure 3.4.5. This is often referred to the case of there being

*no income effects*or

*no wealth effects*present in the demand for

*Q*.

*Figure 3.4.5* Indifference curves without income effects

To see how the assumption of quasi-linear preferences matters, consider the following special case: suppose that the residents also have *identical *quasi-linear preferences over money and the *absence* of Q:

and where we assume u(0) = 0. Then, the set of points where their marginal rates of substitution must be equal must obey:

Since the function u(-) is everywhere increasing, this implies that, along the contract curve we must have:

or:

This situation is illustrated in Figure 3.4.6.

In Figure 3.4.6 it is clear that if both parties have identical quasi-linear preferences, then the invariance version of the Coase Theorem must hold, since, irrespective of the initial allocation, all efficient points lie

on the line q = ^{Q}, so the final allocation after all gains from trade are 2

exhausted always has *Q = Q*. The only difference between legal regimes will be the final allocation of money between the parties.

*Figure 3.4.6* The contract curve with identical quasi-linear preferences

This analysis has just illustrated a very important result: *if individuals have identical quasi-linear preferences and there are no transaction costs, then both the efficiency version and the invariance version of the Coase Theorem hold. ^{8}*

Another way to see this result is to plot the marginal rates of substitution of *Q* (and the absence of *Q*) for money for both the factory and the residents, which is the inverse of the slope of the indifference curves in Figure 3.4.3. For the factory, this marginal rate of substitution is the amount of money that the factory must be given to accept a one unit reduction in *Q*. Therefore, it is also the amount of money that the factory would be willing to pay or give up to get a one unit increase in *Q*. In other words, it measures the factory's *marginal willingness to pay for Q* in terms of money.

With diminishing marginal utility for *Q*, this marginal rate of substitution is decreasing in *Q*.

Similarly, for the residents this marginal rate of substitution is the amount of money that the residents must be given to accept a one unit reduction in *Q - Q*. Therefore, it is also the amount of money that the residents would be willing to pay or give up to get a one unit increase in *Q - Q*. In other words, it measures the factory's *marginal willingness to pay for Q - Q* in terms of money.

With diminishing marginal utility for *Q - Q*, this marginal rate of substitution is decreasing in *Q - Q*.

A special case of quasi-linear preferences has the utility functions taking the quadratic form *u(Q*) = *aQ - bQ*^{2}, and *u(Q - Q*) = *a(Q - Q*) - *b(Q - Q*)^{2}. In this case, the marginal rate of substitution of *Q* (and the absence of *Q*)

*Figure 3.4.7* Marginal willingness to pay under quasi-linear quadratic preferences

for money are linear in *Q* (and the absence of Q), since *u'(Q*) = *a - 2bQ *and *u'(Q - Q*) = *a - 2b(Q - Q*).

This situation is plotted in Figure 3.4.7. The length of the box is the total possible production of *Q*, which is *Q*. There is only one point where the marginal rates of substitution are equal, which is at the point where *Q = Q* /2.

Consider an initial legal rule which states that the factory can produce as much *Q* as it desires, which corresponds to point I in Figure 3.4.3. At this point, the factory has a very low (zero) marginal willingness to pay for additional units of *Q* , since if the marginal utility was positive it would have produced more. But point I also means that the residents enjoy a zero amount of the *absence* of *Q*, which means that with diminishing marginal willingness to pay, their marginal willingness to pay for the first unit must be very high. Thus, we get the two points identified in Figure 3.4.7 as corresponding to point I in Figure 3.4.3.

Note that at these two points, marginal willingness to pay differs considerably between the parties, which implies that substantial possible gains from trade exist at point I, as the UPC in Figure 3.4.8 shows. The same applies to point II. The ways in which these gains from trade are split between the parties is examined further in Chapter 11.

*Figure 3.4.8* The UPC when the parties have identical quasi-linear preferences