The question we now ask is this: does the initial assignment of property rights or legal rights matter, and, if so, how? To answer this question, we compare the two outcomes derived above. From Figure 3.4.3 we can conclude that, in the absence of transaction costs, the efficiency version of the Coase Theorem always holds: as long as initial legal rights are well defined and there are no impediments to trade, parties will reach efficient agreements.

However, it is also clear from Figure 3.4.3 that, in general, the invariance version of the Coase Theorem does not hold, since production of Q is higher in legal regime I than it is in legal regime II. In other words, in the absence of transaction costs the legal regime does not matter for efficiency, but in general, if there is more than one efficient level of production, the legal regime does matter in determining which efficient point is actually reached - even if there are no transaction costs.

Distribution of utilities and the utility possibilities curve

It is also possible to view the same outcome differently, using the utility possibilities curve (UPC), which shows the combinations of utilities that each of the parties can obtain. The UPC corresponding to Figure 3.4.3 is shown in Figure 3.4.4. The utilities or benefits corresponding to the initial assignments of legal rights are shown as points I and II, whereas the utilities corresponding to the final allocations are shown as points e_{I} and e_{n} respectively. Clearly the initial assignment of property rights does not matter for efficiency (as both points e_{I} and e_{n} are on the UPC). On the other hand, the residents clearly prefer regime II to regime I, and the factory prefers regime I to regime II. In other words, the legal regime matters for distribution.

The slope of the utility possibilities curve can be computed as follows.^{7 }Consider a small change in the utility of the residents, du_{R} along the contract curve. For any such change, it must be true for the residents that:

Figure 3.4.4 The utility possibilities curve corresponding to Figure 3.4.3

Dividing both sides by , we obtain:

d(Q - Q)

A similar analysis for the factory shows that:

Now dM_{R} = -dM_{F} and d(Q - Q) = -dQ. Therefore, for the residents we obtain:

where the second equality follows from the equality of marginal rates of

^{dU}R

du_{R} dM_{R}

substitution at points on the contract curve. Since ^{R} = - ^{R}—

du^ d(Q - Q) ^{mrs}Mq _q

and = - ^{dM}F , this means that: dQ MRSM ,q

The slope of the utility possibilities frontier is equal to the (negative of the) ratio of the marginal utilities of money for the parties. In particular, if the marginal utilities of money are constant, then this ratio is constant and the UPC will be a straight line, rather than the bowed shape in Figure 3.4.4.