WORLD ECONOMY

ELSEVIER Japan and the World Economy 7 (1995) 105-111

Aggregation, heterogeneity, and the Coase invariance theorem

Thomas Russell *

Department of Economics, Santa Clara University, Santa Clara, CA 95053, USA

Received March 1993; accepted September 1993

A b s t r a c t

It is possible to view the Coase Theorem as an aggregation- invariance result in which average demand depends solely on average endowments so that redistribution of initial endowments across individuals has no effect on the equilibrium of the economy. Hurwicz shows that the conditions on preferences required for this aggrega- tion to hold are extremely stringent. On the other hand, it follows from the work of Marris and Becker that average demand will depend only on average endowments when individual demands are uniformly distributed over budget sets. We speculate that the Coase theorem will hold in an approximate sense if individual demand types are sufficiently diverse, a result consistent with the "statistical" approach to aggrega- tion of Wald, Hicks, Hildenbrand, and in particular Grandmont.

Keywords: Coase theorem; Statistical aggregation; Diversity

JEL classification: D10; D50; K l l

I . I n t r o d u c t i o n

One version of the Coase Theorem (Coase, 1960) is an invariance result which states that the level of an externality produced in the competitive

* Tel. 408-554-6953. I wish to thank the referee of this paper for clarification of the meaning of the Coase theorem in a general equilibrium context. I would also like to thank M. Adler for helpful discussions.

0922-1425/95/$09.50 © 1995 Elsevier Science B.V. AU rights reserved SSDI 0922-1425(94)00039-5

106 T. Russell/Japan and the World Economy 7 (1995) 105-111

equ i l ib r ium of an economy is not a f fec ted by a r ea l loca t ion of t r a d e a b l e p r o p e r t y r ights in the activi ty which causes the external i ty . 1

Symbolical ly, le t t ing E * r e p r e s e n t the level of ex terna l i ty p r o d u c e d at equi l ib r ium, and le t t ing ei = e l ,e2 . . . . . en r e p r e s e n t some ini t ial a l loca t ion of t r a d e a b l e p r o p e r t y r ights in the ex terna l i ty among n individuals , we have

E * ( e i ) = E * ( e i ) (I)

where ei is any other allocation of property rights satisfying

= = e .

i i

Note that it is not sufficient for the truth of the Coase Theorem that a reallocation of ownership in the rights to the externality leaves the price of these rights unchanged. Ownership of a right confers two quite separate legal benefits, the right to use and the right to prevent others from using. Aggregate demand for externality rights is thus the sum of two more basic demands, the demand of those who wish a positive level of the externality and the demand of those who wish a zero level. A reallocation of ownership rights could leave aggregate demand and therefore the price of these rights unchanged, but could cause changes in the level of the externality since in one case aggregate demand could be due to those who wish to exercise the right to create a positive level of the externality, whereas in the other case aggregate demand could be due to those who wish the externality suppressed. 2

For the Coase Theorem to be true, then, aggregate demand for the externality must depend only on the aggregate quantity of tradeable rights e, and not on the details of the individual endowment vector e i. Thus the Coase invariance theorem fits the framework for classical aggregation set out by Richmond (1976), and it is therefore to be expected that the theorem will hold only under rather stringent rank conditions on individual demands, see Russell (1982). The exact conditions on preferences generating aggregatable demands are set out in Hurwicz (1995).

A priori, the chances that these preference conditions will be met in any actual economy would seem to be rather slim, so it is worth noting that aggregation theorems can also be developed from a quite different point of view. From the early work of Hicks (1939) and Wald (translated in) (1951), to the more recent work of Hildenbrand (1983) and Grandmont (1992), a number of economists have expressed the view that the details of individual demand will not matter for the determination of equilibrium provided

l This version of the theorem is discussed by Hurwicz (1993). Earlier references are provided by him.

2 1 owe this interesting observation to the paper's referee.

T. Russell~Japan and the World Economy 7 (1995) 105-111 107

individuals are sufficiently diverse. Following Grandmont (1992) we call this form of aggregation "statistical aggregation".

In this paper we use a simple model of diverse behavior, first used by Marris (1957) (though more often associated with the name of Becker (1962)), to show how heterogeneity can provide a sufficient condition for invariance. Unlike Hurwicz, we make no assumptions on the shape of individual preferences. ( Indeed Becker does not even assume the existence of individual preference orderings.) Rather we make assumptions on the shape of the distribution of preferences (or, more generally, demands) across the population. We turn now to the precise t reatment of heterogeneity.

2. The model

We consider an exchange economy with m traded goods x j, j = 1,2 . . . . ,m, and one externality in which there are tradeable rights, r e. We denote aggregate endowments of the m goods by to t, j = 1,2,. . . ,m, and we denote the aggregate endowment of the externality rights by e. Let p - -p l ,p2 . . . . . pm be a price vector for the traded goods, and let the price of the externality right be pC. Denote by r+ total demand for the externality right to be used to create a positive level of the externality, and by r~ total demand for the externality right to be used to suppress the externality. We must have

e Jr e = r e . r+ r 0

We assume that the economy consists of n individuals labelled i, i = 1,2 . . . . ,n, and we assume each individual in the economy belongs to one of three classes. The first class we will call class C (for those who could cause the externality) and we will let these individuals be ordered and labelled by c, where c = 1,2 . . . . ,k. The second class we call V (for potential victims of the externality) and we will label these individuals by v, where v = k + 1 . . . . . l. The remaining individuals will be in class U (unaffected by the allocation of externality rights.)

We assume that property rights in the externality are given initially to the members of class C. Let 1 2 m toe,to c . . . . ,to c ,ec, c = 1,2 . . . . ,k, be the endowment of a member of this class. (Obviously for members of classes V and U, the initial endowment of e is zero.) We now calculate the income distribution for this economy.

For an individual i in any of the three classes, a price vector p determines an income level Y/(p) where

Yi( P ) = ~°°JiP j + eiP e. (2) J

The number of individuals with income of Y~ gives the distribution of income. We assume that there are many individuals in each income class, and we will denote the typical income class k by Y~ as defined in (2).

108 T. Russell/Japan and the World Economy 7 (1995) 105-111

In addi t ion where

Hk( P) = (xJ,re l~=rrJpJ + repe = gk( p) ). J

Note again that re = r+e d- r~ .

We assume that income is always fully spent. M a r r i s - B e c k e r assumption.

Assumption 1 (Marris-Becker)

each price vector p de te rmines a budge t hyperp lane H~(p)

(3)

Now comes the crucial

On any budget hyperplane Hk( P) the distribution of total spending is uniformly distributed across all goods, including externality creation x e+ and externality

e 3 suppression x o.

This assumpt ion leads to the following well known result.

Result (Marris-Becker)

Under Assumption 1, aggregate demand for all goods is Cobb-Douglas, with expenditure equally divided across all goods.

It is easy to visualize this result in the plane. Since average d e m a n d on any budget l ine must be at the midpoin t of the line, a change in the price of any one good must genera te an offer curve which is paral lel to the axis of the good whose price is changed. So cross price elasticities of d e m a n d are zero, and own price elasticities are ( - 1 ) i.e. d e m a n d is C o b b - D o u g l a s . Since this is t rue on every budget p lane, aggregate d e m a n d is also C o b b - D o u g l a s , with equal expendi tures on all goods.

Now suppose we reallocate the total quant i ty of proper ty rights in e from the member s of class C to the member s of class V. The exact details of who gains and who loses from this t ransfer are irrelevant, because if we make Assumpt ion 1 both before and after the rights transfer, 4 we have

3 For Becker this was not an assumption on individual behavior which was generated in his model by throwing a dart at the budget hyperplane. With many individuals on each hyperplane, and therefore many darts, smoothing across the budget arises at the group level, where the group is all individuals with the same budget.

4 The assumption that income is spent uniformly both before and after the rights transfer requires that the members of classes C and V are not concentrated at some particular income level. For example, suppose income of both classes is uniformly distributed from 1 to H before the transfer, and suppose that the transfer shifts 1 dollar from each member of class C to each member of class 1/". Then after after the transfer income will be uniformly distributed from 0 to H - 1 in class C and from 2 to H + 1 in class II. With the exception of the extreme tails, the composition of C's and V's in each income class will not be affected by this transfer Of course, the names of the parties in the various income classes are different, but in this case we may expect little change in total spending on all goods.

T. Russell / Japan and the World Economy 7 (1995) 105-111 109

Theorem (Coase)

Under Assumption 1 competitive equilibrium exists, is unique, and is invari- ant with respect to a reassignment of property rights in the externality.

Proof: Existence and uniqueness are guaranteed by the fact that Cobb- Douglas demands satisfy diagonal dominance, see Grandmont (1992). Invari- ance with respect to the reassignment of property rights follows because for those with income Yk, aggregate demand for any good, and in particular aggregate demand for the externality r+, is given by

lYk r+k(p ) - (4) pe

where r+k is aggregate demand for the externality causing activity by all individuals with income Yk and l is a constant.

Thus aggregate demand is

l Y~re+k(p) = ~ p e p e ~ Y k ( P ) " (5) k k k

But total income ~kYk(p) is not changed by a reallocation of endowments at fixed prices p. Thus, in particular, at the original equilibrium level of prices, the level of aggregate demand for the externality is not changed by a reallocation of endowments, so the original equilibrium remains an equilib- rium and the Coase Theorem holds. To compare this result with that of Hurwicz (1993), note that Hurwicz finds conditions so that each individual's demand for r+ is unchanged by an income shift. The approach here looks for conditions so that aggregate demand r+ is unchanged, allowing the possibil- ity that individual demands may change but cancel in aggregate.

3. Remarks

We have demonstrated a proof of a version of the Coase theorem using no assumptions on the shape of individual preferences. Of course, it may be argued that the Marris-Becker assumptions on heterogeneity are a priori just as unlikely to hold as the assumptions derived by Hurwicz on preferences. For example, suppose the externality is second-hand smoke. Redistribute property rights from smokers to non-smokers. Presumably aggregate demand for cigarettes will not be invariant to this income shift. Indeed, in any case in which the externality is clearly associated with systematic preferences for a traded good, (another example might be dog ownership and the obvious externalities associated therewith), we expect aggregate demand shifts as income is redistributed.

110 T. Russell/Japan and the World Economy 7 (1995) 105-111

Nevertheless, in many cases of externalities there seems to be no reason to presume that a reassignment of property rights alters in any significant way the pat tern of demand at any particular income level. Thus, on average, we may expect little change. Moreover, if we a r e prepared to accept "approxi- mate" invariance theorems, the recent f ramework developed by Grandmont (1992) allows us to bound the size of the demand shift associated with a transfer of rights as a function of the degree of heterogeneity in the population. The Marr i s -Becker assumption of uniform distributions is then a limiting case.

Indeed, Grandmont ' s f ramework even goes some limited way to answering one of Samuelson's (1993) concerns with the Coase result. 5 As Grandmont shows, when the population is sufficiently diverse, the standard ta tonnement process

f, = ah( zh( P) ), (6)

Zh(p) = excess demand for good h, Gh(P) = a sign preserving function Gh(O) = 0, G~, > 0 is stable, Grandmont (1992), Corollary 3.4

This does not tell us in Samuelson's small numbers case how to reach the frontier (a duopoly cannot be diverse), but it does say that in the case of a large diverse population we can write down a dynamic process (Eq. (6)), which converges to the same (Pareto-efficient) competitive equilibrium no mat ter what the initial allocation of rights. This would seem to make it interesting to examine the Coase theorem more fully in the Grandmont framework.

References

Becker, G., 1962, Irrational behavior and economic theory, Journal of Political Economy 70, 1-13.

Coase, R.H., 1960, The problem of social cost, Journal of Law and Economics 2, 1-40. Grandmont, J.-M., 1992, Transformations of the commodity space, behavioral heterogeneity, and

the aggregation problem, Journal of Economic Theory 57, 1-35. Hicks, J.R., 1939, Value and capital (Oxford University Press, Oxford). Hildenbrand, W., 1983, On the law of demand, Econometrica 51, 997-1019. Hurwicz, L., 1995, What is the Coase Theorem?, Japan and the World Economy, 7, 49-74. Marris, R.L., 1957, Professor Hick's index number theorem, Review of Economic Studies 25,

25-39. Richmond, 1976, Aggregation and identification, International Economic Review 17, 37-49.

5 In response to the view that the allocation of property rights automatically eliminates deadweight loss, Samuelson asks, "Is there some coherent proof that, without transaction costs, real life rivals must find their way to Pareto-optimal frontiers?"

T. Russell / Japan and the World Economy 7 (1995) 105-111 111

Russell, 1982, Exact aggregation as a corollary of Richmond's Theorem, Economics Letters 9, 311-314.

Samuelson, P.A., 1993, Some uneasiness with the Coase Theorem, Center for U,S. Japan Business Studies, New York University, Working Paper 137.

Wald, A., 1951, On some systems of equations of mathematical economics, Econometrica 19, 368-403.