optimal regulation under fixed rules for income distribution* · pareto optimal allocation can be...

JOURNAL OF ECONOMIC THEORY 45, 65-84 (1988)

Optimal Regulation under Fixed Rules for Income Distribution*

RAJIV VOHRA

Brown University, Providence, Rhode Island 02912

Received February 24, 1986; revised June 2, 1987

It is by now well known that if arbitrary lump sum transfers are feasible, then the optimal regulation of firms involves marginal cost pricing. Here, we consider optimal regulation in an economy in which lump sum taxation is possible but the income distribution is, in some sense, fixed. We show that under certain conditions, production efficiency and marginal cost pricing are desirable. In general, however, this need not be the case. In an economy with increasing returns, it is possible that the second best utility possibility frontier is strictly dominated by the Pareto frontier and none of the marginal cost pricing equilibria lie on the second best frontier. Journal of Economic Literature Classification Numbers: 021, 022, 024. (!:I 1988 Academic Press, Inc

1. INTR~OUCTI~N

In the classical Arrow-Debreu model, marginal cost pricing is suflicient for Pareto optimality. This follows from the First Welfare Theorem given an appropriate definition of “marginal cost prices”.’ Moreover, according to the Second Welfare Theorem, if lump sum transfers are feasible, any Pareto optimal allocation can be sustained as a marginal cost pricing equilibrium. This implies that under smoothness hypotheses, marginal cost pricing is also necessary for Pareto optimality.’ As one would expect, under certain conditions, marginal cost pricing is a necessary condition for Pareto optimality even in the presence of increasing returns. Indeed, this was Hotelling’s [S] argument for regulating natural monopolies to follow marginal cost pricing. A rigorous basis for this argument is now available in the form a generalized Second Welfare Theorem (see, for example,

* This paper is an extensive revision of [ 111 and is very heavily inspired by discussions with Andreu Mas-Colell. I am grateful to M. Ali Khan for his encouragement and help during various stages of this research. Thanks are also due to an Associate Editor and participants of seminars at Brown University, the Indian Statistical Institute, and the London School of Economics for many helpful comments. Financial support from NSF Grants SES 8410229 and SES 8605630 is gratefully acknowledged.

* A formal definition of marginal cost prices is provided below. 2 See also Remark 4.1 below.

65 0022-0531/88 63.00

Copyright 0 1988 by Academic Press. Inc. All rights of reproduction m any form reserved.

66 RAJIV VOHRA

Guesnerie [S, Theorem 1 ] and Khan and Vohra [9, Theorem 11); any Pareto optimal allocation can be sustained through marginal cost pricing even in economies with nonconvex production sets. In other words, if arbitrary lump sum transfers are feasible, optimality can be achieved by regulating firms to follow marginal cost pricing irrespective of whether production sets are convex or not.

The normative significance of marginal cost pricing appears not to be so clear in an economy with increasing returns if arbitrary lump sum taxation is not feasible. Consider a model which is identical to the Arrow-Debreu model except that some firms have nonconvex production sets and the government can regulate firms to follow any feasible production plans that it chooses. As in the Arrow-Debreu model, the profits of firms are distributed to the consumers according to exogenously given rules. Consumers behave competitively at the given market prices and a regulated equilibrium can be defined as an allocation and prices at which all markets clear. Notice that since consumers have exogenously given shares in the various firms, if a regulated firm makes losses these are effectively collected from the consumers in a lump sum manner. But the distribution of these taxes is predetermined; while lump sum taxation is feasible, the government’s ability to redistribute income is restricted. It is for this reason, and this reason alone, that one cannot simply appeal to a general version of the Second Welfare Theorem to assert that optimality can be achieved by regulating firms to follow marginal cost pricing. In fact, it is already known that with fixed rules for income distribution none of the marginal cost pricing equilibria may be Pareto optimal (see Guesnerie [5] and Brown and Heal [2]). While the existence of a marginal cost pricing equilibrium has been established under fairly general conditions (see, for example, Vohra [ 121 and the references therein) its welfare properties have not yet been fully explored.

The aim of this paper is to characterize the optimal form of regulation in an economy with fixed rules for income distribution. As we shall see, the fact that no marginal cost pricing equilibrium may be Pareto optimal suggests that it may be impossible to achieve Pareto optimality. The central issue we shall be concerned with is the characterization of second best optimality under distributional constraints and its relationship to marginal cost pricing. We consider two kinds of income distribution rules: (1) a $xed structure of reuenues under which each consumer owns a fixed proportion of the economy’s net outut; (2) a fixed structure of shares under which each consumer, just as in the Arrow-Debreu model, has given shares in the firms and endowments. Interestingly, the results do depend on which of these rules is followed.

Section 2 specifies the basic model and related notions of equilibrium and optimality.

OPTIMAL REGULATION 61

In Section 3 we deal with optimality in the context of a given social welfare function. Here, the results do not depend in any significant way on the presence of increasing returns. It is reasonable to expect, and we show explicitly, that due to distributional considerations marginal cost pricing may not be an optimal policy. What may be smewhat more surprising is the fact that the desirability of production efficiency or marginal cost pricing depends on the distribution rule being followed. Given a fixed structure of shares, production efficiency may not be desirable. However, if the income distribution is given according to a fixed structure of revenues, production efficiency is always desirable. In this case, any second best equilibrium relative to a Paretian social welfare function can be charac- terized as a marginal cost pricing equilibrium if commodity taxation is feasible.

In Section 4 we consider the situation in which distributional considerations are not relevant. In particular, we examine the possibility of using marginal cost pricing simply to reach the second best utility possibility frontier. Of course, if all production sets are convex, under the classical assumptions, there exists a marginal cost pricing equilibrium which is Pareto optimal and hence second best. It is here that the presence of increasing returns makes a crucial difference in the welfare analysis. As mentioned above, none of the marginal cost pricing equilibria may be Pareto optimal, as has been shown by Guesnerie [5], and Brown and Heal [Z]. More recently, Beato and Mas-Cole11 [l] have shown that none of the marginal cost pricing equilibria may satisfy even aggregate production efficiency. But, since the Beato-Mas-Cole11 example pertains to a fixed structure of shares, and in that case production efficiency is not necessarily desirable, this still leaves open the possibility that some marginal cost pricing equilibria may be on the second best utility possibility frontier. However, we show that this may turn out not to be the case even in economies with a single firm and a fixed structure of revenues. Moreover, it is also possible that there exists an average cost pricing equilibrium which is a second best equilibrium and dominates every marginal cost pricing equilibrium.

The main conclusions are summarized in Section 5.

2. THE MODEL

We consider an economy with m consumers indexed by i, i= 1, . . . . m, and n firms indexed by j, j = 1, . . . . n. Xi E R’ and Xi E Xi refer, respectively, to the consumption set and a consumption plan of consumer i. xjk refers to consumer i’s consumption of commodity k. The utility function of consumer i is denoted ui: Xi H R’. We shall assume that all consumers have continuous

68 RAJIV VOHRA

utility functions. Y, c R’ and yj~ Y, denote the production set and a production plan of the jth firm. yjk is firm j’s production of commodity k. Let x = CT!, xi and y = C;= I yj. The aggregate endowment is denoted w and consumer i’s claim to the net output is denoted wi. For 5, z E R’, z $ z means that Z,>z, for all k, k=l,...,I. R’+={z~R’(z30} is the non- negative orthant of R’ and R’+ + = (z E R’[ z 9 01.

The income distribution is said to be given according to a fixed structure of revenues if, for all i, wi = ai( y + w), where ai > 0 and ET!, ai = 1.

The income distribution is said to be given according to a fixed structure of shares if, for all i, wi = I;=, 8, y.i + oi, where 8, > 0, I?= I 8, = 1 for all j, wi E Xi for all i, and Cy=, oi = w.

Given market prices p E R’, consumer i maximizes utility subject to income Ii = p . wi. The demand correspondence of consumer i is denoted gi(p, wi). We define a regulated equilibrium as an allocation and a vector of prices such that all consumers maximize utility subject to their budget constraints and all markets clear. No restrictions are imposed, at this stage, on the relationship between the production plans of the firms and the market prices, the interpretation being that the government is allowed to regulate all the firms in any way that it chooses. If optimal regulation requires marginal cost pricing, then it is only the firms subject to increasing returns which actually need to be regulated. It may also be worth reemphasizing that throughout this paper we shall be concerned with regulated equilibria in the context of fixed income distribution rules; if redistribution through lump sum transfers is feasible, a complete characterization of Pareto optimality is provided by a generalized Second Welfare Theorem.

A regulated equilibrium consists of consumption plans (Xi), production plans ( yj), and prices p E R’, p # 0, such that

(i) JT,E<&, wi) for all i, i= 1, . . . . m,

(ii) x = Y + 0.

In the recent literature on marginal cost pricing, Clarke’s normal cone is used as a genera1 definition of marginal cost prices. The normal cone characterizes marginal cost prices precisely as those which satisfy the first order conditions for profit maximization. In the case of a firm with a convex production set, therefore, the normal cone is identical to the set of profit maximizing prices. For a definition of the normal cone and a more detailed account of its properties, the reader is referred to Khan and Vohra [9, Section 21. Let N( Yj, y,) denote the normal cone of Yj at y;~ Y,. Henceforth we shall consider any nonzero element of N( Yj, yj) to be a vector of marginal cost prices for firm j when it produces yj.

A marginal cost pricing equilibrium is a regulated equilibrium ((x,), ( yj), p) such that p E N( Y,, yj) for all i.

OPTIMALREGULATION 69

An average cost pricing equilibrium is a regulated equilibrium ((x,), ( y,), p) such that p . y, = 0 for all j.

Whenever a social welfarefunction W((u,)) is specified, we shall assume it to be continuous and monotonically increasing in all the utilities. We can now define the two optimality notions we shall use.

A regulated equilibrium ((-xi), (y,), p) is said to be second best relahe to a social welfare function I#‘( .) if there does not exist another regulated equilibrium ((X,), (j,), p) such that W((u,(X,)) > W((u,(x,)).

A regulated equilibrium ((x,), (y,), p) is said to lie on the second best utility possibility frontier if here does not exist another regulated equilibrium ((Xi), (jj), p) such that ui(.Ui) > ui(xi) for all i, strict inequality holding for at least some i.

3. CHARACTERIZATION OF SECOND BEST EQUILIBRIA RELATIVE TO A SOCIAL WELFARE FUNCTION

If the government’s objective is to maximize a social welfare function over all the regulated equilibria that exist, distributional considerations may, in general, dictate production plans which do not satisfy aggregate production efficiency and prices which are not marginal cost prices. We begin by illustrating this for an Arrow-Debreu economy in Example 3.1. It will be clear from the example that this phenomenon may persist even if one of the production sets is made nonconvex. Example 3.2 is a simple modification of Example 3.1 and shows that even if production efficiency is desirable, due to distributional considerations, marginal cost pricing may not be optimal.

Theorem 3.1 shows that if the distribution is given by a fixed structure of revenues and demand functions are continuous, aggregate production efficiency is always desirable. Remark 3.1 points out that production efficiency in any individual firm is always desirable irrespective of the distribution rule being followed. Remark 3.2 relates this result and its proof to the literature on production efficiency.

Despite Example 3.2, if production efficiency is desirable, as is the case under a fixed structure of revenues, the marginal cost pricing principle turns out to be optimal provided the government has the added instrument of commodity taxation. This is the content of Corollary 3.1 which follows from Theorem 3.1 and Lemma 3.1. Lemma 3.1 guarantees that even if production sets are nonconvex, for any productively efficient production plan there exists a vector of prices which is a vector of marginal costs for each firm at its production plan.

EXAMPLE 3.1. The economy consists of two consumers, two commodities, and two firms. x,~ refers to consumer i’s consumption of

70 RAJIV VOHRA

commodity k and yjk to firms j’s production of commodity k. The data of the economy are as follows:

X,=R$, u,(-~,)=-~,l, WI = (10, 01, e,, = 1, e,2=0,

X2= R2+, u2(x2) = Minh, , x22 ), 02 = (10, O), e2, = 0, e22 = I,

Y,={~‘,ER~I~,,~O;~‘,,+,~~~O},

Y2 = { y2 E R2 I yzl d 0; My,, + ~22 d 0).

This information is sketched in Figs. 3. la and 3.lb. From consumer l’s preferences it is clear that in any regulated equilibrium pi, the price of commodity 1 must be positive. We can, therefore, normalize p, = 1. It is easy to check that the only Walrasian equilibrium is one where p2 = 2/3, y, = (0, 0), y, = (-4, 6), .Y, = (10, 0), and x2 = (6,6). However, a regulated equilibrium can be found where consumer 2’s utility is increased at the expense of consumer 1 by regulating firm 1 to operate at a loss. Suppose production and prices are regulated as follows: y, = (-2.5, 2.5), y,=(-3,4.5), and p=(l,O). Notice that p=(l,O) is not a vector of marginal cost prices for either firm at its given production plan. Now con-

FIGURE 3.la


i -

0

FIGURE 3.lb

sumer l’s income is 7.5 and consumer 2’s income is 7. With xi = (7.5,O) and x2 = (7, 7) this is a regulated equilibrium in which consumer 2’s utility is higher relative to that at the Walrasian equilibrium. We can also verify that this regulated equilibrium lies on the second best utility possibility frontier. If consumer 1 is to be provided with utility level 7.5, then, for p2 > 0, at least 2.5 units of commodity 2 must be produced by firm 1. Since all of commodity 2 is consumed by consumer 2, given the more efficient technology of firm 2, it is not worthwhile to have more than 2.5 units of commodity 2 produced in firm 1. Thus, this regulated equilibrium corresponds to maximizing u2 subject to U, = 7.5. Similarly, it can be shown that maximizing ui subject to u2 = 7 also leads to the same regulated equilibrium. Since this equilibrium is on the second best frontier there exists a social welfare function which dictates that this equilibrium be established. Of course, any production in firm 1 involves aggregate production inefficiency and we have, therefore, shown that if the income distribution is given by a fixed structure of shares, aggregate production efficiency and marginal cost pricing may not be desirable.

EXAMPLE 3.2. The only firm is firm 2 of Example 3.1, o = (20,0), and the consumers have the same preferences as in the previous example. The

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income distribution is given by a fixed structure of revenues and a, = a, = l/2. It is easy to verify that the only Walrasian equilibrium is one where p = (1,2/3), y = (-4,6), X, = (10, 0), and x2 = (6,6). If production is regulated to be ( - 2, 3) and p = (1,4), there is a regulated equilibrium with xi = (15,O) and x2 = (3, 3). Thus by setting prices which are not marginal cost prices U, can be increased at the expense of u2. Similarly, by producing more and reducing p2 it is possible to increase u2 at the expense of u,. This illustrates that even if production efficiency is desirable, due to distributional considerations, the same is not necessarily true of marginal cost pricing.

THEOREM 3.1. Suppose the income distribution is given by a fixed structure of revenues, all utility functions are weakly monotonic, and all production sets satisfy free disposal, in the sense that Yj- R’G Y,. Zf ((xi), (y,),p) lies on the second best utility possibility frontier and ti(p, wi) are continuous functions for all i, then y satisfies aggregate production efficiency.

Proof. Suppose y does not satisfy aggregate production efficiency. Given free disposal, this implies that y lies in the interior of the aggregate production set Y, i.e., there exists E > 0 such that y + EE Y for all EE R’, (11(1 GE. By the continuity of the demand functions, there exists 6 > 0, depending on E, such that for all SE R’ and ll8ll 6 6,

where

X=CXi and -fi=ti(P,ai(.J+J+Q))), for all i.

Choose 6~ 0. Since, by weak montonicity, p B 0 and p # 0, this ensures that p. ai( y + S+ o) > p. a,(y + o) for all i. By weak monotonicity, this means that ui(Xi) > uj(xi) for all i. Let j = y + (X-x). Since IlX --XII GE, j E Y and there exist jje Y, such that cj jj = j. Moreover, X - j = x-y=w, so that ((Xi), (Y,)) is feasible. By weak monotonicity, we also know that p.x=p.(y+o) and p.X=p.(y+S+o). This implies that p.(X--x)=p.6 and p.j=p(y+X-x)=p.(y+S). Thus xi=~,(p,ai(y+S+o))=~,(p,ai(j+w)) for all i which implies that ((Xi), (jj), p) is a regulated equilibrium. Since ui(Xi) > ui(xi) for all i, this contradicts the hypothesis that ((x,), (yj), p) lies on the second best utility possibility frontier. 1

Remark 3.1. From the proof of Theorem 3.1 it should be clear that a similar argument can be used to show that even if income distribution is given by a fixed structure of shares, production efficiency in each individual firm is always desirable.


Remark 3.2. The existing literature (see, for example, Diamond and Mirrlees [4] and Hahn [7]) shows that in an economy in which commodity taxes are used to finance public expenditure, the presence of constant returns or the availability of profit taxes in a convex economy is a sufficient condition for production efficiency to be desirable. Theorem 3.1 shows that in our model, a fixed structure of revenues is a suffkient condition for production efficiency to be desirable even if production sets are nonconvex. It is instructive to compare the proof of Theorem 3.1 with the Diamond-Mirrlees proof. The latter depends on the fact that a small change in prices changes aggregate demand by a small amount which in turn requires a small change in aggregate production. Under constant returns to scale, as in Diamond and Mirrlees [4], or if profits can be taxed, as in Hahn [7], this has no further effect on demands so that utility improving changes in prices are feasible. In our model, not only do demands depend on production plans but also, because of increasing returns, a small change in aggregate production could require a large change in individual production plans3 Under a fixed structure of revenues, demands depend on aggregate production and, therefore, continity arguments can be made. Under a fixed structure of shares this reasoning does not apply since an increase in aggregate production need not increase the income, evaluated at the original prices, of all the consumers. In fact production efficiency may not be desirable, as shown by Mirrlees [lo] for a slightly different model and confirmed by Example 3.1 for our model.

LEMMA 3.1. Suppose all production sets satisfy free disposal and .Y=CjYj, YIE yj f or all j. If y satisfies aggregate production efficiency, in the sense that y belongs to the boundary of Y, then there exists p E R’, p # 0 such that p E N( Y,, yj) for all j.

ProoJ Consider the economy with a single consumer having a consumption set X, and endowment w such that z = y + OE X, and preferences such that (z} + R’+ + is the set of consumption bundles pre- ferred to z and (z} + R’+ is set of bundles at least as good as z. Clearly, (z, (y,)) is a Pareto optimal allocation for this economy. We can now apply Theorem 1 of [9] to assert that there exists p E R’, p #O such that PE N( Y,, y,) for all j. 1

It is worth emphasizing that in the presence of increasing returns, in general, if p E N( Y, y) there is no guarantee that p E N( Y,, y,) for all j. This is brought out very clearly in the Beato-Mas-Cole11 example [ 11. However, Lemma 3.1 ensures that any efficient production plan can be decentralized through some vector of marginal costs.

3 As is the case in the Beato-Mas-Cold example [ 11.

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From Theorem 3.1 and Lemma 3.1 we can immediately deduce the following corollary.

COROLLARY 3.1. Suppose ((x,), (y,), p) is a regulated equilibrium which lies on the second best utility possibility frontier and all the conditions of Theorem 3.1 are satisfied. Then there exists a vector of marginal cost prices p such that the allocation ((x,), (y,)) can be sustained as a marginal cost pricing equilibrium with commodity taxes (p - p) in the sense that:

(a) x, = ri(p, a,( y + 0))) for all i,

(b) p E N( Y,, yj) for all j,

(c) x=y+o.

Thus, if the income distribution is given by a fixed structure of revenues, commodity taxation can be seen as an additional instrument which restores the desirability of marginal cost pricing. Notice that in condition (a) above, given our definition of t,(p, w,), consumers’ profit incomes are evaluated at consumer prices p; Ii = p . ai( y + 0). It may be preferable to consider profits evaluated at producer prices J& in which case the income of consumer i would be Z, = a,(p . y + p . 0). However, if this changes the incomes, the government’s budget will no longer be balanced. The revenue collected would amount to t = (p-p). y. If r is returned to the consumers lump sum, in accordance with the original distribution rule, we would have Ii = a,(p . y + p . w + r) = ai(p . y + p . w ), as in (a) above. Alternatively, we may consider consumers’ profit incomes arising only from the competitive sector. In this case again it is easy to check that if any surplus or deficit is distributed to the consumers according to the original distribution rule this is equivalent to setting Ii as indicated by condition (a). In either case, therefore, Corollary 3.1 is applicable.

Given that in our model lump sum taxation is possible, as long as the distribution rule, given according to a fixed structure of revenues, is followed, it may seem that need for commodity taxation arises solely from distributional considerations. However, as we shall see in Example 4.2, in an economy with increasing returns it may not even be possible to achieve economic efficiency without such taxation. This underscores the view of Brown and Heal [2] that in the presence of increasing returns judgments on elliciency and equity cannot be regarded as separable.

4. CHARACTERIZATION OF EQUILIBRIA THAT LIE ON THE SECOND BEST UTILITY POSSIBILITY FRONTIER

If all production sets are convex, under the classical assumptions there exists a Walrasian equilibrium. Of course, this is also a marginal cost


pricing equilibrium and, since it is Pareto optimal, it lies on the second best utility possibility frontier. Thus, in the absence of distributional considerations, marginal cost pricing is optimal. If all production sets are not convex and the income distribution is fixed, none of the marginal cost pricing equilibria may be Pareto optimal. This was demonstrated by Guesnerie [S] through an example with a fixed structure of revenues and by Brown and Heal [2] through an example with a fixed structure of shares. The fact that some marginal cost pricing equilibrium may not be Pareto optimal is easy to see in an example with one consumer and a production possibility curve which is not concave to the origin. However, examples in which none of the marginal cost pricing equilibria are Pareto optimal must necessarily be ones in which there are many consumers. This is so because for a single consumer economy the question of income distribution being fixed or not is irrelevant. And, given the existence of a Pareto optimal allocation, the general version of the Second Welfare Theorem now implies that such an allocation can be sustained as a marginal cost pricing equilibrium.4 Given this observation, it is easy to understand why Brown and Heal [3, Proposition 23 have also been able to show that if the income distribution is given by a fixed structure of revenues and all preferences are homothetic, so that, by the Eisenberg theorem, they can be aggregated, there exists a Pareto optimal marginal cost pricing equilibrium.

Remark 4.1. If none of the marginal cost pricing equilibria is Pareto optimal, then, given the income distribution rule, the ability to achieve Pareto optimality through regulation is severely restricted. Suppose ((x,), (,;)) is a Pareto optimal allocation. By the generalized Second Welfare Theorem, this allocation can be supported by marginal cost prices. If at this allocation, any consumer has smooth preferences or if any firm has a production set with a smooth boundary, these (normalized) support prices must be unique. Thus, marginal cost pricing is necessary for Pareto optimality. If no marginal cost pricing equilibrium is Pareto optimal, no other form of regulation can achieve Pareto optimality either. In this case, therefore, the second best utility possibility frontier is strictly dominated by the Pareto frontier.

We shall now show that in an economy with increasing returns, none of the marginal cost pricing equilibria may lie on the second best utility possibility frontier. Given Theorem 3.1 it would suffice to construct an example in which the income distribution is given by a fixed structure of

4 See also Brown and Heal [2, Section 61, Beato and Mas-Cole11 [l. Remarks 2 and 31. and Guesnerie [6, Section III A].

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revenues and none of the marginal cost pricing equilibria is productively efficient.’ But this would still not resolve the issue if there is a single firm in the economy. We, therefore, present examples with a single firm so that, given Remark 3.1, production efficiency is no longer an issue. It is clearly preferable to construct an example with a fixed structure of revenues since this can also be seen as one with a fixed structure of shares. We do this in Example 4.2.6 However, we begin by considering in Example 4.1 an economy with a fixed structure of shares since this shows in a very simple way that not only may no marginal cost pricing equilibria lie on the second best frontier but every one of these may actually be dominated by an average cost pricing equilibrium which lies on the second best frontier.

EXAMPLE 4.1.

X,=R:, %(x*)=x12, ml= (0, lo), 81=1,

X,=R:, ~,(x,)=4logxzl +x22, 02 = (20, Oh 8, = 0,

Y={y~R~Iy,<O;y,+y,<0 if y,> -16 and lOy,+y2+14460

if y, < -16).

Given consumer l’s preferences, in looking for marginal cost pricing equilibria we can normalize p2 = 1. Now, marginal cost prices must be such that p, = 1 if y, < 16, pi E [l, lo] if y, = 16 and pi = 10 if y, > 16. If there is a marginal cost pricing equilibrium with y, > 16, we must have I, = 10 + lOy, + y,. Given Y, this implies that I, = -134~0, so there cannot exist a marginal cost pricing equilibrium with y, > 16. If y=(-16,16) and p=(l, 1) w e h ave a marginal cost pricing equilibrium where x1 = (0, 10) and x2 = (4, 16). It is easy to see that this is the only marginal cost pricing equilibrium in this economy.

The consumption possibilities of consumer 2 when xi = (0, 10) are sketched in Fig. 4.1. It is clear that the marginal cost pricing equilibrium is not Pareto optimal. Moreover, given ur = 10 we can increase consumer 2’s utility by considering a regulated equilibrium which is in fact an average cost pricing equilibrium. Let y = (- 18, 36), p = (2, l), x1 = (0, lo), and xq = (2, 36). This is a regulated equilibrium in which U, = 10 but u2 = 4 log 2 + 36 > 4 log 4 + 16. It is also clear from Fig. 4.1 that, given the

5 Note that the Beato-Mas-Colell example [l] on production inelfciency pertains to an economy with a fixed structure of shares. In [ 1 l] a similar result was established for an economy with a fixed structure of revenues.

6 While Guesnerie’s example [5] is one with a fixed structure of revenues, it does not s&ice for our puroses since in that example all regulated eqilibria are also marginal cost pricing equilibria; all marginal cost pricing equilibria lie on the second best frontier.


FIGURE 4.1

income distribution rule, this average cost pricing equilibrium lies on the second best utility frontier.

EXAMPLE 4.2. There are two consumers, two commodities, and one firm and the income distribution is given by a fixed structure of revenues:

o = (6, 16), a, = l/4, a2 = 314,

X,=R:, Ul =x11 +x,2,

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for u2 6 30, u2 = 6x?, + xZ2

for 3O<u,<31.5, zd2= 6.~1 + ~22, ifx,,<3;

(106 - (10/3) u2) x2, +x2? + IO@, - 30), otherwise

foruza31.5, u2= 6.~ + x22 3 if xl1 < 3;

x21 + x22 + 1.5, otherwise.

if .)J, d -2).

The preferences of the consumers and the production possibility curve are illustrated in Figs. 4.2a, 4.2b, and 4.2c, respectively. Notice that consumer 2 has nonhomothetic preferences. The indifference curve of consumer 2 through (3, 12) has slope - 6 and all indifference curves below it are parallel to it. All indifference curves have slope -6 if x2! < 3. For u2 3 31.5 and xyzZ > 3 all indifference curves has slope - 1. For u2 between 30 and 31.5, and xZ2 > 3, the slopes of the indifference curves vary linearly, going from -6 to - 1.

Since preferences are monotonic, in searching for marginal cost pricing equilibria we can normalize p2 = 1. Marginal cost prices are as follows:

FIGURE 4.2a


0

FIGURE 4.2b

P,E[O,~O] if y,=O, p,=O if -2<y,<O, pI~[0,5] if y,= -2, and p, = 5 if y, < -2. Suppose there is a marginal cost pricing equilibrium with y, 6 -2. It is easy to check that in this case I, +Z2 =4pl + 16 and Z2 = 3p, + 12. This means that consumer 2’s budget line must pass through (3, 12), where the slope of the indifference curve is -6 (see Fig. 4.2b). For p, E [0, 51, which must be the case for marginal cost pricing with y, < -2, this implies that consumer 2 will consume only commodity 1 and we get xal = 3 + 12/p, > 4. But, by the feasibility condition of equilibrium, xzl > 4 implies that y, > -2, a contradiction. Clearly then, the only possible marginal cost pricing equilibria must involve y = (0,O) and x = (6, 16). In this case I, + Zz = 6p, + 16 and I, = 4.5~~ + 12. This means that consumer

64?,45:1-6

80 RAJIV VOHRA

j-

I

0 4 6 -)xll+x2l

FIGURE 4.2~

2’s budget line must pass through the point (4.5, 12), where the slope of the indifference curve is - 1 (see Fig. 4.2b). If p, < 1 both consumers consume only commodity 1 and the aggregate demand for commodity 1, x,r + xzl = (6p, f 16)/p,. Since p1 < 1, x,* +x1, > 6 which contradicts the fact that x = (6, 16). If pr > 1 then consumer 1 will consume only commodity 2. Since consumer 2’s budget line passes through (4.5, 12), consumer 2 will consume at most 3 units of commodity 1 and spend the rest of the income on ‘commodity 2 (see Fig. 4.2b). Thus, the aggregate demand for commodity 2, x12+xz2>6p1 + 16-3~~. Since pr > 1, .x1* + xz2 > 16, which again is impossible given x = (6, 16). The only marginal cost pricing equilibrium is one with y = (0,O) and p1 = 1. In this case we have I, = 5.5, Z2 = 16.5. Consumer 1 is indifferent between consuming either commodity. Consumer 2’s budget line passes through (3, 13.5) and, for x2, >, 3 it is identical to the indifference curve corresponding to u2 = 31.5. Thus con-


sumer 2’s utility maximization involves consuming at least 3 units of commodity 1 and spending the rest of the income on commodity 2. x, = (3,2.5) and x2 = (3, 13.5) are two possible demand vectors. There are of course, many other combinations of demands which are consistent with p, = 1 and y = (0,O) but they are all equivalent in terms of U, and us. Thus all the marginal cost pricing equilibria in this economy yield U, = 5.5 and 2.42 = 31.5.

Now consider the following regulated equilibrium which involves average cost pricing: let y = (- 3, 5), x = (3,21), and p = (5/3, 1). In this case I, = 6.5, I, = 19.5 X, = (0, 6.5) and x2 = (3, 14.5) are demands for the two consumers which add up to (3, 21). Thus, this is a regulated equilibrium. Moreover, it dominates every marginal cost pricing equilibrium since U, = 6.5 and u2 = 32.5.

5. CONCLUSION

In this section we conclude by summarizing the results on the welfare analysis of marginal cost pricing. In general, marginal cost pricing cannot be considered optimal if the income distribution is given either by a fixed structure of revenues or by a fixed structure of shares. As we saw in Exam- ples 3.1 and 3.2, this is the case even in convex economies if distributional considerations are relevant. The relationship between the Pareto frontier and the second best utility possibility frontier can, therefore, be represented as in Fig. 5a. In economies with increasing returns none of the marginal cost pricing equilibria may be on the Pareto frontier. Moreover, by Remark 4.1, it may then be impossible to attain Pareto optimality. Given Examples 4.1 and 4.2, every marginal cost pricing equilibrium may be dominated by an average cost pricing equilibrium which lies on the second best frontier. This is illustrated in Fig. 5b.

There are some cases in which marginal cost pricing is desirable. If the income distribution is given by a fixed structure of revenues and commodity taxation is feasible, then according to Corollary 3.1 any second best allocation can be supported through marginal cost pricing. If distributional considerations are not relevant then again if the income distribution is given by a fixed structure of revenues and preferences are homothetic, there exists a Pareto optimal marginal cost pricing equilibrium, as shown by Brown and Heal [3]. Yet other cases in which marginal cost pricing is desirable are discussed in a recent survey of Guesnerie [6, Section IV C]. In Table I we present a summary of the welfare results concerning production efficiency and marginal cost pricing under various assumptions about the income distribution.

A A

Marginal Cost Pricmg Equilibrium Marginal Cost Pricmg Equilibrium

Second Best Frontier Second Best Frontier

Poreto Frontier Poreto Frontier

+ UI + UI 0 0

CONVEX ECONOMY CONVEX ECONOMY

FIGURE 5a

A A

Marginal Cost Prlcmg Equllibrlum Marginal Cost Prlcmg Equllibrlum

Averoge Cost Prlclng Equlllbrlum Averoge Cost Prlclng Equlllbrlum

m m

NONCONVEX ECONOMY NONCONVEX ECONOMY

FIGURE 5b FIGURE 5b


TABLE1

Summary of Welfare Results

Distribution Rules

Welfare properties of

Aggregate production efficiency Marginal cost pricing

Arbitrary lump sum transfers

Fixed structure of shares

Desirable (generalized second Desirable (generalized Second Welfare Theorem, Guesnerie Welfare Theorem) [S], Khan and Vohra [9])

Not necessarily desirable Not necessarily Pareto optimal“ (Example 3.1) (Brown and Heal [Z])

Fixed structure of revenues

Desirable (Theorem 3.1 )

Not necessarily on second best frontier’ (Example 4.1)

Not necessarily on production possibility frontier” (Beat0 and Mas-Colell [ I])

Not necessarily Pareto optimal“ (Guesnerie [5 1). unless preferences are homothetic (Brown and Heal [3])

Not necessarily on second best frontier” (Example 4.2). unless commodity taxation is available (Corollary 3.1)

Not necessarily on production possibility frontier” (Vohra [ll])

“These results depend crucially on the presence of increasing returns.

REFERENCES

1. P. BEATO AND A. MAS-COLELL. On marginal cost pricing with given tax-subsidy rules, J. Econ. Theory 37 (1985), 356365.

2. D. J. BROWN AND G. M. HEAL, Equity, efficiency and increasing returns, Reu. Econ. Stud. 46 (1979), 471485.

3. D. J. BROWN AND G. M. HEAL, The optimality of regulated pricing: A general equilibrium analysis, in “Lecture Notes in Economics and Mathematical Systems,” Vol. 244, “Advan- ces in Equilibrium Theory” (C. D. Aliprantis, 0. Burkinshaw, and N. J. Rothman, Eds.), Springer-Verlag, Berlin, 1985.

4. P. A. DIAMOND AND J. A. MIRRLEES, Optimal taxation and public production, I. Produc- tion etliciency, Amer. Econ. Rev. 61 (1971), S-27.

5. R. GUESNERIE, Pareto optimality in non convex economies, Econometrica 43 (1975), l-29. 6. R. GUESNERIE, First best allocation of resources with non convexities in production,

mimeo, 1984. 7. F. HAHN, On optimum taxation, J. Econ. Theory 6 (1973), 96106. 8. H. HOTELLING, The relation of prices to marginal costs in an optimum system,

Econometrica 7 (1939). 151-155.

84 RAJIV VOHRA

9. M. ALI KHAN AND R. VOHRA. An extension of the second welfare theorem to economies with nonconvexities and public goods, Quart. J. Econ. 102 (1987), 223-241.

10. J. A. MIRRLEES, On producer taxation, Rev. Econ. Bud. 39 (1972), 105-111. 11. R. VOHRA, Second best equilibria in economies with increasing returns and fixed income

distribution, Brown University mimeo, 1984. 12. R. VOHRA, On the existence of equilibria in economies with increasing returns, in “Special

Issue on Increasing Returns,” (B. Cornet, Ed.), J. Math. Econ., in press.

optimal regulation under fixed rules for income distribution* · pareto optimal allocation can be...

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