Optimal provision of public goods through Nash equilibria

JOURNAL

OF ECONOMIC

THEORY

23, 334-347

(1980)

Optima! Provision of Public Goods through Nash Equilibria MIKIO Faculty

of Economics, Received

Toyama December

NAKAYAMA

University, 22, 1978;

3190 revised

Gojiiku,

Toyama

December

City

930, Japan

3, 1979

INTRODUCTION

In recent studies of exchange economies, the concept of a Nash equilibrium has come to play a central role to consider how trading can be organized under informational decentralization. In these studies, achieving Pareto optimality through Nash equilibria is the main purpose of investigation. Postlewaite and Schmeidler [6] have presented a nonWalrasian exchange model in which Nash equilibrium allocations are approximately Pareto optimal. Wilson [9] has shown the existence of a Nash equilibrium which attains a Pareto optimal allocation in a model analogous to competitive bidding, and fruther showed that under replication this allocation converges to a Walrasian allocation. Schmeidler [7] has proved, in a pure exchange model, that a Nash equilibrium allocation do coincide with a Walrasian allocation. In this paper, we shall, consider whether such results can be extended to the case including public goods. Groves and Ledyard [ 1] have presented, in a general equilibrium setting with public goods, an allocation mechanism under which Pareto optimal allocations are attained through Nash equilibria. But, we are interested more in the equivalence between Nash equilibria and Lindahl equilibria. A very general result has been obtained by Hurwicz [3], where it is shown under certain continuity and convexity assumptions imposed on outcome functions that every interior Nash equilibrium allocation which is both Pareto optimal and individually rational must coincide with a Lindahl allocation. The outcome function to be defined in the present paper yields an equivalence between a Lindahl equilibrium and a Nash equilibrium with positive production of a public good, but it turns out that a Nash equilibrium with no production is also existent simultaneously under some circumstances. This latter property of our outcome function does not follow one of the requirements of Hurwicz [3] that every Nash equilibrium allocation be both Pareto optimal and individually rational, 334 0022.0531/80/060334-14$02.00,‘0 Copyright All rights

C 1980 by Academic Press, Inc. of reproduction in any form reserved.

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335

which is needed in the proof of the result that a Lindahl allocation is a Nash equilibrium allocation. Our outcome function consists of several rules for production and financing its cost. Each individual is asked to propose his demand of the quantity of a public good and his share in the cost of its production. Given these proposals, the function then determines the level of production and a cost share distribution to individuals. For expository convenience, we confine ourselves to the case with one public good and one private good. But, the extention to the case with more than one public good is easy as will be noted later.

2.

THE

ECONOMY

Let N= { 1, 2,..., n}, (n > 3) be the set of all individuals. We assume that there are two commodities in this economy, one of which is a public good and the other is a private good that is thought of as money. Let Ii > 0 be the quantity of initial money endowment of i E N. Initial quantity of a public good is assumed to be zero. Let q > 0 be the quantity of the public good produced, and c(q), its cost in terms of money. We assume that c(q) is linear and increasing in q > 0, so that it is represented by cq, where c > 0 is a constant. An allocation is a vector (q, xi,..., xn) that satisfies q > 0 and cq + cj,, xi = LEN I,, where xI is the quantity of money which individual i retains after the payment for the public good. When xi < 0, the individual i is assumed to be in debt by the amount Ixt I. Preference of individual i over all (4, Xl ,***, x,J is represented by a utility function u,(q, xi) defined on R + X R, where R is the real line and R, is the nonnegative part of R. We assume that for all q > 0 and x > 0, u,(q, x) is continuous, monotone increasing and quasiconcave, that is, (monotonicity)

4 > 4, 2 2 x and C&f) # (q, x) imply Ui(qp-f) > u,(q, x), (quasiconcavity) Ui(~, 2) > Ui(q, X) and 0 < cz< 1 imply Ui(U4 + (1 - a)q, UZi+ (1 - U)X) > U,(q, X)a Moreover, we put the following assumption. ASSUMPTION

1.

Zf X < 0, then for all q > 0 and x > ,f, ui(q, x) > ui(q, 2).

336

MIKIO

NAKAYAMA

This assumption is interpreted to mean that if individual i were to get into debt, he would choose a smaller size of the deficit. A stronger assumption is seen in Schmeidler [7]. Now, let r = (ri ,..., r,) be a vector satisfying zjEN rj = 1 and rj > 0 for all j E N. Then, we call the pair (r, q) a Linduhl equilibrium if for every i E N, O
Ui(q, Ii - ricq) > U,(~TIi - ric&

for all CJ> 0 such that r,cq < Ii.

Thus, given cost share ratios r the quantity q of the public good in equilibrium is the one that maximizes the utility of each individual under each budget constraint. Note that ri > 0 for all i E N in equilibrium, due to the monotonicity of ui.

3. THE OUTCOME FUNCTION Following the procedure of the strategic approach to economic equilibrium.we construct an allocation mechanism, the outcome function, by which the outcome of the strategic behavior of individuals is determined. For this purpose we let each individual report both this demand of the level of the public good and his share in the cost of production. Given these proposals, it is then assumed that according to some rule a p,articular agent, which is thought of as a government, determines the production level and a cost share ratio to each individual. Every individual is assumed to know the rules, and the government knows the initial distribution of money. Formally, let Si be the set defined by Si = {Si = (ri, qi) E R + X R, : ricqi < Zi}.

Si is the set of strategies of individual i. Let s = (s,,..., s,) E S = s, x **. x S, be a selection of strategies. Then we denote by q(s) the quantity of the public good produced, and by ri(s) the cost share ratio to individual i, so that the resulting allocation is represented. by (q(s), 1, - rl(s) cds),..., 1, - r,(s) 4s)). F or each selection s E S, we define q(s) by q(s) = \‘ rjqj, ,I

If riqi = 0 for all i E N, then ri(s) = 0 for all i E N.

PUBLIC

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If riqi > 0 for some i E N, we define ri(s) as follows. Let T(S) and T’(s) be subsets of N defined, respectively, by T(s) = {h E N: q,, = q(s)}, TV

for each j E N.

= {h E N: qh = qj}

Then we distinguish between three cases: Case I.

If Tj(s) = N for all j E N, then Ti(S) = 1 - x

rj +

for all i E N.

izi

Case ZZ. If there exists a k E N such that (A) T(S) = {k}, or (B) k @ T(s) and Tj(s)=N{k} for alljE N- {k} then rjqj Ti(S)

+

14j

-

4Cs)l

-

Cj+i

c

jeN

‘jqj

rj(ds)

-

4j)

for i = k,

=

‘i9i-

y-&

I qk

-

q(s)1

+

&

IZj+k

rj(ds)

-

4j)

for all i # k.

= CjsN

‘jqj

Case 111. If neither Case I nor Case II holds, then

rjqj + I4j - q(s)1- +

EjEN

Iqj - 4Cs)l for all i E N.

ri(s) = CjcN

rjqj

In Case I, every individual is demanding the same quantity of the public good. The resulting ratio takes its minimum when the reported ratios sum to 1. Note that the resulting ratios always sum to 1 when rl + s.. + r, 2 1, but is larger than 1 when r, + -.e + r,, < 1. Hence in the latter case the total budget constraint

“

,z

rj(S)

Cq(S)

<

\‘

,=

Zj

(1)

may not be satisfied. In this case, we assume that the government covers the deticit and individuals are getting into debts to the government. This lack of balance of the total budget will be removed later. In Case II, the individual k is the one who is distinguished from the others. He is the only individual whose demand of the public good is either equal to q(s) or not conformed with all of the others.

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Note that rk(s) can be rewritten as rk(S) = 1 - 2 rj + I qk - q(s)1 4s)

j#k

*

Thus if condition (A) holds, his resulting ratio is independent of his proposal rk, and if qk # q(s) it becomes larger. The ratios of all the others are simply adjusted so as to guarantee the equality ri(s) + ... + r,,(s) = 1.

In Case III, no such distinguishable individual exists and the adjustment works in such a way that the ratio becomes smaller as the demand of the public good gets nearer to q(s). Before completing the description of the mechanism, we note that for some selection of strategies the outcome function may assign a negative consumption to some individual, i.e., Ii - ri(s) cq(s) < 0. To deal with the infeasibility we assume that the deficit means a debt of the individual to other individual or the government. As mentioned before, the imbalance of budget can occur in the aggregate in Case I. This total infeasibility can be removed by a slight modification of the rule (Section 7). Yet, it would be useful to note that every individual is endowed with the option to avoid the imbalance of the total budget, because he can always shift his strategy to generate a state in Case II which is totally feasible and better for himself. Moreover, it will turn out that at Nash equilibria the budget is both totally and individually feasible. For a detailed discussion about the infeasibility, see Schmeidler [ 71. Finally, let us recall the definition of a Nash equilibrium. Given s E S and pi E Si for some i E N, we denote by (s / i, Si) the selection of strategies s’ c S, where sj = sj for all j # i, and si = di. Then, a selection s E S is called a Nash equilibrium if for any i E N there exists no strategy Si E Si such that U,(q(S

1i, ~ii), Ii - ri(s 1i, fi)

We will occasionally Lindahl equilibrium.

4.

NASH

Cq(S

1i, Si)) >

Ui(q(S),

Ii - ri(s)

denote by NE a Nash equilibrium,

EQUILIBRIA

WITH

POSITIVE

Cq(S)).

and by LE a

PRODUCTION

In this section we consider the case with positive production of the public good. To begin with, we have the following theorem. THEOREM 1. Let (r:,..., r,*, q*) be an LE with q* > 0. Then s* E S is an NE, where ST = (r:, q:), qT = q* for each i E N.

PUBLIC

ProoJ for all 1 - Cjzi Now, strategy

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By the definition of an LE, we have Cjshr rj* = 1 and r? > 0 j E A? Hence, q(s*) = CjeN r?qT = q*, and therefore ri(s*) = r? = rf for all i E N, because Case I holds in s*. suppose the conclusion is false, and there exists an i E N and a si = (ri, qi) E Si such that Ui(q(S* / i, si), Ii - ri(S* ( i, si)

Cq(S*

/ i, si))

> Ui(q(S*)? Ii - ri(S*) Cq(S*)) = Ui(q*, Ii - rTcq*).

(2)

Since ri # rf or qi # qr, it follows that ri(s* / i, si) > rf, because Case I or Case II applies to (s* ] i, si). Then if Ii - ri(s* 1i, si) cq(s* 1i, si) > 0, we have Ii - rfcq(s*

I i, si) > 0

and ui(q(s* I i, si), Ii - rTcq(s* I i, si)) > ui(q*, Ii - rTcq*) by (2) and the monotonicity of ui. But, this contradicts the assumption that PT,..., rz,q*) is an LE. Next, suppose Ii - ri(s* ) i, si) cq(s* / i, si) < 0. Note that q(s* 1i, si) > 0. Then, by Assumption 1 we have Ui(q(S* 1i, si), 0) > Ui(q(s* 1i, SJ, Ii - ri(s* 1i,

Si) Cq(S*

I i, si)).

(3)

Now, let S be defined by rTcq = Ii. Then, ricqi < Ii = r:cq and q* < q. Hence, q(s* ) i, si) = cj,i rTqj+ riqi < (1 - rT)q + r:q. Then, by the monotonicity of ui we have ~~(4, Ii - r:cq) = u,(& 0) > u,(q(s* I i, si), 0). But, then, (2) and (3) imply ui(q, Ii - rTcq> > ui(q*, Ii - rTcq*), which is a Q.E.D. contradiction. The Lindahl equilibrium can be shown to exist under our assumptions (see, e.g., Kaneko [5]). Thus, by Theorem 1, the existence of a Nash equilibrium is guaranteed. Now, we state the converse of this theorem. 2. Let s* E S be an NE with q(s*) > 0. Then, q(s*) = qf and = rT > 0 for all i E N. Moreover, (rl(s*),..., r,Js*), q(s*)) is an LE.

THEOREM

ri(s*)

To prove this, we need several lemmas. LEMMA

true.

1. Let s* E S be an NE with q(s*) > 0. Then, the followings

are

340

MIKIO

(a) (b) Moreover, (c)

NAKAYAMA

If Case Z applies to s*, then T(s*) = N. Zf Case II applies to s*, r? = 0 for all j # k. CaseZII

then T(s*) = {k)

for

some k E N.

does not apply to s*.

Proof of (a). Suppose the contrary, and let qT # q(s*) for some i E N. Then we have q(s*) = CjeN $47 = CjaN rj*qT # cj,, ryq(s*). Hence, CjsN rj* # 1, and therefore

ri(s*) > 1 - x ri*. jzi

Then,

letting r,qi = rTqT, qi = q(s*) and si = (ri, qi), we have T(s* 1i, si) = {i}. Hence, we have q(s* 1i, si) = q(s*) and ri(s* 1i, si) = 1 Cjzi r? ( ri(s*). But, by monotonicity of ui, this leads to a contradiction that s* is not an NE. Proof of (b). Suppose for some i E N, condition (B) of Case II holds in s*. As in the proof of (a), we show that there exists a strategy si = (ri, qi) satisfying q(s* 1i, si) = q(s*) and ri(s* 1i, si) < ri(s*). To show this, let us define si so that it satisfies riqi = r?qT, 0 < 1qi - q(s*)I < 1s: - q(s*)I and qi # q? for all j # i. Then, we have q(s* / i, si) = q(s*). And, since condition (B) still holds in (s* ) i, s,), we have

rits* I i , s,) = r:qT + l4i - dS*)I c

JEN

Cj+i

$+W*)

-

Si*>

<

ri(S*).

‘j%T

Hence we have a contradiction as before, and therefore condition (B) does not hold in s*. Hence, condition (A) must hold, i.e., T(s*) = {k} for some k E N.

Next, we show that t-7 = 0 for all j # k. Suppose there exists an i E N such that i + k and rT > 0. It will suffice to show that q(s* I i, si) = q(s*) and ri(s* 1i, si) < r,(s*) for some si = (ri, qi) E S;. Let us define si by riqi= rTqT, qi#q(s*) and ri ( rT. Then, we have q(s* 1i, si) = q(s*) and T(s* 1i, si) = T(s*) = {k). Hence, by definition, rTsT + --LTj~i,j,xri*(4(s*)-q~)+~ri(s(s*)-si) n_ 1 r&s* I i, si) = <

cl@*) ri(S*).

Proof of(c). Suppose Case III applies to s*. Then, at least, we must have qT # q(s*) for some i E N. But, for this i E N, we can choose si = (ri, qi) E Si such that r,qi = r-747 and 0 < (qi - q(s*)I < /qT - q(s*)j,

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and

that Case III still q(s* ( i, si) = q(s*) and

ri(s* I i, si) =

applies

to

(s* 1i, si).

Hence,

we

have

46”)

Q.E.D.

< ri(S*)*

LEMMA 2. Let s* E S be a selection of strategies ST= (rf, qf) such that q(s*) > 0 and rT > 0 for only one individual i. Then, s* is not an NE.

ProoJ Suppose s* is an NE. Then, by Lemma 1, only Case I or (A) of Case II is possible. In either case, we have q(s*) = q:, ri(s*) = rT = 1 and rj(s*) = r; = 0 for all j # i. Let us choose, for somej # i, sj = (rj, qj) so that it satisfies qj = qT + rjqj, 0 < rjqj < ZJc and qj # qz for all h # j. Then, we have T(s* I j, sj) = {j}. Hence, we have q(s* 1j, sj) = qj > qT = q(s*) and rj(s* 1j, sj) = rj(s*) = 0. Then, by monotonicity, s* can not be an NE, which is a contradiction. Q.E.D.

Now, we can partially production.

characterize a Nash equilibrium

with positive

PROPOSITION 1. Let s* E S be an NE with q(s*) > 0. Then, qT = q(s*) for all i E N. Moreover, CjsN r? = 1 and rf = ri(s*) for all i E N.

Proof: Since s* is an NE with q(s*) > 0, it follows from Lemma 2 that rT > 0 for at least two individuals. Then, by Lemma 1, we have T(s*) = N. Hence the results follow. Q.E.D.

To our goal, we further need the following two lemmas. LEMMA 3. Let s* E S be a selection of strategies ST= (t-7, q?) such that xjEN rf = 1 and q? = qT for all j, i E N. Take any i E N, and let Q > 0 satisfy

-rTqT < Q - q(s*) < Zi/c - $4:

(4)

Then, there exists a strategy s, = (r,, qi) E S, that satisfies q(s* 1i, si) = Q and ri(s* I i, s,) = ri(s*).

If Q = q(s*), then trivially si = (r:, 4:) is the required one. Q # q(s*). Define si = (rl, qJ by qi = Q and ri = IQ - (1 - $1 ds*)I/Q. Then, riqi = Q - q(s*) + rTqT because qT = q(s*) by assumption. By (4), we have si E Si. Proof:

Suppose

342

MIKIO

Now,

NAKAYAMA

consider (s* ] i, si). Then, we first have q(s* 1i, si) = x

r?qT + riqi = (1 - r:) q(s*) + ri Q = Q.

j+i

Next, note that T(s* ] i, si) = (i). Then, by definition, ri(s* I i, si) = 1 - 1 rj* +

I9i

LEMMA

4.

-

q(S*

j+i

dS*

I k

si>l

we have = rT = ri(s*).

Q.E.D.

I 4 si)

Let s* E S be an NE with q(s*)

> 0. Then, ri(s*)

> 0 for all

i E N. Proof By Proposition 1, we have CjCN r? = 1, and q? = q(s*), r? = rj(s*) for all jE N. Suppose the conclusion is false, and let rT = ri(s*) = 0 for some i E N. Then, by Lemma 3, we have a strategy si = (ri, qi) such that q(s* I i, si) = Q and ri(s* I i, si) = ri(s*) = 0, whenever Q > 0 and 0 < Q - q(s*) 0 for all je N and cj,, rj(s*) = 1. Suppose (r,(s*),..., r,(s*), q(s*)) is not an LE. Then, there must exist an i E N and q > 0 such that Ui(qy

Ii-

rrcq)

Ii - ri(S*)Cq(S*)) = ui(q*, Ii - rTcq*), > Ui(q(S*),

(5)

and

By the quasiconcavity

of ui, we have

ui(Uq + (1 - U) q*, Ii - rTc[aq + (1 - U) q*]) > Ui(q*, Ii - rTcq*) for any a such that 0 < a < 1. Rewriting this, we have Ui(Q, Ii - rTcQ> > u,(q*, Ii - rTcq*),

(6)

where Q = q* - u(q* -4). Note that q* #q, and q* > q if rfcq* = Ii. Hence, choosing a > 0 sufficiently small we can make Q satisfy condition (4) in Lemma 3. Hence, by Lemma 3, there exists a strategy si E Si such that q(s* ] i, si) = Q and ri(s* ] i, si) = ri(s*). Then (6) and (5) imply that s* is not Q.E.D. an NE, which is a contradiction.

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PUBLIC GOODS AND NASH EQUILIBRIA

Theorem 2 and Theorem 1 together imply an equivalence of a Lindahl equilibrium and a Nash equilibrium with positive production. Thus, it remains to be examined if there exists a Nash equilibrium with zero production. We consider this question in the next section.

5. NASH EQUILIBRIA

WITH ZERO PRODUCTION

For each i E N, let sp be the quantity of a public good that satisfies O ui(q, Ii - Cq)

for all q > 0 such that cq < Ii,

Further, let No = (i E N: u,(O, Zi) > ui(qp, Ii - cqp)}. Members of No are those individuals who will not be the first to bear the cost of production. The existence of a Nash equilibrium with zero production will depend upon the existence of such individuals. PROPOSITION

2. Let s* E S be an NE with q(s*) = 0. Then ND = N.

ProojI Supposethe contrary, and let i E N - No. Then, qp > 0. For this i, let us choose si = (T,.,qi) such that qi = qp and ri = 1. Clearly, si E Si. Then, we have q(s* 1i, si) = q: > 0 in each of the three casesI, II and III. We show that li(s* 1i, si) < 1 in each case. CaseI. ri(s* / i, si) = 1 - Cj+i ri* + (n - 1)/n jC+i ri* + 1 - 1 ] = 1, since q; = qp > 0 for all j E N and therefore C+ i r: = 0. Case II.

ri(s* 1i, si) = 1 - JJjti ri* + /qi - qp//qp < 1, since cjgi rj* > 0.

CaseZZZ. ri(s* I i,sJ=

[qp- l/n C,Jqj-&]I/@

< 1.

Hence, in each case we have

Ui(q(S* I i, Si), Ii - li(S* / i, Si) Cq(S* / i, Si)) > Ui(qp, Zi - Cqp) > ui(S(s*)l li - rj(s*) c9(s*)), which is a contradiction.

Q.E.D.

We now prove a partial converse of this proposition. PROPOSITION 3. Let s* E S satisfy rj* = 0 for all j E N, and q,? = qT > 0 for all j, i E N. Then s* is an NE, if W = N.

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MIKIO NAKAYAMA

Proof: Consider any i E N and any si = (rir qi) E Si with riqi > 0. Then, by assumption and by the definition of qp, we have

ui(S(s*>T Ii - ri(s*> Cq(S*)) = U,(O, li) 2 ui(qp 9Ii - @) > ui(q(s* 1i, So9Ii - Cq(S* ( i, si))Hence it suffices to show that ri(s* ] i, si) > 1, which is verified as follows. CaseZ. ri(s*Ii,si)=l-~j+iri*+(n-l)/n]C,irj*+ri-l]>l. CaseZZ. ri(s* 1i, si) = 1 - C+, rj* + jql - riqil/riqi

> 1.

CaseZZZ. Does not hold in (s* ] i, si).

Q.E.D.

Propositons 2 and 3 together imply, in particular, that 0 E S is a Nash equilibrium if and only if No = N. Since the latter condition only requires that no individual be willing to be the only one who bears the cost, it is compatible with the existence of a Lindahl equilibrium with positive production. Thus, two kinds of Nash equilibria can exist: one with positive production of a public good, and the other with zero production. 6. SOME REMARKS

ON THE LITERATURE

In the literature, several models to obtain equivalence theorems between Nash equilibria and Lindahl (Walras) equilibria have appeared. Here, we shall comment on some of them. As mentioned already, Hurwicz [3] has obtained very general results about the equivalence of Nash and Lindahl (Walras) equilibria for a wide class of environments (preferences, endowments or technology). However, strictly speaking, Theorem 1 in the present paper is not a corollary of these general results, for under our outcome function a non-Pareto optimal NE with zero production can exist (Proposition 3). It may also be noted that our outcome function does not necessarily satisfy the convexity assumption exactly as stated by Hurwicz [3].’ The convexity requires that for each i E N and each (n - 1)-tuple of strategies sji( = (s, ,..., si-, , si+, ,..., s,), the set of attainable allocations of i be a convex set. However, in the proof, Hurwicz does not really use the convexity except for the strategy (n - 1)-tuple s*u( such that (sr, s*)‘() is an NE. Our outcome function satisfies the convexity in this sense, which is seen from Lemma 3 and the fact that ri(s* 1i, si) > ri(s*) = rT for all si E S,. example, s”( = (s2 ,...,s”) such that r2 = . . . = rn = 0 and that = qn = 0. Then, it can be shown that given s)*‘, any proper convex combination of the two allocations (q2, I, - r,(s’, , s”() cq2) and (q3, I, - rl(s;, s’l’) cq,), where s; = (1, q2) and s’,’ = (1, q3). is not attainable for individual 1. ’ Consider,

O
for

PUBLIC GOODS AND NASH EQUILIBRIA

345

Hurwicz has presented another outcome function to obtain the equivalence (see, Hurwicz [2]). In this model, the quantity of a public good is given by the average of the desired quantities, and the tax to each individual is computed from the prices chosen by the other individuals and is corrected by some quadratic terms. But, under this outcome function, it can be seen that given s*ji(, where (ST, s*ji() is an NE, individual i may have no strong incentive to choose his NE strategy sI”, because every individual who is able to choose a positive price is then made indifferent as to the choice of his price. On the contrary, in our model, every individual necessarily chooses his NE strategy, when the indifference curve is strictly convex there. Moreover, the NE strategy n-tuple itself constitutes the Lindahl equilibrium in our model, whereas this is generally not the case in the Hurwicz’s model. Schmeidler [7] has proposed, in a pure exchange economy, an outcome function such that a Nash equilibrium allocation and a Walrasian allocation both coincide. The strategy of each individual is taken to be a price vector and a net trade compatible with the price vector. The outcome to each individual is given by his net trade minus the net trade averaged over all individuals proposing the same price vector. This makes the outcome always balanced (totally feasible). However, the Schmeidler’s outcome function seems to depend upon the absence of externalities: the possibility to define a consistent outcome of a trade within any potential subgroup of individuals. In the presence of public goods, the very externality may prevent us from getting a well-defined outcome of a trade and production that could be performed within a potential subgroup independently of the actions outside the group, unless some particular rules for financing and consuming public goods are prescribed. In this respect. Kalai et al. [4] have presented an interesting mechanism, but their Nash equilibrium allocation does not coincide with a Lindahl allocation, Unlike in models of a pure exchange, it is necessary in the presence of public goods to device a machanism by which the possibility of consuming public goods without bearing the cost can be excluded. In particular, to get an equivalence of Nash and Lindahl equilibria, it seems necessary to have such an outcome function that the price of a public good to each individual can be, in some cases, independent of his own proposal. The above quoted Hurwicz’s outcome function with quadratic terms has this property for all selections of strategy, and ours, for the selection in Case II. 7. CONCLUDING

REMARKS

We have shown that our outcome function has the property that a Nash equilibrium with positive production of a public good generates precisely a Lindahl equilibrium, and that a Nash equilibrium which does not generate a

346

MIKIO

NAKAYAMA

Lindahl equilibrium is necessarily the one with zero production. Thus, under our mechanism, individuals are motivated to choose the quantity of a public good and the cost share ratio which together constitute a Lindahl equilibrium, when a positive production is to be carried out. To obtain these results, we have allowed infeasible outcomes away from equilibria. While in such a partial equilibrium model with money the infeasibility is less troublesome than in a general equilibrium setting, it would be desirable to remove, at least, the total infeasibility with respect to the budget constraint given by (1). There would be many ways to remove the total infeasibility. But, the simplest and yet a consistent way to do this is, perhaps, to stop entirely the production whenever the condition

\’ rj(s) = I j-5 is violated. Note that the condition is stronger than (1). By the definition of Ye in Case I, this occurs if and only if Y, + . . . + I, < 1. Hence, we can modify the rule in Case I as follows. n-l r-!(S) = 1 - L ‘j + 7 jti

4Cs) =x

\’ rjI FN

1 I

for all

i E N,

if K’ rj> ,S

rjqj

jeN

=o

1,

otherwise.

Under this modification, we need only change the hypothesis of an NE in Proposition 2 into “with $4: = 0 for all i E N.” All the results obtained are then left unchanged. However, the characterization of the Nash equilibria becomes weaker, because the possibility of Nash equilibria with zero production and CjeN rjqj > 0 simultaneously remains. We conclude with a remark about the extention of the model to the case of more than one public good. By construction, we need only define the rule for each public good. Then, all the assertions will be extended with the appropriate modifications needed in stating in the vectorial forms. In particular, the words “positive production” and “zero production” are understood in coordinate-wise.

ACKNOWLEDGMENTS I am grateful to M. Kaneko and a referee Responsibility for errors is, of course, mine.

of this journal

for several

helpful

comments.

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REFERENCES I. T. GROVES AND J. LEDYARD, Optimal allocation of public goods: a solution to the “free rider” problem, Econometrica 45 (1977), 783-809. 2. L. HURWICZ. Outcome functions yielding Walrasian and Lindahl allocations at Nash equilibrium points, Reu. Econ. Sfudies 46 (1979), 217-22.5. 3. L. HURWICZ, On allocations attainable through Nash equilibria, J. Econ. Theory 21 (1979), 140-165. 4. E. KALAI, A. POSTLEWAITE. AND J. ROBERTS, A group incentive compatible mechanism yielding core allocations, J. Econ. Theory 20 (1979), 13-22. 5. M. KANEKO. The ratio equilibrium and a voting game in apublic goods economy, J. Econ. Theory 16 (1977). 123-136. 6. A. POSTLEWAITE AND D. SCHMEIDLER. Approximate efficiency of non-Walrasian Nash equilibria, Econometrica 46 (1978), 127-135. 7. D. SCHMEIDLER. Walrasian analysis via strategic outcome function, Econometrica. in press. 8. L. S. SHAPLEY, Noncooperative general exchange, in “Theory and Measurement of Economic Externalities” (S.A.Y. Lin, Ed). Academic Press, New York, 1976. 9. R. WILSON, Competitive exchange, Economefrica 46 (1978), 577-585.