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Journal of Public Economics 71 (1999) 97–119

Testable restrictions of Pareto optimal public goodprovision

*Susan K. SnyderDepartment of Economics, Virginia Polytechnic Institute & State University,

Blacksburg, VA 24061-0316, USA

Received 30 November 1996; received in revised form 31 March 1998; accepted 20 April 1998

Abstract

This paper is a theoretical examination of the testable restrictions of Pareto optimality inan economy with public goods. In contrast to other methods for deriving tests for theoptimality of public good provision, ad hoc assumptions about individual public gooddemand are not required. The testable restrictions derived in this paper are defined over afinite series of aggregate-level data, production data, and individual after-tax income.Despite the relatively sparse data requirements and the minimal assumptions aboutpreferences and technology, these restrictions are not vacuous. Thus they provide a strongtheoretical basis for empirical work that tests the Pareto optimality hypothesis in a broadvariety of public good economies. 1999 Elsevier Science S.A. All rights reserved.

Keywords: Public goods; Pareto optimality; Quantifier elimination; Lindahl equilibrium;Revealed preference

JEL classification: D51; H41

1. Introduction

Economic theory predicts that, in general, public goods will be provided at asuboptimal level by private mechanisms, whereas informational problems will tendto lead to inefficiencies when public goods are provided by the public sector. Thusmodels of efficient public good provision are often seen more as normative

*Tel.: 11-540-2314431; fax: 11-540-2315097; e-mail: [email protected]

0047-2727/99/$ – see front matter 1999 Elsevier Science S.A. All rights reserved.PI I : S0047-2727( 98 )00053-X

98 S.K. Snyder / Journal of Public Economics 71 (1999) 97 –119

prescriptions rather than positive descriptions of how public goods are allocated(Laffont, 1988). Whether a particular allocation is efficient, however, is ultimatelyan empirical question.

This paper focuses on the empirical implications of Pareto optimal behavior inan economy with public goods. It demonstrates a method to test whether publicgood provision is consistent with Pareto optimality without having individual-specific information about public good preferences. The only assumptions requiredare that agents have preferences representable by continuous, strictly monotonic,strictly quasiconcave utility functions defined over private and public goodconsumption. Additionally, the only individual-specific information required isindividual after-tax income (or income net of public good contributions).

This paper derives testable restrictions of Pareto optimal public good provisionfor an economy with constant returns to scale technology and two traders for twoobservations of aggregate production of private and public goods, prices, aggregateendowments, aggregate taxes, and individual after-tax incomes. These testablerestrictions are nonparametric, nonstochastic conditions defined over discreteobservations of data. In form, they are analogous to revealed preference tests forindividual demand data such as the Weak Axiom of Revealed Preference.Implementing the tests is simply a matter of checking whether the data satisfy aseries of polynomial inequalities.

The testable restrictions are derived by adapting the techniques demonstrated inBrown and Matzkin’s work (Brown and Matzkin, 1996) on the empiricalimplications of equilibrium models. They prove that nonvacuous testable restric-tions of competitive equilibrium exist on the equilibrium manifold; that is, one cantest equilibrium behavior on a series of observations of prices and individualendowments from an economy. A key element in their proof is the application ofthe algebraic method of quantifier elimination. This method allows the systematicderivation of necessary and sufficient conditions for a given set of observablevariables to be consistent with the model, even when there exist importantunobservable variables in the model’s equilibrium conditions. In the case of aneconomy with public goods, Lindahl prices are inherently unobservable, butthrough quantifier elimination we can derive the necessary and sufficient con-ditions of the model without observing them.

If these restrictions are satisfied by a set of data, we say the set of data isrationalized: there exists an economy such that the data are consistent with Paretooptimal public good provision. If the restrictions are not satisfied, there does notexist such an economy; the data are not consistent with Pareto optimal public goodprovision. Thus these are necessary and sufficient conditions; they describe all theempirical implications of the model given our assumptions about what data are

1observable.

1If there are more than two observations these restrictions can still be used to refute Paretooptimality; if any pair of observations fails to satisfy the restrictions, the data are not consistent withPareto optimal public good provision. They are no longer sufficient conditions, however.

S.K. Snyder / Journal of Public Economics 71 (1999) 97 –119 99

The relatively sparse data requirements of these restrictions means that they willbe of use in a broad variety of contexts. How will we interpret the results of thesetests when applied to data? The relatively few assumptions that go into creatingthese testable restrictions make a negative result more robust than results thatdepend on a particular parametric specification of preferences or technologies.Thus if we cannot rationalize the data, this would provide some evidencesupporting the numerous theories that predict the free-rider problem or in-formational problems will lead to inefficiencies in public good provision.

If a set of data is rationalized with these tests, a number of interpretations arepossible. We could view this as evidence that the free-rider problem or in-formational problem has been overcome by the particular group at hand. We couldfurther interpret this as support for some sort of cooperative behavior.

While these restrictions are for a general equilibrium model with production,one can use a subset of the restrictions to test the Pareto optimality of aggregatedemand. This is likely to be more appropriate for many empirical situations. Weneed only two observations of the aggregate consumption and prices of privategoods and public goods, aggregate taxes or contributions, aggregate before-taxincome, and individual private good expenditures, to test for Pareto optimalprovision of public goods in conjunction with optimal individual private goodchoices.

These tests should provide a useful counterpart to existing empirical work ontesting Pareto efficiency of public good provision. Empirical tests of the efficiencyof public good provision have generally proceeded with additional assumptionsabout the form of individual public good demand. Sandler and Murdoch (1990)apply a formal test of efficiency of private provision of public goods by examiningdefense spending by NATO members. Khanna (1993) applies similar methods to

2examine the supply of agricultural research by American states. Using econo-metric specification of log–linear demand for public goods, both papers find littlesupport for efficient behavior.

The next section presents the public competitive equilibrium model thatprovides the framework for the derivation of tests of efficient public goodprovision. The testable restrictions of this model are derived in Section 3. Section4 discusses some general implementation issues and applies the restrictions ofPareto optimal public good provision to derive tests of efficient householdbehavior.

2. Public competitive equilibrium

In a Lindahl equilibrium, each individual must pay taxes based on theirmarginal benefit of the public good. There may be other tax schemes, however,that are compatible with Pareto optimality in the presence of public goods—and

2See also McGuire and Groth (1985) for a theoretical basis for the tests used in both these papers.


because redistribution is so often a conflating issue in tax schemes, we can study amuch broader range of situations by allowing that redistribution may exist. A lessrestrictive alternative to the Lindahl equilibrium is the politico-economic equilib-rium, first suggested by Wicksell (Musgrave and Peacock, 1958) and formalizedby Foley (1967) as the public competitive equilibrium. In this model, consumersbehave competitively with respect to private goods. Public goods, however, areprovided by the public sector. The public sector chooses public goods andlump-sum taxes to finance them such that there is no other public sector proposal

3that would be preferred by every individual.Foley (1967) proves that public competitive equilibria are Pareto optimal, and

there exist prices and taxes such that, given endowments, a Pareto optimalallocation is a public competitive equilibrium (see also Milleron (1972)). Testingwhether data from an economy are consistent with public competitive equilibriumwill provide a way of testing whether the economy has reached a Pareto optimalallocation of public goods.

2.1. The Model

Consider an economy with T consumers, K pure public goods and J private4goods. Each consumer t has preferences over public and private goods represented

by a continuous, strictly monotonic, strictly quasiconcave utility functionK JU (x ; y ), with x [ R , y [ R . Consumption of the public goods is equal acrosst t t t 1 t 1

all consumers (there is no free disposal): x 5 x for t 5 1, . . . , T, while aggregatet

consumption of private goods is equal to the sum of private consumption:Ty 5 o y .t51 t

JConsumers have endowments of private goods v [ ... R , for t 5 1, . . . , T,t 1

with aggregate endowment represented by v. We assume these endowments can beused to produce public goods and private goods, with net output of private goodsrepresented by z 5 y 2 v, and output of public goods represented by x. We assumepublic goods are not used in production.

Production technology is assumed to be constant returns to scale. Formally, theset of all technically possible production plans is Z, and we assume:

A1. Z is a closed, convex cone.A2. If 0 ± (x; z) [ Z then (x; z) $⁄ 0.

3While agents can choose different tax schemes within the model with a unanimous vote, there is nospecification as to how the status quo tax scheme is chosen. Thus rather than being a true equilibriummodel, the public competitive equilibrium is more a description of public sector Pareto optimalitycombined with private sector competitive equilibrium (see Milleron (1972)).

4This description of the model is adapted from Foley (1970).


TDefinition 1. A Feasible Allocation is a set of vectors (x; hy j ) such thatt t51

(x; y 2 v) [ Z.

There exists a government that can purchase public goods for the use of theeconomy’s members and also has the power to tax members to pay for the publicgoods and to redistribute income (taxes can be negative). Each consumer makes atax payment (or receives a subsidy) of t .t

It is straightforward to show that the equilibrium conditions of the publiccompetitive equilibrium model are the same as a Lindahl equilibrium withtransfers. Thus if the economy is at a Pareto optimal allocation, there exist Lindahlprices such that consumers act as if they were choosing public and private goodssubject to prices and a full income budget constraint. Government choice is thenreduced to a balanced-budget condition.

In the following definition of a public competitive equilibrium the prices qt

correspond to the Lindahl prices. We also distinguish between monetary transfers(t ) and full income transfers (S 5 t 2 q x ).t t t t t

Definition 2. A Public Competitive Equilibrium is a feasible allocationT T(x; hy j ), prices (q; p) $ 0, and taxes ht j such that consumers, producers andt t51 t t51

the government act as if:

1. Each consumer solves the maximization problem:

max U (x ; y ) s.t. q x 1 py # pv 1 St t t t t t t t(x ; y )t t

2. Producers solve the maximization problem:

max(q; p) ? (x; z) s.t. (x; z) [ Z(x ;z)

T3. The government chooses ht j such that:t t51

T

a. qx 5O ttt51

b. t 5 q x 2 S t 5 1, . . . , Tt t t

4. Public good restrictions are satisfied:

a. x 5 x t 5 1, . . . , Tt

T

b. O q 5 qtt51


3. Testable restrictions

Testable restrictions of equilibrium models are most often developed throughthe method of infinitesimal comparative statics. For example, the endogenousvariables of a model are written as functions of the exogenous variables; the modelimplies certain properties for these functions, and we can then test whether a set ofdata is consistent with the model by testing whether the data are consistent withthese properties. It is well-known that the standard model of competitiveequilibrium possesses no interesting comparative statics properties of this sort (see,for example, Kehoe (1987)).

In contrast, Brown and Matzkin (1996) show that there exist nonparametrictestable restrictions of general equilibrium models defined over discrete observa-tions of data. They show these testable restrictions exist through extending theresults of revealed preference theory, developed for the theory of individual andfirm-level optimization, to general equilibrium models. Their method makes use ofthe discrete nature of observable data and the algebraic techniques that can beapplied as a result.

Rather than making assumptions about the particular form of individualpreferences in the economy, Brown-Matzkin generate all the restrictions onaggregate behavior in equilibrium that result from the nonnegativity of theunobserved individual demand. Using the duality of public good and private goodmodels, we can generate restrictions of aggregate public good provision inequilibrium that result from the assumption of nonnegativity of the unobservablepersonalized public good prices. This is a substantive assumption. We haveassumed there is no free disposal of public goods; therefore limiting personalizedprices to be nonnegative implies that in equilibrium individual public goodmarginal benefit from each public good must be nonnegative. That is, at themargin, consumers always get positive benefit from every public good—there areno ‘‘public bads’’ from any consumer’s standpoint. Without the nonnegativityassumption, however, there will generally be no interesting restrictions on

5aggregate public good demand (see Chiappori (1990)).This paper will follow the method used in Brown-Matzkin to derive non-

parametric testable restrictions of the public competitive equilibrium model. Theirmethod has two parts: define the equilibrium conditions of the model as a set ofpolynomial inequalities; then apply the algebraic technique of quantifier elimina-tion to rewrite the equilibrium conditions to involve only variables that we canobserve. These resulting conditions will be the testable restrictions of the model.

The public competitive equilibrium model has essentially three components:individual utility maximization conditions; profit maximization conditions; andsumming-up conditions. We can rewrite the equilibrium conditions of the model as

5Alternatively, one could derive nontrivial restrictions by assuming free disposal of public goods,which implies that the individual marginal benefit of any public good will be nonnegative.


a series of polynomial inequalities in the model’s variables, without makingfurther assumptions about utility or production functions, by applying Afriat’stheorem and its variants. Afriat’s theorem states that there exists a finite set ofpolynomial inequalities (‘‘Afriat inequalities’’) defined over discrete observationsof individual consumption data and other unobservable variables (such as utilitylevels) that provide necessary and sufficient conditions for the data to have beengenerated by the maximization of a nonsatiated utility function (Afriat, 1967, alsoVarian, 1982).

Thus in describing a public competitive equilibrium’s empirical implications,we can replace the utility maximization condition with an equivalent set ofpolynomial inequalities, the Afriat inequalities. Analogous results for profitmaximization allow us to replace the profit maximization condition with equiva-lent polynomial inequalities as well. Doing so will depend on the assumption thatwe can observe a series of allocations of the economy in question where individualpreferences and production technologies do not change over time.

The significance of rewriting the equilibrium conditions as polynomials is thatwe can then apply the technique of quantifier elimination to show that testablerestrictions exist given the data we can observe. Given a finite set of polynomialinequalities in observable and unobservable variables, the Tarski-Seidenbergtheorem proves that there exists an equivalent finite set of polynomial inequalitiesthat does not involve the unobservable variables, and also provides a finitealgorithm for computing the equivalent system through quantifier elimination(Tarski, 1951). Quantifier elimination is the process of eliminating ‘‘quantified’’variables—in our case, the unobservable variables are quantified in that theyappear in conjunction with the existential quantifier (‘‘there exists’’). Quantifierelimination can be used to show that unobservable variables (such as Lindahlprices) can be ‘‘eliminated’’ from the equilibrium conditions of the model, leavingbehind nonvacuous, nonparametric testable restrictions of the public competitiveequilibrium model involving data we can potentially observe.

The Afriat inequalities are equivalent to the Generalized Axiom of RevealedPreference (GARP), a generalization of Samuelson’s Weak Axiom of Revealed

6Preference (WARP) (see Varian (1982)). Thus one can use the Afriat inequalitiesor GARP to create testable restrictions of a model with utility maximization; theadvantage of GARP for our purposes is that it leads to linear quantifier eliminationproblems, while the Afriat inequalities lead to bilinear problems. For our tests wewill make use of a theorem due to Matzkin and Richter (1991) that shows there isa slight modification of the Afriat inequalities that is equivalent to Houthakker’sStrong Axiom of Revealed Preference (SARP), the transitive version of Samuel-son’s Weak Axiom. The difference between SARP and GARP is essentially SARPrestricts demand to be single-valued.

On the production side, the Weak Axiom of Profit Maximization (WAPM)

6Afriat derived equivalent conditions he called ‘‘cyclical consistency’’.


provides necessary and sufficient conditions for profit maximizing behavior in the7form of polynomial inequalities defined over a discrete series of data. We assume

technology is constant returns to scale; therefore profits must be zero. Thusnecessary and sufficient conditions for there to exist a production set defined byconstant returns to scale technology such that producers maximize profits subjectto feasibility constraints defined by this production set combine the WAPM withthe zero profit condition (Hanoch and Rothschild (1972), see also Varian (1984),theorem 6).

If we had a data set that contained all these variables, then we could use theseconditions to test whether the data are consistent with public competitiveequilibrium. It is not possible to observe all these variables, however, particularlythe Lindahl prices, q . Additionally, it may be difficult to observe highlyt

disaggregate data, such as individual consumption of private goods.Theorem 1 presents the public competitive equilibrium model in polynomial

r r r rform. Let after-tax income for consumer t in period r be defined as I 5p v 2t ,t t tr r Tand, in general, let the notation hI j represent hI j .t t t51

r r r r r r rTheorem 1. Let the collection kx , y , q , p , v , hI j, t l of nonnegative vectorst

of variables be given for r51, . . . , R. Then there exist continuous, strictlymonotonic, strictly concave utility functions hU j and a closed convex conical,t

negative monotonic production set Z such that these data are consistent with ar Rseries of Public Competitive Equilibria for the economy (hU j, Z, hv j ), if andt t r51

r ronly if there exist khy j$0, hq j$0l such that:t t

r r r r r1. kx , y , q , p , I l satisfy the Strong Axiom of Revealed Preference and thet t tr r rindividual budget constraint, p y 5I , for each consumer t.t t t

r r r r r2. kx , y , q , p , v l satisfy profit maximization and technology restrictions:

r r r r r r r s s sa. 0 5 (q ; p ) ? (x ; y 2 v ) $ (q ; p ) ? (x ; y 2 v ) for

r, s 5 1, . . . , R, r ± s.r r r rb. If y 2 v ± 0 then y 2 v $⁄ 0 for r 5 1, . . . , R.

r r r3. kx , q , t l satisfy the public good budget constraint:

r r rq x 5 t r 5 1, . . . , R.

7For origins of WAPM see Varian (1984).


r r r r r r r r4. ky , q , p , hy j, v , hI j, hq j, t l satisfy:t t t

Tr ra. O y 5 y r 5 1, . . . , R.t

t51

Tr rb. O q 5 q r 5 1, . . . ,R.t

t51

Tr r r rc. O I 5 p v 2 t r 5 1, . . . , R.t

t51

Proof. Follows directly from Afriat’s theorem, Matzkin and Richter (1991); Varian(1984), theorem 6.

Comment 1. Note that if the conditions of the theorem are satisfied, there existstrictly concave utility functions consistent with a public competitive equilibrium,though we only assumed strict quasiconcavity up to this point. This derives fromAfriat’s theorem: if a finite series of individual consumption data is consistent withmaximization of a nonsatiated utility function, it is also consistent with maxi-mization of a concave, continuous, monotone utility function. Violations ofconvexity cannot be detected with finite data. Similarly, assuming strict concavityin this context is no more restrictive than assuming strict quasiconcavity. Forsimilar reasons, the addition of negative monotonicity of the production set is nomore restrictive than our previous assumptions, given finite data sets.

Comment 2. The strong axiom of revealed preference can be written as a finite setof disjunctions of polynomial inequalities. All the other conditions of the theoremare also finite sets of polynomial inequalities or equations.

Theorem 1 provides a statement of public competitive equilibrium behavior inthe form of a finite set of polynomial inequalities defined over a finite set of

r rvariables. Moreover, we have explicitly assumed that khy j, hq jl, the Lindahlt tr r rprices and individual private good consumptions, are unobserved, while kx , y , q ,

r r r rp , v , hI j, t l, the public good consumptions, aggregate private good consump-t

tions, market prices, aggregate endowments, individual after-tax incomes andaggregate tax payments, are observed. Thus theorem 1 presents the equilibriumconditions of public competitive equilibrium behavior as a finite set of polynomialinequalities in observed and unobserved variables. What remains is to eliminate theunobservable variables from the system.

There are three possible outcomes when quantifier elimination is applied to asystem such as the one in theorem obsthm. One possibility is the equivalentsystem reduces to 1;0, meaning it is impossible to observe data consistent with


8the model; the theory is empirically inconsistent. Another possibility is the systemreduces to 1 ; 1, meaning it is impossible to observe data not consistent with themodel; the theory imposes no testable restrictions on the set of observablevariables. The third possibility is the system reduces to a finite set of polynomialinequalities involving the observable variables. It is possible to observe data thatsatisfy these inequalities, and it is possible to observe data that do not satisfy theseinequalities. In this case we say testable restrictions of the model exist, given theset of observable variables (Brown and Matzkin, 1996).

While Tarski-Seidenberg provides a finite algorithm for quantifier elimination, itis doubly exponential and so is not practical. Fourier-Motzkin elimination is awidely-known algorithm that is applicable to linear systems such as the ones in

9this problem; however, it is also doubly exponential in application. Lacking anypractical quantifier elimination algorithms to apply to this problem, we will focuson very simple models of public competitive equilibrium. We will derive thetestable restrictions of this model for two observations of an economy with two

10agents, a and b, and any finite number of public and private goods. For the caseof one public good and one private good, we will also show that these restrictionsare nonvacuous.

In the following lemma we consider the consumption side of this problem:

r r r r r rLemma 1. Let the collection kx , y , q , p , I , I l of nonnegative variables bea b

given for r51, 2. Then there exist continuous, strictly monotonic, strictlyquasiconcave utility functions U such that this set of data is consistent with eacht

consumer satisfying the Weak Axiom of Revealed Preference ( WARP) and theirindividual budget constraint, Lindahl prices summing to the market price, andindividual private good consumptions summing to aggregate private goodconsumption if and only if:

For some r, s51, 2, r±s, either:

rs r rs r r r s r s rI [E . I AND E . I AND (q ; p ) ? (x 2 x ; y 2 y ) . 0]a a b b

or:rs r sr sII [E . I AND E . I ]a a b b

rs s r r s s s rWhere E 5max m ?(x 2x )1p n s.t. p n 5I , 0#n #y , 0#m #q .t n,m t

8Empirical inconsistency is distinct from the traditional use of ‘‘inconsistency’’ in generalequilibrium theory, see Snyder (1995).

9Fourier-Motzkin is a technique similar to Gaussian elimination. For some explication of the methodand its uses, see Dantzig and Eaves (1973).

10The restrictions could be used with more than two consumers, but the usual problems ofrepresentative agent models are introduced. See Kirman (1992).


Proof. See Appendix A.

Comment. WARP is equivalent to SARP for two observations.

One could think of these conditions as analogous to WARP. WARP testsindividual behavior for utility maximization; these conditions test a combination ofaggregate behavior (public good demand, private good demand) and individualbehavior (after-tax income) for individual utility maximization.

It is well-known that individual utility maximization does not imply collectivebehavior that resembles utility maximization, and that can be seen quickly fromthese restrictions. Note that if efficiency required collective behavior to resemblethat of an individual utility maximizer, then aggregate demand would satisfyWARP. The last restriction in condition I implies that aggregate consumption inperiod r is not revealed preferred to aggregate consumption in period s—if thiscondition is satisfied, then aggregate demand satisfies WARP. For aggregatedemand to satisfy WARP is not a sufficient condition for efficiency, however; theother restrictions in condition I must also be satisfied. Moreover, for aggregatedemand to satisfy WARP is not a necessary condition; if the data satisfy conditionII aggregate demand does not have to satisfy WARP. Thus restricting individualdemand to satisfy WARP does not imply aggregate demand satisfies WARP.

Interpretation of these restrictions in terms of the usual demand restrictions weare familiar with is difficult because these restrictions depend not on aggregatedemand alone but on the (observed) income distribution. These restrictions simplydefine bounds such that it is possible for each individual to be satisfying the axiomof revealed preference. The nonvacuousness of the restrictions comes from thenonnegativity restrictions on individual consumption and Lindahl prices. Giventhis level of observability, it is always possible for each individual to bemaximizing utility given any data; it is not always possible, however, for bothindividuals to be maximizing utility in equilibrium (that is, there may be individualprivate valuations of the public good such that utility maximization is satisfied, butthe individual private valuations of the public good will not sum up to the publicgood price).

Fig. 1 provides an example of data that do not satisfy the restrictions. Weobserve public good consumption; by observing prices and each consumer’sincome we can derive private good consumption for each consumer. This picturerepresents a situation where both consumers get equal amounts of the private good,and by definition both consume the same amount of the public good. The picture

1 1represents consumer a’s situation: a’s period 1 consumption is plotted at (x , y )a2 2and at (x , y ) in period 2. The picture would look exactly the same if we insteada

plotted consumer b’s consumption. We observe the price of the private good butwe do not observe the Lindahl price of the public good for each consumer. Thuswe are free to choose the slopes of the price lines for each consumer, with therestriction that the Lindahl prices must be nonnegative and must sum to the


Fig. 1. Data that do not satisfy the restriction.

observed market public good price (normalized to 1). Then the data represented bythis picture are consistent with the public good restrictions only if we can drawfeasible price lines such that each consumer satisfies WARP.

1 1 1 2 2 2Let r 5q /p be the slope of budget line 1, and r 5q /p be the slope ofa a1 1 2 2budget line 2. Then 0#r #1/p and 0#r #1/p . The limit-cases are drawn in

Fig. 1. Note that for all feasible public good prices, consumer a’s period 2consumption is revealed preferred to her period 1 consumption. Thus the only way

1this consumer can satisfy WARP is if we choose q such that her period 1a

consumption is not revealed preferred to her period 2 consumption. This ispossible; for example, a Lindahl price of 0, or close to 0, would satisfy thisrequirement. In general, it is always possible to choose Lindahl prices such thatWARP is satisfied for one consumer.

Suppose we choose a low enough Lindahl price for consumer a in period 2 sothat her period 2 consumption bundle will no longer be revealed preferred to herperiod 1 consumption bundle. With the data illustrated here, a Lindahl price of 1 /2is too high—at this price, a’s period 1 consumption is revealed preferred to her

1period 2 consumption. Thus q must be less than 1/2. But because the Lindahla

prices must sum to 1, this means choosing a Lindahl price for consumer b that ishigher than 1/2. However, consumer b faces the same situation as consumer a—bmust have a Lindahl price less than 1/2 in order to satisfy WARP. Thus while there


exist Lindahl prices such that either consumer can satisfy WARP, there do not existLindahl prices such that each consumer can satisfy WARP with Lindahl prices thatsum to the market price for the public good. The data represented by this picture

11are not consistent with Pareto optimal public good choice.The following theorem uses lemma 1 to derive testable restrictions of the public

competitive equilibrium model.

r r r r r r r rTheorem 2. Let the collection kx , y , q , p , v , I , I , t l of nonnegativea b

vectors of variables be given for r51, 2. Let:

r r r r r rD 5 kx , y , q , p , I , I l1 a b

r r r r rD 5 kx , y , q , p , v l2

r r r r r r rD 5 kx , q , p , v , I , I , t l3 a b

Then there exist continuous, strictly monotonic, strictly concave utility functionshU j and a closed convex conical, negative monotonic production set Z such thatt

these data are consistent with a series of Public Competitive Equilibria for ther reconomy (hU j, Z, hv j ), if and only if:t t r51

1. D satisfy the conditions of lemma 1.1

2. D satisfy profit maximization and technology restrictions:2

r r r r r r r s s sa. 0 5 (q ; p ) ? (x ; y 2 v ) $ (q ; p ) ? (x ; y 2 v ) for r, s 5 1, 2, r ± s.r r r rb. If y 2 v ± 0 then y 2 v $⁄ 0 for r 5 1, 2.

3. D satisfy the following constraints for r51, 2:3

r r ra. t 5 q x

r r r r rb. I 1 I 5 p v 2 ta b

Proof. Condition 3 is straightforward, as it involves only observable variables.Condition 2 follows from theorem 1, and condition 1 follows directly from lemma1.

Comment. Since the lemma 1 conditions are nonvacuous for the case of onepublic good and one private good, the testable restrictions of the publiccompetitive equilibrium are nonvacuous for that case also.

11See Appendix B for a numerical example of data that are not consistent with Pareto optimality.


Theorem 2 provides the testable restrictions of public competitive equilibrium,given our assumptions about the observability of the model’s variables. Thus theyform testable restrictions of Pareto optimality in a general equilibrium model withpublic goods.

There may be situations where it is clearly not appropriate to test for efficiencyin a general equilibrium model, but it may be appropriate to test for efficiency ofaggregate demand. Suppose we had a group of consumers who faced exogenouspublic and private good prices, and who were taxed (or made contributions) for thecollective purchase of public goods. We could test the Pareto optimality of thegroup’s choice of public and private goods using a subset of the restrictions of themodel of public competitive equilibrium—we simply exclude the productionsector restrictions on technology and profit maximization. In this nonparametric,revealed preference based method of deriving testable restrictions, it is irrelevantthat prices are endogenous in the public competitive equilibrium model andexogenous in the aggregate demand model.

r r r r r r r rCorollary 1. Let the collection kx , y , q , p , v , I , I , t l of nonnegativea b

vectors of variables be given for r51, 2. Let:

r r r r r rP 5 kx , y , q , p , I , I l1 a b

r r r r r r rP 5 kx , q , p , v , I , I , t l2 a b

Then there exist continuous, strictly monotonic, strictly concave utility functionshU j such that this set of data is consistent with Pareto optimal demand for thet

r reconomy (hU j, hv j ), if and only if:t t r51

1. P satisfy the conditions of lemma 1.1

2. P satisfy the following constraints for r51, 2:2

r r ra. t 5 q x

r r r r rb. I 1 I 5 p v 2 ta b

4. Implementation and application

4.1. Issues in implementation

One advantage of these tests is that they require little data to implement. To testefficiency of demand, we require two observations over time of an economy’sallocations or a group’s consumption choices, prices, and the individual after-taxincomes. For example, we could use these conditions to test whether police


protection in a town is Pareto optimal with two observations (perhaps of annualdata) of the number of police officers, the town’s aggregate consumption of privategoods, the ‘‘price’’ of a police officer (perhaps measured by salary or inferredfrom expenditures on police protection and number of officers used), aggregate taxcollection, aggregate income, and expenditure on private goods for two types ofcitizens of the town. Or, we could test whether an interest group purchases aPareto optimal amount of advertising for a candidate it wishes to elect. To test thisproposition we might use data consisting of two observations (perhaps monthly) ofinterest group expenditures on advertising, the quantity of advertising produced(perhaps measured by an estimate of the number of people who viewed theadvertising), the group members’ aggregate consumption of private consumptiongoods, aggregate contributions, aggregate income, and expenditures on privategoods for two types of interest group members.

The assumption that preferences and technology don’t change over time iscrucial for these tests; additionally, we assume that consumption and productionchoices are independent over time. Alternatively, one could apply the restrictionsto a cross-sectional analysis, where it would be necessary to assume that membersof two groups or economies have the same preferences, while having differentendowments or facing different prices.

The tests as formulated in this paper are nonstochastic. While this makes thetests easy to apply, it also creates problems in the interpretation of situations wheredata do not satisfy the restrictions. If the restrictions are not satisfied, it is not clearwhether we should interpret this as a rejection of the model, or as evidence forsome measurement error, or as evidence that there is some stochastic element inbehavior unaccounted for. These tests may be particularly useful as specificationtests in preparation for doing further econometric work.

Strictly speaking, the tests derived in this paper are applicable only to modelswith two consumers. One could assume a model with two representativeconsumers, each representing a distinct group within the economy, though the lackof microfoundations for the representative consumer model may make this an

12unappealing assumption.

4.2. Intra-household decision-making

One possible application of this work is in the area of household decision-making. Traditionally households have been treated as a single utility-maximizingagent (the ‘‘unitary’’ model of household behavior). Recent years have seen thedevelopment of numerous models of household decisions that are based on

12The use of a representative consumer is consistent with individual utility maximization only withhighly restrictive assumptions; for example, to avoid making further restrictions (such as homotheticity)on the form of individual preferences, one would have to assume identical preferences and identicalincomes within groups (see Kirman and Koch (1986)).


individual behavior within the household, modeling outcomes as the result ofeither cooperative or noncooperative bargaining between household members (see,for example, McElroy and Horney (1981); Manser and Brown (1980); Lundbergand Pollak (1994)). Testing whether these models adequately explain householdbehavior can be difficult, however, as we generally observe market decisions at thehousehold level rather than at the individual level—we generally do not observethe division of goods within the household.

Chiappori (1988) proposed an extremely general model of cooperative bargain-ing within the household (the ‘‘collective rationality’’ model), stipulating only thatoutcomes be Pareto optimal. Individuals within the household get utility from theirown consumption and leisure, but their consumption and leisure may also generatepositive externalities for the other person in the household. One could think of thecollective rationality model as a simple model of Pareto optimality with publicgoods, as each individual’s consumption and leisure become public goods withinthe household.

Chiappori (1988) shows that there are nonvacuous testable restrictions of thecollective rationality model on data that can feasibly be observed: household-levelconsumption, individual labor supplies, prices and wages. Chiappori’s tests are inthe form of finding whether a set of bilinear inequalities has a solution—if theprogram has a solution, then the data satisfy the model, if not, the data do notsatisfy the model.

By applying corollary 1 to this problem, we can derive equivalent testablerestrictions as a set of polynomial inequalities defined only over the observablevariables. In other words, Chiappori’s program will have a solution if and only ifthese polynomial inequalities are satisfied.

To illustrate, consider the following variation of Chiappori’s model. A house-rhold is composed of two members, a and b. Let C denote the household-level

quantity of some consumption good in period r, which has a price normalized tor rone. Let L and L denote, respectively, the leisure of the members of thea b

r rhousehold, and w and w their wages. Assume labor supply is observable fora b

each member of the household, and all time not spent supplying labor to themarket is leisure time. The household also receives exogenous nonlabor income

rY .Assume that individual consumption generates externalities within the house-

hold; that is, each individual’s consumption, c and c , is a public good within thea b

household. Each individual has preferences representable by strictly monotone,strictly quasiconcave utility functions: U (c , c , L ).t a b t

r r r r r rCorollary 2. Let the collection kC , Y , L , L , w , w l of nonnegativea b a b

variables be given for r51, 2. Then there exist continuous, strictly monotonic,strictly quasiconcave utility functions U such that this set of data is consistentt

with Pareto optimal demand (or the collective rationality model) if and only if:r r r r r r

;r 5 1, 2 C 5 Y 1 w , 1 w , ANDa a b b


Table 1Data not consistent with collective rationality

Year 1 Year 2

Husband’s wage 5.00 19.00Wife’s wage 3.00 16.00Non-labor income 10 270.00 0.00Husband’s hours worked 2000.00 500.00Wife’s hours worked 2000.00 500.00

For some r, s51, 2, r±s, either:

s r s r r s r s r rI [C 1 w L . w L AND C 1 w L . w L ANDa a a a b b b b

s r r s r r s rC 2 C 1 w (L 2 L ) 1 w (L 2 L ) . 0]a a a b b b

or:s r s r r r s r s sII [C 1 w L . w L AND C 1 w L . w L ]a a a a b b b b

Proof. See Appendix C.

For example, suppose we observe two years of a household’s labor supply13behavior, hourly wages, and nonlabor income. If we assume the household

follows a balanced budget in spending, we can solve for consumption levels inboth years, and apply the restrictions above to test for Pareto optimal behaviorwithin the household.

14Suppose we observe the data in Table 1. This set of data is not consistent withthe collective rationality conditions as given above. In other words, this behavior isnot consistent with Pareto optimal behavior within the household. On the otherhand, the set of data in Table 2 is consistent with collective rationality.

Note that the difference between these examples is that in the first case, both

Table 2Data consistent with collective rationality

Year 1 Year 2

Husband’s wage 5.00 19.00Wife’s wage 3.00 16.00Non-labor income 10 270.00 0.00Husband’s hours worked 2000.00 2000.00Wife’s hours worked 2000.00 2000.00

13Data sets of this sort are available, for example, from the Bureau of Labor Statistics’ NationalLongitudinal Surveys.

14In this example inflation will be ignored; in practice, one could use general data on prices levels,such as CPI data, to normalize the prices.


husband and wife face a large wage increase, but cut back their hours worked,while in the second case they face the same wage increase but remain full-timeworkers.

Empirical work on intra-household allocation has been performed with paramet-ric specifications of preferences; for work specific to the Chiappori model, seeBrowning et al. (1994). Rejection of any model in a parametric framework couldmean rejection of the fundamental model, or rejection of the parametric spe-cification. The tests described above do not require any specification of prefer-ences, and so lead to the broadest possible tests of the model. Their applicationshould provide a useful counterpart to the existing empirical work on intra-

15household allocation.

5. Conclusion

This paper has developed formal tests of efficiency of public good provisionbased on revealed preference theory. In contrast to existing methods of testing, thetests developed here are nonparametric: for example, the assumptions aboutpreferences are that they can be represented by continuous, strictly monotonic,strictly quasiconcave, utility functions. Thus these tests are extremely broad;rejection of Pareto optimality is not a rejection of a particular parametricspecification of preferences. As such they should provide a useful counterpart toexisting work on tests of Pareto efficient public good provision.

Additionally, using the technique of quantifier elimination, we are able to derivetests that require mostly aggregate level data—no individual characteristics arerequired to be observed except after-tax income. Thus there should be a broadvariety of situations where it is feasible to obtain the data required to test whetheran optimal amount of public goods has been provided.

The Tarski-Seidenberg theorem tells us that it is theoretically possible to derivetestable restrictions for an economy with any finite number of consumers orobservations. The lack of practical quantifier elimination algorithms makes thisdifficult, however. The development of quantifier elimination algorithms that arespecifically designed for use with economic models—particularly models centeredaround simple individual utility maximization and profit maximization (or costminimization) behavior—appears to be a potentially useful area of research incomputational economics.

The methods outlined here are also capable of accommodating more complexmodels of public good provision. The theory could be extended to derivenonparametric testable restrictions of more general models of economies withexternalities, local public goods and club goods. Thus the techniques outlined in

15For work applying similar types of nonparametric tests of household behavior to householdconsumption and labor supply data, see Snyder (1997).


this paper should provide a strong theoretical basis for nonparametric hypothesistests and specification tests in a broad variety of situations where private goods,public goods, and goods that are some mixture of the two are present.

Acknowledgements

This paper is a revision of VPI working paper [E95-32. I would like to thankHans Haller, Peter Hammond, Kevin Reffett, Dale Thompson, and seminarparticipants at VPI, the 1996 Stanford Institute for Theoretical Economics, and theFall 1996 Midwest Mathematical Economics Meetings for valuable comments.This paper was greatly improved by the comments of the editor and an anonymousreferee. I would particularly like to thank Don Brown for suggesting the topic andfor many enlightening conversations. All errors remain my responsibility.

Appendix A

Proof of lemma 1. We must prove that if and only if the Pareto optimal publicgood demand restrictions are satisfied, there exists a solution of the following

r rsystem of inequalities over kq , y l:t t

r r r r r s r s'r, s, r ± s s.t. q x 1 p y , q x 1 p y t 5 a,b (1)t t t t

r r rp y 5 I t 5 a, b; r 5 1, 2 (2)t t

r r ry 1 y 5 y r 5 1, 2 (3)a b

r r rq 1 q 5 q r 5 1, 2 (4)a b

Condition Eq. (1) means that WARP is satisfied for each consumer. Eachconsumer has two ways of satisfying WARP (period 1 consumption not revealedpreferred to period 2 consumption or period 2 consumption not revealed preferredto period 1 consumption), thus there are four ways (not mutually exclusive) forboth consumers to satisfy WARP. Thus conditions (1)–(4) are equivalent to thedisjunction of four linear programs, with each one of these programs corre-sponding to one of the four different ways in which WARP can be satisfied. Wewill prove the lemma by showing that if and only if the public good consumptionrestrictions are satisfied will there exist a solution to at least one of these programs.

Case 1. Here we prove that if and only if condition I holds, then WARP can besatisfied by individual a’s period r consumption not being revealed preferred to his


period s consumption and individual b’s period r consumption not being revealedpreferred to his period s consumption for some r, s, r±s.

r rNecessary. Suppose hq , y j, r51, 2, satisfies conditions Eqs. (1)–(4) in thisa ar r r r r s r s r r s rway. Then it must be true that q x 1p y ,q x 1p y ⇔I ,q (x 2x )1a a a a a a

r s r rs r r r r r sp y . This implies I ,E . Similarly it must be true that q x 1p y ,q x 1a a a b b br s r r s r r s r rsp y ⇔I ,q (x 2x )1p y . This implies I ,E ; also this condition can beb b b b b b

r r r r s r r s r r s r s r s r r srewritten p y 2I ,q (x 2x )2q (x 2x )1p y 2p y ⇔q (x 2x )1p y ,a a a a ar r s r r s r r s r r s rI 1q (x 2x )1p ( y 2y ). This is only possible if q (x 2x )1p ( y 2y ).0.a

r rsSufficient. Suppose condition I holds. Then since I ,E , there exists somea ar s r r s r r sfeasible hq , y j such that I ,q (x 2x )1p y . Rewriting this condition, wea a a a a

r r r r s r sget q x 1I ,q x 1p y , which states that a’s period r consumption is nota a a a

revealed preferred to her period s consumption. Similarly, there also exists somer sfeasible hq , y j such that b’s period r consumption is not revealed preferred tob b

her period s consumption. To see that there exist feasible prices and consumptionssuch that conditions (3) and (4) are also satisfied (or that can satisfy WARP for

r s r r s r reach consumer simultaneously), note that q (x 2x )1p ( y 2y ).0 implies [I ,ar r s r r s r rsI 1q (x 2x )1p ( y 2y )] is a nonempty interval. Let e be the value of thea a

rs r sminimization problem corresponding to E . The value of the expression q (x 2a ar r s rs rs sx )1p y ranges continuously in the interval [e , E ] as we vary y from 0 toa a a as r r r rsy and as we vary q from 0 to q . Since I ,E , these intervals have aa a a

s s r r sˆ ˆ ˆnonempty intersection. Thus there exist feasible hq , y j such that I ,q (x 2a a a ar r s r s r r s r r sˆ ˆ ˆx )1p y (satisfies WARP for consumer a) and q (x 2x )1p y ,I 1q (x 2a a a ar r s rx )1p ( y 2y ) (satisfies WARP for consumer b).

Case 2. Here we prove that if and only if condition II holds, then WARP can besatisfied by individual a’s period r consumption not being revealed preferred to hisperiod s consumption and individual b’s period s consumption not being revealedpreferred to his period r consumption for some r, s, r±s.

r rNecessary. Suppose hq , y j, r51, 2, satisfies conditions (1)–(4) in this way.a ar s r r s r rs rThen it must be true that q (x 2x )1p y .I . This implies E .I . It musta a a a a

s r s s r s sr salso be true that q (x 2x )1p y .I . This implies E .I .b b b b b

rs rSufficient. Suppose condition II holds. Then E .I implies there exist somea ar r r s r r s r r s r s r r r rfeasible hq , y j such that q (x 2x )1p y .I , or q x 1p y .q x 1p y .a a a a a a a a a

sr s s s s rSimilarly, E .I implies there exist some feasible hq , y j such that q x 1b b b b bs r s s s sp y .q x 1p y . Then conditions (1) and (2) are satisfied; to see that thereb b b

r rexist hq , y j such that conditions (3) and (4) are also satisfied (or that cana as rsatisfy WARP for each consumer), note that any values of q , y will satisfya a

r sWARP for consumer a, and any values of q , y will satisfy WARP for consumerb bs r r sb. Thus choose q , y so that WARP is satisfied for consumer b, and q , y sob b a a


r s r sthat WARP is satisfied for consumer a, and then choose values for q , q , y , yb a a b

that satisfy conditions (3) and (4).

Appendix B

Proof that nonvacuous testable restrictions of Pareto optimality exist

The following examples prove that lemma 1 provides nonvacuous testablerestrictions that can neither be always satisfied nor never satisfied for the casewhen there is one public good and one private good.

Example of an economy that satisfies the restrictions:

r r r r r r r(data given in the order D 5 kx , y , q , p , I , I l)a b

1D 5 k1, 10, 1, 2, 6, 14l

2 1 1 1] ] ]D 5 9, 1, 1, , ,K L2 4 4

Example of an economy that does not satisfy the restrictions:

1D 5 k1, 10, 1, 2, 10, 10l

2 1 1 1] ] ]D 5 9, 1, 1, , ,K L2 4 4

Appendix C

Proof of corollary 2. This follows almost directly from lemma 1, with somedifferent assumptions about the observable variables. We cannot observe individualafter-tax or after-contribution incomes within the household; however we doobserve private good ‘‘expenditures’’ through observing wages and labor / leisurechoices. A more serious difference is we no longer observe the public good levels.Aggregate (household) consumption, C, is not a public good, but individualconsumptions, c and c , are. We cannot observe individual consumptions, so wea b

cannot observe the amount of each public good provided in the household. We doobserve aggregate consumption, however, and because consumption must benonnegative, this provides restrictions on the possible values of c and c . Definea b

the following maximization problems:


rs s r r sE 5 max m(d 2 g ) 1 n(C 2 C 1 g 2 d ) 1 w Lt t tn,m,d,g

0 # n # 1 0 # m # 1s.t. s r0 # d # C 0 # g # C

These are the same maximization problems seen in lemma 1 except these havetwo additional variables representing the individual consumption choices, makingthe problems bilinear.

Applying lemma 1, the testable restrictions are:

rs r r rs r rI [E . w L AND E . w L ANDa a a b b b

s r r s r r s rC 2 C 1 w (L 2 L ) 1 w (L 2 L ) . 0]a a a b b b

or:

rs r r sr s sII [E . w L AND E . w L ]a a a b b b

rs s r sSolving the maximization problems, we find E 5C 1w L .t t t

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