the economic efficiency of policy reform and … · reform and partial market liberalization under...

THE ECONOMIC EFFICIENCY OF POLICYREFORM AND PARTIAL MARKET

LIBERALIZATION UNDER TRANSACTIONCOSTS

Jean-Paul Chavas* and Zohra Bouamra Mechemache†

*Taylor Hall, University of Wisconsin, Madison, WI, USA, and†INRA-ESR, Department of Economics, Castanet Tolosan cedex,

France

ABSTRACT

The article presents an integrated analysis of the effects of domestic andtrade policy reform on resource allocation and welfare under transac-tion costs. It develops a general multiagent, multicommodity model,where transaction costs are the costs of resources used in the exchangeprocess. The influence of domestic and trade policy (including bothprice and quantity instruments) on distorted market equilibrium isanalysed. Alternative concepts of distorted equilibrium are presentedand investigated. They provide a basis for evaluating the effects ofmultilateral partial market liberalization on resource allocation andwelfare under transaction costs. New conditions are derived underwhich multilateral policy reforms generate Pareto improvements.

Keywords: distortions, market liberalization, multilateral, policyreform, welfareJEL classification numbers: F13, D51, D61

I. INTRODUCTION

The efficiency of competitive markets and trade is well known (e.g.,Allais, 1943, 1981; Arrow and Debreu, 1954; Debreu, 1959; Luenberger,

Correspondence: Jean-Paul Chavas, Taylor Hall, University of Wisconsin, Madison, WI5376, USA. Tel: þ1 608 261 1944; Fax: þ1 608 262 4376; Email: [email protected]

# Blackwell Publishing Ltd and the Board of Trustees of the Bulletin of EconomicResearch 2006. Published by Blackwell Publishing, 9600 Garsington Road, Oxford OX42DQ, UK and 350 Main Street, Malden, MA 02148, USA

Bulletin of Economic Research 58:3, 2006, 0307–3378DOI: 10.1111/j.0307-3378.2006.00241.x

161

1992b, 1994). It has generated a dominant view among economists thatfull market liberalization is desirable. However, partial moves towardsmarket liberalization may not be welfare improving, because theyinvolve ‘second best’ situations. Indeed, current domestic and tradepolicies often impose significant market distortions from taxes, tariffsand subsidies, as well as quotas that restrict trade and productionactivities. Attempts to undertake reform of international trade policyunder the auspices of the General Agreement on Tariffs and Trade(GATT) and the World Trade Organization (WTO) have not been easy.While tariffs have been progressively reduced for many sectors over thelast few decades, non-tariff barriers are still commonly used. This hasstimulated much research on the effects of price instruments (i.e., tariffs,subsidies, taxes) and quantity instruments (i.e., production and tradequotas) on resource allocation and welfare. The effects of tariff reformhave been studied by Bruno (1972), Lloyd (1974), Hatta (1977b),Fukushima (1979), Wong (1991) in a small open economy, Dixit(1986) in a large open economy and Foster and Sonnenschein (1970),Dixit (1975) and Hatta (1977a) in closed economies. More recently,Diewert and Woodland (2004) examined the gains from trade and thewelfare effects of tax/tariff policy changes. Studies of trade liberalizationwith tariffs and quotas include Anderson and Neary (1992) in a smallopen economy and Neary (1995) in a large open economy.

There is a need to extend previous research on policy reform in atleast three directions. First, previous research has investigated the effectsof tariffs and quotas on trade and welfare (e.g., Vousden, 1990;Turunen-Red and Woodland, 1991, 2000; Anderson and Neary, 1992;Neary, 1995). However, the effects of domestic and trade policy ofteninteract with each other. This suggests a need to expand previousanalyses to include the joint implications of both domestic and tradepolicy. For example, in the analysis of the impact of WTO reforms, it isimportant to take domestic policies into account, as most countries useboth trade and domestic instruments to regulate their markets (e.g., thecase of the agricultural sector). Second, except for Turunen-Red andWoodland (1991, 2000), previous research has often focused on a two--country analysis and considered only a limited number of policy instru-ments. However, market liberalization often involves multilateralnegotiations among many nations (e.g., WTO negotiations) where eachspecific tradable or non-tradable commodity may be regulated throughseveral policy instruments. This suggests a need to develop a generalequilibrium model of a distorted world economy consisting of an arbi-trary number of agents engaged in trading an arbitrary number ofcommodities, under domestic and trade policy involving both subsi-dies/tariffs and quotas. Third, previous work on policy reform hastypically assumed that market exchange is costless. This makes it diffi-cult to explain the presence of non-traded goods, which is often assumed

162 BULLETIN OF ECONOMIC RESEARCH

# Blackwell Publishing Ltd and the Board of Trustees of the Bulletin of Economic Research 2006.

to be exogenous (e.g., Hatta, 1977b; Fukushima, 1979). Yet, non-tradedgoods can arise because trade is costly. Transaction costs in trade maytake many forms: transportation cost over space, information cost, etc.In this context, one expects trade to take place only when its benefit islarger than its cost. While the effects of transaction costs on marketequilibrium have been noted (e.g., Hadley and Kemp, 1966; Woodland,1968), their interaction effects with distortionary policy and efficiencyhave not been explored. This suggests the need to introduce transactioncosts in a general equilibrium model under policy distortions.This article proposes an integrated framework to investigate the eco-

nomic and welfare implications of multilateral partial reforms of bothdomestic and trade policies (including price instruments as well as quantityinstruments) under transaction costs in general equilibrium. We definetransaction costs as costs that arise whenever resources are used in theprocess of exchanging goods among agents. The introduction of transac-tion costs in the analysis exhibits several desirable characteristics. First, weallow transaction costs to vary among agents. For example, to the extentthat they increase with the distance between trading agents, transactioncosts can be expected to be higher in international trade (when traders arein different countries) than in domestic markets (when market participantsare from the same country). Second, our analysis provides an endogenoustreatment of what are the traded versus non-traded goods, depending onthe magnitude of exchange costs. This can help explain the existence of‘local markets’ Third, in general equilibrium, the transaction costs arethemselves endogenous and can be affected by changes in economic policy.For example, market liberalization may contribute to reducing the cost ofresources used in exchange, which would further stimulate (beyond theeffects of reducing tariffs/quotas) the development of markets and increasethe benefits from trade. This suggests significant interactions betweenpolicy, transaction costs, market activities and welfare. Capturing sucheffects is a major motivation for our approach.While we expect transactioncosts to have a negative effect on trade incentives, their interactions withdistortionary domestic and trade policy as they affect resource allocationand welfare remain poorly understood. Our approach provides a newconceptual framework to investigate these issues.Some previous analyses of market liberalization have focused on

small changes in policy instruments (e.g., Vousden, 1990; Turunen-Redand Woodland, 1991, 2000; Neary, 1995). Our approach adds to thisliterature by considering discrete changes in policy instruments. Ouranalysis relies on Luenberger’s benefit function and its use in generalequilibrium analysis (Luenberger, 1992a, 1992b, 1995). We extendLuenberger’s general equilibrium analysis by considering price andquantity distortions, by investigating the associated distorted marketequilibrium and by studying the implications of domestic and tradepolicy for resource allocation and welfare under transaction costs.

POLICY REFORM AND PARTIAL MARKET LIBERALIZATION 163


A significant problem in market liberalization is that it is often part ofa second-best strategy. In this context, the reduction or elimination of asubset of distortions in a competitive equilibrium may not be welfareimproving. While free trade is efficient under competitive markets, in thepresence of trade barriers, a partial move towards free trade may actu-ally reduce welfare (Bhagwati, 1958; Johnson, 1967; Woodland, 1982;Falvey, 1988; Diewert et al., 1989; Vousden, 1990; Diewert andWoodland, 2004). A key result from this literature is that a proportionalreduction in all tariffs is typically welfare improving. Anderson andNeary (1992) and Neary (1995) extended this analysis to include bothtariffs and quotas in an open economy. The welfare analysis of multi-lateral trade policy reform is presented by Turunen-Red and Woodland(1991, 2000). Here we extend Turunen-Red and Woodland (1991, 2000)by considering price instruments as well as quantity instruments used inboth domestic and trade policy. This is of particular interest whendomestic policy affects the distortionary effects of trade policy. Forexample, there are situations where domestic production quotas canhelp reduce the distortionary effects of export subsidies (e.g.,Bouamra-Mechemache et al., 2002). This stresses the importance of anintegrated analysis of the effects of domestic and trade policy. Finally,we go beyond Neary (1995) by focusing on multilateral policy reform. Inparticular, we derive general conditions that imply that partial marketliberalization is welfare improving under transaction costs.

The article is organized as follows. Sections II and III develop ageneral equilibrium model of an economy under transaction costs,trade policy distortions (including both tariffs and quotas) and domesticpolicy distortions (including taxes, subsidies and production quotas).The model distinguishes between production agents and consumers. Itincludes an arbitrary number of commodities and agents trading witheach other. Transaction costs are associated with resources used in theexchange process. In this context, the influence of domestic and tradepolicy (including both tariffs and quotas) on distorted market equili-brium is analysed. Section III presents the distorted market equilibriumunder domestic and trade policy. Alternative characterizations of dis-torted equilibrium are presented and investigated in Section IV. Theyprovide a basis for analysing the effects of market liberalization ongeneral equilibrium resource allocation and welfare under transactioncosts. This is the topic of Section V. New conditions are derived underwhich partial multilateral policy reforms generate Pareto improvements.

II. PRELIMINARIES

Consider a global economy consisting of m commodities and n economicagents. We distinguish between two mutually exclusive groups of agents:



consumers and production units. LetNc be the set of consumers andNs the setof production units. The set of all agents isN ¼ Nc [ Ns ¼ {1, 2, . . ., n}. Theith consumer chooses a consumption bundle xi ¼ (xi1, . . ., xim) 2 Xi � Rm,i 2 Nc. The elements of xi are positive for commodity consumed andnegative for commodity produced (e.g., labour). We assume that thefeasible set Xi is closed, convex, has a lower bound and a non-emptyinterior, i 2 Nc. The ith consumption unit has a preference relationrepresented by the utility function ui(xi), i 2 Nc. The utility functionui(xi) is assumed continuous, non-decreasing and quasi-concave1 on Xi,i 2 N.The allocation of m goods among the n agents also involves produc-

tion and trading activities. For the ith production unit, i 2 Ns, theproduction activities yi ¼ (yi1, . . ., yim) are chosen from the transforma-tion set Yi � Rm, consisting of all commodity bundles that can beproduced. In the simplest case, Yi consists in a single point representingthe initial endowment for the ith agent. More generally, we use theconvention that elements of the vector yi measure netputs, i.e., outputswhen positive and inputs when negative. The set Yi is assumed non-empty and closed, i 2 Ns.Trade involves the vector t ¼ ftijk: i, j 2 N; k ¼ 1,…, mg 2 Rmn2 .

For outputs, tijk is the non-negative quantity of the kth commoditytraded from agent i to agent j. When i ? j, tijk � 0 is the quantity ofthe kth commodity ‘sold’ or ‘exported’ by agent i to agent j, orequivalently the quantity ‘purchased’ or ‘imported’ by the jthagent from the ith agent. When i ¼ j, this includes tiik, the quantityof the kth commodity that the ith agent trades with itself. We considerthe case where trade can be costly and involves the use of resources.Let z ¼ (z1, z2, . . ., zn), where zi ¼ (zi1, . . ., zim) 2 Rm is the vector ofcommodities used by the ith agent in trading activities, i 2 N. Thetrading activities (�z, t) are chosen from the transformation setZ � Rmn�Rmn2 consisting of all feasible points involving trade t andthe associated vector z.2 Thus, (�z, t) 2 Z, where the notation ‘�z’ isused to reflect that the zs are inputs in the trading process. We assumethat the set Z is closed and that (0, 0) 2 Z, i.e., the absence of tradecan take place without using any resources. Below, we will interpretthe cost of z as ‘transaction costs’ associated with exchange amongthe agents. Also, we make the following assumption.

1 A function u: X ! R is quasi-concave if, for all x, x¢ 2 X with u(x) � u(x¢), there holds

u½�xþ ð1� �Þx¢� � uðx¢Þ for all �, 0 � � � 1

Quasi-concavity of the utility function u(x) is equivalent to the convexity of preferences.2 The set Z restricts tijk to be non-negative for outputs and non-positive for inputs.



Assumption A1 (free tiik distribution): If (�z, t) 2 Z, then{ð�z, t¢Þ: tijk¢ ¼ tijk for i ? j, tiik¢ ¼ tiik þ dik, k ¼ 1, . . ., m, i, j 2 N} 2 Zfor all dik.

Assumption A1 states that the ith agent can modify tiik, the quantityof commodity k not subject to trade, without affecting the use ofresources z, for all k ¼ 1, . . ., m, i 2 N. This means that no resourcesz are used when agents consume their own production. In other words,transaction costs are relevant only in the presence of exchange betweendifferent agents.

Because trade can exist between any two agents, each being either aproduction unit or a consumer, it will be convenient to treat all agentssymmetrically. For that purpose, we let Xi ¼ {0} � Rm be the consump-tion set of the ith production unit, i 2 Ns, and Yi ¼ {0} � Rm be theproduction set of the ith consumption unit, i 2 Nc. This means that theonly feasible production for a consumption unit is yi ¼ 0, i 2 Nc, andthat the only feasible consumption for a production unit is xi ¼ 0,i 2 Ns. Note that, labour being one of the m commodities, consumerscan trade labour with production units, which allows for joint produc-tion and consumption choices under a single decision maker (e.g., thecase of household production).

Let x ¼ {xi, i 2 N}, y ¼ {yi, i 2 N}, where x 2 X ¼ X1 �X2 � . . . � Xn, and y 2 Y ¼ Y1 � Y2 � . . . � Yn.

Definition 1: A feasible allocation is defined as a vector (x, y, z, t)satisfying X

j2N tij � yi � zi i 2 N ð1aÞ

and

xi �X

j2N tji i 2 N ð1bÞ

where tij ¼ (tij1, tij2, . . ., tijm), xi 2 Xi, yi 2 Yi, i 2 N, and (�z, t) 2 Z.

Equation (1a) states that the ith agent cannot export more than itsproduction yi net of resources used in trade zi, i 2 N. And Equation(1b) states that the ith agent cannot consume more than it obtains eitherfrom itself (tiik) or from others (�j?i tjik). Note that summing (1a) and(1b) over i yieldsX

i2N xi �X

j2N

Xi2N tij �

Xi2N yi �

Xi2N zi

which implies that aggregate consumption cannot exceed aggregate pro-duction minus aggregate resources used for trading purposes. Next, weincorporate various domestic and trade policy instruments in the model



and investigate their effects on market equilibrium and resourceallocation.

III. POLICY DISTORTIONS AND MARKET EQUILIBRIUM

We consider a market equilibrium where the ith agent can face twoprices for commodity k: pik

s when commodity k is treated as a produc-tion activity and pik

c when commodity k is treated as a consumptionactivity. The corresponding price vectors are ps ¼ pik

s: k ¼ 1,…, m;fi 2 Ng 2 Rmn

þþ for ‘producer prices’, and pc ¼ pikc: k ¼ 1,…, m;f

i 2 Ng 2 Rmnþþ for ‘consumer prices’. Although the case where ps ¼ pc

can be seen as an important special case, the distinction between ps andpc will prove important in policy analysis. In particular, we will showbelow how pi

s and pic can differ for the ith agent in the presence of

distortionary policy.In this article, we focus our analysis on policy distortions generated by

domestic policy as well as trade policy. The policy instruments involveprice instruments (i.e., taxes, tariffs and subsidies) as well as quantityinstruments (i.e., production and trade quotas). Denote by rijk the unittariff (unit subsidy if negative) imposed on tijk for commodity kexchanged from agent i to agent j, k ¼ 1, . . ., m, i, j 2 N. We denotethe unit tariffs/subsidies by the vectors rij ¼ frijk: k ¼ 1,…, mg2 Rm and r ¼ frij: i, j 2 Ng 2 Rmn2 . Partition the set of agents intomutually exclusive groups: N ¼ {D1, D2, . . .}, where Ds is the set ofdomestic agents in the sth country. When i =2 Ds and j 2 Ds, then rijkrepresents an import tariff imposed on the kth commodity by the sthcountry. When i 2 Ds and j =2 Ds, then �rijk is an export subsidyimposed on the kth commodity by the sth country. As such, r measuresprice instruments used in trade policy. Alternatively, if (i, j) 2 Ds withi 2 Ns and j 2 Nc, then rijk represents a domestic tax (subsidy if negative)on the kth commodity, which creates a price wedge between producerprice p s

ik and consumer price p cik . As such, r would reflect domestic tax

and pricing policy. Allowing for differences between domestic consumerand producer prices and thus price distortions in domestic markets,this conceptual framework generalizes the model usually found in theliterature based on the ‘traditional’ GNP general equilibrium frame-work. In general, taxes or tariffs (rijk > 0) tend to increase consumerprices, decrease producer prices and generate budgetary revenue.Alternatively, subsidies (rijk < 0) tend to increase producer price pik

s,decrease consumer price pik

c and involve budgetary cost. The implica-tions of these revenues/costs for welfare analysis will be addressed below.Denote by qijk the quantity quota imposed on the trade flow tijk of the

kth commodity exchanged from agent i to agent j, k ¼ 1, . . ., m, i, j 2 N.For simplicity, we will focus our analysis on output quotas, with



qijk � 0.3 The quota qijk imposes an upper bound on the quantity tradedtijk. Letting qij ¼ (qij1, . . .., qijm), this gives

tij � qij i, j 2 N ð2aÞWe also consider domestic production quotas qyi restricting the pro-

duction of the ith producer. The introduction of domestic productionquotas is relevant, as they can affect the distortionary effects of tradepolicy (e.g., Bouamra-Mechemache et al., 2002). Again, for simplicity,we focus our analysis on output quotas, with qyi � 0 imposing an upperbound on the quantity produced by the ith producer

yi � qyi i 2 Ns ð2bÞWe expect the quotas q ¼ {qij: i, j 2 N; qyi, i 2 Ns} to generate quota rents

to market participants. Denote byQij the unit quota rents associated with thequotas qij and by Qyi the unit quota rents associated with the productionquotas qyi. Then, the vector of quota rents is Q ¼ {Qij: i, j, 2 N; Qyi: i 2 Ns}.The effects of quota rents on welfare will be discussed below. We are inter-ested in evaluating the effects of the policy instruments � ¼ (r, q) on resourceallocation and trade, on the market prices (ps, pc) and on the quota rents Q.

We make the following additional assumption.

Assumption A2 (free g distribution): There exists a numeraire good thatcan be traded between any two agents without using any resource z. Letthis good be the mth commodity, which we call ‘money’. Throughout thearticle, we consider monetary valuation that can be expressed in terms ofunits of the bundle g ¼ ð0, …, 0, 1Þ 2 Rm

þ. We assume that

(1) if (�z, t) 2 Z, then f(� z¢, t): tijk¢ ¼ tijk for all i, j 2 N, k ¼ 1, …,m� 1; tijm¢ ¼ tijm þ dijm for all i, j 2 Ng � Z for all dijm satisfyingtijm þ dijm � 0

(2) rijm ¼ 0, qijm ¼ þ1 for i, j 2 N and qym ¼ þ1, meaning thatneither tariff nor quota exists for the mth commodity.

Note that condition (1) in Assumption A2 states that money (i.e.,commodity m) can be exchanged among agents without incurring anytransaction cost. And condition (2) reflects the fact that our analysisfocuses on pricing and trade policy related to the first m � 1commodities.

Next, we consider the case where all agents are price takers.We focus ouranalysis on the effects of the policy instruments (r, q) onmarket equilibrium.We call the associated equilibrium a distorted market equilibrium. Our

3 Extending the analysis to trade quota restrictions on inputs would be straightforward.Using netput notation, inputs are negative, and input quotas would take the formtijk � qijk � 0, which would restrict the quantity of the kth input traded from agent i toagent j.



objective is to investigate the nature of the distortedmarket equilibrium andthe effects of (r, q) on production decisions y, consumption decisions x,trade activities (z, t), market prices (pc, ps) and quota rentsQ.

Definition 2: An allocation (x*, y*, z*, t*) along with market pricesp*s¼fpis*: pis* �g¼1, pis*2Rm

þ, i 2 Ng, pc*¼fpic*: pic*�g¼1, pic*2Rmþ,

i2Ng and the quota rents Q* � 0 is a distorted market equilibrium if

(1) (x*, y*, z*, t*) is a feasible allocation,(2) for each i 2 Nc and all xi 2 Xi, pi

c* � xi � pic*xi* implies

that uiðxiÞ � ui xi*ð Þ,(3) for each i 2 Ns and all yi 2 Yi,

ðpis* � Qyi*Þ � yi* � ðpsi* � Qyi*Þ � yi(4) for all (�z, t) 2 Z,X

i2N

Xj2N pi

c* � pis* � rij � Qij*

� �� tij* �

Xi2N pi

s* � zi*

�X

i2N

Xj2N pj


� �� tij �

Xi2N pi

s* � zi

(5) for each i2N, pis* � 0, pi

c* � 0, with pis* � ½yi* � zi*�

�j2Ntij*� ¼ 0 and pic* � ½�j2Ntji* � xi*� ¼ 0,

(6) for each i, j 2 N, tij* � qij, Qij* � 0, Qij* � ½qij � tij*� ¼ 0 and,for each i, yi* � qyi, Qyi* � 0 and Qyi* � ½qyi � yi*� ¼ 0.

Condition (1) requires feasibility. Condition (2) represents economicrationality for consumption units. Condition (3) is the profit maximiza-tion behaviour for production units under production quotas. It considersthat firms behave as if they were facing prices pi

s* � Qyi*, showing thatquota rents Qyi* � 0 reduce the incentive to produce. Condition (4)states that trade activities maximize profit under trade policy distortions.When i and j represent agents located in different countries, both thetariffs r and the quotas q act as trade barriers that reduce the profitabilityof trade. Condition (5) states the budget constraint for each agent,whether it is treated as a producer (involving prices ps) or a consumer(involving prices pc).4 Finally, condition (6) imposes the quota constraints(2a) and (2b), with the requirement that the quota rent Q* can be positiveonly if the corresponding quotas are binding.

4 Note that, in the case where ps ¼ pc ¼ p, solving for the term pi* � tij* in condition (5)gives

pi* � tij*¼ pi* �yi* � pi* � zi* � pi* �X

j?itij*

� �¼ pi* �xi* � pi* �

Xj?i

tji*� �

i2N

orpi* �

Xj?i

tji*� �

� pi* �X

j?itji*

� �¼ pi* �yi*� pi* � zi*� pi* �xi* i2N

This can be interpreted as a ‘balance of payment’ constraint which states that, for anyagent i 2 N, the value of net exports must equal profit, minus the cost of trade, minusconsumer expenditures.



Condition (4) has important implications for trade activities underpolicy distortions (r, q). To illustrate, consider the trade cost functionCðt, psÞ ¼ minz �i2Npi

s � zi: ð�z, tÞ 2 Zf g. In the special case whereC(t, ps) is differentiable in t and the kth commodity is an output(tijk � 0), the maximization problem implied by condition (4) yields thefamiliar Kuhn–Tucker conditions with respect to tijk:

pjkc* � pik

s* � @C=@tijk � rijk � Qijk* � 0 for tijk* � 0 ð3aÞand

pjkc* � pik

s* � @C=@ijk � rijk � Qijk*� �

� tijk* ¼ 0 ð3bÞEquations (3) show how trade policy generates price distortions through

the tariffs/subsidies rijk and the quota rents Q*ijk. In the context of acompetitive market equilibrium, Equation (3a) implies thatpjk

c* � piks* � @C=@tijk þ rijk þQ*ijk, i.e., that the price difference for

commodity k between agents i and j, pjkc* � pik

s*, cannot exceed themarginal transaction cost, @C/@tijk, plus the price distortion, rijk þQijk*.And when exchange takes place from agent i to agent j for thekth commodity (tijk > 0), then (3a) and (3b) imply that pjk

c*�pik

s* ¼ @C=@tijk þ rijk þQijk*. In this case, the price differencepjk

c* � piks* must equal the marginal transaction cost @C/@tijk plus the

price distortion rijk þQ*ijk. This can be interpreted as the first-ordercondition for profit-maximizing trade under distortionary policy. Forexample, in the absence of transaction costs where @C/@tijk ¼ 0, thenpjk

c* � piks* ¼ rijk þQ*ijk, showing that rijk þQ*ijk acts as a ‘price

wedge’ between consumer price pjkc* and producer price pik

s*. Notethat, in the absence of price distortions (where rijk ¼ 0; Q*ijk ¼ 0), thiswould generate the law of one price: pjk

c* ¼ piks* for all i, j 2 N. This

shows that under competitive markets the law of one price holds only inthe absence of both transaction costs and distortionary policy.Alternatively, when @C/@tijk > 0, transaction costs in (3) create a pricewedge between pjk

c* and piks*. Thus, either policy distortion (rijk ? 0

and/or Q*ijk > 0) or the presence of transaction costs (@C/@tijk > 0)is sufficient to imply that the law of one price fails. Finally, whentransaction costs and price distortions are ‘high enough’ so that@C=@tijk þ rijk þQ*ijk > pjk

c* � piks* for some i and j satisfying

pjkc* � pik

s* � 0, then the incentive to trade disappears as (3b) impliest*ijk ¼ 0. Then, the kth commodity becomes non-traded between agents iand j. If this happened for all agents, this would imply the absence ofmarket for the kth commodity. This illustrates that our general approachtreats the presence and development of markets as endogenous. It showsthe adverse effects that transaction costs and policy distortions can haveon trade and market activities. Alternatively, it stresses the role of lowtransaction costs and market liberalization policies in the creation andfunctioning of competitive markets.



IV. THE UTILITY FRONTIER IN A DISTORTED MARKET EQUILIBRIUM

To analyse the nature and efficiency of the distorted market equilibriumjust defined, it will be useful to explore related concepts of equilibrium forpolicy analysis. Below, we focus on the concepts of zero-maximum equili-brium and Lagrange equilibrium. Both are based on the ‘benefit function’,an aggregate measure of consumer benefits. These concepts are closelylinked with the efficiency of distorted market equilibrium. Luenberger(1992a, 1994, 1995) has investigated the relationship between these alter-native equilibrium concepts under zero transaction costs and in the absenceof policy distortions. Here, we extend Luenberger’s analysis in twoways: (i)we introduce domestic and trade policy distortions in the analysis and (ii)we allow for the presence of transaction costs.To analyse the efficiency effects of distortionary policy, we rely on the

concept of utility frontier.

Definition 3: Under policy � ¼ (r, q), the vector u(x*) ¼ fui (xi*), i 2 Ncgis on the utility frontier of the economy if x* ¼ fxi*: i 2 Ncg is feasible andif there does not exist another feasible x such that u(x) � u(x*),u(x) ? u(x*).

Because domestic and trade policies � ¼ (r, q) generate distortionsthat can adversely affect the efficiency of resource allocation, the utilityfrontier defined above is typically not the Pareto utility frontier.Our objective here is to assess the quantitative and qualitativeeffects of partial policy reforms (represented by changes in �) on thisutility frontier. The following function will prove important in ouranalysis.

Definition 4: Given the reference bundle g 2 Rþm satisfying g ? 0, define

the ith agent’s benefit function as

bi(xi; Ui) ¼max�f� : xi � �g 2 Xi; u(xi � �g) � Uigif xi � �g 2 Xi and u(xi � �g) � U for some �

¼ �1 otherwise

for i 2 Nc. The aggregate benefit function is then defined as

Bðx, UÞ ¼X

i2N biðxi, UiÞ

where x ¼ {xi, i 2 Nc} and U ¼ {Ui, i 2 Nc}.

The benefit function bi (xi, Ui) measures individual consumer benefit(expressed in units of the commodity bundle g) the ith consumer wouldbe willing to give up to obtain xi starting from utility level Ui. Whenthe commodity bundle g has a unit price, the benefit function can be



interpreted as an individual willingness-to-pay measure. And B(x, U)provides a corresponding measure of aggregate consumer benefit. Underthe assumptions that the set Xi is convex for each i 2 N and the functionui (x) is quasi-concave, Luenberger (1992b, pp. 464–6) has shown thatthe benefit function bi (xi, Ui) is concave in xi for i 2 Nc. Then, theaggregate benefit function B (x, U) is concave in x. Next, we presentthe zero-maximum concept that will prove crucial in evaluating theutility frontier.

Definition 5: Under policy � ¼ (r, q), define a maximal equilibrium as anallocation (x, y, z, t) satisfying

Vð�, UÞ ¼ max x,y,z,t

nBðx, UÞ �

Xi2N

Xj2N rij � tij: Eqns:ð1aÞ,ð1bÞ,

ð2aÞ,ð2bÞ;ðx, y � z, tÞ 2 X�Y�Zo

ð4aÞLet

Wð�, UÞ ¼ Vð�, UÞ þX

i2N

Xj2N rij � tij* ð4bÞ

where t* solves the optimization problem in (4a). If, in addition to being amaximal equilibrium, U is chosen such that W(�, U) ¼ 0, then the alloca-tion is zero maximal.

Note that Equations (4a) and (4b) involve the term (�i 2 N �j 2 N

rij � tij), the amount of money associated with the tariffs/subsidies r.This term is subtracted from the aggregate benefit B in (4a). As such,tariffs are treated as an additional cost to exchange commodities amongagents (which reduces the incentive to trade). But, this term is alsoadded in (4b) to reflect that the tariff revenues eventually benefitthe agents that capture them. Next, we establish the relationships betweenzero maximality and the utility frontier (see the proofs in the Appendix).

Proposition 1: Assume that ui(xi) is strictly increasing in the mth commodityxim for at least one consumer. If the feasible allocation (x*, y*, z*, t*) is onthe utility frontier, then it is zero maximal.

Proposition 2: Assume that the utility function ui(xi) is strongly quasi-concave5 for each i 2 Nc, and that the sets Y and Z are convex. If thefeasible allocation (x*, y*, z*, t*) is zero maximal and satisfiesx* 2 int(X), then it is on the utility frontier.

5 A function u: X ! R is strongly quasi-concave if, for all x, x¢ 2 X with u(x) > u(x¢),there holds

u½�xþ ð1 � �Þx¢� > uðx¢Þ for all �, 0 < � � 1



Propositions 1 and 2 establish conditions under which W(�, U) ¼ 0(in a zero-maximal equilibrium) is the implicit equation for the utilityfrontier under policy � ¼ (r, q). This has the following intuitiveinterpretations. First, the set of utilities U satisfying W(�, U) � 0identifies a feasible distribution of welfare among the consumers.Indeed, having W(�, U) < 0 cannot be feasible: it corresponds toB(x*, U) < 0, i.e., to situations where u(xi*) � Ui cannot hold forall i 2 Nc. Thus, the inequality W(�, U) � 0 can be interpreted as theaggregate budget constraint for all agents under distortionary policyand transaction costs. It simply states that aggregate net benefit cannotbe negative, i.e., that all benefits obtained must be feasibly generatedwithin the economy. Second, as investigated earlier, finding W(�,U) > 0 is necessarily below the utility frontier. In this context, we caninterpretW(�, U) as the distributable monetary surplus. This surplus, ifpositive, can always be redistributed costlessly (under Assumption A2)to some non-satiated agent and generate welfare improvements to atleast one agent without making anyone else worse off. It follows thatthe set of U satisfying W(�, U) ¼ 0 traces out the utility frontier undergovernment policy � and in the presence of transaction costs. This is auseful result for empirical analysis to the extent that the surplus func-tion W(�, U) involves monetary measurements, yet it is obtained undergeneral ordinal preferences. Note that the move along the utility fron-tier can take place in several ways. It can involve lump sum transfers(through the tijm) across agents. Or it can involve redistribution acrossagents of profit from production and trade activities, of quota rentsand of revenue/cost generated by tariffs/subsidies.Next, to show the links between the utility frontier and distorted

markets, we want to establish the relationships between zero maximalityand distorted market equilibrium. This is done by consideringa Lagrange equilibrium, which will be used in the next section toevaluate the efficiency implications of policy reform. Forx 2 X, y 2 Y, (� z, t) 2 Z, ps 2 Rmn

þ , pc 2 Rmnþ and Q 2 Rn2

þ , definethe Lagrangian

Lðx, y, z, t, U, ps, pc, Q, �Þ¼ Bðx, UÞ �

Xi2N

Xj2N rij � tij

þX

i2N pis � yi � zi �

Xj2N tij

h i

þX

i2N pic �X

j2N tji � xi

h iþX

i2N

Xj2N Qij � ½qij � tij�

þX

i2N Qyi � ½qyi � yi� ð5Þwhere ps, pc and Q are vectors of Lagrange multipliers associated withconstraints (1a), (1b), (2a) and (2b), respectively, and � ¼ (r, q).



Definition 6: A Lagrange equilibrium is an allocation (x*, y*, �z*,t*) 2 X � Y � Z and a vector ( ps*, pc*, Q*) � 0 which satisfy a saddlepoint of the Lagrangian

L x, y, z, t, U, ps*, pc*, Q*, �ð Þ� L x*, y*, z*, t*, U, ps*, pc*, Q*, �ð Þ� Lðx*, y*, z*, t*, U, ps, pc, Q, �Þ ð6Þ

for all (x, y, �z, t) 2 X � Y � Z and all (ps, pc, Q) � 0, where U ischosen to equal U* satisfying B(x*, U*, g) ¼ 0 and pi

s* � g ¼pi

c* � g ¼ 1, i 2 N.

The variables (ps, pc, Q) in (5) are Lagrange multipliers associated withconstraints (1a), (1b), (2a) and (2b). When the commodity bundle g has aunit price, the benefit function B(x, U) has a monetary interpretation, andthe Lagrange multipliers (ps, pc, Q) have the standard interpretation ofmeasuring the shadow price of the corresponding constraints. In a marketeconomy, ps and pc are then market prices reflecting resource scarcity forsupply and demand facing each agent. And Q measures the quota rentsassociated with quotas q. Next, we examine the close relationships thatexist between the Lagrange equilibrium and the zero-maximum equili-brium (see the proofs in the Appendix).

Proposition 3: Assume that the utility function ui(xi) is strictly increasingin the mth commodity xim for each i 2 Nc. If the feasible allocation (x*,y*, z*, t*) is a Lagrange equilibrium, then it is zero maximal.

Proposition 4: Assume that ui(xi) is quasi-concave in xi and strictlyincreasing in xim for each i 2 N, that the sets X, Y and Z are convexand that there exists a feasible allocation such that the constraints (1a),(1b) and (2) are non-binding.6 If the feasible allocation (x*, y*, z*, t*) iszero maximal, then it is a Lagrange equilibrium.

A distorted equilibrium and a Lagrange equilibrium are closelyrelated, as stated next (see the proof in the Appendix).

6 In the case where the aggregate benefit function B(x, U) is differentiable at (x*, y*, z*,t*) and the maximization problem in (4) has a solution satisfying x* 2 int(X), thenProposition 4 applies under weaker conditions. In this case, the existence of a feasibleallocation where all constraints in (1a), (1b) and (2) are non-binding (Slater’s condition)can be replaced by any of the constraint qualifications identified by Arrow et al. (1961).One of the Arrow, Hurwicz and Uzawa (AHU) constraint qualifications is that allconstraints are linear (see Takayama, 1985, pp. 97–8). Because the constraints (1a), (1b)and (2) are linear, the AHU constraint qualification is always satisfied. This means that,under differentiability and given x* 2 int(X), the maximization problem in (4) is equiva-lent to the saddle-point problem (6) (see Takayama, 1985, theorem 1.D.5, pp. 98–9). Inthis case, Slater’s condition is no longer needed, and Proposition 4 applies without it.



Proposition 5: If the feasible allocation (x*, y*, z*, t*) is a Lagrangeequilibrium, then it is a distorted market equilibrium.

Note the role played by Assumptions (A1) and (A2). Under Assumption(A2), the first inequality in (6) implies that pim

s* ¼ pimc* ¼ pm* for all i 2 N

(otherwise, tijm* or tjim* and thus L(x*, y*, z*, t*, U, ps*, pc*, Q*, �)would be unbounded, a contradiction). Thus, the optimal choice for tijmmeans that the price of themth commodity (money) is the same for all agents.Without loss of generality, it is normalized to be equal to 1, withpm* ¼ pi

s* � g ¼ pic* � g ¼ 1 for all i 2 N. This means that money is used

as a basis for evaluating all welfare measures. And under Assumption (A1),the first equality in (6) implies that pi

c* � pis* � rii þQii* for all i 2 N

(otherwise, tii* would be infinite and L(x*, y*, z*, t*, U, ps*, pc*, Q*, �)would be unbounded, a contradiction). Thus, the optimal choice for tii impliesthat the prices faced by each agent satisfy pi

c* � pis* ¼ rii þQii*, i 2 N. In

the absence of distortionary policy (where rii ¼ 0 and Qii* ¼ 0), this impliesthat pi

c* ¼ pis* for each i 2 N, i.e., that producer prices and consumer prices

become identical for each agent.

Proposition 6: If the feasible allocation (x*, y*, z*, t*) is a distortedmarket equilibrium and B(x*, U*, g) ¼ 0 (where U* ¼ fuiðxi*Þ: i 2 Ncg),then it is a Lagrange equilibrium.

The proof of Proposition 6 is presented in the Appendix. Propositions5 and 6 show that, under some regularity conditions, the concepts ofdistorted market equilibrium and of Lagrange equilibrium are equiva-lent. This relationship will prove useful below in the investigation ofeconomic behaviour under policy distortions.Finally, for completeness, note that the Lagrange equilibrium generates

a useful characterization of the distorted prices ( ps*, pc*) and quota rentsQ* under distortionary policy � ¼ (r, q). When psi � g ¼ pci � g ¼ 1, i 2 N,the first inequality in the saddle-point problem (6) implies profit maximi-zation and expenditure minimization. More specifically, when psi � g ¼ 1,the saddle-point problem (6) implies that

�yiðpis � QyiÞ ¼ supyfðpis � QyiÞ � yi: yi 2 Yig ð7aÞ

where �yiðp si � QyiÞ is the indirect profit function for the ith production

unit, i 2 Ns (see (B1b) in the Appendix). Similarly, it implies that

�Tðps, pc, Q, rÞ ¼ supz, tX

i2N

Xj2N pj

c � pis � rij � Qij

� �� tij�

n

Xi2N pi

s � zi: ð�z, tÞ 2 Zo

ð7bÞ

where �T ( ps, pc, Q, r) is the indirect profit function for trade activities (see(B1c) in the Appendix). It follows that the aggregate profit function can be



defined as �(ps, pc, Q, r)¼P

i2Ns�yi(pi

s � Qyi)þ�T(ps, pc, r, Q). Finally,when p c

i � g¼ 1 and from (B1a) in the Appendix, the saddle-point problem(6) implies that

eiðpc, UiÞ ¼minx pc �xi: uiðxiÞ � Ui, xi 2 Xif g ð8Þwhere ei ( p

c, Ui) is the expenditure function for the ith consumer, i 2 Nc

(Luenberger, 1992b). Then, the aggregate expenditure function can bedefined as E(pc, U)¼�i2Nc

ei(pc, Ui). The profit functions �( ps, pc,

Q, r), �yi( pc) and �T( p

s, pc, Q, r) are each convex in ( ps, pc, Q, r). Andthe expenditure functions E( pc, U) and ei(p

c, Ui) are each concave in pc

(see Berge, 1963; Diewert, 1974). Using this notation, the second inequal-ity in the saddle-point problem (6) implies that (ps*, pc*, Q*) satisfy

Vð�, UÞ ¼ minps, pc,Qf�ð ps, pc, Q, rÞ � Eðpc, UÞþX

i2N

Xj2N Qij � qij

þX

i2N Qyi � qyi: ðps, pc, QÞ � 0g ð9aÞ

where � ¼ (r, q). Let

Wð�, UÞ ¼ Vð�; UÞ þX

i2N

Xj2N rij � tij* ð9bÞ

From Definition 6, if in addition U is chosen to satisfy W(�, U) ¼ 0,then the corresponding allocation is a Lagrange equilibrium. This pro-vides a formulation for equilibrium prices (p*s , p

*c) and quota rents Q*.

Under the conditions stated in Propositions 5 and 6, these are the marketprices and quota rents obtained in a distorted market equilibrium underpolicy instruments � ¼ (r, q). This gives the ‘dual approach’ to marketequilibrium analysis commonly found in the economic literature onpolicy and trade distortions (e.g., Kemp, 1995; Neary, 1995; Diewertand Woodland, 2004). It extends the general equilibrium analysis pre-sented by Luenberger (1992a, 1994) in two ways: (i) it introduces trans-action costs; (ii) it incorporates the effects of pricing policy and quotarestrictions on pricing and resource allocation.

Note that �i 2 N �j 2 N Qij � qij þ �i 2 N Qyi � qyi in (9a) is the aggregatequota rent involving both trade and production activities. It is addedin (9a) to reflect that the quota rents benefit the agents who capture them.Similarly, the aggregate tariff revenue �i2N�j2Nrij � tij* is added in (9b)to reflect that these revenues benefit the agents that receive them.

Propositions 3–5 present formal relationships between three concepts:distorted market equilibrium, Lagrange equilibrium and zero maximal-ity. And Propositions 1 and 2 provide important linkages to the char-acterization of the utility frontier. This is illustrated in Figure 1. Asderived, such relationships hold under transaction costs and distortion-ary domestic and trade policy. The concepts of zero maximality and



Lagrange equilibrium are very useful tools in economic analysis. As firstproposed by Allais (1943, 1981), the concept of zero maximality (and itsclose linkage with the characterization of utility frontier) is intuitive andquite powerful in welfare and efficiency analysis. And the related con-cept of Lagrange equilibrium (and its close linkage with distorted mar-ket equilibrium) can be quite useful and provide additional insights incomparative statics analysis. This is illustrated next in an investigationof the effects of domestic and trade policy on resource allocation.

V. IMPLICATIONS FOR WELFARE AND RESOURCE ALLOCATION

In this section, the concepts of Lagrange equilibrium, zero-maximumequilibrium and zero-minimum equilibrium are used to analyse theeconomic implications of government policy under transaction costs.Clearly, the policy instruments � ¼ (r, q) affect resource allocation.The associated distortions are expected to influence adversely economicefficiency, meaning that the distorted economy is expected not to satisfythe Pareto optimality criterion. This raises two related questions: (i) howto represent the welfare implications for the distorted economy and(ii) how to assess the nature and extent of economic inefficiency due todistortionary domestic and trade policy. To answer these questions, weexamine next the welfare measurements of government policy undertransaction costs.

V.1 Evaluation of a discrete change in policy

We analyse the general case of a discrete change in the policy instru-ments �. We could proceed using any of the equilibrium conceptsdiscussed in Section IV. Keeping in mind the close relationships thatexist between these alternative concepts (see Figure 1), it will be con-venient here to focus on the Lagrange equilibrium. For a given U, let[x*(�, U), y*(�, U), z*(�, U), t*(�, U)] denote an allocation that satisfiesthe saddle-point condition (6). And let Vð�, UÞ ¼ [Bðx*, UÞ � �i2N�j2Nrij � tij*] denote the aggregate net benefit evaluated at x*(�, U),y*(�, U), z*(�, U) and t*(�, U). Then, the following result applies (seethe proof in the Appendix).

Proposition 7: For a given U, assume that a saddle point in (6) holds withsaddle value V(�, U) for all � ¼ (r, q) 2 A. Then, for any �, �¢ 2 A,

Proposition 1 Proposition 4 Proposition 5

Proposition 2 Proposition 3 Proposition 6Zero maximalequilibrium

Lagrange equilibrium

Distorted market

equilibriumUtility frontier

Fig. 1. Relationships among alternative concepts.



Xi2N

Xj2N rij¢ � tij*ð�¢, UÞ � tij*ð�, UÞ

� �

þX

i2N

Xj2N Qij*ð�¢, UÞ � qij¢ � Qij

� �

þX

i2N Qyi*ð�¢, UÞ � qyi¢ � Qyi

� �

�Wð�¢, UÞ � Wð�, UÞXi2N

Xj2N rij � tij*ð�¢, UÞ � tij*ð�, UÞ

� �

þX

i2N

Xj2N Qij*ð�, UÞ � qij¢ � Qij

� �

þX

i2N Qyi*ð�, UÞ � qyi¢ � Qyi

� �ð10Þ

where Wð�, UÞ ¼ Vð�, UÞ þ �i2N�j2N rij � tij*ð�, UÞ.

Proposition 7 provides a lower bound and an upper boundon the change inthe aggregate net benefit [W(�¢ � W(�,U)] evaluated atU. It is very generalin the sense that it applies without restrictions on the setA. It does not requirethe decision rulesx*(�,U), y*(�,U), z*(�,U) and t*(�,U) tobedifferentiablefunctions, nor single value mappings. And it applies to arbitrary discretechanges in the policy instruments � ¼ (r, q). Finally, it considers the jointeffects of price and quantity policy instruments used in both domestic andtrade policy. This provides significant generalizations on previous analyses ofpolicy reform (e.g., Falvey, 1988;Diewert et al., Vousden, 1990; Turunen-Redand Woodland, 1991, 2000; Anderson and Neary, 1992, 1996; 1989; Neary,1995). Also, Proposition 7 includes some intuitive and well-known results asspecial cases. To see that, consider the following corollary.

Corollary 1: For any �, �¢ 2 A,Xi2N

Xj2N rij¢ � rij

� �� tij*ð�¢, UÞ � tij*ð�, UÞ� �

þX

i2N

Xj2N Qij*ð�¢, UÞ � Qij*ð�, UÞ

� �� qij¢ � Qij

� �

þX

i2N Qyi*ð�¢, UÞ � Qyi*ð�, UÞ� �

� qyi¢ � Qyi

� �� 0 ð11Þ

Again, Corollary 1 applies in general for any discrete change in �. Ithas two useful implications. First, consider the case where tariffs arechanged but where quotas are unchanged (q ¼ q¢). Then, (11) becomes

Xi2N

Xj2N rij¢ � rij

� �� tij*ð�¢, UÞ � tij*ð�, UÞ� �

� 0

This means that t*(r, �) is non-increasing in r: ceteris paribus, an increasein tariffs r tends to decrease the corresponding quantities traded. Notethat this intuitive result is obtained without differentiability assumptions.In the special case where the change in tariffs (r¢ � r) is ‘small’ and the



function t*(�, U) is differentiable in r, this implies that [r¢ � r] �[@t*(�, U)/@r] � [r¢ � r] � 0, i.e., the matrix [@t*(�, U)/@r] is symmetric,negative semi-definite. In addition, if (r¢ � r) ? 0 is not in the null spaceof [@t*(�, U)/@r], then [r¢ � r] � [@t*(�, U)/@r] � [r¢ � r] < 0.Second, consider the case where tariffs are now unchanged (r ¼ r¢).

Then, (11) yieldsXi2N

Xj2N Qij*ð�¢, UÞ � Qij*ð�, UÞ

� �� ½qij¢ � Qij�

þX

i2N Qyi*ð�¢, UÞ � Qyi*ð�, UÞ� �

� qyi¢ � Qyi

� �� 0

This means that the quota rent Q*(�, �) is non-increasing in q: ceterisparibus, an increase in quotas q tends to decrease the corresponding quotarents. Again this intuitive result holdswithout differentiability assumptions.In the special case where the change in quotas (q¢ � q) is ‘small’ and thefunction Q*(�, U) is differentiable in q, this implies that [q¢ � q] � [@Q*(�, U)/@q] � [q¢ � q] � 0, i.e., that the matrix [@Q*(�, U)/@q] is symmetric,negative semi-definite. In addition, if (q¢ � q) ? 0 is not in the null space of[@Q*(�,U)/@q], then [q¢ � q] � [@Q*(�, U)/@q] � [q¢ � q] < 0.

V.2 Impacts on the utility frontier

To evaluate the welfare implications of Proposition 7, two attractivechoices for U are possible. First, consider the case where U is chosensuch that aggregate net benefit is zero in situation �: W(�, U) ¼ 0.Then, [W(�¢, U) � W(�, U)] ¼ W(�¢, U) measures the aggregate netincome gain (or loss if negative) associated with a move from � to �¢.In other words, [W(�¢, U) � W(�, U)] ¼ W(�¢, U) is a simple measureof aggregate efficiency gains (‘compensating variation’) generated by apolicy change from � to �¢. And Proposition 7 provides bounds on theseefficiency gains under transaction costs. With this particular choice of U,note that x*(�, U), y*(�, U), z*(�, U) and t*(�, U) correspond to anallocation on the utility frontier under situation �.Second, consider the case where U is chosen such that aggregate

net benefit is zero in situation �¢: W(�¢, U) ¼ 0. Then, [W(�¢, U) � W(�,U)] ¼ �W(�, U) measures the aggregate net income loss (or gain ifnegative) associated with replacing �¢ in favour of �. It follows that[W(�¢, U) � W(�, U)] ¼ �W(�, U) is a simple aggregate efficiency mea-sure (‘equivalent variations’) generated by giving up the exchange environ-ment �¢. With this choice ofU, x*(�¢,U), y*(�¢,U), z*(�¢,U) and t*(�¢,U)are on the utility frontier under situation �¢. With either choice of U, theterm [W(�¢, U) � W(�, U)] can thus be used to evaluate how the utilityfrontier shifts under a policy change from � to �¢. As such, Proposition 7provides a basis to investigate Pareto welfare improving moves.

Proposition 8: For any change from � to �¢ in A, a sufficient condition for[W(�¢, U) � W(�, U)] � (>) 0 is



Xi2N


� �þX

i2N

Xj2N Qij*ð�¢, UÞ�

qij¢ � qij� �

þX

i2N Qyi*ð�¢, UÞ � qyi¢ � qyi� �

� ð>Þ 0 ð12aÞ

and a necessary condition for [W(�¢, U) � W(�, U)] � (>) 0 is

Xi2N


� �þX

i2N

Xj2N Qij*ð�, UÞ�

qij¢ � qij� �

þX

i2N Qyi*ð�, UÞ � qyi¢ � qyi� �

� ð>Þ 0 ð12bÞ

where the weak (strict) inequalities correspond to a weak (strict) Paretowelfare improvement.

Proposition 8 states our main results. They are simple and very general.Again, they apply under both price instruments r and quantity instrumentsq; they consider jointly domestic and trade policy; they allow for discretechange in the policy instruments� ¼ (r, q); they allow for transaction coststhat reduce the incentive to trade; and they hold without differentiabilityassumptions. As such, they are a significant generalization of previouswork (e.g., Turunen-Red and Woodland, 1991, 2000; Anderson andNeary, 1992; Neary, 1995). Interpreting [W(�¢, U ) � W(�, U )] asmeasuring the shift in the utility frontier, Proposition 8 establishesthat (12a) is a sufficient condition for a policy change from � to �¢ to bePareto improving. Expression (12a) states that the term�i2N�j2N rij � [tij*(�¢,U ) � tij*(�,U )], reflecting the change in the aggre-gate value of tariff revenues evaluated at r¢, plus the term�i2N�j2N Qij*(�¢,U) � [qij¢ � qij]þ�i2Nyi*(�¢,U) � [qyi¢ � qyi], reflectingthe aggregate change in quota rents, is non-negative. Given Q* � 0, asufficient condition for the change in the term involving quota rents to benon-negative is that q¢ � q, i.e., that trade and production quota restric-tions be relaxed. Here, we want to stress that this result applies underreform involving both price and quantity policy instruments for domesticas well as trade policy. However, the effects of domestic and trade policyreform on the term involving taxes/tariffs revenue are more complex.Indeed, reducing tariffs/subsidies (where 0� rijk¢� rijk if rijk > 0 andrijk� rijk¢� 0 if rijk < 0) and/or relaxing quotas (q¢ � q) is in general notsufficient to imply that �i2N�j2Nrij � [tij*(�¢,U) � tij*(�,U)] � 0.However, (12a) implies that a sufficient condition for market liberaliza-tion satisfying q¢ � q to be Pareto improving is that it stimulates trade[tijk*(�¢,U) � tijk*(�,U)] for commodities that are subject to tariff(rijk > 0) and reduces trade [tijk*(�¢,U)� tijk*(�,U)] for commoditiesthat are subsidized (rijk < 0). This simple result is quite powerful in thesense that it is intuitive and applies under very general conditions.

It is well known that there are situations where partial market liberal-ization is immiserizing (e.g., Vousden, 1990; Anderson and Neary, 1992;



Neary, 1995). Proposition 8 provides additional information onthis issue. It shows that expression (12b) is a necessary condition for apolicy change from � to �¢ to be Pareto improving. Alternatively, itmeans that, whenever (12b) is not satisfied, then a partial movetoward free markets cannot be efficiency improving. Expression (12b)states that the term �i2N�j2Nrij � [tij*(�¢,U) � tij*(�,U )], reflecting thechange in the aggregate value of tariff revenues evaluated at r, plusthe term �i2N�j2NQij*(�,U ) � [qij¢ � qij]þ �i2NQyi*(�,U ) � [qyi¢ � qyi],reflecting the aggregate change in quota rents, is non-negative. Again,given Q* � 0, a sufficient condition for the change in the term involvingquota rents to be non-negative is that q¢ � q, i.e., that trade and pro-duction quota restrictions be relaxed, a result that applies under bothtariffs and quotas. As before, the effects of domestic and trade policyreform on the term involving tariff revenue are more complex. Indeed,reducing tariffs/subsidies (with 0 � rijk¢ � rijk if rijk > 0 andrijk � rijk¢ � 0 if rijk < 0) and/or relaxing quotas (q¢ � q) is in generalnot sufficient to imply that �i2N�j2N rij � [tij*(�¢,U ) � tij*(�,U )] � 0.When the left-hand side in (12b) becomes negative, then partial marketliberalization necessarily reduces efficiency. This is the situation wherepolicy reform is immiserizing (e.g., Anderson and Neary, 1992; Neary,1995). This is an illustration of the theory of the second best applied topolicy analysis. More specifically, Proposition 8 shows the conditionsunder which partial market liberalization reduces efficiency: with q¢ � q,for policy reform to be immiserizing, (12b) must not hold, implying that�i2N�j2N rij � [tij*(�¢,U ) � tij*(�,U )] must be negative and large. Thissimple result appears new and quite useful. It warns us against domesticand trade policy reform (especially quota reform) that exacerbates thedistorting effects of pricing policy by stimulating exports that aresubsidized and/or reducing imports that are taxed.

V.3 A special case: when t* is differentiable

To relate Proposition 8 to previous literature, consider the special casewhere the change in policy (�¢ � �) is ‘small’, and the function t*(�, U)is differentiable in �. Then, Proposition 8 implies the following result.

Corollary 2: For any small change from� to�¢, [W(�¢,U) � W(�,U)] � (>)0 if and only if

r � ½@t*ð�, UÞ=@r� � ½r¢ � r� þ r � ½@t*ð�, UÞ=@q� � ½q¢ � q�þX

i2N

Xj2N Qij*ð�, UÞ � qij¢ � qij

� �þX

i2N Qyi*ð�, UÞ�

qyi¢ � qyi� �

� ð>Þ 0 ð13Þwhere the weak (strict) inequality corresponds to weak (strict) Pareto welfareimprovement associated with a change from � to �¢.



Expression (13) gives some well-known ‘local results’ about welfareeffects of trade liberalization (e.g., Falvey, 1988; Vousden, 1990;Turunen-Red and Woodland, 1991; Anderson and Neary, 1992;Neary, 1995). To see that, note that the first term in (13) involves thematrix [@t*(�, U)/@r], measuring the effects of tariffs on trade. We haveseen above that [@t*(�, U)/@r] is a symmetric, negative semi-definitematrix. In the case of a proportional decrease in tariffs/subsidies wherer¢ ¼ kr, 0 � k < 1, it follows that the first term in (13) is alwaysnon-negative: r � [@t*(�, U)/@r] � [r¢ � r] � 0.7 Then, there are two simplescenarios where equation (13) is always satisfied. The first scenarioconcerns a proportional tariff/subsidy reduction in the absence ofquotas (where q ¼ q¢ ¼ 1, Q* ¼ 0). The first term in (13) is thennon-negative, whereas the absence of quotas implies that the secondand third terms in (13) vanish. This generates the well-known resultthat, in the absence of quotas, a proportional tariff reduction is always(at least weakly) welfare improving. The second scenario concerns quotarelaxation in the absence of tariffs (where r ¼ r¢ ¼ 0). The absence oftariffs means that the first and second terms in (13) vanish. And anyquota relaxation (q¢ � q) always implies that the third term in (13) isnon-negative, because Q*(�, U) � 0. Thus, in the absence of tariffs,relaxing any quota is always (at least weakly) welfare improving.Finally, note that Equation (13) is obtained without requiring additionalassumptions (such as the ‘rank condition’ used in Turunen-Red andWoodland, 1991, 2000).

What happens to these local results in the more realistic situationwhere both tariffs and quotas are present? Two important findingsfollow from (13). First, it remains true that any proportional tariffreduction is (at least weakly) welfare improving in the presence oftrade and production quotas, provided that these quotas remain con-stant (with q ¼ q¢ < 1). To see that, it suffices to note that the firstterm in (13) is non-negative under proportional tariff reduction, whereasthe second and third terms vanish when q ¼ q¢. This extends a well-known result (e.g., Vousden, 1990, p. 217) to situations covering bothdomestic and trade policy. Second, in a second-best world, it is wellknown that any quota relaxation is not always welfare improving (e.g.,Falvey, 1988). This finding is obtained from Corollary 2: in the presenceof tariffs or subsidies (r ? 0), the second term in (13), r � [@t*(�, U)/@q] � [q¢ � q] (which involves cross-commodity effects of quotason trade), cannot be signed in general. In this case, quota relaxation(q¢ � q) � 0 can interact with tariffs r in such a way that the inequalityin (13) may no longer hold. Note that this indeterminacy remains even if

7 In addition, if the vector r is not in the null space of [@t*(�, U)/@r], then the first termin (13) would become strictly positive: r � [@t* (�, U)/@r] � (r¢ � r) > 0.



tariffs/subsidies remain unchanged (with r ¼ r¢). Thus, in a second-bestworld of partial market liberalization, relaxing production and/or tradequotas in the presence of tariffs does not necessarily generate a Paretoimprovement.In the presence of tariffs, subsidies and quotas, Corollary 2 shows

conditions under which relaxing quotas are efficiency enhancing.8 It alsoshows that these conditions are not always satisfied under partial marketliberalization. In particular, it gives new results that apply to the jointeffects of domestic and trade policy reform. Indeed, expression (13) is notalways positive when r ? 0, as it depends on the effects of quotas ontrade: r � [@t*(�, U)/@q] � [q¢ � q]. Because this includes the effects ofproduction quotas on trade, it suggests that domestic policy has to betaken into consideration in the analysis of policy reform. For example,relaxing production quotas (q¢ � q) in the presence of subsidized exports(rijk < 0) can decrease welfare if this has strong positive impacts onsubsidized trade (with @tijk*(�, U )=@q > 0). This would identify second-best conditions under which partial market liberalization is immiserizing.This is illustrated by Bouamra-Mechemache et al. (2002) in an analy-

sis of partial domestic and trade policy reform in the European Union(EU) dairy sector. The EU dairy sector is of interest because, even afterthe Uruguay round of GATT negotiations and recent EU policyreforms, it is still subject to significant policy distortions: domesticmilk production quotas, domestic subsidies, as well as trade barriers(including import quotas and export subsidies). Bouamra-Mechemacheet al. show that, in the presence of export subsidies, removing milkproduction quotas (with or without decreasing domestic subsidies) inthe EU generates welfare losses for both the EU and the world. Indeed,relaxing production quotas increases EU milk production, which in turnstimulates the EU exports of subsidized dairy products. Thus, relaxingEU milk production quotas in the presence of export subsidies (r < 0)implies that the terms f�i2N�j2Nrij � [tij*(�¢, U ) � tij*(�, U )]g in (12b)and fr � [@t*(�, U )=@q � (q¢ � q)]g in (13) are negative. As long as exportsubsidies are close to their current level, Bouamra-Mechemache et al.find that these quota effects exacerbate the distortionary effects ofexport subsidies and imply a decline in EU and world efficiency. Assuggested by Proposition 8 and Corollary 2, they also find that theremoval of production quotas would become efficiency enhancing forthe EU and for the world if the export subsidies were removed.This shows that the welfare effects of market liberalization become

more complex in the presence of both tariffs/subsidies and quotas in

8 Turunen-Red and Woodland (2000) have shown that, if partial trade liberalization isefficiency enhancing, then a strict welfare improvement can still be attained even withoutlump sum compensation under multilateral policy reform.



domestic and trade policy. For example, in the presence of export tariffs,our analysis points to the importance of the cross-commodity effects ofchanging domestic production quotas on trade. Such effects can lead todecreased efficiency associated with partial market liberalization. Thisillustrates the power of results stated in Proposition 8. They provide thenecessary and the sufficient conditions for domestic and trade policyreform to be welfare improving in a form that is simple and general,allowing for discrete changes in both quantity and price policy instru-ments under general equilibrium and transaction costs.

V.4 Role of transaction costs

To examine the role of transaction costs, consider the case where thefeasible set Z changes. It will be convenient to write it as Z(�), where �is a parameter reflecting the trade technology. We consider a change from� to �¢ such that Z(�¢) �

Z(�). This represents technological progressrelated to the trade technology. Then, transaction cost can be written asC(t, ps, �) ¼ minz �i2Npi

s � zi: (� z, t) 2 Z(�)½ �. Given Z(�¢) �

Z(�), itfollows that C(t, ps, �¢) � C(t, ps, �). This makes it clear that a changefrom � to �¢ corresponds to a decrease in transaction costs. Next, weinvestigate the implications of this decline in transaction costs. The proof issimilar to the one presented in Proposition 7 and is omitted.

Proposition 9: For a given U, assume that a saddle point in (6) holds withsaddle value V(�, �, U) with � ¼ (r, q). Then, for any change from � to �¢,

� C[t*ð�, �, UÞ, ps*ð�, �¢, UÞ, �¢] � C[t*ð�, �, UÞ, ps*ð�, �¢, UÞ, �]f gþX

i2N

Xj2N rij � [tij*ð�, �¢, UÞ � tij*ð�, �, UÞ]

�Wð�, �¢, UÞ � Wð�, �, UÞ� � C[t*ð�, �¢, UÞ, ps*ð�, �, UÞ, �¢]f� C[t*ð�, �¢,UÞ, ps*ð�, �, UÞ, �]gþX

i2N

Xj2N rij � [tij*ð�, �¢, UÞ � tij*ð�, �, UÞ] ð14Þ

where W(�, �, U ) ¼ V(�, �, U )þ �i2N�j2Nrij � tij*(�, �, U ) and V(�,�, U) is given in (4a).

Proposition 9 provides a lower bound and an upper bound in thechange in aggregate net benefit [W(�, �¢, U) � W(�, �, U)] associatedwith a change in transaction costs from � to �¢. Each bound involvestwo terms: the negative of the change in transaction cost, �[C(t, ps, �¢) �C(t, ps, �)], and the change in tax/tariff revenue, �i2N�j2Nrij�[tij*(�, �¢,U ) � tij*(�,�,U )].

First, consider the case where there is no tax or tariff: r ¼ 0. ThenEquation (14) implies W(�,�¢,U ) � W(�, �,U ) � �fC [t*ð�, �,U ),



ps*(�, �¢,U ),�¢) � C [t*(�,�,U ); ps*(�,�¢,U ),�]g. As discussed above,when Z(�¢) � Z(�), we have �[C(t, ps, �¢) � C(t, ps, �)] � 0, implyingthat W(�, �¢, U ) � W(�, �, U ) � 0. This gives the intuitive result thatany reduction in transaction costs contributes to increasing aggregatenet benefit. This illustrates that, in the absence of tax or tariff, reducingtransaction costs is an integral part of economic efficiency.Second, consider the case of pricing policy where tariffs/

taxes or subsidies are present, with r ? 0. Then, Equation (14) impliesW(�, �¢, U) � W(�, �, U) � �fC[t*(�, �, U), ps*(�, �¢, U), �¢] � C [t*(�, �, U), ps*(�,�¢,U), �]g þ�i2N�j2Nrij � [tij*(�, �¢,U) � tij*(�, �,U)].It follows that any change in transaction costs from � to �¢ increasesaggregate benefit withW(�, �¢, U) � W(�, �, U) � 0) if � fC[t*(�,�,U),ps*(�,�¢,U),�¢]�C[t*(�,�,U),ps*(�,�¢,U),�]gþ�i2N�j2Nrij� [tij*(�,�¢,U)�tij*(�,�,U)]�0. We know that Z(�¢) � Z(�) corresponds to a reductionin transaction costs, with �[C(t, ps, �¢) � C(t, ps, �)] � 0. This impliesthat a sufficient condition for a reduction in transaction costs to improveaggregate welfare is that �i2N�j2Nrij�[tij*(�,�¢,U)�tij*(�,�,U)]�0. Thiscondition states that a reduction in transaction costs does not reducethe aggregate net revenue generated by tariffs/taxes (when rij > 0) andsubsidies when (rij < 0). In such situations, any reduction in transactioncosts contributes to increasing aggregate net benefit.When r ? 0, Equation (14) also implies W(�, �¢,U)�

W(�, �,U) � � fC[t*(�, �,U), ps*(�, �¢,U), �¢] � C[t*(�, �,U), ps*(�,�¢, U), �]g þ �i2N�j2Nrij � [tij*(�, �¢, U) � tij*(�, �, U)]. This statesthat W(�, �¢, U) � W(�, �, U) � 0 if �fC[t*(�,�,U),ps*(�,�¢,U),�¢] � C[t*(�,�,U),ps*(�,�¢,U),�g þ �i2N�j2Nrij � [tij*(�,�¢,U)�tij*(�,�,U)]� 0. When Z(�¢) � Z(�), we know that � [C(t, ps, �¢) �C(t, ps, �)] � 0. If follows that [W(�, �¢, U) � W(�, �, U)] can benegative only if the term f�i2N�j2Nrij � [tij*(�, �¢,U)� tij*(�, �,U)]g isnegative and sufficiently large. This could happen when a reduction intransaction costs is associated with a large decline in aggregate netrevenue generated by tariffs/taxes and subsidies, decline which in turncan contribute to increasing the social cost of pricing policy. This isanother example of a second-best scenario where lower transactioncosts could exacerbate the welfare loss associated with pricing and tradepolicy.

VI. CONCLUDING REMARKS

This paper has developed a general equilibrium analysis of the economicand welfare effects of partial market liberalization. It develops a unifiedframework supporting a refined analysis of the effects of domestic andtrade policy reform. First, it considers the market equilibrium of aneconomy distorted by domestic and trade policy (including both



quantity and price instruments) in a multiagent, multicommodity frame-work. Second, it allows for the endogenous determination of traded andnon-traded goods by examining the role of transaction costs occurringwhen goods are exchanged. Finally, our results are general and allow fordiscrete changes in policy instruments. In this context, we investigate thenature of distorted markets and the welfare implications of policyreform.

We derive new results on Pareto improving partial market liberal-ization. We know that, in a second-best world, partial market liberal-ization is not always efficiency improving. This is particularly true ofquota liberalization in the presence of price (tariff and subsidy) distor-tions. We derive simple but general results indicating conditions underwhich policy reform is (or is not) efficiency improving. For example, wefind that a relaxation of domestic production quotas can be immiseriz-ing in the presence of export subsidies, because they exacerbate thedistorting effects of pricing policy. This illustrates the presence ofsignificant interactions effects between domestic and trade policy. Itindicates that policy reform and market liberalization should involvethe joint consideration of domestic and trade policy instruments. Theseresults appear relevant in the evaluation of domestic policy reforms aswell as future WTO trade negotiations.

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APPENDIX

Proof of Proposition 1: The allocation (x*, y*, z*, t*) is feasible. GivenU* ¼ u(x*) in Definition 3, Definition 4 implies that B(x*, U*, g) � 0.Assume that B(x*, U*, g) > 0. Then, the corresponding allocationcannot be on the utility frontier, because the amount of moneyB(x*, U*, g) > 0 can always be feasibly redistributed to the consumerunit that is non-satiated in xm. Thus, B(x*, U*, g) ¼ 0, implying that theallocation is zero maximal.

Proof of Proposition 2: For a feasible allocation (x, y, z, t), zeromaximality implies that B(x, U*, g) � 0. Suppose that x* is not on theutility frontier. Then, there is a feasible x¢ 2 X such that ui(xi¢)�Ui

* forall i 2 Nc with uj(xj¢) > Uj* for some j 2 Nc. Let x ¼ (x* þ x¢)/2. Notethat the feasible set for x is convex when X, Y and Z are convex. Since x*and x¢ are both feasible, it follows that x is also feasible. Furthermore, ifx* 2 int(X), then x 2 int(X). And from the strong quasi-concavity ofui(xi) for all i 2 Nc, ui(xi¢¢) � U* for all i 2 Nc and uj(xj¢¢) > Uj*, imply-ing that x generates a utility improvement over x*. But this contradictsthat (x*, y*, z*, t*) is zero maximal. We conclude that (x*, y*, z*, t*)must be on the utility frontier.

Proof of Proposition 3: From the saddle-point theorem (Takayama,1985, p. 74), the saddle-point problem (6) always implies that (x*, y*,z*, t*) is a solution to the constrained maximization problem inEquation (4), and that the complementary slackness conditions hold(see (B2a), (B2b) and (B2c) below). Equation (8) then implies thatL(x*, y*,z*, t*,U, ps*, pc*,Q*, �)¼B(x*,U)¼�j2N�j2Nrij � tij*. Giventhe choice of U in the Lagrange equilibrium, it follows that B(x*, U*,g) ¼ 0. But Definition 5 implies that, when evaluated at the feasiblepoint (x*, y*, z*, t*, U*), bi � 0, i 2 N. Thus, bi(xi*,Ui*, g)¼ 0 for all



i 2 Nc. Because ui(xi) is strictly increasing in xim for each i 2 Nc, thisimplies from Definition 4 that Ui*¼ ui(xi*) for all i 2 Nn. We concludethat the Lagrange equilibrium is zero maximal.

Proof of Proposition 4: The aggregate B(x, U) benefit function isconcave in x under the quasi-concavity of utility functions ui(xi) andthe convexity of Xi for each i 2 Nc (Luenberger, 1992b). If the sets X, Yand Z are convex, it follows that the maximization problem in (4) is a‘nice’ concave constrained optimization problem: it has a concaveobjective function, linear constraints and a convex feasible set. Theexistence of a feasible point where all constraints in (1a) and (1b) arenon-binding satisfies Slater’s conditions. Under Slater’s conditions, theconcave constrained optimization problem in (4a) always implies thesaddle-point problem (6) (Takayama, 1985, p. 75). Then, from thesaddle-point theorem, the complementary slackness condition (8)holds. Thus, B(x*, U*, g) ¼ 0.It remains to show that we can always choose psi* � g ¼ pci* � g ¼ 1.

When U is chosen such that W(u, �) ¼ 0, the Lagrangean L in (5) canalways be multiplied by a positive constant without affecting the analy-sis. This means that the Lagrange multipliers (ps, pc, Q) are defined up toa positive constant of proportionality. The assumptions that Xi puts noupper bound on xim (from A2) and that ui(xi) is strictly increasing in ximimply that pim

c > 0. Assumption A1 also implies that pims ¼ pjm

c for all i,j 2 N (otherwise, the maximization in (4) with respect to tijm or tjimwould be unbounded, a contradiction). Let pm ¼ pim

s ¼ pimc, i 2 N.

Without loss of generality, we can always choose the normalizationrule pm ¼ 1, or pi

s � g ¼ pic � g ¼ pm ¼ 1. We conclude that a zero-

maximal equilibrium is the Lagrange equilibrium.

Proof of Proposition 5: Satisfying condition (1) in Definition 2(feasibility) is immediate. The first inequality in (6) implies that

xi* 2 argmaxx biðxi, Ui, gÞ � pic* � xi : xi 2 Xi½ � for i 2 Nc ðB1aÞ

yi* 2 argmaxy pis* � Qyi*

� �� yi : yi 2 Yi for i 2 Ns ðB1bÞ

ðz*, t*Þ 2 argmaxz, tX

i2N

Xj2N pj


� �� tij

�X

i2N pis* � zi : ð�z, tÞ 2 Z ðB1cÞ

Given pis* � g ¼ pi

c* � g ¼ 1, Luenberger (1992b, pp. 472–3) has shown thatinfxfp1c* �xi � biðxi, Ui, gÞ: xi 2 Xi ¼ infxfpic* �xi : uiðxiÞ � Ui : xi 2 Xig.Thus, (B1a) implies that pi

c* �xi* � pic* �xi for all xi 2 Xi satisfying

ui(xi) � Ui. This yields condition (2) in Definition 2 of a distortedmarket equilibrium.



Because yi* 2 argmaxy pis* � Qyi*

� �y : yi 2 Yi, y* is feasible.

Expression (B1b) implies that pis* � Qyi*

� �� yi* � pi

s* � Qyi*� �

� yifor all yi 2 Yi, which is condition (3) in Definition 2 of a distortedmarket equilibrium. Condition (4) in Definition 2 is obtained in a similarmanner from (B1c).

Finally, from the saddle-point theorem (Takayama, 1985, p. 74),the saddle-point problem (6) implies the complementary slacknessconditions

pis* � yi* � zi* �

Xj2N tij*

h i¼ 0 ðB2aÞ

pic* �

Xj2N tji* � xi*

h i¼ 0 ðB2bÞ

for i 2 N, and

Qij* � qij � tij*� �

¼ 0 i, j 2 N ðB2cÞwhich satisfy conditions (5) and (6) in Definition 2.

Proof of Proposition 6: Consider the conditions stated in the definitionof the distorted market equilibrium (Definition (2)). Retracing back thesteps presented in the proof of Proposition 5, conditions (2)–(4) implyEquations (B1a), (B1b) and (B1c). And conditions (5) and (6) implyEquations (B2a), (B2b) and (B2c). Combining these results generates thefirst inequality in (6).

Under condition (5), L(x*, y*, z*, t*, U, ps*, pc*, Q*, �) ¼ B (x*,U )�j2N�j2Nrij � tij*. And B(x*, U*, g) ¼ 0 implies that bi(xi*, U*, g) ¼ 0for all i 2 N (because feasibility implies that bi(x, U) � 0 for all i 2 N).It follows that U* ¼ ui(xi*) : i 2 Ncf g.

Note that, for any pis � 0, pi

c � 0 and Q � 0, given that (x*, y*, z*,t*) satisfies (1a), (1b), (2a) and (2b), we haveX

i2Npis � yi* � zi* �

Xi2N tij*

� �þX

i2N pic �

Xj2N tij* � xi*

� �þ

Xi2N

Xj2N Qij* � qij � tij*

� �þX

j2N Qyi* � qyi � yi*� �

� 0

This implies the second inequality in (6). We conclude that (x*, y*, z*,t*) is a Lagrange equilibrium.

Proof of Proposition 7: Letting w ¼ (x, y, z, t) and l ¼ (ps, pc, Q), theLagrangian in (5) can then be written as L(w, �, �, �) ¼ f(w, �, �)þ � � g(w, �) for w 2 W and � � 0. For any � 2 A, from the saddle-point theorem (e.g., Takayama, 1985, p. 74), the saddle-point problem(6) solves the constrained optimization problem (4a) with g(w, �) � 0and implies the complementary slackness conditions �* � g(w*, �) ¼ 0.The first inequality in the saddle point (6) then gives L[w, �*(�),



�] � V(�), for any w 2 W. Choosing w ¼ w*(�¢), we obtain

L½w*ð�¢Þ, �*ð�Þ; �� Vð�Þ ðB3ÞNote that

Vð�¢Þ ¼ f½w*ð�¢Þ, �¢�� f½w*ð�¢Þ, �¢� þ �*ð�Þ � g½w*ð�¢Þ, �¢�because �*ð�Þ � 0 and g½w*ð�¢Þ, �¢� � 0

¼ L½w*ð�¢Þ, �*ð�Þ, �¢� ðB4ÞSumming the two inequalities (B3) and (B4) gives

Vð�¢Þ � Vð�Þ � L½w*ð�¢Þ, �*ð�Þ, �¢� � L½w*ð�¢Þ, �*ð�Þ, �� ðB5Þor, in the context of the Lagrangian (5),

Vð�¢, UÞ � Vð�, UÞ � �X

i2N

Xj2N rij¢ � rij� �

� tij*ð�¢, UÞ

þX

i2N

Xj2N qij¢ � Qij

� ��Qij*ð�, UÞ

þX

i2N qyi¢ � Qyi

� ��Qyi*ð�, UÞ

Using the relationship W(�,U) ¼ V(�, U )þ �i2N�j2Nrij � tij*(�, U),this yields the second inequality in Equation (10).The first inequality isobtained by switching � and �¢ and multiplying by �1.



the economic efficiency of policy reform and … · reform and partial market liberalization under...

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