# Blackwell Publishing Ltd and the Board of Trustees of the Bulletin of Economic
Research 2006. Published by Blackwell Publishing, 9600 Garsington Road, Oxford OX4
2DQ, UK and 350 Main Street, Malden, MA 02148, USA
Bulletin of Economic Research 58:3, 2006, 0307–3378
DOI: 10.1111/j.0307-3378.2006.00241.x
THE ECONOMIC EFFICIENCY OF POLICY
REFORM AND PARTIAL MARKET
LIBERALIZATION UNDER TRANSACTION
COSTS
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 and
trade policy reform on resource allocation and welfare under transaction costs. It develops a general multiagent, multicommodity model,
where transaction costs are the costs of resources used in the exchange
process. The influence of domestic and trade policy (including both
price and quantity instruments) on distorted market equilibrium is
analysed. Alternative concepts of distorted equilibrium are presented
and investigated. They provide a basis for evaluating the effects of
multilateral partial market liberalization on resource allocation and
welfare under transaction costs. New conditions are derived under
which multilateral policy reforms generate Pareto improvements.
Keywords: distortions, market liberalization, multilateral, policy
reform, welfare
JEL 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, WI
5376, USA. Tel: þ1 608 261 1944; Fax: þ1 608 262 4376; Email: jchavas@wisc.edu
161
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1992b, 1994). It has generated a dominant view among economists that
full market liberalization is desirable. However, partial moves towards
market liberalization may not be welfare improving, because they
involve ‘second best’ situations. Indeed, current domestic and trade
policies often impose significant market distortions from taxes, tariffs
and subsidies, as well as quotas that restrict trade and production
activities. Attempts to undertake reform of international trade policy
under 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 the
last few decades, non-tariff barriers are still commonly used. This has
stimulated much research on the effects of price instruments (i.e., tariffs,
subsidies, taxes) and quantity instruments (i.e., production and trade
quotas) on resource allocation and welfare. The effects of tariff reform
have 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 the
welfare effects of tax/tariff policy changes. Studies of trade liberalization
with tariffs and quotas include Anderson and Neary (1992) in a small
open economy and Neary (1995) in a large open economy.
There is a need to extend previous research on policy reform in at
least three directions. First, previous research has investigated the effects
of 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 often
interact with each other. This suggests a need to expand previous
analyses to include the joint implications of both domestic and trade
policy. For example, in the analysis of the impact of WTO reforms, it is
important to take domestic policies into account, as most countries use
both trade and domestic instruments to regulate their markets (e.g., the
case of the agricultural sector). Second, except for Turunen-Red and
Woodland (1991, 2000), previous research has often focused on a two-country analysis and considered only a limited number of policy instruments. However, market liberalization often involves multilateral
negotiations among many nations (e.g., WTO negotiations) where each
specific tradable or non-tradable commodity may be regulated through
several policy instruments. This suggests a need to develop a general
equilibrium model of a distorted world economy consisting of an arbitrary number of agents engaged in trading an arbitrary number of
commodities, under domestic and trade policy involving both subsidies/tariffs and quotas. Third, previous work on policy reform has
typically assumed that market exchange is costless. This makes it difficult to explain the presence of non-traded goods, which is often assumed
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POLICY REFORM AND PARTIAL MARKET LIBERALIZATION
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to be exogenous (e.g., Hatta, 1977b; Fukushima, 1979). Yet, non-traded
goods can arise because trade is costly. Transaction costs in trade may
take many forms: transportation cost over space, information cost, etc.
In this context, one expects trade to take place only when its benefit is
larger than its cost. While the effects of transaction costs on market
equilibrium have been noted (e.g., Hadley and Kemp, 1966; Woodland,
1968), their interaction effects with distortionary policy and efficiency
have not been explored. This suggests the need to introduce transaction
costs in a general equilibrium model under policy distortions.
This article proposes an integrated framework to investigate the economic and welfare implications of multilateral partial reforms of both
domestic and trade policies (including price instruments as well as quantity
instruments) under transaction costs in general equilibrium. We define
transaction costs as costs that arise whenever resources are used in the
process of exchanging goods among agents. The introduction of transaction costs in the analysis exhibits several desirable characteristics. First, we
allow transaction costs to vary among agents. For example, to the extent
that they increase with the distance between trading agents, transaction
costs can be expected to be higher in international trade (when traders are
in different countries) than in domestic markets (when market participants
are from the same country). Second, our analysis provides an endogenous
treatment of what are the traded versus non-traded goods, depending on
the magnitude of exchange costs. This can help explain the existence of
‘local markets’ Third, in general equilibrium, the transaction costs are
themselves endogenous and can be affected by changes in economic policy.
For example, market liberalization may contribute to reducing the cost of
resources used in exchange, which would further stimulate (beyond the
effects of reducing tariffs/quotas) the development of markets and increase
the benefits from trade. This suggests significant interactions between
policy, transaction costs, market activities and welfare. Capturing such
effects is a major motivation for our approach. While we expect transaction
costs to have a negative effect on trade incentives, their interactions with
distortionary domestic and trade policy as they affect resource allocation
and welfare remain poorly understood. Our approach provides a new
conceptual framework to investigate these issues.
Some previous analyses of market liberalization have focused on
small changes in policy instruments (e.g., Vousden, 1990; Turunen-Red
and Woodland, 1991, 2000; Neary, 1995). Our approach adds to this
literature by considering discrete changes in policy instruments. Our
analysis relies on Luenberger’s benefit function and its use in general
equilibrium analysis (Luenberger, 1992a, 1992b, 1995). We extend
Luenberger’s general equilibrium analysis by considering price and
quantity distortions, by investigating the associated distorted market
equilibrium and by studying the implications of domestic and trade
policy for resource allocation and welfare under transaction costs.
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BULLETIN OF ECONOMIC RESEARCH
A significant problem in market liberalization is that it is often part of
a second-best strategy. In this context, the reduction or elimination of a
subset of distortions in a competitive equilibrium may not be welfare
improving. While free trade is efficient under competitive markets, in the
presence of trade barriers, a partial move towards free trade may actually reduce welfare (Bhagwati, 1958; Johnson, 1967; Woodland, 1982;
Falvey, 1988; Diewert et al., 1989; Vousden, 1990; Diewert and
Woodland, 2004). A key result from this literature is that a proportional
reduction in all tariffs is typically welfare improving. Anderson and
Neary (1992) and Neary (1995) extended this analysis to include both
tariffs and quotas in an open economy. The welfare analysis of multilateral 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 in
both domestic and trade policy. This is of particular interest when
domestic policy affects the distortionary effects of trade policy. For
example, there are situations where domestic production quotas can
help reduce the distortionary effects of export subsidies (e.g.,
Bouamra-Mechemache et al., 2002). This stresses the importance of an
integrated analysis of the effects of domestic and trade policy. Finally,
we go beyond Neary (1995) by focusing on multilateral policy reform. In
particular, we derive general conditions that imply that partial market
liberalization is welfare improving under transaction costs.
The article is organized as follows. Sections II and III develop a
general equilibrium model of an economy under transaction costs,
trade policy distortions (including both tariffs and quotas) and domestic
policy distortions (including taxes, subsidies and production quotas).
The model distinguishes between production agents and consumers. It
includes an arbitrary number of commodities and agents trading with
each other. Transaction costs are associated with resources used in the
exchange process. In this context, the influence of domestic and trade
policy (including both tariffs and quotas) on distorted market equilibrium is analysed. Section III presents the distorted market equilibrium
under domestic and trade policy. Alternative characterizations of distorted equilibrium are presented and investigated in Section IV. They
provide a basis for analysing the effects of market liberalization on
general equilibrium resource allocation and welfare under transaction
costs. This is the topic of Section V. New conditions are derived under
which partial multilateral policy reforms generate Pareto improvements.
II.
PRELIMINARIES
Consider a global economy consisting of m commodities and n economic
agents. We distinguish between two mutually exclusive groups of agents:
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POLICY REFORM AND PARTIAL MARKET LIBERALIZATION
165
consumers and production units. Let Nc be the set of consumers and Ns the set
of production units. The set of all agents is N ¼ Nc [ Ns ¼ {1, 2, . . ., n}. The
ith consumer chooses a consumption bundle xi ¼ (xi1, . . ., xim) 2 Xi Rm,
i 2 Nc. The elements of xi are positive for commodity consumed and
negative for commodity produced (e.g., labour). We assume that the
feasible set Xi is closed, convex, has a lower bound and a non-empty
interior, i 2 Nc. The ith consumption unit has a preference relation
represented by the utility function ui(xi), i 2 Nc. The utility function
ui(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 production and trading activities. For the ith production unit, i 2 Ns, the
production activities yi ¼ (yi1, . . ., yim) are chosen from the transformation set Yi Rm, consisting of all commodity bundles that can be
produced. In the simplest case, Yi consists in a single point representing
the initial endowment for the ith agent. More generally, we use the
convention that elements of the vector yi measure netputs, i.e., outputs
when positive and inputs when negative. The set Yi is assumed nonempty and closed, i 2 Ns.
2
Trade involves the vector t ¼ ftijk: i, j 2 N; k ¼ 1, …, mg 2 Rmn .
For outputs, tijk is the non-negative quantity of the kth commodity
traded from agent i to agent j. When i ? j, tijk 0 is the quantity of
the kth commodity ‘sold’ or ‘exported’ by agent i to agent j, or
equivalently the quantity ‘purchased’ or ‘imported’ by the jth
agent from the ith agent. When i ¼ j, this includes tiik, the quantity
of the kth commodity that the ith agent trades with itself. We consider
the 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 of
commodities used by the ith agent in trading activities, i 2 N. The
trading activities (z, t) are chosen from the transformation set
2
Z Rmn Rmn consisting of all feasible points involving trade t and
the associated vector z.2 Thus, (z, t) 2 Z, where the notation ‘z’ is
used to reflect that the zs are inputs in the trading process. We assume
that the set Z is closed and that (0, 0) 2 Z, i.e., the absence of trade
can take place without using any resources. Below, we will interpret
the cost of z as ‘transaction costs’ associated with exchange among
the 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
u½x þ 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.
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Assumption A1 (free tiik distribution): If (z, t) 2 Z, then
{ðz, t¢Þ: tijk ¢ ¼ tijk for i ? j, tiik ¢ ¼ tiik þ dik , k ¼ 1, . . ., m, i, j 2 N} 2 Z
for all dik.
Assumption A1 states that the ith agent can modify tiik, the quantity
of commodity k not subject to trade, without affecting the use of
resources z, for all k ¼ 1, . . ., m, i 2 N. This means that no resources
z are used when agents consume their own production. In other words,
transaction costs are relevant only in the presence of exchange between
different agents.
Because trade can exist between any two agents, each being either a
production unit or a consumer, it will be convenient to treat all agents
symmetrically. For that purpose, we let Xi ¼ {0} Rm be the consumption set of the ith production unit, i 2 Ns, and Yi ¼ {0} Rm be the
production set of the ith consumption unit, i 2 Nc. This means that the
only feasible production for a consumption unit is yi ¼ 0, i 2 Nc, and
that the only feasible consumption for a production unit is xi ¼ 0,
i 2 Ns. Note that, labour being one of the m commodities, consumers
can trade labour with production units, which allows for joint production and consumption choices under a single decision maker (e.g., the
case 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
t y i zi
i2N
ð1aÞ
j 2 N ij
and
xi X
t
j 2 N ji
i2N
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 its
production 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 either
from itself (tiik) or from others (j?i tjik). Note that summing (1a) and
(1b) over i yields
X
X
X
X
X
x
t
y
z
i
ij
i
i2N
j2N
i2N
i2N
i2N i
which implies that aggregate consumption cannot exceed aggregate production minus aggregate resources used for trading purposes. Next, we
incorporate various domestic and trade policy instruments in the model
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POLICY REFORM AND PARTIAL MARKET LIBERALIZATION
167
and investigate their effects on market equilibrium and resource
allocation.
III.
POLICY DISTORTIONS AND MARKET EQUILIBRIUM
We consider a market equilibrium where the ith agent can face two
prices for commodity k: pik s when commodity k is treated as a production activity and pik c when commodity k is treated as a consumption
activity. The corresponding price vectors are ps ¼ fpik s : k ¼ 1, …, m;
c
c
i 2 Ng 2 Rmn
þþ for ‘producer prices’, and p ¼ fpik : k ¼ 1, …, m;
mn
i 2 Ng 2 Rþþ for ‘consumer prices’. Although the case where ps ¼ pc
can be seen as an important special case, the distinction between ps and
pc will prove important in policy analysis. In particular, we will show
below how pi s and pi c 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 involve
price instruments (i.e., taxes, tariffs and subsidies) as well as quantity
instruments (i.e., production and trade quotas). Denote by rijk the unit
tariff (unit subsidy if negative) imposed on tijk for commodity k
exchanged from agent i to agent j, k ¼ 1, . . ., m, i, j 2 N. We denote
the unit tariffs/subsidies by the vectors rij ¼ frijk : k ¼ 1, …, mg
2
2 Rm and r ¼ frij : i, j 2 Ng 2 Rmn . Partition the set of agents into
mutually exclusive groups: N ¼ {D1, D2, . . .}, where Ds is the set of
domestic agents in the sth country. When i 2
= Ds and j 2 Ds, then rijk
represents an import tariff imposed on the kth commodity by the sth
country. When i 2 Ds and j 2
= Ds, then rijk is an export subsidy
imposed on the kth commodity by the sth country. As such, r measures
price instruments used in trade policy. Alternatively, if (i, j) 2 Ds with
i 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 producer
price piks and consumer price pikc . As such, r would reflect domestic tax
and pricing policy. Allowing for differences between domestic consumer
and producer prices and thus price distortions in domestic markets,
this conceptual framework generalizes the model usually found in the
literature based on the ‘traditional’ GNP general equilibrium framework. In general, taxes or tariffs (rijk > 0) tend to increase consumer
prices, 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 implications 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
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qijk 0.3 The quota qijk imposes an upper bound on the quantity traded
tijk. Letting qij ¼ (qij1, . . .., qijm), this gives
tij qij
i, j 2 N
ð2aÞ
We also consider domestic production quotas qyi restricting the production of the ith producer. The introduction of domestic production
quotas is relevant, as they can affect the distortionary effects of trade
policy (e.g., Bouamra-Mechemache et al., 2002). Again, for simplicity,
we focus our analysis on output quotas, with qyi 0 imposing an upper
bound on the quantity produced by the ith producer
yi qyi
i 2 Ns
ð2bÞ
We expect the quotas q ¼ {qij: i, j 2 N; qyi, i 2 Ns} to generate quota rents
to market participants. Denote by Qij the unit quota rents associated with the
quotas qij and by Qyi the unit quota rents associated with the production
quotas 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 interested in evaluating the effects of the policy instruments ¼ (r, q) on resource
allocation 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 that
can be traded between any two agents without using any resource z. Let
this good be the mth commodity, which we call ‘money’. Throughout the
article, we consider monetary valuation that can be expressed in terms of
units 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 satisfying
tijm þ dijm 0
(2) rijm ¼ 0, qijm ¼ þ1 for i, j 2 N and qym ¼ þ1, meaning that
neither 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 any
transaction cost. And condition (2) reflects the fact that our analysis
focuses on pricing and trade policy related to the first m 1
commodities.
Next, we consider the case where all agents are price takers. We focus our
analysis on the effects of the policy instruments (r, q) on market 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 form
tijk qijk 0, which would restrict the quantity of the kth input traded from agent i to
agent j.
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POLICY REFORM AND PARTIAL MARKET LIBERALIZATION
169
objective is to investigate the nature of the distorted market equilibrium and
the effects of (r, q) on production decisions y, consumption decisions x,
trade activities (z, t), market prices (pc, ps) and quota rents Q.
Definition 2: An allocation (x*, y*, z*, t*) along with market prices
c
m
c
c
c
p*s ¼ fpi s *: pi s * g¼1, pi s * 2 Rm
þ , i 2 Ng, p * ¼fpi *: pi * g¼1, pi *2 Rþ ,
i 2 Ng 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 pi c *xi* implies
that ui ðxi Þ ui ðxi *Þ,
(3) for each i 2 Ns and all yi 2 Yi,
ðpi s * Qyi *Þ yi * ðpsi * Qyi *Þ yi
(4) for all (z, t) 2 Z,
X
X
X
c
s
p
*
p
*
r
Q
*
t
*
p s * zi *
i
i
ij
ij
ij
i2N
j2N
i2N i
X
X
X
c
s
p
*
p
*
r
Q
*
t
p s * zi
j
i
ij
ij
ij
i2N
j2N
i2N i
(5) for each i 2 N, pi s * 0, pi c * 0, with pi s * ½yi * zi * j 2 N tij * ¼ 0 and pi c * ½j 2 N tji * 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 economic
rationality for consumption units. Condition (3) is the profit maximization behaviour for production units under production quotas. It considers
that firms behave as if they were facing prices pi s * Qyi *, showing that
quota 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 the
tariffs r and the quotas q act as trade barriers that reduce the profitability
of 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 positive
only 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)
X
X
gives
t * ¼ pi * xi * pi * t *
i2N
pi * tij * ¼ pi * yi * pi * zi * pi * j?i ij
j?i ji
or
pi * X
t *
j?i ji
pi * X
t *
j?i ji
¼ pi * yi * pi * zi * pi * xi *
i2N
This can be interpreted as a ‘balance of payment’ constraint which states that, for any
agent i 2 N, the value of net exports must equal profit, minus the cost of trade, minus
consumer expenditures.
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Condition (4) has important implications for trade activities under
policy distortions (r, q). To illustrate, consider the trade cost function
Cðt, ps Þ ¼ minz fi 2 N pi s zi : ðz, tÞ 2 Zg. In the special case where
C(t, ps) is differentiable in t and the kth commodity is an output
(tijk 0), the maximization problem implied by condition (4) yields the
familiar Kuhn–Tucker conditions with respect to tijk:
pjk c * pik s * @C=@tijk rijk Qijk * 0 for tijk * 0
and
pjk c * pik s * @C=@ijk rijk Qijk * tijk * ¼ 0
ð3aÞ
ð3bÞ
Equations (3) show how trade policy generates price distortions through
the tariffs/subsidies rijk and the quota rents Q*ijk . In the context of a
competitive market equilibrium, Equation (3a) implies that
pjk c * pik s * @C=@tijk þ rijk þ Q*ijk , i.e., that the price difference for
commodity k between agents i and j, pjk c * pik s *, cannot exceed the
marginal transaction cost, @C/@tijk, plus the price distortion, rijk þ Qijk *.
And when exchange takes place from agent i to agent j for the
kth commodity (tijk > 0), then (3a) and (3b) imply that pjk c * pik s * ¼ @C=@tijk þ rijk þ Qijk *. In this case, the price difference
pjk c * pik s * must equal the marginal transaction cost @C/@tijk plus the
price distortion rijk þ Q*ijk . This can be interpreted as the first-order
condition for profit-maximizing trade under distortionary policy. For
example, in the absence of transaction costs where @C/@tijk ¼ 0, then
pjk c * pik s * ¼ rijk þ Q*ijk , showing that rijk þ Q*ijk acts as a ‘price
wedge’ between consumer price pjk c * and producer price pik s *. Note
that, in the absence of price distortions (where rijk ¼ 0; Q*ijk ¼ 0), this
would generate the law of one price: pjk c * ¼ pik s * for all i, j 2 N. This
shows that under competitive markets the law of one price holds only in
the absence of both transaction costs and distortionary policy.
Alternatively, when @C/@tijk > 0, transaction costs in (3) create a price
wedge between pjk c * and pik s *. 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, when
transaction costs and price distortions are ‘high enough’ so that
@C=@tijk þ rijk þ Q*ijk > pjk c * pik s * for some i and j satisfying
pjk c * pik s * 0, then the incentive to trade disappears as (3b) implies
t*ijk ¼ 0. Then, the kth commodity becomes non-traded between agents i
and j. If this happened for all agents, this would imply the absence of
market for the kth commodity. This illustrates that our general approach
treats the presence and development of markets as endogenous. It shows
the adverse effects that transaction costs and policy distortions can have
on trade and market activities. Alternatively, it stresses the role of low
transaction costs and market liberalization policies in the creation and
functioning of competitive markets.
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POLICY REFORM AND PARTIAL MARKET LIBERALIZATION
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IV. THE UTILITY FRONTIER IN A DISTORTED MARKET EQUILIBRIUM
To analyse the nature and efficiency of the distorted market equilibrium
just defined, it will be useful to explore related concepts of equilibrium for
policy analysis. Below, we focus on the concepts of zero-maximum equilibrium and Lagrange equilibrium. Both are based on the ‘benefit function’,
an aggregate measure of consumer benefits. These concepts are closely
linked with the efficiency of distorted market equilibrium. Luenberger
(1992a, 1994, 1995) has investigated the relationship between these alternative equilibrium concepts under zero transaction costs and in the absence
of policy distortions. Here, we extend Luenberger’s analysis in two ways: (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 Nc g
is on the utility frontier of the economy if x* ¼ fxi *: i 2 Nc g is feasible and
if 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 distortions
that can adversely affect the efficiency of resource allocation, the utility
frontier defined above is typically not the Pareto utility frontier.
Our objective here is to assess the quantitative and qualitative
effects of partial policy reforms (represented by changes in ) on this
utility frontier. The following function will prove important in our
analysis.
Definition 4: Given the reference bundle g 2 Rþ m satisfying g ? 0, define
the ith agent’s benefit function as
bi (xi ; Ui ) ¼ max f : xi g 2 Xi ; u(xi g) Ui g
if 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
X
b ðx , Ui Þ
Bðx, UÞ ¼
i2N i i
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 would
be willing to give up to obtain xi starting from utility level Ui. When
the commodity bundle g has a unit price, the benefit function can be
# Blackwell Publishing Ltd and the Board of Trustees of the Bulletin of Economic Research 2006.
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BULLETIN OF ECONOMIC RESEARCH
interpreted as an individual willingness-to-pay measure. And B(x, U)
provides a corresponding measure of aggregate consumer benefit. Under
the assumptions that the set Xi is convex for each i 2 N and the function
ui (x) is quasi-concave, Luenberger (1992b, pp. 464–6) has shown that
the benefit function bi (xi, Ui) is concave in xi for i 2 Nc. Then, the
aggregate benefit function B (x, U) is concave in x. Next, we present
the zero-maximum concept that will prove crucial in evaluating the
utility frontier.
Definition 5: Under policy ¼ (r, q), define a maximal equilibrium as an
allocation (x, y, z, t) satisfying
n
X
X
Vð, UÞ ¼ max x,y,z,t Bðx, UÞ r tij : Eqns:ð1aÞ,ð1bÞ,
i2N
j 2 N ij
o
ð2aÞ,ð2bÞ;ðx, y z, tÞ 2 XYZ
ð4aÞ
Let
Wð, UÞ ¼ Vð, UÞ þ
X
i2N
X
r
j 2 N ij
tij *
ð4bÞ
where t* solves the optimization problem in (4a). If, in addition to being a
maximal equilibrium, U is chosen such that W(, U) ¼ 0, then the allocation 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 among
agents (which reduces the incentive to trade). But, this term is also
added in (4b) to reflect that the tariff revenues eventually benefit
the agents that capture them. Next, we establish the relationships between
zero maximality and the utility frontier (see the proofs in the Appendix).
Proposition 1: Assume that ui(xi) is strictly increasing in the mth commodity
xim for at least one consumer. If the feasible allocation (x*, y*, z*, t*) is on
the utility frontier, then it is zero maximal.
Proposition 2: Assume that the utility function ui(xi) is strongly quasiconcave5 for each i 2 Nc, and that the sets Y and Z are convex. If the
feasible allocation (x*, y*, z*, t*) is zero maximal and satisfies
x* 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
u½x þ ð1 Þx¢ > uðx¢Þ for all ,
0<1
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POLICY REFORM AND PARTIAL MARKET LIBERALIZATION
173
Propositions 1 and 2 establish conditions under which W(, U) ¼ 0
(in a zero-maximal equilibrium) is the implicit equation for the utility
frontier under policy ¼ (r, q). This has the following intuitive
interpretations. First, the set of utilities U satisfying W(, U) 0
identifies a feasible distribution of welfare among the consumers.
Indeed, having W(, U) < 0 cannot be feasible: it corresponds to
B(x*, U) < 0, i.e., to situations where u(xi *) Ui cannot hold for
all i 2 Nc. Thus, the inequality W(, U) 0 can be interpreted as the
aggregate budget constraint for all agents under distortionary policy
and transaction costs. It simply states that aggregate net benefit cannot
be negative, i.e., that all benefits obtained must be feasibly generated
within the economy. Second, as investigated earlier, finding W(,
U) > 0 is necessarily below the utility frontier. In this context, we can
interpret W(, U) as the distributable monetary surplus. This surplus, if
positive, can always be redistributed costlessly (under Assumption A2)
to some non-satiated agent and generate welfare improvements to at
least one agent without making anyone else worse off. It follows that
the set of U satisfying W(, U) ¼ 0 traces out the utility frontier under
government policy and in the presence of transaction costs. This is a
useful result for empirical analysis to the extent that the surplus function W(, U) involves monetary measurements, yet it is obtained under
general ordinal preferences. Note that the move along the utility frontier can take place in several ways. It can involve lump sum transfers
(through the tijm) across agents. Or it can involve redistribution across
agents of profit from production and trade activities, of quota rents
and 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 maximality
and distorted market equilibrium. This is done by considering
a Lagrange equilibrium, which will be used in the next section to
evaluate the efficiency implications of policy reform. For
c
mn
n2
x 2 X, y 2 Y, ( z, t) 2 Z, ps 2 Rmn
þ , p 2 Rþ and Q 2 Rþ , define
the Lagrangian
Lðx, y, z, t, U, ps , pc , Q, Þ
X
X
¼ Bðx, UÞ r tij
i2N
j 2 N ij
h
i
X
X
s
þ
p
y
z
t
i
i
i2N i
j 2 N ij
hX
i X
X
X
c
p
t
x
Q ½qij tij þ
þ
i
ji
i
i2N
j2N
i2N
j 2 N ij
X
Q ½qyi yi þ
i 2 N yi
s
ð5Þ
c
where p , p and Q are vectors of Lagrange multipliers associated with
constraints (1a), (1b), (2a) and (2b), respectively, and ¼ (r, q).
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BULLETIN OF ECONOMIC RESEARCH
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 saddle
point of the Lagrangian
Lðx, y, z, t, U, ps *, pc *, Q*, Þ
Lðx*, y*, z*, t*, U, ps *, pc *, Q*, 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 is
chosen to equal U* satisfying B(x*, U*, g) ¼ 0 and pi s * g ¼
pi c * g ¼ 1, i 2 N.
The variables (ps, pc, Q) in (5) are Lagrange multipliers associated with
constraints (1a), (1b), (2a) and (2b). When the commodity bundle g has a
unit price, the benefit function B(x, U) has a monetary interpretation, and
the Lagrange multipliers (ps, pc, Q) have the standard interpretation of
measuring the shadow price of the corresponding constraints. In a market
economy, ps and pc are then market prices reflecting resource scarcity for
supply and demand facing each agent. And Q measures the quota rents
associated with quotas q. Next, we examine the close relationships that
exist between the Lagrange equilibrium and the zero-maximum equilibrium (see the proofs in the Appendix).
Proposition 3: Assume that the utility function ui(xi) is strictly increasing
in 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 strictly
increasing in xim for each i 2 N, that the sets X, Y and Z are convex
and 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*) is
zero maximal, then it is a Lagrange equilibrium.
A distorted equilibrium and a Lagrange equilibrium are closely
related, 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), then
Proposition 4 applies under weaker conditions. In this case, the existence of a feasible
allocation 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 all
constraints 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 equivalent to the saddle-point problem (6) (see Takayama, 1985, theorem 1.D.5, pp. 98–9). In
this case, Slater’s condition is no longer needed, and Proposition 4 applies without it.
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POLICY REFORM AND PARTIAL MARKET LIBERALIZATION
175
Proposition 5: If the feasible allocation (x*, y*, z*, t*) is a Lagrange
equilibrium, 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 pim c * 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 tijm
means that the price of the mth commodity (money) is the same for all agents.
Without loss of generality, it is normalized to be equal to 1, with
pm * ¼ pi s * g ¼ pi c * 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 * pi s * rii þ Qii * 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 implies
that the prices faced by each agent satisfy pi c * pi s * ¼ rii þ Qii *, i 2 N. In
the absence of distortionary policy (where rii ¼ 0 and Qii * ¼ 0), this implies
that pi c * ¼ pi s * 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 distorted
market equilibrium and B(x*, U*, g) ¼ 0 (where U* ¼ fui ðxi *Þ: i 2 Nc g),
then it is a Lagrange equilibrium.
The proof of Proposition 6 is presented in the Appendix. Propositions
5 and 6 show that, under some regularity conditions, the concepts of
distorted market equilibrium and of Lagrange equilibrium are equivalent. This relationship will prove useful below in the investigation of
economic behaviour under policy distortions.
Finally, for completeness, note that the Lagrange equilibrium generates
a useful characterization of the distorted prices ( ps *, pc *) and quota rents
Q* 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 maximization and expenditure minimization. More specifically, when psi g ¼ 1,
the saddle-point problem (6) implies that
yi ðpi s Qyi Þ ¼ supy fðpi s Qyi Þ yi : yi 2 Yi g
ð7aÞ
where yi ðpi s Qyi Þ is the indirect profit function for the ith production
unit, i 2 Ns (see (B1b) in the Appendix). Similarly, it implies that
nX
X
c
T ðps , pc , Q, rÞ ¼ supz, t
pj pi s rij Qij tij i2N
j2N
o
X
s
p
z
:
ðz,
tÞ
2
Z
ð7bÞ
i
i
i2N
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
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BULLETIN OF ECONOMIC RESEARCH
P
defined as ( ps , pc , Q, r) ¼ i 2 Ns yi ( pi s Qyi ) þ T ( ps , pc , r, Q). Finally,
when pic g ¼ 1 and from (B1a) in the Appendix, the saddle-point problem
(6) implies that
ei ð pc , Ui Þ ¼ minx fpc xi: ui ðxi Þ Ui , xi 2 Xi g
ð8Þ
c
where ei ( p , Ui) is the expenditure function for the ith consumer, i 2 Nc
(Luenberger, 1992b). Then, the aggregate expenditure function can be
defined as E( pc , U) ¼ i 2 Nc ei ( pc , Ui ). The profit functions ( ps, pc,
Q, r), yi( pc) and T( ps, pc, Q, r) are each convex in ( ps, pc, Q, r). And
the expenditure functions E( pc, U) and ei(pc, Ui) are each concave in pc
(see Berge, 1963; Diewert, 1974). Using this notation, the second inequality in the saddle-point problem (6) implies that ( ps *, pc *, Q*) satisfy
Vð, UÞ ¼ minps , pc ,Q fð ps , pc , Q, rÞ Eðpc , UÞ
X
X
þ
Q qij
i2N
j 2 N ij
X
þ
Q qyi : ðps , pc , QÞ 0g
i 2 N yi
where ¼ (r, q). Let
Wð, UÞ ¼ Vð; UÞ þ
X
i2N
X
r
j 2 N ij
tij *
ð9aÞ
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 provides 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 market
prices and quota rents obtained in a distorted market equilibrium under
policy instruments ¼ (r, q). This gives the ‘dual approach’ to market
equilibrium analysis commonly found in the economic literature on
policy and trade distortions (e.g., Kemp, 1995; Neary, 1995; Diewert
and Woodland, 2004). It extends the general equilibrium analysis presented by Luenberger (1992a, 1994) in two ways: (i) it introduces transaction costs; (ii) it incorporates the effects of pricing policy and quota
restrictions 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 aggregate
quota rent involving both trade and production activities. It is added
in (9a) to reflect that the quota rents benefit the agents who capture them.
Similarly, the aggregate tariff revenue i 2 N j 2 N rij 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 maximality. And Propositions 1 and 2 provide important linkages to the characterization of the utility frontier. This is illustrated in Figure 1. As
derived, such relationships hold under transaction costs and distortionary domestic and trade policy. The concepts of zero maximality and
# Blackwell Publishing Ltd and the Board of Trustees of the Bulletin of Economic Research 2006.
POLICY REFORM AND PARTIAL MARKET LIBERALIZATION
Proposition 1
Utility frontier
Proposition 2
Proposition 4
Zero maximal
equilibrium
Proposition 3
Proposition 5
Lagrange
equilibrium
Proposition 6
177
Distorted
market
equilibrium
Fig. 1. Relationships among alternative concepts.
Lagrange equilibrium are very useful tools in economic analysis. As first
proposed by Allais (1943, 1981), the concept of zero maximality (and its
close linkage with the characterization of utility frontier) is intuitive and
quite powerful in welfare and efficiency analysis. And the related concept of Lagrange equilibrium (and its close linkage with distorted market equilibrium) can be quite useful and provide additional insights in
comparative statics analysis. This is illustrated next in an investigation
of 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-maximum
equilibrium and zero-minimum equilibrium are used to analyse the
economic 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 economic
efficiency, meaning that the distorted economy is expected not to satisfy
the Pareto optimality criterion. This raises two related questions: (i) how
to represent the welfare implications for the distorted economy and
(ii) how to assess the nature and extent of economic inefficiency due to
distortionary domestic and trade policy. To answer these questions, we
examine next the welfare measurements of government policy under
transaction costs.
V.1 Evaluation of a discrete change in policy
We analyse the general case of a discrete change in the policy instruments . We could proceed using any of the equilibrium concepts
discussed in Section IV. Keeping in mind the close relationships that
exist between these alternative concepts (see Figure 1), it will be convenient here to focus on the Lagrange equilibrium. For a given U, let
[x*(, U), y*(, U), z*(, U), t*(, U)] denote an allocation that satisfies
the saddle-point condition (6). And let Vð, UÞ ¼ [Bðx*, UÞ i 2 N
j 2 N rij tij *] denote the aggregate net benefit evaluated at x*(, U),
y*(, U), z*(, U) and t*(, U). Then, the following result applies (see
the proof in the Appendix).
Proposition 7: For a given U, assume that a saddle point in (6) holds with
saddle value V(, U) for all ¼ (r, q) 2 A. Then, for any , ¢ 2 A,
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BULLETIN OF ECONOMIC RESEARCH
X
X
tij *ð¢, UÞ tij *ð, UÞ
X
X
Q *ð¢, UÞ qij ¢ Qij
þ
i2N
j 2 N ij
X
Q *ð¢, UÞ qyi ¢ Qyi
þ
i2N yi
i2N
r ¢
j 2 N ij
Wð¢, UÞ Wð, UÞ
X
X
r tij *ð¢, UÞ tij *ð, UÞ
i2N
j 2 N ij
X
X
Q *ð, UÞ qij ¢ Qij
þ
i2N
j 2 N ij
X
Q
*ð,
UÞ
q
¢
Q
þ
yi
yi
yi
i2N
ð10Þ
where Wð, UÞ ¼ Vð, UÞ þ i 2 N j 2 N rij tij *ð, UÞ.
Proposition 7 provides a lower bound and an upper bound on the change in
the aggregate net benefit [W(¢ W(, U)] evaluated at U. It is very general
in the sense that it applies without restrictions on the set A. It does not require
the decision rules x*(, U), y*(, U), z*(, U) and t*(, U) to be differentiable
functions, nor single value mappings. And it applies to arbitrary discrete
changes in the policy instruments ¼ (r, q). Finally, it considers the joint
effects of price and quantity policy instruments used in both domestic and
trade policy. This provides significant generalizations on previous analyses of
policy reform (e.g., Falvey, 1988; Diewert et al., Vousden, 1990; Turunen-Red
and Woodland, 1991, 2000; Anderson and Neary, 1992, 1996; 1989; Neary,
1995). Also, Proposition 7 includes some intuitive and well-known results as
special cases. To see that, consider the following corollary.
Corollary 1: For any , ¢ 2 A,
X
X
r ¢ rij tij *ð¢, UÞ tij *ð, UÞ
i2N
j 2 N ij
X
X
Qij *ð¢, UÞ Qij *ð, UÞ qij ¢ Qij
þ
i2N
j2N
X
0
Q
*ð¢,
UÞ
Q
*ð,
UÞ
q
¢
Q
þ
yi
yi
yi
yi
i2N
ð11Þ
Again, Corollary 1 applies in general for any discrete change in . It
has two useful implications. First, consider the case where tariffs are
changed but where quotas are unchanged (q ¼ q¢). Then, (11) becomes
X
X
t
r
¢
r
*ð¢,
UÞ
t
*ð,
UÞ
0
ij
ij
ij
ij
i2N
j2N
This means that t*(r, ) is non-increasing in r: ceteris paribus, an increase
in tariffs r tends to decrease the corresponding quantities traded. Note
that this intuitive result is obtained without differentiability assumptions.
In the special case where the change in tariffs (r¢ r) is ‘small’ and the
# Blackwell Publishing Ltd and the Board of Trustees of the Bulletin of Economic Research 2006.
POLICY REFORM AND PARTIAL MARKET LIBERALIZATION
179
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 space
of [@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) yields
X
X
Qij *ð¢, UÞ Qij *ð, UÞ ½qij ¢ Qij i2N
j2N
X
þ
Qyi *ð¢, UÞ Qyi *ð, UÞ qyi ¢ Qyi 0
i2N
This means that the quota rent Q*(, ) is non-increasing in q: ceteris
paribus, an increase in quotas q tends to decrease the corresponding quota
rents. Again this intuitive result holds without differentiability assumptions.
In the special case where the change in quotas (q¢ q) is ‘small’ and the
function 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 attractive
choices for U are possible. First, 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 gain (or loss if negative) associated with a move from to ¢.
In other words, [W(¢, U) W(, U)] ¼ W(¢, U) is a simple measure
of aggregate efficiency gains (‘compensating variation’) generated by a
policy change from to ¢. And Proposition 7 provides bounds on these
efficiency gains under transaction costs. With this particular choice of U,
note that x*(, U), y*(, U), z*(, U) and t*(, U) correspond to an
allocation 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 if
negative) associated with replacing ¢ in favour of . It follows that
[W(¢, U) W(, U)] ¼ W(, U) is a simple aggregate efficiency measure (‘equivalent variations’) generated by giving up the exchange environment ¢. With this choice of U, x*(¢, U), y*(¢, U), z*(¢, U) and t*(¢, U)
are on the utility frontier under situation ¢. With either choice of U, the
term [W(¢, U) W(, U)] can thus be used to evaluate how the utility
frontier shifts under a policy change from to ¢. As such, Proposition 7
provides 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
# Blackwell Publishing Ltd and the Board of Trustees of the Bulletin of Economic Research 2006.
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X
BULLETIN OF ECONOMIC RESEARCH
X
X
X
r
¢
t
*ð¢,
UÞ
t
*ð,
UÞ
þ
Q *ð¢, UÞ
ij
ij
ij
i2N
j2N
i2N
j 2 N ij
X
ð12aÞ
qij ¢ qij þ
Q *ð¢, UÞ qyi ¢ qyi ð>Þ 0
i 2 N yi
and a necessary condition for [W(¢, U) W(, U)] (>) 0 is
X
X
X
X
r
¢
t
*ð¢,
UÞ
t
*ð,
UÞ
þ
Q *ð, UÞ
ij
ij
ij
i2N
j2N
i2N
j 2 N ij
X
ð12bÞ
qij ¢ qij þ
Q *ð, UÞ qyi ¢ qyi ð>Þ 0
i 2 N yi
where the weak (strict) inequalities correspond to a weak (strict) Pareto
welfare improvement.
Proposition 8 states our main results. They are simple and very general.
Again, they apply under both price instruments r and quantity instruments
q; they consider jointly domestic and trade policy; they allow for discrete
change in the policy instruments ¼ (r, q); they allow for transaction costs
that reduce the incentive to trade; and they hold without differentiability
assumptions. As such, they are a significant generalization of previous
work (e.g., Turunen-Red and Woodland, 1991, 2000; Anderson and
Neary, 1992; Neary, 1995). Interpreting [W(¢, U ) W(, U )] as
measuring the shift in the utility frontier, Proposition 8 establishes
that (12a) is a sufficient condition for a policy change from to ¢ to be
Pareto improving. Expression (12a) states that the term
i 2 N j 2 N rij [tij *(¢, U ) tij *(, U )], reflecting the change in the aggregate value of tariff revenues evaluated at r¢, plus the term
i 2N j 2N Qij *(¢, U) [qij ¢ qij ] þ i2 N yi*(¢, U) [qyi ¢ qyi ],
reflecting
the aggregate change in quota rents, is non-negative. Given Q* 0, a
sufficient condition for the change in the term involving quota rents to be
non-negative is that q¢ q, i.e., that trade and production quota restrictions be relaxed. Here, we want to stress that this result applies under
reform involving both price and quantity policy instruments for domestic
as well as trade policy. However, the effects of domestic and trade policy
reform on the term involving taxes/tariffs revenue are more complex.
Indeed, reducing tariffs/subsidies (where 0 rijk ¢ rijk if rijk > 0 and
rijk rijk ¢ 0 if rijk < 0) and/or relaxing quotas (q¢ q) is in general not
sufficient to imply that i 2N j 2N rij [tij *(¢,U) tij *(,U)] 0.
However, (12a) implies that a sufficient condition for market liberalization 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 commodities
that are subsidized (rijk < 0). This simple result is quite powerful in the
sense that it is intuitive and applies under very general conditions.
It is well known that there are situations where partial market liberalization is immiserizing (e.g., Vousden, 1990; Anderson and Neary, 1992;
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POLICY REFORM AND PARTIAL MARKET LIBERALIZATION
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Neary, 1995). Proposition 8 provides additional information on
this issue. It shows that expression (12b) is a necessary condition for a
policy change from to ¢ to be Pareto improving. Alternatively, it
means that, whenever (12b) is not satisfied, then a partial move
toward free markets cannot be efficiency improving. Expression (12b)
states that the term i 2 N j 2 N rij [tij *(¢, U) tij *(, U )], reflecting the
change in the aggregate value of tariff revenues evaluated at r, plus
the term i 2 N j 2 N Qij *(, U ) [qij ¢ qij ] þ i 2 N Qyi *(, 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 involving
quota rents to be non-negative is that q¢ q, i.e., that trade and production quota restrictions be relaxed, a result that applies under both
tariffs and quotas. As before, the effects of domestic and trade policy
reform on the term involving tariff revenue are more complex. Indeed,
reducing tariffs/subsidies (with 0 rijk ¢ rijk if rijk > 0 and
rijk rijk ¢ 0 if rijk < 0) and/or relaxing quotas (q¢ q) is in general
not sufficient to imply that i 2 N j 2 N rij [tij *(¢, U ) tij *(, U )] 0.
When the left-hand side in (12b) becomes negative, then partial market
liberalization necessarily reduces efficiency. This is the situation where
policy reform is immiserizing (e.g., Anderson and Neary, 1992; Neary,
1995). This is an illustration of the theory of the second best applied to
policy analysis. More specifically, Proposition 8 shows the conditions
under which partial market liberalization reduces efficiency: with q¢ q,
for policy reform to be immiserizing, (12b) must not hold, implying that
i 2 N j 2 N rij [tij *(¢, U ) tij *(, U )] must be negative and large. This
simple result appears new and quite useful. It warns us against domestic
and trade policy reform (especially quota reform) that exacerbates the
distorting effects of pricing policy by stimulating exports that are
subsidized 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 case
where 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
X
X
X
þ
Q *ð, UÞ qij ¢ qij þ
Q *ð, UÞ
i2N
j 2 N ij
i 2 N yi
ð13Þ
qyi ¢ qyi ð>Þ 0
where the weak (strict) inequality corresponds to weak (strict) Pareto welfare
improvement associated with a change from to ¢.
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Expression (13) gives some well-known ‘local results’ about welfare
effects 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 the
matrix [@t*(, U)/@r], measuring the effects of tariffs on trade. We have
seen above that [@t*(, U)/@r] is a symmetric, negative semi-definite
matrix. In the case of a proportional decrease in tariffs/subsidies where
r¢ ¼ kr, 0 k < 1, it follows that the first term in (13) is always
non-negative: r [@t*(, U)/@r] [r¢ r] 0.7 Then, there are two simple
scenarios where equation (13) is always satisfied. The first scenario
concerns a proportional tariff/subsidy reduction in the absence of
quotas (where q ¼ q¢ ¼ 1, Q* ¼ 0). The first term in (13) is then
non-negative, whereas the absence of quotas implies that the second
and third terms in (13) vanish. This generates the well-known result
that, in the absence of quotas, a proportional tariff reduction is always
(at least weakly) welfare improving. The second scenario concerns quota
relaxation in the absence of tariffs (where r ¼ r¢ ¼ 0). The absence of
tariffs means that the first and second terms in (13) vanish. And any
quota relaxation (q¢ q) always implies that the third term in (13) is
non-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 additional
assumptions (such as the ‘rank condition’ used in Turunen-Red and
Woodland, 1991, 2000).
What happens to these local results in the more realistic situation
where both tariffs and quotas are present? Two important findings
follow from (13). First, it remains true that any proportional tariff
reduction is (at least weakly) welfare improving in the presence of
trade and production quotas, provided that these quotas remain constant (with q ¼ q¢ < 1). To see that, it suffices to note that the first
term in (13) is non-negative under proportional tariff reduction, whereas
the second and third terms vanish when q ¼ q¢. This extends a wellknown result (e.g., Vousden, 1990, p. 217) to situations covering both
domestic and trade policy. Second, in a second-best world, it is well
known that any quota relaxation is not always welfare improving (e.g.,
Falvey, 1988). This finding is obtained from Corollary 2: in the presence
of tariffs or subsidies (r ? 0), the second term in (13), r [@t*(, U)/
@q] [q¢ q] (which involves cross-commodity effects of quotas
on 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 inequality
in (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 term
in (13) would become strictly positive: r [@t* (, U)/@r] (r¢ r) > 0.
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POLICY REFORM AND PARTIAL MARKET LIBERALIZATION
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tariffs/subsidies remain unchanged (with r ¼ r¢). Thus, in a second-best
world of partial market liberalization, relaxing production and/or trade
quotas in the presence of tariffs does not necessarily generate a Pareto
improvement.
In the presence of tariffs, subsidies and quotas, Corollary 2 shows
conditions under which relaxing quotas are efficiency enhancing.8 It also
shows that these conditions are not always satisfied under partial market
liberalization. In particular, it gives new results that apply to the joint
effects of domestic and trade policy reform. Indeed, expression (13) is not
always positive when r ? 0, as it depends on the effects of quotas on
trade: r [@t*(, U)/@q] [q¢ q]. Because this includes the effects of
production quotas on trade, it suggests that domestic policy has to be
taken 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 on
subsidized trade (with @tijk *(, U )=@q > 0). This would identify secondbest conditions under which partial market liberalization is immiserizing.
This is illustrated by Bouamra-Mechemache et al. (2002) in an analysis of partial domestic and trade policy reform in the European Union
(EU) dairy sector. The EU dairy sector is of interest because, even after
the Uruguay round of GATT negotiations and recent EU policy
reforms, it is still subject to significant policy distortions: domestic
milk production quotas, domestic subsidies, as well as trade barriers
(including import quotas and export subsidies). Bouamra-Mechemache
et al. show that, in the presence of export subsidies, removing milk
production quotas (with or without decreasing domestic subsidies) in
the EU generates welfare losses for both the EU and the world. Indeed,
relaxing production quotas increases EU milk production, which in turn
stimulates the EU exports of subsidized dairy products. Thus, relaxing
EU milk production quotas in the presence of export subsidies (r < 0)
implies that the terms fi 2 N j 2 N rij [tij *(¢, U ) tij *(, U )]g in (12b)
and fr [@t*(, U )=@q (q¢ q)]g in (13) are negative. As long as export
subsidies are close to their current level, Bouamra-Mechemache et al.
find that these quota effects exacerbate the distortionary effects of
export subsidies and imply a decline in EU and world efficiency. As
suggested by Proposition 8 and Corollary 2, they also find that the
removal of production quotas would become efficiency enhancing for
the 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 is
efficiency enhancing, then a strict welfare improvement can still be attained even without
lump sum compensation under multilateral policy reform.
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domestic and trade policy. For example, in the presence of export tariffs,
our analysis points to the importance of the cross-commodity effects of
changing domestic production quotas on trade. Such effects can lead to
decreased efficiency associated with partial market liberalization. This
illustrates the power of results stated in Proposition 8. They provide the
necessary and the sufficient conditions for domestic and trade policy
reform to be welfare improving in a form that is simple and general,
allowing for discrete changes in both quantity and price policy instruments under general equilibrium and transaction costs.
V.4 Role of transaction costs
To examine the role of transaction costs, consider the case where the
feasible 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 progress
related to the trade technology. Then, transaction cost can be written as
C(t, ps , ) ¼ minz ½i 2 N pi s zi: ( z, t) 2 Z(). Given Z(¢) Z(), it
follows that C(t, ps, ¢) C(t, ps, ). This makes it clear that a change
from to ¢ corresponds to a decrease in transaction costs. Next, we
investigate the implications of this decline in transaction costs. The proof is
similar to the one presented in Proposition 7 and is omitted.
Proposition 9: For a given U, assume that a saddle point in (6) holds with
saddle value V(, , U) with ¼ (r, q). Then, for any change from to ¢,
fC[t*ð, , UÞ, ps *ð, ¢, UÞ, ¢] C[t*ð, , UÞ, ps *ð, ¢, UÞ, ]g
X
X
þ
r [tij *ð, ¢, UÞ tij *ð, , UÞ]
i2N
j 2 N ij
Wð, ¢, UÞ Wð, , UÞ
fC[t*ð, ¢, UÞ, ps *ð, , UÞ, ¢]
C[t*ð, ¢,UÞ, ps *ð, , UÞ, ]g
X
X
r [tij *ð, ¢, UÞ tij *ð, , UÞ]
þ
i2N
j 2 N ij
ð14Þ
where W(, , U ) ¼ V(, , U ) þ i 2 N j 2 N rij tij *(, , U ) and V(,
, U) is given in (4a).
Proposition 9 provides a lower bound and an upper bound in the
change in aggregate net benefit [W(, ¢, U) W(, , U)] associated
with a change in transaction costs from to ¢. Each bound involves
two terms: the negative of the change in transaction cost, [C(t, ps, ¢) C(t, ps, )], and the change in tax/tariff revenue, i 2 N j 2 N rij [tij *(, ¢, U ) tij *(, , U )].
First, consider the case where there is no tax or tariff: r ¼ 0. Then
Equation (14) implies W(, ¢, U ) W(, , U ) fC [t*ð, , U ),
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POLICY REFORM AND PARTIAL MARKET LIBERALIZATION
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ps *(, ¢, U ), ¢) C [t*(, , U ); ps *(, ¢, U ), ]g. As discussed above,
when Z(¢) Z(), we have [C(t, ps, ¢) C(t, ps, )] 0, implying
that W(, ¢, U ) W(, , U ) 0. This gives the intuitive result that
any reduction in transaction costs contributes to increasing aggregate
net benefit. This illustrates that, in the absence of tax or tariff, reducing
transaction 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) implies
W(, ¢, U) W(, , U) fC[t*(, , U), ps *(, ¢, U), ¢] C [t*
(, , U), ps *(, ¢, U), ]g þ i 2 N j 2 N rij [tij *(, ¢, U) tij *(, , U)].
It follows that any change in transaction costs from to ¢ increases
aggregate benefit with W(, ¢, U) W(, , U) 0) if fC[t*(,,U),
ps *(,¢,U),¢]C[t*(,,U),ps *(,¢,U),]gþi2N j2N rij [tij *(,¢,U)
tij *(,,U)]0. We know that Z(¢) Z() corresponds to a reduction
in transaction costs, with [C(t, ps, ¢) C(t, ps, )] 0. This implies
that a sufficient condition for a reduction in transaction costs to improve
aggregate welfare is that i2N j2N rij [tij *(,¢,U)tij *(,,U)]0. This
condition states that a reduction in transaction costs does not reduce
the aggregate net revenue generated by tariffs/taxes (when rij > 0) and
subsidies when (rij < 0). In such situations, any reduction in transaction
costs 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 i 2 N j 2 N rij [tij *(, ¢, U) tij *(, , U)]. This states
that W(, ¢, U) W(, , U) 0 if fC[t*(, ,U), ps *
(,¢,U), ¢] C[t*(, , U),ps *(,¢,U), g þ i2N j2N rij [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 be
negative only if the term fi 2 N j2 N rij [tij *(, ¢, U) tij *(, , U)]g is
negative and sufficiently large. This could happen when a reduction in
transaction costs is associated with a large decline in aggregate net
revenue generated by tariffs/taxes and subsidies, decline which in turn
can contribute to increasing the social cost of pricing policy. This is
another example of a second-best scenario where lower transaction
costs could exacerbate the welfare loss associated with pricing and trade
policy.
VI.
CONCLUDING REMARKS
This paper has developed a general equilibrium analysis of the economic
and welfare effects of partial market liberalization. It develops a unified
framework supporting a refined analysis of the effects of domestic and
trade policy reform. First, it considers the market equilibrium of an
economy distorted by domestic and trade policy (including both
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BULLETIN OF ECONOMIC RESEARCH
quantity and price instruments) in a multiagent, multicommodity framework. Second, it allows for the endogenous determination of traded and
non-traded goods by examining the role of transaction costs occurring
when goods are exchanged. Finally, our results are general and allow for
discrete changes in policy instruments. In this context, we investigate the
nature of distorted markets and the welfare implications of policy
reform.
We derive new results on Pareto improving partial market liberalization. We know that, in a second-best world, partial market liberalization is not always efficiency improving. This is particularly true of
quota liberalization in the presence of price (tariff and subsidy) distortions. We derive simple but general results indicating conditions under
which policy reform is (or is not) efficiency improving. For example, we
find that a relaxation of domestic production quotas can be immiserizing in the presence of export subsidies, because they exacerbate the
distorting effects of pricing policy. This illustrates the presence of
significant interactions effects between domestic and trade policy. It
indicates that policy reform and market liberalization should involve
the joint consideration of domestic and trade policy instruments. These
results appear relevant in the evaluation of domestic policy reforms as
well as future WTO trade negotiations.
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APPENDIX
Proof of Proposition 1: The allocation (x*, y*, z*, t*) is feasible. Given
U* 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 allocation
cannot be on the utility frontier, because the amount of money
B(x*, U*, g) > 0 can always be feasibly redistributed to the consumer
unit that is non-satiated in xm. Thus, B(x*, U*, g) ¼ 0, implying that the
allocation is zero maximal.
Proof of Proposition 2: For a feasible allocation (x, y, z, t), zero
maximality implies that B(x, U*, g) 0. Suppose that x* is not on the
utility frontier. Then, there is a feasible x¢ 2 X such that ui (xi ¢) Ui * for
all i 2 Nc with uj (xj ¢) > Uj * for some j 2 Nc. Let x ¼ (x* þ x¢)/2. Note
that 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, if
x* 2 int(X), then x 2 int(X). And from the strong quasi-concavity of
ui(xi) for all i 2 Nc , ui (xi ¢¢) U* for all i 2 Nc and uj (xj ¢¢) > Uj *, implying that x generates a utility improvement over x*. But this contradicts
that (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 in
Equation (4), and that the complementary slackness conditions hold
(see (B2a), (B2b) and (B2c) below). Equation (8) then implies that
L(x*, y*, z*, t*, U, ps *, pc *, Q*, ) ¼ B(x*,U) ¼ j 2N j 2 N rij tij *. Given
the 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 feasible
point (x*, y*, z*, t*, U*), bi 0, i 2 N. Thus, bi (xi *, Ui *, g) ¼ 0 for all
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POLICY REFORM AND PARTIAL MARKET LIBERALIZATION
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i 2 Nc. Because ui(xi) is strictly increasing in xim for each i 2 Nc, this
implies from Definition 4 that Ui * ¼ ui (xi *) for all i 2 Nn. We conclude
that the Lagrange equilibrium is zero maximal.
Proof of Proposition 4: The aggregate B(x, U) benefit function is
concave in x under the quasi-concavity of utility functions ui(xi) and
the convexity of Xi for each i 2 Nc (Luenberger, 1992b). If the sets X, Y
and Z are convex, it follows that the maximization problem in (4) is a
‘nice’ concave constrained optimization problem: it has a concave
objective function, linear constraints and a convex feasible set. The
existence of a feasible point where all constraints in (1a) and (1b) are
non-binding satisfies Slater’s conditions. Under Slater’s conditions, the
concave constrained optimization problem in (4a) always implies the
saddle-point problem (6) (Takayama, 1985, p. 75). Then, from the
saddle-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) can
always be multiplied by a positive constant without affecting the analysis. This means that the Lagrange multipliers (ps, pc, Q) are defined up to
a positive constant of proportionality. The assumptions that Xi puts no
upper bound on xim (from A2) and that ui(xi) is strictly increasing in xim
imply that pim c > 0. Assumption A1 also implies that pim s ¼ pjm c for all i,
j 2 N (otherwise, the maximization in (4) with respect to tijm or tjim
would be unbounded, a contradiction). Let pm ¼ pim s ¼ pim c , i 2 N.
Without loss of generality, we can always choose the normalization
rule pm ¼ 1, or pi s g ¼ pi c g ¼ pm ¼ 1. We conclude that a zeromaximal 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 ½bi ðxi , Ui , gÞ pi c * xi : xi 2 Xi for i 2 Nc
ðB1aÞ
ðB1bÞ
yi * 2 argmaxy pi s * Qyi * yi : yi 2 Yi for i 2 Ns
X X
c
s
p
*
p
*
r
Q
*
tij
ðz*, t*Þ 2 argmaxz, t
j
i
ij
ij
i2N
j2N
X
p s * zi : ðz, tÞ 2 Z
ðB1cÞ
i2N i
Given pi s * g ¼ pi c * g ¼ 1, Luenberger (1992b, pp. 472–3) has shown that
infx fp1 c * xi bi ðxi , Ui , gÞ: xi 2 Xi ¼ infx fpi c * xi : ui ðxi Þ Ui : xi 2 Xi g.
Thus, (B1a) implies that pi c * xi * pi c * xi for all xi 2 Xi satisfying
ui(xi) Ui. This yields condition (2) in Definition 2 of a distorted
market equilibrium.
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Because yi * 2 argmaxy pi s * Qyi * y : yi 2 Yi , y*
is feasible.
Expression (B1b) implies that pi s * Qyi * yi * pi s * Qyi * yi
for all yi 2 Yi, which is condition (3) in Definition 2 of a distorted
market equilibrium. Condition (4) in Definition 2 is obtained in a similar
manner from (B1c).
Finally, from the saddle-point theorem (Takayama, 1985, p. 74),
the saddle-point problem (6) implies the complementary slackness
conditions
h
i
X
pi s * yi * zi * t
*
¼0
ðB2aÞ
ij
j2N
pi c * for i 2 N, and
hX
i
t
*
x
*
¼0
i
j 2 N ji
Qij * qij tij * ¼ 0
i, j 2 N
ðB2bÞ
ðB2cÞ
which satisfy conditions (5) and (6) in Definition 2.
Proof of Proposition 6: Consider the conditions stated in the definition
of the distorted market equilibrium (Definition (2)). Retracing back the
steps presented in the proof of Proposition 5, conditions (2)–(4) imply
Equations (B1a), (B1b) and (B1c). And conditions (5) and (6) imply
Equations (B2a), (B2b) and (B2c). Combining these results generates the
first inequality in (6).
Under condition (5), L(x*, y*, z*, t*, U, ps *, pc *, Q*, ) ¼ B (x*,U )
j 2 N j 2 N rij tij *. And B(x*, U*, g) ¼ 0 implies that bi (xi *, U*, g) ¼ 0
for all i 2 N (because feasibility implies that bi(x, U) 0 for all i 2 N).
It follows that U* ¼ fui (xi *) : i 2 Nc g.
Note that, for any pi s 0, pi c 0 and Q 0, given that (x*, y*, z*,
t*) satisfies (1a), (1b), (2a) and (2b), we have
X
X
X
X
s
c
p
y
*
z
*
t
*
þ
p
t
*
x
*
þ
i
i
i
ij
i
ij
i
i2N
i2N
i2N
j2N
X
X
X
Q * qij tij * þ
Q * qyi yi * 0
i2N
j 2 N ij
j 2 N yi
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), the
Lagrangian 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 saddlepoint theorem (e.g., Takayama, 1985, p. 74), the saddle-point problem
(6) solves the constrained optimization problem (4a) with g(w, ) 0
and implies the complementary slackness conditions * g(w*, ) ¼ 0.
The first inequality in the saddle point (6) then gives L[w, *(),
# Blackwell Publishing Ltd and the Board of Trustees of the Bulletin of Economic Research 2006.
POLICY REFORM AND PARTIAL MARKET LIBERALIZATION
191
] V(), for any w 2 W. Choosing w ¼ w*(¢), we obtain
L½w*ð¢Þ, *ðÞ; VðÞ
ðB3Þ
Note that
Vð¢Þ ¼ f½w*ð¢Þ, ¢
f½w*ð¢Þ, ¢ þ *ðÞ g½w*ð¢Þ, ¢
because *ðÞ 0 and g½w*ð¢Þ, ¢ 0
¼ L½w*ð¢Þ, *ðÞ, ¢
ðB4Þ
Summing the two inequalities (B3) and (B4) gives
Vð¢Þ VðÞ L½w*ð¢Þ, *ðÞ, ¢ L½w*ð¢Þ, *ðÞ, ðB5Þ
or, in the context of the Lagrangian (5),
X
X
r ¢ rij tij *ð¢, UÞ
Vð¢, UÞ Vð, UÞ i2N
j 2 N ij
X
X
þ
qij ¢ Qij Qij *ð, UÞ
i2N
j2N
X
þ
qyi ¢ Qyi Qyi *ð, UÞ
i2N
Using the relationship W(,U) ¼ V(, U ) þ i 2 N j 2 N rij tij *(, U),
this yields the second inequality in Equation (10).The first inequality is
obtained by switching and ¢ and multiplying by 1.
# Blackwell Publishing Ltd and the Board of Trustees of the Bulletin of Economic Research 2006.