Theoretical Economics Letters, 2012, 2, 239-251
http://dx.doi.org/10.4236/tel.2012.23044 Published Online August 2012 (http://www.SciRP.org/journal/tel)
LeChatelier Principle and the Effects of Trade Policy
under Induced Innovation
Jean-Paul Chavas
Department of Agricultural and Applied Economics, University of Wisconsin, Madison, USA
Email: jchavas@wisc.edu
Received May 5, 2012; revised June 7, 2012; accepted July 6, 2012
ABSTRACT
This paper explores the effects of trade policy under induced innovation in general equilibrium. The analysis considers
the effects of discrete changes in tariffs and import quotas, allowing for heterogeneous technologies among firms. The
interactions between induced innovation and the effects of trade policy give a set of “LeChatelier effects” comparing
short run versus long run market equilibrium. We investigate how induced innovation can reduce the adverse effects of
tariffs on trade, and influence the effects of quotas on corresponding quota rents. The analysis presents new LeChatelier
results that apply globally, i.e. under any discrete change in trade policy.
Keywords: Trade Policy; Tariffs; Quotas; Induced Innovation; LeChatelier
1. Introduction
Globalization has stimulated much research on the ef-
fects of trade policy (including tariffs and quotas) on
resource allocation and welfare. The relationships be-
tween trade and technology have also been the subject of
much interest (e.g., [1]). Yet, technology can evolve in
response to changes in market conditions. Hicks [2] pro-
posed the idea of induced innovation, where changes in
relative prices stimulate the adoption of technologies that
increase (decrease) the use of inputs that are becoming
cheaper (more expensive). And applied to the output side,
induced innovation means that price changes induce the
adoption of technologies that increase (decrease) the
production of commodities exhibiting higher (lower)
prices. This has stimulated much research on how tech-
nology can adapt to changing resource scarcity (e.g.,
[3-5]). When market prices influence technology, trade
policy would also affect technology choices. Indeed, im-
port tariffs and quotas on specific commodities increase
their prices in domestic markets. And they influence the
prices of all goods through market equilibrium effects.
This suggests that induced innovation would stimulate
the adoption of technologies in response to the changes
in all prices affected by trade policy.
The interactions between technology choice and trade
policy (and its effects on market prices) is relevant when
the process of technology adoption is slow. This can
happen when new technology is embodied in physical or
human capital, as firms that just invested in capital do not
have incentives to adopt an improved technology until
their capital depreciates. Or this can happen when the
adoption of a new technology involves learning cost.
Firms facing lower learning cost are likely to be “early
adopters” of a new technology, while other firms would
be classified as “late adopters”. Technology adoption
being slow, this motivates a need to distinguish between
short run and long run situations. This distinction is at the
heart of “LeChatelier effects” investigated in this paper.
The LeChatelier principle reflects the basic intuition
that restricting choices can lower the ability to make
economic adjustments. It was first introduced in eco-
nomics by Samuelson ([6,7]), who proved that such re-
sults hold “locally”, i.e. for small changes in the neigh-
borhood of a point. Local LeChatelier effects have been
examined in the context of trade by Neary [8] and
Kreickemeier [9]. It is well known that local LeChatelier
results do not necessarily hold globally, i.e. when facing
discrete changes in economic conditions ([7,10,11]). It is
also known that local LeChatelier results can hold glob-
ally under some restrictive assumptions. This raises the
following questions. Is it possible to obtain general im-
plications of the LeChatelier principle without imposing
restrictive assumptions? And what are these implications
in the context of trade policy? Our analysis answers these
two questions in a positive way.
This paper investigates LeChatelier effects under in-
duced innovation in the context of trade policy changes
in general equilibrium. Our LeChatelier results apply
globally to any discrete change in trade policy (including
C
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240
both tariffs and quotas). This is important for two reasons.
First, actual policy changes typically take the form of
large changes in policy instruments. Second, our “global
LeChatelier results” are obtained without imposing re-
strictive assumptions (e.g., without assuming super-
modularity). While we show how “local results” hold as
a special case, obtaining global LeChatelier results is a
key contribution of our analysis.
Capturing the economy-wide effects of trade policy
under induced innovation requires a general equilibrium
model. Following Dixit and Norman [12] and Luenber-
ger [13], we rely on a dual general equilibrium model of
trade. The presence of early adopters and late adopters
requires considering that technology can vary among
firms (e.g., exporting firms versus domestic firms). Our
analysis allows for firm entry/exit and its general equi-
librium effects. The model also considers an arbitrary
number of commodities. This allows for differentiated
products. In this context, the paper studies the effects of
discrete changes in both tariffs and quotas. This is rele-
vant as trade policy reform often means partial market
liberalization that involves the joint effects of tariff and
quotas. Considering discrete changes in trade policy ex-
pands on previous analyses of market liberalization that
focused on small changes in policy instruments (e.g.,
[14-16]).
Our analysis studies market equilibrium under price
and quantity distortions. This provides a basis for evalu-
ating the efficiency gains/losses from any discrete change
in trade policy. The analysis presents conditions under
which a discrete policy change improves aggregate effi-
ciency. It examines the interactions between induced
innovation and the effects of trade policy. These interac-
tions generate a set of “LeChatelier effects” comparing
short run versus long run market equilibrium.
Three important results are obtained. First, we show
that induced innovation tends to reduce the aggregate
welfare loss generated by distortionary trade policy. It
means that previous research that ignored induced inno-
vation has overstated the adverse effects of trade policies.
This result holds under very general conditions. To the
extent that trade policy is motivated by rent seeking be-
havior (which redistributes welfare toward the “rent
seekers”), this also means that induced innovation can
tamper the efficiency losses from rent seeking behavior.
Second, we examine how induced innovation can reduce
the adverse effects of tariffs on trade, providing informa-
tion on how technology choices can moderate the nega-
tive impact of restrictive trade policy on trade. Third, we
study how induced innovation can influence the effects
of quotas on corresponding quota rents. The analysis also
examines the presence of interaction effects between
quotas and tariffs. Importantly, our “global LeChatelier
results” hold without imposing restrictive assumptions
and apply globally, i.e. for any discrete change in trade
policy.
The paper is organized as follows. Section 2 starts
with a discussion of induced innovation at the firm level.
Section 3 presents a dual general equilibrium model of an
economy under trade policy distortions, including both
tariffs and quotas. The model provides a basis for ana-
lyzing the efficiency gains/losses generated by a discrete
change in trade policy. Section 4 introduces the role of
induced innovation. It presents global LeChatelier results
showing how induced innovation interacts with the ef-
fects of both tariff and quota policies. Section 5 discusses
the economic implications for economics and welfare.
While local LeChatelier results apply as a special case,
our global analysis provides new insights in the eco-
nomic analysis of trade policy. Finally, section 6 con-
cludes.
2. Preliminary: Induced Innovations at the
Firm Level
Consider an economy involving a set K = {1, …, K} of
goods produced by a set M = {1, …, M} of firms. Using
the netput notation, the j-th firm produces yj = (y1j, …,
yKj) Yj K, where ykj is the k-th output (k-th input if
negative) of the j-th firm, and Yj is the feasible set repre-
senting the technology available to the j-th firm, j M.
Assume that all firms are price takers and that the set Yj
is bounded and convex, j M. Denote by p = (p1, …, pK)
the vector of prices for the K commodities. Then,
the profit maximizing decisions of the j-th firm facing
prices p are given by1
K
j(p, Yj) = pT yj
*(p, Yj) = maxy {pT yj: yj Yj} (1)
where j(p, Yj) is the indirect profit function, and yj
*(p,
Yj) is the corresponding profit maximizing decision, j
M.
As discussed in the introduction, following Hicks [2],
induced innovation reflects that relative prices can help
guide the innovation process (e.g., [3-5]). Induced inno-
vation stimulates the adoption of technologies that in-
crease (decrease) the use of inputs that are becoming
cheaper (more expensive). And applied to the output side,
induced innovation suggests that price changes stimulate
the adoption of technologies that increase (decrease) the
production of commodities exhibiting higher (lower)
prices. To see that, consider the case where the j-th firm
has the option to choose between T technologies: Yj
1, …,
Yj
T. Denote the set of technology indexes by I = {1, …,
T}. Then, from (1) and for given prices p, the incre-
mental profit obtained by j-th firm switching from tech-
nology Tj
i to Tj
i’ is: jii’ j(p, Yj
i) – j(p, Yj
i’), for i, i’
I.
1In our notation, all vectors are treated as column vectors, and the su-
p
erscript “T” denotes the transpose.
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Technology adoption is typically slow. As discussed in
the introduction, this can happen for at least two reasons:
when technology is embodied in physical or human
capital that depreciates slowly; and when learning about
a new technology is costly. To capture the dynamics of
technology adoption, for the j-th firm, denote the cost of
switching from technology i to i’ after t periods by Cj(i, i’,
t) 0. For the j-th profit maximizing firm, the decision to
switch technology from i to i’ after t periods would de-
pend on the present value of incremental profit jii’ minus
switching cost Cj(i, i’, t). We make the following as-
sumption:
Assumption As1: For any i, i’ I, limt Cj(i, i’, t) =
0.
Assumption As1 states that, over time, the cost of
switching between any two technologies declines toward
zero. Assume for the moment that p is constant. In the
short run, technology adoption decisions can be complex.
The firm would decide to switch from i to i’ when the
incremental profit jii’ is large enough to dominate the
switching cost Cj(i, i’, t). But a positive incremental
profit is not sufficient: the presence of large switching
cost could induce the firm to stay with technology i even
if jii’ > 0.
Under Assumption As1, technology adoption deci-
sions are simpler in the long run (when t ). Under
As1 and profit maximization, the long run technology
decision made by the j-th firm is as follows:
Yj
*(p) = maxY{j(p, Yj
i): i I} (2)
This shows that, in the long run, technology choice
Yj
*(p) in general depends of prices p. This is consistent
with the induced innovation hypothesis.
Now, consider a change in prices from p to p .
It follows that, for the j-th firm in the long run, the profit
maximizing technology would change from Yj
*(p) to
Yj
*(p’). This illustrates that induced innovation involves
the interactions between technology choice and market
prices. These interactions become relevant when the
process of technology adoption is slow. This suggests the
need to distinguish between short run and long run situa-
tions. We define the short run (S) as a situation where a
firm does not have enough time to change their previous
technology. And under As1, we define the long run (L)
as corresponding to situations where a firm has had
enough time to adopt profitable technologies. Implica-
tions of this distinction for economic analysis of trade
policy are investigated in sections 4 and 5 below.
K
In the long run for the j-th firm, note that a price
change from p to p’ induces netput changes from yj
*(p,
Yj
*(p)) to yj
*(p’, Yj
*(p’), j M. This includes two effects:
the direct price effect and the indirect effect of induced
innovation (from Yj
*(p) to Yj
*(p’)). While analyzing each
effect is straightforward, analyzing them jointly is more
challenging. Two aspects of these adjustments are worth
stressing. First, our analysis allows for some firms in M
to be inactive. Indeed, the j-th firm would be completely
inactive under prices p and technology Tj if yj
*(p, Yj) = 0
in (1). Or it could be inactive in some markets (when
ykj
*(p, Yj) = 0 for some k K) while being active in oth-
ers (when yk’
*(p, Yj) 0 for some k’ K – k). As prices
change and technology changes, it follows that the num-
ber of inactive firms in any particular market would also
change. It means that the changes from yj
*(p, Yj
*(p)) to
yj
*(p’, Yj
*(p’), j M, can capture entry/exit processes of
firms in any of the K markets. Importantly, both price
changes (from p to p’) and induced technological inno-
vation (from Yj
*(p) to Yj
*(p’)) can affect entry-exit in
any market.
Second, our analysis allows for technology to vary
across firms as well as over time. The heterogeneity of
technology across firms is captured by defining a feasible
set that is firm-specific (Yj for the j-th firm). This can
reflect the role of agroclimatic and location-specific ef-
fects. And the technological options available in the
process of induced innovation are also firm-specific
((Yj
1, …, Yj
T) for the j-th firm. This can capture hetero-
geneity in human capital across firms. Note that the role
of entry-exit and heterogeneous technology has been
identified in the literature (e.g., [17]). The above discus-
sion indicates that our analysis does capture such effects.
So far, our discussion has been at the firm level. This
provides a building block for the rest of the paper. But it
suffers from a significant drawback: it does not explain
what causes price changes. To resolve this issue, we need
to present the analysis at the aggregate level, where
prices are the outcome of market equilibrium. As dis-
cussed in the introduction, our focus is on the analysis of
trade policy. It means that we need to evaluate the evolu-
tion of prices as the outcome of trade policy reforms.
This is the topic of the next sections.
3. Trade under Policy Distortions
As discussed in Section 2, the economy involves a set K
= {1, …, K} of goods produced by a set M = {1, …, M}
of firms. The j-th firm produces netputs yj = (y1j, …, yKj)
Yj K, where Yj is the feasible set representing the
technology available to the j-th firm, j M. The K goods
are consumed by a set N = {1, …, N} of households. The
i-th household has initial endowment wi = (w1i, …, wKi),
consumes xi = (x1i, …, xKi) and has preferences
represented by the utility function ui(xi), i N. Let x
(x1, …, xN), y (y1, …, yM), and Y Y1 × … × YM. As
noted above, this allows for heterogeneous technologies
among firms. An allocation (x, y) is feasible if it satisfies
x
K
NK
, y Y, and
iN xi iN wi + jM yj (3)
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242
where Equation (3) is the commodity balance equation
stating that aggregate consumption cannot exceed the
aggregate supply of each good.
Throughout the paper, we assume that the set Y is
closed, bounded and convex,2 and that the set {jM Yj +
iN wi} has a non-empty interior. And we as-
sume that the utility function ui(xi) is continuous, non-
satiated and quasi-concave on , i N.
K
K
Under prices p , let production decisions be
made by profit maximizing firms according to Equation
(1). And let
K
ei(p, Ui) = pT xi
*(p, Ui)
= Minx {pT xi: u(xi) Ui, xi } (4)
K
be the expenditure function for the i-th household, where
xi
*(p, Ui) is the corresponding Hicksian demand, i N.
To analyze trade policy, consider that the economy in-
cludes two regions, A and B. Region A has a trade policy
involving a mix of tariffs and quotas on imported goods.
Let with t = (t1, …, tK) be the tariffs on imported goods,
where tk is the import tariff (or export subsidy if tk < 0)
on the k-th product. And let q = (q1, …, qK) K
de-
note the import quotas, where qk the import quota on the
k-th product. While we allow for both tariffs and quotas,
our analysis considers situations where each type of po-
licy instrument applies to different goods. It means that
import tariffs may be imposed on some goods while im-
port quotas are imposed on other goods. In this context,
tk = 0 when the k-th good is not subject to a tariff (or tax),
while qk = when the k-th good is not subject to an im-
port quota, k K. Let N = (NA, NB) and M = (MA, MB)
where Nr is the set of households in region r, and Mr is
the set of firms in region r, r = A, B. The tariffs t apply to
net imports into region A: mA iNA xiiNA wi
jMA yj. Similarly, the quotas q impose the following
trade restricttion:3
iNA xiiNA wijMA yj q (5)
We want to analyze the implications of trade policy
represented by the policy instruments (t, q). Let =
(1, …, K) denote the quota rents associated
with the quota restrictions (5).4 Below, following Dixit
and Norman [12] and Luenberger [13], we present a dual
general equilibrium model of trade and use it to examine
the effects of trade policy (t, q) on prices p, on the quota
rents , on trade, and on welfare. As investigated in pre-
vious literature (e.g., [8,18-22]), the dual approach to
trade policy analysis relies on the profit function j(p, Yj)
in Equation (1) and the expenditure function ei(p, Ui) in
Equation (4).
K
Let g K
be some reference bundle satisfying g
0. Under the price normalization rule pT g = 1, consider
the following minimization problem
V(U, t, q, Y)
= minp, {(p + + t)T iNA wi
+ pT iNB wi + T q
+ jMA j(p + + t, Yj) + jMB j(p, Yj)
iNA ei(p + + t, Ui) – iNB ei(p, Ui)
: pT g = 1, p K
, }, (6)
K
which has solution p*(U, t, q, Y) and *(U, t, q, Y),
where U = (U1, …, UN). Let
W(U, t, q, Y)
V(U, t, q, Y) – tT mA
*(U, t, q, Y), (7)
where mA
*(U, t, q, Y) is the aggregate demand for net
imports into region A defined as
mA
*(U, t, q, Y) iNA xi
*(p*(U, t, q, Y)
+ *(U, t, q, Y) + t, Ui)
iNA wijMA yj
*(p*(U, t, q, Y)
+ *(U, t, q, Y) + t, Yj).
It is clear that V(U, t, q, Y) in Equation (6) and W(U, t,
q, Y) in Equation (7) involve monetary measures. The
function W(U, t, q, Y) in Equation (7) will play a key
role in our analysis. As discussed below, it is a welfare
indicator that will provide a basis to evaluate the eco-
nomic and welfare effects of trade policy. As a starting
point, two key results are stated next (see the proof in the
Appendix).
Lemma 1: Let U {U’: W(U’, t, q, Y) = 0}. Then,
under trade policy (t, ),
1) p*(U, t, q, Y) and *(U, t, q, Y) in Equation (6) are
market equilibrium prices and quota rents, respectively;
2) W(U, t, q, Y) in Equation (7) is a monetary measure
of aggregate benefit, W(U, t, q, Y) being non-increasing
in U.
Lemma 1 includes as a special case competitive mar-
kets in the absence of trade policy (when t = 0 and q = ,
i.e. when there is no tariff and quotas are non-binding).
Then, W(U, 0, , Y) is Allais’ distributable surplus un-
der perfect competition ([23,24]). W(U, 0, , Y) being
non-increasing in U reflects that reaching higher utilities
is typically possible only with a redistribution of the ag-
gregate surplus W, i.e. a reduction in W. When t = 0 and
q = , and following Luenberger ([13,25]), Equation (6)
defines a “minimal allocation”; and Equation (6) along
with U {U’: W(U’, 0, , Y) = 0} define a “zero-
minimal allocation”, i.e. a minimal allocation where all
surplus has been redistributed to consumers. Zero-mini-
mality, competitive equilibrium and Pareto efficiency are
closely related concepts (e.g., [13,25]). It means that
2Note that assuming that Y Y1 × … × YM is a convex set rules out
technologies exhibiting increasing returns to scale. Yet, it allows each
firm to be active (if its productivity is “relatively high”) or inactive (if its
p
roductivity is “low”).
3Note that the trade quotas q place an upper-
b
ound on imports in region
A. When q 0, such quotas impose no restriction on exports from region
A.
4The quota rent k would be positive (zero) when the corresponding
quota restriction is binding (not binding) for the k-th commodity, k
K
.
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p*(U, 0, , Y) are the competitive prices supporting a
Pareto efficient allocation. In addition, the set {U’: W(U’,
0, , Y) = 0} defines the Pareto utility frontier, i.e. the
set of consumer utilities that can be reached under effi-
cient allocations.
Thus, Equations (6) and (7) along with U {U’:
W(U’, t, q, Y) = 0} provide a generalized representation
of a zero-minimal allocation under trade policy (t, q).
First, Equation (6) can be interpreted as a “distorted
minimal allocation” under policy (t, q), with p*(U, t, q,
Y) and *(U, t, q, Y) as the corresponding market equi-
librium prices and quota rents, respectively. Second,
W(U, t, q, Y) in (7) is a measure of aggregate benefit
obtained under policy (t, ). Note that the term [tT mA
*(U,
t, q, Y)] in Equation (7) represents the aggregate revenue
generated by the tariffs t. It means that V(U, t, q, Y) in
Equation (6) can be interpreted as a measure of aggregate
benefit before tariff revenues are redistributed, and that
W(U, t, q, Y) in Equation (7) is a measure of aggregate
benefit after tariff revenues are redistributed.
Third, after choosing U to satisfy W(U, t, q, Y) = 0,
Equations (6) and (7) characterize a “distorted zero-
minimal allocation” under policy (t, q). They also repre-
sent a “distorted market equilibrium” under policy (t, q).
The introduction of trade policy in Equation (6) has two
important effects. First, while p denotes prices in region
B, producers and consumers in region A now face prices
(p + + t). When positive, this means that both import
tariffs t and quota rents contribute to increasing prices
in region A, with ( + t) denoting price wedges between
the two regions. Note that the import tariffs t have direct
effects on agents in region A: they affect aggregate profit
jMA j(p + + t, Yj) as well as aggregate expenditure
iNA ei(p + + t, Ui) in region A. When applied to net
imports into region A, mA, the import tariffs t generate
tariff revenue [tT mA]. As noted above, this tariff revenue
gets redistributed to consumers, as captured by the sub-
traction of [tT mA] in the evaluation of aggregate benefit
W(U, t, q, Y) in Equation (7). In general, tariffs affect
efficiency (as discussed below) as well as the distribution
of welfare (depending on how tariff revenues are redis-
tributed). Second, the term T q in Equation (6) measures
the aggregate quota rent generating income that is even-
tually captured by some agents. This quota rent affects
both efficiency (as discussed below) and the distribution
of welfare (depending on who captures it).
We know that {U’: W(U’, 0, , Y) = 0} defines the
Pareto utility frontier in the absence of trade policy. In an
economy distorted by trade policy, choosing U to satisfy
W(U, t, q, Y) = 0 means that {U’: W(U’, t, q, Y) = 0}
identifies the utility frontier under policy (t, q). In other
words, {U’: W(U’, t, q, Y) = 0} is the set of consumer
utilities that can be reached under distortionary trade pol-
icy. Thus, given U {U’: W(U’, t, q, Y) = 0}, the dis-
torted zero-minimal allocation defined in Equations (6)-
(7) identifies p*(U, t, q, Y) as the market equilibrium
prices in region B, *(U, t, q, Y) as the market equilib-
rium quota rents, and [p*(U, t, q, Y) + *(U, t, q, Y) + t]
as the market equilibrium prices in region A.
Interpreting W(U, t, q, Y) in Equation (7) as a measure
of aggregate benefit under policy instruments (t, q), we
will make use of W(U, t, q, Y) to evaluate the aggregate
welfare effects of trade policy. This includes the effi-
ciency effects of quotas q (and associated quotas rents )
and tariffs t (and associated tariff revenue). With {U:
W(U, t, q, Y) = 0} representing the utility frontier under
trade policy (t, q), the shift in the utility frontier associ-
ated with a policy change from (t, q) to (t’, q’) can be
measured by the associated change in aggregate benefit:
W W(U, tq’, Y) – W(U, t, q, Y), a money-metric
measure of aggregate welfare impact. For a given U,
finding W < 0 means an inward shift in the utility fron-
tier, identifying a Pareto inferior move. And finding W
> 0 means an outward shift in the utility frontier, identi-
fying a potential Pareto improving move. Two particular
choices of U are typically considered in welfare analysis.
First, choosing U to satisfy W(U, t, q, Y) = 0 implies that
W W(U, tq’, Y), corresponding to a “compensating
variation” measure. Second, choosing U to satisfy W(U,
t’, q’, Y) = 0 implies that W –W(U, t, q, Y), corre-
sponding to an “equivalent variation” measure. We pro-
ceed with our analysis below assuming that U follows
one of these two choices.5 On that basis, our analysis of
the aggregate efficiency of trade policy reform will rely
on [W(U, tq’, Y) – W(U, t, q, Y)].
What about distribution effects? W(U, t, q, Y) pro-
vides a measure of aggregate benefit, i.e. it is the sum of
individual benefit across all households. Evaluating how
individual welfare gets distributed is more challenging
for three reasons. First, it involves the distribution of
profit j among households. Any change in profit distri-
bution affects the distribution of welfare among house-
holds. Second, the way tariff revenue is distributed mat-
ters. The redistribution of tariff revenue tT mA is captured
in Equation (7). How this redistribution takes place af-
fects the distribution of welfare among households. Third,
who captures the quota rents * matters. This depends on
how trade policy is implemented. For example, under
“voluntary export restraints”, the quota rents get captured
by exporters. Alternatively, when import quotas are auc-
tioned among exporters, the importing country typically
captures the quota rents. This illustrates how the distribu-
tion of quota rents can affect the distribution of welfare
among households. Since it allows for an arbitrary num-
ber of firms and households, our analysis of aggregate
5Holding Uconstant, it means that our analysis of consumer behavior
below should be interpreted in terms of Hicksian compensated behav-
ior.
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244
efficiency remains valid under alternative distribution
schemes. Each distribution scheme simply involves
choosing a different point U on the utility frontier {U’:
W(U’, t, q, Y) = 0}. In this context, it should be under-
stood that the choice of U depends on the distribution
rules used in economic and trade policy. The analysis of
efficiency presented in this paper is conditional on U.
When using a compensating variation measure, this
means that U satisfying W(U, t, q, Y) = 0 reflects the
distribution rules under trade policy (t, q). Alternatively,
when using an equivalent variation measure, it means
that U satisfying W(U, t’, q’, Y) = 0 reflects the distribu-
tion rules under trade policy (t’, q’).
Next, we analyze the general welfare effects of a
change in trade policy from (t, q) to (t’, q’). Our analysis
will rely on the following result. (See the proof in the
Appendix).
Proposition 1: For any (t, q) and (t’, q’),
*(U, t’, q’, Y)T [q’ – q] + tT [mA
*(U, t’, q’, Y)
mA
*(U, t, q, Y)]
W(U, tq’, Y) – W(U, t, q, Y)
*(U, t, q, Y)T [q’ – q]
+ tT [mA
*(U, t’, q’, Y) – mA
*(U, t, q, Y)]. (8)
Proposition 1 provides a general characterization of
the aggregate effects of trade policy. First, note that
choosing q’ = implies that the quota constraint (5) is
non-binding and that the associated quota rent is zero:
*(U, t’, , Y) = 0 for any t’. Then, choosing t’ = 0 and
q’ = , and for any trade policy (t, q), the first inequality
in equation (8) gives: W(U, 0, , Y) W(U, t, q, Y).
Associating (t’, q’) = (0, ) with competitive markets in
the absence of trade policy, this gives the well-known
result that aggregate welfare is maximized in the absence
of policy distortions, and that perfectly competitive mar-
kets are Pareto efficient. In this context, any trade policy
(t, q) where t 0 and the quotas q are binding is in gen-
eral inefficient and tends to lower aggregate benefit, with
[W(U, 0, , Y) – W(U, t, q, Y)] 0 providing a measure
of the aggregate welfare loss associated with an inward-
shift in the utility frontier.
Second, for any (t, q) and (t’, q’), Equation (8) implies
[*(U, t’, q’, Y) – *(U, t, q, Y)]T [q’ – q]
+ [t’ – t]T [mA
*(U, t’, q’, Y) – mA
*(U, t, q, Y)]
0. (9)
When quotas do not change (q’ = q), Equation (9) be-
comes [t’ – t]T [mA
*(U, t’, q, Y) – mA
*(U, t, q, Y)] 0.
This is the well-known result that any ceteris paribus
increase in import tariffs (t’ > t) tends to reduce imports
mA
*. Similarly, if tariffs do not change (t’ = t), Equation
(9) becomes [*(U, t, q’, Y) – *(U, t, q, Y)]T [q’ – q]
0. Again, this is the well-known result that any ceteris
paribus increase in import quotas (q’ > q) tends to de-
crease the quota rents *. Note that these results are
global as they apply for any discrete change in trade pol-
icy.
Third, Equation (8) presents bounds on the change in
aggregate benefit when trade policy changes from (t, q)
and (t’, q’). From the lower bound in equation (8), it fol-
lows that a sufficient condition for [W(U, t’, q’, Y) –
W(U, t, q, Y)] 0 is
*(U, t’, q’, Y)T [q’ – q]
+ tT [mA
*(U, t’, q’, Y) – mA
*(U, t, q, Y)]
0. (10a)
Thus, Equation (10a) is a sufficient condition for trade
policy reform from (t, q) and (t’, q’) to improve effi-
ciency (by increasing aggregate benefit and thus shifting
up the utility frontier). It applies under general conditions
involving discrete changes in both tariffs and quotas.
This includes as special cases some well-known results.
For example, in situations where there is a move to
eliminating all tariffs (with t’ = 0), then given * 0,
Equation (10a) implies that any scenario where quotas
are relaxed (q’ > q) is efficiency improving. Alterna-
tively, when quotas do not change (q’ = q), then using
Equation (9), Equation (10a) always holds under any
proportional reduction in tariffs (e.g., [14]). This well-
known result (that a proportional decline in all tariffs
tends to be efficiency improving) has guided trade policy
reform supported by WTO over the last decade.
Similarly, from the upper bound in Equation (8), it
follows that a necessary condition for [W(U, t’, q’, Y) –
W(U, t, q, Y)] 0 is
*(U, t, q, Y)T [q’ – q]
+ tT [mA
*(U, t’, q’, Y) – mA
*(U, t, q, Y)]
0. (10b)
It means that Equation (10b) is a necessary condition
for trade policy reform from (t, q) and (t’, q’) to improve
efficiency (by increasing aggregate benefit and shifting
up the utility frontier). Alternatively, finding any situa-
tion where Equation (10b) does not hold implies an ag-
gregate welfare loss. Then, the policy reform from (t, q)
and (t’, q’) cannot be a Pareto improvement. This would
identify rent-seeking behavior. Indeed, such policy re-
form can be a rational move only if it implies a redistri-
bution of welfare toward the “rent seekers” who benefit
at the expense of others (as efficiency and aggregate
benefit decline and the utility frontier shifts down).
4. Trade Policy Analysis under Induced
Innovation
So far, we have explored scenarios of trade policy reform
represented by a change in policy instruments from (t, q)
to (t’, q’). We now consider the case where such policy
Copyright © 2012 SciRes. TEL
J.-P. CHAVAS 245
changes are associated with technological innovation.
Induced innovation was discussed in Section 2 at the firm
level. We showed in Equation (2) how prices affect firm
technology choices in long run equilibrium. The analysis
is now extended to the aggregate level in the context of
trade policy.
We know that the direct effect of import tariffs and
quotas on specific commodities is to increase their cor-
responding prices in domestic markets. But they also
influence the prices of all goods through market equilib-
rium effects. This applies in particular to substitute goods.
Market equilibrium prices of substitute goods tend to
move together. This means that, when the direct effect of
an economic policy is to increase the price of some
goods, induced innovation would help stimulate the ado-
ption of technologies supporting the production of sub-
stitute goods. Such general equilibrium effects are ana-
lyzed below.
From Equations (1) and (6), firms behave so as to
maximize profit, conditional on prices (p + + t) for
firms in MA, and prices p for firms in MB. From Lemma
1 and Equations (A2)-(A3) in the Appendix, this is fully
consistent with the maximization of (distorted) aggregate
benefit. And from Equation (2), the firm profit maximi-
zation motive extends to firm technology choice in the
long run. Denote the technology options available to all
firms by Y = {(Yj
1, Yj
2, …), j M}, where Yj
i is the i-th
technology option available to the j-th firm. It means that,
in the long run, for a given U and under trade policy (t,
q), technology choice are made as follows:6
Y*(U, t, q) argmaxY {W(U, t, q, Y): Y Y}.
Under trade policy reform, this identifies two possible
technology choices: Y*(U, t, q) under policy instruments
(t, q), and Y*(U, t’, q’) under policy instruments (t’, q’).
As discussed in Section 2, this allows heterogeneous
technologies across firms. Perhaps more importantly,
under induced innovation, this allows for entry/exit and
for technology adoption decisions to vary among firms
(e.g., between “domestic firms” in MA and “exporting
firms” in MB).
Consider the case of a policy reform associated with a
change from (t, q) to (t’, q’), with (t, q) (t’, q’). We
focus our attention on situations where induced innova-
tion plays a role, i.e. where Y*(U, t, q) Y*(U, t’, q’).
First, consider the situation before the policy change. It
corresponds to policy instruments (t, q). Assuming that
this trade policy has been in place for an extended period
of time, the associated technology choice is Y*(U, t, q).
Second, consider a policy change from (t, q) to (t’, q’).
There are now two possible scenarios. There is a short
run scenario (S) where firms have not had enough time to
modify their technology, implying that firms are con-
strained to face the original feasible set Y*(U, t, q). And
there is a long run scenario (L) where firms do adjust
their technology and choose Y*(U, t’, q’). What is the
difference between these two scenarios? Answering this
question requires exploring how the effects of policy
change differ between the short run and the long run.
First, consider the short run scenario (S). Denote the
optimal technology chosen under policy (t, q) by YS
Y*(U, t, q). Keeping the feasible set Y in its original state
YS, the short run welfare effects of trade policy reform
can be measured as
WS W(U, t’, q’, YS) – W(U, t, q, YS) (11)
Second, consider the long run scenario (L) and the as-
sociated welfare changes due to a policy change from (t,
q) to (t’, q’). Denote the optimal technology chosen un-
der policy (t’, q’) by YL Y*( U, t’, q’). In the long run,
the welfare effects of trade policy reform can be mea-
sured as
WL W(U, t’, q’, YL) – W(U, t, q, YS) (12)
which allows a switch from technology YS to YL as trade
policy changes from (t, q) to (t’, q’). This raises the
question: how does WL differ from WS? Our analysis
presented below answers this question. In the process, we
will gain new and useful information on how induced
innovation and trade policy interact with each other.7
Our analysis explores the economic and welfare im-
plications of induced innovation. Our main result is
stated next. (See the proof in the Appendix)
Proposition 2 (Global LeChatelier results): For any
policy change from (t, q) to (t’, q’),
0 WLWS
[*(U, t, q, YL) – *(U, t’, q’, YS)]T [q’ – q]
+ tT [mA
*(U, t’, q’, YL) – mA
*(U, t, q, YL)]
tT [mA
*(U, t’, q’, YS) – mA
*(U, t, q, YS)]. (13)
Equation (13) shows that the general implications of
induced innovation for market equilibrium prices, quan-
tities, and welfare. Importantly, these results hold glo-
bally under any discrete changes in trade policy. Equa-
tion (13) gives LeChatelier results related to the effects
of trade policy under induced innovation. These results
appear to be new. They apply globally for any discrete
change from (t, q) to (t’, q’). The economic and welfare
implications of Equation (13) are further discussed be-
low.
While LeChatelier effects have been examined before
7Note that induced innovation involves choosing among technologies,
assuming that they are “on the shelf”. While our discussion below fo-
cuses on this case, the analysis could be easily extended to more general
cases of technological changes. It could be used to show that techno-
logical progress can improve aggregate welfare beyond the gains ob-
tained from induced innovation.
6But some abuse of notation, this maximization problem assumes that
one technology Yji is chosen for each firm, j N.
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246
in the context of trade (e.g., [8,9]), they were developed
for “small changes” in trade policy under differentiability
assumptions. This suggests the existence of a local ver-
sion of Proposition 2. Such local results are presented
next. (See the proof in the Appendix).
Proposition 3 (Local LeChatelier results): Assume that
W(U, t, q, Y) is differentiable in (t, q), that *(U, t, q, Y)
is a continuous function of (t, q), and that mA
*(U, t, q, Y)
is continuously differentiable in (t, q). Then, considering
a small change in trade policy (dt, dq) in the neighbor-
hood of (t, q),
0 dWL – dWS
= [*(U, t, q, YL) – *(U, t, q, YS)]T dq
+ tT [dmA
*(U, t, q, YL) – dmA
*(U, t, q, YS)], (14)
where dmA
*(U, t, q, Y) = [mA
*(U, t, q, Y)/t] dt +
[mA
*(U, t, q, Y)/q] dq.
Proposition 3 is a local version of the LeChatelier re-
sults presented in Proposition 2. Indeed, expression (14)
is a special case of (13) obtained under differentiability
assumptions and considering only a small change in trade
policy (dt, dq) in the neighborhood of (t, q). Comparing
equations (13) and (14), it is clear that the global results
stated in Equation (13) are more general: they apply for
any discrete policy change from (t, q) to (t’, q’), and
without imposing restrictive assumptions (e.g., they ap-
ply without assuming supermodularity or differentiabi-
lity). Implications of these results are discussed next.
5. Implications
Propositions 2 and 3 give information on the welfare
difference from a trade policy change between the short
run (S) and the long run (L): WLWS. From Equa-
tions (13) and (14), this difference has a general lower
bound of 0. It implies that the aggregate welfare effect of
a policy change is always at least as large in the long run
as in the short run. This is a general and intuitive result:
induced innovation tends to generate long run benefits
that are at least as large as the associated short run bene-
fits. In the context of trade policy changes, this means
that the efficiency effects of trade policy changes become
more positive (or less negative) due to induced innova-
tion. Importantly, this result holds under very general
conditions.
The global LeChatelier results given in Equation (13)
provide useful information on the welfare and economic
impact of trade policy. From Equation (13), the welfare
change [WLWS] has a general upper bound equal to:
[*(U, t, q, YL) – *(U, t’, q’, YS)]T [q’ – q] + tT [mA
*(U,
t’, q’, YL) – mA
*(U, t, q, YL)] – tT [mA
*(U, t’, q’, YS) –
mA
*(U, t, q, YS)] 0. This upper bound provides a
measure of the largest possible welfare gain generated by
induced innovation. As discussed below, this upper
bound also provides useful information on the interaction
effects between induced innovation and trade policy re-
form.
First, consider the case where only tariffs change. Then,
when quota policy does not change (q’ = q) and tariff
policy changes from t to t’, Equation (13) gives the fol-
lowing important result:
0 WLWS
tT [mA
*(U, t’, q, YL) – mA
*(U, t, q, YL)]
tT [mA
*(U, t’, q, YS) – mA
*(U, t, q, YS)]. (15)
To interpret Equation (15), consider the case of an in-
crease in tariff, with t’ > t 0. From Equation (9), we
know that, ceteris paribus, any tariff increase tends to
have negative effects on trade mA
*. In this context, Equa-
tion (15) implies that a weighted sum (with tariffs as
weights) of the trade reduction due to higher tariffs tends
to smaller in the long run compared to the short run. This
global LeChatelier effect indicates how tariff reform can
affect trade under induced innovation. To illustrate, let-
ting t t + t and noting that mA
*(U, t, q, YL) = mA
*(U,
t, q, YS), Equation (15) implies that:
tT [mA
*(U, t’, q, YL) – mA
*(U, t’, q, YS)]
tT [mA
*(U, t’, q, YS) – mA
*(U, t, q, YS)]
0, (16)
where the last inequality follows from Equation (9).
Equation (16) implies that a change in import value asso-
ciated with an induced adjustment in technology from YS
to YL, tT [mA
*(U, t’, q, YL) – mA
*(U, t’, q, YS)], is at
least as large as the corresponding short run effect of
tariff change, tT [mA
*(U, t’, q, YS) – mA
*(U, t, q, YS)]
0. This LeChatelier result appears to be new. Note that it
does not imply that induced innovation necessarily re-
duces the adverse effects of tariffs on trade. But Equation
(16) establishes a lower bound on the trade effects, tT
[mA
*(U, t’, q, YL) – mA
*(U, t’, q, YS)], and it states that
this lower bound is non-positive and given by tT
[mA
*(U, t’, q, YS) – mA
*(U, t, q, YS)]. This indicates that
induced innovation could stimulate trade, with tT [mA
*(U,
t’, q, YL) – mA
*(U, t’, q, YS)] > 0, when |tT [mA
*(U, t’,
q, YS) – mA
*(U, t, q, YS)]| is small and tariffs have only
modest effects on trade. Alternatively, this suggests that
induced innovation could possibly reduce trade, with tT
[mA
*(U, t’, q, YL) – mA
*(U, t’, q, YS)] < 0, when |tT
[mA
*(U, t’, q, YS) – mA
*(U, t, q, YS)]| is large and tariffs
have large effects on trade. These results apply globally,
i.e. for any change in t, and without imposing differenti-
ability assumptions.
When there is only a change in tariffs (with dq = 0),
and under differentiability assumptions, Equation (14)
becomes
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J.-P. CHAVAS 247
dWL – dWS
= tT [mA
*(U, t, q, YL)/tmA
*(U, t, q, YS)/t] dt
0. (17)
Equation (17) is a local version of Equation (15). It has
the following implications. Note that, under differenti-
ability, Equation (9) implies that [mA
*(U, t, q, Y)/t] is
a negative semi-definite matrix, a standard result stating
that tariffs tend to have negative effects on trade. This
holds in the short run (when Y = YS) as well as in long
run (when Y = YL). Consider a (small) proportional in-
crease in tariff, where dt = k t > 0 and k is a small posi-
tive scalar. Then, Equation (17) implies that tT mA
*(U, t,
q, YL)/tmA
*(U, t, q, YS)/t] tT 0, i.e. that [mA
*(U,
t, q, YL)/tmA
*(U, t, q, YS)/t] is a positive semi-
definite matrix. It means that, under induced innovation,
the long run negative effects of tariffs on trade, [mA
*(U,
t, q, YL)/t], tends to smaller than its corresponding short
run negative effects, [mA
*(U, t, q, YS)/t]. This is a
standard local LeChatelier result: allowing for adjust-
ments in technology tends to reduce the adverse effects
of tariffs on trade. This is the result obtained by Neary [8]
and Kreickemeier [9]. However, it holds only locally, i.e.
only for small changes in tariffs. To see that this Le-
Chatelier result does not hold globally, it suffices to note
that {tT [mA
*(U, t, q, YL)/tmA
*(U, t, q, YS)/t] dt}
in equation (17) is the local version of the expression on
the left-hand side of (16). But as discussed above, the
left-hand side of Equation (16) can be either positive or
negative. This illustrates the well-known fact that local
LeChatelier results do not necessarily apply globally ([7,
10,11]).
Second, consider the case where only quotas change.
Then, when tariff policy does not change (t’ = t) and
quota policy changes from q to q’, equation (13) gives
the following important result:
0 WLWS
[*(U, t, q, YL) – *(U, t, q’, YS)]T [q’ – q]
+ tT [mA
*(U, t, q’, YL) – mA
*(U, t, q’, YS)]. (18)
Equation (18) shows the complexity of the general
LeChatelier effects associated with a discrete change in
quotas under induced innovation. The right-hand side of
Equation (18) provides an upper bound on the welfare
change [WL - WS] 0. This upper bound involves the
difference between quota rents *(U, t, q’, YS) and *(U,
t, q, YL). Note the interactions between quota effects and
technology: the first quota rent is evaluated at (q’, YS)
while the second is evaluated at (q, YL). And in the
presence of tariffs (when t 0), the upper bound in (18)
also includes the term {tT [mA
*(U, t, q’, YL) mA
*(U, t,
q’, YS)]}. This term reflects changes in tariff revenue
(obtained under tariff t and quota q’) between the short
run (S) and the long run (L). For any discrete change in
quota from q to q’, Equation (18) gives global LeChate-
lier effects showing how quota reform can affect quota
rents and trade under induced innovation.
While Equation (18) generates general implications of
quota reform comparing the short run and the long run, it
does not give sharp predictions on how induced innova-
tion can affect quota rents. Yet, Equation (18) implies
that the quota rents must change in such a way that the
value [*(U, t, q, YL) *(U, t, q’, YS)]T [q q] is at
least as large as tT [mA
*(U, t, q’, YL) mA
*(U, t, q’,
YS)], the negative of the change in tariff revenue between
the short run and the long run. This reflects the presence
of significant interactions between tariffs and quotas in
the evaluation of global LeChatelier effects related to
quota reform.
The analysis simplifies significantly in the absence of
tariffs (when t = 0). Then, the global LeChatelier results
in equation (18) reduce to:
0 WL WS
[*(U, t, q, YL) *(U, t, q’, YS)]T [q q]. (18’)
This implies that [*(U, t, q, YL) *(U, t, q’, YS)]T
[q q] 0, i.e. that any rise in quotas q tends to in-
crease the difference between the long run quota rents
*(U, t, q’, YL) evaluated at q’, and the short run quota
rents *(U, t, q, YS) evaluated at q. These evaluations
involve changes in both technology and quota level,
When considering discrete changes in quota policy, note
that this does not imply that induced innovation alone
(i.e., the switch from YS to YL) necessarily reduces quota
rents.
What about considering small changes in quota policy,
dq? Then, in the absence of tariff changes (with dt = 0)
and under differentiability assumptions, Equation (14)
becomes
dWL dWS = [*(U, t, q, YL) *(U, t, q, YS)]T dq
+ tT [mA
*(U, t, q, YL)/q mA
*(U, t, q, YS)/q] dq
0. (19)
Equation (19) is a local version of Equation (18). Like
(18), Equation (19) shows that local LeChatelier results
associated with quota changes involve interaction effects
between quota policy and tariff policy. Again, when t 0,
Equation (19) does not give precise information on the
effects of changing quotas on quota rents under induced
innovation. Yet, in the simpler case where there is no
tariff (with t = 0), Equation (19) reduces to
dWL – dWS
= [*(U, t, q, YL) – *(U, t, q, YS)]T dq 0. (19’)
Equation (19’) is a local version of (18’). It implies
that [*(U, t, q, YL) – *(U, t, q, YS)]T dq 0, i.e. that
any rise in quotas q tends to increase the difference be-
tween the long run quota rents *(U, t, q, YL) and the
short run quota rents *(U, t, q, YS). We know from
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248
Equation (9) that any rise in quotas tends to reduce quota
rents. This gives the following local LeChatelier result:
when t = 0 and for small changes in quotas, induced in-
novation tends to reduce the corresponding quota rents.
But while intuitive, this local result does not hold glob-
ally. Indeed, comparing Equations (18’) and (19’), this
result does not necessarily hold for arbitrary changes in
quotas. Again, this illustrates that local LeChatelier re-
sults do not necessarily apply globally.
While local LeChatelier results given in Equations (17)
and (19’) are not new (e.g., [8,9]), our investigation has
been innovative in three directions. First, we considered
the case of discrete changes in trade policy. This is rele-
vant as actual policy reforms typically involve large
changes in policy instruments. Second, we have analyzed
the joint effects of tariffs and quotas. Third, we have
shown that global LeChatelier results involve an upper
bound measure of welfare change. This is given in Equa-
tion (15) for tariff changes and Equation (18) for quota
changes. But the general result is the one stated in Pro-
position 2. Indeed, Equation (13) presents the general
implications of trade policy reform under induced inno-
vation, providing useful information on both welfare ef-
fects and economic adjustments in trade and quota rents.
6. Concluding Remarks
This paper has explored the effects of discrete change in
trade policy (including both tariffs and quotas) under
induced innovation in general equilibrium. It examined
the general case where technology and adoption deci-
sions can vary across firms (e.g., domestic versus export-
ing firms). The interactions between induced innovation
and the effects of trade policy give a set of “LeChatelier
effects” comparing short run versus long run market
equilibrium. In contrast with previous research, the
analysis applies globally to arbitrary changes in trade
policy and without imposing a priori restrictions (such as
supermodularity).
We show that induced innovation tends to reduce the
welfare loss generated by distortionary trade policy. It
means that ignoring induced innovation would overstate
the adverse effects of trade policies. When trade policy is
motivated by rent seeking behavior, it also means that
induced innovation can tamper the associated ineffi-
ciency losses. We examine how induced innovation can
influence the adverse effects of tariffs and quotas on
trade. We document how tariffs and quotas can interact
with each other. Our analysis presents the general impli-
cations of trade policy reform under induced innovation,
providing useful information on both welfare effects and
economic adjustments in trade and quota rents.
7. Acknowledgements
I would like to thank Ian Coxhead for useful feedback on
an earlier draft of this paper.
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250
Appendix
Proof of Lemma 1: Note that j(p, Yj) in (1) is convex in
p on , and that ei(p, Ui) in (4) is concave in p on
K
K
.
Thus, the minimization problem (6) is a convex pro-
gramming problem. Luenberger [26] defined the benefit
function as bi(xi, Ui) max
{
: ui(xi
g) Ui, (xi
g) }. The benefit function bi(xi, Ui) is a welfare
measure giving the number of units of the reference bun-
dle g the i-th consumer is willing to give up starting at
point xi to reach utility Ui. Under the quasi-concavity of
ui(xi) on , Luenberger [26] showed that bi(xi, Ui) is
concave in xi, non-increasing in Ui, and satisfies ei(p, Ui)
= minxi {pT xi – bi(xi, Ui): pT g = 1, p }, i N.
Define
K
K
K
L(U, x, y, p, , t, q) = iN bi(xi, Ui)
+ pT [iN wi + jM yjiN xi]
+ T [iNA wi + q + jMA yjiNA xi]
+ tT [iNA wi + jMA yjiNA xi]. (A1)
Using Equations (1)-(4), ei(p, Ui) = minxi {pT xi – bi(xi,
Ui): pT g = 1, p } and assuming that an interior
solution to problem (6) exists, it follows from Rockafel-
lar ([27], pp. 281-283) that solving the convex minimiza-
tion problem (6) is equivalent to finding a saddle-point
(x*, y*, p*, *) Y of L() satis-
fying p*T g = 1, where
K
NK
K
K
L(U, x, y, p*, *, t, q)
L(U, x*, y*, p*, *, t, q)
L(U, x*, y*, p, , t, q), (A2)
for all (x, y, p, ) Y satisfying
pT g = 1, and where V(U, t, q, Y) = L(U, x*, y*, p*, *, t,
q). Interpret p* and * as Lagrange multipliers measuring
the shadow prices of the constraints [iN wi + jM yj
iN xi] 0 and [iNA wi + q + jMA yjiNA xi] 0,
respectively. Under the normalization rule pT g = 1, this
identifies p
* as the market-clearing prices for the K
goods, and * as the unit quota rents associated with the
import quota restriction (5). From the saddle-point theo-
rem ([28], p. 74), (A2) implies the following dual prob-
lem
NK
K
K
V(U, t, q, Y)
= Maxx,y {iN bi(xi, Ui)
+ tT [iNA wi + jMA yjiNA xi]:
iN wi + jM yjiN xi 0,
iNA wi + q + jMA yjiNA xi 0,
x , y Y
}. (A3)
NK
This identifies a feasible allocation satisfying the fea-
sibility constraint (3), the quota constraint (5), x NK
and y Y. And (A3) satisfies profit maximization in (1)
and expenditure minimization in (4). In addition, it fol-
lows from Equations (A3) and (7) that W(U, t, q, Y)
measures the largest feasible aggregate benefit iN bi(xi,
Ui) under policy instruments (t, q). And bi(xi, Ui) being
non-increasing in Ui, Equations (A3) and (7) imply that
W(U, t, q, Y) is non-increasing in U. Finally, choosing U
{U’: W(U’, t, q, Y) = 0} implies that iN bi(xi, Ui) =
0 at the optimum, giving Ui = ui(xi) as the utility level
obtained for each i N.
Proof of Proposition 1: For a given U, let (po, o, xo, yo)
be the solution of the saddle-point problem in (A2) under
(t, q), and (p’, ’, x’, y’) be its solution under (t’, q’).
The second inequality in (A2) implies that
V(U, t’, q’, Y) L(U, x’, y’, po, o, t’, q’). (A4)
And the first inequality in (A2) implies
L(U, x’, y’, po, o, t, q) V(U, t, q, Y). (A5)
Adding Equations (A4) and (A5), and using (A1), we
obtain
V(U, t’, q’, Y) – V(U, t, q, Y)
L(U, x’, y’, po, o, t’, q’) – L(U, x’, y’, po, o, t, q),
= oT [q’ – q] + [t’ – t]T iNA wi + [t’ – t]T jMA yj
– [t’ – t]T iNA xi’,
= oT [q’ – q] – [t’ – t]T mA’, (A6)
where mA iNA xi’ – iNA wijMA yj’. Using
Equation (7), (A6) implies
W(U, tq’, Y) – W(U, t, q, Y)
oT [q’ – q]
+ tT [mA
*(U, t’, q’, Y)] – mA
*(U, t, q, Y)]. (A7)
This gives the second inequality in (8). The first ine-
quality is obtained by switching (t, q) and (t’, q’) in
Equation (A7) and multiplying by (–1).
Proof of Proposition 2: Note that Y*(U, t, q) arg-
maxY {W(U, t, q, Y): Y Y} implies that
W(U, t’, q’, Y*(U, t’, q’))
W(U, t’, q’, Y*(U, t, q)). (A8)
It follows that
W(U, t, q, YL) = W(U, t, q, YS), (A9)
and
W(U, t’, q’, YL) W(U, t’, q’, YS), (A10)
where YS Y*(U, t, q), YL Y*(U, t’, q’). Subtracting
(A9) from (A10) gives the first inequality in (13).
To prove the second inequality in (13), note that Equa-
Copyright © 2012 SciRes. TEL
J.-P. CHAVAS
Copyright © 2012 SciRes. TEL
251
tion (8) implies
W(U, tq’, YL) – W(U, t, q, YL)
*(U, t, q, YL)T [q’ – q] + tT [mA
*(U, t’, q’, YL)
mA
*(U, t, q, YL)],
and
*(U, t’, q’, YS)T [q’ – q] + tT [mA
*(U, t’, q’, YS)
mA
*(U, t, q, YS)]
W(U, tq’, YS) – W(U, t, q, YS).
Adding these two expressions gives the desired result.
Proof of Proposition 3: Under differentiability, the
inequality in (14) follows directly from the first inequal-
ity in (13) when t t and q q. To prove the equal-
ity in (14), let Dmt(U, t, q, Y) and Dmq(U, t, q, Y) de-
note the derivative of mA
*(U, t, q, Y) with respect to t
and q, respectively, evaluated at (U, t, q, Y). Applying
the mean value theorem to mA
*(U, t, q, Y) yields: mA
*(U,
t’, q’, Y) – mA
*(U, t, q, Y) = Dmt(U, t + (1 – ) t’, q
+ (1 – ) q’, Y) [t’ – t] + Dmq(U, t + (1 – ) t’, q + (1
) q’, Y) [q’ – q], for some [0, 1]. Substituting this
result into equation (8) and using the definition of a de-
rivative give
ly that
W(U, t, q, Y)/t = tT [mA
*(U, t, q, Y)/t], (A11)
when t t and q’ = q, and
W(U, t, q, Y)/q = *(U, t, q, Y)T
+ tT [mA
*(U, t, q, Y)/q], (A12)
when t’ = t and q q. Equations (A11) and (A12) are
“envelope-type” results applying locally in the
neighborhood of (t, q). They imp
dW(U, t, q, Y)
= tT [mA
*(U, t, q, Y)/t] dt
+ [*(U, t, q, Y)T + tT (mA
*(U, t, q, Y)/q)] dq,
= *(U, t, q, Y)T dq + tT dmA
*(U, t, q, Y), (A13)
where dmA
*(U, t, q, Y) = [mA
*(U, t, q, Y)/t] dt +
[mA
*(U, t, q, Y)/q] dq. Using (A13) gives the equality
in Equation (14).