Journal of International
GAINS
FROM
Economics 21 (1986) 111-122. North-Holland
TRADE
WITHOUT
LUMP-SUM
COMPENSATION
Avinash DIXIT
Department
of Economics,
Princeton
University,
Princeton,
NJ 08544,
USA
Bergen,
Norway
Victor NORMAN*
Norwegian
School
of Economics
and Business
Administration,
Received June 1985, revised version received July 1985
This paper examines the possibility of designing a free-trade equilibrium that is Pareto superior
to a given autarkic one, using redistributive tools other than lump-sum transfers. It is shown
that (i) if the production frontier allows some non-zero transformation in the neighbourhood of
autarky, and (ii) there is a commodity, pure or composite, for which no two consumers trade on
opposite sides of the market, then taxes and subsidies on goods and factors suffice for the
purpose. If uniform poll subsidies are available, then condition (ii) is not needed. Such policies
are compatible with incentive-compatibility constraints, while lump-sum transfers are not.
1. Introduction
In Dixit and Norman (1980, pp. 79-80) we sketched a method by which
aggregate gains from trade could be distributed among consumers in a
Pareto improving way using only commodity taxes or subsidies. The idea
was that exposing domestic producers to free trade would yield the aggregate
gains on the production side, and these could be beneficially distributed by
lowering the price of any consumer good. The concept and the procedure
seemed too simple to need any formal demonstration. However, Kemp and
Wan (1986) have chided us for this laxness and cast doubt on the result. We
are happy to respond.
In section 2 we will state and prove the formal result, and respond to
some of Kemp and Wan’s incidental comments. In section 3 we will
concentrate on their examples. In section 4 we will discuss the information
requirements of the alternative policies, and make some concluding remarks.
*We are extremely grateful to Hugo Sonnenschein, who supplied the slick proof of theorem 1
that replaces our earlier clumsy attempts. We have benetited from correspondence with Wilfred
Ethier, and from access to his notes [Ethier (1983)]. We also thank Gene Grossman for useful
discussions on the subject.
0022-1996/86/%3.50 0 1986, Elsevier Science Science Publishers B.V. (North-Holland)
112
A. Dixit
and
K Norman,
Gains from
trade
2. A formal result
The basic construct is the economy’s free-trade aggregate consumption
possibility envelope,’ invented by Baldwin (1948). It is shown in fig. 1. At
each point P on the domestic production possibility frontier DD, we add the
rest of the world’s offer PQ at the relative price corresponding to the slope of
DD at P. The locus TT of such aggregate supply points Q is the desired
envelope. It lies outside DD, touching the latter at the rest of the w,orld’s
autarky price.
GOOD 2
GOOD 1
Fig. 1
‘This should not be confused with the better-known Baldwin Envelope for a country levying
a monopoly tariff. That is the outer envelope obtained by sliding the rest of the.world’s offer
curve along the home country’s transformation curve. Since this lies outside the free trade
envelope TX our results concerning gains from free trade apply a fortiori to gains from trade
with an optimal tariff.
A. Dixit
and K Norman,
Gainsjrom
113
trade
Now we generalize this idea and make it more precise. Let D be the
economy’s technological possibility set. We assume this to be a closed convex
cone, containing the origin but no semi-positive net output vectors, and
admitting free disposal. These are all standard assumptions of competitive
equilibrium theory; that of constant returns, if necessary, can be made true by
defining artificial factors that are repositories for pure profits. Let x(p) be the
domestic supply correspondence; for any p in the unit simplex, this is the set
of maximizers of p * x over D. By constant returns, p *x =O for any x in x(p)
Let s(p) be the rest of the world’s net supply correspondence. We assume this
to be upper hemi-continuous
and bounded below, which are standard
assumptions for the existence of competitive equilibrium. For balanced trade,
we have p* s=O for any s in s(p). Some commodities may be non-tradeable;
the corresponding components in s(p) will be identically zero.
The set T of aggregate consumption possibilities is then defined as the set
of vectors which, for some price p, can be obtained by adding the domestic
production to the foreign net supply, and free disposal if needed. Thus,
T={clcgx+2
for some XEX(P), sEs(p) for some p}.
The idea that trade enlarges the set of aggregate consumption
is made precise in the following result.
Theorem I.
(1)
possibilities
D is a subset of T
Since D allows free disposal, it is enough to prove that any boundary
point x0 of D must lie in lY To this end, we construct an artificial economy
with the same production possibilities and the same net supply from the rest
of the world, but different demand. This is supposed to come from an
aggregate preference function that is L-shaped based on x0, as shown in
fig. 2. More precisely, the utility function is
Prooj:
U(C)=min(q-XP).
I
(2)
If the price vector p is normalized to the unit simplex, and Z denotes money
income,
then the demand functions corresponding to the stipulated utility
function are
(3)
Ci(p,Z)=Xp+(Z-p’X”).
Let p’ be an equilibrium
price vector for the artificial economy, i.e.
c(p’, Z’) 5x’ + s1 complementary
to p’ 10,
(4)
A.
114
Dixit and K Norman, Gainsfrom
trade
GOOD 2
//’
/
X”
/’
/
GOOD 1
Fig. 2
where x’Ex($),
s’~s(p’)
and therefore, by (3),
and (0=) Z’~max{p’~x~x~D}.
Then Z’zp’*x”,
Ci(pl,I’)hXf.
(5)
Putting (4) and (5) together and using the definition (l), we have x0 E ‘I:
At this point we make two remarks about the theorem.
Remark 1. The artificial economy was constructed solely in order to prove
the existence of the price vector pl. Nothing about the consumption
distribution or welfare in the actual economy will make use of the aggregate
utility function (2) in the artiticial one. However, there is a natural economic
interpretation for the construction. Since we are interested in establishing
A. Dixit
and V. Norman,
Gains from
trade
115
aggregate production gains, the obvious approach is to see what would
happen if there were no pure exchange gains, i.e. if the actual preferences
were replaced by fixed-coefficient ones.
Remark 2. In view of the worries about the existence of equilibria in
distorted economies that are expressed by Kemp and Wan (1986, footnote 2),
we should emphasize that Theorem 1 relies only on the standard ArrowDebreu existence theorem for the undistorted case.
Using Theorem 1, we can immediately show that a government empowered to use only commodity taxation can implement a free-trade
outcome that is not worse in the Pareto sense than the autarky equilibrium.
This is shown in the following.
Corollary I. Let x0 be the aggregate output in autarky, co the aggregate
consumption, and p” the corresponding price vector. Let p’, x1 and s1 be as in
Theorem 1. Then there exists a free-trade equilibrium in which (i) consumer
prices are p” and producer prices are pl, with commodity tax rates t’ =p”-p1
in specijic form, (ii) the government buys and throws away the vector g’=
x1 +s’-co
of commodities, and (iii) all consumers are exactly as well off as
under autarky.
Proof
Let ch(p,I,,) denote the demand function of consumer h. With
constant returns to scale there are no pure profits, and with the government
restricted to commodity taxation there are no transfer payments. Thus, I,,=0
in autarky as well as in free trade. Consumption vectors are ch(po,O) and
utilities are the same in both equilibria. Since c”sxo, we have g’=
x1 + s1-co 2x’ +s’ -x0 2 0. Market clearance in the free trade situation is
guaranteed by the construction of g’. The government’s budget balance is a
consequence:
= -pl .I ch(po, 0) by constant returns and trade balance
= (PO -P’)
*1
Ch(PO, 0)
using the consumers’ budget constraints.
Remark 3. Given the tax rate vector t’, there may be other free-trade
equilibria. But there can be non-uniqueness even with lump-sum transfers;
see Kemp and Wan (1986, example 4). Further, even unique equilibria can be
unstable; see Scarf (1960). Thus, there is never a reliable market adjustment
process, and policy design must assume that the government can announce and
enforce the desired consumer and producer prices directly. Then it can pick a
Pareto-better outcome, and avoid others, when there are multiple equilibria.
116
A. Dixit
and
K Norman,
Gains from
trade
Next we consider the possibility of strictly positive aggregate production
gains from trade. To this end, we have
Corollary 2. Let x0 be the aggregate output vector in autarky, and p” the
corresponding price vector. If(i) p” is not an admissible free trade equilibrium
price vector, and (ii) p#p” implies p* x>p- x0 for x~x(p), then there exists p1
such that,for selections x1 ox
and s1 us,
we have x”<<xl +s’.
Proof: This proceeds as in Theorem 1 to obtain the existence of p’ which
satisfies (4). We cannot have p’ =p” by condition (i). Then, by condition (ii),
we have I’=p’ .x1 >p’ *x0. Using (3), we see that (5) holds with strict
inequality for each i.
Remark 4.
Correspondingly,
in the construction
of Corollary
1 we can take
g’ >>o.
Remark 5. p” may turn out to be an admissible free trade equilibrium price
vector for either of two reasons. First, we may have OES(P’), i.e. our
country’s autarkic price vector is one at which the rest of the world does not
want to trade. This is the familiar case of point A in fig. 1. The second
reason is that D may have a sufficiently large flat region around x0 to absorb
s(p”), i.e. there exists x1 EX(P’) such that x1 +s’ =x0, for s1 ES(P’). In .this
case, trade merely causes an exactly offsetting adjustment in domestic
production, leaving aggregate consumption unchanged. Both are clearly
extreme cases.
Remark 6. Condition
(ii) reflects the possibility of increasing the real
national product by adapting the production plan to the new trading prices.
Theoretically, this condition is fulfilled so long as there is any neoclassical
transfoimation possibility along the production frontier in a neighbourhood
of x0. Empirically, numerous studies, e.g. Magee (1972) for unilateral tariff
cuts, and Cline et al. (1978) for multilateral reforms, show very large
production-side gains from free or more liberal trade.
We turn to the examination of when, and how, positive aggregate
production gains can be distributed to achieve a Pareto superior outcome
using commodity taxation alone. We use the above equilibrium labelled 1 as
an intermediate staging-post, and construct another free-trade equilibrium
labelled 2 which is actually Pareto superior to the initial autarky equilibrium
0. This two-step procedure is purely for analytical convenience; in practice
we would implement the final equilibrium labelled 2 directly.
We use the following condition from Weymark (1979), generalizing
Diamond and Mirrlees (1970):
A. Dixit
and K Norman,
Gains from
trade
117
Condition W There exists at least one commodity, either pure or a Hicksian
composite, such that in autarky some consumers are net buyers of it and none
are net sellers, or vice versa.
This yields the following result proved by Weymark:
Lemma I. If pO>>O, and Condition W is satisfied, then there exists a Pareto
improving direction of change in consumer prices away from autarky, i.e. a
vector R such that for suficiently small positive scalars a, the consumption
vector ch(po + CLR,0) is at least as good as ch(po, 0) for all h and better for some h.
The idea is very simple. Suppose the commodity whose existence is
assumed in Condition W is such that some consumers are net buyers while
others are on the borderline between buying and selling. A reduction in its
price benefits the former class. Any change benefits the latter class, but for a
small price change this benefit is of the second order. Similarly, if the
commodity is such that some consumers are net sellers while others are on
the boundary, a rise in its price is Pareto improving.
Only one slight caveat is necessary. Weymark avoids boundary problems
by assuming that the initial price vector p” is strictly positive. If the
commodity is a free good we cannot lower its price any further, while if all
other goods are free we cannot raise its relative price any further. But these
are extreme cases without practical interest. For sake of precision, we state
the condition as
Condition W’. There exists at least one commodity, either pure or Hicksian
composite, such that in autarky one of the following holds: (i) some consumers
are net buyers and none are net sellers of it, and it is not a free good, and (ii)
some consumers are net sellers and none are net buyers of it, and it is not the
only valuable good.
Then Lemma 1 holds with Condition
without requiring p” ~-0.
W replaced by Condition
w’,
Remark 7. In any moderately sophisticated production economy, there are
several manufactured consumption goods of which no consumer has any
endowment at all. Thus, satisfaction of Condition W’ is very easy.
Remark 8. Even in the case of a commodity bought by some consumers
and sold by others, if the government is able to tax such sales, in effect
regarding seller consumers just like firms for this transaction, then such a
commodity tax system suffices for Pareto superiority. On this see Ethier
(1983). The case of labor raised by Kemp and Wan can be handled in this
way.
118
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Dixit and K Norman,
Gains/ram trade
Remark 9. In Dixit and Norman (1980) the standard model of trade theory
is used. There all consumers sell factors and buy goods, and Condition W is
automatically satisfied for every commodity. But the result holds under the
much weaker Condition W’, as we proceed to show.
To remind readers, the idea is that if positive aggregate production gains
exist, they can be beneficially distributed to consumers. For the former we
have Corollary 2, while Condition W’ is relevant for the latter. We also need
continuity of demand functions. Thus we have
Theorem 2. Suppose the conditions of Corollary 2 are satisfied, and Condition
w’ holds in autarky. Assume the consumer demand functions are continuous
in prices. Then there exists a free-trade equilibrium with commodity taxation
that is Pareto superior to autarky.
Prooj
Let p’, etc. be as in Corollary
2, and pZ=po+an
as in Lemma
1.
Define
g2 =g’ -1
{Ch(P2,0)--Ch(PO,O)}
=x1 +s’ --~Ch(P2,0).
(6)
By continuity, and since g’ ~0, we can choose CI sufficiently small, i.e. p2
sufficiently close to p”, to ensure g2 2 0.
We claim that the situation where consumer prices are p2, producer prices
are p’, government purchases are g2 which are then thrown away, and
commodity tax rates are defined in specific form by t2 =p2 -p’, is a freetrade equilibrium
Market-clearance is guaranteed by (6), and implies the
government’s budget balance:
by reasoning similar to that employed in the proof of Corollary 1. The
equilibrium is Pareto superior to the autarky equilibrium by construction.
Remark 10. Kemp and Wan (1985) believe that for this procedure to work
it is necessary to rule out inferior goods. That is wrong; normality or
otherwise of goods is nowhere relevant in the proof.
Remark II. Kemp and Wan (1985) claim that the procedure is wasteful.
True, but all that is at stake here is the existence of a Pareto improving
A. Digit
and V. Norman,
Gains
from
trade
119
scheme of commodity taxation with free trade. It is always open to us to
choose p2 so as to minimize or, if possible, eliminate this waste of g2.
Purposive choice of x1 further enlarges the possibilities. The fact that
suitably designed commodity tax schemes can do even better than the one
constructed in Theorem 2 merely strengthens the claims .of our approach,
and weakens Kemp and Wan’s attempted refutation of the possibility of
Pareto improvement using commodity taxation.
Remark 12. Uniform poll grants can be used in the same way. Now, in the
second step, we keep consumer prices at p”, but give a uniform poll grant
small enough to keep g2 2 0. An exact analogue of Theorem 2 goes through
without requiring even the minimal condition w’.
3. Kemp and Wan’s examples
Example 1 has a pure exchange economy in which the Diamond-Mirrlees
or Weymark conditions are not satisfied. In fact the conditions can never be
satisfied in an autarkic pure exchange economy, although Kemp and Wan
do not seem to have realized this. When all the consumers’ net trade vectors
sum to zero, there must be consumers on opposite sides of the market for
every commodity. However, we do not live in a pure exchange economy, so
the example is irrelevant for discussion of policy in the real world.
Example 2 has a model with no production transformation possibility in
response to the price change from autarky to trade. This case is already
covered in Dixit and Norman (1980, p. 72): ‘if the indifference curves or the
production frontiers have kinks at the relevant points, it may be impossible
to change the consumption or production patterns to take advantage of the
changed prices. But otherwise we would expect the [revealed preference type]
inequalities to be strict. The same remarks apply to all the analyses of gains
from trade that follow.’ Later, when we consider commodity taxes on p. 79,
there is only pertinent inequality, namely eq. (20). which expresses producers’
profit maximization. Its being a strict inequality is exactly Condition (ii) of
Theorem 1. A more important point is that empirical studies show more
than enough transformation possibilities (supply elasticities) in the real world.
Example 3 has a large Ricardian economy which has zero gain from free
trade because its constant producer price sets the free trade price. It could
gain by improving its terms of trade. A consumption tax on the imported
good has just that effect. This is true, but has nothing to do with the issue
that is at stake here, namely the relative abilities of different instruments to
redistribute aggregate gains when such gains are known to be positive.
Example 4 is marred by some unfortunate errors. The coordinates of the
point C in their fig. 3(a) are (5.3,20), but the Engel curve as defined does not
pass through that point. The first component of the endowment vector
120
A. Dixit
and
t! Norman,
Gains from
trade
should be 8 instead of 5.3. The offer curve of fig. 3(b) is also wrong. The
segment C”B’ should be deleted.
On substantive matters, they keep consumer prices at (l/3,1/3,1/3),
and
trace out the Engel curve by varying lump-sum income. In the context of
trade, this becomes the offer curve when the revenues associated with the
distortions between consumer and producer prices are handed back to the
consumer as a lump sum. Thus, the example can be turned around into an
argument as to how multiple equilibria can plague a lump-sum scheme.
However, with labor supply totally inelastic, there is no difference between
lump-sum transfers and wage subsidies. Therefore the correct conclusion is
that the example is irrelevant to the question of the relative powers of the
two kinds of redistributive instruments.
The real nature of the example can best be understood by constructing the
free-trade aggregate consumption possibility envelope for it. This is done in
fig. 3. Much of Kemp and Wan’s notation is preserved, but the scale is
xi!
0
Xl
Fig. 3
A. Dixit
and I! Norman,
Gainsfrom
trade
121
changed to improve the clarity. The envelope is c@y. The portion cl? is
traced out when home production is fixed at C, and foreign offers vary along
their offer curve as the relative price takes on all values less than one. The
portion 18 has home production varying along CB while the foreign trade
offer vector remains fixed corresponding to the relative price of 1. In the
portion 8y, home production is at B, and the consumption is obtained by
adding foreign offers for steeper relative price lines.
The home country’s indifference curve through C is II. Its slope at C is
- 1 because the Engel curve for this price ratio passes through C. Now we
see at once that there are free trade points better than autarky; they lie along
the arc Cw of the envelope. But all of them have home production fixed at
C, that is to say, only pure exchange gains are available through trade. In
other words, Condition (ii) of our Theorem 1 is not satisfied. We already saw
how distributive problems can arise in such a case. We also argued that the
absence of production gains is not supported by facts about the real world.
4. Concluding remarks
In the conclusion to their paper, Kemp and Wan ask why commodity
taxes and subsidies are ‘weaker’ than lump-sum transfers. Specifically, they
ask for an explanation of the difference in the informational requirements of
the two. This point was most clearly discussed by Hammond (1979). The
difference arises in the willingness of individuals in the economy to supply
the relevant information truthfully, knowing how each scheme proposes to
use it. The net benefit to each individual under a lump-sum transfer scheme
depends importantly on his own characteristics. He therefore has a strong
incentive to manipulate his behaviour so as to mislead the planner about these
characteristics and secure a larger net transfer. Commodity taxes or subsidies, and poll grants or taxes, are not individual-specific, and the rates
depend only on the distribution of characteristics in the population. In a
large economy, each individual has a negligible effect on this distribution,
and therefore has no incentive to manipulate his behavior. In other words,
commodity taxation and poll grants are incentioe-compatible, lump-sum
transfers are not.
Finally, we can recognize a basic difference between our approach and that
taken by Kemp and Wan. Their aim is to investigate all logical possibilities,
including pure exchange economies and ones without any production
transformation
possibilities, regardless of the empirical relevance of such
constructs. Our aim is at once more humble and more practical; we are
willing to accept assumptions that are borne out by empirical studies about
the real world. In that setting, the existence and the size of aggregate
production gains from trade is of unquestionable importance. Furthermore,
the weighty objections to trade come from the owners of specific factors who
122
A.
Dixit
and
V. Norman,
Gainsfrom trade
stand to lose from production adjustments. The demonstration that such
losses can be compensated by means of a policy that is more easily
implementable than the traditional lump-sum transfers will, we hope, allow
economies to realize the large aggregate gains that are available through
freer trade.
In conclusion, we quote Paul Samuelson: ‘All economic analysis needs
careful auditing; hence, nothing but good can come from’ criticisms like
Kemp and Wan’s. Is this spirit, we thank them for forcing us to make our
argument precise. We believe that it has emerged that much stronger from
the exercise.
References
Baldwin, R.E., 1948, Equilibrium in international trade: A diagrammatic analysis, Quarterly
Journal of Economics, November, 62(5), 748-762.
Cline, W.R. et al., 1978, Trade negotiations in the Tokyo Round: A quantitative assessment
(Brookings, Washington, DC).
Diamond, P.A. and J.A. Mirrlees, 1971, Optimal taxation and public production, American
Economic Review 61,8-27 and 261-278.
Dixit, A. and V. Norman, 1980, Theory of international trade (James Nisbet, Welwyn, UK).
Ethier, W.J., 1983, Commodity taxes and the gains from trade, manuscript.
Hammond, P.J., 1979, Straightforward individual incentive compatibility in large economies,
Review of Economic Studies 47(2), 263-282.
Kemp, M.C. and H.Y. Wan, Jr., 1986, Gains from trade with and without lump-sum
compedsation, Journal of International Economics, this issue.
Magee, S.P., 1972, The welfare effects of restrictions on U.S. trade, Brookings Papers on
Economic Activity, 645-707.
Scarf, H.E., 1960, Some examples of global instability of competitive equilibrium, International
Economic Review 1, 157-172.
Weymark, J.A., 1979, A reconciliation or recent results in optimal taxation theory, Journal of
Public Economics 12(2), 171-189.