Is There an Optimal Non-Zero Level of Protectionism?
Zagros Madjd-Sadjadi
Department of Economics
University of the West Indies, Mona
&
Abdullahi Abdulkadri
Department of Economics
University of the West Indies, Mona
JEL Codes: C7, F10, P16
Abstract: Game theory is used to model actions of countries engaged in trade and examines
equilibrium outcomes resulting when each country is faced with the choice of engaging in bilateral
free trade or tariff, or unilateral free trade or tariff. The likely outcomes are evaluated under the
different scenarios that the players are absolute gains maximizers or relative gains maximizers.
The Nash equilibria for these strategic games are then evaluated to determine if they are Pareto
superior to other outcomes of the game. Various modifications of this basic model were proposed
and analyzed. First we allowed the two countries to allocate different weights to payoffs from
being an absolute gains maximizer or relative gains maximizer. We then extended the model to
include four trading countries instead of two. The Nash equilibrium of free trade obtained for the
base model remained robust for most of the various modifications introduced into the model,
although with restrictions on the payoff parameters in some instances. Some of these
modifications include adding a negotiation cost for free trade, introducing sequential moves by
countries and incomplete information. Thus the model provides testable hypotheses to assist in
determining why defection from free trade occurs in the real world.
*Canadian Economics Association Annual Meeting, June 4, 2004, Ryerson Polytechnic, Toronto. Please
contact the corresponding author at zagros.madjdsadjadi@uwimona.edu.jm
Introduction
Often times each discipline carries with it only part of the answer to a question that can
best be seen as analogous to a jigsaw puzzle. Neoclassical economists look around the world
and may wonder why there isn’t more free trade while political scientists, especially of the Realist
school of international relations, may wonder why nations trade at all. After all, think political
scientists, is not security important and are not nations interested in maximizing power, ensuring
their viability as a separate sovereign power by ensuring that they are self-sufficient? At the
same time, wonder economists, is not it obvious that the gains from trade are just as evident at
the international level as they are at the household level and when did anyone argue that
households should be self-sufficient?
Yet both disciplines suffer from the acute disease of myopia. Economists often ignore
the role that power plays in formulating government policy by assuming that countries seek
1
absolute gains in trade. Such a naïve interpretation cannot explain why countries do not engage
2
in total free trade and may leave one believing that state actors are not utility maximizers. It also
makes states little more than entities concerned with the economic well-being of its citizens. Yet
countries are not like firms. While stockholders of a corporation can freely dump the shares of a
company that is not doing well, citizens of a country cannot change citizenships with the same
modicum of time or expense. At the same time, while firm takeovers are part and parcel of the
corporate world and their relative merits are determined based upon the benefit to the
shareholder, country takeovers are almost always viewed as hostile no matter what the stated
intentions of the aggressor. Indeed, it is a staple of political science that countries seek to
1
Though they may find some comfort that there is a group of theorists within international
relations called neoliberals who generally take the same perspective. Robert O. Keohane (1988)
is the leading exponent of this perspective.
2
Of course, whenever utilities are interdependent, the first welfare theorem may not hold. Thus
competitive markets may not bring about Pareto efficient allocations. Examples of this in
economics include externalities and public goods. In a sense, free trade agreements are at least
partially public goods because they often provide rules of origin that allow for non-partner
countries to benefit as well. They, therefore, contribute externalities and if they provide positive
externalities, they can be underprovided in the competitive marketplace. In addition, total free
trade may not occur because of asymmetries in market power between countries. Most
economists, however, treat this, as a special case. Indeed, the classic theory of free trade is
almost as close to a “free lunch” as economics can get.
2
maximize power to one extent or another in their dealings with other nations. The reality is that
states probably are concerned with both power and plenty.
3
On the other hand, many scholars in political science argue that relative gains
4
maximizers will not engage in free trade. After all, think such realist and neo-realist scholars,
cooperation between states that view each other as competitors is inherently nonsensical
5
because every interaction is a zero-sum game.
Yet around the world countries appear to be
increasingly trading sovereignty for economic and social well-being through the promotion of
international organizations and agreements that transfer sovereignty to third parties such as the
World Trade Organization, the World Court, the United Nations, the European Community, the
Caribbean Community and NAFTA’s trilateral dispute panels.
Do Relative Gains Maximizers Cooperate?
Although there have been many attempts to model power, the easiest is to build a model
of relative gains.6 Under conditions of relative gains, the utility of each country is interdependent.
This paper will demonstrate that even relative gains maximizers will find it in their interest to
engage in cooperation under certain specified criteria. It begins with the standard economic
assumptions about free trade and then progressively alters them. When the gains from trade are
equally distributed and decisions are made simultaneously, then the countries can be expected to
3
See Viner (1948) for a viewpoint as to why the accumulation of wealth and power are
complementary.
4
The terms “realist” and “neo-realist” are used here to describe the so-called “realist” and “neorealist” schools of international relations scholars. A representative realist is Robert Gilpin (1977),
while the primary proponent of neo-realism (also called structural realism) is Kenneth Waltz
(1979). The key difference is that realists tend to assume power maximization while neorealists
argue that countries seek to balance power. Realists and neo-realists tend to believe that either
interpretation would transform cooperation into a zero-sum (or, at best, constant sum) game.
5
For example, Greico (1988) argues that positional concerns dominant international politics so
cooperation can be achieved only for a very narrow range of objectives. However, this position
does not take into consideration the relative gains that can be made at the expense of those who
are not engaged in cooperation. On the other hand, Snidal (1991) uses an iterated prisoner’s
dilemma approach to demonstrate that relative gains maximization is not a hindrance to
cooperation even in zero-sum situations when the number of countries is large.
6This is not to say that these gains are monetary ones. As Kirshner (1997) has noted, there are
“non-economic” reasons for actions, such as security concerns. Later in this note, there will be
an allowance for the fact that there may be both absolute and relative components of utility in a
free trade regime.
3
be playing the following “game”, where all the assumptions listed follow directly from the
neoclassical argument for free trade:
A>0
(assumption that free trade provides absolute gains over no free trade)
B>0
(assumption that unilateral free trade policy provides absolute gains)
A > B > 0 (assumption that total free trade provides more gains than unilateral free trade)
B>C
(assumption that unilateral free trade is preferred to unilateral protectionism—note that
unilateral protectionism may be preferred to bilateral protectionism or vice versa)
ABSOLUTE GAINS
FREE TRADE
TARIFFS
FREE TRADE
(A,A)
(C,B)
RELATIVE GAINS
TARIFFS
(B,C)
(0,0)
FREE TRADE
(0,0)
(C-B,B-C)
ABSOLUTE GAINS
FREE TRADE
TARIFFS
FREE TRADE
(+,+)*
(+/-,+)
TARIFFS
(B-C,C-B)
(0,0)
RELATIVE GAINS
TARIFFS
(+,+/-)
(0,0)
FREE TRADE
(0,0)*
(-,+)
TARIFFS
(+,-)
(0,0)
*Nash Equilibrium
Numerical Examples:
ABSOLUTE GAINS
FREE TRADE
TARIFFS
FREE TRADE
(10,10)*
(2,5)
RELATIVE GAINS
TARIFFS
(5,2)
(0,0)
FREE TRADE
(0,0)*
(-3,3)
TARIFFS
(3,-3)
(0,0)
*Nash Equilibrium
Notice that even under relative gains, the general trend is towards free trade. If both
countries currently have free trade, neither will choose to move away from this position because
to do so will engender a minimum of a one period loss unless the other side also chooses tariffs.
7
To do so would result in less “gain” than a free trade position. On the other hand, there is
incentive for countries using tariffs to move to free trade given that they would gain at least a one
period advantage (unless the other side also moved to free trade) and would not be penalized at
all if both sides moved to free trade.
7
It does not matter at this point if the game is finite or infinite.
4
Yet such a scenario is simply not realistic. Countries are interested in both relative and
absolute gains8 and the restrictive assumptions about the relative rankings of options make free
trade a foregone conclusion. In liberalizing these assumptions, suppose that the relative and
absolute gains division depends on a factor, q, which represents the weight given to absolute
gains. Then 1-q will be the weight given to relative gains. This leads to the following:
1 >= q >= 0
Gains from free trade with the other side having free trade:
qA + (1-q)0 = qA > 0
Gains from trade under free trade with other side having protectionism:
(q)B + (1-q)(B-C) = qB + B - qB + qC - C = B + qC - C = B-(1-q)C > 0
Gains from trade under protectionism with other side having free trade:
(q)C + (1-q)(C-B) = qC + C - qC - B + qB = C + qB - B = C-(1-q)B
ABSOLUTE/RELATIVE GAINS
FREE TRADE
TARIFFS
FREE TRADE
(qA,qA)
(C-[1-q]B,B-[1-q]C)
TARIFFS
(B-[1-q]C,C-[1-q]B)
(0,0)
Now if we drop the assumptions about the benefits or costs of free trade and the relative
9
rankings, we find that a Nash equilibrium free trade regime exists for all A > (C-[1-q]B)/q where
q ≠ 0 and q ≠ 1. This equilibrium is unique under the additional condition that B-[1-q]C > 0.
Or equivalently, if solving for q:
qA > C-(1-q)B
qA > C-B+qB
q(A-B) > C-B
B-C+Cq > 0
Cq > C-B
Where 0 ≤ q ≤ 1 :
q > (C-B)/(A-B) where (A-B) ≠ 0
and
q > (C-B)/C where C ≠ 0
8 See Powell (1991) for a discussion on why both relative and absolute gains are important to
states.
9
If we fail to relax these assumptions we find that countries that value both relative and absolute
gains generate same results as before with total free trade being the only pure Nash solution.
One important aspect of relaxing these conditions is that any possible rank order is now possible.
As such, this represents the general case and no normative implications are present.
5
This generates the necessary and sufficient conditions for Pure Nash equilibria in free
trade regimes. A Pure Nash equilibrium free trade regime exists for all relative gains maximizers
(q = 0) where B > C (unilateral free trade is preferred to protectionism). A Pure Nash equilibrium
free trade regime exists for all absolute gains maximizers (q = 1) where A > C (universal free
trade is preferred to unilateral protectionism) and this equilibrium is unique where B > C.
Negotiating Costs
The next modification is to add a discounted payoff over perpetuity, along with a
10
negotiating cost that is paid up front, n.
Given a known interest rate, r, then the discounted
payoff of a perpetuity is always benefit/r. Under this condition, free trade is not always an optimal
policy. Under non-Nash equilibrium free trade conditions, the present value payoff from free
trade must be greater than the present value payoff from defection. To this end, the payoff
determination depends crucially on what the other country is doing (engaging in free trade or
protectionism):
If the other country is engaging in protectionism, then the benefit is from unilateral free trade:
free trade benefit = (B-[1-q]C)/r
free trade “cost” = n (negotiating cost)
So if (B-[1-q]C)/r > n, the country will implement free trade. This solution depends on the
interest rate, so interest rates must be “low enough” in order to have free trade. The “free trade”
interest rate will be:
r < (B-[1-q]C)/n
10
This negotiating cost could be the cost of negotiating a free trade agreement or it could be the
cost of negotiating side agreements with domestic interests. One question that may arise is why
the benefit is discounted but not the cost. The way to solve this is to think of whether or not you
should invest $1000 or spend it. The cost is paid up front but the benefit is generated far into the
future. In some cases, the benefits from free trade may not be immediately realizable but instead
accrue only in the future. In fact, it may also be the case that free trade actually lowers utility for
the country in the short run (the 30,000 people who lose their jobs in sugar manufacturing make a
lot more noise politically than the masses who just saw the cost of their sugar decrease by 2
cents per pound, even though the economic benefit of cost reduction outweighs the economic
cost.
6
This means that if a country is a relative gains maximizer (q =0), we get r < B-C/n. If the
country is an absolute gains maximizer (q= 1), we get r < B/n.
On the other hand, if the other country is committed to free trade, the gain from trade is
between the benefit from bilateral free trade and unilateral protectionism:
Free trade benefit = (qA – (C-[1-q]B))/r
Free trade “cost” = n (negotiating cost)
So if qA/r – (C-[1-q]B) > n then the country will implement free trade. This solution depends
on the interest rate so interest rates must be “low enough” in order to have free trade. The ‘free
trade’ interest rate will be:
qA > r(n + C-[1-q]B)
r < qA/((n + C-[1-q]B))
At the limit, if the other country is committed to free trade:
where q = 0, then r < 0
and where q = 1, then r < A/(n + C)
Thus if the gains from trade are equally divided and the other country is committed to free
trade, relative gains maximizers will institute free trade only when real interest rates are negative,
while absolute gains maximizers will institute free trade only when they are sufficient low, i.e., the
gains from trade dominate over the cost associated with the transition. An important implication
is that even absolute gains maximizers will not implement free trade when real interest rates are
“too high.”
Unequal Terms of Trade
Now the assumption will be that the gains from trade differ between countries, such that
the second country’s gains are multiplied by a factor k to arrive at the first country’s gains.
Furthermore, the preference for absolute versus relative gains may differ between countries such
that country 1 has a factor q but country 2 has a factor p. This results in the following:
ABSOLUTE GAINS
FREE TRADE
TARIFFS
FREE TRADE
(kA,A)
(kC,B)
RELATIVE GAINS
TARIFFS
(kB,C)
(0,0)
FREE TRADE
(kA-A,A-kA)
(kC-B,B-kC)
7
TARIFFS
(kB-C,C-kB)
(0,0)
ABSOLUTE/RELATIVE GAINS
Free Trade (country 1) under free trade (country 2)= q(kA)+(1-q)(kA-A)= qkA+kA-qkA-A+qA =
kA+qA-A = (k+q-1)A
Free Trade (country 2) under free trade (country 1)= p(A)+(1-p)(A-kA) = pA+A-pA+pkA-kA =
A+pkA-kA = (pk+1-k)A
Free Trade (country 1) under non-free trade (country 2) = q(kB)+(1-q)(kB-C) =
qkB + kB - C -qkB + qC = kB-C+qC
Non-Free Trade (country 2) under free trade (country 1) = p(C)+(1-p)(C-kB) =
pC+C-kB-pC+pkB = C+pkB-kB
Non-Free Trade (country 1) under free trade (country 2) = q(kC)+(1-q)(kC-B) =
qkC+kC-B-qkC+qB = kC-B+qB
Free Trade (country 2) under Non-Free Trade (country 1) = p(B)+(1-p)(B-kC) =
pB+B-kC-pB+pkC = B-kC+pkC
Non-Free Trade (country 1) under non-free trade (country 2) = 0
Non-Free Trade (country 2) under non-free trade (country 1) = 0
FREE TRADE
TARIFFS
FREE TRADE
([k+q-1]A,[pk+1-k]A)
(kC-B+qB,B-kC+pkC)
TARIFFS
(kB-C+qC,C-kB+pkB)
(0,0)
In this case, free trade will be a unique Nash equilibrium only when when:
kA+ (q-1)A > kC + (q-1)B
kB + (q-1)C > 0
B + (p-1)kC > 0
and A + (p-1)kA > C + (p-1)kB
This will be true when:
k >((1-q)A)/(A-C)
k > ((1-q)C)/B
k > B/(1-p)C
and k > (C-A)/(A+(1-p)B)
Using the first country as the test case:
For absolute gains maximizers (q=1), this means that free trade will be preferred whenever:
k > 0 (there are gains to trade)
For relative gains maximizers (q=0), free trade is preferred whenever:
k> A/(A-C) and k > C/B
For such countries, assuming the gains from trade are positive (k>0), the more they receive
gains from trade relative to the other country, the more likely that they will accept free trade.
8
Thus the terms of trade matter to the relative gains maximizer while they do not to the absolute
gains maximizer. But, more critically, the terms not only have to be in favour of the relative gains
maximizer but even if the terms of trade are favourable, the relative gains maximizer may not
always institute free trade. For example, the gains from free trade must be greater by at least the
factor k than any perceived gains under unilateral protectionism. Similarly, the gains from free
trade must be greater than the ratio of the gains from unilateral protectionism divided by the gains
11
from unilateral free trade.
Of course, if the gains from unilateral protectionism are negative and
the gains from unilateral free trade are positive, relative gains maximizers will institute free trade
as well unless they are not receiving “enough” of the terms of trade. In such a case, a relative
gains maximizer will still not institute free trade unless k> A/(A-C), meaning that the less the harm
from engaging in protectionism, the greater the terms of trade must tend towards equality for the
relative gains maximizer to engage in trade. In other words, enemies will only trade if the terms
of trade are relatively balanced but allies will trade much more frequently. Thus, the United
States, which has little perceived benefit from trade with Cuba would require a trade agreement
that provides it with very close to an even split in the terms of trade (or a favourable terms of
trade agreement) but it would not require such an advantage vis a vis Canada, where the desire
for relative gains is probably much less. Similarly, Cuba will likely demand a very even split in the
terms of trade vis a vis the USA but not with Canada. This helps to explain why the United States
is the predominant country imposing trade sanctions against “enemies” and vice versa because
the relative gains from trade matter to these countries.
Free Trade Agreements
Now, an assumption will be made that there are four countries in the world and two of
them are considering a free trade agreement. A further supposition will be made that there is an
equilization of trade among all of the countries. Now what happens? Well, in this case, each of
the absolute gains will be divided in half and the relative gains, instead of being simply the
11
It should be pointed out there can be gains from unilateral protectionism. After all, if one
country is engaged in free trade then both countries can benefit, although the benefit may not be
as great as if both countries had free trade policies. All this means is unilateral protectionism is
preferred to bilateral protectionism (tariffs by both parties).
9
subtraction of the other side’s gains from the current gain will also account for the absolute gain
against the other two non-participating countries, according to the following formula:
relative gains = gains by country 1 - gains by country 2 + absolute gains by country 1 x 2 (for the
relative gains against the other two countries)/3 (the number of other countries).
ABSOLUTE GAINS
FTA
NO FTA
FTA
(5,5)
(-5,2.5)
NO FTA
(2.5,-5)
(0,0)
RELATIVE GAINS
FTA
(3 1/3,3 1/3)
(-10 5/6, 9 1/6)
NO FTA
(9 1/6, -10 5/6)
(0,0)
In this case, under both absolute and relative gains, the Free Trade Agreement is
beneficial and produces a Nash equilibrium.
However, while absolute gains and losses are
halved in such a situation, the relative gains payoffs will actually increase for those in the free
trade agreement while the benefits from unilateral free trade and unilateral protectionism will
decrease. Thus, if gains from trade are no longer divided equally but are instead distributed
unequally, there will be greater cooperation as the number of states increases. Thus, while
countries outside of the free trade zone will always want in, countries in the free trade zone may
not always want them to join. Instead, the number of countries allowed to join the free trade zone
will increase until a maximum is reached where the benefits from cooperation with some partners
are exactly balanced by the benefits from exclusion of others.
In general, where A would be the total gain generated if all countries were free traders
and where there are a total of n countries and with p countries being the number outside the free
trade zone and assuming that all countries are equal in size and that there are equally divided
gains from trade, the countries inside the free trade zone will allow other countries to join as long
as:
A((n-p+1)/n)((p-1)/n-1) > A(n-p)/n)(p/n-1)
This reduces to: p > (n-1)/2
Thus, even relative gains maximizers will seek free trade up to a certain optimal point
when the payoffs are evenly distributed among countries.
Of course, if there are great
differences in payoffs between countries, cooperation may still not occur. This does not mean
that worldwide agreements will fail. To the contrary, if countries cannot exclude other countries
10
from joining (such as the WTO), we might see a “snowball” effect once the threshold is reached.
That is because the greater the number of countries that are party to a cooperative agreement,
the greater the value of that agreement for those who are not part of it in a realist world. Indeed
one of the great ironies of the realist argument is that in seeking relative gains, countries will also
not want to be put in a relative disadvantage.
Thus, international cooperation may actually
increase as the number of countries already committed to a policy increases. This effect will be
seen only under conditions of relative gains, not absolute gains, because under the former
countries care about the relative positioning of themselves vis a vis other countries while under
absolute gains, they do not.
Returning to the case of differences existing between the preferences for relative and
absolute gains by country but including the idea that each country has the same payoff functions,
the following results are generated:
ABSOLUTE/RELATIVE GAINS
FREE TRADE
CURRENT SYSTEM
FREE TRADE
(qA,pA)
(C-[1-q]B,B-[1-p]C)
CURRENT SYSTEM
(B-[1-q]C,C-[1-p]B)
(0,0)
A Nash equilibrium free trade solution exists for only a subset of the results in the first
absolute/relative gains maximization problem. Thus, if preferences differ, a Nash equilibrium free
trade solution exists in all cases where:
Where 0 ≤ q ≤ 1 :
q > (C-B)/(A-B) where (A-B) ≠ 0
and
q > (C-B)/C where C ≠ 0
and
Where 0 ≤ p ≤ 1 :
p > (C-B)/(A-B) where (A-B) ≠ 0
and
p > (C-B)/C where C ≠ 0
If one side is a relative gains maximizer (q = 0) and the other side is an absolute gains
maximizer (p = 1), Nash equilibrium free trade regimes exist if and only if B > C and A > C, and
the equilibrium is unique.
11
What if the gains from trade are not equally distributed? In that case, we have the
following game:
ABSOLUTE GAINS
FREE TRADE
TARIFFS
FREE TRADE
(a1, a2)
(c1,b2)
RELATIVE GAINS
TARIFFS
(b1,c2)
(0,0)
FREE TRADE
TARIFFS
(a1-a2, a2-a1) (b1-c2,c2-b1)
(c1-b2,b2-c1) (0,0)
For the absolute gains maximizer, we still have the case where if the gains from total free
trade dominate, the country will choose free trade.
However, things have gotten a little more complicated for the relative gains maximizer. In this
case, the more the relative differences between the two countries, the less likely that they will
agree to free trade. This is because for a Nash equilibrium free trade regime to exist, it must be
true that:
a1-a2 > c1-b2 and a2-a1 > c2-b1
Rearranging terms, this implies that:
a1- c1 > a2-b2 and a2-c2 > a1-b1
Therefore, the gains that each country makes from total free trade over unilaterial
protectionism must be greater than the gains that the other country gets from going from
unilateral free trade to total free trade.
However, this does not guarantee a unique Nash
equilibrium. A unique Nash equilibrium also requires the following condition be true:
b1 - c2 > 0 and b2 - c1 > 0
Therefore, b1 > c2 and b2 > c1 .
This implies that the gains from unilateral free trade for each country must be greater than
the gains from unilateral protectionism for the other country. Now, going back to our original
equations, we know that, for a free trade regime to exist:
a1 - a2 > c 1 – b2
a2 - a1 > c2 - b1
Suppose that the country that is relatively better off is country 1. Then a1 - a2 > 0 and a2 - a1
< 0. If a2 - a1 < 0 and a2 - a1 > c2 - b1, then c2 - b1 < 0. Therefore b1 > c2. So one of the conditions
is true. Therefore, the only question is whether the other condition is true. In that case:
12
a1 - a2 > c 1 – b2
However, if c1 – b2 is greater than 0 (implying that c1 > b2 which means that the gains from
unilateral protectionism for one country exceed the gains from unilateral free trade for the other
country) we do not have a unique pure Nash Equilibrium.
What about the mixed strategy case? In that case, the payoff for Row is c(a1 - a2) + (1-c) (b1c2) + c(c1 – b2)=ca1-ca2+b1-c2-cb1+cc2=cc1–cb2.
Therefore:
c(a1-a2-b1+c2-c1+b2)= c2-b1
Therefore:
c = (c2 - b1)/(( a1 - a2) + (b2 - b1) + (c2 - c1))
This implies that we have a mixed strategy Nash equilibrium whenever:
(c2 - b1) > 0 and (a1 - a2)- (c1 – b2) > 0
OR
(c2 - b1) < 0 and (a1 - a2)- (c1 – b2) < 0
This implies that for column to have a mixed strategy, it must be that free trade is
preferred to tariffs by row when column chooses free trade and that column receives more benefit
than row when row has unilateral free trade OR if column receives less benefit than row when
row has unilateral free trade, then tariffs must be preferred to free trade by row when column
chooses free trade.
Sequential Moves by Countries
Until now, we have only considered cases in which countries take decisions on whether to
engage in free trade or to impose tariff without knowing what their trading partner or competitor
has decided to do. In reality, such decisions are made sequentially and the outcome of a trade
negotiation may be markedly altered when one of the countries has committed, a priori, to either
a free trade or to a tariff regime. There may also be a first-mover advantage or disadvantage, as
the case may be, as one country realizes a higher (in the case of first-mover advantage) or lower
13
(in the case of first-mover disadvantage) payoff when it first commits to a regime regardless of
what the other country does.
For absolute gains maximizers, a sub-game perfect equilibrium with a bilateral free trade
outcome will always be attained under the following conditions whenever country 1 moves first:
a1>c1
a2>c2
and
b2>0.
These require that Country 1’s gains from bilateral free trade be higher than its gains from
unilateral tariff (a1>c1), and country 2’s gains from bilateral free trade be higher than its gains from
unilateral tariff (a2>c2), in addition to country 2 having strictly positive gains from unilateral free
trade (b2>0).
The gains to country 1 from unilateral free trade (b1) has no effect on this
equilibrium whenever country one makes the first move.
However, when we consider the case of relative gains maximizers, a more interesting
result emerges. A sub-game perfect equilibrium with a bilateral free trade outcome will only exist
under the following conditions whenever country 1 moves first:
a1-a2 > c1-b2
a2-a1 > c2-b1
and
b2-c1>0.
These require that country 1’s gains from switching from unilateral tariff to bilateral free
trade be higher than that of country 2’s gains from switching from unilateral free trade to bilateral
free trade (a1-c1>a2-b2) and country 2’s gains from switching from unilateral tariff to bilateral free
trade be higher than country 1’s gains from switching from unilateral free trade to bilateral free
trade (a2-c2>a1-b1). Also, country 2‘s gains from unilateral free trade must be higher than country
1’s gains from unilateral tariff (b2>c1). The implication of these conditions for the sub-game
perfect equilibrium bilateral free trade regime is that relative gains maximizers will always have
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zero payoff whenever their gains from free trade are identical, otherwise one country derives a
strictly positive payoff while the other derives a strictly negative payoff, although the country with
a negative payoff will still be better off under bilateral free trade than any other outcome.
Incomplete Information and International Cooperation
The final modification adds incomplete information to the mix. We assume that countries
can correctly estimate the gains they will receive but imperfectly estimate the gains made by
others. In this case, the actual benefit is listed using lower case but the perceived benefits are
listed using upper case:
ABSOLUTE GAINS
FREE TRADE TARIFFS
RELATIVE GAINS
FREE TRADE
TARIFFS
FREE TRADE (a1 , a2)
(b1 , c2)
(a1 - A2, a2 – A1)
(b1 - C2, c2 – B1)
TARIFFS
(0,0)
(c1 - B2, b2 – C1)
(0,0)
(c1 , b2)
When will we institute free trade in this case? Now the results become even more cloudy.
In general, there is a single unique Nash equilibrium of free trade when:
a1 - A2 > c1 - B2 and b1 - C2 > 0 and b2 – C1 > 0 and a2 – A1 > c2 – B1
In this case, perceptions form reality. In cases where countries believe that they are
gaining relative to other countries, they will institute free trade but when they believe that they are
doing worse relative to other countries, they will not. This will only harm cooperation when
countries overestimate the gains from trade accruing to others. Of course, as the number of
countries increases, relative gains must be measured against both countries within and outside
the free trade area.
Conclusion
The models provided in this article improve upon the existing literature by providing the
general form solutions to the game-theoretic problem of relative gains maximizers and the ability
to cooperate on the international level. It shows that when rank ordering of options is relaxed,
relative gains maximization can result in a lack of free trade.
Additionally, when there are
transaction costs associated with implementing trade agreements, there may be little reason to
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pursue free trade. Indeed, one could argue that as trade agreements multiply, the gains from
concluding additional agreements will decrease and the costs will increase. This is because one
would likely reach agreements on those issues that are most beneficial to both sides first and
would defer negotiation on more difficult matters that will likely cost more in terms of negotiating
costs. Thus, since the marginal cost of implementing trade agreements is rising as more trade
agreements are reached and since the marginal benefit from those trade agreements is falling,
this implies that there is some optimal level of protectionism that is not zero (and, by extension,
an optimal level of free trade that is not complete). This should not be a surprise conclusion to
economists who are used to seeing arguments in favour of optimal non-zero levels of pollution
and crime.
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