EXPORT SUBSIDIES, REACTION FUNCTIONS,
AND THE CONVEXITY OF DEMAND
Thomas E. Duggan
B.A., Colorado State University, 1988
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillment
of the requirements for the degree of
Master of Economics
Department of Economics
t 4 a.
This thesis for the Master of Economics
Thomas E. Duggan
has been approved for the
Department of Economics
Steven R. Beckman
Duggan, Thomas E. (M.A. Economics)
Export Subsidies, Reaction Functions, and the Convexity
Thesis directed by Assistant Professor Steven R. Beckman
Export subsidies can be an attractive policy
component from a domestic standpoint. Whether export
subsidies are in fact beneficial or detrimental depends
on the type of market structure present and also on the
convexity of demand. In the case of perfect competition,
export subsidies are always destructive to domestic
When a foreign monopolist is present, an optimal
tariff exists when demand is concave or linear. However,
when demand reaches a certain degree of convexity the
optimal government policy is to subsidize imports.
In an intuitive paper, Brander and Spencer claim
that under the condition of Cournot duopoly an export
subsidy is always welfare-enhancing for the domestic
market unless demand is "very convex. A limiting
assumption that Brander and Spencer use in their analysis
of export subsidies is strategic substitution which gives
rise to downward-sloping reaction functions. When
strategic complementarity is assumed, the reaction
functions are upward-sloping. Upward-sloping reaction
functions result in the overturning of Brander and
Spencer's results. This thesis demonstrates that the
degree of convexity under which strategic complementarity
is present is so small that virtually any degree of
convexity can lead to the overturning of Brander and
The form and content of this abstract are approved. I
recommend its publication.
I. INTRODUCTION .................................... 1
II. PERFECT COMPETITION ............................. 5
The Effects of an Export Subsidy on a
Small Country Under Perfect Competition ... 5
Income Redistribution Effects
of an Export Subsidy............................9
The Model for the Optimal Tariff...............19
Demand Structure: Tariff or Subsidy? .... 22
IV. Cournot Duopoly.................................3 0
Why Strategic Complementarity
Cannot Be Considered Perverse ................ 50
BIBLIOGRAPHY .......................................... 60
International competitiveness has increasingly come
to the fore over the last decade. It is commonly felt
that countries are not doing enough to enhance their
international competitiveness. In several prominent
industries, American companies facing stiff competition
from Japan have sought and received government
protection. American workers have also lobbied for job
protection from inexpensive third-world labor. Workers
demand action from their politicians who in turn have
attempted to enact various protectionist policies to
protect American jobs.
It is not only the competitiveness of the U.S. worker
that is at the heart of our international concerns; we
are also concerned about the competitiveness of U.S.
products. U.S. producers are constantly complaining to
the government that not enough is being done to enhance
the international competitiveness of American goods.
Again, various policies have been adopted to attempt to
correct the situation, as evidenced by the seemingly
infinite number of tariffs and quotas in place to cure
the U.S. merchandise trade deficit.
The American public, along with a majority of
politicians, believe that other governments, such as
Japan, have been undertaking policies that support their
own products and workers at the expense of the U.S. The
U.S. public wants our government to adopt similar
policies to combat Japan. There is a definite trade war
being waged in the international marketplace. The U.S.
is making some headway on the front lines, but the
general consensus in America is that more weapons are
needed in this trade war.
If we accept the assumption of the need for
additional trade weapons, the question becomes what
policies should be undertaken. The politicians are
turning to economists for the answers, and the economists
are cranking out policy prescriptions. The literature is
becoming increasing intensified in micro-based
international trade. Two outstanding economists are
leading the way in the drive for more helpful trade
policies. James A. Brander and Barbara J. Spencer have
been leading the way in the theoretic use of tariffs and
subsidies designed to confer the international advantage
on the domestic market.
In a revealing 1985 article Brander and Spencer
demonstrate that an export subsidy awarded to a domestic
Cournot duopolist can be an effective weapon by
conferring the advantage of Stackleberg leadership on the
home Cournot duopolist. The article appears convincing
at first glance. However, Brander and Spencer make some
rather limiting assumptions that I do not believe to be
theoretically justifiable. They assume two Cournot
duopolists, one domestic and one foreign, who produce for
consumption in a third, consumption-only, market.
Brander and Spencer claim that given these assumptions
and any relevant set of demand conditions the optimal
government policy is to subsidize exports.
Brander and Spencer's conclusion hinges on what I
believe to be an unreliable assumption. That assumption
is strategic substitution. This is the idea that as the
output of one firm increases the marginal profit of the
other firm is decreased. Brander and Spencer claim that
this assumption holds for most economically relevant
cases, and is violated only if demand is 'very' convex.
Given convex demand, however, strategic complementarity
is a real possibility.
The objectives of this thesis are two-fold. The
first, and main, objective is to demonstrate that the
existence of strategic complementarity cannot be
considered economically perverse and therefore must be
thoroughly addressed. The second objective of this
thesis is to undertake an in-depth study of export
subsidies with respect to the three market conditions of
perfect competition, imperfect competition, and monopoly.
Chapter II of this thesis analyzes the government's
choices of an export subsidy versus a tariff in a
perfectly competitive international market. Chapter III
of this thesis analyzes the optimal government policy to
maximize the domestic welfare when the home market
imports a good produced by a foreign monopolist. Chapter
IV, the most relevant section of this thesis, will
demonstrate that strategic complementarity cannot be
safely ruled out. It also will address the comparative
static results that are obtained when the assumption of
strategic complementarity is invoked. Chapter V contains
the conclusions of this thesis.
The Effects of an Export Subsidy
on a Small Country Under
An export subsidy is a payment given to a home
exporter by the home government. It is well known that
subsidies in a perfectly competitive market do more harm
than good. But, if one of the groups hurt is foreign,
perhaps subsidies of exporters may be nationally
beneficial. In the case of a small country, as I assume,
the export subsidy will not affect the world price of the
good. This is so because the home country's exports are
a minuscule portion of the total world supply.
Therefore, a small country's trade policy has no effect
on world prices. The effect of the export subsidy is
illustrated in Figure 2.1.
Figure 2.1 The effects of an export subsidy on a small
country under perfect competition.
Before the subsidy, the home country produces the
amount 0-QS. With the world price of the good at WP,
domestic consumption of the good is the amount 0-QDH.
Exports of the home country, then, are the amount QDH-QS.
Now the domestic government offers an export subsidy to
the home firm. The export subsidy shifts out the home
firm's supply curve from S to S'. The vertical distance
between S and S' is the amount of the export subsidy.
The export subsidy brings additional production into the
market that would not otherwise be present in the absence
of the subsidy. All of this extra production is exported
to foreign countries. Exports increase by the amount QS-
QS'. The export subsidy does not alter the consumer's
choice of consumption in the home country because they
still buy at the world price. They continue to consume
the amount O-QDH.
This policy of subsidizing exports has both costs
and benefits that must be weighed so that the net effects
for the home country can be determined. The cost of the
export subsidy to the home government is equal to the
dollar amount of the subsidy, the vertical difference
between S and S', multiplied by the amount of exports.
This cost is shown by areas a+b.
While the subsidy costs the home government, it
benefits the home producers. Producers benefit when the
price they receive exceeds costs. Since the supply curve
reflects costs, the area between price and supply is
known as producers' surplus. At the margin, price equals
cost in competition, but the intramarginal producers may
be extracting rent. That is, the producers' surplus
arises from the intramarginal producers' willingness to
supply the exports for a lower price than they actually
receive, which is the world price plus the export
subsidy. The marginal producer receives no producer's
surplus because the world price plus, the.subsidy was just
sufficient to induce this producer to supply the product.
Drawing in inefficient producers results in a net
welfare loss from the subsidy. The export subsidy
increases home producers' surplus by the amount of area a
in figure 2.1. This is the increase in revenue over and
above the increase in costs where costs are represented
by the supply curve. Note that the benefit is smaller
than the government's cost of a + b. The net welfare
loss is the area b. This weighting assumes that a dollar
of benefits to producers is equal to a dollar of costs to
the government, which may not be true due to opportunity
costs. This welfare loss arises from the extra
production that the export subsidy encourages. In the
absence of the subsidy the extra production is not
present because it is inefficient. Therefore, the export
subsidy encourages inefficient production. The cost to
society of subsidizing this inefficient production is
shown by area b in figure 2.1. An export subsidy is not
an attractive trade policy to a small exporting country
under perfect competition it results in a net welfare
loss to society.
There is an exception to this general rule. If the
home government is producing something that has side
benefits not paid for by the consumers, it may well be
beneficial to subsidize. For example, there may be
unemployment in some region so that the added resources
used and paid for are not really a cost to society.
Income Redistribution Effects
of an Export Subsidy
Even though an export subsidy is not welfare
enhancing, there may be certain factors of production
that gain from an export subsidy. Stolper and Samuelson
(1941) demonstrate that tariffs hurt some groups and
benefit others. Since one of the usual arguments for
protection is that it increases home wages, it is useful
to know when the argument is correct. Their argument is
readily adapted to subsidies because they analyze the
impact of a price change on rewards. The price change
could be due to tariffs, subsidies, technology, or other
The model will be presented in two stages. In the
first stage, assume firms use inputs in the same
proportion regardless of the prices of inputs. That is,
assume Leontief production functions. In the second
stage, input price changes will alter input ratios, but
all the properties derived will be local. Throughout,
two goods are produced by two inputs. Perfect
competition is also assumed, so that price is equal to
cost per unit of production and cost equals wages times,
input use per unit of production.
Pi^BnWi + Bi2w2 (1)
P2=B21w1 + B22w2 (2)
Pi = price of good i.
Bij = input of factor j needed to produce 1 unit of good i
and is fixed by technology.
wi = payment to factor i.
p = cost per unit of production.
This model can be rewritten in matrix form.
Pi "feu ^12 Wi
_P2. .^21 tr to to 1_ .W2_
an alternate expression is:
p = Bw
p = vector of prices
B = matrix of input requirements
w = vector of factor payments
A solution of the matrix system for the vector w will
allow inferences to be made about the effect of an export
subsidy on factor payments.
w = B'1 P
Wj" = i/ (^11^22 ^12^2l) ^22 ^12 Pi'
W2. ."^21 fell. .P2.
W1 ^22Pl t>12P2
W2 t)llP2 t*2lPl
The determinant, D, >0 if good 1 is factor 1 intensive
and good 2 is factor 2 intensive.
fell > fe21
b12 b22 this implies that bnb22 > b21b12
Therefore, if good 1 is factor 1 intensive and good 2 is
factor 2 intensive, the determinant is strictly positive.
Given this positive determinant, the following conditions
dwi 22 >0 dwx = ^fel2
dpi D dp2 D
dw2 = fen >0 dw2 = r^2i
dp2 D dpi D
These expressions say that as the price of a good
increases the reward to the factor used most intensively
in the production of that good also increases, while the
reward to the other factor declines.
The algebra of this model can also be shown in a
diagram. In figure 2.2, the lines represent the amounts
the firm can afford to pay the two factors. The more one
factor is paid the less the other can be paid. This
explains the downward slope of the two lines. The slope
reflects the relative use of each factor. If one factor
is used heavily, then a small change in its reward
requires a large change in the other factors reward.
Where the two lines cross, factor payments equal costs in
both industries. If both factors are mobile between
industries, this is the only possible equilibrium.
Higher rewards to the same factor in one industry would
encourage relocation of the factor to the industry
with the higher reward. This relocation would continue
until rewards between industries for the same factor are
Figure 2.2 The redisributive effects of an export
subsidy offered to only one factor of production
Pi = bnW, + b12W2 CO II : b21Wi + ^22^;
Wi = _Pi_ 12 ^2 Wx = -P2- + 22
bn bn ^21 ^21
This diagram shows that good 1 is factor 1 intensive
zhzz > Z&iz
If the price of good 1 rises, the reward to factor 1
rises while the reward to factor 2 falls. This is
consistent with the matrix algebra conclusions.
The effects of an export subsidy on the rewards that
each factor receives can now be analyzed. The price that
each producer receives is equal to P. This is also the
cost per unit of production. Now assume that the home
government offers an export subsidy to the producers of
good 1 and not good 2. This will have no effect on the
world price of the good because this country is so small
that it can not alter world prices. However, the export
subsidy will have the effect of increasing the price, P,
that each producer receives by the amount of the subsidy.
This will have the effect of increasing the reward to the
factor of production used most intensively in the
production of good 1, as shown by the condition dw^dp!
>0. Simultaneously, the reward to factor 2 falls as the
price of good 1 increases, due to the export subsidy, as
shown by the condition dw2/dp1 <0.
An export subsidy will, then, have redistributive
effects if only offered to the production of one good.
The redistributive effects are to increase the reward to
the factor of production that is used most intensively in
the production of the good that is offered the export
subsidy. The reward to the factor that is used less
intensively in the production of the good that is offered
the export subsidy is decreased.
So far, it is assumed the change in factor reward
left input use unchanged. Input substitution can be
allowed using Shepard's Lemma. Shepard's Lemma is shown
by the form:
c(w,x) wv* = 0. (8)
c(w,x) = a cost function and is the minimum cost of
producing a certain bundle of outputs x, given factor
prices w. Let v=inputs. Producers choose inputs in
order to minimize the cost function. When input choices
are optimal, v = v*, and c(w,x) = wv*. At any other input
combination, v', c(w,x) < wv' since any other input
combination must be cost increasing. This function
reaches a minimum at 0 because minimum costs are always
equal to or less than an optimal choice of wv*. The cost
function is equal to wv* only when v, the chosen amount
of inputs, is optimal. The first order condition for
<5c(w,x) v* = 0 if v is optimal. (9)
This implies v* = <5cCw.x^ (10)
Now let x = vector of ones, then v = inputs per unit of
output. With perfect competition still being assumed,
the price of a good is equal to its minimum cost of
production, as shown below:
p = c (w,x) .
Totally differentiating this condition:
Sw Sw matrix B.
Substituting the input requirement matrix B into the
expression, the expression simplifies to:
dp = B dw.
Rearranging to solve for the vector of factor payments,
dw = B"1 dp. (11)
This proof assumes small changes in the vectors of prices
and factor payments. If these small changes are not
assumed the differential becomes only a crude
approximation. This matrix expression has the same form
as the matrix from the Leontief case. Therefore,
allowing input substitution does not change the local
properties. However, if the price and input changes are
large, it is possible the sign of the determinant
reverses. This is known as factor intensity reversal.
Therefore, an export subsidy offered to a home firm
will not be optimal for the whole society because the
cost to society, the export subsidy, is not offset by the
increase in producers' surplus. The loss comes about
because of the inefficient production that the export
However, certain sectors of the economy can benefit
from the export subsidy if the subsidy is offered only to
the producers of one good and not to the production of
all other goods. As shown above, the factor that is used
most intensively in the production of the good that is
offered the export subsidy will benefit from the
introduction of an export subsidy.
In their 1984 paper entitled "Trade Warfare: Tariffs
and Cartels," Brander and Spencer offer policy
prescriptions to national governments designed to allow
the extraction of rents from a foreign exporting
monopolist. Brander and Spencer effectively demonstrate
that the optimal policy is either a specific tariff or
subsidy. Whether a specific tariff or import subsidy is
optimal depends on the underlying demand conditions.
Brander and Spencer conclude that a subsidy is optimal if
demand is 'very convex,1 as with constant elasticity
demand. Otherwise, claim Brander and Spencer, a specific
tariff will be optimal.
The Model for the Optimal Tariff
The model is the same one that Brander and Spencer
use in their paper. Assume, along with Brander and
Spencer, that the foreign monopolist produces for
consumption in the home market. Further assume that the
foreign monopolist is small compared to the home economy
so that partial equilibrium is appropriate. Domestic
demand for the imported good arises from a utility
function that can be approximated by the form
U = u (X) + m, (1)
where X is the consumption of the imported good and m is
the expenditure on other goods. As Brander and Spencer
note, this approximation for both positive and normative
analysis assumes away a number of theoretical
difficulties, such as income effects, aggregation
problems, and second-best problems induced by other
distortions in the economy. I will accept Brander and
Spencer's reasoning for assuming away these theoretical
difficulties. Brander and Spencer explain that these
problems are difficult but reasonably well understood and
trying to deal with them would obscure the focus of their
Inverse demand is the derivative of the utility of
consuming the imported good, u(X)
p = u1(X); P' < 0, (2)
where p represents the price of the good. Consumer
surplus, the difference between what consumers are
willing to pay for the good and what they actually pay,
u(X) pX, is a measure of the benefit to the consumers
from consuming the good X. The net domestic gain from
importing the good, with specific tariff t, is shown by:
G(t) = u(X) pX + tX. (3)
This domestic gain function has two interesting
components: consumer surplus and tariff revenue. It is
assumed that one dollar of consumer surplus counts the
same as one dollar of tariff revenue. The amount of good
X that will be purchased by domestic consumers depends on
the utility from consuming the good as well as on the
level of the tariff levied on the foreign monopolist.
The optimal specific tariff can be found by
differentiating the domestic gain function with respect
to the tariff, dG/dt, and setting it equal to zero (with
subscripts denoting derivatives):
Gt = u'Xt ptX -pXt + tXt + X = 0.
Using (2) and rearranging:
Gt = -X(pt 1 tXt/X) =0. (4)
Letting ii = -txt/ which is the elasticity of imports with
respect to the tariff, (4) can be rewritten as:
1 Pt = M- (5)
Condition (5) shows when the tariff imposed on the
foreign monopolist will be optimal, given that the second
order condition, Gtt < 0, is satisfied.
The elasticity of imports with respect to the tariff,
and pt, the effect of the tariff on the price of the
good that consumers must pay for the good, must sum to
one at the optimum. The term pt also shows the reduction
in consumer surplus that accompanies a tariff increase.
The elasticity of imports with respect to the tariff, /x,
is likely to be less than one because as the tariff is
increased, the price of the good is also likely to rise.
In other words, pt is positive.
Expression (4) can be solved for t which will yield
the tariff that maximizes the net domestic gain.
t* = X(pt 1)/Xt (6)
Expression (6) is the optimal tariff condition where t*
is the optimal tariff. The optimal tariff, t*, can also
be shown as a function of the producer's price, p t.
Differentiating the producer's price with respect to the
qt = Pt 1*
Substituting that result back into (6) gives the optimal
tariff as a function of producer's price.
t* = Xqt/Xt (6')
With Xt being negative and X being positive, from
(6), a positive optimal tariff requires that pt < 1.
This is equivalent to the price of the good rising by
less that the amount of the tariff. If pt > 1, the price
of the good rises by more than the amount of the tariff.
In that case, the optimal tariff is negative and imports
should be subsidized. Similarly, looking at (61) if the
effect of the tariff is to reduce the producer's price,
qt < 0, the optimal tariff, t*, is positive. If, on the
other hand, the effect of the tariff is to increase the
producer's price, qt > 0, then the optimal tariff is
again negative and imports should be subsidized.
As this section shows, whether a specific tariff or
subsidy is optimal for the net domestic gain depends upon
the effect of the tariff on the consumer's or producer's
price of the good. The next question that needs to be
answered is What is the effect of the tariff on the
producer's and consumer's price of the good?
Demand Structure: Tariff or Subsidy?
Brander and Spencer set up this section of their
paper by assuming that the foreign monopolist produces
goods for sale in the importing country's market only.
This makes it possible to handle conditions of non-
constant marginal cost conditions rather easily.
Therefore, the variable profit for the foreign monopolist
in the domestic market is:
7r = Xp(X) c (X) tX. (7)
With c' denoting marginal cost, the first-order condition
for a profit maximization by the foreign monopolist is:
7rx = p + p'X-c' t = 0. (8)
To ensure that this level of output is a true profit
maximization the second-order condition needs to be
7TXX = 2p' + Xp' 1 c' < 0.
With the second-order condition being satisfied,
Brander and Spencer rearrange the second-order condition
ttxx = p' (2 + R) c" <0, (9)
R = Xp''/p'. (10)
R is a variable which measures the relative curvature
of the demand curve. When R is positive, as when p' 1 is
negative, the inverse demand curve, p(X), is concave to
the origin. A linear inverse demand curve is associated
with R = 0. For R to be equal to 0, the slope of the
demand must be constant along the curve as is the case
with p'' =0. A convex demand curve, which is of special
interest in this thesis, is shown by a R < 0. When R is
negative, p'1 must be positive, as it is with a convex
In order to find the value of pt, I must first solve
for Xt, the effect of the tariff on the amount of
imports. This can be done by totally differentiating
(8), the first order condition for profit-maximization,
with respect to X and t.
Xt = 7TxxdX + 7Txtdt = 0
Xt = dx/dt = -7Txt/7Txx
Because 7rxt = -1 and substituting (9) for 7rxx, the
expression simplifies to:
Xt = lAxx = 1/[p' (2 + R) c"] < 0. (11)
Therefore, Xt is negative by the second order condition.
This means that an increase in the tariff will reduce the
amount of imports sold in the domestic market.
Knowing that pt = p'Xt, from (11)
Pt = P'/[P' (2 + R) c"]
pfc = l/[(2 + R) c''/P'] (12)
Now using the results obtained in (11) and (12),
the optimal tariff, (6), becomes
t* = X(l/[(2 + R) c"/P'] 1)/ 1/ [ P1 (2 +R) -c' ] .
Simplifying the complex equation, a meaningful expression
t* = -p'X(l + R c 1 1 /p 1) . (13)
Focusing on (12) and (13), if marginal cost is
constant, c' 1 = 0, then pt < 1 when R > -1. This means
that t* > 0, that there is a positive optimal tariff,
when demand is concave, linear, or slightly convex (R > -
1) to the origin. Still assuming constant marginal cost,
I find that pt = 0 when the variable R = -1. This result
argues that when the demand curve is of a certain degree
of convexity, R = -1, there is no optimal tariff, t* = 0.
The case of the most interest in thesis occurs when pt >
1. With constant marginal cost, pt > 1 when R < -1. The
variable R < -1 occurs when demand is "sufficiently"
convex to the origin. Therefore, when demand is
sufficiently convex to the origin, R < -1, the optimal
tariff is negative imports should be subsidized.
This result is interesting because a convex demand
curve can lead to three very different policy
prescriptions. When demand is slightly convex, R > -1,
linear, or concave to the origin, there exists a positive
optimal tariff. A convex demand curve with a slightly
higher degree of convexity illustrates the case where
there is no optimal tariff or subsidy. This occurs where
R = -1. This case seems unlikely because the relative
curvature must be exactly equal to -1 along all relevant
portions of the demand curve. This case of "well-
behaved" demand does not seem feasible over the relevant
portions of the demand curve. However, there is a range
of degree of convexity where the optimal policy
prescription is to subsidize imports. This range of
convexity is -2 < R < -1. The possibility that the
demand curve will fall within this range of convexity is
sufficiently great that this degree of convexity cannot
be ruled out and must, therefore, be analyzed.
The case for subsidizing imports can be improved by a
better understanding of what it means for R < -1. The
slope of demand is p'. Concurrently, the slope of the
marginal revenue schedule is Xp11 + 2p'. The difference
between the slope of demand and the slope of the marginal
revenue schedule is
mr' p' = Xp'1 + p1.
Another way to show this expression is
p'(R + 1).
Only when mr' p' is positive, will R < -l An
alternative way of expressing this idea is that only when
demand falls more steeply than marginal revenue will R
< -1. Therefore, with constant marginal cost and demand
being steeper than marginal revenue, the optimal
government policy will be to subsidize imports. The
subsidy actually causes the price of the imported good to
fall by more than the subsidy. The gain in consumer
surplus is greater than the cost of the subsidy. This is
illustrated in Figure 3.1. The import subsidy shifts
down the constant marginal cost schedule of the
monopolist. To maximize profits, the monopolist
increases output from X0 to where marginal revenue is
equal to marginal cost. The increased output lowers the
price of the good from p0 to px. The cost of the subsidy
to the home government is the lower shaded area. The
increase in consumer surplus is shown by the upper shaded
area. As drawn, the increase in consumer surplus is
greater than the cost of the subsidy. The import subsidy
maximizes the social welfare of the home country.
Figure 3.1 Effects of an import subsidy on domestic
This result is particularly interesting because the
welfare of the foreign country is also increased along
with the domestic country's welfare. In trade wars, one
country's gain is usually made at the expense of the
other. This policy, by increasing the domestic welfare
of both countries, is optimal from a normative point of
view. There are likely to be some positive externalities
between the two countries from this "win-win" policy.
For instance, the two countries are likely to get along
better politically after this policy. By two countries
getting along better politically, there will be a more
favorable international climate in which the countries
conduct their business. Therefore, with demand being
sufficiently convex, an import subsidy is the optimal
policy not just for the domestic country but for the
other country as well. For that reason, Brander and
Spencer conclude that this policy cannot be regarded as a
pure 'rent extracting' policy.
Brander and Spencer (1985) demonstrate that a
government can alter the noncooperative rivalry between a
domestic and foreign Cournot duopolist. If the home
government offers an export subsidy, claim Brander and
Spencer, the advantage of Stackleberg leadership can be
conferred upon the home Cournot duopolist. Their results
appear convincing at first glance. However, their
results hinge upon the unreliable assumption of strategic
substitution. Strategic substitution is present when the
increase in output of one firm reduces the marginal
profit of the other firm. When this unreliable
assumption is relaxed Brander and Spencer's policy
recommendations are drastically changed.
The model of export subsidies under imperfect
competition is Brander and Spencer's with a few minor
changes. The changes will be noted as they are
encountered. Firm behavior is modelled as a simple
Cournot duopoly. A Cournot duopoly is a market situation
where each firm assumes the other firm's output will not
change. Then, each firm chooses a profit maximizing
level of output for itself. The duopoly consists of one
domestic firm and one foreign firm. The assumption is
that both firms produce identical products that are
exported to a third, nonproducing, country. There is no
consumption of the good in the producing countries.
Another important assumption made by Brander and Spencer
is that the governments understand the market structure
of the industry and are able to use this information to
set plausible export subsidies before the firms choose
their output levels.
The domestic firm produces quantity x and the
foreign firm produces quantity y. The profit function of
the domestic firm is
7r(x,y;s) = xp(x+y) c(x) + sx, (1)
where c is the variable cost, s is a per-unit subsidy,
and p(x+y) is the world price of the good. Profit
maximization is obtained when the first-order derivative
of the profit function with respect to output equals
zero. The first-order condition for a profit
?rx = xp,+p-cx+s = 0. (2)
The notation is the same as Brander and Spencer's where
subscripts are used to denote derivatives except for p',
which is the derivative of inverse demand. To make sure
that output level is indeed a true profit maximization
the following second-order condition needs to be
'xx=2p,+ xp"-cxx<0. (3)
The first-order condition, 7rx = 0, is a necessary
condition for profit maximization. Where the first-order
derivative is equal to 0, the profit function has either
reached a maximum or a minimum. Once the necessary
condition for profit maximization has been satisfied, the
sufficient condition that < 0 must also be satisfied.
This condition says that the slope of the function is
increasing at a decreasing rate. With the first-order
derivative of profit with respect to output equal to 0,
7rx=0, and the second-order derivative of profit with
respect to output being negative, tt^cO, the profit
function has reached a relative maximum. Therefore, the
necessary and sufficient conditions for a profit
maximization have been met. The profit function of the
foreign firm, tt*, is:
7T* = yp(x+y) c(y) . (l*)
The first-order condition for profit maximization for the
foreign firm is
7r*y = yp' + p c*y = 0. (2*)
Just as with the home firm, the second-order condition
for profit maximization must be checked to make sure that
this level of output is indeed a profit maximizing level
of output. The second-order condition for the foreign
firm, then, is
= 2p' + yp" c^ < 0. (3*)
Brander and Spencer also use the following
Tty,y = P + XP ' < 0 ; 7T*yX = p'+yp',<0 (4)
^xx ^ ^xy' ^yy^^yx* (5)
Strategic substitution, condition (4), implies that as
the output of one firm increases, the marginal profit of
the other firm decreases. Strategic substitution, then,
gives rise to downward sloping reaction functions which
is a graphical depiction of how one firm reacts to output
changes of the other firm. At this point in their paper,
Brander and Spencer make the argument that condition (4)
is a fairly standard regularity condition in
noncooperative models, but it can be violated if demand
is 'very' convex. They go on to argue that from the
standard second-order conditions, which they use in their
paper, and from condition (4), condition (5) always holds
if marginal cost is nondecreasing. Only if marginal cost
falls more steeply than demand can it be violated.
Condition (5) implies that own effects of output on
marginal profit dominate cross effects.
Strategic complementarity is the idea that as the
output of one firm increases, the marginal profit of the
other firm also increases. Strategic complementarity,
which reverses the signs on condition (4), does not help
strengthen condition (5). It definitely serves to weaken
it. However, the Routh-Hurwitz stability condition,
which Brander and Spencer also use, keeps strategic
complementarity from overriding condition (5). In other
words, strategic complementarity does not change the sign
of the determinant. The stability conditions will be
discussed in detail in a later part of this thesis.
Therefore, in my analysis own-effects continue to
dominate cross effects condition (5) continues to hold.
To find the effects of the subsidy, I will use all
of Brander and Spencer's conditions, except for condition
(4), strategic substitution. To bring out the argument,
I will replace condition (4) with condition (6),
7T = p' + xp1 > 0;
= p' + yp'' > 0.
Conditions (2) and (2*), the first-order conditions
of the two firms, show each firm's profit maximizing
output decision. Therefore, total differentiation of (2)
and (2*) will give the output changes, dx and dy, in
response to a subsidy offered by the domestic government.
Total differentiation of (2) and (2*) yields:
7TxxdX + Tr^dy + 7TxsdS =0 (7)
7r*yxdx + 7r*yydy + 7r*ysds = 0. (7")
7rxs equals 1, from (2) because a change in the subsidy
will lead to a proportionate change in the marginal
profit of the domestic firm. 7r*ys equals 0, from (2),
because a change in the domestic subsidy will leave
foreign marginal profit unchanged, ceteris paribus.
Because 7rxs=l and 7r*ys=0, the equation can be put into
matrix form and the solutions for dx/ds and dy/ds are
1 5: i ^xy -ds
* * * 0
^xx^ yy K yx^xy ^ yx TTxx.
^ HxxTT yy 7T yUtTTjjy > 0
Solution of the matrix system shows:
xs = dx/ds = -7r*yy/D >0 (9)
Ys = dY/ds = ir*yyJD > 0. (9*)
The sign of (9 ) is ambiguous because it depends on the
sign of 7r*yx:
?r*yX = P' + YP'
The ambiguity arises from the fact that 7r*yx can be
positive or negative. I will explain in a later section
of this thesis the conditions under which 7r*yx will be
positive or negative.
An increase in the unilateral export subsidy
increases the output of the home firm as expressed in
equation (9). This is not a very shocking result. It is
just what one would expect to happen when an export
subsidy is offered. But the result expressed in (9*) is
unexpected. This result says that when a government
offers an export subsidy the output and marginal profit
of the foreign firm is also increased. This is the
strategic complementarity result. Brander and Spencer's
limiting assumption of strategic substitution gives the
opposite result. They claim that an increase in the
domestic subsidy will reduce the output of the foreign
firm. A discussion of reaction functions will further
aid in understanding the results in (9) and (9*) .
Reaction functions graphically depict how one firm
chooses its output level in relation to another firm.
The slopes of the reaction function are derived from
totally differentiating the first order conditions for
profit maximization. Totally differentiating (2) and
7TxxdX + TTxydy + 7TxsdS = 0
7r*yxdx + CT^dy + 7r*ys ds = 0.
To derive the slope of the reaction functions ignore the
terms with the subsidy attached to them because they are
shift parameters. Now rearrange the above equations to
derive the slopes of the reaction functions.
TTxydy = -7TxxdX
dy = rlLxxdx
dv = 7rx >0 = slope of the home firm's (10)
dx 7reaction function.
ir*yydy = 7T*yxdx
dy = z2L^yxdx
dv* = -7r* >0 = slope of the foreign firm's (10*)
dx 7T ^ reaction function.
These mathematical relations depict the output choice of
a firm in reaction to an output choice of the other firm.
These mathematical relations are the slopes of the
reaction functions. The slopes of the reaction functions
depend on the assumptions that are made about the way the
firms react to changing levels of output. If strategic
substitution is assumed, then the reaction functions
unambiguously slope downward. If and ir*yx are both
negative, as is the case with strategic substitution,
then dy/dx and dy*/dx are both strictly negative. This
says that if the output of one firm is increased then the
output of the other firm is decreased. In other words,
strategic substitution gives rise to downward sloping
reaction functions. On the other hand, if and 7r*yx are
both positive, as in the case of strategic
complementarity, then dy/dx and dy*/dx are both positive.
With the assumption of strategic complementarity the
reaction function are upward sloping. Thus, with
strategic complementarity being assumed in this thesis,
the reaction functions will be upward sloping. For a
discussion of downward sloping reaction functions see
Brander and Spencer (1985).
Which reaction function, home or foreign, slopes more
steeply will determine whether or not the system is
stable. Beginning at point 1 in Figure 4.1, the home firm
is on its reaction function. Given the home firms
output decision of xl, the foreign firm chooses output
level y2. Given the output decision y2, the home firm
chooses x3. As the reaction function shows, the system
is in fact a stable system. In the stable case, dy/dx >
dy*/dx, the home firm's reaction function is relatively
steeper than the foreign firm's reaction function.
Now, to demonstrate the opposite case, make the slope
of the foreign firm's reaction function relatively
steeper than that of the home firm's. Then reexamine the
stability of the system. Looking at Figure 4.2, begin at
point 1 on the foreign firm's reaction function. Given
the foreign firm's output decision of yl, the home firm
chooses output level x2. Given x2, the foreign firm
chooses output level y3. The system is clearly unstable.
Therefore, in order to have an stable system of upward-
sloping reaction functions the following condition must
hold: dy/dx > dy*/dx. This condition means that the home
firm's reaction function must posses a relatively steeper
slope than that of the foreign firm in order for the
system to be stable. This is a necessary condition for
reaction function stability.
Figure 4.1 Stable reaction function case
Figure 4.2 Unstable reaction function case
This stability condition can be related to the
= 7T 7T 7T 7T
' xx'* yy '* xy'* yx
The determinant can be rearranged:
7rxx,r*yy > ^xy^^yx from condition (5) .
Dividing through by a negative number gives:
Multiplying both sides of the expression by a negative
ZlLxx ^ ZlLj-yx
TTxy W yy
An alternative expression from (10) and (10*) is:
dy > dv*
Therefore, a positive determinant is a necessary
stability condition for the system. Strategic
complementarity, which implies upward sloping reaction
functions, shows that own-effects must dominate cross-
effects for the system to be stable.
Now that it has been shown that own-effects must
dominate cross-effects for the system to be stable, the
effect of the export subsidy can be discussed. The
export subsidy increases exports of the home firm as
shown by (9). The subsidy increases exports of the home
country because it lowers the marginal cost of production
of the home firm. With a lower marginal cost, the home
firm commits itself to a higher reaction function. This
can be shown diagrammatically. In Figure 4.3, the export
subsidy will shift out the home firm's reaction function.
As the figure shows, the output of the home firm is
increased. The figure also shows that the output of the
foreign firm is also increased. This result is entirely
consistent with (9*) which says that an increase in the
domestic export subsidy will increase the output of the
foreign firm. Strategic complementarity, then,
drastically alters Brander and Spencer's, comparative
static effects of the export subsidy on the output of the
home and foreign firms. The reversal of Brander and
Spencer's result is important because it shows that a
unilateral export subsidy will actually increase the
output of both firms. Brander and Spencer's result is
that the export subsidy increases the output of the home
firm while the foreign firm's output contracts.
Brander and Spencer now introduce what they call
Proposition 1 into their analysis. Proposition 1
summarizes the comparative static effects of the
unilateral domestic subsidy on prices and profits.
Figure 4.3 The effect of an export subsidy on reaction
Proposition 1. An increase in the domestic subsidy will
(i) lower the world price of the good;
(ii) increase domestic profit; and
(iii) reduces foreign profit.
I will now examine each of the results of Proposition 1
with the assumption of strategic complementarity
replacing strategic substitution.
(i) The change in the world price of the good that
results from an increase in the domestic subsidy is given
by the slope of inverse demand times the change in total
Ps = dp/ds = p' (xs + ys)
= p (7r*yx 7T*yy)/D <0 from (9) and (9*)
Strategic complementarity serves to further strengthen
this result. This makes sense because as the domestic
subsidy is increased, both domestic and foreign output
are increased. As output of both firms is increased,
while holding world demand constant, the world price of
the good must fall in order for the consumers to purchase
the extra output. Therefore, an increase in the domestic
subsidy will lower the world price of the good.
(ii) From total differentiation of 7r with respect to s,
7TS = d7r/ds = 7TxXs + 7Tyys + Sir/Ss
since 7rx=0 by (2), Sw/Ss = x by (1), 7ry = xp', and ys >0
tts = xp'ys + x. (11)
This result is ambiguous because ys >0 by (9*). With ys
>0, either term in equation (10) can dominate. If the
second term dominates, as it does in Brander and
Spencer's analysis, then 7rs >0. If this is the case,
then Brander and Spencer's conclusion will be correct; an
increase in the domestic subsidy will increase domestic
profit. But if the first term dominates, then ns <0.
However, because of the possibility of either sign
dominating, I conclude that the result is ambiguous.
(iii) From (1*), (2*), and (9) and 7T*X = yp':
7T* = d7T V ds = 7TxXs + 7T*yys
since 7r*y = 0 from (2*) ,
K = YP'xs <0. (12)
This result is not reversed by the assumption of
strategic complementarity. An increase in the domestic
subsidy unambiguously lowers the profit of the foreign
A unilateral domestic subsidy, therefore, lowers the
world price of the good, has an ambiguous effect on
domestic profit, and serves to lower foreign profit.
Brander and Spencer then arrive at what they call a
surprising result. They claim that the subsidy actually
increases domestic welfare net of the subsidy. With all
production for export, domestic surplus, G, net of the
subsidy, is the profit of the domestic firm minus the
cost of the subsidy:
G(s) = 7r(x,y;s) sx. (13)
Differentiating the gain with respect to the subsidy
Gs = 7TS x sxs ,
and substituting (12) for ns and simplifying,
Gs = xp'ys sxs. (14)
In the absence of the subsidy (s=0) Gs is clearly
negative. This implies that a marginal increase in the
subsidy will lower domestic welfare net of the subsidy.
This is important because it reverses Brander and
Spencer's claim that an export subsidy will actually
increase domestic welfare net of the subsidy. In my
analysis, the assumption of strategic substitution gives
the opposite result of Brander and Spencer an export
subsidy does not increase domestic welfare net of the
If this analysis is correct, that an export subsidy
does not increase domestic welfare net of the subsidy,
then what is the optimal policy for a government? By
setting Gs = 0 and solving for s, the optimal government
subsidy is obtained:
s = xp'ys /xs <0 by (9) and (9*). (15)
This result is very striking. The optimal export subsidy
is actually negative. With strategic complementarity
being assumed, the optimal government policy to maximize
domestic surplus is an export tax. This leads to an
overturning of Brander and Spencer's Proposition 2.
Proposition 2. The domestic country has a unilateral
incentive to offer an export subsidy to the domestic
With the optimal government subsidy being a negative
subsidy (an export tax), Proposition 2 is overturned.
The idea that an export subsidy will allow the domestic
firm to capture a larger share of profitable
international markets does not seem to be attractive from
the domestic point of view. While the export subsidy
does enable the home firm to increase its output, it is
not in the interest of domestic welfare to do so. With
an export subsidy, the noncooperative equilibrium in
inter-firm rivalry is not altered in favor of the
domestic firm as Brander and Spencer claim it to be.
Proposition 3. The optimal export subsidy, s, moves the
industry equilibrium to what would, in the absence of a
subsidy, be the Stackleberg leader-follower position in
output space with the domestic firm as leader.
In a Cournot duopolist situation, each firm takes the
other firm's output as constant. But if one firm, say
the domestic firm, can make an output decision before the
other firm, the foreign firm, that takes into account the
response of the foreign firm, the domestic firm is said
to be a Stackleberg leader. Furthermore, the foreign firm
believes that it cannot influence the output decision of
the domestic firm. Suppose, now, that the domestic firm
is a Stackleberg leader. From differentiating (1), the
first-order condition for profit maximization of the
domestic Stackleberg leader is:
7rx(x,y;0) + 7ry(x,y;0)dy/dx (16)
where dy/dx is the slope of the foreign firm's reaction
But, dy/dx = from (10*) .
We know from (9) and (9*) that xs = -7T*/D, ys = 7r*yx/D.
Now dividing ys by xs gives
Ys/xs = 7r*yx/-7iy. (17)
Substituting for 7rx and ny and condition (17) back into
expression (16), we get the following result:
xp' + p cx + (xp'ys/xs).
Because (xp'ys/xs) = s, the optimal subsidy, (18)
xp1 + p cx + s.
Comparing this simplified expression to expression (2),
the first order conditions for a profit maximization for
the domestic Cournot duopolist, we find that the
conditions are identical. This proves Brander and
Spencer's Proposition 3.
Why Strategic Complementarity Cannot
Be Considered Perverse
Brander and Spencer's conclusion that the foreign
firm will contract in response to a domestic subsidy, ys
<0, hinges upon the assumption of strategic substitution,
7r*yx <0 and 7TXy <0. Brander and Spencer (1985) justify this
assumption with "This is a fairly standard regularity
condition in noncooperative models, but it can be
violated by feasible demand structures, in particular if
demand is very convex" (p.86,87). They seem to recognize
the possibility of the occurrence of strategic
substitution but go on to say "If one wishes to consider
cases in which the conditions are violated, 'perverse'
comparative static properties and policy implications can
be obtained. This is of some interest, but we focus on
structures which satisfy (4) and (5), since they include
most economically relevaint cases". This argument appears
convincing, but leaves much to be desired. For instance,
how convex is very convex? Brander and Spencer do not
address this question. Secondly, how are they to know
that their conditions include most economically relevant
cases without giving equal treatment to the so called
'perverse' comparative statics?
The possibility of strategic complementarity cannot
be safely ruled out. Strategic complementarity is shown
by positive cross output partial derivatives. These
cross output partials are shown in condition (6) in the
previous section of this chapter:
TTxy = P' + XP' >0 TCyy. = P t J P ' > 0 . (6)
This condition is similar to the slope of a marginal
revenue curve of a single firm:
MR' = 2p' + xp''. (17)
A provocative article written by Formby, Layson, and
Smith (1982), exposes the possibility of an upward
sloping marginal revenue schedule. In equation (17), the
slope of the marginal revenue curve, the first term is
always negative by the law of demand. The second term,
however, can be positive, zero, or negative depending on
the shape of the demand schedule. If demand is linear,
p'' = 0, the second term will be equal to zero. If
demand is concave, p'' <0, the second term will be always
negative. In both of these cases the first term
dominates and the marginal revenue schedule will be
downward sloping. However, if demand is convex, p'1 >0,
the second term in (17) will be positive. Furthermore,
if the second term dominates, the marginal revenue
schedule will be upward sloping. The requirements for an
upward sloping marginal revenue schedule are not at all
stringent. Any convex demand curve can give rise to an
upward sloping marginal revenue schedule.
Formby, Layson, and Smith show that parallel shifts
in demand are all that is needed to demonstrate their
point. If demand is shifted out in a parallel fashion,
p1 and p'' remain unchanged. If that is the case, with x
increasing and p'1 remaining constant, the second term,
xp'1, is getting larger. With xp'1 increasing and p'
staying constant, the second term will eventually
dominate, giving an upward sloping marginal revenue
schedule. Therefore, the existence of an upward sloping
marginal revenue schedule can never be safely ruled out.
This compelling argument can be directly adapted to
condition (6) in order to show that strategic
complementarity is not 'perverse1 and, therefore, cannot
be safely ruled out. In my analysis, I will demonstrate
my point using only n*yx from (6), since that is of most
interest in this thesis.
= P' + yp' ' (18)
In (18), p', the slope of the demand curve, is negative
by the law of demand. If demand is convex, which I
assume it to be, p'1 is positive because.the slope is
increasing at a decreasing rate. Therefore, the
possibility of (18) being positive cannot be ruled out.
If demand shifts in a parallel fashion, then the second
term in (18), yp'1, is becoming more and more positive
and strategic complementarity becomes more likely. When
the second term dominates, (18) becomes positive and
strategic complementarity is present.
There are other possibilities that require no
transformations to obtain the result of strategic
complementarity. Since there are only two producing
firms, the total world supply comes from these two firms.
This may be an extremely large amount of output. If that
amount of output is large enough, then the second term of
(18). will dominate and strategic complementarity becomes
the rule rather than the exception. Assume that p' = -
1.0, which is very feasible. Assume also that p'1 =
.005, which is a very slight degree of convexity. For
strategic complementarity to be present, output must only
exceed 200 units. It is quite likely that the output of
the foreign firm will exceed 200 units. If I set (18)
equal to zero and solve for y I will get the minimal
output necessary for strategic complementarity to be
7r*yx = 0 = p' + yp"
y = -p'/P'' (19)
Therefore, for strategic complementarity to be present
the following condition must be met:
Y > "P'/P'' (20)
Strategic complementarity, then, seems to be more likely
when the foreign firms are producing all the output of a
good that the world will consume. Therefore, by ruling
out strategic complementarity as economically perverse,
Brander and Spencer may actually have ruled out the most
economically relevant case. They themselves may very
well be studying perverse economic cases. The most
important consequence is that they have recommended an
export subsidy when, in fact, an export tax is in the
best interest of the domestic country.
This thesis contains an analysis of trade subsidies
under three types of market structure: perfect
competition, imperfect competition, and monopoly. The
results vary with the type of market structure under
consideration and the assumptions made about the behavior
of the market participants. In some cases the results
are quite sensitive to the assumptions.
In the chapter on subsidizing the exports of a
domestic perfectly competitive producer, the results
obtained do not represent any break from mainstream neo-
classical tradition. This is due to my basic agreement
with the assumptions needed to handle the case of perfect
competition. Assuming price-taking behavior by
individual producers leads to the conclusion that an
export subsidy is not in the best interest of the welfare
of society. The increase in producers' surplus is less
than the cost of the export subsidy. This result
coincides with the textbook literature in international
In a slightly more involved section of the Chapter, I
demonstrated that an export subsidy offered only to the
producer of one good and not the other will have some
redistributive effects in the economy. Specifically, the
subsidy will redistribute wealth from the factor of
production that is used less intensively to the factor
that is used more intensively. I then generalized this
local result using Shepard's Lemma and obtained exactly
the same result.
In Chapter III of this thesis, I analyzed Brander and
Spencer's ground-breaking paper on tariff warfare. Their
paper convincingly argues that the optimal government
policy to extract rents from a foreign exporting
monopolist is either a specific tariff or subsidy
depending on the convexity of demand. When demand is
concave, linear, or slightly convex to the origin, the
optimal government policy is to impose an optimal tariff,
according to Brander and Spencer. However, when demand
is sufficiently convex the optimal government policy,
claim Brander and Spencer, is to impose a negative
tariff an import subsidy.
Brander and Spencer define what constitutes slightly
or sufficiently convex demand by introducing the variable
R that measures the relative curvature of the demand
curve. They effectively show that the optimal government
policy is to impose an import tax when R > -1. They also
provide a compelling argument that the optimal government
response to a foreign monopolist is to offer an import
subsidy. When demand is sufficiently convex, R < -1, the
government maximizes the domestic gain function by
Their paper convinces me that their results are
correct. The concreteness of their paper is best shown
by the fact that they laid their assumptions out before
they began their theorizing. This results in the best
approach to scientific theorizing. They obtained their
results according to their pre-specified assumptions
rather than modifying their assumptions to fit their
preconceived notions of what their results were to be. I
feel that their paper represents the scientific approach
The problem with the assumptions that other
economists have used in their model building has not been
much of a problem thus far in the thesis. However, in
the Chapter on imperfect competition there is one
assumption that I feel is not theoretically justifiable.
In their thought-provoking 1985 paper, Brander and
Spencer attempt to demonstrate that given all
theoretically relevant demand conditions, the optimal
policy for a domestic government to undertake to confer
the advantage of Stackleberg leadership on the home firm
is to subsidize exports. They mention that perverse
comparative static results may be obtained if demand is
'very' convex. They propose that their paper includes
all economically relevant cases. However, Brander and
Spencer do not explain what they mean by demand being
'very' convex. The assumption that I consider not to be
theoretically justifiable, and which their results hinge
upon, is strategic substitution.
It has been demonstrated that strategic substitution
is an unreliable assumption and that strategic
complementarity is a real possibility with convex demand
and must be dealt with. When the assumption of strategic
complementarity is invoked, Brander and Spencer's results
are overturned. With the assumption of strategic
complementarity being utilized, the optimal government
policy, as demonstrated in Chapter IV, is an export tax.
What I wish to deal with in this conclusion is the
problem of assumptions in economic theorizing.
Brander and Spencer assume that strategic
substitution holds for all economically relevant cases.
In their 1981 paper, Formby, Layson, and Smith
demonstrate the existence of an upward sloping marginal
schedule. They conclude that an upward sloping marginal
revenue schedule may be present when a theoretician
believes it to be downward sloping. They discuss the
possibilities of multiple equilibria. In a similar
argument, it has been shown that strategic
complementarity may be the assumption that must be dealt
Economists, as well as other scientists, should learn
a lesson from this. The results of the model should be
in accord with the assumptions of the model. The
assumptions should not only hold where it is convenient
for the theorist.
Brander, James A., and Barbara J. Spencer. 1984. Trade
warfare: Tariffs and cartels. Journal of
International Economics 16: 227-42.
_________. 1985. Export subsidies and international
market share rivalry. Journal of International
Economics 18: 83-100.
Formby, J.P., S. Layson, and W.J. Smith. 1982. The law
of demand, positive sloping marginal revenue, and
multiple profit equilibria. Economic Inquiry
Stolper, W., and P. A. Samuelson. 1941. Protection and
real wages. Review of Economic Studies 9: 58-73.
The following paper was submitted to the Journal of
International Economics and is a direct result of the
Reaction Functions and International
Market Share Rivalry
Duggan, Steven R. Beckman and W. James Smith
Student, Assistant Professor and Professor
Department of Economics
University of Colorado, Denver
Campus Box 181
P.0. Box 173364
Denver, Colorado 80217-3364
Reaction functions and international market share rivalry
International competition between Japan and the U.S.
has been the subject of considerable debate over the last
two decades. A focal point in this debate is
subsidization of Japanese firms by the Japanese
government. Such subsidies, it is popularly argued,
place U.S. firms at a competitive disadvantage and pose a
threat to U.S. economic welfare. In response, the U.S.
Congress has been petitioned to provide similar aid to
U.S. firms to counter Japanese industrial policy.
Economists traditionally have offered little
theoretical support for subsidies. International trade
models of perfect competition suggest that subsidies
benefit foreign consumers at the expense of domestic
taxpayers, a policy not likely to garner long-term
political endorsement. This view has recently been
subject to serious challenge. In an important
contribution, Brander and Spencer (1985) (henceforth BRS)
employ a non-cooperative duopoly model to demonstrate
that government subsidies not only enhance domestic
international competitiveness but increase domestic
economic welfare as well.
BRS's argument has justifiably attracted widespread
attention. The implications for policy are obvious and
significant. As is the case of most analyses of this
kind, however, BRS recognize the necessity of
restrictions on the structure of demand, which if
violated reverse their policy conclusions. The critical
assumption is that of downward sloping reaction
functionsan assumption routinely employed in non-
cooperative models [see, for example, BRS (1985, p. 86),
Cheng (1988, p. 755), Eaton and Grossman (1986, p. 390)
and Helpman and Krugman (1989, p. 89)]. It is only
violated if, in BRS's words, "demand is very convex."
The accepted position [BRS (1985, p. 87)], is that such
demand structures do not represent "the most economically
In this paper, we demonstrate that any degree of
convexity from near linear to extremely large curvature
is compatible with reversals of BRS's results.
Counterexamples are shown to exist for all constant
elasticity demand schedules and for any section on a
specific demand schedule. Moreover, simple comparative
static changes such as shifts in demand, marginal cost or
the subsidy itself serve to overturn BRS's assumption and
policy conclusions. Thus, even if BRS's assumptions are
originally fulfilled, there can be no assurance that they
will remain so as basic economic circumstances change.
Potential damage from subsidies emerges as a much more
likely possibility than has heretofore been recognized.
2. A brief overview of BRS1s model
Following BRS, consider an international duopoly which
serves a third market. Domestic output and profit are
denoted by x and 7r, foreign output and profit by y and n
7r(x, y; s) = xp(x+y) c(x) + sx
tt*(x, y; s) = yp(x+y) c*(y), (l)
c(x) is variable cost, s the per unit subsidy and x+y
total industry output, q. Assuming Cournot reactions,
the first-order conditions for profit maximization are:
7TX = p + xp cx + s = 0 7T* = p + yp' Cy = 0 (2)
where subscripts denote derivatives. Second derivatives
*xx = 2p' + xp" cxx < 0 7!yy = 2p' + yp" c^ <0 (3)
and cross partials:
TTxy = p' + xp" <0 7TyX = p1 + yp" <0. (4)
The effect of the subsidy is obtained by totally
?rxxdx + Ti^dy = -7rxsds (5)
7TyXdx + 7iyydy =0. (6)
Observing that ttxs = 1 we have:
dx/ds = xs = -7iyy/D > 0 dy/ds = ys = 7ryx/D (7)
where D is the determinant of the system. Stability
requires that D be positive [see Bulow, Geanakoplos and
Klemperer (1985 pg. 43)]. The sign of xs is strictly
positive. The sign of ys depends on 7ryx, which from (6)
also determines the sign of the reaction function, yx,
for the foreign firm.
In BRS's model, the gain (G) or loss from subsidization
equals the home firm's profit net of the subsidy, that
G = 7r (x, y; s) sx. (8)
Differentiating G with respect to s to determine the
effect of the subsidy on social welfare, we have:
Gs = ttxxs + 7ryys + dw/ds x sxs = xp'ys sxs. (9)
The second equality follows from the fact that wx is zero
from first-order conditions, 7ry = xp', and dir/ds = x.
For s = 0, the sgn (Gs) = -sgn (ys) If ys < 0, the
foreign firm's reaction function is negatively sloped,
and subsidies result in national gains.1 If ys is
positive, however, the gain in profits is less than the
cost of subsidies. In this case, subsidies diminish
domestic welfare. The sign of ys depends on the sign of
Tr^, which is thus pivotal to the case for subsidization.
3. Stringency of non-standard cases
BRS equate economic relevance of their case for
subsidies with a negative cross partial of the foreign
firm's profit function. It is in this context that BRS's
restriction on convexity takes on meaning. Examination
of 7TyX = p' + yp" verifies that the cross partial is a
function of curvature of demand, but, we stress, not
solely a function of curvature. The point to be
emphasized is that BRS1s assumption is not merely a
simple restriction on curvature of demand, as it has been
interpreted, but a simultaneous restriction on several
variables. The relationship between these variables when
BRS's restriction holds and when it is violated raises an
interesting question concerning the meaning of the
statement "economic relevance". It also raises the
question of what are the characteristics of the sets
defined on (p', p", y) for which subsidies are beneficial
and for which they prove harmful.
Fig. 1 shows projections of these sets. The locus of
points for which 7r*x = 0 forms the dividing line between
the two sets. Setting 7r*x= p' + p"y = 0, we have:
y(-Vp') = l/P" or (p") (-l/p') = l/y. (10)
The first equation is shown in Fig. la which assumes a
fixed value for p" and traces out values of p' and y
which satisfy 7r*x = 0. Fig. lb fixes y. In both cases,
the loci are rectangular hyperbolas.
In Fig. la, the area under the curve identifies the set
of points consistent with BRS's conclusions on subsidies.
The area above the curve identifies the set of points (y,
p') inconsistent with BRS's conclusions. If we fix
demand conditions at, for example, p'= -.02 and p" =
.001, BRS's results pertain for values of y along the
bounded line segment ab, that is, 0 to 20 units of
output. Values of y in the unbounded set beyond 20 units
to +oo, however, are inconsistent with their results. In
short, "large" values of output tend to overturn BRS's
results. For later reference, we denote the critical
value of y which separates normal from "perverse" values
as v. If y > v, BRS's conclusions are reversed. If y <
v, BRS's conclusions follow.
Fig. lb illustrates that the term "very convex" is
arbitrary when viewed across the entire set of economic
outcomes and that any degree of curvature from near zero
to plus infinity is consistent with reversals of BRS's
results. Consider the entire set of possible economic
outcomes and note that the functional relationship in lb
is asymptotic to both axes. It immediately follows that,
for any given y, one can always choose a p" arbitrarily
Figure l.a Critical values of 1/p1 and y for p'1 = .001
small (and a corresponding p') which will reverse BRS's
conclusions regardless of the specific quantitative
meaning given to the restriction "very convex" demand
structure. In Fig. lb, y is fixed at 100 units. If p'
= -.04, the degree of curvature which BRS define as "very
convex" is p" = .0004. Any degree of curvature beyond
.0004 violates BRS's assumption. If, however, p' = -.02,
BRS's definition of "very convex" curvature changes to p"
= .0002. Similar to the previous case, any degree of
curvature beyond .0004 reverses BRS's conclusions. This
new range includes degrees of curvature between .0002 and
.0004 which were previously not considered "very convex"
but now are. One can easily verify that, given the
limits of p', zero and plus infinity, the term "very
convex" has the same range, zero to plus infinity. BRS's
stricture of "very convex", thus, does not and cannot
refer in general to a specific degree of curvature. The
quantitative meaning of the term "very convex demand
structure" varies from problem to problem spanning the
entire range of curvature. The conclusion of benefit
from subsidies appears to dictate, case-by-case, the
demand conditions necessary for the result's validityan
uneasy situation methodologically and one we are
unprepared to accept.2
Figure l.b Critical values of 1/p' and p'' for y = 100
4. Implicit limitations on comparative static exercises
The endogenous nature of BRS's assumption may not be as
damaging to the argument for subsidies as it might first
appear. This would be the case if BRS's conclusions on
subsidies, once established, could be shown to remain
intact as basic economic circumstances change. In this
event, subsidies could be safely implemented once the
conditions for net benefit were empirically verified. If
the validity of BRS's conclusions is sensitive to basic
changes, however, one can never be certain of continuing
benefit from subsidies. This section analyzes the
question. We start with a point for which BRS's
assumptions are fulfilled and then subject the system to
standard comparative static shocks to determine whether
their assumptions and conclusions continue to hold.
To preserve the spirit of BRS's assumption on convexity
and without loss of generality, we consider only standard
comparative static shocks which do not alter the
magnitude of curvature. In this case, the initial
restriction of "very convex" does not change quantitative
meaning throughout the exercise.
The boundary condition between benefit and damage from
subsidies is defined by 7r*x =0, that is,
pi + Vp" = 0.3 (11)
The critical value, v, facilitates comparisons with
values of y. We define an initial point for which y < v.
If after the shock y > v, a violation of Brander and
Spencer's assumption endogenously occurs. It is
important to note that the critical value, v, is defined
for given levels of p' and p". It is the maximum value
of output for which subsidies do not reduce net social
gain. V is a function of q alone (where q = x + y)
because p1 and p" are functions solely of industry
output. In addition v is an indirect function of y
through y's effect on q. Both v and y will change as
comparative static changes are introduced. The relative
magnitudes of change determine whether BRS's assumption
Let z represent any exogenous shift parameter. The
change in the critical value, v, in response to a change
in z is given by:
vz = vqqz = vq(xz + yz) . (12)
Determination of the effects of z on foreign and domestic
outputs, xz and yz, is straightforward. Focusing on vq by
differentiating (11), we have:
p"dq + vqp"dq + vp--dq =0 (13)
vq = (-p" vp-")/p". (14)
The first term, p"/p", reflects the fact that increases
in industry output in the presence of convex demand
curves reduce p' and therefore v at a rate of p". As the
reduction is weighted by p", the term equals -1. Changes
in q also may alter curvature. This is reflected in the
second term which could either be positive, negative or
zero. Neither theory nor BRS restrict the sign of p-".
By our assumption of curvature-preserving change, p-" is
zero. In this case,
vq = -1, (15)
that is, the critical value changes by the same amount as
industry output but in the opposite direction. Any
comparative static change which increases industry output
will reduce v equivalently.
A reduction in the foreign firm's marginal cost, an
increase in demand, and a subsidy are considered in turn.
Assume small changes and therefore that the level of
foreign output, y, is close to v. That is, y + e = v, e
arbitrarily small and positive. Economic changes are
then introduced and the relative magnitudes of y and v
again compared. If y becomes larger than v, then, 7r*x =
p' + yp" > p' + vp" = 0 and BRS's results are overturned.
Standard comparative static exercises establish that:
xz = dx/dc* = -tt^/D and yz = dy/dc* = 7rxx/D <0. (16)
From (3) (4) and (12) we have,
vz = dv/dc* = -(p' c^J/D.
Assuming costs are nondecreasing (the conditions for a
natural monopoly are not present), dv/dc* >0. As a
result, if an innovation reduces foreign costs, y
increases and v declines. A reduction in foreign
marginal cost thus violates BRS1s assumption on convexity
even though the change is curvature preserving. The
shift thus reverses the effect of the subsidy from
benefit to damage.
A parallel shift in demand produces a similar result.
Letting prices be shifted vertically by some positive
amount denoted by "a", we have:
pn = a + p(x + y) (18)
where pn denotes the new price. The first and second
derivatives of the profit functions are unaltered by the
shift. Under these conditions nxa = i and 7r*a = 1.
Xa = (*xy " flyy)/D II Â£ - p' + c^]/D, (19)
Ya = (-Kxx + 7T*x)/D = [p"(y x) - p + cxx]/D, and (20)
Va = -(-2p' + cJy + cXx)/D. (21)
Assuming cxx and Cyy > 0, va < 0. Each firm's output
response depends, in part, on its relative size. If the
foreign firm is larger, foreign output expands so that y
> v.* Again the shock reverses the effect of policy.
Finally, consider a subsidy. Rewriting (7) in more
xs = ~(2p1 + yp" c^)/D and ys = (yp" + p')/D. (22)
vs = ~qs = (P' c^)/D. (23)
The reduction in v is unambiguously larger than the
reduction in y (assuming Cyy > 0) Formally, vs < ys or
(p1 c^)/D < (yp" + p')/D reduces to -c^ < yp", a
condition which holds for constant or rising marginal
cost and convex demand. The introduction of the subsidy
itself thus tends to reverse the effect of policy.
The value of y may initially not be close to the value
of v. In this case, ys may well be negative after the
subsidy changes. However, as the above argument
demonstrates, the subsidy moves the value of y closer to
the value of v. Changes in the level of the subsidy at
some critical point will transform its effects from
benefit to harm. In short, subsidies, whether beneficial
or not, always carry the potential for damage, a fact
which cannot be assumed away.
A simple but important point emerges from the above
considerations. Within the set of possible convex demand
schedules, marginal cost schedules and levels of subsidy,
there exists a subset for which subsidies are beneficial.
There also exists a second set for which subsidies prove
harmful. Simple transformations link these two sets,
transforming an element of one set into an element in the
other. As a result, to in effect assume one set away, is
not only to assume away a subset of what we find to be
theoretically permissible and economically viable demand
and cost conditions but to place severe restrictions on
the nature of the transformations permissible on the
complementary set as well.5 For example, the magnitude
of parallel shifts in demand or shifts in marginal cost
or changes in the level of the subsidy itself would be
strictly limited. We find restrictions on such elemental
transformations too stringent to be generally acceptable.
The above observations relate to a more general
methodological point. BRS's assumption that the foreign
firm reduces output in the face of a subsidy is a
behavioral assumption, not one founded upon optimization.
It is important to keep in mind that the primal concepts
are the assumption of optimization and demand and cost
conditions and not the reaction of the firm. There is a
fundamental distinction to be drawn between behavioral
assumptions and those founded upon optimization.
Behavioral assumptions are in general unstable, changing
as behavior changes in response to simple economic shocks
or the policy proposed. To place restrictions on the
primal concepts, de facto making them secondary to an
assumed direction of reaction by the foreign firm,
invites serious error. As the example of the rational
expectations literature reminds us, behavioral
assumptions are a function of the policy regime in place
(and we would add other primal conditions) and must be
reevaluated as economic circumstances change. Such
assumptions inherently limit the generality of the
5. A specific example and general proof for constant
elasticity demand functions
An example serves to underscore the restrictiveness of
the assumption on convexity. Consider the following
industry inverse demand schedule:
p = 20000 q"-8 (24)
with a price elasticity of 1.25. The schedule is graphed
in Figure 2 for the relevant output range of the example.
As examination of the Figure shows the demand schedule is
"nearly" linear over the relevant range. The degree of
curvature is quite "small" ranging from .000104 to
.000115. We assume that the initial market shares for
Figure 2 Industry Demand
the domestic and foreign firms are 400 and 600 units
respectively. This implies marginal costs of $54.14 and
$41.40, assumed constant throughout. Industry market
price and output initially stand at $79.62 and 1,000
The domestic firm faces a cost disadvantage. If it is
successful in making its case for a subsidy, its marginal
cost shifts downward. Assume a five percent subsidy,
lowering domestic marginal cost by $2.71 to $51.44. As
BRS predict, domestic output rises from 400 to 434.28 and
domestic profit from $10,191.55 to $11,260.96, a gain of
$1,069.41. Foreign output, however, rather than
decreasing as BRS assume, rises from 600 to 602.30 as
shown in Figure 3.7 The cost of the subsidy totals
$1,175.60. The net result is a reduction in domestic
economic welfare in the amount of $106.19.
We would emphasize that the example is in no way
special. The initial industry output chosen is 1,000
units, a modest amount compared to the outputs of
companies engaged in trade. Constant marginal costs are
empirically relevant. Constant elasticity demand
schedules are widely employed in empirical studies. The
parameters and functions are selected to correspond and
be sympathetic to BRS's assumptions. Curvature in the
390 i t i i i \ i i i "* * * * * 1 1
595 596 597 598 599 600 601 602 603 604
Figure 3 Reaction functions
range of .000104 to .000115 is not generally considered
"very convex". The degree of elasticity (1.25) is not in
any way inordinate.
Regardless of how reasonable counterexamples may be,
the question of whether they are "very" special remains.
A generalization of the result serves to address this
issue. Any constant elasticity demand schedule and any
section on that demand schedule will generate
counterexamples similar to the one above. Proposition 1
formally demonstrates the point. We denote marginal
revenue for x and y by rx and r* respectively. The proof
depends upon two lemmas.
Lemma 1: If p = aq"b then 5q < v < q V q.
Proof: From (11), v = -p'/P" which for this demand
specification is q/(b+l). Because 0 < b < 1, .5q < v < q.
Lemma 2: If x < y < q and p = a(x + y)~b then rx > r* > r
> 0 V q.
Proof: Note that rq = (l-b)aq~b = p + p'q>0Vq.
Similarly, rx = p +AP'x and r* = p + p'y. It immediately
follows that rx > r* > rq > 0 V q.
Proposition 1: For any inverse demand function
representable by p = a(x + y)"b where 0 < b < 1 and a > 0,
c* and cx can be selected so that yx > 0 and Gs < 0 for any
given q > 0.
Proof: From lemma 2, marginal revenues for both firms, rx
or r*, are everywhere positive. From lemma 1 for any
output level .5q < v < q. Thus it is always possible to
select y such that .5q < v < y < q and x = q y by
setting cx = rx and c* = r*. For any given q we have y >
v. Therefore 7r*x = p' + yp" > o, implying that both yx
and ys > 0 while Gs < 0 (from (4), (6), (7), and (9)).
Brander and Spencer's results on subsidies can thus be
reversed for the full range of constant elasticity demand
schedules. This strongly suggests that the conditions
for reversing the positive effect of subsidies are not at
all stringent. It is difficult to conclude that such
elemental demand and cost conditions are not included
among "the most economically relevant", particularly at
the theoretical level at which BRS's analysis is
The decline in international competitiveness of U.S.
firms over the last decade has become a major focus of
government policy. Protectionism, hostile negotiation
and threats of retaliation have all been employed in an
effort to bolster the position of domestic corporations.
These policies are not without costs to domestic economic
welfare and good-will of trading partners. Subsidies
recently have emerged in the theoretical literature as an
attractive alternative. BRS show that not only do
subsidies strengthen international competitiveness of
domestic firms but do so while enhancing domestic
economic welfarea policy clearly to be preferred.
This paper questions whether the theoretical case
for subsidies is as strong as BRS indicate. Cogent
argument suggests that it is not. We demonstrate that
any degree of convexity from near zero to plus infinity
is compatible with detrimental effects from subsidies.
This finding stands in direct contradiction of the BRS
assumption that subsidies prove harmful only if demand is
"very convex". "Large" outputs are also shown to violate
BRS's assumptions. Upper limits on output do not provide
an economically compelling argument for imputing
generality to results. Neither, we find, do upper limits
We provide counterexamples to BRS's results using very
elemental demand and cost conditions. Specifically, we
demonstrate that for constant marginal cost and any
constant elasticity demand schedule and any section on a
particular constant elasticity demand schedule, reversals
of BRS's results occur. Finally, we show that even if
subsidies are initially beneficial, elementary shifts in
demand, marginal cost or changes in the level of the
subsidy itself produce reversals in the subsidy's
beneficial results. Subsidies thus always carry the
potential to diminish domestic welfare. When evaluating
options to strengthen U.S. international competitiveness,
this inherent potential warrants serious attention.
A general methodological point merits repetition. The
difficulty with the cross partial restriction lies in its
de facto restriction on optimizing behavior. Behavioral
assumptions by their very nature artificially limit the
set of possible economic actions. Behavioral assumptions
are not founded upon primal principles such as
optimization behavior. They cannot support the level of
generality critical to such policy issues as the decline
of U.S. international competitiveness. As a corollary,
statements ascribing degrees of economic relevance to
particular cases without benefit of empirical
verification are misleading at best. Such statements
impart a subtle distorting and chilling effect to
empirical investigation. Cloaked in terms of acceptable
generality, they predispose researchers at the margin to
discount some results, search more assiduously for others
and allocate scarce research resources to other endeavors
if economically "relevant" results are not forthcoming.
Such statements tend to forestall critical empirical
investigation upon which policy must ultimately depend.
Brander, James A. and Barbara J. Spencer, 1985, Export
subsidies and international market share rivalry,
Journal of International Economics 18, 83-100.
Brander, J. A. and B. J. Spencer, 1988, Unionized
oligopoly and international trade policy, Journal
of International Economics 24, 217-234.
Bulow, J. I., Geanakoplos J. D. and Paul D. Klemperer,
1985, Multimarket oligopoly: strategic substitutes
and complements, Journal of Political Economy 93,
Cheng, L. K., 1988, Assisting domestic industries under
international oligopoly: the relevance of the
nature of competition to optimal policies, The
American Economic Review 78, 746-58.
Eaton, J. and G. M. Grossman, 1986, Optimal trade and
industrial policy under oligopoly, Quarterly Journal
of Economics 101, 383-406.
Formby, J.P., Layson S. and W. J. Smith, 1982, The law of
demand, positive sloping marginal revenue, and
multiple profit equilibria, Economic Inquiry 20,
Helpman, E. and P. R. Krugman, 1989, Trade policy and
market structure (The MIT Press, Cambridge).
Keynes, John Maynard, 1936, The general theory of
employment interest and money, (Harcourt, Brace and
World Inc., New York).
Lucas, R. E. Jr., 1976, Econometric policy evaluation: a
critique, in: R. E. Lucas, Studies in business cycle
theory, (MIT press, Cambridge) 104-130.
Ricardo, David, 1817, The principles of political economy
and taxation, (J.M. Dent and Sons, London).
1. BRS (1988) use essentially the same method to analyze
unionized oligopolies. They show that unions act as a tax
on home firms, and since they believe home subsidies are
optimal, conclude that unions reduce welfare. Note that if
the effect of home subsidies is reversed, then the effect
of home unions is also reversed. Therefore the following
discussion is relevant to subsidies and unions. A detailed
discussion of the union case is available from the authors.
The authors also recognize that the critique presented in
this paper applies to duopoly models in general.
2. Questioning received opinion is a delicate task.
Keynes (1936, pg. v) found it necessary to "ask
'^forgiveness if, in the pursuit of sharp distinctions, my
controversy is itself too keen". Ricardo (1817, preface)
"found it necessary to advert more particularly to those
passages in the writings of Adam Smith from which he"
[Ricardo] "sees reason to differ". We echo these
sentiments. The reader is urged to seek out particularly
those sections of BRS's writings which we do not discuss.
Although we critically examine the assumption of
convexity, we would emphasize that we greatly admire
3. These conditions similar to those under which
marginal revenue possesses a positive slope. Formby,
Layson and Smith (1982) demonstrate that such
possibilities cannot be ruled out and indeed are likely.
Simple shifts transform demand schedules associated with
downward sloping marginal revenue into ones with positive
sloping marginal revenue curves.
4. The result also provides a dynamic motive for a
subsidy. It may be rational to subsidize a home firm in
a growing industry permitting it to capture a larger
share of an expanding market.
5. BRS realize that they assume away perfectly viable
demand structures. They do not further analyze them on the
grounds of economic relevance. BRS do not appear to be
aware of the comparative static restrictions which this
assumption imposes or the arbitrary nature of the term
"very convex". Further extensions and treatments of BRSs
work have tended to even further disregard the cases which
overturn the result.
6. Lucas (1976) makes the same point as a critique of
econometric practice. If the econometric model is grounded
in behavioral assumptions, proper forecasting requires
rederiving optimal behavior in the new policy environment.
7. The reaction functions were found using the solve
procedure in TSP. From the first order condition we know
x = -(p cx)/p*. The corresponding TSP program line,
assuming a subsidy, is x = -(20000*(x+y)~(-.8)-
51.435432)/((-16000*(x+y)"(-1.8)). If this program line
is called by the solve command from a workfile containing
y values, TSP will iteratively solve for x values.