Discussion Paper No. 718

QUOTAS UNDER DYNAMIC
BERTRAND COMPETITION

Kaz Miyagiwa
andYuka Ohno

August 2008

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Quotas under dynamic Bertrand competition

Kaz Miyagiwa*
Osaka University and Emory University
andYuka Ohno
Deloitte Tax LLP

Abstract
We present a new model of dynamic Bertrand competition, where a quota is treated as anintertemporal constraint rather than as a capacity constraint as is common in the literature.
The firm under a quota then can still vary the rates of exports over time provided that its
total sales within the period do not exceed the quota. We show that a quota results in higher
prices than a tariff of equal imports. We also show that firms never play mixed strategies,
which contrasts from the result from a one-shot game, in which the only equilibrium undera quota is in mixed strategies.
JEL Code: F13
Keywords: Dynamic Bertrand competition, quotas, tariffs, differential game

* Corresponding author. The Institute of Social and Economic Research, Osaka University, 6-1 Mihogaoka,

Ibaraki, Osaka 567-0047, Japan. E-mail: kmiyagiwa@gmail.com. We thank participants at the
Hitotsubashi COE/RES Conference on International Trade and FDI, and Economics Department seminars at
the University of Texas-Arlington and Osaka University for comments and suggestions. We alone areresponsible for any errors.

1. Introduction
This paper presents a new model of dynamic Bertrand competition under a quota.
The model reflects the idea that an exporter facing a quota behaves quite differently from afirm facing a capacity constraint. A quota specifies how much a firm is allowed to export
within a given period, say, one year. A firm under the quota can still vary the rates of export
over the course of the year as long as its annual total exports do not exceed the quota. In
contrast, a capacity-constrained firm is subject to the production limit per unit of time, say,
due to the plant size, and therefore does not face a dynamic constraint like a firm under thequota. This distinction between a quota and a capacity constraint is lost in the literature,
however. Virtually every dynamic model of oligopoly examining the effect of quotas, nodoubt influenced by the capacity constraint literature (e.g., Kreps and Sheinkman, 1983),
treats a quota as a capacity constraint.
The objective of the present paper is therefore to explore the nature of a quota as adynamic quantity constraint. To that end we consider Bertrand (price-setting) competition
between a foreign firm and a domestic firm in the domestic market during a given period.
Firms produce differentiated goods and choose prices continuously over time during tomaximize total profits within the period. In addition, the foreign firm is subject to a quota
during that period. We solve this model for a subgame-perfect equilibrium.
Our analysis makes two main contributions to the literature. A first is to theliterature on the relative effect of tariffs and quotas. It is well know that tariffs and quotas
are equivalent under perfect competition in the sense that replacement of a tariff with anequal-import quota leaves the domestic price intact. Over the last quarter-century the validityof this proposition has been questioned under various forms of market structure.1 In static
settings Itoh and Ono (1982, 1984), Harris (1985), and Krishna (1989) showed for

1 See, e.g., Helpman and Krugman (1989) and Brander (1995) for surveys of early work.

2
Bertrand duopoly that the quota results in a higher domestic price than the tariff while
Hwang and Mai (1988) reaffirmed the equivalence of the two policy instruments for
Cuournot duopoly.
The relative effect of tariffs and quotas has also been examined in dynamic contexts.
Rotemberg and Saloner (1989) analyzed a collusive equilibrium in a repeated-game setting
and found that a quota results in a lower domestic price than a tariff of equal imports. The
reason for this surprising result is that under the quota the foreign firm is prevented frompunishing the home firm as harshly as under the tariff when the latter cheats. Hence, theequilibrium price must be set lower to curb the home firm’s temptation to cheat and torestore collusion. Dockner and Haug (1990) considered a dynamic Cournot model withslow price adjustment, and showed that a quota results in higher prices than the quota as theprice adjust to the equilibrium. However, why the market remains out of equilibrium is not
explained. Miyagiwa and Ohno (1995, 1999) re-examined the infant-industry argument forprotection, showing that a quota generally delays adoption or innovation of new technology
relative to a tariff. All these authors assumed the quota binding at each point in time, thereby
ignoring the possibility of intra-period variations in prices, which is the focus of this paper.
An exception is the work of Miyagiwa and Ohno (2001). Treating quotas as dynamic
constraints and examining the intra-period variations in prices and output for Cournot
duopoly, they found that the equilibrium price is higher under the quota, showing that thestatic result of Hwang and Mai (1988) does not extend to a dynamic setting.
A second contribution this paper makes to the literature concerns the fact thatmodels of oligopoly under a quota may not possess equilibria in pure strategies. A basic
reference is Krishna (1989), who showed that for differentiated Bertrand competition the
only Nash equilibrium under the quota is in mixed strategies. In repeated-game setting,Rotemberg and Saloner (1989) also showed that the stage game has homogeneous-goods

3
Bertrand firms playing mixed strategies in equilibrium. The absence of pure-strategy
equilibrium under quotas is not unique to Bertrand competition, however. Reitzes andGrawe (1994) showed that Cournot duopoly has the unique mixed-strategy equilibrium
under market-share quotas, i.e., when the foreign firm is constrained not to exceed theprescribed fraction of sales in the domestic market.
The logic underling these results cannot be faulted. Some researchers havenonetheless expressed skepticism about mixed-strategy equilibriums. The following
statement by Brander (1995, p. 1436) summarizes this sentiment: “The value of Krishna is
not so much that it is likely to be a literal description of an actual outcome. The paper’s
contribution is that it focuses attention on the idea that a VER (and, by extension, any trade
policy instrument) can have important effects through the effects on imperfectly competitive
rivalries between firms.” Our model throws light on this issue. The above-mentioned
models possess no pure-strategy Nash equilibrium under the quota because the game isstatic (Krishna 1989, Reitzes and Grawe 1994) or equivalently a quota is binding at every
instant (Rotemberg and Saloner 1989). Using Krishna (1989)’s model as a benchmark, weshow that, when a quota serves as a dynamic constraint, firms never play mixed strategies.
The next section reviews the Krishna model, which serves as a benchmark. Section
3 sets out the dynamic Bertrand model, showing the absence of mixed strategy equilibrium
under a quota. Section 4 presents additional properties of the equilibrium, and shows that
normally a quota leads to higher price trajectories than the equal-import tariff. Section 5
reconsiders the facilitating nature of quotas. Concluding remarks are in the final section.

2. The one-shot price-setting game under the quota
This section provides a brief review of the Krishna (1989) model, pointing out its
essential features that will be useful in the next section. The Krishna model is an application

4of the standard differentiated-goods Bertrand duopoly model, where a domestic and aforeign firm play a one-shot price-setting game in the domestic market. Let P and p denotethe prices of the domestic and the foreign good, respectively. (Notationally, we use capitalletters to denote the domestic firm and lower case to denote the foreign firm.) Write thedemand function for the domestic and the foreign good as X(P, p) and x(P, p), respectively.
Assume that demands are downward sloping (XP < 0 and xp < 0), and goods are substitutes(Xp > 0 and xP > 0), where the subscripts denote partial derivatives.
Suppose that free trade prevails. The foreign firm takes P as given and chooses p to
maximize the profit:v(P, p) ≡ px(P, p) – c(x(P, p)),where c(.) is the foreign firm’s cost function, with c’ > 0 and c” ≥ 0 (primes denote
derivatives). The domestic firm takes p as given and chooses P to maximize the profit:
V(P, p) ≡ PX(P, p) – C(X(P, p)),where C(.) is the domestic firm’s cost function, with C’ > 0 and C” ≥ 0. The first-ordercondition:(1)

vp(P, p) = x(P, p) + pxp(P, p) - c’[x(P, p)]xp(P, p) = 0

implicitly defines the foreign firm’s best-response function, p = bs(P). The domestic firm’s
best-response function, P = Bs(p), obtains analogously from the first-order condition:(2)

VP (P, p) = X(P, p) + PXP (P, p) - C’[X(P, p)]XP (P, p) = 0.

Assume that v(P, p) is strictly concave in p and V(P, p) is strictly concave in P so that thebest-response functions are indeed “functions”.
Solving the two best-response functions simultaneously yields the one-shot Nashequilibrium denoted by (Ps, ps). (We call this the “static” Nash equilibrium later and use

5
the superscript s to denote it). The best-response functions and the Nash equilibrium (Ps,ps) under free trade are shown in Figure 1. The best-response functions slope upward sinceprices are strategic complements as is usually assumed. The intersection point S of the twobest-response functions corresponds to the Nash equilibrium. The usual conditions are
invoked to ensure the uniqueness of the equilibrium.2 The equilibrium outputs and profits
obtain by substituting Ps and ps into each firm’s demand and objective functions. Let Xs ≡X(Ps, ps) and xs ≡ x(Ps, ps) denote, respectively, the domestic and the foreign firm’s static
equilibrium output.
Suppose now that there is a quota set at ~q = xs, the equilibrium quantity of imports
under free trade. The effect is seen in Figure 1.3 The curve qq* represents the locus of P
and p such that the demand for the foreign product just satisfies the quota; that is, ~q = x(P,p). By design it necessarily goes though S. Given P, if the foreign firm charges a price lessthan the price that satisfies this equation, the demand for the foreign product exceeds thequota. This shortage for the foreign good is assumed to vanish through consumer arbitrage.
That is, those consumers lucky enough to get the foreign good at a lower price resell theirpurchased units at the market-clearing price implied by the locus qq*. Therefore, the foreign
firm would not have any incentive to set a price below the locus qq*. Hence, its bestresponse function consists of the part of qq* to the right of the bs schedule and the part ofbs below the locus qq*, as indicated by the thick line with a distinct kink at the point S. The
domestic firm’s best-response function under the quota (also shown in bold) is not even2

| vpp| > | vpP| and | VPP| >| VPp| guarantee the equilibrium to be unique and stable.

3 Figure 1 duplicates Krishna's Figure 3, where the quota is set at the free-trade level of imports.

6
continuous, consisting of the part of its original best-response curve Bs to the right of ^p andthe horizontal line at PH up to ^p. The horizontal best response function is again due toconsumer arbitrage, which raises any price p set by the foreign firm to the market-clearing
price on qq*. Then the home firm best-responds by setting the price equal to PH, where its
iso-profit curve is tangent to qq*. The effect on the domestic firm’s demand therefore isexactly what it would be if the foreign firm directly charged pA instead of ^p; i.e., the demand
for the domestic good is given by X(PH, pA). This holds true whenever p is below (to theleft of) qq*. When the foreign firm sets ^p, however, the home firm can earn the same profit
by responding with price PL or PH as shown in the figure. For any price higher than ^p, thehome firm’s best responses are indicated by the best-response function Bs(p).
Since the two best-response functions do not intersect each other, there is no Nashequilibrium in pure strategies. However, there is a unique Nash equilibrium in which thedomestic firm randomizes between PH and PL and the foreign firm charges ^p. Since both
PH and PL are higher than Ps, and ^p is higher than ps, we conclude that the quota raises
prices of both firms. This conclusion holds even when the quota is set equal to the volumeof imports under the existing tariff. That is, the quota raises prices of both the foreign andthe domestic firm relative to the tariff.
Notice that, since the domestic firm randomizes between PH and PL, its profit fromeach action must be identical, as indicated by the iso-profit curve. Thus, the domestic firm
earns a greater profit under the quota. The foreign firm exports the same quantity under thequota as under free trade but its price is higher now. Therefore, the foreign firm also
benefits from the quota (as long as the quota is not too restrictive).

7
3. The quota as a dynamic constraint
3.1 Setup
We now consider the model of differentiated-goods Bertrand duopoly in a infinitetime setting, where time t flows continuously from zero to infinity. Divide the time set [0,∞) into infinitely many non-overlapping unit intervals called years, that is,[0, ∞) = [0, 1) ∪ [1, 2) ∪ [2, 3) ∪ ….
X(P(t), p(t)) and x(P(t), p(t)) now denote instantaneous demand functions at time t. The
demand functions are stationary over time. Finite reservation prices exist and quantities
demanded are finite for all (p, P). The Markov-perfect equilibrium is used as the solution
concept to preserve the properties of the one-shot game.
Under free trade, firms choose prices simultaneously at each instant t to maximize
profits over an infinite time horizon. Since there are no “state variables”, the model isstationary in the sense that subgames starting at any t ∈ [0, ∞ ) are identical. Therefore, theequilibrium outcome at any t is exactly the same as that in the one-shot Bertrand game. Thatis, in the Markov-perfect equilibrium the foreign firm produces xs units and charges ps
while the domestic firm produces Xs units and sells them at ps at each t ∈ [0, ∞).

3.2 Quotas
Suppose that the foreign firm is subject to an annual quota q-. Since a year is a unit
interval, we can write the constraint facing the foreign firm as:
τ + 1

(3)

⌠
⎮
⌡

τ

x(P(t), p(t))dt ≤ q-,

8
for all τ = 0, 1, 2, .... To make the analysis interesting, we assume that the quota is binding,that is, q- is less than the total volume of exports under free trade:
τ + 1

q- ≤

⌠
⎮ xsdt.
⌡τ

The constraint (3) implies that how much more output the foreign firm can export during
the remainder of a given year depends on the total output it has sold to date in that year.
Thus, the past sales become a “state variable” in the dynamic optimization problem facing
the foreign firm. However, the foreign firm cannot usually carry the unused portion of thequota over to the following years. Then, by the property of Markov-perfect equilibrium,
whatever happens during the year has no effect on the subgames beginning in the following
or later years. This fact lets us focus on the first year; i.e., t ∈ [0, 1).
Since we are interested in the subgame perfect equilibrium we solve the model
backward, using dynamic programming techniques. Suppose we are at a given time z ∈ [0,1). Let µ(z) denote the foreign firm’s total output up to a time z < 1, that is:z⌠µ(z) ≡ ⎮⌡ x(P(t), p(t))dt.
0Then, the maximum quantity the foreign firm is allowed to sell in the interval [z, 1) is equal
to q- - µ(z). Thus, the foreign firm’s quantitative constraint at time z is written as:
1

(4)

⌠
⎮ x(P(t), p(t))dt ≤ q- - µ(z).
⌡z9Subject to this isoperimetric constraint, and taking µ(z) and P(z) as given, the foreign firmchooses p(t) for t ∈ [z, 1) to maximize the sum of profits over the interval [z, 1):1⌠⎮ e – rt{p(t)x(P(t), p(t)) – c(x(P(t), p(t)))}dt⌡zwhere r is the instantaneous rate of interest. The domestic firm faces no quantitative
restriction, so it takes p(z) as given and chooses P(t) to maximize the followingintertemporal profit over t ∈ [z, 1):1⌠⎮ e – rt{p(t)X(P(t), p(t)) – C(X(P(t), p(t)))}dt.
⌡z

3.3 Characterization of the equilibrium under the quota
We first prove the following proposition, which is useful in solving the model.

Proposition 1. Both the firms play pure strategies at each t ∈ [0,1).

We prove this by contradiction. Suppose that a mixed strategy is played at any time t0 ∈ [0,1). When the quantitative restriction does not bind, strict concavity of the flow profit
functions V(p, p) and v(P, p) implies that firms play pure strategies. Therefore, for firms toplay mixed strategies at t0, at least one firm must have a non-concave flow profit function.
Then the discussion developed in section 2 leads to the following result.

10
Result: When the quota binds at t0, at a time t ∈ [t0, t0 + ε] where ε (> 0) is arbitrarily-

small, the domestic firm randomizes between P ≡ PH(t0) and P- ≡ PL(t0) while the foreign
firm charges ^p ≡ ^p(t0). (Strict concavity of the profit function rules out other types of
mixed-strategy equilibria).

Let Ω(P, p^, t0) denote the domestic firm’s future profit when it charges the price P-

during [t0, t0 + ε]. When it randomizes between P and P- , the domestic firm’s future profits

must be identical; that is, Ω(P, ^p, t0) = Ω(P- , ^p, t0). We show that this is a contradiction.
-

Begin with the case in which the domestic firm charges P during [t0, t0 + ε]. Since
the quota is binding and consumer arbitrage takes place, the effect on the domestic firm’s
demand is exactly what it would be if the foreign firm charged some arbitrage price pA (≥ ^p).4 Thus, during the interval [t0, t0 + ε] the domestic firm makes the profit:
t0 + ε⌠ - r(t - t0) - - A⌡ e{PX(P, p ) - C[X(P, pA)]}dtt0- = ε{PX(P, pA) – C(X(P, pA))},
where we used the fact that for small ε

4 The actual value of pA is different from that in Section 2.

11
t + ε

⌠
⎮ f(z)dz = f(t)ε.
⌡tThen, the foreign firm reaches the quantitative limit q- by the time t0 + ε, and will make nomore sales after that. This will allow the domestic firm to charge the monopoly price Pm andearn the monopoly profit for the rest of the year.. Let Φm(t0 + ε) denote the discounted sumof the monopoly profits earned in the interval [t0 + ε, 1). Now, we can write the profit from-charging P as:

-

-

-

-

Ω(P, ^p, t0) = ε{PX(P, pA) – C(X(P, pA))} + Φm(t0 + ε).
Consider next the case in which the domestic firm charges the low price P- in [t0, t0+ ε]. The domestic firm’s profit during this short interval is given by
ε{P- X(P- , ^p) – C(X(P- , ^p))}.
In this case the foreign firm will not use up its quota by time t0 + ε but continue to engage
in duopolistic competition for some time after (t0 + ε). Let Φ(t0 + ε) denote the discounted
sum of the profits to the domestic firm in the interval [t0 + ε, 1). Then, the domestic firm’s
future profit is given by:Ω(P- , ^p, t0) = ε{P- X(P- , ^p) – C(X(P- , ^p))} + Φ(t0 + ε).
Comparing the profits after t0 + ε, we have that12
Φ(t0 + ε) < Φm(t0 + ε)since Φm(t0 + ε) sums up all the monopoly profits between t0 + ε and 1.5 Therefore, for εarbitrarily small, we haveΩ(P, ^p, t0) > Ω(P- , ^p, t0),-

implying that the domestic firm strictly prefers charging P to charging P- . But this-contradicts the initial hypothesis that the domestic firm randomizes between P and P- in theinterval [t0, t0 + ε]. Since t0 is arbitrary, this contradiction proves that the domestic firm does
not randomize at t ∈ [0, 1).6
To complete the proof, we must show that the quota is not used up at t < 1 in a purestrategy equilibrium, either. To see this, suppose the contrary; that is, suppose that theforeign firm makes the final sale of x units at t < 1. Should it postpone the sale of one unitof output to t + ε instead, the foreign firm would lose p'(x)x + p(x) - c'(x) in profit at time t
but would gain p(0) - c'(0) at time t + ε. For an arbitrarily small ε, therefore, such
postponement would always increase the foreign firm's intertemporal profit. The proof of
Proposition 1 is now complete.

5 The difference between Φm(t0 + ε) and Φ(t0 + ε) may be small if the quota is used up shortly after t0 +ε or if the rates of exports are low. However small, the difference, Φm(t0 + ε) - Φ(t0 + ε), does not vanish
when ε goes to zero, which is sufficient to produce a desired contradiction.
6 This result is not affected by the size of r (< ∞).

13
As we already noted, a mixed-strategy equilibrium outcome emerges only at the
moment when the quota is all used up. But Proposition 1 says that a mixed-strategy
outcome is never observed at t < 1. These two results lead to the next proposition.

Proposition 2: In the equilibrium the foreign firm does not use up its quota before the endof the year. That is:µ(t) < q- for all t < 1
andlim µ(t) = q-.

t→ 1

4. Equilibrium paths under the quota
Now we solve the model. Propositions 1 and 2 let us focus on a pure-strategy
equilibrium. We use dynamic programming techniques to obtain a subgame-perfectequilibrium.7~
Let P [t; j, µ(j)] and ~p [t; j, µ(j)] denote the equilibrium paths for the subgame
starting at time j (< 1) with the initial condition that the foreign firm has sold µ(j) (< q-)between [0, j). Then the intertemporal equilibrium profits for this subgame are given by:
1⌠ - rz ~ ~ ~~ ~Π[j, µ(j)] ≡ ⎮⌡ e {PX(P, p ) – C(X(P, p ))}dz,jfor the domestic firm and,

7 See Kamien and Schwartz (1981, chapter 20) for an exposition of dynamic programming techniques.

14
1⌠ - rz ~ ~ ~~ ~π[j, µ(j)] ≡ ⎮⌡ e {p x(P, p ) - c[x(P, p )]}dz,jfor the foreign firm. Now divide the interval [t, 1) into two disjoint subintervals [t, t + Δt)
and [t + Δt , 1), where Δt is small. Assume that the optimization problem is solved for thesecond subinterval, with the maximum profit from that subinterval denoted by Π[t + Δt, µ(t+ Δt)]. Then, by the principle of optimality we can express the domestic firm’s optimization
problem for the interval [t, 1) as:t + Δt

(5)

max
P

⌠
⎮ e- rz{P(z)X(P(z), p(z)) – C(X(P(z), p(z)))dz⌡t+ Π[t + Δt, µ(t + Δt)].
Note that µ(t + Δt) = µ(t) + Δµ. Since Δt is small, P(z) can be treated as a constant in (5).
Also, for small Δt we have:t + Δt(6)⌠⎮ f(z)dz = f(t)Δt.
⌡t

Using (6) on the first term of (5) and expanding the second term by a Taylor-series
expansion, we can rewrite (5) as:
(7)

max
P

{Δt[PX(P, p) – C(X(P, p))]e- rt

15+ Π(t, µ(t)) + Πt(t, µ(t))Δt + Πµ(t, µ(t))Δµ + h.o.t.},where the h.o.t. represents all the higher-order terms in the Taylor-series expansion.
Since the second and the third term in (7) do not depend on P, they can be ignoredin the optimization problem. Then, dividing through the remaining terms by Δt and taking
the limit Δt → 0, we can rewrite the domestic firm’s objective function as(8)max e- rt[PX(P, p) – C(X(P, p))] + Πµ(t, µ(t))x(P, p),

where x(P,p) is used to replace µ’(t).8 The first-order condition to (8) is given by(9)Ψ ≡ e - rt{X(P, p) + PXP (P, p) - C’(X(P, p))XP (P, p)}+ Π µ(t, µ(t))xP (P, p) = 0,

which defines the domestic firm’s best-response function B(p, t) at time t.
The first-order condition (9) has the following interpretation. The first termmeasures the change in the domestic firm’s profit at time t due to a small increase in P.
Raising P also induces an instantaneous increase in imports (xP > 0), which reduces theremaining quota through the increase in µ(t), which in turn raises the domestic firm’s
profits from sales made during the remainder of the year (Πµ > 0). This dynamic effect of aprice change is captured in the second term on the right-hand side of (9). For an optimum
the domestic firm chooses P to equate these two marginal effects on profits.
The properties of the best-response function B(p, t) can now be examined. We firstcompare B(p, t) with the best-response function Bs(p) from the static game of Section 2.
Notice that the expression in braces in (9) is the same as the first-order condition of the16static game [see eq. (2)]. The second term in (9) is positive by the preceding argument.
Therefore, for a given p the domestic price P must be higher in the dynamic game than in
the static game; i.e., P(t) = B(p, t) > Bs(p). Graphically, B(p, t) lies “outside” Bs(p).
We next examine how the domestic firm’s best-response function shifts over time.
Differentiating (9), while holding p constant, we obtain(10)

dP/dt = -(Πµ t + Π µµx)xP + rΠ µxP
e - r t(2XP + PXPP - C”XP 2 - C’XPP) + Π µxPP.

The denominator is negative by the second-order condition. Lemma 1 in the Appendix
shows that (Πµt + Π µµx)xP is positive. Since Πµ and xP are also positive, we have dP/dt >0, meaning that the domestic firm’s best-response function shifts out over time. To sum theresult so far, the domestic firm’s dynamic best-response function under the quota is locatedoutside of its static-game (and hence free-trade) best-response function, and shifts out overtime.
Turning to the foreign firm, we write its optimization problem as
t + Δt

(11)

max
p

⌠
⎮ e- rz{p(z)x(P(z), p(z)) – c(x(P(z), p(z))}dz⌡t+ π(t + Δt, µ(t + Δt)).9
The procedure used in the case of the domestic firm can be applied to simplify (11) to thefollowing problem:
max e - rt{px(P, p) – c(x(P, p))} + πµ(t, µ(t))x(P, p).
p

8 By definition, µ’(t) = x(P, p). Note also that the h.o.t. vanishes when the limit is taken.

17
The first-order condition is(12)ψ ≡ e - rt{x(P, p) + pxp(P, p) - c’(x(P, p))xp(P, p)}+ π µ(t, µ(t))xp(P, p) = 0,

which yields the foreign firm’s best-response function, p = b(P, t), at time t < 1.
How does b(P, t) compare with the best-response function bs(P) from the one-shotgame in Section 2? The expression in braces in (12) is the same as that in the first-ordercondition in the static game [see eq. (1)], so the position of b(P, t) relative to bs(P) depends
on the second term in (12), which captures the following dynamic effect. A small increase inp decreases the demand for exports (xp < 0), thereby relaxing the quota constraint for theremainder of the year and affecting the foreign firm’s future profit by πµ.
Thus the term (- πµ) may be interpreted as the dynamic shadow price of the quota.10
This shadow price is positive (i.e., π µ < 0) when the quota is binding. The intuition is thatwhen µ increases the foreign firm has less to export for the rest of year and hence earns asmaller profit. Formally, suppose contrarily that πµ ≥ 0, implying that the foreign firm’s
equilibrium profit is increased by a decrease in the (remaining) quota. Then, the foreign firm
would voluntarily give up a portion of the quota until π µ becomes negative. In the absenceof such an option, the foreign firm would export in an instant whatever quantity is needed tomake πµ negative. While such an action may yield a negative profit instantaneously, its

9 By definition π is the equilibrium profit under the quota, so it subsumes the quota constraint (4).
10 See Kamien and Schwartz (1981, p. 240) for this interpretation.

18
effect on total profit is of measure zero and is overwhelmed by an increase in profits due toa smaller quota that it now faces.
Given that πµ < 0, the second term on the right-hand side of (12) is positive,
implying that p = b(P, t) > bs(P); that is, for a given P, the domestic firm charges a higher
price than it does in a static game.
We next examine the transition of the foreign firm’s best-response function overtime. Differentiating (12), while holding P constant, yields:
dp/dt = -(πµ t + π µµx)xp + rπ µxp
e - r t(2xp + pxpp - c”xp2 - c’xpp) + π µxpp.

The denominator is negative by the second-order condition. Lemma 2 in the appendix
shows that the first term of the numerator is positive. Thus, the direction of shift again
depends on the sign of π µ. Since π µ is negative, dp/dt > 0, so the foreign firm’s bestresponse function shifts out over time.
Solving the best-response functions (9) and (12) simultaneously, we can find theequilibrium prices at each t. Let P*(t) and p*(t) denote the equilibrium outcome for the~whole game; that is, P*(t) ≡ P [t; 0, µ(0)] and p*(t) ≡ ~p [t; 0, µ(0)] for t ∈ [0, 1).
We now present some properties of the equilibrium price paths. We alreadyestablished that under the quota the domestic firm’s best-response function is locatedoutside its static-game counterpart, Bs(p), and shifts out over time. We then proved that theforeign firm’s best-response function also is placed on or outside the best-response
function bs(P) and shifts out after some t ≥ 0. These facts lead to the following general
result.

19
Proposition 3: The prices under the quota are higher than what they are under free tradeand increasing over time:
P*(t) > Ps(t), p*(t) > ps(t), P*’(t) > 0, and p*’(t) > 0 for t ∈ [0, 1).

We now ask what happens when the home government imposes a more restrictive
quota. A tighter quota raises the shadow price of the quota at each point in time, thereby
inducing the foreign firm to raise its price at t ∈ [0, 1). Further, since prices are strategic
complements, the domestic price also rises throughout the period. We state this in the next
proposition, while relegating the proof to the appendix (see lemma 3).

Proposition 4: Decreasing the quota raises both firms’ price trajectories.

5. Quotas as facilitating practices
Krishna (1989) showed for a one-shot differentiated-good Bertrand game that boththe foreign and the home firm earn greater profits under a quota than under free trade. This
result holds when there is a tariff; i.e., a quota results in a more collusive equilibrium thanthe equal-import tariff. In this section we examine whether this property extends to the
dynamic setting we consider.
As before, focus on the interval [0,1), and set the quota equal to the quantity ofimports under free trade:
1

(13)

⌠
q-f = ⌡ x(Ps, ps)dt.
0

20
The free-trade prices Ps and ps obviously meet the constraint (13). They fail to satisfy the
first-order conditions (9) and (12), however, and hence no longer constitute a subgameperfect equilibrium.11 Thus, the imposition of the quota at the free-trade level of imports
affects the equilibrium behavior in our model.
Our analysis of section 4 is fully applicable here. Thus, under the quota thedomestic firm’s best-response function is located outside its free-trade position and shifts
out over time. The foreign firm’s best-response function also behaves similarly although it
may be identical to its static counterpart momentarily. With both the best-response
functions being displaced outward from their respective free-trade positions, the equilibrium
prices are higher under the quota than under free trade. Thus, P*f(t) > Ps and p*f(t) > ps for
all t ∈ [0, 1). Then, both firms’ profits are greater under the quota q-f than under free trade.
By an extension, the foreign firm earns a greater profit under a quota than under the equalimport tariff even if the tariff revenue were returned to the foreign firm. In other words, aquota facilitates collusion relative to the equal-import tariff, .

Proposition 5. Both the foreign firm and the domestic firm earn greater profits under aquota than they do under the equal-import tariff.

6. Concluding remarks
This paper explores the fundamental difference between the capacity-constrained
firm and the quota-constrained firm. Quotas limit the total quantity the foreign firm isallowed to export within a given period, say, a year. As a consequence the firm under the

11 To see this, suppose that firms charge Ps , ps at time zero (the moment the quota is imposed). Then thefirst expression on the right-hand side of (9) vanishes. Since Πµ[0, µ(0)]xP > 0 the first-order condition (9)21quota can strategically vary rates of export at every instant as long as the sum of exports
does not exceed the quota. In contrast, capacity constraints are more physical, limiting thefirm’s production ability per unit of time, say, due to the plant size. As a result, the capacityconstrained firm does not face the isoperimetric problem as the firm does under the quota.
This difference is obscured in the literature examining the relative effect of quota, whichanalyzes firms under a quota as capacity-constrained ones.
One benefit from treating the quota as an isoperimetric problem is that firms playpure strategies at all times. This fact is helpful in solving the dynamic profit-maximization
problems under the quota. We develop techniques to solve this type of differential games,and fully characterize the equilibrium path for the quota-constrained Bertrand duopoly
model, the one-shot version of which was considered in Krishna (1989).
We show that the quota results in higher and increasing prices than the tariff ofequal imports. Our result also shows that price variations are greater under the quota as
prices increase over time. Further, if the annual quota is renewed every year, the saw-toothedprice trajectories appear every year: the quota produces price and output cycles while they
are stationary under the tariff.
The techniques we developed are useful in analyzing other models of one-shotgames under the quota that have mixed-strategy equilibriums, e.g., Reitzes and Grawe(1994). We leave such applications of the present analysis to future work.

is not satisfied.

22
Appendix: Proofs of the three lemmas mentioned in the text are presented.
Lemma 1. (Πµt + Π µµx)xP > 0
Proof: First, since goods are substitutes, we have xP > 0. Therefore, the proof is complete if
we show Πµt + Π µµx > 0. The definition of the maximum profit function and the principleof optimality imply~ ~~Π[t, µ(t)] ≡ Δt{PX(P, ~p ) – C(X(P, ~p ))}e- rt + Π[t + Δt, µ(t) + Δµ]~ ~~= Δt{PX(P, ~p ) – C(X(P, ~p ))}e- rt+ {Π[t, µ(t)] + Πt[t, µ(t)]Δt + Πµ[t, µ(t)]Δµ + h.o.t.}
where the equality follows from a Taylor series expansion. Dividing through by Δt andtaking the limit Δt → 0 yields the Bellman equation:
(A1)

~ ~
~
~
- Πt[t, µ(t)] = e- rt{PX(P, ~p ) – C(X(P, ~p ))} + Πµ[t, µ(t)]x(P, ~p ).

Since this holds for all µ, we can differentiate (A1) with respect to µ to obtain:~ ~~~- Πtµ[t, µ(t)] = e- rt{PXp(P, ~p ) - C’(X(P, ~p ))Xp}pµ~~ ~+ Π µµ[t, µ(t)]x(P, ~p ) + Π µ[t, µ(t)]xp(P, ~p )p µ,which simplifies to(A2)~~Πµµ[t, µ(t)]x(P,p) + Π tµ[t, µ(t)]~ ~~~= - {e- rt{PXp(P, ~p ) - C’(X(P, ~p ))Xp} + Π µ[t, µ(t)]xp(P, ~p )}p~ µ.

Rearranging the first-order condition (9) yields

23
~
~
~
~
e- rt{PXP (P, ~p ) - C’(X(P, ~p ))XP (P, ~p )}~~= - Πµ[t, µ(t)]xP (P, ~p ) - e- rtX(P, ~p ).
Substituting this expression, we can express the right-hand side of (A2) as~~- { - e- rtX(P, p) + Πµ[t, µ(t)][xp(P, ~p ) - xP (P, ~p )]}p~ µ
Since we have Πµ > 0, the terms in the braces are negative. Also, in Lemma 3 below, weshow p~ µ > 0. Therefore, we have Πµµx + Πtµ > 0.

❏

Lemma 2. (πµt + π µµx)xp > 0
Proof: Following a procedure similar to the one leading to (A1) in the proof of Lemma 1,
we can obtain the Bellman equation for the foreign firm:
(A3)

~ ~
~
~
~ x(P
- πt[t, µ(t)] = e- rt{p, p ) – c(x(P, ~p ))} + π µ[t, µ(t)]x(P, ~p ).

Differentiating (A3) with respect to µ yields:~ ~~ ~~ - c’(x(P- πtµ[t, µ(t)] = e- rt{p, p ))}xP (P, ~
p )Pµ
~
~ ~
+ π µµ[t, µ(t)]x(P, ~p ) + π µ[t, µ(t)]xP (P, ~
p )Pµ
or
(A4)

~
π tµ[t, µ(t)] + πµµ[t, µ(t)]x(P, ~
p)
~ ~
~ ~
~ - c’(x(P= - {e- rt{p, p ))} + π µ[t, µ(t)]}xP (P, ~p )Pµ.

The expression in braces on the right-hand side of (A4) is positive since rewriting the firstorder condition (12) from the text yields:

24
~ ~
~
~
~ - c’(x(Pe- rt{p, p ))} + π µ[t, µ(t)] = - e- rtx(P, ~p )/xp(P, ~p ) > 0.
~Furthermore, xP > 0 because products are substitutes, and Pµ > 0 as we prove later inLemma 3. We have shown that the right-hand side of (A4) is negative. Lemma 2 thenfollows immediately from the fact that xp < 0.

❏~Lemma 3. ~p µ > 0 and Pµ > 0.
Proof: Differentiating the first-order conditions (9) and (12) simultaneously yields(A5)| D| dP/dµ = - ΠµµxP ψp + π µµxpΨ p| D| dp/dµ = ΠµµxP ψP - πµµxpΨ P

where | D| ≡ ψpΨ P - ψP Ψ p > 0 by the standard stability and uniqueness conditions. We also
know that ΨP < 0, ψp < 0 by the second-order conditions, and Ψ p > 0, ψP > 0 due tostrategic complementarity of prices.
We first show that πµµ < 0. To do so, note that(A6)π(t, µ0)> π(t, µ1) + (µ1 - µ0)e - rt(x + pxp - c’xp)/xp= π(t, µ1) - (µ1 - µ0)πµ(t, µ0)for given t, µ1, and µ0 (µ1 > µ0). The inequality holds for the following reason. Both sidesof the inequality measure profits the foreign firm makes by exporting the quantity q- - µ0
between time t and 1. The left-hand side, π(t, µ0), is by definition the foreign firm’s

25
maximum profit attainable. The first term, π(t, µ1), on the right-hand side of the inequality is
the maximum profit attainable by exporting the smaller quantity q- - µ 1 during this period.
The next term represents the profit resulting from exporting the quantity, µ1 - µ0, at e - rt(x+ pxp - c’xp). However, since this term is not the maximum profit to the foreign firm fromexporting µ1 - µ0 units, the inequality is established. The equality in (A6) follows from thefirst-order condition (12). But (A6) means that π(t, µ) is concave in µ: that is, πµµ < 0.
We next show that Π µµ > 0. Let µ1 and µ0 be given, with µ1 > µ0. Π(t, µ1)
represents the maximum profit the domestic firm can make when q- - µ1 units of the foreign
good are imported between t and 1. If q- - µ0 units were allowed to be imported instead
during the same period, the domestic firm would suffer the loss Π(t, µ1) - Π(t, µ0) > 0. Now
suppose that the domestic firm receives a compensation for this loss in the amount of- e - rt(X + PXP - C’XP )(µ1 - µ0)/xP .
This has the following interpretation. Since xP measures the increase in imports induced bya unit increase in the domestic price, (µ1 - µ0)/xP represents the price increase necessary to
induce the increase in imports by µ1 - µ0. But according to the first-order condition (9), themarginal cost of a price increase (a deviation from optimality) is- e - rt(X + PXP - C’XP ).

26
However, since the profit function is concave, this compensation evaluated at the marginunderestimates the true cost of the discrete price change necessary to induce the increase inimports, µ1 - µ0. Therefore,Π(t, µ1) - Π(t, µ0) > - (µ1 - µ0)e - rt(X + PXP - C’XP )/xP
= (µ1 - µ0)Πµ(t, µ0),
where the equality follows from (9). However, this means that Π(t, µ) is convex in µ, and~hence Πµµ > 0. Using these results in (A5), we have ~p µ > 0 and Pµ > 0.

❏

27
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