Reexamination of Strategic Public Policies

REEXAMINATION OF STRATEGIC PUBLIC POLICIES*

By KAZUHARU KIYONO† and JOTA ISHIKAWA‡

†Waseda University ‡Hitotsubashi University

This paper attempts to reinterpret the familiar approach to strategic public policies from theviewpoint of inefficiencies involved in oligopoly where firms engage in Cournot compe-tition. To this end, we introduce tools called “quasi-reaction functions” and “quasi-supplycurves”. These tools allow us to conduct analyses through use of the standard partial-equilibrium diagram, i.e. the quantity-price plane. We can find the relationship betweenprices and quantities directly and, hence, deal with inefficiencies easily and also suggestpolicies to correct such inefficiencies. Specifically, we reexamine public policies related tomixed-oligopoly, excess entry, technology choices with free entry and exit, and foreignoligopoly.JEL Classification Numbers: F12, F13, L13, L52.

1. Introduction

The late twentieth century could be referred to as the “age of strategic public policies”, withgovernment intervention occurring amid strategic interactions among various types ofprivate and public (or semi-public) agents. von Neuman and Morgenstern’s seminal book(1944) and Nash (1951) provided us with theoretical tools to analyse strategic interactionsamong firms and governments. It was, however, Selten’s (1975) concept of “subgameperfection” that really stimulated the later rapid development of the theoretical underpin-nings of strategic public policies. Spence (1977) and Dixit (1979, 1980) urged economiststo integrate traditional industrial organization and oligopoly theories with game theory, andBrander and Spencer (1984, 1985) made trade theorists rush towards the so-called “stra-tegic trade policies” under international oligopoly.

We have found that game theory is the most effective tool in exploring governmentintervention in oligopolistic markets. However, we often face difficulties in analysis whenusing this tool. In fact, many of the recent papers on strategic public policies start withspecific game-theoretic models, but end up with simulations using Mathematica and thelike instead of completing the qualitative analysis in more general frameworks. Somepieces of “economic” intuition are provided, but they are mostly based on strategic aspectsof interactions among players, such as “strategic competition for rents”, and are oftenlacking the perspective of traditional economic theory.

When we begin studying economics, we learn about three concepts of Pareto efficiencyin (i) consumption, (ii) production and (iii) product mix. Consumption efficiency requiresthat the private marginal benefits of consumption are equal across households. Productionefficiency requires that the private marginal costs are equal across firms. Product mix

* This paper is based on Kazuharu Kiyono’s invited lecture at the Spring Meeting of the Japanese EconomicAssociation held at Kyoto University on 7 June 2009. Because Kazuharu Kiyono passed away beforecompleting this paper, Jota Ishikawa undertook the role of completing it. The authors would like to thankan anonymous referee for useful comments. Jota Ishikawa acknowledges financial support from theMinistry of Education, Culture, Sports, Science and Technology of Japan (MEXT) through a GlobalCenter of Excellence Project and a Grant-in-Aid for Scientific Research (A). Jota Ishikawa dedicates thispaper to the late Professor Kazuharu Kiyono.

bs_bs_bannerThe Japanese Economic Review

The Journal of the Japanese Economic Association

The Japanese Economic Review doi: 10.1111/jere.12009Vol. 64, No. 2, June 2013

201© 2013 Japanese Economic Association

efficiency requires that the social marginal benefits of consumption are equal to the socialmarginal costs.

Whenever a certain allocation is inefficient because any one or more of these require-ments is not met, a comparison with another allocation enables us to decompose (at leastconceptually) the welfare gains and losses associated with a transition from one allocationto another such that we can determine which allocation is associated with greater or smallerlevels of each type of inefficiency. The welfare effects of any policy, including thosetargeted at oligopolies, can and should be expressed in terms of these three types ofinefficiency.

Furthermore, when we review the literature relating to strategic public policies, it is notoften the case that Pigovian taxes and subsidies are fully available to the policy authorities.A government may only be able to regulate entry and exit of firms. There underlies an ideathat the market outcome is that there is a socially undesirable number of firms, whichrepresents another type of inefficiency in the market. Moreover, there may be inefficiencyassociated with technology selection by the market.

In this paper, we attempt to reinterpret the familiar approach to strategic public policiesfrom the viewpoint of inefficiencies involved in oligopoly where firms compete à la Cournot.Although the results themselves are not necessary novel, the paper makes a contribution tothe literature through its use of tools called “quasi-reaction functions” (instead of thestandard reaction functions) and “quasi-supply curves”. These tools allow us to conductanalyses by using the standard partial-equilibrium diagram, i.e. the quantity-price plane. Wecan deal with the relationship between prices and quantities directly and, hence, can handleeconomic surpluses easily. Specifically, we reexamine public policies related to mixed-oligopoly, excess entry, technology choices with free entry and exit, and foreign oligopoly.

We should mention that our quasi-reaction and quasi-supply function approach is thesame in its essence as what Kiyono (1988) once referred to as the “quasi-supply curve” inthe quasi-Cournot oligopoly market or the so-called “Cournot–Ikema curve” devised byIkema (1991) and further elaborated by Ishikawa (1996, 1997).1 In the present study, weexplicitly illustrate production and product-mix inefficiencies in the standard partial-equilibrium diagrams.

The rest of the paper is organized as follows. Section 2 defines quasi-reaction functionsand quasi-supply curves and examines Cournot–Nash equilibrium in the short run. Withthe aid of quasi-supply curves, we show that there are both production and product-mixinefficiencies. Section 3 considers Cournot–Nash equilibrium in the long run (or with freeentry and exit). We first reexamine the excess entry theorem established by Mankiw andWhinston (1986) and Suzumura and Kiyono (1987) and then we consider free-entryequilibrium with multiple technologies. Section 4 applies quasi-supply curves to strategictrade policy (specifically, tariffs) under international oligopoly. Section 5 concludes.

2. Cournot oligopoly in a closed economy

Let us start with an oligopoly market in a closed economy with a set of active firmsN = { }1 2, , ,… n , where the price of the good in question is denoted by p and the total

1 Kiyono (1988) discusses the pass-through, or the change in the import and export prices associated witha change in the exchange rate. Ikema (1991) illustrates Cournot equilibrium on the quantity-price plane.Ishikawa (1997) deals with Stackelberg and Bertrand equilibria as well as Cournot equilibrium. Ishikawa(1996) depicts Cournot oligopsony on the quantity–price plane.

The Japanese Economic Review


output of the good by X. Let U(X) express the gross benefit or utility from consuming Xamount of the good, and let us further assume that the marginal benefit U′(X) is decreasingin consumption, i.e. U″(X) < 0. The inverse market demand function is given by P = PD(X).Because the demand price PD(X) is equal to the marginal benefit of consumption U′(X), thelaw of decreasing marginal benefit ensures ′ <P XD ( ) 0. The consumer surplus is thenexpressed by S(X) = U(X) - PD(X)X.

Let us denote the output of firm i ( )∈N by xi and its total cost function by Ci(xi). Thefirm’s profit is then expressed by

�π ii i i D i i i i i i ix X t P x X x C x t x( , , ) ( ) ( ) ,− −= + − − (1)

where X-i represents the aggregate output of firms other than firm i, and ti is the specific taxon firm i’s output. We impose the following assumption.

Assumption 1: The marginal cost of all active firms is non-decreasing in output, i.e.′′ ≥C xi i( ) 0 for all xi � 0 for i ∈N .

2.1 Short-run Cournot equilibrium

In this subsection, we consider the short-run equilibrium where the number of active firmsis fixed. We ignore the production tax for the moment. We introduce new tools, thequasi-reaction function and the quasi-supply function, to explore Cournot equilibrium.

The short-run Cournot equilibrium requires each active firm to maximize its profit giventhe outputs chosen by the other firms in the industry, which implies the following first-ordercondition for profit maximization to hold:

0 = ∂∂

= + ′ − ′−�π ii i

iD i D i i

x X

xP X x P X C x

( , )( ) ( ) ( ), (2)

where X = xi + X-i.When we express the right-hand side with the function y(xi, X), Assumption 1 implies

ψ x i D i ii x X P X C x( , ) ( ) ( )= ′ − ′′ < 0, so that the implicit function theorem ensures that thebest-response output of firm i is uniquely determined. It gives rise to what we may call the“generalized Cournot reaction function” or the quasi-reaction function, ri(X), which showsfirm i’s profit maximizing output against total output X. It satisfies

r XP X r X P X

P X C r Xi D

iD

D ii

′ = − ′ + ′′′ − ′′( )( )( ) ( ) ( )

( ) ( ).

Note that the firm’s output is a strategic substitute for the others’ in the usual sense if andonly if r i�(X) < 0.2

2 When we express firm i’s reaction function in terms of the aggregate output of the other firms with g i(X-i),it is a solution to ∂p i(g i(X-i), X-i)/∂xi = 0. This reaction function is well-defined when the profitfunction is strictly concave in the own output, which we assume here. Firm i’s output is then a stra-tegic substitute for the others’ if and only if g i�(X-i) < 0. The condition is equivalent to′ +( ) + ′′ +( ) <− − − − −P X X X P X t XD

ii i

ii D

ii i iγ γ γ( ) ( ) ( , ) 0 , which is further equivalent to r i�(X) < 0 with

Assumption 1.

K. Kiyono and J. Ishikawa: Reexamination of Strategic Public Policies


The equilibrium total output in the Cournot–Nash equilibrium, Xe, is a solution to

X r Xe i e

i

=∈∑ ( ).

N

The associated equilibrium output of firm i, which we express as xie , is determined by its

quasi-reaction function, i.e.

x r Xie i e= ( ).

There is an alternative approach to capture the Cournot equilibrium, which we call thequasi-supply curve approach. The first-order condition for profit maximization, Equa-tion (2), shows the price (net of taxes) required for the firm to produce a certain assigned(equilibrium) output, which we call the quasi-supply price of the firm, i.e.′ − ′C x x P Xi i i D( ) ( ) . Because the quasi-supply price depends not only on the firm’s output

but also on the total output, we express it by vi(xi, X), i.e.

v x X C x x P Xii i i i D( , ) ( ) ( ).= ′ − ′

One should note that the difference between the market price and the quasi-supply price isequal to the specific production tax.

One should also note that this quasi-supply price is fully compatible with the standardconcept of the supply price in perfect competition, once we observe that the supply priceof a price-taking firm is just equal to its marginal cost and the firm does not demand anyrent over the marginal cost.3 In the Cournot oligopoly, there is an additional second term,which shows the average rent demanded by the firm to produce the output in question.Therefore, market power as a reason for each firm to earn oligopoly rents raises each firm’ssupply price relative to what it would be with perfect competition.

Because of the technical need to define the quasi-supply price for X = 0, we assume thefollowing:

Assumption 2: lim ( ) .X

DP X→+

′ < +∞0

When the demand function is iso-elastic in the form of P X XD ( ) =− 1

ε , there holdslim ( )X DP X→+ ′ = −∞0 , which creates a problem for defining the quasi-supply price.Because we are interested in illustrating distortions under oligopoly, we impose

3 Let li denote the (quasi-)conjectural variations of firm i, which shows the increase in the total outputexpected when it increases its own output by one unit. The relevant first-order condition for profitmaximization is then given by

0 = + ′ − ′ −P X x P X C x tD i i D i i i( ) ( ) ( ) ,λ

which implies that firm i’s quasi-supply price vi is equal to ′ − ′C x x P Xi i i i D( ) ( )λ . Because li = 0 for aprice-taking firm and li = 1 for a Cournot oligopolist, the quasi-supply price for a price-taker is obtainedby setting li = 0.



Assumption 2. In view of this assumption, whenever we refer to iso-elastic demandfunctions, we mean that the demand is iso-elastic only over the prices relevant for theanalysis.

Now let us find the Cournot equilibrium. Assuming away taxes and subsidies, eachfirm’s supply price should be equalized in equilibrium. Let v denote the equalized supplyprice. We can then solve v = vi(xi, X) with respect to the individual firm’s output, i.e.x i = Si(v, X), and call it firm i’s quasi-supply function. It satisfies:

S v XC x P X

vi

i i D

( , )( ) ( )

,=′′ − ′

1(3)

S v Xx P X

C x P XXi i D

i i D

( , )( )

( ) ( ).= ′′

′′ − ′ (4)

The output decisions of the firms should be consistent in the industry as a whole, i.e.

X S v Xi

i

=∈∑ ( , ).

N

We solve this for v as a function of X, which we express as vS(X). It gives the supply pricefor the industry to produce the total output X, and we call it the industry quasi-supply pricefunction, which is now a function of only the total output.4 Its property is easily capturedby summing the quasi-supply prices over the industry, i.e.

Nv C S v X XP Xii

i

D= ′( ) − ′∈∑ ( , ) ( ).

N

Then the implicit function theorem implies

′ =− ′′

′′ − ′

′′ − ′

=∈

∈

∑

∑v X

x P X

C x P X

C x P X

S

i D

i i Di

i i Di

( )

( )

( ) ( )

( ) ( )

1

1N

N

11

1

−

′′ − ′

+ ′

′

∈

∈

∑

∑

r X

C x P X

P X

i

i

i i Di

D

( )

( ) ( )

( ).N

N

(5)

The associated quasi-supply price curve can be either upward-sloping or downward-sloping. In fact, it is straightforward to establish:

Lemma 1: Suppose that the marginal cost is constant for all the active firms. Then thefollowing hold:

1 ′ = >v X NS ( ) 1 0 if the demand function is linear with a form of P X p XD ( ) = − .

2 ′ = ′ <v X P X NS D( ) ( ) ε 0 if the demand is iso-elastic with a form of P X XD ( ) =− 1

ε wheree is a positive constant.5

4 In Kiyono (1988), a function P X P r XS Di

i( ) ( )= ( )∈∑ N

is referred to as the quasi-supply function.

5 Note ′ = − ′ + ′′ ′{ } = ′ <v X P X N XP X P X P X NS D D D D( ) { ( ) } ( ) ( ) ( )1 0ε , for d P X d XDln ( ) ln′ = − −1 1ε .



The equilibrium total output Xe should then satisfy

P X v XDe

Se( ) ( ).=

We can delineate the equilibrium by using Figure 1. In Figure 1, the industry quasi-supply price curve is shown by the upward-sloping curve v vS S′,6 which intersects the marketdemand curve DD′ at point E. This point E represents the equilibrium with the total outputbeing Xe and the market price being pe. We assume that the industry consists of two firms,L and H. Firm i’s quasi-supply price, vi(xi, X), depends not only on its own output but alsoon the total output.7 Given the equilibrium total output, its quasi-supply price curve as afunction of its own output is shown by the curve civi(i = L, H), where firm L’s output ismeasured rightward from the origin OL (= O) and firm H’s output is measured leftwardfrom the origin OH (= Xe). Because the quasi-supply prices should be equal for the two

6 The same argument applies to the case in which the industry quasi-supply curve is downward-sloping asfar as the equilibrium is stable in the sense of Cournot, as is suggested by the discussion in Subsection 2.2.

7 The firm quasi-supply price curve corresponds to the Cournot–Ikema curve in Ikema (1991) and themonopoly-equilibrium curve in Ishikawa (1997). The industry quasi-supply price curve corresponds to theoligopoly-equilibrium curve in Ishikawa (1997).

C

C¢D¢

H

E

J

L

D

Price, Marginal cost

QuantityOL(=O) E¢¢

E¢

X*OH(=Xe)

exLexH

vs

cL

pe

v¢S vL

vH

cH¢

cSc¢L

cH

K

Product-mix

inefficiency

B*

Production

inefficiency

FIGURE 1. Welfare losses in Cournot oligopoly



firms at the equilibrium, the two quasi-supply price curves cross each other at point E′, theheight of which is just equal to the market price pe. Firm L produces xL

e and firm Hproduces xH

e .

2.2 Short-run stability of Cournot equilibria

It is standard to assume that the short-run output adjustment is subject to the followingprocess:

�x r X xii

i= −( ) ,

where ri(X) = ri(X, 0) represents firm i’s quasi-reaction function with ti = 0. Summing theseoutput adjustments over the industry, we obtain

�X r X Xi

i

= −∈∑ ( ) .

N

The Cournot equilibrium with the total output Xe satisfying X r Xe i e

i=

∈∑ ( )N

, if itever exists, is then stable if the following holds:

Assumption 3: 1> ′∈∑ r Xi

i( )

Nfor all X � 0.

What is the counterpart for this stability condition in our quasi-supply curve approach?In fact, given Assumption 1 we can show that the equilibrium expressed with the demandand quasi-supply curves is stable in the Marshallian sense, i.e. ′ > ′v X P XS D( ) ( ) if and onlyif it is stable in the sense of Cournot.8 In fact, Equation (5) implies:

′ − ′ =−

′′ − ′

′∈

∈

∑∑

v X P Xr X

C x P X

S D

i

i

i i Di

( ) ( )( )

( ) ( )

,1

1N

N

(6)

so that in view of Assumption 1, ′ > ′v X P XS D( ) ( ) if and only 1> ′∈∑ r Xi

i( )

N.

Lemma 2: The short-run Cournot equilibrium is stable in the Marshallian sense, i.e.′ > ′v X P XS D( ) ( ), if and only if it is stable in the Cournot sense, i.e. 1> ′

∈∑ r Xi

i( )

N.

There is one remark in order here as regards the so-called “Cournot–Ikema” curvedeveloped by Ikema (1991) and Ishikawa (1996, 1997), or what Kiyono (1988) once calledthe “quasi-supply curve” in the quasi-Cournot oligopoly market. Basically, they are defined

by P X P r XS Di

i( ) ( )= ( )′

∈∑ N. Even with this formulation, the Marshallian output adjust-

ment process is expressed by

�X P X P XD S= −( ) ( ).

8 The Marshallian adjustment process is given by X = PD(X) - vS(X) within the present framework. Theequilibrium, if it ever exists, is then stable if ′ < ′P X v XD S( ) ( ) .



The associated stability condition is given by

0 1> ′ − ′ = ′ −⎧⎨⎩

⎫⎬⎭

′

∈∑P X P X P X r XD S D

i

i

( ) ( ) ( ) ( ) ,N

which is equivalent to the Cournot stability condition 1> ′∈∑ r Xi

i( )

N. Ikema (1991) and

Ishikawa (1997) do not, however, elucidate the equivalence between the Marshallian andCournot adjustment processes.9 An explanation of the case of the downward-sloping“original” quasi-supply curve is notably absent. Furthermore, as we see below, the presentapproach is more useful for demonstrating production inefficiency in particular.

2.3 Social optimum

Social welfare, �W ( )x is the sum of the consumer, producer and government surpluses,i.e. S X x X t Gi

i i ii( ) ( , , ) ( , )+ +−∈∑ �π

Nt x . The government surplus is given by

G t xi ii( , )t x =

∈∑ N, where t = (t1, . . . , tn) denotes the tax vector, and x = (x1, . . . , xn) the

output vector:

�W U x C xi

i

i i

i

( ) ( ).x = ⎛⎝⎜

⎞⎠⎟−

∈ ∈∑ ∑

N N

Given the set of active firms N , the socially optimal output vector x* ( ( *, , *))= x xn1 …should maximize social welfare, and thus satisfies

0 = ∂∂

= − ′�W

xP X C x

iD i i

( )( ) ( *),

x** (7)

where use was made of PD(X*) = U′(X*) and X xii* ≡

∈∑ *N

. For this condition to ensure

the social optimum, the following condition is sufficient:

′ − ′′ < ≥P X C x XD i i( ) ( *) ,0 0for all

for all firms in N , which are viable at the first-best state. This second-order conditionholds owing to Assumption 1.

The first-order condition in Equation (7) elucidates the following two types of efficiency:

1 Production efficiency. The marginal costs should be equal among firms in the industry.2 Product-mix efficiency. The market price should be equal to the social marginal cost, i.e.

the equalized marginal costs in the industry.

9 Ishikawa (1997) deals with the stability of a Cournot equilibrium but does not investigate the equivalencebetween the Cournot stability and the Marshallian stability. Kiyono (1988) discusses such equivalence toa limited extent.



Oligopoly jeopardizes these two efficiencies achieved in perfect competition due to themarket power each active firm perceives itself to possess. In the following analysis, wedelineate the social costs caused by those inefficiencies.

2.4 Short-run inefficiency in an oligopoly market

Cournot oligopoly involves two familiar types of inefficiency, production inefficiency andproduct-mix inefficiency. Using quasi-supply price curves, we delineate these inefficien-cies in Cournot oligopoly in Figure 1, where the market is served by two firms, L and H.

First, production inefficiency arises from the failure to minimize the total productioncosts in the industry given the total output. In Figure 1, the curve c ci i′ represents themarginal cost of firm i (∈ {L, H}). Equality of the quasi-supply prices between the firmsimplies

′ − ′ = −( ) ′C x C x x x P XL Le

H He

Le

He

De( ) ( ) ( ),

so that the firm with the lower marginal cost should produce more than the other firm.Figure 1 shows the case in which firm L’s marginal cost, E″L, is lower than that of firm H,E″H. Reshuffling the outputs between the two firms with taxes and subsidies then leads tothe gains from minimizing the total production costs: as much as the area HLC. This is theloss from production inefficiency at the Cournot equilibrium.10

Second, even after eliminating production inefficiency, product-mix inefficiencyremains. Note that in Figure 1, the curve cLJcS is the social marginal cost, which representsthe marginal costs equalized over the industry to produce each amount of total output.Because the social marginal cost XeK is lower than the marginal benefit of consumptionXeE measured by the height of the demand curve, an increase in total output furtherenhances social welfare until they become equal at point B*. Welfare increases as much asthe area EKB*, which gives the loss from product-mix inefficiency associated with theCournot equilibrium.

Proposition 1: Given the distribution of firms, there are dual inefficiencies in the Cournotequilibrium, i.e. production inefficiency due to unequal marginal costs within the industryand product-mix inefficiency due to the market power effects of the firms, both of which canbe eliminated through taxes and subsidies on the active firms.

2.5 Inefficiency in mixed oligopoly

The discussion above can be applied to more general modes of output competition once weintroduce (quasi-)conjectural variations. Let li denote the conjectural variation of firm ishowing how much the firm expects total output to increase when it increases its ownoutput by one unit. Assume that these conjectural variations are constant for all activefirms. Then the first-order condition for profit maximization, Equation (2), is now rewrittenas

10 This production inefficiency is often used to derive several seemingly-counterintuitive policy prescrip-tions, such as Lahiri and Ono (1988), who show that subsidizing firms with small market shares lowerssocial welfare. Smaller firms are small because their marginal costs are higher than the others and, hence,an expansion of their outputs aggravates production inefficiency.



0 = + ′ − ′P X x P X C xD i i D i i( ) ( ) ( ),λ

which implies that the firm’s quasi-supply price is given by

v x X C x x P Xii i i i i i D( , , ) ( ) ( ).λ λ= ′ − ′

When li = 0, the firm does not perceive that it possesses market power (i.e. it behaves as aprice-taker), and its quasi-supply price is just equal to the marginal cost, the supply pricein the usual sense.

A specific example for illustrating the use of conjectural variations is the mixed oli-gopoly case in which there are two types of firms in the market, a public firm U seeking tomaximize social welfare and a private firm R trying to maximize its profit.11 We adaptFigure 1 to this mixed duopoly situation, and draw Figure 2.

Because the public firm U chooses its output so as to maximize social welfare, itsmarginal cost is equated with the market price, which means that its marginal cost curve

11 Although the problem was first addressed by De Fraja and Delbono (1989, 1990), an elegant discussionis provided by Matsumura (1998).

Cr

D′

E¢

C¢r

Price, Marginal cost

QuantityOu (=O) E¢¢ Or (= Xe)

exuexr

B*

Cu

pe

A*

D

MPCr

E

vr C¢u

A

B

FIGURE 2. Production inefficiency at the mixed duopoly equilibrium



C Cu u′ is its quasi-supply price curve. In contrast, the private firm has the marginal costcurve C Cr r′ and the quasi-supply price curve Crvr, where the latter is greater than theformer due to the rent required by an oligopolist. The equilibrium for this mixed duopolyrequires the public firm’s marginal cost to be equal to the private firm’s quasi-supply priceat point E′, i.e. at the equilibrium price pe. Because the quasi-supply price of the privatefirm is greater than its marginal cost (given the total output), equilibrium requires the publicfirm’s marginal cost to be greater than the private firm’s marginal cost. The resultingproduction inefficiency is measured by the area ABE′.

Note that this welfare loss due to production inefficiency always exists insofar as thenumber of private firms is exogenously given.

Proposition 2: Given the number of active private firms, there arises a welfare loss fromproduction inefficiency at the mixed oligopoly equilibrium in the sense that the public firmproduces more than is socially desired for minimizing total production costs.

This proposition is the key to understanding that any policy to increase the private firms’output and to decrease the public firm’s output may enhance social welfare. For example,partial privatization of the public firm makes its quasi-supply price greater than its marginalcost and induces it to reduce its output. The resulting decrease in total output increases thewelfare loss from product-mix inefficiency, but such loss may be outweighed by the smallerwelfare loss from production inefficiency.

Complete privatization of the public firm instead might preclude production inefficiencywhen the public firm has the same cost conditions as the private firm. The resulting lossfrom product-mix inefficiency may, however, outweigh such a gain from the recoveredproduction efficiency. That is, society may be worse off than before the privatization.

3. Market failure in selecting viable technologies

Thus far, we have discussed how to describe the short-run equilibrium given the number ofactive firms as well as the two types of inefficiency involved. These inefficiencies can beeliminated when production taxes and subsidies are available.

There is, however, another policy tool available to governments, i.e. direct regulation ofentry and exit. This tool should play an important role if the government cannot employproduction taxes and subsidies for some reason. In fact, as we identify later, the excessentry theorem applies to the Cournot oligopoly market with free entry and exit. Thetheorem claims that the number of firms in the free-entry equilibrium tends to be excessivefrom a social welfare perspective.12 This means that there is another type of inefficiency inoligopoly associated with the number of active firms.

Furthermore, we have already identified from the preceding discussion that the short-runCournot equilibrium allows active firms to have differing cost conditions. One should bewondering at this point whether free entry-exit pressure enables the market to selectefficient technologies even when the government can employ neither taxes nor subsidiesand cannot intervene in the strategic output decisions of active firms; if it does not, there

12 See Mankiw and Whinston (1986) and Suzumura and Kiyono (1987) for the theorem.



may be yet another type of inefficiency associated with technology selection by themarket.13

Because these two inefficiencies are closely related to each other, we extend our theoryunderlying the quasi-supply curves for its full exploration. To elucidate the inefficiencies,we confine ourselves to the case in which the cost function is linear and identical acrossfirms:

C x cx fi i( ) ,= +

where c (>0) is the constant marginal cost, and f the fixed cost. The pair of the marginalcost c and the fixed cost f fully captures the cost condition or production technology usedby an active firm, so that we hereafter refer to it as technology (c, f ), and the firm usingtechnology (c, f ) the “(c, f )-firm”. We also denote its profit function by14

�π( , , , ) ( ) .x X c f P x X x cx fi i D i i i i− −= + − −

3.1 Short-run equilibrium in Cournot competition

We begin with the quasi-reaction function of an active firm. The first-order condition forprofit maximization, Equation (2), is now rewritten as follows:

0 = ∂∂

= + ′ −−�π( , , , )( ) ( ) .

x X c f

xP X x P X ci i

iD i D (8)

Let r(X, c) represent the quasi-reaction function of a firm with marginal cost c. Unlike inSubsection 2.1, we can solve the above first-order condition for xi and obtain the quasi-reaction function as follows:

r X cP X c

P XD

D

( , )( )

( ).= − −

′ (9)

For this best-response output to be well-defined, especially for X = 0, we assume thefollowing:15

Assumption 4: The demand function P = PD(X) satisfies either of the followingconditions:

1 limX→+0PD(X) < +• and lim ( )X DP X→+ ′ < +∞0 .2 The price elasticity of demand e is a positive constant greater than unity.

13 This is the inefficiency explored by Ohkawa et al. (2005) on which this section relies to a great extent.

14 Because firms are identical, superscript i is dropped from �π .

15 When the first condition is satisfied, the demand curve has a positive intercept with a finite slope. Thetypical example is a linear demand function. In the case of the second condition, Equation (9) becomes

r X c X cX( , ) = −⎛⎝⎜

⎞⎠⎟

−ε ε

11

. Either condition ensures limX→+0r(X, c) is well-defined.



The quasi-reaction function satisfies

r X cr X c P X

P XX

D

D

( , )( , ) ( )

( ),= − − ′′

′1 (10)

r X cP X

cD

( , )( )

.=′

<10 (11)

We denote by n(c, f ) the number of active (c, f )-firms and by {n(c, f )} the distributionof active (c, f)-firms. Then given this distribution of firms, the short-run equilibrium totaloutput Xe is governed by16

X n c f r X ce e= ∑ ( , ) ( , ).

One should note that when we disregard the integer problem associated with the number offirms, there are numerous distributions of firms giving rise to the same equilibrium totaloutput Xe.

The associated equilibrium profit is expressed as a function of the total output X and thetechnology used (c, f ) as follows:

π e e De

De

X c fP X c

P Xf( , , )

( )

( ).= −

−( )′

−2

Let us express the (equilibrium) gross profit earned as

F c XP X c

P XD

D

( , )( )

( ),= − −( )

′

2

(12)

so that we can rewrite the above equilibrium profit function as follows:

π e e eX c f F c X f( , , ) ( , )= − (13)

The gross profit function F(c, X) defined by Equation (12) represents the maximumfixed cost that allows a firm with marginal cost c to earn non-negative profits given thetotal output X. Hereafter we refer to it as the viability function. We also letA X c f f F c X( ) = ( ) ≤ ( ){ }, | , , i.e. the set of technologies with which the firm earnsnon-negative profits given the total industry output X.

Any short-run equilibrium is sustainable only when active firms earn non-negativeprofits. That is, active firms use the technologies in A X e( ), which can be shown by (c, f)along or below the viability frontier VV′ (associated with f = F(c, Xe)) in Figure 3.

16 Summing Equation (8) over the active firms at the equilibrium, we obtain

NP X X P X n c f cDe e

De( ) ( ) ( , )+ ′ = ∑ where N n c f= ( )∑ , denotes the total number of active firms. We

can use this condition to find the equilibrium total output Xe. Note that the left-hand side is equal to(N - 1)PD(Xe) + MR(Xe). Therefore, insofar as the industry marginal revenue MR(X) is strictly decreasingin the total output, the equilibrium is unique if it ever exists.



Note that all the technologies in the region VOV′ are not feasible in equilibrium. In fact,there is a constraint due to the feasible technologies. Such a constraint is given by the curveTT′, the technology frontier, which shows the minimum feasible fixed cost for eachmarginal cost. In fact, therefore, the observed viable firms use the technologies in theregion VTB** in Figure 3.

3.2 Free-entry equilibrium with a single technology

The excess entry theorem shows that free entry in Cournot oligopoly entails a sociallyexcessive number of active firms if and only if the outputs are mutually strategic substi-tutes. However, one should note that this result is demonstrated only for the case in whichfirms have the same cost conditions. The question here is therefore what other inefficiencyis likely to arise in addition to the inefficiency associated with an excessive number ofactive firms due to excess entry when there are numerous types of cost conditions available.The above argument implies that this inefficiency is that which is associated with themarket failure in selecting the firms with socially desired technologies. We now demon-strate this result.

We first focus our attention on the free-entry equilibrium. Here, any active firm shouldearn zero profits, which implies that the market price should be equal to its average costs.Thus, unlike in short-run equilibrium, one cannot use average costs for selecting thesocially desirable number of firms.

We then choose any active single technology (c, f ), and consider the free-entry equilib-rium with only firms using this technology. The associated equilibrium number of firmsne(c, f ) is then determined by

T¢

Marginal cost, cO T V

B**

V¢

F

Fixed cost, f

Viable firms

Technology frontier

FIGURE 3. Fixed and marginal costs of viable firms



f F X ce= ( , )

n c fX

r X ce

e

e( , )

( , ).=

The excess entry theorem implies that the second-best number of firms nSB(c, f ) should besmaller than ne(c, f ) when the outputs are strategic substitutes. Let us characterize this asnSB(c, f ).

Proposition 3: Suppose that all active firms have the same technology type (c, f ). Then thenumber of firms in the free-entry equilibrium is greater than the second-best number offirms, i.e. ne(c, f ) > nSB(c, f ) if and only if the outputs of active firms are mutually strategicsubstitutes.

Although the original proof of this excess entry theorem employs a rigorous evaluationof the welfare changes associated with entry regulation, it is actually possible to give asimpler alternative proof by considering the industry total costs subject to the strategicoutput decisions of active firms as follows.

Because all active firms use the same technology (c, f ), given their strategic outputdecision described by the quasi-reaction function r(X, c), the industry can produce the totaloutput with the number of firms given by

n X cX

r X cr ( , )

( , ),= (14)

which we call the required number of active firms. The associated total costs, i.e. theindustry total cost (ITC) with technology (c, f ), is measured by

ITC X c f cXX

r X cf( , , )

( , ).= +

We can then further define the industry average cost (IAC) and the industry marginalcost (IMC) associated with the ITC. Noting nr(X, c) = X/r(X, c), we obtain

IAC X c fITC X c f

Xc

f

r X c( , , )

( , , )

( , ),= = + (15)

IMC X c fITC X c f

Xc

f

r X c fn X c r X cr

X( , , )( , , )

( , , )( , ) ( , ) .= ∂

∂= + −( )1 (16)

In view of Assumption 3, the second term on the right-hand side of Equation (16) isstrictly positive, so that the IMC is also strictly positive for all outputs given a strictlypositive marginal cost c where the segment Oc in Figure 4 is equal to IAC(0, c, f ) =c + f/r(0, c). The two cost curves given technology (c, f) are shown in Figure 4.

There are two remarks in order here. First, as there holds IAC(0, c, f ) = IMC(0, c, f ), thetwo cost curves have the same vertical intercept equal to the marginal cost c. Second,because there holds ITC(X, c, f ) = X · IAC(X, c, f ) by definition, it follows that:



IMC X c f IAC X c f XIAC X c f

X( , , ) ( , , )

( , , ).= + ∂

∂

Then, in view of Equation (15), the IAC is strictly increasing in the total output if andonly if the outputs of active firms are strategic substitutes, i.e. rX(X, c) < 0. The averagevalue is increasing along with the output if and only if the average value is smaller than themarginal value, and, hence, the following lemma holds:

Lemma 3: The IAC curve and the IMC curves satisfy the following relations.

1 IAC(0, c, f ) = IMC(0, c, f ).2 The following conditions are equivalent:

(a) IAC(X, c, f ) < IMC(X, c, f ) for all X > 0;(b) IACX(X, c, f ) > 0 for all X > 0; and(c) rX(X, c) < 0 for all X > 0.

The industry cost curves in Figure 4 are drawn for the case of strategic substitutes. Oneshould note that these two industry cost curves are drawn by assuming that the governmentcannot intervene in any active firm’s strategic output decisions and that it can only regulatethe number of active firms through entry regulation. Free entry and exit make the price

D¢

E¢D

Price

QuantityO XSB Xe

B

c

pe E

Gains fromentry

regulation

IMC

IACpSB

FIGURE 4. Industry average and marginal cost curves, free-entry equilibrium and thesecond-best total output



equal to the IAC, so that the intersection of the demand curve DD′ with the IAC, i.e. pointE, gives rise to the free-entry equilibrium with the price pe and the total output Xe.

Even if the government cannot directly affect the firms’ output decisions through taxesand subsidies, however, it can change the total industry output by means of its direct controlover entry and exit. The second-best total output thus achieved should equate the IMC withthe market price, which is shown by the intersection of the demand curve with the IMCcurve, i.e. point B, with the price pSB and the total output XSB. Because the IMC is greaterthan the IAC, this second-best total output XSB is strictly smaller than output occurring inthe free-entry equilibrium Xe.

How can we demonstrate that the government can enhance social welfare through entryregulation? First, one should note that the required number of active firms associated withXSB, i.e. nr(XSB, c) = XSB/r(XSB, c), is actually the second-best number of firms, nSB(c, f). Theassumption of strategic substitutes implies that the individual output of active firms is greaterat the second-best equilibrium, which further implies (nSB(c, f) =) nr(XSB, c) < ne(c, f ). Thewelfare gain from such an entry regulation is measured by the area BEE′ in Figure 4.

3.3 Second-best choice of technologies

What if there is an enormous range of technologies available in the industry? As discussedabove, active firms may use many possible types of technology subject to the socialtechnology constraint expressed by the technology frontier shown in Figure 3. Even underthe pressure of free entry and exit, several types of technology can be viable, as shown inFigure 5. As each firm should earn zero profits in the free-entry equilibrium using anavailable technology, the viable technologies expressed by the pairs of the marginal costand the fixed costs should be found where the technology frontier TT′ touches the viability

T¢

Marginal cost, cO T V

V¢

F1

Fixed cost, f

Viable firms

Technology frontier

T

F2

F3

FIGURE 5. Viable and second-best technologies



frontier VV′ drawn for the total output in the free-entry equilibrium Xe. In Figure 5, they aregiven by the points along the segment F F1 2 , and point F3.17

More generally, let us denote by Ae the set of technology (c, f ) with which active firmsearn zero profits in the free-entry equilibrium,18 by ne(c, f ) the number of (c, f )-firms in thefree-entry equilibrium, and by { ( , )}

( , )n c fe

c f e∈Aits distribution. Then any distribution of

the number of firms { ( , )}( , )

n c fec f e∈A

is compatible with the total output in the free-entry

equilibrium Xe whenever the distribution satisfies

X n c f r X ce e e

c f e

= ( )∈∑ ( , ) , .

( , ) A

However, from the social point of view, it is better to ensure that the firms with the lowestmarginal cost dominate the market. Let us demonstrate this by comparing the second-bestoptimum given each of the two viable technologies (cL, fL) and (cH, fH), where cL < cH andfL > fH. Hereafter, we refer to the technology (cL, fL) as the lower marginal-cost technologyand (cH, fH) as the higher marginal-cost technology. The following lemma can be proved.19

Lemma 4: dIMC(Xe, c, F(c, Xe))/dc < 0 if either of the following conditions holds:

1 ′′ ≤P XD ( ) 0 holds for all X � 0.2 The demand function is iso-elastic with price elasticity e, and the number of firms with

technology (c, f) in the free-entry equilibrium satisfies n X cr e( , ) > +⎛⎝

⎞⎠2

11

ε.

This lemma implies that the industry marginal cost at total output in the free-entryequilibrium Xe is greater for the lower marginal-cost technology than for the highermarginal-cost technology, as shown in Figure 6, given the conditions in Lemma 4.

In view of Lemma 3, the assumption of strategic substitution implies that each of the IACcurves, IACi for the technology (ci, fi)(i = L, H), starts at ci and is strictly upward-sloping.The two technologies are viable at the free-entry equilibrium, so that the two IAC curvesshould cross the demand curve at the same price pe, the price in the free-entry equilibrium,which is shown by point E.

Figure 6 also shows the IMC curves, IMCi for technology (ci, fi). The discussion in theprevious subsection implies that the second-best optimal total output with either technol-ogy is given by the intersection of the associated IMC with the demand curve, i.e. point Bi

for technology (ci, fi). Social welfare at the second-best optimum is then measured by thearea DciBi. Comparison of the IMC curves tells us that the welfare under the lowermarginal-cost technology is greater than the welfare under the higher marginal-cost tech-nology by as much as the area AcLcH minus the area ABLBH, the sign of which seemsambiguous at first glance.

Because the two technologies are equally viable at the free-entry equilibrium, however,the ITC to produce total output Xe should be the same, which implies that the area below

17 As discussed in Appendix I, an increase in total output shifts the viability frontier downward. Thus, thetotal output in the free-entry equilibrium is obtained by sliding the viability frontier along a change in totaloutput and making it touch the technology frontier.

18 By definition, any c f e,( ) ∈A should satisfy f = F(c, Xe).

19 The proof is given in Appendix II.



the two IMC curves should also be the same. Therefore, the area AcLcH is the same size asthe area AELEH. The area AELEH is larger, however, than the area ABLBH, so that socialwelfare at the second best optimum should be greater under the lower marginal-costtechnology.20

Proposition 4: Given any distinct viable technologies at the free-entry equilibrium, socialwelfare maximized by entry regulation is greater under the lower marginal-cost technologywhen either of the conditions in Lemma 4 holds.

In view of Figure 6, when the government can observe the viable technologies and canselect socially better viable technologies in a strategic way, it should choose only the firmswith technology expressed by point F3 in Figure 5 and should regulate the entry of thesefirms. The market fails in general to choose those socially desirable technologies.

3.4 Much better technologies?

Even when the government can only affect the active firm’s strategic output decisionthrough entry regulation, there may be much better feasible technologies. To demonstratethis possibility, let us consider the iso-welfare curve given the second-best entry regulationsdiscussed in the previous section.

20 Ohkawa et al. (2005) use a different approach to establish that either: (i) a concave demand function or (ii)strategic complementarity implies the assertion in Proposition 4. The present discussion is one of thepossible alternative ways to characterize what technology should be chosen from the viewpoint of socialwelfare.

D¢

EL

D

Price

QuantityO Xe

BL

cL

peE

IMCL

IACL

cH

IACH

IMCH

EH

BHA

XL* *XH

FIGURE 6. Second-best choice of technologies



For this purpose, we first define the maximized social welfare given the technology(c, f ) under the entry regulation, which is expressed by

W c f U X ITC X c fSB

X( , ) max ( ) ( , , ) .

{ }= −{ }

It is straightforward to find W c fcSB ( , ) < 0 and W c ff

SB ( , ) < 0, so that the implicit functiontheorem allows us to define the iso-welfare curve associated with the welfare level W asfollows:

f c W f W c f WSB= = = }ω( , ) { | ( , ) .

Given the conditions in Lemma 4, we know that a decrease in marginal cost along theviability frontier enhances social welfare. It implies that each iso-welfare curve, such asW3** and WB**, cuts the viability frontier from above, as shown in Figure 7. The sociallybest technology given the strategic output decision by active firms is the one where theiso-welfare curve WB** touches the technology frontier TT′, i.e. point B**. In general, thissocially best point B** does not lie along the viability frontier but rather above it, whichimplies that society is better off with inviable technologies under free entry and exit inoligopoly.

Proposition 5: Given the strategic output decision by active firms, the socially besttechnology is not viable at the free-entry equilibrium in Cournot oligopoly.

This result reminds us of the well-known “infant industry protection” argument dis-cussed by Negishi (1962). When the technology in an industry exhibits increasing returns

Marginal cost, c

T¢

O T V

V¢

F1

Fixed cost, f

Viable firms

Technology frontier

T

F2

F3

B**

**WB

**W3

FIGURE 7. Technology selection by the market and the government



to scale, firms with lower marginal costs but greater fixed costs hesitate to enter the market,because their entry leads to a great increase in total output and to a large drop in the marketprice, causing a loss to such firms. From the social point of view, however, the resultingincrease in consumer surplus outweighs the firms’ losses and, hence, they should enter themarket. What is not referred to in Negishi (1962) in an explicit way, however, is that sucha second-best infant industry protection policy might require the social choice of thesecond-best technology as expressed in Proposition 5.

4. Oligopoly in an open economy

In this section, we consider international oligopoly. Specifically, we analyse trade policiesapplied to foreign exporters that face no domestic competitors. The foreign exportersengage in Cournot competition in the domestic market. In this situation, the domesticgovernment acts as a monopsonist and, hence, the marginal purchase cost plays a crucialrole in the determination of optimal policies. Typically, the domestic government imposestariffs to shift rent from the foreign firms to itself. A seminal work is Brander and Spencer(1984), in which a foreign monopolist serves the domestic market. We consider foreignoligopolists in our analysis. We also examine a case in which the domestic country importsfrom multiple countries.

4.1 Strategic tariffs against foreign oligopolists

When the domestic market is served by only foreign exporters and the government imposesa specific tariff t on imports X, domestic welfare is given by

�W X t U X P X X tXD( , ) ( ) ( ) ,= − +

which can be rewritten as

W U X P X t XD= − −{ }( ) ( ) , (17)

where PD(X) - t represents the price paid to the foreign firms, i.e. the international pricefaced by the importing country.

The profit earned by the foreign oligopolist is given by Equation (1) in Section 2.Firm i’s quasi-supply price, i.e. the net-of-tax price required to produce the output, isgiven by

v x X C x x P Xii i i i D( , ) ( ) ( ),= ′ − ′

as before from the first-order condition for profit maximization. When the importingcountry imposes a uniform tariff t on the foreign firms, these quasi-supply prices should beequal, and we can obtain the industry quasi-supply price function vS(X) as before. This isthe price that the importing country must pay for its imports, so that its welfare (17) is nowa function only of total output (or imports) as shown below:

W X U X v X XS( ) ( ) ( ) .= −



Then the change in the welfare associated with the total output is expressed in thestandard way in which the gains from trade are decomposed, i.e.

′ = −{ } − ⋅ ′W X P X v X X v XD S S( ) ( ) ( ) ( ),

where the first term and the second term, respectively, show the trade volume effect arisingfrom the differences between the domestic and foreign prices and the terms-of-trade effect.Given total output in the free-trade equilibrium XF, PD(XF) = vS(XF) holds. Thus, thegovernment should restrict the imports if and only if ′ >v XS F( ) 0, i.e. more imports raisethe international price. Note that the standard theory of optimal tariffs or import regulationapplies here even though the market is imperfectly competitive.

We should also point out that the general rule of welfare maximization requires themarginal benefit of imports, which is equal to the demand price in the importing countryPD(X), to be equal to the marginal purchase cost of imports MPC(X), which is derived fromthe total purchase cost given by TPC(X) = vS(X)X, i.e. MPC X v X Xv XS S( ) ( ) ( )= + ′ , whichis greater than the quasi-supply price if and only if the quasi-supply price is strictlyincreasing in the output, i.e. ′ >v XS ( ) 0. The optimal total output X* should then satisfy

0 = ′ = −W X P X MPC XD( ) ( ) ( ),* * *

which implies the optimal specific tariff rate t* is given by

t P X v X X v XD S D S* * * * *= − = ′( ) ( ) ( ).

Thus, we establish the following proposition.

Proposition 6: When the importing country has no domestic production, the optimalimport tariff is strictly positive if and only if the (equalized) quasi-supply price of theexporting countries is strictly increasing in the total output.

In Figure 8, the free trade equilibrium is given by point F, where the demand curve andthe industry quasi-supply curve intersect. The optimal level of imports is determined bypoint D*, where the MPC curve and the demand curve intersect. To realize this level ofimports, an import subsidy D S* * must be provided. Brander and Spencer (1984) discussthe possibility of import subsidies as the optimal trade policy for the importing countrywhen the domestic market is served by a foreign monopolist. They conclude that theoptimal import tariff can be positive or negative depending on the curvature of the slope ofthe demand curve.21 In our analysis, given the downward-sloping quasi-supply curve, it isclear from Figure 8 that the optimal trade policy is import subsidization.

4.2 Tariff discrimination

A large importing country can exercise monopsony power in the international market.Although the GATT/WTO prohibits such practice, the optimal policy for the monopsonist

21 The elasticity of the slope of the inverse demand function plays a crucial role. For details, see Brander andSpencer (1984) and Ishikawa (2000).



is purchase-price discrimination across the exporting countries.22 This requires the mar-ginal purchase cost from each exporting country to be equal.23

We assume for simplicity that the domestic country imports from two countries, L andH, where each firm has identical technology insofar as it locates in the same country andwe assume also that all outputs in both countries are exported to the domestic country. Inview of Equations (1) or (2), the first-order condition for profit maximization by therepresentative firm in country i (∈ {L, H}) is rewritten as follows:

0 = + ′ − ′⎛⎝⎜⎞⎠⎟ −P X

X

nP X C

X

ntD

i

iD i

i

ii( ) ( ) , (18)

where Xi = nixi represents the total output of country i and ti the specific tariff imposed onimports from country i. The quasi-supply price of exporting country i is then given by

v X X n P X t CX

n

X

nP Xi

i i D i ii

i

i

iD( , , ) ( ) ( ),= − = ′⎛⎝⎜

⎞⎠⎟ − ′ (19)

where one should observe that the quasi-supply price now also depends on the number ofactive firms ni in country i. Equation (19) shows the price that the domestic country mustpay when importing Xi from country i.

22 The following discussion is based on Kiyono (2009).

23 The same results are obtained earlier by Hwang and Mai (1991) and Kiyono (1993).

c¢

D*

D

Price

QuantityO XF

vS (=APC)

pD*

S*

c

D

F

X*

pS*

pF

D¢

MPC

MR

FIGURE 8. Downward-sloping quasi-supply curve and import subsidies as the best trade policy



Given the total output, the purchase cost from country i is then given by

TPC X X n v X X n Xii i

ii i i( , , ) ( , , ) .=

The total purchase cost depends on the import vector (XL, XH), and is given by

TPC X X TPC X X X nL Hi

i i j i

i L H j i

( , ) ( , , ).{ , },

= +∈ ≠∑

Regardless of how much is imported, it is best for the importing country to minimize thetotal purchase cost given any total import volume, which requires the marginal purchasecosts to be equalized between the exporting countries. Given the total import volume, themarginal purchase cost from country i defined by MPCi(Xi, X)(= ∂TPCi(Xi, X)/∂Xi) isexpressed as follows:24

MPC X X v X X n Xv X X n

Xi

ii

i i i

ii i

i

( , ) ( , , )( , , )

.= + ∂∂

Moreover, the equalized marginal purchase costs should be equated with the socialmarginal benefit of imports, which is equal to the domestic price in the importing country,i.e. P = vi (Xi, X, ni) + Xi(∂vi(Xi, X, ni)/∂Xi). Let ti

D represent the discriminatory specificimport tariff on imports from country i. Because t p v X X ni

D ii i= − ( , , ), the above equa-

tion is rewritten as follows:

t Xv X X n

XiD

i

ii i

i

= ∂∂

( , , ).

The property of this discriminatory import tariff policy becomes much clearer when weexpress it in the form of ad valorem tariffs, i.e.

τ iD

ii i

i

v X X n

X= ∂

∂ln ( , , )

ln.

The right-hand side is equal to the inverse of the price elasticity of the constrainedquasi-supply by exporting country i. It is just the same as the optimal tariff formula forperfectly competitive exporters. In this sense, the above formula is a generalized optimaltariff formula.

Let us further elucidate the properties of the optimal discriminatory specific tariff policy.For this purpose, we focus our attention on the case of constant marginal production costs.Let ci denote the constant marginal production cost of exporting country i. Then, in viewof Equation (19), Equation (20) can be rewritten as follows:

MPC X X n v X X n cii i

ii i i( , , ) ( , , ) .= −2 (20)

24 Rigorously speaking, the following is what one may call the “constrained marginal purchase cost”, for itis defined given the total import volume. See Kiyono (2009).



Using p MPC X X n v X X n tii i

ii i i

D= = +( , , ) ( , , ) , we can rewrite the above equation asfollows:

2 2p t c p t cHD

H LD

L−( ) − = −( ) −

2 t t c cLD

HD

H L−( ) = − .

Thus, we obtain the following proposition.

Proposition 7: Suppose that tariff discrimination is possible. If the marginal productioncosts in exporting country i, ci, are constant, then 2 t t c cL

DHD

H L−( ) = − holds and, hence,the exporting country with the lower marginal costs is subject to the higher discriminatorytariff.

4.3 Choice of a free trade agreement partner

The most favored nation clause in the GATT/WTO regulations prohibits any membercountry from imposing discriminatory tariffs on imports from abroad. GATT Article 24,however, provides an exception, and allows tariff discrimination under certain conditions.As long as the conditions in the article are met, a free trade agreement (FTA) can beestablished. In this subsection, we explore the choice of an FTA partner on the welfare ofthe importing country.

As in the last subsection, the domestic country imports from countries L and H. For awelfare comparison between the FTA formation with country H and with country L, weconsider the situation under which the domestic country, initially having an FTA withcountry L, switches to having an FTA with country H, keeping the same total importvolume, XT.25

As is well known, the equilibrium total output in the homogeneous Cournot oligopolywith constant marginal costs is determined solely by the sum of the tariff-inclusive mar-

ginal cost, i.e. n c tk k kk( )+∑ . Thus, the following lemma is immediate.

Lemma 5: Given the number of active firms in exporting countries, n = (nL, nH), the

equilibrium level of total imports, XT, is kept constant if n tk kk∑ is constant.

We focus our attention on the case in which the initial FTA with country L imposes astrictly positive external tariff, tH

L > 0. The import substitution from the old partner L to thenew partner H then requires adjustments in the tariff policies from t tL

HL= ( , )0 to

t tHLH= ( , )0 . In view of Lemma 5, the tariff policies associated with this import substitu-

tion must satisfy

n t n tH HL

L LH= .

As the total output is unchanged at XT, the MPC of each country can be replaced with whatwe may call the constrained MPC, which shows the MPC of each exporting country whenthe total import volume is kept constant.

25 For simplicity, we assume that the domestic country imports from both countries before and afterswitching the partner. For the case without this assumption, see Kiyono (2009).



From Equations (19) and (20), each country’s constrained MPC curve is thus linear andstrictly upward-sloping, as illustrated by Figure 9.26 The line segment OLOH is equal to thetotal imports (given by XT) associated with the domestic price PD in the importing country.The level of imports from country L is measured rightward from point OL, while the levelof imports from country H is measured leftward from point OH. The upward-sloping curvecivi (i = H, L) shows the export price of exporting country i.

The equilibrium of the FTA with country L is shown by point L, where PD = vL holds. Ofthe total imports, OLL′ comes from country L, and OHL′ from country H. The tariff imposedon country H, tH

L , is measured by the difference between its export price and the domesticprice, i.e. t LLH

L = ′′.Now consider the switch of the FTA partner from country L to country H given the total

amount of imports. This requires the export price of country H to be equal to the domesticprice, which is given by point H. The imports from country H increase to OHH′, and thosefrom country L decrease to OLH′ facing the tariff of t HHL

H = ′′.The change in the total purchase costs of imports is measured by the areas ALLLH

(showing the decreased costs) and AHLHH (showing the increased costs). In view ofFigure 9, we can easily verify that the importing country is better off if and only if the sumof country H’s MPC minus that of country L’s MPC at the two FTA equilibria is strictlypositive.27

26 Note that in Equation (19), ′ =C ci i and ′P XD ( ) is evaluated at X = XT.

27 Total purchase costs of imports are minimized when the domestic country imports OLA′ from country Land OHA′ from country H. At point A, the MPCs are equalized between countries L and H.

cH

U

Price

QuantityOL H¢

vL

pD L

cL

A

H

L¢

D¢

vH

Price

HL

HH

U¢¢

D

LH

LL

U¢ OH

MPCL MPCH

E

A¢

UH

UL

H¢¢

L¢¢

FIGURE 9. Choice of exporting countries as free trade agreement partners



Therefore, we obtain the following proposition.

Proposition 8: Suppose that the importing country initially forms an FTA with country L.Then the switching of the FTA partner from country L to country H, while keeping the totalimports constant, makes the importing country better off if c c n n tH L H L H

L> + −( )1 .

Two remarks are in order here. First, if nH = nL, then country H is a better partner as longas the total imports are kept constant. Second, if the uniform tariff is set by the importingcountry in Figure 9, the equilibrium is determined by point U, where vH = vL holds. Thetariff is UU″ and the total purchase costs of imports are less with the uniform tariff thanwith the FTA with country L by the amount ULUHLHLL.

5. Conclusion

We explored production efficiency and product-mix efficiency when firms engage inCournot competition. To this end, we introduced tools called quasi-reaction functions andquasi-supply curves. These tools allow us to conduct analyses easily by using the standardpartial-equilibrium diagram, i.e. the quantity-price plane.

Figure 10 shows how quasi-reaction functions, quasi-supply curves and other conceptsdeveloped in our analysis are related. We first considered the short-run case in which thenumber of firms is fixed. We identified with ease on the quantity-price plane the welfarelosses with the standard Cournot oligopoly. We also depicted production inefficiency in the

<Section 2: Short-run>

<Section 3: Long-runwith identical linear

cost function>

FOC for profit maximizationPD (X) +xiP¢D (X) − C¢i (xi) = 0

Firm’s quasi-supply pricevi (xi, X) = C¢i (xi) − xiP¢D (X)

Quasi-reaction functionx = r (X, c)

Required number of active firmsnr = X/r (X, c)

Industry total cost functionITC (X, c, f ) = cX + nrf

Industry average cost functionIAC (X, c, f ) = c + f/r(X, c)

Industry marginal cost functionIMC (X, c, f ) = c + f (1 −nr(X, c)rX(X, c) /r(X, c))

PD (X) = IAC (X, c, f ) PD (X) = IMC (X, c, f )

Second-best outputXSB

Firm’s quasi-supply functionxi = Si (v, X)

Implicit function theorem

Industry quasi-supply pricev = vs(X)

Equilibrium outputXe

PD (X) = vs(X)

X = S Si (v,X)

FIGURE 10. Quasi-reaction and the related concepts



case of mixed oligopoly. These inefficiencies can be eliminated when production taxes andsubsidies are available. In the long-run equilibrium (i.e. the equilibrium with free entryand exit), using the quasi-reaction function, we reexamined the excess entry theorem andillustrated the situation of excess entry in the standard partial-equilibrium diagram. We alsoconsidered the issue of technology choices, and showed that entry regulations are usefulwhen there is a choice of technology. We then reexamined tariffs as a rent-shifting devicein open economy settings. Specifically, we investigated the case where there is no domesticsupply and foreign firms have different cost structures. Again, quasi-supply curves proveduseful for the analysis.

Although the results themselves are not necessary novel, our analysis makes it possibleto illustrate inefficiencies easily in each case and to gain intuitive insight into the formationof strategic public policies under Cournot oligopoly. Reexamination of other public poli-cies using quasi-reaction functions is left for future research.

Appendix I

Entry dynamics and stability

Consider the market served by firms with the same technology (c, f). As shown inEquation (13), the equilibrium profit of the individual firm given the equilibrium totaloutput X is expressed by

π e X c f F c X f( , , ) ( , ) ,= −

where use was made of the definition of the viability frontier (Equation 12), i.e.

F c XP X c

P XD

D

( , )( )

( ).= − −{ }

′

2

Total output X is the short-run equilibrium output if and only if the number of activefirms n satisfies

X nr X c= ( ), .

In view of Assumption 3, such a total output is uniquely determined given the number ofactive firms n and their marginal cost c. We express this relation with �X n c( , ). Applying theimplicit function theorem, one can obtain:

��

�X n cr X n c

nr X n c cn

X

( , )( , )

( , ),.=

( )− ( ) >1

0

Therefore, more active firms lead to a greater total output if and only if the short-runequilibrium is (at least locally) stable, as stated in Assumption 3.

Let us assume that a change in the number of firms is subject to the following adjustmentrule:

� �n F c X n c f= ( ) −, ( , ) , (21)



i.e. the number of active firms increases if and only if an individual firm earns strictlypositive profit. The free-entry equilibrium is defined as the number of firms under whichthis entry dynamics ceases. We denote the number by ne(c, f).

This free-entry equilibrium is stable when the gross profit is strictly decreasing in thenumber of firms. Because the total output is increasing in the number of firms, however, thestability condition is, in fact, that the gross profit is strictly decreasing in the total output.To derive this condition, it is more convenient to use F c X r X c P XD( , ) ( , ) ( )= −{ } ′2 insteadof the original expression above, the one given by Equation (12). In fact, its partialdifferentiation with respect to the total output X gives rise to the following stabilitycondition:

F c X r X c P X r X cX D X( , ) ( , ) ( ) ( , ) .= ′ −{ } <1 0

Therefore, we have established

Lemma 6: Suppose that all active firms use the same technology (c, f ). Then the free-entryequilibrium resulting from the entry dynamics governed by Equation (21) is stable ifFX(c, X) < 0, i.e. rX(X, c) < 1, holds for all X � 0.

Appendix II

Changes in technology and industry marginal cost

We prove Lemma 4 in this appendix. From Equations (14) and (16), we obtain

IMC X c f cf

r X c

X

r X cr X cX( , , )

( , ) ( , )( , ) .= + −⎧

⎨⎩

⎫⎬⎭

1

A change in the marginal cost c then gives rise to

IMC X c ff

r X cr X c

Xr X c

r X cr X c fc c

Xc( , , )

( , )( , )

( , )

( , )( , )= −

{ }+{ }

−12 3

XXf

r X cr X c

f

r X c P X

Xf

r X c P Xr

Xc

D D

( , )( , )

( , ) ( ) ( , ) ( )

{ }

= −{ } ′

+{ } ′

2

2 31 XX

D

D

X cXf

r X c

P X

P X( , )

( , )

( )

( )+{ }

′′′{ }2 2

∵r X cP X

r X cP X

P X

f

r X

cD

XcD

D

( , )( )

( , )( )

( )

(

=′

= − ′′′{ }

⎛⎝⎜

⎞⎠⎟

= −

1

1

2and

,, ) ( )( , ) ( , )

( , )

( , ) ( )

c P Xn X c r X c

n X c f

r X c P X

D

rX

r

D

{ } ′−{ }

+{ } ′

− −

2

2

1

1 rr X cX ( , ){ }

∵n X cX

r X cr X c

r X c P X

P Xr

XD

D

( , )( , )

( , )( , ) ( )

( )= = − − ′′

′⎛⎝⎜

⎞⎠⎟

= −

and 1

1ff

r X c P Xn X c

D

r

( , ) ( )( , ) .

{ } ′+{ }2

1

Thus, from Equations (9) and (12), there holds



IMC X c F c Xr X c P X

r X c P Xn X cc

ee

De

D

r, , ( , )( , ) ( )

( , ) ( )( ,( ) = + { } ′

{ } ′1

2

2)) .+{ }1 (22)

Similarly, a change in the fixed cost f gives rise to

IMC X c fr X c

n X c r X cfr

X( , , )( , )

( , ) ( , ) .= −{ }11 (23)

In view of Fc(c, Xe) = -2r(Xe, c), Equations (22) and (23) then jointly imply:

dIMC X c F c X

dcn X c n X c r X c

n

ce e

r e r eX

e

r

( , , ( , ))( , ) ( , ) ( , )

(

= + − −{ }

=

2 2 1

XX c r X c

n X cr X c P X

P X

eX

e

r ee

De

De

, ) ( , )

( , )( , ) ( )

( )

1 2

1 2

+( )

= − − ′′′

⎛⎝⎜

⎞⎠⎠⎟ ,

which is strictly negative when ′′ ≤P XD ( ) 0 by virtue of Equation (10), i.e.r X c r X c P X P XX D D( , ) ( , ) ( ) ( )= − ′′ ′1 .

Similarly, in view of Assumption 2, we obtain:

dIMC c F c X

dc

r X c

r c

P X

P

rce e

De

D

( , , ( , )) ( , )

( , )

( )

( )

01

0 02

2

2= + { }

{ }′′

− (( , )

( , )( , )

( )

( )

( , )

( , )

X c

r cn c

P X

P

r X c

r c

er

De

D

e

00 0

0 0

∵ =( )

= ′′

⎛⎝⎜

⎞⎠⎟

22

20

0

0− ′′

+ ′′

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

= ′

P

P X

r X c

r c

P

P X

P

D

De

eD

De

D

( )

( )

( , )

( , )

( )

( )

(( )

( )

( , )

( , )

( )

( )

( ) (X

P

r X c

r c

P

P X

P P Xe

D

eD

De

D De

′− ′

′⎧⎨⎩

⎫⎬⎭

+′ ′

0 0

0 02

)) ( )

( ).

− ′{ }′{ }

⎡

⎣⎢

⎤

⎦⎥

P

P X

D

De

02

When ′′ =P XD ( ) 0, there hold ′ = ′P P XD De( ) ( )0 and rX(X, c) < 0, the latter of which implies

r(Xe, c) < r(0, c). Therefore, the right-hand side of the above equation is strictly positive.The same argument applies to the case for ′ > ′P P XD D

e( ) ( )0 in general where ′ <P XD ( ) 0ensures r(Xe, c) < r(0, c).

Now consider the iso-elastic demand function given by P X XD ( ) =− 1

ε . There thenholds:28

− − ′′′

= − + +⎛⎝

⎞⎠ ∝ +⎛

⎝⎞⎠ −1 2 1

2 11 2

11

r X c P X

P X n X c

eD

e

De r e

( , ) ( )

( ) ( , ) ε εnn X cr e( , ).

Therefore, we establish that even when the demand function is iso-elastic,

n X cr e( , ) > +⎛⎝

⎞⎠2

11

εassures dIMCc(Xe, c, F(c, Xe))/dc < 0.

Final version accepted 17 December 2012.

28 It can be verified easily that 1 1ε + = − ′′ ′XP X P XD D( ) ( ) holds.



REFERENCES

Brander, J. A. and B. J. Spencer (1984) “Trade Warfare: Tariffs and Cartels”, Journal of International Econom-ics, Vol. 16, pp. 227–242.

—— and —— (1985) “Export Subsidies and International Market Share Rivalry”, Journal of InternationalEconomics, Vol. 18, pp. 83–100.

De Fraja, G. and F. Delbono (1989) “Alternative Strategies of A Public Enterprise in Oligopoly”, OxfordEconomic Papers, Vol. 41, pp. 302–331.

—— and —— (1990) “Game Theoretic Models of Mixed Oligopoly”, Journal of Economic Surveys, Vol. 4,pp. 1–17.

Dixit, A. (1979) “A Model of Duopoly Suggesting A Theory of Entry Barriers”, Bell Journal of Economics,Vol. 10, pp. 20–32.

—— (1980) “The Role of Investment in Entry Deterrence”, Economic Journal, Vol. 90, pp. 95–106.Hwang, H. and C. C. Mai (1991) “Optimum Discriminatory Tariffs under Oligopolistic Competition”, Canadian

Journal of Economics, Vol. 24, pp. 693–702.Ikema, M. (1991) Kokusai Fukusen Kyoso no Riron (A Theory Towards An International Duopoly), Tokyo:

Bunshindo. (in Japanese.)Ishikawa, J. (1996) “Diagrammatic Demonstration of Oligopsonies: An Alternative Method”, Hitotsubashi

Journal of Economics, Vol. 37, pp. 90–100.—— (1997) “A Note on the Diagrammatic Demonstration of the Cournot Equilibrium”, Japanese Economic

Review, Vol. 48, pp. 90–100.—— (2000) “Foreign Monopoly and Trade Policy under Segmented and Integrated Markets”, Economic Review

(Keizai Kenkyu), Vol. 51, pp. 321–336.Kiyono, K. (1988) “Price Rigidity and the Competitive Structure of Oligopoly”, unpublished manuscript.—— (1993) “Who Will Be Called Partner? An Importing Country’s Incentive to Form A Free Trade Area”,

Economic Studies Quarterly, Vol. 44, pp. 289–310.—— (2009) “Incentives Towards Economic Integration As the Second-best Tariff Policy”, Asia-Pacific Journal

of Accounting & Economics, Vol. 16, pp. 19–48.Lahiri, S. and Y. Ono (1988) “Helping Minor Firms Reduces Welfare”, Economic Journal, Vol. 98,

pp. 1199–1202.Mankiw, N. G. and M. D. Whinston (1986) “Free Entry and Social Inefficiency”, RAND Journal of Economics,

Vol. 17, pp. 48–58.Matsumura, T. (1998) “Partial Privatization in Mixed Duopoly”, Journal of Public Economics, Vol. 70,

pp. 473–483.Nash, J. (1951) “Non-corporative Games”, Annals of Mathematics, Vol. 54, pp. 286–295.Negishi, T. (1962) “Entry and the Optimal Number of Firms”, Metroeconomica, Vol. 14, pp. 86–96.Ohkawa, T., M. Okamura, N. Nakanishi and K. Kiyono (2005) “The Market Selects the Wrong Firms in the Long

Run”, International Economic Review, Vol. 46, pp. 1143–1165.Selten, R. (1975) “Reexamination of the Perfectness Concept for Equilibrium Points in Extensive Games”,

International Journal of Game Theory, Vol. 4, pp. 22–55.Spence, M. (1977) “Entry, Capacity, Investment and Oligopolistic Pricing”, Bell Journal of Economics, Vol. 8,

pp. 534–544.Suzumura, K. and K. Kiyono (1987) “Entry Barriers and Economic Welfare”, Review of Economic Studies,

Vol. 54, pp. 157–167.von Neuman, J. and O. Morgenstern (1944) The Theory of Games and Economic Behavior, New York: Wiley.



Documents

Reexamination of Strategic Public Policies