European Economic Review 31 (1987) 1159-1171. North-Holland

A PRICE-SETTING SUPERGAME BETWEEN TWOHETEROGENEOUS FIRMS

Hyung BAE*

Dongguk Unitwsity, Seoul, Korea

Received December 1984, final version received April 1986

When firms attempt to tacitly collude, they inevitably face a market sharing problem. Thisproblem is further complicated when firms have asymmetries. This paper studies a price-settingsupergame between two firms with difTerent costs. The paper investigates sustainability of tacitcollusion and solves the market sharing problem under tacit collusion. The resulting equilibriumfrom the supergame is used to study limit pricing, government policies toward entrants, andwelfare change due to technological progress and a change in the time discount rate. The papershows that even costlessly achieved technological progress can decrease social welfare.

1. Introduction

The theory of oligopoly began with Cournot's (1838) duopoly model inwhich each firm chooses an output level to maximize its profits given theother firm's output level. Bertrand (1883) criticized Cournot's use of quantityas the strategic variable, and he used price instead as the strategic variable inthe same model. Either of these models approximates well a competitivemarket under certain conditions. On the other hand, both works have beencriticized for their static nature and because they do not generally givePareto optimal equilibria to the firms. t

This paper analyzes a noncooperative price-setting supergame between twofirms with different costs. A supergame is a countably infinite sequence ofgames played by a fixed set of players. As is well known, the set of (subgame)perfect Nash equilibria [Selten (1965,1975), Kreps and Wilson (1982)] of asupergame is typically large. The paper selects a single equilibrium and usesit to study some topics in industrial organization.

It is assumed that side-payments between the firms are not possible. Thus,the only distribution of profit which can be attained are those generated bythe firms setting prices and producing output for the resulting demand. If

*The author is grateful to William A. Brock, Herman C. Quirmbach, Tom K. Lee, John Rust.Val E. Lambson, Leonard W. Weiss, and two anonymous referees for helpful comments andvaluable suggestions.

ITo simplify exposition, both an equilibrium strategy and an equilibrium outcome are referredto as an equilibrium.

0014-2921/87/$3.50 © 1987, Elsevier Science Publishers B.V. (North-Holland)

1160 II. Bae, A price-setting supergame between two firms

side-payments are possible, the firms would produce the total outputdemanded at minimal cost and then redistribute the revenue. Hence, theanalysis of the paper applies only if side-payments are not allowed for somereason. Only 'tacit' collusion is allowed, although the pair of firms will, forconvenience, be referred to as a cartel in what follows.

This paper also analyzes welfare change due to cost-reducing technologicalprogress. Throughout this paper, social welfare is measured by the sum ofconsumer's surplus and industry profit. The analysis shows that evencostlessly achieved technological progress can lead to a welfare loss in twocases.

When two firms attempt to tacitly collude, they inevitably face a marketsharing problem. This problem is further complicated when the two firmshave asymmetries.

If the cost advantage of the low cost firm against the high cost firm is veryhigh, then the low cost firm may want to engage in exclusionary pricingagainst the high cost firm rather than enforce a cartel with it.2 In this case, adecrease in the high cost firm's cost may press the low cost firm to enforce acartel with the high cost firm. The cost-reducing technological progress of thehigh cost firm then decreases social welfare if the decreased cost of the highcost firm is still higher than that of the ex ante low cost firm, since itincreases both price and industry average cost.

The second case is more complicated. Suppose that the two firms do nothave much asymmetry so that they can enforce a cartel. Equilibrium marketshares and the equilibrium cartel price are likely to depend on the costfunctions of the two firms, The high (low) cost firm wants a higher (lower)cartel price. than does the low (high) cost firm.

Because the low cost firm makes more profit than the high cost firm in acompetitive market, it may have a larger market share than the high costfirm in the cartel. As the cost of the high cost firm decreases, its market shareand its bargaining power in setting the cartel price increase.

At a point where the decreased cost of the high cost firm is still higherthan that of the ex ante low cost firm, the cartel price can increase if theeffect of the high cost firm's increased bargaining power dominates the effectof its decreased cost. In this case, industry average cost may also increasedue to the increased market share of the high cost firm. If both price andindustry average cost increase as the high cost firm's cost decreases, then thecost-reducing technological progress unambiguously decreases social welfare.

In both cases, industry-wide technological progress which yields greatercost reduction to the high cost firm and less cost reduction to the low costfirm may also lead to a welfare loss.

2A firm is said to engage in exclusionary pricing when it sets its price lower than other firms'average production costs in order to drive them out of business.

11. Bae, A price-sellingsupergame between 11\"0 firms 1161

This paper is organized as follows. Section two develops the necessary andsufficient conditions for a sustainable cartel between two firms with differentcosts. Section three develops the equilibrium cartel from the set of sus­tainable cartels.

Section four shows that technological progress can decrease social welfare.In evaluating the welfare change due to a decrease in the cost of the highcost firm, its cost disadvantage elasticity of market share is shown to beimportant.

In section five, welfare change due to a change in the time discount rate isanalyzed. An increase in the time discount rate never increases the equilib­rium price, and in a certain interval of cost difference between the two firms,it decreases the equilibrium price.

Section six analyzes the strategic behavior of an incumbent and an entrantwith an absolute cost disadvantage.' If the sunk cost which the entrant mustpay to enter the market is not very high and entry is quick, then limit pricingby the incumbent for a certain interval of its cost advantage is theequilibrium. In this paper, entry is said to be quick if an entrant can enterthe market and undercut the incumbent's price while the incumbent's price iscommitted.

Government policy towards the entrant is also studied in section six. Thereis a critical value of the entrant's cost disadvantage such that subsidizing theentrant is socially desirable if the cost disadvantage is higher than the criticalvalue, otherwise an entry-prohibiting tax on the entrant is socially desirable.If entry is not quick, then limit pricing cannot be the equilibrium andsubsidizing the entrant is not socially desirable in any casco Section sevencontains conclusions and suggestions for further research.

2. Sustainable cartels between two firms

Suppose two firms produce perfect substitutes with different constant unitcosts, c· and ch where cl~ ch

• Throughout this paper, cl is assumed to be fixedwhile ch is variable, unless otherwise stated. Let the market demand be D(P)and the time discount rate be constant at r where r>O. Throughout thepaper, all prices and costs will, for convenience, be calculated with the subtractionof c'. Let p=P-c· and C=Ch_CI and define d(p) to satisfy d(p)=D(P).

We make the following assumptions on the demand function d(p):

(AI) There exists p>O such that d(p) =0 if p~p and d(p) >0 otherwise,(A2) d'(p) < 0 on (O,p),(A3) 2d'(p)+pd"(p) < 0 on (O,p).

3For simplicity, a potential entrant is referred to as an entrant.

1162 II . Bae, A price-seu lng supergame belweell 1\1'0 firms

(A3) requires that the profit function pd(p) of the low cost firm be strictlyconcave.

Because the two firms are assumed to produce perfect substitutes, themarket demand goes entirely to the low price firm if the two firms setdifferent prices. Hence, under price competition the low cost firm can makepositive profits, but the high cost firm cannot.

This asymmetry makes it less likely to enforce a price cartel which leads toequal market shares. We will therefore analyze the behavior of the two firmswhen they either compete in price or enforce a price market share cartel.

.Let llx(p)=(p-x)d(p) and pm(x)=argmaxllx(p). From (A3), pm(x) is uniqueand dpm(x)/dx>O on (O,p). pm(o) and pm(e) are the monopoly prices of thelow cost firm and the high cost firm, respectively. To simplify the notation,let ll(p) = llo(p),pm = pm(o), and pmc=pm(e).

The one-period price-setting game yields as its only Nash equilibrium amonopoly by the low cost firm with price equal to min (pm, e) in which thehigh cost firm earns zero profit and the low cost firm obtains a profit ofll(min(pm, e)). In a supergame, the one-period game is repeated infinitely, andsince firms are assumed to maximize their present discounted profits,cooperative outcomes can be sustained by trigger strategies.

A trigger strategy is a strategy in which a player stays at a collusiveoutcome unless someone deviates from the (tacitly) agreed-upon outcome,and if someone deviates from the (tacit) agreement at date t, he goes to theone-period game equilibrium at date C+ I and stays there forever. Acollection of all the players' trigger strategies is a (subgame) perfect equilib­rium , whenever the one-period gain to each player from chiselling the (tacit)agreement is .not greater than his capitalized loss due to the consequentbreak-down of the agreement. A supergame has a large set of (subgame)perfect equilibria. which includes the set of trigger-strategy equilibria as asubset. Abreu (1982) characterized the class of (subgame) perfect equilibria ofa supergame.

Let a. be the market share of the high cost firm. Now a cartel is defined tobe an outcome (p, a.) which Pareto dominates the one-period equilibrium. Acartel is said to be sustainable (or enforceable) if it can be induced by a pairof the two firms' trigger strategies which is a (subgame) perfect equilibrium. Ife?;pm, there is no sustainable cartel because the one-period game equilibriumgives monopoly profit to the low cost firm. If e < pm, a cartel is sustainable, ifand only if

and (I)

ll(min(p, pm))-(I-a.)ll(p) ~(I/r)[(I-a.)ll(p)- ll(e)] . (2)

The left hand side of (I) is the maximum gain to the high cost firm from

II. Bae, A price-setting supergame belween IWO firms 1163

chiselling the cartel, and the right hand side is the capitalized loss to thatfirm from the break-down of the cartel. Eq. (2) shows the same expression forthe low cost firm. Hereafter, we assume e < p",

The condition on ex which is necessary and sufficient to enforce a cartel isobtained by rearranging (1) and (2)

where (3)

rn(min(p, pm)) +n(e)

(1 +r)n(p)

The expressione'tp) is the minimum value of ex which prevents the high costfirm from chiselling, and ex'(p) is the maximum value of ex which ensures thatthe low cost firm will stick to the cartel, given p.

For the set of sustainable cartels to be nonempty, it is necessary andsufficient that

(4)

If r» I, (4) cannot hold for any p. If r~ 1, (4) is most likely to hold at p=pmsince exh(p) is minimized and exl(p) is maximized at p= pm. Hereafter, weassume r~ 1.

Eq. (4) together with p=pm gives e~f(r) as a necessary and sufficient con­dition for the existence of a sustainable cartel, where f(r) = Il - 1((1- r)n(pm))with f(r)~pm and tr : is the inverse of Il, If e> f(r), there is no sustainablecartel. If e~f(r), any cartel satisfying (3) is enforceable. From this we cansee that the low cost firm's decision to enforce a cartel or engage in exclusion­ary pricing depends on the magnitude of its cost advantage and the timediscount rate. This agrees with generally accepted theory. [e.g., Scherer(1980, ch. 8)].

Since n'(p»o on (O,pm), it follows that I'(r) <0 on (0,1). This implies thatthe maximum cost differential with which the two firms can enforce a carteldecreases as the time discount rate increases. In other words, a greater degreeof homogeneity is required to enforce a cartel as the time discount rateincreases. The set of enforceable cartels, given e~ f(r), is illustrated in fig. 1by the area surrounded by heavy lines in each case.

3. The best sustainable equilibrium

To obtain a unique equilibrium of the price-setting supergame, which

-z:

Case I: O:! c s g(r) CaSE!" II : g( r) :! C :! f (r )

~

~'"

;::g::.'"::>..

""..,?''"~~.

'"~~s'""'"~~::

ex:

p= pm(cx:c)

_cx:=cx:h( p).'

_r_I+r

p

o

,pmcl---------"f-- - -----------------,

II

~! :I I

pJ~_~~_~~~_._ --~~~~~~~---------i,,------~W'.(PIcx:)= r1(p,cx:) l

, II I, II II II II II II II II II ,I II , _

ex:o _r_l+r

p

__• rcx:=cx: J( p I ~cx:= cx:h( p )-- ...-- ...-. ,

CI_--:"--'IIIIIIII

p

pmc

Fig. I. The set of enforceable cartels and the DSE cartel.

11. Bae, A price-setting supergame between rwofirms 1165

always exists and is Pareto optimal among all trigger-strategy equilibria, thebalanced temptation equilibrium [Friedman (1971), hereafter referred to asBTE] is chosen with its Pareto optimality condition restricted to the set ofsustainable cartels if the set is nonempty. The balanced temptation equilib­rium is a trigger-strategy equilibrium which is Pareto optimal and has thebalanced temptation property, which requires that the firms' ratios of theper-period losses due to break-down of the cartel relative to the maximumone-period gains from chiselling the cartel be the same. If there is nosustainable cartel, the one-period game equilibrium is the only trigger­strategy equilibrium; hence it is chosen. We call the equilibrium employedhere the best sustainable equilibrium (hereafter referred to as BSE).

From (1) and (2), we obtain the necessary and sufficient condition on r toenforce a cartel

r ~min(rh(p, ex), r'(p, ex»), where (5)

I( ) (_1-_ex)_U--,-(p_)-_U_(c_)_r p,« - U(min(p,pm))-(I-ex)U(p)

The expression rh and rl are the per-period losses due to the break-down ofthe cartel, relative .to .the maximum one-period gains from chiselling thecartel for, respectively, the high cost firm and the low cost firm.

The BTE cartel (p,&) requires that the two firms are equally tempted in thesense that rh(p, &) == r'(p, &) and that the cartel should be Pareto optimal to thetwo firms when side-payments are not possible, that is p = pm(&c). A cartelwhich satisfies rh(p, ex) = rl(p, ex) is said to have the balanced temptationproperty. The BTE cartel can be written as a pair of functions of c, using thetwo required conditions

ll(p(c)) - U(c)fJ(c) = pm(&(c)c) and &(c) U(pm)+ U(p(c)) _ U(c)' (6)

The BTE cartel may not be sustainable even though there is a sustainablecartel, yet the BTE cartel is also the BSE cartel whenever it is sustainable.The BTE cartel is sustainable if, and only if, there exists an ex such that

(7)

Eq. (7) is most likely to hold when ex=r/l +r. Eq. (7) together with ex=r/l +r

1166

gives

II. Bae, A price-setting supergame belll'een 111'0firms

al(pm(rc/l +r))~r/l +r. (8)

From (8) the necessary and sufficient condition on c to enforce the BTEcartel can be derived as c ~g(r), where g(r) satisfies al(pm(rg(r)/I + r)) = r/I + r.In addition, g'(r)<O and g(r)<f(r) on (0, I) because &:xI(p)/dp<O on (pm,p)and dpm(x)/dx> 0 on (O,p), where p satisfies p>pm and II(p)=II(c).

When g(r) < c~ f(r), the BTE cartel (p, Ii) is not sustainable even thoughthere is still a nonempty set of sustainable cartels. In this case the BSE cartel(p*,a*) satisfies

a*=r/I +r andal(p*)=r/I +r with p*~p. (9)

The point'B in each case in fig. I is the BSE cartel. From (9), p* = p(r,c)where p(r,c) = II- 1(rII(pm) + II(c)) with p(r,c)~pm. Since II'(p) > 0 on (0, pm) andII'(p) < 0 on (pm,p), it follows that op(r,c)/or, op(r,c)/oc<O when g(r) <c<f(r).That is, if g(r) < c < f(r), then the BSE cartel price decreases with an increasein the cost difference between the two firms and/or with an increase in thetime discount rate.

Now the BSE (p*, a*) of the price-setting supergame between the two firmscan be given for the whole range of c:

( * *)_( 'rn( * ) II(p*)-l1(c) )'f 0< < ()p ,a - p a c'l1(pm)+l1(p*)-II(c) I =c=gr

= (p(r, c), l:r) if g(r)~c~f(r)

=(min(pm,c),O) if c> f(r).

Then the two firms' BSE profits are

IIh(c)=a*(c)IIip*(c)) and III(c)=(I-a*(c))II(p*(c)),

(10)

(II)

where the superscripts h and I denote the high cost firm and the low costfirm respectively.

Fig. 2 shows the graphs of IIh(c) and III(c). A small increase in c at f(r)decreases both firms' profits, because it breaks down the cartel between thetwo firms. This shows the noncooperative nature of the supergame betweenthe two firms.

11. Bae, A price-setting supergame between twofirms 1167

TrIp") -----------.-.~

o flr) c o f(r) c

Fig. 2. Equilibrium profits of the two firms.

4. Welfare decreasing technological progress

In this section, welfare change due to cost-reducing technological progressis analyzed. This technological progress is assumed to be achieved withoutcost, in order to emphasize that even costlessly achieved technologicalprogress can lead to a welfare loss.

Industry average cost is the sum of the two firms' average costs weightedby their market shares, that is ac. Thus, the change in the equilibriumindustry average cost due to a change in e, when O<e<g(r), is"

e dz"d~*(e)ejde=a*(I-I::), where 1::= - -'-.

a* de

Because dpm(x)jdx> 0 on (0, p), the equilibrium price and the equilibriumindustry average cost move in the same direction, as e changes, whenO<e<g(r). When O<c~g(r), cost-reducing technological progress by thehigh cost firm, which decreases c, has two opposite effects on the equilibriumprice and the equilibrium industry average cost. One effect is to reduce thembecause c is decreasing, and the other effect is to increase them because a*(c)is increasing.

If the high cost firm's cost disadvantage elasticity of market share, 1::, isgreater than I, then the latter effect dominates. Hence, a decrease in e increasesboth the equilibrium price and the equilibrium industry average cost if1::> I. Therefore, if O<c~g(r) and 1::> I, then cost-reducing technologicalprogress by the high cost firm decreases social welfare. 5

~Hereafter. the SSE is sometimes referred to as the equilibrium.5A change in :x' (c)c due to a change in c from c' to c" is :x'(c")c"-:x'(c')c' =(c"-c')(:x'(c')+

:x '(c")){I - £)/2. where e= - [(c' +c")(:x*(c") - :x'(t.:')]/[(c" - c')(:x'(c')+:x'(c")] is the arc elasticityof :x'lc). Hence. a decrease in c from c' to c" decreases sociat welfare if and only if £> I.

1168 11. Bae, A price-setting supergame between rwo firms

Fig. 3 provides graphs of the equilibrium price and the equilibriumindustry average cost. A decrease in c, keeping c in the interval (f(r),pm),decreases price and does not change industry average cost, and hence itunambiguously increases social welfare.

a:."c

cg(r) Hr)

/1" I

" I/ I

" I" I" I

" I, I

-.!...- f(r) --------------- ----1+( I1IIII11I,:

cmp

/1, 1

" 1, I, I

" 1, I

" I, I

" I, 1, 1

" 1, I

" 1

Hr)oFig. 3. Equilibrium price and equilibrium industry average cost.

Technological progress by the high cost firm, which reduces c from theinterval (fer), (0) to the interval (0, fer)), unambiguously decreases socialwelfare since it increases both price and industry average cost. This is aresult of the decrease in the high cost firm's cost disadvantage which createsenforceable cartels between the two firms.

A decrease in c, keeping c in the interval (g(r), fer)), decreases price butincreases industry average cost, and hence the welfare effect due to thedecrease in c is ambiguous. Welfare change as a result of technologicalprogress by the high cost firm, keeping c in the interval (O,g(r)), depends onthe high cost firm's cost disadvantage elasticity of market share. If 6> 1, itdecreases social welfare and if 6 < 1, it increases social welfare.

In sum, technological progress by the high cost firm can lead to a welfareloss. Furtherrnore.. industry-wide technological progress which leads to agreater decrease in ch and a smaller decrease in cl may also decrease socialwelfare.

5. Welfare change due to a change in the time discount rate

This section will present a comparative static analysis of welfare changedue to a change in the time discount rate, keeping the assumption thatO<r~ l. Fig. 4 displays graphs of the equilibrium price and the equilibrium

p*

11. Bae, A price-setting supergame be/ween /wofirms

cx:*c

1169

o

II

I-------1---------: - -Y!'----

I •I I

i JtrI ,"'1, , II ...' I

: ,/ II ...' I, , I

.' I...' I I, I I

...' I I

,.,': I" : 'g(r) g(r) f(r) f(r) pm c

A,IIIIIII,IIIII,

g(r') f(r')-<l

g(r) Hr) C

Fig. 4. The SSE cartels under difTerent time discount rates.

industry average cost, given two different time discount rates, r and 1", wherer' > 1', using the fact that 1'(1'), g'(r), and op(r, c)/or are negative.

The increase in the time discount rate from r to 1" changes social welfare if,and only if, g(r') < c~ f(r). If f(r') < c~ f(r), it decreases both price andindustry average cost by breaking down the cartel between the two firms.Hence, social welfare unambiguously increases in this case.

If g(r') < c~ f(r'), it increases industry average cost by increasing the highcost firm's market share to 1"/1 +1", but it decreases price since the BSE cartelgiven the time discount rate l' becomes no longer sustainable. Hence, ifg(r') < c~ f(r'); then the welfare effect of the increase in the time discount rateis ambiguous.

6. Entry, limit pricing, and government policies

This section presents an analysis of the strategic behavior of an incumbentand a potential entrant, at different levels of the potential entrant's costdisadvantage. Let the incumbent be the low cost firm and the entrant be thehigh cost firm. The sunk cost, which firms must pay to enter the market, isdenoted by F.

With the sunk cost, c1 and ch should be interpreted as variable costs, andIl'« should be interpreted as the excess of revenue over variable costs(hereafter referred to as EOR). Quick entry is assumed. The remainder of themodel is the same as in previous sections.

Entry occurs in two cases. One is the case in which the entrant can enforcea cartel with the incumbent, and its capitalized EOR from the BSE cartel isgreater than its sunk cost. The other is the case in which the entrant's one­period EOR from shaving the incumbent's price is greater than its sunk cost.

1170 11. Bae, A price-setting supergame be/weelJ /wofirms

When c;£J(r), the BSE cartel yields positive capitalized net profit(hereafter simply referred to as positive profit) to the entrant if, and only if,(1 +rjr)Iih(c»F. Entry for the one-period gain by shaving the incumbent'sprice p gives positive profit whenever llAp) > F.

Because llh(c) ~(rjl +r)llt(pm) when c;£J(r), entry for one-period gaincannot be the best strategy of the entrant if c;£ J(r). Therefore, when c;£ J(r)the BSE cartel is the equilibrium if (1 +rjr)llh(c) > F, otherwise theincumbent's monopoly pricing is the equilibrium.

When c> J(r), monopoly pricing by the incumbent is the equilibrium ifllApm);£ F. When c> J(r) and HApm) > F, there witt be entry for one-periodgain if the incumbent sets its monopoly price. Hence, the equilibrium is limitpricing by the incumbent, with price p satisfying 7l:t (p)= F and p< pm, in thatcase.

Because llt(pm) is decreasing in c, if the incumbent's monopoly pricing isthe equilibrium at c=J(r), th en it is also the equilibrium for c> J(r). Hence,as c increases the equilibrium can change in three different ways, dependingon the magnitude of F. When the sunk cost F is very high, the equilibrium isthe incumbent's monopoly pricing for the whole range of c. When the sunkcost has an intermediate value, the equilibrium changes from entry tomonopoly pricing as c increases. When the sunk cost is very low, theequilibrium changes from entry, to limit pricing, and to monopoly pricing, asc increases.

The last case seems to agree with the well known theory of Bain (1956).Bain emphasized the incumbent's absolute cost advantage as an importantsource of barriers to entry. He argued that as the existing level of barriers toentry rises, the condition of entry may change from being effectively impeded,to being ineffectively impeded, and to being blockaded.

The impact -o n social welfare of government policies toward the entrantcan now be discussed. When c ~ J(r) , an entry-prohibiting tax on the entrantis socially desirable if (1 +rjr)llh(c) > F, otherwise it has no welfare effect.Subsidizing the entrant may decrease social welfare if (1 +rjr)llh(c) ~ F,otherwise it has no welfare effect, when c;£ J(r).

Hence, if c~ J(r) , an entry-prohibiting tax on the entrant can be sociallydesirable, but subsidizing the entrant may decrease social welfare. This resultstems from the fact that the BSE cartel is worse from society's point of viewthan the incumbent's monopoly. This result agrees with Brock and Scheink­man (1985).

When c> J(r), a lump-sum subsidy to the entrant effectively reduces itssunk cost, and this in turn may decrease the incumbent's limit price, but atax on the entrant, by a similar argument, may increase the incumbent's limitprice. Hence, if c> J(r), subsidizing the entrant can be socially desirable, buttaxing the entrant may decrease social welfare.

In sum, levying an entry-prohibiting tax is socially desirable if c;£J(r),

11. Rae, A price-setting supergame be/I\'een /1\'0 firms 1171

otherwise subsidizing the entrant's sunk cost is socially desirable. If entry isnot quick, limit pricing cannot be an equilibrium, and therefore subsidizingthe entrant is not socially desirable in any case and taxing the entrant neverdecreases social welfare.

7. Conclusions and suggestions for further research

In this paper we have developed a price-setting supergame between twofirms with different costs. In this supergame, the following results werederived: First, cost-reducing technological progress can decrease socialwelfare even when it is achieved without cost. Second, there is a critical valueof the entrant's cost disadvantage such that subsidizing the entrant is sociallydesirable, if the cost disadvantage is higher than the critical value, otherwisean entry-prohibiting tax on the entrant is socially desirable.

This paper [imited its analysis to the two-firm case. Hence, extending thesupergame to an n-firm case is a high priority for further research. AddingR&D rivalry to the first stage of the game could also be a desired extension.

ReferencesAbreu, D., 1982, Repeated games with discounting: A general theory and an application to

oligopoly, Mimeo (Princeton University , Princeton , NJ).Bain, J.S., 1956, Barriers to new competition (Harvard University Press, Cambridge, MA).Bertrand. 1., 1883, Review of L. \Valras, Theorie math ernatique de la richesse socia le, and A.

Cournot, Recherches sur les principes rnathernatiques de la theorie des richesses, Journal desSavants, 499-508 .

Brock. \V.A. and J.A. Scheinkrnan, 1985, Price selling supergames with capacity constraints,Review of Economic Studies 52. 371-382.

Cournot, J., 1883, Review of L. \Valras, Theorie mathematique de la richesse sociale, and A.(Hachcttc, Paris). -

Friedman. I.W.• 197i, A non-cooperative equilibrium for supcrgames , Review of EconomicStudies 38, 1-12.

Kreps, D.M. and R. Wilson. 1982, Sequential equilibria, Econometrica 50, 863-894.Schelling, T.• 1960, The strategy of conflict (Harvard University Press. Cambridge, MAl .Scherer, F.l\f .• 1980, Industrial market structure and economic performance. 2nd cd. (Rand

McNally. Chicago. ILl .Scltcn, R.• 1965, Spieltheoretische Behandlung eines OIigopolomodells mit Nachfragetriigheit,

Zeitschrift Iiir die Gesamte Staatswissensch aft 121,301-324 and 667-689.Selten, R., 1975, Reexamination of the perfectness concept for equilibrium points in extensive

games, International Journal of Game Theory 4, 22-55.