JOURNAL OF ECONOMIC THEORY 55, 441448 (1991)

Cournot Duopoly with Two Production Periods and Cost Differentials*

DEBASHIS PAL

Department -of Economics, University of Wisconsin-Milwaukee, Milwaukee, Wisconsin 53201

Received February 27, 1990; revised March 14, 1991

Saloner (1987) analyzes a Cournot model with two production periods before the market clears. If costs do not vary across periods, any point on the outer envelope of the reaction functions between the firms’ Stackelberg outputs is attainable with a subgame perfect Nash equilibrium (SPNE). Saloner’s analysis is generalized by allowing for cost differences across periods. The continuum of equilibria vanishes for any cost differential. I f costs fall slightly over time there are multiple “leader- follower” equilibria. Otherwise the model generates a unique SPNE where both firms produce their single-period Cournot outputs, in the period when production is cheaper. Journal of Economic Literature Classification Numbers: 022 and 611. 0 1991 Academic Press, Inc.

1. INTRODUCTION

The standard Cournot duopoly model is a one-shot game. Saloner [7] modifies the Cournot model by allowing two production periods before the market clears. In the first period the firms simultaneously choose initial production levels. These outputs become common knowledge and then the firms choose additional second-period outputs simultaneously. The market clears only once, after the second period. Saloner shows that, in this case, any point on the outer envelope of the recation functions between and including the firms’ smallest Stackelberg outputs is attainable with a subgame perfect Nash equilibrium (SPNE). Saloner’s analysis, however, assumes identical costs of production over time.

In this paper, we generalize Saloner’s analysis by allowing cost differen- ces across periods. In reality, the production costs seem likely to vary

*Thanks to Barnali Gupta, Jonathan Hamilton, Richard Romano, David Sappington, Steven Slutsky, and Edward Zabel for their support and helpful comments. I am also very grateful to an anonymous referee and an anonymous associate editor of the Journal of Economic Theory for their insightful comments on an earlier version.

441 0022-0531/91 $3.00

Copyright (7;’ 1991 by Academic Press, Inc. All rights of reproduction in any form reserved

442 DEBASHIS PAL

across periods. Even if the nominal production costs remain the same, allowing for discounting will make the real costs vary over time. When production costs are allowed to vary over time, it turns out that Saloner’s results are “knife-edge” in the sense that the continuum of equilibria vanishes for any cost differential across periods. When production costs vary over time, depending on the cost differential, three kinds of SPNE (in pure strategies) are possible. If production is cheaper in the first period, then there is a unique SPNE at which both firms produce only in the first period, producing their single-period Cournot-Nash quantities. If produc- tion costs are slightly higher in the first period, then there are two SPNE. At each, one firm behaves as a leader producing only in the first period, while the other behaves as a follower and produces only in the second period. However, if production costs are significantly higher in the first period, then again, there is a unique SPNE, but now the firms produce only in the second period. Thus, this analysis also adresses the question of endogenous timing in duopoly games. Whether duopolists play a simultaneous or sequential move game is determined endogenously in this model, rather than being assigned exogenously.

2. THE MODEL

We consider a symmetric duopoly model with two production periods, where the market clears at the end of the second period. In the first period, firms x and y simultaneously produce outputs 4.; and qf, respectively. These outputs become common knowledge and, in the second production period, the lirms simultaneously choose nonnegative outputs q; and qi:.

After the second period, price is determined from the inverse demand func- tion P(q;+ q;+ qf’+ 4:). The firms choose outputs to maximize profits. We assume that within a period the firms have the same constant marginal cost of production. However, production costs vary across periods. Let cj be the marginal cost of production in period j, j = 1,2.

For firm i, define the single-period reaction function’

R’(qk I c) = arg; max Z. (P(z + qk) - c), i # k = x, y.

We assume that these reaction functions are “well-behaved.“’ Let

’ By “single-period reaction function,” we mean the reaction function corresponding to a standard Cournot model. which has only a single production period.

* By “well-behaved,” we mean 0 > [8R’(qk 1 c,)/aq’] > -1 and [iiR’(qk 1 c,)/&,] < 0. The first condition ensures the existence of a unique single-period Cournot-Nash equilibrium. The second condition is used to prove Proposition 3.2. A set of suffkient conditions for R’ func- tions to be “well-behaved” is P(q’ + qk) is strictly positive on some bounded interval (0, Q), on which it is twice continuously differentiable, strictly decreasing, and concave. P(q’ + qk) = 0 for qi+qk>Q.

COURNOTDUOPOLY 443

(N”(c,), N”(c,)) be the unique single-period Cournot-Nash equilibrium outcome given the marginal cost cj, j = 1,2

A strategy for player i specifies an output for period one and an output for period two. Second-period output is a function of the observed (q-F, qf’). We denote the strategy for firm i by pi= (/?I, pk(q.;, qf)), i= x, y.

For simplicity, we assume a unique Stackelberg point exists for each lirm.3 Denote these by S’(c,) = arg, max 2. [P(z + Rk(z I ci)) - c,], i#k=x, y,j= 1,2.

3. RESULTS AND DISCUSSION

The following lemma is used to prove our propositions. Saloner proved this lemma for cl = c2. The result, however, remains valid for c, # c2.

LEMMA 3.1. Given (q.;, qf), the second-period equilibrium fli(q-;, qf), i = x, y, is

P:(q;, 4:‘) = 0

= N’(c,) - q;

if q; 3 R’(ql; I cJ,

d 3 R”W, IcJ

if q; 6 N’(cJ,

q: 6 N”(c2)

=o if q; 2 N’(cz),

s: G Rk(q’ I ~2)

= R’(q’; 1 c2) - q; if q; 6 N’(c,),

q; 2 Nk(cJ, q; 6 R’(q’; I cz).

This lemma says if (q;, q y) lies on or outside the outer envelope of the second-period reaction functions (area “D” in Fig. l), then neither firm produces in the second period. If both firms produce less than the second- period Cournot outputs in the first period (area “A” in Fig. l), then each produces up to its second-period Cournot level. If one exceeds its second- period Cournot output and the other does not (areas “B” and “c” in Fig. l), the latter produces in the second period its best response to the former’s first-period output and the former does not produce. Given a pair of first-period outputs (q-;, qy), Fig. 1 shows the second-period equilibrium outputs (with c, < cz). Firm x does not produce in the second period, whereas firm y’s second-period output equals R’( q; I c2) - q f .

3 A downward sloping linear demand is suficient for the existence of a unique Stackelberg point.

444 DEBASHIS PAL

FIG. 1. The regions of first-period outputs (4;. q{) yield second-period equilibrium out- puts as follows (with c, < cz): If (q;, q;)s A. then q; =N’(c,)-ql, i=x, y. If (9;. q:)ED, thenqi=O, i=x.y. If(q;,q;)EC, thenq;=Oandqi=RY(q;lc,)-q;‘. If(q-T,q{)EB, then q;=R”(q:lc,)-qq; and q;‘=O.

We next briefly review Saloner’s result. It will be useful to understand why the production cost differential plays a crucial role in determining the equilibria in this model.

PROPOSITION 3.1 (due to Saloner). Zf c1 = c2, then any pair of total outputs (q;+q;, q{+qi) which lies on the outer envelope of the reaction functions between (and including) the firms’ Stackelberg outputs is attainable with a SPNE.

ProoJ See Saloner [7]. 1

The intuition behind this result is as follows. Let (x*, y*) be the total output produced by firms x and y, respectively. Assume that (x*, y*) is on the outer envelope formed by R”(q-“) and R-“(q”) and between (and includ- ing) S” and SY.4 Without loss of generality let (x*, y*) be on R”(q”). That is, at (x*, y*) firm x’s total output is more than its single-period Cournot output and firm y’s total output equals its single-period best response to x*. According to the above result (x*, y*) is attainable with a SPNE. Note that, if there were only one production period, then (x*, y*) # (N”, N”) would not be sustainable as a Nash equilibrium outcome since when firm y produces y*, firm x would do better to produce less than x*. In the case of two periods, however, {(q-r=x*, q;=O), (qT= y*, qy=O)} is a SPNE

4 Since cI = c2, R’(qklc,) = R’(qkI c2) and s’(c,) = s’(c2). This allows us to omit the cost parameters.

COURNOTDUOPOLY 445

outcome and (x*, y*) is the corresponding pair of total outputs. Here, unlike the one-period case, firm x cannot gain by choosing q; < x*. If it does, then from Lemma 3.1, firm y would produce qp = R’(q-;) - y* > 0 in the second period and this would lower firm x’s profits below what it gets at (x*, y*). In other words, if firm x deviates from x* by producing less in the first period, then firm y would respond in the second period by produc- ing an additional amount. This threat of second-period response by firm J deters firm x from cutting its output below x* and {(q.; = x*, q; = 0), (qp=y*, q:‘=O)} is sustainable as a SPNE outcome.

Next consider production costs differing across periods. Proposition 3.2 gives the result for c1 < c2.

PROPOSITION 3.2. Zfc, < c2, then the unique SPNE outcome of this two- period Cournot model is ((q-r= N’(c,), q;=O), (qf= N”(c,), ql=O)}.

Proof Since c1 < c2, R’(qk 1 c2) lies inside R’(q“ 1 c,), i # k =x, y. Now consider the outer envelope formed by R-“(q’) ci) and R”(q-‘Jc,). Let (x*, y*) denote any pair of total outputs which is attainable with a SPNE.

Note that (x*, y*) cannot lie either outside or inside this outer envelope. If it lies outside and qi > 0 for either firm, then that firm would do better by unilaterally decreasing output in the second period. If both q; = 0 and qp = 0, then e’th I er firm would do better by decreasing output in the first period. If (x*, y*) lies inside the outer envelope, then at least one of them can gain by producing more in the first period. Hence, we can restrict our attention to points on this outer envelope.

Now, if x* > S’(ci) (or y* >S!(c,)), then firm x (or y) can gain by producing less. If x* > S”(c,) and q; > 0, then firm x would do better by decreasing second-period production. If x* > S-‘(c,) and q’; = 0, then firm x would do better by decreasing first-period output. Similarly if y* > S”(c,), analogous logic applies for firm y. This further restricts the candidate points to the outer envelope between (and including) s”(c, ) and S.F(c, ).

Suppose (x*, y*) lies on this locus. Since ci cc,, if a pair of total out- puts (x*, y*) is attainable with a SPNE, then the corresponding SPNE outcome has to be { (4.: =x*, q; = 0), (qf = y*, q;‘= 0);. Recall that when Cl = c2, all these outcomes are sustainable as SPNE outcomes. However, if c, < c2 and (x*, y*) # (N”(c,), (N?(c,)) 1’ ies on R”(q” 1 c, ), then firm x can do better by producing slightly less than x* in the first period. Since c, < c2, RY(q”lcZ) < R-“(q”I cl). Thus, 3 an E >O such that if q-:=x* -6, then the optimal qy is still zero. Thus firm x can gain by choosing 4-F <x*, and (x*, y*) is not attainable with a SPNE. Similarly if (x*, y*)#(N”(c,), N”‘(c,)) lies on R”(q”‘Ic,), then firm y can gain by choosing q 1’ < y *. The only possibility left is (x*, y*) = (N”(c,), N”(c,)), i.e., the single-period Cournot-Nash outputs given the first-period cost c,

446 DEBASHIS PAL

Clearly {(q;=N”(c,), q;=O), (q;‘=N”(c,),q$=O)I is a SPNE out- come. 1

Proposition 3.2 shows that Saloner’s result disappears when the firms’ costs rise over time. The intuition for this result is as follows. Now firm i cannot be “pegged” to an output q; > N’(c,) in the first period by the belief that firm k (i# k = X, y) will respond in the second period if it produces less in the first period. Since costs are higher in the second period, firm k does not have the incentive to respond in the second period to a small deviation by firm i in the first period. Thus firm i would “cheat” from Saloner’s proposed equilibrium by producing a little less in the first period knowing that no “punishment” would follow. Since this logic applies for any q; >N’(c,), {(q-y=N”(c,), qz=O), (qf=N”(c,), q2)‘=0)) is the unique SPNE outcome.

Next, Proposition 3.3 characterizes the SPNE outcomes when c, > cZ. Lemmas 3.2 and 3.3 are used to prove Proposition 3.3.

LEMMA 3.2. Zf q; > N’(c,), then the optimal q/1 =O, and if q; E [0, N’(c,)], then the optimal qt $ (0, Nk(c2)], i # k =x, y.

Proof: If q{ > Ni(cZ) then’ optimal qi = 0 for any qt. (This follows from Lemma 3.1.) Then since c2 < c,, it is best for firm k to choose q: = 0 and produce its best response in the second period.

If q; E [0, Ni(c2)] and the optimal qf E (0, Nk(c,)], then from the second period’s equilibrium strategies (Lemma 3.1) we have qi = N’(c,) - q{ and q; = Nk(c2) - 41;. Cl early lit-m k can do better by choosing q/; = 0, because C2<CI. I

LEMMA 3.3. A SPNE outcome must satisfy one of the following:

{(q.;=O,q;=N’(c,)),(q;‘=O,qi‘=N-”(c,)))

{(q-;>N”(c,),q;=0), (q1’=O,qr=R“(q;lc,)))

{(q-;=O,q;=R’(qI’Ic,)),(qj>N-“(c,),qz“=O)).

Proof From Lemma 3.2, if q; > Ni(c2) then the optimal qf = 0, and if q; E [0, N’(c,)], then the optimal q/; 4 (0, Nk(c2)], i # k = x, y. This implies that for a SPNE outcome, the first-period outputs must satisfy either (qi > N’(c2), q/; = 0) or (q; = 0, q/; = 0), i # k = x, y. Then from Lemma 3.1, it follows that the corresponding second-period equilibrium outputs are (qi, = 0, q: = R“(qf 1 c?)) and (qi = Ni(cZ). q/; = Nk(c2)), respectively. 1

PROPOSITION 3.3. Zf c, > c2 then 3 a CT > c2 such that:

(1) If c,>c:, then the unique SPNE outcome of this two-period Cournot model is { (4.; = 0, q; = N”(c,)), (qf = 0, 43‘ = N-“(c~))}.

COURNOT DUOPOLY 447

(2) u-c, cc:, then there are two SPNE outcomes, as given by

((q.;=xL(c,), q;=o), (q1’=0, q;=R-‘(xL(c,)Ic,))f

{ (4; = 0, q; = R”(J+(C,) I c,)), (qf = YL(CI 1, 4; = O)}?

where .~~(c,)=arg,maxz~ [P(z+R”‘(zIc2))-c,] and yL(c,)=arg,maxz . [P(z+R”(zIc,))-c,].

(3) if c, = CT, then there are three SPNE outcomes, as given in (1) and (2).

ProojY Suppose q[ = 0. Then for firm x, the optimal q-F is either 0 or greater than N”(c,). (This follows from Lemma 3.2.) If q-F = 0, then in the second period q; = N-‘(c?) and ql= NY(cZ). In this case, firm x’s profit is [n’(x)] = N”(c,)- [P(N”(c,) + N-“(c?)) - c2]. However, if q-r > N”(Q) then the optimal q.3: = arg; max z . [P(z + Ry(z 1 cz)) - c,]. Let us assume xL(cl) uniquely solves this problem, so that optimal q’; =x~(c,).~ Then firm x’s profit is [TC~(X)]=X~(~,)~[P(X~(C~)+R-“(X~(C,)~CJ)-~,]. Hence, optimal

q-y= O i

if [n’(x)] > [7cL(2c)]

X”(CI f if [n’(x)] < [K”(X)].

However, [$(x)1 >< [rcL(s)] oc, 3 c: where c: is such that [z’(x)] = [z’(x)]. That is, N”(c,) . [P(N”(c,)+ N”(c*))-c,] = xL(cT). [P(x”(c:)+ RY(xL(cl*)lcZ))-CT]. Similarly, for q’;=O, optimal qf=O or yL(ci) according as c, ><cF. Hence, if c, > c:, then { (qf =O, qT= N”(c,)), (4: = 0, q; = NY(d)} is the unique SPNE outcome, whereas if ci < cf then both {(q-;=xL(cl), q;=O), (qf=O, qi’=R”(xL(c,)IcZ))} and ((9-T = 0, q; = R”( yL(c,) I c,)), (q;‘= yL(c,), q; = 0)} are SPNE outcomes. Also, for ci = c:, [z’(x)] = [rc”(x)]. Th us all outcomes in ( 1) and (2) can be sustained as SPNE outcomes. 1

The intuition behind this proposition is as follows. By producing in the first period, a firm gets the opportunity to behave as a leader. However, with ci > c2, production is costlier in period one. So a firm will be a leader only if the benefit of being a leader dominates the loss from the use of a costlier technology. As a result, if c, is not “much” higher than c2, we get two equilibria, at each of which one of the firms behaves as a leader and produces the entire output only in the first period. The other behaves as a follower and produces only in the second period. For c, > c:, the gain from behaving as a leader no longer dominates the loss due to the use of costlier technology. Thus both firms produce their best response outputs in the second period and produce their single-period Cournot-Nash quantities (relative to the second period’s costs).

’ A downward sloping linear demand will be sufficient for uniqueness

448 DEBASHiS PAL

4. CONCLUDING REMARKS

One important feature distinguishing different duopoly models is whether firms move simultaneously or sequentially. Much of the traditional duopoly analysis has treated this feature as exogenously given. Recently, recognition has been given to the fact that whether duopolists play a simultaneous or sequential move game should not be exogenous but should be determined endogenously from the modeL6 This duopoly model with two production periods addresses the issue of endogenous timing in duopoly games. For c, < c2 and cl > cf > c2, the model generates the Cournot-Nash outcome as the unique SPNE outcome of the game. However, in the former case the firms produce their entire outputs simultaneously in the first period, whereas in the latter case they produce only in the second period. Stackelberg outcomes are generated endogenously for c2 < cl < CT. In this case, one of the firms behaves as a leader and produces its entire output in the first period. The other behaves as a follower and produces its best response in the second period.’ However, for c1 = c2 and c, = CT, we encounter multiple SPNE, where both Cournot-Nash and Stackelberg outcomes can result endogenously from the model.

REFERENCES

1. M. BOYER AND M. MOREAUX. Being a leader or a follower: Reelections on the distribution of roles in duopoly, fnf. J. Ind. Organ. 5 (1987), 175-192.

2. D. DOWRICK, von Stackelberg and Cournot duopoly: Choosing roles, Rand J. Con. 17 (1986), 251-260.

3. E. GAL-OR, First mover and second mover advantages, Int. Econ. Rev. 26 (1985), 649-652. 4. J. HAMILTON AND S. SLUTSKY. Endogenous timing in duopoly games: Stackelberg or

Cournot equilibria, Games Econ. Behau. 2 (1990). 2946. 5. D. PAL, “Essays on Industrial Organization: Sequential Competition in Duopoly and

Regulation under Incomplete Information,” Ph.D. thesis, University of Florida, August 1990.

6. J. REINGANUM, A two stage model of R & D with endogenous second mover advantages, In!. J. Ind. Organ. 3 (1985), 275-292.

7. G. SALONER, Cournot dupoly with two production periods, J. Econ. Theory 42 (1987). 183-187.

6 See Boyer and Moreaux [ 11, Dowrick [2], Gal-Or [3], Hamilton and Slutsky [4], and Reinganum [6].

’ In the case of a linear demand, a symmetric equilibrium in mixed strategies is derived in Pal [5].