Heterogeneous Strategies in Nonlinear
Duopoly with Product Differentiation∗
Akio Matsumoto

Tamotsu Onozaki

Department of Economics

Faculty of Management & Economics

Chuo University, Tokyo, Japan

Aomori Public College, Aomori, Japanakiom@tamacc.chuo-u.ac.jp

onozaki@bb.nebuta.ac.jp

Abstract
This study constructs a nonlinear duopoly model with product differentiation. Its purpose is twofold; to compare profitability of quantity strategy and price strategy when heterogeneity of production costs
exists and then to show circumstances under which complex dynamics occurs. It is demonstrated that a price strategy may dominate aquantity strategy if the cost difference is large. It is also demonstrated
that a long-run average profit of a price-setter taken along a chaotic
path is negative but that of a quantity setter is larger than its own
equilibrium profit. The results imply that in heterogeneous competition, the price strategy could be favorable when an economy is stable
and is unfavorable when destabilized.

∗

The earlier version of this paper was presented at the 4th international conferenceon Nonlinear Economic Dynamics (NED05) held in Urbino, Italy, 28-30 July 2005. The
authors are indebted to Ferenc Szidarovszky and Shahriar Yousefi for helpful comments.
Akio Matsumoto appreciates financial supports from the Japan Society for the Promotion of Science (Grand-in-Aid for Scientific Research B, 15330037), Tamotsu Onozaki also
appreciates financial supports from the Japan Society for the Promotion of Science (Grandin-Aid for Scientific Research C, 16530123), and both appreciate financial supports from
Chuo University (Joint Research Grant 0382).

1

1Introduction

In the present paper, we consider strategic behavior of economic agents withheterogeneity in a nonlinear system. To this end, we construct a productdifferentiated duopoly model in which heterogeneity and nonlinearity are
involved. Our main aim is to shed light on the following two effects; static
effects on optimal behavior caused by introducing heterogeneity and dynamical effects on a time-path caused by introducing nonlinearity. Three different
kind of heterogeneity is taken into account; heterogeneity (i.e., difference) in
production costs and heterogeneity in strategy (i.e., quantity or price) thateach firm can choose, and heterogeneity in competition by which we meanthat different firms are to take different strategies. For analytical convenience, demand is assumed to be isoelastic when nonlinearity is required.
In the classical duopoly theory, research interest was concentrated on competitions between firms which were homogeneous in the sense that they produce homogeneous goods, taking the same strategy (i.e., quantity or price).
Among considerable efforts devoted to pull the classical model closer to reality, there are two important approaches; one is to introduce the notionof product differentiation, and the other is to consider a mixture of different strategies in a market. Following both research agendas, it has been
demonstrated that a dominant strategy of each firm is to set quantity (price)in a differentiated duopoly if goods are substitutes (complements) [e.g., seeSingh/Vives (1984), Okuguchi (1984) and Cheng (1985).] Although the conclusion is clear, it is derived under circumstances in which production costs
are the same or homogeneous, and demand is linear. So far, not much yet
has been revealed with respect to heterogeneity in production cost. The first
result of this study adds a possibility that the price strategy is dominant if
firms are heterogeneous in cost and goods are substitutes.
Turning our attentions to dynamical analysis, vast amounts of researches
have been devoted to complex dynamics which nonlinear models intrinsicallyexhibit. For an overview of recent developments of complex dynamics in economics, see, e.g., three volumes edited by Rosser (2004) that contain widely
selected topics on complexity in economics. Regarding nonlinear oligopoly
dynamics, see Puu/Sushko (2002) and Chapter 7 of Puu (2003), in whichmainly nonlinear Cournot dynamics is discussed from various points of view.
On the contrary, only a few effort has been devoted to shed light on differentiated duopoly in heterogeneous competition: see Yousefi/Szidarovszky(2005) for a simulation study with a general isoelastic demand and Matsumoto/Onozaki (2005) for a theoretical study with a specific linear demand.
The second result of this study is to reveal the circumstances under which theheterogeneous nonlinear duopoly model exhibits complex dynamics, relying
2

on a simple iterative model.
The paper is organized as follows: Section 2 overviews heterogenous
duopoly under a general form of demand function. Section 3 explains thestatic results obtained in a differentiated duopoly model with linear demand.
Section 4 replaces linear demand with nonlinear demand and construct ourmodel. Section 5 discusses dynamical implications. Section 6 makes concluding remarks.

2

Overview of Duopoly

Two distinct firms exist in a market: one is firm X that produces good x witha unit production cost a and sells goods at price px , and the other is firm Ythat produces good y with a unit production cost b and sells goods at pricepy . Each firm chooses one of two strategies, quantity or price, and maximizes
its profits, taking the competitor’s strategic variable as given. With two firms
and two strategies, there are four possible types of duopoly competition according as which firm takes which strategy. In Cournot-Cournot (henceforthCC) competition, each firm sets quantity, taking its competitor’s output asgiven, and in Bertrand-Bertrand (BB) competition, each firms sets price,
taking its competitor’s price as given. In Cournot-Bertrand (CB) competition, firm X sets quantity, taking py as given while firm Y setz price, taking
x as given. In Bertrand-Cournot (BC) competiton in which strategies areinterchanged, firm X sets price, taking y as given while firm Y sets quantity,
taking px as given. We call the former two types of competitions as homogeneous because both firms follow the same strategy. On the other hand,
we call the latter two types as heterogeneous because the firms take different
strategies.1 In what follows, we present a somewhat general description of
the above heterogeneous competition of duopoly.
Inverse demand functions of firm X and Y are given bypx = f (x, y),(1)py = g(x, y),which are assumed to be monotonically decreasing in x and y. Solving (1)simultaneously with respect to x and y yields direct demands
x = u(px , py ),(2)
y = v(px , py ).
1

To the best of our knowledge, Bylka/Komar (1976) is the first to consider heterogeneous duopoly, what they call, “different mixed oligopolies.”

3

For the sake of convenience, let us solve each equation of (1) with respect tox and y respectively and denote the solutions as
x = f −1 (px , y),(3)−1

y = g (x, py ).
Profit functions are

πx = (px − a)x,(4)πy = (py − b)y.

In CB competition a strategic variable for firm X is quantity and thatfor firm Y is price, so that profit functions (4) should be rewrite, with theaid of (1) and (3), as
πx = (px − a)x = (f (x, y) − a) x = f x, g −1 (x, py ) − a x= Πx (x, py ),
πy = (py − b)y = (g(x, y) − b) y = g x, g −1 (x, py ) − b g −1 (x, py )= Πy (x, py ).
Reaction functions are derived by solving profit-maximizing conditions,∂Πx (x, py )= 0,∂x∂Πy (x, py )= 0,
∂pyand a CB equilibrium is defined as2

xCB = argmax Πx x, pCB
,
y
x


pCB
= argmax Πy xCB , py .
ypy

In BC competition the strategic situation is reversed, so that profit func2
Henceforth, superscript CB or BC is attached to a variable to denote that it is anequilibrium value of the corresponding type of duopoly.

4

tions are
πx = (px − a)x = (f (x, y) − a) x = f f −1 (px , y), y − b f −1 (px , y)
= Πx (px , y),πy = (py − b)y = (g(x, y) − b) y = g f −1 (px , y), y − b y= Πy (px , y).
Reaction functions are also obtained by solving profit-maximizing conditions,∂Πx (px , y)= 0,∂px∂Πy (px , y)= 0,∂yand a BC equilibrium is defined aspBC= argmax Πx px , y BC ,xpx

y BC = argmax Πy pBC
x ,y .
y3

Heterogeneous Strategy with Linear Demand

We recapitulate the essence of the differentiated duopoly model with linear
demand proposed by Singh/Vives (1985) and summarize their main resultsin this section. Linear demand is replaced with nonlinear demand in the next
section.
Inverse demand is assumed to be linear and given bypx = pM − x − θy,(5)py = pM − θx − y,in which θ denotes the degree of product differentiation and pM is a positive
constant indicating the maximum price to be attained. If θ = 1, (5) are
reduced to demand functions for homogenous goods (i.e., perfect substitutes).
If θ = 0, it means the case of independent products in which the duopoly
market turns to be two monopoly markets. A positive (negative) θ implies5that goods are substitutes (complements). Since we get, due to the symmetry,complementarity from substitutability only by changing the sign of θ, we
confine our analysis to the case in which the goods are substitutes in thisstudy. Hence we assume
Assumption 1

0 < θ < 1.

Solving (5) simultaneously with respect to x and y yields direct demands,x=1{(1 − θ)pM − px + θpy } ,1 − θ2y=1{(1 − θ)pM + θpx − py } .
1 − θ2In this section, marginal costs are zero, which is the usual assumption in thecontext of linear model.3 In CB competition, firm X is a quantity-setter andfirm Y is a price-setter. With the aid of the manner described in the previous
section, we get the reaction functions,2(1 − θ2 )x − θpy = (1 − θ)pM

for firm X,

θx + 2py = pM

for firm Y.

Although given in implicit forms, the reaction function of firm X can be seen
to be upward sloping and that of firm Y downward sloping. These reaction
functions intersect only once to yield a CB equilibrium in which outputs andprices are



2−θ
(2 + θ)(1 − θ)
CB
CB
pM ,
pM ,
x ,y
=
4 − 3θ2
4 − 3θ2
CBpCB
x , py=(2 − θ)(1 − θ2 )(2 + θ)(1 − θ)
pM ,
pM
2
4 − 3θ
4 − 3θ2.

Substituting into the profit functions yields CB profits,(2 − θ)2 (1 − θ2 ) 2 (2 + θ)2 (1 − θ)2 2CBCB
π x , πy=pM ,pM(4 − 3θ2 )2(4 − 3θ2 )2
3

The results to be obtained in this section hold if cost are positive and identical.

6

The ratios of each variables are
xCB
2−θ
=
> 1,
CB
y
2 − θ − θ2pCB
2 + θ − θ2
x
=
< 1,
pCB
2+θ
y

(6)

πxCB
4 − 3θ2 + θ3
=
> 1.
πyCB
4 − 3θ2 − θ3
Inequalities imply that firm X produces more output, faces a lower price andmakes more profits.
In BC competition in which firm X is a price-setter and firm Y is aquantity-setter, firm X chooses px , taking y as given while firm Y chooses y,taking px as given. Symmetrically, we get the following ratios of BC variables,xBC
2 − θ − θ2
=
< 1,
y BC
2−θ
2+θ
pBC
x
=
> 1,
BC
py
2 + θ − θ2

(7)

πxBC
4 − 3θ2 − θ3
< 1.
=
πyBC
4 − 3θ2 + θ3
It is noted that all inequalities are reversed, which implies that firm Y produces more output, faces a lower price and makes larger profits in BC competition. From (6) and (7), we conclude that if goods are substitutes, aquantity strategy is preferable in heterogeneous competition, which coincides
with what Singh/Vives (1984) show. The results are then summarized as
Proposition 1 Given Assumption 1, quantity-setter produces more output,
faces a lower prices and makes larger profits than price-setter in CB competition as well as in BC competition.

4

Heterogeneous Competition with Nonlinear Demand

In this section we replace the linear demand with the nonlinear demand andfind out effects on optimal behavior of firms caused by such a replacement.
7

The inverse demand functions of firm X and Y are assumed to be isoelastic,
px =

1
,
x + θy
(8)

1
py =
.
θx + ywhere θ, again, denotes the degree of product differentiation which fulfills
Assumption 1.4 Solving (8) with respect to x and y yields direct demands,


1
1
θ
x=
−
,
1 − θ 2 px py1y=1 − θ2



1
θ
−
px py


.

Assumption 1 assures the normal situation that quantity demanded is negatively related to its own price and does not go to infinity. It also implies thesubstitutability between goods because the cross derivatives are positive.

4.1

Cournot-Bertrand Competition

In this subsection, we consider CB competition in which firm X is a quantitysetter and firm Y is a price-setter. Let us remind that a and b denote constantmarginal costs. Following the procedure described in Section 2, we get the
CB reaction functions,2θpy = a ((1 − θ2 )py x + θ)

for firm X,(9)θxp2y = b

for firm Y,

both of which are defined in (x, py )-space and presented as implicit forms only
for the sake of later analysis. To characterize CB equilibrium in the quantity
space, (x, y), it is convenient to transform (9) using the inverse demand fory asθ(θx + y) = a(x + θy)2
for firm X,(10)2
θx = b(θx + y)
for firm Y.
4

Puu (2003) introduces the same type of isoelastic demand function into a traditional
CC duopoly model and extensively studies its dynamics when goods are homogenous.

8Dividing the first equation of (10) by the second gives, after arrangements,
2
1 + θz.
(11)c(θ + z) =θ+zwhere z = xy and c = ab .
Let us denote the right hand side of (11) by h(z) and the left handside by wCB (z). An intersection of these two functions, if any, which is asolution of (11), determines a CB equilibrium. The function h(z) is positive,
declines monotonically (i.e., h0 (z) < 0), and is bounded from above (i.e.,h(0) = θ12 > 1) as well as below ( lim h(z) = θ2 < 1) under Assumptionz→∞

1. The function wCB (z) has a positive intercept (wCB (0) = c θ > 0) andincreases monotonically. Thus, to ensure the existence of intersection (and,at the same time, its uniqueness) in the nonnegative quadrant, we need to
impose wCB (0) < h(0) or equivalently
Assumption 2 c <1.
θ3

Let us denote the solution of (11) by γ that is the ratio of CB outputs,
γ = γ(θ, c)

(12)


2
CB
where c(θ + γ) = 1+θγ
and γ = xy CB . Since the explicit form of γ(θ, c),θ+γwhich can be constructed, seems to be unmanageable, we do not exhibit it.
Substituting xCB and y CB = γxCB into (10) and arranging terms give
explicit forms of CB outputs in terms of parameters,xCB =θ
θ(θ + γ)=> 0,a(1 + θγ)2b(θ + γ)2(13)

γθ(θ + γ)γθ
y CB ==> 0.
2
a(1 + θγ)b(θ + γ)2
Substituting (13) into (8) yields CB prices and then their ratio as
pCB
=
x

1
,
(1 + θγ)xCB

pCB
=
y

1
,
(θ + γ)xCB

pCB
1 + θγ
y
=
.
CB
px
θ+γ
9

(14)

Finally, substituing (13) and (14) into profit functions yields CB profits andthen their ratio as
1 − θ2,
πxCB =(1 + θγ)2
πyCB =

γ 2 c(θ + γ),
(1 + θγ)2

πyCB
γ2
=
πxCB
1 − θ2(15)

1 + θγθ+γ2.

In what follows we concentrate on determining relative magnitude of CB
variables, which depends upon the configuration of parameters (θ, c). From(12) wCB (γ) = h(γ). Since h(1) = 1, we get
γQ1

according as

wCB (1) R 1.

It is found that from (14)
1 + θγR1θ+γaccording asγ Q 1.

Solving wCB (1) = 1 with respect to c yields a critical value of production
cost ratio denoted by c̄, and under Assumption 1 we obtain
c̄ =

1
< 1.
1+θ

Now it is easy to see the relations with respect to the output ratio and theprice ratio,
pCB
y CB
1 + θγ
y
=
γ
Q
1
and
=
R1
CB
CB
x
px
θ+γ

according asc S c̄,(16)

which implies that the equi-output curve is identical with the equi-price curvein (θ, c)-space. The c = c̄ curve is downward-sloping and divides the parameter region to be considered {(θ, c) | 0 < θ < 1 and 0 < c} into two subregions.
It can be verified that xCB > y CB and pCB< pCB
for (θ, c) in the regionxyabove the curve and the inequality is reversed in the region below. This factmeans that a firm which produces more output selles it at lower prices.
Now let us turn to profit comparison. It is difficult to derive the explicit
form of the equi-profit curve, πyCB = πxCB , due to the complicated expressionof γ. In spite of doing so, we depict the equi-profit curve with the aid
10

of numerical calculations as a bold line in Figure 1(A). It exhibits a Ushaped profile and divides the parameter region into two. Profit contours
with various ratio values of ρ are also illustrated. It is easy to see that
πyCB < πxCB in the upper subregion with ρ < 1 and πyCB > πxCB in the lower
subregion with ρ > 1. Production of firm Y , y CB , is negative in the shaded
region at the upper-right corner of both Figure 1(A) and (B). That region
will be neglected in the further considerations.
To sum up, since the equi-output curve and the equi-profit curve are different and have no intersection within the parameter region to be considered,they divide the region into three subregions labelled as CB1 , CB2 and CB3
in Figure 1(B). The order of magnitude of each firm’s CB variables in eachregion is as follows:CB
CB1 = {(θ, c) | xCB > y CB , pCB< pCB> πyCB },xy , πxCB
CB2 = {(θ, c) | xCB > y CB , pCB< pCB< πyCB },xy , πxCB
CB3 = {(θ, c) | xCB < y CB , pCB> pCB< πyCB }.
xy , πxLet us say that a firm is efficient (inefficient) if its marginal cost is lower(higher). In the region CB1 with c > 1, firm X is efficient, produces more
output, faces lower prices and earns more profits than firm Y . In the region
CB1 with c < 1, it is inefficient, still produces more output and earns more
profits. In the region CB2 in which c < 1, it is inefficient, still produces more
output but earns less profits. Finally, in the region CB3 , firm X is inefficinet,
produces lower output, faces a higher price and makes less profits than firm
Y . In a duopoly situation, the results display the symmetry. Thus from thepoint of view of firm Y , the reversed results applies to firm Y . In the region
CB3 , for example, firm Y is efficient, produces more output, sets lower prices
and makes more profits than firm X. The same is said of regions CB2 andCB1 . We summarize the results of CB competition as follows:
Proposition 2 Firm X (firm Y ) produces more output and makes more
profits in CB1 (CB3 ) whereas firm X produces more output but is inefficientand less profitable while firm Y produces less output but is efficient and more
profitable in CB2 .

***** Figure 1 *****

11

4.2

Bertrand-Cournot Competiton

In this subsection, we consider BC competition in which firm X is a pricesetter and firm Y is a quantitiy-setter. Due to the symmetry between CB
and BC competitions, by replacing x with y, px with py , and a with b in (10),reaction functions in the BC competition areθyp2x = afor firm X
(17)

2

2

θpx = b((1 − θ )px y + θ)

for firm Y

which are defined in (px , y)-space. As in CB competition, we convert BC
equilibrium into the quantity space, so that (17) is rewritten, by substituting
the inverse demand function of firm X, asθy = a(x + θy)2

for firm X,(18)2

θ(x + θy) = b(θx + y)

for firm Y,

which is defined in (x, y)-space. Dividing the first equation of (18) by thesecond and applying again z = xy and c = ab yields
cz=(1 + θz)

1 + θz
θ+z

2
.

(19)

The right hand side of (19) is the same as that in (11) and denoted againby h(z) while the left hand side by wBC (z). An intersection of these twofunctions, if any, which is a solution of (19), determines a BC equilibrium.
Since wBC (z) is monotonically increasing and bounded from above (i.e.,limz→∞ wBC (z) = θc ), we need to impose limz→∞ wBC (z) > lim h(z), or equivz→∞

alently c > θ3 , to ensure a positive solution.5 Thus we require
Assumption 3c > θ3 .

Let us denote a solution of (19) by δ that is the ratio of BC outputs,δ = δ(θ, c)
2BCcδ= 1+θδ
and δ = xy BC . Substituting xBC and y BC = δxBC
where (1+θδ)
θ+δ
into (18) and arranging terms give explicit forms of BC outputs in terms of
5

It should be reminded that h(z) declines monotonically and is bounded from below,
i.e., lim h(z) = θ2 < 1.
z→∞

12parameters,xBC =θ(1 + θδ)
θδ
=
> 0,
2
a(1 + θδ)b(θ + δ)2(20)
2

y BC =

θδδθ(1 + θδ)=> 0.
2a(1 + θδ)b(θ + δ)2

Substituting (20) into (8) yields BC prices and then their ratio as
pBC
=
x

1
,
(1 + θδ)xBC

pBC
=
y

1
,
(θ + δ)xBC

(21)

pBC
1 + θδ
y
=
.
BC
px
θ+δ
By the same reason as in the analysis of CB competition, the wBC (1) = 1
locus is equivalent to the equi-output curve as well as the equi-price curve.
Solving it with respect to c yields a critical value of the production cost ratio
denoted by c̃, and we obtain c̃ = 1 + θ. The following relations are also
derived:
1 + θδ
δ R 1 andQ 1 according as c Q c̃,θ+δwhich implies that firm X produces more output and sets a lower price in theregion above the critical line c = 1 + θ while firm Y produces more output
and faces a lower price in the region below.
Substituing (20) and (21) into the profit functions yields BC profits andthen their ratio as
πxBC =

1,(1 + θδ)2πyBC =

δ 2 (1 − θ2 ),(θ + δ)2πyBC= δ 2 (1 − θ2 )
πxBC



(22)

1 + θδ
θ+δ2.

Although it is, again, difficult to derive the explicit form of the equi-profit
curve, πyBC = πxBC , due to the complicated expression of δ, it is possible, with13

the aid of numerical calculations, to depict the curve that exhibits an inverted
U-shaped profile and divides the parametric region into two subregions, asseen in Figure 2(A). There, σ denotes the profit ratio and thus σ = 1 means
the equi-profit curve. It is easy to see that πyBC < πxBC in the upper subregion
with σ < 1, and πyBC > πxBC in the lower subregion with σ > 1.
To sum up, since the equi-output curve is different from the equi-profit
curves, the whole parameter region is finally divided into three subregions
labeled as BC1 , BC2 and BC3 in Figure 2(B). Production of firm X, xBC ,is negative in the shaded region at the lower-right corner in each of Figure2(A) and (B). That region will be neglected in the further considerations. In
each subregion, the orderings between BC variables are as follows,BC
BC1 = {(θ, c) | xBC < y BC , pBC> pBC< πyBC },xy , πxBC
BC2 = {(θ, c) | xBC < y BC , pBC> pBC> πyBC },xy , πxBC
BC3 = {(θ, c) | xBC > y BC , pBC< pBC> πyBC }.
xy , πxThe same observation as in CB competition applies to the orderings in BC
competition. Thus we summarize the results of BC competition as follows:
Proposition 3 Firm Y (firm X) produces more output and makes more
profits in BC1 (BC3 ) whereas firm Y produces more output but is inefficientand less profitable while firm X produces less output but is efficient and more
profitable in BC2 .

***** Figure 2 *****
According to Proposition 1, a quantity-setter makes larger profit thana price-setter in heterogeneous competitions with the same (i.e., homogeneous) production costs. In this section, we consider the optimal behavior of
duopolists in heterogeneous competitions but with the different (i.e., heterogeneous) production costs. Propositions 2 and 3 indicates that our results
are richer and summarised as follows:
Summary 1 (i) A quantity-setter produces more output and makes more
profits if it is efficient. (ii) A quantity-setter may produce more output andmakes more profits even if it is inefficient. (iii) A price-setter may be more
profitable only if it is efficient.
14

Two remarks are given concerning Summary 1. (i) For consumers ofgood y, the second result is the worst because they buy smaller amounts ofgood y at a higher price and contribute to higher profits of firm Y . (ii) The
qualitatively same results as the summary are shown by Matsumoto/Onozaki(2005) in a linear duopoly model with heterogeneous costs. Although theheterogenenous costs and nonlinear demand are simultaneously introducedin this section, the results obtained here are due to the heterogeneity in costs.

4.3

Profit Comparison in CB and BC Competitions

In this subsection, our main concerns are on the effect on profits caused bya choice of particular strategy, namely, quantity or price. Thus the following
discussion is focused only on profit. In doing so, we compare profits of eachfirm in CB competition with those in BC competition. Such comparison willallow us to see the impact of strategic behavior on profits in a differentiatedmarket.
From (15) and (22), we obtain the profit ratios of each firm across different
competitions as

2
πxCB
1 + θδ
2= (1 − θ ),πxBC
1 + θγ
(23)

2
πyCB
1
γ θ+δ
=
.
BC
2
πy
1−θ
δθ+γ
Since wCB (z) > wBC (z) for all z, and h(z) is decreasing in z, the intersectionof wCB (z) with h(z) is always located at left side of the intersection of wBC (z)with h(z), which imply that γ < δ for z > 0. As seen in (23), this result alone,
however, is not enough to determine profitability of competition. Thus, we
use graphical representation to make comparisons in Figure 3, which is drawn
again with the aid of numerical calculations. The shaded region located ateither the upper-right corner or the lower-right indicates the set of parametersthat violates either Assumption 2 or 3. Thus, these two regions are neglectedin further considerations.
***** Figure 3 *****The πxCB = πxBC curve is upward-sloping and the πyCB = πyBC curve isslightly downward-sloping. It can be confirmed that πxCB < πxBC above theformer equi-profit curve and πyCB < πyBC above the latter equi-profit curve,
and the inequalities are reversed below. The πxCB = πyCB curve and the15

πxBC = πyBC curve are also drawn in Figure 3. By these four equi-profit curves,the parameter region is divided into five subregions, which are labelled as I,
II, III, IV, and V. Since the vertical axis represents the production cost ratio,
the difference in production cost gets larger as the ratio is larger or smaller
than unity. For convenience, we say that cost difference is major in theregions I and II, moderate in the regions III and IV, and minor in the region
V.
In regions I and II in which the cost difference is major, the order ofmagnitude of profits are as follows:
πxCB < πxBC
∨
∨
CB
πy
< πyBCπxCB > πxBC
∧
∧
CB
πy
> πyBC

region I

region II

In region I, both firms prefers BC competition in which they can make more
profits than in CB competition. Within BC competition, firm X is efficientand more profitable. In region II, we have the reverse results; both firms
prefers CB competition in which they can make more profits than in BC
competition. Regarding the selection of competition what one firm intends
is the same as what the other firm intends. In this sense, both firm can becooperative. Within CB competition, firm X is efficient and more profitable.
Firm X takes a price strategy in BC competition and so does firm Y in CB
competition. We summarize these results as
Proposition 4 If the difference in production cost is major, an efficient firm
prefers taking a price strategy and can make more profits in a heterogeneouscompetition while an inefficient firm prefers taking a quantity strategy and its
profit is less than that of the competitor but more than its own profit obtainedin the other competition.
In regions III and IV in which the cost difference is moderate, the orderingof magnitude of profits are as follows:
πxCB > πxBC∨∨
πyCB < πyBC

πxCB > πxBC∧∧
πyCB < πyBC

region III

region IV

In region III, firm X prefers CB competition in which it makes more profits than in BC competion. On the other hand, firm Y prefers BC competition
16

in which it makes more profits than in CB competition. Both firms preferbeing a quantity-setter, which is not allowed in heterogeneous competition.6
However, no matter which competition is selected, efficient firm X is always
more profitable. Thus it can be said that efficiency implies profitability. It
is noted that the results are reversed in the region IV. We summarize theresults in
Proposition 5 If the difference in production cost is moderate, an efficient
firm prefers taking a quanity strategy and can be profitable in a heterogeneouscompetition while an inefficient firm also prefers taking a quantity strategy
but its profit gets smaller if it plays a price-setter.
In region V where the cost difference is minor, the order of magnitude of
profits are as follows:
πxCB > πxBC∨∧
πyCB < πyBC
region V
Similarly as in the case with moderate costs, preference on competition of one
firm differs from that of the other firm. Indeed, on one hand, firm X prefers
CB competition because it can make more profits than in BC competition.
On the other hand, firm Y prefers BC competition because it can make more
profits than in CB competition. Differently from this, however, an inefficient
firm can be profitable in region V. In this sense, efficiency does not necessarily
imply profitability. We summarize as follows:
Proposition 6 If the cost difference is minor, profitability of firm depends
on which competition is selected.
From Propositions 4, 5, and 6, profit comparison between heterogeneouscompetitions are summarizes as follows:
Summary 2 Provided the goods are substitutes and production costs are heterogeneous, a price strategy dominates a quantity strategy if cost difference
is major, and the strategy dominance is reversed if minor or moderate.
6Here is room for discussing homogeneous competition, but we limit our analysis of
this study only to heterogeneous competition.

17

5

Dynamical Analysis of Heterogeneous Competition

In this section, we carry out an analysis of dynamics resulting from heterogeneous competition, based upon a simple iterative model. In Section 5.1,
we reveal the circumstances under which heterogeneous competition exhibits
complex nonlinear dynamics, and further, we consider economic implicationsof nonlinear dynamics from a long-run point of view in Section 5.2.

5.1

Complex Dynamics of CB Competition

Solving the first-order conditions (9) with respect to the decision variables,
each firm obtains an explicit form of its reaction function in CB Competition,
s
!
1
θ
θ
,
−
x=
1 − θ2
apy py
(24)
r
b
py =
.
θx
Although there are some ways to derive dynamical models from the static
reaction functions (24), we concentrate on a simple iterative model by taking
lag of the variables of the right hand sides,7
s
!
θ
1
θ
,
−
x(t + 1) =1 − θ2apy (t) py (t)(25)
spy (t + 1) =

b
.
θx(t)

7
We are now preparing another paper to study dynamical properties of an adaptivemodel such that
s
!
kx
θ
θ
−
x(t + 1) = (1 − kx )x(t) +,1 − θ2apy (t) py (t)sb+ (1 − ky )py (t),py (t + 1) = kyθx(t)where kx and ky are adjustment coefficients and 0 < (kx , ky ) ≤ 1. The model exhibits
Neimark-Sacker bifurcations when (kx , ky ) < 1. The present model is a special case of theabove where kx = ky = 1.

18

Let us linearize (25) in the neighborhood of CB equilibrium to study its
local stability. The Jacobian matrix evaulated at CB equilibrium is 2 21θ2θ + θγ − 10
2(1 − θ2 ) ac(1 + θγ)2 


CB
J
=
 . (26)




2
1 a
2
−
c(1 + θγ)02 θThe eigenvalues of (26) are the solutions of the characteristic equation
λ2 − trJ CB λ + det J CB = 0
where

trJ CB = 0,

det J

CB

2θ2 + θγ − 1.
=4(1 − θ2 )(27)

It is well-known that the equilibrium point is asymptotically stable if andonly if both eigenvalues have modulus smaller than unity. It is also wellknown that the characteristic equation has eigenvalues less than unity inabsolute value if and only if±trJ CB + det J CB + 1 > 0and

1 − det J CB > 0.

(28)

Thus these three conditions define the parameter domain of asymptotic stability of CB equilibrium. To examine the stability, we substitute (27) into
the left-hand sides of the first two inequalities in (28) to have1+2θ2 + θγ − 1> 0,4(1 − θ2 )which always holds for positive a, b, and θ. Regarding the last inequiality,
the det J CB = 1 boundary is given by6θ2 + θγ − 5 = 0.
On the boundary, the eigenvalues are pure imaginary and unity in absolutevalue, which means λ1,2 = e±i2πε , 0 ≤ ε ≤ 1. Therefore,
trJ CB = λ1 + λ2 = 2 cos 2πε.
On the other hand, trJ CB = 0. Thus 2 cos 2πε = 0 from which we derive
ε = 0.25. This implies the emergence of a four-period cycle on this bounday,19

as shown by Sonis (2000). As a set of parameters crosses the boundary, thefixed point is replaced by a four-period cycle, and then proceeds to chaosthrough a period-doubling way.
We need to prevent a solution generated by the model from being negative. Let us denote the reaction functions of firm X and Y (i.e., the first andthe second equation of (24)) by rx (py ) and ry (x), respectively, and let us alsomdenote a maximizer and maximum of rx (py ) by pm
y and x . Then we obtainpm
y = 4aθ

and

xm =

1
.
4a(1 − θ2 )If rx (0) ≥ ry (xm ) holds, then a solution of (25) is non-negative for any
t ≥ 1. Solving the last inequality condition gives the following non-negative
condition for a time-path,θ3.
c≥4(1 − θ2 )Since we assume Assumption 2, the feasible domain of instability is given by
the following inequalities:2

6θ + θγ(θ, c) − 5 > 0,θ3c>,4(1 − θ2 )

and

c<

1
,
θ3

which is a strip-wise region denoted by [U] in Figure 4(A). It can be seen thatcrossing the boundary into this region allows for a period doubling cascadeto chaos. Each color in Figure 4 corresponds to period’s number of cycles as
displayed in the table below the figure; the red area labelled as [S] exhibits a
set of (θ, c) for which trajectories converge to a unique stable fixed point. The
yellow area corresponds to a period-4 cycle, the light blue area to a period-8
cycle, the light red area to a cycle of period-16 and so forth. The black area
corresponds to a cycle of period-17 up to -64 and the white area corresponds
to a cycle of period-over-64 or an aperiodic (including chaotic) orbit.8 CB
output is negative in the light-gray triangle-wise region at the upper-left, anda dynamical solution becomes negative in the dark-gray distorted rectangle
region. An enlargement of the unstable region is shown in Figure 4(B), whichfacilitate taking a view of the aperiodic region. It should be noted that, due
to the symmetry, similar figures are obtained concerning dynamics in BC
competition.
***** Figure 4 *****
8

For practical purposes, it is enough to check up to period-64 cycle in order to detect
aperiodic area.

20

5.2

Long-Run Aspects of Complex Dynamics

It is now well-known that chaotic dynamics has the sensitive dependence on
initial conditions. A consequence of the dependence is that even a slightly
different choice of initial conditions can drastically alter the whole future
evolution of the trajectories. Therefore it is meaningless to investigate individual chaotic trajectories. One way to characterize such chaotic dynamics is
to turn our attention to statistical or long-run behavior of trajectories. Underrather weak mathematical conditions, frequencies of a chaotic trajectory may
converge to a stable density function. Once the explicit form of the density
function is constructed, it can be shown that for any continuous function of avariable, its time average calculated along a chaotic trajectory is equal to its
space average.9 This implies that it is possible to analytically calculate thelong-run average behavior that is independent from a choice of initial conditions. However, it is, in general, difficult to construct an explicit form of thedensity function, even if its existence is mathematically confirmed. We thus
numerically calculate the long-run average behavior over sufficiently a long
period of time as a proxy for the analytical value of the long-run behavior.
Given the simple iterative dynamical sytem, the average profit of a time
path with T -period is defined by
T −1

1X
π̄i =
π(i(0), pj (0))
T t=0i, j = x, y,i 6= j.

We perform numerical simulations of CB competition and then BC competition. Figure 5 illustrates the results of the numerical simulations in which
the degree of product differentiation is measured on the horizontal axis andthe long-run average profits of firm X and Y are on the horizontal axis. For
comparison, graphs of the CB equilibrium profits of both firms are also depicted as slightly downward sloping dashed curves. In each simulation, theaverage is calculated from the last 1000 out of 3000 iterations, and θ is increased in steps of 0.0025 (= 1/400) from θs −k to θe . Here k is an arbitrarily
9

Let xt+1 = φ(xt ) be a dynamical process where xt is a variable at time t. Suppose
the frequencies of the trajectory {xt }∞t=0 converge to a density function, ϕ. Then for any
continuous function f and for any initial point x0 ,
T −1

1 X
f (φt (x0 )) =T →∞ T
t=0
lim

Zf (x)ϕ(x)ds

where φt = φt−1 · φ and φ0 = 1. See, for example, Chapter 8 of Day (1994) for moredetails.

21

small number (k = 0.005 in the simulations), θs is the value for which loss ofstability occurs, and θe is the value for which a trajectory becomes zero.10
It can be observed in Figure 5 that the average profit of each firm isidentical to its CB profit when θ is less than θs , that is, when the dynamicalsystem is stable. In Figure 5(A), the production cost ratio is chosen to beunity for which firm X is more profitable than firm Y . It can be seen thatalong a unstable time-path, the average profit of firm X is larger than its CB
profit while the average profit of firm Y is not only less than its CB profit
but negative for a little larger value than θs . In Figure 5(B), the production
cost ratio is chosen to be 0.5, which means that firm X is inefficient and firm
Y is more efficient. In consequence, CB profit of firmY is larger than thatof firm X. However, as seen in the figure, the average profit of firm Y getsdecreasing very rapidly and sooner or later becomes negative as θ increases
from θs . On the other hand, the average profit of firm X increases in θ andbecomes larger than CB profit of firm X. Due to the symmetry between
CB competition and BC competition, the numerical results obtained here
are reversed in BC competition. Although we do not represent simulation
results in BC competition, we have exactly the same illustrations of long-run
average profits as shown in Figure 5(A) and (B) if we put c̃ = 1 and c̃ = 2
and replace x and y by y and x, and CB by BC. These numerical simulations
indicate the following result:
Proposition 7 When an equilibrium is unstable in heterogenous competitions, a quantity-setting firm is more profitable while the long-run average
profit of a price-setting firm is negative soon after unstability occurs.

***** Figure 5 *****
It can be explained why the long-run average profit of firm Y becomes
negative. By definition, the profit of firm Y taken along a time path ispositive ifpy (t) > b and y(t) > 0.
10

Given c = c̄, θs and θe are defined by
6θs + θs γ (θs , c̄) − 5 = 0

22

and

c̄ =

θe3
.
4(1 − θe2 )Solving the inverse demand of good y for y and substituting the first equationof the dynamical system yields, after arrangements,
s
!
θ
1
θ
y(t) =−,21−θθpy (t)apy (t)which implies the following condition for a positive y(t) :py (t) <a.
θ3

Thus as far as the time path of py stays inside of the interval (b, θa3 ), the profit
of firm Y is positive. However, as can be seen in Figure 6 in which a partof the return map is illustrated, a chaotic path of py is either larger than θa3
or less than b.11 In consequence, the profit taken along the chaotic path isnegative except in a neighborhood of the starting point. On the other hand,since it can be verified that πx (t) = (px (t) − a)x(t) > 0 if py (0) > aθ andx(0) > 0, the long-run average profit of firm X is positive along a chaotic
path.
***** Figure 6 *****
A natural question that arises is what firm Y should do to cope withthis unfavorable situation. There are several possible answers: the first isthat a price-setting firm exits the unstable market because it makes losses.
Since its long-run average profit is much larger than an equilibrium profit,
a quantity-setting firm is in favor of the unstable market and has a strong
reason to remain in the market. In this case, a duopoly market turns out tobe a monopoly market as a result of unstable heterogeneous competition.12
The second is to get rid of the assumption of full information and to reconsider dynamics under boundedly rational circumstances in which the firms
11

In this simulation, we set a = b = 2, θ = 0.9027, x(0) = 0.35 and py (0) = 2.52. 750
iterations are performed and the trajectory obtained after 500 iterations is depicted. The
upper side of the return map is eliminated only for convenience.
12
It is well-known that the isoelastic demand is problematic in a monopoly case. To
remedy this problem, we can consider the following inverse demand function after firm Y
leaves the market,
1
p=x+εwhere ε is a positive autonomous demand. When the monopoly√ firm is required to produce
more than the autonomous demands, its profit is πx = (1 − aε)2 > 0.

23

have only misspecified knowledge of the demand function. In fact, this lineof research has already launched. One interesting approach is proposed by
Bischi et al. (2004) in which a duopoly game with production differentiationis considered in CB competition. Under the assumption of limited information such that the firms do not know the demand function of the market,
a profit maximization problem is solved with the aid of, what they call a
local and monopolistic approximation of the demand function. The result isthat dynamics based on this approach always converges to an equilibrium.
The third is to stabilize chaotic fluctuations through public effort such as
government intervention to the market or individual effort such as changing
the formation of expectations. See Matsumoto (2006) for controlling chaos
in a nonlinear duopoly model. The last is to change a strategy from price toquantity, which implies to change a type of competition from heterogeneity
to homegeneity.

6

Concluding Remarks

In this study, we have two main aims, the first of which is to shed light on theroles of production cost difference on the optimal behavior in a differentiatedduopoly model, and the second of which is to shed light on the roles ofnonlinearity on unstable dynamics possibly generated in the model.
Within a heterogeneous competition, heterogeneities in production costdo matter in the sense that a price strategy can dominate a quantity strategy(roughly speaking) if a price setter is efficient with regard to production
cost. This is not observed in the traditional differentiated duopoly model inwhich homogeneous cost assumption is adapted. Second, it is shown that aheterogeneous competition is a natural consequence of the optimal behaviorof each firm if the cost difference is large. This result is new and is obtainable
only when heterogeneous costs are taken into account.
It have been demonstrated that a homogeneous competition can generatecomplex dynamic involving chaos in duopolistic as well as oligopolistic circumstance in which demand is nonlinear. Turning our attentions to dynamics
of a nonlinear duopoly with product differentiation in a heterogeneous competition, we have shown that heterogeneous competition may follow complex
nonlinear dynamics. In consideration of the result, we have constructed afeasible domain of unstable equilibrium, which provides parametric configurations to give rise to chaotic dynamics. We have also demonstrated thateven if a price-setter can make more profits at equilibrium than a quantity
setter, its long-run average profit taken along the unstable but bounded time
path becomes not only less than the equilibrium profit but also negative.
24

We have focused on the heterogeneous case in which firms are not allowed
to take the same strategy. It is possible to extend our results in the generalcase that involves homogeneous as well as heterogeneous competition.

25

References[1] Bischi, G.I., A.K. Naimzada and L. Sbragia, “Oligopoly Games with
Local Monopolistic Approximation,” mimeo, 2004.
[2] Bylka, S. and J. Komar, “Cournot-Bertrand Mixed Oligopolies,” in M.
Beckmann and H.P. Künzi eds., Warsaw Fall Seminars in Mathematical Economics 1975 (Lecture Notes in Economics and Mathematical
Systems 133), Springer-Verlag, 22–33, 1976.
[3] Cheng, L., “Comparing Bertrand and Cournot Equilibria: a Geometric
Approach,” Rand Journal of Economics 16, 146–152, 1985.
[4] Day, R.H., Complex Economic Dynamics, Vol. I, MIT Press, 1994.
[5] Matsumoto, A., “Controlling the Cournot-Nash Chaos,” forthcoming in
Journal of Optimization Theory and Application 128, 2006.
[6] Matsumoto, A. and T. Onozaki, “Linear Differentiated Duopoly
with Heterogeneous Production Costs in Heterogeneous Competition,”
mimeo, 2005.
[7] Okuguch, K., “Equilibrium Prices in the Bertrand and Cournot
Oligopolies,” Sonderforschungsbereich 303, Bonn University, 1984, andJournal of Economic Theory 42, 128–139, 1987.
[8] Puu, T., Attractors, Bifurcations, and Chaos: Nonlinear Phenomena in
Economics, 2nd Edition, Springer-Verlag, 2003.
[9] Puu, T. and I. Sushko (eds.), Oligopoly Dynamics: Models and Tools,
Springer-Verlag, 2002.
[10] Rosser, J.B., Jr. (ed.), Complexity in Economics, Vol. I–III (The International Library of Critical Writings in Economics 174), Edward Elgar
Publishing, 2004.
[11] Singh, N. and X. Vives, “Price and Quantity Competition in a Differerntiated Duopoly,” Rand Journal of Economics 15, 546–554, 1984.
[12] Sonis, M., “Critical Bifurcation Surfaces of 3D Discrete Dynamics,” Discrete Dynamics in Nature and Society 4, 333–343, 2000.
[13] Yousefi, S. and F. Szidarovszky, “Puu-Type Duopolies with Production
Differentiation: A Simulation Study,” mimeo, 2004.

26

Captions of Figures

Figure 1

Regime classification of CB competitions(A)(B)

Figure 2

Regime classification of BC competitions(A) Profit Contour Curves(B) Division of Parameter Region

Figure 3

Figure 4

Profitability classification in parameter region

Division of the parameter region in CB competition whenkx = ky = 1(A) Division of the parameter region(B) Enlargement of (A)Figure 5Long-run average profits in CB competition(A) c̄ = 1 and πxCB > πyCB(B) c̄ = 0.5 and πxCB < πyCB

Figure 6

Return map of the simple iterative dynamical modelParameters and initial condition are as follows: a = b = 2, θ = 0.9027,
x(0) = 0.35, py (0) = 2.52. The upper side of the return map is eliminated.

27