A naive but optimal route to Walrasian behavior.pdf

7/29/2019 A naive but optimal route to Walrasian behavior.pdf

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Journal of Economic Behavior & Organization

Vol. 52 (2003) 553571

A naive but optimal route to Walrasian behaviorin oligopolies

Weihong HuangNanyang Business School, Nanyang Technological University, Nanyang Avenue, Singapore 639798, Singapore

Received 8 August 2001; accepted 7 March 2002

Abstract

A paradoxical phenomenon where irrationality or ignorance lead to higher profits is shown to

prevail in an oligopolistic market where three firms produce a homogeneous product with identical

technologies. Under the conventional assumption on the convexity of cost function, the profit made

by a naive price-taker will always be higher than or at least as much as the other two rivals, no matter

what kind of strategies the latter may take and whether they form a coalition or not. By sticking

to the price-taking strategy, a firm always gains a positive profit improvement, should any one ofits rivals try to upgrade from price-taker to the Cournotor. With these incentives, a firm prefers to

remain as a naive price-taker.

2002 Published by Elsevier Science B.V.

JEL classification: D14; L13; C73

Keywords: Competitive behavior; Price-taking; Duopoly; Oligopoly; Cournot; Cobweb

1. Introduction

Students of economics are often told that a rational firm uses all relevant information

and that more information leads to higher profits than less. Moreover, they are taught that

a firm exercises non-competitive strategies to maximize profit. In an oligopoly market,

if outputs are the only choice variables, firms are inevitably assumed to adopt sophisti-

cated strategies such as the Cournot or Stackelberg strategy, or to collude so as to achieve

economic efficiency (Varian, 1992). It has long been believed that a Cournot oligopoly

market could converge to a perfect competition only if free entry is allowed and the num-

ber of firms approaches infinite (see for instance, Chamberlin, 1948; Conlisk, 1983;

Frank, 1965; Novshek, 1980), although the convergence issue itself has been questioned

(McManus, 1964; Ruffin, 1971). The implication is that perfect competition is a limiting

E-mail address: [email protected] (W. Huang).

0167-2681/$ see front matter 2002 Published by Elsevier Science B.V.

PII: S 0 1 6 7 - 2 6 8 1 ( 0 2 ) 0 0 1 5 4 - 3


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554 W. Huang / J. of Economic Behavior & Org. 52 (2003) 553571

case of oligopoly and that price-taking is an inferior strategy that firms are forced to adopt

with no better alternatives.

In recent years, maximizing relative profit instead of absolute profit has aroused

the interests of economists from different fields. From evolutionary perspective, Schaffer(1989) demonstrates with a Darwinian model of economic natural selection that if firms

have market power, profit-maximizers are not necessarily the best survivors. In a simple

context with just two quantity-setting firms which have identical and constant marginal

costs, only price-taking behavior is evolutionarily stable. The similar conclusion has been

arrived by Rhode and Stegeman (1995) by applying a stochastic evolutionary approach to

a very specific two-firm context. In Vega-Redondo (1997), it is further argued that, if firms

maximize relative profit, a Walrasian equilibrium can be induced. As he concluded In a fi-

nite oligopoly, Walrasian behavior1 is not optimal if firms are aware of the characteristics of

the game and attempt to maximize their instantaneous payoff. However, if firms objective

function is taken to include survival as primary consideration (for example, if their intertem-

poral discount rate is close to 1) and their survival is linked to relative payoffs (e.g., the larger

is the accumulated profit relative to that of the competitors, the stronger is the firm to launch

and/or repel a predatory campaign), it may well be that the rationality of Walrasian behavior

could be recovered. Schenk-Hoppe (2000) generalized and extended Vega-Redondos ap-

proach by studying an explicitly dynamic evolutionary model of Cournot oligopoly in which

the behavior of the firms is based on imitation of success and experimentation. Meanwhile,

Lundgren (1996) shows that by making managerial compensation depend on relative profits

rather than absolute profits, the incentives for oligopoly collusion can be eliminated.

It is shown in this paper that, in contrast to relative profit maximization, a firm can outper-form its rival by behaving as a naive price-taker, even when its rival behaves as relative profit

maximizer. Therefore, the price-taking strategy is best among all quantity-choosing

strategies in the sense that it always beats its rivals, no matter what strategies the latter may

take.

This finding challenges the usual assumption that greater sophistication in decision mak-

ing is better than less. Instead, price-taking, despite being simple and naive, is actually

the most effective among all possible alternatives. To summarize: (i) a price-taker always

makes a higher or equal profit than its rivals, no matter what strategies the latter may take;

(ii) a price-taker always makes higher profit than the average profit made by a coalition

of its rivals; (iii) by sticking to the price-taking strategy, a firm always gains a positiveprofit improvement, should any one of its rivals adopt an optimal responsethe Cournot

strategy. This establishes an additional routein contrast to free entryfor the transition

from oligopoly to perfect competition.

2. The standard model and strategies

Consider three firms, X, Y and Z that produce a homogeneous product at period t with

output xt, yt, and zt, respectively. It is assumed that

1 Walrasian theory builds upon the central hypothesis that economic agents take prices parametrically, i.e.,

do not consider the possibility of affecting prices through their consumption or production decisions. Walrasian

behavior is identified with output choices by all firms for which profits are maximized at the market-clearing price.


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W. Huang / J. of Economic Behavior & Org. 52 (2003) 553571 555

(i) The firms have identical increasing, strictly convex cost functions C(), with C() > 0and C() > 0.

(ii) The inverse demand for the product is given by pt=

D(qdt ), with D()

0, where qdt

is the quantity demanded.(iii) The market price is determined as to clear the market at every period, i.e., qdt =

xt + yt + zt.The profit of each firm is

q = ptqt C(qt) = qtD(xt + yt + zt) C(qt), where q {x,y,z}.Eachfirmwishestochooseitsoutputsoastomaximizeitsexpectedprofit q = petqtC(qt),the first-order condition for each being that

pet + qtdpetdqt = C(qt), where q {x,y,z}.

In the following, these specifications will be referred to as the standard model.

Of course, pet and dpet/dqt for any firm depend on the actions of each of all the others,

which no one knows in advance. The firms may arrive at different expectations of pet and

dpet/dqt and hence produce at different levels. The following three standard strategies are

compared here:

(1) The naive or price-taking strategy. The production is determined by equating marginal

cost with the expected price pet. Taking firm X, for example, its output xt is determined

by the solution ofpet = C(xt). (1)

Given the simplest naive forecast, pet = pt1, we get a naive, price-taking strategy.(2) The sophisticated or Cournot strategy. A firm presumes a knowledge of the market

demand. It needs to speculate on the outputs of its rivals. So, if firm Y takes this

alternative, its output will be determined by the solution to

D(xet + yt + zet) + ytD(xet + yt + zet) = C(yt). (2)In particular, when qet = qt1, with q = x,z, it will be referred to as the Cournotstrategy. Eq. (2) defines a standard Cournot reaction function for firm Y.(3) The smart or Stackelberg-like strategy. In addition to the market demand function, a

firm knows the strategies its rivals adopt. It is capable of taking into account all possible

reactions from its rivals. Taking firm Z, for example, its output is then determined by

D(xet + yet + zt) + ztD(xet + yet + zt)

1 + dxet

dzt+ dy

et

dzt

= C(zt). (3)

If both firms X and Y react to the market with the standard Cournot reaction functions,

firm Z behaves as a typical leader in a standard Stackelberg game with the assumption

that it makes its decision first, followed by firms X and Y.

The expectations in (1) and (2) are naive in that they are formed based on last periodsinformation only. I do not include more sophisticated expectations such as adaptive ex-

pectation, historical-mean expectation or rational expectations because, so long as the


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expectation itself converges to its realized counterpart, the equilibrium outputs are the same

as that arrived from naive expectations, equilibrium profits remain unchanged. However,

the more sophisticated expectation schemes do alter market stability (see Huang, 1999a).

With this assumption, a standard three-dimensional dynamical process is formed:

xt = fx(xt1, yt1, zt1), yt = fy(xt1, yt1, zt1),zt = fz(xt1, yt1, zt1). (4)

The rest of this paper is directed to the exploration of the relative profits of the three rivals

under different strategical combinations in the strategy space S= {p,c,s}, where p, c ands stand for price-taking, Cournotand Stackelberg-like, respectively. A convention will be

applied so that the first subscript stands for the strategy adopted by firm X, the second by

firm Y and the third by firm Z. For instance, xpcsdenotes the profit earned by firm X when

firm X chooses the simple strategy, firm Y adopts the sophisticated, and firm Z adopts thesmart.

Definition. An output bundle (xt, yt, zt) from the dynamical oligopoly game given by (4)

is said to be viable, if the following inequalities are met:

1. qt > 0 for q {x,y,z}, i.e., positive outputs for all firms;2. q(xt, yt, zt) > 0 for q {x,y,z}, i.e., all firms must make positive economic profits

so as to stay in the market;2

3. 0 < pt

=D(xt

+yt

+zt) 0 (gyx < 0) and vice versa if

gxy < 0 (gyx > 0).

3. More sophisticated profit less!

Smartness may overreach itself!

An ancient Chinese proverb

While upgrading its strategy from the simple, to the sophisticated, and finally to the smart,

a firms economic IQ gradually increases in that more and more information being taken2 The positiveness for profit is only required for the evaluation of the relative profit ratio. It can be replaced with

nonnegativeness.


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into account. It is natural to expect that a firms profit should grow with the growth of

economic IQ, i.e., that a firm taking more sophisticated strategy will make more profit

than a less sophisticated one.

Let us first examine this expectation for a special case.Following the conventional settings (Fisher, 1961), the inverse market demand is assumed

to be linear in the form of

p = D(qdt ) = 1 qdt (5)and the cost function

C(qt) =1

2q2t , (6)

where > 0.

With (5) and (6), the profit of each firm is given by

q = qt(1 xt yt zt) 1

2q2t , q {x,y,z}.

To see the economic meaning of, we first examine the standard Cobweb model, in which

one single firm, say firm X, supplies the output based on the recurrent relation:

xt = MC1(pet) = MC1(pt1) = (1 xt1), (7)where MC1 denotes the inverse function of marginal cost C. The Cobweb model (7)converges to its intertemporal equilibrium x

=/(1

+) if and only if < 1. Recalling

the unity is the demand elasticity, is the supply elasticity, the stability condition can be

rephrased as the market demand elasticity must exceed the supply elasticity. Moreover, the

larger the is, the more unstable is the Cobweb economy. A similar conclusion can be

drawn for the oligopoly game. In fact, if all three firms are price-takers, which is denoted as

(P, P, P), firm qs output is determined from pet1 = C(qt) for q {x,y,z}. Substitutingwith pet = 1 xt1 yt1 zt1, the three identical reaction functions follow:

xt = (1 xt1 yt1 zt1), (8)yt = (1 xt1 yt1 zt1), (9)zt = (1 xt1 yt1 zt1). (10)

A unique viable equilibrium output bundle (xppp, yppp, zppp) results, where qppp = /(1 +3) for q {x,y,z}. The equilibrium profits turn out to be

xppp = yppp = zppp =

2(1 + 3)2 . (11)

The stability of the equilibrium (xppp, yppp, zppp) is measured by the largest modulus of the

eigenvalues, |u|, derived from the Jacobian matrix of the processes of (8):

Jppp =

.


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It is easy to verify that |u| = 3. And hence, the value of reflects the stability of themarket equilibrium.

We next proceed to a comparison of relative profits for several alternative strategy com-

binations.

Case I (P, C, P). If firm Y becomes smarter by taking the Cournot strategy, its output is

then determined by (2), with xet = xt1, zet = zt1 and D() = 1, which leads to astandard Cournot reaction function:

yt =

1 + 2(1 xt1 zt1). (12)

Solving (12) together with (8) and (10) gives us a unique viable equilibrium output bundle,

which leads to the following profits:3

xpcp = zpcp =(1 + )2

2(22 + 4+ 1)2 , y

pcp =(1 + 2)

2(22 + 4+ 1)2 . (13)

What is the relative profit advantage the sophisticatedfirm Y has over its two simpleminded

rivals? It turns out that

gqypcp =

qpcp

y

pcp

1 = 2

1 + 2 > 0 for q = x,z,

i.e., the simpleminded(the price-takers) makes more profit than the more sophisticated(the

Cournotor).Fig. 1(a) shows the profit distributions and the relative profit ratio gqypcp = 2/(1 + 2),

q = x, z, for different values of. We see that gqypcp is a monotonic, increasing function of, which suggests that, the more unstable the market is, the more profit is gained by the

naive over the sophisticated.

Case II (P, C, C). While firm Xs reaction function remains as (8), firm Z imitates firm Y

by upgrading its reaction function to

zt =

1+

2(1 xt1 yt1).

The equilibrium profit distributions turn out to be

xpcc =(1 + )2

2(2 + 4+ 1)2 , y

pcc = zpcc =(1 + 2)

2(2 + 4+ 1)2 , (14)

which leads to

gxqpcc =2

1 + 2 > 0 for q = y,z.

As illustrated in Fig. 1(b), the two sophisticatedfirms still profit less than its naive rival

with the exact the same loss ratio as in Case I.

3 Since it is the equilibrium profits that account, we shall omit the outputs.


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Fig. 1. Non-cooperative strategies: (a) Case I: (P, C, P); (b) Case II: (P, C, C); (c) Case III: (P, C, S); (d) Case

IV: (P, P, S).

Case III (P, C, S). Now assume that firm Z becomes even more intelligent so as to take a

smart strategy in a way that it will take both market demand and the reactions from firms

X and Y into its decision making. In addition, it is further assumed that firm Z is intelligent

enough to accurately estimate the market equilibrium relationship. More precisely, the goal

of firm Z is to maximize the equilibrium profit directly through

D(x + y + z) + zD(x + y + z)

1 + dxdz

+ dydz

= C(z). (15)


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The relationship between equilibrium outputs given by (8) and (12) indicates a pair of

equilibrium reaction sensitivities:

x

z = (1

+)

1 + 3+ 2 ,y

z =

1 + 3+ 2 .Substituting above information into (15) and solving for the optimal equilibrium output z

gives the equilibrium profits as follows:

xpcs =(1 + )2(4+ 22 + 1)2

2(32 + 5+ 1)2(1 + 3+ 2)2 ,

ypcs =(1 + 2)(4+ 22 + 1)2

2(32 + 5+ 1)2(1 + 3+ 2)2 ,

zpcs = (1 + )22(32 + 5+ 1)(1 + 3+ 2) .

It follows that

gxzpcs =2(1 + )2

(32 + 5+ 1)(1 + 3+ 2) > 0

and

gyzpcs =2

1

+2

> 0.

Again, as illustrated in Fig. 1(c), we have xpcs > z

pcs > y

pcs for all > 0.

Case IV (P, P, S). Nowiffirm Ydowngrades back to a price-taker like firm X, the reactions

function (8) leads to the equilibrium reaction sensitivities:

x

z= y

z=

(1 + 2) .

Consequently, the equilibrium profits are

x

pps = y

pps =(3+ 1)2

2(4+ 1)2(1 + 2)2,

z

pps =

2(4+ 1)(1 + 2).

Again, we have

gqzpps =2

(4+ 1)(1 + 2) > 0, where q = x,y.

The relevant results are depicted in Fig. 1(d).

4. Collusion does not work either!

A man of great wisdom often appears slow-witted!

An ancient Chinese philosophical quotation


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From Case I, II, III and IV, a counter-intuitive fact is repeatedly derived for this linear

modela firm taking a naive strategy (a price-taker) beats its rivals all the time, no matter

what strategies the latter may take individually. It is natural to ask whether such disadvanta-

geous status for the smarters against a naivercan be reversed if the former forms a coalition.The answer is negative!

Return to the linear economy and consider a case in which firms Y and Z form a coalition.

The fact of identical cost functions for firms Y and Z suggests that the optimal production

output should be identical, say u, regardless what strategies they adopt and how the profits

made jointly are redistributed between them.

Case V (P, [CC]). 4Firm X sticks to the naive strategy and plans its output according to

xt

=(1

xt

1

2ut

1). (16)

The profit jointly made by the coalition [YZ] is given by

[yz] = 2ut(1 xt 2ut) 1

u2t .

When the coalition takes the sophisticated strategyestimating the price with pet = 1 xt1 2ut, the reaction function is given by

ut =

4+ 1 (1 xt1). (17)

It can be verified that the equilibrium profit is now

xp[cc] =1

2

(1 + 2)2(5+ 1 + 22)2 ,

[yz]p[cc] =

(4+ 1)(5+ 1 + 22)2 . (18)

Let q

p[cc] = 1/2[yz]p[cc] be the average profit of the coalition. Then the relative profit ratio

for the naive against the average profit of the coalition is thus

gx[yz]p[cc] =

xp[cc]

q

p[cc]

1 = 42

4

+1

> 0.

It is surprising to notice the profit earned by the price-taker alone can supersede the total

profits jointly made by the coalition. This case occurs when gx[yz]p[cc] > 2, or equivalently,

> (1 +

6)/2 1.725. The analysis is depicted in Fig. 2(a).

Case VI (P, [SS]). Firm Xs reaction function (16) suggests an equilibrium relation:

x = 1 2u1 + (19)

and dx/du = 2/(1 + ).4 A bracket [] is used to indicate the collusion.


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Fig. 2. Cooperative strategies: (a) Case V: (P, [CC]); (b) Case VI: (P, [SS]).

Therefore, if the coalition [YZ] takes the smart strategy instead, their optimal output level

u are determined from

1 1 2u1 + 2u

u2 + 21 +

1

u = 0, (20)

which gives an equilibrium profit bundle:

xp[ss] =1

2

(1 + 3)2(5+ 1)2(1 + )2 ,

[yz]p[ss] =

(5+ 1)(1 + ) .

Without surprise, the relative profit ratio for the naive against the average profit of the

coalition turns out to be

gx[yz]p[ss] =

xp[ss]

1/2[yz]p[ss]

1 = 42(5+ 1)(1 + ) > 0.

This situation is illustrated in Fig. 2(b).

The coalition does not help!

5. A generic theorem

No matter what strategies the more sophisticated take, either cooperative or non-

cooperative, the naive always gain a relative higher profit. Is this paradoxical phenomenongeneric in the sense that it prevails not just for the linear model but for all oligopoly games

that satisfy the basic assumptions specified in the standard model? If the answer is Yes in


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general, does there exist any strategy (cooperative or non-cooperative) that takes a different

output yet superior to the naive? The following theorem provides definite answers.5

Theorem 1. For the standard model, regardless of whatever strategy might be taken by its

rivals, a firm with naive strategy (a price-taker) would perform better than its rivals in the

sense that a higher profit would be obtained at any viable equilibrium, should the latter

exist and be reached.

Proof. Let firm X be the one that takes the naive strategy so that its output is determined

according to the identity

D(xt1 + yt1 + zt1) = C(xt).

This implies that, at an equilibrium (x, y, z), the market price always equals the marginalcost of firm X: D(x + y + z) = C(x).

Now assume that y x. Then the differences between the profits made by firms X andY is given by

x y = (x y)D(x + y + z) (C(x) C(y))= (x y)C(x) (C(x) C(y)).

It follows from the assumption of C(q) > 0 for all q > 0 that

(x y)C(x) (C(x) C(y)) 0, x, y

or equivalently,

x y, (21)

where the strict inequality holds if and only if x y.The facts that firm Ys strategy is not explicitly specified and that the inequality (21)

holds for any output of firm Z imply the conclusion.

The only explicit assumption made for market structure is the strict convexity for thecost function, which excludes the case of linear cost function. Otherwise, the naive strategy

becomes economically meaningless due to constancy of the marginal costs. As long as

the marginal cost is not constant everywhere, the requirement of strict convexity can be

weakened to convexity.

It is understood in Theorem 1 that when a firm takes a strategy different from the naive

it will produce a different output at an equilibrium. Otherwise, if firm X takes the naive

strategy and firm Y takes an imitation strategy to imitate each move of firm X (either

historical or concurrent moves), i.e., yt = xts, s = 0, 1, 2, . . . , then at an equilibriumfirms X and Y actually achieve the same profits due to the identical cost functions. If such

5 Parts of conclusions in Theorem 1 was presented in Fourth Annual International Business and Economics

Conference held at St. Norbers College in October 2001 (see Huang, 2002 for details).


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strategies are not excluded from comparison, the word better than should be replaced by

no worse than.6

What needs to be emphasized is that the conclusion in Theorem 1 is valid only if a

viable equilibrium is converged. Whether such an equilibrium exists or not, whether it isunique, and whether the convergence can be achieved depends on other firms strategies

and the market structurethe form of demand function and the cost function. The detailed

discussion of this topic is beyond the scope of current research.

The results ofTheorem 1 apply to the oligopolistic markets with more than three firms

where part of the firms behave as price-takers and the rest adopt more sophisticated strate-

gies. The price-takers always perform better than the others, regardless of the distributional

ratio (the percentage of price-takers from total firms). Therefore, if starting with all firms

being Cournotors, there exist incentives for any firm to downgrade to a naive price-taker.

The converse, however, can never be true, i.e., if all firms are currently price-takers, taking

any other strategy by any firm only reduces its profit and benefit the others.

In conclusion, being smarter may not perform better, being naive, however, always do

better.

6. Becoming smarter may not pay!

The optimal tactic against changeable enemy is sticking to your original strategy!

Another Chinese philosophical quotation

It has so far been shown that a price-taker succeeds as a champion among the three

competitors in terms of relative profits, regardless whether its rivals form collusion or not.

As a matter of fact, in the course of updating from a naive to a sophisticated, finally to a

smartstrategy, a firm may not only lose its relative profit advantage status against the others

but also reduces its profit in absolute terms.

This point canbe seen through a particular strategical evolution: (P, P, P) (P, C, P) (P, C, C) (P, [CC]) with the linear oligopoly models applied in the previous sections.

Starting with three price-takers (P, P, P), if firm Y suddenly becomes sophisticated to

adopt Cournot reaction function, the profit gained from changing its strategy is measured

with the improvement ratio defined by

Iyppppcp =

ypcp

y

ppp

1 = (1 + 2 42)2

(4+ 22 + 1)2 0 if 1 = 0.8090,

where y

ppp and y

pcp are from (11) and (13), respectively.

For the two stubborn who remain with the naive strategies, it follows from (11) and (13)

that

Iqppppcp =

qpcp

q

ppp

1 = (2 + 8+ 52)2

(4+

22

+1)2

> 0 for q {x, z}.

6 A relative profit maximizer could turn out producing at the same level as the naiver (see Huang, 2002 for

details).


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What a surprise! A naive firm upgrades to the sophisticatedlevel always help its rivals to

profit more. To the contrary, whether the smarter will gain more profit or not yet depends

on the stability index . If the market is relatively unstable ( > 1), its profit actually

decreases.Conversely, starting with (P, C, P), if > 1, by downgrading to the naive, firm Y

improves its profit and simultaneously punish the other rivals.

What happens if firm Z tries to imitate firm Y?

From (13) and (14), firm Zs improvement is measured by

Izpcppcc =zpcc

zpcp 1 = (1 + 4+

2 23 4)2(1 + 4+ 2)2(1 + )2 0 if 2 = 1.3486.

On the other hand

Ixpcppcc =xpcc

xpcp 1 = (2 + 8+ 3

2)2

(1 + 4+ 2)2 > 0,

Iypcppcc =

ypcc

y

pcp

1 = (2 + 8+ 32)2

(1 + 4+ 2)2 > 0.

It is surprising to see that, firm Z upgrading to the sophisticated strategy will benefit firms

X and Y with same improvement ratio, but improves itself only when < 2. Conversely,

starting with (P, C, C), if the market is relatively unstable ( > 2), by downgrading to

the naive, firm Z benefits from a profit gaining and retaliates its rivals simultaneously. Why

not?

Since both firms Y and Z possess the information set, they are enticed to form a coalition

to improve their profits (in absolute terms). Will it work? From (14) and (18) for q = y, z

Iq

pccp[cc] =

qp[cc]

pccq 1 = (1 + 2 4

3 112)2(5+ 1 + 22)2(1 + 2) 0 if 3 = 0.3740.

The attempt is not in vain only when is in a very limited range ( < 3). When > 3,

the collusion is unstable. However, for the lonely price-taker, it works!

Ixpccp[cc] = xp[cc]

xpcc 1 = 4(82 + 6+ 1 + 23)2

(5+ 1 + 22)2(1 + )2 > 0.

These counter-intuitive results are illustrated in Fig. 3.

The above observations deserve special attention since, if > max{i}3i=1 = 1.3486,from (P, P, P) to (P, C, P) to (P, C, C) to (P, [CC]), any firm that becomes more intelligent

than before always loses profit, while the rest, who wait and see, consistently enjoy a posi-

tively profit growth. The economical implication is, when a market is relative changeable

and unstable (in the sense of higher), keeping a low profile so as to stick to your original

strategy (not necessarily the price-taking strategy) proves to be the most effective strategy.

This unusual phenomenon, however, is not an exceptional incident resulting from the lin-earity assumption, but a generic result under rather broader class of economies that satisfy

the conventional marginal revenue assumption, as to be shown in the following theorem.


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Fig. 3. Evolution of(P, P, P) (P, C, P) (P, C, C) (P, [CC]).

Theorem 2. For the standard model, if the demand is twice differentiable and the marginal

revenue is not flatter than the demand function at any viable equilibrium, i.e.:

D(x + y + z) + qD(x + y + z) 0 for q {x,y,z} (22)by remaining as a price-taker, firm X definitely enjoys a constantly profit growth while its

rivals evolve their strategies from (P, P, P) to (P, C, P) to (P, C, C).

Proof. See Appendix A.

The economic interpretation of(22) is that the marginal revenues to each firm declines

at least as fast as the price (as can be seen by adding D to both sides). This is a classical


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assumption ensuring the convergence of a traditional oligopoly market to a competitive

market if the strict inequality is demanded. Its reasonableness was justified, among others,

by Frank (1963) and Ruffin (1971).

Note that, when two smarter firms form coalition and take Cournot strategy with a naiveexpectation xet = xt1, their expected marginal revenue is given by

M Re(xet, wt)|wt=yt+zt =d

dwtwtD(x

et + wt) = D(xet + wt) + wtD(xet + wt).

Thus, d/dwtM Re(xet, wt) D(xet, wt) for wt = yt + zt would imply

D(x + w) + wD(x + w) 0or equivalently

D(x + 2u) + 2uD(x + 2u) 0 (23)at a viable equilibrium y = z = u.

Condition (23) is a generalization of(22) to the collusion, which stresses that the marginal

revenue for the collusion must decline at least as fast of price. This condition leads to the

following conclusion.

Theorem 3. For the standard model, firm X would enjoy a profit growth while its rivals

evolve their strategies from (P, C, C) to (P, [CC]), should the demand be twice differen-

tiable and satisfy the condition (23).

Proof. See Appendix B.

It needs to emphasize that a condition stronger than (23) would be D 0, which includesall linear and concave demand functions. On the other hand, the condition (23) is weaker

than the condition of D(q) + qD(q) 0 so that it can be satisfied by a much broaderclass of convex demand functions. For instance, for the standard CobbDouglas demand

function D(q) = q, > 0:

D(x+

2u)

+2uD(x

+2u)

=

(1 )u + x

(x + 2u)(

+2)

,

which would be less than zero for all x and u if 1.In this regard, the conclusion drawn in Theorem 3 is generic.

The counter-conventional lazybones benefit paradox suggested in Theorems 2 and 3

affirms the old Chinese proverb quoted at the beginning of this section.

7. Concluding remarks

In summary, we have shown that simple, the naive price-taking is a superior rather than

inferior strategy in an oligopolistic economy.Throughout the analysis all firms are assumed to be identical so to strip away the

non-essentials. It follows from the continuity assumptions on both the market demand and


16/19


cost function that the same conclusion applies even when the firm adopting price-taking

strategy incurs a relatively higher production cost (less-advanced technology) than its rivals.

This is indeed shown to be true in Huang (1999b).

Like most literatures referred in this study, it is the relative profit advantage for the naiveover the sophisticated that has been emphasized so far. But it can be verified that, if all

three firms behave sophisticatedly, the profit made by the previous naive firm increases

in absolute term, i.e., xccc > x

pcc, which often arouse the suspicion about the rationality

of behaving Walrasian. However, for a generalized standard model with m (m 3) firmsproducing a homogeneous product, there do exist situations in which an individual firm can

achieve the dual goal of maximizing the absolute profit and relative profit simultaneously

by changing from Cournot to price-taking. This is especially true when the number of

firms m is large (see Huang (1999b) for details).

Finally, ignorance leading to higher profit is also a generic phenomenon in oligopsony

model (Huang, 2002).

Acknowledgements

I am especially grateful to Richard H. Day for his continuous encouragement and con-

structive comments at various stages of this research. Special thanks are due to Grant A.

Taylor and Mark Pingle for helpful discussions. The usual disclaimer applies.

Appendix A

Proof of Theorem 2. If firm Y takes the simple strategy and the sophisticated strategy, its

reaction functions can be expressed as

D(xt1 + yt1 + zt1) = C(yt) (A.1)

and

D(xt

1

+yt

+zt

1)

+ytD

(xt

1

+yt

+zt

1)

=C(yt), (A.2)

respectively. By introducing a constant , 0 1, (A.1) and (A.2) can be unified into

D(xt1 + yt + (1 )yt1 + zt1)+ytD(xt1 + yt + (1 )yt1 + zt1) = C(yt)

with = 0 indicating the naive and = 1 the sophisticated.Similarly, firm Zs reaction curve can be expressed as

D(xt

1

+yt

1

+zt

+(1

)zt

1)

+ztD(xt1 + yt1 + zt + (1 )zt1) = C(zt)

with = 0 indicating the naive and = 1 the sophisticated.


17/19


Since firm Xs reaction remains as

D(xt1 + yt1 + zt1) = C(xt)

from (C, C, C) to (C, F, C) to (C, F, F), the equilibrium conditions can be unified into

D(x + y + z) = C(x), (A.3)

D(x + y + z) + yD(x + y + z) = C(y), (A.4)

D(x + y + z) + zD(x + y + z) = C(z), (A.5)

where x, y and z are continuous functions of parameters and .

And from (A.3), it follows

D(x + y + z) x

+ y

+ z

= C(x) x

. (A.6)

Since the equilibrium profit of firm X given by

x = xD(x + y + z) C(x)

is a continuous function of and as well, we can evaluate its derivatives as

x

= (D(x + y + z) C(x)) x

+ xD(x + y + z)

x

+ y

+ z

. (A.7)

After substituting (A.6) into (A.7) and applying the identity of D(x + y + z) = C(x),(A.7) yields

x

= xC(x) x

.

By (A.3) and (A.4), we have

C(x) C(y) = yD(x + y + z),

which implies x/ > 0 when

=0 (due to the assumption of C

> 0 and

x

= y).

We proceed to show that the monotonicity of x/ holds for all 0 < 0 1, if thecondition (22) is satisfied.

Taking derivatives with respect to over the both sides of (A.3) and (A.4) gives us the

following identities:x

+ y

+ z

D = C(x) x

,

x

+ y

+ z

D + yD + y

D + y

x

+ y

+ z

D = C(y) y

,

x

+ y

+ z

D + z

D + z

x

+ y

+ z

D = C(z) z


18/19


or equivalently

A

x

y

z

=

0

yD

0

,

where

A = D C(x) D D

D + yD D + yD + D C(y) D + yDD + zD D + zD D + zD + D C(z)

.Therefore, we have

x

= |Ax||A| ,

where |Ax| = y(D)2[D C(z)] and

|A

| =(D

C(

x))

()

(D

C(y))

()

(D

C(z))

()

C(x)((D + yD) ()

(D C(z)) ()

+ (D + zD) ()

(D C(y)) ()

.

By the assumptions that D 0 and D+ qD 0, for q = x,y,z, we have D+yD 0for 0 < 1 and D + zD 0 for 0 < 1, which suggests that

|A| < 0 and |Ax| < 0.Since the above inequalities are independent of the value of, we have actually concluded

that x/ > 0 for all 0 1 and 0 1, i.e., x is a monotonically increasingfunction of for 0 1.

Likewise, it can be established that x is a monotonically increasing function of for

0 < 1, i.e., xpcc > xpcp.

Appendix B

Proof of Theorem 3. At the equilibrium of (P, C, C), the symmetricity of the economyimplies y = z = u with = = 1 in (A.3)(A.5).

D(x + 2u) = C(x), D(x + 2u) + uD(x + 2u) = C(u), (B.1)


19/19


while the equilibrium conditions for (P, [CC]) are given by

D(x + 2u) = C(x), D(x + 2u) + 2uD(x + 2u) = C(u). (B.2)

Again (B.1) and (B.2) can be unified into

D(x + 2u) = C(x), D(x + 2u) + uD(x + 2u) = C(u) (B.3)with 1 2.

After taking derivatives with respect to over both the sides of (B.3) and rearranging,

we obtain

D C(x) 2D

D

+

uD 2D

+2

uD

+D

C(

u)

x

u

=

0

uD

,

which gives

x

= 2u(D

)2

(D C(x)) ()

(D C(u)) () 2C(x)(D + uD) ().

The condition D + 2uD 0 guarantees thatx

= xC x

> 0 for 1 2,

i.e., xp[cc] > x

pcc.

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Frank, C.R., 1965. Entry in a Cournot market. Review of Economic Studies 32, 245250.Huang, W., 1999a. Why Cournot? Working paper. Nanyang Business School, Singapore.Huang, W., 1999b. It is the weak who dominates. Working paper. Nanyang Business School, Singapore.Huang, W., 2002. On the incentive for price-taking behavior. Management Decision, in press.Lundgren, C., 1996. Using relative profit incentives to prevent collusion. Review of Industrial Organization 11

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