endogenous objective function in oligopoly

Version: September 23rd, 2010

1

ENDOGENOUS OBJECTIVE FUNCTION IN OLIGOPOLY

By

Juan Ignacio Guzmán1

Abstract

This paper presents a two-stage delegation game in the context of a non-cooperative

oligopoly. In the first stage, each firm’s principal (owner) competes with each other in

order to maximize profits for their firms, deciding the objective function that their own

agents (managers) should maximize in the second stage. In the second stage, agents

simultaneously select competitive moves to maximize the objective function before

assigned. I show the existence of infinite objective functions that are part of a subgame-

perfect Nash equilibrium, such as the competitive moves resulting are necessarily on the

firms’ contract curve (i.e. the cooperative solution). This result might reposition once

more this curve as the non-cooperative solution to the oligopoly problem, which in

recent decades has been displaced by Pareto inferior solutions such as Cournot,

Bertrand and Stackelberg.

JEL classification: L13, L21, L22

Key Words: Delegation game, Objective function, Oligopoly, Profit maximization

1 College of Engineering and Applied Sciences, Universidad de los Andes. Email: [email protected]. Address: Av. San Carlos de Apoquindo 2200, Las Condes, Santiago, Chile. Tel.: +(56 2) 412-9879. Fax: +( 56 2) 412-9642. I would like to thank Juan-Pablo Montero for his valuable comments on an earlier version of this paper, and seminar participants at the 2010 Annual Meeting of the Chilean Economics Association (SECHI).


2

1. Introduction

The problem of selecting competitive moves or tactics through a process of

profit maximization has been recognized for decades as unrealistic to explain the

behavior of firms in competition (Baumol (1958)). Thus, the management literature

related to competitive strategy recognizes the formation of competitive tactics in the

firms, such as price or quantity, often born of a complex process that begins with the

formation of a competitive strategy (Hax and Majluf (1991), Porter (1980)). In this vein,

perhaps the main factor of a competitive strategy is to identify a goal or objective

function to guide the process of maximizing the profits of the firm.

Therefore, is not surprising that one of the main criticisms of the modeling of

oligopoly as a valid methodology to elucidate the problem facing firms competing in a

market relates specifically to the objective function that pursue them. This critique

argues that, although the objective function of firms will match their profits if they are

rational, in reality – and due to internal friction or bounded rationality –firms must

comply with a different objective function (Cyert and Hedrick (1972)).

In this line, since the 1950s, managerial theory has proposed objective functions

other than profits to recover more realistically the complexity within firms. Cyert and

March (1956) and Baumol (1958) can be considered the seminal works on this topic.2

Maybe one of the most notable developments in this regard relates to the sales

maximization hypothesis (Baumol (1958)).3

Many reasons have been given as valid for the use of an objective function other

than profits. In particular, perhaps the most promising models are those that justify a

correction in the profits (as an objective function) due to strategic delegation between

the principal (or owner) of the firm and the agent (or manager) that runs the firm,

referred to as delegation models.4 In this line of research, Vickers (1985), Fershtman

and Judd (1987) and Sklivas (1987) have proposed as objective functions linear

combinations between profits and sales. The argument behind this is that while rational

firms (represented by the principal) have profits as the objective function, the agents

2 For a review of the literature on the use of objective functions other than profits see Kaneda and Matsui (2003). 3 I could see the importance of this hypothesis in practice, since before getting my current position at the academy a couple of years I worked in a consulting firm with the unique strategic focus of maximizing sales. 4 Another class of models that assume different objective functions is called labor managed firm (Dreeze (1976), Meade (1972)), which is based on maximizing the welfare of employees as the strategic focus of the firm.


3

seek their own utility – which is generally linked to sales or production of the firm

(through an incentive contract).

Despite the significant growth of literature in this vein, the proposal to replace

the maximization of profits by maximizing a linear combination of profits and sales has

not gained greater acceptance among economists. In my view, despite the apparent

attractiveness of an a priori consideration of a more complex objective function, and

therefore more realistic for some, there are two solid arguments that ultimately make

profits to be preferred as the objective function of the firm.

The first has been popularized by Friedman (1953) and is related to the

evolutionary argument in which firms do not only learn how to make things better, and

in regime they end up maximizing profits, but more importantly, if they do not do it

well in the long run, they end up disappearing. Then, under this argument, it seems

reasonable to expect that firms will behave at least as if they were maximizing profits.5

The second argument against considering a different objective function to profits

is no less strong than the first. This lies in the arbitrariness of the objective functions

used to derive the firm´s competitive tactics.6 In fact, an examination within firms

would reveal that the objective function sought by agents usually contains additional

factors to profits and sales of the firm. Thus, if we use the argument given by

proponents of more complex objective functions, then all relevant management

variables should be present in the objective function of a firm, and not just a couple of

them.

Both arguments might seem sufficient to consider profits as the objective

function of rational firms. However, in my opinion, the debate between those who

blindly support the premise that firms maximize profits versus those that refer to more

complex objective functions does not yet appear to be closed.

In line with the above, this paper seeks to contribute to the discussion by closing

up the two positions, while maintaining the assumption of rationality of the firm

(principals) as well as those who make decisions on it (agents). Although this may seem

5 Despite the reasonableness of Friedman's argument at first glance, Dutta and Radner (1999) have shown that profit maximization is not a prerequisite for survival under perfect competition, but certainly is a strategy that ends in the bankruptcy of the firm (while other strategies less "rational" can ensure its survival). 6 Although this might not seem the case when using linear combinations between profits and sales, the fact is that the functional form between this two variables could be infinitely more varied (indeed, any nonlinear function is potentially feasible). Moreover, it is not even clear why the objective function is not complemented with other relevant management indicators, such as market share, cost or sales growth.


4

contradictory at first, this is not necessarily the case as we dissect more deeply the

problem that firms with a principal-agent organization solve in an imperfect market.

I think the apparent contradiction between the neoclassical paradigm of

rationality and the objective function of firms in an oligopoly context can be explained

by differencing between profits (or payoffs in game theory jargon) and objective

function,7 or equivalently between competitive strategy and tactics.

Regarding the conceptual distinction between profit and objective function, it is

possible to note that while the first should be the ultimate goal of the firms in the

markets, the second responds to the guidance of the organization on its way to the

pursuit of this ultimate goal (Simon (1964), Williamson (1964)).

Despite this clear distinction between both concepts, revising the orthodox

theory of the firm, it appears that the objective function is a redundant concept. In fact,

according to Friedman's followers, the only necessary concept to emulate the behavior

of firms in imperfect markets is profits. Thus, the objective function as such has been

removed from most economic theory (and certainly from industrial organization), but

this has not meant that is not cause for serious study in other disciplines. In fact, the

management literature has been more concentrated in the objective function than in the

profits themselves, recognizing in the former the most important control variable that

managers have to guide the organization toward the maximization of the latter.8 This

gives way to argue the existence of a causal relationship between objective function and

profits, the latter being a result of the first.9

On the other hand, the distinction between competitive strategy and tactics (or

moves) is more ambiguous in industrial organization, in particular since the

indiscriminate use of the word strategy in game theory.10 It should be noted that

hereafter the term "competitive" will be ignored deliberately because it is the focus of

industrial organization, and therefore might seem redundant for this discipline.

7 The objective function is commonly called “goal” in the management literature. I preferred to use the concept of objective function in place to enhance the very fact that it can be expressed as a mathematical function (at least at the conceptual level). 8 It is no coincidence that economics enhances the concept of profits, while management enhances the objective function or goal, as the focus of both disciplines are different. Yet it seems obvious that there are areas where both disciplines should gain from intersection. 9 Although it is possible to imagine instances where profits may affect the objective function, this direction of causality is beyond the scope of this paper. 10 In game theory, the concepts of moves, actions, tactics and strategies are often used indistinctly to denote the decisions available to players (or firms). In fact, as Osborne (1971) acknowledges, "most discussions found under the heading of “strategies” are really about moves" (p.539).


5

However, it should be noted the existence of other dimensions where strategy and

tactics are used, which usually concern the field of management.11

Conceptually, the strategy defines the way in which the firm will compete in the

market, being a prelude to the competition itself. In contrast, tactics must be understood

as the competition itself. Thus, in a duopoly market of a homogeneous good where

firms compete in quantities, the tactic should be understood as the move (i.e., selection

of quantity) of the firm in that market. The strategy, on the other hand, refers then to

how firms determine their decisions when their time comes to move. Hence the strategy

can be understood as equivalent to the pursuit of an objective function to guide the firm

in order to compete in the market.12

Just as industrial organization assumes that the moves of a firm in oligopoly

must be derived endogenously, the strategy should also be seen as the result of a

(strategic) interaction anticipated by the firm, and therefore should not be considered

exogenous. Nevertheless, despite the logic of this argument, the exogeneity imposed on

the strategy (assuming that the objective function of a rational firm is that given by

profits) is an almost undisputed paradigm in industrial organization. In fact, I only know

one previous paper that has drawn attention about the confusion in the game theory

literature (and hence in industrial organization) regarding the selection of moves

(tactics) versus strategies (Osborne (1971)).

In light of the results presented in this paper, the conceptual distinction between

profit and objective function as well as between competitive strategy and tactics goes, in

my opinion, to the root of the oligopoly problem. In fact, it appears that this separation –

while not necessary in perfectly competitive markets or monopoly –13 should be the

basis of the oligopoly game in real markets.

With the above distinctions made, the essence of my formulation is to relate the

strategy to the objective function that guide the firm to maximize its profits while the

tactic is associated with moves that are necessary to maximize the firm’s objective

function. Unlike the industrial organization literature to date, in this paper the objective

function that agents will maximize (competing with each other) is endogenous to

11 Examples of these are human resources and corporate social responsibility. 12 Strictly speaking, the strategy not only determines an objective function, but also a set of restrictions. I explicitly omitted these restrictions on the grounds that its exposure contributes little to the development of the theory exposed, complicating the analysis unnecessarily. 13 While it is trivial to show this in the monopoly case, in the case of perfect competition it is possible to justify the use of the objective function equivalent to the profits of firm when using the assumption of free entry.


6

competition. In my opinion, this should be a better representation of how firms compete

in real markets, besides providing a solution to the dilemma between the advocates of

the profit maximization hypothesis and those supporters of more complex objective

functions.

It should be noted that profit maximization remains in my model as the ultimate

goal of firms, and in this sense the theory derived in this paper is embedded in the

neoclassical concept of rationality. That is, unlike the literature of delegation games or

other games involving firms’ bounded rationality in this paper, I assume that the utility

of agents is perfectly aligned with the interests of the principal.14 In fact, I show that it is

precisely the rationality assumption of firms (represented by both principals and agents)

that justifies, in the case of oligopoly, the conceptual separation of profits and objective

functions, or equivalently between tactics and strategies.

In addition to the strategic delegation literature (Vickers (1985), Fershtman and

Judd (1987), Sklivas (1987)), two papers are close to this paper. The first one is

Osborne (1971), who discussed various competitive strategies in a duopoly model in

quantities.15 This author does not discuss explicitly how profit maximization of firms

might not be the only objective function they have, lacking an internal view at the firm

level that I have corrected in this paper by including explicitly a principal-agent

structure. In addition, Osborne (1971) examines a small number of possible strategies,

while in this paper I extend that analysis to an infinite set of objective functions. The

other paper is Kaneda and Matsui (2003), who show that when firms maximize a broad

class of objective functions other than the profits they can eventually gain over what

they had obtained in case of using profits as the objective function.16 Although these

authors separate conceptually the firm's objective function from profits, in my view,

they do not offer a convincing explanation of why that separation must be done. In

addition, their solution suffers from the arbitrariness of the chosen objective function (a

convex combination between profits and other management factors such as sales),

which as it will be shown in this paper cannot be entirely arbitrary but must satisfy a

well defined maximality criterion.

14 This is accomplished by designing a contract for the agent that incentive him to maximize the objective function pursued by the principal. 15 Osborne (1974) extends this result to the case of price competition. 16 In line with these results Alchian (1950) states that realized profits, not profit maximization, is the key of success and viability of a firm.


7

This paper is organized as follows. Section 2 provides the conceptual distinction

between the concepts of profits, objective functions, strategy and tactics in an oligopoly

market and how they relate to profit maximization as described in the orthodox

oligopoly theory. It discusses the conditions required for such a separation to occur as

an endogenous phenomenon of competition among firms. This section concludes that

any subgame-perfect Nash equilibrium leads to the non-cooperative oligopoly solution

to be on the contract curve, which had been anticipated more than eighty years ago by

Chamberlin (1929). Section 3 develops the proposed solution for the very well known

duopoly game of competition in quantities (à la Cournot), linear demand function and

constant and symmetrical marginal costs among firms. Section 4 discusses how the

main result of the paper may help explain some remarkable experiments in duopolies, in

particular by Selten, Mitzkewitz, and Uhlich (1997). I also discuss briefly some

theoretical implications regarding the orthodox oligopoly theory based on the results.

Concluding remarks follow.

2. Endogenous determination of the objective function

The behavior of a firm in the market should not deviate from the desire of the

owner or principal, nor the leadership and capacity of manager or agent who leads it.

With the principal-agent decomposition as a starting point then consider a

market in which there are � firms competing with each other so that the ith firm must

select, simultaneously to other firms, a control vector �� , ��

�, … , ��

��, which consists of decision variables � that directly or indirectly affect the profit of its competitors (such as quantity, price, capacity investment, level of advertising or any

other).17 Assume further that the relevant market is defined by a set of parameters, given

by the vector � ��, ��, … , �� ,18 and the ith firm is defined in turn by an a priori

feature vector that can be represented by �� , ��

�, … , ��

��, where �� is the number of

identification parameters of the ith firm.19, 20

17 I wanted to consider the general case where the firm chooses a vector of moves to demonstrate the robustness of the results, however, much of the literature in industrial organization focuses on models searching for a single (scalar) variable such as price or quantity. Section II develops an example where it is used instead of the move vector a scalar variable equal to the quantity produced by each firm. 18 For example, in the market for a commodity whose inverse demand function is given by ��

��, the vector in question is � ��, ��. 19 In the classical oligopoly theory this vector contains the parameters that describe the cost function of firms.


8

Figure 1 illustrates the process by which the ith firm chooses its strategy and

tactics by selecting an objective function and move, and how these are cleared in the

market to generate profits. As illustrated in this figure, the overall process is separated

into four stages.

In the first stage, the ith firm will decide its strategy through the establishment of

an objective function that the agent should maximize. This function, displayed for the

ith firm by an objective function given by ��, �� , ��, ��, �� , where �� y ��

represent the vectors of characteristics and decision variables of other firms in the

market, respectively. According to the orthodox oligopoly theory, the objective function

is identical to the payoff function, which can be represented conceptually for the ith

firm as ��, �� , ��, �� . In the second stage, the ith firm will choose its move, ��. Internally, this

selection is equivalent to maximizing the objective function imposed by the principal,

and is precisely the agent of the firm who shall ensure that its resources are aligned in

the desired direction. As a result of the maximization process (and assuming the

uniqueness of the solution) the move can be represented as an equilibrium ��.

Then in the third stage, firms compete in the market with their selected moves.

The dependence of the chosen objective function, in addition to the payoff function with

respect to the decision variables of its competitors is what makes the ith firm considers

the strategic interaction to derive both its objective function and a move to compete in

the market.

Finally, as a result of competition, the fourth stage represents the realized profits

of the ith firm that result from having used the chosen objective function and moves, as

well as dependence on the choices made by other market firms. At this stage, realized

profits ��, �� , ��, ��

� correspond to those that ultimately receive the ith firm (or

equivalently its principal).

In line with this conceptual separation of the competitive process in oligopoly,

we can say that the theory of oligopoly in industrial organization has failed to recognize

to date the first stage. That is, the discipline has dismissed the study of strategy

formation (as has been defined in this paper), focusing exclusively on the equilibrium

moves. This criticism was made for the first and only time – at least to my knowledge –

20 Take note that the dimension of vectors �� and �� will not necessarily be the same among different firms in the industry.


9

by Osborne (1971). Nevertheless, although nearly four decades have elapsed since then,

it has not been internalized or even discussed in the literature.

Before defining the game that derives endogenously the objective function used

by a firm to compete in the market, it seems necessary to redefine the concept of

strategic interaction widely used in game theory. In this discipline (and by extension in

industrial organization) strategic interaction arises by the dependence on the players’

payoff functions with respect to actions taken by their competitors. Thus, from the

perspective of the competitive process shown in Figure 1, it seems necessary to

differentiate between strategic and tactical interaction. The former relates to the

dependence of the principal’s objective functions with respect to its competitors’. The

latter corresponds to the dependence of the objective function of each agent on the

moves of its competitors.

2.1 Formulation of the game

Using the process described in Figure 1, it is possible to state the problem facing

each firm in equilibrium. At this point, I decided to maintain the assumption of

rationality of the firms in the market. Thus, the results derived are still compatible with

those given by orthodox oligopoly theory.

Although the previously discussed process consists of four stages, the last 2 lack

any decision from the players, so it is possible to model the principal-agent problem as a

dynamic game with only two stages.

Thus, in the first stage, the principal of the ith firm decides simultaneously with

the rest of its competitors (i.e., other principals) the strategy of the firm through the

imposition of an objective function for the agent. Since rationality is assumed for all

players, the principal’s competitive strategy will maximize the profits earned by the

firm. Theoretically, one can think of this first stage as a strategic stage in which the ith

firm must solve the following problem:21

max ��, �� , ��, �� , ��, �� , (1)

where �� , �� , ��, ��, �� . Notice that in anticipation of the second stage of the game, each principal expects that moves of his agent and competitors will depend on the

established objective functions (i.e., ��, �� and ��, �� ). 21 The problem of the principal should be restricted to the profits of the firm being non-negative, since otherwise the rational move for the firm will be not to participate in the market. As in much of the literature of industrial organization this restriction has been omitted in the interest of clarity of exposition.


10

In the second stage, and observing the objective function selected in the first

stage, the agents will compete with each other by means of establishing the moves that

maximize the objective function assigned by the principal. Theoretically, we can think

of this second stage as a tactical stage in which the ith firm must solve the following

problem:22

max � ��, ��, ��, ��, �� (2)

It should be noted that this problem differs from the orthodox formulation of the

oligopoly problem23 in mainly two ways. First, the firm must choose not only its vector

of moves, but also its objective function. Second, while in the orthodox oligopoly

theory the players are represented by the firm, in this paper (as in most delegation

games) there are two different players at each stage: the principal at the strategic stage

and the agent at the tactical stage. An additional assumption, required to generate the

necessary credibility for the results that follow, is that at the second stage there is no

possibility for the principal to interfere with the agent’s decision. This implies that the

only interference of the principal comes precisely through the imposition of the

objective function in the first stage.

Finally, a key assumption of the problem is the perfect observability that agents

have with respect to their objective functions selected in the first stage of the game,

before committing their move vectors in the second stage. While this may be a

debatable issue, the fact is that there are various arguments for justifying the knowledge

of the objective function by competitors (see for example, Fershtman and Judd

(1987)).24

2.2 Solution

As is standard in dynamic games with a finite number of periods, the solution is

derived by backward induction. Thus, in the second stage of the game, the equilibrium

move vector results from solving the problem (2) for each firm. Assuming that the

22 Strictly speaking, the problem of the agent should be restricted to feasible moves. For example, the price or quantity they decide must not be negative. Again, this restriction is omitted in the interest of clarity of exposition. 23 Whose formulation is simply given by max �� , �� , ��, �� (subject to ��, �� , �� , �� 0 and

�� ). 24 One could even think that part of the firm strategy ought to be precisely to communicate their objective function to the rest of the competitors.


11

objective function is smooth enough as to make use of calculus techniques,25 the

solution to (2) will satisfy:

��, �� , ��, ��

�, �� , (3)

where ��

�� , �

�� , … �

��

�� is the gradient operator. Hence, it is possible to derive a

vector function as the best response of the ith firm, which is �� satisfying (3), from

which it follows that:26

��

��, ��, ��, �� (4)

Note that this vector of best response functions depends not only on the

parameter vectors , �� and ��, or moves chosen by other firms ��, but also its

functional structure depends on the objective function selected by the ith firm, ��. To

emphasize this dependence a superscript has been used. Hence, the Nash equilibrium of

the second stage will be derived intersecting the best response vector functions for the �

firms, and may be denoted by the move vector �� , ��

� , … , � � . It is fundamental to note

that this intersection will depend on the best response curves, which derive their shape

from objective functions selected by the principals in the first stage of the game.27

Anticipating the equilibrium of the second stage, in the first stage of the game

the principals will compete with each other by selecting the objective function that will

maximize the agents in the second stage. However, the problem defined in (1) is not

trivial to solve by traditional methods of calculus, even assuming that the objective and

profits functions are smooth enough. This is because the unknown variable is a function,

not a vector. In fact, the author does not know a technique to solve such problems for

the general case presented in (1), either using calculus of variations or any related

method. Fortunately, it is still possible to find conditions to be satisfied in equilibrium

without solving directly the problem (1), all that is needed is to find the solution to this

problem. Hence, the given solution can be considered as an indirect derivation to

problem (1).

To solve the problem (1), the principal of the ith firm must anticipate the fact

that once its objective function is revealed to competitors, equilibrium move

25 It suffices that the function is continuously differentiable for vector ��. 26 It has been assumed that conditions of the implicit function theorem are satisfied otherwise is not

possible to derive �� from (3).

27 Hence it is clear that vector �� not only depends on the objective function of the ith firm but also on the

objective functions of other firms, i.e., �� , ��.


12

�� , ��

� , … , � � can be derived intersecting best response functions of all firms resulting

from their objective functions.

DEFINITION: Objective functions ��, ��

�, … , � � are part of a Nash equilibrium of the

described game if, for all � � 1,2, … , � and functions ��, the following is satisfied:

�� , �� , ��

��, �� , ��, �� , ��

�� , �� , ��, �� !

�� , �� , ��

��, ��, ��, �� , �

��

�� , ��, ��, �� (5)

According to this notion of equilibrium, the principals must then select their

objective functions ��, ��, ��, ��, �� so that their best response curves allow them to

obtain the maximum profit since the remaining curves of best responses (derived in turn

from the objective functions of other firms) are fixed, and the move vector equilibrium

�� , ��

� , … , � � is obtained by intersecting these best response curves. Take note that

this notion of equilibrium for objective functions is just the extension of Nash

equilibrium (Nash (1951)) for functional spaces.

To better understand the above definition, let us consider as an example the

problem of a duopoly in which each firm have control over a single move (scalar),

which can be called without loss of generality �� and ��, for firms 1 and 2, respectively.

Figure 2 illustrates the best response curves derived from the objective functions

��, �� . The pair ��, ��

� are (Nash) equilibrium objective functions if, after fixing the

best response curve of the competitor (i.e., maintaining a fixed �� ), the ith firm has no

other best response curve that intersects with that of its competitor at a point other than

��, ��

� and to allow it to achieve in the new intersection a greater value for �� than ��.

In terms of Figure 2, one firm would not increase its profit by moving from a best

response curve given by R�

��, ��, ��, �� to another given by R�

��, ��, ��, �� . The same would happen with firm 2 (moving R�

��, ��, ��, �� to R�

��, ��, ��, �� ). As shown graphically in Figure 2, this is equivalent to any of the firms having no

incentives to intersect the best response function of the other firm at some point other

than ��, ��

� , which in turn is equivalent to the fact that for all ��#, ��# and ��##, ��## it is necessary that ��, ��, ��#, ��# ! ��, ��, ��

�, �� and ��, ��, ��##, ��## !

��, ��, ��, ��

� . Otherwise, if there was a best response curve immediately it will

produce a disequilibrium situation in which the principal could take advantage of


13

changing their own best response curve by changing the objective function selected in

the first stage. From here, we are in the presence of an equilibrium in the first stage if

none of the principals have an incentive to deviate from its selected objective function.

LEMMA 1: A necessary condition for (5) to hold is that the best response curve derived

from the objective function for the ith firm, i.e., $�

��%, &� , &��, '�� , is tangent to the

competitor’s iso-profit curves for each of the components of the vector that brings

together all the possible moves of firms, i.e., ' � �'�, '�, … , ' . Namely:

(� �'�� ) $��

��%, &� , &��, '�� · (��%, &��, '�, '�� 0 , (6)

where '�� ) $��

��%, &� , &��, '�� corresponds to the jth component of vector '� )

$�

��%, &�, &��, '�� . Additionally, the region bounded by the best response curve of the

ith firm must be defined for the space where profits of others firms decrease.

PROOF: If the curve defined by ��

��, �� , ��, �� does not cut tangent to the implicit curve ��, ��, ��, �� (constant), or the cut is tangent but it was defined for the space where the profits of the others firms are increasing then it would

contradict (5). Hence, necessarily the curves ��

��, �� , ��, �� and

��, ��, ��, �� must be tangent at �� , ��

� , … , � � . This is equivalent to the

tangent vector of the first curve being perpendicular to the normal vector of the second

curve, from which we obtain precisely equation (6). Q.E.D.

It should be noted that (6) helps to select potential equilibrium best response

curves, and hence the associated objective functions for each of the firms in the market.

Then, finding the equilibrium objective functions is reduced to finding a set of functions

��, ��

�, … , � � such that (6) is satisfied for all � � 1,2, … , �.

Intuition behind this proof can be recovered if we consider again the case of a

duopoly in which each firm have a single control variable, say �� and ��. Figure 3

shows that, for each point ��#, ��# of the plane defined by the Cartesian product �� , ��, it is possible to draw iso-profit curves for firms 1 and 2 that pass through this

point, which is shown in Figure 3 by solid and dashed curves, respectively. Then, select

a function ��, ��, ��, �� for firm 1 that is tangent to the iso-profit curve of firm 2 at


14

that point (i.e., ��, ��, ��, �� , so it cannot intersect at another point in the plane, and another function ��

��, ��, ��, �� for firm 2 to meet the exact opposite. Thus,

assuming that the firm's profits grow in the opposite direction of best response curves of

the other firm, neither of the two firms have an incentive to change its best response

function (if the other firm remains unchanged), since no new intersection will bring

greater profits than those achievable at point ��#, ��# . Note that if there exists an objective functions that satisfy (6), then there will

exist an infinite number of them.28

COROLLARY 1: From (5) or (6) it can be shown that in general the selection of an

objective function equivalent to the first stage profit function of the ith firm (i.e.,

��%, &� , &��, '�, '�� %, &�, '�, '�� ) is not part of a subgame-perfect Nash

equilibrium unless 1) the iso-profit curves of the remaining firms are perpendicular at

the point of tangency with the best response curves, 2) they intersect only at this point

and 3) the profits of the firms grow against the region bounded by their competitor’s

best response curves.

Corollary 1 is essential because, as discussed in a classic example in Section 3, it

not only invalidates the use of the profit function as the objective function in order to

select moves in the classical models of oligopoly (Cournot and Bertrand), but also

allows to test the rationality behind the use of other objective functions, particularly

those proposed in delegation models. What is remarkable about this result is that it is

precisely the desire to achieve maximum profits for firms what pushes the principals to

put into their agents’ hands an objective function that in general will be different from

profits.

(6) is a necessary condition to be satisfied by objective functions in the

subgame-perfect Nash equilibrium. However, it is not a sufficient condition since for

every point ��, ��, … , � in space is possible to build best response curves to ensure this condition, and at the same time the principal will not select it since it does not

necessarily maximize its firm’s profits. Thus, there is another necessary condition,

which together with the former allows to derive a sufficiency condition. This second

28 In fact, if �� satisfy (6) then any function �� with � and � constants such as � � 0 will also

satisfy (6). Besides these functions, there is an infinite class of other non-linear functions �� that also could be sustained in equilibrium.


15

condition can be derived through precisely finding the subgame-perfect Nash

equilibrium.

LEMMA 2: In this oligopoly game, for the moves of each firm, i.e., '��, to be part of the

subgame-perfect Nash equilibrium '�� must belong to the firms’ contract curve:

'�� - . � /�'�, '�� | ∑ ��%, &�, '�, '��

�� 2 , (7)

where �� is the profit made by a monopolist in the market, which satisfies:

�� 345 �6� ��,�� ∑ ��%, &� , '�, '�� (8)

PROOF: Knowing that the second stage of the game will lead players to an equilibrium

given by intersecting their best response curves, the objective functions imposed by the

principals in the first stage are such that not only they can be sustained in equilibrium,

but they also ensure greater possible profits for firms. As the maximum joint profits

firms can do is just ��, any pair ��, �� , ��, ��, �� and ��

� which together lead to

lower profits (i.e., ∑ ��, �� , ��, ��

� �� 7 ��) always will be possible to improve by

at least one of firms, and thus by contradiction this pair cannot be part of a subgame-

perfect Nash equilibrium. Q.E.D.

PROPOSITION 1: In this oligopoly game, the subgame-perfect Nash equilibrium is

given by any pair of objective functions and moves, i.e., ��%, &� , &��, '�, '�� and '�

�,

satisfying the conditions (6) and (7).

PROOF: This is direct from Lemma 1 and Lemma 2. Q.E.D.

COROLLARY 2: The pair defined by the best response function

$�

��%, &�, &��, '�� =/�'�, '�� |��%, &� , '�, '�� %, &� , '�

�, '�� 2 and �'�

�, '�� -

., is a subgame-perfect Nash equilibrium of the described oligopoly game, from where

is possible to derive the equilibrium objective function ��%, &� , &��, '�, '�� .29

What is remarkable about the result prescribed by Proposition 1 is that, before

the Nash equilibrium concept positioned itself as the solution of the oligopoly problem,

for many of the classical economists associated with the study of imperfect markets, the

29 Section II will show how to recover the function ��

� from the best response function.


16

non-cooperative solution was placed on the contract curve, thus coinciding with the

cooperative solution. In fact, more than eighty years ago Chamberlin (1929) conjectured

this solution to the problem of non-cooperative oligopoly:

“If sellers have regard to their total influence upon price, neglecting no phase of

it, the price will be the monopoly one… Independence of the producers and the pursuit

of their self-interest are not sufficient to lower it.” (p. 92) (italics added)

Chamberlin was part of a Pleiad of classical economists, completed by Robinson

(1933), Lewis (1948), Fellner (1949), Wolfe (1953-1954), Henderson (1954) and Stigler

(1964), who conjectured (but were unable to prove) that the contract curve should be the

solution to the (cooperative and non-cooperative) oligopoly problem. Despite the

reputation of many of them, the solution to the problem ended up positioning itself in

Pareto inferior equilibriums, which have dominated the industrial organization literature

over the past 50 years. In fact, Fisher (1989) summarizes well the existing perception of

economists about Chamberlin's conjecture:

“If it were true that joint maximization is an (non-cooperative) equilibrium only

under well-defined special circumstances, we would have a very useful, strong result. I

do not believe this to be the case, however.” (p. 116) (parenthesis added)

To my knowledge, only Osborne (1974) has obtained a solution on the contract

curve as part of a non-cooperative equilibrium (also in a two-stage game), although for

a particular case. I think my results are indeed a natural extension of Osborne’s.

Another relevant factor regarding the solution refers to the selection of moves

that the principals would like to hold onto the contract curve (i.e., ��, ��

� - C). As is clear in the game shown the principals will anticipate in the first stage their agent’s

reactions by choosing where to position themselves (simultaneously to select an

objective function to sustain that position in equilibrium). That is, the solution here

proposed generates a coordination problem, since the principal of the ith firm, in order

to maximize profits has an incentive to choose the point ��

�, �� , where

��, �� , ��, � � ��. If all the principals choose a best response curve in order to

sustain its monopoly move, in the second stage there would be no intersection of these

curves and thus, no equilibrium of the game.30

Remarkably, the coordination problem that arises is itself endogenous to the

examined oligopoly problem. Although, in such cases, it has been consistently

30 Indeed, this occurs for any set of objective functions whose best response curves do not intersect.


17

suggested as a solution the use of focal points (Schelling (1960)), in itself this

coordination problem deserves more attention. This goes beyond, however, the scope

proposed in this paper. Still, I acknowledged that a possible way to solve this problem is

to use a negotiation mechanism, using for example the concept of Nash bargaining

(Nash, 1953). This negotiation can be modeled by incorporating an additional stage

prior to the selection of the objective functions. Notwithstanding this, I have chosen not

to incorporate this stage explicitly in the formulation of my model, in order to separate

the objective function problem from that of negotiation among firms (although, in fact,

the latter problem is part of the former problem).

Before illustrating the result with an example, one last comment about the

solution exposed relates to the credibility generated by the delegation model used.

Obviously, if in the second stage of the game the player was again the principal, then

the non-cooperative equilibrium could be no other than that in which firms maximize

their own profit functions. The role of commitment in the delegation is thus crucial to

generating the solution here derived.

3. Example: duopoly à la Cournot

In order to show the theoretical implications of the result derived in this paper,

this section uses a classic example in oligopoly theory. This corresponds to the case of a

duopoly competing in a market of a homogeneous good where firms simultaneously

choose their production levels to meet demand in a given period. Suppose the firms’

production can be denoted by 9� and 9� for firms 1 and 2, respectively (i.e., �� 9�

and �� 9�). Once committed to such productions, the price clears the market through

an inverse demand function given by :�; � 6 ) <;, where ; � 9� = 9�, and 6 and < are two positive constants (from here � �6, < ). To further simplify the exposition

assume that both firms have constant marginal production costs equal to � (i.e.,

�� ). As in the previous section, this game is divided into two stages: a strategic stage,

in which the principal of each firm selects the objective function that the agent will

maximize, and a tactical stage, in which both agents must choose (once the objective

functions are observed) the production levels that maximize their own objective

function, taking into consideration the tactical interaction with other agents.


18

The solution of this game when firms select only their moves (in this case its

level of production) using profits as the objective function, is undoubtedly one of the

most known and widely accepted results in oligopoly theory. Notwithstanding this, as

shown below, when firms can chose (through a principal) an objective function with

which to derive their production, the Cournot solution is not longer an equilibrium of

the game since the objective function given by the profits cannot be sustained as part of

a subgame-perfect Nash equilibrium.

Consider then the Nash-Cournot solution (>�) when the objective function that the firms selected in the first stage of the game corresponds to profits (i.e.,

��6, <, �, 9�, 9� � ��6, <, �, 9�, 9� ). In this case the best response curves are given by: 9�

�� ?�

��6, <, �, 9��

�� for � � 1,2 , (9)

which are shown graphically in Figure 4 (with thick continuous straight lines). The

equilibrium move associated with the strategy of maximizing profits is given by the

intersection of these curves, whose product is:

9�� 9�

��

�� (10)

It is not difficult to prove, however, that the selection of the objective function

given by profits (i.e., �� 6 ) <�9� = 9� ) � 9� for i � 1, 2) cannot be part of a subgame-perfect Nash equilibrium. Indeed, as shown in Figure 4, if the principal of the

firm 2 selects this objective function for his agent (which is reflected graphically in the

curve of best response of firm 2, i.e., R��6, <, �, 9� ), the optimal for the principal of

the firm 1 would be to choose an objective function whose best response function

intersects the firm 2’s, where it guarantees itself to be the leading firm in the market

(i.e., firm 1 plays à la Stackelberg), with productions given by �9��, 9�

� � A9��

��

��, 9�

� � ��

��. In fact, there will be infinite best response curves for firm 1 that

intersect at this point to the best response curve of firm 2 and, therefore, infinite

objective functions that guarantee firm 1 to become the Stackelberg leader of the

market.

Obviously, any strategy chosen by firm 1 so that the new intersection is more

beneficial for it is not necessarily part of a subgame-perfect Nash equilibrium, because

if the principal of firm 2 anticipates firm 1’s objective function, it can always select an


19

objective function that allows it to achieve a better outcome, and hence, any attempt of

firm 1 to position itself as the Stackelberg leader will be unsuccessful in equilibrium.31

This implies that for this particular game, the selection of an objective function

that is equivalent to firm’s profits will not be part of a subgame-perfect Nash

equilibrium. This undoubtedly has important implications for oligopoly theory, since

much of the theory rests precisely in the fact that if firms are rational their objective

functions must be profits.

Now, since using profits cannot be part of an equilibrium objective function, the

question that arises is whether other functions considered in the literature of managerial

theory can be.

In this line, perhaps the second most widely used objective function after the

profits is the linear combination between profits and sales (BC . In the examined

example this implies that �� D�6 ) <�9� = 9� ) � 9� = �1 ) D �6 ) <�9� =

9� �9� (with 0 ! D ! 1), for � � 1, 2. Hence, the best response curves are given by: 9

�� ?��6, <, �, 9�� !��

�� for � � 1, 2 (11)

Since the intersection of these curves will reach lower profits than those arising

in the Nash-Cournot equilibrium, the principals may choose a different objective

function to improve their profits, and therefore the linear combination of profits and

sales cannot be an objective function that is derived endogenously by the principals.

This limits the conclusions used up to date in strategic delegation models, as indeed

none of their objective functions satisfy the equilibrium condition given by (6).

If the classical solution of the Nash-Cournot game is not longer an equilibrium,

then what is? Lemma 2 predicts that only those objective functions whose best response

curves intersect on the contract curve (i.e., 9� = 9� � 9�) are potentially part of a

subgame-perfect Nash equilibrium. In order to be an equilibrium the solution must also

satisfy that none of the principals have incentives to deviate in their selection of the

objective function (or curve of best response).

Although there are infinite best response curves for firms that meet the sufficient

conditions to derive the subgame-perfect Nash equilibrium, it is possible to derive a set

of equilibrium objective functions using Corollary 2.

31 A model in which it can be explained endogenously that a firm acts as the Stackelberg leader and the best response to another firm is to act as a follower can be found in Basu (1995).


20

An additional difficulty, however, must be solved a priori, which is to determine

the point �9�, 9� on the contract curve that the principals will select. Assuming that

firms are symmetrical, it seems that this point corresponds to the equal division of the

monopoly production (i.e., �9��, 9�

� � A�

�, �

�E). If this were the case, and using

Corollary 2, this point could be sustained as a subgame-perfect Nash equilibrium if the

firms’ best response curves are such that:

?�

��6, <, �, 9�� =F�9�, 9� |��6, <, �, 9�, 9� � �� A6, <, �, �

�, �

�E � ��

�G (12)

Figure 5 shows the iso-profit curves in this particular case. These curves

correspond to the best response curves of the firms in the second stage of the game, and

only tangentially intersect at point �9��, 9�

� � A�

�, �

�E. Hence, this will be the only

Nash equilibrium in the second stage of the game.32

To derive the firms’ objective functions from their best response curves we need

to proceed from (12). Without loss of generality, we will derive firm 1’s objective

function. In this case is clear that:

��

"�) �6 ) <�9� = 9� ) � 9� � 0 (13)

Since the best response curve is derived from the first order condition of (2),

(13) is equivalent to:

#��

#�� 0 (14)

Then it is possible to integrate (13) for 9� to find the general functional form of

the objective function ��6, <, �, 9�, 9� that makes the best response curve as the one

defined by the locus of points �9�, 9� which satisfy (14). Hence: ��

��6, <, �, 9�, 9� � ) ��

�9�

� = �

�<9�

� = ��

"�9� = H��9� , (15)

where H��9� is any function. It is possible to rewrite (15) in terms of the payoff

function and production of a firm as:

��6, <, �, 9�, 9� � 9� I�� ) ��6, <, �, 9�, 9� ) �

�<9�J = H��9� , (16)

where ��

�� is the monopoly profit of the game. By analogy with this derivation

it is possible to obtain the objective function that the principal of the firm 2 delivered to

his agent to maximize the second stage. By symmetry this is given by:

32 Note also that in this case the profits of a firm grow in the opposite direction to the region bounded by the competitor’s best response curve.


21

��6, <, �, 9�, 9� � 9� I�� ) ��6, <, �, 9�, 9� ) �

�<9�J = H��9� (17)

At first glance it is clear that the selection of objective functions as (16) and (17)

does not respond to any structure used in the literature so far. However, the equilibrium

objective functions are depending on both profits and production, two highly significant

variables in the management of any firm (at least for markets where competition is in

quantities). The third variable, the arbitrary function of competitor (i.e., H��9�� ), is actually irrelevant to the formulation, since it is irrelevant to the agent’s optimization

problem.

Therefore, the objective function that each agent must maximize will be equal to a

non-linear combination of profit and production.33 However, this is by no means the

only way that the objective function can be selected by the principal in equilibrium,

because there are infinite best response curves intersecting at �9��, 9�

� � A�

�, �

�E and

satisfying the no-deviation condition in the strategic stage.

4. Practical and theoretical implications

Cournot (1838), Bertrand (1883) and Stackelberg (1934) are now widely

accepted among economists as standard solutions to the problem of non-cooperative

oligopoly. However, empirical studies (real and simulated) in different industries have

not validated the widespread use of any of these solutions (Camerer (2003)). Moreover,

when competition takes place in quantities, some studies have found that – despite not

seeming rational for firms – the solution in which the firm’s production are on the

contract curve tends to more accurately observed in real life than the Nash-Cournot

solution.

For instance, Selten, Mitzkewitz, and Uhlich (1997) conclude that, because the

simulated behavior of a duopoly competing in quantities tends to productions that are

placed on the contract curve (ie, Pareto-optimal), the students representing firms, in this

case, cannot be considered “rational” because they would not be taking their decisions

to optimize their profits.

The result of Selten, Mitzkewitz, and Uhlich (1997) could be reinterpreted in

light of the solution derived here. This reinterpretation requires recognizing that the

33 This contrasts with strategic delegation models which generally use a linear combination between profits and production (or sales). As demonstrated above, however, that combination does not satisfy the condition (6), and therefore cannot be part of an objective function obtained endogenously in equilibrium.


22

one-stage Selten, Mitzkewitz, and Uhlich (1997)’s game is in fact a two-stage game,

with two different players in each of them. Thus, the principals in the Selten,

Mitzkewitz, and Uhlich (1997)’s experiment are students of an economics course, who

must set a strategy that the computer will run against the strategy developed by another

student. In this case the computer could then be considered as the agent.34 From the

derived results it is clear that if the principal (i.e., students) are rational in the neo-

classical sense then the competitive strategy scheduled should be equivalent to selecting

an objective function for the agent (i.e., the computer) so that, as result of competition,

the contract curve is reached as predicted in this paper.

Regarding the theoretical implications of my findings, it appears that the study

of oligopoly should not be modeled as the simplistic structure of a single player (i.e.,

firm) that seeks to maximize profits, for the simple reason that to model the firm in this

form is equivalent to assuming some degree of irrationality for those who are part of the

firm. This had already been warned by many classical economists (being the main

representative of this current Edward Chamberlin).

In this line, Fisher (1989) warns of the impossibility of a single-stage game in an

oligopoly where firms reach the contract curve as part of a non-cooperative solution.

Without contradicting this, as my model arrives at this solution after two stages, it

seems necessary to point out that, in light of the results achieved, to model competition

in oligopoly as a single stage game would probably not be adequate to emulate real

competition among firms. In reality, firms do not arise and disappear at one stage.

Therefore, whenever possible (and in light of my results wise), the principal will take

time, before competing in the market, to define the objective function for an agent to

compete for them.35 This is a remarkable result that at least questions the study of

oligopoly as a static game.

The oligopoly problem proposed here generates an additional problem which I

believe is essential, but not my goal to study in more detail in this paper. This refers to

the coordination or negotiation to be carried out by firms to choose a priori a move on

34 In fact, unlike the real problem facing the firms in the market, it is possible to neglect the potential friction between the principal and the agent as the agent in this case obeys orders without question, and does exactly what it is ordered (in a perfectly rational way). 35 Similarly, the very need for an agent to run the firm should also arise as a strategic requirement according to the results derived here (with the aim of generating credibility.). This result validates the statement by Schelling (1960): “The use of thugs or sadists for the collection of extortion or the guarding of prisoners, or the conspicuous delegation of authority to a military commander of known motivation, exemplifies a common means of making credible a response pattern that the original source of decision might have been thought to shrink from or to find profitless, once the threat had failed” (p. 142-143).


23

the contract curve. This problem arises endogenously in the model reviewed and I think

it is an important recognition that competition in an oligopoly market is more complex

than is generally accepted today (something also intuitive to classical economists).

Another theoretical implication of the results is that whenever there is strategic

interaction (using the language of game theory), the endogeneity of the objective

function must be questioned if we are in presence of a rational firm. Thus, the results

derived here could be extended to situations other than the oligopoly, such as the

durable goods monopoly. If the monopolist had the ability to commit itself, through an

agent, to not reduce prices in the future I have no doubt a different objective function

might fix the problem that generates the monopoly’s failure to take into account all

consumer’s influence on the market.

Finally, although the proposed solution may seem logical when the number of

firms is small enough, it does not seem so for a large number of firms, because the

bargaining problem faced by firms may become intractable in practice when the number

of firms increases, eliminating then any possibility to choose a move on the contract

curve. The divergence of the solution found here between a small and large number of

firms certainly deserves further research.

5. Conclusions

Before undertaking any competitive move, as would be the selection of a price

or quantity to produce, firms in oligopoly will study the market in order to design

(explicitly or implicitly) a competitive strategy (Porter (1980)). The competitive

strategy, so common in management literature, is no more than the determination of the

game that the firm wants to play considering that its competitors will also be in the

same game.

Unfortunately, the management literature has not been able to express

mathematically the competitive strategy pursued by the firms (Caves, 1980), which has

hindered communication with economists working in industrial organization. In this

paper I propose to make equivalent the search of the competitive strategy to determine

an objective function to guide the firm (in fact, the agent) in all matters relating how to

compete in the market.

This paper presents a two-stage delegation game in the context of a non-

cooperative oligopoly. In the first stage, each firm’s principal (owner) competes with


24

each other in order to maximize profits for their firms, deciding the objective function

that their own agents (managers) should maximize in the second stage. In the second

stage, agents simultaneously select competitive moves to maximize the objective

function before assigned.

I show the existence of infinite objective functions that are part of a subgame-

perfect Nash equilibrium. Remarkably, I show that, in general, the selection of an

objective function equivalent to the firm’s profits will not be part of a subgame-perfect

Nash equilibrium. This contradicts virtually everything written to date in oligopoly

theory, at least when it is assumed that firms behave rationally. I also demonstrate that

when the objective function is made endogenous the non-cooperative solution is such

that the moves resulting from these functions are on the firms’ contract curve. This

result should repositions this curve as the non-cooperative solution to the oligopoly

problem, which in recent decades has been displaced by Pareto inferior solutions such

as Cournot, Bertrand and Stackelberg, among others.

The results also suggest further exploration of the field of research that could

lead to intersect the literature of oligopoly theory in industrial organization with the

literature of competitive strategy in management. My impression is that its junction is

not only necessary for both disciplines but the intersection would redirect, at least

partially, the current approach of industrial organization towards a discipline that

enables “to give advice to managers in the way that macroeconomists advise

governments”.36

36 From “An economist takes tea with a management guru” (The Economist, 21 December 1991:91). This article criticizes the industrial organization as a discipline that has only tried to explain the behavior of firms without ever going to propose solutions to them.


25

References

Alchian, A. (1950): “Uncertainty, evolution, and economic theory,” Journal of Political

Economy, 58, 211-221.

Basu, K. (1995): “Stackelberg equilibrium in oligopoly: an explanation based on

managerial incentives,” Economics Letters, 49, 459-464.

Baumol, W. (1958): “On the theory of oligopoly,” Economica, 25, 187-198.

Bertrand, J. (1883): “Review of “théorie mathématique de la richesse socials” and

“reserche sur les principes mathématiques de la théorie des richesses”,” Journal of

Savants, 499-508.

Camerer, C. (2003): Behavioral game theory: experiments in strategic interaction.

Princeton: Princeton University Press.

Caves, R. (1980): “Industrial organization, corporate strategy and structure,” Journal of

Economic Literature, 18, 64-92.

Chamberlin, E. (1929): “Duopoly: value where sellers are few,” Quarterly Journal of

Economics, 44, 63-100.

Cournot, A. (1838): Recherches sur les principles mathematiques de la theorie des

richesses. English edition (ed. N. Bacon): Researches into the mathematical principles

of the theory of wealth. New York: Macmillan, 1987.

Cyert, R. and J. March (1956): “Organizational factors in the theory of oligopoly,”

Quarterly Journal of Economics, 70, 44-64.

Cyert, R. and C. Hedrick (1972): “Theory of the firm: past, present, and future; and

interpretation,” Journal of Economic Literature, 10, 398-412.

Dreeze, J. (1976): “Some theory of labor management and participation,”

Econometrica, 44, 1125-1139.

Dutta, P. and R. Radner (1999): “Profit maximization and the market selection

hypothesis,” Review of Economic Studies, 66, 769-798.

Fellner, W. (1949): Competition among the few. New York: Knopf.

Fershtman, C. and K. Judd (1987): “Equilibrium incentives in oligopoly,” American

Economic Review, 77, 927-940.

Fisher, F. (1989): “Games economists play: a noncooperative view,” RAND Journal of

Economics, 20, 113-124

Friedman, M. (1953): Essays in positive economics. Chicago: Chicago University Press.


26

Hax, A. and N. Majluf (1991): The Strategy Concept and Process: A Pragmatic

Approach. Englewood Cliffs, NJ: Prentice-Hall.

Henderson, A. (1954): “The theory of duopoly,” Quarterly Journal of Economics, 68,

565-584.

Kaneda, M. and A. Matsui. (2003): “Do profit maximizers maximize profits?:

divergence of objective and result in oligopoly,” Working Paper, School of Foreign

Service, Georgetown University.

Lewis, G. (1948): “Some observations on duopoly theory,” American Economic

Review, 38, 1-9.

Meade, J.E. (1972): “The theory of labor-managed firms and of profit sharing,”

Economic Journal, 82, 402-428.

Nash, J. (1951): “Non-cooperative games,” Annals of Mathematics, 54, 286-295.

Nash, J. (1953): “Two-person cooperative games,” Econometrica, 21, 128-140.

Osborne, D.K. (1971): “The duopoly game: output variations,” American Economic

Review, 61, 538-560.

Osborne, D.K. (1974): “A duopoly price game,” Economica, 41, 157-175.

Porter, M. (1980): Competitive strategy: techniques for analyzing industries and

competitors. New York: The Free Press.

Robinson, J. (1933): The economics of imperfect competition. New York: Macmillan.

Schelling, T. (1960): The strategy of conflict. Cambridge: Harvard University Press.

Selten, R., Mitzkewitz, M. and G. Uhlich. (1997): “Duopoly strategies programmed by

experienced players,” Econometrica, 65, 517-555.

Sklivas, S. (1987): “The strategic choice of managerial incentives,” RAND Journal of

Economics, 18, 452-458.

Simon, H. (1964): “On the concept of organizational goal,” Administrative Science

Quarterly, 9, 1-21.

Stackelberg, H. (1934): Marktform und gleichgewicht. Vienna: Springer.

Stigler, G. (1964): “A theory of oligopoly,” Journal of Political Economy, 72, 44-61.

Vickers, J. (1985): “Delegation and the theory of the firm,” Economic Journal, 95, 138-

147.

Williamson, O. (1964): The economics of discretionary behavior: managerial objectives

in a theory of the firm. Englewood Cliffs: Prentice-Hall.


27

Wolfe, J.N. (1953-1954): “The problem of oligopoly,” Review of Economic Studies, 21,

181-192.


28

Figure 1. Representation of the objective function and move selection by the ith

firm in an oligopoly market

Tactics

��, �� , ��, �� , ��

��

m ��, ��, ��, ��

�

Objective

function selection

Move selection Market

competition

Strategy

Profits


29

Figure 2. Equilibrium objective functions for the duopoly case, in which firms

choose a single move

��

��

��, ��, ��, ��

��

��, ��, ��, ��

��, ��, ��, ��

��

��, ��, ��, ��

��

��

��## ��#

��##

��#


30

Figure 3. Iso-profit curves for the duopoly case, in which firms choose a single

move

��

��

��, ��, ��, �� , ��, ��#, ��#

��, ��, ��, �� , ��, ��#, ��#

��#

��# ��

��, ��, ��, ��

��, ��, ��, ��


31

Figure 4. Deviation of the firm 1's objective function for the case in which the firm

2 maximizes its profit in the second stage

9�

9�

$$%��&��6, <, �, 9�

$'%��&��6, <, �, 9�

9�

9�

9��

9��

9��

?��6, <, �, 9�

9� = 9� � 9�

6 ) �<

6 ) �<

0


32

Figure 5. A feasible subgame-perfect Nash equilibrium

9�

9�

$'

%��

?��

9� � 6 ) �2<

9� � 6 ) �2<

9� = 9� � 9�

6 ) �<

6 ) �<

0

45°

9��=

�

�

?��

$$

%��

9��=

�

�

endogenous objective function in oligopoly

Documents