Oligopoly (I): static models .1 Cournot model

The first clear application of modern game-theoretic reasoning to be found in the economic literature appears in Cournot’s (1838) discussion of oligopoly. By now, it has probably become the most paradigmatic model of strategic interaction studied in economics. It is just natural, therefore, that Cournot oligopoly should be the first economic application to be presented in this book.

Let there be n firms operating in a certain market for a homogeneous good, where the consumers’ aggregate behavior is captured by a demand function

F :R+→ R+. (3.1)

This function specifies, for each p∈ R+, the corresponding total demand for the good, F ( p). It will be assumed that the function F(·) satisfies the so-called law of demand, i.e., the total quantity demanded in the market is strictly decreasing in the prevailing price. It is therefore an invertible function, with its corresponding inverse being denoted by P(·). (That is, P(Q) = p ⇔ F(p) = Q.)

Identify each of the firms participating in the market with subindex i ∈ {1, 2, . . . , n}. Every firm i displays a respective cost function

Ci :R+→ R+,

assumed increasing, with Ci(qi) standing for the cost incurred by firm i when it produces output quantity qi.

In the present Cournot context, the decision of each firm concerns solely its output produced, their respective amounts chosen independently (i.e., “simultaneously”) by each of them. Given any output vector q ≡ (q1, q2, . . . , qn) resulting from these independent decisions, the induced aggregate quantity is simply given by Q ≡ q1+ q2+ · · · + qn, which leads to a market-clearing price, P(Q), and the following profits for each firm i ∈ {1, 2, . . . , n}:

πi(q)≡ P(Q)qi − Ci(qi). (3.2)

Note that the above expression implicitly assumes that all output produced by every firm is sold in the market.

72

The above elements define a strategic-form game among the n firms, where each of them has an identical strategy space, Si = R+ (i.e., the set of its possi-ble production decisions), and the payoff functions are identified with the profit functions given in (3.2). In this game, a (Cournot-)Nash equilibrium is any vector q^∗ ≡ (q1^∗, q2^∗,. . . qn^∗) satisfying, for each i= 1, 2, . . . , n, the following conditions:

q_i^∗ ∈ arg max

qi

πi(qi, q_−i^∗ ) (3.3)

or equivalently

∀qi ∈ R+, πi(q^∗) ≥ πi(qi, q_−i^∗ ),

where (qi, q_−i^∗ ) is just the convenient shorthand for the output vector where firm i chooses qi and the remaining firms j = i choose their respective q^∗j.

Let us assume that the function P(·) as well as every Ci(·), i = 1, 2, . . . , n, are differentiable. Then, for the n optimization problems in (3.3) to be simultaneously solved at (q₁^∗, q2^∗,. . . qn^∗), the following first-order necessary conditions (FONC) must hold:

P^(Q^∗)q_i^∗+ P(Q^∗)− Ci^(q_i^∗)≤ 0 (i = 1, 2, . . . , n), (3.4) where Q^∗≡n

i=1q_i^∗and the notation g^(·) stands for the derivative of any arbitrary function g(·) of a single variable with respect to its (only) argument. That is, each firm i must have its respective q_i^∗satisfy the FONC of its individual optimization problem when the other firms are taken to choose their respective equilibrium values q^∗_j, j = i.

Whenever the Nash equilibrium is interior (that is, q_i^∗> 0 for each i = 1, 2, . . . , n), the weak inequalities in (3.4) must apply with equality. Moreover, pro-vided standard second-order conditions hold (e.g., concavity of every firm’s payoff function in its own output), one can ensure that such a system of n equations fully characterizes the set of interior Nash equilibria. In what follows, we assume that those second-order conditions are satisfied (cf. Exercise 3.2) and focus on interior equilibria alone. In this case, (3.4) can be rewritten as follows:

C_i^(q_i^∗)− P(Q^∗)= P^(Q^∗)q_i^∗ (i = 1, 2, . . . , n). (3.5) Verbally, the above conditions can be described as follows:

At equilibrium, the (negative) deviation of each firm’s marginal cost from the market price is proportional to its own output, with the proportionality factor (common to all firms) equal to the slope of the demand function.

To understand the relevance of the previous statement, it must be recalled that, under perfect competition, the prevailing price is assumed to be equal to the marginal cost for each firm. In such a competitive setup, firms do not conceive themselves as market participants of a significant size and, therefore, each of them takes the price prevailing in the market, say ¯p, as independent of its behavior. Consequently,

each firm i maximizes



πi( ¯p, qi)≡ ¯p qi − Ci(qi)

with respect to qi ∈ R+, whose solution ˆqi( ¯p ), if interior, must satisfy the following FONC:

C_i^( ˆqi( ¯p ))= ¯p. (3.6)

Naturally, the prevailing price ¯p must “clear” the market, given the vector of firm outputs ˆq( ¯p )≡ ( ˆq1( ¯p ), ˆq2( ¯p ), . . . , ˆqn( ¯p )) that solves their respective optimization problems. That is, in a perfectly competitive equilibrium, the following “fixed-point condition” must hold:

P

 _n



i=1

ˆ qi( ¯p )

= ¯p.

Note that an analogous condition of market clearing is also required in the Cournot model, as implicitly embodied by the function P(·) in (3.2).

In a heuristic sense, one can interpret the perfectly competitive scenario as a pseudo-Cournotian context in which every firm perceives a totally “elastic” demand function – that is, an inverse demand function that is essentially inelastic so that the price is not affected by the quantity sold. In this case, (3.6) can be seen as a particular case of (3.5). Of course, if the number of firms is finite and the law of demand holds, this perception is wrong. Only when the number of firms is large enough (and, therefore, the weight of each one of them is relatively insignificant) does a very elastic demand function represent a good approximation of the situation faced by each firm. Only then, that is, may the perfect-competition paradigm represent a suitable strategic model of firm behavior and corresponding market performance.

In view of the former discussion, it is natural to conjecture that any discrepancy between Cournot and perfect competition might be understood in terms of the following two factors:

(a) the elasticity of the demand function (i.e., how sensitive is market demand to price changes);

(b) the extent of market concentration (roughly, how many firms enjoy a signi-ficant share of the market).

Focusing, for convenience, on the inverse demand function, its elasticityλ(Q) (i.e., the “inverse elasticity”) is defined in the usual fashion: the relative (marginal) decrease experienced by the market-clearing price for any given relative increase (also marginal) in the quantity sold in the market. That is,

λ(Q) ≡ −d P/P

d Q/Q = −P^(Q)Q P·

On the other hand, a traditional way of measuring market concentration is given by the so-called Herfindahl index. This index is defined over the firms’ vector of

market sharesα ≡ (α1, α2, . . . , αn), where αi ≡ qi

Q

represents the fraction of total output produced by each firm i = 1, 2, . . . , n. Given any such vectorα, its induced Herfindahl index H(α) is given by

H (α) ≡

n i=1

(αi)².

Observe that, if the number of firms n is kept fixed, the function H (·) obtains its maximum, as desired, whenαi = 1 for some firm i. That is, the maximum is attained when the market is “fully concentrated.” In contrast, H (·) obtains its minimum when the market output is uniformly distributed among all firms, i.e., all firms display the same market weight and therefore the market is at its “least concentrated”

state. Thus, in a meaningful sense, the Herfindahl index does appear to reflect an intuitive measure of market concentration.

To attain a clear-cut relationship between concentration, market elasticity, and deviation from perfect competition, it is useful to rewrite (3.5) in the following way:

P(Q^∗)− Ci^(q_i^∗)

P(Q^∗) = −P^(Q^∗) 1

P(Q^∗)q_i^∗ (i = 1, 2, . . . , n). (3.7) The left-hand side of the previous equation expresses, for each firm i, the propor-tional deviation from the “individual” competitive situation where the firm’s marginal cost would coincide with the market equilibrium price. If these relative deviations are added across all firms, each of them weighted by the market share α_i^∗≡ q_i^∗/Q^∗commanded by the respective firm i in a Cournot-Nash equilibrium

This index expresses the weighted-average deviation from perfect competition ob-served in the Cournot-Nash equilibrium q^∗. Adding up the terms in (3.7) and carrying out suitable algebraic manipulations, we have

L(q^∗)= −

which is indeed the sought-after relationship. It reflects, very sharply, the two consid-erations (demand elasticity and market concentration) that were formerly suggested as important to understand the magnitude of any discrepancy between competitive and Cournot equilibria. It indicates, that is, that the average deviation from perfect

competition (as given by Lerner’s index) is simply the product of the two measures we have proposed to measure each of those two factors (i.e., the inverse-demand elasticity and the Herfindahl index, respectively).

Now, we illustrate the previous developments for a particularly simple duopoly context with linear cost and demand specifications (see Exercise 3.3 for a context with more firms). Let there be two firms, i = 1, 2, with identical cost functions:

Ci(qi)= c qi, c > 0. (3.8)

Let the demand function be also linear:

P(Q)= max {M − d Q, 0}, M, d > 0. (3.9)

For an interior Cournot-Nash equilibrium, the first-order conditions (3.5) are par-ticularized as follows:

(M− d(q1^∗+ q2^∗))− c = dqi^∗ ( i = 1, 2). (3.10) Its solution,

q_i^∗ = M− c

3d ( i = 1, 2), (3.11)

defines an interior equilibrium, which exists as long as M > c.

The above computations may be fruitfully reconsidered by focusing on the firms’

best response correspondences (cf. (2.7)), which, in the present oligopoly context, are denoted byηi(·) and receive the customary name of firms’ reaction functions.

For each firm i = 1, 2, its ηi(·) may be obtained from the first-order condition of its individual optimization problem, as this problem is parametrized by each of the hypothetical decisions qj ( j = i) that may be adopted by i’s competitor.

For interior configurations, these reaction functions may be implicitly defined by adapting (3.10) as follows:

(M− d(ηi(qj)+ qj))− c = dηi(qj) ( i, j = 1, 2; i = j).

Including as well the boundary points where one of the firms chooses not to produce, they are found, explicitly, to be of the following form:

ηi(qj)= max



0, M− c

2d − (1/2)qj



. (3.12)

Thus, for example, the optimal monopoly decisions may be obtained from these functions as follows:

q_i^m ≡ ηi(0)= M− c

2d , ( i = 1, 2),

that is, they are simply the optimal reactions of each firm i when the competitor produces no output.

In the present linear context, the reaction functions (3.12) are also linear (for interior configurations). Their intersection obviously defines a Nash equilibrium;

that is, a pair of outputs such that, simultaneously, each one of them is a suitable

η₂(q₁) π1(q₁, q₂) = π-1

q*1 q₁

q₂

π₂(q₁, q₂) = π-₂

q₁^m q2

m

q2*

η1(q₂)

Figure 3.1: Cournot-Nash equilibrium in a linear duopoly.

“reaction” to the other. A graphic illustration of this situation is depicted in Fig-ure 3.1. There, the reaction function of each firm (1 or 2) is identified as the locus of tangency points of its iso-profit curves to the straight lines (horizontal or vertical) associated with each of the outputs (taken as fixed) on the part of the competitor.

In the specific duopoly context where the reaction functions are decreasing and intersect only once (of which the linear case discussed above is just a particular case), its Nash equilibrium enjoys a much stronger foundation than the ordinary one discussed in Subsection 2.2.1. For, in this case, the (unique) Nash equilibrium is also the unambiguous prediction resulting from an iterative elimination of dominated strategies (cf. Section 2.1 and Exercise 2.5).

To verify this claim, it is convenient to use the identification of rationalizable and iteratively undominated strategies established by Theorem 2.6.³³ This result allows us to focus on the firms’ reaction functions and, at each stage, rely on them to discard those strategies that cannot be “rationalized” as a best response to some of the remaining strategies by the opponent.

Proceeding in this fashion, we can first discard, for each firm i = 1, 2, those outputs that exceed the monopoly levels, i.e., those qi such that

qi > qi^m ≡ ηi(0).

These outputs can never be an optimal response to any beliefs over the competitor’s output – or, diagrammatically, they are not on the reaction function of firm i for any possible output of j = i. Once we have ruled out those outputs in the intervals

33 This theorem was stated and proven only for games involving a finite set of pure strategies. However, it is extendable to contexts such as the present one where the spaces of pure strategies are infinite (a continuum).

(q₁^m, ∞) and (q2^m, ∞), we can do the same for those qi that satisfy 0≤ qi < ηi

q^m_j 

= ηi(ηj(0)), (i, j = 1, 2; i = j),

because, having established that firm j = i will not produce beyond the monopoly output, any qi that verifies the above inequality cannot be a best response to any admissible belief on the opponent’s output. Graphically, what this inequality reflects is simply that, if outputs in the interval (q^m_j , ∞) are not allowed on the part of the opponent, there is no output qi ∈ [0, ηi(q^m_j )) that is on the reaction function of firm i. Or, somewhat more precisely,

qi = ηi(qj), qj ≤ q^mj

⇒ qi ∈ ηi

q^m_j  , qi^m



for each i, j = 1, 2 (i = j). Undertaking one further iteration after the elimination of the intervals (q_i^m, ∞) and [0, ηi(q^m_j )), it is immediate to check that, for analogous reasons, we can discard outputs qi that satisfy

qi > ηi(ηj(ηi(0))) (i, j = 1, 2; i = j).

Proceeding indefinitely along this process, it is clear that, in the limit, only the outputs q₁^∗and q₂^∗that define the Nash equilibrium remain undiscarded.

More precisely, q₁^∗and q₂^∗are the only outputs that verify the following conditions:

ηi(ηj(· · · (ηi(ηj(0)))))≤ qi ≤ ηi(ηj(· · · (ηj(ηi(0))))) (i, j = 1, 2; i = j).

for all (finite) alternate compositions of the reaction functions. The above expression embodies, in a compact fashion, the progressively more stringent series of inequal-ities induced by each of the iterations of the process. They can be summarized as follows:

r First iteration: qi ≤ ηi(0);

r Second iteration: qi ≥ ηi(ηj(0));

r Third iteration: qi ≤ ηi(ηj(ηi(0)));

r Fourth iteration: qi ≥ ηi(ηj(ηi(ηj(0)))) . . .

This indefinite shrinkage of admissible outputs is illustrated in Figure 3.2 for the linear case formerly considered in Figure 3.1.

3.1.2 Bertrand model

Almost half a century after Cournot’s work, Bertrand (1883) proposed an alternative model of oligopolistic competition where firms have their prices (instead of outputs) as their decision variable. Again, firms’ decisions in this respect are assumed to be simultaneously adopted by all of them.

Let us focus first on the case where, as postulated in Section 3.1.1, the good produced by every firm is homogeneous. Under these circumstances, it is clear that if the market is “transparent to the eyes” of the consumers and frictionless, any

η₁(q

2)

η₂(q

1)

q*1

q*₂

q1 q2

q^m₁ q2

m

1

2 3

2 3 1

4

4

Figure 3.2: Cournot duopoly – iterative elimination of dominated strategies. (The output intervals discarded in iterations 1–4 are spanned by corresponding arrows.)

equilibrium of the induced game must have all active firms set the same price – all firms demanding a higher price will enjoy zero demand because consumers will buy the homogeneous good only from those that offer the cheapest price.

This gives rise to some especially acute competition among firms that, under quite general conditions, tends to reduce very substantially their profits. In fact, as we shall presently show, there exist paradigmatic conditions in which firms are forced to zero profits at equilibrium, independently of how many of them there are (obviously, as long as there are at least two). Because of its marked contrast with the Cournotian conclusion presented in Section 3.1.1, this state of affairs is often referred to as Bertrand’s paradox.

To illustrate this paradox in its starkest form, consider n (≥2) firms that confront a continuous and nonincreasing demand function F (·) of the sort described in (3.1) and display linear and identical production costs as given in (3.8) for some common marginal cost c> 0. To render the setup interesting, suppose that F(c) > 0, i.e., there is positive demand at the price equal to the marginal cost. We posit that each firm i decides on its respective price pi independently (i = 1, 2, . . . , n), which results in a price vector p≡ (p1, p2, . . . , pn) faced by consumers. As explained, because the good is assumed homogeneous (and the information on market condi-tions perfect), all consumer demand flows to those firms that have set the lowest price.

Given any p, denote θ(p) ≡ min{p1, p2, . . . , pn} and let F(θ(p)) be the total demand induced by such a configuration of firm prices. For simplicity, it will be

supposed that this total demand is uniformly distributed among all firms that have set the minimum priceθ(p). Formally, this defines a strategic-form game for the n firms, where Si = R+is the strategy space of each firm i and the payoffs associated to any strategy (i.e., price) vector p are as follows:

πi( p)= 0 if pi > θ( p)

= ( pi − c) F (θ(p))

#{ j ∈ N : pj = θ(p)} otherwise, (3.13) where #{·} stands for the cardinality of the set in question.

Our first objective is to characterize the (Bertrand-)Nash equilibria of this game.

In this task, the key step is to show that every equilibrium price vector p^∗ must satisfy θ( p^∗)= c, i.e., the minimum price set by firms can neither be (strictly) higher or lower than c. We now discard each of these two possibilities in turn.

On the one hand, it is clear thatθ( p^∗)< c cannot hold. For, in this case, the firms that set the minimum priceθ( p^∗) would earn negative profits and therefore could benefit by unilaterally deviating to a higher price – e.g., if they chose a price equal to c, in which case they would make zero profits.

On the other hand, it can not happen thatθ( p^∗)> c. To see this, let p^∗be some such configuration and assume, for simplicity, that the induced demand F (θ( p^∗))>

0. Consider any of the firms, say firm i, which does not capture the whole demand F (θ( p^∗)). (There must always be at least one firm in this situation, either because its price is higher than θ( p^∗) and therefore its demand is zero, or because it is sharing the total demand F (θ( p^∗)) with some other firm whose price is also equal toθ( p^∗).) If firm i were to deviate to a price “infinitesimally lower” than θ( p^∗), say to some p_i^ = θ( p^∗)− ε for some small ε > 0, it would absorb the whole of the induced market demand F (θ( p^∗)− ε) and obtain profits

πi( p_i^, p^∗_−i)= [(θ( p^∗)− ε − c] F(θ( p^∗)− ε).

Instead, if it does not deviate, its profits are either zero or, if positive, no higher than [θ( p^∗)− c]F (θ( p^∗))

2

because, in the latter case, the set{ j ∈ N : p^∗j = θ( p^∗)} includes firm i and at least one additional firm. Obviously, if ε > 0 is low enough,

πi( p_i^, p^∗_−i)> [θ( p^∗)− c]F (θ( p^∗))

2 .

Therefore, the deviation toward p_i^ would be profitable for firm i , which implies that p^∗cannot be a Nash equilibrium.

Since we have ruled out that θ( p^∗) might be higher or lower than c, only a price vector p^∗that satisfies the equalityθ( p^∗)= c remains a possible equilibrium candidate. In fact, it is straightforward to check (cf. Exercise 3.4) that, if at least two firms set the minimum price in p^∗, this price profile defines a Nash equilibrium of the Bertrand game; Bertrand-Nash equilibria p^∗can be simply characterized by

the following twofold condition:

θ( p^∗)= c

#{ j ∈ N : p^∗_j = θ( p^∗)} ≥ 2. (3.14)

Thus, in equilibrium, all firms in the market (both those that enjoy a positive indi-vidual demand as well as those that do not) attain zero profits.

Under price competition, therefore, the Bertrand-Nash equilibrium gives rise to a fully competitive outcome when firms display common and constant marginal costs. This result contrasts sharply with that obtained in the Cournot model under similar cost and demand conditions (cf. Section 3.1.1). This serves to underscore the idea that, as suggested before, competition in prices (i.e., ´a la Bertrand) typically leads to significantly more aggressive behavior than competition in quantities (`a la Cournot), at least under benchmark conditions.³⁴

Naturally, the analysis turns out to be much less extreme (i.e., the contrast with Cournot competition less “paradoxical”) if the stringent and somewhat unrealistic assumption of good homogeneity is relaxed. Suppose, for example, that each firm is taken to produce a different kind of car, computer, or cooking oil. That is, firms produce different goods, but all of these cover similar consumer needs in a less-than-perfect substitutable manner. Then, if we make the reasonable assumption that consumers’ preferences over the range of differentiated goods are not fully homogenous, some potential for variety arises concerning the range of possible goods sold in the market as well as their corresponding prices. It also becomes natural to posit that the specific demand for any particular good should be gradually (i.e., continuously) responsive to price changes in all prices. Overall, this suggests that, building upon the partially “monopolistic” features brought about by product differentiation, firms might well be able to earn positive profits at equilibrium even under Bertrand (price) competition.

To fix ideas, consider a simple model of oligopolistic competition with differ-entiated products that involves just two “symmetric” firms with linear costs. Thus, each firm i = 1, 2 displays a cost function of the form

Ci(qi)= c qi, c > 0,

which specifies the cost at which firm i may produce any given qi units of its firm-specific (differentiated) good. Concerning the demand side, suppose that each product i (the good produced by firm i ) faces an inverse (also linear) demand function:

Pi(q1, q2)= max{0, M − qi − bqj} (i, j = 1, 2, i = j), (3.15) where M > 0. This formulation embodies the idea that, in general, the demands

34 For other cost and demand scenarios (e.g., when marginal costs are different and/or they are not constant), the conclusions may be much less clear cut. In particular, one may even encounter that Nash equilibria in pure strategies fail to exist due to the abrupt discontinuity induced on the payoff functions by the assumption of good homogeneity (see Exercise 3.5). It can be seen, however, that existence of Bertrand-Nash equilibria in mixed strategies always follows from the general existence results discussed in Section 2.4 (in particular, those

34 For other cost and demand scenarios (e.g., when marginal costs are different and/or they are not constant), the conclusions may be much less clear cut. In particular, one may even encounter that Nash equilibria in pure strategies fail to exist due to the abrupt discontinuity induced on the payoff functions by the assumption of good homogeneity (see Exercise 3.5). It can be seen, however, that existence of Bertrand-Nash equilibria in mixed strategies always follows from the general existence results discussed in Section 2.4 (in particular, those

In document Vega-Redondo - Economics and the Theory of Games (Page 86-97)