Can prices be insensitive to unit cost variations? A game-theoretic alternative to the kinked demand curve explanation

Can prices be insensitive to unit cost variations? A game-theoretic alternative to the kinked demand curve explanation Giorgos Stamatopoulos∗ Minas V...
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Can prices be insensitive to unit cost variations? A game-theoretic alternative to the kinked demand curve explanation Giorgos Stamatopoulos∗

Minas Vlassis†

Abstract We provide a game-theoretic alternative of the kinked demand curve explanation of rigid prices. We analyze a two-stage duopoly where each firm chooses strategically its quantity and its objective (profit or revenue maximization). We identify cases under which firms choose to maximize their revenue, i.e., they choose to ignore their unit costs when adjusting their quantities. In these ranges prices are totally insensitive to unit cost variations. As a corollary, prices in oligopoly can be more rigid than price in monopoly. Keywords: kinked demand curve; revenue maximization; profit maximization JEL: D43, L13, L21.



From the standard oligopoly theory we know that prices are less sensitive to unit cost variations relative to perfect competition. Long time ago, Sweezy (1939) by means of his kinked demand curve apparatus proposed that oligopoly prices can even be totally insensitive to unit cost variations -within a certain range- yet without explicitly deriving that range. At the same time, a better structured -nonetheless similar- prediction was given from the other side of the Atlantic [Hall and Hitch (1939)]. ∗

Department of Economics, University of Crete, 74100 Rethymno, Crete, Greece; email: [email protected]

Department of Economics, University of Crete, 74100 Rethymno, Crete, Greece; email: [email protected]


The main criticism against the model of kinked demand curve was based on its lack of theoretical foundations (regarding the position of the kink and the behavior of firms in the market). A few attempts have been made to offer such foundations. Bhaskar (1988) and Maskin & Tirole (1988) analyzed duopolies with price competition where kinked demand phenomena appear off-equilibrium. More recently, Sen (2004) analyzed a Stackelberg model with cost-asymmetric firms and showed that the leader faces a kinked demand curve. In this paper we provide a game-theoretic explanation of price insensitivity using the standard Cournot model. Our starting point is the assumption that before deciding on their quantities, firms choose strategically their objective functions (as a tool for maximum profit). The candidate objectives are the profit function per se and the revenue function.1 Firms’ choices of objectives determine the price (in)sensitivity as we discuss below. Firms compete in two stages. In the first stage of the interaction, each firm chooses its profit-maximization tool: Namely, it chooses whether it will ignore its unit cost, effectively maximizing a revenue objective, or it will take its unit cost into account, hence maximizing its profit function. In the second stage, firms compete in the product market -by adjusting their quantities- so as to maximize the value of the tool selected in the first stage. If the unit cost is sufficiently low (high) then both firms decide to ignore (take into account) this cost. Hence, they effectively select the revenue (profit) tool. In the first instance, prices prove to be totally insensitive to unit costs variations. For intermediate values of the unit cost, a nonsymmetric equilibrium emerges, where one of the two firms chooses the revenue tool and its opponent chooses the profit tool. In this case, the pass-through of unit costs on to prices -while not zero- once more is lower than what is predicted by the standard Cournot model. A corollary of the above analysis is that prices in oligopoly can be less flexible than the monopolistic price. This last result is in contrast with the standard wisdom in industrial organization. To quote from Rotemberg and Saloner (1987): The relation between industry structure and pricing is a major focus of industrial organization. One of the most striking facts to have emerged about this relationship is that monopolists tend to change their prices less frequently than tight oligopolists. 1

See Baumol (1959), (1962) for a discussion on revenue-maximizing firms.


To understand the last result, we note that a monopolist will always choose the profit over the revenue objective (as the strategic effects of switching to revenue maximization are absent in monopoly). On the other hand, the duopolists (one or both) often choose the revenue objective. Hence, for a robust region of the values of the unit cost, the choices of quantities (and the resulting prices) do not reflect the unit cost. Under this region, therefore, prices in the duopoly will be more rigid than the monopolist’s price.


The framework

Consider a differentiated goods industry with two firms, 1 and 2. Firm i faces the inverse demand function pi = a − qi − γqj ,

i, j = 1, 2, i 6= j


where pi , qi are the price and quantity respectively of firm i, qj is the quantity of firm j, a > 0 and γ ∈ (0, 1] is the differentiation parameter. The production technology is represented by a (common for both firms) linear cost function involving zero fixed cost. The marginal cost is given by c where c < a. Firms’ choices are embedded in a two-stage game. In the first stage, the two firms independently decide on their profit-maximization tool. Namely, each firm chooses whether to maximize its revenue or its profit function. In the second stage, the two firms compete in the market by selecting their quantities -given the outcome of the first stage. We denote this game by G. Remark 1 A distinction must be made between the tool (objective) function and the evaluation function of a firm. To elaborate on this, consider the case where in stage 1, firm i decides to maximize its revenue. Then, its tool function is ui (qi , qj ) = (a − qi − γqj )qi . Let qi∗ = qi∗ (qj ) = argmaxqi ui (qi , qj ) denote its choice. The evaluation of qi∗ is then given by the evaluation function uˆi (qi∗ , qj ) = (a − qi∗ − γqj )qi∗ − cqi∗ . On the other hand, if a firm maximizes its profit, its tool and evaluation functions are the same. The above remark simply says that even if a firm decides to maximize its revenue, what actually matters is its profit: revenue maximization is a tool for obtaining the highest possible profit. Using Remark 1, we identify the sub-game perfect Nash equilibrium (SPNE) outcomes of G. Working 3

backwards, we first present the quantity competition stage of G; we then turn to the stage where the objective functions are chosen.


Equilibrium analysis

Let {P, R} denote the set of choices of each firm in stage 1, where P stands for the choice of profit tool and R stands for the revenue tool. Denote by G(s1 , s2 ) the specific sub-game of G that results when in stage 1 firm i selects the action si ∈ {P, R}, i = 1, 2. Denote the equilibrium quantities and evaluations in G(s1 , s2 ) by qi (s1 , s2 ) and uˆi (s1 , s2 ), i = 1, 2. As an example, consider the case (s1 , s2 ) = (Π, R). Under this case, firm 1 chooses its quantity by maximizing the profit function u1 (q1 , q2 ) = (a − q1 − γq2 − c)q1 , while firm 2 chooses its quantity by maximizing its revenue function u2 (q1 , q2 ) = (a − q2 − γq1 )q2 . Evaluations of the market quantities are given by uˆ1 (Π, R) = (a − q1 (Π, R) − γq2 (Π, R) − c)q1 (Π, R) uˆ2 (Π, R) = (a − q2 (Π, R) − γq1 (Π, R) − c)q2 (Π, R) The following observation is in order. Observation 1. There exist c˜(γ) and c(γ) such that the following hold. (i) G(R, P ) has a Cournot equilibrium with positive evaluations for both firms if c < c˜(γ). (ii) G(R, R) has a Cournot equilibrium with positive evaluations for both firms if c < c(γ). Proof. Appears in the Appendix. In what follows, we pay attention to values of c such that evaluations are all positive, which happens if c < min{c(γ), c˜(γ)}.


Choice of objective functions

Consider now the first stage of G. By the above, the payoff matrix in the first stage is given by P R A = P uˆ1 (P, P ), uˆ2 (P, P ) uˆ1 (P, R), uˆ1 (P, R) R uˆ1 (R, P ), uˆ2 (R, P ) uˆ1 (R, R), uˆ2 (R, R) 4

Given matrix A we can characterize the equilibrium structure of G.With a slight abuse of notation, a SPNE outcome will be denoted by (s1 , s2 ) where si ∈ {P, R}, i = 1, 2. Proposition 1. There exist functions c2 (γ) < c1 (γ) such that the following hold.2 (i) If c2 (γ) < c < c1 (γ), G has a unique SPNE outcome, (R, P ). (ii) If c < c2 (γ), G has a unique SPNE outcome, (R, R). (iii) If c > c1 (γ), G has a unique SPNE outcome, (P, P ). Proof. Appears in the Appendix. Observe by Proposition 1 that whenever the unit cost is sufficiently low (high) then both firms choose the revenue (profit) tool. To grasp the intuition behind this result, consider first the case where c is sufficiently high and focus on a firm that contemplates a deviation from the profit to the revenue tool -given that its opponent selects the profit tool. The deviant firm would produce a higher quantity but its price would fall significantly -in relation to a high unit cost (which is not taken into account when the firm maximizes its revenue but it is considered when the firm evaluates its performance). As a result, its evaluation will be low; hence the firm prefers to stick to the profit tool. Consider now the case where c is sufficiently low and focus on (R,R). If a firm unilaterally deviates from the revenue to profit tool, its quantity would fall and its opponent would be left with a larger market share, produced at a low unit cost. Obviously the deviant firm would then prefer to be in its opponent’s place; hence it does not have an incentive to deviate from the prescribed (R,R) profile. 3.1.1

Price rigidity in duopoly and monopoly

As a result of the equilibrium choices of objectives, prices are given by (2) and (3) below:   

[a + c(1 + γ)]/(2 + γ) p1 =  (2a + γc)/(4 + 2γ)  a/(2 + γ)   

[a + c(1 + γ)]/(2 + γ) p2 = (2a + c(2 − γ 2 ))/(4 + 2γ)   a/(2 + γ) 2

if c > c1 (γ), if c2 (γ) < c < c1 (γ), if c < c2 (γ). if c > c1 (γ), if c2 (γ) < c < c1 (γ), if c < c2 (γ).

The inequality c1 (γ) < min{c(γ), c˜(γ)} holds (see the Appendix).




It will be interesting at this point to compare price rigidities in duopoly and monopoly. To make a meaningful comparison, we allow the monopolist to choose his objective function as well. It is straightforward to see that the monopolist will always choose the profit objective (as no strategic elements are present in a monopoly). Hence the monopolist’s price is pM = (a+c)/2. ∂pm ∂pi < , i = 1, 2, It is easy to see by (2) and (3) that for all c < c1 (γ), ∂c ∂c while the opposite inequality holds for c > c1 (γ).

Appendix Proof of Observation 1. (i) Consider the G(R, P ) game. The best reply function of firm 1 is q1 = (a − γq2 )/2 while that of firm 2 is (

q2 =

(a − γq1 − c)/2, 0,

if q1 < (a − c)/γ, if q1 ≥ (a − c)/γ.


Let c < c˜1 (γ) ≡ a(2 − γ)/2. Then an interior intersection point exists, given by q1 (R, P ) = (2a − γa + γc)/(4 − γ 2 ), q2 (R, P ) = (2a − γa − m1 (m1 − 4c + cγ 2 ) } and 2c)/(4 − γ 2 ). Evaluations are uˆ1 (R, P ) = max{0, (4 − γ 2 )2 (2a − γa − 2c)2 uˆ2 (R, P ) = where m1 = 2a − γa + γc. Note that γ1 > 0 iff (4 − γ 2 )2 c < c˜(γ) ≡ a(2 − γ)/(4 − γ − γ 2 ), where c˜1 (γ) > c˜(γ). Hence, G(R, P ) has positive evaluations for both firms when c < c˜(γ). (ii) Consider the G(R, R) game. Equilibrium quantities are q1 (R, R) = q2 (R, R) = a/(2 + γ). Evaluations are uˆi (R, R) = a(a − 2c − γc)/(2 + γ)2 , i = 1, 2 which are positive iff c < c(γ) = a/(2 + γ). Proof of Proposition 1. By matrix A, note that (R, R) is an equilibrium aγ 2 (2 − γ) outcome if c < ≡ c2 (γ) where c2 (γ) < c(γ). Consider next the 4 pair (R, P ). This is an equilibrium outcome if uˆ1 (R, P ) > uˆ1 (P, P ) and uˆ2 (R, P ) > uˆ2 (R, R). The second inequality holds iff c > c2 (γ) while aγ 2 (2 − γ) ≡ c1 (γ) where c1 (γ) < c˜(γ) (and also the first holds iff c < 4 − γ3 c1 (γ) < c(γ)). Note next that c1 (γ) > c2 (γ) iff γ > 0. Hence (R, P ) is an equilibrium if c2 (γ) < c < c1 (γ). Finally, (P, P ) is an equilibrium if c > c1 (γ).


References 1. Baumol W. J., 1959. Business behavior, Value and Growth, New York: Macmillan. 2. Bhaskar V. 1988. The kinked demand curve: a game-theoretic approach. International Journal of Industrial Organization 6, 373-384. 3. Baumol W. J., 1962. The revenue maximization hypothesis, in M.L. Joseph, N.C Seeber and G.L. Bach eds, Economics Analysis and Policy, Background Readings for Current Issues, Prentice Hall, 220-226. 4. Hall, R. and C. Hitch 1939. Price theory and business behaviour. Oxford Economic Papers 2, 12-45. 5. Maskin E. and J. Tirole 1988. A theory of dynamic oligopoly, II: price competition, kinked demand curves, and Edgeworth cycles. Econometrica 56, 571-599. 6. Rotemberg, J.J. and G. Saloner 1987. The relative rigidity of monopoly pricing. American Economic Review 77, 917-926. 7. Sen D. 2004. The kinked demand curve revisited. Economics Letters 84, 99-105. 8. Sweezy, P. 1939. Demand under conditions of oligopoly. The Journal of Political Economy 4, 568-573.


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