Dynamic Bertrand and Cournot Competition

Dynamic Bertrand and Cournot Competition Effects of Product Differentiation Andrew Ledvina Department of Operations Research and Financial Engineering Princeton University Joint with Ronnie Sircar Princeton-Lausanne Workshop 13 May 2011

History Cournot (1838) was the first game-theoretic analysis of an oligopolistic market. In the Cournot model, firms chose a quantity of mineral water to extract from the ground. P = 1 Q where Q is the aggregate quantity supplied by the firms. Bertrand (1883) criticized the quantity setting assumption. In the Bertrand model, firms set prices, and the firm with the lowest price produces to meet all demand. Q = 1 P where P is the minimum price in the market. Both models assumed goods were homogeneous, e.g. mineral water. See Harris, et. al (2010) for the related homogeneous differential Cournot game. We shall only consider differentiated goods, e.g. soft drinks.

Differential Bertrand: Set-up We shall assume throughout this presentation that there are two firms in the market, i.e. we consider only the duopoly case. Each firm is assumed to have a fixed lifetime capacity of production at time t = 0 given by x i (0). This approach allows us to study the effect of firm size on price decisions. Firms employ Markovian pricing strategies so that p i (t) = p i (x(t)).

Differential Bertrand: Set-up Firms expect the market to demand at a rate q i (p), given the prices p i, i = 1, 2. We assume actual demands are subject to short-term unpredictable fluctuations: q i (t) = q i (p(x(t))) σ i ż i t, where {ż i t } i=1,...,n are correlated white noise sequences. The dynamics of capacity are given by dx i (t) = q i (t)dt, i.e. dx i (t) = q i (p 1 (x(t)), p 2 (x(t)))dt + σ i dw i t. This equation holds only for x i > 0. If x i (t) = 0 for any t, then x i (s) = 0 for all s t: zero is an absorbing boundary such that random shocks cannot resuscitate a firm that has gone out of business.

Differential Bertrand: Profit and Equilibrium Firm i = 1, 2 seeks to maximize its expected discounted lifetime profit { } E e rt p i (x(t))q i (p 1 (x(t)), p 2 (x(t)))1 {xi (t)>0} dt, 0 where r > 0 is a discount rate. An M.P. Nash equilibrium is a pair ( p 1 (x(t)), p 2 (x(t))) such that for i = 1, 2, j i, and for all x(0) R 2 +, { E 0 { E ( ) } e rt pi (x(t))qi pi (x(t)), pj (x(t)) 1 {xi (t)>0} dt 0 ( ) e rt p i (x(t))q i p i (x(t)), pj (x(t)) 1 {xi (t)>0} dt for any Markov Perfect strategy p i of player i. },

Linear Systems of Demand The linear inverse demand system for a duopoly is given by p i (q 1, q 2 ) = α β ( q i + εq j ), for i = 1, 2 and j i. Here, α > 0, β > 0, and ε [0, 1]. The term ε is the degree of product substitutability. ε = 0 implies independent goods, while ε = 1 implies homogeneous goods. We can invert this system (if ε < 1) to obtain the linear demand system q i (p 1, p 2 ) = α β(1 + ε) 1 β(1 ε 2 ) p ε i + β(1 ε 2 ) p j. For a firm with a monopoly, the demand and inverse demand consistent with the above are given by q m (p) = α β 1 β p, and pm (q) = α βq.

Definition of Value Functions in Bertrand Game For Firm i = 1, 2 we define the value function V i (x 1, x 2 ) = { } sup E e rt p i (x(t))q i (p 1 (x(t)), p 2 (x(t))) 1 {xi (t)>0} dt. p i 0 If the value function has sufficient regularity, we can apply dynamic programming arguments for nonzero-sum differential games. See for example Friedman (1971), Başar and Olsder (1995). There is also work towards regularity in differential games in Bensoussan, Frehse (2002), although these apply only in very restrictive settings.

System of HJB Equations in Bertrand Game Define the (shadow) costs S i (x) V i (x) ε V i (x) x i x j The system of HJB equations can then be written LV i q m (p j ) V i x j { ( )} + sup q i (p) p i S i (x) rv i = 0, p i for i = 1, 2 with j i, and where L is the second-order differential operator: L = 1 2 σ2 1 2 + ρσ x1 2 1 σ 2 2 x 1 x 2 + 1 2 σ2 2 2, x2 2 and ρ is the correlation coefficient of the Brownian motions, i.e. E { dwt 1dW t 2 } = ρ dt.

Analysis of PDE in Betrand Game The implicit equilibrium problem in the system of PDEs is a static game: sup {demand (price shadow cost)} price Given the unique static game Nash equilibrium strategies pi ( S 1 (x), S 2 (x) ), and the corresponding equilibrium demands q i, = q i (p1, p 2 ), we can write the PDE system LV i q m (p j ) V i x j + β(1 ε 2 ) ( q i, ) 2 rv i = 0.

Small Degree of Substitutability Asymptotics (σ = 0) We look for an approximation to the solution of the system of PDEs of the form V i (x 1, x 2 ) = v (0) (x i ) + εv i,(1) (x 1, x 2 ) + ε 2 v i,(2) (x 1, x 2 ) + In the case ε = 0, the demand function for Firm i depends only on p i and not p j. This corresponds to the firms having monopolies in independent markets. We therefore expect for small ε to be able to find a solution close to the monopoly problem. Let v M be the value function for a monopolist. We find 1 ( ) α v 2 4β M rvm = 0, v M (0) = 0.

Solution of Monopoly Problem Proposition The value function for the monopoly with σ = 0 is v M (x) = α2 [ ( W e µx 1) 2 + 1], (1) 4βr where µ = (2βr)/α and W is the Lambert W function defined by the relation Y = W(Y )e W(Y ) with domain Y e 1. In terms of the asymptotic expansion, we find v (0) (x i ) v M (x i ).

Asymptotic Expansion at 1st Order We define λ(x) q m ( p ( v M (x))) = 1 ( 2β α v M (x) ). Let A = λ(x 1 ) x 1 + λ(x 2 ) x 2 + r. The 1st order correction satisfies the linear first-order PDE Av i,(1) = λ(x 1 )λ(x 2 ) for i = 1, 2 and j i, with v i,(1) (0, x 2 ) v i,(1) (x 1, 0) 0.

1st Order Value Function Correction Proposition The solution v 1,(1) is given, for x 1 > x 2, by v 1,(1) (x 1, x 2 ) = α2 ( 4β 2 e rλ(x2) (1 + rλ(x 2 )) (2) r ) e rλ(x1) (1 rλ(x 2 )) + e r(λ(x 1)+Λ(x 2 )) 1, where Λ(x) x 0 1 ( ( λ(u) du = 1 r log W e µx 1)), and, for x 2 x 1, by reversing the roles of x 1 and x 2 in Eqn. (2). This is clearly the same for v 2,(1), i.e. v 2,(1) v 1,(1).

2nd Order PDE Problem in Bertrand Game We can perform a similar analysis to obtain a PDE for the second-order correction term. We find again a linear first-order PDE with the same differential operator A. ( ) Av i,(2) = 1 2β λ(x v i,(1) i) + 1 v j,(1) v i,(1) λ(x i ) + λ(x j ) x j 2β x j x i + 1 4β (λ(x j)) 2 1 β (λ(x i)) 2 + 1 v i,(1) v j,(1) + 1 2β x j x j 4β ( ) 2 v i,(1). (3) x i

Solution of 2nd Order PDE Problem in Bertrand Game Proposition The solution of (3) is given, for x 1 > x 2 by [ ( ) v 1,(2) = α2 8β e rλ(x 2) Λ(x 2) 2 2re rλ(x 2) 2 e rλ(x2) 1 + rϕ 1 1 e rλ(x 1) Λ(x 2)(1 ϕ 1 ) 1 e rλ(x 1) + 1 ϕ 1 2r 3 r + 3 ( ) e rλ(x2) (1 ϕ 1 ) 2 1 2r 2r ( e ( rλ(x 2) 1 e )) rλ(x 1) ( e rλ(x2) ϕ 2 1 e rλ(x2) 1 + ϕ 2 1 2ϕ 1rΛ(x 2 )) ]. where ϕ 1 = exp { r (Λ(x 1 ) Λ(x 2 ))}. The solution for x 2 x 1 is given by a similar expression. The solution for Firm 2 can be found by symmetry through the relationship v 2,(2) (x 1, x 2 ) = v 1,(2) (x 2, x 1 ).

Differential Cournot Game with Differentiated Goods We have also analyzed the game in which firms set quantities rather than prices. The structure of the analysis is very similar to the Bertrand game. We are still in the linear duopoly framework, but we work with inverse demand rather than direct demand. Carrying out the same asymptotic analysis, remarkably, v c,(0) v b,(0) and v i,c,(1) v i,b,(1). Therefore, up to a first-order approximation, these two types of competition are identical.

2nd Order PDE Problem in Cournot Game Carrying out the asymptotic analysis to obtain the 2nd order correction, we find a linear first-order PDE which is nearly identical to that obtained in the Bertrand game. ( ) Av i,c,(2) = 1 2β λ(x v i,(1) i) 1 v j,(1) v i,(1) λ(x i ) + λ(x j ) x j 2β x j x i + 1 4β (λ(x j)) 2 + 1 2β (λ(x i)) 2 + 1 v i,(1) v j(1) + 1 2β x j x j 4β ( ) 2 v i,(1). (4) x i

Monopoly and 1st Order Approximation 9 Monopoly Value Function 8 7 6 v m (x) 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 x (a) Monopoly (b) First Order Approximation Figure: v m (x) and v (0) (x 1 ) + εv (1) (x 1, x 2 ).

2nd Order Approximation Surfaces (a) Bertrand (b) Cournot Figure: v (0) (x 1 ) + εv 1,(1) (x 1, x 2 ) + ε 2 v 1,(2) 1 (x 1, x 2 ). Approximate value functions for Bertrand and Cournot games.

Comparing Price and Capacity Paths 5 4.5 4 Capacities (Firm 1) Bertrand Cournot 6 5.5 Bertrand Cournot Prices (Firm 1) 3.5 5 3 4.5 2.5 2 4 1.5 3.5 1 0.5 3 0 0 0.5 1 1.5 2 2.5 3 (a) Capacity Paths 2.5 0 0.5 1 1.5 2 2.5 3 (b) Price Paths Figure: Bertrand and Cournot Paths with 2nd Order Approximation with ε = 0.2

Numerical Analysis: Implicit Finite Difference (a) Bertrand (b) Cournot Figure: Numerical solution of PDE Systems for value functions of Firm 1.

Finite Difference Solution of PDEs: θ tan 1 (x 2 /x 1 ) 3.8 Price as a function of market share 3.4 Price as a function of market share 3.7 3.6 Cournot Bertrand 3.3 3.2 Cournot Bertrand 3.5 3.1 Price for Firm 1 3.4 3.3 3.2 Price for Firm 1 3 2.9 3.1 2.8 3 2.7 2.9 2.6 2.8 0.2 0.4 0.6 0.8 1 1.2 1.4 θ 2.5 0.2 0.4 0.6 0.8 1 1.2 1.4 θ (a) ε = 0.2 (b) ε = 0.4 Figure: Equilibrium price as a function of market share

Conclusion In a dynamic monopoly, the optimal strategy of a firm does not depend on the type of market (i.e. Bertrand or Cournot) For a small degree of substitution, the equilibrium strategies still do not depend on the type of market. Therefore, the choice of strategic variable has little effect on the decisions of firms if they produce a good which is highly differentiated from that of their competitors. At the second-order of our approximation, we see that the value functions are different and therefore the strategic behavior of firms is dependent on the type of market. Price is higher in a Cournot market initially, but as capacities decline at a faster rate in Bertrand, price is eventually higher in Bertrand.

Extra Figures: 2nd Order Surfaces (a) Bertrand (b) Cournot Figure: 2nd order correction functions: v 1,(2) (x 1, x 2 )