Dynamic Analysis of Cournot-Bertrand Equilibria in Duopoly Models

On the Dynamic Analysis of On the Dynamic Analysis of Cournot Cournot- -Bertrand Equilibria Bertrand Equilibria Authors : Authors : Aggey AggeySemenov and Jean Semenov and Jean- -Baptise (Economic Letters 183 (2019)) (Economic Letters 183 (2019)) BaptiseTondji Tondji Presented by : Putriesti Mandasari Presented by : Putriesti Mandasari

Introduction Introduction Static models of product differentiation mostly focus on quantity (Cournot) and price (Bertrand) types of competition. There are sectors where firms engage in mixed (Cournot-Bertrand) competition. 30 out of 70 Japanese industries firms use some form of mixed competition (Flath, 2012). Market for Japanese home electronic industry : Panasonic (quantity competition) ; Sanyo (price competition) (Sato, 1996). Market for small cars : Honda & Subaru (quantity competition) ; Saturn & Scion (price competition) (Tremblay &Tremblay, 2011).

Motivation Motivation Literature on differentiated duopoly with linear demand (Singh & Vives, 1984; and Vives, 2001) found: With symmetric costs, firms prefer Cournot competition. When product are substitutes, firms prefer mixed (Cournot-Bertrand) competition. Firms often compete each other by investing in R & D to reduce production cost (Qiu, 1997). Such dynamic framework changes the post-innovation demand and cost structures, and might affect the market competition.

Research Questions Research Questions What are the comparisons between Cournot and Bertrand's competition in a heterogenous duopoly setting in terms of: Amount of R&D investment (R&D level) Price Quantity Welfare

The Model The Model Duopoly model, 2 firms i = 1, 2 and i j Differentiated goods (q1 and q2 ) = degree of product differentiation (differentiation as ) Consumer s Utility function: U(q1, q2) = (q1+q2) (q12+2 q1 q2 +q22)/2 where > 0 and [0,1] Inverse market demands (linear): Firms may invest in cost-reducing R&D. Pre-innovation costs (constant ): R&D spending { V(xi) } lowers its marginal cost by xi

Assuming R&D spending fs. : v = productivity of the R&D technology ( v means productivity ). xi = amount of R&D investment 2 stages game: Stage 1 : Firms (F1&F2) simultaneously invest in R&D with amount of xi. Firms observe the amount invested by their rivals (Fi observe xj) Stage 2: Firms engage in mixed competition (in the market stage, F1chooses output q1, F2 chooses a price p2) Consider SPNE

Backward induction Stage 2: Equilibrium of the market subgame : Equilibrium of the game :

The equilibrium R&D level: The equilibrium q & p : Denotes:

Guarantees positive investment of R&D for both firms xi > 0 Guarantees positive post-innovation costs ci = c > 0 Needed for second order and stability conditions. If R&D investment is very productive ( ? ), the firms will invest more (xi ) to gain competitive advantage that will lead c=0 or c<0 (post innovation cost). Throughout the remaining paper, we assume that Assumption 1 holds.

Consider & = equilibrium price & quantities in the game without R&D Equilibrium of prices, quantities, and profits in the game with R&D in Mixed competitions : Firm 1 (chooses output /Cournot) Firm 2 (chooses price /Bertrand) Profits

Consider the incentives to invest in R&D. Using FOC & Envelope Theorem : + - - - Strategic effect: a firm s R&D (xi) reduces its production cost (ci), and consequently affects the competitor s strategic choice. Size effect : q1 > q2 The size effects dominates the negative strategic effect. Without innovations: , with innovations : P1 < P2 because: 1. F1 invest more in cost-reduction than F2 x1 > x2 2. Residual demand curve is more elastic for F1 than F2 so the price changes in F1 will have bigger impact on the changes in quantity compares to F2 F1 has a stronger size effect Thus, F1 has a greater incentives to lower the price more efficient higher profit.

Bertrand and Cournot Competition

Assuming this condition holds to guarantee the regularity conditions of the equilibria: The Optimal R&D investments, prices & quantities :

Cournot Model: Strategic effect (+) Residual demand is more inelastic So, the size effect is weaker Bertrand Model : Strategic effect (-) The size effect is stronger

However, the size effect is not enough to overcome the differences in strategic effects xc > xB. The relative importance of positive strategic effect compared to the size effect of F2 x2 < xB. The ranking of investment by F1 depends on the level of substitutability of goods: (-) strategic effect becomes stronger X1 < Xc if high X1 < XB However, the differences in x (level of innovations) are not enough to change the ranking of prices and quantities..

CS, Profits, Welfare

With and Without cost-reducing R&D CSB>CSC For substitute goods and without R&D, Cournot s profit > Bertrand and mixed competition. For substitute goods and with R&D, Profit F1 (mixed) > Cournot s profit since x1 > xc WB > WC since Bertrand competition is always more efficient than Cournot. Mixed competition is in intermediate welfare.

1st & 2ndAssumptions hold, the Social Planners optimal decision is symmetric:

In the socials planner model, the strategic effect vanishes. Although F1 invests more than F2 (x1 > x2) it never overinvests. For close substitute goods, F1 may produce more than the social optimum. The (-) strategic effect is strong, so to overcome it, in equilibrium, F1 produces more than the socially optimum quantity.

Conclusion Conclusion In a 2-stage game of heterogenous duopoly setting : The quantity-setting firm invests more in R&D, has a lower price, and produces higher quantity than the price- setting firm. The mixed-competition yields greater CS & W than Cournot