Stackelberg Equilibrium with Many Leaders and Followers. The Case of Setup Costs

I provide conditions that guarantee that a Stackelberg game with a setup cost and an integer number of leaders and followers has an equilibrium in pure strategies. The main feature of the game is that when the marginal follower leaves the market the price jumps up, so that a leader’s payoff is neither continuous nor quasiconcave. To show existence I check that a leader’s value function satisfies the following single crossing condition: When the other leaders produce more the leader never accommodates entry of more followers. If demand is strictly logconcave, and if marginal costs are both non decreasing and not flatter than average costs, then a Stackelberg equilibrium exists. Besides showi…

Drastic innovation reduces firms’ incentives to create divisions

I study a game in which two firms create independent divisions, then they choose whether to do R&D so as to reduce their divisions’ marginal costs, and then the divisions compete in the market. I provide necessary and sufficient conditions under which the game has an equilibrium in pure strategies, and I show that the game has an equilibrium only if each firm threatens that if the rival creates more divisions it will use R&D to foreclose the market. The case we find in the literature, in which firms flood the market with their divisions, should happen only in industries with low returns to R&D.

R&D WITH SPILLOVERS: MONOPOLY VERSUS NONCOOPERATIVE AND COOPERATIVE DUOPOLY

This paper compares industry profit and R&D propensity for a duopoly conducting either noncooperative or cooperative R&D and a monopoly, using two different basic models of strategic R&D. One postulates spillovers in R&D inputs and predicts that equilibrium joint profit and R&D levels are always larger under monopoly. The other postulates spillovers in R&D outputs and sometimes predicts that joint profit and R&D levels are larger under either of the alternative scenarios. In addition, unlike input spillovers, spillovers in R&D outputs sometimes exert a positive effect on both effective and private noncooperative R&D levels.

Endogenous firm asymmetry and cooperative R&D in linear duopoly with spillovers

In a linear model ofcost reducing R&D/Cournot competition, firm asymmetry is shown to be sustainable as subgame perfect Nash equilibrium with R&D competition only ifthe productivity of research is sufficiently large relative to the benefits from imitation. In such a case, industry-wide cost reduction and firms asymmetry are increasing and decreasing functions of the spillover rate, respectively. In the absence of spillovers, a symmetric joint lab generates higher consumer surplus and social welfare than a pair ofasymmetric competitors. If spillovers are not too small, asymmetric R&D competition is advantageous toconsumers, but not to firms.

Competing R&D Joint Ventures in Cournot oligopoly with spillovers

This paper considers competition between R&D cartels, whereby prospective Cournot competitors coordinate their R&D decisions in order to maximize joint profit. It studies how R&D activity, aggregate profit, consumer surplus, and social welfare vary as the number of competing cartels varies. It also compares equilibrium with second best R&D, and discusses the policy implications of the results. The results show that the effects of R&D cartel competition depend on the welfare criterion adopted and on whether there are cooperative synergies or not.

Un anno di "Journal of Regional Science"

ENDOGENOUS FIRM ASYMMETRY VS SYMMETRY IN OLIGOPOLY MODELS

Endogenous asymmetry and cooperative R&D in linear duopoly with spillovers

In a standard model of R&D followed by linear Cournot competition, firm asymmetry is sustainable as equilibrium with non cooperative R&D if and only if the productivity of research is sufficiently large relative to the benefits of imitation. Increasing spillovers distribute R&D results among asymmetric competitors, causing price, firm asymmetry, and joint profit to reduce. With zero spillovers, a symmetric joint lab dominates asymmetric R&D competition in terms of social welfare and consumer surplus, but is sometimes dominated in terms of joint profit. Raising spillovers encourage symmetric collusion but makes the latter potentially harmful to consumers. (JEL : C72; L13; O32).

Stackelberg equilibrium with multiple firms and setup costs

Abstract I provide conditions that guarantee that a Stackelberg game with a setup cost and an integer number of identical leaders and followers has an equilibrium in pure strategies. The main feature of the game is that when the marginal follower leaves the market the price jumps up, so that a leader’s payoff is neither continuous nor quasiconcave. To show existence I check that a leader’s value function satisfies the following single crossing condition: When the other leaders produce more the leader never accommodates entry of more followers. If demand is strictly logconcave, and if marginal costs are both non decreasing and not flatter than average costs, then a Stackelberg equilibrium ex…

Strictly convex variable cost does not imply U-shaped average cost

Abstract We show that strictly convex variable costs do not imply U-shaped average costs and provide a sufficient condition for U-shaped average costs. As an application we study endogenous entry when firms have market power and they have decreasing average cost but increasing marginal cost.

R&D Reduces Firms' Incentives to Create Divisions

I study a game in which two firms create independent divisions, then they choosewhether to do R&D so as to reduce their divisions' marginal costs, and then the divisions compete in the market. I provide necessary and sufficient conditions under which the game has an equilibrium in pure strategies, and I show that the game has an equilibrium only if each firm threatens that if the rival creates more divisions it will use R&D to foreclose the market. The case we find the literature, in which firms flood the market with their divisions, should happen only in industries with low returns to R&D.

Stackelberg equilibrium with many leaders and followers. The case of zero fixed costs

Abstract I study a version of the Stackelberg game with many identical firms in which leaders and followers use a continuous cost function with no fixed cost. Using lattice theoretical methods I provide a set of conditions that guarantee that the game has an equilibrium in pure strategies. With convex costs the model shows the same properties as a quasi-competitive Cournot model. The same happens with concave costs, but only when the number of followers is small. When this number is large the leaders preempt entry. I study the comparative statics and the limit behavior of the equilibrium and I show how the main determinants of market structure interact. More competition between the leaders …

R&D with spillovers: Monopoly versus noncooperative and cooperative duopoly.

This paper compares industry profit and R&D propensity for a duopoly conducting either noncooperative or cooperative R&D and a monopoly, using two different basic models of strategic R&D. One postulates spillovers in R&D inputs and predicts that equilibrium joint profit and R&D levels are always larger under monopoly. The other postulates spillovers in R&D outputs and sometimes predicts that joint profit and R&D levels are larger under either of the alternative scenarios. In addition, unlike input spillovers, spillovers in R&D outputs sometimes exert a positive effect on both effective and private noncooperative R&D levels.

A Dynamic Model of Open Source Vs Proprietary R&D

Abstract We propose a dynamic model in which firms compete to produce sequential and cumulative innovations, and in which the more firms do research in one sector the more likely it is that one of them innovates. Firms choose research effort and whether to patent innovations or to use an Open Source license like the General Public License. We show that ( i ) patents generate a larger stationary reward but foreclose research within a sector, and that ( ii ) Open Source generates a smaller stationary reward but allows everyone to use the technology, and therefore, by attracting firms to the sector, it induces a faster pace of innovation. We characterize all the equilibria of the model and sho…

ENDOGENOUS TIMING WITH FREE ENTRY

A free entry model with linear costs is considered where firms first choose their entry time and then compete in the market according to the resulting timing decisions. Multiple equilibria arise allowing for infinitely many industry output configurations encompassing one limit-output dominant firm and the Cournot equilibrium with free entry as extreme cases. Sequential entry is never observed. Both Stackelberg and Cournot-like outcomes are sustainable as equilibria however. When the number of incumbents is given, entry is always prevented, and industry output is sometimes larger than the entry preventing level.

Stable sharing rules and participation in pools of essential patents

Abstract For pools of essential patents I study whether a pool's sharing rule is stable against arbitrage, so that the pool's members have no incentive to trade patents. I show that the only stable rule is the numeric proportional rule, which gives each member a share of the pool's profit equal to its share of the pool's patents. I study how the stable rule affects firms' incentives to participate, and I show that firms with few patents tend to remain outside the pool. I look at the trade off between stability and participation, and I show that as trade dilutes their shares, members prefer the stable rule. I consider individual licenses, stand-alone patents, integration, and R&D. The result…

Allocating cost reducing investments over competing divisions

This paper examines a three-stage model of divisionalization where, first, two parent firms create independent units, second, the parent firms allocate cost reduction levels over these units, and third, the resulting units compete in a Cournot market given their current costs of production. The introduction of the cost reduction phase is shown to reduce the incentives toward divisionalization severely, relative to other existing models. Namely, the scope for divisionalization in equilibrium reduces as the marginal cost of the cost reducing investment decreases, and eventually vanishes. A second-best welfare analysis shows that, for any given market structure, the equilibrium investment deci…

Endogenous R&D Symmetry in Linear Duopoly with One-way Spillovers

A duopoly model of cost reducing R&D-Cournot market competition is extended to encompass endogenous timing of R&D investments. Under the assumption that R&D spillovers are zero under simultaneous choices of R&D and only flow from the R&D leader to the follower under sequential choices, sequential and simultaneous play at the R&D stage are compared in order to assess the role of technological externalities in stimulating or attenuating endogenous firm asymmetry. The only timing structure of the R&D stage sustainable as subgame–perfect Nash equilibrium involves simultaneous play and thus zero spillovers.

Endogenous timing with infinitely many firms

Abstract A model with constant marginal costs is considered where firms choose first a period for production and then the amount to produce when competing in the market according to the resulting timing decisions. Multiple equilibria arise allowing for infinitely many industry output configurations encompassing one limit-output dominant firm and the Cournot equilibrium with free entry as extreme cases. At each of these equilibria a firm produces a positive amount only if this firm commits to produce at period one. Both Stackelberg and Cournot-like outcomes are sustainable as equilibria however. When the number of leaders is given, production at subsequent periods is always prevented, and in…

Cumulative innovation, open source, and distance to frontier

We develop a multistage game in which firms do cumulative research and development (R&D) to complete a lengthy process, and we study whether firms patent intermediate results or release them in Open Source. A patent holder obtains a larger reward in the market, but since in equilibrium it forecloses R&D, it remains alone to complete the process and so pays a larger cost than an Open Source firm. We have Open Source equilibria when R&D is highly complementary, R&D costs are large, and firms are sufficiently different and far from the frontier. We identify two market failures, in the forms of free riding and coordination failure, and we discuss public intervention.

A Further Note on Endogenous Spillovers in a Non-tournament R&D Duopoly

This note considers the paper of Poyago-Theotoky (1999) on strategic R&D with endogenous spillovers. It proves through an example that, under R&D collusion, optimality sometimes requires either minimal or asymmetric spillovers. It also provides a simple sufficient condition for optimal spillovers between colluding firms to involve maximal spillovers (i.e., complete sharing of information).