Dynamic Price Competition

We consider the model of price competition for a single buyer among many sellers in a dynamic environment. The surplus from each trade is allowed to depend on the path of previous purchases, and as a result, the model captures phenomena such as learning by doing and habit formation in consumption characterize Markovian equilibria for finite and infinite horizon versions of the model and show that the stationary infinite horizon version of the model possesses an equilibrium where all the sellers receive an equilibrium payoff equal to their marginal contribution to the social welfare.Dynamic competition, Marginal contribution, Markov perfect equilibrium, Common agency

Does the absence of competition in the market foster competition for the market ? A dynamic approach to aftermarkets

In this paper, we investigate dynamic price competition when firms strategically interact in two distinct but interrelated markets : a primary market and an aftermarket, where indirect network effects arise. We set up a differential game of two-dimensional price competition and we conclude that the absence of price competition in the aftermarket (competition in the market) fosters dynamic price competition in the primary market (competition for the market). We also investigate the impact of network sizes on firms’ prices in the primary market concluding that, in equilibrium, larger firms have incentives to compete more fiercely for new ‘uncolonized’ consumers.dynamic competition; differential games; Linear Markov Perfect equilibrium

Does the absence of competition in the market foster competition for the market? A dynamic approach to aftermarkets

In this paper, we investigate dynamic price competition when firms strategically interact in two distinct but interrelated markets: a primary market and an aftermarket, where indirect network effects arise. We set up a differential game of two-dimensional price competition and we conclude that the absence of price competition in the aftermarket (competition in the market) fosters dynamic price competition in the primary market (competition for the market). We also investigate the impact of network sizes on firms' prices in the primary market concluding that, in equilibrium, larger firms have incentives to compete more fiercely for new "uncolonized" consumers.dynamic competition, differential games, Linear Markov Perfect Equilibrium, aftermarkets, network effects.

Dynamic Price Competition

We consider the model of price competition for a single buyer among many sellers in a dynamic environment. The surplus from each trade is allowed to depend on the path of previous purchases, and as a result, the model captures phenomena such as learning by doing and habit formation in consumption characterize Markovian equilibria for finite and infinite horizon versions of the model and show that the stationary infinite horizon version of the model possesses an equilibrium where all the sellers receive an equilibrium payoff equal to their marginal contribution to the social welfare

Dynamic price competition with fixed capacities

Many revenue management (RM) industries are characterized by (a) fixed capacities in the short term (e.g., hotel rooms, seats on an airline flight), (b) homogeneous products (e.g., two airline flights between the same cities at similar times), and (c) customer purchasing decisions largely influenced by price. Competition in these industries is also very high even with just two or three direct competitors in a market. However, RM competition is not well understood and practically all known implementations of RM software and most published models of RM do not explicitly model competition. For this reason, there has been considerable recent interest and research activity to understand RM competition. In this paper we study price competition for an oligopoly in a dynamic setting, where each of the sellers has a fixed number of units available for sale over a fixed number of periods. Demand is stochastic, and depending on how it evolves, sellers may change their prices at any time. This reflects the fact that firms constantly, and almost costlessly, change their prices (alternately, allocations at a price in quantity-based RM), reacting either to updates in their estimates of market demand, competitor prices, or inventory levels. We first prove existence of a unique subgame-perfect equilibrium for a duopoly. In equilibrium, in each state sellers engage in Bertrand competition, so that the seller with the lowest reservation value ends up selling a unit at a price that is equal to the equilibrium reservation value of the competitor. This structure hence extends the marginal-value concept of bid-price control, used in many RM implementations, to a competitive model. In addition, we show that the seller with the lowest capacity sells all its units first. Furthermore, we extend the results transparently to n firms and perform a number of numerical comparative statics exploiting the uniqueness of the subgame-perfect equilibrium.revenue management, bid-prices, subgame-perfect equilibrium.

Conformity based behavior and the dynamics of price competition: a new rational for fashion shifts

This paper deals with dynamic price competition in markets in which the perception of consumers regarding the value of goods depends on the choices of other consumers in the market. In particular, we consider the case in which consumers tend to imitate their peers, generating a conformity effect. In the context of a finite horizon model, we show that conformity based behavior creates new channels of dynamic interaction between firms, changing the nature of price competition. As time evolves, both price strategic complementarity and substitutability may arise along the equilibrium trajectory. This leads to V-shaped equilibrium price paths and oscillating trajectories of market shares. We provide also a new rational for the inversion of fashion trends.dynamic price competition, consumer behavior, conformity, fashion shift

Dynamic Duopoly with Inattentive Firms

This paper analyzes an infinite horizon dynamic duopoly with stochastic demand in which firms face costs of absorbing and processing information. Our main result is that the structure of dates at which firms choose to absorb information differ starkly between price and quantity competition. Firms synchronize their actions under price competition whereas they plan sequentially and in an alternating manner under quantity competition. The reason is that under quantity competition the planning firm reduces the uncertainty in the residual demand curve of the inattentive firm which renders planning less attractive for that firm. The opposite holds true under price competition.Inattentiveness, Price Competition, Quantity Competition, Synchronization

Price competition and convex costs

In the original model of pure price competition, due to Joseph Bertrand (1883), firms have linear cost functions. For any number of identical such price-setting firms, this results in the perfectly competitive outcome; the equilibrium price equal the firms’ (constant) marginal cost. This paper provides a generalization of Bertrand’s model from linear to convex cost functions. I analyze pure price competition both in a static setting - where the firms interact once and for all - and in dynamic setting - where they interact repeatedly over an indefinite future. Sufficient conditions are given for the existence of Nash equilibrium in the static setting and for subgame perfect equilibrium in the dynamic setting. These equilibrium sets are characterized, and it is shown that there typically exists a whole interval of Nash equilibrium prices in the static setting and subgame perfect equilibria in the dynamic setting. It is shown that firms may earn sizable profits and that their equilibrium profits may increase if their production costs go up.Bertrand competition;

Dynamic Price Competition with Price Adjustment Costs and Product Differentiation

We study a discrete time dynamic game of price competition with spatially differentiated products and price adjustment costs. We characterise the Markov perfect and the open-loop equilibrium of our game. We find that in the steady state Markov perfect equilibrium, given the presence of adjustment costs, equilibrium prices are always higher than prices at the repeated static Nash solution, even though, adjustment costs are not paid in steady state. This is due to intertemporal strategic complementarity in the strategies of the firms and from the fact that the cost of adjusting prices adds credibility to high price equilibrium strategies. On the other hand, the stationary open-loop equilibrium coincides always with the static solution. Furthermore, in contrast to continuous time games, we show that the stationary Markov perfect equilibrium converges to the static Nash equilibrium when adjustment costs tend to zero. Moreover, we obtain the same convergence result when adjustment costs tend to infinity.Price adjustment costs, Difference game, Markov perfect equilibrium, Open-loop equilibrium