arXiv subjects

Costis Melolidakis

Publications and source records attributed to Costis Melolidakis.

8 recordsLinked to original sources

arXiv2021

A Class of Distributions for Linear Demand Markets

In this paper, we study distributions that describe markets with linear stochastic demand. We express the price elasticity of expected demand in terms of the mean residual demand (MRD) function of the demand distribution and characterize optimal prices or equivalently, points of unitary elasticity, as fixed points of the MRD function. This leads to economic interpretable conditions on the demand distribution under which such fixed points exists and are unique. In particular, markets with increasing price elasticity of expected demand that eventually become elastic correspond to distributions with decreasing generalized mean residual demand (DGMRD) and finite second moment. DGMRD distributions strictly generalize the widely used increasing generalized failure rate (IGFR) distributions. In real life economic applications, they arise naturally as mixtures of (possibly) IGFR distributions over disjoint intervals. We further elaborate on the relationship of the two classes and link their limiting behavior at infinity. We examine moment and closure properties of the DGMRD distributions that are important in economic applications and illustrate our results with examples.

math.PR↗

arXiv2021

Comparative Statics via Stochastic Orderings in a Two-Echelon Market with Upstream Demand Uncertainty

We revisit the classic Cournot model and extend it to a two-echelon supply chain with an upstream supplier who operates under demand uncertainty and multiple downstream retailers who compete over quantity. The supplier's belief about retail demand is modeled via a continuous probability distribution function F. If F has the decreasing generalized mean residual life (DGMRL) property, then the supplier's optimal pricing policy exists and is the unique fixed point of the mean residual life (MRL) function. This closed form representation of the supplier's equilibrium strategy facilitates a transparent comparative statics and sensitivity analysis. We utilize the theory of stochastic orderings to study the response of the equilibrium fundamentals - wholesale price, retail price and quantity - to different demand distribution parameters. We examine supply chain performance, in terms of the distribution of profits, supply chain efficiency, in terms of the Price of Anarchy, and complement our findings with numerical results.

cs.GT↗

arXiv2021

Measuring Market Performance with Stochastic Demand: Price of Anarchy and Price of Uncertainty

Globally operating suppliers face the rising challenge of wholesale pricing under scarce data about retail demand, in contrast to better informed, locally operating retailers. At the same time, as local businesses proliferate, markets congest and retail competition increases. To capture these strategic considerations, we employ the classic Cournot model and extend it to a two-stage supply chain with an upstream supplier who operates under demand uncertainty and multiple downstream retailers who compete over quantity. The supplier's belief about retail demand is modeled via a continuous probability distribution function F. If F has the decreasing generalized mean residual life property, then the supplier's optimal pricing policy exists and is the unique fixed point of the mean residual life function. We evaluate the realized Price of Uncertainty and show that there exist demand levels for which market performs better when the supplier prices under demand uncertainty. In general, performance worsens for lower values of realized demand. We examine the effects of increasing competition on supply chain efficiency via the realized Price of Anarchy and complement our findings with numerical results.

cs.GT↗

arXiv2021

Monopoly Pricing in Vertical Markets with Demand Uncertainty

Pricing decisions are often made when market information is still poor. In turn, existing theoretical models often reason about the response of optimal prices to changing market characteristics without exploiting all available information about the demand distribution. Our aim is to develop a theory for the optimization and systematic comparison of prices between different instances of the same market under various forms of knowledge about the corresponding demand distributions. We revisit the classic problem of monopoly pricing under demand uncertainty in a vertical market with an upstream supplier and multiple forms of downstream competition between arbitrary symmetric retailers. In all cases, demand uncertainty falls to the supplier who acts first and sets a uniform price before the retailers observe the realized demand and place their orders. Our main methodological contribution is that we express the price elasticity of expected demand in terms of the mean residual demand (MRD) function of the demand distribution. This leads to a closed form characterization of the points of unitary elasticity that maximize the supplier's profits and the derivation of a mild unimodality condition for the supplier's objective function that generalizes the widely used increasing generalized failure rate (IGFR) condition. A direct implication is that optimal prices between different markets can be ordered if the markets can be stochastically ordered according to their MRD functions or equivalently, their elasticities. Using the above, we develop a systematic framework to compare optimal prices between different market instances via the rich theory of stochastic orders. This leads to comparative statics that challenge previously established economic insights about the effects of market size, demand transformations and demand variability on monopolistic prices.

math.OC↗

arXiv2021

Endogenizing the Cost Parameter in Cournot Oligopoly

We study the effects of endogenous cost formation in the classic Cournot oligopoly through an extended two-stage game. The competing Cournot firms produce low-cost but limited quantities of a single homogeneous product. For additional procurements, they may refer to a revenue-maximizing supplier who sets a wholesale price prior to their orders. We express this chain as a two-stage game and study its equilibrium under two different information levels: complete and incomplete information on the side of the supplier about the actual market demand. In the deterministic case, we derive the unique subgame perfect Nash equilibrium for different values of the retailers' capacity levels, supplier's cost and market demand. To study the incomplete information case, we model demand uncertainty via a continuous probability distribution. Under mild assumptions, we characterize the supplier's optimal pricing policy as a fixed point of a proper translation of his expectation about the orders that he will receive from the retailers. If this expectation is decreasing in his price, then such an optimal policy always exists and is unique. Based on this characterization, we are able to proceed with comparative statics and sensitivity analysis, both analytically and numerically. Incomplete information gives rise to market inefficiencies because the supplier may ask for a too high price. Increasing supplier's cost results in increasing wholesale prices, decreasing orders from the retailers and hence decreasing consumer surplus. Increasing retailers' production capacities results in decreasing wholesale prices and increasing consumer surplus. Finally, as the number of second-stage retailers increases, the supplier's profit may initially rise but eventually drops.

math.OC↗

arXiv2021

On the Equilibrium Uniqueness in Cournot Competition with Demand Uncertainty

We revisit the linear Cournot model with uncertain demand that is studied in Lagerlöf (2006)* and provide sufficient conditions for equilibrium uniqueness that complement the existing results. We show that if the distribution of the demand intercept has the decreasing mean residual demand (DMRD) or the increasing generalized failure rate (IGFR) property, then uniqueness of equilibrium is guaranteed. The DMRD condition implies log-concavity of the expected profits per unit of output without additional assumptions on the existence or the shape of the density of the demand intercept and, hence, answers in the affirmative the conjecture of Lagerlöf (2006)* that such conditions may not be necessary. *Johan Lagerlöf, Equilibrium uniqueness in a Cournot model with demand uncertainty. The B.E. Journal in Theoretical Economics, Vol. 6: Iss 1. (Topics), Article 19:1--6, 2006.

econ.TH↗

arXiv2021

On the Mean Residual Life of Cantor-Type Distributions: Properties and Economic Applications

In this paper, we consider the mean residual life (MRL) function of the Cantor distribution and study its properties. We show that the MRL function is continuous at all points, locally decreasing at all points outside the Cantor set and has a unique fixed point which we explicitly determine. These properties readily extend to the parametric family of p-singular, Cantor type distributions introduced by Mandelbrot (1983). The findings offer evidence that, contrary to common perceptions, Cantor-type distributions are tractable enough to be considered for practical applications. We provide such an example from the field of economics in which Cantor-type distributions can be used to model markets with recurrent bandwagon effects and show that earlier anticipated bandwagon effects lead to higher monopolistic prices. We conclude with a simple implementation of the algorithm by Chalice (1991) to plot Cantor-type distributions.

math.PR↗

arXiv2016

On the commitment value and commitment optimal strategies in bimatrix games

Given a bimatrix game, the associated leadership or commitment games are defined as the games at which one player, the leader, commits to a (possibly mixed) strategy and the other player, the follower, chooses his strategy after having observed the irrevocable commitment of the leader. Based on a result by von Stengel and Zamir [2010], the notions of commitment value and commitment optimal strategies for each player are discussed as a possible solution concept. It is shown that in non-degenerate bimatrix games (a) pure commitment optimal strategies together with the follower's best response constitute Nash equilibria, and (b) strategies that participate in a completely mixed Nash equilibrium are strictly worse than commitment optimal strategies, provided they are not matrix game optimal. For various classes of bimatrix games that generalize zero sum games, the relationship between the maximin value of the leader's payoff matrix, the Nash equilibrium payoff and the commitment optimal value is discussed. For the Traveler's Dilemma, the commitment optimal strategy and commitment value for the leader are evaluated and seem more acceptable as a solution than the unique Nash equilibrium. Finally, the relationship between commitment optimal strategies and Nash equilibria in $2 \times 2$ bimatrix games is thoroughly examined and in addition, necessary and sufficient conditions for the follower to be worse off at the equilibrium of the leadership game than at any Nash equilibrium of the simultaneous move game are provided.

cs.GT↗