Pricing with markups in industries with increasing marginal costs

We study a game that models a market in which heterogeneous producers of perfect substitutes make pricing decisions in a first stage, followed by consumers that select a producer that sells at lowest price. As opposed to Cournot or Bertrand competition, producers select prices using a supply function that maps prices to production levels. Solutions of this type of models are normally referred to as supply function equilibria. We consider a market where producers’ convex costs functions are proportional to each other, depending on the efficiency of each particular producer. We provide necessary and sufficient conditions for the existence of an equilibrium that uses simple supply functions that replicate the cost structure. We then specialize the model to monomial cost functions with exponent \(q>0\) , which allows us to reinterpret the simple supply functions as a markup applied to the production cost. We prove that an equilibrium for the markups exists if and only if the number of producers in the market is strictly larger than \(1+q\) , and if an equilibrium exists, it is unique. The main result for monomials is that the equilibrium nearly minimizes the total production cost when the market is competitive. The result holds because when there is enough competition, markups are bounded, thus preventing prices to be significantly distorted from costs. Focusing on the case of linear unit-cost functions on the production quantities, we characterize the equilibrium accurately and refine the previous result to establish an almost tight bound on the worst-case inefficiency of equilibria. Finally, we derive explicitly the producers’ best response for series-parallel networks with linear unit-cost functions, extending our previous result to more general topologies. We prove that a unique equilibrium exists if and only if the network that captures the market structure is 3-edge-connected. For non-series-parallel markets, we provide an example that does not admit an equilibrium on markups.     

On a generalized Cournot oligopolistic competition game

We consider a model whereby players compete for a set of shared resources to produce and sell substitute products in the same market, which can be viewed as a generalization of the classical Cournot oligopolistic competition model, or, from a different angle, the Wardrop type routing model. In particular, we suppose that there are K players, who compete for the usage of resources as well as the sales of the end-products. Moreover, the unit costs of the shared resources and the selling prices of the products are assumed to be affine linear functions in the consumption/production quantities. We show that the price of anarchy in this case is lower bounded by 1/K, and this bound is essentially tight, which manifests the harsh nature of the competitive market for the producers.     

The effect of supplier capacity on the supply chain profit

In this paper, we study the role of capacity on the efficiency of a two-tier supply chain with two suppliers (leaders, first tier) and one retailer (follower, second tier). The suppliers compete via pricing (Bertrand competition) and, as one would expect in practice, are faced with production capacity. We consider a model with differentiated substitutable products where the suppliers are symmetric differing only by their production capacity. We characterize the prices, production amounts and profits in three cases: (1) the suppliers compete in a decentralized Nash equilibrium game, (2) the suppliers “cooperate” to optimize the total suppliers’ profit, and (3) the two tiers of the supply chain are centrally coordinated. We show that in a decentralized setting, the supplier with a lower capacity may benefit from restricting her capacity even when additional capacity is available at no cost. We also show that the loss of total profit due to decentralization cannot exceed 25 % of the centralized chain profits. Nevertheless, the loss of total profit is not a monotonic function of the “degree of asymmetry” of the suppliers’ capacities. Furthermore, we provide an upper bound on the supplier profit loss at equilibrium (compared with the cooperation setting) that depends on the “market power” of the suppliers as well as their market size. We show that there is less supplier profit loss as the asymmetry (in terms of their capacities) increases between the two suppliers. The worst case arises when the two suppliers are completely symmetric.     

Two-period dynamic versus fixed-ratio pricing in a capacity constrained duopoly

This paper analyzes the impact of dynamic and fixed-ratio pricing policies on firm profits and equilibrium prices under competition. Firms that have equal inventories of perfectly substitutable and perishable products compete for customer segments that demand the product at different times. In each period, customers first purchase from the low price firm and then from the high price firm up to their inventories, provided the prices are lower than the maximum they are willing to pay. The main conclusions of this paper are as follows: although dynamic pricing is a more sophisticated policy than fixed-ratio pricing, it may lead to decreased equilibrium profits; under both pricing policies, one firm assumes the role of a low-cost high-output firm while the other assumes the role of a high-cost low-output firm; and, the supply demand ratio has more impact on the outcome of the competition than the heterogeneity in consumer reservation prices.     

Marine inventory routing: shipments planning

This paper addresses a shipments-planning problem faced by producers of large volume liquid bulk products. Producing origins with limited tank storage capacity supply multiple products by ships (or barges) to consuming destinations that also have limited storage capacity. Timing, origin, destination, and product quantities of shipments have to be determined in a manner that minimizes costs and does not violate storage capacity constraints at both ends (neither stopping production at the origins, nor running out of stock at the destinations). A mixed integer-programming model is used to derive cost effective solutions within a few minutes. A cost-based heuristic algorithm is used to assure that acceptable solutions are obtained quickly.     

A differential oligopoly game with differentiated goods and sticky prices

We investigate a dynamic oligopoly game where goods are differentiated and prices are sticky. We study the open-loop and the closed-loop memoryless Nash equilibrium, and show that the latter equilibrium entails a larger level of steady state production as compared to the former; both equilibria entail a larger level of production in steady state than the static game. We also study the effects of price stickiness and product differentiation upon the steady state equilibrium allocation and profits. The per-firm equilibrium output is increasing in both product differentiation and price stickiness, while profits are increasing in both product differentiation and the speed of price adjustment. The steady state social welfare monotonically increases in the speed of price adjustment, and the overproduction entailed by dynamic competition has beneficial effect from a social standpoint.     

Capacity expansion and forward contracting in a duopolistic power sector

The surge in demand for electricity in recent years requires that power companies expand generation capacity sufficiently. Yet, at the same time, energy demand is subject to seasonal variations and peak-hour factors that cause it to be extremely volatile and unpredictable, thereby complicating the decision-making process. We investigate how power companies can optimise their capacity-expansion decisions while facing uncertainty and examine how expansion and forward contracts can be used as suitable tools for hedging against risk under market power. The problem is solved through a mixed-complementarity approach. Scenario-specific numerical results are analysed, and conclusions are drawn on how risk aversion, competition, and uncertainty interact in hedging, generation, and expansion decisions of a power company. We find that forward markets not only provide an effective means of risk hedging but also improve market efficiency with higher power output and lower prices. Power producers with higher levels of risk aversion tend to engage less in capacity expansion with the result that together with the option to sell in forward markets, very risk-averse producers generate at a level that hardly varies with scenarios.     

Cournot vs. Bertrand under relative performance delegation: Implications of positive and negative network externalities

This paper demonstrates that relative-performance based strategic managerial delegation does not lead to the equivalence of Bertrand and Cournot equilibria in the presence of network externalities, regardless of the strength and type of network externalities — positive or negative. In the presence of positive network externalities, under relative-performance based delegation, Bertrand competition yields lower prices and profits, and higher quantities, consumers surplus and welfare than Cournot competition. On the contrary, these rankings are completely reversed in the presence of negative network externalities. It also discusses the endogenous choice of price or quantity contract under delegation in the presence of network externalities.     

Entropy methods for identifying hedonic models

This paper contributes to the literature on hedonic models in two ways. First, it makes use of Queyranne’s reformulation of a hedonic model in the discrete case as a network flow problem in order to provide a proof of existence and integrality of a hedonic equilibrium and efficient computation of hedonic prices. Second, elaborating on entropic methods developed in Galichon and Salanié (Cupid’s invisible hand: social surplus and identification in matching models. Working Paper, 2014), this paper proposes a new identification strategy for hedonic models in a single market. This methodology allows one to introduce heterogeneities in both consumers’ and producers’ attributes and to recover producers’ profits and consumers’ utilities based on the observation of production and consumption patterns and the set of hedonic prices.     

Open versus closed loop capacity equilibria in electricity markets under perfect and oligopolistic competition

We consider two game-theoretic models of the generation capacity expansion problem in liberalized electricity markets. The first is an open loop equilibrium model, where generation companies simultaneously choose capacities and quantities to maximize their individual profit. The second is a closed loop model, in which companies first choose capacities maximizing their profit anticipating the market equilibrium outcomes in the second stage. The latter problem is an equilibrium problem with equilibrium constraints. In both models, the intensity of competition among producers in the energy market is frequently represented using conjectural variations. Considering one load period, we show that for any choice of conjectural variations ranging from perfect competition to Cournot, the closed loop equilibrium coincides with the Cournot open loop equilibrium, thereby obtaining a ‘Kreps and Scheinkman’-like result and extending it to arbitrary strategic behavior. When expanding the model framework to multiple load periods, the closed loop equilibria for different conjectural variations can diverge from each other and from open loop equilibria. We also present and analyze alternative conjectured price response models with switching conjectures. Surprisingly, the rank ordering of the closed loop equilibria in terms of consumer surplus and market efficiency (as measured by total social welfare) is ambiguous. Thus, regulatory approaches that force marginal cost-based bidding in spot markets may diminish market efficiency and consumer welfare by dampening incentives for investment. We also show that the closed loop capacity yielded by a conjectured price response second stage competition can be less or equal to the closed loop Cournot capacity, and that the former capacity cannot exceed the latter when there are symmetric agents and two load periods.     

Semi-Lagrangean approach for price discovery in markets with non-convexities

From standard economic theory, the market clearing price for a commodity is set where the demand and supply curves intersect. Convexity is a property that economic models require for a competitive equilibrium, which is efficient and well-behaved and provides equilibrium prices. However, some markets present non-convexities due to their cost structure or due to some operational constraints that need to be addressed. This is the case for electricity markets where the electricity producers incur costs for shutting down a generating unit and then bringing it back on. Non-convex cost structures can be a challenge for the price discovery process, since the supply and demand curves may not intersect, or if they intersect, the price found may not be high enough to cover the total cost of production. We apply a Semi-Lagrangean approach to find a price that can be applied in the electricity pool markets where a central system operator decides who produces and how much they should produce. By applying the model to an example from the literature, we found prices that are high enough to cover the producer’s total costs, and follows the optimal solution for achieving mining cost in production. The prices are an alternative solution to the price discovery problem in non-convexities economies; in addition, they provide nonnegative profits to all the generators without the use of side-payments or up-lifts, and closes the integrality gap.     

Learning,convergence and economic constraints

This article investigates a partial equilibrium production model with dynamic information aggregation. Firms use observed prices to estimate the unknown model parameter by applying Bayesian learning. In the baseline setting, the demand structure is linear and the noise term is Gaussian. Then, prices and quantities are supported by the real line and convergence of the limited information to rational expectations quantities is obtained. Since a production economy is considered, the economic constraint of non-negative quantities is imposed. This non-negativity constraint and the assumption that signals about demand are only received in periods where production takes place destroy the “optimistic” convergence result observed in the baseline model. With this constraint firms learning an unknown demand intercept parameter exit with strictly positive probability, even when the true value of this parameter would induce production in the full information setting. In a second step, the linear demand structure is replaced by piece-wise linear demand, such that prices become non-negative. Also in this stetting the convergence result of the baseline model does not hold.     

消费者连续统集合下一种投入产出经济的均衡和一般均衡价格

本给出了一个经济模型把跨期间的一般均衡理论和neo-Ricardian的生产价格理论联系起来，并改进了Roce-Anne Oana模型[1][2]。即把消费有限改变消费连续统，并在Rose-Anne Dana模型的基础上，加了条件C.4和C.7，得出了两个重要的结果：（1）在标准条件下，此模型有一均衡使得每一个生产的每期最大利润是相等的；（2）如考虑生产价格是稳定价格；且等于最大利润；则在适当的条件下，这样的价格系统是存在且唯一的，而所定义的均衡的非折扣的价格序列收敛于生产价格。因此均衡价格依赖于技术和消费特征，但在取极限时，有关价格仅依赖于技术。     

Dynamic oligopoly pricing with reference-price effects

We analyze the strategic implications of consumers’ reference-price effects, either symmetric (for loss-neutral consumers) or asymmetric (for loss-averse consumers), in a differentiated oligopoly model where firms compete either in prices (à la Bertrand) or in quantities (à la Cournot) over an infinite time horizon. The dynamic game is specified in continuous time. The solution concept is Markov Perfect Equilibrium. We show how price dynamics in the presence of reference-price effects crucially depends on the nature of market competition. One of the main results of our analysis is that, with loss-averse consumers, there exists an interval of initial reference prices such that firms adopt the same constant-pricing strategy in both the Bertrand and the Cournot games, implying that the distinction between price and quantity competition has no impact on market conduct and performance.     

Price and quality competition: The effect of differentiation and vertical integration

This paper studies an instance of price and quality competition between firms as seen in the recent Internet market. Consumers purchase a product based on not only its price but also its quality level; therefore, two firms compete in determining their prices and quality levels to maximize their profits. Characterizing this competition from a microeconomic viewpoint, we consider two possible business strategies that firms can utilize to overcome the competition—the differentiation and the vertical integration with another complementary firm. We show an interesting result not seen in the well-known Bertrand price competition: not only does the differentiation always increase the firms’ profits, but also it can increase the consumer’s welfare in a quality-sensitive market. We further derive that under some mild conditions the monopolistic vertical integration that excludes the combination-purchase with a competitor’s product is beneficial for both the integrated firm and its consumers.     

Can piracy lead to higher prices in the music and motion picture industries?

Piracy of copyrighted goods has received increased attention in the literature. Much of this research has focused on pricing policies, protection against piracy, and governmental policies in the software industries. In this paper, we focus on pricing policies of producers in the music and motion picture industries. Exact analytical results are difficult to obtain; therefore, we develop an approximating function of the cumulative demand. This enables us to obtain closed-form expressions for the optimal price. Our results show that the existence of piracy in these industries and the lack of positive network externalities may cause monopolists to charge higher prices to optimize profits. These prices increase with increases in the speed of piracy and longer product lifecycles. We demonstrate the accuracy of our demand approximation function using a numerical experiment. We show how a two-price strategy and dual distribution channels may help in reducing the negative effects of piracy. We perform some numerical sensitivity analysis and provide managerial insights.     

Consumer environmental awareness and competition in two-stage supply chains

This paper focuses on the impact of competition and consumers’ environmental awareness on key supply chain players. We consider both the production competition between partially substitutable products made by different manufacturers, and the competition between retail stores. We use two-stage Stackelberg game models to investigate the dynamics between the supply chain players given three supply chain network structures. We find that as consumers’ environmental awareness increases, retailers and manufacturers with superior eco-friendly operations will benefit; while the profitability of the inferior eco-friendly firm will tend to increase if the production competition level is low, and will tend to decrease if the production competition level is high. In addition, higher levels of retail competition may make manufacturers with inferior eco-friendly operations more likely to benefit from the increase of consumers’ environmental awareness. Moreover, as production competition intensifies, the profits of the retailers will always increase, while the profits of the manufacturers with inferior eco-friendly operations will always decrease. The profitability of the manufacturers with superior eco-friendly operations will also tend to decrease, unless consumers’ environmental awareness is high and the superior manufacturer has a significant cost advantage related to product environmental improvement.     

多阶段生产系统最优生产批量和原材料订购决策模型

Sarker和Parija(1996)建立了生产系统最优生产批量和原材料订购决策模型。然而他们的模型仅局限于单阶段生产系统,本文将他们的模型扩展到多阶段生产系统,我们首先建立了使整个多阶段生产系统总成本最小的各阶段最优生产批量、原材料订购批量及阶段之间的运输批量模型,然后分析了原材料订购费、半成品运费及设备安装费的敏感性。最后,我们结合实例综合分析了原材料订购费、半成品运输费和设备安装费的变化及最小值点取整后对原材料订购决策、最优生产批量和总成本的影响。     

Stability analysis of a cobweb model with market interactions

This paper explores the steady-state properties and the dynamic behavior of a generalization of the classical cobweb model. Under fairly general demand and cost functions, producers form naïve expectations about future prices and select their output so as to maximize expected profits. Unlike the traditional setup, producers have the choice between two markets, and tend to enter that which was more profitable in the recent past. Such a switching process implies time-varying aggregated supply schedules, thus representing a further source of nonlinearity for the dynamics of prices. Analytical investigations and the numerical simulation of a particular case with linear demand and supply indicate that such interactions may destabilize otherwise stable markets and generate complex dynamics.     

Risk-averse order policies with random prices in complete market and retailers’ private information

We consider a retailer who orders products before the price for them becomes known. The price is an outcome of perfect competition in a complete market. Since the demand is price sensitive, the uncertainty in prices induces uncertain profits and associated risks. In this paper we show that if the retailer is risk averse and, as a result, selects a utility function of profit to maximize, then his subjective assessment of future prices is affected by the risk attitude. This, in turn, introduces a bias in retailer’s ordering policies. By considering coordinated pricing and ordering policies we derive a relationship between risk aversion, retailer’s subjective (private) assessment and the market implied, risk neutral forecast. This relationship and the induced bias are then illustrated for two typical operations management strategies which involve either inventory considerations or promotions avoiding accumulation of stocks.