Asymmetric outcome in a symmetric dynamic duopoly

In a dynamic duopoly, will an initial asymmetry between firms increase or decrease over time? We examine this issue within a stochastic dynamic alternate-move duopoly model that explicitly accounts for action and reaction between firms. We consider two firms that are symmetric with regard to all primitives such as demand, cost and production functions, and which are subject to the same stochastic environment. The only asymmetry is with regard to their initial capital stocks which in turn asymmetrically influences their current and future profit and investment possibilities. We offer a characterization of the stochastic steady state and its supporting ergodic set for each firm. We are then able to identify the precise restrictions on the initial conditions under which the two firms either converge or diverge in the long run.     

Irreversible investment and R&D spillovers in a dynamic duopoly

Investments in new production processes usually involve a significant amount of R&D, generating spillovers that lowers the second comer's investment cost. We show that these spillovers substantially affect the equilibrium of the dynamic game. Even for low spillover values, the leader delays her investment until the stochastic fundamental has gone past the level such that the follower's optimal strategy is to invest as soon as he attains the spillover. This bears several interesting implications. First, because the follower invests as he benefits from the spillover, in equilibrium the average time delay between the two investments is short, as it should be expected. Second, in case of a major innovation, an optimal public policy requires an intervention in favor of the investment activity; an increase in uncertainty - delaying the equilibrium - calls for higher subsidization rates. Third, numerical simulations show that the spillover reduces the difference between the leader's and the follower's maximum value functions. Accordingly, our model can help generate realistic market betas.     

Dynamic advertising and pricing with constant demand elasticities

In this paper we analyze a stochastic dynamic advertising and pricing model with isoelastic demand. The state space is discrete, time is continuous and the planing horizon is allowed to be finite or infinite. A dynamic version of the Dorfman–Steiner identity will be derived. Explicit expressions of the optimal advertising and pricing policies, of the value function and of the optimal advertising expenditures will be given. The general results will be used to analyze the case of impatient customers. Furthermore, particular time inhomogeneous models and homogeneous ones with and without discounting will be examined. We will study the social efficiency of a monopolist's optimal policies and the consequences of specific subsidies. From a buyer's perspective, our analysis reveals that waiting – when looking at (immediate) expected prices – is never profitable should two or more units be available. But we will also prove that the sequence of average sales prices is monotone decreasing. Moreover, the techniques applied to solve the discrete stochastic advertising and pricing problem will be used to solve a related deterministic control problem with continuous state space.     

Demand relationships and pricing decisions for related products

This paper presents in terms of price changes the theoretical conditions for optimal pricing when products are demand-related. It then suggests an approach which could be used by companies to provide guidance in the pricing of such products over time.     

Demand for durable and nondurable goods,environmental policy and consumer welfare

If a tax on energy affects the demand for a nondurable good, the stock of a durable good, and the proportion of the energy consumption per unit of service of the durable good, an integrated approach for investigating consumer demand for durables and nondurables is required. The purpose of this paper is to employ the concept of a variable expenditure function with quasi-fixed durable goods as arguments in order to derive a demand system for nondurable goods in prices of the nondurables, in the stocks of durables, and in variable expenditure. From the envelope condition desired stocks of durables can be calculated and investment demand for durables can be determined. For an application we choose a variable expenditure function based on the almost ideal demand specification, and evaluate alternative environmental policy measures using the equivalent variation for comparing the welfare effects.     

Privatisation and vertical integration under a mixed duopoly

This paper analyses the privatisation of public firms when private firms may be vertically integrated with their suppliers. We consider a mixed duopoly with a vertically integrated public firm. The private firm bargains the price of the input with its supplier if they are not vertically integrated. We find that for a given bargaining power of the private firm, it vertically integrates with its supplier if goods are weak substitutes. We also find that there is less vertical integration in the mixed duopoly than in the private duopoly. Finally, in general, the public firm is privatised when goods are close substitutes and the bargaining power of the private firm is low enough.     

Bid-ask dynamic pricing in financial markets with transaction costs and liquidity risk

According to the classic no arbitrage theory of asset pricing, in a frictionless market a No Free Lunch dynamic price process associated with any essentially bounded asset is a martingale under an equivalent probability measure. However, real financial markets are not frictionless. We introduce an axiomatic approach of Time Consistent Pricing Procedure (TCPP), in a model free setting, to assign to every financial position a dynamic ask (resp. bid) price process. Taking into account both transaction costs and liquidity risk this leads to the convexity (resp. concavity) of the ask (resp. bid) price. We prove that the No Free Lunch condition for a TCPP is equivalent to the existence of an equivalent probability measure R that transforms a process between the bid price process and the ask price process of every financial instrument into a martingale. Furthermore we prove that the ask (resp. bid) price process associated with every financial instrument is then a R super-martingale (resp. R sub-martingale) which has a càdlàg version.     

Investment and pricing strategies of competing firms: A computational approach

We analyze the classical investment and pricing problem of a dominant firm faced with competition from substitute industries or marginal firms in the same field. The firm owns a finite level of a resource (e.g. the stock of an exhaustible one), the consumption of which is to be divided optimally over a finite planning horizon. The competitors' measures affect the demand for the resource towards the dominating firm. Rising crude oil prices and investments in forms of alternative energy are representative examples of the strategic questions which involve competitive and contradictory interests among firms within an industry. The investment and pricing problem can be solved analytically only with strong, simplifying assumptions. To make the analysis simpler and to relax these restrictions, we combine a series of numerical tools, computerize them, and build up a user-oriented, computerized decision aid, which we call a ‘computerized approach’. We solve the problem under different sets of theoretical assumptions. This chosen incremental theory building allows us to study the theoretical sensitivity of the original problem.     

A differential game in a duopoly with instantaneous incentives

In this paper, we study a differential game in which two competing firms exploit a public renewable resource that is relevant from a landscape point of view. We consider a policy maker that provides an instantaneous incentive to the firms in order to prevent the resource exhaustion during the whole extraction period, which coincides with the harvesting license period. We compute an open-loop Nash equilibrium of the differential game, showing that it coincides with a linear feedback Nash equilibrium. Finally, we compute the value of the incentive that leads to the maximization of social welfare considering the incentive both as a pure transfer and as a cost.     

Robust games: theory and application to a Cournot duopoly model

In this paper, the robust game model proposed by Aghassi and Bertsimas (Math Program Ser B 107:231–273, 2006) for matrix games is extended to games with a broader class of payoff functions. This is a distribution-free model of incomplete information for finite games where players adopt a robust-optimization approach to contend with payoff uncertainty. They are called robust players and seek the maximum guaranteed payoff given the strategy of the others. Consistently with this decision criterion, a set of strategies is an equilibrium, robust-optimization equilibrium, if each player’s strategy is a best response to the other player’s strategies, under the worst-case scenarios. The aim of the paper is twofold. In the first part, we provide robust-optimization equilibrium’s existence result for a quite general class of games and we prove that it exists a suitable value \(\epsilon \) such that robust-optimization equilibria are a subset of \(\epsilon \)-Nash equilibria of the nominal version, i.e., without uncertainty, of the robust game. This provides a theoretical motivation for the robust approach, as it provides new insight and a rational agent motivation for \(\epsilon \)-Nash equilibrium. In the last part, we propose an application of the theory to a classical Cournot duopoly model which shows significant differences between the robust game and its nominal version.     

Formulation and implementation of a pricing policy for municipal parking

The efficient allocation of a municipally-owned resource, like automobile parking space, requires a long-term policy that recognizes the social, technical, economic and political realities of the system. This paper develops such a policy, and reviews the progress in implementing it in a medium size city over a 3 yr period. An analysis of the interim results finds a significant increase in revenue to the City, some progress toward a socially desirable allocation of space and evidence that further progress is continuing.Much of the success in implementing the recommended policy was attributed to the small number of links in the chain of decision-makers and to having essentially the same group of people involved with the project over an extended period of time. With an understandable, well-developed and logical policy to follow, rational decisionmaking could proceed in small steps, thereby building consensus by reducing uncertainty about the ultimate impact of the recommended course of action. This type of policy formulation and implementation is analogous to programmed learning.     

Road pricing in a free society

Improved road pricing would be an indispensable element in a road system run on commercial principles, with charges for road use not co-mingled with taxes, and with no discrimination against private road providers. But pricing imposed by a monopoly supplier, to 'restrain' motorised mobility, is unacceptable in a free society.     

Optimal advertising for selling a product with a nondifferentiable demand function

We consider a firm which seeks the maximum profit by selling a product and assume that the firm can advertise its product in order to improve the firm goodwill and affect the product demand positively. In particular the good demand is zero when the goodwill is less than a fixed threshold. The problem is formulated in terms of optimal control theory. We show that there exists a unique optimal solution and sketch an algorithm to determine it.     

短生命周期产品供应链协调动态激励研究

为了解决短生命周期产品供应链协调的特殊性问题,文中采用博弈理论分析了供应链系统的多周期激励协调问题,从理论上给出了系统协调的最终结果:供应链系统最终处于平衡状态。应用马尔可夫决策过程建立了短生命周期产品供应链协调动态激励模型,实现了有限时段供应链系统多周期总体收益最大。通过数值分析,得到了供应链系统动态激励的帕累托最优策略。动态激励深入挖掘了供应链系统的潜能,在现有文献关注的激励内容的基础上,着重解决了激励过程,从而实现了供应链系统协调静态和动态激励的有机结合。     

On how size and composition of customer bases affect equilibrium in a duopoly with switching costs

Switching costs may facilitate monopoly pricing in a market with price competition between two suppliers of a homogenous good, provided the switching cost is above some critical level. It is also well known that asymmetric size of customer bases makes monopoly pricing more difficult. Adding consumer heterogeneity to the model we demonstrate that also composition of each firms customer base affects pricing, and this composition may aggravate or ease the incentives to break out of the monopoly pricing equilibrium.Received: 16 August 2001, Accepted: 27 September 2004, JEL Classification: D43, L10, L13This research has been sponsored by Telenor, through the Foundation for Research in Economics and Business Administration. We thank participants at the 2002 EARIE conference in Madrid, Matthew Jackson and an anonymous referee for helpful and constructive comments to an earlier version. The usual disclaimer applies.     

Road pricing and the 'new deal for transport'

There are several good reasons why motorists should be charged for using roads, and it is important to understand these and to distinguish between them.
Alternatives, however attractive, fail to stand up to close scrutiny. The change of government attitude showed in the recent White Paper is to be welcomed, but of the three specific proposals, two are questionable. Road pricing should not be used as a major welfare tool.