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 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 in static game. We also study the effects of price stickiness and product differentiation upon the steady state equilibrium allocation. We find that per-firm equilibrium output is increasing in both product differentiation and price stickmess

Dynamic Duopoly with Differentiated Goods and Sluggish Demand

This thesis investigates dynamic Bertrand competition between two firms in a market where the goods are differentiated and demand is sluggish. Unlike homogeneous goods, differentiated goods are not perfect substitutes for each other. Sluggish demand means that there is a delay in the adjustment of demand after price changes. Sluggish demand is a remarkably ignored topic in the economic literature. The competitive situation is modelled as a differential game. The dynamic model employs the demand system as in Singh and Vives 1984 and dynamics as in Wirl 2010. It is shown that the dynamic model has a unique symmetric open-loop Nash equilibrium. The long-term open-loop steady state is compared with the equilibrium point of the static model. The fundamental mathematical theory and solution methods of optimal control theory and differential games that are required in the analysis of the model are also presented in the thesis. As the main result of the analysis of the model, it is shown that when sluggishness of demand is relatively small (i.e. the adjustment of demand after price changes is sufficiently fast), sluggishness of demand increases the market power and profits of the firms in the open-loop steady state compared to the equilibrium point of the static model. After sluggishness of demand exceeds a certain point, the profits of the firms decline below the static equilibrium profits. Moreover, it is shown that product differentiation relaxes price competition also in the presence of sluggish demand, as it does in a static model.Tutkielmassa tarkastellaan kahden yrityksen välistä dynaamista Bertrand-kilpailua markkinoilla, joilla hyödykkeet ovat differoituja ja kysyntä on jäykkää. Toisin kuin homogeeniset hyödykkeet, differoidut hyödykkeet eivät ole täydellisiä korvikkeita toisilleen. Kysynnän jäykkyys tarkoittaa sitä, että kysyntä sopeutuu viiveellä hinnanmuutoksiin. Kysynnän jäykkyys on yllättävän vähälle huomiolle jäänyt aihe taloustieteellisessä kirjallisuudessa. Kilpailutilanne mallinnetaan differentiaalisena pelinä. Dynaamisen mallin kysyntäjärjestelmä on kuten Singh ja Vivesin (1984) ja dynamiikka kuten Wirlin (2010) artikkelissa. Dynaamisella mallilla osoitetaan olevan uniikki symmetrinen open-loop Nash -tasapainostrategia. Pitkän aikavälin open-loop -tasapainopistettä vertaillaan staattisen Bertrand-mallin tasapainopisteen kanssa. Tutkielmassa esitetään lisäksi mallin tarkasteluun vaadittavan optimaalisen kontrolliteorian ja differentiaalisten pelien keskeinen matemaattinen teoria ja ratkaisumenetelmät. Mallin analyysin päätuloksena osoitetaan, että kun kysynnän jäykkyys on verrattain pientä (ts. kysynnän sopeutuminen hinnanmuutoksiin on riittävän nopeaa), kysynnän jäykkyys lisää yritysten markkinavoimaa ja voittoja open-loop -tasapainopisteessä verrattuna staattiseen tasapainopisteeseen. Kun kysynnän jäykkyys ylittää tietyn pisteen, yritysten voitot laskevat staattisten tasapainovoittojen alle. Lisäksi osoitetaan, että tuotedifferointi pehmentää hintakilpailua myös kysynnän jäykkyyden olosuhteissa, aivan kuten staattisessa mallissa

A Differential game approach in the case of a polluting oligopoly

In this paper we propose an oligopolistic market model of pollution, where demand is not linear and firms are revenue maximizers. Additionally we assume that the rate of purification is very small tending to zero and that each firm accumulates a pollution share depending for example on firm’s size. The game ends up with Markov strategies employed by all firms. Our findings show that under conditions it is possible a marginal decrease on the total pollution stock to increase firms’ discounted revenues. A reallocation caused by a uniform decrease in all firms pollution, reorders the marginal change of the pollution stocks in reverse of the original order of the allowed pollution.Non linear strategies; Markov equilibrium; allowed pollution stock.

Pricing and Marketing Rules with Brand Loyalty

Many firms face a dynamic trade-off: if price is reduced, the firm attracts new customers who will yield profits in the future, but it also forgoes the opportunity to squeeze profits now from loyal customers. This paper identifies a rule that represents the optimal resolution of this trade-off, in terms of an intuitive modification to the static Lerner rule. We find that the “effective” price elasticity depends on the discount rate used by the firm, on the rate of depreciation of the clientele through exit from the stock of repeat purchasers, and on a weighted sum of the price elasticities of the flow of entries into the stock of repeat-purchasers and the flow of exits from the stock of repeat-purchasers. None of these factors enter the optimal pricing strategy for a firm facing a conventional demand function with instantaneous adjustment, i.e. where consumers are “fast switchers”, rather than repeat-purchasers. We also find optimal rules for marketing investment and for quality of service, which are extensions of the Dorfman-Steiner conditions. The paper shows that our rules, with suitable modifications, are valid for many market structures, including monopolistic competition, pure monopoly and strong cartels, dominant firms and oligopolists that have full commitment ability. In the case of dominant firms and oligopolists that cannot commit to their strategy paths, these simple optimal pricing and marketing rules do not apply.