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Decentralized decision making is a fact in the modern business world accompanied by extensive research that looks into its consequences for overall firm profits. We study the interactions of decentralized marketing and operations divisions in a corporation and explore their impact on overall firm profits in the case with and without coordination of the two decentralized units. We assume that the marketing department is responsible for the price that influences the demand (sales), and the operations department is responsible for the production rate. We allow for backlogging over time. We model the interdependence involving marketing and operations decisions as a non-cooperative differential game, with the two divisions as strategically interacting players. We find that, without coordination, strategic interactions of marketing and production result in inefficiencies that can quantitatively be substantial. Next, we introduce a dynamic transfer pricing scheme as a coordination device and evaluate if it establishes efficient (first best and fully coordinated) outcomes. We show that if production and marketing play a game with pre-commitment strategies, there exists a dynamic transfer price that efficiently (fully) coordinates decentralized decision making and hence results in Pareto-efficient company profits. If the two decentralized divisions play a game without pre-commitment, dynamic transfer prices can partially coordinate decentralized decision making but fail to fully eliminate overall inefficiencies arising from strategic interactions among decentralized divisions.

Defining speed of diffusion as the amount of time it takes to get from one penetration level to a higher one, we introduce a dynamic model in which we study the link between pricing policy, speed of diffusion, and number of competitors in the market. Our analysis shows that, in the case of strategic (oligopolistic) competition, the speed of diffusion has an important influence on the optimal pricing policy. In particular, we find that higher speeds of diffusion create an incentive to strategically interacting firms to lower their prices.

Leitmann (Ref. 1) introduced coordinate transformations to derive global optima of a class of dynamic optimization problems. We present applications of this method to derive open-loop Nash equilibria for finite-time horizon differential games. The method of coordinate transformations is especially useful in cases where the original game does not satisfy the global curvature conditions normally imposed in sufficient optimality conditions.

We consider a differential game of R&D competition and explore the impact of rivalry on the firms' investment behavior over time. Using closed-loop strategies and hence allowing for strategic interactions among rival firms we show that R&D spending by the individual competitor is increased due to competition in the race for priority. This leads us to argue that competitive encounters enhance R&D activities at the same time as increasing efficiency in the race for a technological breakthrough.

The success of the introduction of a new product in a market is very sensitive to the marketing decision variables adopted by the firm. In the present paper we are concerned with the question of new product advertising in a heterogeneous oligopoly market consisting of N firms. A dynamic game is formulated to model strategic as well as sales interactions in such a market. Optimal advertising strategies are identified as open-loop Nash solutions.

Problems of macroeconomic stabilization policies have been analyzed by means of optimal control theory since the early seventies [see, for example, Kendrick (1981)]. However, with the emergence of the New Classical Macroeconomics, this theoretical framework has been increasingly questioned. In particular, Kydland’s and Prescott’s (1977) claim that government’s stabilization policies derived by optimum control methods might be time-inconsistent when private-sector agents have rational expectations of policy-makers’ behaviour has been regarded as a serious argument against the use of optimum control theory in macroeconomics, although the real-world importance of time-inconsistent behaviour is still controversial. [For an early theoretical critique, see Hughes Hallett (1986a); for empirical analyses, Ireland (1999), Boschen and Weise (2004), among others.] Later research on time-inconsistency indicates that dynamic game theory is a more appropriate tool for analyzing stabilization policy problems because more than one decision-maker has to be considered explicitly. Therefore over the last three decades, a large number of studies of macroeconomic stabilization policies (as well as of other areas in economics) have made use of the concepts and results of the theory of dynamic games.

In this paper, we study the stability properties of the class of capital accumulation games introduced by Fershtman and Muller (Ref. 1). Both discrete and continuous time versions are discussed. It is shown that the open-loop Nash equilibrium solutions for both games are characterized by a general saddle-point property, a result best known from the turnpike literature in optimal growth theory. In the case of zero discount rates, an even stronger result can be derived: As long as the Hessian matrix of the instantaneous profit functions has a quasidominant diagonal, no pure imaginary roots are possible.

For a general linear-quadratic differential game model with two decision-makers, we characterize analytically the open-loop Stackelberg equilibrium solution. This can be interpreted as a model of macroeco-nomic policies, where the government acts as the leader and the private-sector agents act as followers in the game. On the other hand, we consider a linear multivariable dynamic economic system in continuous time, where economic agents have rational expectations with respect to some state variables. From a comparison of the two models, it becomes obvious that they are formally equivalent under some further assumptions. Some consequences of this result for the design of stabilization policies under rational expectations are briefly discussed.

This paper analyzes tariffs and quotas in a differential game model of duopolistic competition. For a homogeneous good, equal import tariffs and quotas are compared with respect to resulting domestic prices. The open-loop and closed-loop (subgame perfect) equilibria are derived. The open-loop game leads to domestic price equivalence; however, the closed-loop game does not, unless firms are myopic or the number of domestic firms becomes large. Thus, we prove that the equivalence for static Cournot competition does not carry over to the dynamic case.

It is well known that non-cooperative behaviour of different policy-makers, either central bank and government on a national level or governments of different countries on an international level, may result in severe overall losses and inefficiencies. On the other hand, policy coordination is difficult to achieve, given the mechanisms of negotiations between several policy-making institutions. A solution to this “trade-off” would be a mechanism inducing individual behaviour that ensures efficient (Pareto-optimal) outcomes equivalent to those achieved by cooperation, but without the need of explicit agreements.

For a general linear-quadratic differential game model of macroeconomic policies with two policy-makers, we derive no-memory non-cooperative feedback Nash equilibrium strategies and Pareto-optimal strategies. Since the non-cooperative equilibrium solution in general is inefficient, policy coordination provides advantages but may not be sustainable, if only memoryless strategies are considered. However, if memory strategies are admitted, then there may exist subgame-perfect efficient noncooperative equilibrium solutions, and efficient stabilization policies may be achieved without policy coordination.

In 1977, Lieber and Barnea presented an investment and pricing differential game. There are two players, a monopoly possessing some exhaustible resource and a price-taking competitor. The monopoly wishes to determine an optimal price, whereas the competitor wishes to choose optimal investment in an alternative production. This paper presents a non-linear extension of the game. The special structure of the game permits a complete solution in the state—costate phase plane as well as in the control phase plane. The monopoly plays an optimal decreasing price strategy and the competitor plays an optimal decreasing investment strategy with an increasing production capacity.

The purpose of the paper is to determine the behavior of Nash optimal open-loop advertising strategies of a dynamic duopoly. The logarithmic two-person nonzero-sum advertising differential game analyzed byJorgensen [1982] is extended to more general nonlinear functionsg (u1,u2) measuring the effectiveness of advertising expendituresu1,u2 of two firms with respect to the development of the market shares over time. It is shown that under the assumption of zero cross effects of the advertising efficiency function the main features of Jorgensen's model are preserved. Furthermore, two models with advertising rates interacting asymmetrically with respect to the marginal advertising effectiveness are studied. For zero salvage values of terminal market shares it is shown that the Nash optimal advertising expenditures of a small firm are always downward sloping, whereas the shape of a large competitor's advertising rate can only be determined under additional assumptions.

For a general linear-quadratic differential game model of macroeconomic policies with two policy-makers, we derive no-memory non-cooperative feedback Nash equilibrium strategies and Pareto-optimal strategies. Since the non-cooperative equilibrium solution in general is inefficient, policy coordination provides advantages but may not be sustainable, if only memoryless strategies are considered. However, if memory strategies are admitted, then there may exist subgame-perfect efficient noncooperative equilibrium solutions, and efficient stabilization policies may be achieved without policy coordination.

A monopoly possesses a finite stock of a resource and wishes to determine an optimal pricing policy. The competitive fringe invests in production capacity and wishes to select an optimal investment rate. Demand towards the monopoly depends on price as well as on the sales rate of the competition. Modelling the situation as a differential game, non-cooperative (Nash and Stackelberg) and cooperative (Pareto) equilibria are determined. Owing to the special structure of the game, these solutions can be found in closed form.

This paper deals with the determination of optimal pricing policies for firms in oligopolistic markets. The problem is studied as a differential game and optimal pricing policies are established as Nash open-loop controls. Cost learning effects are assumed such that unit costs are decreasing with cumulative output. Discounting of future profits is also taken into consideration. Initially, the problem is addressed in a general framework, and we proceed to study some specific cases that are related to models presented in recent literature. Three basic classes of sales dynamics are analyzed: competition with price effects only, competition with price as well as adoption effects, and competition with adoption effects only. In some cases it turns out that results which hold for the monopoly case, carry over to the multi-firm case, in the sense that the qualitative structure of optimal pricing strategies is the same in the monopoly and the oligopoly cases. However, due to competitive interdependencies, differences certainly exist in the levels as well as the rates of change of optimal prices.

This paper identifies some classes ofN-person nonzero-sum differential games that are tractable, in the sense that open-loop Nash strategies can be determined, either explicitly or qualitatively in terms of a phase-diagram portrait. The classes are characterized by conditions imposed on the Hamiltonians. Also, the underlying game structures needed to satisfy these conditions are characterized.

A simple linear dynamic macroeconomic model of the trade-off between unemployment and inflation is considered. Two policy-making institutions, namely the government and the central bank, can exert influence on this economic system, aiming at reducing the rates of unemployment and of inflation by fiscal and monetary policies, with quadratic costs attached to the use of the respective own control variables; in addition, they give different weights to the target variables, which arc assumed to enter linearly into the objective functions. We solve the resulting linear-quadratic differential game both for the non-cooperative feedback Nash solution and for the set of Pareto-optimal solutions. It is shown that both solutions lead to constant strategies over the entire infinite time horizon. In the non-cooperative case, open-loop Nash, feedback Nash, and Stackelberg equilibria coincide. On the other hand, they do not belong to the set of Pareto-optimal outcomes and are therefore inefficient. Sensitivity analyses are provided for the optimal non-cooperative and cooperative strategies.

Stabilization policy is modeled as a linear-quadratic differential game between government and central bank aiming at a reduction of unemployment and inflation with different weights attached to these targets. We derive non-cooperative open-loop Nash equilibrium strategies for both players, showing them to be constant, subgame perfect (feedback Nash equilibrium strategies), and simultaneously Stackelberg equilibrium strategies with either player as the leader. For the same game, cooperative Pareto-optimal strategies are derived, also resulting in constant controls to be interpreted as fixed policy “rules”. It is shown that non-cooperative outcomes in general do not belong to Pareto-optimal ones and therefore are inefficient.