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Many economic problems can be formulated as dynamic games in which strategically interacting agents choose actions that determine the current and future levels of a single capital stock. We study necessary as well as sufficient conditions that allow us to characterise Markov perfect Nash equilibria for these games. These conditions can be translated into an auxiliary system of ordinary differential equations that helps us to explore stability, continuity and differentiability of these equilibria. The techniques are used to derive detailed properties of Markov perfect Nash equilibria for several games including voluntary investment in a public capital stock, the inter-temporal consumption of a reproductive asset and the pollution of a shallow lake.

We consider an economy with a single capital stock that two agents strategically exploit by choosing a consumption profile over an infinite time horizon. We analyze two different games and their corresponding equilibria. In one game firms are able to pre-commit and choose simple time functions as their strategies. In the other game agents are assumed to employ Markov strategies. It turns out that in the case of pre-commitment there exists a threshold level of the capital stock such that if the initial stock is above this threshold, equilibrium consumption converges to the efficient steady state while if the initial condition is below it, the capital stock converges to zero. In case of the Markov equilibrium there exists a unique interior steady state that is globally stable.

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 model in which two agents share access to the same stock of a productive asset and study the implications of non-cooperative consumption behavior over time. Under standard assumptions on the production function we construct infinitely many Nash equilibria, all of which consist of stationary Markovian strategies. These equilibria result in overconsumption so that the uniquely determined long-run asset stock is too low as compared to the cooperative solution. However, in the limit as the discount rate becomes sufficiently small the stead states generated by the Markovian Nash equilibria converge to the efficient steady state.Journal of Economic LiteratureClassification Numbers: C73, O41, Q22.

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.

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.

We examine a model in which two firms strategically compete in a duopolistic product market. Firms produce a homogenous product and face stochastic industry demand. Each firm has a single option either to expand or contract capacity, and hence output. In this setup we analyze the risk characteristics of industries as well as single firms and look at corresponding asset price dynamics. We focus on sequential exercise of options. We find that strategic competition in the product market is risk reducing. Irrespective of expansion or contraction the presence of strategically interacting rivals causes firm's risk to decline. This is the consequence of a simple hedging argument. Moreover, we find that own firm and industry characteristics have opposite risk implications in case of expansion and contraction. Empirical evidence, however, is that a strong negative relationship exists between firm and rival risk measures.

Publisher Summary This chapter discusses the solutions for a general class of capital accumulation games with stability properties of Nash equilibrium. It focuses on the application of turnpike theory to dynamic games. The turnpike results known from the theory of optimal economic growth can be used to derive a general convergence result. They also yield comparative statics and dynamics properties. The key result is the global convergence of the Nash trajectories to a unique steady state provided that the matrix of second order derivatives of the profit functions has a dominant diagonal. This result implies that the block diagonal assumption introduced by Araujo and Scheinkman is satisfied. Thus, comparative statics and dynamics analyses are possible. The chapter presents a model, which is very general in nature and can be applied to various market situations. The chapter discusses the relationship between the equilibrium path of the finite and infinite-horizon games. The finite horizon eqilibrium path stays within a closed neighborhood of the stationary equilibrium.

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.

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.

Summary In this paper we consider a dynamic oligopolistic model where the demand curve is described by a first—order differential equation which relates at each point in time the price of the commodity, the rate of change of the price, and the outputs of all firms in the market. The linear dynamic demand equation is considered as a first—order price adjustment mechanism. Open—loop, feedback and closed—loop Nash equilibria are derived. The corresponding stationary solutions are compared to the static Cournot equilibrium price and the competitive equilibrium price. It is demonstrated that for the “limiting game” (when the constant speed of adjustment reaches infinity) only the stationary open—loop price converges to the static Cournot price. The stationary feedback and a particular closed—loop solution converge to a price which is below the static Cournot price. For the limiting game when the number of firms goes to infinity the Nash equilibrium prices (independent of the information structure assumed) converge to the long—run competitive equilibrium.

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 this paper we study a two-person non-cooperative differential game model of the trade-off between unemployment and inflation with the government as player one and the central bank as the second player. Open-loop Nash and Stackelberg equilibrium solutions are derived analytically; it is shown that they do not coincide. Using numerical examples we study the sensitivity of the optimal open-loop solutions with respect to the parameters for both solution concepts. We show that in the examples used both players prefer the Stackelberg to the Nash equilibrium solution

A general dynamic oligopolistic price-advertising model is formulated and open-loop Nash solutions are derived. As a main result a generalisation of the well known Dorfman — Steiner — Theorem to heterogenous oligopoly markets is presented. Furthermore a detailed discussion of long run equilibrium solutions is given. For an important special case of the general model formulation it is shown that the optimal advertising policies are constant over time and constitute degenerated feedback solutions.

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.

In a simple capital accumulation game we study existence and dynamic properties of the cooperative solution, two kinds of Markov perfect equilibria (MPE), and a trigger strategy equilibrium. In the cooperative solution the stock always converges towards a steady state. This property is shared by our first MPE, whereas the second type of MPE may result in complicated dynamics. A Pareto comparison of both MPE depends on the parameter specifications but can turn out either way. Finally, we show how the players can achieve the efficient payoffs of the cooperative solution as an equilibrium outcome by using trigger strategies.