A stochastic differential game of capitalism

We develop a stochastic differential game of capitalism to analyze the role of uncertainty. In the deterministic game, the firm’s rent is completely taxed away and the firm stops investing completely. In the stochastic game, the government does not tax the firm’s rent completely. The firm posts a positive rate of investment if the firm’s rent exceeds the labor’s income. Although the cooperative solution is indeterminate, cooperation is always Pareto optimal compared to the non-cooperative Markovian Nash equilibrium. For individual rationality, we apply a payoff distribution procedure based on Yeung and Petrosyan (2006) to derive a subgame-consistent solution.     

Existence of equilibrium in a differential game of spatial competition with advertising

We investigate a differential duopoly game with horizontal product differentiation and advertising efforts aimed at increasing market demand, to show that the standard approach to spatial competition fails to produce a pure-strategy price equilibrium in a dynamic game framework. This holds independently of the shape of the transportation cost function. Then, we introduce an endogenous cost associated with the choice of location and characterise the feedback equilibrium, identifying the necessary and sufficient condition for the existence of the pure-strategy (stationary) price equilibrium. The same condition is singled out for the static game where consumer population is constant. Finally, we show that the static game cannot be viewed as a special case of the dynamic one.     

A continuous-time heterogeneous duopoly model with delays

In this paper, the continuous-time version with delays of the model of Tramontana (Econ Model 27(1):350–357, 2010) is presented. The stability of the equilibrium point is discussed by analyzing the distribution of roots of associated characteristic equation. It is found that combining marginal costs with time delays gives rise to different economic scenarios, where stability switches may appear and Hopf bifurcations occur within certain range of parameters.     

Fiscal and monetary policy co-ordination: A differential game approach

In this paper the interaction between the Treasury and the central bank is examined in the case of both cooperative and non-cooperative behaviour. Differential games are used in the framework of a continuous-time econometric model of the Italian economy. The Nash and the Stackelberg non-cooperative equilibrium solutions are computed, and the case for cooperation is analysed by considering the Nash and the Kalai-Smorodinsky bargaining models. It is shown that, in the Italian case, the government has a stronger bargaining position than the central bank. A comparison is then made between the different solutions to show that the drawbacks that emerge from non-cooperation are not simply those depending on the players' payoffs. Other features are in fact considered which constitute a further argument for policy co-ordination.     

Buying cooperation in an asymmetric environmental differential game

We consider a two-player asymmetric differential game of pollution control. One player is non-vulnerable to pollution, or unwilling to consider damages when choosing her production policy in a non-cooperative game. We characterize the feedback-Nash equilibrium and the cooperative solution. We establish conditions under which the vulnerable player can buy the cooperation of the non-vulnerable player to control her emissions. We further use the Nash bargaining solution to allocate the total cooperative dividend between the two players and propose a time-consistent decomposition overtime of the total payoff.     

Dynamic voluntary provision of public goods with uncertainty: a stochastic differential game model

Fershtman and Nitzan (Eur. Econ. Rev. 35:1057–1067, 1991) presented a continuous dynamic public good game and solved the model for feedback Nash equilibria. Wirl (Eur. J. Polit. Econ. 12:555–560, 1996) extended the model and considered nonlinear strategies. Both models do not include uncertainty and hence neglect an important factor in the theory of public goods. We extend the framework of Nitzan and Fershtman and include a diffusion term. We consider two cases. In the first case, the volatility of the diffusion term is dependent on the current level of the public good. This set-up will in principle lead to the same type of feedback strategies computed under certainty. In the second case, the volatility is dependent on the current rate of public good provision by the agents. The results are qualitatively different. We provide a detailed discussion as well as numerical examples. In particular, we show that in both cases uncertainty signifies the free rider effect.     

Advertising a product to face a competitor entry: a differential game approach

We analyze a market in which advertising is the dominant marketing tool to create market share. We assume that an incumbent firm dominates the market during an initial stage, and that a new competitor is going to enter the market. In particular, we analyze the different advertising policies that the incumbent firm can adopt, before and after the entry of the rival. We explore three possible behaviours. In the first scenario the firm knows that the competitor will arrive at a given instant. In the second one we assume the original firm to be surprised, in the sense that it does not anticipate the entry of the opponent either because it does not expect the competitor to arrive, or it is not prepared to react before the entry takes place. Finally, in the third scenario, the original firm knows that the competitor will enter at a constant rate. We characterize a differential game model and compare the firms’ behaviours in a strategic perspective.     

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.     

The local stability of an open-loop Nash equilibrium in a finite horizon differential game

We propose a finite time differential game as a model for some economic processes and derive conditions for the Nash equilibrium solution to be locally asymptotically stable. We adopt the traditional ‘Cournot-reaction function’ notion of stability, which in our (continuous time) model becomes a function-to-function, or trajectory-to-trajectory, mapping. The conditions for stability seem to make economic sense. The equilibrium is less stable if the interaction terms in each period are large, if the game has a long duration, and if the discount rate is small.     

A profit-center game with incomplete information

Profit centers in a firm in multidivisional form agree in the ex ante stage upon a plan about their joint production and profit imputation. The plan is executed in the subsequent two periods of the interim stage: the setup period and the manufacturing period. In the setup period, each center has its private information, but a part of its information is revealed to the other centers through its action. Based on the information endogenously pooled this way, the centers take another round of actions in the manufacturing period. A core plan is defined as a Bayesian incentive-compatible plan of the grand coalition of profit centers, upon which no coalition can improve using its Bayesian incentive-compatible plan. A core plan is called full-information revealing if each center fully reveals its private information in the setup period. Three existence theorems for a full-information revealing core plan are established. The first two theorems impose alternative conditions on returns to scale: (1) the neoclassical convex technology, and (2) increasing returns to scale. In case (2), a stronger condition than Scarf's distributiveness is imposed on the total production set. The third theorem is based on a specific supplier-customer relationship among the divisions. Received: 13 October 1997 / Accepted: 26 July 1999     

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.     

Inference in a synchronization game with social interactions

This paper studies inference in a continuous time game where an agent’s decision to quit an activity depends on the participation of other players. In equilibrium, similar actions can be explained not only by direct influences but also by correlated factors. Our model can be seen as a simultaneous duration model with multiple decision makers and interdependent durations. We study the problem of determining the existence and uniqueness of equilibrium stopping strategies in this setting. This paper provides results and conditions for the detection of these endogenous effects. First, we show that the presence of such effects is a necessary and sufficient condition for simultaneous exits. This allows us to set up a nonparametric test for the presence of such influences, which is robust to multiple equilibria. Second, we provide conditions under which parameters in the game are identified. Finally, we apply the model to data on desertion in the Union Army during the American Civil War, and find evidence of endogenous influences.     

Cournot duopoly and insider trading with two insiders

We study the effect of competition among insiders in an extension of the static Kyle [Kyle, A. (1985). Continuous auctions and insider trading. Econometrica, 53, 1315–1335] model of insider trading introduced by Jain and Mirman (JMC) [Jain, N., & Mirman, L.J. (2002). Effects of insider trading under different market structures. The Quarterly Review of Economics and Finance, 42, 19–39]. In the JMC model competition in the real sector is introduced. In this paper we introduce competition in the stock sector in the JMC model by assuming that there is a manager who is responsible for making the real decisions of the firm as well as an ‘owner’ who has the same information as the manager but has no managerial responsibilities. In this way we can study the interaction between competition in the real sector and competition in the financial sector. We show that the stock price set by the market makers reveals more information than in the JMC model and that the expected equilibrium values of the manager’s profits sometimes decline and sometimes increase depending on the exogenous parameters of the model. Moreover, we prove that due to the competition in the financial sector, the level of output produced by the firm is less than in JMC. Finally, we also study the effect of financial competition in the case in which the market makers receive only one signal and analyze the comparative statics in this case.     

A market game with symmetric limit orders

We extend the market game with symmetric limit orders studied in Weyers (2003, 2004) to a many-good setup. Our limit orders are symmetric in terms of payment and determine a unique consistent price system for every strategy profile. The limit orders studied in the previous literature—see Dubey (1982), Simon (1984) and Mertens (2003)—share none of these properties. It is shown that three mild market-thickness conditions imply that the set of symmetric Nash equilibrium outcomes coincides with the set of price-taking equilibrium outcomes. First, the Dubey and Shubik (1978) refinement is used to eliminate no-trade as an equilibrium. Second, any price-taking equilibrium has trade in each market. Third, there are at least two agents of each type, where a type is determined by preferences and endowments. The last two conditions enable applying the Bertrand argument. This paper thus provides new insights to Bertrand’s (1883) classic critique of Cournot and the associated problem of capacity constraints raised by Edgeworth (1897).     

Multiple equilibria in a firing game with impartial justice

In this paper, we argue that firms' firing strategies and the judicial strategy of dismissed employees depend to a large extent on labor judges' ability to shed light on the various cases. The model is cast as a sequential game with imperfect information featuring firms, employees and labor judges. The judges' error margin increases with the congestion of the judicial system. The game presents multiple equilibria which differ in the frequency of good workers fired for unfair motives and the frequency of unreliable workers who abusively sue firms for unfair dismissal. The probability that the judge sits with the employee appears to be positively related to the ratio between the severance payment for economic dismissal and the company fine for abusive dismissal.     

On the dominance of capitalists leadership in a ‘Feedback-Stackelberg’ solution of a differential game model of capitalism

This paper deals with a modified version of the Lancaster model of capitalism, where it is assumed that there is a cost jointly borne by the two groups of players (workers versus capitalists) and associated with the bargaining of a larger share of consumption for the workers. It is shown that a Feedback- Stackelberg solution, with the capitalists acting as leaders and announcing their investment policy at the beginning of each period, is a solution dominating the Feedback-Nash solution. The paper is also intended to be a tutorial on the Feedback-Stackelberg solution, a concept not so often used by economic modelers.     

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.     

Rebates in a Bertrand game

We study a price competition game in which customers are heterogeneous in the rebates they get from either of two firms. We characterize the transition between competitive pricing (without rebates), mixed strategy equilibrium (for intermediate rebates), and monopoly pricing (for larger rebates).     

Optimal control and differential game models of military expenditure in less developed countries

This paper presents optimal control and differential game models of military expenditure in less developed countries. We emphasise the strategic interconnections between aggressive neighbours and explicitly stress the optimal resource allocation problem between civilian and military expenditure. A major innovation of the analysis is to analyse the asymmetric reaction functions of the two agents and the formalisation of the stylised fact that most LDC arms races have taken place between participants of ‘unequal’ size.