Analytic solutions for Hamilton-Jacobi-Bellman equations

Closed form solutions are found for a particular class of Hamilton- Jacobi-Bellman equations emerging from a differential game among fims competing over quantities in a simultaneous oligopoly framework. After the derivation of the solutions, a microeconomic example in a non-standard market is presented where feedback equilibrium is calculated with the help of one of the previous formulas

On Lie point symmetries in differential games

A technique to determine closed-loop Nash equilibria of n-player differential games is developed when their dynamic state-control system is composed of decoupled ODEs. In particular, the theory of Lie point symmetries is exploited to achieve first integrals of such systems

The Incentive to Invest in Environmental-Friendly Technologies: Dynamics Makes a Difference

The established view on oligopolistic competition with environmental externalities has it that, since firms neglect the external effect, their incentive to invest in R&D for pollution abatement is nil unless they are subject to some form of environmental taxation. We take a dynamic approach to this issue, using a simple differential game to show that the conclusion reached by the static literature is not robust, as the introduction of dynamics shows that firms do invest in R&D for environmental-friendly technologies throughout the game, as long as R&D is accompanied by an output restriction exhibiting a distinctively collusive flavour. We also examine the social planning case and the effects of Pigouvian taxation, to show that there exists a feasible tax rate inducing profit-seeking firms to choose a combination of output and R&D such that the resulting social welfare level is the same as in the first bestpollution, environmental externality, R&D, differential games, social planning

A multi-factor inequality approach to a transfer scheme: the case of Common Agricultural Policy

The purpose of this paper is to propose theoretical foundations on the impact of transfer scheme, e.g. Community Agricultural Policy, on income inequality within European Countries. First, we show that ex-post inequality (in the after-transfer distribution) may increase if either initial aggregate income or the amount of fiscal contributions are sufficiently high. Second according to welfare ordering, we characterize a multi-factor decomposition of the Atkinson index to gauge the impact of each income source on the inequality profile. Third, we introduce a methodology to construct a cooperative game played by different income factors (as net incomes and/or incoming transfers) explicitly measuring the cost of inequality across the population in terms of welfare loss. We finally rely on Banzhaf and Shapley values to determine the marginal contributions of each factor to overall inequality

Government Spending Under Non-Separability: a Theoretical Analysis

In this paper we derive analytic implicit form conditions for the qualitative analysis of government spending multipliers and the optimal level of government spending in presence of non-separability between private and public components of aggregate demand. Using the simplest neo-classical flexible price model with no capital accumulation, we show that Edgeworth dependence is not a suitable condition to utomatically assess the signs of the consumption and income multipliers, for which a more complex analysis must be carried out. We propose a detailed investigation of the form and the characteristics of the involved utility functions, which are crucial to such evaluation. We also show that if Edgeworth complementarity is strong enough, a public spending stimulus can raise at the same time private consumption and real activity. In order to reconcile our general framework with existing literature, we discuss recent examples of non-separable functional forms from the standpoint of our results, and argue that their consistency relies on specific assumptions about steady- state points

On the Feedback Solutions of Differential Oligopoly Games with Hyperbolic Demand Curve and Capacity Accumulation

We characterise the subgame perfect equilibrium of a differential market game with hyperbolic inverse demand where firms are quantity-setters and accumulate capacity over time à la Ramsey. The related Hamilton-Jacobi-Bellman are solved in closed form both on infinite and on finite horizon setups and the optimal strategies are determined. Then, we analyse the feasibility of horizontal mergers in both static and dynamic settings, and find appropriate conditions for their profitability under both circumstances. Static profitability of a merger implies dynamic profitability of the same merger. It appears that such a demand structure makes mergers more likely to occur than they would on the basis of the standard linear inverse demand

Inequality assessment in a dynamic framework with heterogenous agents

We propose a method to carry out an inequality assessment in a dynamic and cross sectional framework, by applying the dynamic version of a suitable inequality index, such as the Gini coefficient, as a function of time. We use our methodology to a setup where the optimal value functions is the individuals’ income flows while the initial conditions characterize their level of wealth. When the Hamilton–Jacobi– Bellman system of equations can be solved in closed form, the monotone path of the income distribution is established. Extending the model according to a government intervention gives the possibility to study, first policy for reducing income inequality under a specific exogenous target, and second to minimise income inequality across individuals

On the Time Consistency of Equilibria in Additively Separable Differential Games

The relationship amongst state-redundancy and time consistency of differential games is investigated. A class of state-redundant games is detected, where the state dynamics and the payoff functions of all players are additively separable w.r.t. control variables. We prove that, in this class of games, open-loop Nash and degenerate feedback Stackelberg equilibria coincide, both being subgame perfect. This allows us to bypass the issue of the time inconsistency that typically affects the open-loop Stackelberg solution