A Dynamic Model of Stochastic Innovation Race: Leader-Follower Case

We provide steps towards analysis of rational behaviors of innovators acting on a market of a technological product. The situation when a technological leader competes with a large number of identical followers is in the focus. The process of development of new generations of the product is treated as a Poisson-type cyclic stochastic process. The technology spillovers effect acts as a driving force of the technological progress. We obtain an analytic characterization of optimal leaders R&D and manufacturing investment policies. Numerical simulations and economic interpretations are presented as well

Needle Variations in Infinite-Horizon Optimal Control

The paper develops the needle variations technique in application to a class of infinite-horizon optimal control problems in which an appropriate relation between the growth rate of the solution and the growth rate of the objective function is satisfied. The optimal objective value does not need to be finite. Based on the concept of weakly overtaking optimality we establish the normal form version of the Pontryagin maximum principle with an explicitly specified adjoint variable. A few illustrative examples are presented as well

Maximum Principle for Infinite-horizon Optimal Control Problems under Weak Regularity Assumptions

The paper deals with first order necessary optimality conditions for a class of infinite-horizon optimal control problems that arise in economic applications. Neither convergence of the integral utility functional nor local boundedness of the optimal control is assumed. Using the classical needle variations technique we develop a normal form version of the Pontryagin maximum principle with an explicitly specified adjoint variable under weak regularity assumptions. The result generalizes some previous results in this direction. An illustrative economical example is presented

The Pontryagin Maximum Principle for Infinite-Horizon Optimal Controls

this paper (motivated by recent works on optimization of long-term economic growth) suggests some further developments in the theory of first-order necessary optimality conditions for problems of optimal control with infinite time horizons. We describe an approximation technique involving auxiliary finite-horizon optimal control problems and use it to prove new versions of the Pontryagin maximum principle. A special attention is paid to behavior of the adjoint variables and the Hamiltonian. Typical cases, in which standard transversality conditions hold at infinity, are described. Several significant earlier results are generalized

A Dynamical Model of Optimal Allocation of Resources to R&D

We provide steps towards a welfare analysis of a two-country endogenous growth model where a relatively small follower absorbs part of the knowledge generated in the leading country. To solve a suitably defined dynamic optimization problem an appropriate version of the Pontryagin maximum principle is developed. The properties of optimal controls and the corresponding optimal trajectories are characterized by the qualitative analysis of the solutions of the Hamiltonian system arising through the implementation of the Pontryagin maximum principle. We find that for a quite small follower, optimization produces the same asymptotic rate of innovation as the market. However, relative knowledge stocks and levels of productivity differ, in general. Thus, policy intervention has no effect on growth rates but may also affect these relative levels. The results are different for not so small follower economies. The present paper provides the rigorous justification for the results presented in Aseev, Hutschenreiter and Kryazhimskii, 2002

Optimal Investment in R&D with International Knowledge Spillovers

We provide steps towards a welfare analysis of a two-country endogenous growth model where a relatively small follower absorbs part of the knowledge generated in the leading country. To solve a suitably defined infinite-horizon dynamic optimization problem a specialized version of the Pontryagin maximum principle had to be applied. For a quite small follower, optimization produces the same asymptotic rate of innovation as the market. However, relative knowledge stocks and levels of productivity differ in the two solutions. Thus, optimal policy intervention has no effect on long-run growth rates but affects these relative levels

Optimal Endogenous Growth with Exhaustible Resources

We study optimal research and extraction policies in an endogenous growth model in which both production and research require an exhaustible resource. It is shown that optimal growth is not sustainable if the accumulation of knowledge depends on the resource as an input, or if the returns to scale in research are decreasing. The model is stated as an infinite-horizon optimal control problem with an integral constraint on the control variables. We consider the main mathematical aspects of the problem, establish an existence theorem and derive an appropriate version of the Pontryagin maximum principle. A complete characterization of the optimal transitional dynamics is given

Optimality Conditions for Discrete-Time Optimal Control on Infinite Horizon

The paper presents first order necessary optimality conditions of Pontrygin's type for a general class of discrete-time optimal control problems on infinite horizon. The main novelty is that the adjoint function, for which the (local) maximum condition in the Pontryagin principle holds, is explicitly defined for any given optimal state-control process. This is done based on ideas from previous papers of the first and the last authors concerning continuous-time problems. In addition, the obtained (local) maximum principle is in a normal form, and the optimality has the general meaning of weakly overtaking optimality (hence unbounded processes are allowed), which is important for many economic applications. Two examples are given, which demonstrate the applicability of the obtained results in cases where the known necessary optimality conditions fail to identify the optimal solutions

Value Functions and Transversality Conditions for Infinite-Horizon Optimal Control Problems

This paper investigates the relationship between the maximum principle with an infinite horizon and dynamic programming and sheds new light upon the role of the transversality condition at infinity as necessary and sufficient conditions for optimality with or without convexity assumptions. We first derive the nonsmooth maximum principle and the adjoint inclusion for the value function as necessary conditions for optimality that exhibit the relationship between the maximum principle and dynamic programming. We then present sufficiency theorems that are consistent with the strengthened maximum principle, employing the adjoint inequalities for the Hamiltonian and the value function. Synthesizing these results, necessary and sufficient conditions for optimality are provided for the convex case. In particular, the role of the transversality conditions at infinity is clarified