This chapter relates the notions of mutations with the concept of graphical derivatives of set-valued maps and more generally links the above results of morphological analysis with some basic facts of set-valued analysis that we shall recall.

Set-valued analysis

Progress report 2010

1. Random Events and Their Probabilities 2. Sequences in Independent Trials 3. Markov Chains 4. Random Valuables and Distribution Functions 5. Numberical Characteristics of Random Variables 6. The Law of Large Numbers 7. Characteristic Functions 8. The Classical Limit Theorem 9. The Theory of Infinitely Divisible Distributions 10. The Theory of Stochastic Processes 11. The Elements of Statistics Appendix

The Theory of Probability.

Since the WWII the largest and fastest growing component of world trade has been the exchange of manufactures between industrialized economies. The emerging of the intraindustry trade has somewhat challenged and cast doubts on the traditional Heckscher-Ohlin-Samuelson (H-O-S) trade model. Economists have some insight into this phenomenon, and attention has accordingly shifted to influence of the product differentiation on the economies of scale. But their research has no little impact on the dominating trade theory. What's more, they cannot give sufficient explanation to the modern trade practices. This paper focuses on the differentiated producer goods and constructs a model yielding international returns to scale. It also employs this model to explore the relations between international returns, the traditional national returns to scale and the factor endowments theory of international trade.

National and International Returns to Scale in the Modern Theory of International Trade

Asymptotic Behaviour and Stability Problems in Ordinary Differential Equations. By L. Cesari. Pp. 271. DM. 68. 1959. (Springer, Berlin)

This monograph is devoted to the theory of the Pontryagin maximum principle as applied to a special class of optimal control problems that arise in economics when studying economic growth processes. The main distinctive feature of such problems is that the control process is considered on an infinite time interval. In this monograph, we develop a new approximation approach to deriving necessary optimality conditions in the form of the Pontryagin maximum principle for problems with infinite time horizon. The attention is focused on the characterization of the behavior of the adjoint variable and the Hamiltonian of a problem at infinity. The approach proposed is applied to the analysis of the problem of optimal economic growth of a technological follower, a country that absorbs, in its technological sector, part of knowledge produced by a technological leader. By optimizing its growth performance, the technological follower dynamically redistributes available labor resources between the manufacturing and research and development (R&D) sectors of the economy. This problem is of independent interest in the endogenous economic growth theory. Moreover, it serves as an illustration of the approximation approach proposed. 

The main results presented in this monograph are new. They generalize and strengthen many previous studies in this direction.

The Pontryagin maximum principle and optimal economic growth problems

This paper extends optimal control theory to a class of infinite-horizon problems that arise in studying models of optimal dynamic allocation of economic resources. In a typical problem of this sort the initial state is fixed, no constraints are imposed on the behaviour of the admissible trajectories at large times, and the objective functional is given by a discounted improper integral. We develop the method of finite-horizon approximations in a broad context and use it to derive complete versions of the Pontryagin maximum principle for such problems. We provide sufficient conditions for the normality of infinite-horizon optimal control problems and for the validity of the 'standard' limit transversality conditions with time going to infinity. As a meaningful example, we consider a new two-sector model of optimal economic growth subject to a random jump in prices. Bibliography: 53 titles.

https://iopscience.iop.org/article/10.1070/RM2012v067n02ABEH004785/pdf

Infinite-horizon optimal control problems in economics

A general constraint aggregation technique is proposed for convex optimization problems At each iteration a set of convex inequalities and linear equations is replaced by a single surrogate inequality formed as a linear combination of the original constraints After solving the simplified subproblem, new aggregation coefficients are calculated and the iteration continues This general aggregation principle is incorporated into a number of specific algorithms Convergence of the new methods is proved and speed of convergence analyzed Next, dual interpretation of the method is provided and application to decomposable problems is discussed Finally, a numerical illustration is given

/pdf/constraint-aggregation-principle-in-convex-optimization-43x8nngxu0.pdf

A. V. Kryazhimskii

Papers

The Pontryagin maximum principle and optimal economic growth problems

Infinite-horizon optimal control problems in economics

Constraint aggregation principle in convex optimization

Adaptive dynamics in games played by heterogeneous populations

Dynamic inverse problems for parabolic systems