Mathematical economics and control theory: Studies in policy optimisation.

Chapter 1 deals with the origin and limitations of mathematical economics and its implications for economic applications of optimal control theory. Using an historical approach, we have proposed a hypothesis on the origin and limitations of classical and modern mathematical economics. Similar hypotheses proposed by Cournot, Walras, von Neumann-Morgenstern and Debreu are shown not to be convincing. Conditions are established under which applications of mathematical methods, in general, and optimal control theory, in particular, may produce economic results of value. Chapter 2 concerns the formation and development of optimal control applications to economic policy optimisation. It is shown that the application of mathematical control theory (as compared with engineering control) may significantly contribute to mathematical economics (as compared to econometrics). The development of optimal growth theory has been examined as an example. Within the context of economic policy optimisation, a critical examination of the recent developments in macroeconomic modelling, the relationship between theory and observation, rational expectations, the Lucas critique and the problem of time- inconsistency is presented. Chapter 3 provides the first illustration of the main theme of the earlier chapters. Using the generalised Hamiltonian in Pontryagin's maximum principle, as well as using Bellman's dynamic programming, we have obtained a number of new results on the mathematical properties of optimal consumption under liquidity constraints. For example, we have demonstrated how the response of optimal consumption to liquidity constraints is conditioned by the consumer's intertemporal elasticity of substitution. Considered as a mathematical structure, this is shown to capture the effects of the following variables on the optimal consumption path; pure preference parameters, the interest rates variations and the structural parameters prevailing in the credit markets. In chapter 4, the dynamic Leontief model, which according to the conditions established in chapter 1, is one of the most successful applications of mathematical methods to economic policy analysis, is first considered as a control problem. We have then obtained the optimal consumption path for deterministic and stochastic dynamic Leontief models with substitute activities which are in turn formulated in deterministic and stochastic environments. Our solution uses Pontryagin's maximum principle. Bellman's method and Astrom's Lemma on stochastic dynamic programming