Applications to Economics

Optimal control theory has been extensively applied to the solution of economic problems since the late sixties. In Sect. 11.1, two capital accumulation or economic growth models are presented. The first is a finite-horizon fixed-end-point model with a stationary population. Its aim is to maximize the present value of the utility of consumption for the society, as well as to accumulate a specified capital stock by the end of the horizon. The second model incorporates an exogenously and exponentially growing population in the infinite horizon setting. The technique of phase diagrams is used to analyze the model. In Sect. 11.2, we formulate and solve an epidemic control model. Certain infectious epidemic diseases are seasonal. Examples are the common cold, the flu, and certain children’s diseases. When it is beneficial to do so, control measures are taken to alleviate the effects of these diseases. Section 11.3 discusses a pollution control model in terms of an economic system in which labor is the only primary factor of production, which is allocated between food and DDT production. The food produced is used for consumption and the DDT produced is used as a secondary factor of production which, along with labor, determines the food output. However, when used, DDT causes pollution, which can only be reduced by natural decay. The objective is to maximize the total present value of the utility of food less the disutility of pollution due to the use of DDT. Sect 11.4 considers an adverse selection model in the form of a game between a seller (the principal) and a buyer (the agent), where the buyer has private information about her willingness-to-pay for the seller’s goods. Here, the principal would like to know the agent’s private information which he cannot learn simply by asking the agent because it is in the agent’s interest to distort the truth. Fortunately, according to the theory of mechanism design, the principal can design a game whose rules can influence the agent to act in the way the principal would like. Thanks, particularly to the revelation principle the principal needs only consider the game in which the agent truthfully reports her private information. Finally, since the number of topics in economics and management science treated in the optimal control framework is so large that all we can do is list some of them in Sect. 11.5 along with selected references. There are many exercises at the end of the chapter.