Daniel Leonard

Optimal control theory and static optimization in economics: Frontmatter

Optimal control theory is a technique being used increasingly by academic economists to study problems involving optimal decisions in a multi-period framework. This textbook is designed to make the difficult subject of optimal control theory easily accessible to economists while at the same time maintaining rigour. Economic intuitions are emphasized, and examples and problem sets covering a wide range of applications in economics are provided to assist in the learning process. Theorems are clearly stated and their proofs are carefully explained. The development of the text is gradual and fully integrated, beginning with simple formulations and progressing to advanced topics such as control parameters, jumps in state variables, and bounded state space. For greater economy and elegance, optimal control theory is introduced directly, without recourse to the calculus of variations. The connection with the latter and with dynamic programming is explained in a separate chapter. A second purpose of the book is to draw the parallel between optimal control theory and static optimization. Chapter 1 provides an extensive treatment of constrained and unconstrained maximization, with emphasis on economic insight and applications. Starting from basic concepts, it derives and explains important results, including the envelope theorem and the method of comparative statics. This chapter may be used for a course in static optimization. The book is largely self-contained. No previous knowledge of differential equations is required.

Optimal Control Theory and Static Optimization in Economics by Daniel Léonard

Optimal control theory is a technique being used increasingly by academic economists to study problems involving optimal decisions in a multi-period framework. This textbook is designed to make the difficult subject of optimal control theory easily accessible to economists while at the same time maintaining rigour. Economic intuitions are emphasized, and examples and problem sets covering a wide range of applications in economics are provided to assist in the learning process. Theorems are clearly stated and their proofs are carefully explained. The development of the text is gradual and fully integrated, beginning with simple formulations and progressing to advanced topics such as control parameters, jumps in state variables, and bounded state space. For greater economy and elegance, optimal control theory is introduced directly, without recourse to the calculus of variations. The connection with the latter and with dynamic programming is explained in a separate chapter. A second purpose of the book is to draw the parallel between optimal control theory and static optimization. Chapter 1 provides an extensive treatment of constrained and unconstrained maximization, with emphasis on economic insight and applications. Starting from basic concepts, it derives and explains important results, including the envelope theorem and the method of comparative statics. This chapter may be used for a course in static optimization. The book is largely self-contained. No previous knowledge of differential equations is required.

Market behavior of rational addicts

Abstract This paper analyzes the phenomenon of addiction, broadly interpreted to encompass the taking of piano lessons (beneficial addiction) as well as of drugs (harmful addiction) by a perfectly informed and rational individual. We distinguish between the addictive commodity and the market good purchased to produce that commodity. Harmful addiction displays all the expected features: a users purchases of the market good increase with time although quitting may be observed in some range of human capital and an increase in addictiveness usually decreases welfare while at the same time the individual buys more of the drug. The opposite is true of beneficial addiction which may be more aptly described as self-education.

Co-state variables correctly value stocks at each instant A proof

Abstract We prove this often mentioned result by showing that the marginal rate of increase of the maximand of a control problem with respect to the level of stock at any instant in the interior of the horizon is the value of the associated costate variable at that instant. This implies as a corollary that the marginal gain to be made over the future is the same as could be made over past and future.

Advantageous reallocations: A constructive example

This note shows how to construct smooth examples of a phenomenon, noted by Gale, in which a pair of agents make themselves better off at the expense of a third agent by reallocating their own endowments. Walrasian stability prevails.

The signs of the co-state variables and sufficiency conditions in a class of optimal control problems

Abstract A class of optimal control problems is identified, for which co-state variables have definite signs and the Maximum Principle is sufficient for optimality. The restrictions defining this class of problems are given an economic interpretation.

Endogenous Changes in Property Rights Regime

The purpose of this article is to present a model of the endogenous evolution of a societys property rights regime. We use an overlapping-generations framework in which capital accumulation takes place. Property rights enforcement is costly. Individuals decide collectively in each period the appropriate level of enforcement and pay taxes to finance it. Poor households have less interest in the enforcement of property rights than rich households. We also consider heterogenous households with different tastes. They differ on their choice of law enforcement. As their wealth and numbers evolve, political power may change sides, hence the property rights regime also evolves.

Trades unions, seniority and unemployment

The paper offers a general-equilibrium model of an economy containing powerful trades unions which set the wage levels to which competitive firms adjust, which have some perception of the bearing of their policies on the economy as a whole and within which seniority and majority-rule prevail. It is shown that increases in the dole or in the size of the workforce increase the level of employment and depress the net wage and that technical improvements have the opposite effect. Several externalities are identified. these make possible less conventional outcomes; for instance, an increase in the dole sometimes may be unambiguously welfare improving.

Legal Wage Constraints and Capital Accumulation

This paper studies the ability of a fully competitive economy to recover from shocks in the long run in the presence of constraints involved by anti-discrimination rules by age, such as the requirement of ?equal pay for equal work? and the restriction against decreasing salaries as workers age. After an external shock in the presence of the second type of constraint, the economy recovers in the long run, while in the presence of the first type of constraint the economy never recovers. Unfortunately when both constraints are enforced the economy is totally unable to recover from an external shock. It stagnates. Therefore the convergence of the economy to a first-best steady state, irrespective of initial conditions or shocks, is compromised.

Technology Transfers and Industry Closures

There has been a shift of manufacturing industries from Organization for Economic Co-operation and Development (OECD) countries to emerging countries. In a competitive global economy increases in productivity in any country are generally welfare-enhancing. The established industrialized countries can suffer from the collapse of some industries, and from the associated increase in unemployment. We model this process and analyze the interactions between various rigidities that cause it, such as the minimum viable scale of an industry or the number of workers who lack the necessary skills to change jobs. When, under free trade, the technology transfer causes the manufacturing industry to collapse in the home country, it experiences a discrete drop in welfare and the price of the manufactured good rises sharply. Further transfers may reverse these results. The optimal level of protection is the minimum size required to operate. Conditions that make supporting an ailing industry worthwhile can be interpreted in several ways but the conclusion is inescapable: technology transfers fundamentally affect arguments for industry protection at home.