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# Handbook of Environmental and Resource Economics

## Edited by Jeroen C.J.M. van den Bergh

This major reference book comprises specially commissioned surveys in environmental and resource economics written by an international team of experts. Authoritative yet accessible, each entry provides a state-of-the-art summary of key areas that will be invaluable to researchers, practitioners and advanced students.

# Chapter 71: Optimal Control Theory in Environmental Economics

## T. Feenstra, H. Cesar and P. Kort

Subjects: economics and finance, environmental economics, environment, environmental economics

## Extract

~ Talitha Feenstra, Herman Cesar and Peter Kort' 1. Introduction Optimal control theory originated as a mathematical tool to solve problems of dynamic optimization. Applying it to economic problems allows the explicit consideration of time.2 This makes it suitable for analysing the intertemporal trade-off between current consumption and future pollution or exhaustion of natural resources that is inherent in many environmental problems. Optimal control models consist of an intertemporal objective that must be optimized subject to a set of dynamic equations which specify how some instruments, or control variables, influence the development of the state variables. Examples of control variables in environmental economics are the level of fossil fuel use, the investment in abatement technology and the energy tax rate. State variables are, for instance, the atmospheric concentration of greenhouse gases and the stock of capital. Control theory provides methods of finding optimal levels of the instruments or control variables at each instant of time. Standard references for optimal control theory include Feichtinger and Hart1 (1986), Kamien and Schwartz (199 1) and Seierstad and Sydsaeter (1987). Typically, with the help of Pontryagin's maximum p r i n ~ i p l eoptimality conditions can be formu,~ lated in terms of the so-called shadow values of the state variables. In Section 2 we present a basic model of intertemporal trade-offs and discuss the conclusions that can be drawn from it. Section 3 examines the assumptions that are needed to arrive at these conclusions, such as constant and positive rate of discount and perfect...