On the Intertemporal Incidence of Externalities

Abstract

In this paper we investigate the shifting incidence over time of a change in the demand for certain types of commodity. Such a change could be the result of the imposition of an excise tax, or of a change in tastes, or of an externality.' Our analysis is in fact quite general, although for expositional purposes we have chosen to discuss a specific, limited problem. The distinctive feature of the analysis is that we assume that there is a component in the capital stock used in the production of some good or service that requires maintenance and is an argument of the producer's revenue function as well as his cost function. We study the intertemporal results of a once-and-for-all change in the environment, and it is in this sense that our analysis is general, although we present it in the context of a simple externality example. The example is that of a resort operator who supplies hotel services which are consumed jointly with the amenities of a natural environment such as a beach or lake, which is exogenous to him. The operator is assumed to supply a certain standard of comfort-in general, a flow of services-generated by an element of his capital stock. This component of capital may be thought of as a level of upkeep of plumbing, paintwork, bed linen and the like. The state of of the resort depreciates unless maintained, so that maintenance will be a control variable for the operator, determined as the solution of an optimal control problem.2 We consider the change in the optimal maintenance path induced by an exogenous change in the environment, and the consequences for consumer welfare. In particular, we consider the following welfare problem: given that the consumer has suffered an irreversible impact loss from the environmental change, does he3 suffer further loss from the induced change in the optimal maintenance path? It is not too surprising to discover that the answer depends on consumer tastes. In the next section of this paper, we sketch the operator's optimal control problem. This is done briefly since the general theory of optimal exit paths, or how to run a bankrupt railroad, is fully analysed elsewhere (Davidson, 1977). In Section II we relate the producer's revenue function directly to the consumer's utility function so that we may investigate welfare effects. The crucial feature of the analysis, as we shall show later, is that, while the producer's problem is one in comparative dynamics, the welfare problem can be reduced to one in comparative statics. Our assumptions are, for the most part, standard and obvious and include as much differentiability as we need of the representative consumer's utility function. In addition we assume (a) that investment in comfort is irreversible, and (b) that comfort and the consumption of accommodation are gross complements in demand. Some other technical assumptions are discussed in the Appendix.