Forschungsgruppe ORCOS
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Economics of Crime: Age-specific Control Models

Many important processes in economics are age-dependent. Examples can be found, among others, in vintage capital theory, population economics, marketing, inventory control of perishable items, and the management of renewable resources. The fact that some processes evolve along a dimension of age as well as time makes analysis difficult. Integral equations and partial differential equation control theory, however, are not contained in the toolkit of most economists. This might explain the fact why in economics duration-dependence is a largely neglected field (which is, for instance, not the case in biology). This is in sharp contrast to the number of interesting problems in economics which rely on age or duration dependence.

The goal of economic policy is to induce agents to change their economic behaviour. In many cases those interventions are targeted to individuals who are homogenous with respect to certain characteristics. In the present proposal we focus on age-structured populations of agents. Age-dependent demand for a fashionable good, say, leads to a market segmentation which is adequately dealt with by an age-specific mix of marketing instruments. Note, ho­wever, that the applicability of age-structured models is much broader, since age can be seen as proxy of various economic characteristics like household and family status, income etc.

The present project intends to debate the relative merits of ‘broadcast’ vs. age-specific inter­ventions. Given the fact that most people who open investment accounts are in their working years, brokerage houses and banks may prefer to pay more per person reached to advertise in a media that is selectively concerned by people in their 30’s.

Sometimes the state variables of dynamic economic models are not only functions of time but also are functions of other dimensions such as duration or spatial coordinates. Such systems are characterized by partial differential equations. In natural sciences, distributed parameter systems have been extensively used, e.g., in heat transfer problems. In economics and mana­gement science almost all applications of optimal control theory are described by ordinary differential equations. There are, however, many socio-economic problems where, in addition to time, other coordinates, mostly age or other characteristics of duration, play a crucial role. And age or, more general, duration since a certain event, can be seen as important proxy for weight, quality, income etc. Thus, distributed parameter systems have a broad field of potential applications.

The question arises why there is such a gap between several interesting age-structured economic problems and the rather short list of distributed parameter control applications which exists up to now. First, probably because most economists are not skilled in dealing with partial differential equations. Second, and more important, the mathematics behind distributed parameter control is rather sophisticated and analytical solutions only exist in exceptionally simple situations. Thus, age-structured optimal control promises a challenging field in computational dynamic economics.

The intention of this research proposal is to help to fill the aforementioned gap. In what follows, we will briefly sketch the state of the art. No attempt is made to be exhaustive, but due to the relatively slow progress in the field we hope to include most of the important papers.

There are several books on distributed parameter control (see, e.g., Butkovsky), but most of them are not easy to read and deal with no economic applications. Derzko et al. (1984) provides an introduction to the field; see also Feichtinger and Hartl (1986, Appendix A.5) for a steep introduction.

A primary field of application circulates around population dynamics. In particular, demo­graphy, population economics, bioeconomics (i.e. the management of renewable resources) contain many problems in distributed parameter control. Gopalsamy (1976) studies age-structured populations with the birth trajectory as boundary control. In are important contri­bution Brokate (1985) analyzes a similar model for birth control and age-specific harvesting, see also Gurtin and Murphy (1981), Murphy and Smith (1990). An excellent overview on bioeconomic applications up to the late seventies is given in the Ph.D. thesis of Muzicant (1980); see also Derzko et al. (1980, 1984) dealing with the optimal management of a cattle ranching problem.

Arthur and McNicoll (1977) include the age structure of a population in planning the fertility of a population. By assuming that the fertility level and the savings rate can be influenced, and taking into consideration welfare trade-offs and social discounting, they try to develop a theory of demo-economic policy. The optimality conditions imply, among others, that policy should balance the life time value of births and capital against social costs of creating them. It should be noted that this approach up to now delivers only preliminary results. Although it is a rather difficult field, it is important enough to be pursued further.

Other promising applications of the age-dependent processes occur in manpower modeling, where the age variable may be interpreted as seniority (or duration in a certain state); see Gaimon and Thompson (1981). Haurie et al. (1984) present a rather general multi-class age-structured framework with interesting applications to social services planning.

Bensoussan et al. study an inventory model where the parameter may be interpreted as quality of a good which decreases with age (in the case of a perishable good like a blood conserve) or increases with age (with quality wine as example); see also Tzafestas (1982) for further applications of distributed optimal control to production/inventory problems.

Robson (1985) analyzes a model for the management of a housing stock. A salient feature of housing is its durability. Although the quality of houses declines over time, this decline is slow. Housing stocks, automobiles or other consumer durables often show a ‘vintage’ struc­ture. We assume that quality and age of a house are inversely related and that quality has a positive impact on value. The maximum principle for partial differential equations provides a powerful tool to get insight into the structure of optimal solutions of vintage models of such kind. Furthermore, capital vintage models might be formulated in a related framework.