Population policy under an arbitrary welfare criterion : theory and issues

Abstract

Many well-defined policy problems can be formally described as follows: Let x(t) be an n-dimensional vector describing the state of a system at time t , and y(t) be an m-dimensional vector of policy instrument variables at time t. Let U = U[x(t),y(t),t] be a measure of the condition or "welfare" of a system in state x and adopting policies y at time t. Then it is possible to evaluate any specified policy {y(t), 0≤t≤T} by its impact on J =∫T0 U dt . In particular, an optimal policy is one that maximizes J subject to whatever constraints on x and y are applicable.Analyzing this formulation by means of the calculus of variations enables one to relate a variation δy(t) in policy to the variation δJ in J that it induces. This relation involves also a set of multiplier functions (analogous to Lagrange multipliers) that can be interpreted as the "shadow prices" of the state variables.In this paper, population policy is analyzed in the above format, with stress on the demographic insights that follow from the variational approach. Particular problems investigated in terms of simple but fairly general models are: the value of a marginal birth (the shadow price of the state variable, population); the conditions under which δy is a policy improvement, i.e., δJ is positive (enabling, inter alia, a rigorous definition of "overpopulation"); and the characteristics of an optimal policy.