ADVANCED TEXTBOOKS IN ECONOMICS VOLUME 1 Editor: C. J. BLISS University of Essex NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM · LONDON AMERICAN ELSEVIER PUBLISHING CO., INC- NEW YORK. VARIATIONAL METHODS IN ECONOMICS G. HADLEY M. C. KEMP 0, 1 > s > 0. For values of k less than k* k > 0; and 9 for values of k greater than k* k < 0. Clearly k(t) approaches k* asymp- 9 totically, whatever the initial value of k (provided only that it is positive); hence the average product of labor approaches (X-*)a, consumption per capita approaches (1 — s)(k*)a, and in the limit all quantities undeflated by population grow at the same rate as the population itself. However we are not at this stage concerned with specific solutions. The important thing to notice is that the solutions, if they exist, depend on the parameters y and s, that these parameters can be influenced by government, and that there is therefore potentially a problem of choosing the optimal y and s. More generally, there is potentially a problem of choosing the optimal time paths y(t) and s(t). But optimal in relation to what criterion? Until the criterion is specified, the problem is not even defined. Collective decisions about the temporal distribution of consumption are of course political decisions, based on compromise. We therefore cannot hope to find a criterion which will carry conviction to everyone. The most that we can hope is that any criterion the implications of which we analyze will be found interesting. It will be assumed, here and later, that it is the stream of consumption C(t) or the stream of consumption per capita c(t) which 'matters'. What we need then is a rule for associating with each function C(t) or c(t) a number ^[C(0] 0Γ ^ΚΟ] which provides a numerical evaluation of the entire consumption stream. The number will be referred to as the utility of the consumption stream. Even if it is agreed that only consumption matters, however, there is still no single compelling rule of association, that is, definition of ^. Among many more-or-less plausible possibilities, economists have been especially interested in the simple or undiscounted sum or integral, over some finite or infinite planning period, of some function U of total or average consumption. The function of consumption is often referred to, not quite appropriately, as an instantaneous utility function. In a special case, the function may be a constant positive multiple of total or average consumption so that, in effect, % is simply the integral of consumption. Let us, for the sake of illustration, choose as our desideratum the simple or un- weighted integral of the instantaneous utility derived from consumption per capita. Now consumption per capita at time t is c(t) = (1 — s(t))y(t). Thus the problem is to find y(t) and s(t) < 1 such that T I l/((l - s(t))y(t), t)dt 0 < T < oo o is a maximum subject to (1-4) and (1-6) and to the requirement that k(t) GROWTH MODELS IN ECONOMICS 5 be non-negative, and given k(0) = k. Notice that we have not at this stage 0 imposed the requirement that s(t) > 0. This problem is typical of those we shall consider in later chapters. To pose the problem it was necessary to provide: (i) A description of the welfare criterion. (In the above example the time horizon and the utility function had to be specified.) (ii) A description of the economy. (In the above example the description is conveyed by (1-4) and (1-6).) Once the problem is properly posed there remains the further problem of solving it; and that is a matter of mathematical technique. The relevant techniques are discussed in the following four chapters.

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