ABSTRACT
In Chapter 1 we started from the familiar textbook analysis, involving indifference curves and a budget line in commodity space, and simply added a consumption time constraint. While our attention was centred on quantities of commodities-the xi-it was of course also true that each equilibrium position considered involved a certain allocation of the total available time between the different consumption activities-the (xi/ri). Here, by contrast, we shall follow at least provisionally the example of unser Schutzheilige (our patron saint) Hermann Heinrich Gossen. In his brilliant but ill-fated work Entwickelung der Gesetze des menschlichen Verkehrs of 1854, Gossen worked, one might say, at the ‘opposite pole’ from the now standard analysis of consumer behaviour. While the latter ignores the consumption time constraint, Gossen began by ignoring the budget constraint and recognizing that even in the Land of Cockaigne there would be an allocation problem, that of allocating one’s time to different activities. Gossen’s formulation of his analysis was rather grand, even grandiose, supposing the agent to allocate, from the beginning, a whole lifetime to different activities. But (some of) the central issues are unchanged if, more modestly, we consider the time-allocation problem facing Vecchio, an elderly man convinced that he has just six months to live. Since he possesses ample stocks of durable consumer goods and has a large pension, he faces no binding budget constraint. But he still has to decide what to do, how to use his time, over his remaining six months. (And the same would be true if Vecchio simply adopted six months as his ‘planning period’, notwithstanding the fact that he expected to live somewhat longer than six months; though this interpretation would lead inexorably into more complex considerations of ‘rolling time horizons’ and so on.)
Gossen on the uses of time On setting aside a number of complications (such as uncertainty, the repetition and sequencing of activities, etc.) and using modern terminology where it will be more clear, one may present a central feature of Gossen’s approach to Vecchio’s problem as follows. Take the utility to Vecchio of his pursuit of activity i to depend only on the time ti devoted to that activity and suppose that his total utility is simply the sum of the various activity-specific utilities. Purely for simplicity, take the utility rendered by activity i to be given by [aiti− so that
(2-1)
Then u in (2.1) is to be maximized subject to the identity (2.2)
where T represents the six months available. It is assumed that T, so that Vecchio is not able to achieve the ‘satiation’ level (ai/bi) in each activity. (Note that, for simplicity, we shall attend only to those ti which are chosen to be positive. And note too that later in this chapter we shall offer an objection to Gossen’s formulation of Vecchio’s problem.)
The first order conditions for maximization are clearly that (2.2) obtain and that the ‘marginal utilities of time use’ should all be equal, say:
ai−biti=θ (2.3) From (2.2) and (2.3),
(2.4) Thus θ is determined and each ti then follows from (2.3). If only t1, t2 and t3 are positive, for example, then
(2.5)
etc. It is readily seen from (2.4) that θ is increasing in each ai but decreasing in each bi. From (2.5), t1 is increasing in a1, b2 and b3 but decreasing in a2, a3 and b1.
