Leisure and consumption in three dimensions

Anne Miller

MPRA Paper from University Library of Munich, Germany

Abstract: The separability rule is unable to distinguish between two commodities fulfilling the same need and those fulfilling different needs, for utilities displaying only diminishing marginal utility. An S-shaped utility, bounded below and above, represents the stages of an individual’s fulfilment of a need, including deprivation (increasing marginal utility), subsistence, sufficiency (diminishing marginal utility) and satiation. A utility function is created by adding two (S-shaped) normal cumulative distribution functions for consumption and leisure, each with a subsistence and an intensity-of-need parameter and satiation at infinity. Its indifference curve map features a straight-line indifference curve separating concave- from convex-to-the-origin indifference curves. The utility function is then maximised subject to a budget constraint to produce consumption demand and labour supply equations. These two functional forms are dependent on only two independent variables – the real wage rate and endowments of unearned consumption. Thus, both consumption demand and labour supply are 3-dimensional figures, which ideally would be presented as 3-D models. The typical demand/supply and Engels diagrams are only two dimensional, representing a dependent variable as a function of only one of its two independent variables, from which the 3-D figure is very difficult to envisage. The aim of this paper is to present the third 2-D diagram for each dependent variable, presented as contours on a map of the real wage rate vs endowments. They highlight the instability of labour and consumption around the intersection of the ‘survival endowment’ and ‘equilibrium wage/price’ created by the straight-line indifference curve.

Keywords: S-shaped cardinal utility includes increasing marginal utility expressing deprivation; additive utilities represent separate needs; dysfunctional poverty causes involuntary unemployment and disequilibrium; labour and consumption contours; instability at the conjunction of ‘survival endowment’ and ‘equilibrium wage’. (search for similar items in EconPapers)
JEL-codes: D11 J22 (search for similar items in EconPapers)
Date: 2025-12-29
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