Leisure and consumption in three dimensions

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.