Utility, theory and theorem: the economic case for a Basic Income by Anne G. Miller Chair Citizen’s Income Trust, UK for Social Policy Association Annual Conference Monday, 14 July, 2014, University of Sheffield

Proposition 1. The leaning S-shaped utility function The individual’s experience of consumption, Xi, of a commodity i (good, service or event) can be represented by a continuous, smooth, single-valued, utility function, that has the shape of a leaning S-shape. It has a minimum of Ui =0, for Xi < 0. It has increasing marginal utility, Ui’, until a point of inflection is reached at Xi = µi. The U-fn then has diminishing marginal utility until satiation is reached, where it has a maximum, Ui = 1, at either finite or infinite consumption. If satiation is reached at finite consumption, a surfeit can occur for increased consumption (and price < 0).

Figure 1.

The leaning S-shaped utility function For 0 < Xi < µi, the consumer experiences deprivation of commodity i. Xi = µi is the subsistence level of consumption for commodity i. µi < Xi < sati, the consumer experiences sufficiency. At Xi = sati, the consumer is satiated. For finite sati, when Xi > sati, the consumer is in surfeit.

Proposition 2. The separability of commodities The utilities of a group of commodities that satisfy the same need are multiplicatively related (with or without dependence). The utilities of groups of satisfiers, each group satisfying a different need, are additively related. It is assumed that there is a finite number of fundamental human needs, and that these are universal and ahistoric. Needs are satisfied by an infinite diversity of culturally-determined satisfiers. We apply this to consumption and leisure (additively related), see Fig 5.

Fig 2. Indifference curve map, for additive utilities, following. Note the following: The straight line indifference curve, AB, separating indifference curves that are concave-to-the-origin from those that are convex-to-the-origin; The triangle OAB is a non-solution space, - corner solutions only. The left hand and lower borders, where the consumer is deprived of X1 and X2 respectively; Both X1 and X2 can take on the characteristics of all of ultra-superior, superior-normal, inferior-normal and inferior Giffen good, depending on its combination with the other good.

Fig 4. Demand curves for additive utilities, following: Note the following: Horizontal axis, demand for X1, with parameter µ1. Vertical axis, real price p1/p2, with parameter. Normal downward sloping demand curves for p1/p2 >, and below. Downward sloping demand curves shifting to the right, for inferior goods; Upward sloping demand curves for Giffen good behaviour.

Fig 4

Fig 5. Consumption - Leisure indifference curves * Horizontal axis = leisure, parameter µ1, leisure constrained to eg 168 hours pw; let this endowment-of-time be labelled Z1. Vertical axis = consumption, parameter µ2. Straight line indifference curve separates concave-to-the-origin from the convex-to-the origin indifference curves. It has slope and represents the relative-intensity-of-need between the two dimensions. It may be thought of as a natural wage. The smaller the the greater the intensity-of-need.

Fig 5 ctd. Consumption- Leisure The left-hand and lower borders represent deprivation of leisure and consumption respectively. Leisure can be all of ultra-superior, superior, inferior, and Giffen. The indifference curve map is divided into areas L, M, N, and R. Z2 is an endowment of unearned consumption measured as the intercept on the ‘axis’ where Leisure = Z1 hrs pw. Z2.p2 = unearned income, eg Basic Income. For a low Z2, ie. 0 < Z2 < C, BI leads to a polarised outcome: ie dysfunctional poverty or high income. This is the economic case for a BI. Ie, Z2 < C can lead to dysfunctional poverty for individuals facing low wages.

Fig 6. Labour supply curves Horiziontal axis measures labour hours, (Z1 - X1), with parameter (Z1- µ1). Vertical axis is p1/p2, (real wages). The areas L, M, N and R from the indifference curve graph can be mapped onto the labour supply curves. R leads to downward-sloping labour supply curves for relatively high wages, to the right - deprived of leisure. The rest are backward-bending labour supply curves. The elastic ones for low prices derive from area L, deprived of consumption. There is an envelope curve below the labour supply curves co-incidental with the border between inferior and superior characteristics. When consumer has gained subsistence consumption, his/her labour supply curves become inelastic.

Labour supply curves ctd. The intercept on the p1/p2 axis represents the reservation wage, the consumer’s minimum acceptable wage-rate. The reservation wage is a U-shaped function of Z2, being highest when p1/p2 =, reaching a minimum when Z2 = µ2, and increasing again for µ2 < Z2 < F.