Utility, theory and theorem:the economic case for a

Basic Incomeby

Anne G. Miller

Chair

Citizen’s Income Trust, UK

for Social Policy Association Annual Conference

Monday, 14 July, 2014,

15.30-17.00

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.

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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.

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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.

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