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,
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  2 /1 .
• Normal downward sloping
demand curves for p1/p2 >  2 /1 ,
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  2 /1 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 =  2 /1, reaching a minimum
when Z2 = µ2, and increasing again for
µ2 < Z2 < F.
