Utility, theory and theorem: the economic case for a Basic Income

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Utility, theory and theoremthe economic case
for a Basic Income

• by
• Anne G. Miller
• Chair
• Citizens Income Trust, UK
• for Social Policy Association Annual Conference
• Monday, 14 July, 2014,
• 15.30-17.00
• University of Sheffield

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Proposition 1.The leaning S-shaped utility
function

• The individuals 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 lt 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 lt 0).

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Figure 1.
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The leaning S-shaped utility function

• For 0 lt Xi lt µi, the consumer experiences
  deprivation of commodity i.
• Xi µi is the subsistence level of consumption
  for commodity i.
• µi lt Xi lt sati, the consumer experiences
  sufficiency.
• At Xi sati, the consumer is satiated.
• For finite sati, when Xi gt sati, the consumer is
  in surfeit.

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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.

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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.

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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 gt
  , 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
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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 lt Z2 lt C, BI leads to a
  polarised outcome ie dysfunctional poverty or
  high income.
• This is the economic case for a BI.
• Ie, Z2 lt C can lead to dysfunctional poverty for
  individuals facing low wages.

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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.

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Labour supply curves ctd.

• The intercept on the p1/p2 axis represents the
  reservation wage, the consumers 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 lt Z2 lt F.