Properties of Demand Functions

1
Properties of Demand Functions

• Comparative statics analysis of ordinary demand
  functions -- the study of how ordinary demands
  x1(p1,p2,m) and x2(p1,p2,m) change as prices
  p1, p2 and income m change.
• Own-Price changes
• How does x1(p1,p2,m) change as p1 changes,
  holding p2 and m constant?

2
p1
Own-Price Changes
Ordinarydemand curvefor commodity 1
Fixed p2 and m.
p1
p1
p1
x1
x1(p1)
x1(p1)
x1(p1)
x1(p1)
x1(p1)
x1(p1)
3
Own-Price Changes

• The curve containing all the utility-maximizing
  bundles traced out as p1 changes, with p2 and m
  constant, is the p1- price offer curve.
• The plot of the x1-coordinate of the p1- price
  offer curve against p1 is the ordinary demand
  curve for commodity 1.

4
Own-Price Changes Cobb-Douglas

• Take
• Then the ordinary demand functions for
  commodities 1 and 2 are
• What does the demand curve look like?
• What does a p1 price-offer curve look like?

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p1
Own-Price Changes
Ordinarydemand curvefor commodity 1 is
Fixed p2 and m.
x1
x1(p1)
x1(p1)
x1(p1)
6
Own-Price Changes Perfect complements

• Take
• Then the ordinary demand functionsfor
  commodities 1 and 2 are
• What does the demand curve look like?
• What is the p1 price-offer curve?

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p1
Own-Price Changes
Ordinarydemand curvefor commodity 1 is
Fixed p2 and m.
p1
x2
p1
m/p2
p1
x1
x1
8
Own-Price Changes Perfect Substitutes

• Take
• Then the ordinary demand functionsfor
  commodities 1 and 2 are
• What does the demand curve look like?
• What is the p1 price-offer curve?

9
p1
Own-Price Changes
Ordinarydemand curvefor commodity 1
Fixed p2 and m.
p1
x2
p2 p1
p1 price offer curve
p1
x1
x1
10
Income Changes

• How does the value of x1(p1,p2,m) change as m
  changes, holding both p1 and p2 constant?
• A plot of quantity demanded against income is
  called an Engel curve.
• A plot of bundles chosen as we vary income is
  called the income-offer curve.
• Draw these two curves for Perfect complements,
  Cobb-Douglas, and Perfect Substitutes.

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Income Changes

• In every example so far the income offer curves
  have all been straight lines for the origin?Q
  Is this true in general?
• A No. Income offer curves are straight lines
  only if the consumers preferences are homothetic.

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Homotheticity

• A consumers preferences are homothetic if and
  only iffor every k gt 0.
• That is, the consumers MRS is the same anywhere
  on a straight line drawn from the origin.

p
p
Û
(x1,x2) (y1,y2) (kx1,kx2)
(ky1,ky2)
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Why the connection?

• Take a budget and choice (x1,x2). Double the
  budget so new budget is 2m.
• Notice if (x1,x2) were in the old budget, (2 x1,
  2 x2) is in the new budget.
• Homotheticity implies that if (x1,x2)gt (x1,x2)
  then (2x1,2x2)gt(2x1,2x2), thus new choice is
  (2x1,2x2).
• Notice if you draw a line from origin to
  (x1,x2) it goes through (2x1,2x2).
• Since these are the choices, the MRS must be the
  same at both points to be tangent to the ICs.

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Homogeneous Utility Functions

• U(k x1,k x2) g(k) u(x1,x2) and g(k)gt0.
• This implies homotheticity!
• U(x1,x2)gtU(y1,y2) gt
• g(k) U(x1,x2)gt g(k) U(y1,y2) gt
• U(k x1,k x2)gtU(k y1,k y2)
• Does Cobb-Douglas, Perfect Substitutes, Perfect
  Complements satisfy this?

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Income Effects -- A Nonhomothetic Example

• Quasilinear preferences are not homothetic.
• For example,

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Income Effects

• A good for which quantity demanded rises with
  income is called normal.
• Therefore a normal goods Engel curve is
  positively sloped.
• A good for which quantity demanded falls as
  income increases is called income inferior.
• Therefore an income inferior goods Engel curve
  is negatively sloped.

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Ordinary and Giffen goods.

• A good is called ordinary if the quantity
  demanded of it always increases as its own price
  decreases.
• If, for some values of its own price, the
  quantity demanded of a good rises as its
  own-price increases then the good is called
  Giffen.

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Cross-Price Effects

• If an increase in p2
• increases demand for commodity 1 then commodity 1
  is a gross substitute for commodity 2.
• reduces demand for commodity 1 then commodity 1
  is a gross complement for commodity 2.
• What happens with Cobb-Douglas preferences?