Demand

1
Demand
2
PROPORTIES 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.

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OWN-PRICE CHANGES

• How does x1(p1,p2,M) change as p1 changes,
  holding p2 and M constant?
• Suppose only p1 increases, from p1 1 to p1 2
  and then to p1 3.

4

OWN-PRICE CHANGES
x2
Fixed p2 and M
p1x1 p2x2 M
p1 1
x1
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OWN-PRICE CHANGES
Fixed p2 and M
x2
p1x1 p2x2 M
p1 1
p1 2
x1
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OWN-PRICE CHANGES
Fixed p2 and M
x2
p1x1 p2x2 M
p1 1
p1 3
p1 2
x1
7
p1
Own-Price Changes
Fixed p2 and M
x2
p1 1
P1 1
x1
x1(p11)
x1(p11)
x1
8
p1
Own-Price Changes
Fixed p2 and M
x2
p1 2
P11
x1
x1(p11)
x1(p11)
x1(p12)
x1
9
p1
Own-Price Changes
Fixed p2 and M
x2
p1 2
P12
P11
x1
x1(p11)
x1(p12)
x1(p11)
x1(p12)
x1
10
p1
Own-Price Changes
Fixed p2 and M
x2
P13
p1 3
P12
P11
x1
x1(p11)
x1(p12)
x1(p13)
x1(p11)
x1(p13)
x1(p12)
x1
11
p1
Own-Price Changes
Ordinarydemand curvefor product 1
Fixed p2 and M
x2
P13
P12
P11
x1
x1(p11)
x1(p12)
x1(p13)
x1(p11)
x1(p13)
x1(p12)
x1
12
p1
Own-Price Changes
Ordinarydemand curvefor product 1
Fixed p2 and M
x2
P13
P12
P11
P1 price offer curve
x1
x1(p11)
x1(p12)
x1(p13)
x1(p11)
x1(p13)
x1(p12)
x1
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OWN-PRICE CHANGES

• The curve containing all the utility-maximizing
  bundles traced out (in x1, x2 space) as p1
  changes, with p2 and M constant, is the price
  offer curve.
• The plot of the x1co-ordinates of the price offer
  curve against p1 is the ordinary demand curve for
  product 1. Q F(P) or X F(P)

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Own-Price Changes

• Taking quantity demanded as given and then asking
  what must be price describes the inverse demand
  function of a good. P F(Q) or P F(X)

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OWN-PRICE CHANGES
A Cobb-Douglas example
is the ordinary demand function and
is the inverse demand function.
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OWN-PRICE CHANGES
A perfect complements example
is the ordinary demand function and
is the inverse demand function.
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OWN-PRICE CHANGES

• What does a p1 price-offer curve look like for
  Cobb-Douglas preferences?
• TakeThen the ordinary demand functions for
  goods 1 and 2 are

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OWN-PRICE CHANGES
and
Notice that x2 does not vary with p1 so
theprice offer curve is flat and the
ordinarydemand curve for product 1 is a
rectangular hyperbola.
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Own-Price Changes
Fixed p2 and M
x2
X2bM/ (ab)p2
X1aM/ (ab)p1
x1
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p1
Own-Price Changes
Fixed p2 and M
x2
X2bM/ (ab)p2
x1
X1aM/ (ab)p1
x1
21
p1
Own-Price Changes
Ordinarydemand curvefor product 1
is X1aM/(ab)p1
Fixed p2 and M
x2
X2bM/ (ab)p2
x1
X1aM/ (ab)p1
x1
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OWN-PRICE CHANGESPERFECT COMPLEMENTS

• What does a p1 price-offer curve look like for a
  perfect-complements utility function?

The ordinary demand functionsfor products 1 and
2 are
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OWN-PRICE CHANGESPERFECT COMPLEMENTS
With p2 and M fixed, higher p1 causessmaller x1
and x2.
As
As
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OWN-PRICE CHANGESPERFECT COMPLEMENTS
x2
Fixed p2 and M
x1
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p1
OWN-PRICE CHANGESPERFECT COMPLEMENTS
Fixed p2 and M
x2
p1 p11
M/p2
p11
x1
x1
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p1
OWN-PRICE CHANGESPERFECT COMPLEMENTS
Fixed p2 and M
x2
p1 p12
p12
M/p2
p11
x1
2
x1
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p1
OWN-PRICE CHANGESPERFECT COMPLEMENTS
Fixed p2 and M.
p13
x2
p1 p13
p12
M/p2
p11
x1
x1
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p1
OWN-PRICE CHANGESPERFECT COMPLEMENTS
Ordinarydemand curvefor product 1 is
Fixed p2 and M
p13
x2
p12
M/p2
p11
x1
x1
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OWN-PRICE CHANGESPERFECT SUBSITUTES

• What does a price-offer curve look like for a
  perfect-substitutes utility function?

Then the ordinary demand functionsfor products 1
and 2 are
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OWN-PRICE CHANGESPERFECT SUBSTITUTES
and
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OWN-PRICE CHANGESPERFECT SUBSTITUTES
Homework Draw the diagrams.
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INCOME CHANGES

• How does the value of x1(p1,p2,M) change as M
  changes, holding both p1 and p2 constant?

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INCOME CHANGES

• A plot of quantity demanded against income is
  called an Engel curve.

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INCOME CHANGES
Fixed p1 and p2
M1 lt M2 lt M3
Engelcurve good 1

Incomeoffer curve
M
x23
M3
x22
M2
x21
M1
x11
x13
x13
x11
x1
x12
x12
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INCOME CHANGES
Engelcurve good 2
M
Fixed p1 and p2
M3
M2
M1 lt M2 lt M3
M1
Incomeoffer curve
x23
x21
x2
x22
x23
x22
x21
x13
x11
x12
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INCOME CHANGES and HOMOTHETIC PREFERENCES

• Demand for each good goes up by the same
  proportion as income
• Income offer curve (also known as the income
  expansion path) is a straight line through the
  origin
• Engel curve is a straight line through the origin

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HOMOTHETICITY

• A consumers preferences are homothetic if and
  only if
• (x1, x2) gt (y1, y2) ? (tx1, tx2) gt (ty1, ty2)
• for all positive t.

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HOMOTHETICITY

• Linear expansion path through the origin

X2
X1
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Income Effects

• A product for which quantity demanded rises with
  income is called normal.
• Therefore a normal products Engel curve is
  positively sloped.

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Income Effects

• A product for which quantity demanded falls as
  income increases is called inferior.
• Therefore an inferior products Engel curve is
  negatively sloped.

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Income Changes Products 1 2 Normal
Engelcurve good 2
M
M3
M2
M1
Incomeoffer curve
x23
x21
x2
M
x22
x23
M3
Engelcurve good 1
x22
M2
x21
M1
x13
x11
x13
x11
x1
x12
x12
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As income changes
Engel curves
M
x2
product 2
x2
M
Inferior ?x1/?Mlt0
product 1
Normal ?x1/?Mgt0
x1
x1
Product 2 Is Normal, Product 1 Becomes Inferior
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ORDINARY PRODUCTS

• A product is called ordinary if the quantity
  demanded always increases as its own price
  decreases.

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ORDINARY PRODUCTS
Fixed p2 and M.
Downward sloping demand curve
x2
p1
p1 price offer curve
Û
Product 1 isordinary
x1
x1
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GIFFEN PRODUCTS

• If, for some values of its own price, the
  quantity demanded of a product rises as its own
  price increases then the product is called a
  Giffen product.

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GIFFEN PRODUCTS
Demand curve has a positively
sloped part
Fixed p2 and M
x2
p1
p1 price offer curve
Û
Product 1 isGiffen
x1
x1
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CROSS PRICE EFFECTS

• If an increase in p2
• increases demand for product 1 then product 1 is
  a gross substitute for product 2.
• reduces demand for product 1 then product 1 is a
  gross complement for product 2.

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CROSS PRICE EFFECTS
A perfect complements example
so
Therefore product 2 is a grosscomplement for
product 1
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CROSS PRICE EFFECTS
p1
Increase the price ofproduct 2 from p21 to
p22and
p13
p12
p11
x1
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CROSS PRICE EFFECTS
Increase the price ofproduct 2 from p21 to
p22and the demand curve for product 1 shifts
inwards-- product 2 is acomplement for product
1.
p1
p13
p12
p11
x1
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CROSS PRICE EFFECTS
A Cobb- Douglas example
so
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CROSS PRICE EFFECTS
A Cobb- Douglas example
so
Therefore product 1 is neither a grosscomplement
nor a gross substitute forproduct 2.
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SUMMARY I

• Changes in income
• Cobb Douglas
• Perfect Substitutes
• Perfect Complements

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SUMMARY II

• Cross Price Changes
• Cobb Douglas
• Perfect Substitutes
• (Be Careful!)
• Perfect Complements