Download Analyzing demand

Changing m
Elasticity
Examples
Changing own price
Changing other good’s price
Analyzing demand
Intermediate Micro
Lecture 6
Chapter 6 of Varian
Examples
Changing m
Elasticity
Examples
Changing own price
Changing other good’s price
Analyzing demand
Can model utility function and decisions
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Even for p, m we don’t observe
Can use demand functions to model comparative statics
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Comparative statics: Studying the effect on the equilibrium
outcome due to a change in parameters.
How does demand change as income increases?
What are the effects on demand of a change in prices?
Categorize goods based on comparative statics
Examples
Changing m
Elasticity
Examples
Changing own price
Changing other good’s price
Review: Demand function
Start with basic consumer decision problem:
maxx1 ,x2 u(x1 , x2 )
s.t.p1 x1 + p2 x2 = m
Leave p1 , p2 , m as parameters, and obtain demand functions
x1 (p1 , p2 , m)
x2 (p1 , p2 , m)
Examples
Changing m
Elasticity
Examples
Changing own price
Changing other good’s price
Examples
Changing m
x1 (p1 , p2 , m)
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Consider effect of ↑ m
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Easiest measure:
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Normal good: x1 is a normal good (at (p1 , p2 , m)) if
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Inferior good: x1 is an inferior good (at (p1 , p2 , m))
dx1
dm
(=
d
dm x1 (p1 , p2 , m))
dx1
dm > 0
1
if dx
dm < 0
Changing m
Elasticity
Examples
Changing own price
Changing other good’s price
Normal vs Inferior
x1 is normal
x2 is normal
x1 is inferior
x2 is normal
Can both goods be inferior?
Examples
Changing m
Elasticity
Examples
Changing own price
Income offer curve
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Income offer curve: A
graph of all optimal
bundles for a given
p1 , p2 , for all values of m
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m varies
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p1 , p2 stay constant
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Plug various values of m
into demand functions,
plot results
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If both goods are normal,
income offer curve is
upward sloping (%)
Changing other good’s price
Examples
Changing m
Elasticity
Examples
Changing own price
Income offer curve
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Income offer curve: A
graph of all optimal
bundles for a given
p1 , p2 , for all values of m
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m varies
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p1 , p2 stay constant
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Plug various values of m
into demand functions,
plot results
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If both goods are normal,
income offer curve is
upward sloping (%)
Changing other good’s price
Examples
Changing m
Elasticity
Examples
Changing own price
Changing other good’s price
Examples
Engel curve
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Engel curve: A graph of the demand for one of the goods, for
all values of m, holding constant p1 , p2
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m varies
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p1 , p2 stay constant
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Plug various values of m into demand function, plot results
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If the good is normal, Engel curve is upward sloping (%)
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The Engel curve never slopes downward
Changing m
Elasticity
Examples
Changing own price
Changing other good’s price
Income offer curve
axes: x1 , x2
Engel curves
←
axes: xi , m
↑
Examples
Changing m
Elasticity
Examples
Changing own price
Changing other good’s price
Elasticity - Not in book!
dxi
dm
∗
m
xi
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Income elasticity of demand: xi ,m =
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This formula is called point (income) elasticity
Percent change in xi relative to the percent change in m
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∆x/x
Non-calculus formula: ∆m/m
Called arc (income) elasticity
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Formula for (instantaneous) percent growth of y due to z:
d
dz ln(y (z))
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xi ,m =
d
ln(x1 (p1 ,p2 ,m))
dm
d
ln(m)
dm
Examples
Changing m
Elasticity
Examples
Changing own price
Changing other good’s price
Why use elasticity - Not in book!
I dxi :
dm
change in xi due to increase in m
dxi
dm
dx
i
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dm
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> 0: increasing in m
< 0: decreasing in m
Scale?
I dxi ∗ m (income elasticity of demand): Same sign
dm
xi
I Note that elasticity (and slope!) can vary with m
as
dxi
dm
Examples
Changing m
Elasticity
Examples
Changing own price
Changing other good’s price
Elasticity-based definitions- Not in book!
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Unit elasticity: when
xi ,m = 1
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xi grows at same rate as
m
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Any ray through the
origin has unit elasticity
Engel curve with unit elasticity
Examples
Changing m
Elasticity
Examples
Changing own price
Changing other good’s price
Homothetic preferences
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Homothetic preferences: A set of preferences with the
property that, if (x1 , x2 ) ∼ (y1 , y2 ), then
(tx1 , tx2 ) ∼ (ty1 , ty2 ), ∀t ≥ 0
Equivalent properties:
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Income offer curves are straight lines through the origin, for
any (p1 , p2 )
Engel curves are straight lines through the origin, for any
(p1 , p2 )
xi ,m for any (p1 , p2 , m), for any good i
Examples
Changing m
Elasticity
Examples
Changing own price
Changing other good’s price
Elasticity-based definitions- Not in book!
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Luxury good: xi for
which xi ,m > 1
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xi grows at faster rate
than m
To identify on Engel
curve
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1. Draw ray from origin
to point
2. If curve crosses ray
from left to right,
good is luxury at this
(p1 , p2 , m)
Engel curve for Ikea furniture
Examples
Changing m
Elasticity
Examples
Changing own price
Changing other good’s price
Elasticity-based definitions- Not in book!
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Necessary good: xi for
which xi ,m < 1
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xi grows at slower rate
than m
To identify on Engel
curve
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1. Draw ray from origin
to point
2. If curve crosses ray
from right to left,
good is ncessary at
this (p1 , p2 , m)
Engel curve for Ikea furniture
Examples
Changing m
Elasticity
Examples
Changing own price
Changing other good’s price
Perfect substitutes
u(x1 , x2 ) = 2x1 + 3x2
m = x1 + 2x2
x1 (1, 2, m) = m, x2 (1, 2, m) = 0
Income offer curve
Engel curve for x1
Examples
Changing m
Elasticity
Examples
Changing own price
Changing other good’s price
Cobb Douglas
u(x1 , x2 ) = x10.4 x20.6
m = 0.5x1 + 1.5x2
x1 (0.5, 1.5, m) = 0.8m, x2 (0.5, 1.5, m) = 0.4m
Income offer curve
Engel curve for x2
Examples
Changing m
Elasticity
Examples
Changing own price
Changing other good’s price
Examples
Quasilinear
u(x1 , x2 ) = ln(x1 ) + 0.25x2
m= x1 + x2
m if m < 4
0
if m < 4
x1 (1, 1, m) =
, x2 (1, 1, m) =
4 if m ≥ 4
m − 4 if m ≥ 4
Income offer curve
Engel curve for x1
Changing m
Elasticity
Examples
Changing own price
Changing p1 : effect on x1
x1 (p1 , p2 , m)
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Consider effect of ↑ pi on xi
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Derivative:
I dxi
dpi
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dxi
dpi
< 0 for all known goods
Giffin good: A good for which
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No documented examples!
dxi
dpi
>0
Changing other good’s price
Examples
Changing m
Elasticity
Examples
Changing own price
Changing other good’s price
Own-price elasticity - Not in book!
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dxi
Own-price elasticity of demand: xi ,pi = − dp
∗
i
% ↓ in xi relative to % ↑ in p1
xi ,pi
xi ,pi = 0
0 < xi ,pi < 1
xi ,pi = 1
1 < xi ,pi < ∞
xi ,pi = ∞
Description
Perfectly
inelastic
Inelastic
Unit elastic
Elastic
Perfectly
elastic
pi
xi
Interpretation
xi
does
not
change when pi
does
xi changes less
than pi does
xi changes by
same % pi does
xi changes more
than pi does
↑ pi ⇒ xi = 0,
↓ pi ⇒ xi = ∞
Examples
Changing m
Elasticity
Examples
Changing own price
Changing other good’s price
Inverse demand
x1 (p1 , p2 , m)
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Take m, p2 as fixed
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Rewrite demand function as x1 (p1 )
Can find inverse demand function: p1 (x1 )
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Gives p1 that causes x1 to be optimal
Only exists if each value x1 optimal only for one p1
Examples
Changing m
Elasticity
Examples
Changing own price
Changing other good’s price
Implications of own-price elasticity - Not in book!
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Expenditure on good 1 = p1 x1
If xi ,pi > (=, <)1
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↑ p1 causes ↓ ( no change, ↑) in p1 x1
xi ,pi
xi ,pi = 0
0 < xi ,pi < 1
xi ,pi = 1
1 < xi ,pi < ∞
xi ,pi = ∞
Description
Perfectly
inelastic
Inelastic
Unit elastic
Elastic
Perfectly
elastic
Interpretation
xi
does
not
change when pi
does
xi changes less
than pi does
xi changes by
same % pi does
xi changes more
than pi does
↑ pi ⇒ xi = 0,
↓ pi ⇒ xi = ∞
Examples
Changing m
Elasticity
Examples
Changing own price
Changing other good’s price
Changing p1 : effect on x2 - Not in book!
x2 =
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m
p1 x1
−
p2
p2
Suppose ↑ p1
I dx2
dp1
> (=, <)0 when xi ,pi > (=, <)1
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Complements: Two goods for which
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Substitutes: Two goods for which
dx2
dp1
dx2
dp1
<0
>0
Examples
Changing m
Elasticity
Examples
Changing own price
Price offer curve
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Price offer curve: A
graph of all optimal
bundles for a given
m, p2 , for all values of p1
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p1 varies, m, p2 constant
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Plug values of p1 into
demand functions, plot
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Complements: POC
upward sloping (%)
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Substitutes: POC
downward sloping (&)
Changing other good’s price
Examples
Changing m
Elasticity
Examples
Changing own price
Price offer curve
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Price offer curve: A
graph of all optimal
bundles for a given
m, p2 , for all values of p1
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p1 varies, m, p2 constant
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Plug values of p1 into
demand functions, plot
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Complements: POC
upward sloping (%)
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Substitutes: POC
downward sloping (&)
Changing other good’s price
Examples
Changing m
Elasticity
Examples
Changing own price
Demand curve
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Demand curve: A graph
of the demand for good
i, for all values of pi ,
holding constant
m, pnot i
Non-Giffin goods:
downward-sloping or flat
Changing other good’s price
Examples
Changing m
Elasticity
Examples
Changing own price
Changing other good’s price
Examples
Perfect substitutes
u(x1 , x2 ) = x1 + x2
10 =p1 x1 + x2


 10
if
p
<
1 
if p1 < 1 
 0
 p1
1
10 − x1 if p1 = 1
, x (p , 1, 10) =
x1 (p1 , 1, 10) =
[0, 10] if p1 = 1
 2 1



10
if p1 > 1
0
if p1 > 1
Price offer curve
Demand curve for x1
Changing m
Elasticity
Examples
Changing own price
Changing other good’s price
Cobb-Douglas
u(x1 , x2 ) = x10.75 x20.25
40 = 2x1 + p2 x2
x1 (2, p2 , 40) = 15, x2 (2, p2 , 40) =
Price offer curve
10
p2
Demand curve for x2
Examples
Changing m
Elasticity
Examples
Changing own price
Changing other good’s price
Quasilinear
u(x1 , x2 ) = ln(x1 ) + x2
10 = p1 x1 + x2
x1 (p1 , 1, 10) = p11 , x2 (p1 , 1, 10) = 9
Price offer curve
Demand curve for x1
Examples