previously, we examined a consumer’s optimal choice under his budget constraint. in this chapter,...
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Previously, we examined a consumer’s optimal choice under his budget constraint.
In this chapter, we will perform comparative static analysis of ordinary demand functions. Namely, how do ordinary demands x1*(p1,p2,m) and x2*(p1,p2,m) change as prices p1, p2 and income m change?
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How does x1*(p1,p2,m) change as p1 changes, holding p2 and m constant?
Suppose only p1 increases, from p1’ to p1’’ and then to p1’’’.
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4
x1
x2
p1= p1’’p1=p1’’’
Fixed p2 and m.
p1 = p1’p1x1 + p2x2 = m
x1*(p1’)
Own-Price Changes
p1 = p1’
Fixed p2 and m.
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x1*(p1’)
p1
x1*(p1’)
p1’
x1*
Own-Price ChangesFixed p2 and m.
p1 = p1’
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x1*(p1’)
x1*(p1’’)
p1
x1*(p1’)
x1*(p1’’)
p1’
p1’’
x1*
Own-Price ChangesFixed p2 and m.
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x1*(p1’’’) x1*(p1’)
x1*(p1’’)
p1
x1*(p1’)x1*(p1’’’)
x1*(p1’’)
p1’
p1’’
p1’’’
x1*
Own-Price ChangesFixed p2 and m.
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x1*(p1’’’) x1*(p1’)
x1*(p1’’)
p1
x1*(p1’)x1*(p1’’’)
x1*(p1’’)
p1’
p1’’
p1’’’
x1*
Own-Price Changes Ordinarydemand curvefor commodity 1Fixed p2 and m.
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x1*(p1’’’) x1*(p1’)
x1*(p1’’)
p1
x1*(p1’)x1*(p1’’’)
x1*(p1’’)
p1’
p1’’
p1’’’
x1*
Own-Price Changes Ordinarydemand curvefor commodity 1
p1 price offer curve
Fixed p2 and m.
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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.
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A good is called an ordinary good if the quantity demanded of it always increases as its own price decreases and vice versa (negatively related), holding all other factors, such as prices, income and preference constant.
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Fixed p2 and m.
x1
x2
p1 price offer curve
x1*
Downward-sloping demand curve
Good 1 isordinary
p1
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If the quantity demanded of a good decreases as its own-price decreases and vice versa (i.e. positively related) holding all other factors constant, then the good is called a Giffen Good.
Note: we need to hold other factors constant. Thus, if the price change is also associated with change in income or preference, then even if there’s a positive relation between price and quantity, it is not characterized as Giffen good.
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Fixed p2 and m.
x1
x2 p1 price offer curve
x1*
Demand curve has a positively sloped part
Good 1 isGiffen
p1
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What does a p1 price-offer curve look like for a perfect-complements utility function?
U x x x x( , ) min , .1 2 1 2
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With p2 and m fixed, higher p1 causes smaller x1* and x2*.
As
As
.),,(),,(21
21*221
*1 pp
mmppxmppx
.,02
*2
*11 p
mxxp
.0, *2
*11 xxp
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Fixed p2 and m.
x1
x2
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p1
x1*
Fixed p2 and m.
Perfect Complements
x1
x2
p1’
p1 = p1’
’
’
m/p2
21
*1 ' pp
yx
21
*1 pp
mx
21
*2
pp
m
x
19
p1
x1*
Fixed p2 and m.
Perfect Complements
x1
x2
p1’
p1’’p1 = p1’’
m/p2
21
*2
" pp
m
x
21
*1 " pp
mx
21
*1 " pp
mx
20
p1
x1*
Fixed p2 and m.
Perfect Complements
x1
x2
p1’
p1’’
p1’’’
p1 = p1’’’m/p2
21
*2
''' pp
m
x
21
*1 ''' pp
mx
21
*1 ''' pp
mx
21
p1
x1*
Ordinarydemand curvefor commodity 1 is
Fixed p2 and m.
Perfect Complements
x1
x2
p1’
p1’’
p1’’’
m/p2 .21
*1 pp
mx
2p
m
21
*1 pp
mx
21
*2
pp
m
x
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What does a p1 price-offer curve look like for a perfect-substitutes utility function?
U x x x x( , ) .1 2 1 2
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and
211
2121
*1 ,/
,0),,(
ppifpm
ppifmppx
.,/
,0),,(
212
2121
*2 ppifpm
ppifmppx
24
Fixed p2 and
Perfect Substitutes
x2
x1
p1
x1*
Fixed p2 and m.
p1’
p1 = p1’ < p2
'1
*1 p
mx
1
*1 p
mx
x2 0*
25
Fixed p2 and m.
Perfect Substitutes
x2
x1
p1
x1*
Fixed p2 and .
p1’
p1 = p1’’ = p2
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Fixed p2 and
Perfect Substitutes
x2
x1
p1
x1*
Fixed p2 and m.
x2 0*
p1’
p1 = p1’’ = p2
x1 0*
p2 = p1’’
2
*2 p
mx
2
*1 p
mx
2
*10
p
mx
27
Fixed p2 and m.
Perfect Substitutes
x2
x1
p1
x1*
Fixed p2 and
2
*2 p
mx
x1 0*
p1’
p1’’’
x1 0*
p2 = p1’’
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Fixed p2 and m.
Perfect Substitutes
x2
x1
p1
x1*
Fixed p2 and
p1’
p2 = p1’’
p1’’’
1
*1 p
mx
2
*10
p
mx
2p
mp1 price offer curve
Ordinarydemand curvefor commodity 1
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Usually we ask “Given the price for commodity 1 what is the quantity demanded of commodity 1?”
But we could also ask the inverse question “At what price for commodity 1 would a given quantity of commodity 1 be demanded?”
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p1
x1*
p1’
Given p1’, what quantity isdemanded of commodity 1?Answer: x1’ units.
x1’
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p1
x1*
p1’
x1’
Given p1’, what quantity isdemanded of commodity 1?Answer: x1’ units.
The inverse question is:Given x1’ units are demanded, what is the price of commodity 1? Answer: p1’
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If an increase in p2, holding p1 constant, 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.
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A perfect-complements example:
so
Therefore commodity 2 is a gross complement for commodity 1.
21
*1 pp
mx
.02
212
*1
pp
m
p
x
34
p1
x1*
p1’
p1’’
p1’’’
Increase the price ofgood 2 from p2’ to p2’’and
'2p
m
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p1
x1*
p1’
p1’’
p1’’’
Increase the price ofgood 2 from p2’ to p2’’and the demand curvefor good 1 shifts inwards-- good 1 is acomplement for good 2.
''2p
m
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For perfect substitutes, how will the quantity demanded x1 change when p2 increases?
Note: it only changes the point where x1 starts to be positive.
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p1
x1*
p1’
p1’’
p1’’’
Generally, if increase the price of good 2 from p2’ to p2’’ and the demand curve for good 1 shifts outwards-- good 1 is asubstitute for good 2.
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How does the value of x1*(p1,p2,m) change as m changes, holding both p1 and p2 constant?
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Income ChangesFixed p1 and p2.
m’ < m’’ < m’’’
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Income ChangesFixed p1 and p2.
x1’’’
x1’’
x1’
x2’’’
x2’’
x2’
m’ < m’’ < m’’’
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Income ChangesFixed p1 and p2.
x1’’’
x1’’
x1’
x2’’’
x2’’
x2’
Incomeoffer curve
m’ < m’’ < m’’’
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A plot of income against quantity demanded is called an Engel curve.
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Income ChangesFixed p1 and p2.
m’ < m’’ < m’’’
x1’’’
x1’’
x1’
x2’’’
x2’’
x2’
Incomeoffer curve
x1*
x2*
y
y
x1’’’x1’’x1’
x2’’’x2’’x2’
m’
m’’
m’’’
m’
m’’
m’’’
Engelcurve;good 2
Engelcurve;good 1
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p1
x1*
p1’
p1’’
p1’’’
When income increases, the curve shifts outward for each given price, if the good is a normal good.
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A good for which quantity demanded rises with income is called normal.
Therefore a normal good’s Engel curve is positively sloped.
Generally, most goods are normal goods.
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A good for which quantity demanded falls as income increases is called inferior.
Therefore an inferior good’s Engel curve is negatively sloped.
E.g.: low-quality goods.
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Income Changes; Goods1 & 2 Normal
x1’’’
x1’’
x1’
x2’’’
x2’’
x2’
Incomeoffer curve
x1*
x2*
y
y
x1’’’
x1’’
x1’
x2’’’
x2’’
x2’
y’
y’’
y’’’
y’
y’’
y’’’
Engelcurve;good 2
Engelcurve;good 1
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Income Changes; Good 2 Is Normal, Good 1 Becomes Inferiorx2
x1
Incomeoffer curve
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Income Changes; Good 2 Is Normal, Good 1 Becomes Inferiorx2
x1x1*
y
Engel curvefor good 1
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Now we compute the equations of Engel curves for the perfectly-complementary case.
The ordinary demand equations are.
21
*2
*1 pp
mxx
U x x x x( , ) min , .1 2 1 2
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Rearranging these equations, we get:
*221
*121
)(
)(
xppm
xppm
Engel curve for good 1
.21
*2
*1 pp
mxx
Engel curve for good 2
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Fixed p1 and p2.
Income Changes
x1
x2
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Income Changes
x1
x2 y’ < y’’ < y’’’
x1’’
x1’
x2’’’
x2’’
x2’
x1’’’
Fixed p1 and p2.
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Income Changes
x1
x2 y’ < y’’ < y’’’
x1’’
x1’
x2’’’
x2’’
x2’
x1’’’ x1*
x2*
y
y x2’’’
x2’’
x2’
y’
y’’
y’’’
y’
y’’
y’’’
Engelcurve;good 2
Engelcurve;good 1
x1’’’
x1’’
x1’
Fixed p1 and p2.
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Income Changes
x1*
x2*
y
y x2’’’
x2’’
x2’
y’
y’’
y’’’
y’
y’’
y’’’
x1’’’
x1’’
x1’
y p p x ( ) *1 2 2
y p p x ( ) *1 2 1
Engelcurve;good 2
Engelcurve;good 1
Fixed p1 and p2.
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Another example of computing the equations of Engel curves; the perfectly-substitution case.
The ordinary demand equations are
U x x x x( , ) .1 2 1 2
211
2121
*1 ,/
,0),,(
ppifpm
ppifmppx
. ,/
,0),,(
212
2121
*2 ppifpm
ppifmppx
57
x2 0* .y p x 1 1*
y y
x1* x2*0
Engel curvefor good 1
Engel curvefor good 2
Suppose p1 < p2
It is the opposite when p1 > p2.
58
In every example so far the Engel curves have all been straight lines.
Q: Is this true in general?
A: No. Engel curves are straight lines if the consumer’s preferences are homothetic.
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A consumer’s preferences are homothetic if and only if(x1,x2) (y1,y2) (kx1,kx2) (ky1,ky2)for every k > 0.
That is, the consumer’s MRS is the same anywhere on a straight line drawn from the origin.
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Quasi-linear preferences
For example,
U x x f x x( , ) ( ) .1 2 1 2
U x x x x( , ) .1 2 1 2
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Income Changes; Quasilinear Utility
x2
x1
x1
~
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Income Changes; Quasilinear Utility
x2
x1
x1
~
x1*
x2*
y
y
x1
~
Engelcurveforgood 2
Engelcurveforgood 1
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Income Changes; Good 2 Is Normal, Good 1 Becomes Inferior
x2
x1x1*
x2*
y
y
Engel curvefor good 2
Engel curvefor good 1
64
If income and prices all change by the same proportion k, how does the consumer demand change? (e.g. same rate of inflation for prices and income)
Recall that prices and income only affects the budget constraint: p1 x1 + p2 x2 = m.Now it becomes: kp1 x1 + kp2 x2 = km. Which clearly gets back to the original one.
Thus, xi (kp1 , kp2 , km) = xi (p1 , p2 , m).65
How do the demand curves and Engel curves look like for a Cobb-Douglas utility?
Recall the Cobb-Douglas utility function:
On solving using the MRS=p1/ p2 and the budget constraint, you will have:
66
U x x x xa b( , ) .1 2 1 2
The ordinary demand functions for commodities 1 and 2 are
121
*1 ),,(
p
m
ba
amppx
.),,(2
21*2 p
m
ba
bmppx
67
Own-Price effect: inversely related to its own price.
Cross-Price effect: no cross price effect
Income Effect: Engle curve is a straight line passing through the origin.
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x1*(p1’’’) x1*(p1’)
x1*(p1’’)
p1
x1*
Own-Price Changes Ordinarydemand curvefor commodity 1 is
Fixed p2 and y.
2
*2
)( pba
bm
x
1
*1 )( pba
amx
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Rearranged to isolate m, these are:
Engel curve for good 1
Engel curve for good 2
.)(
;)( 2
*2
1
*1 pba
bmx
pba
amx
*2
2
*1
1
)(
)(
xb
pbam
xa
pbam
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m
mx1*
x2*
Engel curvefor good 1
Engel curvefor good 2
*1
1)(x
a
pbam
*2
2)(x
b
pbam
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Demand Elasticity measures the percentage change of demand as a result of one percent change in exogenous variable.
Own price elasticity of demand:
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i
i
i
i
ii
iiii dp
dx
x
p
pdp
xdx
/
/
Cross Price Elasticity of demand:
Income Elasticity of demand:
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j
i
i
j
jj
iiji dp
dx
x
p
pdp
xdx
/
/
dm
dx
x
m
mdm
xdx i
i
iimi
/
/
Ordinary Good: Giffen Good:
Gross Substitutes:Gross Complements:
Normal Good:Inferior Good:
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0ii
0ii
0ji
0ji
0mi
0mi
The consumer demand function for a good generally depends on prices of all goods and income.
Ordinary: demand decreases with own priceGiffen: demand increases with own price
Substitute: demand increases with other priceComplement: demand decreases with other price
Normal good: demand increases with incomeInferior good: demand decreases with income
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