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Microeconomic Theory -1- Consumers
© John Riley October 11, 2016
Consumers
A. The simple mathematics of elasticity 2
B. Income and substitution effects 7
C. Application: Labor supply 12
D. Determinants of demand 17
E. Measuring consumer gains and losses 23
F. Elasticity of substitution 36
G. Introduction to choice over time 38
Technical Notes* 45
1. Mathematics of income and substitution effects
2. Super level sets of the aggregated utility function
56 slides
*Not examinable. Will be omitted from the lectures.
Microeconomic Theory -2- Consumers
© John Riley October 11, 2016
A. Elasticity
Consider the figure opposite. A very useful measure
of the sensitivity of y with respect to z is the
proportional rate of change of y with respect to z .
This is called the “arc elasticity”
Arc elasticity =
y
z yy
z y z
z
*
Arc elasticity
Microeconomic Theory -3- Consumers
© John Riley October 11, 2016
A. Elasticity
Consider the figure opposite. A very useful measure
of the sensitivity of y with respect to z is the
proportional rate of change of y with respect to z .
This is called the “arc elasticity”
Arc elasticity =
y
z yy
z y z
z
Consider some other country that measures both y
and z in different units.
Y ay and Z bz . Then y a y and Z b z .
It follows that the arc elasticity is the
same.
Z Y z y
Y Z y z
Arc elasticity
Microeconomic Theory -4- Consumers
© John Riley October 11, 2016
(Point) Elasticity
In theoretical analysis it is helpful to take the limit
and define the (point) elasticity
Elasticity = ( , ) limz y z dy
y zy z y dz
( )
( )
zf z
f z
*
Point elasticity
Microeconomic Theory -5- Consumers
© John Riley October 11, 2016
(Point) Elasticity
In theoretical analysis it is helpful to take the limit
and define the (point) elasticity
Elasticity = ( , ) limz y z dy
y zy z y dz
( )
( )
zf z
f z
Note that 1
lnd dy
ydz y dz
. Therefore
( , ) lnz dy d
y z z yy dz dz
.
Using this formula we can derive the following proposition
Elasticity of products and ratios
The elasticity of a product is the sum of the elasticities. ( , ) ( , ) ( , )xy z x z y z
The elasticity of a ratio is the difference in elasticities ( , ) ( , ) ( , )x
z x z y zy
Point elasticity
Microeconomic Theory -6- Consumers
© John Riley October 11, 2016
Derivation of the sum rule
( , ) lnz dy d
y z z yy dz dz
Consider the elasticity of a product.
( , ) lnd
xy z z xydz
[ln ln ]d
z x ydz
ln lnd d
z x ydz dz
( , ) ( , )x z y z
Microeconomic Theory -7- Consumers
© John Riley October 11, 2016
B. Income and substitution effects on demand
Decomposition of the effects of a price increase
A consumer with endowment facing price vector p
chooses x that solves
0
{ ( ) | }x
Max U x p x p
*
Price increase raises utility
Microeconomic Theory -8- Consumers
© John Riley October 11, 2016
C. Income and substitution effects on demand
Decomposition of the effects of a price increase
A consumer with endowment facing price vector p
chooses x that solves
0
{ ( ) | }x
Max U x p x p
We consider an increase in the price of commodity 1
As depicted, the consumer’s endowment of
commodity 1 is so high that she is a net seller of
this commodity. Therefore the price increase
raises her utility.
Price increase raises utility
Microeconomic Theory -9- Consumers
© John Riley October 11, 2016
Substitution effect
Suppose that the consumer is taxed just enough
that her utility is unchanged.
This is called the compensated price effect.
The new optimum is cx .
Since the relative price of commodity 1 has
risen, demand for commodity 1 falls and
demand for commodity 2 rises.
Compensated effect of the price increase
Microeconomic Theory -10- Consumers
© John Riley October 11, 2016
Income effect
Now give the tax back to the consumer.
A “normal good” is a commodity for which
consumption rises with income. As depicted
both commodities are normal goods so the
income effect on the consumption of both
commodities is positive.
*
Compensated and income effects
Microeconomic Theory -11- Consumers
© John Riley October 11, 2016
Income effect
Now give the tax back to the consumer.
A “normal good” is a commodity for which
consumption rises with income. As depicted
both commodities are normal goods so the
income effect on the consumption of both
commodities is positive.
Total effect
Thus with normal goods, the two effects are
opposing on the commodity for which the
price has increased.
The two effects are reinforcing for the other
commodity.
Compensated and income effects
Microeconomic Theory -12- Consumers
© John Riley October 11, 2016
C. Application: Labor supply
We now interpret commodity 1
as a consumer’s leisure time. The
price of commodity 1 becomes the wage rate
If he does not work he has 1 units of leisure.
If he supplies 1z units of labor, his income is
1 1p z and so his budget constraint is
2 2 1 1p x p z . (*)
*
Labor supply
Microeconomic Theory -13- Consumers
© John Riley October 11, 2016
C. Application: Labor supply
We now interpret commodity 1
as a consumer’s leisure time. The
price of commodity 1 becomes the wage rate
If he does not work he has 1 units of leisure.
If he supplies 1z units of labor, his income is
1 1p z and so his budget constraint is
2 2 1 1p x p z . (*)
Note that his hours of leisure have dropped to
1 1 1x z
Therefore
1 1 1 1 1 1p x p p z (**)
Adding (*) and (**)
1 1 2 2 1 1p x p x p
Labor supply
Microeconomic Theory -14- Consumers
© John Riley October 11, 2016
Budget constraint:
1 1 2 2 1 1p x p x p
The value of his consumption of leisure and the
other commodity cannot exceed the value of his
endowment.
**
Compensated effect of the price increase
Microeconomic Theory -15- Consumers
© John Riley October 11, 2016
Budget constraint:
1 1 2 2 1 1p x p x p
The value of his consumption of leisure and the
other commodity cannot exceed the value of his
endowment.
Substitution effect
The substitution effect of a wage increase (increase in 1p )
is a reduction in demand for commodity 1 and thus an
increase in the labor supply if leisure is a normal good
*
Compensated effect of the price increase
Microeconomic Theory -16- Consumers
© John Riley October 11, 2016
Budget constraint:
1 1 2 2 1 1p x p x p
The value of his consumption of leisure and the
other commodity cannot exceed the value of his
endowment.
Substitution effect
The substitution effect of a wage increase (increase in 1p )
is a reduction in demand for commodity 1 and thus an
increase in the labor supply if leisure is a normal good
Income effect (normal goods)
The income effect is to increase demand for leisure
and thus reduce the supply of labor.
Thus the two effects are offsetting.
It is therefore not surprising that data analysis shows that wage effects on the aggregate supply of
labor are small.
Compensated effect of the price increase
Microeconomic Theory -17- Consumers
© John Riley October 11, 2016
D. Determinants of demand
Consider the standard consumer problem
0{ ( ) | }
xMax U x p x I
To better understand the determinants of demand for commodity i it is helpful to think of the
consumer as solving this problem in two steps.
**
Microeconomic Theory -18- Consumers
© John Riley October 11, 2016
D. Determinants of demand
Consider the standard consumer problem
0{ ( ) | }
xMax U x p x I
To better understand the determinants of demand for commodity i it is helpful to think of the
consumer as solving this problem in two steps.
Step 1:
Suppose that the consumer must consume ix units of commodity i and is given y to spend on the
other 1n commodities. We write the vector of all the other commodities as ix , i.e.
1 1 1( ,..., , ,..., )i i i nx x x x x .
The price vector ip is similarly defined.
Then the consumer solves the following problem.
0{ ( , ) | }
i
i i i ixMax U x x p x y
.
*
Microeconomic Theory -19- Consumers
© John Riley October 11, 2016
D. Determinants of demand
Consider the standard consumer problem
0{ ( ) | }
xMax U x p x I
To better understand the determinants of demand for commodity i it is helpful to think of the
consumer as solving this problem in two steps.
Step 1:
Suppose that the consumer must consume ix units of commodity i and is given y to spend on the
other 1n commodities. We write the vector of all the other commodities as ix , i.e.
1 1 1( ,..., , ,..., )i i i nx x x x x .
The price vector ip is similarly defined.
Then the consumer solves the following problem.
0{ ( , ) | }
i
i i i ixMax U x x p x y
.
Let ix be a solution of this maximization problem. Then the maximized utility is
( , ) ( , ( , ))i i i iu x y U x x p y
Microeconomic Theory -20- Consumers
© John Riley October 11, 2016
Step 2:
Choose ,ix y that solves
,
{ ( , ) | }i
i i ix yMax u x y p x y I
.
It is this second step that we now consider.*
*In the figure the superlevel sets of the aggregated utility function are convex. As long as the
superlevel set of ( )U x are strictly convex, then so are the super level sets of ( , )iu x y . If you are
interested in the proof, see the Technical Note at the end.
Aggregated utility function
Microeconomic Theory -21- Consumers
© John Riley October 11, 2016
( , )ix y solves
,
{ ( , ) | }i
i i ix yMax u x y p x y I
.
It is this second step that we now consider.
Decomposition of the effects of a price increase
A consumer with income I facing a price vector p
chooses ( , )ix y that solves
,
{ ( , ) | }i
i ix y
Max u x y p x y I
We consider an increase in the price of ix .
In the figure B is the maximizer at the initial
price ip and B is the maximizer after the price increase.
Substitution effect
Price increase raises utility
Microeconomic Theory -22- Consumers
© John Riley October 11, 2016
Suppose that the consumer is subsidized just enough
that her utility is unchanged.
This is called the compensated price effect.
The new optimum is the point C .
Since the relative price of commodity i has
risen, demand for commodity i falls and spending
on other goods rises .
Microeconomic Theory -23- Consumers
© John Riley October 11, 2016
Income effect (normal goods)
Now give the tax back to the consumer.
A “normal good” is a commodity for which
consumption rises with income. As depicted
commodity i is a normal good so the
income effect is positive, offsetting the
substitution effect.
*
Price increase lowers utility
Microeconomic Theory -24- Consumers
© John Riley October 11, 2016
Income effect (normal goods)
Now give the tax back to the consumer.
A “normal good” is a commodity for which
consumption rises with income. As depicted
commodity i is a normal good so the
income effect is positive, offsetting the
substitution effect.
Theoretical possibility:
Giffen Good: demand increases with price
An example? The price of fuel oil rises sharply in New England. Enough people who were planning to
winter in Florida can no longer afford to do so. They stay home and demand for fuel oil rises.
Price increase lowers utility
Microeconomic Theory -25- Consumers
© John Riley October 11, 2016
E. Income compensation and consumer surplus
Consider a consumer with income I . At the
initial price ip her choice is ( , )ix y .
Let ( , )M p u be the income that the consumer
requires to remain on her original indifference
curve as ip rises. i.e.
0
( , ) { | ( , ) )}i i i ix
M p u Min p x y u x u u
.
**
Compensated effect of the price increase
Microeconomic Theory -26- Consumers
© John Riley October 11, 2016
E. Income compensation and consumer surplus
Consider a consumer with income I . At the
initial price ip her choice is ( , )ix y .
Let ( , )M p u be the income that the consumer
requires to remain on her original indifference
curve as ip rises. i.e.
0
( , ) { | ( , ) )}i i i ix
M p u Min p x y u x u u
.
Appealing to the Envelope Theorem
The rate at which this income rises with 1p ,
( , )c
i i i
i i
Mp x y x p u
p p
*
Compensated effect of the price increase
Microeconomic Theory -27- Consumers
© John Riley October 11, 2016
E. Income compensation and consumer surplus
Consider a consumer with income I . At the
initial price ip her choice is ( , )ix y .
Let ( , )M p u be the income that the consumer
requires to remain on her original indifference
curve as ip rises. i.e.
0
( , ) { | ( , ) )}i i i ix
M p u Min p x y u x u u
.
Appealing to the Envelope Theorem
The rate at which this income rises with ip ,
( , )c
i i i
i i
Mp x y x p u
p p
In the Figure, with the additional income her consumption choice is C . In the absence of the
compensation her income is I and her consumption choice is B . Thus to be fully compensated for
the price increase the consumer must be paid
( , )iM p u I .
Compensated effect of the price increase
Microeconomic Theory -28- Consumers
© John Riley October 11, 2016
Compensating Variation in income
At the initial price vector no compensation is
necessary so ( , )iM p u I . Then the compensating
income change (called the compensating variation
in income) is
( , ) ( , ) ( , )i i iCV M p u I M p u M p u .
*
Compensated demand price function.
Microeconomic Theory -29- Consumers
© John Riley October 11, 2016
Compensating Variation in income
At the initial price vector no compensation is
necessary so ( , )iM p u I . Then the compensating
income change (called the compensating variation
in income) is
( , ) ( , ) ( , )i i iCV M p u I M p u M p u .
Differentiating and reintegrating the right hand side,
( , ) ( , )i iCV M p u M p u
( , )i i
i i
p p
c
i i i i
ip p
Mdp x p u dp
p
In the figure the compensating variation is the shaded area
to the left of the compensated demand price function.
Compensated demand price function.
Microeconomic Theory -30- Consumers
© John Riley October 11, 2016
The green ordinary demand function is ( , )i ix p I
is also depicted. Assuming that commodity i is a
normal good, income compensation raises
demand. Thus the compensated demand price
function is steeper.
Microeconomic Theory -31- Consumers
© John Riley October 11, 2016
Equivalent variation in income (EV)
At the new higher price the consumer is worse off.
How much would he be willing to pay to have the initial
price restored?
*
Microeconomic Theory -32- Consumers
© John Riley October 11, 2016
Equivalent variation in income (EV)
At the new higher price the consumer is worse off.
How much would he be willing to pay to have the initial
price restored?
Arguing as above, he would be willing to pay.
( , )iEV I M p u where ( , )iu u x y
( , ) ( , ))i iM p u M p u
( , ))i i
i i
p p
c
i i
p p
Mdp x p u dp
p
Microeconomic Theory -33- Consumers
© John Riley October 11, 2016
Which of these measures should we use?
Note that the area to the left of the green ordinary
demand price function lies between the
CV and EV. As Robert Willig showed, for most
such calculations, the difference between
the two areas is small.
Microeconomic Theory -34- Consumers
© John Riley October 11, 2016
Which of these measures should we use?
Note that the area to the left of the green ordinary
demand price function lies between the
CV and EV. As Robert Willig showed, for most
such calculations, the difference between
the two areas is small.
Thus they are both close to the area to the left of
the ordinary demand curve ( , )i ix p I
This area is called the consumer surplus.
In practice this is what economists try to estimate.
*
Microeconomic Theory -35- Consumers
© John Riley October 11, 2016
Which of these measures should we use?
Note that the area to the left of the green ordinary
demand price function lies between the
CV and EV. As Robert Willig showed, for most
such calculations, the difference between
the two areas is small.
Thus they are both close to the area to the left of
the ordinary demand curve ( , )i ix p I
This area is called the consumer surplus.
In practice this is what economists try to estimate.
Remark on the use of survey data
Microeconomic Theory -36- Consumers
© John Riley October 11, 2016
F. Elasticity of substitution
Definition: The elasticity of substitution for commodity i is the compensated elasticity of the
ratio c
c
i
y
x with respect to
ip .
( , )i i
i
yp
x
Remark: CES family
Constant elasticity of substitution utility functions
1
( ) lnn
j j
j
U x a x
, 1
11
1
( )n
j j
j
U x a x
, 1
11
1
( )n
j j
j
U x a x
, 0 1
Elasticity of substitution
Microeconomic Theory -37- Consumers
© John Riley October 11, 2016
Two central results
Proposition: Price elasticity of compensated demand
The own price elasticity of compensated demand is
( , ) (1 )c
i i i ix p k .
Proposition: Decomposition of the own price elasticity of demand
The own price elasticity of demand is a convex combination of the elasticity of substitution and the
income elasticity of demand.
( , ) ((1 ) ( , ))i i i i i ix p k k x I
Remark: If the fraction spent on commodity 1 is small, then the own price elasticity is approximately
equal to i .
Microeconomic Theory -38- Consumers
© John Riley October 11, 2016
G. Introduction to choice over time
Two periods:
Utility 1 2( , )U c c is strictly increasing
Interest rate on savings: r
Income sequence 2
1{ }t ty .
Consumption sequence 2
1{ }t tc
Financial capital sequence 3
1{ }t tK .
**
Microeconomic Theory -39- Consumers
© John Riley October 11, 2016
G. Introduction to choice over time
Two periods:
Utility 1 2( , )U c c is strictly increasing
Interest rate on savings: r
Income sequence 2
1{ }t ty .
Consumption sequence 2
1{ }t tc
Financial capital sequence 3
1{ }t tK .
Period budget constraints:
2 1 1 1(1 )( )K r K y c (1)
3 2 2 2(1 )( )K r K y c (2)
*
Microeconomic Theory -40- Consumers
© John Riley October 11, 2016
G. Introduction to choice over time
Two periods:
Utility 1 2( , )U c c is strictly increasing
Interest rate on savings: 2y
Income sequence 2
1{ }t ty .
Consumption sequence 1c
Financial capital sequence 3
1{ }t tK .
Period budget constraints:
2 1 1 1(1 )( )K r K y c (1)
3 2 2 2(1 )( )K r K y c (2)
To maximize utility 3 0K .
Rewrite the period 2 budget constraint as follows then add (1) and (2’)
2 2 20 K y c (2’)
1 1 1 2 20 (1 )( )r K y c y c
Consumer saves
Microeconomic Theory -41- Consumers
© John Riley October 11, 2016
Rewrite as follows:
2 21 1 1
1 1
c yc K y
r r
1{ } { }t tPV c K PV y .
Note that the higher the interest rate, the
steeper is the life-time budget line.
The relative price of future consumption has fallen.
Consumer saves
Slope = 1+ r
Microeconomic Theory -42- Consumers
© John Riley October 11, 2016
Decomposition into substitution and income effects
Substitution Effect
The relative price of future goods 1/ (1 )r is
lower so first period consumption falls and second
period consumption rises.
**
Consumer saves
Microeconomic Theory -43- Consumers
© John Riley October 11, 2016
Decomposition into substitution and income effects
Substitution Effect
The relative price of future goods 1/ (1 )r is
lower so first period consumption falls and second
period consumption rises.
Income effect
Since the consumer is a saver, a higher interest rate
Makes him better off. Thus income has to be
reduced to keep her on the same level set.
Assuming that goods are normal, giving the income
back leads to higher consumption in both periods.
In particular, first period consumption rises.
*
Consumer saves
Microeconomic Theory -44- Consumers
© John Riley October 11, 2016
Decomposition into substitution and income effects
Substitution Effect
The relative price of future goods 1/ (1 )r is
lower so first period consumption falls and second
period consumption rises.
Income effect
Since the consumer is a saver, a higher interest rate
Makes him better off. Thus income has to be
reduced to keep her on the same level set.
Assuming that goods are normal, giving the income
back leads to higher consumption in both periods.
In particular, first period consumption rises.
Total effect
The two effects are off-setting for first period consumption and hence for first period saving.
Data analytics typically show small interest rate effects on personal saving.
Consumer saves
Microeconomic Theory -45- Consumers
© John Riley October 11, 2016
Class Exercise: Are the effects reinforcing for borrowers?
Consumer borrows
Microeconomic Theory -46- Consumers
© John Riley October 11, 2016
Technical Notes (not examinable)
1. The mathematics of substitution and income effects
Definition: The elasticity of substitution for commodity i is the compensated elasticity of the
ratio c
c
i
y
x with respect to
ip .
( , )i i
i
yp
x
Around the level set as ip increases
c c
i
i i i i
x ydu u u
dp x p y p
1
( ) ( ) 0c c
ii
i i i i
x yu up
p x p y p
.
The consumer equates the marginal utility per
dollar. It follows that
0c c
ii
i i
x yp
p p
.
Elasticity of substitution
Microeconomic Theory -47- Consumers
© John Riley October 11, 2016
In the last slide we showed that
0c c
ii
i i
x yp
p p
We convert this equation into elasticities,
( ) ( ) ( , ) ( , ) 0c c
c ci i ii i i i i
i i i i i
p x p yy yx x x p y p
x p p y p p
.
**
Microeconomic Theory -48- Consumers
© John Riley October 11, 2016
In the last slide we showed that
0c c
ii
i i
x yp
p p
We convert this equation into elasticities,
( ) ( ) ( , ) ( , ) 0c c
c ci i ii i i i i
i i i i i
p x p yy yx x x p y p
x p p y p p
.
Multiply both terms by ip
I .
( , ) ( , ) 0c ci ii i i
p x yx p y p
I I
*
Microeconomic Theory -49- Consumers
© John Riley October 11, 2016
In the last slide we showed that
0c c
ii
i i
x yp
p p
We convert this equation into elasticities,
( ) ( ) ( , ) ( , ) 0c c
c ci i ii i i i i
i i i i i
p x p yy yx x x p y p
x p p y p p
.
Multiply both terms by ip
I .
( , ) ( , ) 0c ci ii i i
p x yx p y p
I I
Define the expenditure share i ii
p xk
p x
. Then 1 1 i i
i
p x yk
I I
Therefore
( , ) (1 ) ( , ) 0c c
i i i i ik x p k y p
Microeconomic Theory -50- Consumers
© John Riley October 11, 2016
In the last slide we showed that
( , ) (1 ) ( , ) 0c c
i i i i ik x p k y p .
Add (1 ) ( , )c
i i ik x p to the first term and subtract it from the second.
Then
( , ) (1 )[ ( , ) ( , )] 0c c
i i i i i ix p k y p x p .
Microeconomic Theory -51- Consumers
© John Riley October 11, 2016
In the last slide we showed that
( , ) (1 ) ( , ) 0c c
i i i i ik x p k y p .
Add (1 ) ( , )c
i i ik x p to the first term and subtract it from the second.
Then
( , ) (1 )[ ( , ) ( , )] 0c c
i i i i i ix p k y p x p .
The term in brackets is the elasticity of substitution, ( , ) ( , ) ( , )c
c c
i i i i ic
i
yp y p x p
x .
Therefore
( , ) (1 ) 0c
i i i ix p k
Proposition: Price elasticity of compensated demand
The own price elasticity of compensated demand is
( , ) (1 )c
i i i ix p k .
Microeconomic Theory -52- Consumers
© John Riley October 11, 2016
Decomposition of the own price elasticity of demand
Let ( , )i ix p I he the consumer’s uncompensated demand for commodity i . In section E we defined
( , ))iM p u to be the income the consumer would need to maintain a constant utility.
Then the consumer’s compensated demand for commodity i is
( , ( , ))c
i i i ix x p M p u .
Differentiating by ip
c
i i i
i i i
x x x M
p p I p
.
*
Microeconomic Theory -53- Consumers
© John Riley October 11, 2016
Decomposition of the own price elasticity of demand
Let ( , )i ix p I he the consumer’s uncompensated demand for commodity i . In section E we defined
( , ))iM p u to be the income the consumer would need to maintain a constant utility.
Then the consumer’s compensated demand for commodity i is
( , ( , ))c
i i i ix x p M p u .
Differentiating by ip
c
i i i
i i i
x x x M
p p I p
.
Appealing to the Envelope Theorem, i
i
Mx
p
.
We therefore have the following result
Slutsky equation
c
i i ii
i i
x x xx
p p I
.
Microeconomic Theory -54- Consumers
© John Riley October 11, 2016
Rewrite the Slutsky equation as follows:
c
i i ii
i i
x x xx
p p I
Converting into elasticities,
1c
i i i i ii i
i i i i i
p x p x xp x
x p x p p I
1 1 1 1 1
1 1 1
( )cp x p x xI
x p I p I
.
Therefore
( , ) ( , ) ( , )c
i i i i i ix p x p k x I
Appealing to our earlier result, we have the following proposition.
Proposition: Decomposition of the own price elasticity of demand
( , ) ((1 ) ( , ))i i i i i ix p k k x I
The own price elasticity of demand is a convex combination of the elasticity of substitution and the
income elasticity of demand.
Remark: If the fraction spent on commodity 1 is small, then the own price elasticity is approximately
equal to i .
Microeconomic Theory -55- Consumers
© John Riley October 11, 2016
2. Aggregated utility function
Proposition: If ( )U x is strictly increasing and the superlevel sets of ( )U x are strictly convex then the
superlevel sets of
( , ) { ( , ) | }i i i i ix i
u x y Max U x x p x y
are also strictly convex.
Proof:
Suppose that 0 0( , )ix y and 1 1( , )ix y are, as depicted,
in the superlevel set {( , ) | ( , ) }i iS x y u x y u . i.e.
0 0( , )iu x y u and 1 1( , )iu x y u (*)
We need to show that for any convex combination,
( , )ix y , ( , )iu x y u .
From (*)
(i) for some 0
ix costing 0y , 0 0( , )i iU x x u and (ii) for some 1
ix costing 1y , 1 1( , )i iU x x u
Microeconomic Theory -56- Consumers
© John Riley October 11, 2016
We argued above that
(i) for some 0
ix costing 0y , 0 0( , )i iU x x u and (ii) for some 1
ix costing 1y , 1 1( , )i iU x x u
Consider ( , )i ix x
, a convex combination
of 0 0( , )i ix x and 1 1( , )i ix x .
Since the superlevel sets of U are strictly convex,
( , )i iU x x u
(*)
Note that from (i) 0(1 ) ix costs 0(1 )y and
from (ii) 1
ix costs 1y . Therefore the convex
combination ix
is feasible with a budget
0 1(1 )y y y .
It follows that
( , ) { ( , ) | } ( , )i ix i
u x y Max U x x p x y U x x
.
Appealing to (*) , ( , )u x y u .
QED