Microeconomic Theory -1- Consumers - UCLA Econ · Microeconomic Theory ... The simple mathematics of elasticity 2 B. Income ... Microeconomic Theory

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.



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



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



(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



(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



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



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



C. Income and substitution effects on demand


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.




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



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



Income effect




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



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



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



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.

**




Budget constraint:

1 1 2 2 1 1p x p x p



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

*




Budget constraint:

1 1 2 2 1 1p x p x p



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.




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.

**





0{ ( ) | }

xMax U x p x I



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

.

*





0{ ( ) | }

xMax U x p x I



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



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



( , )ix y solves

,

{ ( , ) | }i

i i ix yMax u x y p x y I

.

It is this second step that we now consider.


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




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 .







commodity i is a normal good so the

income effect is positive, offsetting the

substitution effect.

*

Price increase lowers utility







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



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

.

**










0

( , ) { | ( , ) )}i i i ix


.

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

*










0

( , ) { | ( , ) )}i i i ix


.

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 .




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.



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.



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.



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?

*



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



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.

*









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



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



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 .



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 .

**




Two periods:


Interest rate on savings: r

Income sequence 2

1{ }t ty .

Consumption sequence 2

1{ }t tc


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)

*




Two periods:


Interest rate on savings: 2y

Income sequence 2

1{ }t ty .

Consumption sequence 1c


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



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



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




Substitution Effect




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




Substitution Effect




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



Class Exercise: Are the effects reinforcing for borrowers?

Consumer borrows



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



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

.

**




0c c

ii

i i

x yp

p p


( ) ( ) ( , ) ( , ) 0c c

c ci i ii i i i i

i i i i i


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

*




0c c

ii

i i

x yp

p p


( ) ( ) ( , ) ( , ) 0c c

c ci i ii i i i i

i i i i i


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




( , ) (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 .




( , ) (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 .



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

.

*



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

.



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 .



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



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

Documents

Microeconomic Theory -1- Consumers - UCLA Econ · Microeconomic Theory ... The simple mathematics of elasticity 2 B. Income ... Microeconomic Theory