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Sample problems and examples in lecture.Download recitation problems before recitation.Complete exercises in recitations.Reorganize web site.Have power point slides available earlier.Overview class at the beginning.

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Book does not have enough examples.

Market Demand

From individual to market demand.

Price elasticity of demand.

Income elasticity of demand.

An example: the Laffer curve.

From Individual to Market demand

Individual ‘s demand function for good 1:

Aggregate demand (market demand) function for good 1:

iii mppxx ,, 2111

n

i

iin mppxmmmppX1

2112,1211 ,,,...,,,

i

Market Demand: Example

Consider 2 consumers of CDs:

Each consumer has the demand function:

Consumers have different incomes:

pmx ii

100$1 m

2,1i

200$2 m

Market Demand: Example

Individual demand functions:

Market demand:

for

for

px 100$1

pX 2300$

px 200$2

100$p

pX 200$ 200$100$ p

Inverse Market Demand: Example

Market demand: for

for

Inverse demand: for

for

pX 2300$ 100$p

pX 200$ 200$100$ p

2/150$ Xp 100$p

Xp 200$ 200$100$ p

Market Demand Curve

Individual demand curves:

Market demand curve:

ix

pp

X

100

200

100 20000

200

100

300100

Aggregation

Q:Is the sum of our demands (aggregate demand) for a good always equal to the demand of one individual whose income is given by the sum of our incomes?

In other words is aggregate demand equal to the demand of some representative consumer who has income equal to the sum of all individual incomes?

Aggregation

A: No, for two reasons:

1. Individuals have different preferences

2. Even if individuals had the same preferences, some goods are necessary goods, and others are luxury goods.

Aggregation: Example with a Necessary Good

1x0

2 consumers with same preferences

Equal income distribution:

Unequal income distribution:

21xmm

100$

10

2010101 X

144$

56$

125.75.195.7121 X

Elasticity

Looking for a measure of how “responsive” individual and aggregate demands are to changes in price and income.This measure is important to determine effects of taxes on prices.

One Candidate

One candidate measure of how “responsive” demand is to price changes is the slope of the demand function (at a given point):

1

2,1211 ,...,,,

p

mmmppX n

Problem with Slope of Demand Function

Example: where represents gallons of gasoline and is the price of

one gallon.

Change units and measure gasoline in quarts (1/4 of gallon). Let represent quarts of gasoline. Demand is:

pXG 100 1Xp

QX

pXQ 4400

Elasticity

Instead of using slope, use price elasticity of demand :

Advantage: independent of units

1

1

1

2,1211 ,...,,,

X

p

p

mmmppX n

Example Cont’d

Demand for gasoline:

Elasticity:

Demand for gasoline:

Elasticity:

pXG 100

p

p

X

p

G

1001

pXQ 4400

p

p

p

p

X

p

Q

1004400

44

Properties of Elasticity

Elasticity changes with demand:

p

p

100

p

X

100

1000

50

50

1

0

Properties of Elasticity

A demand function is elastic if:

A demand function is inelastic if:

A demand function is unit elastic if:

1

1

1

Example: Cobb-Douglas

Demand function:

Slope:

Elasticity:

p

mcx

2p

mc

p

x

12

2

cm

p

p

mc

x

p

p

x

Income Elasticity of Demand

Describes how responsive demand is to changes in individual or aggregate income.

Defined similarly to price elasticity:

1

211 ,,

x

m

m

mppx

Income Elasticity of Demand

Normal goods:

Inferior goods:

Luxury goods:

0

,,

1

211

x

m

m

mppx

0

,,

1

211

x

m

m

mppx

1

,,

1

211

x

m

m

mppx

The Laffer Curve

How do government tax revenue change when the tax rate changes?

The Laffer Curve

If : zero revenues.

If : zero revenues.

There exists a tax rate that maximizes revenues.

0 1

Tax .R

t

0t

1t

*t

*t

The Laffer Curve

Consider a population of identical workers

Each worker earns an hourly wage

Each worker has to pay a tax on his/her wage

Thus a worker’s net hourly wage is:

*wt

*1 wtw

The Laffer Curve

A worker decides how many hours to work according to the following labor supply function:

Tax revenue:

aah wtwx *)1(

htwxT

The Laffer Curve

Tax revenue:

How do revenues change with the tax rate:

htwxT

t

xtwwx

t

T hh

The Laffer Curve

How do revenues change with the tax rate:

Compute:

t

xtwwx

t

T hh

wwtat

wt

t

x aa

h 1))1(())1((

The Laffer Curve

Compare

with

so that

wwtat

x ah 1))1((

twtaw

x ah

1))1(( 1

tw

w

x

t

x hh

1

The Laffer Curve

We know that:

Then:

tw

w

xtwwx

t

xtwwx

t

T hh

hh

1

tw

w

x

t

x hh

1

The Laffer Curve

We want tax revenues to decrease with the tax rate:

This occurs when:

01

t

w

w

xtwwx

t

T hh

tw

w

xtx

hh

1

The Laffer Curve

This occurs when:

Rearrange:

tw

w

xtx

hh

1

t

t

x

w

w

x

h

h

1

The Laffer Curve

Condition:

Compute elasticity of labor supply:

t

t

x

w

w

x

h

h

1

a

a

h

h

wt

wtwta

x

w

w

x

))1((1))1(( 1

a

The Laffer Curve

Thus we have that tax revenues increase when government reduces tax rate if:

Elasticity of labor supply estimated to be at most 0.2Tax rate on labor income is at most 0.5

t

ta

1

The Laffer Curve

Elasticity of labor supply estimated to be at most 0.2

Tax rate on labor income is at most 0.5

Plug into our condition and check that it is not verified:

1

5.0

5.012.0

1

t

ta