中级微观经济学-Ch5

Chapter

5Choice1Structure

of

Chapter

525.1

The

optimal

choice

of

consumers❏5.1.1

Interior

solution（内解）❏5.1.2

Corner

solution（角解）❏5.1.3“Kinky”solution（拐点解）5.2

Consumer’s

demand

function

5.3

Application

of

consumer

theory:Choosing

taxes5.1

The

Optimal

Choice

of

Consumers3

Consumer’s

problem:

maximizing

utilitysubject

to

the

budget

constraint

(Utilitymaximization

problem).The

Optimal

Choicex1x2x1*4x2*x*Properties

of

the

Optimal

Choice5The

optimal

choice:Must

be

on

the

budget

line,

why?Must

be

on

the

highest

indifference

curvethat

touches

the

budget

line.5.1.1.

Interior

Solutions6

Interior

solutions:

If

all

xi*

>

0,

xi*

areinterior

solutions.

In

the

case

of

two

goods,the

optimal

choice

involves

consuming

someof

both

goods,

i.e.,

x1*

>

0

and

x2*

>

0.

Two

equilibrium

conditions

for

interiorsolutions,

(x1*,x2*)

:❏

Budget

condition❏

Tangent

conditionCondition

1:

the

Budget

Is

Exhaustex1x2x1*7x2*(1)

p1x1*

+

p2x2*

=

m.Condition

2:

Tangent

Conditionx1x2x1*8(2)

The

slope

of

theindifference

curve

at(x1*,x2*)

equals

the

slopeof

the

budget

constraint.x2*

MU1/P1

or

MU2/P2

:

Marginal

utiltiy

per

dollaof

expenditure.

Equal

marginal

principle:

Utility

ismaximized

when

the

consumer

hasequalized

the

marginal

utility

per

dollar

spon

all

goods.Why

is

this

an

equilibrium?9Equal

Marginal

PrincipleDisequilibrium

PointDisequilibriumx210x1X1*X’1EquilibriumX’2X2*U2U1Solving

for

Interior

Solution

(method

1)11(x1*,x2*)

satisfies

two

conditions:(1)

Budget

constraint:

p1x1*

+

p2x2*

=

m(2)

Tangency:

MU1/MU2=

P1/P2Solving

these

conditions

simultaneouslyyou

can

get

the

interior

solutions.Solving

for

Interior

Solution

(method

2)12

The

conditions

may

be

obtained

by

using

theLagrangian

multiplier

method,

i.e.,constrained

optimization

in

calculus.How

to

do

it?

see

supplement.5.1.2

Corner

Solutions13

If

some

of

the

xi*

=

0,

then

we

say

there

arecorner

solutions.

In

the

case

of

two

goods,if

either

x1*=0

orx2*=0,then

the

ordinary

demand(x1*,x2*)isat

a

corner

solution

(角解).5.1.3

‘Kinky’

Solutions:Perfect

Complementsx1x2U(x1,x2)

=

min{ax1,x2}x2

=

ax114Perfect

Complementsx1x2U(x1,x2)

=

min{ax1,x2}x2

=

ax115Equilibrium

Conditions16The

optimal

choice,

(x1*,x2*)

satisfiesp1x1*

+

p2x2*

=

m;x2*

=

ax1*.5.2

Solutions

to

the

Consumer’sProblem:

Demand

Function17

The

optimal

choice

is

called

the

consumer’sORDINARY

DEMAND

（一般需求）at

thegiven

prices

and

income.

Ordinary

demand

function

are

denoted

byx1*(p1,p2,m)

and

x2*(p1,p2,m).Example

1:

Cobb-Douglas

Preference

Suppose

that

the

consumer

has

Cobb-Douglasutility

function: ,

and

the

budgetconstraint

is

p1x1+p2x2=m.What

is

the

consumer’s

demand

function?18Example

2:

Quasi-linear

Utility

Functio19

Suppose

a

consumer

has

quasi-linear

utilityfunction:

u(x1,x2)=aln

x1+x2,

a>0.

The

pricesof

x1

and

x2

are

p1

and

p2,

respectively,

andconsumer’s

income

is

denoted

by

m.

Also,x1

and

x2

are

nonnegative.

What

is

theoptimal

consumption

bundle?Example

3:

Perfect

Substitutes20

Suppose

a

consumer’s

utility

function

isu(x1,x2)

=

x1+x2,

the

budget

constraint

isp1x1

+

p2x2

=

m,

what

is

the

consumer’soptimal

choice?If

p1

>

p2x2MRS

=

1Slope

=

-p1/p2

with

p1

>

p2.x121If

p1

<

p2x2MRS

=

1Slope

=

-p1/p2

with

p1

<

p2.x122If

p1

=

p223x1x2Example

4:

Non-Convex

Preferencesx1x2Better24Non-Convex

Preferencesx1x2255.3

Choosing

Taxes:

Various

Taxes26Quantity

tax

on

x:

(p+t)xValue

tax

on

p:

(1+t)p❏

Also

called

ad

valorem

taxLump

sum

tax:

TIncome

tax:❏

Can

be

proportional

or

lump

sumIncome

Tax

vs.

Quantity

Tax27Proposition:

Suppose

the

purpose

of

taxes

isto

raise

the

same

revenue,

then

income

tax

isbetter

than

quantity

tax

for

the

consumer.Analysis28Exercise29

Suppose

that

a

consumer

always

consumes

2spoons

of

sugar

with

each

cup

of

coffee.

If

theprice

of

sugar

is

p1

per

spoonful

and

the

priceof

coffee

is

p2

per

cup

and

the

consumer

ha