Esercizi Microeconomics

tutorial

Ex 8

A

.

a) Pt

oso

O to

5 k

k k Az

th

Hz o o

e )

) (

(

feasts

( ) )

(

e)

(

to s as as

ein

to

e , , ,

,

b) the )

(

N keep

keep

E Pr

Pa

where

is always

all to end

all eeen ,

. . ,

,

always o

earns .

c) 13=0.6

D= 8

o .

Player Kehn Schmidt

's function

utility

2 is :

-

=/ molts ] it cq

x.

's Xia

" o

- -

a)

( .

Ur x , if

)

[

G

Xz 29×1

Xg X

e. Xi

- - max o

- ,

We payoff

peinlxe.nl

the for

to utility

have of

compute each :

( ) 8[ a) 8

to

to o

: =

o -

o -

- .

, )

( )

Cas 9

IS :

S IS s

6 =

e.

- -

.

, )

( [12.5-7.5]=3.5

7. : 7.

S

12.5 8

s e.

-

, )

( 6130 e)

30

: 12

0,30 =

e. -

- .

)

( ( ]

Is As

=

: as

Is 6 o

e.

- .

, that

If knew with

inequity

I 13=0.6

player player end

2 is D= 8

arene e.

.

Pa then

to Pz

Pa

payoff

went PL

his will

must to

to

end max pen

,

. ,

ben beck Is .

Ex tutorial 8

2

.

the Jo [

E

SX

offer amount where S

y e.

=

can

properer any .

,

,

then X SX

responder :

con - )

(

)

( SX

X11 )

payoff sx

accept SX

X = s ;

cone -

• - -

, ,

,

( )

payoff e

resect o

o• en

, .

,

, (

( )

Ui ) )

( 3 0.3

Xi Xs

Xi

Xs

Xi Xs

= Xi mex

- o

-

o

max -

- .

,

, ,

Consider the B

responder

• ,

if payoffs will be

le decline two

- ;

, =/ )

)

(

then

if be

payoff will

offer too So so

A s

s too

e. too

S

= e.

o

o

- .

-

, .

. , .

,

,

then

Utility positive B tee offer

accept

is , )

( )

(

if A XS

payoffs

offers x where

t s

S XB ta

- > s >

e. en - ,

, ,

[

then )

( ]

VB X XB XA

0.3 Mex

Xa XB = B - o

- ,

, 3( )

X X XA

= e. B

B -

- XB

= X 0.3 34A

t e.

B -

0.7 0.3

= X t XA

B too

too

teen Xlt )

plug SX TB

end

in ta in

s

we -

( )

) ( ) ( t

too s

= Loos 3

7- t

o -

e.

.

70

= 3

30

+

, - Is ten

positive B the offer

7 accept

+30

= he 0

, .

If A offer SL XBLXA

S

- e. , )

( (

)

UB XB

3 XA

XB X

=

XA -

-

B

, -1343

Xp 34A

= - too

HXB 34A

= too

- )

then ( XB=sX

X t

XA= end

s

plug -

we 3( ) )

(

( ) tee

4 tooo s

t

= -

. -

hoo -13003

300

= -

s

Foo 300

= > -

B accepts If

when 7003 700093k

only >

Zoe so o

- , ,

3g

B indifferent when -

is s -

B ¥

the offer when

declines c

>

the

Consider A :

•• proper

Since know then

Ig

A 3g

it exactly

will will often

reject he

> c

, .

)

( (

VA ) ( )

XA 3

XA XA

XB X 3 XA

= XB

e.

-

- B -

mex mex

o o

-

, ,

,

when XA

< 5 > XB

7 :

e. , )

) (

(

UA XA

XA XA

3 XB

XB = e. -

-

, X 3

= XB

0.3

XA

e. t

A -

7- 3×13

XA t

= o

e .

. (

) )

)

(

( toes

t t

= too 3

9- s e.

e. -

Fee Foo 3007

= t

- ,

= hee

geo - >

JUA

then If that it

find

do increases

we we decreases s

can es .

y ,

So possible

A prefers low

o es es .

, tutorial 8

Ex 3

.

a) { )

N E defect firm

defect both

is where o

eeen ,

. . .

,

I tutorial

Ex 8

h

.

Ex ut

I exercise

.

A So

preference the bundle

counter

f

it if

for

relation bundle

called tf

ut

is is

u n

convex convex

upper .

, ,

then the

bundle between (

My d) for 11

founder 's

have

's 7h

is h

Y 3h E

2-

end end

n t

E

V

mix +

so our

we -

,

, ,

that (

to ]

belong o a

. , .

So extremes

least

at good

es

averages es

are .

,

then to the to

have find less preferred

where h

Z is end V.

example

an

we

,

Example (

( ) (

) )

be coke

milk

: wine

orange

pitta Juice

may enemas , , ,

, .

Xz

Orange t.o.TN

COUNTER

guice SET NOT

× CONVEX

•

,

mixture H

7th ' y

- • Xt

milk

Ex net

2 exercise

. PERFECT COMPLEMENT

Xz X Xz

lis

Xo Lz

•

• Xz

Xs la

• Xt

north

became preferences easterly

>>

Xz Xs increases - .

,

¥0 the that that

vectors it fellows

're 2 indifference Xi

is Xo

x. Xs

same x -

any curve

on , , .

.

Sd because not

city monotonicity

strong verified because

mento to

verified not

while

Xz preferred X

is

Xs is

is

is

mi .

, ,

preferred it

if commodity

to of

Xs has 2

more

even .

Ex ut

3 exercise

.

a) this '

ulxltlulnll

the the that

function it then

Hp

if if

( (

uld ( )

Ms D=

must ut

u end

and

order ein =3 2

is n we

preserves so we u

u

, ,

,

the f- fly

it 3-133 2+23

f

substitute )

function (

into )

3+27--30

get 2+8

also

end to

=

= =

: =

x =

we .

then the before f-

function ( 43 ( )

) ( )

(

. f-

set

order s

n

x x

u

preserves n

es

euro we . .

,

b) thx lulxlt

) uld

-

- - If

then

Hp

Our that u(H=z the

mid

ufx

) ul ul them function

ut into get

3 plug

is end

s - :

we we we

.

.

, then the

)

f- 33=3 have

( ! thx

fly don't

)

fly

) function

)

9=-6 4=-2

=3 end 2 s

2

aero z

n so

=

- we

-

- - an .

,

. ,

, ,

the

preserve order - OVER

supination

c) FOR

BUNDLES

ALL

(a)

flu ) n

+

= u ; ×

.

In It

this have ( )

bundles bundle

the with bundle

at utility enocieted

2 end x t

we -

can e

euro

every are -

. ,

) the

( to the

) )

( )

(

u( function

verify if

have ul

while ul

end t order

too =

y y

x

= y

-

preserves

we

=

o u u

, .

f- (a) it the

then doesn't

fly

fly )

)

Atx

) s

1+1=2 end order

aero tot

It too

= preserve

= no

= .

, ,

,

Ex tutorial

t

.

A In the

be transitive this

preference to it

if

national

is complete evaluated

has

said end all

example

is consumer

.

bundles She that that but

preferences than it to

end

complete she said

Xvxz X

said X

tank

no -

means

ere ,

,

, .

, ,

then then

preferences

prefer transitive not

her preferences rational

her

to not and

Xa Xz ere

are .

, ,

tutorial

Ex 2

. bad

She the

that between

transitivity

that then by that

it indifferent

XI but said she

the

said is

K xiexz

end she

xz

means

, ,

then not rational

her preferences

X end Xz are

s .

,

tuteiiee

Ex 3

.

In that

the xsyxsyxs.lt doesn't

first represent

have Ann preferences

: Xu > 's

x >

Xu

now we , .

In It

the doesn't represent Ann

have preferences

as Xs

XI

> 's

second Xs

Xz >

tow x xz

: ~

we .

.

In that

the 3rd It represents Ann

have Xuxa Xz

> > Xi

Xs ' 's preferences

Xs

: -

now we . .

.

Ex I exam

. that (

a) Suppose ( )

)

) )=

(

ul )

ufx > text

(

K 2,2 t

Ya

and end t

yah he

suppose -

our -

, . , .

, ,

then ( 1+1/(1+1)=4

) (

2+1/(2+1) rely

the

function )

ulna 9 >

must Ya

order = =

m =

preserve

euro ,

, .

,

We So )

a)

f- ,H= then

( (

fly t

have tlxa.at functions

2 2=4 1=2

t

end >

xx ya

: ya

= - pneoau

- ,

. ,

, ,

the order .

b) So

(f) )

( the

(21/2)=-4 )

,H=

ffxa.lu/= function

file flxe.in

Ours functions end t t don't order

=

-

- - since

preserve

e re ,

,

.

.

then )

fly

is < zs

= o

n -

, . .

= It

o

- . then

c) In

ta I )

Aban

function tlxa.at thank

Our is

= end =

- -

: - -

e re

. . e.

q go [( .

, . * .

, .

m

)

( the

f- functions order

> yay preserve

no , .

,

{ rut

I exercise

x.

A) the function for commodity

Xs 2

end demand t

XL end

e re .

this

budget then

Our it

constraint for

solving

is

in get

Xz

W :

pzxz

core peat

peat - we

- ,

Pats Pak

W - -

P3X3= P B

Pzxz

# X

W =

- 's

-

s pz .

b) to (

find the became the

homogeneity

what this

to demand

have multiply

of end

Xs to

is in end

core p w

we , ,

) )

(

functions for IL

depend end s

on w

p o .

then I the ) then

tf xelp

P

xslilp.tw/=

for + which

set s

: '

x too is same w

. as

-

,

, ,

,

,

homogeneous of

Xs in and

degree

is p

ten W .

For ! I

! then

( )

the

)

Hap which

f

2+13 he

get y is is

aw xr

: - +

+ pm

xr same

we as ,

, ,

homogeneous of degree end

' in

Fero w

p .

EX set

2 exercise

.

(a) ) )

then MLK

ufxa

8. 2h= 16-12=192

192 2-

16 192

=

µ =

- = .

- . .

, -

167=192

192/16

2- = 12

2- =

Ex 3 set

exercise

. it becomes

when two a

function

utility

COBB DOUGLAS

- .

a) the

Is be

utility

her

defined must always

fei 3

and

x xu

> is o

o

, , . .

.

b) xr ,

¥

¢

the

b. is min .

level consumption

of /

.

/

utility

O t

- Xx

t

's

.

If the utility

and

la

X Xue s a :

c is

, " "

⇐¥¥

.

q

utility

O a

- . utility became

O -

xzcs

C- ut

h

x. exercise PERFECT SUBSTITUTES

a) We the utility

it

find

have not

to end

de in

increasing Xi

is xa .

to )

the (

)

Julia it's

gradient

have t

to compute I

do increasing

: - xn

xn vs

no we -

,

,

,

Xy decreasing

end Xz

on .

.

the )

ufx when I K

is K

Xu Xuan

so > t

* s o

o -

- -

, . .

We L

't became

the these perfect substitutes

system

we

can are .

PIT then

We that

know IMRSI

optimal

the must satisfy

bundle where

= past

,

I

IMRSI

have =

we Pz .

We then

MV MV the

end Mur

already computed Mui

with gradient and t

A x

no -

, .

, ,

,

,

I ¥ !

then

that MRS if for

equate

have = =p solve

get

we we

we can

Xz l

I - ,

get I I

Xz=

Xz and Xz=

Pz pz

Xo f

*

: pz .

- -

-

-

then )

H

when

Xz I pact

s

Pa •

> pas

o

o -

- • - .

Past

Our budget constraint for tht

if get

relive

and

is xxx we

pzxz phew xx

.

w

#

p at - we

that )

puff

know have

but t

xx XE

W Pa

Xi

pzxz w

pz

- we w

- we -

- -

- .

, ,

then pulse )

(

( ein t

when t Pa

X1 pre

w pz s

so - w -

- .

c)

b) I

( ) pal to

have ulxa.tv/,w

pale

# put #

end into

x x

=

pa w

pa w - -

.

, ,

, ' -2,02

ttp

fs

ftp./+lt-p4-t-zft-paYXz#

p

w

a - ,

)

( Epi

E

pztpihtt.ph #

=

Ps Pz I Pa

W w -

- -

-

. .

, , E III

Pat

w + -

-

EX further Utep

t exercises on t

'

- )

VIP

pal

Et

w put no ,w

- so PERFECT

A

IS

IT

COMPLEMENT

)

a) { Exa

( )

Loo Ps

w Xz

he

= pies xx

end xn

u min

=

= .

, . .

,

,

BAMBI Xy Xz

a

, A

2 2

Exa h

Xz =

Hila

4¥ les

' la

la

B Batteries

b) to the

A

find the optimal budget

put into

bundle to

got

have XE Exa

we

constraint for

then

( to

) have

Ext

get solve

w

peat =

peat

w

pair so pie

= we we

,

, , I

) 200

( ¥ *

PHIR 8=4

while

8

Xt xx = = '

. w . xa=

= xt=

q .

← w

,

,

Ex further UMP

2 exercises on

.

a) take

If the

tht they bustles

't

random

2

you you

pawns so

say are

can

, , ,

relation complete

isn't .

b) It tear brotha it transitive

of

of

brotha is is

y

x y

is so

x .

, ,

Ex 3 further UMP

exercises on

. ft

tomatoes Sau

Baek WHEN

TIEN 3

b) > .

a. Herst 3

.

T

When p

>

the T

when ' . f-

' Rst

IM = \ BE

nectarines

tutorial

Ex h

.

In

a) take transformation

C- D it the

function function

utility to log of utility

useful

is

, a .

)

dlnfxa.ba/tl3lnfxz-bz)-Vln/Xz subject

63 to s Pax pun

W

- pax pcxctpzx pzxsso

# w . -

- -

, . .

L( xa.xr.xz.cl/=dlnfxa-bdtl3lnfxobal+Vln/xs-bs)til(w-paxa-kxoPsxsl .

:#

c

§ ¥6 d

ill

2¥ ) # ( H

it = I

+

Pa - o

" + - Pt

b

Xt

,

, ,

-

*

"

" "

t

"

! " " "

"

"

"

"

" h h

" "

- -

t%%= t

V f

)

M.p

28 . .

. . .

+

-

2×3 63 by

Xz

Xz - -

§

{ Pak

pit P 3×3=0

so w - -

-

, law . p

a #

)

(

th

We It the 62

plus Xr

record

into : -

eg = 12¥42

Paba

Patt -

µ*Hk pint

(

=P pm -

.

Paba

Pete -

. Tn )

)

Half (

Blpaxa Paba

-

- ( )

13 peba

Peta -

-1362

Pzxz = bat

Petit

4 y - %)

(

PCP t

#

( "

" "

pyxes paba

+

Ba

tf xa.ba/tbr

(

xi v .

then 3rd

first the

the

by into

plugging

do :

we same

)

pzfxs -63 f 1363¥

x =pz¥

)

# ,

ta

a -

Htt

)

) ( )

( h

63 pale bat

Ps Xs - y

, -

Pt(xe

¥1 ball

ix.

(

13/43 )

63 , t pi

=

a

- . htt

Megha ( %)

( )

-63 -

=

x, )

( Paka

.be/

V

63

Xs =

- 134 )

( Habit

V B- b

× = ,

+

, 134

Pa

8×3 ) 63

He be +

= -

.

" .

P3 d

then to #

have the for

into

Xz solve

last

end end

plug Xz xx i

we

, Ba fxa.ba/t6rJ-p/z !

.f¥ bio

I )

plz Hobe

Pee

w +

-

- . .

. . .

. .

)

la (

By

By )

Va f d

⑨ Paba Pak Paba 1363=0

Pata

pet

- Pex

- + t

- - -

,

Bpa V

tdpzbz

.be

13 ype.ba 21363

" A + LW

Peta

pets Pata

t

t + =

-

- .

ps.x.fh.t.PH/=Ppaba-dprbztVPtba-dpzbztdw

)

( ¥

( )

)

) (

afpzbz 13+8