ECO206Y5 Final 2012W CarolynPitchik

8/12/2019 ECO206Y5 Final 2012W CarolynPitchik

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UNIVERSITY

OF

TORONTO MISSISSAUGA

APRIL 2 12 FINAL EXAl\lINATION

EC0206Y5Y

Carolyn Pitchik

Duration 3 hours

No Aids

Allowed

The University

of

Toronto Mississauga and you, as a student, share a commitment to aca-

demic integrity. You are reminded that you may be charged with an academic offence for

possessing any unauthorized aids during the writing of an exam, including but not limited to

any electronic devices with storage, such as cell phones, pagers, personal digital assistants

PDAs), iPods, and MP3 players. Unauthorized calculators and notes

are

also not permitted.

Do not have any of these items in your possession in the area of your desk. Please turn the

electronics off and put all unauthorized aids with your belongings at the front of the room

before the examination

begins

If any

of

these items are kept with you during the writing

of

your exam, you may

be

charged with an academic offence. A typical penalty may cause you

to fail the course.

Please note, you

C NNOT

petition to

re-write

an examination once the exam has begun

Instructions: You have 3 hours to complete this test. Please write your answers in the space

provided below each question. Make sure you justify your answers. A simple yes or no is

not sufficient. The number in parentheses beside each question is the worth of its correct

answer including explanation. Please fill in the information requested above. This test has

38 pages.

This examination consists

of

six questions which

are

equally weighted.

all questions.

For

instructor's

use only

#

1

2

3

4

5

You

must answer

6

Grade

Out of 7 7 17

17

17

17


2/7

EC0206Y5Y page2of38

Answer each

of the following SIX

equally weighted questions below. You

must show your

work

1. (17points)Answereachofthe followingquestions

shov.ring

allyourreasoning.

(a) Suppose

that

Gabriella'spreferencesoverx andy are representedbythe utility

function

U x, y) =x +

15)2

y + 10)3

whichisdefinedoverthepositiveorthant and that atypicalindifferencecurve

{x + 15)2(y+ 10)3=C

defines

y

=

h x)

asadifferentiablefunctionof

x

in

the

positiveorthant.

i. Use the Implicit FunctionTheorem to find out how a typical indifference

curveofthis function looks like and describe the "better than" setin the

positive orthant. Youdo not have to graph

the

function y = h x). You

just need to use the ImplicitFunctionTheorem to inferwhether the curve

y

=

h x) implicitlydefinedabove increases ordecreasesasx increasesand

whether itisconcaveorconvex. Showyour reasoning.

ii. IsGabriella's utility functionquasi-concaveon the positive orthant? Show

yourreasoning.

(b)

If

Xin'svaluefunctionsatisfies

v (P 1)

=

201

(OPy,Pz, 4

+20

+

Px Jy pz

then what is Xin'sMarshallian demand x (Px,Py,Pz' ) for good x whenPx =

py=pz =1and1 25?

(c) If

Deepa'svaluefunctionsatisfies

{

i

1S

10

(Px -

Py)

PII

(I+lOp", + lOp

1)2

V (PX,PYl 1) if

1;::::

lOmax{px-

Py,Py

- Px}

4p",Pll

lO(I+10p",)

p",

if

1S 10(py

Px)

and

herexpenditurefunctionsatisfies

Pv u-lOO)

10

if

U < .!.QQzk

- Pll

p.,P., u) ; {

(4Pxp

y

U)1/2

-

lOpx

lOpy if

U

:: max{ l ~ l p r

,

}

",

p,, u-100)

i

10 -

P:r:

then whatisthe minimumthat Deeparequiresincompensationforapriceincrease

fromPx = 1to Px 4whenI =20, Py

=

I?

Examination continued

on

page 3


3/7

EC0206Y5Y page 8 of

38

2. (17 points) Demand for hockey tickets at hockey matches in the Prairie town of Saska

toon fluctuate with the weather.

The

effect of the weather on a hockey rink's wealth

depends on the skill level of the hockey rink's set of teams. With probability 1 2 the

wealth level of a hockey rink with High level AAA players is 8100 dollars while with

probability 1 2 it is 4900 dollars. \Vith probability

1 3 the

wealth level of a hockey rink

with Fun level B players is 8100 dollars while with probability 2 3 it

is

4900 dollars.

1 2

The Bernouilli utility of any hockey rink owner is u w) = w

/ .

Half of the hockey

rink owners are those with High level AAA players and half are those with un level

B players.

(a) Wnat

is

the expected wealth of a hockey rink ov:.'1ler with High level AAA players?

(b) Rink-Op Insurance offers contracts

that

payout Y for a price of b. f represents

the value of the loss and

11

represents the probability of a loss, then what

is

the

algrebraic relationship between general Y, b Land 11 in a fair contract? in a full

contract? For

both

types of hockey rink owners, using the actual probabilities

and losses given in the question, illustrate in one diagram, the fair contracts and

the full contracts in state space in which the horizontal axis represents wealth in

good times and the vertical axis represents wealth in bad times. For each type,

illustrate the expected utility indifference curve through the no-insurance bundle.

(c)

Illustrate the maximum

Mh

that a High-level AAA hockey rink owner

is

willing

to pay for full insurance in your diagram and

state

whether

and

why A1h is greater

or smaller than the fair price of full insurance? \\;nat happens to Mh if the prob

ability of good times decreases for H-Ievel AAA hockey rink owners? Illustrate

your answer and give some intuition.

(d) Suppose

that

the contract A which pays out Y = 3200 and has price b =

1600

is currently on offer by every insurance firm. Show whether this contract A

would earn positive, negative, or zero profits. Also, illustrate whether there exists

another contract B

that

would attract consumers away from A and would earn

the firm offering B positive profits.

Examination

continued on

page 9


4/7

EC0206Y5Y page 14 of 38

3. (17 points) Ten year old Krippa has a planning horizon of three days. Her preferences

over dollars spent on consumption today (denoted by

Co),

dollars spent on consumption

tomorrow (denoted by

Cd

and dollars spent on consumption on the day after tomorrow

(denoted by

C

2

) are represented by the utility function u(Co

C

1

,

C

2

) =

C

O

C

1

C

2

. Krippa

receives

A

= $3

3

4

today, $0 tomorrow, and A =

$3

3

4

on the day after tomorrow.

There is no uncertainty in the model. Krippa can borrow and save from her brother

Marcin at

an interest rate fb = s = 1/3. In what follows, you may use the fact

that

when preferences are represented by

u (XI,

X2, X3)

=

XIX2X3 and prices are represented

by the vector

(PI,

P2,

P3)

and income

is

I, Marshallian demand

is xi (PI, P2, P3,

1

I/3Pi.

(a) Write down Krippa's budget constraint over her consumption today, tomorrow

and the day after tomorrow.

(b) vvnat is her optimal consumption plan over today, tomorrow and the day after

tomorrow?

(c) For each day, state how much she borrows or saves?

Examination

continued

on

page

5


5/7

EC0206Y5Y page

20

of 38

4. (17 points) Steve's company Applenut is the only company

that

supplies gourmet

apples. Steve

oV\.ns

three

separate technologies

that

can

be

used

to

prepare his gourmet

apples. The cost of preparing

ql

kilos of gourmet apples using technology 1 is

q

C1(ql)

=

where

F

> 0 is the fixed cost of operating technology 1

The

cost

of

preparing q kilos

of gourmet apples using technology 2 is

20

The cost of preparing q3 kilos of gourmet apples using technology 3 is

Inverse demand for gourmet apples is

78

- f if q

780

P(q)

= {

o

if

q

780

where

q

is the aggregate quantity produced.

(a) Explain your answers to the following two questions. (i) Vv'hat is Steve's aggregate

production of gourmet apples if he produces

qi

2:

0 units of gourmet apples for

i

1 2

3?

(li)

\\,That is Steve's profit as a function of

qb

q

and

q3?

(b) For each i

=

1 2 3 suppose that Steve is producing qi 0 units of gourmet

apples using technology

i.

Explain your answers to the follo'\\ring two questions.

(i) What is Steve's marginal revenue of selling another unit of gourmet apples?

(ii) For each i = 1 2 3

what

is Steve's marginal cost

of

producing another uni t of

gourmet apples using technology

i?

(c) Suppose that = O Explain your answers to the following two questions. (i)

What are Steve's short- run profit-maximizing levels of

q ,

q

and

q3? (li) What is

the short-run profit-maximizing price that Steve charges?

(d) Vvnat are Steve's short-run profit-maximizing levels

qI,

q and q3

if

=

3200?

Explain your answer.

Examination

continued on page 2


6/7

EC0206Y5Y page

26

of 38

5. (17

points) Suppose

that the

aggregate inverse demand for lentils is

220 - Q if 0 $ Q $ 220

P Q)

{

o

if 220 Q

where Q

is

the aggregate quantity produced by all firms in

the

industry. Suppose that

there are exactly n firms in the industry. irm i = 1, ... ,n faces the cost function

The firms compete as in Cournot with each firm i

=

1, ... ,n making its choice of

q

simultaneously. Explain your answers to the following questions.

(a) Assuming

that

all firms produce positive quantities,

what

is

the

Cournot-Nash

equilibrium as a function of

n

(b) Given the firms produce positive quantities in

the

Cournot Nash equilibrium,

what is the price? \\That is the profit of each firm?

(c) As n increases to

00

what happens to the Cournot Nash equilibrium price and

profits of each firm?

Examination continued on

page

7


7/7

EC0206Y5Y page 32 of 38

6. (17 points) l\rpita, Dominique and Jin are best friends who are enrolled in a cornmon

program

at

UTM. UTM issues

90

points to each student

to

be used either

to

purchase

solution time (denoted by S)

to

search for on-line solutions

to

past tests or to pur

chase personal lesson time (denoted by

L)

in swimming, basketball and chess at UTM

recreational clubs. AIpita, Dominique and Jin share their test solutions but purchase

recreational lessons for personal use at UTM clubs. The UTM per unit price of S is 3

points per unit. The UTM price of L is 2 points per unit. Each student has preferences

over Sand L that are represented by the utility function u S, L) =

S

10L.

(a) \Vhat are the Nash equilibrium (or laissez-faire) purchase and consumption plans

of AIpita, Dominique and Jin? Make the calculations and illustrate your answer

in a diagram.

(b) \\;'hat are

the

socially efficient consumption plan levels? Make

the

calculations

and illustrate your answer

in

a diagram.

(c)

Use a diagram to indicate what effect a decrease in the number of individuals

would have on the Nash equilibrium production of public and private goods. You

do not have

to

actually make any calculations. \\7bat is the intuition

for

your

answer?

Examination continued on

page

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ECO206Y5 Final 2012W CarolynPitchik