# ECON 305 Study Guide - Midterm Guide: Utility Maximization Problem, Budget Constraint, Utility

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Published on 20 Apr 2013

Department

Economics

Course

ECON 305

Professor

Midterm -2B practice questions

The exam is out of 100 points.

Grading:

1. Omid

2. Sepideh

3. Emanuel

4. Bill

5. Edouard

1. Suppose Gilligan is on an island by himself and lives only for two periods. He has a

fruit tree that can produce

y110

units of fruit in period 1 and

y210

units of fruit

in period 2. The fruit rots each period if not consumed and cannot be saved. Gilligan

has a utility function given by

U(c1,c2)c1c2

0.5

a) Given that Gilligan cannot trade with another person, what is his optimal

level of consumption of fruit in each of the two periods? How much should he

save in each period? (5 points)

Answer: Gilligan cannot save, so he should consume as much as possible of his

endowment each period. He will set

c110

c210

5 points for correct answer. 0 otherwise.

b) Now suppose Gilligan is visited by an explorer, Jack, who is willing to engage

in trade with Gilligan. Jack wants to lend to Gilligan in Period 1 at a gross

interest rate R, but wants to be able to borrow from Gilligan in Period 2.

Would Gilligan accept this deal? Why or why not? Explain in detail. (5 points)

Answer: You can easily see that Gilligan values consumption relatively more in

Period 1 than Period 2. This means he would want to borrow today and repay

tomorrow. This is what Jack is offering, so Gilligan should accept the deal.

5 points for correct answer (accept/reject deal + explanation. 2 points for

correct answer (accept/reject deal).

2. Anne has the following two-period utility function.

2

5.0

121 2),( ccccU

Anne is considering going to university in Period 1. This will cost her an amount ,

which must be paid in Period 1. Anne cannot work while she is attending university.

Q

After completing university, she will earn . For what level of is it worthwhile

for her to attend university? Assume that if she does not go to university, she will

earn in each of Periods 1 and 2. Also assume that she can borrow at a gross

interest rate R.

(20 points)

Answer: Anne must compare the utility she gets from going to university and only

working in Period 2 to the utility she gets from working in both periods and earning

each period.

If she goes to university, her utility can be solved for as

RQ

R

y

RQ

R

y

R

RQ

R

y

R

U

RQ

R

ycQRyc

R

c

R

Solving R

c

cQ

R

yR

c

cc

R

c

cQ

R

y

ccL

1

12

11

2

1

1

,

0:

01:

0:

2

2

2

2

5.0

2

212

*

2

2

2

1

2

1

2

2

5.0

11

2

1

2

2

5.0

1

-5 points for finding c1 and c2 properly. Allow for part marks.

-3 additional points for plugging into U and finding the utility level.

If she works continuously, her utility level can be determined as follows:

y2

y2

y

y

R

Ry

R

Ry

R

R

yRy

R

U

R

yRycyRyc

R

c

R

Solving R

c

c

R

y

y

R

c

cc

R

c

c

R

y

yccL

1

)1(

1

)1(

2

)

1

(

1

2

)

1

()(

1

,

0:

01:

0:

2

2

5.0

2

2

1

*

2

2

2

1

2

1

2

5.0

11

2

12

5.0

1

-5 points for finding c1 and c2 properly. Allow for part marks.

-3 additional points for plugging into U and finding the utility level.

She will go to university if

RQRyy R

RyRQ

R

y

)1(

1

)1(

1

2

2

-4 points for comparing the utility levels and solving for y2.

3. Grace is deciding how much to work and consume today and tomorrow. She has

the following utility function:

U(c1,c2,n1,n2)ln(c)3ln(1n)

ln(c)3ln(1n)

Grace earns

y1z1n1(1

1)

in Period 1 and

y2z2n2(1

2)

in Period 2. That is,

she earns labour income, but must pay a fraction of it each period to the government

at a pre-specified tax rate.

z1

and

z2

are exogenous productivity parameters. She

can also borrow and save in international financial markets at a gross interest rate

R. (30 points)

## Document Summary

Grading: omid, sepideh, emanuel, bill, edouard, suppose gilligan is on an island by himself and lives only for two periods. He has a fruit tree that can produce units of fruit in period 2. The fruit rots each period if not consumed and cannot be saved. How much should he save in each period? (5 points) Answer: gilligan cannot save, so he should consume as much as possible of his endowment each period. 0 otherwise: now suppose gilligan is visited by an explorer, jack, who is willing to engage in trade with gilligan. Jack wants to lend to gilligan in period 1 at a gross interest rate r, but wants to be able to borrow from gilligan in period 2. Answer: you can easily see that gilligan values consumption relatively more in. This means he would want to borrow today and repay tomorrow. This is what jack is offering, so gilligan should accept the deal.