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

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Published on 20 Apr 2013

Department

Economics

Course

ECON 305

Professor

Midterm – 2A with solutions

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

y23

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)c1

0.5 c2

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?

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

endowment each period. He will set

c110

c23

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 your result with

the help of a diagram.

Answer: You can easily see that Gilligan would want to consume relatively more in

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

tomorrow. This is not the offer that Jack is offering so Gilligan should not accept this

deal.

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

U(c1,c2)c1

0.5 c2

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

Q

,

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

After completing university, she will earn

y2

. For what level of

y2

is it worthwhile

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

earn

y

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

interest rate R.

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

y

each period.

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

RQy

R

RQ

R

y

R

RQ

R

y

R

U

RQ

R

yQcRyc

R

c

R

Solving

Q

R

c

c

R

yR

c

cc

R

c

cQ

R

y

ccL

2

2

2

5.0

2

212

*

2

2

22

1

2

1

2

2

5.0

11

2

1

2

2

5.0

1

25.0

25.05.0

25.025.0

25.0

)(

25.0

2

,

0:

01:

05.0:

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

Lc1

0.5 c2

y y

Rc1c2

R

c1:0.5c1

0.5

0

c2:1

R0

:y y

Rc1c2

R0

Solving,

R

c122

20.25

R2

c2

*y R(y c1)y R(y 0.25

R2)

U0.25

R2

0.5

y R(y 0.25

R2)

0.5

Ry (1R)0.25

R

0.25

Ry (1R)

She will go to university if

RQRyy

Ry

R

RQy

R

)1(

)1(

25.025.0

2

2

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)ln(1n)

ln(c)ln(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.

a) Write out Grace’s intertemporal budget constraint.

Answer: Grace’s budget constraint is given by

## Document Summary

Midterm 2a with solutions: suppose gilligan is on an island by himself and lives only for two periods. He has a units of fruit fruit tree that can produce in period 2. The fruit rots each period if not consumed and cannot be saved. Answer: gilligan cannot save, so he should consume as much as possible of his endowment each period. He will set: 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. Explain your result with the help of a diagram. Answer: you can easily see that gilligan would want to consume relatively more in. This means he would want to save fruit today and borrow tomorrow.