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

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Published on 20 Apr 2013
School
Simon Fraser University
Department
Economics
Course
ECON 305
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
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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:
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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
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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.

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