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Midterm

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Economics

ECON 305

Luba Patterson

Winter

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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 y 110 units of fruit in period 1 andy2 3 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
0.5
U(c 1c 2c 1 c 2
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
c2 3
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(c ,c )c 0.c
1 2 1 2
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
0.5 y2 c2
L c1 c2 Q c 1
R R
c :0.5c 0.5 0
1 1
c2:1 0
R
y2 c2
: c1 Q 0
R R
Solving,
R
2 2 0.25
c1 2 R 2
* 0.25
c2 y 2R(c Q1 y 2 R RQ
0.5
0.25 0.25
U 2 y 2 RQ
R R
0.5 0.25
y2 RQ
R R
0.25
y 2 RQ
R
If she works continuously, her utility level can be determined as follows:
L c 0.5c y y c c2
1 2 R 1 R
0.5
c1:0.5c1 0
c2:1 R 0
y c2
:y c 1 0
R R
Solving,
R
c 2 2 0.25
1 R 2
c y R(y c ) y R(y 0.25 )
2 1 R 2
0.5
0.25 0.25
U 2 y R(y 2 )
R R
0.5 0.25
y(1R)
R R
0.25
y(1R)
R
She will go to university if
0.25 y RQ 0.25 y(1 R)
R 2 R
y2 y(1 R) RQ
3. Grace is deciding how much to work and consume today and tomorrow. She has
the following utility function:
U(c 1c 2n 1n 2 ln(c) ln(1 n) lnc) ln(1 n)
Grace earns y1 z 1 11 )1 in Period 1 andy 2 z 2 (2 ) 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.1 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 z n (1 ) c
z1 1(1 1 2 2 2 c1 2
R R
b) Write out Grace’s choice problem. (ie. What is she maximizing given certain
constraints, and what must she choose?)
Answer: Grace’s objective is to maximize her utility
U(c 1c 2n 1n 2 ln(c) ln(1 n) lnc) ln(1 n)
subject to the budget constraint:
z n (1 ) c
z1 1(1 1 2 2 2 c 1 2
R R
by choosing optimal levels of consumption in periods 1 and 2, and labour
supplies in period 1 and 2.
c) Using a Langrangian, set up Grace’s optimization problem and take first
order conditions.
L ln(c)ln(1n) ln(c)ln(1n) z n (1 ) z2 2(1 )2 c c2
1 1 1 R 1 R
1
c1: 0
c1
c2: 0
c2 R
n : 1 z (1 )0
1 1n 1 1
1
z2(1 2
n 2 0
1n 2 R
: z n (1 ) z2 2(1 )2 c c2 0
1 1 1 R 1 R
d) Find Grace’s intertemporal Euler equation for consumption. How will Grace’s
intertemporal decision be affected by an increase in the interest rate?
Answer: To find the Euler equation use the first order conditions for
Consumption in Periods 1 and 2. 1
c1: 0
c1
1
c1
c2: 0
c 2 R
c Rc
2 1
Increasing the interest rate will cause Grace to consume relatively m

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