Eco11, Fall 2008
Simon Board
Economics 11: Solutions to Homework 2
October 28, 2008
Due date:
Tuesday 28th October.
Instructions:
You are required to write up your solution separately and independently, al
though you are encouraged to discuss and work in groups. Please write your name, student ID
number, and the name of your TA on the front page of the assignment that you hand in. Also,
please put boxes around your final answer to each part.
1. Budget Sets with Price Discounts
There are two goods:
x
1
and
x
2
. The seller of
x
2
charges
p
2
= 2. The seller of
x
1
offers price
discounts. If the agent buys
x
1
<
10 she pays
p
1
= 2 for every unit. If the agent buys
x
1
≥
10
she pays
p
1
= 1 for every unit (including the first 10).
(a) Suppose
m
= 30. Draw the agent’s budget set.
(b) Assume the agent has monotone preferences.
Do preferences exist such that the agent
chooses (i)
x
1
= 3, (ii)
x
1
= 7 and (iii)
x
1
= 12? Explain your answers.
Solution
(a) The budget line is given by
x
2
= 15

x
1
for
x
1
<
10
= 15

1
2
x
1
for
x
1
≥
10
As a result, there is a discontinuity at
x
1
= 10.
(b) (i) Yes.
(ii) No. Because of the discontinuity, the agent can obtain more of both goods.
(iii) Yes.
1
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Eco11, Fall 2008
Simon Board
2. Intertemporal Choice with Differential Interest Rates
An agent allocates consumption across two periods. Let the consumption in period
t
be
x
t
, and
the income in period
t
be
m
t
. The agent’s utility is
u
(
x
1
, x
2
) = ln(
x
1
) +
3
4
ln(
x
2
)
The agent is poor in period 1 but wealthy in period 2. In particular, she has income
m
1
= 3 in
period 1 and
m
2
= 4 in period 2.
(a) Suppose the agent can borrow and save at interest rate
r
= 1
/
3, so $1 in period 1 is worth
$(1+1/3) in period 2. Sketch the agent’s budget constraint. Solve for her optimal consumption.
(b) Suppose the agent can still save at
r
= 1
/
3, but can only borrow at
r
= 1
/
2. Sketch the
agent’s budget constraint. Solve for her optimal consumption.
(c) Suppose the agent can still save at
r
= 1
/
3, but can only borrow at
r
= 1.
Sketch the
agent’s budget constraint. Solve for her optimal consumption. [Hint: Beware of kinks.]
Solution
(a) Writing everything in terms of period 1 money, the agent’s budget constraint is
m
1
+
3
4
m
2
=
x
1
+
3
4
x
2
The tangency condition states that
MRS
=
p
1
/p
2
. That is,
4
3
x
2
x
1
=
4
3
which implies
x
1
=
x
2
. The LHS of the budget equation is 6. Using this, we obtain
x
*
1
=
x
*
2
=
24
/
7. Since
x
*
1
>
3, the agent borrow in period 1.
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 Fall '08
 cunningham
 Economics, X1, Simon Board

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