ECON 5113 Microeconomic Theory
Winter 2015
Test 1
January 30, 2015
Answer ALL Questions
Time Allowed: 1 hour 20 minutes
Instruction:
This is a closedbook exam.
No mobile
phones or calculators are allowed. Please write your an
swers on the answer book provided. Use the rightside
pages for formal answers and the leftside pages for your
rough work.
Answers should be provided in complete
and readable essay form, not just in mathematical for
mulae and notations. Remember to put your name on
the front page.
You can keep the question sheet after
the test.
1. Suppose that a consumer’s preference relation
%
on a
consumption set
X
✓
R
n
+
is complete and transitive.
(a) Define the following induced relations on
X
:
(a) “is strictly preferred to”,
,
(b) “is indi
↵
erent to”,
⇠
.
(b) Explain if
is complete, reflexive, transitive,
circular, symmetric, asymmetric, or antisym
metric.
(c) Show that
⇠
is an equivalence relation.
2. Suppose that a consumer’s preference relation satis
fies A1, A2, A3, A4, and A5’ (but not A5, see ap
pendix A).
(a) Explain if the solution to the utility maximiza
tion problem still exists.
(b) If yes,
is the choice for the optimal bundle
unique?
(c) Illustrate the answers above with a diagram with
a twogood case.
3. Aminata buys two goods, xigua and yam every week.
Her utility function is given by
U
(
x, y
) = 2
x
+ log(
y
+ 4)
.
The market prices for xigua and yam are $2 and $1
per kg respectively. Aminata’s weekly budget for food
is $5.
(a) Set up Aminata’s grocery shopping as a Kuhn
Tucker optimization problem.
(b) Find the optimal bundle.
(c) State the condition of the marginal rate of sub
stitution at Aminata’s optimal bundle in rela
tion with the market prices.
4. A consumer’s utility function is given by
U
(
x
1
, x
2
) =
Ax
↵
1
x
1

↵
2
,
A >
0
,
0
↵
1
.
(a) Set up the expenditure minimization problem
and find the expenditure function
E
(
p
, u
).
(b) A cost of living index is defined as
I
(
p
0
,
p
1
, u
0
) =
E
(
p
1
, u
0
)
E
(
p
0
, u
0
)
,
where
p
0
and
p
1
are prices in period 0 and 1
respectively, and
u
0
is the utility level in period
0. Show that the index for our consumer is in
dependent of the utility level.
5. Consider the Slutsky equation
@
d
i
(
p
, y
)
@
p
j
=
@
h
i
(
p
, u
)
@
p
j

d
j
(
p
, y
)
@
d
i
(
p
, y
)
@
y
.
State and prove the law of demand. You can assume
the properties of the expenditure function listed in
appendix B.