1
10th April 2006
Instructions
: Answer 7 of the following 9 questions. All questions are of equal
weight. Indicate clearly on the
f
rst page which questions you want marked.
1. Let
X
be a choice set and
(
B
,C
(
·
))
be a choice structure on
X
.
(a) Using mathematics, de
f
ne the weak axiom of revealed preference (WARP).
Assume
B
1
∈
B
,
B
2
∈
B
,
{
a, b
}
⊂
B
1
,
{
a, b
}
⊂
B
2
.ThenWARPsay
s
a
∈
C
(
B
1
)
,
b
∈
C
(
B
2
)
⇒
a
∈
C
(
B
2
)
.
(b) Assume a twogood model and a pricetaking consumer whose behaviour is
consistent with WARP. Prove that the substitution e
f
ect (Slutsky method) of a
reduc
t
ioninthep
r
iceo
fgood1canno
tleadtoareduc
t
ioninthecon
sump
t
iono
f
good 1.
Diagram from Econ 201. Note: no indi
f
erence curves or other references to the
preferencebased approach are permitted here.
2. Consider a pricetaking agent who likes consumption
c
and leisure
l
.Anyt
ime
notspentatleisureisspentworkingandthetimeendowmentis
T
. Denotethepriceof
consumption by
p
, the price of leisure by
w
and the value of the person’s endowment
of money by
m
.L
e
t
c
(
p, w, m, T
)
and
l
(
p, w, m, T
)
be Marshallian demands, and
h
c
(
p, w, u, T
)
and
h
l
(
p, w, u, T
)
be Hicksian demands. Fill in the following tables for
u
=
c
+2
l
1
/
2
,
0
≤
l
≤
T.
Thebudgetconstra
intis
pc
+
wl
=
wT
+
m.
c
(
p, w, m, T
)
l
(
p, w, m, T
)
Range
wT
+
m
p
−
p
w
p
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h
c
(
p, w, u, T
)
h
l
(
p, w, u, T
)
Range
u
−
2
p
w
p
2
w
2
p
w
≤
min
±
u
2
,T
1
/
2
²
0
u
2
4
u
2
≤
p
w
and
u
2
≤
T
1
/
2
u
−
2
T
1
/
2
TT
1
/
2
≤
p
w
and
u
2
>T
1
/
2
3. Consider the twostate world of Andy and Brian. In the good state each has a
wealth of
100
; in the bad state each has a wealth of
50
, assuming they do not trade
with each other. Let the probability of the good state be
3
/
4
.
The only way in which
they di
f
er is Andy is risk averse and Brian is risk neutral. Describe as precisely
as you can the Pareto e
ﬃ
cient allocations for these two people. Now suppose there
are many Andys and an equal number Brians. Describe as precisely as you can the
competitive equilibrium of this economy.
Draw an Edgeworth rectangle with wealth in the good state along the horizontal
axis and wealth in the bad state along the vertical axis. The dimensions of the
rectangle are
200
by
100
and the endowment point is right in the middle. Put Andy’s
origin in the lower left corner and Brian’s origin in the upper right corner. Andy is
risk averse so his indi
f
erence curves are convex to his origin. Brian is risk neutral
so his indi
f
erence curves are straight lines; in fact, they have a numerical slope
equal to the probability of the good state over the probability of the bad state,
which is
3
/
4
over
1
/
4
or
3
. The tangencies trapped between the Andy and Brian
indi
f
erence curves through their endowment points form the core – that is, the
Pareto e
ﬃ
cient allocations where neither person is worse o
f
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 Fall '08
 Burbidge,John
 Game Theory, Nash, good state, proportional earnings tax

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