Department of Economics
The Ohio State University
Midterm Questions and Answers–Econ 808
Profs. Levin and Peck
March 7, 2002
1. (30 points)
In the village of Debreuvia, there are 150 consumers and one physical com-
modity per state of nature. For i = 1, .
.. , 150, consumer i is a von Neumann-
Morgenstern expected utility maximizer with (Bernoulli) utility of certain con-
sumption given by
u
i
(
x
i
) = log(
x
i
)
.
For i = 1, .
.. , 100, consumer i lives in the highlands, and has an endowment
of
1
is all states of nature.
For i = 101, .
.. , 150, consumer i lives in the ‡ood plain.
When a ‡ood
occurs, all consumers living in the ‡ood plain receive an endowment of
0
.T
h
e
probability of a ‡ood is
1
10
. When a ‡ood does not occur (probability
9
10
), all
consumers living in the ‡ood plain receive an endowment of
1
.
(a) (10 points) If the entire village of Debreuvia trades state-contingent
commodities, de…ne a competitive equilibrium for this economy.
(b) (20 points) Calculate the competitive equilibrium price vector and allo-
cation.
Answer:
(a) There are two states of nature, corresponding to a ‡ood (state 1) and
no ‡ood (state 2). A competitive equilibrium is a price vector,
(
p
1
;p
2
)
,andan
allocation,
(
x
1
i
;x
2
i
)
150
i
=1
, such that
(i) for i = 1, .
.. , 100,
(
x
1
i
2
i
)
solves
max[
1
10
log(
x
1
i
)+
9
10
log(
x
2
i
)]
subject to
p
1
x
1
i
+
p
2
x
2
i
·
p
1
+
p
2
;
(ii) for i = 101, .
.. , 150,
(
x
1
i
2
i
)
solves
max[
1
10
log(
x
1
i
9
10
log(
x
2
i
)]
subject to
p
1
x
1
i
+
p
2
x
2
i
·
p
2
;
(iii) markets clear:
150
X
i
=1
x
1
i
·
100
150
X
i
=1
x
2
i
·
150
:
1