Econ 51: Midterm Exam Solutions
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Short Questions (28 points, 7 points each)
(1) Let there be two states of the world
s
= 1
,
2, and let
x
and
y
be the consumptions
in two states 1 and 2, respectively. Does the utility function
u
(
x, y
) = 6 + 5
x
2
+
7
y
2
+14
y
+10
x
represent a vNM preferences? Why or why not? If it does represent
a vNM preferences, what are the implied probabilities of each state?
Answer:
The function represent a vNM preferences because, by rearranging,
u
(
x, y
) =
5
12
v
(
x
) +
7
12
v
(
y
) where
v
(
x
) = 12
x
2
+ 24
x
+ 6, which is of the vNM
utility function.
The implied probabilities for states 1 and 2 are 5/12 and 7/12,
respectively.
(2) There are three states
s
= 1
,
2 with probabilities
π
and 1

π
. The agent’s vNM
utility function is
u
(
x, y
) =
π
(

x

3
) + (1

π
)(

y

3
), where
x
≥
0 and
y
≥
0 are
consumptions in states 1 and 2, respectively. Is this agent riskaverse? Why or why
not?
Answer:
This agent is riskaverse. To see this, let her utility function be
v
(
x
) =

x

3
and note that
v
0
(
x
) = 3
x

4
,
v
00
(
x
) =

12
x

5
<
0
.
(3) Suppose that there are two kinds of goods, xylophones (
x
) and yogurt (
y
), and two
individuals, A and B. Utility functions of the individuals are given by
u
A
(
x, y
) =
2
x
+
y
and
u
B
(
x, y
) =
x
+ 2
y
, respectively. There are 7 xylophones and 16 yogurts
in the economy. Draw the contract curve in the Edgeworth box.
Answer:
The contract curve in this economy is composed of the two edges – the
bottom and the right. See Figure 1.
(4) Country A can produce 4 apples or 9 bananas with one unit of labor while country
B can produce 2 apples or 2 bananas with one unit of labor. Goods are freely traded
between countries A and B, although immigration is impossible, so people in each
country can work only in their home country. Now consider the following statement:
“Because A has a better technology than B for producing both apples and bananas,
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