API 111 / Economics 2020a / Government 2005 / HBS 4401
Midterm Examination
November 8, 2004
1.
[30 points]
Consider a twocommodity world and consumer with the following utility
function:
(
)
(
)
2
1/ 2
1/2
1
2
1
2
,
u x
x
x
x
=
+
The prices of goods 1 and 2 are p
1
and p
2
, respectively, and the consumer has initial
wealth w.
a)
State the consumer’s utility maximization problem [3 points]
b)
Will this consumer’s Walrasian demand functions satisfy Walras’ Law?
Explain
your answer? [3 points]
c)
Derive the consumer’s Walrasian demand functions and indirect utility functions.
[12 points]
d)
Suppose that the consumer has wealth w = 260.
Initially, prices are (p
1
,p
2
) =
(6,8).
Suppose prices change to (p'
1
,p'
2
) = (5,12).
What is the (exact) equivalent
variation of this price change? [12 points]
2.
[15 points]
Suppose
a rational consumer’s Hicksian demand curves are given by:
(
)
(
)
2
1
1
1
1
1
2
2
2
2
,
,
p u
h
p u
k
a
p
p u
h
p u
k
a
p
=
+
=
+
where a
1
, a
2
, k
1
, and k
2
are constants.
a)
What must be true of the constants a
1
, a
2
, k
1
and k
2
?
Explain your answer. [10
points]
b)
What is the consumer’s Walrasian demand for good 1? [5 points]
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3. [30 points] Online betting services such as the Iowa electronic markets and
Tradesports.com have gained tremendously in popularity in recent years.
A typical trade
in these markets is a purchase of a share in one political candidate for a price
p
which
will be worth $1 if the candidate wins and $0 if the candidate loses.
Suppose that candidates 1 and 2 each have probability ½ of winning an election, but that
the price for a share of candidate 1 is
p
< ½ and that the price for a share of candidate 2 is
1 
p
> ½.
a) Suppose that a riskaverse consumer with wealth
W
is deciding whether to invest in
shares of candidate 1.
Assume that the consumer has VNM utility function for wealth
u
with
u’(w)
> 0,
u’’(w)
< 0.
Write down the optimization problem for this consumer.
[5 points]
b) Does the consumer’s (expected) utility maximizing investment consist of a positive
number of shares of candidate 1, no shares of candidate , or is it impossible to tell without
knowing more about
u
?
Explain your answer.
[5 points]
For parts c) through e), suppose that exit polls will be revealed during the course of
voting.
With probability 1/2, the exit polls will reveal that candidate 1 has a 75% chance
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 Fall '08
 Avery
 Utility, exit poll, Utility maximization problem

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