Homework assignment
Lecturer: Dmytro Vikhrov
Due date: May 20, 2014 (before the final examination)
Problem 1
You know that U(x1, x2) = x1 − x−1
2 , prices of goods x1 and x2 are p1 and p2, and w is the
available budget.
1. Find the Marshallian demand functions for both goods and check whether they are normal
or inferior, Giffen or non-Giffen. Are both goods complements or substitutes?
2. Compute elasticities of each good with respect to p1 and p2. Is any of the goods luxury?
3. Obtain the indirect utility function, derive the cost function and the Hicksian demand.
4. Setup the cost minimization problem and verify that the Hicksian demand in the point
above is correct. Compute the income and substitution effects for both goods separately
and show that the Slutsky equation holds. How is x1 different from x2. Draw IE and SE
for both goods in separate graphs.
Problem 2
Let X denote a random variable, then V ar(X) = E [X − E(X)]2
. Show this can also be
written as V ar(X) = E X2 − [E(X)]2
.
Problem 3
Let x1, x2, ..., xn be a sample from some distribution. The sample variance can be computed
as s2 = 1
n−1
n
i=1 (xi − ¯x)2
. Show that this is equivalent to s2 = 1
n−1
n
i=1 x2
i −
( n
i=1 xi)2
n .
Problem 4
Revenues reported last week from nine stores franchised by an international clothier are
listed in the table below. Based on these figures alone, in what range might the company
expect to find the average revenue of all of its stores? That is, compute a 95% confidence
interval: Prob −α0.025 ≤
¯Y −µ
S/
√
n
≤ α0.025 = 0.95.
Hint: First, compute ¯Y and S from the given sample using the formula in Problem 3.
Then, find the critical value, α0.025, from the t-statistics table (9 degrees of freedom). Finally,
re-arrange the terms in brackets to compute the confidence interval for µ.
Problem 5
Compute the maximum likelihood estimator of θ for the following density functions:
1. p(k, θ) = θk(1 − θ)1−k, k = {0, 1}, 0 < θ < 1. You are given 8 data points: X1 = 1,
X2 = 0, X3 = 1, X4 = 1, X5 = 0, X6 = 1, X7 = 1, X8 = 0.
2. f(y, θ) = θe−θy, y ≥ 0. You are given four data points: Y1 = 8.2, Y2 = 9.1, Y3 = 10.6,
Y4 = 4.9.
1
store ID revenue, thousand USD
1 40
2 48
3 60
4 15
5 50
6 80
7 50
8 36
9 16
10 89
3. f(y, θ) = θ
2
√
y e−θ
√
y. You are given four data points: Y1 = 6.2, Y2 = 7, Y3 = 2.5, Y4 = 4.2.
Problem 6
Suppose that random variables X and Y have a joint density function f(x, y) = 2
3(x + 2y),
0 ≤ x ≤ 1, 0 ≤ y ≤ 1. Find:
1. Find marginal densities, f(x) and f(y).
2. Find E(X + Y).
Problem 7
Suppose that the economy is populated by two consumers: A and B with respective utility
functions: UA(x1, x2) = x
1
4
1 x
3
4
2 and UB(x1, x2) = x
2
3
1 x
1
3
2 . Endowments are (eA
1 , eA
2 ) = (1, 1) and
(eB
1 , eB
2 ) = (3, 2). Find the vector of competitive allocations and draw your findings in the
Edgeworth box.
Problem 8
Production in a capital abundant country, K
L > 1, is described by Y = KαL1−α. The
country welfare function is W = rK + wL, where r and w are the competitive rental and wage
rates. The country chooses between two policies: importing capital and importing labor. Rank
these two policies.
Problem 9
An object is to be sold on a first-price sealed-bid auction. There are two risk-neutral bidders
with private values for that object, vi, vi ∼ G[0, ¯v], where G is a cumulative distribution function.
1. Find agent’s expected utility from bidding r when the agent’s private value is v.
2. Find the bidding function b∗(v), show that b∗(v) < v and ∂b∗(v)
∂v > 0.
Suppose that the owner of the object considers selling the object on the second-price auction.
1. How does the bidding behavior of bidders change? Draw a graph.
2. Compare the expected revenue in both auctions.
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