Problem sets
Microeconomics
Autumn 2014
Consumer theory. Deadline: 4.12.2014
1. Suppose that the set of consumption bundles X is finite and the preference
relation satisfies completness and transitivity. Show that there is a
utility function u : X → representing .
2. Consider two utility functions u(x1, x2) and v(x1, x2), where v = f(u) and
f is strictly increasing function. Derive the first-order conditions for the
utility maximization problem and show that Marshallian demands will be
the same for u and v
3. Find the Marshallian and Hicksian demands for the following utility functions.
You can suppose that there are only two goods when deriving
Hicksian demands.
• Cobb-Douglas u(x1, . . . , xn) = xa1
1 . . . xan
n
• Leontieff u(x1, . . . , xn) = min{a1x1, . . . , anxn}
• Perfect substitutes u(x1, . . . , xn) = a1x1 + . . . + anxn
4. Let the utility function of a consumer be in quasilinear form u(x1, x2) =
x1 + f(x2). Normalize p1 = 1. Derive the Marshallian demand and interpret
it in terms of wealth effects.
5. Suppose that an indirect utility function has the Gorman form u(p, w) =
a(p) + b(p)w. Show that the wealth expansion curves are linear. Show
that homothetic and quasilinear preferences are special case of Gorman
form indirect utility function.
6. Show that Hicksian demand function is homogenous of degree zero in
prices. Use this fact and Euler’s theorem to prove that
n
j=1
∂Hi
∂pj
pj = 0
Interpret this in terms of Slutsky matrix.
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7. Show that if complemets and substitutes are defined in terms of Marshallian
demand derivatives, goods could be complements on the basis of the
sign od ∂Di
∂pj
and substitutes on the basis of the sign of
∂Dj
∂pi
. Give precise
conditions under which this occurs.
Production and general equilibrium. Deadline:
18.12.2014
1. Consider a production function that exhibits decreasing returns to scale.
Show that cost function is strictly convex in output, i.e. c(p, αy) > αc(p, y)
for any α > 1
2. Consider a two-input production function that employs capital (K) and
labor (L) as the two inputs. Find the supply function, input demand
functions, conditional input demand function and cost function for the
following production functions:
• Cobb-Douglas f(K, L) = Kα
Lβ
where α, β > 0
• Leontieff f(K, L) = (min{αK, βL})a
where α, β > 0 and a ∈ (0, 1)
• f(K, L) = (αK + βL) where α, β > 0 and a ∈ (0, 1)
3. Do all homogeneous production functions of whatever degree have a)
marginal products and b) marginal technical rates of substitution which
are independent of the level of output?
4. Consider a two-consumer (a and b) and two-good exchange economy. Let
ua
(xa
1, xa
2) = (xa
1)1/3
(xa
2)2/3
, ωa
= (3, 1), ub
(xb
1, xb
2) = (xb
1)1/3
(xb
2)2/3
and
ωb
= (1, 1). Take good 1 as numeraire. Find competitive equilibria nad
graph it in the Edgeworth box. Find the set of interior pareto efficient
allocation and graph it in the Edgeworth box.
5. Consider an economy with two consumers, two firms, and two commodities:
time and consumption good. The consumers have to sleep 12 hours
a day and hence each consumer is endowed with 12 hours of waking
time and has preferences over consumption (x1) and leisure (x2) given
by the utility function u1(x1
1, x1
2) = (x1
1)3/4
(x1
2)1/4
forconsumer 1 and
u1(x2
1, x2
2) = (x1
2)1/4
(x2
2)3/4
for consumer 2. Each firm produces the consumption
good (y1) out of labor (−y2) using the production possibility set
characterized by y1 − 2
√
−y2 ≤ 0. Each consumer owns one of the two
firms.
• Solve for the competitive equilibrium in this economy when the wage
rate is normalized to 1 (including the price of consumption, consumption
bundles and production plans).
• Find the set of interior Pareto-efficient allocations in this economy.
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6. Summarize the roles played by the continuity and zero-degree homogeneity
of the excess demand functions and by Walras’ Law in the proof of
existence of general equilibrium.
Decision under uncertainty. Deadline: 9.1.2015
1. Person A is a expected utility maximizer with Bernoulli uitlity function
v(y) =
√
y where y is income. She is asked to enter a business, which
involves 50-50 chance of an income 900 or 400 and so the expected value
of the income is 650.
• If asked to pay a fair price of 650 in order to take a part in the
business, would she accept?
• What is the largest sum of money she would be prepared to pay to
take part in the venture?
2. Consider the following utility functions:
• Show that v(y) = −e−ay
exhibits constant absolute risk aversion.
• Show that v(y) = a + by1−ρ
exhibits constant relative risk aversion.
3. Consider a strictly risk-averse decision maker who has an initial wealth of
w but who runs a risk of loss of D crowns. The probability of loss is π.
The decison maker can buy an insurance. One unit of insurance costs q
crowns and pays 1 crown if the loss occurs. The decision maker’s problem
is to choose the optimal level if α.
• Suppose that the price of an insurance is fair, ie. q = π. Show that
the decision maker insures completely, i.e. α = D
• Show that if insurance is not fair, then the individual will not insure
comletely.
4. Suppose that two individuals have utility functions u(y) and v(y) where
v(y) = f(u(y)) is an increasing concave transformation of u. Show that he
individual with the utility function v(y) has greater absolute and relative
risk aversion. Use Arrow-Pratt approximation to show that she will also
have larger risk premium.
5. Suppose that utility function u(y) represents preferences satisfying the
axioms of expected utility. Show that v(y) = a+bu(y) represents the same
preferences. Give an example, which shows that non-linear transformation
of u(y) does not represent the same preferences.
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