Problems for the second seminar
ECON5200 Advanced Microeconomics – Fall semester 2009.
Solutions to the problems will be presented at the seminars in week 39.
Please direct any question to Kjell Arne Brekke (Room ES1032, Tel 228 41169,
E-mail: k.a.brekke@econ.uio.no)
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Mas-Colell et al Problem 5.E.1
Mas-Colell et al Problem 10.C.1.
Mas-Colell et al Problem 10.C.2.
Mas-Colell et al Problem 10.D.2.
Problem 5
Consider an economy with three commodities, and where consumer i’s preferences are
represented by the utility function u ( x1 , x2 , x3 ) = x1 + φi ( x2 ) + ψ i ( x3 ) with φi () and ψ i ()
differentiable, increasing and strictly concave, and where firms produce commodities 2
and 3 with 1 (the numeraire good) as input, whith technologies (cost functions) cl (ql ) for
commodity l=2,3, where cl (ql ) is increasing and convex. (All firms have the same
technologies). Set the price of the numeraire equal to 1.
a) Derive the conditions characterizing the market equilibrium.
b) Show that the equilibrium price of commodity 2 can be determined independently
of commodity 3.
c) Show that welfare changes in the market for commodity 2 is determined by the
Marshallian aggregate surplus for that market.
Problem 6
Consider lotteries with three possible consequences, win -100, 0 or +100, and
with two states of the world = f! 1 ; ! 2 g: A decisionamaker with preferences
satisfying all the axioms in Blume et al is indi¤erent no lottery and the lottery
that pay +100 with probability 2/3 and -100 with probability 1=3;independent
of the state of the world.
a) Calibrate the utility function such that u( 100) = 1; u(0) = 0 and …nd
u(100)
The same person is also indi¤erent to a lottery L1 that in state ! 1 gives a
lottery: -100 with 10% probability and +100 with 90% probability and 0 in
state ! 2 and L2 that gives +100 (with certainty) in state ! 2 and 0 in state ! 1
b) What is the decision makers subjective probability measure that the state
of the world is ! 1 ?
Problem 7
Suppose the set of consequences contain two elements: c1 :You get an
umbrella; c2 :You get sunglasses. Suppose that there are two states of the
world ! 1 :it rains, and ! 2 : The sun shines. You prefer c1 to c2 in state ! 1 and
you prefer c2 in state ! 2 . Does these preferences satisfy the axioms of Blume
et al. If not, what assumptions are violated?
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