Olivier Bochet
Advanced Micro (Micro II)
Fall 2011
Homework 3
This homework should be solved alone in a group of up to 4 people. It
should be submitted in class (and stapled!) on Tuesday, December 13th. Late
submissions will not be accepted. If you work in a group, please submit only
one version with your names on it.
Please write neatly. Show all work.
Exercise 1: Quasilinear Preferences and Exchange
Consider a 2 agents-2 goods exchange economy. The consumption set of
each agent is R2+ . Initial endowments are ω1 = (0, a) and ω2 = (5, 2 − a) where
0 ≤ a ≤ 2. Agents’ preferences are represented by
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ui (x1i , x2i ) = (x1i ) 2 + x2i
1) What is the set of Pareto efficient allocations of this exchange economy?
2) For each value of a, compute the Walrasian equilibria of the economy.
Exercise 2: Neutral goods
Consider an exchange economy with n agents and L goods and where the
consumption set is RL+ . For each consumer i, his preferences are represented
by ui (x1i , ..., xLi ) = xii .
1) Consider the case n = L. Show that regardless of the initial endowments
of the economy, there is a unique Pareto efficient allocation. Next show that
this conclusion does not extend to the case n < L.
2) Consider the case n = L = 3 but where preferences for agents 1 and 2
are represented by,
ui (x1i , x2i , x3i ) = xii for i = 1, 2
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but preferences for agent 3 are represented by,
u3 (x13 , x23 , x33 ) = x13 x23 x33
Agents’ endowments are given respectively by, ω1 = (0, 0, 1), ω2 = (0, 0, 1)
and ω3 = (1, 1, 0). Compute the set of efficient allocations and the unique
Walrasian equilibrium of this economy.
Exercise 3: Private ownership economy
Consider a one-producer-one consumer economy (the single agent is both
a producer and a consumer). There are two goods. Good 1 is used as input
in the production process. Good 2 is output. The aggregate endowment is
ω̄ = (2, 2). The production set of the firm is described by
Y = {y ∈ R2 : y2 ≤ −y1 , 0 ≤ y2 ≤ 2} ∪ {y ∈ R2 : 0 ≤ y2 ≤ 2, y1 ≥ 2}
Preferences are represented by,
u(x1 , x2 ) = x2
1) Represent graphically Y +{ω̄} and compute the set of efficient production
plans y.
2) Compute the set of efficient allocations (y, x)
3) Find the Walrasian Equilibria
Exercise 4: Non-Differentiable Preferences
Consider an exchange economy with two agents and two goods. The consumption set of each agent is R2+ . The endowments are ω1 = ( 32 , 12 ) and
ω2 = ( 12 , 32 ). Hence the aggregate endowment is ω̄ = (2, 2).
Preferences of agent 1 and 2 are represented by the following utility functions,
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x11 , x21
u1 (x11 , x21 ) = min
2
u2 (x12 , x22 ) = min {2x12 , x22 }
For each of the following questions, you should complement your answers
with Edgeworth boxes whenever appropriate.
1) Compute the set of Pareto efficient allocations.
2) Find all the Walrasian equilibria (be precise).
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Exercise 5: Lexicographic and non-convex preferences
Consider an exchange economy with 2 agents and 2 goods and where the
consumption set is R2+ . The aggregate endowment is ω̄ = (1, 1). Agent 1 has
lexicographic preferences over good 1 while agent 2 has preferences represented
by,
u2 (x12 , x22 ) = (x12 )2 + (x22 − 1)2
1) Compute the set of efficient allocations
2) Find all the price quasi-equilibrium. Establish whether these are price
equilibrium and connect your results with the welfare theorems.
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