Microeconomics I 2022
Problem Set 4
You may form a group of at most three (individual work is fine, too). Submit one solution per group(write the
names of all the members of the group) to: utmicro2018[at]gmail.com (change [at] to @)1 Type your answer (Latex
is preferable; if you cannot type, write down the answer neatly).
You don’t have to prove the theorem that is already proved in the past homework.
Due: 6/28
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Assume X = RL+, p 0 and x(p,M) 0. Suppose all the axioms and differentiability.
Suppose I = {1, 2}.
Suppose that an indirect utility function vi is written as
vi(p,M
i) = ai(p) + b(p)M i,
where a1, a2, b is differentiable real-valued functions on RK++.
(1) Show that
x1(p,M1)
M1
=
x2(p,M2)
M2
(2)Show that the expenditure function ei is written as
ei(p, u
i) = c(p)ui + di(p)
1Questions about the lectures or homework may also be sent to this email address.
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for some c(p), di(p). (Define c(p) and di(p) appropriately in your solution.)
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Let X be the set of consumption bundles and consider a pure exchange economy:
E = I, (ωi)i∈I , ( i)i∈I ,
where I is the set of agents, ωi ∈ X is agent i’s endowment, and i is i’s preference, which depends only on i’s own
consumption. Some preferences can be represented by utility functions. In such a case, let ui : X → R be i’s utility
function that represents i.
(1)Let I = {A,B}. Find offer curves and the set of competitive equilibria for the following endowments and
preferences/utility functions (if exists).
(a) ui(x1, x2) = min{x1, x2}, ωi = (ωi1, ωi2), ωik > 0 for all k = 1, 2 and all i ∈ I;
(In (a), You can assume ωA1 + ωB1 > ωA2 + ωB2)
(b) uA(x1, x2) = x1x2 and B is lexicographic, i.e.,
x = (x1, x2) B y = (y1, y2) if
x1 > y1, or [x1 = y1 ∧ x2 ≥ y2],
ωA = (ωA1, ωA2) = (2, 0) and ωB = (ωB1, ωB2) = (0, 2)
(2) Let I = {1, · · · , n}, and X = {ω1, · · · , ωn} with ωi being the endowment of agent i ∈ I. In other words,
in this economy, each agent owns and consumes one unit of indivisible good. Agent i ∈ I has a strict preference
over X, i.e., j = k implies ui(ωj) = ui(ωk) for all i, j, k ∈ I. Denote the price of ωi by pi (i ∈ I). Then, agent i’s
problem is given by
max
xi∈X
ui(xi) s.t. pk ≤ pi( if xi = ωk for k ∈ I)
Show that in a competitive equilibrium, for all i, j,∈ I, if agent i consumes good ωj , then pi = pj holds.
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3Consider a pure exchange economy:
E = I, (ωi)i∈I , (ui)i∈I ,
with I = {A,B} and the number of goods being 2, i.e., K = 2. Assume that preferences satisfy all the axioms
including strong monotonicity. On the other hand, the set X of possible consumption bundles is restricted as indi-
cated in the following. (To be precise, the strict convexity of the preference is modified as follows: for all x, y ∈ X
and all λ ∈ (0, 1), z = λx + (1 λ)y ∈ X) implies ui(z) > min{ui(x), ui(y)}, i = A,B.) In the sequel, you may
assume the existence of competitive equilibrium for any endowment.
(1) Suppose that goods are indivisible, i.e., X = N2+ = {0, 1, 2, · · · }2. Do the first and second fundamental theorems
hold Prove or disprove, respectively.
(2) Suppose that good 1 is indivisible, while good 2 is divisible, i.e., X = N+ × R+. Do the first and second
fundamental theorems hold Prove or disprove, respectively.
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Consider a pure exchange economy:
E = I, (ωi)i∈I , (ui)i∈I ,
with I = {1, · · · , n} and the number of goods being K ≥ 2. Suppose ωi ∈ RK++ for all i. (RK++ ≡ {y ∈ RK |y 0})
(1) Assume
∑
i ωik = 1 for all k = 1, · · · ,K.
Suppose that every agent has a utility function:
ui(xi) =
K∑
k=1
vi(xik),
where vi : R+ → R is strictly increasing, strictly concave, continuously differentiable. You can ignore boundary
solutions in this question.
Prove or disprove that this economy has at most one competitive Equilibrium.
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(2)Suppose n = K = 2 and normalize p1 = 1.
Suppose that u1(x) = αx1 + x2 and u2(x) = βx1 + x2, where α > β > 0.
Find a necessary and sufficient condition for the existence of competitive equilibrium.
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