EC2210
UNIVERSITY OF WARWICK
Summer Examinations 2015/16
Mathematical Economics 1B
Time Allowed: 1.5 Hours
Answer TWO questions only. All questions carry equal weight. Answer each question in a
separate answer booklet.
Read carefully the instructions on the answer book provided and make sure that the particulars
required are entered on each answer book. If you answer more questions than are required and
do not indicate which answers should be ignored, we will mark the requisite number of answers
in the order in which they appear in the answer book(s): answers beyond that number will not
be considered.
1. Consider a two-consumer (
i = 1
, 2) exchange economy where consumption sets are given
by
Xi =
R2 ,
i = 1
, 2 and preferences can be represented by the following utility functions:
+
u1(
x11
, x12) =
x11
u2(
x21
, x22) =
x21
x22
Now assume that the economy has the aggregate endowment vector
ω = (2
, 2) where the
initial endowments of the consumers are
ω1 = (1
, 1),
ω2 = (1
, 1).
(a) Prove that
p 0 in any Walrasian equilibrium for this economy.
(10 marks)
(b) Find a Walrasian equilibrium for this economy.
(15 marks)
Now consider a general economy specified by {(
Xi, %
i)}
I , {
Y
, ω with an allocation
i=1
j }
J
j=1
denoted (
x, y), wealth levels (
w1
, ..., wI) and prices
p = (
p1
, ..., pL).
(c) Define what is meant by a price-equilibrium with transfers and state any difference
between this and a Walrasian equilibrium.
(10 marks)
(d) State then prove the First Welfare Theorem of Economics.
(15 marks)
(Continued overleaf)
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EC2210
2. Consider a two-consumer (
i = 1
, 2) exchange economy where the aggregate endowment is
given by
ω = (1
, 1). Consumption sets are given by
Xi =
R2 ,
i = 1
, 2 and preferences can
+
be represented by the following utility functions:
√
√
u1(
x11
, x12) =
x11 +
x12
u2(
x21
, x22) =
x21
(a) Define what is meant by locally non-satiated preferences.
(5 marks)
(b) Are consumer 2’s preferences locally non-satiated? Provide a proof to accompany
your answer.
(10 marks)
Let
x = [(0
, 1)
, (1
, 0)]. Denote prices
p = (
p1
, p2) and assume for the rest of this question
that
p ∈
2
R .
+
(c) Prove that
x is a Pareto optimal allocation.
(10 marks)
(d) Does there exist a price vector
p 6= (0
, 0) and wealth levels
w1
, w2 ≥ 0, such that
(
p, x) is a price-equilibrium with transfers? If yes, find one. If no, explain why.
(10
marks)
(e) Does there exist a price vector
p 6= (0
, 0) and wealth levels
w1
, w2 ≥ 0, such that
(
p, x) is a price quasi-equilibrium with transfers? If yes, find one. If no, explain why.
(10 marks)
(f) In a few sentences, provide a comparison of your answers to parts (d) and (e) making
reference to a welfare theorem.
(5 marks)
3. Consider a two-good economy (good 1 and good 2) where there are two firms (firm 1
and firm 2). Firm
j produces good
j using capital (
Kj) and labour (
Lj) as inputs of
production,
j = 1
, 2. There is a total amount of capital and labour in the economy given
by ¯
K, ¯
L = 1 respectively. Assume the prices for the two consumption goods and factor
inputs [(
p1
, p2)
, (
w, r)] 0 throughout the question.
(a) Define what is meant by a technically efficient factor allocation in this economy.
(5
marks)
The firms are characterised by the following production functions:
q
f1(
L1
, K1) =
L1
K1
for
L1
, K1 ≥ 0
f2(
L2
, K2) =
L2 +
K2
for
L2
, K2 ≥ 0
(b) Find the set of all technically efficient factor allocations and draw it in an appropriate
diagram. In the diagram, select one such efficient allocation and show the isoquants
of each firm that pass through it.
(20 marks)
(Question 3 continued overleaf)
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EC2210
Suppose there are also two individuals in the economy, Mr.1 and Mr.2. Mr.1 owns firm 1
and has 1 unit of labour. Mr.2 owns firm 2 and has 1 unit of capital. Consumption sets
are
X
2
i = R
, i = 1
, 2 and preferences are given by:
++
ui(
xi1
, xi2) = log(
xi1) + log(
xi2)
(c) Define a Walrasian equilibrium for this economy.
(10 marks)
(d) Find a Walrasian equilibrium for this economy.
(15 marks)
(End)
3