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ECON 101- Midterm Exam Guide - Comprehensive Notes for the exam ( 81 pages long!)


Department
Economics
Course Code
ECON 101
Professor
wallace
Study Guide
Midterm

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UCLA
ECON 101
MIDTERM EXAM
STUDY GUIDE

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Economics 101 — Problem Set 6 Solutions: GE,
Risk, and Uncertainty
Troy Smith
Spring 2017
1. General Equilibrium in an exchange economy
Consider a pure exchange economy with two consumers, 1 and 2, and two goods,
xand y. Consumer 1’s initial endowment is ω1= (1,0), that is, 1 unit of good
xand 0 units of good y. Consumer 2’s initial endowment is ω2= (0,1), that
is, 0 units of good xand 1 unit of good y. Consumer i’s utility function (for
i= 1,2) is ui(xi, yi) = xiyi, where (xi, yi) represents i’s consumption bundle.
(a) Show this economy (with some indifference curves and the initial endow-
ments) in an Edgeworth box diagram.
Suggested Answer:
Edgeworth Box.
(b) Write down the equations that describe the Pareto efficient allocations.
Identify them in the Edgworth box. Is the initial endowment point Pareto
efficient? Why or why not?
Suggested Answer:
The Pareto optimal points in this example are points of tangency between
indifference curves of the two consumers. Tangency requires that con-
sumer 1’s marginal rate of substitution equal consumer 2’s marginal rate
of substitution. Consumer i’s marginal rate of substitution is
MRSi=M U i
x
MUi
y
=yi
xi
Setting the two consumers’ marginal rates of substitution equal, and then
substituting 1 x1for x2and 1 y1for y2, gives
y1
x1
=y2
x2
=1y1
1x1
This leads directly to
1
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x1=y1
Therefore the set of Pareto optimal points, that is, the contract curve, is
simply the upward sloping diagonal of the box diagram, from consumer
1’s origin to consumer 2’s origin. The initial point Wis obviously not
efficient; it’s not on the contract curve. In fact any move from Winto
the interior of the Edgeworth box diagram would make both consumers
better off.
2. Expected Value and Variance Proofs
Let Xbe a random variable taking the value xiwith probability pi, for i=
1, ..., n. Let Ybe a random variable taking the value yiwith probability pj, for
j= 1, ..., n.Xand Yare independent. Let aand bbe real numbers. Prove the
following:
(a) E[aX +b] = aE[X] + b
Suggested Answer:
E[aX +b] =
n
X
i=1
(axi+b)pi
=a
n
X
i=1
xipi+b
n
X
i=1
pi
=aE[X] + b
(b) V ar[X] = E[X2]E[X]2
Suggested Answer:
V ar[X] =
n
X
i=1
(xiE[X])2pi
=
n
X
i=1
(x2
i2xiE[X] + E[X]2)pi
=
n
X
i=1
x2
ipi2E[X]
n
X
i=1
xipi+E[X]2
n
X
i=1
pi
=E[X2]2E[X]E[X] + E[X]2
=E[X2]E[X]2
2
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