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Demand: Consumer Choice
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c. The consumer chooses the bundle (x1 , x2 ) which satises x1 /x2 = p1 /p2
and costs w. (Does the utility function u(x) = x2 + x2 rationalize the
1
2
consumers behavi
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Lecture Five
Proof:
Assume that both x and y (where x = y ) are solutions to the consumers problem B(p, w). Then x y (both are solutions to the same
maximization problem) and x + (1 )y B(p, w
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Demand: Consumer Choice
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nding an optimal bundle is equivalent to solving the problem
maxxB(p,w) u(x). Since the budget set is compact and u is continuous,
the problem has a solution.
To empha
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LECTURE 5
Demand: Consumer Choice
The Rational Consumers Choice from a Budget Set
In Lecture 4 we discussed the consumers preferences. In this lecture we adopt the rational man paradigm in discu
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Consumer Preferences
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Problem 7. (Easy )
We say that a preference relation satises separability if it can be represented
by an additive utility function, that is, a function of the type u(x) =
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Problem Set 4
Problem 1. (Easy )
Characterize the preference relations on the interval [0, 1] that are continuous and strictly convex.
Problem 2. (Easy )
Show that if the preferences satisfy con
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Consumer Preferences
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Figure 4.5
Differentiable preferences.
Examples:
The preferences represented by 2x1 + 3x2 are differentiable. At
each point x, v (x) = (2, 3).
The preferences represente
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Lecture Four
Proof:
In the problem set you will prove that every preference relation that
is monotonic, continuous, and quasi-linear in commodity 1 satises that for every (x2 , . . . , xK ) t
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Consumer Preferences
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Figure 4.4
Quasi-linear (in good 1) preferences.
the preferences are homothetic, x ( t (x), . . . , t (x) and thus
u( x) = t (x) = u(x).
Let us now consider an additional
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References
133
Radner, R. (1993). The organization of decentralized information processing. Econometrica 61: 11091146.
Reny, P. J. (2001). Arrows theorem and the Gibbard-Satterthwaite theorem:
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Lecture Four
Figure 4.3
Homothetic preferences.
Homothetic Preferences:
A preference
(See g. 4.3.)
is homothetic if x
y implies x
y for all 0.
The preferences represented by k=1,.,K xk k , w
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References
Jehle, G., and P. J. Reny (1997). Advanced Microeconomic Theory. Boston:
Addison-Wesley.
Kahneman, D. (2000). Evaluation by moments: Past and future. In
Choices, Values, and Fra
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Consumer Preferences
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As usual, the above property also has a stronger version:
Strict Convexity:
The preference relation satises strict convexity if for every a
b y , a = b and (0, 1) imply t
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References
In the electronic version of the book, available online at
http:/arielrubinstein.tan.ac.il/micro1/, some of the references contain the
links to the electronic versions of the articl
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Lecture Four
Figure 4.2
Two denitions of convexity.
Convexity 2:
The relation
satises convexity 2 if for all y , the set AsGood (y ) =
cfw_z X |z y is convex.
(Recall that a set A is convex
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Review Problems
129
cfw_A, B, L, where A and B are prizes and L is the lottery which yields
each of the prizes A and B with equal probability. Each member
has a strict preference over the thre
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Consumer Preferences
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M = (maxk cfw_xk , . . . , maxk cfw_xk ) is at least as good as x. Both 0 and M are
on the main diagonal. By continuity, there is a bundle on the main
diagonal that is in
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Review Problems
Problem 8 (Tel Aviv 1998)
A consumer with wealth w = 10 must obtain a book from one of
three stores. Denote the prices at each store as p1 , p2 , p3 . All prices are
below
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Lecture Four
Figure 4.1
satisfy continuity if for all a, b X , if a b, then there is an > 0
such that x y for any x and y such that d (x, a) < and d (y , b) < .
Existence of a Utility Represe
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Review Problems
127
good 1 for f (x) units of good 2, or he can exchange y units of good
2 for g (y ) units of good 1. Assume the consumer can only make one
exchange.
1. Show that if the excha
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Consumer Preferences
41
Monotonicity:
The relation satises monotonicity if for all x, y X ,
if xk yk for all k, then x y , and
if xk > yk for all k, then x y .
In some cases, we will further ass
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Review Problems
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good 1 for f (x) units of good 2, or he can exchange y units of good
2 for g (y ) units of good 1. Assume the consumer can only make one
exchange.
1. Show that if the excha
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Consumer Preferences
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Monotonicity:
The relation satises monotonicity if for all x, y X ,
if xk yk for all k, then x y , and
if xk > yk for all k, then x y .
In some cases, we will further ass
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Review Problems
He is indifferent between a distribution that is fully concentrated in location 1 and one which is fully concentrated in
location 2.
1. Show that the only preference relati
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LECTURE 4
Consumer Preferences
The Consumers World
Up to this point we have dealt with the basic economic model of
rational choice. In this lecture we will discuss a special case of the
rational
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Review Problems
125
Problem 2 (Princeton 2001)
A consumer has to make his decision before he is informed whether
a certain event, which is expected with probability , happened or
not. He assig
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Choice
39
Let X be a nite grand set. A list is a nonempty nite vector of elements
in X . In this problem, consider a choice function C to be a function that
assigns to each vector L =< a1 , . .
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Review Problems
The following is a collection of questions I have given in exams
during the last few years.
Problem 1 (Princeton 2002)
Consider a consumer with a preference relation in a world
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Lecture Three
Problem 3. (Easy )
Check whether the following two choice functions satisfy WA:
C(A) = cfw_x A| the number of y X for which V (x) V (y ) is at least |X |/2,
and if the set is em
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Social Choice
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Problem 3. (Easy )
Assume that the set of alternatives, X , is the interval [0, 1] and that each
individuals preference is single-peaked, i.e., for each i there is an alterna