Chapter 1 Preferences and Utility
1.
Fundamental Assumptions
The rst fundamental assumption that we make about people is that they know what
they like: they know their preferences among the set of things. If a person is given a
choice between x and y he can say (one and only one sentence is true):
1.
He prefers x to y
2. He prefers y to x
3. He is indifferent between the two.
This is the axiom of completeness. It seems reasonable enough.
The second fundamental assumption is the axiom of transitivity. The assumpştion has
four parts:
1.
If a person prefers x to y and prefers y to z, then he prefers x to z.
2. If a person prefers x to y and is indifferent between y and z, then he prefers x to z.
3. If a person is indifferent between x and y and prefers y to z, then he prefers x to z.
4. If a person is indifferent between x and y, and is indifferent between y and z, then he
is indifferent between x and z.
2.
Best Alternatives and Utility Functions
Asking "Would you prefer x to y" will never get you a measure of utility with well-de ned
units, a zero, and other nice mathematical properties. But it will allow you to nd
alternatives that are at least as good as all others, and, remarkably, it will allow you to
construct a numerical measure to re ect tastes. The determination of best alternatives
and the construction of a measure of satisfaction are both made possible by the
completeness and transitivity assumptions on preferences. Therefore, the theory of
preferences, with those two assumptions, is connected to and is a generalization of, the
old-fashioned nineteenth-century theory of utility.
3.
The Formal Model of Preferences
Before we can proceed, we need to introduce some notation. Let x and y be two
alternatives. We consider a group of people who are numbered 1, 2, 3, and so on. To
symbolize the preferences of the ith person we write xRiy for "i prefers x is at least as good
as y"; xPiy for "i prefers x to y"; and xliy for "i is indifferent between x and y.”
xPiy if xRiy and not yRix
xliy if xRiy and yRix
In words: Person i prefers x to y if he thinks x is at least as good as y but he does not think
y is at least as good as x. And i is indifferent between x and y if he thinks x is at least as
good as y and he thinks y is at least as good as x. Now our fundamental axioms of
completeness and transitivity are formally put this way:
Completeness. For any pair of alternatives x and y, either xRiy or yRiX.
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fi
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Transivity. For any three alternatives x,y, and z, if xRiy and yRiz., then xRiz.
3.
Decisions under Uncertainty and Expected Utility
.
.
.
Chapter 2 Barter Exchange
1.
Introduction
This chapter will be about such swapping, or barter exchange. In order to analyze barter
exchange, we will construct a model in which a group of people exchange bundles of
goods among themselves.
2.
Allocations
The set of alternatives is now a set of distributions or allocations of goods in an economy.
✦
There are n people, numbered 1,2,…,n. Usually a person is indexed with the letter i.
Often we’ll let n = 2, in which case we are talking about what happens when there are
only two people (like Adam and Eve exchanging fruit in the Garden of Eden).
✦
There are m different goods. Typically we index a good with the letter j, so the goods
are numbered j = 1,2,…,m. This is the entire list of goods. In some context, there might
be only one (m = 1) or two (m = 2) goods.
some more important notation;
✦
We let xij be person i’s quantity of good j.
✦
We let xi person i’s bundle or vector of goods xi. =(xi1, xi2,…, xim); so xi shows m things, i’s
quantity of good 1, i’s quantity of good 2, …, i’s quantity of good m.
✦
We de ne x = (x1, x2, …, xn). Now x shows person 1’s bundle of goods, person 2’s bundle
of goods, …., person n’s bundle of goods.
In the theory of exchange there is no production; wha goods are available are ther ein
the beginning. We let wij be person i’s starting or initial quantity of good j. Similarly, wi is
his initial bundle, and w is the initial list of bundles of goods. The total of hood j available
must be
w1j + w2j + w3j + ... + wnj =
The symbol
n
“
∑
i=1
n
∑
i=1
wij .
wij”
Is short-hand for “summation of the wij’s, where i ranges from 1 to n.”
An allocation is a list of bundles of goods x with totals of goods consistent with the totals
initially available. This means
n
∑
i=1
xij must be equal to
n
∑
i=1
wij
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for every j , that is, every good. We call the set of allocations a, and a is formally written as
follows:
n
a = { x | xij a ≥ for all i,j and
∑
i=1
xij =
n
∑
i=1
wij for all j } .
The set of alternatives in the theory of exchange is a.
One of the simplest examples of an exchange economy involves only two people and
one good; so n = 2 and m = 1. Let the total quantity of the good initially available,w11 + w21,
be equal to 1. Then a is the set of all pairs (x11 + x21) such that x11 + x21 ≥ 0 and x11 + x21 = 1. The
set of allocations in this economy can be easily diagrammed. To picture a, draw a line
segment one unit long. Choose a point x on the line segment, and let the distance from
the lefthand end of the line to x represent the person i's quantity of the good x11, and let
the distance from the righthand end of line x represent person 2's quantity of the good
x21. Now x11 + x21 ≥ 0 and x11 + x21 = 1, so every such point represents an allocation, and,
conversely, every allocation can be represented by such a point or division of the line.
3.
The Edgeworth Box Diagram
The most useful example of an exchange economy is one in which there are two people
and two goods. This economy's set of allocations can be illustrated in an Edgeworth box
diagram.
4. Pareto Optimal Allocations and the
Core
An allocation x is not Pareto optimal if there is
another allocation y such that
ui(y) ≥ ui(x) for all i = 1,2,...,n
ui(y) ≥ ui(x) for at least one i .
If there is no such alternative, x is a Pareto
optimal, or ef cient allocation.
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Let's consider an illustration in an Edgeworth
box diagram. When there are just two traders
and two goods, and both traders have convex-
shaped smooth indifference curves like the curves in Figure 2.3, the Pareto optimal
allocations are the points, like x, y, and z, at which indifference curves of the two people
are tangent. At these points, it is impossible to make one party better off without hurting
the other. Point X, however, is blocked by person 1, since u1(w1) > u1(x1). Similarly, z is
blocked by 2. The core is the locus of Pareto optimal points, such as y, lying on or within
the lens-shaped area bounded by the indifference curves passing through w.
5.
Algebraic Examples
How to calculate the Pareto optimal and
core allocations in an exchange economy.
u1 = x11x12
u2 = x21 + x22 .
In other words, person I's utility level is the
product of the quantities of the two
goods he has, person 2's utility level is
equal to the amount of good 1 he has plus
twice the amount of good 2 he has. Let
the initial allocation be w1 = (1/2,1/2), w2 =
(1/2,1/2). Each starts with 1/2 unit of each
good.
In this case, person 1's indifference curves
are hyperbolic (because x11 x12 = constant
is the formula for a hyperbola) and person
2's indifference curves are straight lines. In
order to proceed, we need to
nd
expressions for the marginal rates of
substitution or the absolute values of the
slopes of the two people's indifference
curves.
The loss in utility equals the gain in utility, in absolute value,
▵ xi1 . MU of good 1 for i
=
▵ xi2 . MU of good 2 for i
From which it follows that
▵ xi2
▵ xi1
MRS for person i =
=
MU of good 1 for i
.
MU of good 2 for i
MRS person 1 = MRS person 2. This gives
x12
1
1
= or x12 = x11
x11
2
2
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Graphically, this is the straight line from person i’s origin to point z. This straight line
segment gives part of the set of Pareto optimal allocations since any move from a point
on it (like x or y) must make someone worse off.
But there are Pareto optimal allocations
other than these tangency points.
Consider, for instance, the point w. In
order to make person 1 better off, starting
at w, we would have to move above the
hyperbolic indifference curve for person 1
going to w. This would make person 2
worse off. That is, there is no way to make
everyone as well off and at least one
better off. In fact, all the points on the
right-hand side of the box above point z
are nontangency Pareto optimal
allocations.
Where are the core allocations in this
example? Any core allocation must also
be Pareto optimal, so the core allocations
lie somewhere on the lines we've already
identi ed as Pareto optimal. But a core allocation must not be blocked by person 1, or by
person 2. Person 1 would block any allocation that gives him less utility than w1, or any
allocation below his hyperbolic indifference curve through w. Similarly, person 2 would
block any allocation that gives him less utility than w1, or any allocation below (with
respect to his origin) his straight-line indifference curves through w. Consequently, the
core is the locus of points on the straight line between points x and y, including the
endpoints x and y.
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In this example, then, we would expect barter exchange between persons 1 and 2 to
move the economy from w to a point on the line segment from x to y.