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Part I
The Consumer and the Demand function
Chapter 1
The Utility Function
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Outline of the chapter
1.
2.
3.
4.
5.
The consumer.
The utility function.
The indifference curve.
The optimal consumer’s choice.
Exercises
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1. The consumer
a/ Definitions
The consumer is an economic agent who buys goods
from supplier in order to maximize her satisfaction under
her budget constraint.
• The consumer is an individual/household/community.
• The relationship between the consumer and the supplier
is characterized by two facts:
1. The company pays the consumer a salary/remuneration
in exchange of his job/effort.
2. The consumer acquires some of the goods produced by
the company.
• The consumer’problem :
How does she maximize her utility under her budget
constraint?
•
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What is a good?
There are two definitions of the word ‘‘good’’:
1.
The goods can be acquired and sold. This definition
does not take into account the free goods such as the
air and the water of the sea.
2.
The production of goods is unlimited. However, many
goods such as work of art does not satisfy this
definition.
In this course, we consider just the goods which are not
free and can be produced in a unlimited way.
There are many kinds of goods:
* The consumption/supply/intermediate goods.
* Substitute/complement goods.
* Bad/Neutral goods.
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What is a consumption bundle?
•
•
•
•
We call the objects of the consumer choice
consumption bundle.
The bundle is a complete list of the goods and services
that are involved in the choice problem that we are
investigating.
We want not only a complete list of the goods that the
consumer might consume but also a description when,
where and under what circumstances
they would
become available.
Hypothesis : There are many bundles in the economy.
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We consider an economy with n goods, a
consumption bundle Pi is given by:
Pi=(x1i,x2i,…,xni)
where xji is the amount of the good j (j=1…n) in the
bundle of the consumer i.
• In this chapter, we consider a bundle with just
two goods, one of them is calling ‘‘all the other
goods’’.
• We focus in the tradeoff between the two goods
and use a two-dimensional diagrams.
• Consider the bundle X given by:
X=(x1,x2)
•
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-
•
•
•
b/ The consumer preferences
Consider the consumption bundles A and B, the
consumer can rank them as to their desirability.
That is, the consumer can determine that one of the
consumption bundles is strictly better than the other, or
decide that she is indifferent between the two bundles.
If A is strictly preferred to B, we write
If the consumer is indifferent between the two bundles A
and B, we write
If the consumer prefers or is indifferent between the two
bundles, we say that she weakly prefers A to B and write
. This just says that the consumer thinks that A is
al least as good as the bundle B.
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Assumptions about Preferences
•
Economists usually make some assumptions about the
‘‘consistency’’ of consumers’ preferences.
•
We cannot have the situation where A is strictly preferred to B and
at the same time B is striclty preferred to A.
•
We usually make some assumptions about how the preferences
relations work.
•
Some of the assumptions about preferences are so fundamental
that we can refer to them as ‘‘axioms’’ of consumer theory.
•
There are three axioms
1- Complete
Any two bundles can be compared :
2- Reflexive
Any bundle is at least as good as itself:
3- Transitive
Consider three bundles A, B and C:
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2. The utility function
A utility function is a way of assigning a number to every
possible consumption bundle such that more-preferred bundles
get assigned larger number than less-preferred bundles.
•
Consider an economy with n goods, the utility function U is
written:
U=U(x1, x2, x 3,…xn)
Where xi is the amount of good i=1…n acquired by the consumer.
•
There are some theories of the utility function:
1/ The ordinal utility
•
The only property of a utility assignement that is important is
how it orders the bundles of goods.
•
The magnitude of the utility function is only important insofar as
it ranks the different consumption bundles; the size of the utility
difference between any two consumption bundles does not
matter.
•
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Consider for example the following table:
Different ways to assign utilities
Bundle
U1
U2
U3
A
3
17
-1
B
2
10
-2
C
1
0.002
-3
The consumer prefers A to B and B to C.
The three functions describe the same preferences because they all
have the property that A is assigned a higher number than B, which
in turn is assigned a higher number than C.
Since only the ranking of the bundles matters, there can be no
unique way to assign utilities to bundles of goods.
U2 is a monotonic transformation of U1 : a monotonic transformation
is a way of transforming one set of numbers into another set of
numbers in a way that preserves the order of the numbers.
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Consider the utility function U, we typically represent a monotonic
transformation f(U).
f transforms each number x into some other number f(x) in a way that
preserves the order of the numbers in the sense that x1 >x2 implies
that f(x1)>f(x2).
A monotonic transformation or a monotonic function are essentially the
same thing.
If
, this implies that U(A)>U(B).
If
, this implies that U(A)=U(B).
If A:=(x1A, x2A, …xnA) and B:=(x1B, x2B, …xnB). We have:
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2/ the cardinal utility
•
This theory gives a significance to the magnitude of utility.
In this theory, we offer the consumer a choice between two bundles
and see which one is chosen.
We just assign a higher utility to the chosen bundle than to the
rejected bundle.
Any assignment will be a utility function. But how do we tell if a
person likes one bundle twice as much as another?
There are many definitions for this kind of assignments :
* I like one bundle twice as much as another, if I am willing to pay
twice as much as for it.
* I like one bundle twice as much as another, if I am willing to run
twice as far to get it.
•
•
•
•
Knowing how much larger doesn’t add anything to the description
of the consumer’s choice : in the cardinal utility, we have just to
know which is the preferred bundle.
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c- The marginal utility and the Marginal Rate of
Substitution (MRS)
The marginal utility measures the rate of change of the
utility ∆U associated with a small change in the amount of
good i ( ).
It is written:
When
converges to 0, the marginal utility is given by:
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The marginal utility is decreasing:
Gossen (1854) points out that the rate of increase of the
utility decreases when the consumer increases her
consumption of the good xi.
In other words, the marginal utility decreases with the
quantity of the good xi.
Example :
Consider an individual who is thirsty in the desert.
The first glass of water gives her the highest utility.
The utility of the second glass is lower than the first one but it still has a
positive utility.
.
.
The 10th glass has a negative utility because she cannot drink water.
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The MRS
The MRS is the rate at which the consumer is just willing to substitute
some amount of the good 2 for one unit of the good 1; the level of
utility is the same.
It is given by the ratio of the marginal utilities of the 2 goods 1 and 2
and written:
Note that we have 2 over 1 on the left-hand side of the equation
and 1 over 2 on the right-hand side. Don’t be confused!
The algebraic sign of the MRS is negative, since if you get more of the
good 1 you have to get less of the good 2 to keep the same level of
utility.
However economists typically refer to the MRS by its absolute value,
that is as a positive number.
We follow this convention as long as no confusion will result.
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3. The indifference Curve (IC)
•
•
•
•
The IC represents all the consumption bundles which
provide the same level of utility.
We cannot represent this curve in an economy with more
than 2 goods.
Consider the following utility function: U(x1, x2 ).
We use a three-dimensional diagram to represent this
curve.
U(x1, x2)
x2
U2
U1
U2
U1
x1
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Hereafter, we will consider the two-dimensional diagram of the IC.
The vertical axis measures the amount of the good 2
The horizental axis measures the amount of the good 1.
x2
U2
U1
The ICs satisfy some conditions to represent the
x preference relations.
1
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Consider the following utility function:
U(x,y)=3x+4y
•
How do we represent the IC of this function in the diagram (x,y)?
•
Consider the IC with the level of utility u (a number). It illustrates all
the bundles providing the utility u.
•
So we can write:
u=3x+4y
which implies that:
y=(u/4) -(3/4)x
•
This is the equation of a line with negative slope –(3/4). If u=4, it is
represented in the following way: A(0,1) and B(4/3,0)
•
y
A
Slope=-(3/4)
x
B
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Properties of the consumer’s preferences
a/ The monotonicity of preferences
The consumer is always willing to acquire more of both goods. We are going to
examine situation before the satiation point (the most preferred bundle).
The monotonicity implies that:
The ICs cannot cross, otherwise, we have a contradiction:
x2
X
Z
Y
x1
It is easy to check that X and Y lie on different Indifference curves, they cannot
have equal utilities.
More of both goods is better bundle for the consumer, less of both goods
represents a worse bundle.
x2
Better
Bundles
Worse
Bundles
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x1
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b/ Convex preferences
According to this property, an average bundle is at least as good as or
strictly preferred to each of the two extreme bundles.
This weighted average bundle has the average amount of good 1 and
the average amount of good 2 that is present in the two bundles.
y
yA
A
C
yC =αyA +βyB
U2
B
yB
xA
xC =αxA +βxB
xB
U1
x
The bundle C is strictly preferred to the extreme bundles A and B.
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c/ Substitution of the goods
If the goods are substitutes, the consumer care only about the total number of
the units of the goods, not their qualities (color, size, packaging..).
The ICs are straight line
Example: U(x, y)=x+2y
y
Slope=-(1/2)
x
If the goods are complements, such as the right and the left shoes, the
consumer always wants to consume the goods in fixed proportion to each
other.
y=(5/4)x
y
Consequently, the ICs are L-shaped.
Example : U(x, y)=min{5x, 4y} there is no MRS.
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C
A
U2
U1
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4. The optimal consumer’s choice
The budget constraint/line (BC)
The consumer has an income R (revenue, salary, remuneration, fortune).
We suppose that we can observe the prices of the goods (p1, p2).
The variables p1, p2 and R are exogenous variables.
D denotes the amount of money spent by the consumer to acquire the
quantities x1 and x2 of the goods 1 and 2.
D=p1 x1 +p2 x2
where p1x1 and p2x2 are the amount of money the consumer is spending on
good 1 and 2.
The amount of money spent on the two goods in no more than the total
amount the consumer has to spend.
R≥D
R≥ p1 x1 +p2 x2
The consumer’s affordable consumption bundles are those that don’t cost
any more than R.
We call this set of affordable consumption bundles at prices (p1,p2) and
income R the budget set.
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However, the consumer spends all her revenue in consumption of the goods 1
and 2 to maximize her utility.
The BC is therefore written:
R= p1 x1 +p2 x2
(1)
These are the bundles that just exhaust the consumer’s income.
How do we depict graphically the BC?
The equation (1) is written:
x2=(R/p2)-(p1/p2)x1
It is the equation of a line with a negative slope -(p1/p2).
x2
R/p2
R/p2 is the amount of money spent in the
consumption of the good 2 if the consumer
Budget
set
doesn’t acquire the good 1.
R/p1
x1
The budget set consists of all bundles that are offordable at a given
prices and income.
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An increase of the income causes a parallel shift outward of the budget line.
If good 1 becomes more expensive, then the BC becomes steeper.
The consumer’s program is to maximize her utility under the BC.
There are 2 methods to solve for the optimal choice of the consumer:
1/ The graphic method
The objective of the consumer is to find the bundle in the budget set that is on
the highest IC.
We restrict our attention to bundles of goods that lie on the budget line.
The optimal consumer’s choice is given by the position where the IC U2 is
tangent to the budget line. The optimal bundle is the bundle C.
x2
R/p2
X*
A
C
B
2
U3
U1
x*1
U2
x1
R/p1
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2/ The algebraic method
The consumer’s program is given by:
There are 2 algebraic methods to solve for this program:
The substitution method
Consider the budget line with the equation : x2=(R/p2)-(p1/p2)x1 = x2(x1)
We substitute x2 for its expression into the utility function. The latter is
therefore written:
U(x1 , x2 )= U(x1 , x2(x1) )=U(x1 , (R/p2)-(p1/p2)x1 )
The first order condition of x*1 is given by:
dU(x1 , (R/p2)-(p1/p2)x1 )/dx1=0 which implies that: Ux1 /Ux2 =p1 /p2
The marginal rate of substitution is equal to the price ratio.
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The second order condition is given by
This condition enables us to ensure that x*1 is a maximum.
The Lagrangien method
Given the consumer’s program:
We write the lagrangian function:
L(x1 , x2, λ)=U(x1 , x2 )+λ(R-p1 x1 –p2x2)
The first order conditions of L are given by:
The equations (1) and (2) imply that: RMS=Ux1 /Ux2 =p1 /p2
Which means that the absolute slope of the BC (p1/p2) is equal to the
slope of the IC (Ux1/Ux2).
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This condition gives the optimal amounts of the two goods:
x*1 and x*2.
The equation (3) is the equation of the budget line.
The second order of the langrangian is given by:
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