Consumer Theory
September 8, 2016
Consumer Theory
Consumer Theory
Building Blocks
The consumers problem is to choose a consumption bundle,
out of all feasible bundles, that is most preferred.
There are four building blocks of the consumers problem:
The consumption set
The feasible set
The preference relation
The behavioural assumptions
Each is conceptually distinct from the others.
This basic structure of consumer theory is extremely general,
and so, very flexible.
By specifying the form each of these takes in a given problem,
many different situations involving choice can be formally
described and analysed.
Consumer Theory
Consumer Theory
Consumption & Feasible Set
The consumption set X is the set of all possible consumption bundles.
The consumption set, X , represent the set of all alternatives, or complete
consumption plans, that the consumer can conceive - whether some of them will
be achievable in practice or not. It is also called the choice set.
We will assume X is Rn+ , the set of all possible non-negative quantities of n
goods.
Let each commodity be measured in some infinitely divisible units.
Let xi R represent the number of units of good i .
We assume that only non - negative units of each good are meaningful.
There is finite, fixed but an arbitrary number n of different goods. We call
x = (x1 , . . . , xn ) a consumption bundle or a consumption plan.
ASSUMPTION 1.1 Properties of the Consumption Set, X
The minimal requirements on the consumption set are
1. X Rn+
2. X is closed.
3. X is convex.
4. 0 X.
The feasible set B is a subset of X , representing the set of consumption bundles
achievable with the consumers resources (i.e. wealth).
Consumer Theory
Consumer Theory
Axiomatizing Preferences
The consumers preferences determine which bundles are more preferred.
We will characterize preferences axiomatically :
In this method of modelling, minimum number of meaningful and distinct
assumptions are set forth to characterise the structure and properties of
preferences.
The rest of the theory then builds logically from these axioms, and
predictions of behaviour are developed through the process of deduction.
These axioms are intended to give formal mathematical expression to
fundamental aspects of consumer behaviour and attitudes towards the
objects of choice.
Together, they formalise the view that the consumer can choose and that
choices are consistent in a particular way.
Formally, we represent the consumers preferences by a binary relation, %,
defined on the consumption set: let x 1 ; x 2 be any two elements of the
consumption set X . We say x 1 % x 2 if x 1 is at least as good as x 2 .
Consumer Theory
Consumer Theory
Axioms of Consumer Choice
Axiom 1: Completeness. For all x 1 , x 2 X , either x 1 % x 2 or
x2 % x1
Axiom 1 implies that there are no non-comparabilities; that is, there
are no bundles that cannot be compared to each other.
Axiom 2: Transitivity: for any three elements x 1 , x 2 , and x 3 , if
x 1 % x 2 and x 2 % x 3 then x 1 % x 2
Axiom 2 implies that preferences are consistent. (Lecture Notes in
Microeconomic Theory Ariel Rubinstein)
This is necessary for any discussion of preference maximisation, i.e.
if not transitive, then there might be set of bundles X with no best
elements.
A binary relation % is called a rational preference relation if it satisfies
the axioms of completeness and transitivity.
Consumer Theory
Consumer Theory
Strict Preference & Indifference Relation
The two preference relation, strict preference and indifference, are
induced by the preference relation %.
We can define strict preference relation, as:
x 1 x 2 if and only if x 1 % x 2 and x 2 x 1 .
We can define indifference relation, as:
x 1 x 2 if and only if x 1 % x 2 and x 2 % x 1 .
Consumer Theory
Consumer Theory
Axioms of Consumer Choice
Using these two supplementary relations, we can establish something very concrete
about the consumers ranking of any two alternatives. For any pair x 1 and x 2 , exactly
one of three mutually exclusive possibilities holds: x 1 x 2 , or x 1 x 2 , or x 1 x 2 .
Figure: Hypothetical preferences satisfying Axioms 1 and 2.
Consumer Theory
Consumer Theory
Sets Derived from %
Given a preference relation %, we can define sets relative to a
given point x 0
Define the set % (xx 0 ) = {xx |xx X , x % x 0 }
This is the set of all bundles that are at least as good as x 0
Similarly, we can define :
- (xx 0 ) = {xx |xx X,x - x 0 }, the no better than set.
(xx 0 ) = {xx |xx X,x x 0 }, the worse than set.
(xx 0 ) = {xx |xx X,x x 0 }, the preferred to set.
(xx 0 ) = {xx |xx X,x x 0 }, the indifference set.
Consumer Theory
Consumer Theory
Continuity
Axiom 3: Continuity. For all x Rn+ ; % (xx ) and - (xx ) are closed (i.e. contains its
boundary).
The continuity axiom guarantees that sudden preference reversals do not occur.
- (xx ) is a closed set as if {xx i } is a sequence of consumption bundles that are all
at least as good as a bundle y , and if {xx i } converges to some bundle x , then
x must also be at least as good as y , i.e. no discontinuity in the % relation.
Note if - (xx ) was an open set, then there may be sequences {xx i } that
converges to a point on the boundary which is not in the set, i.e. the preference
suddenly flips to y x .
(xx ) is an open set as the sequences can converges to a point on the boundary
where x % y .
If x y , and z is close to x , then z must also be strictly preferred to y . This
would not be the case in a closed set if x was on the boundary
Consumer Theory
Consumer Theory
Local Non-satiation
The next two axioms are alternatives:
Axiom 4 : Local non-satiation. For all x 0 Rn + and for all
> 0, there exists some x B (xx 0 ) such that x x 0 , where
B (xx 0 ) is a ball of radius centred on x 0 .
This axiom says that from any point x 0 , you can always find a
path that leads to a strictly more preferred bundle.
Figure: Hypothetical preferences satisfying Axioms 1, 2, 3 and 4 .
Consumer Theory
Consumer Theory
Strict Monotonicity
Axiom 4: Strict monotonicity. For all x 0 , x 0 :
If x 0 x 1 (i.e. every component of x 0 is at least as large as in
x 1 ), then x 0 % x 1 .
If x 0 x 1 (i.e. every component of x 0 is at least as large as
in x 1 , and there is one strictly greater), then x 0 x 1 .
This is a stricter condition that local non-satiation: it says that from any point
x 1 , any path in a direction that increases a component of x 1 leads to a strictly
more preferred bundle.
In the consumer problem, Axioms 4 and 4 guarantee that the chosen bundle
will lie on the budget line (therefore, we can use the Lagrange method with an
equality condition).
Figure: Hypothetical preferences satisfying Axioms 1, 2, 3 and 4.
Consumer Theory
Consumer Theory
Convexity and Strict Convexity
Axiom 5: Convexity. If x 1 % x 0 , then txx 1 + (1 t)xx 0 % x 0 for all t [0, 1].
Axiom 5: Strict Convexity: If x 1 x 0 , then txx 1 + (1 t)xx 0 x 0 for all
t (0, 1).
Axiom 5 implies that the set % (xx 0 ) is convex for any x 0 . Axiom 5 implies that
it is strictly convex.
We shall see later that this implies the utility function is (strictly) quasiconcave
if (strict) convexity is satisfied.
For the rest of this course, unless specified, we will usually assume Axiom 4
(strict monotonicity) and 5 (strict convexity).
Consumer Theory
Consumer Theory
Convexity and Strict Convexity
Figure: Hypothetical preferences satisfying Axioms 1, 2, 3, 4 and 5 or 5 .
There are at least two ways we can intuitively understand the implications of convexity for consumer tastes.
Suppose x 1 x 2 .
x 1 contains a proportion of the good x2 which is relatively extreme, compared to the proportion of x2 in
the other bundle x 2 .
The bundle x 2 , by contrast, contains a proportion of the other good, x1 , which is relatively extreme
compared to that contained in x 1 .
Any convex combination of x 1 and x 2 , such as x t , will be a bundle containing a more balanced
combination of x 1 and x 2 .
Axiom 5 or Axiom 5 forbids the consumer form preferring such extremes in consumption
Axiom 5 requires that any such relatively balanced bundle as x t be no worse than either of the two
extremes between which the consumer is indifferent.
Axiom 5 requires that the consumer strictly prefer any such relatively balanced consumption bundle.
Consumer Theory
Consumer Theory
Convexity and Strict Convexity
Another way to describe the implications of convexity for consumers tastes focuses
attention on the curvature of the indifference sets.
When X = R2+ , the (absolute value of the) slope of an indifference curve is
called the marginal rate of substitution of good 2 for good 1.
This slope measures the rate at which the consumer is just willing to give up
good 2 per unit of good 1 received. Thus, the consumer is indifferent after the
exchange.
If preferences are strictly monotonic, any form of convexity requires the
indifference curves to be at least weakly convex-shaped relative to the origin.
This is equivalent to requiring that the marginal rate of substitution not increase
as we move from bundles such as x 1 towards bundles such as x 2 .
Axiom 5 requires the rate at which the consumer would trade x2 for x1 and
remain indifferent to be either constant or decreasing as we move from
north-west to south-east along an indifference curve.
Axiom 5 requires that the rate be strictly diminishing.
Consumer Theory
Consumer Theory
The Utility Function
We can summarize preferences with a utility function u(.),
that assigns a number for every consumption bundle x .
u(xx 0 ) u(xx 1 ) if and only if x 0 % x 1
u(xx 0 ) > u(xx 1 ) if and only if x 0 x 1
Theorem 1.1: If % is complete, transitive, continuous and
strictly monotonic, there exists a continuous real-valued
function, u : Rn+ R which represents %.
Consumer Theory
Consumer Theory
The Utility Function
Proof:
Let the relation % be complete, transitive, continuous and strictly monotonic.
Let e (1, . . . , 1) Rn+ be a vector of ones, and consider the mapping
u : Rn+ R defined so that the following condition is satisfied:
u(xx )ee x (1)
In words, (1) says take any x in the domain Rn+ and assign to it the number u(xx ) such
that the consumption bundle, u(xx )ee , is ranked indifferent to x .
Two questions immediately arise.
First, does there always exist a number u(xx ) satisfying (1)?
Second, is it uniquely determined, so that u(xx ) is a well-defined function?
Consumer Theory
Consumer Theory
The Utility Function
To settle the first question, fix x Rn+ and consider the following two
subsets of real numbers:
A {t 0|tee % x }
B {t 0|tee - x }.
Note that if t A B, then t e x , so that setting u(xx ) = t would
satisfy (1).
Thus, the first question would be answered in the affirmative if we show
that A B is guaranteed to be non-empty.
This is precisely what we shall show.
Consumer Theory
Consumer Theory
The Utility Function
Proof of does there always exist a number u(xx ) satisfying (1)?
According to Exercise 1.11, the continuity of % implies that both A
and B are closed in R+
By strict monotonicity, t A implies t A for all t t.
Consequently, A must be a closed interval of the form [t, ).
Similarly, strict monotonicity and the closedness of B imply that B
must be a closed interval of the form [0, t ].
Now, for any t 0, completeness of % implies that either tee % x or
tee - x , that is, t A B. But this means that
R+ = A B = [0, t ] [t, ).
We conclude that t t so that A B 6= .
Consumer Theory
Consumer Theory
The Utility Function
The second question: Is u(xx ) satisfying (1) uniquely determined?
We show that there is only one number t 0 such that tee x .
But this follows transitivity: if t1e x and t2e x , then by
transitivity of , t1e t2e .
By strict monotonicity, it must be the case that t1 = t2 .
Hence, for every x Rn+ , there is exactly one number, u(xx ), such
that (1) is satisfied.
Consumer Theory
Consumer Theory
The Utility Function
We now show that this utility function, which assigning each bundle in X a number,
represent the preferences %.
Consider two bundles x 1 and x 2 , and their associated utility numbers u(xx 1 ) and u(xx 2 ),
which by definition satisfy u(xx 1 )ee x 1 and u(xx 2 )ee x 2 . Then we have the following:
x1 % x2 (2)
u(xx 1 )ee x 1 % x 2 u(xx 2 )ee (3)
1 2
u(xx )ee % u(xx )ee (4)
u(xx 1 ) u(xx 2 ) (5)
(2) (3) follows by definition of u.
(3) (4) follows from the transitivity of %, the transitivity of and the
definition of u.
(4) (5) follows from the strict monotonicity of %.
Together, (2) through (5) imply that (2) (5).
Hence, x 1 % x 2 if and only if u(xx 1 ) u(xx 2 ).
Consumer Theory
Consumer Theory
The Utility Function
It remains to show that the utility function u : Rn+ R representing % is continuous.
By theorem A1.6, it suffices to show that the inverse image under u of every open ball in R is open in Rn+ . This is
equivalent to showing that u 1 ((a, b)) is open in Rn+ for every a < b.
Now,
1 n
u ((a, b)) = {xx R+ |a < u(xx ) < b}
n
= {xx R+ |aee u(xx )ee bee }
n
= {xx R+ |aee x bee }.
The first equality follows from the definition of the inverse image.
The second from the monotonicity of %.
The third from u(xx )ee x and Exercise 1.4.
Rewriting the last set on the right-hand side gives:
1
u ((a, b)) = (aee ) (bee ). (6)
By continuity of %, the sets - (aee ) and % (bee ) are closed in X = Rn+ .
Consequently, the two sets on the right-hand side of (6), being the complements of these closed sets, are
open in Rn+ .
Therefore, u 1 ((a, b)), being the intersection of two open sets in Rn+ , is open in Rn+ .
Consumer Theory
Consumer Theory
Invariance to Positive Monotonic Transforms
The utility function u(.) representing % is not unique; there
are infinitely many possibilities
Suppose we have a strictly increasing function f : R R, and
let v (xx ) = f (u(xx )). Then v (.) is also a utility function that
represents %, and vice versa.
Consumer Theory
Consumer Theory
Properties of Preferences and Utility Functions
Suppose % is represented by u : Rn+ R. Then:
1. u(xx ) is strictly increasing if and only if % is strictly
increasing.
2. u(xx ) is quasiconcave if and only if % is convex.
3. u(xx ) is strictly quasiconcave if and only if % is strictly
convex.
As we will see, strict quasiconcavity guarantees that the consumer
problem has a unique solution (most preferred bundle).
Consumer Theory
