Appunti Microeconomics

Consumer theory

The traditional approach to consumer behaviour is to assume that the consumer has well-defined

preferences over all the alternative bundles, and he attempts to select the most preferred bundle from

those which are available.

There are 4 building blocks in any model of consumer choice:

- The consumption set or commodity space which is the set of all alternatives the consumer can

conceive;

- The feasible set which is the set of all alternatives that are conceivable and realistically obtainable

for the consumer;

- The preference relation that informs about the consumer's tastes for different objects of choice;

- The behavioural assumption, the guiding principle the consumer uses to make final choices

selecting the most preferred available alternative in the light of his personal tastes.

We let the consumption set, (capital X) represents the set of all alternatives that the consumer can

conceive unfettered (ANFETED, privo di limiti) by the consideration of his situation.

We assume that there is a fixed number of different goods called (capital L).

∈ ( )

Let represent the number of units of good (small x

subscript i belongs to R).

Only non-negative units of each good are meaningful (are feasible) and having no units is always possible.

= ( , … , )

We let be a vector containing different quantities of each of the commodities and we call

1 +

∈

a consumption bundle represented by a point .

Consumer preferences are characterised axiomatically, and we represent the consumer’s preferences by a

.

binary relation defined on the consumption set, We require only that the consumer can make binary

comparisons, by examine two bundles at a time and decide.

So, we need 2 axioms:

- Completeness;

- Transitivity.

Completeness, for any bundles in the consumer has the ability to evaluate the alternatives and, so, he

.

can decide if the bundles is at least as good as or if is at least as good as

, , , ≥ ≥ ≥ .

Transitivity, for any 3 elements, if then

These 2 axioms imply that the consumer can completely rank any finite number of elements in the

consumption set from the best to the worst, possibly with some ties (pareggi). These axioms define what it

means to be rational, so, a preference relation is called a rational preference if it is complete and transitive.

Two additional preference relations that we use are: ≥ ,

If is at least as good as.

- Strict preference relation, when bundle is at least as good as > ,

If is preferred to.

; ≤ ,

bundle but bundle is not as good as bundle If no better than.

< ,

If worse than.

- Indifference relation, when bundle is at least as good as bundle ~,

If indifferent to.

~ ~

,so, we can write it down as and and say that is

.

indifferent to and is indifferent to

.

The indifference set containing is the set of all bundles that are indifferent to

The indifference set capture the trade-offs the consumer is willing to make among various

commodities.

Two other assumptions about preferences that we use are: desirability (axioms of monotonicity, strong

monotonicity and local non-satiation) and convexity.

When representing preferences over ordinary consumption goods, we express the view that “wants" are

unlimited by the axioms of monotonicity and strong monotonicity. In economics, a consumer’s preference

,

are said to be monotonic if, given bundles the consumer prefers bundle that has more of all

> .

goods and that is A consumer’s preference is said to be strongly monotonic if, given consumption

,

bundles the consumer prefers the bundle that has more of at least one good, and has not less in

≥ .

any other good, that is

In the figure bundles like are strictly preferred to because it contains the same quantity of as the

1

,

bundle but more of commodity .

2

If preferences are monotone or strongly monotone a consumer will choose a bundle on the boundary of

the budget set, that is the set of bundles the consumer can afford, given her wealth and the prices of

+ +

{

= ⊂ : ∗ ( ℎ ) ≤ } =

the various commodities and is the budget set.

,

.

Capital B, subscript prices and wealth is defined by bundle that is a proper subset of

ℎ.

the commodity space such that is not greater than the

The axiom of local non-satiation states that for any bundle of goods there is always another bundle of

∈ > 0

goods closed that is preferred to it. So, if is the consumption set, then for any and for every

‖

∈ − ‖ ≤ . () =

there exists a bundle such that and is preferred to

.

Local non-satiation does not exclude that the preferred alternative may contain less of some commodities,

so it doesn’t imply that giving the consumer more of everything necessarily makes him better off (più ricco).

It allows the possibility that some commodities are “bads”, however, it is not possible for all commodities

= 0)

to be “bads” if preferences are non-satiated because in this case no consumption at all (the point

would be a satiation point. {

( ) = ∈ : ~}

Local non-satiation if all commodities are goods ensure that is:

- downward sloping;

- and thin. ,

If the indifference curves were thick, then there would be points such as where in his nearby all points

.

are indifferent to And since there is no strictly preferred point in this region, this could be a violation of

local non-satiation.

With the (indifference set) we can define an upper counter set which is the set of all bundles that are

{

, = ∈ : ≥ },

at least as good as while the lower counter set is the set of all bundles that are

{

, = ∈ : ≥ }.

no better than The indifference curve containing is the intersection between

(capital U subscript x) and the .

The axiom of convexity states that a preference relation is convex if the upper counter set of is convex,

so, if there are 2 bundles, and that are at least as good as then a convex combination of bundles

.

and is at least as good as It means that averages are at least as good as extremes.

The first figure shows us that if you combine the bundle and you are still in the upper counter set, and

;

so you have a bundle which is at least as good as while in the second figure, you see that indifference

curve is not convex so if you combine the two bundles then you have a combination that lies in the lower

counter set, and this is a violation of the axiom.

The upshot (risultato) of the convexity and non-satiation assumptions is that the indifference sets must be

1. thin;

2. downward sloping;

3. bowed upward (so, convex). ≠ ), >

Strict convexity is a preference relation in which for any distinct bundles and (and and

> , .

so if you combine the 2 bundles you will obtain another bundle which is better than So, imposing

strict convexity on preferences strengthens the requirement of convexity to say that averages are strictly

better than extremes. = 2

Implications of the different properties for indifference curves if and there are only “goods”:

1. If the preference relation is complete, it means that there is an indifference curve through every

possible bundle;

2. If the preferences are non-satiated, then bundles on the indifference curve farther from the origin

are preferred to those on indifference curves closer to the origin;

3. Transitivity implies that the indifference curves cannot intersect;

4. Non-satiation monotonicity implies that the indifference curves are downward sloping and cannot

be thick;

5. Strict convexity implies that indifference curves are bowed upward, so they are convex.

,

In this case the consumer is indifferent between and but also between and while the axiom of

, .

transitivity implies that the consumer must be indifferent between and but has more of than

1

If two (indifference curves) intersect, the same bundle ( in the example) offers two different levels of

satisfactions, which is not possible.

A utility function is useful to summarizing the information contained in the consumer's preference relation

and It allows us to examine consumer behaviour using calculus rather than set theory.

() ∈ .

A utility function is a function that assigns a number to every consumption bundle

≥,

Utility function represents the preference relation if for all bundles that are in the commodity space, the

() ≥ () ≥ .

utility function if and only if Thus, a utility function represents a preference relation if

it assigns higher numbers to preferred bundles.

If the preference relation is rational (so complete and transitive) and continuous, then there exists a

continuous utility function representing that preference relation.

The assumption that preferences are rational agrees with how we think consumers should behave while

the assumption that preferences are continuous is a only technical assumption, it is needed for the

arguments to be mathematically rigorous, but it imposes no real restrictions on consumer behaviour.

Utility is an ordinal concept, so, if there is a utility function representing preferences of a consumer, this

function is not unique, because the numbers assigned to the bundles matters only in an ordinal sense.

Thus, if we modify these numbers, for example if we:

- multiply all the numbers by 2;

- or add 6 to them;

- or take the square root.

The numbers assigned to the indifference curves after the transformation would still represent the same

preferences.

A first implication of utility function is that it is invariant to positive monotonic transforms because what all

we require of the preference relation is that his rankings between bundles be meaningful, then all any

utility function representing that relation is capable of conveying to us his ordinal information.

1 21−

() =

For example, consider the Cobb-Douglas utility function , that can be not easy to analyse, so

we apply the log to the original function so we will get this which is easier , so

(x) (x).

represents the same preferences as

First law of log:

= + ;

-

( ) = − ;

-

= ∗ .

-

A second implication of the ordinal nature is that the difference between the utility of two bundles doesn't

mean anything.

() − () = 7 () − () = 14,

For example, if and it doesn't mean that going from consuming

.

to consuming is twice as much of an improvement than going from to The number assigned by a

utility function to bundles has no significance.

≥ ():

Let be represented by a utility function, then

1. Is strictly increasing if and only if is strictly monotonic;

2. Is quasiconcave if and only if is convex;

3. Is strictly quasiconcave if and only if is strictly convex.

Imposing quasiconcavity is convenient because, if strict, guarantees that the function has unique maximum

on a closed set. If quasiconcavity is not strict, there is a convex set of maxima.

Ex. 1 to 3 and Tutorial 1 Utility maximisation problem

We assume that the consumer is motivated to choose the most preferred feasible alternative according to

∗ ∗

∈ ≥ ∈

his preference relation. So, the consumer seeks the bundle such that for all where

is the feasible set. To be able to use calculus we recast the consumer's problem in terms of a utility function

+

(),

that is complete, continuous, strictly monotone and strictly convex in .

In the 2 goods case, preferences like these can be represented by an indifference map whose level sets are

non-intersecting, strictly convex away from the origin, and increasing north-easterly.

Our concern is with an individual consumer operating within a market economy. A market economy is an

economic system in which transactions between agents are mediated by markets. So, there is a market for

.

each commodity and in these markets a price prevails for each commodity We suppose that prices are

> 0

strictly positive, so for each commodity and we assume the individual consumer is an insignificant

force on every market. ≫ 0,

Formally we take the vector of market prices, as fixed from the consumer’s point.

≥ 0.

The consumer is endowed with a fixed money income which is Because the purchase of units of

commodity at price per unit requires an expenditure of , the requirement that expenditure not

∗ ≤ .

exceed income can be stated as We summarise these assumptions by specifying the feasible set,

+

→ {

, = ⊂ , ∗ ≤ }.

called the budget set

In the 2 goods case, consists of all bundles lying inside or on the boundaries.

Under our assumptions total expenditure must not exceed income. The consumer’s problem be cast as the

problem of maximising the utility function subject to the budget constraint.

max () . . ∗ ≤ ,

Formally the utility maximisation problem is written as: so consumer wants

∈

+

to maximize the utility function respect all components of the bundle under the budget constraint.

,

The budget set, is a non-empty, closed, bounded, and thus compact subset of . By the Weistrass

()

theorem, we are therefore assured that a maximum of over exists.

Because is convex and the objective function is strictly quasiconcave (because is convex), the maximiser

∗

of the utility function over is unique. Because preferences are strictly monotonic the solution will

satisfy the budget constraint, lying on the boundary of the budget set.

()

In order to do calculus, we assume the arguments on are both differentiable, so we can explore the

∗ − ≤ 0

demand behaviour. If we rewrite the constraint as and then form the Lagrangian, we obtain

(, ) = () − [ ∗ − ].

∗ ∗ ∗ ∗

≥ 0 ( , )

If is strictly positive, then, there exists a such that satisfy the FOC:

()

From the first order condition, the solution tell us that where the derivative of with respect to is

0 ,

greater than is the marginal utility with respect to good and at the optimum it is proportional to price

for all goods.

→ →

= = = = = =

So, at the optimum and , then and and

first terms is MRS in absolute value(marginal rate of substitution which is the rate at which the consumer is

willing to give up one good in exchange for another good, keeping utility constant) while the right side is

1

= − ∗ ).

the absolute value of the slope of the budget line (in the 2-good case 2

2 2

So, at an interior optimum, the absolute value of the MRS between any two goods must be equal to the

ratio of the goods’ prices.

Therefore, in the two-good case, the optimality conditions require that the slope of the indifference curve

∗ ∗

through be equal to the slope of the budget constraint, and that lie on the budget line.

Example COBB-DOUGLAS SLIDE

Suppose that we have found the utility maximizing point, x*. What we really found?

∗

1. First, note that is a function of prices and wealth (it would have been different if and/or

were different); ,

2. Because it is unique for given values of and we can view the solution to the as a function

from the set of prices and wealth to the set of quantities.

∗ ∗

(,

= ), = 1, … , = (, ).

So, we write or in a vector notation

∗

We call the consumer's Walrasian demand function.

Let's illustrate the relationship between the consumer's maximization problem and his demand behaviour

in the two-good case: Assume that the consumer faces

1′ 2′

prices and , and has wealth

∗

, so the optimal bundle is

0

( ′, ′, .

1 2 0) 1′

’’,

But if price goes up, it is in

1

∗

=

this case the optimal it is

1′′

( , ′, ).

2 0

The budget line B