consumer preferences - didatticaweb

Elisa Battistoni Lecture note 01: Consumer’s choice

Consumer preferences In analysing consumer’s behaviour, we start from the hypothesis that he/her chooses the best combination

of goods and services among those that are available.

When making a choice the consumer takes into account a set of goods and services, named basket of goods.

Therefore, baskets of goods are the object of a consumer’s choice. They are a complete list of goods and

services: for each good/service availability conditions are specified (When is the good available? Where?

Under what circumstances? And so on.). This specification is very important. In fact, the same good/service

can be perceived in a different way in different situations. As an example, the same umbrella is perceived in

a different way when the weather is sunny or when the weather is cloudy.

Therefore, we can consider as different goods/services the same good/service available in different

situations.

Preference relationships Let us consider two baskets made up by only two goods – good 1 and good 2. The two baskets are (𝑥1, 𝑥2)

and (𝑦1, 𝑦2).

We have:

1. (𝑥1, 𝑥2) ≻ (𝑦1, 𝑦2) ⟹ (𝑥1, 𝑥2) is strictly preferred to (𝑦1, 𝑦2). Anytime the consumer has the possibility

to choose between these two baskets, he/she will definitely choose (𝑥1, 𝑥2);

2. (𝑥1, 𝑥2) ∼ (𝑦1, 𝑦2) ⟹ (𝑥1, 𝑥2) is indifferent to (𝑦1, 𝑦2). The consumer gains the same satisfaction from

both baskets, so when he/her has the possibility to choose, sometimes he/she will choose (𝑥1, 𝑥2) and

some other times he/she will choose (𝑦1, 𝑦2);

3. (𝑥1, 𝑥2) ≽ (𝑦1, 𝑦2) ⟹ (𝑥1, 𝑥2) is weakly preferred to (𝑦1, 𝑦2). In other words, consumer is at least as

much satisfied by (𝑥1, 𝑥2) as by (𝑦1, 𝑦2).

Preferences allow the consumer to rank all the baskets he/she can access and to choose the best for him/her.

Each consumer has his/her own preference structure: therefore, the ranking is not necessarily the same for

all consumers.

As we are considering baskets made up by only two goods/services, we can represent them on a Cartesian

plane (Figure 1). In this plane good 1 is represented on the horizontal axis and good 2 on the vertical one. As

quantities of goods cannot be negative, we consider only the first quarter of the Cartesian plane. In the

following picture the basket A is represented on the Cartesian plane, along with its two components on the

axes.

Figure 1 – Representing baskets on a Cartesian plane.

x2

x1

A

x1A

x2A


Preference relationship properties Given three baskets (𝑥1, 𝑥2), (𝑦1, 𝑦2), and (𝑧1, 𝑧2), preference relationships have three main properties:

1. Transitivity property

If (𝑥1, 𝑥2) ≻ (𝑦1, 𝑦2) and (𝑦1, 𝑦2) ≻ (𝑧1, 𝑧2) then

(𝑥1, 𝑥2) ≻ (𝑧1, 𝑧2)

2. Reflexive property

(𝑥1, 𝑥2) ≽ (𝑦1, 𝑦2)

A basket is at least as desirable as itself (or an identical basket)

3. Completeness property

Given two baskets (𝑥1, 𝑥2) and (𝑦1, 𝑦2), then

(𝑥1, 𝑥2) ≽ (𝑦1, 𝑦2) or (𝑦1, 𝑦2) ≽ (𝑥1, 𝑥2) or (𝑥1, 𝑥2) ∼ (𝑦1, 𝑦2)

It is always possible to make a choice between two baskets.

Indifference curves Given a basket A, it is possible to find an infinite number of baskets that provide the consumer with the same

level of satisfaction. All these baskets can be represented on the Cartesian plane by a geometric locus, called

indifference curve. Therefore, the indifference curve contains all and only the baskets that provide the same

level of satisfaction to the consumer.

An indifference curve is the geometric locus of all the points (𝑥1, 𝑥2) in the Cartesian plane that

represent indifferent baskets.

Given a basket A it is also possible to find an infinite number of baskets that provide the consumer with a

higher satisfaction and that are strictly preferred to A; in the same way, it is possible to identify a number of

baskets to which A is strictly preferred to. Correspondingly, an indifference curve divides the Cartesian plane

in two areas: the one of strictly preferred baskets, and the one containing the baskets which A is strictly

preferred to.

Let us suppose that we leave the basket A and move to a strictly preferred basket B: correspondingly, we are

moving from a point to another in the Cartesian plane. Once we are in B, we can find once again an infinite

number of baskets that are indifferent to B: once again, these baskets lie on an indifference curve, different

from that of A. Moreover, we can find again a number of baskets which B is strictly preferred to, and an

infinite number of baskets strictly preferred to B.

This reasoning can be repeated an infinite number of times. Therefore:

There exists an infinite number of indifference curve for a consumer, each containing all and

only the baskets that provide the same level of satisfaction. The set of all the indifference curves

is called indifference map. Each consumer has his/her own indifference map, describing his/her

preference structure.

Regular preferences We talk about regular preferences when two hypotheses are satisfied:

1. The more is the better

The consumer always prefers a basket containing at least the same quantity of one good and an additional

quantity of the other good. In Figure 2, given a basket 𝐴 = (𝑥1, 𝑥2) the area of the Cartesian plane in which

we can seek to find all the baskets strictly preferred to A is represented with a “+”, whereas the area in which

we can look for all the baskets which A is strictly preferred to is represented by a “”.


Figure 2 – Areas in which we can look for baskets strictly preferred to A and for baskets which A is strictly preferred to.

It comes that the baskets that are indifferent to A can be found left-upward and right-downward with respect

to the position of A. In other words,

the indifference curve passing through A has to be downward sloping.

Therefore, the first hypothesis of regular preferences provides information about the slope of indifference

curves. Nonetheless, we still do not have any information about concavity. There are three possibilities,

represented in Figure 3 and the second hypothesis allows understanding which of the three is the correct

one.

(a) (b) (c)

Figure 3 – Possible concavities of an indifference curve.

2. An intermediate basket C is strictly preferred to the end baskets A and B

Let us consider two baskets 𝐴 = (𝑥1, 𝑥2) and 𝐵 = (𝑦1, 𝑦2) lying on the same indifference curve – so, they

provide the same satisfaction – and their weighted average C

𝐶 = (𝑡𝑥1 + (1 − 𝑡)𝑦1, 𝑡𝑥2 + (1 − 𝑡)𝑦2) with 𝑡 ∈ [0,1]

Basket C is an intermediate basket between the two end baskets A and B and it lies on the segment linking

A and B. Changing the value of t in the range [0,1], means moving C towards one of the extremes: in

particular, if t=0 then CA, whereas if t=1 then CB (Figure 4).

The only configuration for indifference curve that allows C to be on a higher indifference curve with respect

to that of A and B is in Figure 5(a). In the other two configurations – Figure 5(b) and Figure 5(c) – basket C

provides respectively the same satisfaction than A and B – lying on the indifference curve to which they

belong – and a lower satisfaction than A and B – lying on a lower indifference curve.

x2

A

x1 x1A

x2A

+

A x2A

x2

x1 x1A

A

x2

x2A

x1 x1A

A

x2

x2A

x1 x1A


Figure 4 – Basket C is an intermediate basket between end baskets A and B.

(a) (b) (c)

Figure 5 – Possible concavities of an indifference curve and satisfaction coming from basket C.

Therefore,

the indifference map for regular preferences is characterized by downward sloping and convex

indifference curves

as shown in Figure 6. Each indifference curve of the map defines a set of strictly preferred baskets, that lies

rightward and upward with respect to the curve; moreover, also a set of weakly preferred baskets can be

identified, coming from the union of the set of strictly preferred baskets and of the indifference curve.

Figure 6 – An indifference curve for regular preferences.

Theorem

Two indifference curves related to different levels of satisfaction cannot intersect.

A

C

B

x2A

x2B

x2C

x2

x1A x1

B x1C x1

A

x2

x1

B

C

x2 A

x1

B

C

x2 A

x1

B C

x2

x1

Set of strictly preferred baskets


Proof

Let us suppose that two indifference curves related to different levels of satisfaction are intersecting, as

shown in Figure 7. We can identify three baskets – A, B, and C – with basket C corresponding to the

intersection point between the curves, whereas baskets A and B belong to different curves. Finally, let us

suppose that A lies on the highest indifference curve (conclusions do not change if B belongs to the highest

indifference curve).

Figure 7 – Two intersecting indifference curves with different levels of satisfaction and three baskets over them.

As A and C belong to the same indifference curve, they are indifferent: 𝐴~𝐶

As C and B belong to the same indifference curve, they are indifferent: 𝐶~𝐵

Consequently, for the transitivity property it has to be also 𝐴~𝐵. Nonetheless, this two baskets lie on

different curves, corresponding to different levels of satisfaction: in particular, in our hypothesis 𝐴 ≻ 𝐵.

The two conditions cannot be valid at the same time. Therefore, we have reached an absurd conclusion,

which neglects our thesis.

Therefore,

the indifference map for regular preferences is characterized by an infinite number of downward

sloping, convex and never intersecting indifference curves (Figure 8).

Figure 8 – An indifference map for regular preferences.

x2

x1

A

B

C

x2

x1


The Marginal Rate of Substitution (MRS) The Marginal Rate of Substitution (MRS) represents the ratio with which the consumer is willing to substitute

one good with the other in the basket, so as to remain with the same level of satisfaction. Therefore, the

question the MRS answers to is: “If I renounce to a quantity x2 of the second good in my basket, which

quantity x1 do I have to gain in order to remain with the same satisfaction (i.e. to remain on the same

indifference curve)?”.

The marginal rate of substitution between good 1 and good 2 is indicated with MRS1,2 and is represented by

the ratio between the variations of the two goods in the basket. As the variations of the two goods have

opposite signs – the quantity of one good in the basket has to increment, whereas the quantity of the other

good diminishes – the MRS has always a negative sign.

𝑀𝑅𝑆1,2 =∆𝑥2∆𝑥1

< 0

Having in mind the representation of baskets and preferences in the Cartesian plane, we note that the first

good is represented on the horizontal axis, whereas the second one is represented on the vertical axis.

Therefore, ∆𝑥2

∆𝑥1 is the incremental ratio for indifference curves of two-good baskets and – as a consequence –

it represents their slope. The sign of the MRS is consistent with the fact that – under regular preferences –

indifference curves are downward sloping.

As the slope of an indifference curve is not constant (unless it is a line), the MRS changes its value as we

change the point (the basket) on the curve. In particular, for convex curves – as it is the case for regular

preferences – the MRS decreases its value as x1 increases. This means that the higher the quantity the

consumer has of a good, the more he/her will be willing to give it back to increase the other good in the

basket (remaining with the same satisfaction). In Figure 9, the same decrease in the second good produces

different increases in the first one moving from the basket A or from the basket B. In particular, when the

consumer has a very scarce quantity of x1 but many units of x2 (basket A), to increase the quantity of the

scarce good in his/her basket of a x1 he/her is willing to give back a x2. But when the consumer has a large

quantity of x1 – and, as a consequence, few units of x2 (basket B) – to renounce to the same x2 he/she wants

back a larger increase x1 of the first good, precisely because he/she is renouncing to a scarce good in his/her

basket.

Figure 9 – Variations of the MRS along a convex indifference curve.

x1

x2

A

x1

B

x2

x2

x1


In its general formulation as incremental ratio, the MRS represents an average slope for the indifference

curve. If we need to know the slope of the curve in a single point, we can make a very small variation x1,

taking the first derivative of the indifference curve.

𝑀𝑅𝑆1,2 = lim∆𝑥1→0

∆𝑥2∆𝑥1

=𝑑𝑥2𝑑𝑥1

< 0

The MRS represents, in this formulation, the instant slope of the indifference curve and it is negative, being

the curve downward sloping.


Utility Utility is a means to give value to the satisfaction a consumer gains from a basket. A utility function assigns

a value to each basket in a way that respects consumer’s preference relationship.

Therefore, given two baskets (𝑥1, 𝑥2) and (𝑦1, 𝑦2) we have:

1. (𝑥1, 𝑥2) ≻ (𝑦1, 𝑦2) ⟺ 𝑢(𝑥1, 𝑥2) > 𝑢(𝑦1, 𝑦2) A basket (𝑥1, 𝑥2) is strictly preferred to (𝑦1, 𝑦2) if and only if it provides a strictly higher utility than

(𝑦1, 𝑦2);

2. (𝑥1, 𝑥2) ∼ (𝑦1, 𝑦2) ⟺ 𝑢(𝑥1, 𝑥2) = 𝑢(𝑦1, 𝑦2) A basket (𝑥1, 𝑥2) is indifferent to (𝑦1, 𝑦2) if and only if it provides the same utility than (𝑦1, 𝑦2);

3. (𝑥1, 𝑥2) ≽ (𝑦1, 𝑦2) ⟺ 𝑢(𝑥1, 𝑥2) ≥ 𝑢(𝑦1, 𝑦2) A basket (𝑥1, 𝑥2) is weakly preferred to (𝑦1, 𝑦2) if and only if it provides a at least the same utility than

(𝑦1, 𝑦2).

Utility has an ordinal meaning: in other words, it allows ranking the baskets basing on the satisfaction they

provide to the consumer. As all the baskets on the same indifference curve provide the same level of

satisfaction, they have the same value of utility. In a similar way, as a basket on a higher curve provides a

higher satisfaction, it will also have a higher value of utility. Therefore, the utility function assigns a value to

each indifference curve of the map, so that at a higher curve corresponds a higher value of utility (Figure 10).

Figure 10 – An indifference map and its values of utility.

There can be many ways to assign utility values to the same set of baskets and – therefore – to indifference

curves: it is not important which way we choose to assign values, provided that we respect the ranking among

baskets. Therefore, if u(x1,x2) is a utility function describing consumer’s preferences, every v(u(x1,x2)) – with

v monotonically increasing function of u – will be a utility function for the same consumer’s preferences. The

utility function is not unique.

Marginal Utility (MU) Let us consider a basket (x1,x2) with its value of utility u(x1,x2) and let us suppose that we change the level

of a good – say good 1, but the same holds for good 2 – of a x1 without changing the other good.

Consequently we move from basket (x1,x2) to basket (x1+x1,x2). As good 1 has changed and good 2 has

not, utility changes too – we are not on the same indifference curve – up to u(x1+x1,x2) (Table 1).

Basket Utility

(x1,x2) u(x1,x2) (x1+x1,x2) u(x1+x1,x2)

Table 1 – Changing good 1 in the basket: baskets and their utility.

x2

x1

u1 u2>u1

u3>u2


The variation u1 in utility due to the variation of good 1 in the basket is

∆𝑢1 = 𝑢(𝑥1 + ∆𝑥1, 𝑥2) − 𝑢(𝑥1, 𝑥2)

We cannot know if u1 is big or small (in absolute values) unless we compare it with the variation in good 1

that has caused the variation in utility. In this way, we obtain a relative variation in utility that is

∆𝑢1∆𝑥1

=𝑢(𝑥1 + ∆𝑥1, 𝑥2) − 𝑢(𝑥1, 𝑥2)

∆𝑥1

As u() is a function of x1, ∆𝑢1

∆𝑥1 represents the incremental ratio of the utility function with respect to good 1.

If we consider a very small variation in good 1 – so x1 approaches 0 – we obtain the first partial derivative

of utility with respect to good 1. This is called marginal utility of good 1 (MU1) and it represents the variation

in utility that comes from a variation in good 1, when good 2 does not change.

𝑀𝑈1 = lim∆𝑥1→0

∆𝑢1∆𝑥1

= lim∆𝑥1→0

𝑢(𝑥1 + ∆𝑥1, 𝑥2) − 𝑢(𝑥1, 𝑥2)

∆𝑥1=𝜕𝑢(𝑥1, 𝑥2)

𝜕𝑥1

If the good 1 in the basket increases, also utility will increase: therefore, the marginal utility of good 1 will be

positive. In the same way, if the good 1 in the basket decreases, also utility will decrease: therefore, the

marginal utility of good 1 will be positive once again.

The same reasoning can be made with respect to good 2. Let us consider a basket (x1,x2) with its value of

utility u(x1,x2) and let us suppose that we change the level of good 2 of a x2 without changing the level of

good 1. Consequently we move from basket (x1,x2) to basket (x1,x2+x2). As good 2 has changed and good

1 has not, utility changes too – we are not on the same indifference curve – up to u(x1,x2+x2) (Table 2).

Basket Utility

(x1,x2) u(x1,x2) (x1,x2+x2) u(x1,x2+x2)

Table 2 – Changing good 2 in the basket: baskets and their utility.

The variation u2 in utility due to the variation of good 2 in the basket is

∆𝑢2 = 𝑢(𝑥1, 𝑥2 + ∆𝑥2) − 𝑢(𝑥1, 𝑥2)

whereas the relative variation in utility is

∆𝑢2∆𝑥2

=𝑢(𝑥1, 𝑥2 + ∆𝑥2) − 𝑢(𝑥1, 𝑥2)

∆𝑥2

As u() is a function of x2, ∆𝑢2

∆𝑥2 represents the incremental ratio of the utility function with respect to good 2.

If we consider a very small variation in good 2 – so x2 approaches 0 – we obtain the first partial derivative

of utility with respect to good 2. This is called marginal utility of good 2 (MU2) and it represents the variation

in utility that comes from a variation in good 2, when good 1 does not change.


∆𝑢2∆𝑥2

= lim∆𝑥2→0

𝑢(𝑥1, 𝑥2 + ∆𝑥2) − 𝑢(𝑥1, 𝑥2)

∆𝑥2=𝜕𝑢(𝑥1, 𝑥2)

𝜕𝑥2


If the good 2 in the basket increases, also utility will increase: therefore, the marginal utility of good 2 will be

positive. In the same way, if the good 2 in the basket decreases, also utility will decrease: therefore, the

marginal utility of good 2 will be positive once again.

Summing up:

the marginal utility of a good represents the variation in utility coming from an infinitesimal

variation in the level of that good, when the level of the other good does not change. The

marginal utility of a good is always positive.

Marginal Utilities and the Marginal Rate of Substitution The marginal rate of substitution between two goods can be expressed in terms of their marginal utilities.

Let us consider two indifferent baskets, A=(x1,x2) and B=(x1+x1,x2+x2). As A and B are indifferent, they

lie on the same indifference curve and they provide the same utility to the consumer. Moving from A to B,

the quantities of both goods in the basket change. The consumer can move from A to B in two ways: onto

the segment that links the two points; or splitting the passage in two stages – from A to C, and from C to B.

If the consumer moves onto the segment, he/she will move with an average slope that corresponds to the

MRS1,2 of the indifference curve

𝑀𝑅𝑆1,2 =∆𝑥2∆𝑥1

If the consumer splits the passage in two stages, he/she will have to combine the changes in his/her utility

coming from the two stages to obtain the total effect.

It is necessary to note that the basket C lies on a higher indifference curve than A and B – and therefore it

has a higher utility (Figure 11). Moreover, it contains the same quantity of good 1 as B, and the same quantity

of good 2 as A. Therefore, studying the two stages separately, we will be able to vary only one good at a time,

leaving unchanged the other one. In this way, for each stage we will obtain a change in utility due to the

variation of only one good and – as a consequence – linked to the marginal utility of that good.

Figure 11 – Passing from A to B moving onto the segment or splitting the movement in two stages.

As the total effect coming from the movement onto the segment and from the two-stage movement has to

be the same, at the end of the reasoning we will be able to link the marginal rate of substitution to the

marginal utilities of the goods.

x2

A x2

B x2+x2

C

x1 x1+x1 x1


Now, let us analyse the two-stage passage.

1. First stage – from A=(x1,x2) to C=(x1+x1,x2)

In this stage only the good 1 in the basket changes: in particular, it increases. As the basket C contains

the same quantity of the good 2 as A, and a higher quantity of good 1 it provides a higher utility to the

consumer. As a consequence, the consumer has a total variation u1 in his/her utility given by

∆𝑢1 = 𝑢(𝑥1 + ∆𝑥1, 𝑥2) − 𝑢(𝑥1, 𝑥2)

and a relative variation ∆𝑢1

∆𝑥1 that is

∆𝑢1∆𝑥1

=𝑢(𝑥1 + ∆𝑥1, 𝑥2) − 𝑢(𝑥1, 𝑥2)

∆𝑥1

As we have already seen, when x1 approaches 0 the previous formula can be interpreted as a partial

derivative, and, from an economic point of view, as the marginal utility of the first good in the basket

MU1


∆𝑢1∆𝑥1

= lim∆𝑥1→0

𝑢(𝑥1 + ∆𝑥1, 𝑥2) − 𝑢(𝑥1, 𝑥2)

∆𝑥1=𝜕𝑢(𝑥1, 𝑥2)

𝜕𝑥1

Therefore, for each infinitesimal increase 𝜕x1 in the basket, consumer’s utility increases of MU1. As the

total variation of x1 in the basket is x1, the correspondent total variation in the utility is u1 equal to

∆𝑢1 = 𝑀𝑈1∆𝑥1 > 0

2. Second stage – from C=(x1+x1,x2) to B=(x1+x1,x2+x2)

In this stage only the good 2 in the basket changes: in particular, it decreases. As the basket B contains

the same quantity of the good 1 as C, and a lower quantity of good 2 it provides a lower utility to the

consumer. As a consequence, the consumer has a total variation u2 in his/her utility given by

∆𝑢2 = 𝑢(𝑥1 + ∆𝑥1, 𝑥2 + ∆𝑥2) − 𝑢(𝑥1 + ∆𝑥1, 𝑥2)

and a relative variation ∆𝑢2

∆𝑥2 that is

∆𝑢2∆𝑥2

=𝑢(𝑥1 + ∆𝑥1, 𝑥2 + ∆𝑥2) − 𝑢(𝑥1 + ∆𝑥1, 𝑥2)

∆𝑥2

As we have already seen, when x2 approaches 0 the previous formula can be interpreted as a partial

derivative, and, from an economic point of view, as the marginal utility of the second good in the basket

MU2


∆𝑢2∆𝑥2

= lim∆𝑥2→0

𝑢(𝑥1, 𝑥2 + ∆𝑥2) − 𝑢(𝑥1, 𝑥2)

∆𝑥2=𝜕𝑢(𝑥1, 𝑥2)

𝜕𝑥2

Therefore, for each infinitesimal decrease 𝜕x2 in the basket, consumer’s utility decreases of MU2. As the

total variation of x2 in the basket is x2, the correspondent total variation in the utility is u2 equal to

∆𝑢2 = 𝑀𝑈2∆𝑥2 < 0


Combining the two stages, the consumer passes from basket A to basket B, remaining onto the same

indifference curve: therefore, the total variation in the utility u has to be null (the utility does not vary). This

variation can be expressed as the sum of the variations of the two stages

∆𝑢 = ∆𝑢1 + ∆𝑢2 = 𝑀𝑈1∆𝑥1 +𝑀𝑈2∆𝑥2 = 0

Rearranging the previous equation we obtain

∆𝑥2∆𝑥1

= −𝑀𝑈1𝑀𝑈2

Therefore, we have

𝑀𝑅𝑆1,2 =∆𝑥2∆𝑥1

= −𝑀𝑈1𝑀𝑈2

As marginal utilities are always positive, the marginal rate of substitution is always negative, as we were

expecting.


The budget line The indifference map and the utility function represent what a consumer would like to have, but this does

not necessarily corresponds with what the consumer can have. In particular, the baskets a consumer can

access depend on the prices of the goods and on consumer’s budget.

Let us hypothesise that the first good has a price p1 and the second good has a price p2. Let us also hypothesise

that the consumer has a budget m that he/she can spend to buy the two goods.

We can define budget set the set of all the baskets (x1,x2) that consumer can access with a budget equal to

m and prices p1 and p2, respectively.

If the consumer buys x1 units of the first good at a price p1 he/she will pay p1x1. In the same way, if he/she

buys x2 units of the second good at a price p2 he/she will pay p2x2. The consumer can decide to buy all the

quantities of the two goods that make him/her pay no more than his/her income m. Therefore, the budget

set is made up by all the baskets (x1,x2) for which the following relationship holds

𝑝1𝑥1 + 𝑝2𝑥2 ≤ 𝑚

The line corresponding to the baskets that run out all the income is called budget line

𝑝1𝑥1 + 𝑝2𝑥2 = 𝑚

The budget set and the budget line can be represented on a Cartesian plane (Figure 12), with the two goods

on the axes. Clearly, we will consider only the first quarter of the plane, as it is a non-sense considering

negative values for quantities. In this plane, the budget line is a downward sloping line.

Figure 12 – Budget line and budget set for a consumer with budget equal to m when prices for goods are p1 and p2.

If the consumer wants to buy only the first good, the maximum quantity he/she can buy can be obtained by

the budget line, forcing x2 to zero

𝑝1𝑥1 + 𝑝2𝑥2 = 𝑚

𝑝1𝑥1 + 𝑝20 = 𝑚

𝑝1𝑥1 = 𝑚

𝑥1 =𝑚

𝑝1

x2

Budget set

x1

𝑥2 =𝑚

𝑝2

𝑥1 =𝑚

𝑝1


In the same way, if the consumer wants to buy only the second good, the maximum quantity he/she can buy

can be obtained by the budget line, forcing x1 to zero

𝑝1𝑥1 + 𝑝2𝑥2 = 𝑚

𝑝10 + 𝑝2𝑥2 = 𝑚

𝑝2𝑥2 = 𝑚

𝑥2 =𝑚

𝑝2

These two values represent the horizontal and the vertical intercept of the budget line with the Cartesian

axes, respectively (Figure 12).

The slope of the budget line can be obtained simply by expressing x2 in terms of x1. We obtain

𝑝1𝑥1 + 𝑝2𝑥2 = 𝑚

𝑥2 =𝑚

𝑝2−𝑝1𝑝2

𝑥1

Therefore, the slope of the budget line is −𝑝1

𝑝2 and it is negative.

Possible movements of the budget line Let us consider an initial situation in which the consumer has an income m and the prices of the goods are

respectively p1 and p2. The initial budget line will have equation

𝑝1𝑥1 + 𝑝2𝑥2 = 𝑚

Variations in the income level Let us suppose that the income passes from m to m’>m. As the prices for goods have remained the same,

the slope of the budget line has not changed: therefore, the final budget line has to be parallel to the initial

one.

Nonetheless, having changed the value of income the horizontal and vertical intercepts have to be different

from the initial ones. In particular

Horizontal intercept

{𝑝1𝑥1 + 𝑝2𝑥2 = 𝑚′𝑥2 = 0

{𝑝1𝑥1 = 𝑚′𝑥2 = 0

{𝑥1 =

𝑚′

𝑝1𝑥2 = 0

Vertical intercept

{𝑝1𝑥1 + 𝑝2𝑥2 = 𝑚′𝑥1 = 0

{𝑝2𝑥2 = 𝑚′𝑥1 = 0

{𝑥2 =

𝑚′

𝑝2𝑥1 = 0


As m’>m in the final situation both intercepts have a higher value than in the initial one. Therefore, the

budget line has moved up- and right-ward, with the same slope.

In a similar way, if the income decreases passing from m to m’’<m, the budget line moves down- and left-

ward, with the same slope (Figure 13).

Figure 13 – Movements of the budget line due to variations in the income level.

Summing up:

When the income level changes and the ratio between prices remains the same, the budget line

shifts. If the income level decreases the budget line will shift towards axes; if the income level

increases the budget line will shift upward and rightward.

Variations in price levels Let us suppose that p1 increases up to p1’>p1, whilst p2 and m remain the same. Consequently, the budget

line equation passes from

𝑝1𝑥1 + 𝑝2𝑥2 = 𝑚

to

𝑝1′𝑥1 + 𝑝2𝑥2 = 𝑚

Figure 14 – Movements of the budget line due to variations in p1 levels.

x2

𝑚′𝑝2

𝑚𝑝2

𝑚

𝑝1

𝑚′′𝑝2

x1 𝑚′′

𝑝1

𝑚′

𝑝1

m>0

m<0

x2

𝑚𝑝2

𝑚

𝑝1 x1

𝑚

𝑝1′ 𝑚′

𝑝1′′

𝑝1 < 0

𝑝1 > 0


As in the final situation p1’>p1 the slope changes, passing from −𝑝1

𝑝2 to −

𝑝1′

𝑝2. In this way, also the horizontal

intercept changes, whilst the vertical one remains the same. In particular, it will have a lower value.


{𝑝1′𝑥1 + 𝑝2𝑥2 = 𝑚

𝑥2 = 0

{𝑝1′𝑥1 = 𝑚

𝑥2 = 0

{𝑥1 =

𝑚

𝑝1′

𝑥2 = 0

Vertical intercept

{𝑝1′𝑥1 + 𝑝2𝑥2 = 𝑚

𝑥1 = 0

{𝑝2𝑥2 = 𝑚𝑥1 = 0

{𝑥2 =

𝑚

𝑝2𝑥1 = 0

Similarly, let us suppose that p1 decreases down to p1’’<p1, whilst p2 and m remain the same. Consequently,

the budget line equation passes from

𝑝1𝑥1 + 𝑝2𝑥2 = 𝑚

to

𝑝1′′𝑥1 + 𝑝2𝑥2 = 𝑚

As in the final situation p1’’<p1 the slope changes, passing from −𝑝1

𝑝2 to −

𝑝1′′

𝑝2. In this way, also the horizontal

intercept changes, whilst the vertical one remains the same. In particular, it will have a higher value.


{𝑝1′′𝑥1 + 𝑝2𝑥2 = 𝑚𝑥2 = 0

{𝑝1′′𝑥1 = 𝑚

𝑥2 = 0

{𝑥1 =

𝑚

𝑝1′′

𝑥2 = 0

Vertical intercept

{𝑝1′′𝑥1 + 𝑝2𝑥2 = 𝑚𝑥1 = 0

{𝑝2𝑥2 = 𝑚𝑥1 = 0

{𝑥2 =

𝑚

𝑝2𝑥1 = 0

The corresponding movements of the budget line are represented in Figure 14.

If variations occure in price p2 levels, whilst p1 and m remain the same we will have an analogous result and

corresponding movements of the budget line will be represented in Figure 15.


Figure 15 – Movements of the budget line due to variations in p2 levels.

Summing up:

When all parameters remains the same and the price of one good changes the intercept on the

corresponding axis will change too. If the price increases the intercept moves towards the origin

of the axes, whilst if the price decreases it will move away from the origin.

x2

𝑚𝑝2′′

𝑚𝑝2

𝑚

𝑝1

𝑚𝑝2′

x1

𝑝2′′ < 0

𝑝2′ > 0


Consumer’s optimal choice The budget line allow a consumer to separate affordable baskets from non-affordable ones, given prices and

the income. Now, we have to try to identify which is the best basket to choose for consumption among all

affordable baskets.

To this end, we consider consumer’s indifference map and his/her budget line. Among all baskets in the

budget set, the consumer will choose the one providing the highest utility, i.e. the one lying on the highest

indifference curve that can be reached given the budget line. Therefore, the choice will be the basket in the

tangency point between one of his/her indifference curves and the budget line: each other basket in the

budget set provides, indeed, a lower level of utility, lying on a lower indifference curve.

Figure 16 – Optimal basket choice.

In Figure 16, the basket A is affordable for the consumer, providing him/her a utility u1. Utility can still be

improved, moving on a higher indifference curve. Nonetheless, if we choose basket B – providing a utility

u3>u1 – we will not be able to afford such an expense, because our budget is limited: so basket B will not be

affordable. The optimal choice, therefore, is to choose basket C, providing utility u2, with u1<u2<u3.

The optimal basket C lies both on the budget line and on one of the indifference curves and in C the

indifference curve and the budget line must have the same slope, being C a tangency point.

The general equations for slopes of an indifference curve and of the budget line are:

Slope of an indifference curve

Δ𝑥2Δ𝑥1

= 𝑀𝑅𝑆1,2 = −𝑀𝑈1𝑀𝑈2

Slope of the budget line

Δ𝑥2Δ𝑥1

= −𝑝1𝑝2

The slope of a general indifference curve changes its value from point to point, whilst the slope of the budget

line remains always constant. Therefore, generally the two slopes will have different values, safe for the

tangency point. So, equalling the two slopes is a condition to determine the tangency point – and to find out

the optimal basket – and having a tangency point – an optimal basket – means that the slopes are the same.

We have a necessary and sufficient condition to find consumer’s optimal basket.

Δ𝑥2Δ𝑥1

= 𝑀𝑅𝑆1,2 = −𝑀𝑈1𝑀𝑈2

= −𝑝1𝑝2

𝑀𝑈1𝑀𝑈2

=𝑝1𝑝2

x2

x1

u3>u2

u2>u1 u1

x1*

x2*

A

B

C


Therefore, the two coordinates of the optimal basket C must satisfy this condition. Moreover, as the optimal

basket belongs to the budget line, they will also have to satisfy its equation. Therefore, we can find both

coordinates of C solving the following system of equations

{

𝑀𝑈1𝑀𝑈2

=𝑝1𝑝2

𝑝1𝑥1 + 𝑝2𝑥2 = 𝑚

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