Post on 11-May-2022
Elisa Battistoni Lecture note 01: Consumerβs choice
Consumer preferences In analysing consumerβs 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 consumerβs 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 β good 1 and good 2. The two baskets are (π₯1, π₯2)
and (π¦1, π¦2).
We have:
1. (π₯1, π₯2) β» (π¦1, π¦2) βΉ (π₯1, π₯2) is strictly preferred to (π¦1, π¦2). Anytime the consumer has the possibility
to choose between these two baskets, he/she will definitely choose (π₯1, π₯2);
2. (π₯1, π₯2) βΌ (π¦1, π¦2) βΉ (π₯1, π₯2) is indifferent to (π¦1, π¦2). The consumer gains the same satisfaction from
both baskets, so when he/her has the possibility to choose, sometimes he/she will choose (π₯1, π₯2) and
some other times he/she will choose (π¦1, π¦2);
3. (π₯1, π₯2) β½ (π¦1, π¦2) βΉ (π₯1, π₯2) is weakly preferred to (π¦1, π¦2). In other words, consumer is at least as
much satisfied by (π₯1, π₯2) as by (π¦1, π¦2).
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 β Representing baskets on a Cartesian plane.
x2
x1
A
x1A
x2A
Elisa Battistoni Lecture note 01: Consumerβs choice
Preference relationship properties Given three baskets (π₯1, π₯2), (π¦1, π¦2), and (π§1, π§2), preference relationships have three main properties:
1. Transitivity property
If (π₯1, π₯2) β» (π¦1, π¦2) and (π¦1, π¦2) β» (π§1, π§2) then
(π₯1, π₯2) β» (π§1, π§2)
2. Reflexive property
(π₯1, π₯2) β½ (π¦1, π¦2)
A basket is at least as desirable as itself (or an identical basket)
3. Completeness property
Given two baskets (π₯1, π₯2) and (π¦1, π¦2), then
(π₯1, π₯2) β½ (π¦1, π¦2) or (π¦1, π¦2) β½ (π₯1, π₯2) or (π₯1, π₯2) βΌ (π¦1, π¦2)
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 (π₯1, π₯2) 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 π΄ = (π₯1, π₯2) the area of the Cartesian plane in which
we can seek to find all the baskets strictly preferred to A is represented with a β+β, whereas the area in which
we can look for all the baskets which A is strictly preferred to is represented by a ββ.
Elisa Battistoni Lecture note 01: Consumerβs choice
Figure 2 β 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 β 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 π΄ = (π₯1, π₯2) and π΅ = (π¦1, π¦2) lying on the same indifference curve β so, they
provide the same satisfaction β and their weighted average C
πΆ = (π‘π₯1 + (1 β π‘)π¦1, π‘π₯2 + (1 β π‘)π¦2) with π‘ β [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 β Figure 5(b) and Figure 5(c) β basket C
provides respectively the same satisfaction than A and B β lying on the indifference curve to which they
belong β and a lower satisfaction than A and B β 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
Elisa Battistoni Lecture note 01: Consumerβs choice
Figure 4 β Basket C is an intermediate basket between end baskets A and B.
(a) (b) (c)
Figure 5 β 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 β 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
Elisa Battistoni Lecture note 01: Consumerβs choice
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 β A, B, and C β 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 β 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: π΄~πΆ
As C and B belong to the same indifference curve, they are indifferent: πΆ~π΅
Consequently, for the transitivity property it has to be also π΄~π΅. Nonetheless, this two baskets lie on
different curves, corresponding to different levels of satisfaction: in particular, in our hypothesis π΄ β» π΅.
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 β An indifference map for regular preferences.
x2
x1
A
B
C
x2
x1
Elisa Battistoni Lecture note 01: Consumerβs choice
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: βIf 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)?β.
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 β the quantity of one good in the basket has to increment, whereas the quantity of the other
good diminishes β the MRS has always a negative sign.
ππ π1,2 =βπ₯2βπ₯1
< 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, βπ₯2
βπ₯1 is the incremental ratio for indifference curves of two-good baskets and β as a consequence β
it represents their slope. The sign of the MRS is consistent with the fact that β under regular preferences β
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 β as it is the case for regular
preferences β 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 β and, as a consequence, few units of x2 (basket B) β 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 β Variations of the MRS along a convex indifference curve.
x1
x2
A
x1
B
x2
x2
x1
Elisa Battistoni Lecture note 01: Consumerβs choice
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.
ππ π1,2 = limβπ₯1β0
βπ₯2βπ₯1
=ππ₯2ππ₯1
< 0
The MRS represents, in this formulation, the instant slope of the indifference curve and it is negative, being
the curve downward sloping.
Elisa Battistoni Lecture note 01: Consumerβs choice
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 consumerβs preference relationship.
Therefore, given two baskets (π₯1, π₯2) and (π¦1, π¦2) we have:
1. (π₯1, π₯2) β» (π¦1, π¦2) βΊ π’(π₯1, π₯2) > π’(π¦1, π¦2) A basket (π₯1, π₯2) is strictly preferred to (π¦1, π¦2) if and only if it provides a strictly higher utility than
(π¦1, π¦2);
2. (π₯1, π₯2) βΌ (π¦1, π¦2) βΊ π’(π₯1, π₯2) = π’(π¦1, π¦2) A basket (π₯1, π₯2) is indifferent to (π¦1, π¦2) if and only if it provides the same utility than (π¦1, π¦2);
3. (π₯1, π₯2) β½ (π¦1, π¦2) βΊ π’(π₯1, π₯2) β₯ π’(π¦1, π¦2) A basket (π₯1, π₯2) is weakly preferred to (π¦1, π¦2) if and only if it provides a at least the same utility than
(π¦1, π¦2).
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 β An indifference map and its values of utility.
There can be many ways to assign utility values to the same set of baskets and β therefore β 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 consumerβs preferences, every v(u(x1,x2)) β with
v monotonically increasing function of u β will be a utility function for the same consumerβs 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 β say good 1, but the same holds for good 2 β 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 β we are not on the same indifference curve β up to u(x1+x1,x2) (Table 1).
Basket Utility
(x1,x2) u(x1,x2) (x1+x1,x2) u(x1+x1,x2)
Table 1 β Changing good 1 in the basket: baskets and their utility.
x2
x1
u1 u2>u1
u3>u2
Elisa Battistoni Lecture note 01: Consumerβs choice
The variation u1 in utility due to the variation of good 1 in the basket is
βπ’1 = π’(π₯1 + βπ₯1, π₯2) β π’(π₯1, π₯2)
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
βπ’1βπ₯1
=π’(π₯1 + βπ₯1, π₯2) β π’(π₯1, π₯2)
βπ₯1
As u() is a function of x1, βπ’1
βπ₯1 represents the incremental ratio of the utility function with respect to good 1.
If we consider a very small variation in good 1 β so x1 approaches 0 β 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.
ππ1 = limβπ₯1β0
βπ’1βπ₯1
= limβπ₯1β0
π’(π₯1 + βπ₯1, π₯2) β π’(π₯1, π₯2)
βπ₯1=ππ’(π₯1, π₯2)
ππ₯1
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 β we are not on the same indifference curve β up to u(x1,x2+x2) (Table 2).
Basket Utility
(x1,x2) u(x1,x2) (x1,x2+x2) u(x1,x2+x2)
Table 2 β 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
βπ’2 = π’(π₯1, π₯2 + βπ₯2) β π’(π₯1, π₯2)
whereas the relative variation in utility is
βπ’2βπ₯2
=π’(π₯1, π₯2 + βπ₯2) β π’(π₯1, π₯2)
βπ₯2
As u() is a function of x2, βπ’2
βπ₯2 represents the incremental ratio of the utility function with respect to good 2.
If we consider a very small variation in good 2 β so x2 approaches 0 β 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.
ππ2 = limβπ₯2β0
βπ’2βπ₯2
= limβπ₯2β0
π’(π₯1, π₯2 + βπ₯2) β π’(π₯1, π₯2)
βπ₯2=ππ’(π₯1, π₯2)
ππ₯2
Elisa Battistoni Lecture note 01: Consumerβs choice
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 β 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
ππ π1,2 =βπ₯2βπ₯1
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 β 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 β as a consequence β linked to the marginal utility of that good.
Figure 11 β 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
Elisa Battistoni Lecture note 01: Consumerβs choice
Now, let us analyse the two-stage passage.
1. First stage β 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
βπ’1 = π’(π₯1 + βπ₯1, π₯2) β π’(π₯1, π₯2)
and a relative variation βπ’1
βπ₯1 that is
βπ’1βπ₯1
=π’(π₯1 + βπ₯1, π₯2) β π’(π₯1, π₯2)
βπ₯1
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
ππ1 = limβπ₯1β0
βπ’1βπ₯1
= limβπ₯1β0
π’(π₯1 + βπ₯1, π₯2) β π’(π₯1, π₯2)
βπ₯1=ππ’(π₯1, π₯2)
ππ₯1
Therefore, for each infinitesimal increase πx1 in the basket, consumerβs 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
βπ’1 = ππ1βπ₯1 > 0
2. Second stage β 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
βπ’2 = π’(π₯1 + βπ₯1, π₯2 + βπ₯2) β π’(π₯1 + βπ₯1, π₯2)
and a relative variation βπ’2
βπ₯2 that is
βπ’2βπ₯2
=π’(π₯1 + βπ₯1, π₯2 + βπ₯2) β π’(π₯1 + βπ₯1, π₯2)
βπ₯2
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
ππ2 = limβπ₯2β0
βπ’2βπ₯2
= limβπ₯2β0
π’(π₯1, π₯2 + βπ₯2) β π’(π₯1, π₯2)
βπ₯2=ππ’(π₯1, π₯2)
ππ₯2
Therefore, for each infinitesimal decrease πx2 in the basket, consumerβs 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
βπ’2 = ππ2βπ₯2 < 0
Elisa Battistoni Lecture note 01: Consumerβs choice
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
βπ’ = βπ’1 + βπ’2 = ππ1βπ₯1 +ππ2βπ₯2 = 0
Rearranging the previous equation we obtain
βπ₯2βπ₯1
= βππ1ππ2
Therefore, we have
ππ π1,2 =βπ₯2βπ₯1
= βππ1ππ2
As marginal utilities are always positive, the marginal rate of substitution is always negative, as we were
expecting.
Elisa Battistoni Lecture note 01: Consumerβs choice
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 consumerβs 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
π1π₯1 + π2π₯2 β€ π
The line corresponding to the baskets that run out all the income is called budget line
π1π₯1 + π2π₯2 = π
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 β 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
π1π₯1 + π2π₯2 = π
π1π₯1 + π20 = π
π1π₯1 = π
π₯1 =π
π1
x2
Budget set
x1
π₯2 =π
π2
π₯1 =π
π1
Elisa Battistoni Lecture note 01: Consumerβs choice
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
π1π₯1 + π2π₯2 = π
π10 + π2π₯2 = π
π2π₯2 = π
π₯2 =π
π2
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
π1π₯1 + π2π₯2 = π
π₯2 =π
π2βπ1π2
π₯1
Therefore, the slope of the budget line is βπ1
π2 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
π1π₯1 + π2π₯2 = π
Variations in the income level Let us suppose that the income passes from m to mβ>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
{π1π₯1 + π2π₯2 = πβ²π₯2 = 0
{π1π₯1 = πβ²π₯2 = 0
{π₯1 =
πβ²
π1π₯2 = 0
Vertical intercept
{π1π₯1 + π2π₯2 = πβ²π₯1 = 0
{π2π₯2 = πβ²π₯1 = 0
{π₯2 =
πβ²
π2π₯1 = 0
Elisa Battistoni Lecture note 01: Consumerβs choice
As mβ>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 mββ<m, the budget line moves down- and left-
ward, with the same slope (Figure 13).
Figure 13 β 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 p1β>p1, whilst p2 and m remain the same. Consequently, the budget
line equation passes from
π1π₯1 + π2π₯2 = π
to
π1β²π₯1 + π2π₯2 = π
Figure 14 β Movements of the budget line due to variations in p1 levels.
x2
πβ²π2
ππ2
π
π1
πβ²β²π2
x1 πβ²β²
π1
πβ²
π1
m>0
m<0
x2
ππ2
π
π1 x1
π
π1β² πβ²
π1β²β²
π1 < 0
π1 > 0
Elisa Battistoni Lecture note 01: Consumerβs choice
As in the final situation p1β>p1 the slope changes, passing from βπ1
π2 to β
π1β²
π2. In this way, also the horizontal
intercept changes, whilst the vertical one remains the same. In particular, it will have a lower value.
Horizontal intercept
{π1β²π₯1 + π2π₯2 = π
π₯2 = 0
{π1β²π₯1 = π
π₯2 = 0
{π₯1 =
π
π1β²
π₯2 = 0
Vertical intercept
{π1β²π₯1 + π2π₯2 = π
π₯1 = 0
{π2π₯2 = ππ₯1 = 0
{π₯2 =
π
π2π₯1 = 0
Similarly, let us suppose that p1 decreases down to p1ββ<p1, whilst p2 and m remain the same. Consequently,
the budget line equation passes from
π1π₯1 + π2π₯2 = π
to
π1β²β²π₯1 + π2π₯2 = π
As in the final situation p1ββ<p1 the slope changes, passing from βπ1
π2 to β
π1β²β²
π2. In this way, also the horizontal
intercept changes, whilst the vertical one remains the same. In particular, it will have a higher value.
Horizontal intercept
{π1β²β²π₯1 + π2π₯2 = ππ₯2 = 0
{π1β²β²π₯1 = π
π₯2 = 0
{π₯1 =
π
π1β²β²
π₯2 = 0
Vertical intercept
{π1β²β²π₯1 + π2π₯2 = ππ₯1 = 0
{π2π₯2 = ππ₯1 = 0
{π₯2 =
π
π2π₯1 = 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.
Elisa Battistoni Lecture note 01: Consumerβs choice
Figure 15 β 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
ππ2β²β²
ππ2
π
π1
ππ2β²
x1
π2β²β² < 0
π2β² > 0
Elisa Battistoni Lecture note 01: Consumerβs choice
Consumerβs 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 consumerβs 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 β 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 β providing a utility
u3>u1 β 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
Ξπ₯2Ξπ₯1
= ππ π1,2 = βππ1ππ2
Slope of the budget line
Ξπ₯2Ξπ₯1
= βπ1π2
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 β and to find out
the optimal basket β and having a tangency point β an optimal basket β means that the slopes are the same.
We have a necessary and sufficient condition to find consumerβs optimal basket.
Ξπ₯2Ξπ₯1
= ππ π1,2 = βππ1ππ2
= βπ1π2
ππ1ππ2
=π1π2
x2
x1
u3>u2
u2>u1 u1
x1*
x2*
A
B
C
Elisa Battistoni Lecture note 01: Consumerβs choice
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
{
ππ1ππ2
=π1π2
π1π₯1 + π2π₯2 = π