Constraints, Optimal Choices, and Demand in Microeconomics

Introduction to Microeconomics

MAS
EVA
PAX
NGE
TIBI
LISTA
MAR
MEUS
CE
R
CARI
M
Università
Ca'Foscari
Venezia
ET2020
Microeconomics - 1
Group A - K

Julien Bergeot
jul.bergeot@unive.it
1IN
DOMO FOSC
ART
Università
Ca'Foscari
Venezia

Introduction
. If you ask any student or
professor in San Giobbe
about their favourite ever
meal,
probably
nobody
would say the one in the
cafeteria
· Whereas a cafeteria lunch is
plain, it is also affordable and
there
are
long
queues
around noon
. Consumer choices is all
about constraints and trade-
offs
2

The Budget Constraint

IN
DOMO FOSC
ART
Università
Ca'Foscari
Venezia

Defining the Budget Constraint

The Budget Constraint
· A consumer can afford a bundle of goods when the total cost
she is paying is lower than her income M in a given time unit
· We say that a bundle (X, Y) is affordable at prices Px and Py if
the following budget constraint is satisfied,
PxX + PYY ≤ M
· The left-hand side of this equation is total cost
. When the total cost is equal to M, all income goes into
purchasing these two goods
4

Understanding the Budget Line

IN
DOMO FOSC
ART
Università
Ca'Foscari
Venezia

The Budget Line
· Associated to a budget constraint,
there is a budget line
. All bundles on the budget line
have the same total cost, equal to
available income
PxX + PyY = M -> Y = =_ Xx
PY
Px
PY
. This is the equation of a straight
line in (X, Y), with vertical intercept
equal to ,
M
horizontal intercept
PY
M
equal to -, and slope -
Px
Px
PY
Good Y
M
Py
Budget line
M
Good X
Px
5

Budget Line Example

IN
DOMO FOSCO
ART
Università
Ca'Foscari
Venezia

The Budget Line: Example
Py = 1, Px = 2, M = 80

Bundle	Y	X	Total Exp
A	0	40	80
B	20	30	80
D	40	20	80
E	60	10	80
G	80	0	80

Y
G
80
E
60
D
40
B
20
A
X
0
10
20
30
40

IN
DOMO FOSCO
ART
Università
Ca'Foscari
Venezia

The Budget Line: Example
Py = 1, Px = 2, M = 80

Bundle	Y	X	Total Exp
A	0	40	80
M /PY
E
B	20	30	80
D	40	20	80
E	60	10	80
G	80	0	80

Y
G
80
Budget line
2X + Y = 80 -> Y = 80-2X
60
D
40
B
20
A
X
0
10
20
30
40
M /Px

Economic Interpretation of Intercepts

DOMO FOSCO
ART
Università
Ca'Foscari
Venezia

The Budget Line: Example
· The intercept on the X axis is
M/Px. the intercept on the Y
axis is M / PY
· What
is
the
intercepts'
economic interpretation?
Y
M /Py
G
80
Budget line
2X + Y = 80 -> Y = 80-2X
E
60
D
40
B
20
A
X
0
10
20
30
40

Market Rate of Substitution

DOMO FOSCO
ART
Università
Ca'Foscari
Venezia

The Budget Line: Example
· The intercept on the X axis is
M/Px. the intercept on the Y
axis is M / PY
· What
is
the
intercepts'
economic interpretation?
. On
the
budget line
the
(market) rate of substitution
between goods is constant
and equal to its slope -Px / PY
Y
G
80
Budget line
2X + Y = 80 -> Y = 80-2X
On this budget line, consumer
can substitute goods at
ΔΥ/ΔΧ = - Ρχ/Ργ = - 2
E
60
D
40
B
20
A
X
0
10
20
30
40

Affordable and Unaffordable Bundles

DOMO FOSCO
ART
Università
Ca'Foscari
Venezia

The Budget Line: Example
· The intercept on the X axis is
M/Px. the intercept on the Y
axis is M / PY
· What
is
the
intercepts'
economic interpretation?
. On
the
budget line
the
(market) rate of substitution
between goods is constant
and equal to its slope -Px / Py
· The budget line separates the
bundles that can be afforded
from those that cannot be
afforded
Y
G
80
E
60
unaffordable
bundles
D
40
affordable
bundles
B
20
A
X
0
10
20
30
40

Changes in the Budget Line

Impact of Income on Budget Line

IN
DOMO FOSCO
ART
Università
Ca'Foscari
Venezia

Changes in the Budget Line: Income
· Consider
a
budget
line
corresponding
to
total
income M1, i.e. Y = M1 _ PX x
M1
PY
PY
· If
the
income
of
the
consumer
increases,
the
budget line shifts right
. Note that the slope has not
changed
Good Y
M2
PY
all these bundles
become affordable
M1
PY
BL1
BL2
M1
M2
Good X
Px
Px
11

IN
DOMO FOSCO
ART
Università
Ca'Foscari
Venezia

Changes in the Budget Line: Income
· Consider
a
budget
line
corresponding
income M2
to
total
· If
the
income
of
the
consumer decreases,
budget line shifts left
the
. Note that the slope has not
changed
Good Y
M2
PY
M
PY
BL1
BL2
M1
M2
Good X
Px
Px
12

Impact of Price Ratio on Budget Line

IN
ART
DOMO FOSCE
Università
Ca'Foscari
Venezia

Changes in the Budget Line: Price Ratio
· Consider a budget line with
slope
- Px / PY
and
corresponding to income M1,
Y =™1 _PXX
PY
PY
· If
the
prices
decrease,
keeping the slope constant
(i.e they increase in the
same
proportions),
the
budget line shifts right
. The consumer has the same
income as before, but can
now afford more goods
Good Y
Pý
all these bundles
become affordable
M
PY
BL2
BL1
M1
M1
Good X
Px
13

IN
DOMO FOSCO
ART
Università
Ca'Foscari
Venezia

Changes in the Budget Line: Price Ratio
· Consider a budget line with
slope
- Px / PY
and
corresponding to income M1
· If
the
prices
increase,
keeping the ratio of prices
constant, the budget line
shifts left
. The consumer has the same
income as before, but can
now afford fewer goods
Good Y
M.
PY
1
PV
BL1
BL2
M1
M1
Good X
′
Px
14

Impact of Good X Price on Budget Line

IN
DOMO FOSCO
ART
Università
Ca'Foscari
Venezia

Changes in the Budget Line: Price
· Consider a budget line with
slope
- Px / PY
and
corresponding to income M1
. If the price of good X
decreases, the budget line
pivots right (keeping
the
vertical intercept unchanged)
. The consumer has the same
income as before, but can
now afford more goods
Good Y
M
PY
all these bundles
become affordable
BL2
BL1
M1
M1
Good X
′
Px
15

IN
DOMO FOSCO
ART
Università
Ca'Foscari
Venezia

Changes in the Budget Line: Price
· Consider a budget line with
slope
- Px / PY
and
corresponding to income M1
. If the price of good X
increases, the budget line
pivots
left
(keeping the
vertical intercept unchanged)
. The consumer has the same
income as before, but can
now afford fewer goods
Good Y
M
PY
BL1
BL2
M1
M1
Good X
′
Px
16

Impact of Good Y Price on Budget Line

IN
DOMO FOSCO
ART
Università
Ca'Foscari
Venezia

Changes in the Budget Line: Price
· Consider a budget line with
slope
- Px / PY
and
corresponding to income M1
. If the price of good Y
decreases, the budget line
pivots up
(keeping the
horizontal
intercept
unchanged)
. The consumer has the same
income as before, but can
now afford more goods
Good Y
M1
PV
M
all these bundles
become affordable
Py
BL2
BL1
M1
Good X
Px
17

IN
DOMO FOSCO
ART
Università
Ca'Foscari
Venezia

Changes in the Budget Line: Price
· Consider a budget line with
slope
- Px / PY
and
corresponding to income M1
. If the price of good Y
increases, the budget line
pivots down
(keeping the
horizontal
intercept
unchanged)
. The consumer has the same
income as before, but can
now afford fewer goods
Good Y
M
Py
M
PV
BL1
BL2
M1
Good X
Px
18

Summary of Income and Price Variations

IN
DOMO FOSC
ART
Università
Ca'Foscari
Venezia

Income and Prices Variations
· Changes in a consumer income modify the intercepts but not
the slope of the budget line
o An income increase shifts the budget line outwards
o An income decrease shifts the budget line inwards
· Changes in the price of a good modify the slope of the budget
line without changing the intercept of the good whose price has
not changed
o The budget line rotates outwards if the price of one good decreases
(more bundles can be afforded)
o The budget line rotates inwards if the price of one good increases (fewer
bundles can be afforded)
· If income and prices have the same relative variation (double,
half, .... ) the budget line stays the same
19

Budget Line Exercise

IN
DOMO FOSCA
ART
Università
Ca'Foscari
Venezia

Exercise: Budget Line
· For dinner, Mike consumes cheese and potatoes. The price of
cheese is 1.0 euro for a hg. The price of potatoes is 0.5 euro
for a hg. Knowing that Mike can spend 5 euro, define the set of
affordable bundles
. How does the set change if Mike can spend twice as much?
What if instead potatoes become more expensive at P=1?
· Assume now that, given that cheese is scarce in Venice, the
mayor has imposed that each citizen can buy only 2 hg of
cheese. Find the set of affordable bundles
20

Consumer's Optimal Choices

IN
DOMO FOSCE
ART
Università
Ca'Foscari
Venezia

Principle of Optimal Choice

Consumer's Optimal Choices
. We now have the two fundamental elements to try to
understand constrained consumption choices: consumer's
preferences and budget constraint
· Principle of optimal choice: among all possible bundles, a
consumer chooses the affordable bundle which, considering his
preference ordering over bundles, has the highest rank (i.e.
gives her the most satisfaction)
· Graphically, this implies
o Choosing a bundle on the budget line, as those below the
line do not satisfy the non satiation property
o On the budget line, choose the bundle on the "left-most"
indifference curve
22

Unaffordable Choices Explained

IN
DOMO FOSC
ART
Università
Ca'Foscari
Venezia

Unaffordable Choices
. Given the income and the
prices, the bundle D cannot
be afforded by this consumer
Good Y
D
M
indifference curve
PY
budget line
0
M
Good X
Px

IN
DOMO FOSC
ART
Università
Ca'Foscari
Venezia

Unaffordable Choices
. Given the income and the
prices, the bundle D cannot
be afforded by this consumer
. All the bundles above the
budget
line
cannot
be
afforded by this consumer,
given the prices Px and Py
and her income M
Good Y
All these bundles
are not affordable
M
PY
0
M
Good X
Px

Affordable Choices and Satisfaction

IN
DOMO FOSC
ART
Università
Ca'Foscari
Venezia

Affordable Choices
. Given the income and the
prices, the bundles A and B
can be afforded by this
consumer
· But bundle B is on a higher
indifference curve, and so it
provides more satisfaction to
the consumer than bundle A
· Is bundle B providing the
most satisfaction possible?
Good Y
M
PY
B
A
0
M
Good X
Px

Sub-Optimal Choices and Trade-offs

IN
DOMO FOSC
ART
Università
Ca'Foscari
Venezia

Sub-Optimal Choices
· Bundle B cannot be optimal,
as there exists affordable
bundles that are preferred
. For example, this consumer
could use the market to
move from B to C, trading
away some good Y (of which
she has in abundance) with
good X (of which she has
little)
· Is bundle C optimal?
Good Y
M
PY
B
C
0
M
Good X
Px

Optimal Choice: Interior Solution

IN
DOMO FOSCO
ARI
Università
Ca'Foscari
Venezia

Optimal Choice: Interior Solution
. The consumer can keep
moving to higher indifference
curves until the indifference
curve
is
tangent
to
the
budget
line
(tangency
condition)
· Bundle E is optimal, as there
is no affordable bundle that
is preferred
· In E, market and consumer
have the same substitution
rate between the two goods
(since the lines have the
same slopes)
Good Y
M
Py
E
0
M
Good X
Px

IN
DOMO FOSCE
ART
Università
Ca'Foscari
Venezia

Optimal Choice: Interior Solution
· Given a set of bundles
{(X, Y)},
the bundle (X*,Y*) is chosen
when
the
consumer's
MRS
XY
computed in (X*, Y*) is
equal to the market substitution
rate
given
by
the
price
ratio
Px/ PY
· MRSxy (X*, Y*) measures how the
consumer wants to substitute X
with Y in (X*, Y*)
· The price ratio Px/Py measures
how the market
allows
to
substitute X with Y
Good Y
slope of the indifference
curve in (X*, Y*)
is
- MRSxy (X*, Y*)
M
Py
slope of the budget line is
- Px /PY
E
∗
-
0
∗
M
Good X
Px