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Lecture 1 - Budget Constraint

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EC270 Lecture Notes
Consumer Budget Constraints, Budget Sets
Choice and
Constraints on Choice
The BUDGET Set…Feasible Choice
Economic Rationality

The principal behavioral postulate is that a
decision maker chooses its most preferred
alternative from those available.

The available choices constitute the choice
set.
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EC270 Lecture Notes
Consumer Budget Constraints, Budget Sets
Consumption Choice Sets

A consumption choice set is the collection
of all consumption choices available to the
consumer.

What constrains consumption choice?

Budgetary, time and other resource limitations.
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Budget Constraints

A consumption bundle containing



x1 units of commodity 1,
x2 units of commodity 2
and so on up to
xn units of commodity n

is denoted by the vector (x1, x2, … , xn).

Commodity prices are p1, p2, … , pn.
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EC270 Lecture Notes
Consumer Budget Constraints, Budget Sets
Budget Constraints

Q: When is a consumption bundle (x1, … , xn)
affordable at prices p1, … , pn?

A: When
p1x1 + … + pnxn  m
where m is the consumer’s (disposable)
income.
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Budget Constraints

The bundles that are only just affordable form
the consumer’s budget constraint.

This is the set
{ (x1,…,xn) | x1  0, …, xn  0 and
p1x1 + … + pnxn = m }

These are the bundles ON, rather than
INSIDE the budget constraint
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EC270 Lecture Notes
Consumer Budget Constraints, Budget Sets
Budget Constraints

The consumer’s budget set is the set of all
affordable bundles;

B(p1, … , pn, m) =
{ (x1, … , xn) | x1  0, … , xn  0 and
p1x1 + … + pnxn  m }

The budget constraint is the upper boundary
of the budget set.
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Budget Set and Constraint for Two
Commodities
x2
m /p2
p2x2 = m
Budget constraint is
p1x1 + p2x2 = m
p1x1 = m
m /p1
x1
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EC270 Lecture Notes
Consumer Budget Constraints, Budget Sets
Budget Set and Constraint for Two
Commodities
x2
Budget constraint is
p1x1 + p2x2 = m.
m /p2
Not affordable
Just affordable
Affordable
m /p1
x1
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Budget Set and Constraint for Two
Commodities
x2
Budget constraint is
p1x1 + p2x2 = m.
m /p2
the collection
of all affordable bundles.
Budget
Set
m /p1
x1
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EC270 Lecture Notes
Consumer Budget Constraints, Budget Sets
Budget Set and Constraint for Two
Commodities
x2
p1x1 + p2x2 = m
m /p2
p2x2 = - p1x1 + m
x2 = - (p1/p2)x1 + m/p2
so slope is -p1/p2
Budget
Set
m /p1
x1
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Budget Constraints
x2
Opp. cost of an extra unit of
commodity 1 is p1/p2 units
foregone of commodity 2.
-p1/p2
+1
x1
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EC270 Lecture Notes
Consumer Budget Constraints, Budget Sets
Budget Constraints
x2
Opp. cost of an extra unit of
commodity 1 is p1/p2 units
foregone of commodity 2.
And
the opp. cost of an extra
+1
unit of commodity 2 is
-p2/p1
p2/p1 units foregone
of commodity 1.
x1
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Budget Constraints

If n = 3 what do the budget constraint and the
budget set look like?
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EC270 Lecture Notes
Consumer Budget Constraints, Budget Sets
Budget Constraint for Three
Commodities
x2
m /p2
p1x1 + p2x2 + p3x3 = m
m /p3
m /p1
x3
x1
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Budget Set for Three Commodities
x2
m /p2
{ (x1,x2,x3) | x1  0, x2  0, x3  0 and
p1x1 + p2x2 + p3x3  m}
m /p3
m /p1
x3
x1
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EC270 Lecture Notes
Consumer Budget Constraints, Budget Sets
Budget Sets & Constraints;
Income and Price Changes

The budget constraint and budget set depend
upon prices and income.

What happens as prices or income change?
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Budget set change: income m increases
x2
Original
budget set
x1
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EC270 Lecture Notes
Consumer Budget Constraints, Budget Sets
Higher income gives more choice
x2
New affordable consumption
choices
Original and
new budget
constraints are
parallel (same
slope).
Original
budget set
x1
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Budget set change: income m decreases
x2
Original
budget set
x1
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EC270 Lecture Notes
Consumer Budget Constraints, Budget Sets
Budget set change: income m decreases
x2
Consumption bundles
that are no longer
affordable.
New, smaller
budget set
Old and new
constraints
are parallel.
x1
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Budget Constraints - Income Changes


Increases in income m shift the constraint
outward in a parallel manner, thereby
enlarging the budget set and improving
choice.
Decreases in income m shift the
constraint inward in a parallel manner,
thereby shrinking the budget set and
reducing choice.
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EC270 Lecture Notes
Consumer Budget Constraints, Budget Sets
Budget Constraints - Income Changes


No original choice is lost and new choices are
added when income increases, so higher
income cannot make a consumer worse off.
An income decrease may (typically will) make
the consumer worse off.
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Budget Constraints - Price Changes


What happens if just one price decreases?
Suppose p1 decreases.
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EC270 Lecture Notes
Consumer Budget Constraints, Budget Sets
p1 decreases from p1' to p1"
x2
m/p2
-p1'/p2
Maximum
affordable amount
of x1 rises (budget
buys more x1 at
smaller p1)
Original
budget set
m/p1'
m/p1" x1
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p1 decreases from p1' to p1"
x2
m/p2
New affordable choices
-p1'/p2
Original
budget set
m/p1'
m/p1" x1
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EC270 Lecture Notes
Consumer Budget Constraints, Budget Sets
p1 decreases from p1' to p1"
x2
m/p2
New affordable choices
-p1'/p2
Original
budget set
Budget constraint
pivots; slope flattens
from -p1'/p2 to
-p1"/p2
-p "/p
1
m/p1'
2
m/p1" x1
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Budget Constraints - Price Changes


Reducing the price of one commodity pivots
the constraint outward.
No old choice is lost and new choices are
added, so reducing one price cannot make
the consumer worse off.
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EC270 Lecture Notes
Consumer Budget Constraints, Budget Sets
Budget Constraints - Price Changes

Similarly, increasing one price pivots the
constraint inwards, reduces choice and may
(typically will) make the consumer worse off.
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Uniform Ad Valorem Sales Taxes



An ad valorem sales tax levied at a rate of
5% increases all prices by 5%, from p to
(1+0.05)p = 1.05p.
An ad valorem sales tax levied at a rate of t
increases all prices by tp from p to (1+t)p.
A uniform sales tax is applied uniformly to
all commodities.
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EC270 Lecture Notes
Consumer Budget Constraints, Budget Sets
Uniform Ad Valorem Sales Taxes
A uniform sales tax levied at rate t changes
the constraint from
p1x1 + p2x2 = m
to
(1+t)p1x1 + (1+t)p2x2 = m
i.e.
p1x1 + p2x2 = m/(1+t).
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Uniform Ad Valorem Sales Taxes
x2
m
p2
p1x1 + p2x2 = m
m
p1
x1
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EC270 Lecture Notes
Consumer Budget Constraints, Budget Sets
Uniform Ad Valorem Sales Taxes
x2
m
p2
m
( 1  t ) p2
p1x1 + p2x2 = m
p1x1 + p2x2 = m/(1+t)
m
( 1  t ) p1
m
p1
x1
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Uniform Ad Valorem Sales Taxes
x2
m
p2
m
( 1  t ) p2
Equivalent income loss is
m
m
 1 t  m
= m

1 t
 1 t  1 t
 m  mt  m 
=

 1 t

 t 
=
m
1+ t 
m
( 1  t ) p1
m
p1
x1
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EC270 Lecture Notes
Consumer Budget Constraints, Budget Sets
Uniform Ad Valorem Sales Taxes
x2
m
p2
m
( 1  t ) p2
A uniform ad valorem
sales tax levied at rate t
is equivalent to a uniform
income tax levied at rate
t
1+ t
Equivalent
income loss
 t 

m
1
+
t


m
( 1  t ) p1
m
p1
x1
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The Food Stamp Program


Food stamps are coupons that can be legally
exchanged only for food.
How does a commodity-specific gift such as a
food stamp alter a family’s budget constraint?
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EC270 Lecture Notes
Consumer Budget Constraints, Budget Sets
The Food Stamp Program

Suppose m = $100, pF = $1 and the price of
“other goods” is pG = $1.

The budget constraint is then
F + G =100.
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The Food Stamp Program
G
F + G = 100: before stamps.
100
100
F
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EC270 Lecture Notes
Consumer Budget Constraints, Budget Sets
The Food Stamp Program
G
F + G = 100: before stamps.
100
Budget set after 40 food
stamps issued.
40
100 140
F
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The Food Stamp Program
G
F + G = 100: before stamps.
100
Budget set after 40 food
stamps issued.
The family’s budget
set is enlarged.
40
100 140
F
40
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EC270 Lecture Notes
Consumer Budget Constraints, Budget Sets
The Food Stamp Program

What if food stamps can be traded on a black
market for $0.50 each?
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The Food Stamp Program
G
120
100
F + G = 100: before stamps.
Budget constraint after 40
food stamps issued.
Budget constraint with
black market trading.
Black market trading
makes the budget
set larger again.
40
100 140
F
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EC270 Lecture Notes
Consumer Budget Constraints, Budget Sets
If the government gave you a subsidy of $200 per month
that you had to spend on housing and if you could
spend the remainder of your income in any way you
wished…
…the effect of the subsidy would differ from the effect of
a $200 per month unrestricted increase in your income
only if
A. housing were an inferior good for you.
B. housing were a normal good for you.
C. you would spend less than $200 per month on housing when
you received the unrestricted $200 monthly increase in
your income.
D. you would spend more than $200 per month on housing
when you received the unrestricted $200 monthly
increase in your income.
Other Goods
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If consumer buys fewer than $200/PH
units of housing, then optimum occurs
somewhere here [only if unrestricted grant]
If consumer buys at least $200/PH
units of housing, then consume a
combination of Housing and
Other Goods somewhere here,
with unrestricted grant or
$200 tied grant.
$200/pH
Housing
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EC270 Lecture Notes
Consumer Budget Constraints, Budget Sets
If the government gave you a subsidy of $200 per month
that you had to spend on housing and if you could
spend the remainder of your income in any way you
wished…
…the effect of the subsidy would differ from the effect of
a $200 per month unrestricted increase in your income
only if
A. housing were an inferior good for you.
B. housing were a normal good for you.
C. you would spend less than $200 per month on housing
when you received the unrestricted $200 monthly
increase in your income.
D. you would spend more than $200 per month on housing
when you received the unrestricted $200 monthly
increase in your income.
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Budget Constraints - Relative Prices




“Numeraire” means “unit of account”.
Suppose prices and income are measured in
dollars.
Say p1=$2, p2=$3, m = $12.
Then the constraint is
2x1 + 3x2 = 12.
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EC270 Lecture Notes
Consumer Budget Constraints, Budget Sets
Budget Constraints - Relative Prices


If prices and income are measured in cents,
then p1=200, p2=300, m=1200 and the
constraint is
200x1 + 300x2 = 1200,
the same as
2x1 + 3x2 = 12.
Changing the numeraire changes neither the
budget constraint nor the budget set.
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Budget Constraints - Relative Prices



The constraint for p1=2, p2=3, m=12
2x1 + 3x2 = 12
is also 1×x1 + (3/2)x2 = 6,
the constraint for p1=1, p2=3/2, m=6.
Setting p1=1 makes commodity 1 the
numeraire and defines all prices relative to p1;
i.e. 3/2 is the price of commodity 2 relative to
the price of commodity 1.
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EC270 Lecture Notes
Consumer Budget Constraints, Budget Sets
Budget Constraints - Relative Prices

Any commodity can be chosen as the
numeraire without changing the budget set or
the budget constraint.
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Budget Constraints - Relative Prices




p1=2, p2=3 and p3=6 
price of commodity 2 relative to commodity
1 is 3/2,
price of commodity 3 relative to commodity
1 is 3.
Relative prices are the rates of exchange
of commodities 2 and 3 for units of
commodity 1.
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EC270 Lecture Notes
Consumer Budget Constraints, Budget Sets
Will spends his entire income on 8 sacks of acorns
and 8 crates of butternuts.
The price of acorns is 9 dollars per sack and his
income is 96 dollars.
He can just afford a commodity bundle with A
sacks of acorns and B crates of butternuts that
satisfies the budget equation
A. 9A + 6B = 96
B. 18A + 6B = 192
C. 8A + 8B = 96
D. 9A + 6B = 192
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Will spends his entire income on 8 sacks of acorns
and 8 crates of butternuts.
The price of acorns is 9 dollars per sack and his
income is 96 dollars.
He can just afford a commodity bundle with A
sacks of acorns and B crates of butternuts that
satisfies the budget equation
Budget Eqn: 9A +PBB = 96
Spends 9×8 = $72 on acorns
Left with $96 – $72 = $24 to
spend on 8 crates butternuts
PB must be $24/8 = $3
Budget const: 9A + 3B = 96
Equivalent to 18A + 6B = 192
A. 9A + 6B = 96
B. 18A + 6B = 192
C. 8A + 8B = 96
D. 9A + 6B = 192
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EC270 Lecture Notes
Consumer Budget Constraints, Budget Sets
Shapes of Budget Constraints


Q: What makes a budget constraint a straight
line?
A: A straight line has a constant slope and
the constraint is
p1x1 + … + pnxn = m
so if prices are constants then a
constraint is a straight line.
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Shapes of Budget Constraints



But what if prices are not constants?
E.g. bulk buying discounts, or price penalties
for buying “too much”.
Then constraints will be curved.
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EC270 Lecture Notes
Consumer Budget Constraints, Budget Sets
Shapes of Budget Constraints Quantity Discounts


Suppose p2 is constant at $1 but that p1=$2
for 0  x1  20 and p1=$1 for x1>20.
Then the constraint’s slope is
- 2, for 0  x1  20
-p1/p2 =
- 1, for x1 > 20
{
and the constraint is
55
x2
100
Shapes of Budget Constraints Quantity Discounts
m = $100
Slope = - 2 / 1 = - 2
(p1=2, p2=1)
Slope = - 1/ 1 = - 1
(p1=1, p2=1)
20
50
80
x1
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EC270 Lecture Notes
Consumer Budget Constraints, Budget Sets
x2
100
Shapes of Budget Constraints Quantity Discounts
m = $100
Slope = - 2 / 1 = - 2
(p1=2, p2=1)
Slope = - 1/ 1 = - 1
(p1=1, p2=1)
20
50
x1
80
57
x2
Shapes of Budget Constraints Quantity Discounts
m = $100
100
Budget Constraint
Budget Set
20
50
80
x1
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EC270 Lecture Notes
Consumer Budget Constraints, Budget Sets
Shape of Budget Constraint with a
Quantity Penalty
x2
Budget
Constraint
Budget Set
x1
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Application: Budget Constraint with
Quantity Penalty
x2
Suppose x1 is consumption($) NOW,
x2 is consumption($) in FUTURE,
Consumer/economic agent has
income $m1 NOW and $m2 in FUTURE
m2
Can lend at interest rate rL,
but must pay rH (>rL) to borrow
Intertemporal (over time)
budget set looks like this
m1
x1
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EC270 Lecture Notes
Consumer Budget Constraints, Budget Sets
Shapes of Budget Constraints
One Price Negative



Commodity 1 is stinky garbage. You are
paid $2 per unit to accept it;
i.e. p1 = − $2.
p2 = $1.
Income, other than from accepting
commodity 1, is m = $10.
Then the constraint is
− 2x1 + x2 = 10 or x2 = 2x1 + 10.
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Shapes of Budget Constraints
One Price Negative
x2
x2 = 2x1 + 10
Budget constraint’s slope is
-p1/p2 = -(-2)/1 = +2
10
x1
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EC270 Lecture Notes
Consumer Budget Constraints, Budget Sets
Shapes of Budget Constraints
One Price Negative
x2
Budget set is
all bundles for
which x1  0,
x2  0 and
x2  2x1 + 10.
10
x1
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More General Choice Sets

Choices are usually constrained by more
than a budget; e.g. time constraints and other
resources constraints.

A bundle is available only if it meets every
constraint.
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EC270 Lecture Notes
Consumer Budget Constraints, Budget Sets
More General Choice Sets
Other Stuff
At least 10 units of food
must be eaten to survive
Food
10
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More General Choice Sets
Other Stuff
Choice is also budget
constrained.
Budget Set
10
Food
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EC270 Lecture Notes
Consumer Budget Constraints, Budget Sets
More General Choice Sets
Other Stuff
Choice is further restricted by
a time constraint.
10
Food
67
More General Choice Sets
So what is the choice set?
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EC270 Lecture Notes
Consumer Budget Constraints, Budget Sets
More General Choice Sets
Other Stuff
10
Food
69
More General Choice Sets
Other Stuff
10
Food
70
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EC270 Lecture Notes
Consumer Budget Constraints, Budget Sets
More General Choice Sets
Other Stuff
The choice set is the
intersection of all of
the constraint sets.
10
Food
71
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