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Lecture 2 - Theory of the
Consumer; Budget Constraint
The budget constraint Definition
Notations and assumptions
Properties of the budget set/constraint
Effects of changes in income and prices
Representing alternative policies in budget lines
Some observations
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Some notations:
spendcanconsumermoneyofamount
2goodofprice
1goodofprice
2goodofnconsumptio
1goodofnconsumptio
2
1
2
1
m
p
p
x
x
Consumption Bundle List of n numbers
that indicate how much the consumer is
choosing to consume of n goods. In the case
of two goods, list of two numbers that
indicate how much the consumer ischoosing to consume of good 1 and good 2.
Denoted by:
Xxx or),( 21
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Budget line or budget constraint: the set of
bundles that cost exactly m:
i.e.,
{ (x1,,xn) | x1 0, , xn and
p1x1 + + pnxn m }.
The consumers budget setis the set of all
affordable bundles;
B(p1, , pn, m) =
{ (x1, , xn) | x1 0, , xn 0 and
p1x1 + + pnxn m } The budget line or budget constraint is the
upper boundary of the budget set.
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In the case of two goods, the budget constraint istherefore:
spending on good 1:
spending on good 2:
Budget constraint requires that the total amount ofmoney spent on the two goods be no more than the
total amount of money the consumer has to spend
Budget set set of affordable consumption bundlesat price
mxpxp 2211
11xp
22xp
),( 21 pp
Two good assumption
General enough if we interpret one good as acomposite good
We can think of good 2 as the amount of money that isbeing spent on other goods. The price if good 2 isassumed to be one since the price of one peso is onepeso.
Thus:
This representation is just a special case where theprice of good 2 is equal to 1. We can say about thebudget constraint will hold under the composite goodinterpretation
mxxp 211
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x2
x1
Budget constraint is
p1x1 + p2x2 = m.
m /p1
m /p2
x2
x1
Budget constraint is
p1x1 + p2x2 = m.m /p2
m /p1
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x2
x1
Budget constraint is
p1x1 + p2x2 = m.
m /p1
Just affordable
m /p2
x2
x1
Budget constraint is
p1x1 + p2x2 = m.
m /p1
Just affordable
Not affordable
m /p2
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x2
x1
Budget constraint is
p1x1 + p2x2 = m.
m /p1
Affordable
Just affordable
Not affordable
m /p2
x2
x1
Budget constraint is
p1x1 + p2x2 = m.
m /p1
Budget
Set
the collection
of all affordable bundles.
m /p2
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x2
x1
p1x1 + p2x2 = m is
x2 = -(p1/p2)x1 + m/p2so slope is -p1/p2.
m /p1
Budget
Set
m /p2
If n = 3 what do the budget constraint and
the budget set look like?
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x2
x1
x3
m /p2
m /p1
m /p3
p1x1 + p2x2 + p3x3 = m
x2
x1
x3
m /p2
m /p1
m /p3
{ (x1,x2,x3) | x1 0, x2 0, x3 0 and
p1x1 + p2x2 + p3x3 m}
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We can rearrange the equation of the budget line to yield:
Equation of a straight line that gives the consumption of good 2that would just satisfy the budget constraint for units of good 1consumed. Slide 8
Vertical intercept:
Slope:
1
2
1
2
2 xp
p
p
mx
2pm
2
1
p
p
Intercepts:
Interpretation: measures how much of each good can
be obtained if only that good is consumed
To derive the vertical intercept, just set equal to
zero and solve for
To derive the horizontal intercept, just set equal tozero and solve for
To derive the budget line, just connect these two
intercepts.
1x
2x
2x
1x
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Slope of the budget line: Slope of budget line measures the rate at which
the market is willing to substitute good 1 forgood 2. If you consume more of good 1, you haveto give up some good 2 in order to satisfy yourbudget constraint.
Suppose that the consumer is going to increaseconsumption of good 1:
2goodofncons'inchange
1goodofncons'inchange
2
1
x
x
Budget line should be fulfilled before and
after changes in consumption:
this just indicates that the change in the total
value of consumption should be zero.
0
:yieldssecondthefromfirstthegsubtractin
)()(
2211
222111
2211
xpxp
mxxpxxp
mxpxp
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Solving for , the rate at which good 2 can
be substituted for good 1 while still satisfying the
budget constraint gives:
this is just the slope of the budget line
this is negative because and should
have opposite signs.
2
1
1
2
p
p
x
x
12xx
2
x1
x
Slope of the budget line is also interpreted as the
opportunity costof consuming good 1. In order to
consume more of good 1, you have to give up some
consumption of good 2.
The true cost of economic cost of more good 1 is the
opportunity to consume more of good 2, and it is theslope of the budget line.
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For n = 2 and x1 on the horizontal axis,
the constraints slope is -p1/p2. What
does it mean?
Increasing x1 by 1 must reduce x2 by
p1/p2.
2
1
2
12
p
mx
p
px
x2
x1
Slope is -p1/p2
+1
-p1/p2
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1
x2
x1
+1
-p1/p2
Opp. cost of an extra unit ofcommodity 1 is p1/p2 units
foregone of commodity 2.
x2
x1
Opp. cost of an extra unit of
commodity 1 is p1/p2 units
foregone of commodity 2. And
the opp. cost of an extra
unit of commodity 2 is
p2/p1 units foregone
of commodity 1.
-p2/p1
+1
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Income changes
An increase in income will change m andnot the slope.
An increase in income - parallel shiftoutward of the budget line, therebyincreasing the budget set
A decrease in income - parallel inward shift
of the budget line, thereby decreasing thebudget set
To draw the new curve, just derive the newintercepts and then connect the lines again.
Original
budget set
x2
x1
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Original
budget set
New affordable consumptionchoices
x2
x1
Original and
new budget
constraints are
parallel (same
slope).
Original
budget set
x2
x1
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x2
x1
New, smaller
budget set
Consumption bundles
that are no longer
affordable.
Old and new
constraints
are parallel.
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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Price changes
Changes in prices will result in changes in theslope of the budget line
Case 1: Price of one good decreases, price ofgood 2 and income are constant.
Change in p1 will shift the horizontal interceptoutward because for the same income, we canpurchase more of good 1
vertical intercept is unchanged since you can buy thesame amount of good 2 as before if good 1consumption is zero.
Therefore budget line is flatter.
Original
budget set
x2
x1
m/p2
m/p1 m/p1
-p1/p2
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Original
budget set
x2
x1
m/p2
m/p1 m/p1
New affordable choices
-p1/p2
Original
budget set
x2
x1
m/p2
m/p1 m/p1
New affordable choices
Budget constraint
pivots; slope flattens
from -p1/p2 to-p1/p2
-p1/p2
-p1/p2
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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.
Similarly, increasing one price pivots the
constraint inwards, reduces choice and may
(typically will) make the consumer worse
off.
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Case 2: Price of both goods increase by the
same factor E.g., prices of good 1 and good 2 double
horizontal and vertical intercepts shift inward by .
The budget curve shifts inward and it is like dividing income
by one half.
0for,2
0for,2
22
1
2
2
2
1
1
2211
xp
mx
xp
mx
mxpxp
Multiplying both prices by a constant factor
is just like dividing income by the same
constant factor
t
mxpxp
mxtpxtp
mxpxp
2211
2211
2211
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Multiplying prices and income by the same factor
will not change the budget line at all:
1
2
1
2
2
2
11
2
2
1122
2
2211
,
,
,forsolving
xp
p
p
mx
tp
xtp
tp
tmx
xtptmxtp
x
tmxtpxtp
Case 3: Price and income changes together
Suppose m decreases andp1 andp2 increase
Intercepts must decrease and the budget line
shifts inward.
Changes in the slope depends on the amount ofthe change inp1 andp2
Ifp2 increases by more than p1, -p1 /p2 decreases,
the budget line becomes flatter
Ifp1 increases by more than p2, budget line is
steeper
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Numeraire
The same budget set can be represented by setting
one of the prices or income to some fixed value and
adjusting the other variables accordingly.
Setting the variables in terms of the price of one
good, suppose p2, the budget line is therefore:
Setting the variables in terms of income, we just
divide everything by m
2
21
2
1
p
mxx
p
p
122
1
1 x
m
px
m
p
Numeraire
In the first case, the price ofp2 is pegged at 1,
and in the second case m is pegged at 1.
Numeraire refers to the price that is set to one.
The numeraire price is the price relative to
which we are measuring other prices and
income.
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Government imposes policies that affect the
consumers budget constraint.
Some of these policies include:
Taxes both quantity and value
Subsidies both quantity and value
Lump-sum taxes or subsidies
Rationing constraints
Taxes
quantity taxes
paid for each unit of the good that is purchased
increases the price that is paid for the good
Quantity tax: t
value taxes
tax on the price of the good, expressed as a percentage
of the price
most common example are sales taxes
Value tax :
tp
p)1(
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Subsidies
Subsidies decrease the effective price paid for
the good.
Quantity subsidies
given based on the amount that is purchased
Quantity subsidy:s
Value subsidies
given based on the price of the good being subsidized
Value subsidy:
sp
p)1(
Lump sum tax or subsidy: A fixed amount
of money that is given (in the case of
subsidies) or taken away (in the case of
taxes) regardless of the consumers
behavior.
taxes shift the budget line inward
subsidies shift the budget line
outward
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Rationing constraints: level of consumptionis fixed to be less than some amountFigure 2.4
Rationing constraint combined with taxesand subsidies has the effect of changing the prices faced by
the consumer for certain goods beyond acertain amount consumed Figure 2.5 conc
slide 29
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slide 29
Q: What makes a budget constraint a
straight line?
A: A straight line has a constant slope and
the constraint is
p1x1 + + pnxn = mso if prices are constants then a constraint
is a straight line.
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But what if prices are not constants?
E.g. bulk buying discounts, or price penalties
for buying too much.
Then constraints will be curved.
Suppose p2 is constant at P1 but that p1=P2
for 0 x1 20 and p1=P1 for x1>20.
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Suppose p2 is constant at P1 but that p1=P2
for 0 x1 20 and p1=P1 for x1>20. Then
the constraints slope is
- 2, for 0 x1 20
-p1/p2 =
- 1, for x1 > 20
and the constraint is
{
m =
P100
50
100
20
Slope = - 2 / 1 = - 2
(p1=2, p2=1)
Slope = - 1/ 1 = - 1(p1=1, p2=1)
80
x2
x1
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m =P100
50
100
20
Slope = - 2 / 1 = - 2
(p1=2, p2=1)
Slope = - 1/ 1 = - 1
(p1=1, p2=1)
80
x2
x1
m =
P100
50
100
20 80
x2
x1
Budget Set
Budget Constraint
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x2
x1
Budget Set
Budget
Constraint
Commodity 1 is stinky garbage. You are
paid P2 per unit to accept it; i.e. p1 = - P2.
p2 = P1. Income, other than from
accepting commodity 1, is m = P10.
Then the constraint is- 2x1 + x2 = 10 or x2 = 2x1 + 10.
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10
Budget constraints slope is
-p1/p2 = -(-2)/1 = +2
x2
x1
x2 = 2x1 + 10
10
x2
x1
Budget set is
all bundles for
which x1 0,
x2
0 and
x2 2x1 + 10.
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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.
Food
Other Stuff
10
At least 10 units of food
must be eaten to survive
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Food
Other Stuff
10
Budget Set
Choice is also budget
constrained.
Food
Other Stuff
10
Choice is further restricted by a
time constraint.
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So what is the choice set?
Food
Other Stuff
10
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Food
Other Stuff
10
Food
Other Stuff
10
The choice set is the
intersection of all of
the constraint sets.
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Perfectly balanced inflation, one in which all prices and incomesrise at the same rate doesnt change anybodys budget set andthus cannot change anybodys optimal choice: e.g. incomeindexation or wage increases are a way of assuring that theoptimal choices do not change, no adjustments in consumptionare necessary
Income increases with prices remaining the same leaves theconsumer at least as well off as before
If one price declines and all others remain the same, then theconsumer must be at least as well-off.
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