economics 102 lecture 2 budget constraint rev

7/28/2019 Economics 102 Lecture 2 Budget Constraint Rev

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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


p1x1 + p2x2 = m.m /p2

m /p1


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x2

x1


p1x1 + p2x2 = m.

m /p1

Just affordable

m /p2

x2

x1


p1x1 + p2x2 = m.

m /p1

Just affordable

Not affordable

m /p2


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x2

x1


p1x1 + p2x2 = m.

m /p1

Affordable

Just affordable

Not affordable

m /p2

x2

x1


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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1

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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1

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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1

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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1

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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1

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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1

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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2

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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3

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.

economics 102 lecture 2 budget constraint rev

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