Varian Chapter02 Budget Constraint

8/10/2019 Varian Chapter02 Budget Constraint

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Chapter Two

Budgetary and Other

Constraints on Choice


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Consumption Choice Sets

A consumption choice setis thecollection of all consumption choices

available to the consumer.What constrains consumption

choice?

Budgetary, time and otherresource limitations.


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Budget Constraints

A consumption bundlecontaining x1units of commodity 1, x2units of

commodity 2 and so on up to xnunitsof commodity n is denoted by thevector (x1, x2, , xn).

Commodity prices are p1

, p2

, , pn

.


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Budget Constraints

Q: When is a consumption bundle(x1, , xn) affordable at given prices

p1, , pn?


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Budget Constraints

Q: When is a bundle (x1, , xn)affordable at prices p1, , pn?

A: Whenp1x1+ + pnxnmwhere mis the consumers(disposable) income.


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Budget Constraints

The bundles that are only justaffordable form the consumers

budget constraint. This is the set

{ (x1,,xn) | x1 0, , xn andp1x1+ + pnxn=m }.


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Budget Constraints

The consumers budget setis the setof all affordable bundles;B(p

1, , p

n, m) =

{ (x1, , xn) | x10, , xn 0 andp1x1+ + pnxnm}

The budget constraint is the upperboundary of the budget set.


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Budget Set and Constraint for

Two Commoditiesx2

x1

Budget constraint isp1x1+ p2x2= m.

m /p1

m /p2


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Two Commoditiesx2

x1


m /p2

m /p1


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Two Commoditiesx2

x1


m /p1

Just affordable

m /p2


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Two Commoditiesx2

x1


m /p1

Just affordableNot affordable

m /p2


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Two Commoditiesx2

x1


m /p1

Affordable

Just affordableNot affordable

m /p2


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Two Commoditiesx2

x1


m /p1

BudgetSet

the collectionof all affordable bundles.

m /p2


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Two Commoditiesx2

x1

p1x1+ p2x2= m isx2= -(p1/p2)x1+ m/p2

so slope is -p1/p2.

m /p1

BudgetSet

m /p2


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Budget Constraints

If n = 3 what do the budget constraintand the budget set look like?


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Budget Constraint for Three

Commodities

x2

x1

x3

m /p2

m/p1

m/p3

p1x1 + p2x2+ p3x3= m


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Budget Set for Three

Commodities

x2

x1

x3

m /p2

m/p1

m/p3

{ (x1,x2,x3) | x10, x2 0, x30 andp1x1+ p2x2+ p3x3m}


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Budget ConstraintsFor n = 2 and x

1on the horizontal

axis, the constraints slope is -p1/p2.What does it mean?

x pp

x mp

2 12

12

=


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Budget ConstraintsFor n = 2 and x

1on the horizontal

axis, the constraints slope is -p1/p2.What does it mean?

Increasing x1by 1 must reduce x2by

p1/p2.

x pp

x mp

2 12

12

=


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Budget Constraints

x2

x1

Slope is -p1/p2

+1

-p1/p2


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Budget Constraints

x2

x1

+1

-p1/p2

Opp. cost of an extra unit ofcommodity 1 is p1/p2units

foregone of commodity 2.


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Budget Constraints

x2

x1

Opp. cost of an extra unit ofcommodity 1 is p1/p2units

foregone of commodity 2. And

the opp. cost of an extraunit of commodity 2 is

p2/p1 units foregone

of commodity 1.

-p2/p1

+1


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Budget Sets & Constraints;

Income and Price Changes

The budget constraint and budgetset depend upon prices and income.What happens as prices or incomechange?

d h b d d b d


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How do the budget set and budget

constraint change as income m

increases?

Original

budget set

x2

x1


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Higher income gives more choice

Original

budget set

New affordable consumptionchoices

x2

x1

Original andnew budget

constraints areparallel (sameslope).


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H d h b d d b d


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constraint change as income m

decreases?x2

x1

New, smaller

budget set

Consumption bundlesthat are no longer

affordable.Old and newconstraints

are parallel.


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Budget Constraints - Income

Changes

Increases in income mshift theconstraint outward in a parallelmanner, thereby enlarging thebudget set and improving choice.


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Changes

Increases in income mshift theconstraint outward in a parallelmanner, thereby enlarging thebudget set and improving choice.

Decreases in income mshift theconstraint inward in a parallelmanner, thereby shrinking thebudget set and reducing choice.


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Changes

No original choice is lost and newchoices are added when incomeincreases, so higher income cannotmake a consumer worse off.

An income decrease may (typicallywill) make the consumer worse off.


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Budget Constraints - Price

Changes

What happens if just one pricedecreases?

Suppose p1decreases.

H d th b d t t d b d t


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constraint change as p1decreases

from p1to p1

?

Original

budget set

x2

x1

m/p2

m/p1 m/p1

-p1/p2



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from p1to p1

?

Original

budget set

x2

x1

m/p2

m/p1 m/p1

New affordable choices

-p1/p2



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from p1to p1

?

Original

budget set

x2

x1

m/p2

m/p1 m/p1

New affordable choices

Budget constraintpivots; slope flattensfrom -p1/p2to

-p1/p2

-p1/p2

-p1/p2


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Changes

Reducing the price of onecommodity pivots the constraintoutward. No old choice is lost andnew choices are added, so reducingone price cannot make the consumerworse off.


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Changes

Similarly, increasing one price pivotsthe constraint inwards, reduceschoice and may (typically will) makethe consumer worse off.


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UniformAd ValoremSales Taxes

An ad valoremsales tax levied at arate of 5% increases all prices by 5%,from p to (1+0

05)p = 1

05p.

An ad valoremsales tax levied at arate of t increases all prices by tpfrom p to (1+t)p.

A uniform sales tax is applieduniformly to all commodities.


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A uniform sales tax levied at rate tchanges the constraint from

p1x1+ p2x2= m

to (1+t)p1x1+ (1+t)p2x2= m


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A uniform sales tax levied at rate tchanges the constraint from

p1x1+ p2x2= m

to (1+t)p1x1+ (1+t)p2x2= mi.e.

p1x1+ p2x2= m/(1+t).


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UniformAd ValoremSales Taxesx2

x1

m

p2

m

p1

p1x1+ p2x2= m


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x1

m

p2

m

p1

p1x1+ p2x2= m

p1x1+ p2x2= m/(1+t)

m

t p( )11

mt p( )1 2


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x1

mt p( )1 2

m

p2

m

t p( )11

m

p1

Equivalent income lossis

m mt

tt

m

=

1 1


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x1

mt p( )1 2

m

p2

m

t p( )11

m

p1

A uniform ad valorem

sales tax levied at rate tis equivalent to an incometax levied at rate t

t1.


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The Food Stamp Program

Food stamps are coupons that canbe legally exchanged only for food.

How does a commodity-specific giftsuch as a food stamp alter a familysbudget constraint?


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Suppose m = $100, pF= $1 and theprice of other goods is pG= $1.

The budget constraint is thenF + G =100.


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The Food Stamp ProgramG

F100

100

F + G = 100; before stamps.


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F100

100

F + G = 100: before stamps.


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F100

100


Budget set after 40 food

stamps issued.

14040


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F100

100


Budget set after 40 food

stamps issued.

140

The familys budgetset is enlarged.

40


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What if food stamps can be traded ona black market for $0.50 each?


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F100

100


Budget constraint after 40food stamps issued.

140

120

Budget constraint withblack market trading.

40


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F100

100


Budget constraint after 40food stamps issued.

140

120

Black market tradingmakes the budget

set larger again.

40

B d t C t i t R l ti


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Budget Constraints - Relative

Prices

Numeraire means unit of

account.

Suppose prices and income aremeasured in dollars. Say p1=$2,p2=$3, m= $12. Then the constraintis

2x1+ 3x2= 12.

B d t C t i t R l ti


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Prices

If prices and income are measured incents, then p1=200, p2=300, m=1200and the constraint is

200x1 + 300x2= 1200,the same as

2x1+ 3x2= 12.

Changing the numeraire changesneither the budget constraint nor thebudget set.

B dget Constraints Relati e


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PricesThe constraint for p1=2, p2=3, m=12

2x1+ 3x2 = 12is also 1.x1+ (3/2)x2= 6,

the constraint for p1=1, p2=3/2, m=6.Setting p1=1 makes commodity 1 thenumeraireand defines all prices

relative top1; e.g.3/2 is the price ofcommodity 2 relative to the price ofcommodity 1.

Budget Constraints Relative


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Prices

Anycommodity can be chosen asthe numeraire without changing thebudget set or the budget constraint.

B d C i R l i P i


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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 tocommodity 1 is 3.

Relative prices are the rates of

exchangeof commodities 2 and 3 forunits of commodity 1.


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Shapes of Budget Constraints

Q: What makes a budget constraint astraight line?

A: A straight line has a constantslope 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 pricepenalties for buying too much.

Then constraints will be curved.



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Shapes of Budget Constraints -

Quantity Discounts

Suppose p2is constant at $1 but thatp1=$2 for 0 x120 and p1=$1 forx1>20.



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Quantity Discounts

Suppose p2is constant at $1 but thatp1=$2 for 0 x120 and p1=$1 forx1>20. Then the constraints slope is

- 2, for 0 x120-p1/p2=

- 1, for x1> 20

and the constraint is

{



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with a Quantity Discountm= $100

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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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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50

100

20 80

x2

x1

Budget Set

Budget Constraint



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with a Quantity Penalty

x2

x1

Budget Set

BudgetConstraint



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One Price Negative

Commodity 1 is stinky garbage. Youare 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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One Price Negative

10

Budget constraints slope is-p1/p2= -(-2)/1 = +2

x2

x1

x2= 2x1+ 10



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One Price Negative

10

x2

x1

Budget set isall bundles for

which x10,x20 andx22x1+ 10.


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More General Choice Sets

Choices are usually constrained bymore than a budget; e.g. timeconstraints and other resources

constraints.

A bundle is available only if it meetseveryconstraint.


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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 budgetconstrained.


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Food

Other Stuff

10

Choice is further restricted bya time constraint.


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So what is the choice set?


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Food

Other Stuff

10


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Food

Other Stuff

10


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Food

Other Stuff

10

The choice set is theintersection of all ofthe constraint sets.

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Varian Chapter02 Budget Constraint