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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Budget Set and Constraint for
Two Commoditiesx2
x1
Budget constraint isp1x1+ p2x2= m.
m /p2
m /p1
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Budget Set and Constraint for
Two Commoditiesx2
x1
Budget constraint isp1x1+ p2x2= m.
m /p1
Just affordable
m /p2
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Budget Set and Constraint for
Two Commoditiesx2
x1
Budget constraint isp1x1+ p2x2= m.
m /p1
Just affordableNot affordable
m /p2
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Budget Set and Constraint for
Two Commoditiesx2
x1
Budget constraint isp1x1+ p2x2= m.
m /p1
Affordable
Just affordableNot affordable
m /p2
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Budget Set and Constraint for
Two Commoditiesx2
x1
Budget constraint isp1x1+ p2x2= m.
m /p1
BudgetSet
the collectionof all affordable bundles.
m /p2
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Budget Set and Constraint for
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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How do the budget set and budget
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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Budget Constraints - Income
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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Budget Constraints - Income
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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How do the budget set and budget
constraint change as p1decreases
from p1to p1
?
Original
budget set
x2
x1
m/p2
m/p1 m/p1
-p1/p2
H d th b d t t d b d t
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How do the budget set and budget
constraint change as p1decreases
from p1to p1
?
Original
budget set
x2
x1
m/p2
m/p1 m/p1
New affordable choices
-p1/p2
H d th b d t t d b d t
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How do the budget set and budget
constraint change as p1decreases
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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Budget Constraints - Price
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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Budget Constraints - Price
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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UniformAd ValoremSales Taxes
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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UniformAd ValoremSales Taxes
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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UniformAd ValoremSales Taxesx2
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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UniformAd ValoremSales Taxesx2
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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UniformAd ValoremSales Taxesx2
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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The Food Stamp Program
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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The Food Stamp ProgramG
F100
100
F + G = 100: before stamps.
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The Food Stamp ProgramG
F100
100
F + G = 100: before stamps.
Budget set after 40 food
stamps issued.
14040
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The Food Stamp ProgramG
F100
100
F + G = 100: before stamps.
Budget set after 40 food
stamps issued.
140
The familys budgetset is enlarged.
40
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The Food Stamp Program
What if food stamps can be traded ona black market for $0.50 each?
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The Food Stamp ProgramG
F100
100
F + G = 100: before stamps.
Budget constraint after 40food stamps issued.
140
120
Budget constraint withblack market trading.
40
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The Food Stamp ProgramG
F100
100
F + G = 100: before stamps.
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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Budget Constraints - Relative
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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Budget Constraints - Relative
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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Budget Constraints - Relative
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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Shapes of Budget Constraints
But what if prices are not constants?
E.g.bulk buying discounts, or pricepenalties for buying too much.
Then constraints will be curved.
Shapes of Budget Constraints
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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.
Shapes of Budget Constraints
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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. Then the constraints slope is
- 2, for 0 x120-p1/p2=
- 1, for x1> 20
and the constraint is
{
Shapes of Budget Constraints
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Shapes of Budget Constraints
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
Shapes of Budget Constraints
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Shapes of Budget Constraints
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
Shapes of Budget Constraints
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Shapes of Budget Constraints
with a Quantity Discountm= $100
50
100
20 80
x2
x1
Budget Set
Budget Constraint
Shapes of Budget Constraints
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Shapes of Budget Constraints
with a Quantity Penalty
x2
x1
Budget Set
BudgetConstraint
Shapes of Budget Constraints
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Shapes of Budget Constraints -
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.
Shapes of Budget Constraints
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Shapes of Budget Constraints -
One Price Negative
10
Budget constraints slope is-p1/p2= -(-2)/1 = +2
x2
x1
x2= 2x1+ 10
Shapes of Budget Constraints -
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Shapes of Budget Constraints -
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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More General Choice Sets
Food
Other Stuff
10
At least 10 units of food
must be eaten to survive
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More General Choice Sets
Food
Other Stuff
10
Budget Set
Choice is also budgetconstrained.
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More General Choice Sets
Food
Other Stuff
10
Choice is further restricted bya time constraint.
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More General Choice Sets
So what is the choice set?
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More General Choice Sets
Food
Other Stuff
10
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More General Choice Sets
Food
Other Stuff
10
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More General Choice Sets
Food
Other Stuff
10
The choice set is theintersection of all ofthe constraint sets.