L3 Constraints.ppt

8/9/2019 L3 Constraints.ppt

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Modeling Budgetary and

Other Constraints on Choice


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Start modeling consumer behavior:•

..to build something similar to a model of demandbehavior we just saw/used

(in apartment-market example)


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..to build something similar to a model of demandbehavior we just saw/used

(in apartment-market example)

Start by introducing some “precise language”


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Q: When is a consumption bundle (x1, … , xn) affordableat given prices (p1, … , pn)?

(i.e., how to express this concisely?)


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Q: When is a consumption bundle (x1, … , xn) affordableat given prices (p1, … , pn)?

• A: Whenp1x1 + … + pnxn ! m

where m is the consumer ’s (disposable) income.

• (n is the number of commodities in a bundle)

• (e.g.: n=2, then p1x1 + p2x2 ! m)


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Budget Constraints•

The bundles that are only just affordable form theconsumer’s budget constraint

•

That is, the set:

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


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Budget Set•

The consumer’s budget set is the set of all affordablebundles:

B(p1, … , pn, m) =

{ (x1, … , xn) | x1 " 0, … , xn " 0 andp1x1 + … + pnxn ! m }

•

Budget constraint is the upper boundary of the budget set.


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Budget Set and Constraint for 2 Commoditiesx2

x1


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x1

Budget constraint isp1x1 + p2x2 = m

m /p1

Just affordable

m /p2


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x1


m /p1

Just affordable

Not affordable

m /p2


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x1


m /p1

Affordable

Just affordable

Not affordable

m /p2


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x1


m /p1

BudgetSet

the collectionof all affordable bundles

m /p2


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x1

p1x1 + p2x2 = mis the same as

x2 = -(p1 /p2)x1 + m /p2 so slope is -p1 /p2

m /p1

BudgetSet

m /p2


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Budget Constraints•

Can we extend this?

•

Suppose n = 3 (i.e. three commodities in the bundle)

• What do the budget constraint and the budget set looklike?


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Budget Constraint for Three Commodities

x2

x1

x3

m /p2

m /p1

m /p3

p1 x 1 + p2 x 2 + p3 x 3 = m


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Budget Set for Three Commoditiesx2

x1

x3

m /p2

m /p1

m /p3

{ (x 1,x 2 ,x 3 ) | x 1 ! 0, x 2 ! 0, x 3 ! 0 and

p1 x 1 + p2 x 2 + p3 x 3 " m}


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Understanding Budget Constraints

•

For n = 2 and x1 on the horizontal axis, theconstraint’s slope is -p1 /p2 . What does it mean?

x2 =!

p1

p2

x1 +m

p2


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Understanding Budget Constraints

• For n = 2 and x1 on the horizontal axis, theconstraint’s slope is -p1 /p2 . What does it mean?

• Staying within budget:

to increase x1 by 1 must reduce x2 by p1 /p2

x2 =!

p1

p2

x1 +m

p2


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Budget Constraintsx2

x1

Slope is -p1 /p2

+1

-p1 /p2


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Budget Constraintsx2

x1

+1

-p1 /p2

Opp. cost of an extra unit ofcommodity 1 is p1/p2 units

foregone of commodity 2.


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Budget Sets: Income and Price Changes

•

Notice: budget set (and budget constraint) depend onprices and income

• (i.e., p’s and m)

•

What happens when prices or income change?

• (i.e., can we model it somehow using budget

constraints/sets?)


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How does the budget set change as

income, m, increases?

Original

budget set

x2

x1


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Higher income gives more choice

Original

budget set

Newly affordable consumptionchoices

x2

x1

Original and

new budgetconstraints areparallel(same slope)


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How do the budget set and budget

constraint change as income m decreases?

Original

budget set

x2

x1


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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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Original

budget set

x2

x1

m/p2

m/p1 m/p1

-p1 /p2

How does the budget set change as p1

decreases from p1

to p1

?


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Original

budget set

x2

x1

m/p2

m/p1 m/p1

New affordable choices

-p1 /p2


decreases from p1

to p1

?


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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


decreases from p1

to p1

?


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•

An example to practice modeling changes in the budget

•

..and may even learn something not obvious.

Uniform Ad Valorem Sales Taxes


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•

“uniform” = applied uniformly to all commodities (say, 2)

•

“ad valorem” sales tax levied at a rate of 5% (i.e. 0.05)

increases all prices by 5%,i.e. from p to (1+0.05)p = 1.05p

• ..if at a rate of t, increases all prices by t p,

from p to (1+t )p.

•

Let’s see how budget constraint incorporates that..



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• A uniform sales tax levied at rate t changes theconstraint from

p1x1 + p2x2 = m to

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



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• A uniform sales tax levied at rate t changes theconstraint from

p1x1 + p2x2 = m to

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



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x2

x1

m

p2

m

p1

p1x1 + p2x2 = m


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x2

x1

m

p2

m

p1

p1x1 + p2x2 = m

p1x1 + p2x2 = m/(1+t )

m

t p( )1 1

m

t p( )1 2


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x2

x1

m

t p( )1 2

m

p2

m

t p( )1 1

m

p1

So: uniform ad valorem sales tax at rate t is equivalent to anincome tax at rate

t

1+ t


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Budget Constraints - Relative Prices

• “Numeraire” = “unit of account”

•

Suppose prices and income are measured in dollars.• Say p1=$2, p2=$3, m = $12.

•

Then the constraint is2x1 + 3x2 = 12.


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•

If prices and income are measured in cents,

• then p1=200, p2=300, m=1200 and the constraint is200x1 + 300x2 = 1200,

the same as2x1 + 3x2 = 12.

• Changing the numeraire changes neither the budgetconstraint nor the budget set.


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•

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

•

2x1 + 3x2 = 12

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

i.e. the constraint for p1=1, p2=3/2, m=6.

Setting p1=1 makes commodity 1 the numeraire anddefines all prices relative to p1;

e.g. 3/2 is the price of commodity 2 relative to the priceof commodity 1.


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Notice:

any commodity can be chosen as the numeraire without

changing the budget set or the budget constraint.


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Shapes of Budget Constraints

•

Q: What is a necessary condition for a budget constraintto be a straight line?


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•

Q: What is a necessary condition for a budget constraintto be a straight line?

•

A: Constant prices. A straight line has a constant slope and the constraint is

p1x1 + … + pnxn = m so if prices are constants then the constraint is a straightline.


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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.


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Shapes of Budget Constraints - Quantity

Discounts•

Suppose p2 is constant at $1 but p1=$2 for 0 ! x1 ! 20 andp1=$1 for x1>20.


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Discounts•

Suppose p2 is constant at $1 but p1=$2 for 0 ! x1 ! 20 andp1=$1 for x1>20. Then the constraint’s slope is

- 2, for 0 ! x1 ! 20-p1/p2 =

- 1, for x1 > 20and the constraint is..{


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Discountsm = $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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Discountsm = $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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Discountsm = $100

50

100

20 80

x2

x1

Budget Set

Budget Constraint


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Shapes of Budget Constraints with a Quantity

Penaltyx2

x1

Budget Set

Budget

Constraint


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Shapes of Budget Constraints - One Price

Negative• Suppose commodity 1 is smelly garbage


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•

You are paid $2 per unit to accept it:

p1 = - $2 p2 = $1


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•


p1 = - $2 p2 = $1•

Income, other than from accepting commodity 1:

m = $10


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•


p1 = - $2 p2 = $1•

Income, other than from accepting commodity 1:

m = $10

•

Then the constraint is- 2x1 + x2 = 10 [or: x2 = 2x1 + 10]


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Negative

10

Budget constraint

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

x2

x1

x2 = 2x1 + 10


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More General Choice Sets

•

Choices can be 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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Food

Other Stuff

10

At least 10 units of foodmust 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 the

intersection of all ofthe constraint sets.

L3 Constraints.ppt

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