l3 constraints.ppt
TRANSCRIPT
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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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Start modeling consumer behavior:•
..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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Start modeling consumer behavior:•
Q: When is a consumption bundle (x1, … , xn) affordableat given prices (p1, … , pn)?
(i.e., how to express this concisely?)
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Start modeling consumer behavior:•
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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Budget Set and Constraint for 2 Commoditiesx2
x1
Budget constraint isp1x1 + p2x2 = m
m /p1
Just affordable
m /p2
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Budget Set and Constraint for 2 Commoditiesx2
x1
Budget constraint isp1x1 + p2x2 = m
m /p1
Just affordable
Not affordable
m /p2
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Budget Set and Constraint for 2 Commoditiesx2
x1
Budget constraint isp1x1 + p2x2 = m
m /p1
Affordable
Just affordable
Not affordable
m /p2
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Budget Set and Constraint for 2 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 2 Commoditiesx2
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
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
Budget constraint pivots:
slope flattens from -p1
/p2 to -p1 /p2-p1 /p2
-p1
/p2
How does the budget set change as p1
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..
Uniform Ad Valorem Sales Taxes
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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
Uniform Ad Valorem Sales Taxes
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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 ).
Uniform Ad Valorem Sales Taxes
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Uniform Ad Valorem Sales Taxes
x2
x1
m
p2
m
p1
p1x1 + p2x2 = m
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Uniform Ad Valorem Sales Taxes
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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Uniform Ad Valorem Sales Taxes
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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Budget Constraints - Relative Prices
•
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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Budget Constraints - Relative Prices
•
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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Budget Constraints - Relative Prices
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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Shapes of Budget Constraints
•
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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Shapes of Budget Constraints
•
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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Shapes of Budget Constraints - Quantity
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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Shapes of Budget Constraints - Quantity
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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Shapes of Budget Constraints - Quantity
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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Shapes of Budget Constraints - Quantity
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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Shapes of Budget Constraints - One Price
Negative• Suppose commodity 1 is smelly garbage
•
You are paid $2 per unit to accept it:
p1 = - $2 p2 = $1
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Shapes of Budget Constraints - One Price
Negative• Suppose commodity 1 is smelly garbage
•
You are paid $2 per unit to accept it:
p1 = - $2 p2 = $1•
Income, other than from accepting commodity 1:
m = $10
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Shapes of Budget Constraints - One Price
Negative• Suppose commodity 1 is smelly garbage
•
You are paid $2 per unit to accept it:
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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Shapes of Budget Constraints - One Price
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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More General Choice Sets
Food
Other Stuff
10
At least 10 units of foodmust be eaten to survive
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More General Choice Sets
Food
Other Stuff
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Budget Set
Choice is also budgetconstrained.
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More General Choice Sets
Food
Other Stuff
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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
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The choice set is the
intersection of all ofthe constraint sets.