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Budgetary and Other Constraints on Choice

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Consumption Choice Sets
uA consumption choice set is the collection of all consumption choices available to the consumer.
uWhat constrains consumption choice?
–Budgetary, time and other resource limitations.

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Budget Constraints
uA consumption bundle containing x1 units of commodity 1, x2 units of commodity 2 and so on up to
xn units of commodity n is denoted by the vector  (x1, x2,  … , xn).
uCommodity prices are p1, p2, … , pn.

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Budget Constraints
uQ: When is a consumption bundle
(x1, … , xn) affordable at given prices p1, … , pn?

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Budget Constraints
uQ: When is a bundle (x1, … , xn) affordable at prices p1, … , pn?
uA: When
          p1x1 + … + pnxn £ m
where m is the consumer’s (disposable) income.

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Budget Constraints
uThe bundles that are only just affordable form the consumer’s budget constraint.  This is the set
{ (x1,…,xn) | x1 ³ 0, …, xn ³ 0 and
                   p1x1 + … + pnxn = m }.

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Budget Constraints
uThe consumer’s budget set is the set of all affordable bundles;
B(p1, … , pn, m) =
{ (x1, … , xn) | x1 ³ 0, … , xn ³ 0 and
                       p1x1 + … + pnxn £ m }
uThe budget constraint is the upper boundary of the budget set.

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Budget Set and Constraint for Two Commodities
x2
x1
Budget constraint is
p1x1 + p2x2 = m.
m /p1
m /p2

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Budget Set and Constraint for Two Commodities
x2
x1
Budget constraint is
p1x1 + p2x2 = m.
m /p2
m /p1

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Budget Set and Constraint for Two Commodities
x2
x1
Budget constraint is
p1x1 + p2x2 = m.
m /p1
Just affordable
m /p2

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Budget Set and Constraint for Two Commodities
x2
x1
Budget constraint is
p1x1 + p2x2 = m.
m /p1
Just affordable
Not affordable
m /p2

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Budget Set and Constraint for Two Commodities
x2
x1
Budget constraint is
p1x1 + p2x2 = m.
m /p1
Affordable
Just affordable
Not affordable
m /p2

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Budget Set and Constraint for Two Commodities
x2
x1
Budget constraint is
p1x1 + p2x2 = m.
m /p1
Budget
Set
    the collection
    of all affordable bundles.
m /p2

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Budget Set and Constraint for Two Commodities
x2
x1
p1x1 + p2x2 = m  is
     x2 = -(p1/p2)x1 + m/p2
     so slope is -p1/p2.
m /p1
Budget
Set
m /p2

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Budget Constraints
uIf n = 3 what do the budget constraint and 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) | x1 ³ 0, x2 ³ 0, x3 ³ 0 and
                p1x1 + p2x2 + p3x3 £ m}

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

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Budget Constraints
uFor n = 2 and x1 on the horizontal axis, the  constraint’s slope is -p1/p2.  What does it mean?
uIncreasing x1 by 1 must reduce x2 by p1/p2.

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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 of
    commodity 1 is p1/p2 units
         foregone of commodity 2.

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Budget Constraints
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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Budget Sets & Constraints; Income and Price Changes
uThe budget constraint and budget set depend upon prices and income.  What happens as prices or
income change?

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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 consumption
choices
x2
x1
Original and
new budget
constraints are
parallel (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 bundles
that are no longer
affordable.
Old and new
constraints
are parallel.

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Budget Constraints - Income Changes
uIncreases in income m shift the  constraint outward in a parallel manner, thereby enlarging the
budget set and improving choice.

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Budget Constraints - Income Changes
uIncreases in income m shift the  constraint outward in a parallel manner, thereby enlarging the
budget set and improving choice.
uDecreases in income m shift the  constraint inward in a parallel manner, thereby shrinking the
budget set and reducing choice.

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Budget Constraints - Income Changes
uNo original choice is lost and new choices are added when income increases, so higher income
cannot make a consumer worse off.
uAn income decrease may (typically will) make the consumer worse off.

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Budget Constraints - Price Changes
uWhat happens if just one price decreases?
uSuppose p1 decreases.

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How do the budget set and budget constraint change as p1 decreases from p1’ to p1”?
Original
budget set
x2
x1
m/p2
m/p1’
m/p1”
-p1’/p2

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How do the budget set and budget constraint change as p1 decreases from p1’ to p1”?
Original
budget set
x2
x1
m/p2
m/p1’
m/p1”
New affordable choices
-p1’/p2

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How do the budget set and budget constraint change as p1 decreases from p1’ to p1”?
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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Budget Constraints - Price Changes
uReducing 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.

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Budget Constraints - Price Changes
uSimilarly, increasing one price pivots the constraint inwards, reduces choice and may (typically
will) make the consumer worse off.

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Uniform Ad  Valorem Sales Taxes
uAn ad valorem sales tax levied at a rate of 5% increases all prices by 5%, from p to (1+0×05)p =
1×05p.
uAn ad valorem sales tax levied at a rate of t increases all prices by tp from p to (1+t)p.
uA uniform sales tax is applied uniformly to all commodities.

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Uniform Ad  Valorem Sales Taxes
uA uniform sales tax levied at rate t changes the constraint from
                 p1x1 + p2x2 = m
to
           (1+t)p1x1 + (1+t)p2x2 = m

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Uniform Ad  Valorem Sales Taxes
uA uniform sales tax levied at rate t changes the constraint from
                 p1x1 + p2x2 = m
to
           (1+t)p1x1 + (1+t)p2x2 = m
i.e.
              p1x1 + p2x2 = m/(1+t).

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Uniform Ad  Valorem Sales Taxes
x2
x1
p1x1 + p2x2 = m

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Uniform Ad  Valorem Sales Taxes
x2
x1
p1x1 + p2x2 = m
p1x1 + p2x2 = m/(1+t)

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Uniform Ad  Valorem Sales Taxes
x2
x1
Equivalent income loss
is

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Uniform Ad  Valorem Sales Taxes
x2
x1
A uniform ad valorem
sales tax levied at rate t
is equivalent to an income
tax levied at rate

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The Food Stamp Program
uFood stamps are coupons that can be legally exchanged only for food.
uHow does a commodity-specific gift such as a food stamp alter a family’s budget constraint?

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The Food Stamp Program
uSuppose m = $100, pF = $1 and the price of “other goods” is pG = $1.
uThe budget constraint is then
              F + G =100.

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The Food Stamp Program
G
F
100
100
F + G = 100; before stamps.

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The Food Stamp Program
G
F
100
100
F + G = 100: before stamps.

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The Food Stamp Program
G
F
100
100
F + G = 100: before stamps.
Budget set after 40 food
stamps issued.
140
20%
40

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The Food Stamp Program
G
F
100
100
F + G = 100: before stamps.
Budget set after 40 food
stamps issued.
140
The family’s budget
set is enlarged.
40

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The Food Stamp Program
uWhat if food stamps can be traded on a black market for $0.50 each?

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The Food Stamp Program
G
F
100
100
F + G = 100: before stamps.
Budget constraint after 40
   food stamps issued.
140
120
Budget constraint with
    black market trading.
40

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The Food Stamp Program
G
F
100
100
F + G = 100: before stamps.
Budget constraint after 40
   food stamps issued.
140
120
Black market trading
    makes the budget
        set larger again.
40

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Budget Constraints - Relative Prices
u“Numeraire” means “unit of account”.
uSuppose prices and income are measured in dollars.  Say p1=$2, p2=$3, m = $12.  Then the
constraint is
               2x1 + 3x2 = 12.

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Budget Constraints - Relative Prices
uIf prices and income are measured in cents, then p1=200, p2=300, m=1200 and the constraint is
         200x1 + 300x2 = 1200,
the same as
                 2x1 + 3x2 = 12.
uChanging the numeraire changes neither the budget constraint nor the budget set.

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Budget Constraints - Relative Prices
uThe constraint for p1=2, p2=3, m=12
                   2x1 + 3x2  = 12
is also     1.x1 + (3/2)x2 =   6,
the constraint for p1=1, p2=3/2, m=6.  Setting p1=1 makes commodity 1 the numeraire and defines all
prices relative to p1; e.g. 3/2 is the price of commodity 2 relative to the price of commodity 1.

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Budget Constraints - Relative Prices
uAny commodity can be chosen as the numeraire without changing the budget set or the budget
constraint.

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Budget Constraints - Relative Prices
up1=2, p2=3 and p3=6  Þ
uprice of commodity 2 relative to commodity 1 is 3/2,
uprice of commodity 3 relative to commodity 1 is 3.
uRelative prices are the rates of exchange of commodities 2 and 3 for units of commodity 1.

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Shapes of Budget Constraints
uQ: What makes a budget constraint a straight line?
uA: A straight line has a constant slope and the constraint is
              p1x1 + … + pnxn = m
so if prices are constants then a  constraint is a straight line.

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Shapes of Budget Constraints
uBut what if prices are not constants?
uE.g. bulk buying discounts, or price penalties for buying “too much”.
uThen constraints will be curved.

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Shapes of Budget Constraints - Quantity Discounts
uSuppose p2 is constant at $1 but that p1=$2 for 0 £ x1 £ 20 and p1=$1 for x1>20.

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Shapes of Budget Constraints - Quantity Discounts
uSuppose p2 is constant at $1 but that p1=$2 for 0 £ x1 £ 20 and p1=$1 for x1>20.  Then the
constraint’s slope is
                 - 2,  for 0 £ x1 £ 20
-p1/p2 =
                 - 1,  for x1 > 20
and the constraint is
{

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Shapes of Budget Constraints with a Quantity Discount
m = $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 with a Quantity Discount
m = $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 with a Quantity Discount
m = $100
50
100
20
80
x2
x1
Budget Set
Budget Constraint

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Shapes of Budget Constraints with a Quantity Penalty
x2
x1
Budget Set
Budget Constraint

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Shapes of Budget Constraints - One Price Negative
uCommodity 1 is stinky garbage.  You are paid $2 per unit to accept it; i.e. p1 = - $2.  p2 = $1.
Income, other than from accepting commodity 1, is m = $10.
uThen 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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Shapes of Budget Constraints - One Price Negative
10
x2
x1
     Budget set is
    all bundles for
  which x1 ³ 0,
x2 ³ 0 and
x2 £ 2x1 + 10.

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More General Choice Sets
uChoices are usually constrained by more than a budget; e.g. time constraints and other resources
constraints.
uA 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 food
must be eaten to survive

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More General Choice Sets
Food
Other Stuff
10
Budget Set
Choice is also budget
constrained.

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More General Choice Sets
Food
Other Stuff
10
Choice is further restricted by a 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 the
intersection of all of
the constraint sets.