Two
Budgetary Chapter and Other Constraints on Choice
Consumption Choice Sets
A consumption choice set is the collection of all consumption choices available to the consumer. What constrains consumption choice? Budgetary, time and other resource limitations.
Budget Constraints
A 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 n). Commodity prices are p1, p2 n.
Budget Constraints
Q: When is a consumption bundle (x1 n) affordable at given prices p1 n?
Budget Constraints
Q: When is a bundle (x1 affordable at prices p1 A: When p1x1 m nxn where m (disposable) income.
n)
n?
Budget Constraints
The bundles that are only just budget constraint. This is the set
{ (x1
n)
| x1 0 p1x1
n
and m }. nxn
Budget Constraints
budget set is the set of all affordable bundles; B(p1 n, m) = { (x1 0 0 and n) | x1 n p1x1 m} nxn The budget constraint is the upper boundary of the budget set.
x
2
Budget Set and Constraint for Two Commodities
Budget constraint is p1x1 + p2x2 = m.
m /p2
m /p1
x1
x
2
Budget Set and Constraint for Two Commodities
Budget constraint is p1x1 + p2x2 = m.
m /p2
m /p1
x1
x
2
Budget Set and Constraint for Two Commodities
Budget constraint is p1x1 + p2x2 = m.
Just affordable
m /p2
m /p1
x1
x
2
Budget Set and Constraint for Two Commodities
Budget constraint is p1x1 + p2x2 = m.
Not affordable Just affordable
m /p2
m /p1
x1
x
2
Budget Set and Constraint for Two Commodities
Budget constraint is p1x1 + p2x2 = m.
Not affordable Just affordable Affordable
m /p2
m /p1
x1
x
2
Budget Set and Constraint for Two Commodities
Budget constraint is p1x1 + p2x2 = m. the collection of all affordable bundles. Budget Set
m /p2
m /p1
x1
x
2
Budget Set and Constraint for Two Commodities
p1x1 + p2x2 = m is x2 = -(p1/p2)x1 + m/p2 so slope is -p1/p2. Budget Set
m /p2
m /p1
x1
Budget Constraints
If n = 3 what do the budget constraint and the budget set look like?
Budget Constraint for Three Commodities
x2
m /p2
p1x1 + p2x2 + p3x3 = m
m /p3
x3
m /p1
x1
Budget Set for Three Commodities
x2
m /p2
{ (x1,x2,x3) | x1 0, x2 0, x3 0 and p1x1 + p2x2 + p3x3 m}
m /p3
x3
m /p1
x1
Budget Constraints
For n = 2 and x1 on the horizontal -p1/p2. What does it mean?
x2
p1 x1 p2
m p2
Budget Constraints
For n = 2 and x1 on the horizontal -p1/p2. What does it mean?
x2
p1 x1 p2
m p2
Increasing x1 by 1 must reduce x2 by p1/p2.
Budget Constraints
x2
Slope is -p1/p2
-p1/p2
+1
x1
Budget Constraints
x2
Opp. cost of an extra unit of commodity 1 is p1/p2 units foregone of commodity 2.
-p1/p2
+1
x1
Budget Constraints
x2
Opp. cost of an extra unit of commodity 1 is p1/p2 units foregone of commodity 2. And the opp. cost of an extra +1 unit of commodity 2 is -p2/p1 p2/p1 units foregone of commodity 1. x1
Budget Sets & Constraints; Income and Price Changes
The budget constraint and budget set depend upon prices and income. What happens as prices or income change?
How do the budget set and budget constraint change as income m x2 increases?
Original budget set
x1
Higher income gives more choice
x2
New affordable consumption choices Original and new budget constraints are parallel (same slope). Original budget set x1
How do the budget set and budget constraint change as income m x2 decreases?
Original budget set
x1
How do the budget set and budget constraint change as income m x2 decreases?
Consumption bundles that are no longer affordable. New, smaller budget set Old and new constraints are parallel.
x1
Budget Constraints - Income Changes
Increases in income m shift the constraint outward in a parallel manner, thereby enlarging the budget set and improving choice.
Budget Constraints - Income Changes
Increases in income m shift the constraint outward in a parallel manner, thereby enlarging the budget set and improving choice. Decreases in income m shift the constraint inward in a parallel manner, thereby shrinking the budget set and reducing choice.
Budget Constraints - Income Changes
No original choice is lost and new choices are added when income increases, so higher income cannot make a consumer worse off. An income decrease may (typically will) make the consumer worse off.
Budget Constraints - Price Changes
What happens if just one price decreases? Suppose p1 decreases.
How do the budget set and budget constraint change as p1 decreases x2 from p1 to p1 ?
m/p2 -p1
2
Original budget set
m/p1
m/p1
x1
How do the budget set and budget constraint change as p1 decreases x2 from p1 to p1 ?
m/p2 -p1 New affordable choices
2
Original budget set
m/p1
m/p1
x1
How do the budget set and budget constraint change as p1 decreases x2 from p1 to p1 ?
m/p2 -p1 New affordable choices
2
Original budget set
Budget constraint pivots; slope flattens from -p1 2 to -p1 2 -p
1 2
m/p1
m/p1
x1
Budget Constraints - Price Changes
Reducing 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.
Budget Constraints - Price Changes
Similarly, increasing one price pivots the constraint inwards, reduces choice and may (typically will) make the consumer worse off.
Uniform Ad V a lo r em Sales Taxes
An ad valorem sales tax levied at a rate of 5% increases all prices by 5%, from p to (1+0 05)p = 1 05p. An ad valorem sales tax levied at a rate of t increases all prices by tp from p to (1+t)p. A uniform tax sales is applied uniformly to all commodities.
Uniform Ad V a lor em Sales Taxes
A uniform sales tax levied at rate t changes the constraint from p1x1 + p2x2 = m to (1+t)p1x1 + (1+t)p2x2 = m
Uniform Ad V a lor em Sales Taxes
A 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).
Uniform Ad V a lo r em Sales Taxes
x2
m p2
p1x1 + p2x2 = m
m p1
x1
Uniform Ad V a lo r em Sales Taxes
x2
m (1 t ) p2
m p2
p1x1 + p2x2 = m
p1x1 + p2x2 = m/(1+t)
m (1 t ) p1
m p1
x1
Uniform Ad V a lo r em Sales Taxes
x2
m (1 t ) p2
m p2
Equivalent income loss is m t m m 1t1t
m (1 t ) p1
m p1
x1
Uniform Ad V a lo r em Sales Taxes
x2
m (1 t ) p2
m p2
A uniform ad valorem sales tax levied at rate t is equivalent to an income t tax levied at rate
1t
.
m (1 t ) p1
m p1
x1
The Food Stamp Program
Food stamps are coupons that can be legally exchanged only for food. How does a commodity-specific gift
budget constraint?
The Food Stamp Program
Suppose m = $100, pF = $1 and the G = $1. The budget constraint is then F + G =100.
G 100
The Food Stamp Program
F + G = 100; before stamps.
100
F
G 100
The Food Stamp Program
F + G = 100: before stamps.
100
F
G 100
The Food Stamp Program
F + G = 100: before stamps.
Budget set after 40 food stamps issued.
40
100 140
F
G 100
The Food Stamp Program
F + G = 100: before stamps.
Budget set after 40 food stamps issued.
set is enlarged. 40 100 140 F
The Food Stamp Program
What if food stamps can be traded on a black market for $0.50 each?
G 120 100
The Food Stamp Program
F + G = 100: before stamps. Budget constraint after 40 food stamps issued. Budget constraint with black market trading.
40
100 140
F
G 120 100
The Food Stamp Program
F + G = 100: before stamps. Budget constraint after 40 food stamps issued. Black market trading makes the budget set larger again. 40 100 140 F
Budget Constraints - Relative Prices
Suppose prices and income are measured in dollars. Say p1=$2, p2=$3, m = $12. Then the constraint is 2x1 + 3x2 = 12.
Budget Constraints - Relative Prices
If 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. Changing the numeraire changes neither the budget constraint nor the budget set.
Budget Constraints - Relative Prices
The 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.
Budget Constraints - Relative Prices
Any commodity can be chosen as the numeraire without changing the budget set or the budget constraint.
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 to commodity 1 is 3. Relative prices are the rates of exchange of commodities 2 and 3 for units of commodity 1.
Shapes of Budget Constraints
Q: What makes a budget constraint a straight line? A: A straight line has a constant slope and the constraint is p1x1 nxn = m so if prices are constants then a constraint is a straight line.
Shapes of Budget Constraints
But what if prices are not constants? E.g. bulk buying discounts, or price
Then constraints will be curved.
Shapes of Budget Constraints Quantity Discounts
Suppose p2 is constant at $1 but that p1=$2 for 0 x1 20 and p1=$1 for x1>20.
Shapes of Budget Constraints Quantity Discounts
Suppose p2 is constant at $1 but that p1=$2 for 0 x1 20 and p1=$1 for x1>20 - 2, for 0 x1 20 -p1/p2 = - 1, for x1 > 20 and the constraint is
{
x2
100
Shapes of Budget Constraints with a Quantity Discount
m = $100
Slope = - 2 / 1 = - 2 (p1=2, p2=1) Slope = - 1/ 1 = - 1 (p1=1, p2=1)
20 50
80
x1
x2
100
Shapes of Budget Constraints with a Quantity Discount
m = $100
Slope = - 2 / 1 = - 2 (p1=2, p2=1) Slope = - 1/ 1 = - 1 (p1=1, p2=1)
20 50
80
x1
x2
100
Shapes of Budget Constraints with a Quantity Discount
m = $100 Budget Constraint
Budget Set
20 50
80
x1
x2
Shapes of Budget Constraints with a Quantity Penalty
Budget Constraint
Budget Set
x1
Shapes of Budget Constraints One Price Negative
Commodity 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. Then the constraint is - 2x1 + x2 = 10 or x2 = 2x1 + 10.
Shapes of Budget Constraints One Price Negative
x2 x2 = 2x1 + 10
-p1/p2 = -(-2)/1 = +2 10
x1
Shapes of Budget Constraints One Price Negative
x2
Budget set is all bundles for which x1 0, x2 0 and x2 2x1 + 10.
10
x1
More General Choice Sets
Choices are usually constrained by more than a budget; e.g. time constraints and other resources constraints. A bundle is available only if it meets every constraint.
More General Choice Sets
Other Stuff
At least 10 units of food must be eaten to survive
10
Food
More General Choice Sets
Other Stuff Choice is also budget constrained.
Budget Set
10
Food
More General Choice Sets
Other Stuff Choice is further restricted by a time constraint.
10
Food
More General Choice Sets
So what is the choice set?
More General Choice Sets
Other Stuff
10
Food
More General Choice Sets
Other Stuff
10
Food
More General Choice Sets
Other Stuff The choice set is the intersection of all of the constraint sets.
10
Food