Chapter 2 Budget Constraint

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Chapter 2Budget Constraint
2
Consumption Theory

• Economists assume that consumers choose the best
  bundle of goods they can afford.
• In this chapter, we examine how to describe what
  a consumer can afford the budget constraint, and
  budget set.
• What is best for a consumer, or the preference on
  the possible consumption bundles, will be
  discussed in the next chapter.

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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, , xn).
• Commodity prices are p1, p2, , pn.

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Budget Constraints

• Q When is a consumption bundle (x1, , xn)
  affordable at prices p1, , pn?
• A When total expenditure is smaller than
  income p1x1 pnxn mwhere m is
  the consumers (disposable) income.
• That is, the consumers affordable consumption
  bundles are those that dont cost more than m.

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Budget Constraints

• The bundles that are only just affordable form
  the consumers budget constraint. This is the
  set (x1,,xn) x1 ³ 0, , xn ³ 0 and
  p1x1 pnxn m .

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Budget Constraints

• The consumers budget set is the set of all
  affordable bundlesB(p1, , pn, m) (x1, ,
  xn) x1 ³ 0, , xn ³ 0 and
  p1x1 pnxn m
• The budget constraint is the upper boundary of
  the budget set.

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Budget Set and Constraint for Two Commodities
x2
Budget constraint is p1x1 p2x2 m.
m /p2
x1
m /p1
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Budget Set and Constraint for Two Commodities
x2
Budget constraint is p1x1 p2x2 m.
m /p2
x1
m /p1
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Budget Set and Constraint for Two Commodities
x2
Budget constraint is p1x1 p2x2 m.
m /p2
Just affordable
x1
m /p1
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Budget Set and Constraint for Two Commodities
x2
Budget constraint is p1x1 p2x2 m.
m /p2
Not affordable
Just affordable
x1
m /p1
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Budget Set and Constraint for Two Commodities
x2
Budget constraint is p1x1 p2x2 m.
m /p2
Not affordable
Just affordable
Affordable
x1
m /p1
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Budget Set and Constraint for Two Commodities
x2
Budget constraint is p1x1 p2x2 m.
m /p2
the collection of all affordable bundles.
Budget Set
x1
m /p1
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Budget Set and Constraint for Two Commodities
x2
p1x1 p2x2 m is x2 -(p1/p2)x1 m/p2
so slope is -p1/p2.
m /p2
Budget Set
x1
m /p1
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Budget Constraint for Three Commodities
x2
p1x1 p2x2 p3x3 m
m /p2
m /p3
x3
m /p1
x1
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Budget Set for Three Commodities
x2
(x1,x2,x3) x1 ³ 0, x2 ³ 0, x3 ³ 0 and
p1x1 p2x2 p3x3 m
m /p2
m /p3
x3
m /p1
x1
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Budget Constraints

• For n 2 and x1 on the horizontal axis, the
  constraints slope is -p1/p2. What does it
  mean?
• Holding income m constant, increasing x1 by 1
  unit must reduce x2 by p1/p2 units.

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Budget Constraints
x2
Slope is -p1/p2
-p1/p2
1
x1
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Budget Constraints
x2
Opp. cost of an extra unit of commodity 1 is
p1/p2 units foregone of commodity 2.
-p1/p2
1
x1
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Budget Constraints
x2
The opp. cost of an extra
unit of commodity 2 is
p2/p1 units foregone
of commodity 1.
1
-p2/p1
x1
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Budget Sets Constraints as Income or Price
Changes

• The budget constraint and budget set depend upon
  prices and income.
• When prices and incomes change, the set of goods
  that a consumer can afford changes as well. How
  do these changes affect the budget constraint and
  budget set?

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How do the budget set and budget constraint
change as income m increases?
x2
Original budget set
x1
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Higher income gives more choice
x2
New affordable consumptionchoices
Original and new budget constraints are parallel
(same slope).
Original budget set
x1
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How do the budget set and budget constraint
change as income m decreases?
x2
Original budget set
x1
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How do the budget set and budget constraint
change as income m decreases?
x2
Consumption bundles that are no longer affordable.
Old and new constraints are parallel.
New, smaller budget set
x1
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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.
• The slope p1 / p2 does not change.

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Budget Constraints - Income Changes

• When income increases, no original choice is lost
  and new choices are added, so higher income
  cannot make a consumer worse off.
• When income decreases, a consumer may (typically
  will) be worse off, as one can no longer afford
  some of the bundles anymore.

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Budget Constraints - Price Changes

• What happens to the budget constraint and budget
  set if one of the prices changes?
• For example, consider a situation in which p1
  decreases while p2 and income remain unchanged.

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How do the budget set and budget constraint
change as p1 decreases?
x2
m/p2
-p1/p2
Original budget set
x1
m/p1
m/p1
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How do the budget set and budget constraint
change as p1 decreases?
x2
m/p2
New affordable choices
-p1/p2
Original budget set
x1
m/p1
m/p1
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How do the budget set and budget constraint
change as p1 decreases?
x2
m/p2
New affordable choices
Budget constraint pivots slope flattens
from -p1/p2 to -p1/p2
-p1/p2
Original budget set
-p1/p2
x1
m/p1
m/p1
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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.
• Similarly, increasing one price pivots the
  constraint inwards, reduces choice and may
  (typically will) make the consumer worse off.

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Tax and Subsidy

• Quantity/per-unit tax price increases from p to
  pt.
• Quantity/per-unit subsidy price decreases from p
  to p-s.
• Ad valorem/value tax price increases from p to
  (1t)p.
• Ad valorem/value subsidy price decreases from p
  to (1-s)p.

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Uniform Ad Valorem Sales Taxes

• A uniform sales tax levied at rate t on all goods
  changes the constraint from p1x1
  p2x2 mto (1t)p1x1 (1t)p2x2
  mi.e. p1x1 p2x2 m/(1t).

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Uniform Ad Valorem Sales Taxes
x2
p1x1 p2x2 m
x1
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Uniform Ad Valorem Sales Taxes
x2
p1x1 p2x2 m
p1x1 p2x2 m/(1t)
x1
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Uniform Ad Valorem Sales Taxes
x2
Equivalent income lossis
x1
37
Uniform Ad Valorem Sales Taxes
x2
A uniform ad valoremsales tax levied at rate
tis equivalent to an incometax levied at rate
x1
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Lump-Sum Tax and Subsidy

• Lump-sum tax government tax a fixed sum of
  money, T, regardless of individuals behavior.
• This is equivalent to a decrease in income by T,
  implying an inward parallel shift of budget line.
• Similarly, lump-sum subsidy S implies an outward
  parallel shift of budget line corresponding to an
  amount S.

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The Food Stamp Program

• Food stamps are coupons that can be legally
  exchanged only for food.
• How does a commodity-specific gift such as a food
  stamp alter a familys budget constraint?
• Here we assume one of the two goods is food.

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The Food Stamp Program

• Suppose m 100, pF 1 and the price of other
  goods is pG 1.
• The budget constraint is then F G
  100.

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The Food Stamp Program
G
F G 100 before stamps.
100
F
100
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The Food Stamp Program
G
F G 100 before stamps.
100
Budget set after 40 foodstamps issued.
F
100
140
40
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The Food Stamp Program
G
F G 100 before stamps.
100
Budget set after 40 foodstamps issued.
The familys budgetset is enlarged.
F
100
140
40
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The Food Stamp Program

• What if food stamps can be traded on a black
  market for 0.50 each?

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The Food Stamp Program
G
F G 100 before stamps.
120
Budget constraint after 40 food stamps issued.
100
Black market trading makes the budget
set larger again.
F
100
140
40
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Budget Constraints - Relative Prices

• How does the unit of account affect the budget
  constraint and budget set?
• Suppose prices and income are measured in
  dollars. Say p12, p23, m 12. Then the
  constraint is 2x1 3x2 12.

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Budget Constraints - Relative Prices

• If prices and income are measured in cents, then
  p1200, p2300, m1200 and the constraint is
  200x1 300x2 1200,the same as
  2x1 3x2 12.
• Changing the unit of account changes neither the
  budget constraint nor the budget set.

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Budget Constraints - Relative Prices

• The constraint for p12, p23, m12
  2x1 3x2 12 can also be written as
• 1x1 (3/2)x2 6,the
  constraint for p11, p23/2, m6.
• Setting p11 makes commodity 1 the numeraire and
  defines all prices relative to p1.
• For example, 3/2 is the price of commodity 2
  relative to the price of commodity 1.

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Budget Constraints - Relative Prices

• Multiplying all prices and income by any constant
  k does not change the budget constraint
• kp1x1 kp2x2 km.
• Any commodity can be chosen as the numeraire
  without changing the budget set or the budget
  constraint.
• It is also clear from the graph that, only the
  ratios p1/p2, m/p1 and m/p2 are relevant to the
  budget line and budget set.

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Shapes of Budget Constraints

• Q What makes a budget constraint a straight
  line?
• A A straight line must have a constant slope.
  The constraint is p1x1
  p2x2 m.So if prices are constants, the
  constraint is a straight line.

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Shapes of Budget Constraints

• But what if prices are not constants?
• For example, bulk buying discounts, or price
  penalties for buying too much.
• Then constraints will be curved or have kinks.

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Shapes of Budget Constraints - Quantity Discounts

• Suppose p2 is constant at 1 but that p12 for 0
  x1 20 and p11 for x1gt20.

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Shapes of Budget Constraints - Quantity Discounts

• Suppose p2 is constant at 1 but that p12 for 0
  x1 20 and p11 for x1gt20.
• Then the constraints slope is -
  2, for 0 x1 20-p1/p2 -
  1, for x1 gt 20
• Also assume m100.

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Shapes of Budget Constraints with a Quantity
Discount
x2
m 100
Slope - 2 / 1 - 2 (p12, p21)
100
Slope - 1/ 1 - 1 (p11, p21)
x1
80
50
20
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Shapes of Budget Constraints with a Quantity
Discount
x2
m 100
Slope - 2 / 1 - 2 (p12, p21)
100
Slope - 1/ 1 - 1 (p11, p21)
x1
80
50
20
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Shapes of Budget Constraints with a Quantity
Discount
x2
m 100
100
Budget Constraint
Budget Set
x1
80
50
20
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Shapes of Budget Constraints with a Quantity
Penalty
x2
Budget Constraint
Budget Set
x1
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Whats Next?

• The budget set describes what consumption bundles
  are affordable to the consumers.
• The next chapter will introduce preference, which
  describes the ordering of what a consumer likes
  among the consumption bundles.
• Then we can combine both preference and budget
  constraint to analyze consumers choice.