Chapter Two

1
Chapter Two

• Budget Constraint

2
Budget Constraints

• Focusing on decisions of an individual buyer.
• How does this buyer decide what to purchase?
• Two things relevant in determining choice
• What buyer can afford given a limited income and
  prices of goods Budget Constraint (Ch. 2)
• What the buyer likes Preferences (Ch. 3)

3
Budget Constraints

• Suppose there is some set of goods that a
  consumer can purchase.
• What constrains consumption choice?
• Assume that prices of goods and income are given.
  
• Any affordable consumption choice must satisfy
• Total Expenditures Total Income

4
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).
• For example Assume there are only two goods
• apples (good 1) and oranges (good 2)
• Then if x1 4 and x2 6, then this means that 4
  apples and 6 oranges are purchased and consumed.

5
Budget Constraints

• Commodity prices are denoted by
• p1, p2, , pn.
• Let m denote the consumers available income for
  purchasing these goods.

6
Budget Constraints

• When is a consumption bundle (x1, , xn)
  affordable at given prices p1, , pn and income
  m?
• Total Expenditures Total Income

7
Budget Constraints

• Total Expenditures Total Income
• Given prices (p1, , pn) of the goods and income
  m, we can write the budget constraint as.
• p1x1 pnxn m

8
Budget Constraints

• The consumers 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

9
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 .
• Just affordable means that the consumer spends
  all of his income.

10
Budget Constaints

• Assume that the consumer only purchases two
  goods.
• Then the budget constraint can be written as p1x1
  p2x2 m.
• Notice, we could draw this constraint by putting
  x2 on the vertical axis, and x1 on the horizontal
  axis. To do so rewrite the constraint as x2 m/
  p2- (p1/p2 ) x1

11
Budget Constaints

• The budget constraint has the form of a line.
• x2 m/ p2 - (p1/p2 ) x1
• x2 is the vertical axis good
• x1 is the horizontal axis good
• m/ p2 is the vertical intercept
• - p1/p2 is the slope of the line

12
Budget Set and Constraint for Two Commodities
x2
Budget constraint is p1x1 p2x2 m. Or x2
m/ p2 - (p1/p2 )x1
m /p2
x1
m /p1
13
Budget Set and Constraint for Two Commodities
x2
Budget constraint is p1x1 p2x2 m. Or x2
m/ p2 - (p1/p2 )x1
m /p2
Just affordable
x1
m /p1
14
Budget Set and Constraint for Two Commodities
x2
Budget constraint is p1x1 p2x2 m. Or x2
m/ p2 - (p1/p2 )x1
m /p2
Not affordable
Just affordable
x1
m /p1
15
Budget Set and Constraint for Two Commodities
x2
Budget constraint is p1x1 p2x2 m. Or x2
m/ p2 - (p1/p2 )x1
m /p2
Not affordable
Just affordable
Affordable (but dont spend all income.
x1
m /p1
16
Budget Set and Constraint for Two Commodities
x2
Budget constraint is p1x1 p2x2 m. Or x2
m/ p2 - (p1/p2 )x1
m /p2
the collection of all affordable
bundles includes all points on and below
the budget constraint line.
Budget Set
x1
m /p1
17
Budget Constraints

• If n 3 what do the budget constraint and the
  budget set look like?

18
Budget Constraint for Three Commodities
x2
p1x1 p2x2 p3x3 m This is a plane instead
of a line.
m /p2
m /p3
x3
m /p1
x1
19
Budget Set for Three Commodities
x2
The Budget SET (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
20
Budget Constraints Slope

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

21
Budget Constraints Slope
x2

• Slope is -p1/p2
• Increasing x1 by 1 must
• reduce x2 by p1/p2.

-p1/p2
1
x1
22
Budget Constraints Slope
x2
Opportunity cost of an extra unit of
commodity 1 is p1/p2 units foregone of
commodity 2.
-p1/p2
1
x1
23
Budget Constraints Slope
x2
The opp. cost of an extra unit
of commodity 2 is p2/p1
units foregone of
commodity 1.
1
-p2/p1
x1
24
Budget Constraints Slope

• The slope of the budget constraint is the
  opportunity cost of 1 unit of good 1 (in terms of
  the number of units of good 2 that must be given
  up)
• 1/slope of the budget constraint is the
  opportunity cost of 1 unit of good 2 (in terms of
  the number of units of good 1 that must be given
  up)

25
Example 2.1Budget Constraints

• Example 2.1
• Joe consumes text books (T) and beer (B).
• The price of a text book is 100 and the price of
  a beer is 2. Joe has 1000 available to spend.
• Draw Joes budget set and constraint.

26
Example 2.1 continuedBudget Constraints

• Joes budget set can be written as
• 100T2B 1000
• Or B 500-50T
• Joes budget constraint is
• B 500-50T

27
Example 2.1 continuedBudget Constraints
B
Joes budget constraint B 500-50T
500
Joes budget set B 500-50T
T
10
28
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?

29
How do the budget set and budget constraint
change as income m increases from m to m?
x2
m/p2
Original budget set
x1
m/p1
30
Higher income gives more choice
x2
New affordable consumptionchoices
m/p2
Original and new budget constraints are parallel
(same slope).
m/p2
Original budget set
x1
m/p1
m/p1
31
Higher income gives more choice

• Increases in income m shift the constraint
  outward in a parallel manner, thereby enlarging
  the budget set and improving choice availability.
• No original choice is lost and new choices are
  added when income increases, so higher income
  cannot make a consumer worse off.

32
How do the budget set and budget constraint
change as income m decreases from m to m?
x2
m/p2
Original budget set
x1
m/p1
33
Lower income gives less choice
x2
m/p2
Consumption bundles that are no longer affordable.
m/p2
Old and new constraints are parallel.
New, smaller budget set
x1
mp1
m/p1
34
Lower income gives less choice

• Decreases in income m shift the constraint
  inward in a parallel manner, thereby shrinking
  the budget set and reducing choice availability.
• An income decrease may (typically will) make the
  consumer worse off.

35
Budget Constraints - Price Changes

• What happens if just one price decreases?
• Suppose p1 decreases.

36
How do the budget set and budget constraint
change as p1 decreases from p1 to p1?
x2
m/p2
-p1/p2
Original budget set
x1
m/p1
m/p1
37
How do the budget set and budget constraint
change as p1 decreases from p1 to p1?
x2
m/p2
New affordable choices
-p1/p2
Original budget set
x1
m/p1
m/p1
38
How do the budget set and budget constraint
change as p1 decreases from p1 to p1?
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
39
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.

40
Budget Constraints - Price Changes

• Similarly, increasing one price pivots the
  constraint inwards, reduces choice and may
  (typically will) make the consumer worse off.

41
Taxes and Budget Constraints

• The government may impose taxes on consumers when
  they purchase goods.
• What we will see is that taxes cause a
  contraction in the budget set making fewer
  consumption bundles affordable for purchase. This
  will typically make consumers worse off.

42
Taxes and Budget Constraints

• Types of taxes
• quantity or per unit tax
• Value or ad valorem tax
• Lump sum tax on income
• Taxes change the shape of the budget constraint
  by either changing prices or disposable income.

43
Taxes and Budget Constraints

• A quantity or unit tax is a tax per unit
  purchased.
• For example, a tax of 1 per gallon of gasoline
• If gasoline normally costs 2.00 per gallon, then
  the consumer must now pay a total of 3 per
  gallon (assuming the price does not change).
• After a per unit tax of t, then the total price
  that a consumer pays increases from p to (pt).

44
Taxes and Budget Constraints

• A value or ad valorem tax is a tax on the value
  of the purchased item (expressed as a percentage)
• For example, a 20 tax on wine implies that if
  wine costs 10 per bottle, then for each bottle
  purchased a tax of .2102 is imposed.
• The total price a consumer pays for each bottle
  will be 12 per bottle.
• In general, after an ad valorem tax of t, the
  total price of the taxed good paid by the
  consumer will rise from p to p(1t).

45
Taxes and Budget Constraints

• No matter whether the tax is a per unit or ad
  valorem tax, the price of the taxed item
  increases.
• Suppose there are only two goods and a per unit
  tax of t is imposed on good 1.

46
Taxes and Budget Constraints
x2
Before the tax, the budget constraint is given
by p1x1 p2x2 m
m/p2
x1
m/p1
47
Taxes and Budget Constraints
x2

m/p2
After the tax, the budget constraint is given
by (p1t)x1 p2x2 m
x1
m/(p1t)
m/p1
48
Uniform Ad Valorem Sales Taxes

• What if the same ad valorem tax is levied
  uniformly across all goods?
• A uniform sales tax is the same tax applied
  uniformly to all commodities.
• An ad valorem sales tax levied at a rate of t
  increases all prices by tp from p to (1t)p.

49
Uniform Ad Valorem Sales Taxes

• A uniform sales tax levied at rate t changes the
  constraint from p1x1 p2x2
  mto (1t)p1x1 (1t)p2x2 m

50
Uniform Ad Valorem Sales Taxes

• A uniform sales tax levied at rate t changes the
  constraint from p1x1 p2x2
  mto (1t)p1x1 (1t)p2x2 mi.e.
  p1x1 p2x2 m/(1t).
• This has the same effect as a decrease in income.

51
Uniform Ad Valorem Sales Taxes
x2
p1x1 p2x2 m
m/p2
x1
m/p1
52
Uniform Ad Valorem Sales Taxes
x2
p1x1 p2x2 m
m/p2
p1x1 p2x2 m/(1t)
m/((1t)p2)
x1
m/p1
m/((1t)p1)
53
Uniform Ad Valorem Sales Taxes

• A uniform ad valorem sales tax levied at rate t
  is equivalent to an incometax levied at rate
  t/(1t)

54
Lump sum income tax

• A lump sum income tax is where the government
  makes a consumer pay a certain amount
  regardless of what they purchase.
• (Assume that the pre-tax level of income does not
  change)
• For example, if the government imposes a lump sum
  income tax of 100, then the new budget
  constraint will be p1x1 p2x2 m-100

55
Subsidies and Budget Constraints

• The government may also choose to subsidize
  consumption of a good.
• A consumption subsidy is the opposite of a tax
  the government pays consumers when they purchase
  goods.

56
Subsidies and Budget Constraints

• Types of Subsidies
• Per unit subsidy
• Ad valorem subsidy
• Lump sum income subsidy
• Subsidies will expand the budget constraint
  making more consumption choices affordable to
  consumers.

57
Subsidies and Budget Constraints

• A quantity or per unit subsidy of s dollars will
  lower the per unit amount paid by the consumer
  from p to (p-s)
• An ad valorem subsidy s (expressed as a percent
  of the price) will lower the amount paid by the
  consumer per unit from p to p(1-s)
• A lump sum subsidy is a flat amount of money that
  is given to consumers regardless of what or how
  much they purchase.

58
Subsidies and Budget Constraints

• For example Suppose the government decides to
  subsidize computers by offering a 25 ad valorem
  subsidy.
• The pre-subsidy price that consumers were paying
  was 1000.
• Now the total that consumers pay (net of the
  subsidy) is 1000-.251000 750

59
Example 2.2 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?

60
Example 2.2 The Food Stamp Program

• Consider the two good example where consumers
  purchase food (F) and all other goods are lumped
  into one category (G).
• Suppose m 100, pF 1 and the price of other
  goods is pG 1.
• The budget constraint is then F G
  100.

61
Example 2.2 The Food Stamp Program
G
F G 100 before food stamps are issued.
100
F
100
62
Example 2.2 The Food Stamp Program

• Now assume that the government offers each family
  food stamps worth 40.
• Draw the new budget constraint of a typical
  family.

63
Example 2.2 The Food Stamp Program
G
F G 100 before stamps.
100
F
100
64
Example 2.2 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
65
Example 2.2 The Food Stamp Program

• What if food stamps can be traded on a black
  market for 0.50 each?
• This will mean that families can actually sell
  their food stamps to buy other goods instead of
  food.

66
Example 2.2 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
67
Budget Constraints - Relative Prices

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

68
Budget Constraints - Relative Prices

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

69
Budget Constraints - Relative Prices

• The constraint for p12, p23, m12
  2x1 3x2 12 is also 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 e.g. 3/2 is
  the price of commodity 2 relative to the price of
  commodity 1.

70
Budget Constraints - Relative Prices

• Any commodity can be chosen as the numeraire
  without changing the budget set or the budget
  constraint.

71
Shapes of Budget Constraints

• What makes a budget constraint a straight line?
• A A straight line has a constant slope and the
  constraint is p1x1 pnxn
  mso if prices are constants then a constraint
  is a straight line.

72
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 non-linear.

73
Example 2.3Quantity Discounts

• Suppose p2 is constant at 1 but that p12 for 0
  x1 20 and p11 for x1gt20. m100
• This is an example of a quantity discount where
  the price drops for good 1 once a certain
  quantity is purchased.
• Note In this example, we assume that the first
  20 units of x1 cost 2 and the rest after that
  cost 1.

74
Example 2.3Quantity 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
• Draw the Budget Constraint

75
Example 2.3Quantity Discounts
x2
m 100
Slope - 2 / 1 - 2 (p12, p21)
100
60
Slope - 1/ 1 - 1 (p11, p21)
x1
80
50
20
76
Example 2.3Quantity Discounts
x2
m 100
Slope - 2 / 1 - 2 (p12, p21)
100
Slope - 1/ 1 - 1 (p11, p21)
x1
80
50
20
77
Example 2.3Quantity Discounts
x2
m 100
100
Budget Constraint
Budget Set
x1
80
50
20
78
Shapes of Budget Constraints with a Quantity
Penalty
x2
A quantity penalty where the price of good 1
increases after a certain amount Is purchased.
Budget Constraint
Budget Set
x1
79
Shapes of Budget Constraints - One Price Negative

• We could also extend our analysis to derive
  budget constraints when the price of one good is
  negative ie when consumers are paid to consume
  these goods.

80
Example 2.4 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 budget constraint is - 2x1 x2 10
  or x2 2x1 10.

81
Example 2.4 One Price Negative
x2
x2 2x1 10
Budget constraints slope is -p1/p2 -(-2)/1 2
10
x1