Chapter 2
Budgets
Intermediate Microeconomics:
A Tool-Building Approach
Routledge, UK
© 2016 Samiran Banerjee
Commodity Space
• Two goods case: good 1 and good 2
• Quantities are denoted by x1 and x2
• Generally assume goods are divisible
x2
X
A = (4, 3) is a
commodity bundle
The commodity space is
the entire non-negative
quadrant
A
3
0
4
x1
Competitive Budgets
• Main feature: Price per unit is always constant
• Per unit prices are denoted by p1 and p2
• Income is denoted by m
• Budget constraint: p1x1 + p2x2 ≤ m
Expenditure
on good 1
Expenditure
on good 2
• Budget line: p1x1 + p2x2 = m
Rewriting,
x2 =
Vertical intercept
p1
m
x1
–
p2
p2
Absolute value of the
slope of the budget
|Slope of budget| =
ratio of prices
Competitive budget example 1
• p1 = $2 per unit
• p2 = $1 per unit
• m = $10
Vertical
intercept
Budget
line
x2
m/ p2
(0, 10)
R
T
Budget
set
S
(5, 0)
0
m/ p1
x1
Horizontal
intercept
Competitive budget example 2
• 3 goods
• p1 = p2 = p3 = $2 per unit
• m = $20
x3
Intercept
for good 3
Intercept
for good 1
(10, 0, 0)
x1
Budget “line”
(surface)
(0, 0, 10)
(0, 10, 0)
Intercept
for good 2
x2
(The tetrahedron defined by the space between the budget surface and
the origin is the budget set.)
Competitive budget: Change in p1
• m = $10
• p2 = $1 per unit
• p1old = $2 per unit, falls to p1new = $1.25
x2
10
New budget
Old budget
– 1.25
0
5
8
x1
Competitive budget: Change in p2
• m = $10
• p1 = $2 per unit
• p2old = $1 per unit, rises to p2new = $2
x2
10
Old budget
5
–1
New budget
0
5
x1
Competitive budget: Change in m
• p1 = $2 per unit
• p2 = $1 per unit
• mold = $10, falls to mnew = $6
x2
10
Old budget
6
–2
New budget
0
3
5
x1
Endowment budget
• Person i’s endowment: ωi = (4, 3)
• pa = $1 per unit
• pb = $2 per unit
“omega”
The value of this
endowment at these
prices is $10:
bananas
($1 x 4) + ($2 x 3) = $10
5
ωi
3
– 0.5
0
4
10
apples
Endowment budget: price change
• Person i’s endowment: ωi = (4, 3)
• pa = $1 per unit
• pb falls from $2 per unit to $1
The value of this
endowment at these
prices is $7:
bananas
($1 x 4) + ($1 x 3) = $7
7
5
New budget
Old budget
ωi
3
–1
0
4
– 0.5
7
10
apples
Budget line pivots about the endowment point!
Non-Competitive Budgets
• Main feature: Price per unit is NOT always constant
• Some examples:
– price discounts on incremental purchases
– price discounts with bulk purchases
– buying and selling at different prices
– food stamps
– coupons
• Assume goods are divisible!
Incremental price discounts
• p1 = $10 per unit
up to 6 units
• p2 = $6 per unit
• m = $120
x2
• p1 = $6 per unit
beyond 6 units
• p2 = $6 per unit
• m = $120
Trade off
is 5 units
of good 2
for 3 units
of good 1
20
– 5/ 3
A
10
x2
Trade off
is 1 unit of
good 2 for
1 unit of
good 1
20
A
10
–1
0
6
x1
0
6
16
x1
Buy price ≠ Sell price
• ωk = (10, 10): Ms. k’s endowment of Euros and US dollars
• p$ = €0.80 per $ (a dollar can be bought for €0.80)
• p€ = $1.25 per € (a Euro can be bought for $1.25)
Starting from ωk,
one Euro can be
sold for $1.25.
All 10 Euros can
be sold for $8.
$
18
– 0.8
10
ωk
–1.25
0
10
18
€
Starting from ωk,
one dollar can be
sold for €0.80. All
10 dollars can be
sold for €8.
Food stamps
• Price of food: p1 = $5 per unit
• Price of clothing: p2 = $5 per unit
• m = $100
• Government provides 4 stamps (divisible): 1 stamp = 1
unit of food
Clothing
Budget line
without
food stamps
20
A
B
10
Shaded budget
set with food
stamps
C
0
10
20
24
Food
BOGO* coupons
• p1 = p2 = $1 per unit
• m = $10
• Buy one whole unit of good 2 and get one free
x2
11
D
Budget line
after using
coupon
C
2
1
0
*Buy one get one
Cannot redeem
coupon since
x2 < 1 in this range
B
A
9 10
x1