ECON2101 Intermediate Microeconomics
Budgets
Aleksandra Balyanova
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Introduction to decision theory
In any decision making problem, there are two fundamentally different things to take
into account: what is feasible, and what is desirable.
We begin by addressing the first element. We will
1. see some simple examples, then
2. specialise our framework to the setting of a competitive market.
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Examples of feasible sets
• You have 8 hours
before your
microeconomics
final exam
• You can spend
each hour sleeping
or studying
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Examples of feasible sets
• Your
microeconomics
and
macroeconomics
final exams are
both tomorrow and
you have 10 hours
left to study
• Each hour spent
studying for the
Micro exam adds
10 marks to your
grade, each hour
spent studying for
the Macro exam
adds 5 marks to
your grade
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Budget constraint in market setting
In a market setting with L goods (commodities), a consumption bundle is an
L-dimensional vector
x = (x1, x2, . . . , xL)
where
• x` ≥ 0 represents the quantity of commodity ` in the consumption vector x;
• p` ≥ 0 represents the market price of good `; and
• p = (p1, . . . , pL) represents the vector of prices.
The consumption space X is the collection of all available bundles.
If goods are perfectly divisible, the consumption space is X = RL+
Even in a market setting, consumption choices are restricted by various considerations,
but we focus on budgetary restrictions.
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Budget constraint in market setting
A consumer (Anne) is endowed with (disposable) income m > 0.
If Anne can afford to buy x at prices p, it must be that
p1x1 + . . . + pLxL ≤ m
Given prices p (p1, . . . , pL) and income m > 0, the collection of all affordable bundles
forms Anne’s budget set:
B(p,m) =
{
x ∈ X : p1x1 + . . . + pLxL ≤ m
}
.
The bundles that are exactly affordable belong to the budget line: the outer boundary
of the budget set.
• If L = 2, the budget line is a line segment.
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Two goods case
We will focus on the two-goods case for the remainder of our analysis of the consumer
problem. Why?
• A two-good world allows us to capture a fundamental trade-off: whenever you
buy some of one good, you give up the possibility of buying some of another
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Two goods case
In the two-good case we have X = R2+, x = (x1, x2) and p =(p1, p2).
Thus, B(p1, p2,m) = {(x1, x2) ∈ X, : p1x1 + p2x2 ≤ m}.
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Two goods case
Write the equation for the budget line as x2 = mp2 −
p1
p2 x1.
The slope of the budget line captures the opportunity cost in terms of good 2 of
increasing consumption of good 1 by 1 unit.
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Comparative statics: income
• A change in income,
keeping relative prices fixed,
does not change the slope
of the budget line
• The budget line moves
parallel to the original
budget line
• higher income ⇒ line
moves out
• lower income ⇒ line
moves in
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Comparative statics: income
• An ad valorem sales tax at a
rate t increases the price
from p1 to (1+ t)p1.
• A uniform sales tax is
applied uniformly to all
goods.
• Old budget line:
p1x1 + p2x2 = m.
• New budget line:
(1+ t)p1x1 + (1+
t)p2x2 = m.
• Relative prices don’t
change.
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Comparative statics: prices
• Increasing the price of one
good with respect to others
pivots the budget line
inward.
• In the picture, the price of
good 1 changes from p′1 to
p′′1 > p′1.
• Both income m and the
price of good 2 remain fixed.
• This would also be the
change caused by a tax on
only one good, where
p′′1 = (1+ t) p′1
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The numeraire
• For any k > 0, p1x1 + . . . + pLxL ≤ m corresponds to the same budget as
kp1x1 + . . . + kpLxL ≤ km
• Intuition: scaling up all prices and income by the same factor does not
increase or decrease what you can afford
• We can choose k to normalise one good’s price to equal 1 (the numeraire)
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Bulk discounts
Sometimes vendors offer bulk discounts, e.g. “buy two for less than twice the price of
one”
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Bulk discounts
Bulk discounts can also take the form of receiving “money off” your total if you spend
more
In both of these cases, the bulk discount creates a discontinuity in the price ratio.
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Bulk discounts: a simple example
• You have m = 100 that you
can spend on books or other
goods
• The price of other goods is
1, while chocolates cost $2
each if you buy 20 or less,
and $1 each if you buy more
than 20
• The kink in the price ratio
occurs at chocolates= 20
• “Buy 20 units for $40. Buy
40 units for only $60 and
SAVE $20!!”.
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Multiple constraints
• Choices in the real world are
constrained by more than
budgetary restrictions.
• Time constraints.
• Technological and
physical constraints
(e.g., indivisible
goods).
• Regulations and law
provisions.
• In general ---i.e., not just in
competitive market settings
--- a choice bundle is
feasible (or affordable) only
if it meets every constraint
imposed by the
environment.
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Budget sets: exercise 1
• You spend money on books and other goods
• Your initial income is m = 100, and pg = pb = 1. Your budget set is therefore B
= {(xb , xg ) : xb + xg ≤ 100}.
• You are given a gift card for $40 to spend at your local bookstore.
Graphically depict your budget in the following two scenarios:
1. A secondary market is available to exchange your gift card for actual dollars at a
1:1 rate
2. No secondary market is available
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Budget sets: exercise 1
The case with no secondary market:
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Budget sets: exercise 1
The case with a secondary market:
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