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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