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日期：2023-01-07 11:13

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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