# COMMERCE 4FP3 Lecture Notes - Lecture 9: And1, Opportunity Cost, Ordinal Utility

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31 Jan 2013

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Preference Theory and Derivation of Demand

CONSUMER CHOICE THEORY (DEMAND)

BUDGET CONTRAINTS:

- Suppose that the consumer (household) consumes only two goods (X and Y).

Given the Prices of the two goods (PX, PY) and the consumer’s income (m), the possible

quantities purchasable by the consumer (Xo, Yo) are constrained by

PXXo + PYYo ≤ m

e.g., Suppose that a student has a budget for coffee and milk of $100/month. If the price of

Coffee (X say) is $1/cup and the price of Milk (Y) is $2/litre, the budget constraint is

$1*QC + $2*QM ≤ $100

Note: We can define one of the goods (Y say) as a composite commodity, representing all goods

other than X.

Budget Line:

Definition: The maximum combination of two commodities purchasable by a consumer given

the prices of the two commodities and the consumer’s money income

Rearranging the budget equation for the assumption that all income is spent gives the Budget line

Yo = m/PY – PX/PY *X0

e.g. The budget line for Coffee and Milk with Milk as the Y commodity is

→ QM = $100/$2 – $1/$2*QC = 50 – 0.5QC

Since the Opportunity Cost of X = -dY/dX

(Recall: Opportunity Cost = the best foregone option)

→ Opportunity Cost of X = -dY/dX = PX/PY (= -slope of the budget line)

(=> dY = -PX/PY *dX)

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Preference Theory and Derivation of Demand

e.g. The opportunity cost of a unit of Coffee = -(-0.5) = 0.5 Milk

i.e., one unit of Coffee costs ½ litre of Milk

Q

Y

Q

X

m/P

Xo

m/P

Y

Budget Line Constraint

slope = -P

Xo

/P

Y

m/P

X1

'

slope = -P

X1

/P

Y

- The diagram above shows the budget line for m, PXo, PY, and the budget line that ensues

given a fall in the price of X to PX1. A change in one of the prices causes a rotation of the

budget line around the intercept of the commodity whose price is unchanged

Note: A change in Income causes a parallel shift in the budget line: an increase in income shifts

it out and an increase in income shifts it in.

PREFERENCES AND INDIFFERENCES CURVES

For any two consumption bundles (X0, Y0) and (X1, Y1), define an individual’s preference

ranking of each of the two bundles by

1. (X1, Y1) > (X0, Y0) means the individual prefers (X1, Y1) to (X0, Y0)

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Preference Theory and Derivation of Demand

2. (X1, Y1) ~ (X0, Y0) means the individual is indifferent between (X1, Y1) and (X0, Y0)

Preference Assumptions

1. Completeness

For every pair of consumption bundles (X1, Y1) and (X0, Y0), (X1, Y1) > (X0, Y0) or (X0, Y0)

> (X1, Y1) or (X1, Y1) ~ (X0, Y0)

i.e. The consumer prefers one or other bundle or is indifferent between them. In short the

consumer does not make contradictory choices.

2. Transitivity

(X1, Y1) ≥ (X0, Y0) and (X0, Y0) ≥ (X2, Y2) => (X1, Y1) ≥ (X2, Y2)

(where ≥ means either preferred to or indifferent to)

3. Non-Satiation (More is always preferred to Less)

If both X1 or Y1 are at least equal to X0 or Y0 respectively and at least one of X1 or Y1 is a

greater amount than X1 or Y1 respectively, then

(X1, Y1) ≥ (X0, Y0)

This assumption is sometimes called the monotonicity of preferences

4. Convex

The less one has of a good, the more one requires of the other good in exchange to remain

indifferent.

[Assumption 1 follows from the assumption that economic agents are rational

Assumption 2 is actually an assumption since transitivity need not necessarily hold. Indeed,

transitivity for groups of individuals often does not hold.

Assumption 3 and 4 are actually assumptions about well-behaved indifference curves.

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