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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, Y ) and the consumer’s income (m), the possible
quantities purchasable by the consumer (X o Yo) are constrained by
PXX o P YY≤ o
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*Q C $2*Q ≤ M100
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/P –YP /PX*XY 0
e.g. The budget line for Coffee and Milk with Milk as the Y commodity is
→Q = $100/$2 – $1/$2*Q = 50 – 0.5Q
M C C
Since the Opportunity Cost of X = -dY/dX
(Recall: Opportunity Cost = the best foregone option)
→Opportunity Cost of X = -dY/dX = P /P X= Yslope of the budget line)
(=> dY = -P XP YdX)
- 1 - 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 Budget Line Constraint
m/P Y
slope = -P /P X1 Y
slope = -P /PXo Y
Q X
m/P Xo m/P 'X1
- The diagram above shows the budget line for Xo PY, P , and the budget line that ensues given
a fall in the price of X1to P . 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 0X ,0Y ) and1(X1, Y ), define an individual’s preference
ranking of each of the two bundles by
1. (1 , 1 ) > 0X 0 Y ) means the individual pre1er1 (X ,0Y 0 to (X , Y )
- 2 - Preference Theory and Derivation of Demand
2. (X 1 Y )1~ (X , 0 ) m0ans the individual is indifferent between (X , Y ) and (X ,1Y ) 1 0 0
Preference Assumptions
1. Completeness
For every pair of consumption bundles (X , Y ) and (1 , Y1), (X , Y 0 > (0 , Y )1or 1X , Y ) >0 0 0 0
(X ,1Y )1or (X , Y1) ~1(X , Y )0 0
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
(X 1 Y )1≥ (X , Y 0 and0(X , Y ) ≥ 0X ,0Y ) => (X2, Y2) ≥ (X , Y1) 1 2 2
(where ≥ means either preferred to or indifferent to)
3. Non-Satiation (More is always preferred to Less)
If both X or1Y are 1t least equal to X or Y res0ectivel0 and at least one of X or Y is a 1 1
greater amount than X or Y respectively, then
1 1
(X ,1Y )1≥ (X , Y 0 0
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.
- 3 - Preference Theory and Derivation of Demand
Assumption 3 may also not hold particularly beyond certain amounts and Assumption 4 is likely
but not obvious.]
Indifference Curves
Definition: Combinations of two commodities between which the consumer is indifferent (or that
give the same level of utility)
Indifference Curves
Q Y
Q X
Marginal Rate of Substitution (MRS) of Y for X
Definition: The amount of Y that the individual will give up for an increase in X while remaining
indifferent to the combinations of X and Y
→ -ΔY/ΔX (or –dY/dX in calculus notation) where Y and X are quantities
→ the negative of the slope of an indifference curve (for Y on the vertical axis)
(Since substitution implies giving up one of the commodities, MRS is a positive number)
The Preference Assumptions =>
1. Indifference Curves are negatively sloped (Non-satiation)
i.e., an increase in one commodity => a decrease in the other for possible indifference
- 4 - Preference Theory and Derivation of Demand
=> dQ /YQ < X or the MRS > 0
-> Indifference curves are not positively sloped (i.e., they don’t curl back)
2. Indifference Curves

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