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MGMT 4F90
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Professor Cottrel
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Management

MGMT 4F90

Professor Cottrel

Fall

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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 (X o Y o 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 ≤ $M00
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 /PX1 Y
slope = -P /P Xo 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 (0 ,0Y ) and1(X1, Y ), define an individual’s preference
ranking of each of the two bundles by
1. (1 , 1 ) > 0X ,0Y ) means the individual pre1er1 (X , 0 )0to (X , Y )
- 2 - Preference Theory and Derivation of Demand
2. (X 1 Y )1~ (X , Y0) m0ans the individual is indifferent between (X , Y ) and (X , Y1) 1 0 0
Preference Assumptions
1. Completeness
For every pair of consumption bundles (X , Y ) and (X1, Y 1, (X , Y )0> (X0, Y ) 1r (X1, Y ) 0 0 0 0
> (X ,1Y ) 1r (X , Y1) ~ 1X , 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 , 0 ) => (X2, Y 2 ≥ (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 resp0ctivel0 and at least one of X or Y is a 1 1
greater amount than X or Y r1spectiv1ly, then
(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 do not intersect

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