Download Lecture 5

ECO - ECONOMICS
2011
Effect of price changes on Marshallian
demand
A simple change in the consumers budget
(i.e., an increase or decrease in Y) involves
a parallel shift of the feasible consumption
set outward or inward from the origin. Since
this shift preserves the price ratio (p1/p2),
it typically has no effect on the consumer’s
MRS (U1/U2) unless the chosen bundle is
either initially or ultimately at a corner solution.
1
A rise in the price of one good holding constant both income and the price of other
goods has economically more complex effects:
1.
Income Effect:
Budget set shifts in-
ward toward the origin for the good whose
price has risen. Consumer is now effectively
poorer.
2. Substitution(Price) Effect: Slope of the
budget set changes; consumer faces a different set of market trade-offs.
2
Although both shifts occur simultaneously,
they are conceptually distinct and have potentially different implications for consumer
behavior.
Income Effect
What is the impact of an inward shift in the
budget set in a 2-good economy (Q1, Q2) :
1. Total consumption? [Falls]
2. Utility? [Falls]
3. Consumption of q1? [Answer depends on
normal, inferior]
4. Consumption of q2? [Answer depends on
normal, inferior]
3
Substitution Effect
What happens to consumption of q1 if p1/p2
↑ but utility stays constant?
E
∂q1
Sign ∂p |U =U0 ?
1
D
Provided that the axiom of diminishing MRS
∂q1
applies, we have ∂p
|
<0
1 U =U0
In words, holding utility constant, the substitution effect is always negative.
4
Types of Goods
The fact that the substitution effect is always negative but the income effect has an
ambiguous sign gives rise to three types of
goods:
1 > 0 ∂q1 |
1. Normal good: ∂q
∂Y
∂p1 U =U0 < 0
For this type of good, a rise in its price and
a decline in income have the same effectless consumption
5
1 < 0 ∂q1 |
2. Inferior good: ∂q
∂Y
∂p1 U =U0 < 0
For this type of good, the income and substitution effects are countervailing. A rise in
price reduces real income, increasing consumption through the income effect even
while reducing it through the substitution
effect.
6
3.
Strongly inferior good (Giffen good):
∂q1
∂q1
<
0
∂Y
∂p1 |U =U0 < 0.
Here, the income effect dominates the substitution effect (in some range).
Hence, rise in price causes the consumer
to buy more - demand is upward sloping.
The price increase reduces demand due to
the substitution effect holding utility constant, but the consumer is effectively so
much poorer due to the income loss that
her demand for the inferior good rises.
7
Marshallian and Hicksian demand
Marshallian demand curves are conventional
market/individual demand curves.
They answer the question: Holding income
and other prices constant, how does quantity of good Q1 demanded change with p1?
We denote this as D1(p1; p2; Y )
8
Marshallian demand curves implicitly combine income and substitution effects.
They are ‘net’ demands that sum over these
two conceptually distinct behavioural responses
to price changes.
9
One can also conceive of a demand curve
that is composed solely of substitution effects.
This is called Hicksian demand and it answers the question: Holding consumer utility constant, how does the quantity of good
q1 demanded change with p1: We denote
this demand function as h1(p1; p2; Ū )
10
Hicksian demand is also called compensated
demand. This name follows from the fact
that to keep the consumer on the same indifference curve as prices vary, one would
have to adjust the consumer’s income, i.e.,
compensate them.
For the analogous reason, Marshallian demand is called uncompensated demand.
11
Relationship between Compensated and
Uncompensated demand
Expenditure function, E(p1; p2; Ū ), gives the
minimum expenditure necessary to obtain
utility Ū given prices p1; p2
For any chosen level of utility Ū , the following identity will hold:
h1(p1; p2; Ū ) = D1(p1; p2; E(p1; p2; Ū ))
12
For any chosen level of utility, compensated
and uncompensated demand must equal one
another.
Fix prices at p1, p2. Fix utility at Ū
Use the expenditure function to determine
the income Ȳ necessary to attain utility Ū
given p1, p2.
It must be the case that
h1(p1; p2; Ū ) = D1(p1; p2; Y )
13
Differentiating the above equality with respect to p1 gives:
∂h1
∂D1
∂d1 ∂E
=
+
∂p1
∂p1
∂Y . ∂p1
1 = ∂h1 − ∂D1 . ∂E
Rearranging, ∂D
∂p1
∂p1
∂Y ∂p1
∂D1
∂Y = income effect on demand for good
q1
∂E ?
What is ∂p
1
14
Expenditure minimization problem: £ = p1q1+
p2q2 + λ(Ū − U (q1, q2))
Solution to this problem will have the following Langragian multipliers:
1 = p2
λ = MpU
M U2
1
From the Envelope Theorem, at the solution to this problem, λ = ∂E
∂ Ū
In other words, relaxing the minimum utility
constraint by one unit, raises expenditures
by the ratio of prices to marginal utilities.
15
∂E ? Holding utility constant,
But what is ∂p
1
how do optimal expenditures respond to a
minute change in the price of one good?
The answer is:
∂E = q
1
∂p1
q1 and q2 are optimally chosen, hence a
minute change in p1 or p2 will not affect the
optimal quantity consumed of either good
holding utility constant.
16
But a price increase will change total expenditures (otherwise utility is not held constant!)
Since the consumer is already consuming q1
units of the good 1, a rise in price of good 1
raises total expenditures needed to maintain
the same level of utility by q1 (Shephard’s
lemma)
∂E equal
Question: Is the q1 obtained from ∂p
1
to h1(.) or D1(.); i.e., compensated or uncompensated demand?
17
1 = ∂h1 − ∂D1 . ∂E
Recall: ∂D
∂p1
∂p1
∂Y ∂p1
∂E , we get the Slutsky equaSubstituting for ∂p
1
tion:
∂D1
∂h1
∂D1
=
−
∂p1
∂p1
∂Y .q1
The difference between the uncompensated
demand response to a price change (the
left-hand side) is equal to the compensated
demand response (1st term on RHS) minus the income effect scaled by the effective
change in income due to the price change
18
The size of the income effect on total demand for good q1 in response to a change
in p1 depends on the amount of q1 that the
consumer is already purchasing.
For a large q1, an increase in p1 has a large
income effect, and vice versa.
19
Effect of a rise in p1 in a 2-good economy:
Consumption of q1
Consumption of q2
Consumer utility
Marshallian
Substitution Income +/Substitution Income +/-
Hicksian
Substitution Income 0
Substitution Income 0
0
20
Uncompensated demand and the indirect utility function
V = maxx1,x2 U (X1, X2) s.t. p1X1 + p2X2 ≤
Y
£ = U (X1, X2) + λ(Y − p1X1 + p2X2)
∂ £ = M U − λp = ∂ £ = M U − λp =
1
1
2
2
∂X1
∂X2
∂£ = Y − p X + p X = 0
1 1
2 2
∂λ
By the envelope theorem for constrained
problems,
∂ £ = ∂V = M U2 = M U1 = λ
∂Y
∂Y
p2
p1
21
By a similar argument,
∂V = ∂ £ = −λX
1
∂p1
∂p1
Intuition: The utility cost of a one unit price
increase is = the additional monetary cost
(which is simply equal to X1, the amount
you are already consuming, times one) multiplied by the shadow value of additional income. We thus get
∂V (P,Y )/∂P
= X(P, Y ) (Roy’s identity)
∂V (P,Y )/∂Y
22
Welfare Evaluation of Price Changes
Let consumer have fixed wealth Y > 0, and
initial price vector is p0.
Impact on consumer’s welfare of a change
from p0 to p1?
If we know consumer’s preferences, then
D
sign V (p1, Y ) − V (p0, Y
E
directly tells us the
impact
23
Money metric indirect utility functions give
us welfare change expressed in money units.
Take any IUF V (., .), choose an arbitrary
price vector p̄ >> 0 and consider the function
E(p̄, V (p, Y )) - wealth required to reach utility level V (p, Y ) when prices are p̄. Then,
E(p̄, V (p1, Y )) − E(p̄, V (p0, Y )) gives a measure of welfare change in money units
24
Let U 0 = V (p0, Y ) and U 1 = V (p1, Y ); note
that E(p0, U 0) = E(p1, U 1) = Y .
Equivalent variation: EV (p0, p1, Y ) = E(p0, U 1)−
E(p0, U 0) = E(p0, U 1) − Y
The amount of income that, if taken away,
would lower that consumer’s utility by the
same amount as the price increase is the
equivalent variation, EV
25
and Compensating variation: EV (p0, p1, Y ) =
E(p1, U 1) − E(p1, U 0) = Y − E(p1, U 0)
The amount of money that would fully compensate the consumer for a price increase is
the compensating variation, CV
26