Lecture 5 - Choice, Demand. Normal and
Inferior Goods
14.03 Spring 2003
Agenda
1. Expenditure function
2. Relationship between Expenditure function and Indirect utility function
3. Indirect Utility function
4. Demand functions
5. Changes in incomes and Engel’s law
6. Income and substitution effects
7. Normal, inferior and Giffen goods
8. Compensated and uncompensated demand (Hicksian, Marshallian)
9. Irish Potato famine
Roadmap:
[Chart 34]
1
Axioms of consumer
preference
Chart 34
Dual
Primal
Max U(x,y)
s.t. pxx+ pyy < I
Min pxx+ pyy
s.t. U(x,y) > U
Indirect Utility function
U*= V(px, py, I)
Expenditure function
E*= E(px, py, U)
Marshallian demand
X = dx(px, py, I) =
(by Roy’s identity)
Hicksian demand
X = hx(px, py, U) =
(by Shepard’s lemma)
∂E
−
∂p x
−
1
∂V / ∂p x
∂V / ∂I
Slutsky equation
Theory of consumer choice
1.1
Utility maximization subject to budget constraint
Ingredients:
• Utility function (preferences)
• Budget constraint
• Price vector
Consumer’s problem
Maximize utility subjet to budget constraint
Characteristics of solution:
• Budget exhaustion (non-satiation)
• For most solutions:
psychic tradeoff = monetary payoff
2
• Psychic tradeoff is MRS
• Monetary tradeoff is the price ratio
From a visual point of view utility maximization corresponds to the following
point:
[Graph 35]
Graph 35
y
IC2
IC3
IC1
B
C
A
D
x
What’s wrong with some of these points?
We can see that A P B, A I D, C P A. Why should one choose A then?
The slope of the indifference curves is given by the MRS.
1.1.1
Interior and corner solutions
There are two types of solution to this problem.
1. Interior solution
2. Corner solution
[Graph 36]
3
Graph 36
y
Typical case
x
The one below is an example of a corner solution. In this specific example
the shape of the indifference curves means that the consumer is indifferent to
the consumption of good y. Utility increases only with consumption of x.
[Graph 37]
Graph 37
y
x
[Graph 38]
4
Graph 38
y
x
In the graph above preference for y is sufficiently strong relative to x that
the the psychic tradeoff is always lower than the monetary tradeoff.
This must be the case for many products that we don’t buy.
Another type of ”corner” solution results from indivisibility.
[Graph 39]
5
Graph 39
y
10
I = 500
px= 450
py= 50
1
x
Why can’t we draw this budget set, i.e. conect dots?
This is because only 2 points can be drawn. This is a sort of ”integer
constraint”. We normally abstract from indivisibility.
Going back to the general case, how do we know a solution exists for consumer, i. e. how do we know the consumer can choose?
We know because of the completeness axiom. Every bundle is on some
indifference curve and can therefore be ranked: A I B, A Â B, B Â A.
1.1.2
Mathematical solution
Mathematics:
maxU (x, y)
x,y
s.t. px x + py y
L
∂L
1.
∂x
∂L
2.
∂y
∂L
3.
∂λ
≤ I
= U (x, y) + λ(I − px x − py y)
= Ux − λpx = 0
= Uy − λpy = 0
= I − px x − py y = 0
6
Rearranging 1. and 2.
−
Ux
px
=
Uy
py
This means that the psychic tradeoff is equal to the monetary tradeoff between the two goods.
3. states that budget is exhausted.
Also notice that:
Ux
px
Uy
py
= λ
= λ
So what is the meaning of λ?
1.1.3
Interpretation of λ, the Lagrange multiplier
∂U/∂x1
∂U/∂x2
∂U/∂xn
=
= ... =
=λ
p1
p2
pn
At the utility-maximizing point, the next dollar spent on each good yields
the same marginal utility.
So what is ∂U
∂I ? Return to Lagrangian:
L
∂L
∂x
∂L
∂y
∂L
∂λ
∂L
∂I
= U (x, y) + λ(I − px x − py y)
= Ux − λpx = 0
= Uy − λpy = 0
= I − px x − py y = 0
µ
¶ µ
¶
∂x
∂x
∂y
∂y
=
Ux
− λpx
+ Uy
− λpy
+λ
∂I
∂I
∂I
∂I
By substituting λ =
We conclude that:
Ux
px
and λ =
Uy
py
both expressions in parenthesis are zero.
∂L
=λ
∂I
We could have known this without calculations from the envelope theorem.
λ equals the ”shadow price” of the budget constraint, i.e. it expresses the
quantity of utils that could be obtained with the next dollar of consumption.
7
This shadow price is not uniquely defined. It is defined only up to a monotonic
transformation.
Corner Solution: unusual case
When at a corner solution, consumer buys zero of some good and spends the
entire budget on the rest.
What problem does this create for the Lagrangian?
[Graph 40]
Graph 40
y
U0
U1 U2
x
The problem is that a point of tangency doens’t exist for positive values of
y.
Hence we also need to impose ”non-negativity constraints”: x ≥ 0, y ≥ 0.
This will not be important for problems in this class, but it’s easy to add
these constraints to the maximization problem.
8
1.1.4
Lagrangian with Non-negativity Constraints
max U (x, y)
s.t. px x + py y ≤ I
y ≥ 0
L = U (x, y) + λ(I − px x − py y) + γ(y − s2 )
∂L
1.
= Ux − λpx = 0
∂x
∂L
2.
= Uy − λpy + γ = 0
∂y
∂L
3.
= −2sγ = 0
∂s
Point 3. implies that γ = 0, s = 0, or both.
1. s = 0, γ 6= 0 (since γ ≥ 0 then it must be the case that γ > 0)
(a)
Uy − λpy + γ
Uy
py
Ux
px
= 0 −→ Uy − λpy < 0
< λ
= λ
Combining the last two expressions:
Ux
px
>
Uy
py
This consumer would like to consume even more x and less y, but she
cannot.
2. s 6= 0, γ = 0
Uy − λpy + γ
Uy
py
= 0 −→ Uy − λpy = 0
Ux
=
=λ
px
Standard FOC, here the non-negativity constraint is not binding.
3. s = 0, γ = 0
Same FOC as before:
px
Ux
=
py
Uy
Here the non-negativity constraint is satisfied with equality so it doesn’t
distort consumption.
9
1.2
Indirect Utility Function
For any:
• Budget constraint
• Utility function
• Set of prices
we obtain a set of optimally chosen quantities.
x∗1
x∗n
= x1 (p1 , p2 , ..., pn , I)
...
= xn (p1 , p2 , ..., pn , I)
So when we say
max U (x1 , ..., xn ) s.t. P X ∗ ≤ I
we get as a result:
max U (x∗1 (p1 , ..., pn , I), ..., x∗n (p1 , ..., pn , I))
⇒ U ∗ (p1 , ..., pn , I) ≡ V (p1 , ..., pn , I)
which we call the ”Indirect Utility Function”. This is the value of maximized
utility under given prices and income.
So remember the distinction:
Direct utility: utility from consumption of x1 , ..., xn
Indirect utility: utility obtained when facing p1 , ..., pn , I
Example:
max U (x, y)
s.t. px x + py y
L
∂L
∂x
∂L
∂y
∂L
∂λ
= x.5 y .5
≤ I
= x.5 y .5 + λ(I − px x − py y)
= .5x−.5 y .5 − λpx = 0
= .5x.5 y −.5 − λpy = 0
= I − px x − py y = 0
We obtain the following:
10
.5x−.5 y .5
.5x.5 y −.5
=
px
py
1
px x = py y = I
2
I
I
x∗ =
, y∗ =
2px
2py
λ =
Half of the budget goes to each good.
Let’s derive the indirect utility function in this case:
µ
¶.5 µ
¶.5
I
I
I
I
U(
,
)=
2px 2py
2px
2py
What is this indirect utility function good for?
1. It saves us time in recalculating utility for every set of prices and budget
constraints.
2. It is used in theory as we will see when studying prices indices.
3. It allows us to calculate consumer demand as a function of prices and
income.
1.3
Dual Problem of Utility Maximization
Most economic problems have a dual, an inverse problem.
For example, the dual of choosing output in order to maximize profits is
minimizing costs at a given output level: cost minimization is the dual of profit
maximization.
1.3.1
Expenditure function
Consumer’s problem: maximize utility subject to a budget constraint. What is
the dual of this problem?
It’s minimizing expenditure subject to a utility constraint (i.e. a level of
utility you must achieve)
This yields the ”expenditure function”: the minimum expenditure required
to attain a given utility level.
Setup of the dual
1. Start with:
s.t. px x + py y
11
max U (x, y)
≤ I
2. Solve for x∗ , y ∗ ⇒ z ∗ = U (x∗ , y ∗ ) given px , py , I.
z is the indirect utility.
3. Now solve the following problem:
min px x + py y
s.t.U (x, y) ≥ z ∗
1.3.2
Graphical representation of dual problem
[Graph 41]
Graph 41
y
U = z*
x
The dual problem consists in choosing the lowest budget set tangent to a
given indifference curve.
Example:
min E
s.t. x.5 y .5
= px x + py y
≥ Up
where Up comes from the primal problem.
¡
¢
L = px x + py y + λ Up − x.5 y .5
12
∂L
∂x
∂L
∂y
∂L
∂λ
= px − λ.5x−.5 y .5 = 0
= py − λ.5x.5 y −.5 = 0
= Up − x.5 y .5 = 0
µ ¶.5
µ ¶.5
py
px
∗
=
Up , y =
Up
px
py
x∗
E∗
.5
= 2Up p.5
x py
How do solutions to Dual and Primal compare?
1.3.3
Relation between Expenditure function and Indirect utility
function
Let’s look at the relation between expenditure function and indirect utility
function.
V (px , py , I0 )
E(px , py , U0 )
V (px , py , E(px , py , U0 ))
E(px , py , V (px , py , I0 ))
=
=
=
=
U0
I0
U0
I0
Expenditure function and Indirect Utility function are inverse one of the
other.
Let’s verify this in the example we saw above.
Recall that primal gave us factor demands x∗p , yp∗ as a function of prices and
income (not utility).
Dual gave us expenditures (budget requirement) as a function of utility and
prices.
¶.5 µ
¶.5
µ
I
I
I
I
, yp∗ =
, U∗ =
2px
2py
2px
2py
Now plug these into expediture function:
x∗p =
µ
¶.5 µ
¶.5
I
I
.5
E =
p.5
x py = I
2px
2py
Finally notice that the multipliers are such that the multiplier in the dual
problem is the inverse of the multiplier in the primal problem.
∗
λP
=
λD
=
Ux
Uy
=
px
py
px
py
=
Ux
Uy
13
1.4
Demand Functions
So far we have been solving for:
• utility as a function of prices and budget
• expenditure as a function of prices and utility
We haven’t yet talked about individual demand functions, but obviously
implicitly we have found demand schedules.
1.4.1
Marshallian demand
In our previous example where:
U (x, y) = x.5 y .5
we derived:
I
px
I
y(px , py , I) = .5
py
x(px , py , I) = .5
In general we will write these demand functions (for individuals) as:
x∗1
x∗2
x∗n
= d1 (p1 , p2 , ..., pn , I)
= d2 (p1 , p2 , ..., pn , I)
...
= dn (p1 , p2 , ..., pn , I)
We call this ”Marshallian” demand after Alfred Marshall who first drew
demand curves.
1.4.2
Hicksian demand
Similarly we derived that:
x(px , py , U ) =
y(px , py , U ) =
14
µ
µ
py
px
px
py
¶.5
¶.5
Up
Up
In general we will write these demand functions (for individual) as:
x∗1,c
x∗2,c
x∗n,c
= h1 (p1 , p2 , ..., pn , U )
= h2 (p1 , p2 , ..., pn , U )
...
= hn (p1 , p2 , ..., pn , U )
This is called ”Hicksian” or compensated demand after John Hicks. Note
that this demand function takes utility as an argument, not income. This turns
out to be an important distinction.
Demand curves for individual goods come from the following:
[Graph 42]
I/py
Graph 42
dx(px,py,I)
I/px
So a demand function is a set of tangency points between indifference curves
and budget set holding I and py (all other prices) constant.
Now that we have a notion of individual demand, we want to study responses
to:
15
• Changes in income
• Changes in prices
But aren’t these two things really the same? No, it turns out that changes
in prices are more complex to analyze than changes in income.
1.5
Changes in income
[Graph 43]
Graph 43
y
I3
I2
I1
U3
U2
U1
x
What happens to the slope of the budget set as income rises/falls? Nothing
happens as the slope is determined by relative prices, which remain unchanged.
Furthermore the optimality condition remains unchanged:
Ux
px
=
Uy
py
This might not be true if we were at a corner solution before the change in
income.
Notice that in the graph above:
∂x
∂y
> 0,
>0
∂I
∂I
16
1.5.1
Inferior goods
But does consumption of every good increase with income?
Even Spam, rides on the T, frozen pizza?
[Graph 44]
Graph 44
y
I2
I1
I3
x3 x2 x1
x
This is the case of an ”inferior good”.
Obviously a good cannot be inferior over the entire income range.
And also the two goods cannot be inferior in the same income range, because
of non-satiation.
1.5.2
Engel’s Law 1857
Annual Income
F ood Expenditure
$250 − $300 $450 − $600 $750 − $1, 000
62%
55%
50%
A consistent empirical relationship can be observed: the proportion of income spent on food declines as income rises.
Q: What is the proportion of income that Americans spend on food on
average?
17
A: 15% (Business Week, 01/02/2000)
Does this mean that food is an inferior good?
[Graph 45]
Graph 45
Food expenditure
Engel curve
Income
Notice that:
∂F
> 0,
∂I
∂2F
(∂I)2
18
<0