Utility Maximization Consumer Behavior Utility Maximization Indirect Utility Function The Expenditure Function Duality Comparative St
Consumer Theory: Utility Maximization
Juan Manuel Puerta
October 20, 2009
Utility Maximization Consumer Behavior Utility Maximization Indirect Utility Function The Expenditure Function Duality Comparative St
Introduction
In production theory we considered profit maximization given
the firm’s technological constraints
We will use an analogous framework in order to understand
consumer’s utility-maximizing behavior and derive her demand
functions.
Our first goal is to find a “utility” function that captures the
preferences of the individual, in order to do so, we will start
defining a preference relation that tells us how the individual
ranks the available bundles. Then, we will study under which
conditions the existence of a utility function is ensured.
Utility Maximization Consumer Behavior Utility Maximization Indirect Utility Function The Expenditure Function Duality Comparative St
Consumer Preferences
We consider a consumer facing to the decision of consuming a
bundle that is in his Consumption Set, X ∈ <k
We will assume that X is closed and convex.
We will use an analogous framework in order to understand
consumer’s utility-maximizing behavior and derive her demand
functions.
We assume the consumer has preferences over the elements of
the consumption set. E.g. x y means that the x is at least as
good as y for the consumer.
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Consumer Preferences (cont.)
The main goal is that this weak preference relation orders the
bundles according to their desirability. In order to do so, we need
the preference relation to satisfy the following properties
1
2
3
Complete. For all x and y in X, either x y, or x y, or both.
Transitive. If x y and y z, then x z for all x, y, z in X.
Reflexible. For all x in X, x x (Redundant)
If a preference relation satisfies these properties, we say it is
rational
Implications of these properties. Are they reasonable? †
We could also define the strict preference relation, x y,
meaning that “x is strictly preferred to y”
Utility Maximization Consumer Behavior Utility Maximization Indirect Utility Function The Expenditure Function Duality Comparative St
Consumer Preferences (cont.)
The main goal is that this weak preference relation orders the
bundles according to their desirability. In order to do so, we need
the preference relation to satisfy the following properties
1
2
3
Complete. For all x and y in X, either x y, or x y, or both.
Transitive. If x y and y z, then x z for all x, y, z in X.
Reflexible. For all x in X, x x (Redundant)
If a preference relation satisfies these properties, we say it is
rational
Implications of these properties. Are they reasonable? †
We could also define the strict preference relation, x y,
meaning that “x is strictly preferred to y”
Utility Maximization Consumer Behavior Utility Maximization Indirect Utility Function The Expenditure Function Duality Comparative St
Consumer Preferences (cont.)
The main goal is that this weak preference relation orders the
bundles according to their desirability. In order to do so, we need
the preference relation to satisfy the following properties
1
2
3
Complete. For all x and y in X, either x y, or x y, or both.
Transitive. If x y and y z, then x z for all x, y, z in X.
Reflexible. For all x in X, x x (Redundant)
If a preference relation satisfies these properties, we say it is
rational
Implications of these properties. Are they reasonable? †
We could also define the strict preference relation, x y,
meaning that “x is strictly preferred to y”
Utility Maximization Consumer Behavior Utility Maximization Indirect Utility Function The Expenditure Function Duality Comparative St
Consumer Preferences (cont.)
The main goal is that this weak preference relation orders the
bundles according to their desirability. In order to do so, we need
the preference relation to satisfy the following properties
1
2
3
Complete. For all x and y in X, either x y, or x y, or both.
Transitive. If x y and y z, then x z for all x, y, z in X.
Reflexible. For all x in X, x x (Redundant)
If a preference relation satisfies these properties, we say it is
rational
Implications of these properties. Are they reasonable? †
We could also define the strict preference relation, x y,
meaning that “x is strictly preferred to y”
Utility Maximization Consumer Behavior Utility Maximization Indirect Utility Function The Expenditure Function Duality Comparative St
Consumer Preferences (cont.)
Rational preferences rule out many strange situations but they
are not enough to get a smooth utility function that will allow
maximization.
Example: Lexicographic preferences (x1 , x2 ) (y1 , y2 ) if x1 ≥ y1
or if x1 = y1 , x2 ≥ y2 †
Continuous: The preference relation on X is continuous if it is
preserved under limits. That is, for any sequence of pairs
n
n
n
n
{(xn , yn )}∞
n=1 with x y for all n, x = limn→∞ x ,y = limn→∞ y ,
we have x y. Equivalently, is continuous if, for all x, both the
upper and lower contour sets are convex, i.e. {y ∈ X : y x} and
{y ∈ X : y x} are convex sets.
Utility Maximization Consumer Behavior Utility Maximization Indirect Utility Function The Expenditure Function Duality Comparative St
Consumer Preferences (cont.)
Comment Lexicographic preferences do not satisfy continuity. To
see thus take xn = (1 + 1n , 1) and yn = (1, 2 + 1n ). For all n,
xn yn . But then, x = limn→∞ xn = (1, 1) and similarly
y = limn→∞ yn = (1, 2). But then, x y and continuity does not
hold.
This is so because
x1 = (2, 1), x2 = (1.5, 1), x3 = (1.33, 1), x4 = (1.25, 1)... and
y1 = (1, 3), y2 = (1, 2.5), y3 = (1, 2.33), y4 = (1, 2.25)...
Utility Maximization Consumer Behavior Utility Maximization Indirect Utility Function The Expenditure Function Duality Comparative St
Desirable Properties of Preference Relations: Monotonicity
Monotonicity. The preference relation on X is monotone if
x ∈ X and y x implies y x. is weakly monotone if y ≥ x,
implies y x. is strongly monotone if y ≥ x and y , x imply
that y x
Local Non-satiation. The preference relation on X is locally
non-satiated if for every x ∈ X and every > 0, there is y ∈ X
such that k y − x k≤ and y x.1
Intuition behind the monotonicity assumptions.†
1
k y − x k is the euclidean distance between points x and y, i.e.
P
k y − x k= [ L`=1 (x` − y` )2 ]1/2
Utility Maximization Consumer Behavior Utility Maximization Indirect Utility Function The Expenditure Function Duality Comparative St
Desirable Properties of Preference Relations: Convexity
Convexity. The preference relation on X is convex if the upper
contour set {y ∈ X : y x} is convex,i.e. if y x and z x, then
ty + (1 − t)z x for t ∈ (0, 1)
Strict Convexity. The preference relation on X is strictly
convex if y x and z x and y , z, then ty + (1 − t)z x for
t ∈ (0, 1)
Intuition behind the convexity assumptions.†
Utility Maximization Consumer Behavior Utility Maximization Indirect Utility Function The Expenditure Function Duality Comparative St
Existence of Utility Function
Proposition. Suppose that the rational preference relation on X
is continuous. Then there is a continuous utility function u(x)
that represents , i.e. a function that u : X → < such that
u(x) ≥ u(y) ⇐⇒ x y.
Sketch of the proof of existence for a monotone preference
relation. †
Interpretation of the utility function: cardinal vs. ordinal
Utility Maximization Consumer Behavior Utility Maximization Indirect Utility Function The Expenditure Function Duality Comparative St
As in production theory, we assume consumers have a goal,
which is to choose the most preferred bundle she can afford.
Let m be the amount of money available to the consumer. The
prices of the k available goods is given by p = (p1 , p2 , ..., pk ).
The set of available bundles for the consumer is given by:
Bp,m = {x ∈ X : px ≤ m}
Then, the utility maximization problem is expressed as,
maxx u(x) subject to px ≤ m and x ∈ X
Utility Maximization Consumer Behavior Utility Maximization Indirect Utility Function The Expenditure Function Duality Comparative St
There are a number of features of the Utility Maximization problem
that are interesting to consider in detail:
(1) Existence of equilibrium. We use the theorem of the
maximum (continuous objective function over a compact range).
1
2
If we assume that preferences are rational and continuous, the
utility function is continuous.
Compact constrain set. If p 0 and m ≥ 0, boundedness follows.
We need the price of every good to be greater than 0 in order to
avoid the obvious problem that people that the demand for a free
good may be unbounded. Closedness, follows from the fact that
B includes its boundary.
(2) Note that the maximizing choice is independent of the
functional choice used for representing preferences. If f and g
both represent the same preferences, for any two bundles that
f (x) ≥ f (y) then g(x) ≥ g(y), so the optimal choices would be
exactly the same.
Utility Maximization Consumer Behavior Utility Maximization Indirect Utility Function The Expenditure Function Duality Comparative St
(3) If all prices and income change by a positive constant, then
the solution to the problem is the same. This is other form of
saying that the solution to this problem is homogeneous of
degree 0.
1
Intuition. Assume x∗ is a solution to the problem with prices and
income (p, m). Then when prices and income are (tp, tm), the
budget set looks like
Bp,m = {x ∈ X : tpx ≤ tm} = {x ∈ X : px ≤ m} because t > 0. But
then the budget set is just like the one in the original problem.
(4) If we add a monotonicity assumption, we will ensure that all
income is spent at the optimum. The proof is easy, assume the
weakest monotonicity assumption, i.e local non-satiation (LNS).
Assume that contrary to my assumption, at the optimum x∗ and
p.x∗ < m. Then, LNS ensures that there is a sufficiently close
bundle x0 , that is both strictly preferred to x∗ and affordable, i.e
x0 x∗ and px0 < m. But then x∗ is not a utility maximizing
bundle.
Utility Maximization Consumer Behavior Utility Maximization Indirect Utility Function The Expenditure Function Duality Comparative St
(5) We will often need to assume that the solution to the Utility
Maximization Problem (UMP) is unique. It turns out that strict
convexity ensures uniqueness.
1
Proof. Assume not. x∗ and x∗∗ both solve the UMP. Then px∗ ≤ m
and px∗∗ ≤ m. Take x0 = tx∗ + (1 − t)x∗∗ , then
tpx∗ + (1 − t)px∗∗ ≤ tm + (1 − t)m = m. So x0 is feasible.
Now we have to show that x0 yields higher utility. Since both x∗
and x∗∗ are solutions to the UMP, x∗ x and x∗∗ x for all x in
Bp,m . But then, strict convexity implies that x0 x. Since x0 is
feasible at (p, m) and it is strictly preferred to any bundle in Bp,m ,
x∗ and x∗∗ cannot solve the UMP; A contradiction.
Utility Maximization Consumer Behavior Utility Maximization Indirect Utility Function The Expenditure Function Duality Comparative St
Utility Maximization Problem: L = u(x) − λ(px − m)
First order conditions:
∂u(x)
∂xi
− λpi = 0 for i = 1, 2, ..., k
As in the case of cost minimization, a useful economic
interpretation is found from dividing the ith by the jth condition.
∂u(x∗ )
∂xi
∂u(x∗ )
∂xj
=
pi
pj
Economic interpretation of the first order conditions †
Geometric interpretation of the first order conditions †
Utility Maximization Consumer Behavior Utility Maximization Indirect Utility Function The Expenditure Function Duality Comparative St
FOC could be expressed as Du(x) being proportional to p †
SOC for this problem require that the hessian of the utility
function is negative semi-definite for all the vectors h that are
orthogonal to prices. That is, for all h that do not increase overall
2
expenditure ph = 0, ht D u(x∗ )h ≤ 0
As usual, SOC could be written in terms of the bordered Hessian.
0 −p −p 1
2 −p1 u11 u12 > 0
−p u
u22 2
21
0 −p1 −p2 −p3 −p1 u11 u12 u13 <0
−p2 u21 u22 u23 −p3 u31 u32 u33 etcetera.
Note the particular sign convention for a maximization problem
(Cf. cost minimization).
Utility Maximization Consumer Behavior Utility Maximization Indirect Utility Function The Expenditure Function Duality Comparative St
Let the function that expresses the maximized utility as a
function of prices and income be the Indirect Utility Function,
υ(p, m).
Properties of the Indirect Utility Function
1
2
3
4
υ(p, m) is non-increasing in p; i.e. for any p0 ≥ p,
υ(p0 , m) ≤ υ(p, m). Similarly, it is non-decreasing in m. If
preferences are locally non-satiated, υ(p, m) is strictly increasing
in m (PROOF: HOMEWORK).
υ(p, m) is homogeneous of degree zero in (p, m).
υ(p, m) is quasiconvex in p; that is {p : υ(p, m) ≤ k} is a convex
set. So if υ(p, m) ≤ ū and υ(p0 , m) ≤ ū, then υ(p00 , m) ≤ ū, for
p00 = tp + (1 − t)p0 and t > 0.
υ(p, m) is continuous at all p 0, m>0.
Utility Maximization Consumer Behavior Utility Maximization Indirect Utility Function The Expenditure Function Duality Comparative St
Proof:
υ(p, m) is non-increasing in p. †
υ(p, m) is homogeneous of degree zero in (p, m) †
υ(p, m) is quasiconvex in p †
Continuity is a result of the theorem of the maximum. Technical.
Utility Maximization Consumer Behavior Utility Maximization Indirect Utility Function The Expenditure Function Duality Comparative St
Proof:
υ(p, m) is non-increasing in p. †
υ(p, m) is homogeneous of degree zero in (p, m) †
υ(p, m) is quasiconvex in p †
Continuity is a result of the theorem of the maximum. Technical.
Utility Maximization Consumer Behavior Utility Maximization Indirect Utility Function The Expenditure Function Duality Comparative St
Proof:
υ(p, m) is non-increasing in p. †
υ(p, m) is homogeneous of degree zero in (p, m) †
υ(p, m) is quasiconvex in p †
Continuity is a result of the theorem of the maximum. Technical.
Utility Maximization Consumer Behavior Utility Maximization Indirect Utility Function The Expenditure Function Duality Comparative St
Proof:
υ(p, m) is non-increasing in p. †
υ(p, m) is homogeneous of degree zero in (p, m) †
υ(p, m) is quasiconvex in p †
Continuity is a result of the theorem of the maximum. Technical.
Utility Maximization Consumer Behavior Utility Maximization Indirect Utility Function The Expenditure Function Duality Comparative St
If υ(p, m) is strictly increasing in m, then we can solve m as a
function of utility and prices. This is what we call the
expenditure function, e(p, ū).
Similarly, the expenditure function could be obtained as the
solution of the following expenditure minimization problem
(EMP),
e(p, ū) = minx px
subject to u(x) ≥ ū
As in the case of the indirect utility function, the expenditure
function has certain useful properties
Utility Maximization Consumer Behavior Utility Maximization Indirect Utility Function The Expenditure Function Duality Comparative St
Properties of the Expenditure Function
1
2
3
e(p, ū) is nondecreasing in p; i.e. for any p0 ≥ p,
e(p0 , ū) ≥ e(p, ū).
e(p, ū) is homogeneous of degree one in p.
e(p, ū) is concave in p. So e(p00 , ū) ≥ te(p, ū) + (1 − t)e(p0 , ū) for
p = tp + (1 − t)p0 and t > 0
4
e(p, ū) is continuous at all p 0.
5
If h(p, ū) is the expenditure minimizing bundle necessary to
achieve utility level ū at prices p, then
hi (p, ū) =
∂e(p,ū)
∂pi
for i = 1, 2, ..., k,
assuming the derivative exists and pi >0.
Utility Maximization Consumer Behavior Utility Maximization Indirect Utility Function The Expenditure Function Duality Comparative St
Proof
Properties (1)-(4) are analogous to the properties of the cost
function, which were proven already. Homework: PROOF.
Property (5) is an application of the envelope theorem in the
expenditure minimization problem:
e(p, ū) = minx px
subject to u(x) ≥ ū
Note that we can also write the EMP as e(p, ū) = ph(p, ū)
At the optimum, the envelope theorem implies that
∂e(p,ū)
∂pi
= xi |xi =hi (p,ū) = hi (p, ū)
As in all the previous examples, it is possible to derive the result
without using the envelope theorem. Just start from the definition
of EMP and use the FOC’s and the fact that utility has to remain
constant.
Utility Maximization Consumer Behavior Utility Maximization Indirect Utility Function The Expenditure Function Duality Comparative St
Hicksian vs. Marshallian Demands
We have introduced two alternative ways of defining consumer
demand. We could think of the consumer demand of good i as a
function of her income m and the price vector p. This is the
so-called Marshallian Demand. We will often denote it as
xi (p, m)
An alternative way of thinking about the demand of good i is the
Compensated Demand or Hicksian Demand. This is the
solution to the EMP, and tells us the demand for good i given the
price vector p and a constant utility level ū, i.e. hi (p, ū). The
term compensated comes from the idea that since ū is fixed, this
function pure effects and not changes in demand due to changes
in income.
Utility Maximization Consumer Behavior Utility Maximization Indirect Utility Function The Expenditure Function Duality Comparative St
Hicksian vs. Marshallian Demands
For example, imagine that the price of good i goes up. Then
there are 2 effects: On the one hand the individual is poorer
because he can buy fewer goods. On the other hand, one good
increased its price relative to the rest. While marshallian demand
will mix the two effects, hicksian demand will isolate the pure
“substitution” effect (We will come back to this later)
Note that while marshallian demand is observable are prices and
income are observable, hicksian demand is not. Why?
Utility Maximization Consumer Behavior Utility Maximization Indirect Utility Function The Expenditure Function Duality Comparative St
Idea
Our goal is to relate the Utility Maximization Problem with the
Expenditure Minimization Problem. In that way we could, for
example, derive relationships between hicksian and marshallian
demands.
We need to show that the solution to
maxx u(x) s.t. px = m
and
minx px s.t. u(x) ≥ ū
are equal.
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Duality
Proposition: Duality Assume the locally-non satiated preferences are represented by the continuous utility function u(x). Let p 0.
Then
1
2
If x∗ is a solution to the UMP at prices p and income m > 0, then
x∗ is a solution of the EMP when prices are p and the utility level
is ū = u(x∗ ). Furthermore, the optimal expenditure e(p, ū) is
equal to m.
If x∗ is a solution to the EMP at prices p and utility ū > u(0),
then x∗ is a solution of the UMP when prices are p and income is
m = px∗ . Furthermore, the maximum utility level of the UMP is
exactly ū
Utility Maximization Consumer Behavior Utility Maximization Indirect Utility Function The Expenditure Function Duality Comparative St
Some Useful Identities
As a consequence of the duality result, we can establish a series
of identities.
1
2
3
4
e(p, υ(p, m)) ≡ m
υ(p, e(p, ū)) ≡ ū
hi (p, υ(p, m)) ≡ xi (p, m)
xi (p, e(p, ū) ≡ hi (p, ū)
The 4th identity is of particular interest as it relates the
unobservable hicksian demand with the observable marshallian
demand.
Utility Maximization Consumer Behavior Utility Maximization Indirect Utility Function The Expenditure Function Duality Comparative St
Roy’s Identity
Proposition. Let x(p, m) be the marshallian demand function,
then
∂υ(p,m)
∂p
i
xi (p, m) = − ∂υ(p,m)
for i = 1, 2, ..., k
∂m
provided that the expression in the right-hand side is well defined
and that pi > 0 and m > 0
Proof †.
1
2
3
First. Use υ(p, e(p, ū)) ≡ ū and the 5th property of the expenditure
function derived above, ∂e(p, ū)/∂pi ) = hi (p, ū).
Second. Envelope theorem argument. Derive u(x) − λ(px − m)
with respect to pi and m, evaluate at the optimum, and combine to
find the result.
Third. Use υ(p, m) = u(x(p, m)), FOC and the budget constraint.
Note that as a by-product in proof (2) and (3), it is shown that
λ = ∂υ(p, m)/∂m. So, the lagrange multiplier simply represents
the marginal utility of income.
Utility Maximization Consumer Behavior Utility Maximization Indirect Utility Function The Expenditure Function Duality Comparative St
Roy’s Identity
Proposition. Let x(p, m) be the marshallian demand function,
then
∂υ(p,m)
∂p
i
xi (p, m) = − ∂υ(p,m)
for i = 1, 2, ..., k
∂m
provided that the expression in the right-hand side is well defined
and that pi > 0 and m > 0
Proof †.
1
2
3
First. Use υ(p, e(p, ū)) ≡ ū and the 5th property of the expenditure
function derived above, ∂e(p, ū)/∂pi ) = hi (p, ū).
Second. Envelope theorem argument. Derive u(x) − λ(px − m)
with respect to pi and m, evaluate at the optimum, and combine to
find the result.
Third. Use υ(p, m) = u(x(p, m)), FOC and the budget constraint.
Note that as a by-product in proof (2) and (3), it is shown that
λ = ∂υ(p, m)/∂m. So, the lagrange multiplier simply represents
the marginal utility of income.
Utility Maximization Consumer Behavior Utility Maximization Indirect Utility Function The Expenditure Function Duality Comparative St
Roy’s Identity
Proposition. Let x(p, m) be the marshallian demand function,
then
∂υ(p,m)
∂p
i
xi (p, m) = − ∂υ(p,m)
for i = 1, 2, ..., k
∂m
provided that the expression in the right-hand side is well defined
and that pi > 0 and m > 0
Proof †.
1
2
3
First. Use υ(p, e(p, ū)) ≡ ū and the 5th property of the expenditure
function derived above, ∂e(p, ū)/∂pi ) = hi (p, ū).
Second. Envelope theorem argument. Derive u(x) − λ(px − m)
with respect to pi and m, evaluate at the optimum, and combine to
find the result.
Third. Use υ(p, m) = u(x(p, m)), FOC and the budget constraint.
Note that as a by-product in proof (2) and (3), it is shown that
λ = ∂υ(p, m)/∂m. So, the lagrange multiplier simply represents
the marginal utility of income.
Utility Maximization Consumer Behavior Utility Maximization Indirect Utility Function The Expenditure Function Duality Comparative St
Roy’s Identity
Proposition. Let x(p, m) be the marshallian demand function,
then
∂υ(p,m)
∂p
i
xi (p, m) = − ∂υ(p,m)
for i = 1, 2, ..., k
∂m
provided that the expression in the right-hand side is well defined
and that pi > 0 and m > 0
Proof †.
1
2
3
First. Use υ(p, e(p, ū)) ≡ ū and the 5th property of the expenditure
function derived above, ∂e(p, ū)/∂pi ) = hi (p, ū).
Second. Envelope theorem argument. Derive u(x) − λ(px − m)
with respect to pi and m, evaluate at the optimum, and combine to
find the result.
Third. Use υ(p, m) = u(x(p, m)), FOC and the budget constraint.
Note that as a by-product in proof (2) and (3), it is shown that
λ = ∂υ(p, m)/∂m. So, the lagrange multiplier simply represents
the marginal utility of income.
Utility Maximization Consumer Behavior Utility Maximization Indirect Utility Function The Expenditure Function Duality Comparative St
Money metric utility function
Idea: Given prices p, how much money should we give to the
consumer in order for him to be as well off as he was when
consuming bundle x?
Mathematically, it is simply the solution of minz pz such that
u(z) ≥ u(x)
Alternatively, we may simply write it as m(p, x) = e(p, u(x))
Note that it is not obvious that m(p, x) is a utility function.
For u(x) = ū fixed, it follows that m is like a expenditure
function, i.e. monotonic, homogeneous and concave in p.
Graphic example. †
Utility Maximization Consumer Behavior Utility Maximization Indirect Utility Function The Expenditure Function Duality Comparative St
However, it is not obvious that m is a utility function. In order to
do so, it should be the case that for any x and y in X, if x y
then u(x) ≥ u(y). It is easy to see that this will hold if e(p, ū) is
strictly increasing in u.
1
2
Monotonicity in ū. If the preference relation is complete,
transitive, continuous and locally-non satiated, then the
expenditure minimization function e(p, u) will be strictly
increasing in u, i.e. if u0 > u, then e(p, u0 ) > e(p, u)
Proof: Homework!
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Money Metric Indirect Utility Function
The idea is to answer how much money should one get when
prices are p in order to have the same utility than when prices
where q and income was m. µ(p, q, m) = e(p, υ(q, m))
Graphic example. †
Note that as in the case of the MMUF, this is just a monotonic
transformation of a indirect utility function and, thus, an indirect
utility function itself.
A nice feature of both MMUF and MMIUF is that the both
depend only on observable parameters.
Utility Maximization Consumer Behavior Utility Maximization Indirect Utility Function The Expenditure Function Duality Comparative St
Examples
Cobb Douglas utility function (Homework)
CES utility function †
Utility Maximization Consumer Behavior Utility Maximization Indirect Utility Function The Expenditure Function Duality Comparative St
Income Changes
Income Expansion Path (IEP): The locus of utility-maximizing
bundles that results from changing m while leaving p fixed.
From the IEP we can derive, for each good, its demand as a
function of income (with constant prices). These are the Engel
Curves. Depending on the shape of the Engel Curves a goods
can be classified as:
1
2
3
Unit income elasticity. The IEP and the Engel curve are a line that
goes through the origin. The proportions of each good consumed
remain constant when income varies.
Luxury Good and Necessary Goods. The consumer expands the
consumption of both goods when income increase, but he
increases proportionally the consumption of one good (the luxury
good) relative to the other good (the necessary good).
Inferior and Normal Goods. As income increases, the IEP could
bend backwards, meaning that the consumer is consuming
actually less (in absolute terms) of one good. These are the
Inferior goods. If the demand of a good increases when income
increases, we say this is a Normal good.
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Income Expansion Paths
Source: Varian, Microeconomic Analysis, 2nd Edition, p. 117
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Price Changes
Now we want to see what happens if we keep income fixed and
p2 fixed. Let p1 change and see how the optimal choices of both
goods change. The locus of the tangencies between the budget
lines constructed in this way and the indifference curves is called
offer curve
If the demand of one good decreases when its own price
decreases, so that the demand curve is positively sloped at some
point, then we say this is a Giffen good.
Utility Maximization Consumer Behavior Utility Maximization Indirect Utility Function The Expenditure Function Duality Comparative St
The Slutsky Equation
We have mentioned before that there are 2 types of demand
functions, the marshallian demand that takes prices and income
as arguments and, the hicksian demand, that takes prices as given
and changes income so as to satisfy a given utility level. That’s
why it is often called compensated demand
Imagine the price of good i pi changed. Would there any relation
between the changes in the hicksian and marshallian demand? It
turns out that this is the case. This is the Slutsky equation
slutsky equation.
∂xj (p,m)
∂pi
=
∂hj (p,υ(p,m))
∂pi
−
∂xj (p,m)
∂m xi (p, m)
Utility Maximization Consumer Behavior Utility Maximization Indirect Utility Function The Expenditure Function Duality Comparative St
Proof
Let x∗ maximize utility at (p, m), and let u∗ = u(u∗ ). It is
identically true that,
hj (p, u∗ ) ≡ xj (p, e(p, u∗ ))
Take the derivative with respect to pi
∂hj (p,u∗ )
∂pi
=
∂xj (p,m)
∂pi
∂e(p,u∗ )
∂pi
+
∂xj (p,m) ∂e(p,u∗ )
∂m
∂pi
And now use the fact that
= hi (p, u∗ ) (see properties of
the expenditure function above). And further note that in
equilibrium hi (p, u∗ ) = xi (p, m) since m = e(p, u∗ )
Substituting and rearranging you obtain the result,
∂xj (p,m)
∂pi
=
∂hj (p,u∗ )
∂pi
−
∂xj (p,m)
∂m xi (p, m)
Utility Maximization Consumer Behavior Utility Maximization Indirect Utility Function The Expenditure Function Duality Comparative St
Interpretation of the Slutsky Equation
The slutsky equation decomposes the effect of a change in prices
into 2 effects
1
2
Substitution Effect. The first term in the RHS captures the “pure”
change in demand of good j when the price of good i changes.
The term “pure” means that income is changed so as to keep
utility unchanged. We know that for own-price, this term has to
be strictly negative (why?).
Income Effect. The second term in the RHS captures the change
in demand of good j due to the “income” effect of the change in
price pi
Graphical interpretation of the Slutsky equation †
Slutsky Equation for all goods and all price changes (Matrix
notation). 2
Dp x(p, m) = Dp h(p, u) − Dm x(p, u)x(p, u)t
2
The Gradient Dm x and quantity vector x are assumed column vectors as usual.
Utility Maximization Consumer Behavior Utility Maximization Indirect Utility Function The Expenditure Function Duality Comparative St
Interpretation of the Slutsky Equation
The slutsky equation decomposes the effect of a change in prices
into 2 effects
1
2
Substitution Effect. The first term in the RHS captures the “pure”
change in demand of good j when the price of good i changes.
The term “pure” means that income is changed so as to keep
utility unchanged. We know that for own-price, this term has to
be strictly negative (why?).
Income Effect. The second term in the RHS captures the change
in demand of good j due to the “income” effect of the change in
price pi
Graphical interpretation of the Slutsky equation †
Slutsky Equation for all goods and all price changes (Matrix
notation). 2
Dp x(p, m) = Dp h(p, u) − Dm x(p, u)x(p, u)t
2
The Gradient Dm x and quantity vector x are assumed column vectors as usual.
Utility Maximization Consumer Behavior Utility Maximization Indirect Utility Function The Expenditure Function Duality Comparative St
Interpretation of the Slutsky Equation
The slutsky equation decomposes the effect of a change in prices
into 2 effects
1
2
Substitution Effect. The first term in the RHS captures the “pure”
change in demand of good j when the price of good i changes.
The term “pure” means that income is changed so as to keep
utility unchanged. We know that for own-price, this term has to
be strictly negative (why?).
Income Effect. The second term in the RHS captures the change
in demand of good j due to the “income” effect of the change in
price pi
Graphical interpretation of the Slutsky equation †
Slutsky Equation for all goods and all price changes (Matrix
notation). 2
Dp x(p, m) = Dp h(p, u) − Dm x(p, u)x(p, u)t
2
The Gradient Dm x and quantity vector x are assumed column vectors as usual.
Utility Maximization Consumer Behavior Utility Maximization Indirect Utility Function The Expenditure Function Duality Comparative St
Properties of the demand function
From the result that Dp e(p, u) = h(p, u) you can find a number of
properties of the hicksian and marshallian (through the Slutsky
equation) demands.
1
2
3
4
The substitution matrix Dh(p, u) = D2 p e(p, u) is negative
semidefinite due to the concavity of the expenditure function.
The substitution matrix is symmetric implying ∂hi /∂pj = ∂hj /∂pi .
Again this comes from the simetricity of the expenditure function
Hessian.
The compensated own-price effect is non-positive ∂hi /∂pi ≤ 0.
This follows from negative semidefiniteness of the Hessian.
∂x (p,m)
∂x (p,m)
The substitution matrix ( j∂pi + j∂m xi (p, m)) is negative
semidefinite (Slutsky equation).
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Comparative Statics Using FOC
Take the two-good case. From first order conditions, you have
these identities.
p1 x1 (p1 , p2 , m) + p2 x2 (p1 , p2 , m) ≡ m
∂u(x1 (p1 ,p2 ,m),x2 (p1 ,p2 ,m))
∂x1
− λp1 ≡ 0
∂u(x1 (p1 ,p2 ,m),x2 (p1 ,p2 ,m))
∂x2
− λp2 ≡ 0
As we did earlier (e.g. cost minimization), we can take derivative
with respect to one price (say p1 and rearrange in matrix form to
get.

  ∂λ   
 0 −p1 −p2   ∂p1  x1 
  ∂x1   

−p1 u11 u12   ∂p1  =  λ 
−p2 u21 u22  ∂x2 
0
∂p1
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As in the previous case (Cf. cost minimization), the matrix on
the left-hand side is the bordered hessian. As in the previous
case, we assume a non-degenerate maximum so that H > 0.
Solving for ∂x1 /∂p1
∂x1
∂p1
=λ
∂x1
∂p1 =
0
−p
2
0 x1 −p2
−p1 λ u12
−p2 0 u22
H
−p2 u22 H
− x1
−p1 u12
−p2 u22
H
The two terms in this expression kind of looks like the Slutsky
equation. It turns out that this can be established. We will show
that the second term is the “income” term in the Slutsky
equation.
Utility Maximization Consumer Behavior Utility Maximization Indirect Utility Function The Expenditure Function Duality Comparative St
Differentiating the equations with respect to m.
  ∂λ   

 0 −p1 −p2   ∂m
 −1
  ∂x1   

−p
u
u

11
12 
  ∂m  =  0 
 1
 2
0
−p2 u21 u22 ∂x
∂m
So again by Cramer’s rule we can solve for ∂x1 /∂m
−p
u
1
12
−p
2 u22 ∂x1 /∂m =
H
So that,
∂x1
∂p1
=λ
0 −p2
−p2 u22
H
1
− x1 ∂x
∂m
This starts to look like the Slutsky equation.
It turns out that you
−p2 0
−p2 u22 1
can prove that ∂h
=
λ
(Homework!)
∂p1
H
Utility Maximization Consumer Behavior Utility Maximization Indirect Utility Function The Expenditure Function Duality Comparative St
Summary
UMP
EMP
Slutsky Equation
xi (p, m)
xi (p
, e(
Roy’s Identity
∂xj (p,m)
∂pi
p, ū
)) =
=
∂hj p,ū)
∂pi
hi (p
, ū)
−
h i(
∂xj (p,m)
∂m xi
(p,
p, υ
i
xi = − ∂υ(p,m)/∂p
∂υ(p,m)/∂m
υ(p, m)
υ(p, e(p, ū)) = ū
e(p, υ(p, m)) = m
m))
hi (p, ū)
)
p, m
i(
x
=
∂e(p,ū)
∂pi
= hi (p, ū)
e(p, ū)