Consumer Theory: Utility Maximization - Personal … MaximizationConsumer BehaviorUtility...

Utility Maximization Consumer Behavior Utility Maximization Indirect Utility Function The Expenditure Function Duality Comparative Statics

Consumer Theory: Utility Maximization

Juan Manuel Puerta

October 20, 2009


Introduction

In production theory we considered profit maximization giventhe firm’s technological constraints

We will use an analogous framework in order to understandconsumer’s utility-maximizing behavior and derive her demandfunctions.

Our first goal is to find a “utility” function that captures thepreferences of the individual, in order to do so, we will startdefining a preference relation that tells us how the individualranks the available bundles. Then, we will study under whichconditions the existence of a utility function is ensured.


Consumer Preferences

We consider a consumer facing to the decision of consuming abundle 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 understandconsumer’s utility-maximizing behavior and derive her demandfunctions.

We assume the consumer has preferences over the elements ofthe consumption set. E.g. x � y means that the x is at least asgood as y for the consumer.


Consumer Preferences (cont.)

The main goal is that this weak preference relation � orders thebundles according to their desirability. In order to do so, we needthe preference relation to satisfy the following properties

1 Complete. For all x and y in X, either x � y, or x � y, or both.2 Transitive. If x � y and y � z, then x � z for all x, y, z in X.3 Reflexible. For all x in X, x � x (Redundant)

If a preference relation satisfies these properties, we say it isrationalImplications of these properties. Are they reasonable? †

We could also define the strict preference relation, x � y,meaning that “x is strictly preferred to y”



Rational preferences rule out many strange situations but theyare not enough to get a smooth utility function that will allowmaximization.

Example: Lexicographic preferences (x1, x2) � (y1, y2) if x1 ≥ y1or if x1 = y1, x2 ≥ y2 †

Continuous: The preference relation � on X is continuous if it ispreserved under limits. That is, for any sequence of pairs{(xn, yn)}∞n=1 with xn � yn for all n, x = limn→∞ xn,y = limn→∞ yn,we have x � y. Equivalently, � is continuous if, for all x, both theupper and lower contour sets are convex, i.e. {y ∈ X : y � x} and{y ∈ X : y � x} are convex sets.



Comment Lexicographic preferences do not satisfy continuity. Tosee thus take xn = (1 + 1

n , 1) and yn = (1, 2 + 1n ). For all n,

xn � yn. But then, x = limn→∞ xn = (1, 1) and similarlyy = limn→∞ yn = (1, 2). But then, x � y and continuity does nothold.

This is so becausex1 = (2, 1), x2 = (1.5, 1), x3 = (1.33, 1), x4 = (1.25, 1)... andy1 = (1, 3), y2 = (1, 2.5), y3 = (1, 2.33), y4 = (1, 2.25)...


Desirable Properties of Preference Relations: Monotonicity

Monotonicity. The preference relation � on X is monotone ifx ∈ 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 implythat y � x

Local Non-satiation. The preference relation � on X is locallynon-satiated if for every x ∈ X and every ε > 0, there is y ∈ Xsuch that ‖ y − x ‖≤ ε and y � x.1

Intuition behind the monotonicity assumptions.†

1‖ y − x ‖ is the euclidean distance between points x and y, i.e.‖ y − x ‖= [

∑L`=1(x` − y`)2]1/2


Desirable Properties of Preference Relations: Convexity

Convexity. The preference relation � on X is convex if the uppercontour set {y ∈ X : y � x} is convex,i.e. if y � x and z � x, thenty + (1 − t)z � x for t ∈ (0, 1)

Strict Convexity. The preference relation � on X is strictlyconvex if y � x and z � x and y , z, then ty + (1 − t)z � x fort ∈ (0, 1)

Intuition behind the convexity assumptions.†


Existence of Utility Function

Proposition. Suppose that the rational preference relation � on Xis continuous. Then there is a continuous utility function u(x)that represents �, i.e. a function that u : X →< such thatu(x) ≥ u(y)⇐⇒ x � y.

Sketch of the proof of existence for a monotone preferencerelation. †

Interpretation of the utility function: cardinal vs. ordinal


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. Theprices 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


There are a number of features of the Utility Maximization problemthat are interesting to consider in detail:

(1) Existence of equilibrium. We use the theorem of themaximum (continuous objective function over a compact range).

1 If we assume that preferences are rational and continuous, theutility function is continuous.

2 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 toavoid the obvious problem that people that the demand for a freegood may be unbounded. Closedness, follows from the fact thatB includes its boundary.

(2) Note that the maximizing choice is independent of thefunctional choice used for representing preferences. If f and gboth represent the same preferences, for any two bundles thatf (x) ≥ f (y) then g(x) ≥ g(y), so the optimal choices would beexactly the same.


(3) If all prices and income change by a positive constant, thenthe solution to the problem is the same. This is other form ofsaying that the solution to this problem is homogeneous ofdegree 0.

1 Intuition. Assume x∗ is a solution to the problem with prices andincome (p,m). Then when prices and income are (tp, tm), thebudget set looks likeBp,m = {x ∈ X : tpx ≤ tm} = {x ∈ X : px ≤ m} because t > 0. Butthen the budget set is just like the one in the original problem.

(4) If we add a monotonicity assumption, we will ensure that allincome is spent at the optimum. The proof is easy, assume theweakest monotonicity assumption, i.e local non-satiation (LNS).Assume that contrary to my assumption, at the optimum x∗ andp.x∗ < m. Then, LNS ensures that there is a sufficiently closebundle x′, that is both strictly preferred to x∗ and affordable, i.ex′ � x∗ and px′ < m. But then x∗ is not a utility maximizingbundle.


(5) We will often need to assume that the solution to the UtilityMaximization Problem (UMP) is unique. It turns out that strictconvexity ensures uniqueness.

1 Proof. Assume not. x∗ and x∗∗ both solve the UMP. Then px∗ ≤ mand px∗∗ ≤ m. Take x′ = tx∗ + (1 − t)x∗∗, thentpx∗ + (1 − t)px∗∗ ≤ tm + (1 − t)m = m. So x′ is feasible.Now we have to show that x′ yields higher utility. Since both x∗and x∗∗ are solutions to the UMP, x∗ � x and x∗∗ � x for all x inBp,m. But then, strict convexity implies that x′ � x. Since x′ isfeasible 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 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 economicinterpretation is found from dividing the ith by the jth condition.

∂u(x∗)∂xi

∂u(x∗)∂xj

=pipj

Economic interpretation of the first order conditions †

Geometric interpretation of the first order conditions †


FOC could be expressed as Du(x) being proportional to p †SOC for this problem require that the hessian of the utilityfunction is negative semi-definite for all the vectors h that areorthogonal to prices. That is, for all h that do not increase overallexpenditure ph = 0, htD2u(x∗)h ≤ 0As usual, SOC could be written in terms of the bordered Hessian.∣∣∣∣∣∣∣∣∣

0 −p1 −p2−p1 u11 u12−p2 u21 u22

∣∣∣∣∣∣∣∣∣ > 0

∣∣∣∣∣∣∣∣∣∣∣0 −p1 −p2 −p3−p1 u11 u12 u13−p2 u21 u22 u23−p3 u31 u32 u33

∣∣∣∣∣∣∣∣∣∣∣ < 0

etcetera.Note the particular sign convention for a maximization problem(Cf. cost minimization).


Let the function that expresses the maximized utility as afunction of prices and income be the Indirect Utility Function,υ(p,m).Properties of the Indirect Utility Function

1 υ(p,m) is non-increasing in p; i.e. for any p′ ≥ p,υ(p′,m) ≤ υ(p,m). Similarly, it is non-decreasing in m. Ifpreferences are locally non-satiated, υ(p,m) is strictly increasingin m (PROOF: HOMEWORK).

2 υ(p,m) is homogeneous of degree zero in (p,m).3 υ(p,m) is quasiconvex in p; that is {p : υ(p,m) ≤ k} is a convex

set. So if υ(p,m) ≤ u and υ(p′,m) ≤ u, then υ(p′′,m) ≤ u, forp′′ = tp + (1 − t)p′ and t > 0.

4 υ(p,m) is continuous at all p � 0, m>0.


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.


If υ(p,m) is strictly increasing in m, then we can solve m as afunction of utility and prices. This is what we call theexpenditure function, e(p, u).

Similarly, the expenditure function could be obtained as thesolution of the following expenditure minimization problem(EMP),

e(p, u) = minx pxsubject to u(x) ≥ u

As in the case of the indirect utility function, the expenditurefunction has certain useful properties


Properties of the Expenditure Function

1 e(p, u) is nondecreasing in p; i.e. for any p′ ≥ p,e(p′, u) ≥ e(p, u).

2 e(p, u) is homogeneous of degree one in p.3 e(p, u) is concave in p. So e(p′′, u) ≥ te(p, u) + (1 − t)e(p′, u) for

p = tp + (1 − t)p′ and t > 04 e(p, u) is continuous at all p � 0.5 If h(p, u) is the expenditure minimizing bundle necessary to

achieve utility level u at prices p, then

hi(p, u) =∂e(p,u)∂pi

for i = 1, 2, ..., k,

assuming the derivative exists and pi>0.


Proof

Properties (1)-(4) are analogous to the properties of the costfunction, which were proven already. Homework: PROOF.

Property (5) is an application of the envelope theorem in theexpenditure minimization problem:

e(p, u) = minx pxsubject to u(x) ≥ u

Note that we can also write the EMP as e(p, u) = ph(p, u)

At the optimum, the envelope theorem implies that∂e(p,u)∂pi

= xi|xi=hi(p,u) = hi(p, u)

As in all the previous examples, it is possible to derive the resultwithout using the envelope theorem. Just start from the definitionof EMP and use the FOC’s and the fact that utility has to remainconstant.


Hicksian vs. Marshallian Demands

We have introduced two alternative ways of defining consumerdemand. We could think of the consumer demand of good i as afunction of her income m and the price vector p. This is theso-called Marshallian Demand. We will often denote it asxi(p,m)

An alternative way of thinking about the demand of good i is theCompensated Demand or Hicksian Demand. This is thesolution to the EMP, and tells us the demand for good i given theprice vector p and a constant utility level u, i.e. hi(p, u). Theterm compensated comes from the idea that since u is fixed, thisfunction pure effects and not changes in demand due to changesin income.


Hicksian vs. Marshallian Demands

For example, imagine that the price of good i goes up. Thenthere are 2 effects: On the one hand the individual is poorerbecause he can buy fewer goods. On the other hand, one goodincreased its price relative to the rest. While marshallian demandwill 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 andincome are observable, hicksian demand is not. Why?


Idea

Our goal is to relate the Utility Maximization Problem with theExpenditure Minimization Problem. In that way we could, forexample, derive relationships between hicksian and marshalliandemands.

We need to show that the solution to

maxx u(x) s.t. px = m

and

minx px s.t. u(x) ≥ u

are equal.


Duality

Proposition: Duality Assume the locally-non satiated preferences �are represented by the continuous utility function u(x). Let p � 0.Then

1 If x∗ is a solution to the UMP at prices p and income m > 0, thenx∗ is a solution of the EMP when prices are p and the utility levelis u = u(x∗). Furthermore, the optimal expenditure e(p, u) isequal to m.

2 If x∗ is a solution to the EMP at prices p and utility u > u(0),then x∗ is a solution of the UMP when prices are p and income ism = px∗. Furthermore, the maximum utility level of the UMP isexactly u


Some Useful Identities

As a consequence of the duality result, we can establish a seriesof identities.

1 e(p, υ(p,m)) ≡ m2 υ(p, e(p, u)) ≡ u3 hi(p, υ(p,m)) ≡ xi(p,m)4 xi(p, e(p, u) ≡ hi(p, u)

The 4th identity is of particular interest as it relates theunobservable hicksian demand with the observable marshalliandemand.


Roy’s Identity

Proposition. Let x(p,m) be the marshallian demand function,then

xi(p,m) = −

∂υ(p,m)∂pi

∂υ(p,m)∂m

for i = 1, 2, ..., k

provided that the expression in the right-hand side is well definedand that pi > 0 and m > 0Proof †.

1 First. Use υ(p, e(p, u)) ≡ u and the 5th property of the expenditurefunction derived above, ∂e(p, u)/∂pi) = hi(p, u).

2 Second. Envelope theorem argument. Derive u(x) − λ(px − m)with respect to pi and m, evaluate at the optimum, and combine tofind the result.

3 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 thatλ = ∂υ(p,m)/∂m. So, the lagrange multiplier simply representsthe marginal utility of income.


Money metric utility function

Idea: Given prices p, how much money should we give to theconsumer in order for him to be as well off as he was whenconsuming bundle x?

Mathematically, it is simply the solution of minz pz such thatu(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) = u fixed, it follows that m is like a expenditurefunction, i.e. monotonic, homogeneous and concave in p.

Graphic example. †


However, it is not obvious that m is a utility function. In order todo so, it should be the case that for any x and y in X, if x � ythen u(x) ≥ u(y). It is easy to see that this will hold if e(p, u) isstrictly increasing in u.

1 Monotonicity in u. If the preference relation � is complete,transitive, continuous and locally-non satiated, then theexpenditure minimization function e(p, u) will be strictlyincreasing in u, i.e. if u′ > u, then e(p, u′) > e(p, u)

2 Proof: Homework!


Money Metric Indirect Utility Function

The idea is to answer how much money should one get whenprices are p in order to have the same utility than when priceswhere 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 monotonictransformation of a indirect utility function and, thus, an indirectutility function itself.

A nice feature of both MMUF and MMIUF is that the bothdepend only on observable parameters.


Examples

Cobb Douglas utility function (Homework)

CES utility function †


Income Changes

Income Expansion Path (IEP): The locus of utility-maximizingbundles that results from changing m while leaving p fixed.From the IEP we can derive, for each good, its demand as afunction of income (with constant prices). These are the EngelCurves. Depending on the shape of the Engel Curves a goodscan be classified as:

1 Unit income elasticity. The IEP and the Engel curve are a line thatgoes through the origin. The proportions of each good consumedremain constant when income varies.

2 Luxury Good and Necessary Goods. The consumer expands theconsumption of both goods when income increase, but heincreases proportionally the consumption of one good (the luxurygood) relative to the other good (the necessary good).

3 Inferior and Normal Goods. As income increases, the IEP couldbend backwards, meaning that the consumer is consumingactually less (in absolute terms) of one good. These are theInferior goods. If the demand of a good increases when incomeincreases, we say this is a Normal good.


Income Expansion Paths

Source: Varian, Microeconomic Analysis, 2nd Edition, p. 117


Price Changes

Now we want to see what happens if we keep income fixed andp2 fixed. Let p1 change and see how the optimal choices of bothgoods change. The locus of the tangencies between the budgetlines constructed in this way and the indifference curves is calledoffer curveIf the demand of one good decreases when its own pricedecreases, so that the demand curve is positively sloped at somepoint, then we say this is a Giffen good.

Source: Varian, Microeconomic Analysis, 2nd Edition, p. 118


The Slutsky Equation

We have mentioned before that there are 2 types of demandfunctions, the marshallian demand that takes prices and incomeas arguments and, the hicksian demand, that takes prices as givenand changes income so as to satisfy a given utility level. That’swhy it is often called compensated demand

Imagine the price of good i pi changed. Would there any relationbetween the changes in the hicksian and marshallian demand? Itturns out that this is the case. This is the Slutsky equationslutsky equation.

∂xj(p,m)∂pi

=∂hj(p,υ(p,m))

∂pi−

∂xj(p,m)∂m xi(p,m)


Proof

Let x∗ maximize utility at (p,m), and let u∗ = u(u∗). It isidentically 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

+∂xj(p,m)∂m

∂e(p,u∗)∂pi

And now use the fact that ∂e(p,u∗)∂pi

= hi(p, u∗) (see properties ofthe expenditure function above). And further note that inequilibrium 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)


Interpretation of the Slutsky Equation

The slutsky equation decomposes the effect of a change in pricesinto 2 effects

1 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 keeputility unchanged. We know that for own-price, this term has tobe strictly negative (why?).

2 Income Effect. The second term in the RHS captures the changein demand of good j due to the “income” effect of the change inprice pi

Graphical interpretation of the Slutsky equation †

Slutsky Equation for all goods and all price changes (Matrixnotation). 2

Dpx(p,m) = Dph(p, u) − Dmx(p, u)x(p, u)t

2The Gradient Dmx and quantity vector x are assumed column vectors as usual.


Properties of the demand function

From the result that Dpe(p, u) = h(p, u) you can find a number ofproperties of the hicksian and marshallian (through the Slutskyequation) demands.

1 The substitution matrix Dh(p, u) = D2pe(p, u) is negative

semidefinite due to the concavity of the expenditure function.2 The substitution matrix is symmetric implying ∂hi/∂pj = ∂hj/∂pi.

Again this comes from the simetricity of the expenditure functionHessian.

3 The compensated own-price effect is non-positive ∂hi/∂pi ≤ 0.This follows from negative semidefiniteness of the Hessian.

4 The substitution matrix ( ∂xj(p,m)∂pi

+∂xj(p,m)∂m xi(p,m)) is negative

semidefinite (Slutsky equation).


Comparative Statics Using FOC

Take the two-good case. From first order conditions, you havethese identities.

p1x1(p1, p2,m) + p2x2(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 derivativewith respect to one price (say p1 and rearrange in matrix form toget. 0 −p1 −p2

−p1 u11 u12−p2 u21 u22

∂λ∂p1∂x1∂p1∂x2∂p1

=

x1λ

0


As in the previous case (Cf. cost minimization), the matrix onthe left-hand side is the bordered hessian. As in the previouscase, we assume a non-degenerate maximum so that H > 0.

Solving for ∂x1/∂p1

∂x1∂p1

=

∣∣∣∣∣∣∣∣∣∣∣0 x1 −p2−p1 λ u12−p2 0 u22

∣∣∣∣∣∣∣∣∣∣∣H

∂x1∂p1

= λ

∣∣∣∣∣∣∣∣ 0 −p2−p2 u22

∣∣∣∣∣∣∣∣H − x1

∣∣∣∣∣∣∣∣ −p1 u12−p2 u22

∣∣∣∣∣∣∣∣H

The two terms in this expression kind of looks like the Slutskyequation. It turns out that this can be established. We will showthat the second term is the “income” term in the Slutskyequation.


Differentiating the equations with respect to m. 0 −p1 −p2−p1 u11 u12−p2 u21 u22

∂λ∂m∂x1∂m∂x2∂m

=

−100

So again by Cramer’s rule we can solve for ∂x1/∂m

∂x1/∂m =

∣∣∣∣∣∣∣∣ −p1 u12−p2 u22

∣∣∣∣∣∣∣∣H

So that,

∂x1∂p1

= λ

∣∣∣∣∣∣∣∣ 0 −p2−p2 u22

∣∣∣∣∣∣∣∣H − x1

∂x1∂m

This starts to look like the Slutsky equation. It turns out that you

can prove that ∂h1∂p1

= λ

∣∣∣∣∣∣∣∣ 0 −p2−p2 u22

∣∣∣∣∣∣∣∣H (Homework!)


Summary

xi(p,m) hi(p, u)

υ(p,m) e(p, u)

UMP EMP

Slutsky Equation∂xj(p,m)∂pi

=∂hjp,u)∂pi−

∂xj(p,m)∂m xi

e(p, υ(p,m)) = m

υ(p, e(p, u)) = u

Roy’s Identity

xi = −∂υ(p,m)/∂pi∂υ(p,m)/∂m

∂e(p,u)∂pi

= hi(p, u)

xi (p, e(p, u)) = hi (p, u) h i(p,υ(p,m

)) = x i(p,m)

Consumer Theory: Utility Maximization - Personal … MaximizationConsumer BehaviorUtility...

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