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Advanced Microeconomics

Konstantinos Serfes

September 24, 2004

These notes have borrowed content and notation from the following two textbooks: i) Microeconomic Theory byA. Mas-Colell, M.D. Whinston and J. Green and ii) Advanced Microeconomic Theory by G.A. Jehle and P.J. Reny.You should study these notes in conjunction with the above two textbooks. These lectures notes are likely to containa number of errors.

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1 CONSUMER THEORY

Formulate the consumers utility

maximization problem subject to a budget

constraint

Solve the consumers problem to

derive the demand functions

Study the properties of the demand

functions

A QUICK OVERVIEW OF CONSUMER

THEORY

There are four building blocks in any model of consumer choice: i) the consumption set, ii) the

preference relation, iii) the feasible set, and iv) the behavioral assumption.

The consumption set depicts all the options that are available to the consumer (whether these

options are aordable of not). The preference relation puts some structure on how the consumer

ranks the dierent alternatives available to him. The feasible set includes all the options from

the consumption set that are also aordable (in a sense that will be made precise later). Finally,

the behavioral assumption is about how the consumer chooses among his alternatives, given his

preferences. In particular, we will assume that the consumer chooses the best alternative.

We will study each one of the four building blocks separately and at the end we will put them

together to pose the consumers constraint maximization problem.

The consumption set X

We assume that there are L commodities. Each commodity ` is measured in some innitely

divisible unit. Let x` 2 R+ (non-negative) represent the number of units of good `. We denote

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a vector of the L goods (consumption bundle) by x = (x1; x2;:::;xL) 2 RL+:1 We make thefollowing assumptions about X.

1.; 6

= X

RL

+

:

2. X is closed.2

3. X is convex.3

4. 0 2X:

First, we assume that the consumption set is a non-empty (i.e., it contains at least one element)

subset of an Ldimensional Euclidean space. Assumption 2 is needed for technical reasons that

will become clear later on (RL+ for instance is closed). The third assumption ensures that X has no

holes in it and no missing parts (i.e., averages of any two bundles in the set also belong in the set).

For example, if the goods were not perfectly divisible (say each good is measured only in integers

0, 1, 2, see gure 1), then X would not be convex. Finally, assumption 4 allows for the possibility

of consuming nothing.

x1

x2

0 1 2

1

2

Figure 1: A non-convex consumption set

1 We will use bold letters to denote vectors.2 For a denition of a closed set go to pages 943-944 of Mas-Colell.3 For a denition go to page 946 of Mas-Colell.

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Preferences

A preference relation is denoted by %. This is a binary relation on X: Ifx and y are in X and

x % y, then we say that x is at least as good as y. Next, we list the basic axioms that a preference

relation must satisfy.

AXIOM 1: (Completeness). For all x and y in X, either x % y, or y % x.

This axiom basically says that the consumer, when he is confronted with two dierent consump-

tion bundles, he must be able to say which one he prefers more, or that he is indierent between the

two. What is ruled out by this axiom is a situation where the consumer cannot rank two bundles. A

deeper inspection should convince you that this axiom is absolutely necessary for consumer theory.

After all, we want the consumer to choose the best bundle for him. But to choose the best element

of X, he must be able to make pairwise comparisons.

AXIOM 2: (Transitivity). For any three elements x;y and z in X; x % y and y % z, then

x % z:

Without the transitivity axiom there may not exist a best consumption bundle in X, even when

pairwise comparisons are allowed for. For example, suppose that X = fx;y; zg, i.e., it consists ofonly three consumption bundles. Further assume that axiom 2 is violated. More specically, x % y

and y % z, but z x. You can easily check that there is no best element in X: If the consumerhas to choose between x and y he will choose x. Between y and z he will choose y. But between x

and z he will choose z! We made a full cycle. Without being able to identify a set of best elements,

consumer theory does not have much predictive power.

Denition 1: The preference relation % on the consumption set X is rational, if it satises

axioms 1 and 2.

Denition 2: (Strict preference relation). The binary relation on X is dened as follows:x y if and only ifx % y and y x:

The phrase x y is read, x is strictly preferred to y.

Notice that is not complete as it cannot compare indierent bundles.

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Denition 3: (Indierence relation). The binary relation on X is dened as follows:

x y if and only ifx % y and y % x:

The phrase x y is read, x is indierent to y.

Denition 4: (Sets in X derived from the preference relation). Let x be a consumption bundle

in X. Relative to this point we can dene the following subsets of X:

1: % (x) = fy 2 X : y % xg , called the at least as good as set.2: - (x) = fy 2 X : y - xg , called the no better than set.3: (x) = fy 2 X : y xg , called the worse than set.4: (x) = fy 2 X : y xg , called the preferred to set.5: (x) = fy 2 X : y xg , called the indierence set.

Axioms 1 and 2 have put very few restrictions on the preferences. In particular, the induced

from % sets (i.e., 1-5 above) can practically admit a wide range of shapes (see gure 2 where

X = R2+). For instance, the indierence set may be thick, the at least as good set may not be

closed or it may not be convex.

x

>(x)~(x)

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Utility Functions

Denition 5: (Utility function) A function u : X ! R is a utility function representingpreference relation %, if for all x;y

2X,

x % y if and only if u(x) u(y):

We want to construct a function which, given any two consumption bundles, will assign a greater

number to the bundle that the consumer prefers more. Dealing with utility functions is much more

convenient (because functions can be dierentiated, integrated etc.) than dealing with preferences.

However, we have to demonstrate that such a transition (i.e., from preferences to utility functions

that are consistent with these preferences) is possible and without any loss of generality. This is

what we will eventually prove.

The utility function u is not unique. For any strictly increasing function f : R ! R, f(u(x)) is anew utility function representing the same preferences [see exercise 1.B.3]. For example assume that

there are only two goods and each good is measured only in integers f0; 1; 2g : Moreover, assumethat the preferences are dened as follows:

x y if and only if x1 + x2 > y1 + y2

andx y if and only if x1 + x2 = y1 + y2:

Observe that % as dened above are complete and transitive. One utility representation is:

u (0; 0) = 0; u (0; 1) = u (1; 0) = 1; u(2; 0) = u(0; 2) = u(1; 1) = 2

u(2; 1) = u(1; 2) = 3 and u(2; 2) = 4:

The above function is indeed a utility function. But it is not unique. Any monotonic transfor-

mation of the utility function preserves the rankings. If you take, for example, the logarithm (or

the square root) then you obtain another utility function which represents the same preferences.

What matters is the ranking of the consumption bundles not the actual number that the function

assigns to a bundle.

Properties of the utility function that are invariant to a strictly increasing transformation are

called ordinal.

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Cardinal, on the other hand, are not preserved under all such transformations.

The preference relation induced by u, as we demonstrated above, is an ordinal property. The

numerical values assigned by u are cardinal. I will list more ordinal and cardinal properties later

in the course.

An important issue in consumer theory is to identify the axioms that would guarantee that for

any preference relation % which satises these axioms there exists a utility function representing

these preferences. What the next proposition says is that completeness and transitivity are the

minimum set of axioms that must be imposed on %. Without these two axioms it is impossible to

construct a utility function.

Proposition 1: A preference relation % can be represented by a utility function only if it isrational.

Proof: We want to prove the following. Suppose that there exists u : X ! R representing %.Then % must be complete and transitive.

Suppose by way of contradiction that this is not true. That is, there exists such a u but

preferences are not complete and transitive. So, they may be either incomplete, or intransitive, or

both.

Completeness: Take any x and y in X such that the consumer cannot compare. Then, u(x)

cannot be compared with u(y). But u is a real-valued function and u(x) and u(y) are real numbers,

which can be ranked. Contradiction.

Transitivity: Take any x, y and z in X such that x % y and y % z but z x: This implies thatu(x) u(y) and u(y) u(z). But since u is a real-valued function, it must be that u(x) u(z).Contradiction.

In other words, rational preferences is a necessary condition for the existence of a utility functionrepresenting these preferences. However, rationality of preferences in general is not sucient for

them to be represented by a utility function. As we will shortly see we need more axioms to

guarantee that. Nonetheless, we can ask the following question.

Question: Under what conditions rationality of preferences is also a sucient condition for

the existence of a utility function?

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Exercise 1.B.5.: Show that if X is nite and % is a rational preference, then there exists

u : X ! R that represents %.

You can easily see this and a proof is not needed. The point is that niteness of the consumption

set and rationality of preferences are sucient conditions for the existence of a utility function. As

it will become evident later, when X is not nite, this assertion is not necessarily true (when for

example L = 2 and X = [0; 1] [0; 1]).

We will return to preferences again to impose more axioms.

The feasible set

Let p = (p1; p2;:::;pL) 2 RL++ denote a vector of prices and w 2 R++ denote the consumersincome. We assume that p is a row vector and x a column vector. A competitive budget is dened

as,

Bp;w = fx 2 X : p x =p1x1 +p2x2 + +pLxL wg :

The set,

Bp;w = fx 2 X : p x = wg

is called the budget hyperplane. In two dimensions (i.e., when L = 2) the feasible set is the shaded

area in gure 3 below.

x1

x2

w/p1

w/p2 Slope = - (p1/p2)

Bp,w

{x in RL+: px = w}

Figure 3

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Claim: The competitive budget Bp;w is a convex set.

The above assertion can be proved as follows: Take y and z in Bp;w. We have to show that

x0 = y + (1 )z 2 Bp;w,

where 2 [0; 1] : More precisely, we must show two things: i) x0 2 X and ii) p x0 w.

Clearly x0 2 X since by assumption X is convex. Also we have,

p y w and p z w;

which implies,

p

y

w and (1

)p

z

(1

)w:

By adding the two inequalities we obtain,

p y + (1 )p z = p y + p (1 )z = p (y+(1 )z) =p x0 w + (1 )w = w;

which completes the proof.

Behavioral assumption

The consumer chooses the bundle that is most preferred among the ones that are in the feasible

(budget) set.

We continue by imposing more axioms on the preferences.

AXIOM 3 (Monotonicity):

Strong monotonicity: Ifx and y 2 X such that y x and y 6= x, then y x. (y x meansthat every coordinate ofy is weakly greater than the corresponding coordinate ofx, but the

assumption y 6= x prevents all of them from being equal).

Monotonicity: Ifx and y 2 X such that y x then y x (y x means that every coordinateofy is strictly greater than the corresponding coordinate ofx).

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Locally non-satiated: For all x 2 X and for all " > 0 there exists a y 2 X such that,

ky xk " and y x;

where ky xk is the Euclidean norm, i.e., ky xk = qPL`=1 (x` y`)2:To see the dierence between the three monotonicity concepts consider rst the preferences

which are represented by a Cobb-Douglas utility function: u = x1x2: These preferences are strongly

monotone, monotone and locally non-satiated. A bundle that contains more of only one good is

strictly preferred to the initial bundle.

Now consider preferences which are represented by a Leontief utility function: u = min fx1; x2g :

These preferences are monotone and locally non-satiated, but not strongly monotone. A new bundlemust contain more of both goods to be strictly preferred.

The following is true (PROVE IT AS AN EXERCISE):

Strong monotonicity ) Monotonicity ) Local non-satiation.

Therefore, strong monotonicity is the strongest assumption and local non-satiation is the weak-

est. For most of the subsequent analysis local non-satiation is enough.

Question: What does each one of the monotonicity assumptions imply about the shape of the

indierence curves?

With local non-satiation we eliminate thick indierence curves (gure 4). With monotonicity we

can guarantee that the indierence curves are not thick, do not have any increasing parts (at part

are allowed; gure 5)) and the preferred to set is above the indierence curve and the worse

than set below. Finally, with strong monotonicity we obtain indierence curves which are not

thick, are always (strictly) downward sloping (gure 6) and the preferred to set is above the

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indierence curve and the worse than set below. [Show why the above claims are true]

x

x1

x2

Figure 4

x

x1

x2

Figure 5

x

x1

x2

Figure 6

AXIOM 4 (Continuity):

For all x 2 X, the set % (x) (i.e., the at least as good as set, or the upper contour set) and- (x) (i.e., the no better than set, or the lower contour set) are closed in X.

A set is closed in a particular domain if its complement is open in that domain. So, if% (x)

is closed in X then its complement (x) is open in X: The continuity axiom guarantees thatsudden preference reversals do not occur. The continuity axiom can equivalently be expressed

using sequences of bundles, rather than open and closed sets. We can then say that % (x) is closedif for any sequence yn 2% (x) such that yn ! y, we have y 2% (x).

Notice that if both % (x) and - (x) are closed, then so is (x) since (x) =% (x)\ - (x)and an intersection of closed sets is closed. This implies that the continuity axiom rules out open

areas in the indierence set like the one in gures 4,5 and 6.

We need both the upper and the lower contour sets to be closed. Take a look at the preferences

as given in gure 7 (no monotonicity assumption is satised in this example). The indierence set

is closed (the dark grey ball). The strictly lower contour set (the light grey ball) does not contain

its boundary points. Everything else is the strictly upper contour set. Notice that the set (x) isclosed. The set - (x), however, is not closed. Notice that - (x) = (x)[ (x) which is the unionof a closed with an open set which does not have to be closed. Nevertheless, the upper contour set

% (x) is closed.

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I can nd a converging sequence yn 2- (x) such that yn ! y and y is located on the dottedcircumference of the light grey ball. In this case the limit point of yn which is y belongs in (x)(sudden preference reversal).

x1

x2

x

~(x)

(x)

y yn

Figure 7

AXIOM 5 (Convexity):

Weak convexity: Preferences % on X are convex if for all x, the set % (x) is a convex set, that is,

ify 2% (x) and z 2% (x), then w =y + (1 )z 2% (x), for all 2 [0; 1] :

Strict convexity: Preferences % on X are strictly convex if for all x, the set % (x) is a strictly

convex set, that is, ify 2% (x) and z 2% (x) and y 6= z then w = y + (1 )z 2 (x), forall 2 (0; 1) :

Convexity of preferences implies that the consumer prefers average bundles to extreme ones.

Another interpretation, is that the consumer has a taste in favor of diversication (see Chapter 6

of the main text).

When L = 2 the absolute value of the slope of an indierence curve is called the marginal rate

of substitution (MRS), which measures how many units of good 2 the consumer is willing to give

up for a marginal unit of good 1, such that he remains indierent after the exchange. If preferences

are monotonic (which implies that the upper contour set is above and the lower contour set below)

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convexity implies that the amount of good 2 that the consumer is willing to give up for an extra

(marginal) unit of good 1 decreases as the amount of good 1 increases (see gure 8). In particular

we have,

Convexity ) Diminishing (MRS)Strict convexity ) Strictly diminishing (MRS).

x1

x2

x

y

z

w

Figure 8: Strictly convex preferences and diminishing MRS

EXISTENCE OF A UTILITY FUNCTION

A central question in consumer theory is: what is the minimum set of axioms that we should

impose on preferences to guarantee the existence of a utility function representing these prefer-

ences? As we have seen in the previous lecture rationality of preferences is an absolutely necessary

assumption. Without it we can guarantee the non-existence of a utility function. We also saw that

if X is nite, rationality of preferences guarantees the existence of u.

First, we oer an example where X is not nite, preferences are rational but not continuous,

and it is impossible to construct a utility function.

Assume that L = 2. Dene % as follows,

x % y if either: i) x1 > y1 or ii) x1 = y1 and x2 y2

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This is known as lexicographic preference relation. (Draw an indierence curve).

Claim: These preferences do not satisfy the axiom of continuity. They are however rational,

strongly monotone and strictly convex [see exercise 3.C.1].

I will show that the lower contour set is not closed. Let y = (2; 3) and yn = (2 1=n; 3) : Alsolet x = (2; 2) : Clearly yn ! y as n ! 1 and yn 2- (x) for all n. But y =2- (x), since y x: Asimilar argument can show that the upper contour set is not closed either.

Question: Can we nd a u : X ! R that represents lexicographic preferences?

Without any loss of generality, assume that X = [0; 1] [0; 1] : Suppose that there exists au : X

!R which represents the lexicographic preferences.

x1

x2

x1

1

1

y

y

z

z

x

x1

x

Figure 9

The proof follows gure 9 closely. It can be easily seen that u(y) > u(y0

). Since rationals aredense we can always nd a rational between two real numbers. Hence, we can nd a bundle x

such that u(x) is a rational number [denoted by u(x) = r(x)] and u(y) > r(x) > u(y0

): We also

have u(z) > u(z0

) and by the same logic there exists another rational, denoted by r(x0

) such that,

u(z) > r(x0

) > u(z0

): Moreover, u(z0

) > u(y) [since bundle z0

contains more of good 1 than bundle

y] and therefore,

u(z) > r(x0

) > u(z0

) > u(y) > r(x) > u(y0

):

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Notice that x0

1 > x1 and r(x0

) > r(x): I have constructed a one-to-one function from the set of

reals (the xs) to the set of rationals (the r(x)s). But this is a mathematical impossibility since

the set of reals is an uncountable set while the set of rationals is countable (the rationals can be

put into a one-to-one correspondence with the set of natural numbers which is also countable; thereals cannot). They do not have the same cardinality and therefore it is not plausible to construct

a one-to-one mapping. Therefore, a utility function does not exist.

Theorem 1 (Existence of a continuous utility function): If % are complete, transi-

tive, continuous and strongly monotone, then there exists a continuous real-valued function (utility

function) u : X ! R representing %.

Idea of the proof: The intuition behind the proof can be grasped by studying gure 10.

x1

x2

x

u(x)e

~(x)

u(x)

u(x)

1

1 e

45o

~(y)

y

u(y)e

Figure 10

Fix a bundle x: Find all the indierent to x bundles (indierence curve). Based on the axioms

this set exists and it is not thick. Assign a number to all these bundles. A dierent bundle y that

is strictly preferred to x must lie on a higher indierence curve. Assign a higher number to y (andto the bundles indierent to y). This is how we construct a utility function which represents these

preferences. The function is also continuous, because there are no jumps in the preferences and

therefore no jumps in the numerical assignments.

Rigorous proof: We will prove that at least one such utility function exists. Let

e (1; 1;:::; 1) 2 RL++

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and consider the mapping u : X ! R dened so that the following is satised,

u(x) e x. (*)

Question 1: Does there always exist a number u(x) satisfying (*) ?

Question 2: Is it uniquely determined so that u(x) is a well-dened function ?

We attempt to answer the rst question. Fix x 2 X and consider the following two subsets ofthe real numbers:

A ft 0 : t e % xg and B ft 0 : t e - xg .

If t 2 A \ B, then t e x, so setting u(x) = t would satisfy (*). Thus, we have to show

that A \ B 6= ;.

Continuity of %: This implies that A and B are closed sets

Strong monotonicity of %: t 2 A ) t0 2 A for all t0 t: In other words, A must be a closedinterval of the form [t

, 1). Similarly, B must be a closed interval of the form [0; t].

Completeness of %: This implies that either t e % x or t e - x, that is t 2 A [ B: But thismeans that R+ = A [ B = [t

, 1) [ [0; t].

Therefore, t

t so that A \ B 6= ;.

Now lets answer the second question. We must show that there exists only one number t 0such that t e x.

If t1 e x and t2 e x then by transitivity t1 e t2 e. By strong monotonicity it must bethat t1 = t2: So for every x 2 X there exists exactly one number u(x) such that

u(x) e x.

Next, we have to show that u (as we constructed it) represents %.

Consider two bundles x and y and u(x), u(y) which by denition must satisfy

u(x) e x and u(y) e y:

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We have

x % y () u(x) e x % u(y) e ytransitivity

()u(x)

e %u(y)

e

strong monotonicity () u(x) u(y).

Now it remains to be shown that u is continuous. We must show that the inverse image under

u of every open ball in R is open in X.

u1 ((a; b)) = fx 2 X : a < u(x) < bgmonotonicity = fx 2 X : a e u(x) e b eg

u(x) e x and transitivity = fx 2 X : a e x b eg=

(a

e)

\ (b

e):

% and - are closed sets and therefore and are open. u1 ((a; b)) is the intersection of twoopen sets and hence it is also open.

Exercise 3.C.4. Exhibit an example of a preference relation that is not continuous but is

representable by a utility function.

Let X = R+ and dene u() : X ! R by,

u = 8 1any in [0; 1], if x = 1:

Denote by % the preference relation represented by u(). We will show that % is not continuous,by showing that either the upper or the lower contour set is not closed. First, suppose that u(1) > 0:

Let

xn = 1 1n

0.

xn 2- (0) and it converges to x = 1 =2- (0) since 1 0. Thus, the lower contour set is notclosed.

Second suppose that u(1) < 0: Let

xn = 1 +1

n 2.

xn 2% (2) and it converges to x = 1 =2% (2) since 2 1. Thus, the upper contour set is notclosed.

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1.1 Properties of preferences and utility functions

Let % be represented by u : X ! R: Then,

1. u(x) is strictly increasing if and only if% is strongly monotonic.

2. u(x) is quasi-concave if and only if% is convex.

3. u(x) is strictly quasi-concave if and only if% is strictly convex.

Homotheticity: A continuous % is homothetic if and only if it admits a utility function that ishomogeneous of degree one.

A function f : Rn ! R is homogeneous of degree one if f( x) = f(x), for any 2 R++.

A function h : Rn ! R is homothetic if it can be written as g(f(x)) where g : R ! Ris strictly increasing and f : Rn ! R is homogeneous of degree 1.

If preferences are homothetic then the slopes of the level sets ofu (Level set = fx 2 X : u(x) = kg)are unchanged along any ray through the origin (see gure 11). As it will become evident later on,

this implies that if preferences are homothetic, the goods are not inferior goods.

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x1

x2

x

2x

y

2y

Figure 11

Example: The preferences represented by a Cobb-Douglas utility function, u = xa1xb2, a > 0 and

b > 0 are strongly monotonic, strictly convex and homothetic. The Cobb-Douglas utility function

is strictly quasi-concave, strictly increasing and homothetic.

The fact that u is strictly increasing is easy to check. To verify that it is strictly quasi-concave,

the bordered Hessian matrix must be negative denite [see Mas-Colell p.938], i.e.,u11 u12 u1u21 u22 u2u1 u2 0

> 0:The latter is true if and only if

2u1u2u12 [u1]2 u22 [u2]2 u11 > 0:

Now you can check that the following is true,

2u1

u2

u12

[u1

]2 u22

[u2

]2 u11

= ab(a + b)x(3a2)

1x(3b2)

2> 0:

Put it dierently, we require u to be strictly concave (i.e., to have a negative denite Hessian

matrix D2u(x)) not everywhere but in the subspace fz 2 X : ru(x) z = 0g, where,

ru(x) =

@u

@x1;

@u

@x2

;

is the gradient vector (see gure 12).

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x1

x2

Gradient vector

z

x

Figure 12

So the condition says that u must be strictly concave when we move in such a way that the

level of the function does not change.

Another way to prove the strict quasi-concavity of u is to show that the upper contour set is

strictly convex. The upper contour set is,nx 2 X : xa1xb2 k

o:

The indierence curve isx2 =

k

xa1

1=b:

Dierentiate the above function twice and you will see that the outcome is positive which

indicates that the indierence curve is strictly convex and the upper contour set as a consequence

is also convex.

To prove that preferences are homothetic you have to show that preferences admit a utility

function which is homogeneous of degree 1. Right now, it is not necessarily true that u = xa1xb2 is

homogeneous of degree 1. But all we have to show is that % admit a homogeneous of degree 1 utility

function. Now remember that any strictly monotonic transformation of u leaves the preferences

unchanged. Transform u as follows,

~u =

xa1xb2

1=(a+b):

Finally verify that ~u is homogeneous of degree 1,

~u(t x) =

(tx1)a (tx2)

b1=(a+b)

= ta=(a+b)tb=(a+b)

xa1xb2

1=(a+b)= t(a+b)=(a+b)~u = t~u:

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Exercise: Consider% which are represented by the following utility function: u =p

x2+p

x1x2:

Are these preferences: i) monotonic, ii) convex and iii) homothetic?

Solution: Preferences are clearly strongly monotonic. They are also strictly convex. To see this

compute the determinant of the bordered Hessian matrix, which yields (verify the result),

:06255p

x1x2 + 4x1p

x2 +p

x2x1

px2

px1x2

> 0:

To prove that preferences are homothetic, we must show that the slope of the indierence curves

is constant along a straight line emanating from the origin. In other words, the following set of

bundles

(x1; x2) 2 X :M U1

M U2

= kforms a straight line. MUi = @u=@xi (marginal utility) and the ratio of the marginal utilities is the

marginal rate of substitution. This would imply that the locus where the slope of the indierence

curves is constant is a straight line.

M U1M U2

= k )12 (x1x2)

1=2 x212x1=22 +

12 (x1x2)

1=2 x1= k )

12x1=22 +

12 (x1x2)

1=2 x112 (x1x2)

1=2 x2=

1

k

)p

x1 + x1x2

=1

k) x2 = k

px1 + x1

Preferences are not homothetic since the slope of the indierence curves is not constant on a

straight line starting from the origin.

In the remaining of the course we will assume (without proving it) that the utility function is

dierentiable (that is, the right and left derivatives at any point are equal). Notice that continuity

does not guarantee dierentiability. A standard example is the function y = jxj which is clearlycontinuous, but at zero is not dierentiable. The left derivative is 1 and the right +1. To ensurethe dierentiability of the utility function we have to assume that preferences are smooth (in some

sense that will not be made precise). Basically, one thing that is not allowed is indierence curveswith kinks. For a rigorous treatment see G. Debreu, Smooth preferences, Econometrica (1972).

Given that we can dierentiate u, we can formally introduce the following concepts.

Marginal utility of good `:M U (x) =

@u(x)

@x`:

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Marginal rate of substitution between goods i and j:

M RSij(x) = @u(x)=@xi@u(x)=@xj

:

Now we are ready to state and solve the consumers utility maximization problem.

1.2 The utility maximization problem (UMP)

The consumer takes the market prices p and his income w as given and chooses the consumption

bundle x which gives him the highest utility provided that the chosen bundle is aordable. Formally,

maxx2X

u(x)

subject to : p x w:

The solution to this problem is known as Marshallian (Mas-Colell calls them Walrasian)

demand functions and are represented as,

x(p; w) =

0BBBB@

x1(p; w)

xL(p; w)

1CCCCA

2 RL+:

The vector x(p; w) shows how many units of each good the consumer will choose so that he

maximizes his utility given the vector of prices p and his income w:

First, we will prove that a solution to the UMP exists.

Proposition: If p 0 and u is continuous, then the UMP has a solution.

Proof: If p 0 then the budget set Bp;w = fx 2 X : p x wg is both closed and bounded.

Bounded: 0 x` w=p`. There exists a clear upper and lower bound to the consumption ofeach good.

Closed: Take any sequence xn 2 Bp;w such that xn ! x. We must show that x 2Bp;w. Supposeby way of contradiction that this is not the case, i.e., x =2Bp;w. This means that p x > w. Since

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the inner product p x is continuous in x, I can nd an ~n (large enough) and an associated bundlex~n very close to x, such that p x~n > w. But this is a contradiction, since we have assumed thatxn 2 Bp;w for any n.

By the Heine-Borel theorem (which states that a subset of RL is compact if and only if it is

closed and bounded) Bp;w is a compact set.

Now we can apply the Maximum Theorem, which states that a real-valued continuous function

on a compact set attains its maximum and minimum.

In our case the function is the utility function (which is continuous if axioms 1-4 hold) and is

dened on the budget set which is compact. The maximum is attained by the solution to the UMP

which is the demand functions.

1.2.1 Demand functions and comparative statics

Denition: x(p; w) is homogeneous of degree zero in (p; w) if, x(p; w) = x(p; w), for all

p; w and > 0.

This says that if all prices and income increase (or decrease) by the same proportion, then the

optimal demand should not change. This is like a synchronized ination which raises all prices and

incomes by the same percentage.

Denition: x(p; w) satises the Walras Law if, p x(p; w) = w, for all p; w:

This says that the consumer will spend all his income. As we will see below, this is clearly true

when preferences are locally non-satiated.

Denition (Income eects): A commodity ` is normal at (p; w) if,

@x`(p; w)

@w 0;

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and inferior if,@x`(p; w)

@w 0:

If every commodity is normal at all (p; w), then we say that the demand is normal.

Below, I present the income (wealth) eects in vector representation,

Dwx(p; w) =

0BBBB@

@x1(p; w)=@w

@xL(p; w)=@w

1CCCCA 2 RL:

Denition (Price eects): A commodity ` is ordinary at (p; w) if,

@x`(p; w)

@p` 0;

and Gien if,@x`(p; w)

@p` 0:

If a goods demand curve is everywhere downward sloping in the goods own price, then the

good is ordinary. Otherwise, it is Gien.

Below, I present the price eects in matrix representation,

Dpx(p; w) =

0BBBBBB@

@x1(p; w)=@p1 @x1(p; w)=@pL

@xL(p; w)=@p1 @xL(p; w)=@pL| {z }

1CCCCCCA

LL matrix

:

Denition (Price elasticities): Good `s elasticity with respect to good ks price is dened

as,

"`k(p; w) =@x`(p; w)

@pk

pkx`(p; w)

:

It measures good `s percentage change following a marginal percentage change in good ks

price. Ifk = `, then we call it own price elasticity, while ifk 6= ` it is called cross price elasticity.

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Denition (Income elasticities): Good `s elasticity with respect to income w is dened as,

"`w(p; w) =@x`(p; w)

@w

w

x`(p; w):

It measures good `s percentage responsiveness to a small percentage change of income.

Elasticities do not depend on the units chosen for measuring commodities and for this reason

they provide a unit-free way of capturing demand responsiveness. They are used extensively in

applied work.

Proposition: Suppose that u is a continuous and strictly quasi-concave utility function rep-

resenting locally non-satiated preferences. Then the Marshallian demand functions x(p;w) satisfy

the following properties:

1. They are homogeneous of degree zero in (p;w):

2. They satisfy the Walras Law.

3. x(p; w) consists of a single element, i.e., it is a function rather than a correspondence (multi-

valued function).

Proof:

1. We have to show that x(p; w) = x(p; w), for all p; w and > 0. Multiply each price and

income by a scalar . First notice that the utility is not aected by . Simply does not

enter u. Second observe that

Bp;w = fx 2 X : p x wg = Bp;w = fx 2 X : p x wg :

The budget set remains unchanged. Therefore, does not change the maximization problem

in any way and consequently the solution must be the same, regardless of the .

2. Suppose by way of contradiction that p x(p; w) < w. Then there must be another bundle yvery close to x such that: i) (by continuity of the inner product) p y < w and ii) (by localnon-satiation) y x. So I found an aordable bundle which is strictly preferred to x. Butthis contradicts the optimality ofx.

3. I must show that x(p; w) is single-valued. In other words, for any (p; w), there is only one

value of each good which solves the consumers maximization problem. Suppose otherwise,

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i.e., there are two bundles q and y with q 6= y such that both belong to x(p; w). This meanstwo things: i) both bundles are aordable and ii) there is no other aordable bundle that is

strictly preferred to these two. Let

z = q + (1 )y; 2 (0; 1) :

Since the utility function is strictly quasi-concave, preferences are strictly convex and a convex

combination of q and y (which is the new bundle z) must be strictly preferred, i.e., z qand z y: But z is also aordable, i.e.,

p z = p (q + (1 )y) w + (1 )w = w:

Hence, q and y cannot be optimal at the same time. This contradiction leads us to conclude

that x(p; w) is single-valued.

1.2.2 Implications of homogeneity and Walras Law for price and wealth eects

N The homogeneity of degree zero implies that for all (p; w),

L

Xk=1"`k(p; w) + "`w(p; w) = 0; for ` = 1;:::;L. (*)

To see this x one good, say good `. The above equation says that the sum of all the price (own

and cross) and income elasticities is zero. If all prices and income increase by 1%, the demand of

good ` will remain unaected. The same holds for all goods.

Proof: Homogeneity of degree zero means,

x(p; w) x(p; w) = 0; > 0:

Dierentiate the above expression with respect to ;

LXk=1

@x`(p; w)

@(pk)pk +

@x`(p; w)

@(w)w = 0; ` = 1;:::;L;

and set = 1 to obtain (the above equation holds for any > 0 and it will certainly hold for

= 1),LX

k=1

@x`(p; w)

@(pk)pk +

@x`(p; w)

@(w)w = 0; ` = 1;:::;L:

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If we divide both sides of the above equation by x`(p; w) and using the elasticity formulas we

obtain (*).

N The Walras Law implies that for all (p; w);

(Cournot aggregation)

LXk=1

bk(p; w)"k`(p; w) + b`(p; w) = 0; for ` = 1;:::;L, (**)

which says that total expenditure cannot change in response to a change in prices, and

(Engel aggregation)LX

k=1

bk(p; w)"`w(p; w) = 1; for ` = 1;:::;L, (***)

which says that total expenditure must change by an amount equal to any wealth change. bk(p; w) =

pkxk(p; w)=w is the budget share of the consumers expenditure on good ` given the prices and

income.

Proof: From Walras Law we have,

p x(p; w) = 0:

Dierentiating the above with respect to prices we obtain,

L

Xk=1pk

@xk(p; w)

@p`+ x`(p; w) = 0; for ` = 1;:::;L.

By multiplying both sides by p`=w and multiplying and dividing all the terms except the last

one by xk(p; w), we obtain,

LXk=1

pkxk(p; w)

xk(p; w)

@xk(p; w)

@p`

p`w

+x`(p; w)p`

w= 0; for ` = 1;:::;L;

which yields (**).

A similar argument can be used to prove (***).

1.2.3 Solving the UMP

The UMP,

maxx2X

u(x)

s.t. : p x w

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is a nonlinear programming problem with one inequality constraint. From now on and due to a

monotonicity assumption on the preferences we will assume that the budget constraint is satised

with equality. The Lagrangian of the above problem is,

L(x) = u(x) + (w p x) :

The Kuhn-Tucker necessary conditions say that ifx(p; w) is a solution to the UMP, then there

exists a Lagrange multiplier 0 such that,@L(x)

@x`=

@u(x)

@x` p` 0; and x`

@L(x)

@x`= 0; ` = 1;:::;L; (1)

and@L(x)

@= (w p x) 0; and @L(x

)

@= 0: (2)

(2) will always be satised with equality since preferences are monotonic, i.e.,

@L(x)

@= (w p x) = 0: ((20))

If the solution is interior, i.e., x(p; w) 0; then (1) is also satised with equality (see gure13),

@L(x)

@x`=

@u(x)

@x` p` = 0: ((10))

(10

) gives the famous condition that at the optimum M RSij = pi=pj: In this case the gradientvector is proportional to the price vector.

x1

x2

w/p1

w/p2

Slope = - (p1/p2) = -MRS12(x*)x*

p

u(x*)

Figure 13: Interior solution

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Corner solutions, however, where at least one good is zero cannot be ruled out a priori. In such

a case, (1) becomes @L(x)=@x` 0 (which means that by lowering x` the objective increases, butthe decrease of x` has to stop at zero) and x

` (@L(x)=@x`) = 0 since x` = 0 (see gure 14).

x1

x2

w/p1

w/p2

Slope = -MRS12(x*)

x*

u(x*)

pp

Figure 14: Corner (boundary) solution

If the utility function is quasi-concave, then the Kuhn-Tucker rst order conditions are not only

necessary but also sucient.

EXAMPLES

1. Cobb-Douglas. u = xa1xb2: The solution to the UMP

max(x1;x2)

u = xa1xb2

s.t. : p1x1 +p2x2 = w

can take one of the following four forms: i) both goods are strictly positive (interior solution), ii)

and iii) only one is strictly positive and iv) both goods are zero (boundary solutions). The Lagrange

function is,

L = xa1xb2 + (w p1x1 p2x2) :

Since the Cobb-Douglas function is strictly quasi-concave the rst order K-T conditions are

necessary and sucient.

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Case 1 . x1 > 0 and x2 > 0. The K-T conditions must be satised with equality,

@L

@x1= 0 ) axa11 xb2 p1 = 0; (1)

@L@x2

= 0 ) bxa1xb12 p2 = 0; (2)@L

@= 0 ) w p1x1 p2x2 = 0: (3)

Divide (1) by (2) to obtain,ax2bx1

=p1p2

: (4)

Solve (4) with respect to x2 and plug it into (3) and solve for x1. This yields,

x1 =aw

(a + b)p1: (5)

Plug (5) back into (4) to obtain,

x2 =bw

(a + b)p2: (6)

Case 2 . x1 > 0 and x2 = 0: This is impossible to be optimal since the utility even when one

good is zero is zero and the consumer can clearly do better than that. For the same reason the

remaining cases will not be optimal.

(5) and (6) are the Marshallian demand functions.

2. Perfect substitutes. u = ax1 + bx2: As before, the solution to the UMP

max(x1;x2)

u = ax1 + bx2

s.t. : p1x1 +p2x2 = w

can take one of the following four forms: i) both goods are strictly positive (interior solution), ii)

and iii) only one is strictly positive and iv) both goods are zero (boundary solution). The Lagrange

function is,

L = ax1 + bx2 + (w p1x1 p2x2) :Since the utility function is quasi-concave the rst order K-T conditions are necessary and

sucient.

Case 1 . x1 > 0 and x2 > 0: The K-T conditions must be satised with equality,

@L

@x1= 0 ) a p1 = 0; (1)

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@L

@x2= 0 ) b p2 = 0; (2)

@L

@= 0 ) w p1x1 p2x2 = 0: (3)

By looking at the ratio (1)/(2) which is a=b = p1=p2 we can say that case 1 is valid only when

a

b=

p1p2

:

In any other case, the K-T conditions are not satised with equality. In this case the optimal

bundle must satisfy (3) and it is not unique.

Case 2 . x1 > 0 and x2 = 0: The K-T conditions are,

@L@x1= 0 ) a p1 = 0; (4)

@L

@x2< 0 ) b p2 < 0: (5)

From (4) = a=p1 which if it is plugged into (5) yields the condition under which case 2 is

valid,

b ap1

p2 < 0 ) p2p1

>b

a: (6)

The optimal solution in this case is,

x1 =w

p1and x2 = 0:

Case 3 . x1 = 0 and x2 > 0: The K-T conditions are,

@L

@x1< 0 ) a p1 < 0; (7)

@L

@x2= 0 ) b p2 = 0: (8)

From (8) = b=p2 which if it is plugged into (7) yields the condition under which case 3 isvalid,

a bp2

p1 < 0 ) p2p1

:

w

p1 ; if

p2

p1 >b

a2 [0;w=p1] ; if p2p1 = ba0; if p2p1 :

w

p2 ; if

p2

p1

ba

3. Concave preferences. u = ax21 + bx22. The Lagrange function is,

L = ax21 + bx22 + (w p1x1 p2x2) :

In this case, the utility function is not quasi-concave and the K-T conditions are only necessary.

We can guess the solution and then verify the K-T conditions.

Case 1. x1 > 0 and x2 > 0: This case will yield a minimum, not a maximum, since the worse

than set is now convex (see g. 15).

x1

x2

(c/a).5

(c/b).5

Better than

set

Figure 16: Non-convex preferences

Hence the solution will be a boundary solution.

The solution is,

x1 =

8>:

wp1

; if p1p2 p

ab

and x2 =

8>:

wp2

; if p1p2 >p

ab

0 or wp2 ; ifp1p2

=p

ab

0; if p1p2 0: Hence, if the non-negativity constraints are not explicitly included then at

the optimum if a variable is zero the partial of the Lagrange function with respect to this variable

must be negative.

Claim: The Lagrange multiplier (on the budget constraint) gives the marginal value of relax-

ing the constraint (by increasing the consumers income).

Proof: Let x(p; w) 0. The change in the utility from a marginal increase in w is given by,Ou(x(p; w)) Dwx(p; w) (*)

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But from the K-T rst order conditions (for interior maximum) we have,

Ou(x(p; w)) = p (**)

Plug (**) into (*),

Ou(x(p; w)) Dwx(p; w) = pDwx(p; w)

which after using the Engel aggregation condition (p Dwx(p; w) = 1) becomes,

Ou(x(p; w)) Dwx(p; w) = :

1.2.4 Continuity and dierentiability of the Marshallian demand correspondences(MDC)

Continuity Recall that when preferences are strictly convex then the solution is a function not a

correspondence (multi-valued function). But when preferences are not strictly convex (e.g. perfect

substitutes) then the solution is in general a correspondence.

First, we will introduce a new notion of continuity (called upper hemicontinuity) that is used

whenever we have a correspondence and it coincides with the standard notion of continuity for

functions when the correspondence becomes a function.

Denition (upper hemicontinuity): A MDC x(p; w) is upper hemicontinuous (uhc) at

(p; w) if whenever (pn; wn) ! (p; w) as n ! 1, xn 2 x(pn; wn) for all n and x = limxn; we have,

x 2 x(p; w).

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p

x(p)

pbar pn

xbar

xn

Figure 18: A correspondence that is NOT uhc

In gure 18 we depict a correspondence that is not uch. The sequence xn is in the correspondence

for all n but the limit x is not.

Proposition (uhc of the MDC): Suppose that u is a continuous utility function representing

locally non-satiated preferences. Then the solution to the UMP x(p; w) is an uhc correspondence

at all (p; w) 0.

Proof: By way of contradiction suppose that the demand correspondence x(p; w) is not uhc.

Take a sequence (pn; wn) ! (p; w) and a sequence xn such that xn 2 x(pn; wn) for all n butlimxn = ~x =2 x(p; w). This implies the following about ~x: i) either it is not aordable, or ii) it isaordable but there exists another aordable bundle with a strictly higher utility. We will rst

show that it is indeed aordable.

Observe that pn

xn

wn for all n (since xn

2x(pn; wn) and therefore it must be aordable).

By taking limits as n ! 1pn xn wn

# # #p ~x w

Hence, ~x 2Bp; w, i.e., it is aordable.

It must be then that there exists at least one other aordable bundle y (p y w) such that

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u(y) > u(~x): By the continuity of u there exists a bundle z close to y such that p z < w and

u(z) > u(~x): (*)

If now n is large enough pn z wn. Hence, z 2 Bpn;wn and we must have u(xn) u(z) becausexn 2 x(pn; wn). Taking limits as n ! 1 the continuity of u implies that u(~x) u(z) contradicting(*).

Remark: Ifx(p; w) is a function, then the above proposition says that the Marshallian demand

functions are continuous.

Dierentiability We assume that u is twice continuously dierentiable and

ru(x)

6= 0 for all x:

Suppose that x(p; w) 0. This is the solution to the system of L + 1 equations in L + 1 unknowns,

ru(x) p = 0w p x = 0:

By the implicit function theorem (IFT) (Theorem M.E.1 p.941) we know that the dierentiabil-

ity ofx(p; w) depends on the Jacobian matrix of the system having a non-zero determinant. The

Jacobian of the system is,

D2u(x) ppT 0 Example: Suppose L = 2. The Lagrangian is,

L = u(x1; x2) + (w p1x1 p2x2):

The system of the 3 equations is,

@u

@x1 p1 = 0

@u

@x2 p2 = 0w p1x1 p2x2 = 0:

The Jacobian of this system is, 0B@

@2u@x21

@2u@x1@x2

p1@2u

@x2@x1@2u@x22

p2p1 p2 0

1CA

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Since the F.O.C.s hold (i.e., @u=@x` = p`) the above matrix has a non-zero determinant if

and only if the determinant of the bordered Hessian

@2u@x21

@2u@x1@x2

@u@x1

@2u@x2@x1@2u@x22

@u@x2@u@x1

@u@x2

0

6= 0The Cobb-Douglas utility function, for example, has a bordered Hessian with a strictly positive

determinant and we can conclude that the demand functions must be dierentiable.

In general, and following the same logic as in the example above, the Jacobian is non-zero i

the bordered Hessian

D2u(x) ru(x)

ru(x) 0

6= 0:

This condition means that the indierence set through x has a non-zero curvature at x (it is

not even innitesimally at). This condition is a slight strengthening of quasi-concavity.

1.2.5 The indirect utility function

The indirect utility function measures the highest utility the consumer can attain for any given

prices and income, i.e.,

v(p; w) = u(x(p; w))

where x(p; w) are the Marshallian demand functions. This is the value function of the UMP.

Proposition (properties of the indirect utility function): The IUF is,

1. Homogeneous of degree zero in (p; w):

2. Strictly increasing in w and non-increasing in p` for any `.

3. Quasi-convex, that is, the set f(p; w) : v(p; w) vg is convex for any v.

4. Continuous in (p; w):

Proof:

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1. Since the demand functions are homogeneous of degree zero it follows that,

v(p; w) = u(x(p; w)) = u(x(p; w)) = v(p; w):

2. Easy to see.

3. Consider (p; w) and (p0; w0) such that v(p; w) v and v(p0; w0) v. Consider

(p00; w00) =

p + (1 )p0; w + (1 )w0 for 2 [0; 1] :We need to show that v(p00; w00) v: We have to show that for any x such that p00 x w00 we

must have u(x) v. Notice rst that if, p00 x w00 then,

p x + (1 )p0 x w + (1 )w0:

Thus, either px w or p0x w0 or both. If the rst inequality holds, then u(x) v(p; w) v.If the second inequality holds then u(x) v(p0; w0) v:

4. When preferences are strictly convex, we proved that the demand functions are continuous.

Hence v(p; w) = u(x(p; w)) is a composition of two continuous functions and it should be contin-

uous. Note that v(p; w) is continuous even when preferences are not strictly convex (Maximum

Theorem). We will not present a more general proof.

1.3 The expenditure minimization problem

The expenditure minimization problem (EMP) can be stated as follows,

minxp x

subject to : u(x) u.

So, in this problem the consumer chooses the cheapest consumption bundle provided that he

obtains a level of utility at least u. We denote the solution to the EMP by h(p; u). This is an L 1

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vector and is called the Hicksian (or compensated) demand functions. The minimized expenditure

is denoted by e(p; u) = p h(p; u) and is called the expenditure function.

Proposition (comparison between the UMP and the EMP):

i) If x is optimal in the UMP when wealth is w > 0, then x is optimal in the EMP when the

required utility level is u(x). Moreover, the minimized expenditure level in this EMP is exactly w.

ii) If h is optimal in the EMP when utility is u > u(0), then h is optimal in the UMP when

the wealth is p h. Moreover, the maximized utility level in this UMP is exactly u.

Proof: Straightforward.

What is implied by the above proposition is the following relationship between indirect utility

and expenditure functions,

e(p; v(p; w)) = w and v(p; e(p; u)) = u:

x1

x2

w/p1

w/p2

u1

h = x

w/p1

u2

Figure 19

price of good 1

h, x

p1

x(p,w)

h(p,u1)

p1

h(p,u2)

Figure 20

Figures 19 & 20 depict the relationship between the UMP and the EMP graphically. Fix (p; w)

and solve both problems. The solutions coincide (h = x). If p1 decreases the bundle that solves

the EMP must lie on the same indierence curve. The bundle that solves the UMP in general

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lies on a higher indierence curve. The Marshallian (x(p; w)) and Hicksian (h(p; u)) demands

intersect at the point where v(p; w) = u. That is, at the point where the maximum utility that

the consumer can attain given the prices and income is equal to the least utility that the consumer

must attain when he minimizes his expenditure. At this point the same consumption bundle solvesboth problems. At any other point the two solutions diverge (as gure 20 clearly illustrates). Now

if u is not xed but it varies with (p; w) such that v(p; w) = u (see gure 20) then we obtain the

following duality result between the Marshallian and the Hicksian demand functions,

x(p; w) = h(p; v(p; w)):

Similarly, if w is not xed but it varies with (p; w) such that e(p; u) = w, then we obtain the

second duality between the Marshallian and the Hicksian demand functions,

h(p; u) = x(p; e(p; u)).

Example

Let u = xa1x1a2 : The Lagrange function is,

L = p1x1 +p2x2 + (u xa1x1a2 ):

Since the constraint set is strictly convex the rst order conditions are also sucient and the

solution will be in the interior.

@L

@x1= 0 ) p1 = axa11 x1a2 ;

@L

@x2= 0 ) p2 = (1 a)xa1xa2 ;

@L

@= 0 ) u = xa1x1a2 :

Solving the above system of equations we obtain the Hicksian demand functions,

h1 = u ap2(1 a)p1(1a)

and h2 = u (1 a)p1ap2 a

: (1)

The expenditure function is,

e(p; u) = p1h1 +p2h2 =

aa(1 a)a1pa1p1a2 u: (2)Checking for the duality

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Recall that the Marshallian demand functions are,

x1 =aw

p1and

(1 a)wp2

. (3)

Therefore, the Cobb-Douglas indirect utility function is,

v(p; w) =

aw

p1

a(1 a)wp2

1a: (4)

Now it must be the case (verify it) that if we plug (4) into (1) (in place of u) we obtain (3).

And if we plug (2) in (3) (in place of w) we obtain (1).

Also, if we set (2) equal to w and solve with respect u we will obtain the indirect utility function.

Conversely, if we set (4) equal to u and solve with respect w we will obtain the expenditure function.

Proposition (properties of the expenditure function):

i) Homogeneous of degree 1 in p.

ii) Strictly increasing in u and non-decreasing in p` for all `.

iii) Concave in p.

iv) Continuous in p and w.

Proof:

i) We need to show that e(p; u) = e(p; u). Let h be the bundle that solves the EMP at

(p; u). Now multiply all prices by . First note that the constraint set fx 2 X : u(x) ug is notaected. The objective function p x gets only re-scaled and hence whatever was optimal underp x must be optimal under (p) x. Thus,

e(p; u) = (p) h = e(p; u):

ii) Proof straightforward.

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iii) By the denition of a concave function, we have to show the following. Fix p and p0.

e(p+(1 )p0; w) e(p; w) + (1 )e(p0; w), for any 2 [0; 1] :

Let

p00 = p+(1 )p0,and assume that h00 is optimal at p00. That is,

e(p00; w) = p00 h00 = p h00 + (1 )p0 h00

e(p; u) + (1 ) e(p0; u):

The last inequality follows from the fact that h00 is not the expenditure minimizing bundle at

p and p00

:

iv) It follows from Berges maximum theorem. The idea is similar to the continuity of the

indirect utility function.

Proposition (properties of the expenditure function):

i) Homogeneous of degree 1 in p.

ii) Strictly increasing in u and non-decreasing in p` for all `.

iii) Concave in p.

iv) Continuous in p and w.

Proof:

i) We need to show that e(p; u) = e(p; u). Let h be the bundle that solves the EMP at

(p; u). Now multiply all prices by . First note that the constraint set fx 2 X : u(x) ug is not

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aected. The objective function p x gets only re-scaled and hence whatever was optimal underp x must be optimal under (p) x. Thus,

e(p; u) = (p)

h = e(p; u):

ii) Proof straightforward.

iii) By the denition of a concave function, we have to show the following. Fix p and p0.

e(p+(1 )p0; u) e(p; u) + (1 )e(p0; u), for any 2 [0; 1] :

Let

p00 = p+(1 )p0,

and assume that h00 is optimal at p00. That is,

e(p00; u) = p00 h00 = p h00 + (1 )p0 h00

e(p; u) + (1 ) e(p0; u):

The last inequality follows from the fact that h00 is not the expenditure minimizing bundle at

p and p0:

iv) It follows from Berges maximum theorem. The idea is similar to the continuity of the

indirect utility function.

Proposition (properties of the Hicksian demand functions): For all p 0, h(p; u) hasthe following properties:

i) Homogeneous of degree zero in p,

h(p; u) = h(p; u), for all > 0.

ii) No excess utility: For all h 2 h(p; u), u(h) = u.

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iii) Convexity/Uniqueness: If % are convex, then h(p; u) is a convex-set (i.e., the solution

might be a correspondence, but for any xed set of parameters the set of solutions is a convex set).

In other words, the correspondence is convex-valued. If % are strictly convex, then h(p; u) is a

singleton (the solution is a function, not a correspondence).

Proof: We will prove only iii). The rst two parts are easy.

First assume that preferences are convex. We want to show that h(p; u) is convex-valued. Fix

(p; u). We need to show that for any two h0 and h00 that belong in h(p; u),

h000 = h0 + (1 )h00 2 h(p; u), for all 2 [0; 1] :

Since h0 and h00 2 h(p; u), it must be that

p h0 = p h00 = w;

and u(h0) u, u(h00) u. That is, both bundles satisfy the constraint and attain the same levelof minimum expenditure. Since preferences are convex, or u is quasi-concave, it must be (by the

denition of a quasi-concave function) that, u(h000) min fu(h0); u(h00)g u. Moreover,

p h000 = p h0 + (1 )p h00 = w + (1 )w = w:

Therefore, h000 2 h(p; u):

Now assume that preferences are strictly convex. We have to show that h(p; u) consists of only

one element. Suppose by way of contradiction that this is not the case and that there are two

bundles that solve the EMP problem, i.e., h0 2 h(p; u) and h00 2 h(p; u). It must be that (as weargued above)

p h0 = p h00 = w;and u(h0) u, u(h00) u. Since preferences are strictly convex,

u(h000) > minfu(h0); u(h00)g u;

where h000 = h0 + (1 )h00 2 h(p; u), with 2 (0; 1). Now note that,

p h000 = p h0 + (1 )p h00 = w + (1 )w = w:

Since u(h000) > u and p h000 = w and by the continuity of u we can nd a bundle z in theneighborhood ofh000 such that u(z) u and p z < w. This contradicts the initial assumptionthat both h0 and h00 are optimal:

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Proposition (compensated law of demand): For all p0, p00 the following holds,

p00 p0 h(p00; u) h(p0; u) 0. (*)

Proof: From the denition of the expenditure function we have,

p00 h(p00; u) p00 h(p0; u);

and

p0 h(p00; u) p0 h(p0; u):

By subtracting the above two inequalities we obtain (*).

For example, when L = 2 (*) can be written as follows,

0B@ (p001 p01) , (p002 p02)| {z }

(12 vector)

1CA

0BBB@ [h1(p

00; u) h1(p0; u)][h2(p00; u) h2(p0; u)]| {z }

(21 vector)

1CCCA =

p001 p01

h1(p

00; u) h1(p0; u)

+p002 p02

h2(p

00; u) h2(p0; u) 0:

What this proposition says is that the Hicksian demand functions are downward sloping in each

goods own price. If, for instance, only one price changes then the above implies that,

p00` p0`

h`(p

00; u) h`(p0; u) 0:

This is not necessarily true for the Marshallian demand functions, e.g. when a good is Gien

then the Marshallian demand is upward sloping. We will introduce a modied law of demand for

Marshallian demand functions later.

1.3.1 Relationships between demand indirect utility and expenditure functions

Proposition (relationship between Hicksian and expenditure function): For all p and

u the Hicksian demand h(p; u) is the derivative vector of the expenditure function with respect to

prices, i.e.,

h(p; u) = rpe(p; u):

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Proof: Suppose p changes. Using the chain rule the change in expenditure can be written as,

rpe(p; u) = rp [p h(p; u)] = h(p; u) + [p Dph(p; u)]T : (*)

Recall that the rst order condition for an interior solution to the EMP is,

p = ru(h(p; u)): (**)

By substituting (*) into (**) we obtain,

rpe(p; u) = h(p; u) + [ru(h(p; u)) Dph(p; u)]T : (***)

Since the constraint u(h(p; u)) = u holds for all prices in the EMP, we must have ru(h(p; u)) Dph(p; u) = 0: Given this (***) becomes,

rpe(p; u) = h(p; u).

Proposition (more properties of the Hicksian demand functions): Suppose that %

are strictly convex and locally non-satiated and h(; u) is continuously dierentiable at (p; u) anddenote its derivative L

L matrix by Dph(p; u). Then,

i) Dph(p; u) = D2e(p; u).

ii) Dph(p; u) is a negative semi-denite matrix.

iii) Dph(p; u) is a symmetric matrix. The symmetry implies that @h`=@pk = @hk=@p` for any

k; `.

iv) Dph(p; u) p = 0:

Proof:

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i) From the previous proposition we know that,

rpe(p; u) = h(p; u):

If we dierentiate it again with respect to p we get the result.

ii) + iii) We proved in a previous proposition that e(p; u) is a concave function and there-

fore it must have a negative semi-denite and symmetric Hessian matrix, D2e(p; u). but from i)

Dph(p; u) = D2e(p; u) and the result follows.

iv) In a previous proposition, we showed that h(p; u) is homogeneous of degree zero in p, i.e.,

h(p; u) h(p; u) = 0, for all > 0.

By dierentiating the above with respect to and setting = 1 we obtain, Dph(p; u) p = 0.

For example, when L = 2 the above becomes,

@h1@p1

p1 +@h1@p2

p2 = 0;

@h2@p1

p1 +@h2@p2

p2 = 0:

Denition (substitutes and complements): Two goods ` and k are net substitutes at (p; u)

if @h`=@pk 0 and net complements if @h`=@pk 0. If, on the other hand, @x`=@pk 0, then wesay that good ` is a gross substitute for good k, and if @xk=@p` 0, then we say that good k is agross substitute for good `. Similarly, if@x`=@pk 0, then we say that good ` is a gross complementfor good k, and if @xk=@p` 0, then we say that good k is a gross complement for good `.

Remark 1: While the Dph(p; u) matrix is symmetric and therefore @h`=@pk = @hk=@p` for

any k; `, the Dpx(p; u) matrix (as it will become clear later) is not symmetric. Hence, it may very

well be the case that @x`=@pk

0 while @xk=@p`

0.

Remark 2: When there are only two goods the above imply that they must be net substitutes.

With more than two goods the above implies that every good must have at least one net substitute

(WHY?).

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1.3.2 Income and substitution eects

When the price of a good declines there are at least two conceptually separate reasons why we expect

some change in the quantity demanded. First that good becomes relatively cheaper compared toother goods. We would expect the consumer to substitute the relatively cheaper good for the now

relatively more expensive ones. This is the substitution eect. At the same time, however, when

the price of a good declines, the consumers real income (purchasing power) increases which allows

him to change his purchases of all goods in an optimal way. This is the income eect. Intuition

tells us that we can decompose the total eect of a price change into these separate conceptual

categories. We will follow the method proposed by Hicks (1939). It starts with the observation

that the consumer achieves some level of utility at the original prices. The substitution eect is the

(hypothetical) change in the consumption that would occur if prices were to change to their new

levels but the maximum utility were kept the same as before the price change. This can eectively

happen if the consumer gets compensated for a price increase or gets de-compensated for a price

decrease. The income eect is the residual eect. This decomposition is captured by the Slutsky

equation.

Proposition (Slutsky equation): For all (p; u) and u = v (p; u) we have,

@h` (p; u)

@pk=

@x` (p; u)

@pk+

@x` (p; u)

@wxk (p; u) , for all ` and k.

or equivalently in matrix notation,

Dph (p; u) = Dpx (p; u) + Dwx (p; u) x (p; u)T :

Proof: Fix (p; w) and let u = v(p; w): We know that w = e(p;u). From the fact that h(p; u) =

x(p; e(p; u)) we have that

h`(p; u) = x`(p; e(p; u)).

Dierentiate the above expression with respect to pk.,

@h`(p; u)

@pk=

@x`(p; e(p; u))

@pk+

@x`(p; e(p; u))

@w

@e(p; u)

@pk:

Using the fact that h` = @e=@p`, the above becomes,

@h`(p; u)

@pk=

@x`(p; e(p; u))

@pk+

@x`(p; e(p; u))

@whk(p; u):

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Finally, and since w = e(p;u) and hk(p; u) = x(p; e(p; u)) = x(p; w) we have,

@h`(p; u)

@pk=

@x`(p; w)

@pk+

@x`(p; w)

@wxk(p; w):

If we re-arrange the terms of the Slutsky equation as follows,

@x`(p; w)

@pk| {z }Total eect

=@h`(p; u)

@pk| {z }Substitution eect

@x`(p; w)@w

xk(p; w)| {z }Income eect

;

we can obtain the total eect as the sum of the substitution and the income eect.

Consider the own price eect,

@x`(p; w)

@p`| {z }Total eect

=@h`(p; u)

@p`| {z }Substitution eect

@x`(p; w)@w

x`(p; w)| {z }Income eect

:

We have already proved (see the compensated law of demand) that the substitution eect is

negative, i.e., as p` % h` & and vice vera. We can deduce from the above the following relationships,

If good ` is normal (i.e., @x`=@w 0), then good ` must be an ordinary good (i.e., @x`=@p`

0). This the law of demand which says that if a good is normal, then the demand curve isdownward sloping. The converse is not necessarily true. That is, if a good is ordinary it is

not necessarily normal, it may be inferior.

If good ` is Gien (i.e., @x`=@p` 0), then it must be inferior. The reverse is not necessarilytrue. An inferior good might not be Gien.

The Slutsky substitution matrix can be written as,

S(p; w) =0BB@

@h1(p;u)

@p1 @h1(p;u)

@pL.... . .

...@hL(p;u)

@p1 @hL(p;u)@pL

1CCA =

=

0BB@

@x1(p;w)@p1

+@x1(p;w)

@w x1(p; w) @x1(p;w)@pL +@x1(p;w)

@w xL(p; w)...

. . ....

@xL(p;w)@p1

+@xL(p;w)

@w x1(p; w) @xL(p;w)@pL +@xL(p;w)

@w xL(p; w)

1CCA

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Because S(p; w) = Dph(p; u) (with u = v(p; w)) then from Proposition (more properties

of the Hicksian demand functions) above it follows that the substitution matrix must be: i)

negative semi-denite, ii) symmetric and iii) satisfy S(p; w) p = 0:

Proposition (Roys identity): Suppose that the indirect utility v is dierentiable at (p; w) 0. Then,

x (p; w) = 1rwv (p; w)rpv (p; w) :

That is, for every ` = 1;:::;L

x` (p; w) = @v (p; w) =@p`@v (p; w) =@w

:

Proof: Fix u = v (p; w). Since v(p; e(p; u)) = u, when we dierentiate it with respect to p andset p = p we obtain,

rpv (p; e(p; u)) + @v (p; e(p; u))@w

rpe(p; u) = 0:

Since rpe(p; u) = h(p; u) we have,

rpv (p; e(p; u)) + @v (p; e(p; u))@w

h(p; u) = 0:

Since w = e(p; u) and h(p; u) = x(p; w) we can obtain,

rpv (p; w) + @v (p; w)@w

x(p; w) = 0: (*)

Rearranging (*) we obtain the desired result.

1.3.3 A comprehensive example

Consider the constant elasticity of substitution (CES) utility function: u = (xr1 + xr2)1=r, where

r 2 (1; 1]. When r = 1 the indierence curves become linear (i.e., perfect substitutes). When

r ! 0 then the utility function represents the same preferences as a Cobb-Douglas (u = x1x2)utility function. Finally, when r ! 1 the indierence curves become right angles (i..e, Leontiefpreferences, or perfect complements).

1. Solve the UMP.

The Lagrange function is,

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L = (xr1 + xr2)1=r + (w p1x1 p2x2) :

First observe that the utility function is quasi-concave (strictly for r < 1). To see this set

(xr1 + xr2)

1=r = c and solve for x2 to obtain an indierence curve, i.e.,

x2 = (cr xr1)1=r > 0:

Dierentiate x2 with respect to x1 twice to obtain,

d2x2dx21

= (1 r)h

(cr xr1)(12r)

r x2(r1)1 + (c

r xr1)(1r)r xr21

i 0:

Therefore, preferences are convex and the rst order necessary Kuhn-Tucker conditions are also

sucient. @L

@x1= 0 ) 1

r(xr1 + x

r2)

(1=r)1 rxr11 p1 = 0; (1)

@L

@x2= 0 ) 1

r(xr1 + x

r2)

(1=r)1 rxr12 p2 = 0: (2)

(1)/(2) yields,

x1 = x2

p1p2

1=(r1): (3)

Plug (3) into the budget constraint and solve for x2. After some algebra we obtain,

x2 =wp

1=(r1)2

pr=(r1)1 +p

r=(r1)2

: (4)

Now plug (4) back into (3) to obtain,

x1 =wp

1=(r1)1

pr=(r1)1 +p

r=(r1)2

: (5)

(4) and (5) are the Marshallian demand functions.

To simplify the notation let k = r=(r 1): Then, k 1 = 1=(r 1): The Marshallian functionscan be rewritten as

x1 =wpk11

pk1 +pk2

and x2 =wpk12

pk1 +pk2

: (6)

Note that,

limr!1

x1 = limr!1

x2 =w

p1 +p2;

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which is the solution to the UMP with a Leontief utility function (u = min fx1; x2g).

If we now plug (6) back into the utility function we obtain the indirect utility function,

v(p; w) = w hpk1 +pk2i1=k : (7)First, note that (7) is % in w: Also, it is & in p1 and p2, i.e.,

@v

@p1= w

pk1 +p

k2

(1+k)=kp(k1)1 < 0 and

@v

@p2= w

pk1 +p

k2

(1+k)=kp(k1)2 < 0:

Moreover, it is homogeneous of degree zero in (p; w), i.e.,

v(p; w) = wh

(p1)k + (p2)

ki1=k

= wh

kpk1 +p

k2

i1=k=

= (w)k1=k hpk1 +pk2i1=k = 1v(p; w) = v(p; w):Finally, we also know that v(p; w) is quasi-convex in income and prices but it is tedious to verify

it.

2. Solve the EMP.

The Lagrange function is,

L = p1x1 +p2x2 + u (xr1 + xr2)1=r :The F.O.C. are,

@L

@x1= 0 ) p1 1

r(xr1 + x

r2)(1=r)1 rxr11 = 0; (8)

@L

@x2= 0 ) p2 1

r(xr1 + x

r2)(1=r)1 rxr12 = 0: (9)

(8)/(9) yields,

x1 = x2

p1p2

1=(r1): (10)

Plug (10) into the utility constraint u = (xr1 + xr2)1=r and solve for x2. After some algebra we

obtain,

h2 = upk1 +p

k2

(1k)k

pk12 : (11)

Now plug (11) back into (10) to obtain,

h1 = upk1 +p

k2

(1k)k

pk11 : (12)

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(11) and (12) are the Hicksian demand functions. If we plug (11) and (12) back into

p1x1 +p2x2 we obtain (after some simplications) the expenditure function,

e(p; u) = upk1 +pk21=k

: (13)

The expenditure function is clearly increasing in u and increasing in prices, i.e.,

@e

@p1= u

pk1 +p

k2

(k1)=kp(k1)1 > 0 and

@e

@p2= u

pk1 +p

k2

(k1)=kp(k1)2 > 0:

It is also homogeneous of degree 1 in prices, i.e.,

e(p; u) = u

(p1)k + (p2)

k1=k

= u

k1=k

pk1 +pk2

1=k= u

pk1 +p

k2

1=k= e(p; u):

The expenditure function is concave in prices, but it is tedious to verify it with this example.

Numerical example:

The graph below plots x1 and h1 against p1 when w = 10; u = 3; p2 = 5 and r = :5: Notice

that the two demand functions intersect at p1 = 10: At this point v(p; w) = u. The Marshallian

demand function is above the Hicksian for p1 < 10:

Figure 21: Hicksian and Marshallian demand functions

3. Verify all the dualities.

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a. Duality between indirect utility and expenditure functions.

e(p; v(p; w)) = w: (14)

andv(p; e(p; u)) = u: (15)

These conditions imply that for a xed vector p, e(p;) and v(p;) are inverses to one another.Lets see this in our example. From (13) we have

upk1 +p

k2

1=k= w =) u = w

hpk1 +p

k2

i1=k;

same as (7) the indirect utility function. Similarly, we can obtain the expenditure function from

the indirect utility function. From (7) we have,

w hpk1 +pk2i1=k = u =) w = upk1 +pk21=k ;same as (13), the expenditure function.

b. Duality between Marshallian and Hicksian demand functions.

h(p; u) = x(p; e(p; u));

and

x(p; w) = h(p; v(p; w)):

The Marshallian demand functions are,

x1 =wpk11

pk1 +pk2

and x2 =wpk12

pk1 +pk2

;

and the Hicksian are,

h1 = upk1 +p

k2

(1k)k

pk11 and h2 = upk1 +p

k2

(1k)k

pk12 :

Plug the indirect utility function v(p; w) = w

pk1 +pk2

1=kinto the Hicksian demand func-

tions in place of u. This yields (we only look at good 1),

h1 = wh

pk1 +pk2

i1=k pk1 +p

k2

(1k)k

pk11 =wpk11

pk1 +pk2

= x1:Now plug the expenditure function e(p; u) = u

pk1 +p

k2

1=kinto the Marshallian demand func-

tions in place of w. This yields (again we only look at good 1)

x1 =upk1 +p

k2

1=kpk11

pk1 +pk2

= upk1 +p

k2

(1k)k

pk11 = h1:

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c. h(p; u) = re(p; u).

With two goods the above duality can be written as,

h1h2

= @e@p1

@e@p2

!:

@e

@p1=

@

upk1 +p

k2

1=k@p1

= upk1 +p

k2

(1k)k

pk11 = h1;

and

@e

@p2=

@

upk1 +p

k2

1=k@p2

= upk1 +p

k2

(1k)k

pk12 = h2:

4. Slutsky substitution matrix

The substitution matrix is,

S(p1; p2; w) =

@h1@p1

@h1@p2

@h2@p1

@h2@p2

!=

@x1@p1

+ @x1@w x1,@x1@p2

+ @x1@w x2@x2@p1

+ @x2@w x1,@x2@p2

+ @x2(p;w)@w x2

!=

= 0B@wpk21 p

k

2(k1)

(pk1

+pk

2)2 ,

wpk11 pk12 (k1)

(pk1

+pk

2)2

wpk11 pk12 (k1)(pk1+pk2)

2 ,wpk1p

k22 (k1)

(pk1+pk2)2

1CAFirst note that the diagonal elements are negative, verifying the compensated law of demand.

Moreover S(p1; p2; w) is symmetric, i.e., @h1=@p2 = @h2=@p1. Also S(p1; p2; w) is negative semi-

denite, i.e., the diagonal elements are negative and the determinant is zero.

We also know that S(p; w) p = 0; i.e.,

0B@wpk21 p

k

2(k1)

(pk

1

+pk

2)

2 ,

wpk11 pk12 (k1)

(pk

1

+pk

2)

2

wpk11 pk12 (k1)(pk1+pk2)

2 ,wpk1p

k22 (k1)

(pk1+pk2)2

1CA p1p2 = 00 :

5. Substitutes and complements

Net. Using the Hicksian demand functions we can see that the two goods are net substitutes,

i.e., @h1=@p2 = @h2=@p1 0. This holds regardless of the value of k.

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Gross. By taking the cross partial derivatives of the Marshallian demand function we obtain,

@x1@p2

=@x2@p1

wp(k1)2 p

(k1)2 k

pk1 +p

k2

2 .

Note that it is not necessarily true that @x1@p2 =@x2@p1

, although in this example it happens to be the

case. Recall that k = r=(r 1) and r 2 (1; 1]: Hence, ifr 0, k 0 and the two goods are grosssubstitutes. If, on the other hand, r 0, then k 0 and the two goods are gross complements.

This conrms what we would expect to obtain, since when r ! 1 the two goods are perfect(gross) complements (u = min fx1; x2g), while at r = 1 they are perfect (gross) substitutes (u =x1 + x2).

6. Roys identity

x1 = @v=@p1@v=@w

= wpk1 +p

k2

(1+k)=kp(k1)1

pk1 +pk2

1=k = wpk11pk1 +pk2 :x2 = @v=@p2

@v=@w= w

pk1 +p

k2

(1+k)=kp(k1)2

pk1 +pk2

1=k = wpk12pk1 +pk2 :END OF EXAMPLE.

Dualities The gure below summarizes all the dualities.

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UMP EMP

h(p,u)Slutsky equation

v(p,w)

Roysidentity

e(p,u)

h=

e

e(p,v(p,w)) = w

v(p,e(p,u)) = u

h(p,u)=x(p,e(p,u)) x(p,w

)=h(p,v

(p,w))

DUAL PROBLEMS

x(p,w)

Figure 22: Dualities

If we start from the UMP, we can solve it to derive the Marshallian demand functions. Then

we can derive the indirect utility function. If we invert it we obtain the expenditure function. By

dierentiating the expenditure function, we obtain the Hicksian demand functions. Conversely, if

we start from the EMP, we can solve it to derive the Hicksian demand functions. Using them we

obtain the expenditure function, whose inverse is the indirect utility function. Using Roys identity

we derive the Marshallian demand functions.

1.3.4 An important conclusion

Based on our results so far we can say that: If the Marshallian demand functions x(p; w) are

generated by preferences which satisfy our AXIOMS 1-5 and are continuously dierentiable then

they must satisfy the following,

1. Homogeneous of degree zero.

2. Satisfy Walras Law (and as a consequence Cournot and Engel aggregation).

3. Substitution matrix S(p; w) which is: i) negative semi-denite, ii) symmetric and iii) satisfy

S(p; w) p = 0:

When researchers try to estimate a system of demand equations they want to know whether it

is: i) consistent with rational behavior and ii) an outcome of a well-dened utility maximization

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problem. One way is to start from a well-dened utility function derive the Marshallian demands

and then estimate them. This however is restrictive and one may want to begin from a system of

demand functions which t the problem under investigation better, rather than to begin from a

utility function. Is it true that this system is compatible with utility maximization? The answeris yes, as long as the system satises 1., 2. and 3. above. All one has to do is to check for these

three properties. In other words, 1.,2., and 3. are not only necessary but are also sucient for the

existence of rational generating preferences. This is the converse to the conclusion above and is

called the integrability problem, which we will study next.

1.4 Integrability

Theorem (integrability theorem): A continuously dierentiable system x(p; w) is the sys-

tem of demand functions generated by some strictly increasing, strictly quasi-concave and continu-

ous utility function, if and only if the system satises: Walras Law and the substitution matrix is

symmetric and negative semi-denite.

In other words, Walras Law, symmetry and negative semi-deniteness are not only the con-

sequences of preference-based demand theory, but also are all of its consequences. So, if we start

with a system of equations, we will be certain that it is utility generated if it satises Walras Law,

symmetry and negative semi-deniteness.

Next, I demonstrate, with the aid of an example, how we can go from the demand functions

back to utility.

Consider the following system,

x1 =w

p1and x2 =

(1 )wp2

: (1)

Is it compatible with utility maximization? If so, what is the utility function?

The answer to the rst question is armative, by invoking the integrability theorem. Verify

that (1) satises Walras Law, and the Slutsky substitution matrix is symmetric and negative

semi-denite.

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Lets now nd the utility which generated this demand system.

N Recovering the expenditure function from the demand system

Fix w and let u = v(p1; p2; w): Given this we know that x(p; w) = h(p; u): Using h(p; u) =

rpe(p; u) we obtain,

@e(p1; p2; u)

@p1= h1(p1; p2; u) = x1(p1; p2; w) =

w

p1=

e(p1; p2; u)

p1; (2)

and

@e(p1; p2; u)

@p2= h

2(p

1; p

2; u) = x

2(p

1; p

2; w) =

(1 ) wp2

=(1 ) e(p1; p2; u)

p2: (3)

We should nd e(p1; p2; u) that solves (2) and (3). We are looking for a solution to the system

of partial dierential equations. We can re-write it as,

@ln [e(p1; p2; u)]

@p1=

p1and

@ln [e(p1; p2; u)]

@p2=

(1 )p2

:

The above can be written as,

ln [e(p1; p2; u)] = lnp1 + c(p2; u); (4)

and

ln [e(p1; p2; u)] = (1 ) lnp2 + c(p1; u): (5)

We must choose c(p1; u) and c(p2; u) such that (4) and (5) hold simultaneously. This yields,

ln [e(p1; p2; u)] = lnp1 + (1 ) lnp2 + c(u);

or,

e(p1; p2; u) = p1p

(1)2 c(u): (6)

We must ensure that e is increasing in u so we can choose c(u) to be any strictly increasing

function in u.

This is the expenditure function when u = x1 x12 (Cobb-Douglas). Recall that with this

utility,

e(p1; p2; u) =h

(1 )(1)i

p1p(1)2 u: (7)

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Now if we choose c(u) =h

(1 )(1)i

u then (6) is equivalent to (7). Note that it does not

matter what c(u) we end up choosing since we know that implied demand behavior is independent

of strictly increasing transformations.

N Recovering the indirect utility function from the expenditure function

From (7) we have,h (1 )(1)

ip1p

(1)2 u = w =) v = wp1 p(1)2 C;

where C is the constant.

N Recovering the utility function from the indirect utility function

Idea: Fix x0, and let u(x0) denote the utility the consumer attains, if he consumes x0. The

following inequality must hold for any p,

v(p;p x0) u(x0):

The above inequality says that when the income is p x0 = w0, then the highest utility that theconsumer can attain is at least u(x0), since x0 does not have to be the best bundle at these prices

and income. Moreover, there will be one price vector p0 such that,

v(p0;p0 x0) = u(x0):

In other words when the v(p;p x0) is minimized it is equal to the utility at x0. This can bedone for any x and therefore we can recover the utility if we solve,

u(x) = minp

v(p;p x).

Since v is homogeneous of degree zero in p and w we have: v(p;p x) = v(p; (p x)). Now

if you take = 1= (p x) we have v(p;p x) = v(p= (p x) ; 1). The above minimization problemcan be equivalently stated as,

u(x) = minp

v(p; 1)

subject to: p x = 1:

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In our example,

min1p1 p(1)2

s.t. : p1x1 +p2x2 = 1:

Recall that v is quasi-convex in prices and income. Hence, this minimization problem can be

solved by solving the rst order necessary and sucient Kuhn-Tucker conditions. The Lagrange

function is,

L = p1 p(1)2 + (p1x1 +p2x2 1) :

The F.O.C.s are,@L

@p1= p11 p(1)2 + x1 = 0 (1)

@L

@p2

=

(1

)p1 p

(1)12 + x2 = 0: (2)

(1)/(2) yields,p2

(1 )p1 =x1x2

=) p2 = (1 )p1x1x2

: (3)

Plug (3) into p1x1 +p2x2 = 1 to obtain,

p1x1 +(1 )p1x1

x2x2 = 1: (4)

Solving (4) with respect to p1 yields,

p1 =

x1: (5)

Plug (5) back into (3) to obtain,

p2 =1

x2: (6)

Finally, plug (5) and (6) into the objective function p1 p(1)2 . This yields,

x1

1

x2 (1)

=x1 x

(1)2

(1 )(1)

= Ax1 x(1)2 ;

where A = (1 )(1). This utility function represents Cobb-Douglas preferences.

So, we started from a system of demand functions and we recovered the utility that generated

this demand system.

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1.5 Welfare analysis

Consider a consumer with preferences % which satisfy axioms 1-5. Suppose that prices are

initiall