Microeconomics 1 Lecture notes - uniroma1.it · D. Tosato – Appunti di Microeconomia – Lecture Notes of Microeconomics - a.y. 2014-2015 Pagina 2 4.1. The utility maximization

D. Tosato – Appunti di Microeconomia – Lecture Notes of Microeconomics - a.y. 2014-2015 Pagina 1

Sapienza University of Rome. Ph.D. Program in Economics a.y. 2014-2015

Microeconomics 1 – Lecture notes

4. The utility maximization problem

4.1 Utility maximization problem (UMP)

4.2 Properties of the Walrasian demand functions/correspondences

4.3 Examples of demand functions

4.4 The indirect utility function and its properties

4.5 Examples of indirect utility functions

Appendix 4.A.1 The Constant Elasticity of substitution (CES) utility function

Appendix 4.A.2 Discrete choice analysis

After the study in Lecture Note 1 of consumer preferences and their representation by a

numerical function, the utility function, after the definition in Lecture Note 2 of the analytical

notions of concavity and quasi concavity, and after the presentation in Lecture Note 3 of the

analytical techniques for the solution of optimization problems we now turn to the study of

consumer behavior as expressed by the Walrasian, or Marshallian, demand functions. We

assume that the consumer behaves rationally, in the sense that he chooses a commodity

bundle which is optimal according to his preferences and subject to his budget constraint. The

possibility of representing continuous preferences by means of a utility function, defined up to

a positive monotonic transformation, makes it possible to formulate the consumer problem in

the analytical terms of the maximization of his utility function subject to constraints. While

the analytical formulation of the maximization problem refers to the general case of 𝐿

commodities, graphical representations are, as usually, confined to the manageable two-

commodity case.

The utility maximization problem, which has already been formulated in Lecture Note 3,

Section 3.4, is represented here in section 4.1, with special attention to the graphical

illustration of the internal and the boundary solutions. The properties of the Walrasian

demand functions, which represent the solution of the set of equations which define the

critical points of the Lagrangean, are analyzed in section 4.2. Several examples of derivation

of demand functions from commonly used utility functions, together with their properties, are

presented as solved exercises in section 4.3. The final sections 4.4 and 4.5 are respectively

dedicated to the definition of the indirect utility function and to its properties, as well as to the

study of the indirect utility functions associated with the examples considered in the previous

section 4.3. Appendix 4.A dedicates special attention to the CES utility function.


4.1. The utility maximization problem (UMP)

Assume that 𝑢(𝑥) is a continuous, twice differentiable utility function representing monotone,

convex preferences defined on the nonnegative orthant of the commodity space. Let 0p

be the vector of prices of the 𝐿 commodities and 𝑤 > 0 the wealth of the consumer. The

consumer budget set is then

(4.1) 𝐵(𝑝,𝑤) = {𝑥 ∈ ℝ+𝐿 |𝑝 ⋅ 𝑥 ≤ 𝑤}

The budget set is, therefore, a non empty, convex and compact subset of the non negative

orthant of the commodity space ℝ+𝐿

: the shaded area in Fig. 4.1. The north-east boundary AB

of the budget set is the budget line, representing the subset of commodity bundles which, at

given prices, exhaust the consumer wealth. The linearity of the budget line reflects the price

taking assumption: consumers operate in perfectly competitive markets. The normal to the

budget line is the price vector 𝑝 = (𝑝1, 𝑝2) in the two-commodity example of Fig. 4.1.

As indicated in Lecture Note 3, Section 3.4, the utility maximization problem can be

formulated in the following analytical terms

(4.2) max 𝑢(𝑥)

𝑥

𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝑝 ⋅ 𝑥 − 𝑤 ≤ 0 and 𝑥 ≥ 0

Given the assumptions made on the utility function and the constrain set, Weierstrass theorem

establishes the existence of a solution to the UMP (see Lecture Note 3, Section 3.1).

As explained in Section 3.4 of Lecture Note 3, in the statement of the Lagrangean function we

can disregard an explicit reference to the nonnegativity constraint on the variables and to the

associated multipliers. We have accordingly

(4.3) 𝐿(𝑥, 𝜆) = 𝑢(𝑥) − 𝜆(𝑝 ⋅ 𝑥 − 𝑤)

Using the Kuhn-Tucker conditions, the critical values of the utility function 𝑢(𝑥) must satisfy

the following conditions involving the first derivatives of the Lagrangean with respect to the

vector of the variables 𝑥 and to the Lagrangean multiplier 𝜆

(4.4)

∂𝐿

∂𝑥= ∇𝑥𝐿 = ∇𝑢(𝑥∗) − 𝜆∗𝑝 ≤ 0

𝑥∗ ⋅ [∇𝑢(𝑥∗) − 𝜆∗𝑝] = 0𝑝 ⋅ 𝑥∗ − 𝑤 = 0

The second order necessary condition for a maximum is that the Hessian matrix of the second

order partial derivatives of the utility function be negative semidefinite in the subspace

defined by the budget set. The second order sufficient condition for a maximum is that the

Hessian of the utility function be negative definite in the subspace defined by the budget set.


This second order sufficient condition is satisfied if the utility function is strictly

quasiconcave.1

Panels (a) and (b) of Fig. 4.1 illustrate, with reference to the two-commodity case, an interior

solution and a boundary solution. In the interior solution (i.e. with 𝑥∗ ≫ 0) the vector of

partial derivatives of the utility function is proportional to the price vector; in the boundary

solution we have instead ∇𝑢(𝑥∗) ≤ 𝜆∗𝑝. This latter situation is depicted in Fig 4.1 Panel (b)

with 𝜆∗𝑝2 >∂𝑢(𝑥∗)

∂𝑥2. In the interior solution, the marginal rate of substitution between any two

commodities l and k 𝑀𝑅𝑆𝑙,𝑘(𝑥∗) =

𝑢𝑙(𝑥)

𝑢𝑘(𝑥) is equal to the price ratio

𝑝𝑙

𝑝𝑘; in the boundary solution

we have instead 𝑀𝑅𝑆𝑙,𝑘(𝑥∗) >

𝑝𝑙

𝑝𝑘.

2 The internal solution is thus characterized by the

tangency condition between the slope of the indifference curves and the slope of the budget

line. This tangency condition defines the wealth-consumption expansion line of the consumer,

i.e. the path of his optimal consumption choices at the various levels of wealth. Among these

possible optimal choices, the optimal solution of the UMP is therefore represented by that

particular point of the wealth-consumption expansion path that intersects the budget line and

thus satisfies the wealth constraint, as required by the final line of (4.4).

Fig. 4.1 Panel (a) – Interior solution Fig. 4.1 (Panel b) – Boundary solution of

the UMP the UMP

If the indifference curves are smooth, the solution of the system of relations (4.4) is unique

and can therefore be expressed as a function of the parameters (𝑝,𝑤)

(4.5) 𝑥∗ = 𝑥(𝑝,𝑤)

𝜆∗ = 𝜆(𝑝,𝑤)

1 See Lecture Note 2, Section 2.4 “The role of concavity and quasiconcavity in optimization problems”.

2 It is useful to remember that the marginal rate of substitution was defined in Lecture Note 1 as a positive quantity.

𝑥2 𝑥2

𝑥1 𝑥1

∇𝑢(𝑥∗) = 𝜆 𝑝

(𝑥∗) = x(p,w)

B(p,w)

p

B(p,w)

(𝑥∗) = x(p,w)

𝜆∗𝑝

∇𝑢(𝑥∗)


The relation 𝑥∗ = 𝑥(𝑝,𝑤) represents the vector of the Walrasian demand functions, the

consumer’s optimal choice, given his preferences and his budget constraint, in principle

observable in the market. Given wealth, the Walrasian demand functions express a relation

between commodity prices, the independent variables, and the quantities of the various

commodities, which are the dependent variables. The demand functions 𝑥∗ = 𝑥(𝑝,𝑤) thus

directly reflect the assumption that the consumer is a price taker in the market and optimally

adjusts his consumption choice to market prices. Not infrequently these demand function are

called Marshallian demands. Marshall’s thought experiment is, however, different from

Walras’, in the sense that the question that he posed is “at what prices would a consumer be

willing to buy a certain quantity of the different commodities, given his wealth”. In this type

of thought experiment the role of dependent and independent variable is reversed with respect

to Walras’ approach. The function defined is, actually, the inverse demand function 𝑝∗ =

𝑝(𝑥,𝑤).3 We stick here to the Walrasian approach and call the functions 𝑥∗ = 𝑥(𝑝,𝑤) the

Walrasian demands.

4.2 Properties of the Walrasian demand functions

If indifference curves are strictly convex, as in Fig. 4.1, Panels (a) and (b), the UMP has a

unique solution: the vector of Walrasian demand functions 𝑥∗ = 𝑥(𝑝,𝑤). If indifference

curves are, on the contrary, convex, but not strictly convex, they may contain linear segments

and the UMP may have solution on those segments. In this case, to the same set (𝑝,𝑤) there

corresponds the convex set of 𝑥∗ values in that linear segment. The solution is then a multi-

valued relation, namely the convex correspondence 𝜙(𝑝,𝑤). A specific solution must in this

instance be indicated as an element of this correspondence, 𝑥∗ ∈ 𝜙(𝑝,𝑤).4

We will now consider two sets of properties of demand functions: the first refers to properties

that are inherent in the derivation of these demands as the outcome of a maximization process

and do not depend, therefore, on prices and wealth; the second reflects comparative statics

properties arising from changes in ,p w . A classification of commodities follows from the

reaction of the demand functions of the various commodities to changes in ,p w .

Proposition 4.1 The Walrasian demand functions are

1/ Homogeneous of degree zero in prices and wealth: 𝑥(𝛼𝑝, 𝛼𝑤) = 𝑥(𝑝,𝑤) for 𝛼 > 0

3 The inverse demand function is the analytical tool commonly used to determine the profit maximizing solution in the

market regimes of monopoly and oligopoly. Note that moving from the direct to the inverse demand function requires

inverting the former. 4 Since the set of (𝑝, 𝑤) ∈ (ℝ++

𝐿 , 𝑅+) pairs that can give rise to a non unique solution is of measure zero, we will

henceforth disregard this possibility and confine to footnotes the generalizations needed in order to take account of it.

MWG’s approach is formulated in terms of demand correspondences.


Since a proportional change in all prices and wealth does not affect the budget set, the

consumer’s optimal choice is unchanged. In economic terms, this means that our assumptions

model the behavior of a consumer that is free of money illusion, that is that responds only to

real changes of the environment.

2/ Walras’ Law: Walrasian demand functions exhaust the available wealth: 𝑝 ⋅ 𝑥(𝑝, 𝑤) = 𝑤

Note that, because of the assumption of monotone preferences, in solving the UMP we have

imposed the condition that wealth be fully utilized. The optimal solution, as a tangency point

between the budget line and the highest attainable indifference curve, must then necessarily

be on the budget line. This property is also called, for obvious reasons, the adding-up restriction.

3/ The optimal solution is unique if, as assumed, preferences are strictly convex.5

4/ Walrasian demand functions are continuous.6

This property follows from Berge’s Theorem of the maximum (see MWG, p. 963, JR, p.505,

Varian, p.506).

5/ Walrasian demand functions are differentiable if the utility function represents smooth

preferences

As indicate in Lecture Note 1, Section 1.5, smooth preferences imply a technical

strengthening of the condition that 𝑢(𝑥) is strictly quasiconcave: the determinant of the

Bordered Hessian cannot be zero at any 𝑥 ∈ ℝ++𝐿 .

7

Some properties of Walrasian demand functions associated with changes in prices and wealth

can be derived from the homogeneity property and Walras’ Law when the Walrasian demand

functions are differentiable

The homogeneity restriction implies that, for all 1,...,l L , a proportionate change in ,p w

5 If the solution is a multi-valued correspondence, then the set of solutions is convex.

6 If Walrasian demand relations are correspondences, then they are upper hemicontinuous.

7 This is a very technical point. Katzner (1968) presents it as a condition on the property of indifference curves, for

instance, the indifference curve 1 1( ..., ; , )L L Lx x x x p w expressing commodity Lx as a function of the quantities of all

the other commodities. We have shown in Lecture Note 2, Section 2.1.B that a strictly quasiconcave utility function

represents strictly convex preferences. Given that a possible definition of strict quasiconcavity of u x implies that the

Hessian determinant of the indifference curve ;Lx is positive definite for all 1 1,..., Lx x x , Katzner shows, by

means of a counter example, that the converse is not necessarily true. There may, in fact, exist a particular ,p w at

which demand functions are not differentiable. MWG (pp. 94-95) approach the problem of differentiability of the

demand functions ,x p w using the implicit function theorem and the properties of the Jacobian matrix of the system

of the 1L first order conditions of the UMP. Differentiability requires that the Jacobian matrix have a non zero

determinant and conclude that the demand functions ,x p w are differentiable if and only if the determinant of the

Bordered Hessian of the utility function u x is different from zero at ,p w .


(4.6)

1

, ,0

Ll l

k

k k

x p w x p wp

p w

leaves the demand of good l unchanged.

The properties following from Walras’ Law are presented in the form of a Proposition.

Propositions 4.2 If the Walrasian demand functions are differentiable and satisfy Walras’

Law, the for all 𝑝 ≫ 0 and 𝑤 > 0 we have

(4.7) ∑ 𝑝𝑙𝐿𝑙=1

∂𝑥𝑙(𝑝,𝑤)

∂𝑤= 1

(4.8) ∑ 𝑝𝑙𝐿𝑙=1

∂𝑥𝑙(𝑝,𝑤)

∂𝑝𝑘+ 𝑥𝑘(𝑝,𝑤) = 0

From Walras’ Law, 𝑝 ⋅ 𝑥(𝑝, 𝑤) = 𝑤, differentiating with respect to 𝑤 we immediately obtain

(4.7).

To simplify the derivation of (4.8) assume first 𝐿 = 2. Taking the partial derivatives of

Walras’ Law with respect to 𝑝1 and 𝑝2, we obtain

(4.9) 𝑥1(𝑝,𝑤) + 𝑝1

∂𝑥1(𝑝,𝑤)

∂𝑝1+ 𝑝2

∂𝑥2(𝑝,𝑤)

∂𝑝1= 0

𝑥2(𝑝,𝑤) + 𝑝1

∂𝑥1(𝑝,𝑤)

∂𝑝2+ 𝑝2

∂𝑥2(𝑝,𝑤)

∂𝑝2= 0

Generalizing to 𝐿 > 2 and in a compact matrix notation (4.9) can be rewritten as

(4.10) 𝑥(𝑝, 𝑤) + [𝐷𝑝𝑥(𝑝, 𝑤)]𝑇𝑝 = 0

where

(4.11) [𝐷𝑝𝑥(𝑝,𝑤)] = [

∂𝑥1(𝑝,𝑤)

∂𝑝1. . .

∂𝑥1(𝑝,𝑤)

∂𝑝𝐿. . . . . .∂𝑥𝐿(𝑝,𝑤)

∂𝑝1. . .

∂𝑥𝐿(𝑝,𝑤)

∂𝑝𝐿

]

is the matrix of the price effects on the demand functions.

The economic implication is that the rearrangements in purchases caused by changes in

wealth – property (4.7) – and in prices – property (4.8) – do not violate the budget constraint:

(4.7) reflects the fact that the change in total expenditure must be equal to the change in

wealth, whereas (4.8) expresses the fact that a change in prices cannot change total

expenditure if wealth is unchanged.

The study of the change in the quantity demanded due to changes in own price, in the price of

a different commodity and in wealth can be conveniently expressed in terms of the notion of

demand elasticity, defined as the ratio of the percentage change in quantity demanded and the


percentage change of a particular variable. Since the notion of elasticity is independent of the

units of measurement, the use of the elasticity to measure the reaction of demand to the

change of prices or wealth is to be preferred to the use of the derivative. The parameters to be

estimated in empirical studies of demand analysis are, generally, the demand elasticities. The

signs and the numerical values of the various elasticities lead to a standard classification of

the different goods.

Starting with the elasticity of demand with respect to a change in its own price

(4.12) ln

ln

lll

l

x

p

we have, according to the sign of ll , the distinction between ordinary and Giffen goods: the

own elasticity of demand is negative for the former and positive for the latter group. Ordinary

goods respect the Law of Demand, namely the inverse relation between the quantity

demanded of a commodity and its own price. Giffen goods represent a violation of the Law of

Demand; for some ranges of the own price there is a direct, rather than an inverse, relation

between demand and price.8 The analytical construction of the theory of consumer’s behavior

does not exclude such possibility as we will show in Lecture Note 7.

The cross elasticity of demand measures the percentage change in the quantity demanded of

commodity l in response to a percentage change in the price of commodity k

(4.13) ln

ln

llk

k

x

p

A positive value of lk implies that commodities are gross substitutes: an increase in the price

of commodity k determines an increase in the quantity demanded of commodity l. A negative

value implies, on the contrary, that commodities l and k are gross complements in

consumption.9

8 These goods are named after the Scottish economist Sir Robert Giffen, to whom Alfred Marshall attributed this idea

in his Principles of Economics (1920, 8th

ed., p. 132). Potatoes during the Irish Great Famine have long been considered

the main example of a Giffen good. This idea has, however, been recently challenged. Giffen goods are often associated

with low quality products, with the idea that their consumption is reduced when income rises at given prices and may

actually increase when real income falls due to a price rise. The close connection in theory between Giffen goods and

income will be examined below, when dealing with the wealth elasticity of demand. It should, nonetheless, be observed

that the argument based on the existence of a quality scale in the supply of a commodity and the change in the chosen

quality as a function of income is based on the critical assumption that different varieties can be put together in the

definition of a given commodity, whereas a rigorous theoretical approach requires that different varieties be classified

just as many different commodities. 9 The adjective “gross” refers to demand functions which are wealth uncompensated, as Walrasian demand functions

are. In the general case of 2L commodities, the term gross substitutes refers to a much more restrictive property of

Walrasian demands, namely to a situation in which the increase in the price of commodity k increases the consumption

of all other commodities. In the Appendix 7.A of Lecture Note 7 the implications of the definitions of gross substitutes

and complements are further analyzed with the help of the substitution and the wealth effects of a cross price change.


The elasticity of demand with respect to wealth (income)

(4.14) ln

ln

llw

x

w

measures the reaction of demand to a change in wealth. If lw is positive commodities are

called normal, if lw is negative inferior. Normal commodities are further distinguished in

luxuries and necessities, commodities belong to the class of luxuries if the elasticity of

demand with respect to wealth is greater than 1 – implying a percentage increase in demand

greater than the percentage increase in wealth – to the category of necessities if the elasticity

is positive, but less than 1.

Let, furthermore,

(4.15) ,

l

l lp x p ww

w

be the budget share of commodity l. Budget shares and price and wealth elasticities just

defined make it possible, as can be verified, to express the relations (4.6)-(4.8) in the

following equivalent forms

(4.6’) 0lk lw

k

(4.7’) 1l lw

l

w

(4.8’) 0 for 1,...,k kl l

l

w w l L

4.3 Examples of demand functions

We will consider, with reference to the two-commodity case, three examples of demand

functions: Cobb-Douglas, Stone-Geary and quasilinear, which are derived from utility

maximization. We define in each case the demand functions and verify their properties; using

the properties of the Bordered Hessian, we show, when appropriate, that the utility functions

are quasiconcave. The Appendix is dedicated to the study of the properties of the constant

elasticity of substitution (CES) utility function and of the resulting demand functions.

In the Almost Ideal Demand System (AIDS) proposed by Deaton and Muellbauer (1980)

demand functions are derived from an approach of expenditure minimization. They will be

briefly considered in that context


4.3.1 Cobb-Douglas utility function

Assume that preferences are represented by the Cobb-Douglas utility function

(4.16) 𝑈(𝑥1, 𝑥2) = 𝑥1𝛼𝑥2

𝛽 with 𝛼, 𝛽 > 0

It can be easily verified that this utility function represents strictly monotone and strictly convex

preferences. The Cobb-Douglas function is homogeneous of degree (𝛼 + 𝛽): if the variables

are multiplied by a common proportionality factor 𝑡 the function is multiplied by a factor 𝑡𝛼+𝛽

(4.17) 𝑈(𝑡𝑥1, 𝑡𝑥2) = 𝑡𝛼+𝛽𝑈(𝑥1, 𝑥2) with 𝛼, 𝛽 > 0

As shown in Lecture Note 2, Sections 2.3.A.2 and 2.3.B.2. (4.14) is strictly concave, concave and

quasiconcave respectively if (𝛼 + 𝛽) is less than, equal to or greater than one.

Applying the logarithmic transformation, the same preferences can be represented in the convenient

form

(4.18) 𝑢(𝑥1, 𝑥2) = 𝛼ln𝑥1 + 𝛽ln𝑥2

The critical values of 𝑥1 and 𝑥2 , subject to the wealth constraint, are determined (see Lecture Note 3)

as part of the solution of the Kuhn-Tucker conditions of the associated Lagrangean function

(4.19)

𝑢1(𝑥) − 𝜆𝑝1 =𝛼

𝑥1− 𝜆𝑝1 ≤ 0

𝑢2(𝑥) − 𝜆𝑝2 =𝛽

𝑥2− 𝜆𝑝2 ≤ 0

𝑥∗[∇𝑥𝑢(𝑥) − 𝜆∗𝑝] = 0𝑝 ⋅ 𝑥∗ − 𝑤 = 0

where the first two weak inequalities admit of the possibility of a zero consumption of one of the

commodities, the third line is, strictly speaking, the Kuhn-Tucker condition and the last line the wealth

constraint. If prices and wealth are strictly positive, as assumed, the problem of utility maximization

with a Cobb-Douglas function has an interior solution. Eliminating 𝜆 from the first two conditions we

obtain a linear relation between the optimal consumption of the two commodities along the wealth-

consumption expansion path

(4.20) 𝑥1 =𝛼

𝛽

𝑝2

𝑝1𝑥2

Making the appropriate substitution in the budget constraint, we can solve for the optimal

consumption of commodity 2 and then, using (4.20), of commodity 1. Finally from either the first or

the second equation in (4.19) we can obtain the value of the multiplier. The solution of the UMP is

therefore10

(4.21) 𝑥1∗ =

𝛼

𝛼+𝛽

𝑤

𝑝1, 𝑥2

∗ =𝛽

𝛼+𝛽

𝑤

𝑝2, 𝜆∗ =

𝛼+𝛽

𝑤

10

We have previously stated that 𝜆 measures the marginal utility of wealth. With the utility function taken into

consideration, the marginal utility of wealth is diminishing. We will come back to this point after introducing the value

function of the problem.


With regard to the properties of these demand functions:

1/ homogeneity of degree zero is immediate;

2/ Walras’ Law: substituting in the budget constraint, we obtain

(4.22) 𝑝1𝑥1∗ + 𝑝2𝑥2

∗ = 𝑝1 (𝛼

𝛼+𝛽

𝑤

𝑝1) + 𝑝2 [

𝛽

𝛼+𝛽

𝑤

𝑝2] = 𝑤

3/ unique is immediate

4/ continuous is immediate

5/ differentiable because the utility function is strictly quasiconcave (actually strictly concave).

The determinant of the Bordered Hessian of the utility function (4.18)

(4.23) 𝑑𝑒𝑡𝐻𝐵(𝑢(𝑥)) = det

[ −

𝛼

𝑥12 0

𝛼

𝑥1

0 −𝛽

𝑥22

𝛽

𝑥2

𝛼

𝑥1

𝛽

𝑥20 ]

2 2

2 2 2 2 2 21 2 1 2 1 2x x x x x x

is strictly positive for all 2x thus showing that the function is strictly quasiconcave

11 and

the demand function differentiable. The same conclusion concerning the property of the utility

function (4.18) could have been reached simply noting that a linear combination of concave

function – as the ln functions are – is concave.

We can also determine the elasticity of substitution of the Cobb-Douglas utility function. The

elasticity of substitution measures the rate of change of the ratio 𝑥2

𝑥1 between the quantities

demanded and the rate of change of the marginal rate of substitution 𝑀𝑅𝑆1,2 =𝛼

𝛽

𝑥2

𝑥1, which is

the ratio of the marginal utilities.12

We obtain

(4.24) 𝜎1,2 =𝑑ln(𝑥2 𝑥1⁄ )

𝑑ln𝑀𝑅𝑆1,2=

(𝑑𝑥2𝑥2

−𝑑𝑥1𝑥1

)

(𝑑𝑥2𝑥2

−𝑑𝑥1𝑥1

)= 1

Let us take a closer look at some implications of the Cobb-Douglas demand functions (4.21).

The optimal expenditure on commodity l - *

l lp x , 1,2l - is a constant fraction of wealth and,

11

Note that the property of strict concavity refers to the Cobb-Douglas representation of preferences (4.12). When the

same preferences are represented by the Cobb-Douglas function (4.10), the properties are different as shown in the

examples of Lecture Note 2. 12

As indicated in the Appendix of this Note, the elasticity of substitution can also be used to distinguish between

commodities that are substitutes or complements in consumption: 1lk identifies a relation of substitution, while

1lk indicates a relation of complementarity in consumption.


as a consequence, the expenditure shares *

l ll

p xw

w are constants, independent of prices and

wealth and, in line with the adding-up restriction (Walras’ Law), add to 1.

These are very strong and restrictive properties, very rarely observed in empirical analysis

over periods of time of some length. Typically the wealth share of the expenditure on food

and beverages is declining, while the share of the expenditure on amenities is rising. The

Stone-Geary demand functions as well as the Almost Ideal System of demand functions aim

at removing these rigidities.

4.3.2 The Stone-Geary utility function

Maintaining the convenient assumption of a two-commodity space, the Stone-Geary utility

function has the following form13

(4.25) 𝑢(𝑥) = (𝑥1 − 𝛾1)𝛼(𝑥2 − 𝛾2)

𝛽

where the vector of constants 𝛾 = (𝛾1, 𝛾2) stands for predetermined subsistence levels of

consumption. Defining the variables 𝑦𝑙 = 𝑥𝑙 − 𝛾𝑙, (𝑙 = 1,2), which represent consumption

levels in excess of subsistence, (4.25) becomes the standard Cobb-Douglas

(4.26) 𝑢(𝑦) = 𝑦1𝛼𝑦2

𝛽

With the product 𝑝 ⋅ 𝛾 indicating the expenditure necessary to buy the subsistence

consumption and with 𝑤′ = 𝑤 − 𝑝 ⋅ 𝛾 > 0 the supernumerary wealth, the wealth constraint is

now

(4.27) 𝑝 ⋅ 𝑦 − 𝑤′ = 0

Following the usual technique of utility maximization subject to an equality constraint and

assuming an interior solution, from (4.21) the demand functions are

(4.28) 𝑦1∗ =

𝛼

𝛼+𝛽

𝑤′

𝑝1, y2

∗ =𝛽

𝛼+𝛽

𝑤′

𝑝1

Reverting back to the original variable and substituting for 𝑤′, we obtain the Stone–Geary

demand functions

13

The Stone-Geary utility function originates in a brief comment made by Geary(1949-1950) on an earlier paper by

Klein and Rubin (1947-1948), the scope of which was to determine an appropriate price index for a situation in which,

due to the presence of rationing, the standard Laspeyre price index could overstate price inflation. Klein and Rubin

produced a system of demand function in which, differently from the usual Cobb-Douglas approach, demands depend

on all prices and not only on the own price. Geary then produced the utility function (4.19), thus showing that Klein and

Rubin’s demand functions could be rationalized by the usual constrained utility maximization approach. Using Geary’s

utility function, Stone (1954) reformulated the Walrasian demands into expenditure functions on the various

commodities introducing the notion of supernumerary money income as the income (expenditure) exceeding the level

necessary to buy the subsistence quantities.


(4.29) 𝑥1∗ = 𝛾1 +

𝛼

𝛼+𝛽

𝑤−𝑝⋅𝛾

𝑝1, 𝑥2

∗ = 𝛾2 +𝛽

𝛼+𝛽

𝑤−𝑝⋅𝛾

𝑝2

Through the subsistence expenditure 𝑝 ⋅ 𝛾 all demand functions now depend on all prices. We

have, for instance, *1 2

2 1

x

p p

. This means that an increase in the cost of the

subsistence level of commodity 2 would reduce the demand of commodity 1 through the

reduction of the supernumerary income w . From this point of view, all commodities are

complements in consumption.

In his econometric study of British demand, Stone actually estimated the expenditure

functions derived from (4.29)

(4.30) for 1,...,l l l l lp x p a w p l L

where the la are the appropriate. This system of L equations represents Stone’s linear

expenditure system.

The properties of the Stone-Geary utility function are obviously the same as those of the

Cobb-Douglas function. Note, in particular, that the homogeneity and the adding-up

restriction are satisfied.

4.3.3 Quasilinear utility function

Suppose that the utility function has the following form

(4.31) 𝑈(𝑥) = 𝑢(𝑥1) + 𝑥2

The linearity in commodity 2, with 𝑢(𝑥1) generally non linear, explains the denomination of

quasilinear utility function.14

Assume that the marginal utility of commodity 1 is always positive and decreasing - 𝑢′(𝑥1) >

0 and 𝑢′′(𝑥1) < 0 for all 𝑥1 ≥ 0 - and that commodity 2 is measured in money terms, so that

𝑥2 is to be interpreted as the quantity of money reserved for the purchase of all other

commodities different from 𝑥1. The price of commodity 2 is thus set equal to 1, namely

𝑝2 = 1. The budget set, with prices and wealth strictly positive, is accordingly

14

This form of the utility function reflects Hick’s (1939, p. 33) construction of a composite commodity. Suppose that

there are in fact other 𝐿 − 1 commodities besides commodity 1. This collection of goods “can always be treated as if

they were divisible units of a single commodity so long as their relative prices … [are] … unchanged”. These goods can

thus “be lumped together into one commodity ‘money’ or ‘purchasing power in general’”.


(4.32) 𝐵(𝑝, 𝑤) = {(𝑥1, 𝑥2) ∈ ℝ+2 |𝑝1𝑥1 + 𝑥2 ≤ 𝑤}

The marginal rate of substitution 𝑀𝑅𝑆1/2 = 𝑢′(𝑥1) reveals the peculiarity of the quasilinear

utility function: the slope of the indifference curves depends only on the quantity of

commodity 1; the indifference curves and are, therefore, vertical displacement one of the

other as shown in Fig. 4.2.

Fig. 4.2 – Indifference curves of a utility function quasi linear in commodity 2

The Lagrangean function for the UMP is

(4.33) 𝐿(𝑥1, 𝑥2, 𝜆) = 𝑢(𝑥1) + 𝑥2 − 𝜆(𝑝1𝑥1 + 𝑥2 − 𝑤)

The critical values are determined by the solution of the following set of relations

(4.34)

∂𝐿(𝑥1,𝑥2,𝜆)

∂𝑥1= 𝑢′(𝑥1

∗) − 𝜆∗𝑝1 ≤ 0 with equality if 𝑥1∗ > 0

∂𝐿(𝑥1,𝑥2,𝜆)

∂𝑥2= 1 − 𝜆∗ ≤ 0 with equality if 𝑥2

∗ > 0

∂𝐿(𝑥1,𝑥2,𝜆)

∂𝜆= 𝑝1𝑥1

∗ + 𝑥2∗ = 𝑤

Note that, because of the assumption that wealth is strictly positive, the consumption of at

least one commodity must be positive. If we assume that 𝑢(𝑥1) meets the Inada conditions -

the marginal utility of commodity 1 becomes infinitely large as 𝑥1 tends to zero and infinitely

small if 𝑥1 becomes infinitely large – then the consumption of commodity 1 is always positive

in the optimal solution of the UMP.15

The non negativity constraint on the consumption of

15

See Chen-Ici Inada (1963).

𝑥2

𝑥1 𝑥1


commodity 2 may, however, be binding in the optimal solution. This circumstance requires

some care in the determination of the solution.

We have to consider two distinct situations

(4.35) 𝑤 − 𝑝1𝑥1∗ {

>≤

} 0

They correspond to interior solutions and boundary solutions, respectively.

If 𝑤 − 𝑝1𝑥1∗ > 0, the UMP admits of an interior solution with the consumption of both

commodities strictly positive and the relations (4.34) are all satisfied as strict equalities. We

have

(4.36) 𝑥1∗ = 𝑢′

−1(𝑝1), 𝑥2∗ = 𝑤 − 𝑝1𝑥1

∗, 𝜆∗ = 1

with the quantity of commodity 2 being determined by the budget constraint, 𝜆∗ by the second

equation in (4.34) and the quantity of commodity 1 by the first.

If, on the contrary, 𝑤 − 𝑝1𝑥1∗ ≤ 0, we have a boundary solution with the wealth

constraint implying 𝑥2∗ = 0; the entire wealth is then used for the consumption of commodity

1, 𝑥1∗ =

𝑤

𝑝1. This has an important implication for the value of the multiplier 𝜆∗. Suppose that

*1 1 0w p x so that the second condition in (4.34) is now satisfied as a strict inequality. The

value of 𝜆 must, therefore, be determined by the first order condition on commodity 1. We

have

(4.37) 𝜆∗ =𝑢′(𝑥1

∗)

𝑝1=

𝑢′(𝑤

𝑝1)

𝑝1

Since 𝑢′(⋅) is by assumption a decreasing function of its argument, it is a decreasing function

of w , whatever the value of 𝑝1. This means that the marginal utility of wealth is the larger the

smaller the quantity of wealth. Note finally that, if 𝑤 = 𝑝1, 𝑥2∗ = 0 is a limiting interior

solution, so that we may solve for 𝜆∗ from the first order condition of commodity 2 and thus

obtain 𝜆∗ = 1. We can conclude that

(4.38) 𝜆∗ {≥=

} 1as𝑤 − 𝑝1𝑥1∗ {

≤>

} 0

This matches the intuition that, at least after a point, the marginal utility of wealth increases as

wealth is further and further reduced.16

16

A numerical example may help. Let 𝑢(𝑥1) = ln𝑥1 so that 𝑢′(𝑥1) = 1𝑥1

⁄ . Consider the boundary solution 𝑥1∗ = 𝑤

𝑝1⁄ .

Substituting in the first order condition we obtain 𝜆∗ = 1𝑤⁄ which clearly exhibits the inverse relation between the

marginal utility of wealth and its quantity.


The second order condition for a maximum is satisfied if the utility function is strictly

quasiconcave, that is if the determinant of the bordered Hessian is strictly positive for all

strictly positive values of (𝑥1, 𝑥2) that are solutions of the UMP. Let, for instance, 𝑢(𝑥1) =

ln𝑥1. The Bordered Hessian is:

(4.39) 𝐻𝐵(𝑥1, 𝑥2) = [−[(𝑥1)

−2] 0 (𝑥1)−1

0 0 1(𝑥1)

−1 1 0]

The function is strictly quasiconcave since

(4.40) (−1)2 [−[(𝑥1)

−2] 0 (𝑥1)−1

0 0 1(𝑥1)

−1 1 0] = (−1)2(𝑥1)

−2 > 0

Note that the function is concave, but not strictly concave, since the quadratic form 𝑧 ⋅ 𝐻(𝑥)𝑧

is equal to zero for all vectors 𝑧 because det𝐻(𝑥) = 0.

4.3The indirect utility function and its properties

As the solution of the UMP shows, the consumer’s optimal behavior depends on market

prices and on his personal wealth: the Walrasian demand functions are, in fact, the analytical

expression of this dependence. It follows that the utility level attained by the consumer when

he optimally chooses his consumption bundle also depends of the (𝑝, 𝑤) pair. We define the

function relating utility to prices and wealth as

(4.41) 𝑣(𝑝,𝑤) = 𝑢(𝑥(𝑝, 𝑤))

and call it the indirect utility function.

Proposition 4.3 The indirect utility function is

1. Homogeneous of degree zero

This property follows from the homogeneity of degree zero of Walrasian demand functions

Since 𝑢(𝑥(𝛼𝑝, 𝛼𝑤)) = 𝑢(𝑥(𝑝, 𝑤)) from (4.36) we immediately have 𝑣(𝛼𝑝, 𝛼𝑤) = 𝑣(𝑝,𝑤)

2. Strictly increasing in w and non increasing in 𝑝𝑙 for any l

An increase in w leads to a parallel outward shift of the budget line and, therefore, to the

attainment of a higher indifference curve. With regard to the effect of a price increase, we

must distinguish between the internal and the boundary solution. If the UMP has an interior

solution, an increase in the price, say of commodity 2, leads to an inward rotation of the

budget line and thus pushes the consumer onto a lower indifference curve: indirect utility


diminishes. If the UMP has, instead, a boundary solution with 𝑥2∗ = 0 , then an increase in the

price of commodity 2 would not change the optimal solution. The consumer would remain on

the initial indifference curve; the indirect utility would be unchanged.

3. Quasiconvex; that is the lower contour set of the indirect utility function

(4.42) 𝐼−(𝑝, 𝑤) = {𝑝,𝑤 ∈ ℝ++𝐿 × ℝ+|𝑣(𝑝, 𝑤) ≤ �̅�}

is quasiconvex for any �̅�.

We will consider two approaches to the proof of this statement.

The first approach takes the price space as the natural context in which the definition of

quasiconvexity must be verified. As we have seen in Lecture Note 2, in order to establish that

𝑣(𝑝,𝑤) is quasiconvex we have to show that the Hessian matrix 𝐻(𝑝, 𝑤) is positive

semidefinite in the linear subspace ∇𝑣(𝑝,𝑤) ⋅ 𝑧 = 0. But, due to the presence of three

variables – two prices and wealth – the number of conditions to be verifies is large and

numerical conclusions may be difficult. The dimensions of the problem can, however, be

reduced from three to just two taking advantage of the homogeneity of degree zero of the

budget line. This means that a proportional change in prices and wealth leaves the budget set

unaltered. Let us then take as our proportionality factor 1

𝑤. Dividing all the terms of the wealth

constraint by 𝑤 and defining the normalized price vector 𝜋 = (𝜋1, 𝜋2) = (𝑝1

𝑤,𝑝2

𝑤), we obtain

the normalized budget line 𝜋 ⋅ 𝑥 − 1 = 0 and the indirect utility function 𝑣(𝜋; 1). Our task, at

this point, is to show that the lower contour set of is quasiconvex for any �̅�.

Let 𝜋, 𝜋′ ∈ 𝐼−(𝜋) and assume 𝑣(𝜋; 1) = 𝑣(𝜋′; 1) = �̅�. We must show that their convex

combination 𝜋′′ = 𝛼𝜋 + (1 − 𝛼)𝜋′satisfies the definition of quasiconvexity 𝑣(𝛼𝜋 +

(1 − 𝛼)𝜋′; 1) ≤ �̅�. From property 2 above, we know that the indirect utility function is non

increasing in p, hence in 𝜋; more generally, a decreasing function of 𝜋. We are therefore in

the case considered in Lecture Note 2, Section 2.2.C, in which we have shown, applying the

definition of quasiconvexity, that it is the lower contour set of the function which is convex.

Fig. 4.3 reproduces, with adaptation to the case under consideration, Fig. 2.6, Panel (a) of

Lecture Note 2. In the nonnegative (𝜋1, 𝜋2) quadrant, the level set 𝐼(𝜋0) = �̅� is depicted, for

convenience as a smooth curve, as well as the points 𝜋, 𝜋′ ∈ 𝐼(𝜋) and their convex

combination 𝜋′′ = 𝛼𝜋 + (1 − 𝛼)𝜋′. Since 𝑣(𝜋; 1) increased moving in the direction of the

origin – the lower are commodity prices, given wealth, the grater the quantities of

commodities that can be purchased and correspondingly greater the level of utility and

indirect utility reached - 𝑣(𝜋′′; 1) ≤ �̅� is in the region above. 𝑣(𝜋; 1) is, therefore,

quasiconvex and so is 𝑣(𝑝,𝑤) for all 𝑤 > 0.


Fig. 4.3 – Lower contour set of the indirect utility function 𝒗(𝝅)

The second approach considers different budget lines in the commodity space and determines

the properties of a convex combination of them. Let (𝑝, 𝑤) and (𝑝′, 𝑤′) be two price-wealth

pairs such that 𝑣(𝑝,𝑤) ≤ �̅� and 𝑣(𝑝′, 𝑤′) ≤ �̅�. In the diagram of Fig. 4.4 we actually assume

that these conditions are satisfied with the equal sign. Let 𝑥(𝑝, 𝑤) and 𝑥(𝑝′, 𝑤′) be the optimal

choices with respect to the corresponding budget sets 𝐵(𝑝,𝑤) and 𝐵(𝑝′, 𝑤′). By construction,

these optimal choices are on the same indifference curve at the points in which the budget

lines are tangent to the indifference curve and thus attain the same utility level �̅� = �̅�. Let

(𝑝′′, 𝑤′′) = (𝛼(𝑝,𝑤) + (1 − 𝛼)(𝑝′, 𝑤′)) = (𝛼𝑝 + (1 − 𝛼)𝑝′, 𝛼𝑤 + (1 − 𝛼)𝑤′), where the last

equality follows from the property of homogeneity of degree zero of the budget set. In Fig.

4.4, we have assumed, for graphical convenience, 𝛼 ∈ (0,1), so that any point 𝑥′′ ∈ 𝐵(𝑝′′, 𝑤′′)

lies below the indifference curve �̅� = �̅�.17

We conclude, on the basis of the definition of

quasiconvexity, that the set (4.42) is quasiconvex.

17

With 𝛼 = 1, 𝑥′′ coincides with 𝑥and with 𝛼 = 0, 𝑥′′ coincides with 𝑥′. In these cases, 𝑣(𝑝′′, 𝑤′′) = �̅�.

𝑥2

𝜋2

𝜋1

𝑥1

𝑥1

𝜋 𝜋0

𝜋’’ 𝜋’’

𝜋’

0I)(v

vI 0

),( wpx

),'( wpx

),'( wpB

vu

),( wpB

)'',''( wpB


Fig. 4.4 – Quasiconvexity of the indirect utility function

4. Continuous and differentiable in 𝑝 and 𝑤

Since 𝑢(𝑥) is, by assumption, a continuous, twice differentiable function and the demand

functions 𝑥(𝑝, 𝑤) is continuous and differentiable, so is 𝑣(𝑝,𝑤) = 𝑢(𝑥(𝑝, 𝑤)).

The differentiability of the indirect utility function in the arguments 𝑝 and 𝑤 leads to two

important results.

Proposition 4.4 If the indirect utility function is differentiable at all (𝑝, 𝑤) ≫ 0, then we have

(4.43) ∂𝑣(𝑝,𝑤)

∂𝑤= 𝜆

(4.44) ∂𝑣(𝑝,𝑤)

∂𝑝𝑙= −𝜆𝑥𝑙(𝑝, 𝑤)𝑙 = 1, . . . , 𝐿

Proof. Assume 𝐿 = 2 and write 𝑣(𝑝,𝑤) = 𝑢(𝑥1(𝑝, 𝑤), 𝑥2(𝑝, 𝑤)). Differentiating with respect

to 𝑤 we have

(4.45) ∂𝑣(𝑝,𝑤)

∂𝑤=

∂𝑢(𝑥)

∂𝑥1

∂𝑥1

∂𝑤+

∂𝑢(𝑥)

∂𝑥2

∂𝑥2

∂𝑤= 𝜆𝑝1

∂𝑥1

∂𝑤+ 𝜆𝑝2

∂𝑥2

∂𝑤= 𝜆

where the second equality follows from the first order conditions of utility maximization and

the third from property (4.7) of Walrasian demand functions.

Differentiating 𝑣(𝑝, 𝑤) = 𝑢(𝑥1(𝑝, 𝑤), 𝑥2(𝑝, 𝑤)) with respect to 𝑝𝑙 we obtain

(4.46) ∂𝑣(𝑝,𝑤)

∂𝑝𝑙= ∑

∂𝑢(𝑥)

∂𝑥𝑘

𝐿𝑘=1

∂𝑥𝑘

∂𝑝𝑙= 𝜆 ∑ 𝑝𝑙

𝐿𝑘=1

∂𝑥𝑘

∂𝑝𝑙= −𝜆𝑥𝑙(𝑝, 𝑤)𝑙 = 1, . . . , 𝐿

where the second equality follows from the first order conditions of utility maximization and

the third from property (4.8) of Walrasian demand functions.

A more direct and elegant proof can be obtained applying the Envelope Theorem, which is

presented in the appendix of the Lecture Note 6.

4.5 Examples of indirect utility functions

We have examined in Section 4.3 the Cobb-Douglas and quasi linear utility functions;18

we

turn now to consider the corresponding indirect utility functions. We derive these functions

and verify their properties.

18

The definition of the indirect utility function of the Stone-Geary utility and its properties can be easily deduced from

those of the Cobb-Douglas function.


4.5.1 Cobb-Douglas indirect utility function

Substituting the demand functions (4.10) 𝑥1∗ =

𝛼

𝛼+𝛽

𝑤

𝑝1and𝑥2

∗ =𝛽

𝛼+𝛽

𝑤

𝑝2 in the Cobb-Douglas

utility function (4.7) 𝑢(𝑥1, 𝑥2) = 𝛼ln𝑥1 + 𝛽ln𝑥2, we obtain the indirect utility function

(4.47) 𝑣(𝑝,𝑤) = [𝛼ln𝛼

𝛼+𝛽+ 𝛽ln

𝛽

𝛼+𝛽] + (𝛼 + 𝛽)ln𝑤 − 𝛼ln𝑝1 − 𝛽ln𝑝2

Let us verify the properties.

1. Homogeneity of degree zero: immediate

2. Strictly increasing in w and non increasing, actually decreasing in 𝑝: immediate

3. Quasiconvex in 𝑝, given 𝑤

For given 𝑤, we proceed substantially as with the general proof above and consider the

dependence of 𝑣(𝑝, 𝑤) only on prices 𝑝 and write (4.47) as 𝑣(𝑝;𝑤) = 𝐾 − (𝛼ln𝑝1 + 𝛽ln𝑝2)

with K a constant. Since (𝛼ln𝑝1 + 𝛽ln𝑝2) is concave, as a linear combination of concave

functions, −(𝛼ln𝑝1 + 𝛽ln𝑝2) is convex and, therefore, also quasiconvex.

To show that 𝑣(𝑝,𝑤) is quasiconvex, we can proceed to verify that the Hessian matrix

𝐻(𝑣(𝑝;𝑤)) is positive semidefinite in the linear space ∇𝑝𝑣(𝑝;𝑤) ⋅ 𝑧 = 0. Since the Bordered

Hessian 𝐻𝐵(𝑣(𝑝; 𝑤)) = −𝐻𝐵(𝑢[𝑥]), quasiconvexity follows from (4.17) with a simple

change of variables – from 𝑥 to 𝑝 - and of sign.

4. Continuous and differentiable in 𝑝 ≫ 0 and 𝑤 > 0: immediate from the definition (4.43)

Given the differentiability of the indirect utility function we can verify the properties (4.39)

and (4.40). We have

(4.48) ∂𝑣(𝑝,𝑤)

∂𝑤=

𝛼+𝛽

𝑤= 𝜆

(4.49) ∂𝑣(𝑝,𝑤)

∂𝑝1= −

𝛼

𝑝1= −

𝛼

𝑝1

𝛼+𝛽

𝛼+𝛽

𝑤

𝑤= −

𝛼+𝛽

𝑤(

𝛼

𝛼+𝛽

𝑤

𝑝1) = −𝜆𝑥1(𝑝, 𝑤)

and similarly with regard to the derivative with respect to 𝑝2.

4.5.2. Indirect utility function of the quasilinear utility function

Assume the following explicit form

(4.50) 𝑈(𝑥) = ln𝑥1 + 𝑥2


for the quasilinear utility function. In order to derive a basic property of the indirect utility

function we cannot maintain the assumption of a fixed price of commodity 2. Let accordingly

𝑝2 be its price. With a variable 𝑝2, we can rewrite the previous solution (4.31) of the UMP as

(4.51) 𝑥1∗ = 𝑥1(𝑝) =

𝑝2

𝑝1, 𝑥2

∗ = 𝑥2(𝑝, 𝑤) =𝑤

𝑝2− 1, 𝜆∗ = 2

which with 𝑤 > 1 is an interior solution. Substituting these demand functions in the

quasilinear utility function (4.46), we obtain the indirect utility function

(4.52) 𝑣(𝑝,𝑤) = ln𝑝2

𝑝1+

𝑤

𝑝2− 1

Let us verify the properties.

1. Homogeneity of degree zero: immediate

2. Strictly increasing in w and decreasing in 𝑝1: immediate. To show the dependence on 𝑝2,

derive 𝑣(𝑝,𝑤) with respect to 𝑝2; we have

(4.53) ∂𝑣(𝑝,𝑤)

∂𝑝2=

1

𝑝2−

𝑤

𝑝22 =

1

𝑝2(1 −

𝑤

𝑝2)

which is negative if 𝑤 > 𝑝2 as required for the assumed internal solution

3. Quasiconvex in 𝑝, given 𝑤

To show that 𝑣(𝑝,𝑤) is quasiconvex, we can proceed to verify that the Hessian matrix

𝐻(𝑣(𝑝;𝑤)) is positive semidefinite in the linear space ∇𝑝𝑣(𝑝;𝑤) ⋅ 𝑧 = 0. Since the Bordered

Hessian is

(4.54) 𝐻𝐵(𝑣(𝑝;𝑤)) =

[

1

𝑝12 0 −

1

𝑝1

01

𝑝22 (−1 +

𝑤

𝑝2)

1

𝑝22 (−1 +

𝑤

𝑝2)

−1

𝑝1

1

𝑝2(−1 +

𝑤

𝑝2) 0 ]

We have

(4.55) det𝐻𝐵(𝑣(𝑝;𝑤)) = −(1

𝑝1)2 1

𝑝22 (−1 +

𝑤

𝑝2) − (

1

𝑝1)2 1

𝑝23 (−1 +

𝑤

𝑝2)2

< 0

thus showing that 𝑣(𝑝,𝑤) is effectively quasiconvex, actually strictly quasiconvex.19

4. Continuous and differentiable in 𝑝 ≫ 0 and 𝑤 > 0: immediate from the definition (4.48)

19

Remember that, as stated in Lecture Note 2 Definition 2.20, with a function of just two variables – the prices (𝑝1, 𝑝2)

- and just one constraint – the wealth constraint – the function is quasiconvex if the determinant of the Bordered hessian

is non positive.


Given the differentiability of the indirect utility function we can verify the properties (4.39)

and (4.40). We have

(4.56) ∂𝑣(𝑝,𝑤)

∂𝑤= 1 = 𝜆

(4.57) ∂𝑣(𝑝,𝑤)

∂𝑝1= −

1

𝑝1= −𝜆𝑥1(𝑝1)

and from (4.53)

(4.58)

*2

2 2 2

, 11 ,

v p w wx p w

p p p

Appendix. 4.A The Constant Elasticity of substitution (CES) utility function

Assume that strictly monotone and convex preferences are represented by the utility function

(4.A1) 𝑈(𝑥) = (𝑥1𝜌

+ 𝑥2𝜌)1

𝜌⁄

with 0 ≠ 𝜌 ≤ 1.

Panel (a): 𝝆 = 𝟏, 𝝈 = +∞ Panel (b): 𝝆 = 𝟎, 𝝈 = 𝟏 Panel (c): 𝝆 = −∞,𝝈 = 𝟎

Fig. 4.A1 – Indifference curves of the CES function for alternative values of 𝝆

Panels (a), (b) and (c) of Fig. 4.A1 depict the form of a typical indifference curve generated

by the extreme values of 𝜌, namely 𝜌 = {1,0, −∞}. When 𝜌 = 1 (Panel (a)), the indifference

curve is a straight line showing that commodities are in this case perfect substitutes. When

𝜌 = 0, the CES function is undefined; considering, however, the limit as 𝜌 approaches zero,

𝑥2 𝑥2 𝑥2

𝑥1 𝑥1 𝑥1

1 0lim lim


the CES function becomes a Cobb-Douglas (see Panel (b)).20

Finally, when 𝜌 approaches

−∞, the CES becomes a Leontief utility function (see Panel (c)), which shows that the two

goods are perfect complements. Since the elasticity of substitution of the CES function is, as

subsequently shown, related to the parameter 𝜌 by the equality 𝜎 =1

1−𝜌 , the three situations

can be alternatively described using the notion of elasticity of substitution, respectively as

𝜎 = {+∞, 1,0}. Fig. 4.A2 joins the previous three diagrams into a single one and highlights

the range of values of the elasticity of substitution in the various parts of the diagram as well

as the intuitive meaning of substitution.

Fig. 4.A2 – Varying values of the elasticity of substitution for the CES function

In order to determine the properties of the CES function it is convenient to work, as we have

done with the Cobb-Douglas function, with the logarithmic transformation of (4.A1), namely

with the function

(4.A2) ln𝑈(𝑥) =1

𝜌ln(𝑥1

𝜌+ 𝑥2

𝜌) =

1

𝜌ln𝑢(𝑥)

20

Considering the more general form of CES function 𝑈(𝑥) = (𝛼𝑥1𝜌

+ (1 − 𝛼)𝑥2𝜌)1

𝜌⁄ and taking logarithms we have

ln𝑈(𝑥) =ln(𝛼𝑥1

𝜌+ (1 − 𝛼)𝑥2

𝜌)

𝜌

Using L’Hopital’s rule, we have

ln𝑈(𝑥) = lim𝜌→0

ln(𝛼𝑥1𝜌+(1−𝛼)𝑥2

𝜌)

𝜌= lim

𝜌→0

𝛼𝑥1𝜌ln𝑥1+(1−𝛼)𝑥2

𝜌ln𝑥2

(𝛼𝑥1𝜌+(1−𝛼)𝑥2

𝜌)

= 𝛼ln𝑥1 + (1 − 𝛼)ln𝑥2 = ln𝑥1𝛼𝑥2

1−𝛼

whence 𝑈(𝑥) = 𝑥1𝛼𝑥2

1−𝛼 which is the Cobb-Douglas that was examined in the preceding section.

𝑥2

𝑥1

1

01

0

1


where 𝑢(𝑥) = (𝑥1𝜌

+ 𝑥2𝜌). With 0 ≠ 𝜌 < 1, the UMP has an interior solution so that the Lagrangean

associated with the problem is

(4.A3) 𝐿(𝑥, 𝜆) =1

𝜌ln𝑢(𝑥) − 𝜆(𝑝 ⋅ 𝑥 − 𝑤)

Following Lagrange’s method, the critical values of 𝑥1 and 𝑥2 are determined as part of the solution of

the following set of relations

(4.A.4)

∂𝐿

∂𝑥1=

1

𝜌

∂ln𝑢(𝑥)

∂𝑥1− 𝜆𝑝1 =

𝑥1𝜌−1

𝑢(𝑥)(1/𝜌)−1 − 𝜆𝑝1 = 0

∂𝐿

∂𝑥2=

1

𝜌

∂ln𝑢(𝑥)

∂𝑥2− 𝜆𝑝2 =

𝑥2𝜌−1

𝑢(𝑥)(1/𝜌)−1 − 𝜆𝑝2 = 0

∂𝐿

∂𝜆= 𝑝1𝑥1 + 𝑝2𝑥2 − 𝑤 = 0

Eliminating 𝜆 from the first two conditions and rearranging terms, we have

(4.A5) 𝑥1 = (𝑝1

𝑝2)

1𝜌−1⁄

𝑥2

The relation (4.A5) represents the wealth-consumption path, which, as in (4.14), is a linear

relation between the levels of consumption of 𝑥1 and 𝑥2 in the optimal solution of the UMP.

The Walrasian demands are then obtained substituting (4.A5) in the budget constraint21

and

the resulting optimal 𝑥2∗ back into (4.A5):

(4.A6)

𝑥1∗ =

𝑝1

1𝜌−1⁄

𝑝1

𝜌𝜌−1⁄

+𝑝2

𝜌𝜌−1⁄

𝑤

𝑥2∗ =

𝑝2

1𝜌−1⁄

𝑝1

𝜌𝜌−1⁄

+𝑝2

𝜌𝜌−1⁄

𝑤

Letting 𝑟 = 𝜌/(𝜌 − 1), we can simplify the notation and rewrite (4.A6) as

(4.A7)

𝑥1∗ =

𝑝1𝑟−1

𝑝1𝑟+𝑝2

𝑟 𝑤

𝑥2∗ =

𝑝2𝑟−1

𝑝1𝑟+𝑝2

𝑟 𝑤

Let us not prove that the CES function (4.A2) is concave. The first order derivatives are

(4.A8) ∇𝑥ln𝑈(𝑥)𝑇 = [𝑥1𝜌−1

𝑢(𝑥)

𝑥2𝜌−1

𝑢(𝑥)]

The Hessian matrix is therefore

21

We obtain 𝑝1𝑥2 (𝑝1

𝑝2)1/(𝜌−1)

+ 𝑝2𝑥2 = 𝑥2 (𝑝1𝜌/(𝜌−1)

+ 𝑝2𝜌/(𝜌−1)

) 𝑝2−1/(𝜌−1)

= 𝑤


(4.A9) 𝐻ln𝑈(𝑥) = [−

𝜌𝑥1𝜌−2

[(1−𝜌)𝑢(𝑥)+𝜌𝑥1𝜌]

[𝑢(𝑥)]2−𝜌

(𝑥1𝑥2)𝜌−1

[𝑢(𝑥)]2

−𝜌(𝑥1𝑥2)𝜌−1

[𝑢(𝑥)]2−

𝜌𝑥2𝜌−2

[(1−𝜌)𝑢(𝑥)+𝜌𝑥2𝜌]

[𝑢(𝑥)]2

]

To check for concavity, let us first note that the leading principal minors of all permutations

are non positive, actually strictly negative for 𝜌 ≠ 0.We further have

(4.A10)

det𝐻ln𝑈(𝑥) ∝ [(𝜌 − 1)𝑢(𝑥) − 𝜌𝑥1𝜌][(𝜌 − 1)𝑢(𝑥) − 𝜌𝑥2

𝜌] − 𝜌(𝑥1𝑥2)

𝜌 =

= (𝜌 − 1)[(𝜌 − 1)𝑢(𝑥)2 − 𝜌𝑢(𝑥)(𝑥1𝜌 + 𝑥2

𝜌) + 𝜌(𝑥1𝑥2)

𝜌] =

= (𝜌 − 1)[(𝜌 − 1)𝑢(𝑥)2 − 𝜌𝑢(𝑥)2 + 𝜌(𝑥1𝑥2)𝜌] =

= (𝜌 − 1) [−(𝑥1𝜌 + 𝑥2

𝜌)2+ 𝜌(𝑥1𝑥2)

𝜌]

where the proportionality factor eliminated in the first line is 𝜌2 (𝑥1𝑥2)𝜌−2

𝑢(𝑥)4> 0. In order to

establish the concavity of the CES ln𝑈(𝑥) we must show that the determinant in (4.A10) is

nonnegative. If 𝜌 = 1, the determinant in (4.A10) is equal to zero; if 𝜌 ≤ 0, the term in square

bracket is negative, hence the determinant is positive; if, finally 0 < 𝜌 < 1, the term in square

bracket [−𝑥12𝜌 − (2 − 𝜌)(𝑥1𝑥2)

𝜌 − 𝑥22𝜌

] is certainly negative. We may conclude that the

function is concave – strictly concave of 𝜌 < 1 - and thus also quasiconcave and strictly

quasiconcave if 𝜌 < 1.

We will later solve for the multiplier form the indirect utility function.

With regard to the properties of the CES demand functions:

1/ homogeneity of degree zero.

Assume that prices and wealth are both multiplied by a common factor 𝑡. Then

(4.A.11) 𝑥1(𝑡𝑝, 𝑡𝑤) =(𝑡𝑝1)

𝑟−1

(𝑡𝑝1)𝑟+(𝑡𝑝1)

𝑟 (𝑡𝑤) =𝑡⋅𝑡𝑟−1𝑝1

𝑟−1

𝑡𝑟(𝑝1𝑟+𝑝1

𝑟)𝑤 = 𝑥1(𝑝,𝑤)

2/ Walras Law: substituting in the budget constraint: immediate

3/ unique: immediate

4/ continuous: immediate

5/ differentiable because the utility function is strictly quasiconcave (actually strictly concave). We

then have that the CES demand functions satisfy the Law of Demand:

(4.A12) ∂𝑥1

∗

∂𝑝1= −

𝑝1𝑟−2[𝑝1

𝑟+(1−𝑟)𝑝2𝑟 ]

(𝑝1𝑟+𝑝2

𝑟)2 𝑤

Let us now determine the elasticity of substitution of the CES utility function, already defined in

equation (4.24) of this Note. Given the marginal rate of substitution 𝑀𝑅𝑆1,2 = (𝑥2

𝑥1)𝜌−1

, we have


(4.A13) 𝜎1,2 =𝑑ln(𝑥2 𝑥1⁄ )

𝑑ln𝑀𝑅𝑆1,2=

(𝑑𝑥2𝑥2

−𝑑𝑥1𝑥1

)

(𝜌−1)(𝑑𝑥1𝑥1

−𝑑𝑥2𝑥2

)=

1

1−𝜌

We conclude that the Cobb-Douglas is also a CES function with 𝜌 = 0.

Turning to the CES indirect utility function, substituting the optimal solutions (4.A7) in the

utility function (4.A1), we have

(4.A14) 𝑉(𝑝,𝑤)) = 𝑈(𝑥∗) = [[

𝑝1𝑟−1

𝑝1𝑟+𝑝2

𝑟 𝑤]𝜌

+ [𝑝2𝑟−1

𝑝1𝑟+𝑝2

𝑟 𝑤]𝜌

]

1

𝜌

=

= 𝑤(𝑝1𝑟 + 𝑝1

𝑟)−

1

𝑟

where the final equality follows from the definition 𝑟 = 𝜌/(𝜌 − 1). The indirect utility

function associated with the logarithmic transformation (4.A2) is

(4.A15) 𝑣(𝑝,𝑤) = ln𝑈(𝑥∗) = ln𝑤 −1

𝑟ln(𝑝1

𝑟 + 𝑝2𝑟)

With regard to the properties of this indirect utility function, we have

1. Homogeneous of degree zero:

(4.A16)

𝑣(𝑡𝑝, 𝑡𝑤) = ln𝑡𝑤 −1

𝑟ln [(𝑡𝑝1)

𝑟+ (𝑡𝑝2)

𝑟] =

= ln𝑡 −1

𝑟ln𝑡𝑟 + ln𝑤 −

1

𝑟ln [(𝑝1)

𝑟+ (𝑝2)

𝑟] =

= ln𝑡 −1

𝑟𝑟ln𝑡 + 𝑣(𝑝,𝑤) = 𝑣(𝑝,𝑤)

where the last step follows from the definition 𝑟 = 𝜌/(𝜌 − 1).

2. Strictly increasing in w and non increasing, actually decreasing in 𝑝: immediate.

3. Quasiconvex, actually convex, in 𝑝, given 𝑤.

Using the same approach as in Proposition 3.3, we can rewrite the CES indirect utility

function (4.A15) in terms of the vector of normalized prices 𝜋 = (𝜋1, 𝜋2) as

(4.A17) 𝑣(𝜋) = −1

𝑟ln(𝜋1

𝑟 + 𝜋2𝑟)

And note that, since the ln function is concave, -ln is convex and, therefore, quasiconvex.

Alternatively, we must verify that the Hessian matrix 𝐻(𝑣(𝑝;𝑤)) is positive semidefinite for

all 𝑤:

(4.A18) 𝐻𝑝𝑣(𝑝;⋅) =

[ 𝑝1𝑟−2[(1−𝑟)ℎ(𝑝)+𝑟𝑝1

𝑟 ]

[ℎ(𝑝)]2𝑟

(𝑝1𝑝2)𝑟−1

[ℎ(𝑝)]2

𝑟(𝑝1𝑝2)

𝜌−1

[ℎ(𝑝)]2

𝑝2𝑟−2[(1−𝑟)ℎ(𝑝)+𝑟𝑝2

𝑟 ]

[ℎ(𝑝)]2 ]


This matrix is very similar to the matrix in (4.A9) after substituting 𝑥 with 𝑝 and 𝑢(𝑥) with

ℎ(𝑝) and changing all sign from minus to plus. Note first that the leading principle minors of

all permutations are strictly positive for 𝜌 < 1. To show that the determinant of 𝐻𝑝𝑣(𝑝;⋅) is

also positive one has to perform similar algebraic simplifications as the ones previously

carried out in expression (4.A10).

4. Continuous and differentiable in 𝑝 ≫ 0 and 𝑤 > 0: immediate from the definition (

4.A15)

Given the differentiability of the indirect utility function we can determine the Lagrangean

multiplier

(4.A19) 𝜆 =∂𝑣(𝑝,𝑤)

∂𝑤=

1

𝑤

and verify Roy’s identity

(4.A20) ∂𝑣(𝑝,𝑤)

∂𝑝1= −

𝑝1𝑟−1

𝑝1𝑟+𝑝2

𝑟 = −1

𝑤

𝑝1𝑟−1

𝑝1𝑟+𝑝2

𝑟 𝑤 = −𝜆𝑥1(𝑝,𝑤)

References

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Hicks, J.R. (1939), Value and Capital, 2nd

edition 1945, Oxford, Clarendon Press

Inada, C-I (1963), “On a Two-Sector Model of Economic Growth: Comments and a

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Katzner, D. W. (1968), “A Note on the Differentiability of Consumer Demand Functions”,

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Microeconomics 1 Lecture notes - uniroma1.it · D. Tosato – Appunti di Microeconomia – Lecture Notes of Microeconomics - a.y. 2014-2015 Pagina 2 4.1. The utility maximization