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 twocommodity 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.
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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 p 0
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 ∈ ℝ𝐿+ |𝑝 ⋅ 𝑥 ≤ 𝑤}
The budget set is, therefore, a non empty, convex and compact subset of the non negative
orthant of the commodity space ℝ𝐿+ : 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
∗
= ∇𝑥 𝐿 = ∇𝑢(𝑥 ∗ ) − 𝜆∗ 𝑝 ≤ 0
𝑥 ⋅ [∇𝑢(𝑥 ∗ ) − 𝜆∗ 𝑝] = 0
𝑝 ⋅ 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.
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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
𝑢 (𝑥)
𝑝
commodities l and k 𝑀𝑅𝑆𝑙,𝑘 (𝑥 ∗ ) = 𝑢 𝑙 (𝑥) is equal to the price ratio 𝑝 𝑙 ; in the boundary solution
𝑘
we have instead 𝑀𝑅𝑆𝑙,𝑘
(𝑥 ∗ )
𝑝𝑙
>𝑝 .
𝑘
𝑘
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).
𝑥2
𝑥2
∇𝑢(𝑥 ∗ ) = 𝜆 𝑝
p
(𝑥∗ ) = x(p, w)
(𝑥∗ ) = x(p, w)
𝜆∗ 𝑝
∇𝑢(𝑥 ∗ )
𝑥1
B(p, w)
𝑥1
B(p, w)
Fig. 4.1 Panel (a) – Interior solution
the UMP
Fig. 4.1 (Panel b) – Boundary solution of
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)
1
2
𝑥∗ = 𝑥(𝑝, 𝑤)
𝜆∗ = 𝜆(𝑝, 𝑤)
See Lecture Note 2, Section 2.4 “The role of concavity and quasiconcavity in optimization problems”.
It is useful to remember that the marginal rate of substitution was defined in Lecture Note 1 as a positive quantity.
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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 multivalued 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) ∈ (ℝ𝐿++ , 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.
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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 ∈ ℝ𝐿++ .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 l  1,..., L , a proportionate change in  p, w 
5
If the solution is a multi-valued correspondence, then the set of solutions is convex.
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 xL  xL ( x1..., xL 1 ; p, w) expressing commodity xL 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
6
Hessian determinant of the indifference curve xL  ;  is positive definite for all x   x1 ,..., xL 1  , 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 L 1 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  .
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L
(4.6)
 pk
k 1
xl  p, w  xl  p, w 

0
pk
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)
∑𝐿𝑙=1 𝑝𝑙
∂𝑥𝑙 (𝑝,𝑤)
∂𝑤
(4.8)
∑𝐿𝑙=1 𝑝𝑙
∂𝑥𝑙 (𝑝,𝑤)
+ 𝑥𝑘 (𝑝, 𝑤)
∂𝑝𝑘
=1
=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)
∂𝑥2 (𝑝,𝑤)
+
𝑝
2
∂𝑝1
∂𝑝1
∂𝑥1 (𝑝,𝑤)
∂𝑥2 (𝑝,𝑤)
𝑥2 (𝑝, 𝑤) + 𝑝1 ∂𝑝 + 𝑝2 ∂𝑝
2
2
𝑥1 (𝑝, 𝑤) + 𝑝1
=0
=0
Generalizing to 𝐿 > 2 and in a compact matrix notation (4.9) can be rewritten as
(4.10)
𝑇
𝑥(𝑝, 𝑤) + [𝐷𝑝 𝑥(𝑝, 𝑤)] 𝑝 = 0
where
(4.11)
∂𝑥1 (𝑝,𝑤)
∂𝑥1 (𝑝,𝑤)
.
.
.
∂𝑝1
∂𝑝𝐿
[𝐷𝑝 𝑥(𝑝, 𝑤)] = [. . . . . .
∂𝑥𝐿 (𝑝,𝑤)
∂𝑥 (𝑝,𝑤)
. . . 𝐿∂𝑝
∂𝑝1
𝐿
]
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
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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)
ll 
 ln xl
 ln pl
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)
lk 
 ln xl
 ln pk
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 L  2 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.
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The elasticity of demand with respect to wealth (income)
(4.14)
lw 
 ln xl
 ln 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)
wl 
pl xl  p, w 
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’)
lk  lw  0
k
(4.7’)
 wllw  1
l
(4.8’)
 wkkl  wl  0
for l  1,..., 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
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4.3.1 Cobb-Douglas utility function
Assume that preferences are represented by the Cobb-Douglas utility function
(4.16)
𝛽
𝑈(𝑥1 , 𝑥2 ) = 𝑥1𝛼 𝑥2 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 ) = 𝑡 𝛼+𝛽 𝑈(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)
𝛼
− 𝜆𝑝1
𝑥1
𝛽
𝜆𝑝2 = 𝑥 − 𝜆𝑝2
2
∗
𝑢1 (𝑥) − 𝜆𝑝1 =
≤0
𝑢2 (𝑥) −
≤0
∗ [∇
𝑥 𝑥 𝑢(𝑥) − 𝜆 𝑝] = 0
𝑝 ⋅ 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 wealthconsumption expansion path
(4.20)
𝛼𝑝
𝑥1 = 𝛽 𝑝2 𝑥2
1
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)
𝛼
𝑤
𝛽
𝑤
𝑥1∗ = 𝛼+𝛽 𝑝 , 𝑥2∗ = 𝛼+𝛽 𝑝 , 𝜆∗ =
1
2
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.
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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)
𝛼
𝑤
𝛽
𝑤
𝑝1 𝑥1 ∗ + 𝑝2 𝑥2 ∗ = 𝑝1 (𝛼+𝛽 𝑝 ) + 𝑝2 [𝛼+𝛽 𝑝 ] = 𝑤
1
2
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)
𝛼
− 𝑥2
𝛼
0
𝛽
𝑥1
𝛽
2
𝑥2
1
(4.23)
𝑑𝑒𝑡𝐻 𝐵 (𝑢(𝑥)) = det 0
is strictly positive for all x 
2

− 𝑥2
𝛼
𝛽
[𝑥 1
𝑥2

 2
x12 x22

 2
x12 x22
    

x12 x22
0]
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
𝛼𝑥
demanded and the rate of change of the marginal rate of substitution 𝑀𝑅𝑆1,2 = 𝛽 𝑥2 , which is
1
the ratio of the marginal utilities.12 We obtain
(4.24)
𝜎1,2 =
𝑑𝑥
𝑑𝑥
𝑥2
𝑥1
2
1
𝑑ln(𝑥2 ⁄𝑥1 ) ( 𝑥2 − 𝑥1 )
=
𝑑ln𝑀𝑅𝑆1,2 (𝑑𝑥2 −𝑑𝑥1 )
=1
Let us take a closer look at some implications of the Cobb-Douglas demand functions (4.21).
The optimal expenditure on commodity l - pl xl* , l  1, 2 - 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:  lk  1 identifies a relation of substitution, while
 lk  1 indicates a relation of complementarity in consumption.
D. Tosato – Appunti di Microeconomia – Lecture Notes of Microeconomics - a.y. 2014-2015 Pagina 10
pl xl*
as a consequence, the expenditure shares wl 
are constants, independent of prices and
w
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)
𝛽
𝑢(𝑦) = 𝑦1𝛼 𝑦2
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)
𝛼 𝑤′
𝛽 𝑤′
1
1
𝑦1∗ = 𝛼+𝛽 𝑝 , y2∗ = 𝛼+𝛽 𝑝
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.
D. Tosato – Appunti di Microeconomia – Lecture Notes of Microeconomics - a.y. 2014-2015 Pagina 11
(4.29)
𝛼 𝑤−𝑝⋅𝛾
, 𝑥2∗
𝑝1
𝑥1∗ = 𝛾1 + 𝛼+𝛽
𝛽 𝑤−𝑝⋅𝛾
𝑝2
= 𝛾2 + 𝛼+𝛽
Through the subsistence expenditure 𝑝 ⋅ 𝛾 all demand functions now depend on all prices. We
x1*
  2 

have, for instance,
   . This means that an increase in the cost of the
p2     p1 
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)
pl xl  pl  l  al  w  p    for l  1,..., L
where the al 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’”.
D. Tosato – Appunti di Microeconomia – Lecture Notes of Microeconomics - a.y. 2014-2015 Pagina 12
(4.32)
𝐵(𝑝, 𝑤) = {(𝑥1 , 𝑥2 ) ∈ ℝ2+ |𝑝1 𝑥1 + 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.
𝑥2
𝑥1
𝑥1
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(𝑝1 𝑥1 + 𝑥2 − 𝑤)
The critical values are determined by the solution of the following set of relations
∂𝐿(𝑥1 ,𝑥2 ,𝜆)
(4.34)
∂𝑥1
∂𝐿(𝑥1 ,𝑥2 ,𝜆)
∂𝑥2
∂𝐿(𝑥1 ,𝑥2 ,𝜆)
∂𝜆
∗
= 𝑢′(𝑥1 ) − 𝜆∗ 𝑝1 ≤ 0 with equality if 𝑥1∗ > 0
= 1 − 𝜆∗ ≤ 0 with equality if 𝑥2∗ > 0
= 𝑝1 𝑥1∗ + 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).
D. Tosato – Appunti di Microeconomia – Lecture Notes of Microeconomics - a.y. 2014-2015 Pagina 13
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
>
𝑤 − 𝑝1 𝑥1∗ { } 0
≤
(4.35)
They correspond to interior solutions and boundary solutions, respectively.
If 𝑤 − 𝑝1 𝑥1∗ > 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
𝑥1 ∗ = 𝑢′−1 (𝑝1 ), 𝑥2 ∗ = 𝑤 − 𝑝1 𝑥1 ∗ , 𝜆∗ = 1
(4.36)
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 − 𝑝1 𝑥1∗ ≤ 0, we have a boundary solution with the wealth
constraint implying 𝑥2∗ = 0; the entire wealth is then used for the consumption of commodity
𝑤
1, 𝑥1 ∗ = 𝑝 . This has an important implication for the value of the multiplier 𝜆∗ . Suppose that
1
w
p1 x1*
 0 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 − 𝑝1 𝑥1∗ { } 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
A numerical example may help. Let 𝑢(𝑥1 ) = ln𝑥1 so that 𝑢′(𝑥1 ) = 1⁄𝑥1 . Consider the boundary solution 𝑥1∗ = 𝑤⁄𝑝1 .
Substituting in the first order condition we obtain 𝜆∗ = 1⁄𝑤 which clearly exhibits the inverse relation between the
marginal utility of wealth and its quantity.
16
D. Tosato – Appunti di Microeconomia – Lecture Notes of Microeconomics - a.y. 2014-2015 Pagina 14
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)
−[(𝑥1 )−2 ] 0 (𝑥1 )−1
𝐻 𝐵 (𝑥1 , 𝑥2 ) = [0
]
0 1
(𝑥1 )−1
1 0
The function is strictly quasiconcave since
(4.40)
(−1)2
−[(𝑥1 )−2 ] 0 (𝑥1 )−1
[0
] = (−1)2 (𝑥1 )−2 > 0
0 1
(𝑥1 )−1
1 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
D. Tosato – Appunti di Microeconomia – Lecture Notes of Microeconomics - a.y. 2014-2015 Pagina 15
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 ∈ ℝ𝐿++ × ℝ+ |𝑣(𝑝, 𝑤) ≤ 𝑣̅ }
is quasiconvex for any 𝑣̅ .
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
1
unaltered. Let us then take as our proportionality factor 𝑤. Dividing all the terms of the wealth
𝑝
𝑝
constraint by 𝑤 and defining the normalized price vector 𝜋 = (𝜋1 , 𝜋2 ) = ( 𝑤1 , 𝑤2 ), 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 𝑣̅ .
Let 𝜋, 𝜋′ ∈ 𝐼 − (𝜋) and assume 𝑣(𝜋; 1) = 𝑣(𝜋′; 1) = 𝑣̅ . We must show that their convex
combination 𝜋′′ = 𝛼𝜋 + (1 − 𝛼)𝜋′satisfies the definition of quasiconvexity 𝑣(𝛼𝜋 +
(1 − 𝛼)𝜋′; 1) ≤ 𝑣̅ . 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(𝜋 0 ) = 𝑣̅ 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) ≤ 𝑣̅ is in the region above. 𝑣(𝜋; 1) is, therefore,
quasiconvex and so is 𝑣(𝑝, 𝑤) for all 𝑤 > 0.
D. Tosato – Appunti di Microeconomia – Lecture Notes of Microeconomics - a.y. 2014-2015 Pagina 16
𝜋2
𝜋0
𝜋
𝜋’’
 
I 0
v( )
𝜋’
 
I 0 v
𝑥1
𝜋1
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) ≤ 𝑣̅ and 𝑣(𝑝′, 𝑤′) ≤ 𝑣̅ . 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 𝑢̅ = 𝑣̅ . 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 𝑢̅ = 𝑣̅ .17 We conclude, on the basis of the definition of
quasiconvexity, that the set (4.42) is quasiconvex.
𝑥2
x( p, w)
x( p' , w)
u v
B( p, w)
B( p' , w)
B( p' ' , w' ' )
17
𝑥1
With 𝛼 = 1, 𝑥′′ coincides with 𝑥and with 𝛼 = 0, 𝑥′′ coincides with 𝑥′. In these cases, 𝑣(𝑝′′, 𝑤′′) = 𝑣̅ .
D. Tosato – Appunti di Microeconomia – Lecture Notes of Microeconomics - a.y. 2014-2015 Pagina 17
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)
(4.44)
∂𝑣(𝑝,𝑤)
∂𝑤
∂𝑣(𝑝,𝑤)
∂𝑝𝑙
=𝜆
= −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
∂𝑥
∂𝑥
= 𝜆𝑝1 ∂𝑤1 + 𝜆𝑝2 ∂𝑤2 = 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
= ∑𝐿𝑘=1
∂𝑢(𝑥) ∂𝑥𝑘
∂𝑥𝑘 ∂𝑝𝑙
∂𝑥
= 𝜆 ∑𝐿𝑘=1 𝑝𝑙 ∂𝑝𝑘 = −𝜆𝑥𝑙 (𝑝, 𝑤)𝑙 = 1, . . . , 𝐿
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.
D. Tosato – Appunti di Microeconomia – Lecture Notes of Microeconomics - a.y. 2014-2015 Pagina 18
4.5.1 Cobb-Douglas indirect utility function
𝛼
𝑤
𝛽
𝑤
Substituting the demand functions (4.10) 𝑥1∗ = 𝛼+𝛽 𝑝 and𝑥2∗ = 𝛼+𝛽 𝑝 in the Cobb-Douglas
1
2
utility function (4.7) 𝑢(𝑥1 , 𝑥2 ) = 𝛼ln𝑥1 + 𝛽ln𝑥2 , we obtain the indirect utility function
(4.47)
𝛼
𝛽
𝑣(𝑝, 𝑤) = [𝛼ln 𝛼+𝛽 + 𝛽ln 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 ∇𝑝 𝑣(𝑝; 𝑤) ⋅ 𝑧 = 0. Since the Bordered
Hessian 𝐻 𝐵 (𝑣(𝑝; 𝑤)) = −𝐻 𝐵 (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)
(4.49)
∂𝑣(𝑝,𝑤)
∂𝑤
∂𝑣(𝑝,𝑤)
∂𝑝1
=
𝛼+𝛽
𝑤
=𝜆
𝛼
𝛼 𝛼+𝛽 𝑤
= −𝑝 = −𝑝
1
1
=−
𝛼+𝛽 𝑤
𝛼+𝛽
𝑤
𝛼
𝑤
(𝛼+𝛽 𝑝 ) = −𝜆𝑥1 (𝑝, 𝑤)
1
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
D. Tosato – Appunti di Microeconomia – Lecture Notes of Microeconomics - a.y. 2014-2015 Pagina 19
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)
𝑝
𝑤
𝑥1 ∗ = 𝑥1 (𝑝) = 𝑝2 , 𝑥2 ∗ = 𝑥2 (𝑝, 𝑤) = 𝑝 − 1, 𝜆∗ = 2
1
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)
𝑝
𝑤
𝑣(𝑝, 𝑤) = ln 𝑝2 + 𝑝 − 1
1
2
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
𝑤
1
𝑤
= 𝑝 − 𝑝2 = 𝑝 (1 − 𝑝 )
2
2
2
2
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 ∇𝑝 𝑣(𝑝; 𝑤) ⋅ 𝑧 = 0. Since the Bordered
Hessian is
1
(4.54)
1
0
𝑝12
1
𝐵
𝐻 (𝑣(𝑝; 𝑤)) = 0
𝑝22
1
[− 𝑝1
1
𝑝2
−𝑝
1
𝑤
(−1 + 𝑝 )
2
(−1 +
𝑤
𝑝2
)
1
𝑝22
𝑤
(−1 + 𝑝 )
2
0
]
We have
(4.55)
1 2 1
det𝐻 𝐵 (𝑣(𝑝; 𝑤)) = − (𝑝 )
1
𝑤
1 2 1
(−1 + 𝑝 ) − (𝑝 )
𝑝2
2
2
1
𝑤 2
(−1 + 𝑝 ) < 0
𝑝3
2
2
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.
D. Tosato – Appunti di Microeconomia – Lecture Notes of Microeconomics - a.y. 2014-2015 Pagina 20
Given the differentiability of the indirect utility function we can verify the properties (4.39)
and (4.40). We have
∂𝑣(𝑝,𝑤)
(4.56)
∂𝑤
∂𝑣(𝑝,𝑤)
(4.57)
∂𝑝1
=1=𝜆
1
= − 𝑝 = −𝜆𝑥1 (𝑝1 )
1
and from (4.53)
v  p, w  1 
w
*

1       x2  p, w  
p2
p2  p2 
(4.58)
Appendix. 4.A The Constant Elasticity of substitution (CES) utility function
Assume that strictly monotone and convex preferences are represented by the utility function
𝜌
𝜌
𝑈(𝑥) = (𝑥1 + 𝑥2 )
(4.A1)
1⁄
𝜌
with 0 ≠ 𝜌 ≤ 1.
𝑥2
𝑥2
𝑥2
lim   0
 1
𝑥1
Panel (a): 𝝆 = 𝟏, 𝝈 = +∞
lim   
𝑥1
Panel (b): 𝝆 = 𝟎, 𝝈 = 𝟏
𝑥1
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,
D. Tosato – Appunti di Microeconomia – Lecture Notes of Microeconomics - a.y. 2014-2015 Pagina 21
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
1
subsequently shown, related to the parameter 𝜌 by the equality 𝜎 = 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.
    1
1   0
𝑥2
 0
 1
  
𝑥1
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)
1
𝜌
1
𝜌
ln𝑈(𝑥) = 𝜌 ln(𝑥1 + 𝑥2 ) = 𝜌 ln𝑢(𝑥)
1⁄
𝜌
𝜌
𝜌
Considering the more general form of CES function 𝑈(𝑥) = (𝛼𝑥1 + (1 − 𝛼)𝑥2 )
𝜌
𝜌
ln(𝛼𝑥1 + (1 − 𝛼)𝑥2 )
ln𝑈(𝑥) =
𝜌
Using L’Hopital’s rule, we have
20
𝜌
ln𝑈(𝑥) = lim
𝜌→0
𝜌
ln(𝛼𝑥1 +(1−𝛼)𝑥2 )
𝜌
𝜌
= lim
𝜌→0
𝜌
𝛼𝑥1 ln𝑥1 +(1−𝛼)𝑥2 ln𝑥2
𝜌
𝜌
(𝛼𝑥1 +(1−𝛼)𝑥2 )
and taking logarithms we have
= 𝛼ln𝑥1 + (1 − 𝛼)ln𝑥2 = ln𝑥1𝛼 𝑥21−𝛼
whence 𝑈(𝑥) = 𝑥1𝛼 𝑥21−𝛼 which is the Cobb-Douglas that was examined in the preceding section.
D. Tosato – Appunti di Microeconomia – Lecture Notes of Microeconomics - a.y. 2014-2015 Pagina 22
𝜌
𝜌
where 𝑢(𝑥) = (𝑥1 + 𝑥2 ). With 0 ≠ 𝜌 < 1, the UMP has an interior solution so that the Lagrangean
associated with the problem is
1
𝐿(𝑥, 𝜆) = 𝜌 ln𝑢(𝑥) − 𝜆(𝑝 ⋅ 𝑥 − 𝑤)
(4.A3)
Following Lagrange’s method, the critical values of 𝑥1 and 𝑥2 are determined as part of the solution of
the following set of relations
∂𝐿
∂𝑥1
1 ∂ln𝑢(𝑥)
− 𝜆𝑝1
∂𝑥1
=𝜌
𝑥
𝜌−1
= 𝑢(𝑥)(11/𝜌)−1 − 𝜆𝑝1 = 0
𝜌−1
𝑥
∂𝐿
1 ∂ln𝑢(𝑥)
= 𝜌 ∂𝑥 − 𝜆𝑝2 = 𝑢(𝑥)(21/𝜌)−1
∂𝑥2
2
∂𝐿
= 𝑝1 𝑥1 + 𝑝2 𝑥2 − 𝑤 = 0
∂𝜆
(4.A.4)
− 𝜆𝑝2 = 0
Eliminating 𝜆 from the first two conditions and rearranging terms, we have
𝑥1 =
(4.A5)
1
𝑝1 ⁄𝜌−1
(𝑝 )
𝑥2
2
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 𝑥∗2 back into (4.A5):
𝑥∗1 =
1⁄
𝜌−1
𝜌
𝑝1
⁄𝜌−1
𝑝1
(4.A6)
𝑥∗2
𝜌
⁄𝜌−1
𝑤
+𝑝2
1⁄
𝜌−1
=
𝜌
𝑝2
⁄𝜌−1
𝑝1
𝜌
⁄𝜌−1
𝑤
+𝑝2
Letting 𝑟 = 𝜌/(𝜌 − 1), we can simplify the notation and rewrite (4.A6) as
𝑝𝑟−1
𝑥∗1 = 𝑝𝑟1+𝑝𝑟 𝑤
1
(4.A7)
𝑥∗2 =
2
𝑝𝑟−1
2
𝑤
𝑝𝑟1 +𝑝𝑟2
Let us not prove that the CES function (4.A2) is concave. The first order derivatives are
𝑥
𝜌−1 𝜌−1
𝑥
∇𝑥 ln𝑈(𝑥)𝑇 = [ 𝑢1(𝑥) 𝑢2(𝑥) ]
(4.A8)
The Hessian matrix is therefore
21
𝑝
1/(𝜌−1)
We obtain 𝑝1 𝑥2 ( 1)
𝑝2
𝜌/(𝜌−1)
+ 𝑝2 𝑥2 = 𝑥2 (𝑝1
𝜌/(𝜌−1)
+ 𝑝2
−1/(𝜌−1)
) 𝑝2
=𝑤
D. Tosato – Appunti di Microeconomia – Lecture Notes of Microeconomics - a.y. 2014-2015 Pagina 23
𝜌−2
−
(4.A9)
𝜌𝑥1
𝜌
[(1−𝜌)𝑢(𝑥)+𝜌𝑥1 ]
[𝑢(𝑥)]2
𝐻ln𝑈(𝑥) = [
−𝜌
(𝑥1 𝑥2
)𝜌−1
−𝜌
−
[𝑢(𝑥)]2
(𝑥1 𝑥2 )𝜌−1
[𝑢(𝑥)]2
]
𝜌−2
𝜌
𝜌𝑥2 [(1−𝜌)𝑢(𝑥)+𝜌𝑥2 ]
2
[𝑢(𝑥)]
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) − 𝜌𝑥𝜌1 ][(𝜌 − 1)𝑢(𝑥) − 𝜌𝑥𝜌2 ] − 𝜌(𝑥1 𝑥2 )𝜌 =
= (𝜌 − 1)[(𝜌 − 1)𝑢(𝑥)2 − 𝜌𝑢(𝑥)(𝑥𝜌1 + 𝑥𝜌2 ) + 𝜌(𝑥1 𝑥2 )𝜌 ] =
= (𝜌 − 1)[(𝜌 − 1)𝑢(𝑥)2 − 𝜌𝑢(𝑥)2 + 𝜌(𝑥1 𝑥2 )𝜌 ] =
2
= (𝜌 − 1) [−(𝑥𝜌1 + 𝑥𝜌2 ) + 𝜌(𝑥1 𝑥2 )𝜌 ]
where the proportionality factor eliminated in the first line is 𝜌2
(𝑥1 𝑥2 )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
2𝜌
𝜌
bracket [−𝑥2𝜌
1 − (2 − 𝜌)(𝑥1 𝑥2 ) − 𝑥2 ] 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
𝑟−1
(4.A.11)
𝑥1 (𝑡𝑝, 𝑡𝑤) =
(𝑡𝑝1 )
𝑟
𝑡𝑤) =
𝑟(
(𝑡𝑝1 ) +(𝑡𝑝1 )
𝑡⋅𝑡𝑟−1 𝑝𝑟−1
1
𝑤
𝑡𝑟 (𝑝𝑟1 +𝑝𝑟1 )
= 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)
∂𝑥∗1
∂𝑝1
=−
𝑟
𝑟
𝑝𝑟−2
1 [𝑝1 +(1−𝑟)𝑝2 ]
2
(𝑝𝑟1 +𝑝𝑟2 )
𝑤
Let us now determine the elasticity of substitution of the CES utility function, already defined in
𝜌−1
𝑥
equation (4.24) of this Note. Given the marginal rate of substitution 𝑀𝑅𝑆1,2 = (𝑥2 )
1
, we have
D. Tosato – Appunti di Microeconomia – Lecture Notes of Microeconomics - a.y. 2014-2015 Pagina 24
(4.A13)
𝑑ln(𝑥2 ⁄𝑥1 )
𝑑ln𝑀𝑅𝑆1,2
𝜎1,2 =
=
𝑑𝑥
𝑑𝑥
( 2− 1 )
𝑥2
𝑥1
𝑑𝑥1 𝑑𝑥2
(𝜌−1)( − )
𝑥1
𝑥2
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∗ ) =
𝑝𝑟−1
[[ 𝑟1 𝑟 𝑤]
𝑝1 +𝑝2
𝜌
1
𝑝𝑟−1
+ [𝑝𝑟2+𝑝𝑟 𝑤]
1
2
𝜌 𝜌
] =
1
−𝑟
= 𝑤(𝑝𝑟1 + 𝑝𝑟1 )
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)
1
𝑣(𝑝, 𝑤) = ln𝑈(𝑥 ∗ ) = ln𝑤 − ln(𝑝1𝑟 + 𝑝2𝑟 )
𝑟
With regard to the properties of this indirect utility function, we have
1.
Homogeneous of degree zero:
𝑟
1
𝑟
𝑣(𝑡𝑝, 𝑡𝑤) = ln𝑡𝑤 − 𝑟 ln [(𝑡𝑝1 ) + (𝑡𝑝2 ) ] =
(4.A16)
1
1
𝑟
𝑟
= ln𝑡 − 𝑟 ln𝑡𝑟 + ln𝑤 − 𝑟 ln [(𝑝1 ) + (𝑝2 ) ] =
1
= ln𝑡 − 𝑟 𝑟ln𝑡 + 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)
1
𝑣(𝜋) = − 𝑟 ln(𝜋1𝑟 + 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)
𝐻𝑝 𝑣(𝑝;⋅) =
𝑟
𝑝𝑟−2
1 [(1−𝑟)ℎ(𝑝)+𝑟𝑝1 ]
2
[ℎ(𝑝)]
𝜌−1
[𝑟
(𝑝1 𝑝2 )
[ℎ(𝑝)]2
𝑟−1
𝑟
(𝑝1 𝑝2 )
[ℎ(𝑝)]2
𝑟
𝑝𝑟−2
2 [(1−𝑟)ℎ(𝑝)+𝑟𝑝2 ]
2
]
[ℎ(𝑝)]
D. Tosato – Appunti di Microeconomia – Lecture Notes of Microeconomics - a.y. 2014-2015 Pagina 25
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 𝐻𝑝 𝑣(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
𝑝𝑟−1
1 𝑝𝑟−1
= − 𝑝𝑟1+𝑝𝑟 = − 𝑤 𝑝𝑟1+𝑝𝑟 𝑤 = −𝜆𝑥1 (𝑝, 𝑤)
1
2
1
2
References
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