Consumer Theory[edit]
The aim of this section is to explain a fundamental problem in economics, the derivation of a
consumer’s demand function, in a very simple way. The article is organized as follows:

Conceptual review of assumptions in demand theory

Description of the Utility Maximization Problem

Derivation of the Expenditure Minimization Problem

Relationship between both problems
Assumptions[edit]
The consumer theory assumes that the consumer is rational. This implies that his preferences
satisfy the following properties: 1. They are complete; that is, given any set of possible bundles of
goods, the consumer is always capable of deciding which one is preferable to the others and then
ranking them in terms of preference.
2. They are reflexive; it means that any bundle is at least as good as itself.
3. They are transitive; meaning that if a bundle
preferable to a third bundle
bundle
is preferred to a bundle
, then it is implied that the first bundle
, and this bundle
is
will be preferred to the
.
4. They are continuous; there are no big jumps in the ranking of alternatives.
The fulfillment of these properties ensures that consumer’s preferences are consistent and can be
represented by an utility function,
such that if bundle
is preferred to bundle
,
then
The locus of all bundles that give a certain level of utility to the consumer constitutes an indifference
curve (or level curve), which is the usual way of representing preferences. Nevertheless, in spite of
these four properties, there is still the possibility of having “special cases” such as the existence of
perfect substitutes or perfect complements, among others, which lead to special shapes for the
indifference curves. For avoiding these cases, two additional properties are assumed:
5. Preferences are monotonic, or “more is preferred to less”; this implies that, given any set of two
bundles, if one of them contains at least as much of all goods and more of one good than the other,
then the first bundle will be preferred to the second.
6. Preferences are convex; that is, any combination of two equally preferable bundles will be more
desirable than these bundles.
These five properties confer a special shape to level curves: they are downward slopping and
convex.
Utility Maximization Problem[edit]
This section develops the Utility Maximization Problem (UMP) for the simplest case of only two
goods. The model can easily be generalized to
Assume that there are two goods,
and
consumer has a fixed amount of income,
goods.
, whose prices are
and
, respectively. The
, for spending on consumption, and his preferences are
represented by a generic utility function,
, with
. The consumer’s
,
aim is to obtain the maximum possible utility but he is constrained by his level of income. He cannot
spend more than
1
, thus he faces a budget constraint:
Formally, the problem can be formulated as follows:
Max
subject to
And it can be solved by the Lagrange Multipliers method:
Max
The first order conditions (FOC) are:
Note that conditions
and
imply that
. That is, the marginal rate of substitution
(MRS) must be equal to the relation of prices, and it means that the indifference curve must be
tangent to the budget constraint.
The second order conditions (SOC) are:
It can be demonstrated that the SOC imply that indifference curves are convex. The reciprocal is
true only for the case of two goods.
The solutions to the FOC are
written as
,
,
,
.They depend on prices and income, thus they can be
,
.The functions
the Marshallian Demand Functions. They represent the amount of goods
and
and
,< are
, that the
consumer is willing to purchase given their prices, income and tastes.
Another concept that emerges from the UMP is the Indirect Utility Function, and it can be obtained
by replacing the Marshallian demands into the utility function. By definition, it also is a function of
prices and income, then it can be written
as
. Intuitively, it represents the
maximum utility that the consumer can achieve for any given values of
.
Note that, because of the Envelop Theorem, it must be the case that
. It
implies that the Lagrange multiplier can be thought as the marginal utility of income. That is, it
represents the rate of change of the maximum utility that is derived from an infinitesimal rise in
income.
1 Striclty, the constraint is
, but the monotonicity assumption ensures that he will spend all his income.
Expenditure Minimization Problem[edit]
The Expenditure Minimization Problem (EPM) is the dual problem of the UMP and it can be thought
as follows. Consider a consumer who gets utility through the consumption of the two goods. In this
case, there is no restriction on the income to be spent, but the consumer must be on a certain level
curve,
. Given this constraint, his objective is to reach this indifference curve with the minimum
possible expenditure. Therefore, the problem is:
Min
subject to
Again, this constrained optimization can be solved by the Lagrange Multipliers method:
Min
The FOC of this program are:
And the SOC are:
Note that conditions
and
imply the same tangency condition than the UMP:
. In
this program, it means that the expenditure function must be tangent to the indifference curve
Solving equations
functions
to
and
gives the optimal levels of
,
,
.
.The demand
are the Hicksian (or Compensated) Demand Functions. Note that these
demands depend on prices and the utility level, therefore they are denoted
,
,
.
The function resulting from replacing the Hicksian demands into the expenditure function gives the
minimum expenditure necessary to reach
for any given values of
. It is called
the Indirect Expenditure Function and is
denoted
.
Again, the Lagrange multiplier has a special interpretation. The Envelop Theorem implies
that
, meaning that the Lagrange multiplier represents the rate of
change of the expenditure function given a change in the utility level to reach.
Relationship Between UMP and EMP[edit]
References[edit]
Nicholson, W "Microeconomic Theory"
Chiang, A. Métodos fundamentales de economía matemática. Mc Graw-Hill. 2006.
Silberberg, E. The Structure of Economics. McGraw Hill, 3rd Edition. 2000.
Varian, H. Microeconomía intermedia. Antoni Bosch, 5ta edición. 2004.
Varian, H. Microeconomic Analysis. W.W. Norton, 3rd Edition. 1992.
Marshallian demand function
In microeconomics, a consumer's Marshallian demand function (named after Alfred Marshall)
specifies what the consumer would buy in each price and wealth situation, assuming it perfectly
solves the utility maximization problem. Marshallian demand is sometimes called Walrasian
demand (named after Léon Walras) or uncompensated demand functioninstead, because the
original Marshallian analysis ignored wealth effects.
According to the utility maximization problem, there are L commodities with prices p. The consumer
has wealth w, and hence a set of affordable packages
where
function
is the inner product of the prices and quantity of goods. The consumer has a utility
The consumer's Marshallian demand correspondence is defined to be
If there is a unique utility maximizing package for each price and wealth situation, then it is called
the Marshallian demand function. See the utility maximization problem entry for a discussion of
this definition.
Marshallian demand function Example
If there are two commodities, and the consumer has the
utility function
(Cobb-Douglas), he chooses to spend half of its income on
each commodity, and its Marshallian demand function is the following:
In general, an agent with Cobb-Douglas preferences, given by
constant share of his income in order to buy each of the two commodities, as follows:
If we are in a more general case, i.e.
, will use a
, we have:
Marshallian Demand Functions : Some Examples
http://dept.econ.yorku.ca/~sam/4000/eg_demand.html
i Fixed Coefficients
If the utility function is
u(x) = min (ax1,bx2)
then either u/x1 = 0 ( if ax1bx2 ), or u/x2 = 0 ( if ax1bx2 ), or the utility
function is not differentiable ( if ax1 = bx2 ).
If the price of each good is positive, then the consumer's optimal choice of
consumption bundle x*(p1,p2,M) must be located at the kink in the L-shaped
indifference curve, at which ax1 = bx2. [Why? If ax1 bx2, for example, then the firstorder conditions would be u1 = 0 = p1 and u2 = b = p2 : the first condition says  = 0
and the second condition says 0. Or, more intuitively, why waste money on
good 1, when added consumption of the good does not make you any better off? ]
So the Marshallian demands satisfy the condition ax1 = bx2, and the budget line
equation p1x1 + p2x2 = M. The first condition says that
a
x2 = x1
b
Substituting this definition in the equation of the budget line,
a
p1x1 + p2
x2 = M
b
or
b
x1 =
M
p1 b + p2 a
which is the Marshallian demand function for good # 1. Therefore the Marshallian
demand function for good #2 is
a
x2 =
M
p1b + p2a
ii Perfect Substitutes
If
u(x) = a1x1 + a2x2 + anxn
then
u
xi
= ai
for each good i. The marginal rate of substitution between any two goods i and j is
ai/aj, which is a constant, independent of the quantities consumed of the goods. The
indifference curves between any two goods are straight lines. Mathematically, the
first-order conditions
u
x1
= a1 = p1
and
u
x2
= a2 = p2
could both hold only if a1/a2 = p1/p2, which would happen by coincidence. Usually, the
consumer will choose to be at a corner solution, spending all her money on the good i
for which ai/pi is highest. That is, if
a1/p1 a2/p2 an/xn
for example, then she will choose
M
x1 =
p1
and xi = 0 for every i 1. Those would be her Marshallian demands. Only if there
was a tie, so that, for example
a1
a2
=
p1
p2

a3

p3
an
pn
would she choose to consume positive quantities of more than one good. In this case,
the slope of her indifference curve between x-1 and x2 equals the slope of her budget
line. Her Marshallian demands are not unique : any (x1,x2,x3,,xn) with p1x1 + p2x2 =
M, and x3 = x4 =  = xn = 0 would be tied for most preferred among the bundles
which she could afford.
iii Quasi-Linear Preferences
(iiia)
__
u(x1,x2,x3) = x1 + 2 x2 + lnx3
In this case, the three first-order conditions are
1 = p1
(1)
1
__ = p2
x2
1
= p3
(2)
(3)
x3
Equation (1) can be used to substitute for  in equations (2) and (3) :
1
p2
=
x2
(2)
p1
1
p3
=
x3
(3)
p1
which can be re-arranged into
p1
)2
x2 = (
(2)
p2
p1
x3 =
(3)
p3
which are the Marshallian demand functions for goods #2 and #3.
Since
M - p2x2 - p3x3
x1 =
p1
therefore
M p1
x1 = - - 1
p1 p2
is the Marshallian demand function for good #1. This expression is only correct if the
person's income M is high enough so that M (p1)2/p2 - p1 ; otherwise x1 would be
negative. [ left to the reader : what would Marshallian demands be if income M were
lower than this? ]
In this example, the quantities demanded of goods 2 and 3 were independent of the
person's income M. Increases in income are all spent on good #1. This property holds
whenever a person has quasi-linear preferences. If
u(x) = x1 + f(x2,x3,,xn)
then the first-order conditions to the consumer's utility maximization problem are
1 = p1
f
xi
= pi
i = 2,,n
If the first equation is used to substitute 1/p1 for  in the remaining n-1 equations, then
the first-order conditions for x2,x3,,xn are n-1 equations in the n-1 unknowns
x2,x3,,xn. That means they can be solved without reference to the budget condition,
or to the income level M. This property holds only if preferences are quasi-linear.
(iiib)
u(x1,x-2,x3) = x1 + lnx2 +ln(x2+x3)
Here the first-order condition on x1 again implies that
1
=
p1
so that the other two first-order conditions can be written
1
1
p2
+
x2
=
(2)
x2 + x3
1
p1
p3
=
x2 + x3
(3)
p1
Substitution of equation (3) into equation (2) yields
1
=
x2
p2 p3
p1 p1
(2)
or
p1
x2 =
(2)
p2-p3
which is the Marshallian demand function for good #2. Since equation (3) can be
written
p1
x2 + x3 =
(3)
p3
then equation (2) implies that
p1(p2-2p3)
x3 =
(3)
p3(p2-p3)
which is the Marshallian demand function for good #3. Substitution into the budget
constraint then implies
M
x1 =
-2
p1
is the Marshallian demand function for good #1. These demand functions are only
valid when M2p1 and when p22p3. [ left to the reader : what happens when
these inequalities are not satisfied? ]
Here quantity demanded of good #3 depends on all three prices : but quasi-linearity
implies that quantity demanded of good #2 and of good #3 is independent of income
M.
iv Cobb-Douglas Preferences
If
u(x1,x2,,xn) = x1a1x2a2 xnan
then it is simplest to use the monotonic transformation U(x) = ln[u(x)] to get
U(x) = a1lnx1 + a2 lnx2 + an lnxn
First-order conditions are now
ai
= pi
i = 1,,n
xi
These equations can then be written
ai
pi xi =

i = 1,,n
From the budget constraint
p1x-1 + p2x2 + pn xn =
a1 + a2 + an
=M

so that
1

M
=
a1 + a2 + an
Substituting back in the original first-order conditions,
ai
xi =
M
a1 + a2 + + an pi
i = 1,2,,n
which are the Marshallian demand functions in this case. With Cobb-Douglas
preferences, quantity demanded of each good does depend on income M : in fact
quantity demanded of each good is proportional to income. But quantity demanded of
each good depends only on the price of that good, and not on the prices of any of the
other goods. In this case, the proportion of her income that the person spends on good
i, [(pixi)/ M] equals
ai
a1 + a2 + an
which is a constant - independent of income and of all the prices. If a person had
Cobb-Douglas preferences, then the proportion of her income which she spent on
food, or on housing, would depend only on her tastes, and would not change with her
income, or with the prices of food or housing.
v CES Preferences
Done in the textbook.
vi Stone-Geary Preferences
For simplicity, I choose the utility function
U(x) = b1 ln(x1-s1) + b2ln(x2-b2)+ +bnln(xn-sn)
with
b1 + b2 + + bn = 1
The first-order conditions are
bi
= pi
i = 1,2,,n
xi -si
or
bi
pi xi = pi si +

Using the budget constraint p1x1 + p2x2 + pnxn = M and the fact that the bi's sum to
1, therefore
n
M=
1
p s + 
j j
j=1
so that
n
1

=M-
p s
j j
j=1
which means that
n
bi
xi = si +
[M pi
p s ]
j j
j=1
is the Marshallian demand function for good #i. These demand functions are only
valid if the person has enough income to pay for her ``required'' consumption levels
si : if M  j = 1n pjsj.
File translated from TEX by TTH, version 2.00.
On 21 Jan 2002, 13:53.