Advanced Microeconomics
Topic 3: Consumer Demand
Primary Readings: DL – Chapter 5; JR - Chapter 3; Varian, Chapters 7-9.
3.1 Marshallian Demand Functions
Let X be the consumer's consumption set and assume that the X = Rm+. For a given price vector p
of commodities and the level of income y, the consumer tries to solve the following problem:
max u(x)
subject to px = y
xX

The function x(p, y) that solves the above problem is called the consumer's demand function.
 It is also referred as the Marshallian demand function. Other commonly known names
include Walrasian demand correspondence/function, ordinary demand functions,
market demand functions, and money income demands.
 The binding property of the budget constraint at the optimal solution, i.e., px = y, is the
Walras’ Law.
 It is easy to see that x(p, y) is homogeneous of degree 0 in p and y.
Examples:
(1) Cobb-Douglas Utility Function:
m
u ( x)   xi i ,  i  0, i  1,..., m.
i 1
From the example in the last lecture, the Marshallian demand functions are:
xi 
where
i y
.
 pi
  i 1 i .
m
(2) CES Utility Functions:
u( x1 , x2 )  ( x1  x2 )1 / 
(0    1)
Then the Marshallian demands are:
x1 (p, y ) 
p1r 1 y
;
p1r  p 2r
x 2 (p, y ) 
p 2r 1 y
,
p1r  p 2r
where r = /( -1). And the corresponding indirect utility function is given by
v(p, y)  y( p1r  p2r ) 1 / r
Let us derive these results. Note that the indirect utility function is the result of the utility
maximization problem:
max ( x1  x 2 )1 / 
x1 , x2
subject to p1 x1  p 2 x 2  y
Define the Lagrangian function:
1
L( x1 , x2 ,  )  ( x1  x2 )1 /    ( p1 x1  p2 x2  y)
The FOCs are:
L
 ( x1  x 2 ) (1 /  ) 1 x1 1  p1  0
x1
L
 ( x1  x 2 ) (1 /  ) 1 x 2 1  p 2  0
x 2
Eliminating , we get
L
 p1 x1  p 2 x 2  y  0

1 /(  1)

 p1 
 x1  x2  

 p2 

 y  p1 x1  p2 x2
So the Marshallian demand functions are:
x1  x1 (p, y ) 
p1r 1 y
p1r  p 2r
x 2  x 2 (p , y ) 
p 2r 1 y
p1r  p 2r
with r = /(-1). So the corresponding indirect utility function is given by:
v(p, y )  u ( x1 (p, y ), x 2 (p, y ))
 y ( p1r  p 2r ) 1 / r
3.2 Optimality Conditions for Consumer’s Problem
First-Order Conditions
The Lagrangian for the utility maximization problem can be written as
L = u(x) - ( px - y).
Then the first-order conditions for an interior solution are:
u (x)
 pi i; i.e. u (x)  p
xi
px  y
2
(1)
Rewriting the first set of conditions in (1) leads to
MRS kj 
MU j
MU k

pj
pk
, j  k,
which is a direct generalization of the tangency condition for two-commodity case.
x2
u(x1, x2 = u
slope = - MRS21
slope = - p1/p2
x1
Sufficiency of First-Order Conditions
Proposition: Suppose that u(x) is continuous and quasiconcave on Rm+, and that (p, y) > 0. If u if
differentiable at x*, and (x*, *) > 0 solves (1), then x* solve the consumer's utility maximization
problem at prices p and income y.
Proof. We will use the following fact without a proof:
 For all x, x'  0 such that u(x')  u(x), if u is quasiconcave and differentiable at x, then
u(x)(x' - x)  0.
Now suppose that u(x*) exists and (x*, *) > 0 solves (1). Then,
u(x*) = *p,
px* = y.
If x* is not utility-maximizing, then must exist some x0  0 such that
u(x0) > u(x*) and px0  y.
Since u is continuous and y > 0, the above inequalities implies that
u(tx0) > u(x*) and p(tx0) < y
for some t  [0, 1] close enough to one. Letting x' = tx0, we then have
u(x*)(x' - x) = (*p)( x' - x) = *( px' - px) < *(y - y) = 0,
which contradicts to the fact presented at the beginning of the proof since u(x1) > u(x*).
Remark


Note that the requirement that (x*, *) > 0 means that the result is true only for
interior solutions.
Roy's Identity
Note that the indirect utility function is defined as the "value function" of the utility maximization
problem. Therefore, we can use the Envelope Theorem to quickly derive the famous Roy's
identity.
Proposition (Roy's Identity?): If the indirect utility function v(p, y) is differentiable at (p0, y0)
and assume that v(p0, y0)/ y  0, then
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v(p 0 , y 0 )
pi
xi (p 0 , y 0 )  
,
v(p 0 , y 0 )
y
i  1,..., m.
Proof. Let x* = x(p, y) and * be the optimal solution associated with the Lagrangian function:
L = u(x) - ( px - y).
First applying the Envelope Theorem, to evaluate v(p0, y0)/ pi gives
v(p, y ) L(x*, *)

  * xi* .
pi
pi
But it is clear that * = v(p, y)/ y, which immediately leads to the Roy's identity. 
Exercise

Verify the Roy's identity for CES utility function.
Inverse Demand Functions
Sometimes, it is convenient to express price vector in terms of the quantity demanded, which leads
to the so-called inverse demand functions.

the inverse demand function may not always exist. But the following conditions will
guarantee the existence of p(x):
 u is continuous, strictly monotonic and strictly quasiconcave. (In fact, these conditions
will imply that the Marshallian demand functions are uniquely defined.)
Exercise (Duality of Indirect and Direct Demand Functions):
(1) Show that for y = 1 the inverse demand function p = p(x) is given by:
u (x)
xi
p i ( x)  m
, i  1,..., m.
u (x)
xj

j 1 x j
(Consult JR, pp.79-80.)
(2) Show that for y = 1, the (direct) demand function x = x(p, 1) satisfies
v(p,1)
pi
xi (p,1)  m
, i  1,..., m.
v(p,1)
pj

p j
j 1
(Hint: Use Roy’s identity and the homogeneity of degree zero of the indirect utility
function.)
3.3 Hicksian Demand Functions
Recall that the expenditure function e(p, u) is the minimum-value function of the following
optimization problem:
e(p, u )  minm p  x
xR 
for all p > 0 and all attainable utility levels.
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s.t. u (x)  u,
It is clear that e(p, u) is well-defined because for p  Rm++, x  Rm+, px  0.
If the utility function u is continuous and strictly quasiconcave, then the solution to the above
problem is unique, so we can denote the solution as the function xh(p, u)  0. By definition, it
follows that
e(p, u) = pxh(p, u).

xh(p, u) is called the compensated demand functions, also commonly known as Hicksian
demand functions, named after John Hicks when he first discussed this type of demand
functions in 1939.
x2
u(x1, x2) = u
slope = - p10 / p20
x2h ( p10 , p20 , u)
slope = - p11 / p20
x 2h ( p11 , p 20 , u )
x1h ( p10 , p 20 , u)
x1h ( p11 , p 20 , u )
x1
p1
p 10
Hicksian demand function
p 11
x1h ( p10 , p 20 , u)
x1h ( p11 , p 20 , u )
x1
Remarks
1. The reason that they are called "compensated" demand function is that we must
impose an artificial income adjustment when the price of one good is changing while
the utility level is assumed to be fixed.
2. It is important to understand that, in contrast with the Marshallian demands, the
Hicksian demands are not directly observable.
As usual, it should be no longer a surprise that there is a close link between the expenditure
function and the Hicksian demands, as summarized in the following result, which is again a direct
application of the Envelope Theorem..
Proposition (Shephard's Lemma for Consumer): If e(p, u) is differentiable in p at (p0, u0) with
p0 > 0, then,
xih (p 0 , u 0 ) 
e(p 0 , u 0 )
,
pi
Example: CES Utility Functions
5
i  1,..., m.
u( x1 , x2 )  ( x1  x2 )1 / 
(0    1)
Let us now derive the Hicksian demands and the corresponding expenditure function.
min {p1x1 + p2x2}
subject to
( x1  x2 )1 /   u
The Lagrangian function is
L( x1 , x2 ,  )  p1 x1  p2 x2   (( x1  x2 )1 /   u)
Then the FOCs are:
L
 p1   (( x1  x 2 )1 /  1 x1 1  0
x1
L
 p 2   (( x1  x 2 )1 /  1 x 2 1  0
x 2
Eliminating , we get
L
 u  ( x1  x 2 )1 /   0

1 /(  1)
p 
x1  x 2  1 
 p2 
u  ( x1  x 2 )1 / 
From these, it is easy to derive the Hicksian demand functions given by:
x1h (p, u )  u ( p1r  p 2r ) (1 / r )1 p1r 1
x2h (p, u )  u ( p1r  p 2r ) (1 / r )1 p 2r 1
where r = /(-1). And the expenditure function is
e(p, u)  p1 x1h (p, u)  p2 x2h (p, u)  u( p1r  p2r )1 / r .
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Alternatively, since we know that the indirect utility function is given by:
v(p, y)  y( p1r  p2r ) 1 / r ,
then by using the identity
xv(p,
2
e(p, u)) = u,
we can find the expenditure function, i.e.,
e(p, u )( p1r  p2r ) 1 / r  u
 e(p, u )  u ( p1r  p 2r )1 / r
3.4 Slutsky Equation
Recall that (last lecture) under certain regularity conditions on the utility function, the indirect
utility function v(p, y) and the expenditure function e(p, u) satisfy the following identities:
(a) e(p, v(p, y)) = y.
(b) v(p, e(p, u)) = u.
x1
IE points corresponding to the optimal solutions of
Furthermore, we have shown thatSEthe demand
both optimization problems are identical. This result can be expressed into the following
interesting identities between Marshallian demands and Hicksian demands:
x(p, y) = xh(p, v(p, y))
xh(p, u) = x(p, e(p, u))
which hold for all values of p, y and u.
The second identity leads to a classic differentiation relation between Hicksian demands and
Marshallian demands, known as Slutsky equation.
Proposition (Slutsky Equation): If the Marshallian and Hicksian demands are all well-defined
and continuously differentiable, then for p > 0, x > 0,
xi (p, y ) xih (p, u ) xi (p, y )


 x j (p, y ),
p j
p j
y
where u = v(p, y).
Proof. It follows easily from taking derivative and applying Shephard's Lemma. 
Substitution and Income Effects

The significance of Slutsky equation is that it decomposes the change caused by a price
change into two effects: a substitution effect and an income effect.
 The substitution effect is the change in compensated demand due to the change in
relative prices, which is the first item in Slutsky equation.
 The income effect is the change in demand due to the effective change in income
caused by the price change, which is the second item in Slutsky equation.
 The substitution effect is unobservable, while the income effect is observable.
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Question: From the above diagram (also know as Hicksian decomposition), can you see crossing
property between a Marshallian demand function and the corresponding Hicksian demand? (Hint:
there are two general cases.)
Slutsky Matrix
The substitution effect between good i and good j is measured by
sij 
xih (p, u )
, i, j
p j
So the Slutsky matrix or the substitution matrix is the mm matrix of the substitution items:
 x h (p, u ) 
S  [ sij ]   i

 p j 
The following result summarizes the basic properties of the Slutsky matrix.
Proposition (Substitution Properties). The Slutsky matrix S is symmetric and
negative semidefinite.
Proof. By Shephard’s Lemma (for consumer), we know that
sij 
h
xih (p, u )  2 e(p, u )  2 e(p, u ) x j (p, u )



 s ji
p j
pi p j
p j pi
pi
Hence S is symmetric. It is evident that S is the Hessian matrix of the expenditure function e(p, u).
Since we know that e(p, u) is concave, so its Hessian matrix must be negative semidefinite. 
Since the second-order own partial derivatives of a concave function are always nonpositive, this
implies that sii  0, i.e.,
sii 
xih (p, u )
 0, i
pi
which indicates the intuitive property of a demand function: as its own price increases, the
quantity demanded will decrease. You are reminded that this is a general property for Hicksian
demands.
For the Marshallian demands, note that by Slutsky equation,
xi (p, y ) xih (p, u ) xi (p, y )


 xi (p, y ).
pi
pi
y
Then for a small change in pi, we will have the following:
xi 
xi (p, y )
x h (p, u )
x (p, y )
pi  i
pi  i
 xi (p, y )pi .
pi
pi
y
The first item, capturing the own price effect of the Hicksian demands, is of course nonpositive.
The sign of the second item depends on the nature of the good:
 Normal good: xi(p, y)/ y > 0.
 This leads to a normal Marshallian demand function: it is decreasing in its own
price.
 Inferior good: xi(p, y)/ y < 0.
 When the substitution effect still dominates the income effect, the resulting
Marshallian demand is also decreasing in its own price.
 When the substitution effect is dominated by the income effect, it will lead to a
Giffen good, that is, its demand function is an increasing function of its own
price.
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Because of Slutsky equation, the Slutsky matrix (i.e., the substitution matrix) also has the
following form that is in terms of Marshallian demand functions.
 x h (p, u )   xi (p, y ) xi (p, y )

S  [ sij ]   i

x j (p, y )

y
 p j   p j

We will get back to the above Slutsky matrix in the next lecture when we discuss the integrability
problem.
3.5. The Elasticity Relations for Marshallian Demand Functions
Definition. Let x(p, y) be the consumer’s Marshallian demand functions. Define
i 
xi (p, y )
y
,
y
xi (p, y )
 ij 
xi (p, y ) p j
,
p j
xi (p, y )
si 
p i xi (p, y )
.
y
Then
1. i is called the income elasticity of demand for good i.
2.  ij is called the price elasticity of the demand for good i with respect to a price change in
good j.  ii is the own-price elasticity of the demand for good i. For i  j,  ij is the
cross-price elasticity.
3. si is called the income share spent on good i.
The following result summarizes some important relationships among the income
shares, income elasticities and the price elasticities.
Proposition. Let x(p, y) be the consumer’s Marshallian demand functions. Then
1. Engel aggregation:
m
s
i 1
i i
 1.
2. Cournot aggregation:
m
s 
i 1
i
ij
 s j ,
j  1,..., m.
Proof. Both identities are derived from the Walras’ Law, namely, the fact that the budget is tight or
balanced:
y = px(p, y) for all p and y.
(A)
To prove Engel aggregation, we differentiate both sides of (A) w.r.t. y:
m
1   pi
i 1
m
xi (p, y) m pi xi (p, y) xi (p, y)
y

  sii ,
y
y
y
xi (p, y) i 1
i 1
as required.
To prove Cournot aggregation, we differentiate both sides of (A) w.r.t. pj:
9
0   pi
i j
x j (p, y )
xi (p, y )
 x j (p, y )  p j
p j
p j
m
  x j (p, y )   pi
i 1
xi (p, y )
.
p j
Multiplying both sides by pj/y leads to

p j x j (p, y )
y
m

i 1
m
pi xi (p, y )
p x (p, y ) xi (p, y ) p j
pj   i i
y p j
y
p j xi (p, y )
i 1
m
  s j   si  ij
i 1
as required too. 
3.6 Hicks’ Composite Commodity Theorem
Any group of goods & services with no change in relative prices between themselves may be
treated as a single composite commodity, with the price of any one of the group used as the price
of the composite good and the quantity of the composite good defined as the aggregate value of
the whole group divided by this price. Important use in applied economic analysis.
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Additional References
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Arrow, K. J. (1951, 1963) Social Choice and Individual Values. 1st Ed., Yale University Press,
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Becker, G. S. (1962) "Irrational Behavior and Economic Theory," Journal of Political Economy,
70, 1-13.
Cook, P. (1972) "A One-line Proof of the Slutsky Equation," American Economic Review, 42, 139.
Deaton, A. and J. Muellbauer (1980) Economics and Consumer Behavior. Cambridge University
Press, Cambridge.
Debreu, G. (1959) Theory of Value. John Wiley & Sons, New York.
Debreu, G. (1960) "Topological Methods in Cardinal Utility Theory," in Mathematical Methods in
the Social Sciences, ed. K. J. Arrow and M. D. Intriligator, North Holland, Amsterdam.
Diewert, W. E. (1982) "Duality Approaches to Microeconomic Theory," Chapter 12 in Handbook
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Amsterdam.
Gorman, T. (1953) “Community Preference Fields,” Econometrica, 21, 63-80.
Hicks, J. (1946) Value and Capital. Clarendon Press, Oxford, England.
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Marshall, A. (1920) Principle of Economics, 8th Ed. MacMillan, London.
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Pollak, R. (1969) "Conditional Demand Functions and Consumption Theory," Quarterly Journal
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Roy, R. (1942) De l'utilite. Hermann, Paris.
Roy, R. (1947) "La distribution de revenu entre les divers biens," Econometrica, 15, 205-225.
Samuelson, P. A. (1938) "A Note on the Pure Theory of Consumer's Behavior," Econometrica, 5,
61-71, 353-354.
Samuelson, P. (1947) Foundations of Economic Analysis. Harvard University Press,
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Sen, (1970) Collective Choice and Social Welfare. Holden Day, San Francisco.
Stigler, G. (1950) "Development of Utility Theory," Journal of Political Economy, 59, parts 1 & 2,
pp. 307-327, 373-396.
Varian, H. R. (1992) Microeconomic Analysis. Third Edition. W.W. Norton & Company, New
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Wold, H. and L. Jureen (1953) Demand Analysis. John Wiley & Sons, New York.
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