We provide a necessary and sufficient condition for goods to be normal when utility functions are differentiable and strongly quasi-concave. Our condition is equivalent to the condition proposed by Alarie et al. (1990), but it is easier to check: it only requires to compute the minors associated with the border column (or row) of the bordered Hessian matrix of the utility function.normal goods; bordered hessian

Ennio Bilancini

Leonardo Boncinelli

English

Research Papers in Economics

 
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WORKING PAPER SERIES 
 
 
 
Preferences and Normal Goods: 
An Easy-to-Check 
Necessary and Sufficient Condition 
 
 
Ennio Bilancini and Leonardo Boncinelli 
 
 
Working Paper 42 
 
February 2010 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
www.recent.unimore.it Preferences and Normal Goods:
An Easy-to-Check
Necessary and Su cient Condition 
Ennio Bilancini† Leonardo Boncinelli‡
Abstract
We provide a necessary and su cient condition for goods to be normal
when utility functions are di erentiable and strongly quasi-concave.
Our condition is equivalent to the condition proposed by Alarie et al.
(1990), but it is easier to check: it only requires to compute the minors
associated with the border column (or row) of the bordered Hessian
matrix of the utility function.
JEL classiﬁcation: D11
Keywords: Normal Goods; Bordered Hessian
 We would really like to thank Ernesto Savaglio and one anonymous referee for their
useful comments and suggestions. All mistakes remain ours.
†Department of Economics, University of Modena and Reggio Emilia, Viale Berengario
51, 43 ovest, 41100 Modena, Italy, e-mail: bilancini@unisi.it, tel.: +39 059 205 6843,
fax: +39 059 205 6947.
‡Department of Economics, University of Siena, Piazza San Francesco 7, 53100 Siena,
Italy. e-mail: boncinelli@unisi.it., tel.: +39 0577 235058 6843, fax: +39 0577 232661.
11 Introduction
The present paper is devoted to the problem of characterizing normality
of goods in terms of the properties of the utility function. Leroux (1987)
raised the issue providing a su cient condition for all goods to be normal
when preferences can be represented by di erentiable strongly quasi-concave
utility functions.1 His condition is based on the Hessian and the bordered
Hessian of the utility function. Although the condition is only su cient, and
therefore possibly dependent on the utility representation, Leroux (1987)
showed that indeed it is a property of the preference order.
Fisher (1990) provided a necessary and su cient condition for a good to
be normal in terms of the expenditure function. Essentially, his condition
says that a good is normal if and only if, holding constant utility, an increase
in the price of the good would decrease the marginal utility of income.
Since the condition is on the expenditure function, Leroux’s problem of
characterizing normality only in terms of the utility function was however
not solved.
Alarie et al. (1990) provided a necessary and su cient condition for all
goods to be normal in terms of the derivative of shadow prices with re-
spect to quantities. Moreover, they showed that their condition can be
restated in terms of a suitable Allais matrix, implying that it can be tested
against observed consumption choices. They also provided an interpreta-
tion of Leroux’s su cient condition: it requires any two goods to be Allais
complements.2
We pick up the original Leroux’s problem of characterizing normality of
goods only in terms of the utility function. More precisely, we provide an
easy-to-check condition which only requires to compute the minors associ-
ated with the border column (or row) of the bordered Hessian matrix. This
is obtained by looking at the problem of the consumer who faces a marginal
increase in income as the problem of ﬁnding, in the consumption space, the
direction along which pairwise marginal rates of substitution do not change.
While our condition cannot be directly tested against observed behavior,
it has the advantage of involving only ﬁrst and second derivatives of the
utility function and requiring a few simple computations to be checked.
Moreover, our condition independently characterizes the normality of each
good so that, if one is only interested in the normality of a subgroup of
goods, calculations are even simpler.
The paper is organized as follows. In next section we provide the re-
1In an earlier contribution Chipman (1977) has shown that if a strongly concave and
twice-di erentiable utility function is such that all commodities are complements according
to the Auspitz-Lieben-Edgeworth-Pareto deﬁnition then all goods must be normal.
2The work by Quah (2007) about comparative statics under constrained optimization
has provided a condition for normality of goods in a rather general framework. However,
such a condition is only su cient.
1quired preliminaries (notation, assumptions and a suitable description of
the consumer problem). In section 3 we present our necessary and su cient
condition, we illustrate it by means of a graphical representation with two
goods, and we provie a simple example which shows that our condition may
be much simpler to check than the equivalent one by Alarie et al. (1990).
Section 4 brieﬂy summarizes our contribution.
2 Preliminaries
2.1 Assumptions and Notation
Following Leroux (1987) and Alarie et al. (1990) we conduct our analysis in
a neighborhood of x , interior point of the consumption set X   Rn
+. More-
over, let U : X   R be twice continuously di erentiable in a neighborhood
of x  and let U represent a preference order on X.
Vectors are column vectors. Denote with tx the transpose of a vector
x   X. Let G and H be the gradient vector and the Hessian matrix of U,
respectively. The bordered Hessian of U is
˜ H  
 
H G
tG 0
 
We assume that U is strongly increasing, i.e. G > 0, and strongly quasi-
concave, i.e. ˜ H is negative deﬁnite.
Finally, we indicate with |A| the determinant of a matrix A, with A 1
its inverse and with Ai,j its (i,j)-minor, i.e. the sub-matrix obtained by
deleting the i-th row and the j-th column of A.
2.2 The consumer problem
Let p   Rn be the price vector and R   R be the individual income. Then,
the consumer problem is
max
x
U(x) s.t. tpx = R (1)
Let x  be the solution to problem (1). From ﬁrst order conditions (FOCs)
we get
G(x ) = kp, k   R+ \ {0} (2)
Now, suppose that R is increased by an arbitrarily small amount. Strong
quasi-concavity of U grants that the new optimum is unique and fully de-
termined by the FOCs. Moreover, the gradient G of the new optimum lies
in the one-dimensional sub-space of Rn identiﬁed by the span of p. There-
fore, by di erentiating (2) along the direction that goes from x  to the new
optimum, i.e.  x / R, we have that
2H(x )
 x 
 R
= skp = sG(x ), s   R (3)
The linear system (3) imposes that a marginal movement along  x / R
produces a proportional change in marginal utilities.
Let us deﬁne the (n + 1)-dimensional vector
˜ x  
 
 
 
 x 
 R
 s
 
 
 
Then, (3) and the fact that utility must increase in the new optimum imply
that
˜ H(x )˜ x =
 
0
c
 
, 0   Rn,c > 0 (4)
3 Result
3.1 Condition
We are now ready to state our necessary and su cient condition for nor-
mality.
Proposition 1 Let x  be the maximizer of U(x) subject to tpx = R. Then
the i-th good is normal at x , i.e. ˜ xi   0, if and only if
| ˜ Hi,n+1| = | ˜ Hn+1,i|
 
  0 if i is even
  0 if i is odd
(5)
Strict normality of the i-th good, i.e. ˜ xi > 0, obtains if and only if the
inequality holds strictly.
Proof. We know that strong quasi-concavity of U implies that ˜ H has full
rank (see e.g., Barten and Bohm, 1982, Theorem 11.2). Therefore, system
(4) can be solved by inverting ˜ H. Given the symmetry of ˜ H we set
˜ H 1 =
 
B b
tb  
 
, b   Rn,    R (6)
From (6) and (4) we get ˜ x = (cb,c ). Since c > 0, we have that the i-th
element of ˜ x, with 1   i   n, has the same sign of the i-th element of b.
The i-th element of b has the same sign of | ˜ Hi,n+1|/| ˜ H| if (i + n + 1)
is even and has opposite sign if (i + n + 1) is odd. Moreover, since ˜ H is
negative deﬁnite, the sign of | ˜ H| is negative if n is odd and positive if n is
even. Note that, when i is odd, (i + n + 1) is odd if and only if n is odd;
3instead, when i is even, (i+n+1) is odd if and only if n is even. From these
observations and the symmetry of ˜ H 1 the result follows.  
Of course, the value of | ˜ Hn+1,i| is not invariant to positive monotone trans-
formations of U. However, Proposition 1 implies that the sign of | ˜ Hn+1,i| is
indeed a characteristic of the preference order.
3.2 A graphical representation with two goods
Admittedly, the statement of condition (5) in Proposition 1 is not easily
interpretable. A brief illustration of the case with two goods will help to
give the geometric intuition behind our result.
When n = 2 Proposition 1 states that good 1 is normal if and only if
 2U
 x1 x2
 U
 x2
 
 U
 x1
 2U
 x2 x2
  0 (7)
where x1 and x2 denote the quantities of, respectively, good 1 and good
2. We note that (7) can be rewritten as an inequality relating the rate
of change of the marginal utility of good 1 with the rate of change of the
marginal utility of good 2 when we increase the consumption of good 2, i.e.:
 2U
 x1 x2
 U
 x1
 
 2U
 x2 x2
 U
 x2
(8)
Inequality (8) says that good 1 is normal at x  if and only if a marginal
increase in the consumption of good 2 does not modify the marginal rate of
of substitution between the two goods in favor of good 2. In other words, at
x  an increase in the consumption of good 2 should not make the gradient
of the utility function rotate counter-clockwise. This is shown in more detail
in ﬁgure 1.
3.3 Example
Let us illustrate the ease of use of our condition with respect to the one by
Alarie et al. (1990) with the following example. Consider n = 3 and the
following utility
U(x) = x
 
1x 
2 + x
 
3 (9)
where   > 0,   > 0 and   > 0 are such that U is strongly quasi-concave.3
The bordered Hessian of U(x) is
3Note that, for every   > 0 and   > 0 there exists ˆ   such that for every     (0, ˆ  )
function U is strongly quasi-concave.
4 
 
x1
x2
 
 
 
 
 
 
 
 
 
 
 
 
 
B
 
x 
 
    G
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
   
B 
 
ˆ x
 
   
G  
      G   
 
    
G 
Figure 1: Normality of good 1. An increase in income raises the budget line
from B to B . The maximizer in B is x , with G its gradient. Suppose
that we spend the additional income on good x2 only, and we reach ˆ x on
B . If the gradient at ˆ x is counter-clockwise rotated with respect to G, like
in G , then moving further north-west along the budget line B  increases
utility and x1 turns out to be non-normal. Instead, x1 proves normal if the
gradient is like in G   or G    (in the latter case x2 is strictly normal).
˜ H =
 
 
   
   
   
   
 
 
 (    1)x
(  2)
1 x 
2   x
(  1)
1 x
(  1)
2 0  x
(  1)
1 x 
2
  x
(  1)
1 x
(  1)
2  (    1)x
 
1x  2
2 0  x
 
1x
(  1)
2
0 0  (    1)x
(  2)
3  x
(  1)
3
 x
(  1)
1 x 
2  x
 
1x
(  1)
2  x
(  1)
3 0
 
   
   
 
   
   
 
 
(10)
Now, if we use the condition by Alarie et al. (1990), in order to check whether
goods are normal we have to: (i) calculate the Jacobian of the vector of
shadow prices, (ii) calculate the inverse of one of its non-singular square
sub-matrices of order 2, (iii) eliminate from the column which has not been
used to build the sub-matrix the element whose index coincides with that
of the column itself, (iv) use the resulting column vector to post-multiply
the previously inverted matrix, (v) check the sign of the resulting values.
5Since the vector of shadow prices is of the form mG/twG, where m   R and
w   Rn identify the chosen normalization, such calculations may be tedious.
On the contrary, if we apply our condition (5) we get
 x 
1
 R
  0   (1    )x
(2  1)
1 x
(2  2)
2 [    (    1)]   0       1 (11)
 x 
2
 R
  0   (1    )x
(2  2)
1 x 
2x
(2  1)
2 [(    1)    ]       1 (12)
 x 
3
 R
  0   x
(2  2)
1 x
(2  2)
2 [(    1)(    1)     ]   0     +     1(13)
which immediately show that concavity of U in x3 implies normality of both
good 1 and good 2, while normality of good 3 requires concavity of U in
(x1,x2).
4 Conclusions
In this paper we have provided a necessary and su cient condition for goods
to be normal when utility functions are di erentiable and strongly quasi-
concave. The interest in our condition stems from the fact that it only
requires to compute the minors associated with the border column (or row)
of the bordered Hessian matrix which is obtained from the ﬁrst and second
derivatives of the utility function. Equivalent conditions such as the one
proposed by Alarie et al. (1990) typically require more cumbersome calcu-
lations.
References
Yves Alarie, Camille Bronsard, and Pierre Ouellette. Preferences and normal goods: A
necessary and su cient condition. Journal of Economic Theory, 51:423–430, 1990.
Anton Petrus Barten and Volker Bohm. Consumer theory. In K. J. Arrow and M. D.
Intriligator, editors, Handbook of Mathematical Economics Vol.II. North-Holland, Am-
sterdam, 1982.
John S. Chipman. An empirical implication of auspitz-lieben-edgeworth-pareto comple-
mentarity. Journal of Economic Theory, 14:228–231, 1977.
Franklin M. Fisher. Normal goods and the expenditure fucntion. Journal of Economic
Theory, 51:431–433, 1990.
Alain Leroux. Preferences and normal goods: A su cient condition. Journal of Economic
Theory, 43:192–199, 1987.
J. K.-H. Quah. The comparative statics of constrained optimization problems. Economet-
rica, 75:401–431, 2007.
6RECent Working Papers Series 
 
 
 
The 10 most RECent releases are:  
 
No. 42  PREFERENCES AND NORMAL GOODS: AN EASY-TO-CHECK NECESSARY AND 
SUFFICIENT CONDITION (2010) 
E. Bilancini and L Boncinelli 
  
No. 41  EFFICIENT AND ROBUST ESTIMATION FOR FINANCIAL RETURNS: AN APPROACH 
BASED ON Q-ENTROPY (2010) 
D Ferrari and S. Paterlini 
 
No. 40  MACROECONOMIC SHOCKS AND THE BUSINESS CYCLE: EVIDENCE FROM A 
STRUCTURAL FACTOR MODEL (2010) 
M. Forni and L.Gambetti  
 
No. 39  RENT SEEKERS IN RENTIER STATES: WHEN GREED BRINGS PEACE (2010) 
K. Bjorvatny and A. Naghavi 
 
No. 38  PARALLEL IMPORTS AND INNOVATION IN AN EMERGING ECONOMY (2010) 
A. Mantovani and A. Naghavi 
 
No. 37  THE RECENT PERFORMANCE OF THE TRADITIONAL MEASURE OF CORE INFLATION IN 
G7 COUNTRIES (2009) 
L. Lo Bue and A. Ribba 
 
No. 36  EDUCATION AND WAGE DIFFERENTIALS BY GENDER IN ITALY (2009) 
T. Addabbo and D. Favaro 
 
No. 35 
 
SINGLE-VALUEDNESS OF THE DEMAND CORRESPONDENCE AND STRICT CONVEXITY 
OF PREFERENCES: AN EQUIVALENCE RESULT (2009) 
E. Bilancini and L. Boncinelli 
 
No. 34 
 
SIGNALLING, SOCIAL STATUS AND LABOR INCOME TAXES (2009) 
E.Bilancini and L. Boncinelli 
 
No. 33 
 
ON SOME NEGLECTED IMPLICATIONS OF THE FISHER EFFECT (2009) 
A. Ribba 
 
 
 
The full list of available working papers, together with their electronic versions, can be found on 
the RECent website: http://www.recent.unimore.it/workingpapers.asp 
 

6RECent Working Papers Series The 10 most RECent releases are:

An empirical implication of auspitz-lieben-edgeworth-pareto complementarity.

Normal goods and the expenditure fucntion.

Preferences and normal goods: A necessary and su cient condition.

Preferences and normal goods: A su cient condition.

The comparative statics of constrained optimization problems.

Preferences and Normal Goods: An Easy-to-Check Necessary and Sufficient Condition

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Preferences and Normal Goods: An Easy-to-Check Necessary and Sufficient Condition

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