While the Slutsky matrix and duality theory have been used to establish that constant elasticity demand functions imply unitary income elasticities, zero cross price elasticities and own price elasticities equal to minus one, this note shows that these results can also be straightforwardly derived from the simple assumption that demand functions satisfy the budget constraint with strict equality.

Andrés Vázquez

English

Research Papers in Economics

ON CONSTANT ELASTICITIES OF DEMAND 
Andrés Vázquez 
Consejo Superior de Investigaciones Científicas 
Resumen: Mientras la matriz de Slutsky y la teoría de la dualidad 
se han utilizado para demostrar que las funciones de 
demanda con elasticidades constantes implican: elasti-
cidades de renta unitarias, elasticidades cruzadas de la 
demanda nulas y elasticidades directas iguales a menos 
uno; esta nota demuestra que estos resultados se pueden 
obtener también directamente suponiendo simplemente 
que se verifica la ecuación de balance con estricta igualdad. 
Abstract: While the Slutsky matrix and duality theory have been 
used to establish that constant elasticity demand func-
tions imply unitary income elasticities, zero cross-price 
elasticities and own-price elasticities equal to minus 
one, this note shows that these results can also be stright-
forwardly derived from the simple assumption that de-
mand functions satisfy the budget constraint with strict 
equality. 
1. Introduction 
The easy simplicity of the elasticity concept and the fact that elasticities 
are pure numbers have led economists to see their estimation as a primary 
aim of empirical studies. In econometric studies of consumer demand it 
has long been a common practice to assume that demand are elasticities 
constant. While this simple assumption facilitates the parametric estima-
tion of demrnd elasticities (since it means that the underlying demand 
functions must be log-linear), it also imposes restrictions upon the mag-
nitude of the elasticity values.
1 Slutsky equations, as well as duality 
1 Pareto (1911, pp. 613-616) was early in pointing out that the Marshallian constant 
demand curve is inadmissible, except in the special case when elasticity is unity. 
EEco, 13,1, 1998  73 74 ESTUDIOS ECONÓMICOS 
theory, have been used to fully describe the framework within which such 
demand funtions are consistent with the theoretical restrictions implied 
by the theory of consumer choice. In particular, it turns out that income 
elasticities will be unitary, the own-price elasticities will be equal to -1, 
and the cross-price elasticities will all equal to zero, see, e.g., Basmann 
et al. (1973), Dimasi and Schap (1985), Samuelson (1965), and Willig 
(1976). The purpose of this note is to show that these results can also be 
derived by simply assuming that demand functions satisfy the budget 
constraint with strict equality. 
2. Implications of Constant Elasticities of Demand 
Let the standard neoclassical twice-differentiable demand functions be 
x. = x.(p,y), where x. denotes the quantity demanded of the ith good 
(i = I, ... , it), p = (pj, ... , pn) is a vector representing the prices, and y 
is the consumer's income. Then, 
n 
i= 1 
and 
n 
XS.= 1, (2) 
!= 1 
where S. stands for the relative share of the ith good in total income 
(expenditure), i. e., S. = x.p./y (i = 1, ... , ri). 
Next, denote the'elastic'ity of the ith good with respect to price p. by 
r\.. and with respect to income by r\iy. Then, since demand functions are 
homogeneous of degree zero in prices and income, Euler's theorem 
allows us to write the aggregation condition 
n 
5>.- = -V d=U...,n) (3) 
./'= i 
first derived by Slutsky (1915), and then by Hicks (1937) and Schultz 
(1938) from the second order conditions for utility maximization. Notice 
that (3) establishes ELASTICITIES OF DEMAND 75 
PROPOSITION 1. If all cross-price elasticities of demand of a commodity 
vanish, then its own-price demand elasticity is numerically equal to its 
income elasticity. 
On the other hand, partial differentiation of (1) with respect to p. 
yields, after some manipulations, Cournots aggregation condition 
n 
£S.r,.. = -S., (7=1, ... ,n) (4) 
i = 1 
which can also be written as 
n 
Vc^ - - S(r\ +1) (i = 1 n) 
ft; ' ~ ' " ""' 
from which we have, 
PROPOSITION 2. If the cross-demand elasticities of all goods with respect 
to a given price pj are all zero, then the own-price demand elasticity of 
the jth good must be - 1. 
Finally, by partially differentiating (1) with respect to y, we get the 
Engel aggregation condition
2 
n 
^Sp =1. (5) 
i = i 
We now present the following theorem: 
THEOREM 1. If the price elasticities of demand for a good are all constant, 
then the cross-price demand elasticities all vanish and the own-price 
demand elasticity equals minus one. 
PROOF. By assuming that x[i} constant, and noting that 
dSi/dpj = Si/pj(T)ij+ 8y), where 8,-, is the Kronecker symbol (i. e., 
&ij = 1, if i = j, and 5. = 0, if / * j), partial differentiation of (4) with 
respect to Pj yields 
2 Notice that (2), (3) and (4) imply (5). 76 ESTUDIOS ECONÓMICOS 
n 
X S.(TI..)
2 + S.(y\.. + l)
2 = 0, (j = 1, ... , n) (6) 
i* 1 
which, for S. > 0, implies r\ = - 1, and T|.. = 0 (/ *J). • 
COROLLARY 1. // the price demand elasticities are constant, then the 
income demand elasticities become unity. 
PROOF. This follows immediately from theorem 1 and proposition 1. 
THEOREM 2. If the income elasticities are all constant, then they must 
all be equal to 1. 
PROOF Using (2), (5) can be rewritten as 
n 
IS-(%-D = O, 
;= l 
whose partial differentiation with respect to y, assuming r\. = constant, 
and noting that,
3 dS. /dy = S. / y(r| - 1 ), gives * 
n 
^Sfy\.y- 1)
2 = 0, (7) 
;= l 
which implies ri.y = 1 (/ = 1, ... , ri) U 
THEOREM 3. If the price demand elasticity of a good is independent of 
income, then the good's income elasticity is also independent of price, 
and vice-versa. 
PROOF. Since 
8TI.. pd
zx. i 
dy xdpdy y '« W 
and 
3 Incidentally, this derivative indicates the tautology of Engel's law, according to 
which the proportion of income devoted to necessary goods (0 < -n,. < 1) decreases as 
income increases, while the opposite holds for luxury goods (n,- > 1)'* See, for example, 
Derycke (1964, pp. 129-130), and Wold and Jureen (1953, pp. 323). ELASTICITIES OF DEMAND 77 
3%- yd
2*/ _ 1 
we have the duality relation 
3ri 3n. 
31ny ain/j.' 
which establishes that 
3T).. 3TI. 
V
L=°«V
Z=
0
 (¿,7=1,2,... ,«) 
3y 3p 
COROLLARY 2. (i) If the price demand elasticities of a good are all 
independent of the income, then the good's income elasticity is constant, 
(ii) If this holds for all goods, then income elasticities are all unity. 
PROOF, (i) is immediate from theorem 3 and the aggregation condition 
(3); (ii) then follows directly from theorem 2. 
3. Conclusions 
In econometric studies of consumer demand it has long been postulated 
that demand elasticities are constant. This simple assumption facilitates 
the estimation of demand elasticities in parametric form because, as first 
noted by Moore (1926), demand functions must then be of the form
4 
n 
xrA.y%Ylp^, (¿,7=1,2,... ,«) (8) 
7=1 
and taking logarithms leads to the log linear representation: 
n 
\nx. = InA. + ii lny + ti £ Inp. (i = 1, 2, ... , n) 
7= 1 
However the assumption of constant demand elasticities also im-
poses severe restrictions upon the own elasticity values. In this note we 
have shown that without resorting to duality theory or the integrability 
4 A similar specification was used in Wold and Juréen (1953, pp. 3, 105). 78 ESTUDIOS ECONÓMICOS 
conditions, and by simply assuming that demand equations satisfy the 
budget constraint with strict equality, r\iy = 1, ^¡¡ = - 1, and T).. = 0 for 
i * j. Substitution of these results into (8) gives us the correct form of 
the underlying demand function 
x. = A.yp-
x (9) 
or, arranging, x.p./y = A, where A. = constant, 0 < A. < 1, and 
n I I I I I 
X = 1. Notice that (9) simply says that the expenditure on each com-
mArlitv is constant fraction r\f thf* KnHcrpt With thic fnpt \r\ minH thf*  moauy is a constant rraction oi me ouuget. wun tnis lact in minu, ine 
potential gains in estimation from choosing constant elasticities in em 
pineal demand works must then be balanced against the rather unattrac-
tive structure that this assumption imposes on the underlying demand 
function. Of course, the same conclusions apply input demand functions 
in the theory of production under the formally parallel assumption of cost 
minimization.
5 
References 
Basmann, R. L., R. C. Battalio and J. H. Kagel (1973). Comment on R. P. 
Byron's, "The Restricted Aiken Estimation of Sets of Demand Relation", 
Econometrica, vol. 41, pp. 365-370. 
Derycke, P. H. (1964). Élasticité et Analyse Économique, Paris, Éditions Cujas. 
Dimasi, J. A. and D. Schap (1985). "The Appropriate Specification of Constant 
Elasticity Demand Functions", Social Choice Welfare, vol. 2, pp. 89-94. 
Hicks, J. R. (1937). Théorie Mathématique de la Valeur en Régime de Libre 
Concurrence, Paris, Hermann et Cie. 
Moore, H. L. (1926). "Partial Elasticity of Demand", Quarterly Journal of 
Economics, vol. 40, pp. 393-401. 
Pareto, V. (1911). "Economie Mathématique", Encyclopédie des Sciences 
Mathématiques, vol. 4, pp. 591-640. 
Samuelson, P. A. (1965). "Using Full Duality to Show that Simultaneously 
Additive Direct and Indirect Utilities Implies Unitary Price Elasticity of 
Demand", Econometrica, vol. 33, pp. 781-796. 
Schultz, H. (1938). The Theory and Measurement of Demand, Chicago, The 
University of Chicago Press. 
5 For details, see Vázquez (1972), and Vázquez and Puu (1973). ELASTICITIES OF DEMAND 79 
Slutsky, E. (1915). "Sulla teoria del bilancio del consumatore", Giornale degli 
economisti, vol. 51, pp. 1-26. 
Vázquez, A. (1972). "Input Demand Functions in the Theory of Production", 
Rivista Internazionale di Scienze Economiche e Commerciali, voi. 19, pp. 
931-953. 
, and T. Puu (1973). "Factor Demand Functions in the Long-run Equilib-
rium", Rivista Internazionale di Scienze Economiche e Commerciali, voi. 
20, pp. 1209-1229. 
Willig, R. D. (1976). "Integrability Implications for Locally Constant Demand 
Elasticities", Journal of Economic Theory, vol. 12, pp. 391-401. 
Wold, H. and Juréen, L. (1953), Demand Analysis, New York, John Wiley and 
Sons. i 

5 For details, see Vázquez

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Economie Mathématique&quot;,

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Factor Demand Functions in the Long-run Equilibrium&quot;,

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Partial Elasticity of Demand&quot;,

The Appropriate Specification of Constant Elasticity Demand Functions&quot;,

The Restricted Aiken Estimation of Sets of Demand Relation&quot;,

The Theory and Measurement of Demand,

Théorie Mathématique de la Valeur en Régime de Libre Concurrence, Paris, Hermann et Cie.

Using Full Duality to Show that Simultaneously Additive Direct and Indirect Utilities Implies Unitary Price Elasticity of Demand&quot;,

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On constant elasticities of demand

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