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Functional forms and multivariate risk independence
67
Economics Letters 40 (1992) 67-70 North-Holland
Functional forms and multivariate risk independence Theodore
M. Horbulyk
The Universityof Calgary, Calgary, Alb., Canada Received 18 May 1992 Accepted 4 August 1992
This paper derives a test for multivariate risk independence that employs the normalized, quadratic restricted profit function [Diewert and Ostensoe (1988)]. This flexible functional form is better suited to this purpose than the modified translog originally proposed by Epstein (1980).
Empirical economists are continually seeking methods to test theoretical hypotheses, such as those which describe a firm’s behavior under uncertainty. An important paper by Epstein (19801 generates hypotheses with can be used to perform an empirical test for multivariate risk independence (RI, hereafter). One application of such hypotheses would be to test aspects of producer behavior under uncertainty, where third-order properties of the firm’s variable profit function are associated with risk independence in a two-period model. Epstein proposes a functional form which is flexible to the third order, and a test for RI which is based on a zero parameter restriction. This paper makes the following points. The specific test proposed by Epstein (1980) is not valid for the functional form he presents. Therefore, we derive a test for another flexible functional form, the normalized, quadratic restricted profit function described by Diewert and Ostensoe (19881, and show why this test will be more useful than its predecessor. Epstein provides an example which employs a two-period producer model, where the firm’s variable profit function is given by g(q; x1. The vector x = (xi,. . . , XJ represents (ex ante) factors which are chosen subject to uncertainty about prices, q = (ql,. . . , q,), that will prevail for the (ex post) factors of production and outputs, y, in the second period. Epstein asserts (correctly) that ‘the behavior of an expected profit maximizing producer is determined by all first- and second-order derivatives of g and by all third-order derivatives of the form gX,9k4j’(1980, p. 983). He proposes a modified translog variable profit function In g( q; x) = Cai
In qi + i Cyii
+ Cpi Correspondence to: Theodore Canada, T2N lN4. 01651765/92/$05.00
In qi In qj + Caij
In xi++zCij
M. Horbulyk, Department
In qi In xi
1n xi In xj + $Sijk In xi In qj lnq,.
(1)
of Economics, The University of Calgary, Calgary, Alberta,
0 1992 - Elsevier Science Publishers B.V. Ah rights reserved
68
T.M. Horbulyk / Functional forms and multivariate rtik independence
Subject to specified symmetry and homogeneity restrictions, Epstein then asserts (incorrectly) that ‘risk independence corresponds to simple zero parameter restrictions’, namely, ‘4 RI x if and only if aij = aijk = 0 for all i, j, and k’ [Epstein (1980, p. 983)l. It is well known from work of Epstein (1977, 1978, 1980) and Hartman (1976) that the property of the (expected-profit-maximizing) firm’s third-order derivatives which would be sufficient for RI, is g = 0. ’ However, close inspection of the variable profit function (1) will show that the condx~i$n aij = sijk = 0 (V i, j, k) is not sufficient to ensure g, 4*4j= 0 (and risk independence). ’ As a consequence, the functional form (1) is not well suited io an empirical test of RI. Empirical work based on this modified translog form would ordinarily employ Hotelling’s Lemma to derive linear share equations which have considerably fewer parameters to estimate than (1) itself. However, the implementation of a test for RI in this case will require the direct estimation of eq. (1). This may prove problematic given the larger number of coefficients to be estimated. The (net supply) share equations derived from (1) take the form 3 Si = (qi.ji(q; = a,+
x))/g(q;
x)
= a In g(q;
CYij In qj+ CSij In
Xj+
x)/a
In
C6ijk
In
4i
Xi
In qk Vi=l,...,m.
A sufficient (but not necessary) condition to ensure gX,qkq,= 0 is sij = aijk = 0 (V i, j, k) and ag/ax, = 0 (V k). This will be the case when Slj = pi = c5ij = arjk = 0 (V i, j, k), where a test of zero restrictions on pi and 4ij would require direct estimation of (1). Another flexible functional form, the normalized, quadratic restricted profit function described by Diewert and Ostensoe (1988), appears to be more useful for undertaking such a test. 4 It is given by
( xjxI> +
+ CPicliC
i
+
C &ijqixj i
j
+ ~&q;~h,~ i
j
Cbjlp
i
+ + cb,i
I
Xl
Pi4j
Xl
+ Cciqi. i
Application of Hotelling’s Lemma to the variable profit function (3) gives a set of i linear net supply equations, y^;(q; x) 2 0. Unlike the modified translog case, only (i - 1) of these equations are similar in form, noting that the net supply equation corresponding to the numeraire price has
’ Alternatively, if the sign of a3g(q; x)/ax,aq,aqj is positive, then a firm will increase (reduce) its use of x, when there is a mean-preserving spread, m.p.s. (reduction in the dispersion) of q. ’ If one chose instead to evaluate alternative third-order derivatives of the form #ln g(q; x)/ah x,aln q,aln qj, then such derivatives would be identically zero under the linear restriction Brjk = 0 (V i, j, k). However, there is not necessarily any link between the sign of these alternative third-order derivatives and those of the form gxzqkq,, where it is the latter which have been shown to characterize firm response to changing price variability. 3 Symmetry of cross-price terms has been employed to collect like terms in deriving (2). 4 Two forms are presented by the authors, where one exhibits constant returns to scale and the other does not. The latter, which nests or imbeds the former, is described here. See DuPont (1990) for an application of this functional form, and for a description of its other features and limitations.
TM. Horbulyk / Functional forms and multivariate risk independence
69
relatively more terms. The equations to be estimated are ’
(4)
+flizbjx’
i
++bOh Xl
V i=&...,m.
+ci Xl
In estimating the system of i equations (41, (5), homogeneity of degree zero in prices is imposed implicitly through normalization by ql. As well, the following additional restrictions are typically imposed. aik = ski
Vi=1
,..., m; k=l,...,
m,
(6)
alk = ail = 0
Vi=1
,..., m; k=l,...,
m,
(7)
bj, = b,
Vj=l,...,
n;l=l,...,
n,
b,, = bj, = 0
Vj=l,...,
n;l=l,...,
It.
and
(8) (9)
See Diewert and Ostensoe (1988) for details. The third-order derivatives, gxiqkqj,are evaluated as follows:
a3g(q;x) axjag:
a%(q; X) axjaqf
G1(q; x) =
axjaql
=
am; axjaqi
c
=aj
&iky)
4
“j =aii
i
41
(10)
Vj=l,...,II,
k
i
i
Vi=2
,..., m, Vj=l,...,
n.
(11)
Given that aj and prices, qi, are strictly positive, and subject to restrictions (6)-(91, a sufficient test of risk independence corresponds to zero parameter restrictions, such that q RI x if aii = aik = 0 for all i and k. If non-zero, the sign of aii (i = 2,. . . , m) signs gxlqi4, and will signal the optimal direction of firm response in xi to a marginal m.p.s. in qi. 5 The coefficients to be estimated would ordinarily include ‘yi, aik, pi, bjl, c,,, bj, b,, and ci. However, Diewert and Ostensoe (1988, p. 44) indicate that aj and pi (czj > 0 and pi > 0) are pre-specified. DuPont (1990, p. 31) cites Diewert and Wales (1987) to support arbitrarily defining czj = l/Pj, where f is the ex ante (fixed) factor vector for the first observation. Similarly, pi may be preset by the researcher such that pi = l/&, where 6 is the price vector for the first observation.
70
TM Horbulyk / Functional forms and multivariate risk independence
References Diewert, W.E. and L. Ostensoe, 1988, Flexible functional forms for profit functions and global curvature conditions, in: W.A. Barnett, E.R. Berndt and H. White, eds., Dynamic econometric modeling: Proceedings of the third international symposium in economic theory and econometrics (Cambridge University Press, Cambridge) 43-51. Diewert, W.E. and T.J. Wales, 1987, Flexible functional forms and global curvature conditions, Econometrica 55, 43-68. DuPont, D.P., 1990, Rent dissipation in restricted access fisheries, Journal of Environmental Economics and Management 19, 26-44. Epstein, L.G., 1977, Essays in the economics of uncertainty, Unpublished Ph.D. dissertation (University of British Columbia, Vancouver). Epstein, L.G., 1978, Production flexibility and the behaviour of the competitive firm under price uncertainty, Review of Economic Studies 45, 251-261. Epstein, L.G., 1980, Multivariate risk independence and functional forms for preferences and technologies, Econometrica 48, 973-985. Hartman, R., 1976, Factor demand with output price uncertainty, American Economic Review 66, 675-681.
Economics Letters 40 (1992) 67-70 North-Holland
Functional forms and multivariate risk independence Theodore
M. Horbulyk
The Universityof Calgary, Calgary, Alb., Canada Received 18 May 1992 Accepted 4 August 1992
This paper derives a test for multivariate risk independence that employs the normalized, quadratic restricted profit function [Diewert and Ostensoe (1988)]. This flexible functional form is better suited to this purpose than the modified translog originally proposed by Epstein (1980).
Empirical economists are continually seeking methods to test theoretical hypotheses, such as those which describe a firm’s behavior under uncertainty. An important paper by Epstein (19801 generates hypotheses with can be used to perform an empirical test for multivariate risk independence (RI, hereafter). One application of such hypotheses would be to test aspects of producer behavior under uncertainty, where third-order properties of the firm’s variable profit function are associated with risk independence in a two-period model. Epstein proposes a functional form which is flexible to the third order, and a test for RI which is based on a zero parameter restriction. This paper makes the following points. The specific test proposed by Epstein (1980) is not valid for the functional form he presents. Therefore, we derive a test for another flexible functional form, the normalized, quadratic restricted profit function described by Diewert and Ostensoe (19881, and show why this test will be more useful than its predecessor. Epstein provides an example which employs a two-period producer model, where the firm’s variable profit function is given by g(q; x1. The vector x = (xi,. . . , XJ represents (ex ante) factors which are chosen subject to uncertainty about prices, q = (ql,. . . , q,), that will prevail for the (ex post) factors of production and outputs, y, in the second period. Epstein asserts (correctly) that ‘the behavior of an expected profit maximizing producer is determined by all first- and second-order derivatives of g and by all third-order derivatives of the form gX,9k4j’(1980, p. 983). He proposes a modified translog variable profit function In g( q; x) = Cai
In qi + i Cyii
+ Cpi Correspondence to: Theodore Canada, T2N lN4. 01651765/92/$05.00
In qi In qj + Caij
In xi++zCij
M. Horbulyk, Department
In qi In xi
1n xi In xj + $Sijk In xi In qj lnq,.
(1)
of Economics, The University of Calgary, Calgary, Alberta,
0 1992 - Elsevier Science Publishers B.V. Ah rights reserved
68
T.M. Horbulyk / Functional forms and multivariate rtik independence
Subject to specified symmetry and homogeneity restrictions, Epstein then asserts (incorrectly) that ‘risk independence corresponds to simple zero parameter restrictions’, namely, ‘4 RI x if and only if aij = aijk = 0 for all i, j, and k’ [Epstein (1980, p. 983)l. It is well known from work of Epstein (1977, 1978, 1980) and Hartman (1976) that the property of the (expected-profit-maximizing) firm’s third-order derivatives which would be sufficient for RI, is g = 0. ’ However, close inspection of the variable profit function (1) will show that the condx~i$n aij = sijk = 0 (V i, j, k) is not sufficient to ensure g, 4*4j= 0 (and risk independence). ’ As a consequence, the functional form (1) is not well suited io an empirical test of RI. Empirical work based on this modified translog form would ordinarily employ Hotelling’s Lemma to derive linear share equations which have considerably fewer parameters to estimate than (1) itself. However, the implementation of a test for RI in this case will require the direct estimation of eq. (1). This may prove problematic given the larger number of coefficients to be estimated. The (net supply) share equations derived from (1) take the form 3 Si = (qi.ji(q; = a,+
x))/g(q;
x)
= a In g(q;
CYij In qj+ CSij In
Xj+
x)/a
In
C6ijk
In
4i
Xi
In qk Vi=l,...,m.
A sufficient (but not necessary) condition to ensure gX,qkq,= 0 is sij = aijk = 0 (V i, j, k) and ag/ax, = 0 (V k). This will be the case when Slj = pi = c5ij = arjk = 0 (V i, j, k), where a test of zero restrictions on pi and 4ij would require direct estimation of (1). Another flexible functional form, the normalized, quadratic restricted profit function described by Diewert and Ostensoe (1988), appears to be more useful for undertaking such a test. 4 It is given by
( xjxI> +
+ CPicliC
i
+
C &ijqixj i
j
+ ~&q;~h,~ i
j
Cbjlp
i
+ + cb,i
I
Xl
Pi4j
Xl
+ Cciqi. i
Application of Hotelling’s Lemma to the variable profit function (3) gives a set of i linear net supply equations, y^;(q; x) 2 0. Unlike the modified translog case, only (i - 1) of these equations are similar in form, noting that the net supply equation corresponding to the numeraire price has
’ Alternatively, if the sign of a3g(q; x)/ax,aq,aqj is positive, then a firm will increase (reduce) its use of x, when there is a mean-preserving spread, m.p.s. (reduction in the dispersion) of q. ’ If one chose instead to evaluate alternative third-order derivatives of the form #ln g(q; x)/ah x,aln q,aln qj, then such derivatives would be identically zero under the linear restriction Brjk = 0 (V i, j, k). However, there is not necessarily any link between the sign of these alternative third-order derivatives and those of the form gxzqkq,, where it is the latter which have been shown to characterize firm response to changing price variability. 3 Symmetry of cross-price terms has been employed to collect like terms in deriving (2). 4 Two forms are presented by the authors, where one exhibits constant returns to scale and the other does not. The latter, which nests or imbeds the former, is described here. See DuPont (1990) for an application of this functional form, and for a description of its other features and limitations.
TM. Horbulyk / Functional forms and multivariate risk independence
69
relatively more terms. The equations to be estimated are ’
(4)
+flizbjx’
i
++bOh Xl
V i=&...,m.
+ci Xl
In estimating the system of i equations (41, (5), homogeneity of degree zero in prices is imposed implicitly through normalization by ql. As well, the following additional restrictions are typically imposed. aik = ski
Vi=1
,..., m; k=l,...,
m,
(6)
alk = ail = 0
Vi=1
,..., m; k=l,...,
m,
(7)
bj, = b,
Vj=l,...,
n;l=l,...,
n,
b,, = bj, = 0
Vj=l,...,
n;l=l,...,
It.
and
(8) (9)
See Diewert and Ostensoe (1988) for details. The third-order derivatives, gxiqkqj,are evaluated as follows:
a3g(q;x) axjag:
a%(q; X) axjaqf
G1(q; x) =
axjaql
=
am; axjaqi
c
=aj
&iky)
4
“j =aii
i
41
(10)
Vj=l,...,II,
k
i
i
Vi=2
,..., m, Vj=l,...,
n.
(11)
Given that aj and prices, qi, are strictly positive, and subject to restrictions (6)-(91, a sufficient test of risk independence corresponds to zero parameter restrictions, such that q RI x if aii = aik = 0 for all i and k. If non-zero, the sign of aii (i = 2,. . . , m) signs gxlqi4, and will signal the optimal direction of firm response in xi to a marginal m.p.s. in qi. 5 The coefficients to be estimated would ordinarily include ‘yi, aik, pi, bjl, c,,, bj, b,, and ci. However, Diewert and Ostensoe (1988, p. 44) indicate that aj and pi (czj > 0 and pi > 0) are pre-specified. DuPont (1990, p. 31) cites Diewert and Wales (1987) to support arbitrarily defining czj = l/Pj, where f is the ex ante (fixed) factor vector for the first observation. Similarly, pi may be preset by the researcher such that pi = l/&, where 6 is the price vector for the first observation.
70
TM Horbulyk / Functional forms and multivariate risk independence
References Diewert, W.E. and L. Ostensoe, 1988, Flexible functional forms for profit functions and global curvature conditions, in: W.A. Barnett, E.R. Berndt and H. White, eds., Dynamic econometric modeling: Proceedings of the third international symposium in economic theory and econometrics (Cambridge University Press, Cambridge) 43-51. Diewert, W.E. and T.J. Wales, 1987, Flexible functional forms and global curvature conditions, Econometrica 55, 43-68. DuPont, D.P., 1990, Rent dissipation in restricted access fisheries, Journal of Environmental Economics and Management 19, 26-44. Epstein, L.G., 1977, Essays in the economics of uncertainty, Unpublished Ph.D. dissertation (University of British Columbia, Vancouver). Epstein, L.G., 1978, Production flexibility and the behaviour of the competitive firm under price uncertainty, Review of Economic Studies 45, 251-261. Epstein, L.G., 1980, Multivariate risk independence and functional forms for preferences and technologies, Econometrica 48, 973-985. Hartman, R., 1976, Factor demand with output price uncertainty, American Economic Review 66, 675-681.