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A globally concave, symmetric, flexible cost function
Economics Letters North-Holland
31 (1989) 211-214
A GLOBALLY
CONCAVE,
211
SYMMETRIC,
FLEXIBLE
COST FUNCTION
*
Arthur LEWBEL Brand&
University,
Waltham,
Received Accepted
27 January 1989 12 April 1989
MA 02254, USA
This paper describes a cost function that is globally concave, flexible, and treats all prices symmetrically. this ‘CSFLEX’ cost function includes a new technique for proving flexibility of models.
The derivation
of
Diewert and Wales (1987) showed that common flexible forms like the translog lose flexibility when global concavity is imposed. They construct cost functions that are globally concave and flexible, but treat one input differently from the others, i.e., asymmetrically. This note describes a variation of McFadden’s (1978) cost function similar to Diewert and Wales’ models that is flexible, globally concave, and treats all input symmetrically. It also uses a new implicit function theorem technique to prove flexibility. Let y be a firm’s positive output, t be any technical change measure, p = ( pI,. . . , p,,)’ be a positive vector of input prices, and C(p, y, t) be a twice continuously differentiable cost function. Summations are from 1 to n, unless otherwise indicated. Subscripts that index inputs can take on any value from 1 to n. Let c,= aclap,,c,,=a2c/(ap, ap,), c,= aclay,c,,= a2c/(ap, ay),c,= acjat,and c,,= a2C/(ap, at). Cost functions must satisfy the price derivative restrictions forall
CC,p,=C,CC,,p,=O
j,
andC,,=C,,
forall
i, j,
i
I
0)
and y and t derivative restrictions C C,,.p, = CY, CjY = CY, for all I
i,
C C,,p, = C,, i
C,, = C,,
for all
i,
and C, y = cV1.
(2)
The final theory restriction on C is concavity, requiring the matrix of C!, derivatives be negative semidefinate. By Diewert (1974), C( p, y, t) is flexible if, for any twice continuously differentiable cost function C*( p, y, t) evaluated at any admissable p, y, and f, there exist values of the parameters of * This research
0165-1765/89/$3.50
was funded
in part by the National
Science Foundation,
0 1989, Elsevier Science Publishers
through
B.V. (North-Holland)
grant
number
SES-8712787.
212
A. Lewbel /A
C( p, y,t)
globally concaue, symmetric, flexible cost function
that make
(3) hold for all i and j. Define the Symmetric, C(P,
Flexible
Y, t> =0.5X k
+
(SFLEX)
C
C
i#k
j#k
cost function
St,PIP,PklY
by
+ Cb,yPiY’
Cb,rPzV
i
(Cb,p,i(1+ b,t + byy2+ bd2y), i
where, e.g., ‘C; + k’ means the sum over all i from 1 to n except i = k. In equation (4) sip bi, b,, b,, b,,, b,,, and b,, for all i and j are parameters s IJ = s,, for all i and j.
to be estimated,
with
Theorem 1. The SFLEX cost function given by eq. (4) is linearly homogenous in all prices, treats all prices symmetrically, and is flexible for all prices in the positive orthant except on a set of measure zero. then the Let S be the matrix of parameters s,, for all i and j. If S is negative semidefinate, SFLEX cost function is globally concave, because eq. (4) is then the sum of linear and globally concave functions in p. Define the globally Concave, Symmetric, Flexible cost function (CSFLEX) by eq. (4) where s,~ for each i and j is replaced by (5)
where a,, = 0 for all i >j and the matrix A of a,, constants is estimated instead of S. Since S = -A’A and A is triangular, eq. (5) is the Cholesky decomposition of S, which exists if and only if S is negative semidefinite. So, when S is negative semidefinite, the SFLEX and CSFLEX are identical. Theorem 2. The CSFLEX cost function is linearly homogeneous in all prices, globally concave, treats all prices symmetrically, and is flexible for all prices in the positive orthant except on a set of measure zero. Theorem 2 proves that the SFLEX does not lose its flexibility when global From Shephard’s lemma, the SFLEX factor shares f, for each input i are f, = ( C
C
k#I
j#k
“tjPzPI1)
+b,, + b,,t + bi( y-’ The CSFLEX
- o.5(
concavity
is imposed.
,Ilj kIisjXPjPkP;2)
+ b,ty-’
(6)
+ b,y + b,,t2).
factor shares are eq. (6) with s,, is replaced
with -&ahrah,.
A. Lewbel / A globally concaue, symmetnc, flexible cost function
213
The CSFLEX can be converted into a consumer demand model by taking y to be utility, solving the cost function for y, and substituting the result into factor share equations. This is simplified when by = 0. Proof of theorem I. Homogeneity flexibility. By eq. (4), C,, = for all i #j,
(
s,,
c k#i,j
and symmetric
Pkl ) - (p;kP*P;q
treatment
- ( c
of factors
r,kPkp,lj
is obvious.
What remains
is
(7)
k+/
and for all i:
(8)
Let c” be the vector of Cl, for all i and j such that 2 I i
for some functions g,. Since p, # 0, this determinant is a non-degenerate n(n - 1)/2 degree polynomial in l/p,, and so equals zero only at the measure zero set of points where l/p, is a root of this polynomial. Since this determinant is non-zero almost everywhere, we can implicitly solve for 2 in terms of C*, S,, and p, for any chosen values of S,, thereby getting C= C*. Combining this result with eq. (1) holding for both C and C * yields C,, = CiT for all i and j, Now consider other C derivatives. Excluding the measure zero set where b’p = 0 or y = 0, let b,, = C,,*/(yb’p) to get C,, = C,: and let by = C$/(2b’p) to get C,., = CT,, both as functions of b. Next, C,, = b,,y + b,bi + 2b,,b,,y for all i and C,, = (C,bi,p,,j + Zb,,b’p, so replace C,, with C,: and replace CtY with C,:, then solve these n + 1 equations for b, and each b,,. This solution exists as long as (Y,# 0 for all i, pi > 0 for all i, and y > 0. Now C,f equals b,, plus other terms, so choose b,,. to solve CY = C;,* conditional on the values of all the other parameters. Next choose b, for all i to make C, = C,* for all i. The only remaining equalities required, CY= CT, C, = C,*, and C = C *, follow automatically from eqs. (1) and (2) holding for all cost functions. Proof of Theorem 2. The CSFLEX imposes the Cholesky decomposition S = -A’A on SFLEX, making S negative sernidefinate, so the CSFLEX is globally concave. It remains to show that this restriction maintains flexibility.
214
A. Lewbel / A globally concaue, symmetric, flexible cost function
Begin as in Theorem 1, obtaining 6= F(L?, S,, P). Define 2 as the vector of a,, for all i and j such that 2 I i sj 2 n, and let A, = (a,,, . . . , a,,). Using S = -A’A, construct the vector function G such that s”= G(J, A,). Note S, = a,,A,, so S, doesn’t depend on 2. Substituting these expressions into F yields the vector function H defined by c”= F(G(A, A,), a,,A,, P) = H(a, A,, P). Both d and 2 have n(n - 1)/2 elements, so by the implicit function theorem we can solve for a as a function of c”, A,, and p if the Jacobian ClH/ak is non-singular. Now, i3H/ak = (aF,QS”)(aG/aA”), SO tlH/ak is non-singular if i3F/X? and aG/aa are both non-singular. Theorem 1 proved dF/as” is non-singular, so now show that aG/a~ is also non-singular. Since A is triangular, s,, equals the sum of akrak, over all k from 1 to the minimum of i and j. Therefore for any si, in S, s,, does not depend on ahk when h > i or k > i, so the Jacobian matrix aG/ak is triangular, and so is non-singular if all its diagonal elements are non-zero. These diagonal elements are as,,/aa,, for 2 I i 5 j I n, which equal a,, when i #j and 2a,, when i = j. Therefore, if a,, # 0 for all i, then aG/ak is non-singular, and hence aH/aa is non-singular everywhere except on a set of measure zero. By the implicit function theorem, we can let c”* = C and implicitly solve for A” in terms of c”*, A,, and p, for whatever values of A, that we choose. Combining this result with eq. (1) holding for both C and C *, yields C,, = C,,* for all i and j. The remainder of the proof is identical to that of Theorem 1.
References Diewert, W.E., 1974, Applications of duality theory, in: M.D. Intriligator and D.A. Kendrick, eds., Frontiers of quantitative economics, vol. II (North-Holland, Amsterdam) 1066171. Diewert, W.E. and T.J. Wales, 1987, Flexible functional forms and global curvature conditions, Econometrica 55, 43-68. McFadden, D., 1978, The general linear profit function, in: M. Fuss and D. McFadden, eds., Production economics in production economics: A dual approach to theory and applications, vol. 1 (North-Holland, Amsterdam) 269-286.
31 (1989) 211-214
A GLOBALLY
CONCAVE,
211
SYMMETRIC,
FLEXIBLE
COST FUNCTION
*
Arthur LEWBEL Brand&
University,
Waltham,
Received Accepted
27 January 1989 12 April 1989
MA 02254, USA
This paper describes a cost function that is globally concave, flexible, and treats all prices symmetrically. this ‘CSFLEX’ cost function includes a new technique for proving flexibility of models.
The derivation
of
Diewert and Wales (1987) showed that common flexible forms like the translog lose flexibility when global concavity is imposed. They construct cost functions that are globally concave and flexible, but treat one input differently from the others, i.e., asymmetrically. This note describes a variation of McFadden’s (1978) cost function similar to Diewert and Wales’ models that is flexible, globally concave, and treats all input symmetrically. It also uses a new implicit function theorem technique to prove flexibility. Let y be a firm’s positive output, t be any technical change measure, p = ( pI,. . . , p,,)’ be a positive vector of input prices, and C(p, y, t) be a twice continuously differentiable cost function. Summations are from 1 to n, unless otherwise indicated. Subscripts that index inputs can take on any value from 1 to n. Let c,= aclap,,c,,=a2c/(ap, ap,), c,= aclay,c,,= a2c/(ap, ay),c,= acjat,and c,,= a2C/(ap, at). Cost functions must satisfy the price derivative restrictions forall
CC,p,=C,CC,,p,=O
j,
andC,,=C,,
forall
i, j,
i
I
0)
and y and t derivative restrictions C C,,.p, = CY, CjY = CY, for all I
i,
C C,,p, = C,, i
C,, = C,,
for all
i,
and C, y = cV1.
(2)
The final theory restriction on C is concavity, requiring the matrix of C!, derivatives be negative semidefinate. By Diewert (1974), C( p, y, t) is flexible if, for any twice continuously differentiable cost function C*( p, y, t) evaluated at any admissable p, y, and f, there exist values of the parameters of * This research
0165-1765/89/$3.50
was funded
in part by the National
Science Foundation,
0 1989, Elsevier Science Publishers
through
B.V. (North-Holland)
grant
number
SES-8712787.
212
A. Lewbel /A
C( p, y,t)
globally concaue, symmetric, flexible cost function
that make
(3) hold for all i and j. Define the Symmetric, C(P,
Flexible
Y, t> =0.5X k
+
(SFLEX)
C
C
i#k
j#k
cost function
St,PIP,PklY
by
+ Cb,yPiY’
Cb,rPzV
i
(Cb,p,i(1+ b,t + byy2+ bd2y), i
where, e.g., ‘C; + k’ means the sum over all i from 1 to n except i = k. In equation (4) sip bi, b,, b,, b,,, b,,, and b,, for all i and j are parameters s IJ = s,, for all i and j.
to be estimated,
with
Theorem 1. The SFLEX cost function given by eq. (4) is linearly homogenous in all prices, treats all prices symmetrically, and is flexible for all prices in the positive orthant except on a set of measure zero. then the Let S be the matrix of parameters s,, for all i and j. If S is negative semidefinate, SFLEX cost function is globally concave, because eq. (4) is then the sum of linear and globally concave functions in p. Define the globally Concave, Symmetric, Flexible cost function (CSFLEX) by eq. (4) where s,~ for each i and j is replaced by (5)
where a,, = 0 for all i >j and the matrix A of a,, constants is estimated instead of S. Since S = -A’A and A is triangular, eq. (5) is the Cholesky decomposition of S, which exists if and only if S is negative semidefinite. So, when S is negative semidefinite, the SFLEX and CSFLEX are identical. Theorem 2. The CSFLEX cost function is linearly homogeneous in all prices, globally concave, treats all prices symmetrically, and is flexible for all prices in the positive orthant except on a set of measure zero. Theorem 2 proves that the SFLEX does not lose its flexibility when global From Shephard’s lemma, the SFLEX factor shares f, for each input i are f, = ( C
C
k#I
j#k
“tjPzPI1)
+b,, + b,,t + bi( y-’ The CSFLEX
- o.5(
concavity
is imposed.
,Ilj kIisjXPjPkP;2)
+ b,ty-’
(6)
+ b,y + b,,t2).
factor shares are eq. (6) with s,, is replaced
with -&ahrah,.
A. Lewbel / A globally concaue, symmetnc, flexible cost function
213
The CSFLEX can be converted into a consumer demand model by taking y to be utility, solving the cost function for y, and substituting the result into factor share equations. This is simplified when by = 0. Proof of theorem I. Homogeneity flexibility. By eq. (4), C,, = for all i #j,
(
s,,
c k#i,j
and symmetric
Pkl ) - (p;kP*P;q
treatment
- ( c
of factors
r,kPkp,lj
is obvious.
What remains
is
(7)
k+/
and for all i:
(8)
Let c” be the vector of Cl, for all i and j such that 2 I i
for some functions g,. Since p, # 0, this determinant is a non-degenerate n(n - 1)/2 degree polynomial in l/p,, and so equals zero only at the measure zero set of points where l/p, is a root of this polynomial. Since this determinant is non-zero almost everywhere, we can implicitly solve for 2 in terms of C*, S,, and p, for any chosen values of S,, thereby getting C= C*. Combining this result with eq. (1) holding for both C and C * yields C,, = CiT for all i and j, Now consider other C derivatives. Excluding the measure zero set where b’p = 0 or y = 0, let b,, = C,,*/(yb’p) to get C,, = C,: and let by = C$/(2b’p) to get C,., = CT,, both as functions of b. Next, C,, = b,,y + b,bi + 2b,,b,,y for all i and C,, = (C,bi,p,,j + Zb,,b’p, so replace C,, with C,: and replace CtY with C,:, then solve these n + 1 equations for b, and each b,,. This solution exists as long as (Y,# 0 for all i, pi > 0 for all i, and y > 0. Now C,f equals b,, plus other terms, so choose b,,. to solve CY = C;,* conditional on the values of all the other parameters. Next choose b, for all i to make C, = C,* for all i. The only remaining equalities required, CY= CT, C, = C,*, and C = C *, follow automatically from eqs. (1) and (2) holding for all cost functions. Proof of Theorem 2. The CSFLEX imposes the Cholesky decomposition S = -A’A on SFLEX, making S negative sernidefinate, so the CSFLEX is globally concave. It remains to show that this restriction maintains flexibility.
214
A. Lewbel / A globally concaue, symmetric, flexible cost function
Begin as in Theorem 1, obtaining 6= F(L?, S,, P). Define 2 as the vector of a,, for all i and j such that 2 I i sj 2 n, and let A, = (a,,, . . . , a,,). Using S = -A’A, construct the vector function G such that s”= G(J, A,). Note S, = a,,A,, so S, doesn’t depend on 2. Substituting these expressions into F yields the vector function H defined by c”= F(G(A, A,), a,,A,, P) = H(a, A,, P). Both d and 2 have n(n - 1)/2 elements, so by the implicit function theorem we can solve for a as a function of c”, A,, and p if the Jacobian ClH/ak is non-singular. Now, i3H/ak = (aF,QS”)(aG/aA”), SO tlH/ak is non-singular if i3F/X? and aG/aa are both non-singular. Theorem 1 proved dF/as” is non-singular, so now show that aG/a~ is also non-singular. Since A is triangular, s,, equals the sum of akrak, over all k from 1 to the minimum of i and j. Therefore for any si, in S, s,, does not depend on ahk when h > i or k > i, so the Jacobian matrix aG/ak is triangular, and so is non-singular if all its diagonal elements are non-zero. These diagonal elements are as,,/aa,, for 2 I i 5 j I n, which equal a,, when i #j and 2a,, when i = j. Therefore, if a,, # 0 for all i, then aG/ak is non-singular, and hence aH/aa is non-singular everywhere except on a set of measure zero. By the implicit function theorem, we can let c”* = C and implicitly solve for A” in terms of c”*, A,, and p, for whatever values of A, that we choose. Combining this result with eq. (1) holding for both C and C *, yields C,, = C,,* for all i and j. The remainder of the proof is identical to that of Theorem 1.
References Diewert, W.E., 1974, Applications of duality theory, in: M.D. Intriligator and D.A. Kendrick, eds., Frontiers of quantitative economics, vol. II (North-Holland, Amsterdam) 1066171. Diewert, W.E. and T.J. Wales, 1987, Flexible functional forms and global curvature conditions, Econometrica 55, 43-68. McFadden, D., 1978, The general linear profit function, in: M. Fuss and D. McFadden, eds., Production economics in production economics: A dual approach to theory and applications, vol. 1 (North-Holland, Amsterdam) 269-286.