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Underestimation in the Leontief model
171
bonomics Letters 18 (1985) 171-174 \lorth-Holland
UNDERESTIMATION IN THE LEONTIEF MODEL Sjur D. FLAM Chr. Michelson Institute,
N - 5036 Fantoft, Norway
Lars THORLUND-PETERSEN Norwegian Received
School of Economics and Business Administration, 4 December
A unifying coefficients
N- 5035 Bergen - Sandviken,
Norway
1984
criterion for underestimation be moment-associated.
of the Leontief
inverse
is given. The condition
imposed
is that the input-output
1. Introduction
In a Leontief model several stochastic specifications are known to imply underestimation of the Leontief inverse. For example, Simonovits (1975) assumes independent technological coefficients whereas Lahiri (1983) assumes that these coefficients are biproportionally distributed. Both of these specifications are particular cases verifying a more general condition on the consumption matrix to be presented below. The condition imposed is a weakened version of one used by Milgrom and Weber (1982) in connection with auction theory. Econometric literature speaks about the technological coefficients being in error. This reflects imperfections in measurement and observations. However, the coefficients may equally well be genuinely random; thus our results are not confined to applications in econometrics. We consider a stochastic Leontief model as described by a random consumption matrix A and a random final demand vector y with non-negative entries aij, yj, i,j = 1,. . . , n. It is assumed throughout that every aij, y/ has finite mean. The maximal non-negative characteristic root of a non-negative square matrix B is denoted X(B). Furthermore, the Rayleigh quotient of a square matrix C is given as the real number
P(C)= suP,zo{xCx/xx].
(1)
Being the pointwise supremum of linear functions, p is a convex function on the space of n X n matrices. We need the following relation between X and p: Lemma 1.
For any non-negative square matrix B one has p(B) 2 X(B) with equality for B symmetric.
The inequality follows from inserting an eigen-vector corresponding to X(B) in (1). Thus suppose B is symmetric. Then equality is a consequence of results in the theory of symmetric matrices [see Bellman (1960, ch. 7)]. Q.E.D. Proofi
0165-1765/85/%3.30
6 1985, Elsevier Science Publishers
B.V. (North-Holland)
172
S.D. Flrim, L. Thorlund-Petersen
/ Underestimation
in Leonrief model
Let E denote the expectation operator. Then if B is a random non-negative and symmetric m one has, by Jensen’s inequality,
EA(B) 2 A(EB). This result, however, is of no direct interest to our problem, as a consumption matrix ca reasonably be assumed to be symmetric. Thus we must impose more appropriate conditions or matrix in order to obtain results on underestimation.
2. Moment-associated
variables
Suppose z,, . . . , z, are non-negative random variables all having finite mean. Then these varia are called moment-associated if for all non-negative integers v,, . . . , v,,,, E(z;Y.:z~
)2
(Ez~)“‘...(Ez,)“~.
We remark that if (3) holds not merely for moments but then these variables are associated in the sense of Milgrom are independent or identical, then (3) holds. Furthermore, of variables has non-negative covariance. In fact, the latter that (3) holds for all y1 + . . . + vm= 2.
for all increasing functions of z,, . . . , and Weber (1982). Clearly, if the varial moment-association implies that any 1 condition is equivalent to the requirem
Lemma 2. If A = (aij) is a random matrix, y = ( yj) is a random vector such that all the aij, y, non-negative moment-associated variables, then for k = 0, 1, 2,. . . , EAk r (EA)~,
EAky 2 (EA) kEy,
E(AA’) L (EA)(EA’),
(4M5),
where A’ denotes the transpose of A. Proof: Elements of Ak, Ak y and AA’ are sums of moments of the a,, and yi. Thus the abc Q.E.D. inequalities follow from moment-association. Remark. In order to establish (6) it suffices to assume that any pair of elements from A h non-negative covariance.
3. Underestimation We can now present our main results. Theorem 1.
Let A be a non-negative random square matrix satisfying (6). Then
EXZ(A)2A2(EA).
S. D. Fliim, L. Thorlund-Petersen / Underestimation in
Proof.
Leontief
By Lemmas 1 and 2, together with Jensen’s inequality, Q.E.D.
173
model
one has EX’(A) = Eh(AA’) =
Ep( AA’) 2 p((EA)(EA’)) = X’(EA).
In order to establish underestimation of the Leontief inverse, Simonovits (1975) assumes that the aij are independent and Lahiri (1983) assumes the biproportional specification aij = rJijsj with r,, sj independent. These conditions both imply that the ujj are moment-associated variables. Thus the results of Simonovits and of Lahiri are contained in the following theorem: Theorem 2. Suppose A is a non-negative random square matrix and y a non-negative random vector. Then (4) implies that ExAk 0
(8)
2 c (EA)! 0
In particular, if xFAk is integrable, then (4) entails that both (I - A)-’ defined almost sure& and that
and (I - EA)-’
are well
(9)
E(Z-A)-‘>(I-EEA)?
Furthermore, integrability of CTAk together with (4) and (5) implies that the following integrals exist and satisfy the inequality E(I-
A)-‘yr
(I-
EA)-‘Ey.
Finally, given (6) then a sufficient condition for existence of (I - EA)-’
00) is that EX2(A) < 1.
Proof Since the above series are non-negative, (8) follows from (4) and the Monotone Convergence Theorem. Inequalities (9) and (10) are proved in a similar manner. The last sentence of Theorem 2 is a consequence of Theorem 1. Q.E.D.
4. Concluding remarks
We have demonstrated that if the data of a stochastic Leontief model satisfy a certain moment condition, then the Leontief inverse and the gross output vector will be underestimated by the corresponding deterministic counterpart, as described by (9) and (10). It is tempting to relate the underestimate X(EA) < (EX~( A))“~ to similar estimates of the maximal growth rate of the economy r(A). In fact, r satisfies the relation (1 +r(A))-’
=X(A),
[see Gale (1956, Theorem 6)]. However, r is a convex decreasing function of X, thus it does not seem possible to state in general which of the numbers Er(A) and r(EA) is the largest.
174
S. D. Fbm, L. Thorlund-Petersen
/
Underestimation in Leontiefmodel
References Bellman, R.E., 1960, Introduction to matrix analysis (McGraw-Hill, New York). Gale, D., 1956, The closed linear model of production, in: H.W. Kuhn and A.W. Tucker, eds., Linear inequalities and I systems (Princeton University Press, Princeton, NJ) 285-303. Lahiri, S., 1983, A note on the underestimation and overestimation in stochastic input-output models, Economics Lettc 361-366. Milgrom, P.R. and R.J. Weber, 1982, A theory of auctions and competitive bidding, Econometrica 50, 1089-1122. Simonovits, A., 1975, A note on the underestimation and overestimation of the Leontief inverse, Econometrica 43, 493
bonomics Letters 18 (1985) 171-174 \lorth-Holland
UNDERESTIMATION IN THE LEONTIEF MODEL Sjur D. FLAM Chr. Michelson Institute,
N - 5036 Fantoft, Norway
Lars THORLUND-PETERSEN Norwegian Received
School of Economics and Business Administration, 4 December
A unifying coefficients
N- 5035 Bergen - Sandviken,
Norway
1984
criterion for underestimation be moment-associated.
of the Leontief
inverse
is given. The condition
imposed
is that the input-output
1. Introduction
In a Leontief model several stochastic specifications are known to imply underestimation of the Leontief inverse. For example, Simonovits (1975) assumes independent technological coefficients whereas Lahiri (1983) assumes that these coefficients are biproportionally distributed. Both of these specifications are particular cases verifying a more general condition on the consumption matrix to be presented below. The condition imposed is a weakened version of one used by Milgrom and Weber (1982) in connection with auction theory. Econometric literature speaks about the technological coefficients being in error. This reflects imperfections in measurement and observations. However, the coefficients may equally well be genuinely random; thus our results are not confined to applications in econometrics. We consider a stochastic Leontief model as described by a random consumption matrix A and a random final demand vector y with non-negative entries aij, yj, i,j = 1,. . . , n. It is assumed throughout that every aij, y/ has finite mean. The maximal non-negative characteristic root of a non-negative square matrix B is denoted X(B). Furthermore, the Rayleigh quotient of a square matrix C is given as the real number
P(C)= suP,zo{xCx/xx].
(1)
Being the pointwise supremum of linear functions, p is a convex function on the space of n X n matrices. We need the following relation between X and p: Lemma 1.
For any non-negative square matrix B one has p(B) 2 X(B) with equality for B symmetric.
The inequality follows from inserting an eigen-vector corresponding to X(B) in (1). Thus suppose B is symmetric. Then equality is a consequence of results in the theory of symmetric matrices [see Bellman (1960, ch. 7)]. Q.E.D. Proofi
0165-1765/85/%3.30
6 1985, Elsevier Science Publishers
B.V. (North-Holland)
172
S.D. Flrim, L. Thorlund-Petersen
/ Underestimation
in Leonrief model
Let E denote the expectation operator. Then if B is a random non-negative and symmetric m one has, by Jensen’s inequality,
EA(B) 2 A(EB). This result, however, is of no direct interest to our problem, as a consumption matrix ca reasonably be assumed to be symmetric. Thus we must impose more appropriate conditions or matrix in order to obtain results on underestimation.
2. Moment-associated
variables
Suppose z,, . . . , z, are non-negative random variables all having finite mean. Then these varia are called moment-associated if for all non-negative integers v,, . . . , v,,,, E(z;Y.:z~
)2
(Ez~)“‘...(Ez,)“~.
We remark that if (3) holds not merely for moments but then these variables are associated in the sense of Milgrom are independent or identical, then (3) holds. Furthermore, of variables has non-negative covariance. In fact, the latter that (3) holds for all y1 + . . . + vm= 2.
for all increasing functions of z,, . . . , and Weber (1982). Clearly, if the varial moment-association implies that any 1 condition is equivalent to the requirem
Lemma 2. If A = (aij) is a random matrix, y = ( yj) is a random vector such that all the aij, y, non-negative moment-associated variables, then for k = 0, 1, 2,. . . , EAk r (EA)~,
EAky 2 (EA) kEy,
E(AA’) L (EA)(EA’),
(4M5),
where A’ denotes the transpose of A. Proof: Elements of Ak, Ak y and AA’ are sums of moments of the a,, and yi. Thus the abc Q.E.D. inequalities follow from moment-association. Remark. In order to establish (6) it suffices to assume that any pair of elements from A h non-negative covariance.
3. Underestimation We can now present our main results. Theorem 1.
Let A be a non-negative random square matrix satisfying (6). Then
EXZ(A)2A2(EA).
S. D. Fliim, L. Thorlund-Petersen / Underestimation in
Proof.
Leontief
By Lemmas 1 and 2, together with Jensen’s inequality, Q.E.D.
173
model
one has EX’(A) = Eh(AA’) =
Ep( AA’) 2 p((EA)(EA’)) = X’(EA).
In order to establish underestimation of the Leontief inverse, Simonovits (1975) assumes that the aij are independent and Lahiri (1983) assumes the biproportional specification aij = rJijsj with r,, sj independent. These conditions both imply that the ujj are moment-associated variables. Thus the results of Simonovits and of Lahiri are contained in the following theorem: Theorem 2. Suppose A is a non-negative random square matrix and y a non-negative random vector. Then (4) implies that ExAk 0
(8)
2 c (EA)! 0
In particular, if xFAk is integrable, then (4) entails that both (I - A)-’ defined almost sure& and that
and (I - EA)-’
are well
(9)
E(Z-A)-‘>(I-EEA)?
Furthermore, integrability of CTAk together with (4) and (5) implies that the following integrals exist and satisfy the inequality E(I-
A)-‘yr
(I-
EA)-‘Ey.
Finally, given (6) then a sufficient condition for existence of (I - EA)-’
00) is that EX2(A) < 1.
Proof Since the above series are non-negative, (8) follows from (4) and the Monotone Convergence Theorem. Inequalities (9) and (10) are proved in a similar manner. The last sentence of Theorem 2 is a consequence of Theorem 1. Q.E.D.
4. Concluding remarks
We have demonstrated that if the data of a stochastic Leontief model satisfy a certain moment condition, then the Leontief inverse and the gross output vector will be underestimated by the corresponding deterministic counterpart, as described by (9) and (10). It is tempting to relate the underestimate X(EA) < (EX~( A))“~ to similar estimates of the maximal growth rate of the economy r(A). In fact, r satisfies the relation (1 +r(A))-’
=X(A),
[see Gale (1956, Theorem 6)]. However, r is a convex decreasing function of X, thus it does not seem possible to state in general which of the numbers Er(A) and r(EA) is the largest.
174
S. D. Fbm, L. Thorlund-Petersen
/
Underestimation in Leontiefmodel
References Bellman, R.E., 1960, Introduction to matrix analysis (McGraw-Hill, New York). Gale, D., 1956, The closed linear model of production, in: H.W. Kuhn and A.W. Tucker, eds., Linear inequalities and I systems (Princeton University Press, Princeton, NJ) 285-303. Lahiri, S., 1983, A note on the underestimation and overestimation in stochastic input-output models, Economics Lettc 361-366. Milgrom, P.R. and R.J. Weber, 1982, A theory of auctions and competitive bidding, Econometrica 50, 1089-1122. Simonovits, A., 1975, A note on the underestimation and overestimation of the Leontief inverse, Econometrica 43, 493