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New results on rational random behavior
201
Economics Letters 2 (1979) 201-204 0 North-Holland Publishing Company
NEW RESULTS ON RATIONAL
John P. BOWMAN, Kenneth
RANDOM BEHAVIOR *
LAITINEN and Hem? THElL
University of Cllicago, Chicago, IL 60637, Received
4 June
USA
1979
This letter deals with conditions under which it is preferable to buy a zero amount of information; with a proportionality of the information and loss variances associated with two different decision distributions; and with a limiting &i-square distribution of the decision maker’s loss when the marginal cost of information is small.
1. Introduction
Let x = [x1 .. . xk]’ be a vector of variables controlled by a decision maker (assumed to be continuously variable over a region J); let Z(x, X) be the decision maker’s loss function, satisfying I(x, x) = 0 if x = X, I(x, Y) > 0 if x f X, where x, x E J. The theory of rational random behavior assumes that the decision maker does not know X and behaves randomly according to a prior decision distribution with density function p,,( ) unless he decides to acquire information. ’ We define the information received when this distribution is transformed into that with density function p( ) as
I=Jp(x)loga J
PO (xx>
dxl ...dxk.
Let ~(4 be the cost of buying information 1. Then dc/dI is the marginal cost of information, which we assume to be positive and non-decreasing (d’c/d? > 0). The criterion is a minimum of c(Q t c where lis the expected loss, t= JZ(x, x)p(x)
dxl
. dxk,
(2)
J * Research supported in part by NSF Grant SOC76-82718. ’ For example, a housewife does not know the prices of all consumer goods and hence does not know the optimal quantities which she would obtain from utility maximization if she knew all prices. These optimal quantities are then elements of X, while the randomness represented by the decision distribution is justified by the housewife’s uncertainty as to the values of the prices. We assume PO(X) > 0 for each x Ed.
J.P. Bowman et al. /Rational random behavior
202
yielding the solution p(x)=Kpe(x)exp
-w C ( where K is a proportionality dc/dI at the solution (3). ’
if xEJ, (3) I coefficient and c’ is the marginal cost of information
2. When does it pay not to buy information? From (3). log K = log P(x)/po (x) + Z(x, X)/c’, yielding log K = Z f UC’ after multiplication by p(x) and integrating over x. Hence (3) implies I = _ &x, x> - I P(X) log __ -PO(X)
We multiply
C’
(4)
this by p. (xx>and integrate over x. This yields
where 3
10
PO(X) ‘JPOW log-P(X)
dx, . . . dxk )
J
lo
=JZ(x, x)po(x)
dxl ... dxk.
(7)
The value of the criterion function c(Z) t fat po( ), when no information is acquired, equals 17,if c(0) = 0. It thus follows from (5) that it pays to buy no information if and only if
It~,
(8)
If c( ) is continuous at each I > 0 with positive dc/dI and non-negative d2 c/d12, then c(I) < c’l; this and lo > 0 contradict (8) so that buying no information is not 2 This dc/dl depends on p( ) via (1). The result (3) was proved by Barbosa (1975); for a more elementary proof see Theil (1979, ch. 7). The solution (3) represents a local minimum and does not exclude the possibility that acquiring no information (I = 0, implying the selection of the prior decision distribution) is a superior choice; this matter is pursued in section 2. Note that (3) resembles the Bayesian derivation of a posterior density function. The similarity is indeed close, but there is a difference in that the exponent in (3) contains the unknown constant X which does not have a Bayesian counterpart. 3 I +I0 is Kullback’s (1959, p, 6) measure of the divergence between the distributions with density functions p( ) and po( ).
J.P. Bowman et al. /Rational
203
random behavior
a superior policy. However, this conclusion may change when c( ) is discontinuous at I = 0 and jumps from zero to a positive amount of fixed costs (co) for a small positive I. For a linear cost function, c(IJ = co + ~‘1, I > 0, (8) is satisfied when IO < co/c’, implying that it is better not to buy information when the fixed costs are sufficiently high relative to the marginal cost.
3. Variances of information
and loss
When we square both sides of (4) and then multiply x, we obtain
1
1 2 p(X) dX, . .. d.X, = 4 c
J[lk
by p(x) and integrate over
x) - ti”p(x)
Next, by adding (4) and (5) and then squaring and multiplying grating over x, we obtain
__ J[log!o(x)
-I0
J
P(X)
1
&I
.. . kk.
(9)
J
2po(x> dx, ... dxk =$ J[Z(x, T) -
by pa(x) and inte-
tol*Po(X) dxl ... dxk. (10)
J
The integral on the right in (10) is the variance of the loss distribution which is associated with the prior decision distribution, and that in (9) is the loss variance associated with the solution (3). The left sides of (9) and (10) are analogous variances of information. 4 The equations state that the information variance is proportional to the loss variance, for both (3) and the prior distribution, and that the proportionality constant is equal to the squared reciprocal of the marginal cost of information at the solution (3), in both cases.
4. The asymptotic
loss distribution
If Z(x, 2) has a zero gradient and a positive definite Hessian matrix A at x =x, a Taylor expansion yields I(x,x)=~(x-~)“4(x-~)tu03,
(11)
where O3 is a remainder term containing cubic powers of the elements ofx - X. If it is also true that pa(x) is differentiable at x =X, then the density function (3) con-
4 These information variances are measures of the degree to which two distributions differ. Such variances were considered by Kullback (1959, p. 77) and, in a univariate context, by Theil (1972, pp. 10-11, 92-93).
204
J.P. Bowman
et af. / Rafional
random
behavior
verges, as c’ -+ 0, ’ to p(x)aexp(-i(x-Z)‘($A)(x-Z)]
,
(12)
which is the density function of the normal decision distribution with mean vector x and covariance matrix c’A_’ . This asymptotic decision distribution is independent of po( ), reflecting the fact that if the marginal cost of information is sufficiently small, the decision maker acquires information to such a degree that his actions become independent of previous knowledge. 6 It follows from (11) and (12) that as c’ + 0, 21(x, %)/c’ converges in distribution to the quadratic form (X - .?Q’[(l/c’)A] (X - Z) and that this form is distributed as x2 (k), k being the number of decision variables. Therefore, the asymptotic loss distribution of rational random behavior is a multiple $c’ of x2 (k). This is a generalization of an earlier result obtained for a multiproduct firm which operates under competitive conditions, with the loss function interpreted as maximal minus actual profit. Laitinen and Theil(1978) proved that if the firm engages in rational random behavior (for a small marginal cost of information), the second differential of its profit has a random component which is distributed as a negative multiple of a x2 variate. The number of degrees of freedom of this variate equals the total number of inputs and outputs less one; the subtraction of one results from the technology constraint on the inputs and outputs. References Rarbosa, F. de H., 1975, Rational random behavior:
Extensions and applications, Doctoral dissertation (University of Chicago, Chicago, IL). Kullback, S., 1959, Information theory and statistics (Wiley, New York). Laitinen, K. and H. Theil, 1978, Supply and demand of the multiproduct firm, European Economic Review 11, 107-154. Theil, H., 1972, Statistical decomposition analysis with applications in the social and administrative sciences (North-llolland, Amsterdam). Theil, H., 1979, The system-wide approach to microeconomics (University of Chicago Press, Chicago, IL). 5 The marginal cost of information depends on the prices of certain goods and services which the decision maker buys in order to acquire his information. In the limiting process c’ -+ 0 we imagine that these prices converge to zero. For a proof of the convergence of (3) to (12) see Theil (1979, ch. 8). Random behavior generated by (12) is formally equivalent to large-sample maximum likelihood estimation of the vector x (with a large sample interpreted as a small marginal cost of information). 6 Since the covariance matrix c’A_’ converges to zero as c’ + 0, (12) implies that the decision made converges in probability to X, which agrees with intuition. Also, for any po( ) which takes a positive value for each x EJ, Io increases beyond bounds as c’ * 0 when p(x) in (6) is interpreted as p(x). This means that the left-hand side of (8) also increases beyond bounds. What happens on the right depends on the assumptions made on the limiting behavior of c( ), but it seems plausible that weak assumptions are sufficient to ensure that (8) is violated when c’ + 0, thus implying the superiority of the asymptotic normal decision distribution (12) over the prior distribution.
Economics Letters 2 (1979) 201-204 0 North-Holland Publishing Company
NEW RESULTS ON RATIONAL
John P. BOWMAN, Kenneth
RANDOM BEHAVIOR *
LAITINEN and Hem? THElL
University of Cllicago, Chicago, IL 60637, Received
4 June
USA
1979
This letter deals with conditions under which it is preferable to buy a zero amount of information; with a proportionality of the information and loss variances associated with two different decision distributions; and with a limiting &i-square distribution of the decision maker’s loss when the marginal cost of information is small.
1. Introduction
Let x = [x1 .. . xk]’ be a vector of variables controlled by a decision maker (assumed to be continuously variable over a region J); let Z(x, X) be the decision maker’s loss function, satisfying I(x, x) = 0 if x = X, I(x, Y) > 0 if x f X, where x, x E J. The theory of rational random behavior assumes that the decision maker does not know X and behaves randomly according to a prior decision distribution with density function p,,( ) unless he decides to acquire information. ’ We define the information received when this distribution is transformed into that with density function p( ) as
I=Jp(x)loga J
PO (xx>
dxl ...dxk.
Let ~(4 be the cost of buying information 1. Then dc/dI is the marginal cost of information, which we assume to be positive and non-decreasing (d’c/d? > 0). The criterion is a minimum of c(Q t c where lis the expected loss, t= JZ(x, x)p(x)
dxl
. dxk,
(2)
J * Research supported in part by NSF Grant SOC76-82718. ’ For example, a housewife does not know the prices of all consumer goods and hence does not know the optimal quantities which she would obtain from utility maximization if she knew all prices. These optimal quantities are then elements of X, while the randomness represented by the decision distribution is justified by the housewife’s uncertainty as to the values of the prices. We assume PO(X) > 0 for each x Ed.
J.P. Bowman et al. /Rational random behavior
202
yielding the solution p(x)=Kpe(x)exp
-w C ( where K is a proportionality dc/dI at the solution (3). ’
if xEJ, (3) I coefficient and c’ is the marginal cost of information
2. When does it pay not to buy information? From (3). log K = log P(x)/po (x) + Z(x, X)/c’, yielding log K = Z f UC’ after multiplication by p(x) and integrating over x. Hence (3) implies I = _ &x, x> - I P(X) log __ -PO(X)
We multiply
C’
(4)
this by p. (xx>and integrate over x. This yields
where 3
10
PO(X) ‘JPOW log-P(X)
dx, . . . dxk )
J
lo
=JZ(x, x)po(x)
dxl ... dxk.
(7)
The value of the criterion function c(Z) t fat po( ), when no information is acquired, equals 17,if c(0) = 0. It thus follows from (5) that it pays to buy no information if and only if
It~,
(8)
If c( ) is continuous at each I > 0 with positive dc/dI and non-negative d2 c/d12, then c(I) < c’l; this and lo > 0 contradict (8) so that buying no information is not 2 This dc/dl depends on p( ) via (1). The result (3) was proved by Barbosa (1975); for a more elementary proof see Theil (1979, ch. 7). The solution (3) represents a local minimum and does not exclude the possibility that acquiring no information (I = 0, implying the selection of the prior decision distribution) is a superior choice; this matter is pursued in section 2. Note that (3) resembles the Bayesian derivation of a posterior density function. The similarity is indeed close, but there is a difference in that the exponent in (3) contains the unknown constant X which does not have a Bayesian counterpart. 3 I +I0 is Kullback’s (1959, p, 6) measure of the divergence between the distributions with density functions p( ) and po( ).
J.P. Bowman et al. /Rational
203
random behavior
a superior policy. However, this conclusion may change when c( ) is discontinuous at I = 0 and jumps from zero to a positive amount of fixed costs (co) for a small positive I. For a linear cost function, c(IJ = co + ~‘1, I > 0, (8) is satisfied when IO < co/c’, implying that it is better not to buy information when the fixed costs are sufficiently high relative to the marginal cost.
3. Variances of information
and loss
When we square both sides of (4) and then multiply x, we obtain
1
1 2 p(X) dX, . .. d.X, = 4 c
J[lk
by p(x) and integrate over
x) - ti”p(x)
Next, by adding (4) and (5) and then squaring and multiplying grating over x, we obtain
__ J[log!o(x)
-I0
J
P(X)
1
&I
.. . kk.
(9)
J
2po(x> dx, ... dxk =$ J[Z(x, T) -
by pa(x) and inte-
tol*Po(X) dxl ... dxk. (10)
J
The integral on the right in (10) is the variance of the loss distribution which is associated with the prior decision distribution, and that in (9) is the loss variance associated with the solution (3). The left sides of (9) and (10) are analogous variances of information. 4 The equations state that the information variance is proportional to the loss variance, for both (3) and the prior distribution, and that the proportionality constant is equal to the squared reciprocal of the marginal cost of information at the solution (3), in both cases.
4. The asymptotic
loss distribution
If Z(x, 2) has a zero gradient and a positive definite Hessian matrix A at x =x, a Taylor expansion yields I(x,x)=~(x-~)“4(x-~)tu03,
(11)
where O3 is a remainder term containing cubic powers of the elements ofx - X. If it is also true that pa(x) is differentiable at x =X, then the density function (3) con-
4 These information variances are measures of the degree to which two distributions differ. Such variances were considered by Kullback (1959, p. 77) and, in a univariate context, by Theil (1972, pp. 10-11, 92-93).
204
J.P. Bowman
et af. / Rafional
random
behavior
verges, as c’ -+ 0, ’ to p(x)aexp(-i(x-Z)‘($A)(x-Z)]
,
(12)
which is the density function of the normal decision distribution with mean vector x and covariance matrix c’A_’ . This asymptotic decision distribution is independent of po( ), reflecting the fact that if the marginal cost of information is sufficiently small, the decision maker acquires information to such a degree that his actions become independent of previous knowledge. 6 It follows from (11) and (12) that as c’ + 0, 21(x, %)/c’ converges in distribution to the quadratic form (X - .?Q’[(l/c’)A] (X - Z) and that this form is distributed as x2 (k), k being the number of decision variables. Therefore, the asymptotic loss distribution of rational random behavior is a multiple $c’ of x2 (k). This is a generalization of an earlier result obtained for a multiproduct firm which operates under competitive conditions, with the loss function interpreted as maximal minus actual profit. Laitinen and Theil(1978) proved that if the firm engages in rational random behavior (for a small marginal cost of information), the second differential of its profit has a random component which is distributed as a negative multiple of a x2 variate. The number of degrees of freedom of this variate equals the total number of inputs and outputs less one; the subtraction of one results from the technology constraint on the inputs and outputs. References Rarbosa, F. de H., 1975, Rational random behavior:
Extensions and applications, Doctoral dissertation (University of Chicago, Chicago, IL). Kullback, S., 1959, Information theory and statistics (Wiley, New York). Laitinen, K. and H. Theil, 1978, Supply and demand of the multiproduct firm, European Economic Review 11, 107-154. Theil, H., 1972, Statistical decomposition analysis with applications in the social and administrative sciences (North-llolland, Amsterdam). Theil, H., 1979, The system-wide approach to microeconomics (University of Chicago Press, Chicago, IL). 5 The marginal cost of information depends on the prices of certain goods and services which the decision maker buys in order to acquire his information. In the limiting process c’ -+ 0 we imagine that these prices converge to zero. For a proof of the convergence of (3) to (12) see Theil (1979, ch. 8). Random behavior generated by (12) is formally equivalent to large-sample maximum likelihood estimation of the vector x (with a large sample interpreted as a small marginal cost of information). 6 Since the covariance matrix c’A_’ converges to zero as c’ + 0, (12) implies that the decision made converges in probability to X, which agrees with intuition. Also, for any po( ) which takes a positive value for each x EJ, Io increases beyond bounds as c’ * 0 when p(x) in (6) is interpreted as p(x). This means that the left-hand side of (8) also increases beyond bounds. What happens on the right depends on the assumptions made on the limiting behavior of c( ), but it seems plausible that weak assumptions are sufficient to ensure that (8) is violated when c’ + 0, thus implying the superiority of the asymptotic normal decision distribution (12) over the prior distribution.