Comparing iterative planning procedures

JOURNAL

OF COMPARATIVE

ECONOMICS

Is,‘@

l-499

( 199 1)

COMMENT Comparing Iterative Planning Procedures’ P. CHAIWER Indian Statistical Institute, New Delhi 110 016, India AND

T. R. KUNDU Kurukshetra University, Kurukshetra 132 I1 9, India Received May 10, 1990; revised December 7, 1990

Chander, P., and Kundu, T. R.-Comparing

Iterative Planning Procedures

This comment reconsiders the comparison of E. Malinvaud’s (in E. Malinvaud and M. 0. L. Bacharach, Eds., Activity Analysis in the Theory of Growth and Planning, pp. 170-208. New York: St. Martin’s Press, 1967) and P. Chander’s (Econometrica 46,4:761-777, July 1978) iterative planning procedures recently presented by S. J. Clark (.I. Cornp. Econom. 13, 1:61-84, Mar. 1989). It demonstrates clearly the effect that increasing the number of produced goods has on the costs of these procedures. It considers explicitly the setup costs of putting in place the institutional requirements of these procedures. It also shows that Chander’s procedure can be suitably modified so as to have a positive interpretation as well, in that it can be claimed to correspond to the competitive price adjustment process. J. Comp. Econom., Sept. 1991,15( 3), pp. 49 l-499. Indian Statistical Institute, New Delhi 110 0 16, India; and Kurukshetra University, Kurukshetra 132 119, India. o 1991 Academic Press, Inc.

Journal of Economic Literature Classification Numbers: C63, 02 1, P2 1.

1. INTRODUCTION Clark ( 1989) presents a comparison of Malinvaud’s ( 1967 ) and Chander’s ( 1978b) planning procedures, each designed to operate in environments known to satisfy the assumptions of Leontief-Samuelson technology. Following a method, with due modifications, due to Hurwicz ( 1972), Clark ’ The authors acknowledge valuable suggestions by an anonymous referee of this Journal. 491

0147-5967191 $3.00 Copyright 0 I99 I by Academic Press, Inc. All rights of reproduction in any form reserved.

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simulates the two procedures on a computer and evaluates them for each environment and for each specification of their organizational or operational costs using a given utility function.’ In order to appreciate the significance of this comparison, it may be useful to first clarify why simulations are necessary for resolving which procedure is better. We proceed by introducing some notation. Let TM and T, denote respectively the number of iterations in which Malinvaud’s and Chander’s procedures converge, and let (Yand p denote respectively the cost incurred at each iteration of Malinvaud’s and Chander’s procedures, except at the last iteration of the latter, when tasks involved are the same as under the former. The total operating costs under the two procedures then amount to

and

&=/3(T,-

l)+a,

respectively. Clark ( 1984) proves that

TM G T,. Since Malinvaud’s procedure is informationally intensive, one also has, to begin with, ff 2 p.

(1) and computationally

more (2)

Given ( 1) and (2)) it cannot be resolved whether x&f sx,.

(3)

In fact, examples can be constructed so that X, > Xc or Xi,, < Xc. As argued by Clark, the best way to resolve (3) is by means of simulations. Note that if a! # p, then there is really nothing to be resolved. There is no need for any simulations in that case.3 Moreover, as noted below, aa(n+

l)&

where n is the number of produced goods in the economy. Clark, however, ignores this relationship between (Yand /3 and sometimes even assumes (Yto be less than /3. Consequently, his Tables 8 and 10 include a large number of entries that are irrelevant. ’ The idea of such a comparison may be traced back to Chander ( 1978a). The comparison became all the more compelling after Clark ( 1984) established the following result: Chander’s procedure converges slowly compared with Malinvaud’s procedure, though the latter is informationally and computationally more intensive. 3 Also note that if either Thl = 1 or Tc = 1, then, by ( 1 ), X, =ZXc. Therefore, simulations are needed only if 7’M > 1 and Tc > 1.

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Another aspect of the comparison put forward by Clark is that it is rather incomplete. The comparison is performed in terms of operational or variable costs only. It does not take into account a procedure’s fixed costs, i.e., those costs that do not depend on how many times the procedure is operated, but are incurred once and for all. Typically these costs might include those of restructuring institutions, creating channels of information, and providing for computational and other technical facilities, depending on the operational requirements of the procedure. As we note in Section 3, the two procedures differ significantly in respect to these costs. Hence, they should also be taken into account when one is deciding which procedure is better. The purpose of this comment is twofold: first, to provide a correct view of the comparison by deleting the aforementioned irrelevant entries from Clark’s Tables 8 and 10 and then to make the comparison more complete by assessing the procedures further in terms of their fixed costs. 2. OPERATIONAL COSTS: A RE-COMPARISON In this section, we note that (Y2 (y1+ 1)p and reconsider the comparison in Clark in light of this relationship. Clark looks at (Yand fi as though these were independent of each other, but this view of operational costs of the procedures is not correct. In fact, for the class of procedures of which Malinvaud’s and Chander’s procedures are special cases, it has formally been established that informationally, Chander’s procedure is the most efficient one (Chander, 1983). So, one has the following condition, to start with: a! > p. In order to get a better approximation of the exact magnitude by which (Y exceeds & one should simply take note of the difference between the informational and computational requirements of the two procedures. Under Malinvaud’s procedure, at each iteration t , firms collectively send an 12by n matrix A’ and an n-dimensional vectorf’ to the center, and the center calculates fi* = f’[Z - A’]-’ and sets p*+’ = $‘.” In Chander’s procedure, at each iteration t (t # rc), firms collectively send an n-dimensional vector qf , and the Center simply sets p’+’ = q’. Now let us assume that the cost of transmitting an n-dimensional vector message is y(n) and the cost of inverting an n by n matrix is 6(n) (20). Then P = r(n), and CY= (n + 1)7(n) + 6(n) = (n + 1)p + 6(n). (4) Thus (Y 2 (n + l)@. And since, in Clark, n = 3, in particular, we have a 3 4p. (5) Clark claims that the dependence of (Y on p and y1does not affect the consequent choice of a best procedure in Tables 8 and 10. To quote him, “In 4 Unless explicitly stated, our notation is the same as that in Clark ( 1989).

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TABLE 1

B

0.0015

0.0030

0.0045

0.0060

0.0075

0.0090

0.0105

0.0120

0.0135

0.0150

0.0003 0.0006 0.0009 0.0012 0.0015 0.0018 0.002 1 0.0024 0.0027 0.0030

c

c C

c c C

c c C C

c c C C

c c C C

c c c c

c c c c

c c c c

c c c c

M

c M

c C M

c c C M

c c C C M M

c c C C C M

c c C C C C

0 Table 8 in Clark (1989) after the elements a i 48 have been deleted. M indicates that Malinvaud’s procedure is best, and C indicates that Chander’s procedure is best. Scores Procedure

Original

Revised

Malinvaud’s Chander’s

45% 55%

11.30% 88.70%

terms of the framework of this paper it is perfectly acceptable to allow (Yand B to depend on n. When faced with choosing a best procedure then if n is already determined so are (Yand p; thus Table 8 or 10 still determines which procedure is best.” Clark thus seems to believe that the specification of particular values for (Yand /3 in these tables is consistent with ( 5 ) . This is however not true. A scrutiny of these values reveals that several of them are not consistent even with (2). Tables 8 and 10 in Clark’s work therefore need to be reconsidered. In the first place, all those elements in these tables which correspond to (Yand @ values such that (Y< @must be excluded. Further, in view of ( 5 ) , all elements corresponding to (Yand /3 such that (Y< 4@should also be excluded. Accordingly, we present here in Tables 1 and 2 the respective truncated versions of Tables 8 and 10 in Clark ( 1989) obtained after deleting all such entries. The contrast between the original and the revised score of a procedure under Tables 1 and 2 indicates that deletion of the irrelevant entries changes the comparison significantly, moving it more in favor of Chander’s procedure. Note that in sorting out the irrelevant entries we have assumed that s(n) > 0 and thus in effect ignored the computational costs under Malin-

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COMMENT

TABLE 2 Bns~ PROCEDUREJUDGEDBYMINIMUMUTILITP

P

0.0015

0.0030

0.0045

0.0060

0.0075

0.0090

0.0105

0.0120

0.0135

0.0150

0.0003 0.0006 0.0009 0.0012 0.0015 0.0018 0.002 1 0.0024 0.0027 0.0030

M

c M

c C M

c c C M M

c c C C M M

c c C C M M M

c c C C c M M M

c c c C c M M M M M

c c c C c C M M M M

c c c C c C C M M M

’ Table 10 in Clark (1989) after the elements (Y -z 40 have been deleted. M indicates that Malinvaud’s procedure is best, and C indicates that Chander’s procedure is best.

Procedure

Original

Revised

Malinvaud’s Chander’s

63% 37%

40.30% 59.70%

vaud’s procedure. If 6( n) > 0, as it always should be, it will further move the comparison in favor of Chander’s procedure. A word about which of the two alternative approaches, i.e., Table 1 or 2, is more appropriate for making the final choice may also be in order. Table 2, which uses the criterion of minimum utility, presumes risk aversion on the Center’s part. This is clearly a strong assumption. The Center is not an individual, but a panel of individuals, usually with varying attitudes toward risk. Table 1, using the criterion of expected utility, therefore may be more appropriate for making the final choice. Given the specification of costs in (4)) one can also see the effect of IZ on the choice of procedure. From (4)) we have

which tends to infinity with n irrespective of the behavior of 6( n) and y(n). Furthermore, given the implicit assumption that the increase in y(n) is proportional to the increase in n, and the fact that the increase in 6(n) is more than proportional to the increase in n, the ratio a/P should indeed increase

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with n all the way to infinity. This consideration, when taken in conjunction with the evidence suggesting that the size of n makes only a minute difference to the speed of convergence of each procedure (Clark 1989, p. 8 1)) makes Malinvaud’s procedure even less attractive when n is large, thus justifying our claim in Chander and Kundu ( 198.5).5 3. FIXED

COSTS

We now assessthe two procedures in terms of their fixed costs as defined earlier. We do this by examining how the price adjustment processes under the two procedures are institutionally different from each other. It can easily be appreciated that price adjustment as per Malinvaud’s iteration, P t+1 = j’ =f’[l-

A’l-1,

t=

1,2,...,

involves a substantial change in the initial institutional setup, i.e., the setup obtained in the absence of an explicit procedure. At each iteration t, prices are set so as to make (A’, f’) just break even. Thus the price adjustment is completely centralized, with all prices being revised simultaneously and individual firms playing no independent role whatsoever in setting the prices. By all means, this represents a very special kind of price adjustment arrangement, which would hardly be observed in the absence of the procedure. On the other hand, price adjustment as per Chander’s iteration, P

t+1 - 4,

t=

1,2,...,

involves no significant change in the initial institutional setup. At each iteration t firms collectively send the vector q’ of minimum average costs that they are able to achieve at the prices of the previous iteration, and the center simply sets p’+’ = q’.6 Thus prices are virtually determined by individual firms themselves, rather than by the Center, whose role in the price adjustment as such is purely fictitious. It is in fact perfectly admissible for the Center to let firms themselves carry out the price adjustment process by using only horizontal information flows. Accordingly, at the end of iteration t , each firm communicates the minimum average cost of production not to ’ To give an idea about the size of the n that actual planning procedures have to deal with, Clark cites the following example. In the preparation of the 198 I - 1985 Five Year Plan for the USSR, Gosplan, the Central Planning Authority, constructed roughly 400 material balances; for the 198 1 annual plan the number was more than 2000 (Kushnirsky, 1982, p. 112). 6 Note that as the price of each good is revised by setting it equal to the minimum average cost of production, each production unit can really be seen as an independent agent revising the price of the good it produces. Thus the price adjustment is perfectly in accordance with the competitive price adjustment, where any profits in excess of the minimum average cost are eliminated through forces of competition.

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the Center, but to all other firms. These are used as input prices at the beginning of iteration t + 1. This suggests that the simultaneous revision of prices as specified in Chander is only a formal description of the price adjustment process. It can be easily seen that at no iteration do prices need to be revised simultaneously or even in any specific order. To this end, consider the difference equation system7 p):+’ = p’a: + f i < p’a, +fi P:”

PZ

=

(pi+‘,

P:,

ph..

G (P:“,

~5,

P:,

=

pi+‘,

. . . , PC%P~,,

P:“,

. . . , PZ,

(pi+‘,

G (P:“,

for all( a,, fi)cs,

. , pi)&

+f

. . . , p%

+fi

i

for all(a,,f,)ts,

it,

. . . , pt)&,

+f

PA

. . . , p’nh-,

+A,

PL,,

L,

for p;+’

=

Wak-l,fk-I)+I

(p:+‘, pi+‘, . . . ,~~~,,~:,~:,,,...,p:,)a:+f:

G (P:+‘,

pi+‘,

. . . , pfkf,,

pZ,pi+ly

. . . , pL)a,

+fk

for all(ak,fk)e.sk P%

=

(pi+‘,

P:+‘,

. . . , PZPL

G (P:‘“,

pi+‘,

. . . , pi?,

PL

PL

. . . ,pS4+,

+f

PL

. . . pi&+,

+fk+l for

pi+’ = (pi+‘, p:“, G (p:“‘,p;+‘,

. . . , p’,‘_‘,, pi)ai

+

. . . , Cl,

+f,

pf>a,

i+,

all

(ak+,

yfk+l

)a+l

f i

for all(a,,f,)~s,,

wheret= 1,2,..., and p1 >, p’A” + f” for some (A’, f O). Note that for each t , the ordering of firms and goods is purely arbitrary and the iteration can be stopped at any of the n-substeps.* Then it can easily be shown that the sequence { pr } generated by this system is nonincreasing and converges to the equilibrium price vector p* (for details, see Kundu ( 1985 )). ’ Notation: (i) (ak,fk) denotes an activity to produce good k, k = 1,2, . . . , n, where uk andf, refer respectively to the kth column and the kth component of A and Sin (A, /) ; (ii) s, denotes the set of all activities available to produce good k. * As a strictly rigid cyclic process with all the n-substeps being performed together in a fixed cyclic order, the above system incidentally defines what is popularly known in the literature as the Gauss-Seidel process.

498

CHANDER

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Thus, prices can be revised separately in an arbitrarily chosen order, by setting each equal to the respective minimum average cost of production determined at the latest prices. In light of the considerations discussed above it should be clear that the price adjustment under Chander’s procedure imposes no significant institutional constraint. The institutional setup that is necessary for executing Malinvaud’s procedure is in any case sufficient for Chander’s procedure. To sum up this section, the two procedures differ substantially in respect to their institutional requirements, the fixed costs of Chander’s procedure being the minimal. 4. CONCLUDING

REMARKS

We have demonstrated the effect that increasing the number of produced goods has on the costs of the two procedures. We have also considered explicitly the setup costs for putting in place the institutional requirements for these procedures. This, however, does not exhaust the list of costs appropriately compared in such procedures. In particular, we have not considered the costs of enforcing the prescribed rules and verifying the truthfulness of the messages between the firms and the Center or among the firm~.~ In Chander’s procedure, each firm is required to report to the Center its average cost at the current prices, which it knows will be set equal to the price of its product in the next iteration. Thus there is an obvious incentive for each firm to overstate its average cost. However, the scope for doing so is limited as the average cost at an iteration cannot be claimed to be higher than that at the preceding iteration. In the case of Malinvaud’s procedure, each firm is required to report the input coefficients of its cost-minimizing production activity at the current prices and it knows that prices at the next iteration will be set to be equal to the average cost corresponding to this activity. Each firm therefore will have an incentive to overstate the input coefficients of its cost-minimizing activity and will be able to achieve the same effect as in the case of Chander’s procedure. Though the costs of enforcement and verification are an important aspect of planning procedures, their inclusion, as made clear above, does not affect the comparison in the present case. For this reason we have restricted ourselves to a comparison in terms of operational and fixed costs only. REFERENCES Chander, P., “The Computation 1978a.

of Equilibrium

Prices.” Econometrica 46, 3:123-726, May

9 Our attention was drawn to this by an anonymous referee of this journal.

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Chander, P., “On a Planning Process Due to Taylor.” Econometrica 46,4:76 i-777, July 1978b. Chander, P., “On the Informational Size of Message Spaces for Efficient Resource Allocation Processes.” Econometrica S&4:919-938, July 1983. Chander, P., and Kundu, T. R., “A Note on a Class of Planning Procedures.” Technical Report 8508, Indian Statistical Institute, 1985. Clark, Simon J., “Informational and Performance Properties of a Class of Iterative Planning Procedures.” Rev. Econom. Stud. 51,4:6 15-63 1, Oct. 1984. Clark, Simon J., “Comparing Iterative Planning Procedures: A Proposed Method and Some Numerical Results.” J. Comp. Econom. 13, 1:61-84, Mar. 1989. Hurwicz, Leonid, “On Informationally Decentralized Systems.” In Charles B. McGuire and Roy Radner, Eds., Decision and Organisation, pp. 297-336. Amsterdam and London: North Holland, 1972. Kundu, T. R., “A Characterization of Computational Procedures for Equilibrium Prices in the Generalized Leontief Model.” Sankhya Ser. B-2 47: 259-27 1, 1985. Kushnirsky, Fyodor I., Soviet Economic Planning, 1965-1980. Boulder, Co: Westview Press, 1982. Malinvaud, E., “Decentralised Procedures for Planning.” In E. Malinvaud and M. 0. L. Bacharach, Eds., Activity Analysis in the Theory of Growth and Planning, pp. 170-208. New York: St. Martin’s Press, 1967.