informational and performance properties of a class of iterative planning procedures

The Review of Economic Studies, Ltd.

Informational and Performance Properties of a Class of Iterative Planning ProceduresAuthor(s): Simon ClarkSource: The Review of Economic Studies, Vol. 51, No. 4 (Oct., 1984), pp. 615-631Published by: Oxford University PressStable URL: http://www.jstor.org/stable/2297782 .

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Review of Economic Studies (1984) LI, 615-631 0034-6527/84/00410615$02.00

? 1984 The Society for Economic Analysis Limited

Informational and Performance

Properties of a Class of Iterative

Planning Procedures SIMON CLARK

University of Edinburgh

This paper analyses a class of iterative planning procedures that can be applied in environments describable by the Leontief-Samuelson technology. Members of the class are distinguished by the extent of the communication of technical information from firms to the Centre. All members of the class are monotonic and convergent, but the speed and finiteness of convergence is shown to depend critically on the extent of the transfer of technical information throughout each procedure. The paper thus establishes a trade-off between the informational and performance properties of a class of resource allocation mechanisms, taking environmental coverage as given.

1. INTRODUCTION

The literature on iterative planning procedures has produced a large number of theoretical models, each with different properties. A useful heuristic framework for comparing procedures is to consider a three way classification of their properties: (i) performance properties, relating to convergence, monotonicity, feasibility and the like; (ii) informational properties, relating to the nature, extent and use of messages between agents; (iii) environmental properties, relating to the kind of economic environment in which procedures can operate and achieve a desired result.

A question that now naturally arises is: does there exist some sort of efficiency frontier in a space with appropriate measures of these properties on the axes? One approach to this question is to identify the properties of known procedures in an attempt to locate points on the efficiency frontier (see for example, Table 6.1 in Cave and Hare (1981), page 114). Alternatively, one may ask questions of the type: if a procedure is to operate in a certain environment, and is to reach a certain performance standard, is there a limit to the "size" or "complexity" of the messages that must be sent between agents? Examples of this second method can be found in Hurwicz, (1959) and Mount and Reiter (1974). Mount and Reiter concentrate on analysing the trade-off between environmental coverage and informational requirement, taking performance as given (specifically, the property they take as given is that of Pareto Satisfactoriness (Mount and Reiter (1974) page 32)). This can be interpreted as taking a cross-section of the efficiency frontier at a particular point on the "performance" axis.

Clearly, one can examine different cross-sections of the efficiency frontier. In this paper I look at the trade-off between performance and information transfer: I take the environment as given, and see how the transfer of more or less information between agents affects the performance properties of a given type of planning procedure. The performance properties I analyse are those of speed and finiteness of convergence, and the environment considered is that describable by the Leontief-Samuelson technology, i.e. I assume constant returns to scale, no joint production, one primary good and no externalities or public goods.

615



616 REVIEW OF ECONOMIC STUDIES

Under these conditions on the environment, the Nonsubstitution Theorem allows a restatement of the planning problem in terms of finding a unique equilibrium vector of commodity prices, the price of the scarce primary factor being taken as numeraire. The collection of activities that minimise production costs at these prices constitutes an optimal technique that in turn minimises the use of the scarce factor for any given vector of final demands. Final demand can therefore be determined when the equilibrium price vector or, as I shall show, an approximation to it has been found.

The procedures described and investigated in this paper are iterative methods of finding the equilibrium vector and involve a sequence of messages between the Centre and firms. Except at the end of the procedure, the message of the Centre is always one of prices, whereas the messages of firms involve costs and some technical information. Different members of this class are distinguished by the extent of this transfer of technical information, which has critical implications for the speed and finiteness of convergence to the equilibrium price vector.

2. TECHNOLOGY, PRICES AND PLANS

2.1. Technological assumptions

According to the model of technology adopted here, each good can be produced by a number of activities, subject to constant returns to scale. There is no joint production and only one primary (i.e. non-producible) resource, which I call labour.

Let there be n goods, where, for simplicity, good j is produced only by firm j (j= 1, 2, .. ., n). Each firm has available to it a finite set of production activities. A technique is a collection of n activities, one producing each good and is represented by a pair (A,f), where A = (ai) is a square matrix of order n, f = (fi) is a row vector in R', and aij and f are respectively the input of good i and the amount of labour required to produce a gross output of one unit of good j. The set of all available techniques is denoted by S; it is the product of the n sets of activities available to firms and is clearly finite.

The technique (A, f) is said to be productive if [I-A] has a non-negative inverse, in which case for a vector y of net outputs the necessary production vector x using that technique is given by x = [I - A]-'y, and the labour required is fx =f[I - A]-'y. I denote by Q the set of all productive techniques and I assume it to be non-empty. It is clear that if (A,f) e S, then A ?0 and f-' O.' I shall in fact assume something stronger than this, namely that all activities use positive am6unts of labour. Formally

f>0 forall(A,f)eS. (1)

2.2. Prices and the Nonsubstitution Theorem

The Nonsubstitution Theorem is a well known result of the assumptions made so far, and it is of fundamental importance to the planning procedures discussed in this paper. To draw out its full power, it is useful to use the framework of Chander (1978). Here, and throughout the rest of the paper, the wage rate is set equal to one.

Definition 1. p is the price vector associated with the productive technique (A,f) if pA +f = p, i.e. if p =f[I -A]-'.



CLARK ITERATIVE PLANNING PROCEDURES 617

Definition 2. The price vector p is an equilibrium price vector if p=pA+f forsome (A,f)eQ (2) p pA+f forall (A, f)e S.

The main characteristics of the Leontief-Samuelson technology may now be described.

Theorem 1 (Nonsubstitution Theorem). There exists an optimal technique (A*, f*) E Q such that f*[I -A*]-lyf[IA]-ly for all (A,J)e Q, and allyE R+.

A proof of this version of the theorem is given in Chander (1974). Now, let p* be the price vector associated with (A*,f*). Then two corollaries follow from Theorem 1 almost immediately (they are proved in Chander (1978)).

Corollary 1. p* is an equilibrium price vector.

Corollary 2. (i) p*'f[I-A]- for all (A,J )e Q; (ii) p* =f[I-A]-' if and only if (A, f) is also an optimal technique.

Corollary 2 shows that the equilibrium price vector is unique even though there may be more than one optimal technique.

2.3. The planning problem

To complete the picture, I assume (i) that there is a central planning agency, "the Centre", which wishes to maximise u(y), a continuous and strictly increasing function of net outputs, (ii) that the available endowment of labour is fixed at L and that this is known by the Centre, (iii) that the Centre has virtually no knowledge of the technical capabilities of firms, and (iv) that all firms are honest and obedient.

A plan can be represented by a triple (x, (A, f), y) denoting output levels, the technique to be used, and consumption levels and it is feasible if x - Ax ' y, fx ' L, (A f) E S, and x, y, 0. The task facing the Centre is to choose a feasible plan to maximise u(y), but its problem is that it starts off in ignorance of what techniques are in S. The procedures discussed in this paper are designed to overcome this ignorance.

Consider what the Centre could do if it knew p*. By asking firms for the coefficients of the activities that minimise costs at p*, it can construct A* and f*, since by definition p*A* +f* p*A +f for all (A, f) e S. It then follows that if y* maximises u(y) subject to p*y ' L and y ? 0, then (x*, (A*, f*), y*) is an optimal plan, where x* = [I -A*]- y. Thus, the planning problem can be reduced to finding p*: once this is known, constructing the plan is a relatively simple affair. The procedures of Malinvaud (1967) and Chander (1978), and those discussed in the paper, are all methods of finding p*, but they differ in the amount of technical information that is sent by firms to the Centre. This is explained in the next section, where those properties common to all the procedures are discussed. In Section 4 intra-class differences are examined. It is here that the trade-off between information transfer and the speed and finiteness of convergence to p* is examined.

3. THE PROCEDURES AND THEIR COMMON PROPERTIES

Firstly, an explanation of the notation used. The superscript of any symbol refers to the time or stage of the procedure, and the subscript refers, where necessary, to a particular




member of this class of planning procedures. For example, by pM I mean the vector of prices sent out at stage t under Malinvaud's procedure. (A',f') is the technique chosen at stage t: since it turns out that it is productive, it has an associated price vector, denoted by ^'t.

3.1. How the procedures work

The general class of procedures whose properties we are going to investigate may be described as follows. At stage t, the Centre sends out to all firms a vector of prices pt, and asks them to minimize costs at those prices. The n cost minimising activities thus chosen define a technique (At,ft), and the vector of costs achieved may be written as qt = ptAt +ft. (At,ft) minimises average costs at prices pt, so

qt= ptAt +ft ? 'ptA +f for all (A,f)E S. (3)

The type of message from the Centre to the firms is invariant over the class of procedures, and it is the type of reply from firms to the Centre that distinguishes one member of the class from another. The message of each firm is composed of two parts: firstly, the minimum cost that it can achieve at prices pt; secondly, technical information relating to the non-labour coefficients of the activity chosen at stage t. How much technical information depends on the rules of each particular procedure. Under Malinvaud's procedure firms send back all their coefficients, so the Centre knows qt and At. Under Chander's procedure, firms report qt and nothing else. Clearly there is room for intermedi- ate cases. Other possible rules for the transfer of technical information are:

(a) "Report only the coefficients of the following inputs...." (b) "Report only the coefficient of the good you produce yourself." (c) "Report the coefficient of the input that contributes most to costs." (d) "Report the coefficients of the inputs that contributes more than x% to costs." (e) "Report no technical information before stage h, report all coefficients then and

thereafter."

To formalise: the matrix At can be written as the sum of two non-negative matrices, Rt and Nt:

Rt+Nt=At; R't-0, N't0. (4)

The message of the i-th firm to the Centre is composed of two parts: firstly, the minimum cost that it can achieve at prices pt (i.e. the i-th element of qt), and secondly, the i-th column of Rt.2

Clearly there are an infinite number of ways in which any matrix At can be divided according to (4). A complete categorisation of possible rules is probably impossible, but the following should be noted: for a particular procedure to be well defined, each firm must know what message it has to send to the Centre when it has chosen a cost-minimizing activity. Since no messages are sent between firms, each firm's message to the Centre is independent of the activities chosen by other firms at the same stage. Each procedure is thus distinguished by its set of n rules, one for each firm; a rule for firm i defines unambiguously the message that firm i must send to the Centre at any particular stage. In addition, it is clear from the examples above that the Centre must know the source of each message. Thus I have dispensed with "anonymity", often cited as an essential requirement of decentralised planning procedures.




From the firm's messages at stage t, the Centre knows q' and R'. It can calculate the vector (p'N' +f'), since from (3) and (4) this is equal to q' -p'R'. Prices to be sent out at the next stage are calculated according to the formula

p = pt Rt +ptNt +ft, (5)

or equivalently, since it can be shown that [I - R t] is non-singular,

pt+1 =(ptNt +ft)[I - Rt] 1. (6)

Chander's and Malinvaud's procedures can now be shown to be, in a sense, the two extreme members of this class of procedures.

Under Chander's procedure RC = 0 and Nt = At. In this case (6) reduces to

pC' =

qt (7)

Under Malinvaud's procedure RM = At and Nt =0. In this case (6) reduces to

PM' =f Mf1 -AM] (8)

i.e. t+1 At 4

PM = PM.

The similarity of form of equations (6) and (8) suggests the following interpretation of the mechanics of price formation: at each stage the Centre attempts to approximate p', and the contribution to costs of non-reported coefficients, ptNt, is treated in the same way as direct labour costs, ft. The more information it has (the larger are the elements of Rt given At) the better the approximation (this is formally proved in Theorem 4).

All procedures are started on the assumption that the Centre knows of a set of prices p' ?0, at which at least one activity for each good is weakly profitable, and this vector is the first one to be sent out by the Centre.5 Formally

p' ?pp'A0+f0 forsometechnique(A?,f0). (9)

Equations (3), (4), (5) and (9) define a sequence of prices {p'} (t = 1, 2,. . .) and an associated sequence of techniques {(At, ft)} (t = 0, 1, 2,. . .). The following properties can be established.

Theorem 2. (i) The sequence {p'} (t = 1, 2,. . .) is non-increasing, non-negative and converges to p*;

(ii) All techniques in the sequence {(At, ft)} (t = 0, 1, 2, . . .) are productive.

Corollary 3. If at any stage pt = p t , then p' = p*.

Corollary 4. For some finite number r, t - r implies that Tt is an optimal technique.

The proofs of these propositions are given in Appendix 1. They are rather tedious, and rely essentially on the Debreu-Herstein inversion lemma and the manipulation of vector inequalities.6'7 For later use note the following relations:

t> qt>

t+1> __t p*>O. (10) p =q p

Interestingly the inequality 1y _ ftt+l does not hold in general, although it does, of course, under Malinvaud's procedure.




3.2. Feasibility of arbitrary truncation

Corollary 3 enables the Centre to know when p* has been found, so that it can start the process of plan construction outlined earlier. However, there is no assurance that p* will be found within a finite period, even though firms continue after some point to choose activities that constitute an optimal technique. It is important therefore to investigate the properties of these procedures when they are truncated at some stage T such that pT 0 p*.

The last price message to firms is p '; but the final message of the k-th firm to the Centre is now the k-th column of At, and the k-th element offT i.e. the Centre indicates to each firm that the procedure is being truncated, and that it requires all the coefficients of the most recently chosen technique. The Centre now has the knowledge to construct a feasible plan. Let yT maximise u(y) subject to pTy C L and y -0, and set xT=

[I-a AIyT. Since (A fT) is productive, x=_ 0. Furthermore, fx =f [--Alv = ATy T.< Ty T X TfT T p y 'p y L In short, the plan (x, (A',fT), y ) is feasible.

3.3. Monotonicity

As prices never increase at any stage, delaying truncation cannot reduce the utility of the final plan, i.e. u(yT)?u(yT+I). The procedures are thus monotonic. In fact, u(yT)< u(yT+l) if pT 0 p*, so there is always some advantage to be gained by delaying transaction if the sequence of prices has not yet converged. This property of strict monotonicity follows from (i) the assumption that u( ) is increasing in all its arguments and (ii) the result that pT=pT+I implies that they both equal p*. Finally, since {pT} tends to p*, then {u(yT)} tends to u(y*), so the procedures are convergent (see Malinvaud, (1967) p. 178).

4. INTRACLASS DIFFERENCES

Up to this point, the properties that I have discussed have been true of this entire class of planning procedures and I now proceed to examine how different members of the class differ from each other, and what consequences the differences have.

In fact, they differ in one basic way: the amount of information that is sent to the Centre at each stage. The transfer of information is unlikely to be costless and for a more information intensive procedure to be used, it must be shown that there is some offsetting advantage. It is not the purpose of this paper to provide a complete analysis of the costs and benefits of the transfer of information in the use of these procedures, but merely to suggest how these advantages might be obtained to a greater or lesser degree.

4.1. The speed of convergence

Before comparing different procedures, I shall adopt the following convention to ensure that the ceteris paribus assumption holds. (a) the procedures under comparison are operated in identical technological environments; in particular, the number of goods (and firms) are the same, and the technology available to firms is the same, irrespective of the procedure used; (b) all procedures start from the same initial price vector, i.e. p' =p for any two procedures A and B. This convention is adopted throughout the paper.

With these two assumptions, one possible method is to construct the indirect utility function V(p) = sup [u(y): py ? L, y ?0]; the benefits of procedure A could then be said to be at least as great as those of procedure B if V(pA) > V(pB) for all t. This would




certainly be the case if pA- pB for all t, but the sign of V(pA) - V(pB) if pA and pB are not comparable in a vector sense clearly depends on the form of u(y).

In order to compare procedures independently of the utility function, I adopt the following stronger criterion: the benefits of procedure A are no less than, and possibly greater than, those of procedure B if for any initial price vector pl =i P, and for any technology satisfying the assumptions of Section 2.1, pt -pB for all t.

I now proceed to a proper comparison of procedures. The first result is the following theorem.

Theorem 3. No procedure has benefits less than Chander's procedure.

Proof Let procedure A be any procedure. Then it is sufficient to prove that if PA --pt, then PA+I <=Pc 1, since by assumption PA =PC. It follows that PA 'PC for all t.

Consider the vector of minimum average costs achieved at stage t under procedure A.

qt = p At +fAt-?ptA+f forall(A,f)e S. In particular, qt ? ptA' +Jfc i.e. the minimum cost achieved by using technique (AA, fA)

is no more than the cost achieved by using (At fJc) which is what firms choose under Chander's procedure. (Different techniques are of course chosen in response to different prices). By assumption, pbt ? pA t so qA - ptAt +fC. But p A' +f'c = qC - ptc l, so qA_

ps'. From (12), p'+1 qA and therefore pt+) _p"1. This proves the theorem.8 j|

It is tempting to think that there is a dual to Theorem 4, i.e. that the benefits of Malinvaud's procedure are no less than those of any other procedure, so that PM -=PB

for all t and for any procedure B. However, this is not necessarily so. Nevertheless, I prove some weaker results, which go some way to offset this, and they are presented as Theorems 4, 5 and 6.

Theorem 4. At any stage t of a procedure, if pt has already been determined, an increase (decrease) in any element of Rt will not increase (decrease) any element of pt+I.

Proof Equation (5) may be rewritten as

pt+I[I-Rt]=ptNt +,P.( l

Consider now a variation in the rules of the procedure, so that R' + Et is reported when (At, ft) is the cost minimising technique, where 0 c Rt +E' F At. Then at the next stage there would be a different price vector, say pt+l, defined by

jt+[I-Rt -Et]=pt[Nt-Et]+ft. (12)

Rewriting (11) by subtracting pt +Et from both sides gives

Pt +I[I - Rt - E t] = p tN t- p t + E t+ft (13)

Since 0-Rt+Et-At and (At,ft) is productive then [I-Rt-Et]-' exists and is non- negative (see Lemma 1 of Appendix 1). Then (12) and (13) together yield

t+1 -_ t+1 t t+1)Et[I-Rt_Et]- .

I have already shown that pt ? pt+Is so if Et is non-negative then pt+l ?fit+l and if Et is non-positive, then Pt+ ?P t+l. This proves the theorem. II




Two interesting points now follow. Firstly, it can be seen that if p' =pt+l, then E', however large, will have no effect. This is because ptl = p*, and prices can fall no further. Secondly, if at any stage t the full matrix of technical coefficients is sent to the Centre, i.e. R' = A', then the maximum possible reduction in prices will have been achieved. It is this that suggests that under Malinvaud's procedure the price vector at any stage is always as low, and possibly lower, than under any other. However, Theorem 4 took as its starting point a given vector p'. If one compares two procedures, procedure A and Malinvaud's, and if pM <--p it is not necessarily true that p"' - _pPA+; there is the interesting possibility that Malinvaud's procedure may, at some stage, be "overtaken" by another. An example of this is given in Appendix 2.

A further result of Theorem 4 is that it confirms the inequality pt'+ ?ft in (12), and because of this it is possible to construct a more efficient method of plan formulation. As before, the final price message to firms is p', and the final messages of firms to the Centre enable it to assemble AT and fT. The Centre now has the information to formulate the plan X (A ,fT),5T) where 9T maximises u(y) subject to 'Ty L and y _ 0, and AT ATAY L, adAt

x =[I-AT] y . As u(y) is increasing in all its arguments, pT Y=L, and therefore fTXT = L, i.e. this plan ensures full employment; this can not be guaranteed if pT rather than AT is used in the utility maximisation problem.

Clearly, u(AT)?> u(yT). Against this possible gain must be set the extra cost of calculating p^T, but this only involves n vector products, since [I - AT]-I must be calculated anyway. A further point is that since it is not generally true that pA, itpt

A then V(pt) does not necessarily increase at each stage, so procedures using the revised truncation method are not always monotonic. This would appear to be a trivial disadvantage, since V(p^') dominates V(p'), which also ensures that the revised procedures are convergent.9

4.2. The finiteness of convergence

I have shown that all these procedures result in convergence to the equilibrium price vector, but it is clearly of interest to know whether this ever occurs after a finite number of stages. It must be stressed that all the results of this section depend critically on the assumption that the set S, and hence Q, is finite.

Malinvaud defines a procedure X as finite if there exists a finite number t that p' is equal to p* (Malinvaud (1967) p. 178, my notation). But this definition is ambiguous. Whether a particular procedure actually does result in finite convergence may depend not only on the rules of the procedure but on other circumstances as well: the initial price vector, for example, or the activities actually available to firms. In order to be rid of this ambiguity, I put forward the following definitions.

Definition 3. A procedure X is assuredly finite if, for any technology satisfying the assumptions of Section 2.1 and for any initial price vector p. satisfying (9) there exists a number t such that p' equals the equilibrium price vector of that technology. A

procedure is potentially infinite if it is not assuredly finite.

Definition 4. A procedure X is potentially finite if, for some technology satisfying the assumptions of Section 2.1 and for some initial price vector p' satisfying (9), there exists a number t such that p' equals the equilibrium price vector of that technology. If a procedure is potentially finite only if the initial price vector is the equilibrium price vector then it is trivially potentially finite; otherwise it is non-trivially potentially finite.




With these definitions I can now be more precise about an important advantage of always reporting all non-labour coefficients.

Theorem 5. Malinvaud's procedure is assuredly finite.

A proof is given in Malinvaud (1967) on p. 195. I now show that there is an important subclass of procedures whose rules may allow

for almost all technical coefficients to be reported, but which are nevertheless potentially infinite. It was mentioned earlier that an exhaustive taxonomy of procedural rules was likely to prove impossible. The examples (a) to (e) that were given were fairly straightfor- ward, but clearly one could think of more complex rules. All I have required so far is that the rules be unambiguous and that 0?_ R' _ A'. I now concentrate on a class of rules, of which the examples (a) and (b) are members, for which strong results are possible. I start with another definition.

Definition 5. A procedure has activity based rules if to each technique (A,f) it is possible to assign a matrix R, 0 R A, such that whenever (A, f) is chosen during that procedure, R is the matrix of reported technical information.

This excludes the possibility that although the same technique is chosen at different stages, the technical information sent to the Centre is different. The rules of examples (c) to (e) are therefore not activity based, but the rules of Chander's and Malinvaud's procedures are. I now prove the following result:

Theorem 6. Apartfrom Malinvaud'sprocedure, allprocedures with activity based rules are potentially infinite.

Proof. Consider a procedure X with activity based rules. If procedure X is not Malinvaud's, then its reporting rules must allow for the

possibility that there is a technology with at least one activity whose coefficients are not completely reported. If this were not so, procedure X would be identical to Malinvaud's procedure. I take such an activity, assume without loss of generality that it produces good one, and denote it by a1. I can construct a productive technique (AO,f0) by adding to a1 n - 1 further activities a2, a3, ..., an specifying that the coefficients of each of these n - 1 activities are all positive. This is possible since I can make all these coefficients arbitrarily close to zero. Corresponding to this technique (AO,f?) there is an R-matrix derived according to the rules of procedure X. However, I have not specified how the rules of procedure X apply to the activities a2, a3, ..., a,n only that the coefficients of a1 are not completely reported. Consider therefore a subsidiary procedure Y, in which a 1 is treated according to the rules of procedure X, and a2, a3, ., an are treated according to the rules of Malinvaud's procedure i.e. they are fully reported. Because the assumption that the coefficients of a2, a3, ..., an are all positive and fully reported by procedure Y, the matrix [I - RO] can be partitioned in the following way.

[I -R0]-1=[ z Z

where z is a scalar, z, and Z2 are n - 1 dimensional row and column vectors respectively, Z3 is a square matrix of order n - 1, and any zero elements of [I - R?] are in the




vector Z2- I have chosen the activity a I so that the first column of No has at least one non-zero element. Then the matrix No [I - R0]-1 must have at least one non-zero diagonal element and cannot therefore be nilpotent (recall that No and [I - R']-' are both nonnegative).

Consider now the sequence of prices generated under procedure Y if (A0,fj) is the only productive technique. By construction, procedure Y has activity based rules; therefore

t+l -

(pt N +f?)[I-Ro]-1 (t = 12, 2...)

and by repeated substitution one can obtain pt+ = p'[B0]t +,t-'f?[I - R?][B?]' (t = 1, 2,...) (14)

where

Bo = N?[I -R]-l.

Similarly, since the equilibrium price vector is defined by p* -p*No +p*R? +f?, so that p* =(p*No +f) [I-R0]-', then

p= p*[BO]' +E t-f0[I - R]-'[BO] (t = 1, 2,...).

Hence, pyt+ =p* if and only if

(p - p*)[BO]t = O.

Suppose now that p > p*. Then -y+ =p for some number t only if [BQ]t equals the null matrix, i.e. only if Bo is nilpotent. But I have shown that Bo is not nilpotent; therefore procedure Y will not converge in finite time in circumstances where (A0,f) is the only productive technique and p, > p*. To complete the proof, I show that procedure X will not converge in finite time in the same circumstances. To do this is sufficient to prove that if p' _ py then pt+I > p=+l; by taking pl = pl, it follows that p, $ p* implies pt $ p*.

By construction, R? ' R?, so from Theorem 4,

PY= (pyN +f)[I-RR]-' _ (ptN? +f)[I-RR]-'.

If py c pt, then

(ptNo +f)[I - R?]' ? (pt No +f)[I -Ro]- = p=t

so pt+l pt+l. Since procedure X can be taken to be any procedure with activity based rules that is not Malinvaud's this completes the proof of the theorem. ||

Remark. It is not necessarily the case that Bo> Bo, but the theorem does imply that Bo cannot be nilpotent: were this not so, then t 2 n would imply [B?]t = 0 and pt+l

equal to p*. This theorem must be interpreted with special care: although a procedure might not

converge after a finite number of stages under one set of circumstances it might under another set, as shown in the special case constructed in Theorem 7.

Theorem 6 refers only to procedures with activity based rules because the sequence of price vectors can be expressed by (14) only if there is no change in the information transmitted when the same technique is chosen at each stage. In order to examine whether other types of procedures are potentially infinite one must proceed case by case. For particular procedures with specified rules, finding circumstances in which convergence is not finite may not prove too difficult: for example, it is clear that the procedures whose rules are given by (c) and (d) on page 8 are potentially infinite. On the other hand the




rules given by (e) give rise to procedures that are assuredly finite, since after a certain number of stages such procedures are essentially the same as Malinvaud's. In short, whether a procedure is potentially infinite or not depends critically on its rules. This is in sharp contrast to the next result.

Theorem 7. All procedures are non-trivially potentially finite.

Proof I have only to show that there are circumstances in accordance with Definition 4 in which Chander's procedure results in finite convergence, since by Theorem 3 any other procedure will result in finite convergence in the same circumstances.

Consider a technology that only has one productive technique (AO,f0). Then the sequence of prices under Chander's procedure is given by

p t+1 p* +(pl._p*)[AO]t (15)

I can clearly construct (A?,f?) such that A? is nilpotent. Then pt+ = p* for some number t ? n even if p' > p*. This proves the theorem.'0 11

Theorem 7 does not contradict Theorem 6, but indicates that only when more structure is imposed on the technology or when the procedures are specified in more detail will stronger conclusions be available.

So far, this section of the paper has concentrated on technologies with only one productive technique. I now provide a more general characterisation of the sequence of price vectors, of which equations (14) and (15) are special cases. The only restriction I impose is that the optimal technique (A*, f*) be unique. Corollary 4 says that the economy will "lock onto" (A*, f*) after a finite number of stages; let the last stage at which (A*, f*) is not chosen be stage r - 1, so that (At,ft) = (A*,f*) if t'? r. Then

pt+l = (pNt +f*)[I-Rt]-l, Nt +Rt = A* (t r).

By repeated substitution it is possible to obtain

pt+l =pr fJ_t= B' +f* t=r ([I - R]' Ht=j+l Bk) (t? r)

where B' = N'[I - R']-. Similarly, since p =(p*Nt +f*) [I - Rt]-' for any t'? r then

p* rlt__ B +f* Et=r ([I-Rj]-' fIt+ Bk)

so that pt+l =p* if and only if

(Pr _p*) ft=t Bi = 0. (16)

In considering a procedure with activity-based rules then fI7=r B' can be written as [B*]t-r+l; nilpotency of B* is then a sufficient condition for finite convergence. It is not a necessary condition, however, unless pr > p*; one cannot say a priori whether this will be the case or not.

Equation (16) shows how important the reporting rules are for determining the finiteness of convergence. It also confirms a virtue of Malinvaud's procedure suggested in Theorem 5; since BtMr20 for all t, then Pr =pr+2=P* and the procedure will in herem5 ine B 0foral t the P =PM p tepoeuewl immediately go into the final process of plan construction. Of course, procedures can be designed so that the Centre checks from time to time to see if (A*,f*) has been located; this would avoid the unnecessary expense embodied in (16) if the B"s have few zero elements, while not requiring the transfer of the complete input matrix at every stage.




5. THE TRANSFER OF LABOUR COEFFICIENTS

In this paper, I do not consider the transfer of information relating to labour coefficients. This is undoubtedly a limitation, and I now discusss very briefly the possible effects of lifting this restriction. Does direct knowledge of ft allow the Centre to make a better approximation to pt? (if a formula for price change produces a vector lower than ", there is no way of ensuring that successive techniques are productive). Under Malinvaud's procedure, pt+l = pft so it has no need for further information (of course, there is really none to be had, since if At = Rt it can calculate ft from qt, pt and Rt). With other procedures, however, some improvement may be possible." Let prices be generated in the following way: the Centre receives qt Rt and ft as messages, and calculates ptNt q t -ptR t -ft. It then solves

p pt+lRt +AtptNt +ft

where At = maxi= I... (qt/pt) 1. It is easy to verify that using information about ft in this way is similar in effect to transferring more technical information via R: given pt no element of pt+' increases (although overtaking can not be ruled out) and if the method is applied to Chander's procedure, then the benefits of the amended procedure are no less than those of the original. These two conclusions parallel the results of Theorems 5 and 4 respectively.

A slightly more sophisticated method is to set A =maxi (p+'/pf). The Centre must now solve for pt+I and At simultaneously. Briefly, this can be found in the following way. Let hi be a vector equal to the i-th column of [I-Rt]-' i=l,...,n. Define

ai = ptNthi and ,8i =fthi. Then At is the solution to A = maxi (Aai +j3j)/pt, and this is given by maxi 13i/(pt - ai). The solution for pt+l follows immediately.

This second method of using ft produces a lower value of A t and therefore a greater fall in prices, although the caveat regarding overtaking still applies. If applied to Chander's procedure then the benefits are not only greater than those of the original procedure, but

greater than when A is set equal to maxi (qt/pt).

6. CONCLUSION

There are unlikely to be any easy answers to the general question of how the convergence properties of iterative planning procedures vary with the amount of information transmitted between agents. By adopting the assumptions of the Leontief-Samuelson technology, I have simplified the analysis and derived some concrete results: on the whole, more information intensive procedures lead to more rapid convergence and a greater likelihood of finite convergence, although there are exceptions to this rule.

The results presented here can be seen as part of an exercise in a cost-benefit analysis of alternative economic systems. Although I have not discussed the costs of these procedures at all (and only some of the benefits) there can be little doubt that more information intensive procedures are, in some sense, no less costly, and probably more costly. Then the claim that for the class of procedures considered here there exists a

trade-off between performance and informational properties can be translated into more familiar language: in order to secure greater benefits from a resource allocation mechanism, greater costs must be incurred. Whether similar results can be established for different

types of procedures, perhaps operating in different environments, is a question that only future research can answer.




APPENDIX 1

Lemma 1. If A is a non-negative square matrix such that [I-A]-' exists and is non-negative, and if 0? R A, then [I - R]1 exists and is non-negative and [I -R]- _

[I - A]'.

The proof follows immediately from a consideration of the power-series expansion of [I-A]-I.

Theorem 2. (i) The sequence { pt} (t = 1, 2,...) is non-increasing, non-negative, and converges to p*.

Proof. I prove first that p = ? p2 0. By definition (A', f') minimises costs at prices pI, so p A1 +fl ' p1A0 +f0 and (9) therefore implies p1A1 +f'?_ p1. Since fl > 0, p' and A' satisfy the requirements of the inversion lemma (see Note 6) i.e. pI> p1A1, so that [I-AT'='-'0 and since A1'?R', by Lemma 1, [I-R1]-'?0. But p p 'Al +fl= p1R' +p1N1 +f1 so that p1 ?(p'N1 +f1) [I-R']'. This last expression defines p2 and is clearly non-negative. Hence p = p2 ?

I now show that if pt-' ? pt ? O,then pt, ?pt+1 _O. The technique (At,ft) minimises costs at prices pt so ptAt+ft'_p'At'l+ft'. By hypothesis pt-`-pt, and Nt'=-'0, so ptAt +ft '_ptRt'-1 + pt ' Nt-1 +f '-I. The right hand side of this last inequality defines pt, and since ft >0, the existence and non-negativity of [I - At]-, and hence [I - R']-1, are assured by the inversion lemma. Since pt'' ptRt+ptNt+ft, then pt'_ (ptNt+ft) [I- Rt]-1. This last expression defines pt+l and is clearly nonnegative, so pt' pt 0> implies p t > p t+ 1 >

Since pt ' ptAt +ft, and [I - At]- ' 0, then pt '-ft[I - A]-' = p'. But'from Corollary 2, p^t _ p* so { p '} is a non-increasing sequence bounded below by p*. It therefore converges to alimit, say F,andPF 'p*. Now p"'1 =pt"1R' +p'Nt +f' _=p'R' +p'N' +f' - q' ' p'A +f for all (A, f)e S. In particular, pt+1 C ptA* +f*. Since p is the limit of the sequence {pt}, then p-PA*+f*; this implies Fj' f*[I-A*]- =p*. Since F '?p* it must be the case that p = p*; this completes the proof of part (i) of the theorem.

Remark. The string of inequalities in equation (10) can be inferred from this proof, as follows:

pt = ptRt-1 + pt -1 Nt-1 +ft-1 (by definition)

_?ptRt +ptNt-1 +ft-1 (since p t1 pt)

_ ptRt +ptNt +ft = qt (by definition of (At, ft))

_p Rt + pt Nt +ft (since pt ?>ptp)

= pt+l (by definition)

-P'?At +ft (Since pt +pt1' and At Rt + Nt).

This last inequality implies pt+' ?>ft[I -At]-I = p^. From Corollary 2, p' 'p*, sop ' q' pt + 1 > p^ t >p*

p + l ptepr

(ii) All techniques in the sequence { (A t,ft)}I (t = 0, 1, .. .) are productive.




Proof. The existence and non-negativity of [I -A']-' have been shown for t = 1 and t ?-2 in the first and second paragraphs respectively of the proof of part (i) of the theorem. By assumption, p = p'A0 +f 0; since fo > O, p > p'A0; hence [I-A]0-' 1> 0? This proves part (ii) of the theorem. ||

Corollary 3. If a t any stage pt = p+1, then p' = p*.

Proof. I have already established that pt+1 C-p'A +f for all (A,f)e S and by definition pt+1 =p t+1Rt +ptNt +ft. If pt =pt+1 then these may be rewritten as

p _ p'tA +f for all (A, f) E S

p =ptAt +f'.

Since (At,ft) e Q, it can be seen from (2) that these equations define pt as the equilibrium price vector p*. 11

Corollary 4. For somefinite number r t -r implies that (At,ft) is an optimal technique.

Proof I denote by a?ik the k-th activity for producing good i (k must be finite), and by Yik(P) I mean the average cost of producing good i using aik evaluated at the price vector p = (pi).

By definition of the optimal price vector p*, pP -< Yik(P*) for all aik in Si, and a?ik iS

part of an optimal technique if and only if pP = Yik(P*)- Consider now a comparison of the cost of producing i using aik with the cost using aij, where aik is part of an optimal technique and aij is not, so that

pP = Yik(P ) < Yij(P

It is clear that for a sufficiently small increase in all prices above their levels in p*, this inequality will still hold. In other words, I can find a vector eijk >0 such that p*_ p _ p* + ijk implies Yik(P) < yij(p). From this it follows that for each good i, there exists a vector e'> O such that if p* ' p _ p* + e', then the activity which minimises Yik(P) iS part of an optimal technique.

Writing e'= (e'), I define the vector e = (eh) so that eh= mini=l,...,n(e'h), h= 1, 2, ... ., n; then e > 0. { pt} converges monotonically to p*, so there exists a finite number r such that t ? r implies p* pt' p* + e. But e has been constructed so that if p*c ptC p* + e, then (At, ft) is composed only of activities which are parts of optimal techniques; this in turn means that (At,ft) is itself an optimal technique, and the Corollary is proved. 11

APPENDIX 2

The following example shows how Malinvaud's procedure may be "overtaken" by another, which I call procedure X, when both start from the same initial price vector and both are operated in the same technological environment.

There are two goods, 1 and 2, each producible by three activities. The input and labour requirements of the six activities are given in Table I.

Table II shows the unit costs (the costs of producing one unit of gross output) of each activity at different prices. The prices shown are those that will be relevant for selecting the cost minimising activities at the various stages of the two procedures. Throughout, the wage rate is taken to be one, as in the text.




TABLE I

Input of Input of Labour Gross output Activity good 1 good 2 requirement (one unit)

all 7 0 53 Good I a12 2 0 4 Good I

1 3

a13 2 Good 1 a21 6 20 Good 2

a22 4 0 4 Good 2 a23 0 4 3 Good 2

TABLE II

Unit costs at various prices '

Prices

Activity (210,210) (10,24) (40,30) (4, 4)

a,1 1033 10 24 723

a12 109 9 24 63 a!13 1334 132 234 43 a21 55 24 25 202

a22 564 6 14 56

a23 552 9 1O 4

Note: 1. Wage rate equals one throughout.

I take an initial price vector of (210, 210). Then Table II shows that the cost minimising technique is composed of activities a1 and a2 I. This technique is chosen under both procedures, i.e.

(A', f) = (A',f4

Malinvaud's procedure

Under Malinvaud's procedure, p2M is calculated according to (8), which yields

PM= (53, 20) [ ] =(10, 24).

Firms now choose new activities at these prices, and Table II shows that these are a12 and a22-

Again, (8) shows how P3 is calculated, and

PM = (4, 4) [0 1] =(8, 6).

Procedure X

The second price vector of procedure X, p2, is calculated according to (6).

2= (P1 N 1 +fx)[I - RX]'.




The rule that distinguishes procedure X gives us a matrix N'(=A' - R') for any matrix A', and for the purposes of this example I specify that

The vector p2 is now given by

P2 = ((210 210) [516 +(53 2)) [ ?

= (40,30).

As Theorem 4 predicts, px _ pM and Table II shows that the activities chosen at prices 2

PX are a13 and a23- I specify

[- 1-] I100 100

and the vector p3 is given by

I_ _194019 1

p3X=((40,30) 1 X +(2,3) 73 19 100 100 -200 25

This is approximately (5-64, 4-34), So p3 <P3 . The final column of Table II shows that A2 equlibiu

p2 is the equilibrium price vector so (Ax, f2) is the optimal technique, and it can be verified that p4 = p2, although this is not shown. The example is rather forced (the reporting rules of procedure X have been chosen for the convenience of the example and have no other significance) but the general proposition in the text is shown to be true.

First version received April 1983; final version accepted March 1984 (Eds.). I would like to thank Professor M. Morishima, Dr. J. Broome, Mr. D. A. R. George and two referees of

this journal for much helpful advice and comment during the preparation of this paper.

NOTES 1. I adopt the following convention for vector and matrix comparisons. x> y means xi > y, for all i

where x = (xi) and y = (yi). x _ y means xi ' yi for all i. 2. As a referee has pointed out to me, this leaves no scope for aggregation of technical information.

Reporting of coefficients excludes rules like "report the value of total energy costs per unit of output". This is an interesting area for research.

3. Morishima (1959) also studies a process of price formation equivalent to Chander's although it is embedded in a more complex model, and suggested as the basis of a planning procedure elsewhere (Morishima (1976) p. 247).

4. Malinvaud (1967) has the Centre receiving ft and At, but not qt. Given pt and At, knowledge of ft and q' amount to virtually the same thing: the trivial difference is that under our interpretation, the Centre has to calculate qt -ptAt. I briefly discuss in the conclusion to this paper the possible effects of the Centre receiving information about ft when At is not completely known.

5. Malinvaud (1967) starts his procedure on the assumption that the Centre knows of at least one productive technique, and the first price vector is one associated with a known productive technique. Chander's assumption is the one that I have adopted. I have resolved this by assuming that the productive technique known by Malinvaud's Centre is weakly profitable at the prices known by Chander's Centre.

6. The inversion lemma states that if A is a non-negative square matrix of order n, and there exists a vector p > 0 such that p > pA, then (I - A) has a non-negative inverse, where I is the identity matrix of order n. A proof is given in Debreu and Herstein (1953), in particular Lemma* and Theorem III*.

7. Corollary 4 depends on the assumption that the set S is finite. The other propositions do not. 8. Theorem 3 contradicts a footnote in Chander (1978) p. 774, note 24 where he states that the relative

speeds of convergence of his and Malinvaud's procedure depend on the number of activities. Theorem 3 states that Chander's procedure can never converge faster, irrespective of the number of activities.




9. This revised method of plan construction generalises the one given by Malinvaud, whereas that presented in Section 3 is a generalisation of the truncation procedure given by Chander.

10. Theorem 7 shows that a remark made by Chander is incorrect. He states that if p is not the equilibrium price vector, then the sequence { pt} is strictly declining (3, pp. 768-769, our notation). If this were so, convergence could never be finite.

11. The rest of this section outlines an ingenious suggestion made by an anonymous referee, to whom I am very grateful.

REFERENCES CAVE, M. and HARE, P. (1981) Alternative Approaches to Economic Planning (London: Macmillan). CHANDER, P. (1978), "On a Planning Process Due to Taylor", Econometrica, 46, 761-777. CHANDER, P. (1974), "A Simple Proof of the Non-Substitution Theorem", Quarterly Journal of Economics,

88, 698-701. DEBREU, G. and HERSTEIN, I. N. (1953), "Non-negative Square Matrices", Econometrica, 21, 597-607. HURWICZ, L. (1960), "Optimality and Informational Efficiency in Resource Allocation Processes", in K. J.

Arrow, S. Karlin and P. Suppes (eds.), Mathematical Methods in the Social Sciences (Stanford: Stanford University Press) 27-46.

MOUNT, K. and REITER, S. (1974), "The Informational Size of Message Spaces", Journal of Economic Theory, 8, 161-192.

MALINVAUD, E. (1967), "Decentralized Procedures for Planning", in E. Malinvaud and M. 0. L. Bacharach (eds.), Activity Analysis in the Theory of Growth and Planning (New York: St. Martin's Press) 170-208.

MORISHIMA, M. (1959), "Some Properties of a Dynamic Leontief System with a Spectrum of Techniques", Econometrica, 27, 626-637.

MORISHIMA, M. (1976) The Economic Theory of Modern Society (Cambridge: Cambridge University Press).



informational and performance properties of a class of iterative planning procedures

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