IX. The Continuous Process of Production

CHAPTER IX THE CONTINUOUS PROCESS OF PRODUCTION FOR a number of practical and historical reasons, we shall base our descriptions of the continuous process of pro duction on the input function rather than the output function. It was in the form of an input The use of the In,ut function that the time dimension was first function and Ita limitations explicitly introduced into the theory of production by W. S. Jevons. 1 And it is probably the approach which is mQre easily comprehended. This very fact, however, is associated with certain pitfalls to which the use of the input function easily leads. It is essentially an approach to the problem of capital from a cost angle. Such an approach is. even tolerably adequate only under strictly stationary conditions, and the dangers attaching to its use have already been pointed out. It does, how ever, help to elucidate a number of important relation ships, and we shall use it here with the attached warning that it may prove misleading in certain connections if used incautiously. But so long as we confine ourselves to the consideration of stationary conditions we can dis regard these difficulties.

The problems which we want to study with the help of this concept are first, the relationships that exist under stationary conditions between the stock of non-per manent resources (or, what in these conditions amounts to the same thing, the stock of" intermedi~te products ") 1 Dr. Marschak, 1933, has suggested that for this reason we should speak of the" Jevonian Investment Figure". Although I have, on an earlier occasion, myself used the terms" investment function" and " invf3stment curve" in this context, the terms" input function " and "input curve" now seem to me to be preferable by reason both of their brevity and of the analogy to the term " output function ". 113 114 I nvestment in a Simple Economy PT. II and the range of periods for which current input is in vested; and secondly, the interconnections between the different ways in which the input function can be used to describe a continuous and stationary process of production.

Let us commence by considering the result of the continuous repetition of an investment for a given period in the simplest case imaginable: the "point input The result of con-point output" case. We may assume that tlnuouslyrepeatedln-input is continuously applied to start some vestment In the sImplest (" point Input - natural process of fermentation or growth point output") case which, without requiring any further application of labour, will yield a certain product after a given interval of time. We assume in other words that any quantity of input applied at a moment of time will result after a fixed interval in a definite pro duct at another moment of time, and that input is applied, and consequently output matures, continuously at a constant rate. This may be conveniently illustrated by a diagram (Fig. 5). If we measure time along the ordinate Ot and quantities of input and output (the latter of course in such units as are the product of a u!lit of input) along the abscissa Or, any constant rate of flow through time of input or output will be represented by a straight line of appropriate slope (e.g. OP). For any interval on the ordinate measuring time, say T1T 2 , the corresponding segment on the abscissa R1R2 will give us the quantity of input invested during that interval. If we assume that this input is invested for constant periods, say of the length OT l' we can represent the resulting flow of output by a parallel sloping line T 1 Q. The product of any input invested at any point of time and shown by the point on the line OP corresponding to the appropriate point on the time-axis, will be indicated by a point with the same abscissa (i.e. directly above the former) on the line TIQ.

Every process leading from a moment's investment to the product of the investment may then be represented by CR. IX The Continuous Process of Production 115 a vertical line connecting the two sloping lines. Since we assume that similar processes are started continuously at every moment of time, we have to conceive of the whole area between the two sloping lines as being com pletely filled by such vertical lines den.oting individual processes. This means that every horizontal line drawn from any point corresponding to any moment of time (e.g. 1 1 2) will T4~------------------~~ T3~----------~ T, Fo--------,r o R, FIG. 5 Q p r cut those vertical lines at a series of points which will include every possible point from the beginning to the end of. a single process. Translating back from diagrammatic to real terms, this means that where a process is continuously repeated all the;, successive stages or phases through which each individual process passes will also coexist "Synchronised" pro at any moment of time. This conclusion ductlon is important for the understanding of the subsequent analysis. Another way of formulating it is as follows: on the ~ssumption of stationary conditions in which investm~nt is carried on continuously, the complete 116 Investment in a Simple Economy PT. II description of the historical process of production as it proceeds in time is also, and at the same time, a complete description of all the different stages of different indi vidual processes which exist simultaneously at anyone moment of time. Or, as the same thing has sometimes been expressed, l at anyone moment of time we find all the phases through which the process of production passes" synchronised" or going on at the same time.

This relationship becomes slightly more complicated if, instead of assuming one single investment period, we consider a process where investments are spread over the C II I continuous range of investment periods on nuous Dvest .. men! over a range described by the input function. It is only of perIods in this form that we can obtain a really useful picture of the relations which exist in the real world. The diagram which we shall use in this connection has to be drawn in a three-dimensional system of co ordinates. The horizontal r-axis in the plane of the paper again measures the rate at which input becomes available, while the t-axis, moving backwards into space, measures time. In the horizontal plane tOr formed by these two axes, the area enclosed by the input curve RQa is shown shaded. So far the diagram corresponds to Fig. 1 above: the whole strip enclosed between Ot and RR1R2 . . . and moving backwards into time representing the expected stream of output. And the shaded portion of this strip shows that part of the stream of products which is already provided for by the" inchoate wealth" or " intermediate products" existing at zero hour.

Everything in the base plane refers therefore to final products, present or future. In order to be able to show the stock of "intermediate products" (the transitory form which the input takes on its way to the final con sumable product) in the same diagram, a third or 8-axis is introduced. The quantities of such intermediate pro1 This term is due to J. B. Clark, The Distribution of Wealth (1899), chap. xx.

CR. IX The Oontin'uou8 Process of Production 117 ducts existing at any moment are ranged along this per pendicular axis according to the "stage" they have reached in the process. Intermediate products which are very near consumption or in a late stage RepresentalioD of tbe (goods of a relatively low order) are stock of Inlermedlate producl$ exlsllng al a shown near the base, while those belong-moment of time ing to earlier stages are shown correspondingly higher up in the triangular figure. The connection between the individual items in this FIG. 6 stock of capital goods existing at zero hour and the contribution to future income which they are expected to yield is shown by the slanting planes which Repre.entation of tbe connect points on the perpendicular triangle process In lime with the corresponding points on the horizontal triangle. At the base the two triangles coincide, expressing the fact that the whole of the product to be consumed at the initial moment must at this moment exist in finished form. If we go a little higher up in the perpendicular triangle, 118 I nve8tment in a Simple Economy PT. II SaOR, say to the goods in the earlier stage represented by the line SIPI, these w.ill be goods which constitute part of the product expected at time TI . Similarly the amount of intermediate products at the still earlier stage indicated by the line S2P2' represents part of the product that will mature at T 2' It will be noticed that in both cases and in the cases of all the planes that can be visualised as coming between - the intermediate products already existing at the initial moment 0 represent only part of the total product of the moment for which they are destined, and a part which decreases as we go on to earlier stages. Thus the quantity of intermediate goods S2P2 represents only the small part T 2Q2 of the product at T2 ; and much the larger part, indicated by Q2R2' has yet to be created by the further application of input which will only become available in the interval between 0 and T2.

The rate at which this input will be applied can be read off from the curves PIR1 and P2R2 • When we reach still higher stages, e.g. that shown at the top of the diagram, we find that very little of the product of the relatively distant moment Ta i;o in exist The range or Invest-ence at the initial moment, and that nearly ment periods may all of that product will have to be c"eated exlend Inlo the In- .' deftnlie fuiure by the application of input in the interval between 0 and T 3 • The diagram has, however, deliber ately not been continued, as it conceivably might have been, to a point where all the future product has to be created by the application of input in the future. Under actual conditions we shall always find that however far we may look into the future, some small part of the product of that future moment will already be provided for by sonie part of the existing nonpermanent equip ment. This part will tend to become negligible as we go far into the future, but it will not disappear altogether within any time we can conveniently represent in the diagram. Account has been taken of this by leaving the diagram open at the top, so to speak, and by similarly CR. IX The Contimwu8 Proces8 of Production 119 indicating that the shaded triangle in the base plane is bounded by a curve which approaches Ot asymptoti cally.

Special interest attaches to the topmost of the three slanting planes in the diagram - the not quite complete, curvilinear triangle, SaTaRaPa. As will easily be seen, it reproduces the input curve in an inverted The Input curv.ln Its form with the narrow part of the tri-Inverted form angular figure which it bounds pointing towards us and the base pointing away from us. In this form the curve has a special significance. It describes the actual process of applying the input, the rate at which it will be applied from now onwards till the moment T 3 in order to produce the output of that, moment. In other words, it describes the continuous input which leads to the output of the moment Ta.l Later on we shall sometimes make use of this inverted form of the input curve in discussing the case where we have to deal with a stock of goods in process which are the result of the actual duration of the pro cess of production (as distinguished from the durability of the goods). For the present, however, it is only necessary to be aware of the relation of this aspect of the figure to the aspects that have already been dis cussed.

The whole solid body of the figure can obviously be conceived as being made up of an infinite number of planes, similar to the one that has just been described. Anyone of the perpendicular planes thus The meaning of the represents a cross-section through an solid infinite number of these slanting curvilinear triangles each of which represents a single process. This means 1 It was in this form that I first used the triangular figures in Prices and Production. But since in this inverted form the input function has sometimes been interpreted as " backward-looking". it should perhaps be emphasised that even in this form it is "forward-looking" as it refers to all the investments that will have to be made from to-day onwards in order to obtain the product that will mature at Borne future date.

120 Investment in a Simple Economy PT. II that each of the perpendicular triangles which represent the stock of intermediate products existing at the moment concerned is shown to consist of all the different· stages through which the different processes going on at that moment are passing. In this respect the figure corre sponds to the simpler diagram used previously (Fig. 5, p. 115), except that we are now considering the case, not of a single uniform investment period, but of a continuous range of investment periods of different lengths. Since we are considering stationary "conditions we again have, of course, identity between each of the various phases through which the process passes in time and each of the various "stages" represented by the stock of inter mediate products existing at a moment of time. In fact, the whole solid body represents all the phases through which all the processes now under way, or to be started at any time in the future, will pass. Any perpen dicular cross-section of the solid, representing the stock of intermediate products at the corresponding moment, will be like every other in that it will consist of the same combination of intermediate goods. But. no individual good will be in the same position at two successive moments. As time passes, i.e. as we move in the diagram backwards into space, each good progresses downwards to more advanced stages, its place being taken by other goods which are simultaneously advancing in like manner from still earlier stages. All these movements, all these transformations, are of course effected by the continuous application of current input, which is constantly becoming available and being combined with the stock of inter mediate goods in the different stages. The distribution of the total flow of input at anyone moment between the different stages is shown by the vertical curve P 3R, the continuous application of input in anyone process by the slanting curve P 3R3' and the distribution in time of the marginal additions to the product of the input applied at the initial moment by the curve RQ3.

CH. IX The Continuou8 Proce88 of Production 121 There are thus three fundamental aspects 1 of the input function. In its original form it describes the time distribution of the output due to a moment's input; it can also be used (as in the perpendicular The three funda triangles of our figure) to describe the stock mental aspects of the f . t d' t d t . t' t Input function o m erme la e pro uc s eXls mg a any moment of time; and finally it can be used in an inverted form (as in the slanting planes) to describe the process leading up to the product maturing at a moment of time. Under stationary conditions all these triangles would be exactly similar. They are all carved out, so to speak, from the more complete picture of the total process given by the solid figure, and it is only in relation to this that their significance can be fully understood. So long as we confine ourselves to stationary conditions it really does not matter which aspect we use, but it has been found convenient to treat the one shown in the base plane :1S the basic one, because of the use that will be made of it later. For certain purposes, however, the interpretation of the function in its inverted form, that is as a descrip tion of the historical process leading to the output of a particular moment, may be more instructive.

The foregoing exposition should have made clear what is meant when it is said that the picture given by anyone of the inclined planes represents no more than an abstract description of the continuous process of The relation between real life The" quantity" of product in the tim. rat •• shown • In the diagram and which each process results, and which is concrete quantities indicated by the horizontal line in the base plane (e.g. TaRa), is really only a time rate at which the product matures at that moment, and not an actual quantity. It becomes an actual quantity only if we multiply it by time. and so obtain, in place of a rate of flow at a moment of time, the amount produced during a period of time. In 1 A fourth aspect in which the input function is represented in the diagram, namely by the surface PaRR. will be discussed below, p. 195.

122 Investment in a Simple Economy PT. 11 order to make our shadow-processes into something con crete we should therefore have to replace our merely two-dimensional planes by corresponding "slices" of some definite thickness. What period of time we take as the unit is, of course, entirely arbitrary, and the actual thickness of the " slice " will depend on this choice. In describing a stationary continuous process it is convenient to make the unit-period as short as possible and ultimately to go to the limit where this dimension disappears. But we must never forget that this method of isolating a particular aspect (the time rate of flow) is a mathematical device, a process of abstraction, and that the result assumes concrete meaning only when we reintroduce the dimension which we have disregarded. The appearance of the input curve in its inverted form, in which it is particularly useful for depicting a single process in the narrower sense of that term, offers a welcome The Input funcllon opportunity to add a few words on the as a description 01 meaning of this curve and the factors which time-consuming processes determine its shape. 1 The more difficult question of how durable goods can be fitted into the general scheme will be reserved for the next chapter.

I begin with the question of the shape of the input curve. So long as we confine ourselves to considering the process of production of a particular commodity, there is Its h I j I little reason for supposing that the curve is sapenasnge branch-process 01 more likely to be concave, as I have drawn production 't th 'th t' ht It 1, an el er s ralg or convex. seems just as probable that, in anyone process, relatively more will be invested in the early stages than in the late stages, or that investment will proceed at a constant rate through1 It is important to remember that the description of the process here runs in terms of physical quantities. In consequence we have either to assume that we have to deal with the production of only one commodity and that only one homogeneous factor is being used, or, if we want to apply the diagram to cases where more than one commodity and more than one factor are involved, we have to assume that the relative values are known and constant.

CR. IX The Continuou8 Process of Production 123 out, as that more will be invested in the later stages than in the earlier ones. The question takes on a somewhat different com plexion, however, when we remember that even the pro cess leading up to a particular commodity is not usually linear, but will as a rule consist of many 114 sbape In tbe separate branches of different lengths which oomplote proeess of producllon 01 one gradually join up together to form the commodity main stream. In order to obtain the input function for the complete process, we must of course make a summa tion at each stage of all the input invested at the same mom~nt (that is, in that same stage) in all the various branches of the process. Beginning with the one which starts earliest, we shall, as we progress to later stages, have to include more and more of these branch processes which have for a time been going on simultaneously but separately. Now, even if input is applied at a constant rate in each of these sub-processes, the aggregate effect must be that, as the number of such sub-processes which are going on simultaneously increases, the rate at which input is applied in the process as a whole will tend also to increase.

This tendency for the input curve to be concave will be even more marked if we use the curve to describe the rate at which input is applied, not merely in a single process leading up to a particular product, 114 shape lor tbe but in all the different processes going on system as • whole in the economic system. The total length of the pro cesses in the different industries will of course vary widely. Some of them will be very long and will begin at a time when no other preparations are yet being made for the product of the time when they will finally mature. As we pass to· the next stage of these processes, some other processes will be starting up, and so on as we get nearer to consumption. The total number of individual processes going on parallel will constantly increase as we proceed to later stages. And this will mean that, even if input is 124 Investment in a Simple Economy PT. II applied at a constant rate in anyone process, for the economic system as a whole the rate at which input is applied at the successive stages will constantly increase, i.e. the aggregate curve will be concave.

It is perhaps reasonable to assume that the amount of input which is applied to the stock of intermediate products in each stage will bear a constant propor tion to the amount of those intermediate products (or, in more popular but more inexact terminology, that the proportion between capital and labour will be roughly the same in all stages). In this case the input curve will be some kind of exponential curve. The second question is the meaning of the concept of " the rate at which input is invested" in the course of the process. The term is deliberately vague. It may The unlls in terms or cover two different things. It may refer to which input is meas-the value of the input which is invested at ured each point of the process, or it may refer to the physical quantities. If the input used is homogeneous and of one kind, the same curve will describe both. But the situation is different if, as will of course usually be the case, different kinds of input are used in the same process. In this case a single input curve cannot be drawn in physical terms at all; and it can only be drawn in terms of value if we assume given relative values for the different kinds of input. The shape of this curve will depend on these relative values and will change with every change in them.

It will cease to be a technological datum which describes the technical character of the process in physical terms. For this purpose we should have to start with separate input functions each of which describes the rate at which one particular factor is applied. These functions would all belong together and would have to be considered jointly in order to obtain a complete description of the particular type of process. Indeed it will be groups of input functions of this type which we shall have to use later in our dis cussion of the productivity of investment.

CR. IX The Continuous Process of Production 125 I It is worth mentioning in conclusion that it is possible to modify the diagram used in this chapter so as to show certain consequences of changes either in the investment periods or in the amount of input that Application of the becomes available for investment. An diagram to the repre. . h· . d Id sentation of changes InCreaSe In t e Investment perlo s wou be shown by a gradual increase in the height of the figure, and an increase in population or any other increase in the amount of input would be shown by an increase in the width of its base. It would then be possible to trace with precision the effect of any such change on the stream of products, an effect which would only manifest itself after periods varying with the varying investment periods. But since all conclusions drawn from this con struction would have to be based on a rather artificial assumption, namely, that only the investment periods of the input could be changed and that all the intermediate products were so completely specific that they could only be used as originally intended, it would be of very limited usefulness.