VIII. The Output Function and the Input Function

This part of our way leads through a rather arid tract where the profit which we derive from our labour will for some time be difficult to see. And it is not surprising that nearly all of our predecessors, anxious to get on to what are the more interesting problems, were satisfied with a few simple generalisations about the" period of produc tion ", and proceeded, without really analysing the nature and interrelationship of the various time-intervals involved, to consider their relation to the productivity of investment. We shall see later that this procedure almost inevitably leads to muddles and confusions which are very difficult to clear up at a later stage. In view of this experience we shall do well, before we approach the problem of the pro ductivity of investment at all, patiently to explore all the types of relationships with which we shall have to deal. 8 The first three chapters of this Part will accordi~gly 97 98 Investment in a Simple Economy PT. II be devoted entirely to describing the formal character of the various possible relationships between the stock of nonpermanent resources existing at a moment of time, the stream of income expected from this stock, and the way in which the current input is being invested. These relationships will here be considered merely as techno logical facts which arise out of the circumstance that pro duction takes time. Our task here will be essentially to provide a convenient way of describing the possible rela tionships in a manner which will assist in the later treat ment of the economic problems involved. In the present chapter in particular we shall consider the various waysin which quantities of input and quantities of output may be related in isolation. In the following chapter we shall see how the technique evolved here helps us to describe a continuous process of production in all its 'aspects, and in Chapter X certain peculiarities connected with durable goods will be separately considered.

And not until the completion of this preliminary task shall we then be prepared to study the effect of the different productivity of different forms of invest ment on the choice of a particular investment structure. These relationships between the productivity of the different forms of investment, the particular investment structure that will be adopted under different conditions, the uniform rate of interest that will characterise a state of equilibrium, and the value of the capital goods in existence will occupy us for the greater portion of this part of the investigation. For a considerable part of the way (Chapters XI-XV) we shall try to concentrate on the effects of the productivity of investment on the in vestment structure by making special assumptions which will enable us more or less to disregard the psychological element of "time-preference" which forms of course an essential part of the complete picture. This element will be introduced in the last two chapters of this Part.

Throughout this part of the book we shall adhere to CR. vnI Output 'Function and Input Function 99 a number of simplifying assumptions. Until in Part III we explicitly introduce the market, it will be assumed that we have to deal with a closed economic system in which all economic actiVity is directed by a single Simplifying &ssump will and according to a coherent plan." We tiona shall deal, that is, either with the economy of an isolated individual, or with that of a communist society where all economic activity is directed by a dictator. Until we get to the two final chapters of this Part, it will further be assumed that the available resources are to be used to produce an output stream of unvarying size for an indefinite or perhaps infinite period. We shall not for the moment go into the question of the exact meaning of a constant income stream. For our present purpose we shall simply assume that the output for which the dictator plans consists at successive dates either of constant quanti~ies of one homogeneous commodity, or at least of constant proportions of various commodities, so that it can be measured in physical units. During the next few chapters we shall also disregard the considera tions which will have to be taken into account in order that the greatest possible output stream may be obtained.

All these economic or value problems will have to be taken up systematically from Chapter XI onwards. At the moment we shall simply assume that one particular plan has been decided upon for using the stock of resources with which our society is provided. Similarly we shall for the time being assume that, within this production plan, each separate unit of available input is expected to make a definite and determinable contribution to the output stream of the future. How the magnitude of these specific contributions is to be determined, that is, on what principle particular parts of the future output stream can be attributed to lI?articular units of input, is also a question which must wait for later discussion. 1 1 It will later be seen that, in the discussion of the economic problems involved, we need not necessarily know the connection between all the 100 I nve8tment in a Simple Economy PT. II It is probable that the stock of nonpermanent resources existing at any moment will embody a very con siderable part of the output of the immediate future and The stock of capital at any moment repre sents definite contri butions to the Income expected at dlllerent luture dates a constantly diminishing proportion of the output of more and more distant future dates. Nearly all the output of the very next moment will already be in existence in the form of intermediate, semi-finished products or in the form of durable goods which will con tinue to render services for some time to come. The part of the output of the immediate future which is not yet in existence in some such form (that is, as what Professor Taussig has described as "inchoate wealth ") will be added in the interval by the use of some part of the input which is applied during that time.

As we look forward to more distant future dates, the part of the total final output which is already available in an inchoate form as nonpermanent resources will become smaller and smaller, and the part which has yet to be provided for (by the application of input which does not become disposable until a later date) will become correspondingly larger and larger. The more distant the future date, the smaller will be the part of the output of that date which can be said to be already provided for. But although this share of future output will become very small when we look towards the very distant future, it is doubtful whether, within any period in which we are at all interested, it will vanish completely and whether some of the" nonpermanent" resources will not cease to make contributions only at a date in which we are not really interested. But since we are using the concept of non permanent not in an absolute sense but with reference individual units of input and the corresponding units of output, or all the individual investment periods, but that it will prove sufficient if we know those affected by marginal changes. But for the preserit purpose of constructing an apparatus for the description of the technological relationships involved it will be convenient to retain the assumption stated in the text.

CR. VIII Output Function and Input Function 10 1 to the period for which the person in question plans, this problem need not trouble us further. The position can be conveniently represented by a simple diagram. In Fig. 1 the horizontal or r-axis meas ures quantities of output and the vertical or t-axis time. l The two parallel vertical lines Ot and RQ Diagrammatic repre indicate the expected output stream the .enlallon of the two , portions of Ihe outpul distance between these two lines represent-slream ing its constant size. The base line represents the pre sent, and the two vertical lines may be conceived to T: Q extend indefinitely into T2 \ the future. The area \ under the curve T 2R re presents the part of the future output which is already provided for in the form of some kind of , , , , non -permanent resources. The curve itself has been P :::-::. R, drawn concave on the T, - - - - - -plausible assumption that the proportion of the output of increasingly o R FIG.

r distant future dates which is already provided for, will diminish at a decreasing rate. Under stationary conditions we should find a similar situation at every subsequent moment. The part of the stock of nonpermanent resources which had been con sumed in the meantime would have been The curve describing replaced by the application of current pure the time distribution of the returns from input during the interval. The dotted current Input curve '1\Rl gives the situation as it would appear after a short interval. The area TlORR l represents the amount 1 For reason of convenience in the construction of some of the later more complicated diagrams for which this will have to serve as a basis, it has been found expedient to represent time not, as is the usual practice, along the abscissa, but along the ordinate.

lO2 Investment in a Simple Economy PT. II that has been consumed during the interval aT 1 and the area between the two curves T2R and TaRl shows the output which is expected to accrue in the future from the investment (in instruments and other nonpermanent resources) of the input that became available during the interval OTl. For certain purposes it is useful, instead of referring to the contribution made by the input invested during some definite period of time, to refer to the marginal increment due to the application of input at a particular moment of time. l This can be shown in the diagram by making the interval between the two horizontal lines smaller and smaller until they finally coincide. In place of the interval between the two curves we then have the single curve T2R. The slope of this curve at any point represents the addition to the future income stream (at the corresponding point of time) which is due to the pure input applied at moment O.

The concept of the product due to the input at a moment of time is of course an altogether unrealistic, purely abstract concept. Input can be applied, and The use 01 curves In output will mature, only during a finite this and later connec-interval of time. But the concept of a rate tiona Involves the abstract ooncept of a of flow at a moment of time is a convenient time rate of How mathematical device for expressing the volume of the flow independently of the assumption of a period of particular length. It helps us to isolate certain 1 This use of the concept of the marginal increment may at first appear somewhat unfamiliar, but it is quite in conformity with the strict meaning of the term. We have to deal here with small variations in one quantity (the stream of output) relative to the change in another quantity (the stream of input). In the more familiar application of the concept of a marginal increment it is usually assumed that the quantity varies at a given moment of time, i.e. that instead of one quantity another slightly greater or smaller quantity is given at that moment. In the present case the independent variable (input) is a flow in time which varies not in width but in length. The marginal increment of output is consequently due, not to the fact that more input is being applied at anyone moment, but to the fact t.hat the stream of input is applied over a somewhat longer period.

CR. VIII Output Function and Input Function 103 aspects of continuous processes, and enables us to deter mine the size of the concrete magnitudes involved for any period of time we may choose. We shall repeatedly make use of such curves (and the corresponding functions), which refer only to time rates and not to actual quantities. So long as we keep in mind that they are only artificial devices in:tended to describe certain aspects of an essenti ally continuous process, the fact that they do not refer directly to something tangible need be no objection to their use. The distribution in time of the product of a moment's input can thus be represented by a curve: the curve that bounds the area representing the part of the future income which is due to all the non•. The output curve permanent resources already III eXIstence at the given moment. The curve indicates, as we have seen, the marginal increment of this area due to the application of the moment's input. Its ordinates (the distances from the base) describe the full range of different periods for which we have to wait for the different units of the output which are due to a moment's input. And its slope shows the rate at which the product of that input matures at the corresponding dates. 1 In many respects this curve (which we shall call the output curve or the curve representing the output function) is one of the most fundamental magnitudes that are necessary for .describing the capitalistic process of pro duction.

The diagram we have been discussing was originally introduced, it will be remembered, not to show the output curve (i.e. the time distribution of the output due to a moment's input) but to show the time distribution of the product of the stock of nonpermanent resources existing 1 Strictly speaking, the rate at which output matures is measured by the inverse value of the slope: the rate becoming smaller as the slope becomes steeper (and therefore larger in algebraic terms) and approaching zero as the curve tends to become perpendicular (that is, as the slope becomes" infinite ").

104 Investment in a Simple Economy PT. II at the given date. 1 The amount of the output due to a moment's pure input which will mature at each succes sive date is shown only indirectly by the slope of the I t t tf curve. This is due to the fact that the curve n erpre a on as a cumulative frequency shows the time distribution of this output distribution • I t· £: h· U d t t· In a cumu a Ive as Ion. n er S a lonary conditions, the total of all the units of output due to a moment's input must be equal to the total output matur ing at a moment (that is, to the distance between the two vertical lines). And for any future moment the part of the line to the right of the curve in Fig. 1 (for instance P RI at the moment T 1 ) gives us the portion of this total which has already accrued , and the part of the line to the left (TIP) gives us the portion which has still to accrue. The curve may therefore be regarded as a cumulative frequency curve (or ogive - cumulated downwards) 2 representing the part of the product of a given moment's input which remains invested beyond any particular date.

Although this manner of representation is in some ways more instructive, and will be used extensively in what follows, it will facilitate the understanding of the Th It tl exact meaning of the output curve if we e same s ua on represented by a simple show the same time distribution of the frequency curve d f .. . h pro uct 0 a moment's Input In a way WhlC is more directly appropriate to this purpose, i.e. by a simple (non - cumulative) frequency curve. We now measure along the abscissa (Fig. 2), not the part of the input at 0 which has not yet matured, but the rate at which that output will mature at any moment (i.e. the magnitude repre~ented by the negative slope of the output curve in Fig. 1). In this way we obtain a vertical strip which directly represents the shape of the output stream 1 The quantity of product which is yielded by this stock at each successive moment is shown by the abscissa of the curve at the corre sponding· point.

2 The student who experiences difficulties at this point is advised to refer to any textbook of statistics for a fuller explanation of the relation between a simple and a cumulative frequency curve.

CH. VIII Output Function and Input Function 105 due to a moment's input. It will be seen without difficulty that this strip will be of rectangular shape if the output function is linear and the output "curve" therefore a straight line, that it will be of decreasing width upwards if the output curve is concave, and that it will itself be concave if the slope of the output curve decreases at a decreasing rate. These three cases are represented by the diagrams marked a, b, and c respectively, in Fig. 2. (The t t T p T o R r a b FIG. 2 r t T r c relationship between the two sets of curves is the general one between a simple frequency curve and the ogive, i.e. the former represents the first derivative of the latter.) It is advisable immediately to contrast the concepts of the output curve or output function w:th another closely related and no less important concept which is easily confused with it. As has just been explained, the output curve describes the range of periods for which we have to wait The description of the range 01 periods dur ing which we have to walt (or tbe dlnerent for the different units of output which are the units o( output must be supplemented product of a moment's input. This is not the same thing as the range of periods for which we have to wait for the products of different units of input of equal size.1 The two curves representing these ranges of 1 As in the case of output we shall have to assume for our present purposes either that input is completely homogeneous or that, if it is composed of services of different resources, these are always used in constant proportions.

106 Investment in a Simple Economy PT. II periods would be identical for any given process only if equal quantities of input always yielded equal quantities of output, no matter what the period for which these different units of input were invested. But this is evi dently not the case. Although we have not yet system atically considered the productivity of investment or the source of interest, we know that they exist and we ought therefore to leave room for them in our diagrams. In general terms the significance of these factors for our present purpose is that units of input which are invested for longer periods will yield a larger product than those which are invested for shorter periods. There are, there fore, two ways of representing the range of waiting periods according as we use units of input or units of output as units of reference. The difference between the two ways of looking at the range of waiting periods is due to the fact that in the first case we take units of input (or factor units) and in the second case we take units of output (or commodity units) as our units of reference. If we speak in terms of units of output, the share of the total product for which we have to wait a comparatively long time will clearly be larger than the share of total input for whose product we have to wait an equally long time.

This distinction is a little difficult to grasp. But it - by a description of the range of periods for which we have to walt for the products of different unlts of Input is so important for what follows that it is necessary to be quite clear about it. Perhaps it will be easier if we restate the difference by beginning with a definition of the second of the two curves, the input curve, or the curve representing the input function. 1 In order to draw this curve we require a system of co-ordinates in which the abscissa, instead of measuring quantities of output as in the former diagram, measures quantities of input (or such quantities of output as are 1 This is the same function (and curve) which in an earlier publication (1934b) I have discussed under the name of "investment func tion " (or curve).

CR. VITI 'Output Function and Input Function 107 due to these quantities of input), and the ordinate as before measures time, the present being indicated by the zero point. The points on the curve (which may again be represented by Fig. 1) will then show The oonstruotlon of the points of time at which the product of the Input cune particular .parts of the total input applied at zero hour will mature. The general principle of the arrangement is of course again that of a cumulative frequency curve. The abscissa indicates the quantities of input which are invested beyond any of the periods shown along the. ordinate. The slope of the curve so obtained describes the rate at which the products of equal units of input mature at different points of time. The whole curve thus gives us a description of the complete range of periods for which the services of the different units of input are invested.

In the sense in which the term" invested" is used here all input is invested, although some of it (the part which is shown at the extreme right of the base of our figure) will be invested only for very short, • •• .' All Input applied Is and In the lImitIng case zero, perIods. But here desorlbed as since it would be entirely arbitrary to fix being Invested some minimum interval which must elapse between the application of the input and the maturing of the product before we can speak of the input's being invested, and since in fact only a negligible part of the input can be consumed immediately it becomes available, it is on the w~ole more consistent to speak of all input as being invested. In any case the input curve must be under stood to refer to all input used, whether it is being in vested in the usual sense of the word or used in current production. If some part of the input actually serves consumption the moment it becomes available (as will be the case with some personal services), this will be shown by the input curve coinciding for some distance on the right with the base line. The same applies, mutatis mutandis, to the output curve.

108 Investment in a Simple Economy PT. II The amount of input whose product will mature at any moment will not be proportional to the amount of output (due to that input) which will mature at the same The dllIerenee bemoment. The reason is that the size of the tween the o~tput product will depend not only on the amount curve and the mput curve of the input but also on the time for which it has been invested. In order to obtain the value of the output due to a particular amount of input, compound interest for the period of investment has to be added to the value of that input. This means that the propor tional share of aggregate product of a given input which will mature in the more distant future will be larger than the proportional share of the input which is invested for these longer periods. In terms of our curves this means that the input curve will be steeper at the top than the output curve, showing that towards the end of the range of investment periods the rate at which the product of given units of input matures will fall off more rapidly than the rate at which the output (measured in terms of its own) becomes available.

If we measure input as well as output in terms of value, we can show both curves on the same diagram. The expected total output at any date will consist partly of the value equivalent of the input whose product matures at that moment and partly of interest. If total output already provided for for each date is .shown by the output curve TV2 , we can divide the output expected at each moment of time into these two parts and obtain thus a second curve, TV1. The horizontal distance between this curve and the ordinate gives us for each moment of time the value of the input whose product matures at that moment, while the horizontal distance, at the same point, between the new curve and the output curve, gives us the additional value of the output due to interest accrued on the value of the input. We have then in the same diagram two descriptions of the time distribution of the output due to a given input: CH. VIII Output Function and Input Function 109 one in terms of the products of given units of input (in factor terms) and the other in terms of units of output (commodity terms). Of these two curves the second is of course the output curve and the first the input curve.

The important point, however, is the difference between the shapes of these two curves. The proportional addition due to interest (compared with the value of the input on which it accrues) will of course become larger and larger T o Output curve Input curve Value of pure input FIG. 3 V1 Interest V2 as we go further into the future; or, in terms of the dia gram, the distance between the input curve and the output curve, 'although it will become ahsolutely smaller as we move upwards along the ordinate, 'will become relatively larger compared with the abscissa of the input curve. This means that towards the top of our diagram the curvature of the input curve will become greater (or its steepness will increase more rapidly than that of the output curve). This expresses the fact that as we move further into the future the rate at which products of given units of input mature WIll decrease more rapidly than the rate at which given units of output mature.

110 Investment in a Simple Economy PT. II The difference between the two curves can perhaps be seen more clearly if we again use the non-cumulative form of representation. Let us first assume that the investTh diU ment periods of different units of input are e erence restatedlntermsofnon-spread evenly over the whole range, so cumulative curves that the products of the given units of input will mature at a constant rate. Measured in terms of the units of input to which it is due, the stream of output can then be represented by the single rectangular strip shown before and indicated by ORPT in Fig. 4. But the products of equal units of input maturing at different t t t t T PST P s T T ORrORrORrORr abc d FIG. 4 dates will not be of equal size. In order to obtain the magnitude of the output in terms of units of its own we shall have to add compound interest, at the ruling rate, for the time for which the different units of input have been invested. Assuming the rate of interest to be given, we can depict this on the diagram by adding to the abscissa at each point a quantity correspondin.g to com pound interest on the initial quantity of input invested for the periods shown along the ordinate. The result is the compound interest curve RS, and the output stream corresponding to an even distribution of the investment periods of the different units of input will be of the shape indicated by ORST. Instead of starting from a linear input curve and deriving the shape of the output stream CR. VIII Output Function and Input Function III from it, we might of course start from a constant output stream and deduct interest from it in order to derive the rate at which products of equal units of input will mature.

Starting from a constant output stream this process of discounting gives us the distribution of input over the different investment periods which is indicated by the strip marked ORPT in Fig. 4 (b). Figs. 4 (c) and 4 (d) illustrate the two other types of case which were considered before. Later on we shall have to discuss the nature of the relationship between the input curve and the output curve, and their mutual interdependence. We shall then see that in certain cases, particularly where it is the duration of the process of produc tion with which we have to deal, the input function is the fundamental magnitude from which we have to start, and the out Both the Input and tb. output curv. are required (or the dIs cussion 01 the economic problems Involved put curve can only be derived from it by construction. In other cases, particularly those of durable goods, the output function is the initial datum from which we have to start, and the input curve has to be constructed from it. In the former case (the" continuous input - point output" case) we know when particular units of input are invested and when the total product of a process matures.

This means that we know how long we have to wait for the product of particular units of input invested. But as we do not always know what share of the product is due to each of these units of input, we may be unable to decide on technical grounds how long we have to wait for par ticular units of output. In the second case (the "point input - continuous output" case) we know when all the input has been invested. in a particular process, and when the particular units of the product of that process mature, but we do not necessarily know how long we have to wait for the product of particular units of input. And in the real world, where "continuous input continuous output" cases are the rule, the situation is, of course, still more complicated.

112 Investment in a Simple Economy PT. II We shall see that in a great many cases only one of the two magnitudes is directly given a,nd that the other has to be derived by a process of discounting or accumulation. Either may, however, And it is only by such methods of convert serve asa basis lor the ing one into the other that we are able to schematic descripllon olthe continuous pro-arrive at a complete picture of the whole cess 01 production structure of investment in terms of either the input function or the output function. For the present, however, we shall neglect these difficulties and shall make use of only one of these. two concepts for describing the process as a whole.