XVI. The Marginal Value Product of Investment: The Problem of Attribution (Imputation)

CHAPTER XVI THE 'MARGINAL VALUE PRODUCT OF INVESTMENT: THE PROBLEM OF ATTRIBUTION (IMPUTATION) IN the cases we considered in Chapter XV we were able to build up the relationship between the input function and the output function from the connection between Particular Input lunc !ions ara uniquely correlaie.! with par ticular output luno lions only If physical marginal product 01 every unit 01 Input can be Isolated individual units of input and individual units of output, which were assumed to be known. Given the shape of the input function, and the marginal product of every small unit of input to which it re ferred, we were able to derive from it the shape of the output function. And since it wa,s assumed that we knew the amount of input on the co-operation of which each particular part of the output depended, we were able similarly to derive the input function from the output function. This connection between the two functions was independent of the rate of interest and was based on the known physical marginal productivity of the different units of input. In these cases particular input functions and particular output functions were uniquely correlated in the sense that one, and only one, output function would fit a particular input function and vice versa. The rate of interest came into the discussion only to the extent that, in equilibrium, such an input function would have to be chosen that the relation between the value of any unit of input and the value of the dependent unit of output would correspond to the rate of interest ruling in the system. Changes in the output function could be brought about only by changes in the input function and if the connection between any particular input function and the corresponding output function was 202 CR. XVI Marginal Value Product of Investment 203 not dependent on the rate of interest (or complex of interest rates). And the input function (or rather the different input functions of the different kinds of input) and the output function could both be regarded as techno logical data which were given independently of the rate of interest or of any other value phenomenon which could only be the resultant of equilibrium.

We must now pass on to the cases where, in place of this technologically given connection between individual units of input and individual units of output, all that is given is a connection between aggregates Tho easo whore only of input an~ aggregates of output. It is the relation between aggregates of Input convenient to proceed. immediately to the andaggregatosofoutextreme opposite of the cases so far con-put Is known sidered, i.e. to the cases where nothing more than the connection between the aggregates is given. And we shall leave for later consideration the intermediate cases where it is possible, on technological grounds, to establish a connection between at least some parts of the input and some parts of the output. There are two main cases to be considered here. The first is that of goods in process where the shape of the input function is rigidly fixed by technical conditions, and where, in consequence, W<l know how long Main Instances to be we have to wait for the product of different considered parts of the input, but cannot say, on technological grounds, what parts of the product are due to the different units of input, or how long we have to wait for the different parts of the output. The second is that of durable goods whose durability cannot be varied, and where, accordingly, we know exactly how long we have to wait for the different parts of the services the goods will render, but do not know to \Vhat portions 01' the input each part of these services is to be attributed, or how long we have to wait for the product of different portions of the input.

In the first case the input function is known and the output function is unknown; in the second case the output 204 Investment in a Simple Economy PT. n function is known and the input function unknown. The problem in the first case is how to allocate a given aggre gate of output among the different units of input invested for a known range of periods, and the problem in the second case is how to allocate a given aggregate of input between the units of output which are due to it and which mature at a known rate over a known period of time. The first of these two cases presents no real difficulties. We have a product of given size which is due to the investment of a given aggregate of input over a given t. Time - consuming range of periods. Under stationary con processes with an ditions the product maturing at a moment Input lunctlon of Invariable shape of time as the result of a series of investments made for a range of periods preceding that moment (described by the inverted input curve) will be equal to the total product maturing over a range of periods and due to the investments made at a moment of time (described by the original input curve). We can there fore make either of these two aspects of the process the basis of our discussion. For the sake of conformity with the treatment of the next case we choose the second aspect, i.e. the representation by means of the input curve in its original form, which was also used for the foundation of Fig. 16 above.

Equilibrium requires that the total value of the pro duct shall be equal to the value of the input plu8 interest on every unit of the input, for the known period for The relallon between value of Input and value 01 output Is ad JOIied by varying the total quantity 01 out put which it is invested, at a rate which is equal to that ruling in the system. Since with given technique, the quantity of the product obtainable from investment of given quantities of input is a datum, the only way in which this equality can be brought about is by varying the total quantity produced (which implies varying the total quantity of input invested in that line of production). Every increase in the quantity produced in the line of production concerned will have two effects; OB. XVI Marginal Value Product of Investment 205 it will decrease the value of a unit of output, and it will increase the value which units of input have in alternative uses. Thus, by varying the total quantity produced, any desired proportion between the total value of the input and the total value of the output can be brought about.

At a given rate of interest the quantity which can profit ably be invested in this line of production will therefore be uniquely. determined. But we ne~d not assume that the rate of interest is given independently of the magnitude of the output in this particular liIie of production. The condition that the rate of interest here should be equal to that ruling else where takes account of any possible effect of changes in the scale of production in this industry on the rates of interest elsewhere; it therefore gives us the general condition of equilibrium. The idea that the value of the product should be sufficient, and only just sufficient, to cover the value of all the units of input plus interest for their respective investment periods obviously implies some The general problem " ideal" allocation of the total product of attribution (Impubetween the co-operating units of input, tatlOD) i.e. an attribution of particular "ideal" shares of the product to particular units of input. To use a concept which has been traditionally applied in economics to this sort of attribution, we may talk about,the imputation of definite parts of the value of the product to the different units of input co-operating in its production. And this process of imputation, which enables us to connect par ticular quantities of input with definite quantities of output, also enables us to construct from a given input function the corresponding output function (i.e. a descrip tion of the range of periods during which we have to wait for the quantities of output which are due to the invest ments made at a moment of time). The output function so obtained is not, of course, in any sense a physical datum, a magnitude which can be found in the real 206 1 nvestment in a Simple Economy PT. II world: it is a calculating device, a " construct", which, given equilibrium conditions, can be derived from the data. The fact that we can express the equilibrium con ditions in the form of an output function is, however, not without significance. For, as we have already observed, in real life the relevant relationships are in some cases given in the form of output functions, and in others sometimes in the form of input functions, and in consequence we are only able to give a comprehensive picture if we are able to convert one into the other.

The circumstance that the problem of attributing par ticular parts of the output to particular units of input cannot be solved by reference to the physical dependence The determination of the "marginal value product" analogous to other cases of Dxed coemclents of pro duction of the former on the latter, but has to be solved by means of imputation of value, is, of course, not peculiar to our present case. It applies generally to all cases where the proportion in which different kinds of input are combined is not continuously variable. In all such cases of fixed coefficients of production it is impossible to determine a physical marginal product, i.e. the quantity of the product which depends on the co-operation of a small unit of input. All that we can do is to determine a "marginal value product", that is, that part of the value of the product which it must be possible to assign to a unit of a factor in order to render its employment profitable.

The case of a rigid input function, as well as the case of an invariable output function which we have still to consider, are only special instances of the general pheno menon of " constant coefficients of production ". Their only peculiarity is that it is not the proportions in which physically different factors can be used in one process of production that are rigidly fixed, but the . proportions between the quantities of factors that can be applied at different stages of the process. It follows that the prob lems to which these cases give rise are to be solved along CR. XVI Marginal Value Product of Investment 207 the same lines as other instances of the more general case of fixed coefficients. We turn now to our second case, that of durable goods with fixed and invariable durability. In order to avoid complications arising out of the use of such durable goods in production we shall here confine our 2. Durable goods with attention to the case of durable con-IIxe<llenglhs 011110 sumers' goods. It will also be convenient to begin once more with the case of the "ideal" durable good, the production of which takes no time, and which therefore.

corresponds to the theoretical" point input - continuous output" case. In this case the investment of a given quantity of input at a particular moment yields a stream of services of a given and invariable time shape. This means that while we know how long we have to wait for e'ach unit of the services of the good, we do not know to what portion of the input invested these units are due, and consequently how long we have to wait for the products of the different units of the input invested. In other words, the output function is given, and the input function has to be derived from it. The question to be answered is to what "ideal" portions of the input invested at a moment of time the different segments of the resultant output stream have to be attributed. Equilibrium requires that the value of the output stream, each part being discounted for the relevant period at a rate of interest equal to that ruling elsewhere in the system, shall be equal to the value of the input invested. And the required relationship between the aggregate value of the input invested and the value of the output stream can be brought about by varying the total volume of output. The discounted value of any small part of the output stream maturing during a certain time interval gives us the " ideal" portion of the input, which may be said to be invested for the relevant period.

The principle of the solution is exactly parallel to that applying in the previous case. But the present case is so 208 I nve~tment in a Simple Economy PT. IJ impqrtant that it is worth while illustrating its t>ignific ance by considering one or two special instances in greater detail. We shall ask first, what will be the effect of a change in the rate of interest on the shape of the input function which we derive from a given output function? and, secondly, what, at a given rate of interest, will be the shapes of the input functions belonging to different output functions? Both questions can be conveniently answered by means of a diagram similar to Fig. 16 above. FIG. 18 In Fig. 18 the strip marked 8(t) represents the shape of the output stream (in this case assumed to be of constant volume through time) expected from a given durable Enect 01 rale 01 10-good, and shown as a simple (not cumu teres! on shape 01 lative) frequency distribution The amount (constructed) Input • curve of input used in the production of the good is indicated by the distance ORl along the r-axis. Values are shown, as in the former diagram, along the perpen dicular v-axis, and the growth at a given rate of compound interest will therefore again be represented by an upward sloping curve in any plane parallel to vOt. For the pur poses of diagrammatic representation it is useful to assume that the value of the services of the good (the output stream) is given, although actually it is of course variable in just the same way as the value of the input.

CH. XVI Marginal Value Product of Investment 209 The value of the output stream at a moment of time (i.e. as a time rate) is then shown by the height of the strip which describes its shape. In order for the discounted value of the output stream to be equal to the value of the input, it must be possible to exhaust the total value of the output stream by allotting to each small unit of input a segment of the output stream such that the discounted value of that segment is equal to the value of the unit of input. Or, since the value of all units of input must grow, during the time that they remain invested, at the same compound rate of interest, the product of the different units of input must be presumed to mature at such a rate that the value of the input whose product matures during any interval, plus compound interest for the period for which it has been invested, will be equal to the given value of the output stream during that interval. In geometrical terms this means that the slope of the input curve RITI at any point must be such that the product of this slope times the height of the interest surface will be equal to the given size of the output stream at this point.

As will be seen from the diagram, with a constant income stream and the given rate of interest represented by the fully drawn interest surface V 2QT J', we obtain the concave input function represented by the fully drawn curve RIT l' The meaning of this is, of course, that since a greater amount of interest accrues on the "ideal" shares of the input which are invested for the longer periods than on those which are invested for the shorter periods, a larger part of the output stream has to be attributed to the former than to the latter: or, what amounts to the same thing, that where the income stream is constant the product of given units of input must be conceived to mature at a slower rate during the later part of the life of the good than during the earlier part. (The whole relationship is the same as that shown in a simpler manner in Fig. 4 (b) on p. llO, and the reader who finds 15 210 I nvestment in a Simple Economy PT. II the present more complete representation difficult will do well to refer back to that earlier diagram.) The diagram also depicts the effect of a rise in the rate of interest, the dotted curves indicating the interest sur face V 1Q'T" 1 corresponding to the higher rate of interest and the new input function shown as dotted curves. It will be seen that the effect is to make the input curve more concave, ioe. to attribute the services maturing later to a greater amount of input and the services maturing earlier to a smaller amount, and to reduce the value of the input relatively to the value of the output. (The diagram shows only a decrease in the value of the input, but it is clear that the decrease in its value relative to the value of its product will be brought about partly by a fa~l in the value of the input and partly by a rise in the value of the output.) Lastly, Fig. 19 shows the shape of the input functions, which, at a given rate of interest, correspond to output functions of different shapes. The strips marked 81 , 82, Inftuence of shape of and 83 represent three (simple) output.

output fUnction on functions which increase at a constant shape of input function at given rate of interest rate, remain constant throughout, and decrease at a constant rate respectively. The curves marked rP1' rP2' and rP3 represent the corre sponding (cumulative) input functions. It will be noticed that a tendency for the output function to decrease makes the input function more concave, and that a tendency for the output function to increase makes the input function less concave, or may (if the output function increases at a rate greater than the rate of interest) even make it (partially or entirely) convex. In order for the input function to be linear it would be necessary for the output function to increase in geometrical progression at a rate equal to the prevailing rate of interest. In order to complete the analysis we ought now to extend our argument to two further cases. The first is the " continuous input - continuous output" case correCH. XVI Marginal Value Product of Investment 211 sponding to durable goods which are the product of a 'time-consuming process of production. The second case, . or group of cases, includes durable goods which do not render final services directly without further The more oompIlcaled collaboration from other factors but give cases off different amounts of these services according to the amount of co-operating factors used, and durable goods which give off different amounts of final services according to the intensity with which they are used.

The latter group of cases includes, of course, not only a T' 1 FIG. 19 great many durable consumers' goods but also all durable producers' goods. Although these cases present no really new problems, their actual analysis is so complicated that it is hardly possible to give it in any detail without resorting to an elaborate mathematical apparatus, and they are consequently best left to more specialised studies. In order to give some indication of the kind of difficulty which arises, and of the general principle under-. lying its solution, a few words may be added on the " continuous input - continuous output" Tb" II I e con nuous ncase. Here the difficulty is that it is no pul-oonllnuous oul• put" C&lO longer sufficient to be able to attribute definite parts of the output to ~, ideal" portions of-the input; we require to know what part of the output is to 212 Investment in a Simple Economy PT. II be attributed to the concrete quantities of input that are invested at particular dates. But it is not possible to say that some particular unit of input invested in -the course of the production of the durable good concerned contri butes only to the services which that good renders at a particular moment or during a particular small interval of time. What we have to do in order to obtain a clear picture of the value relationship is to attribute to each of the different units of input invested at different stages of the production of the durable good some small part of the services which the durable good will render at all moments of its life. This means that, in order to dis tribute the value of the services rendered by the good among the units of input invested in it at different dates, we shall have, so to speak, to slice the output stream longitudinally, and to attribute to the different units of input slices of different thickness according as they have been invested earlier or later in the process of production.

This relationship, which cannot be more than suggested in words, obviously transcends the possibility of dia grammatic representation even in three dimensions; it can be adequately described only in terms of mathematics. There remains, however, a more important problem which must be dealt with before we conclude this chapter. We have confined our attention so far in the chapter to Parllal rigidities cases where either the input function or the output function was rigidly fixed and could not be varied in any way. It was mentioned near the beginning, as a point reserved for later consideration, that between the case of continuous variability of the input function and the case where either the input function or the output function was of an invariable shape, there were important intermediate cases. These include the case where the known input function is only partially variable, i.e. where the rate at which input is applied in the course of the process can be varied at certain points, or for some part of the input, but not at others. This case, OK. XVI Marginal Value Product of Inve8tme~t 213 although a source of additional complications, does not require any further comment. A more interesting case which merits further discussion is where the shape of the known output function allows of limited variation because the durability of the good to which it refers may be changed by altering the amount of input invested in that good. This may serve as an instance of the· general case where a definite physical marginal product can be determined for some part of the input but not for all.

The peculiarity of a durable good with variable durability is that it will not usually be possible to vary the shape of the stream of services obtained from it in any arbitrary manner; the effects of varying the amount of input invested in the good will be mainly concentrated at the end of the stream of services derived from the good. Exactly how the additions Changing the length of life of a durable good: the time ells trlbutlon of the result 01 a marginal Invest ment to the stream.of services from any durable good which are due to the investment of additional input will be dis tributed in time will t depend on the initial shape of this stream T, of services (and the T change in this shape, if any, which is caused by the addi tional investment). t T, T t T, T If the stream is ORrORr ORr of constant shape, a b and simply becomes longer when more FIG. 20 c input is invested in the good, the marginal increment will all be at the end of that stream, as is shown in diagram (a) of Fig. 20. The two other cases that are most likely to occur are those where the stream of services decreases at a constant rate and at a decreasing rate respectively.

Assuming the general shape of the stream to remain the 214 Investment in a Simple Economy PT. II same with the additional input as without, the marginal additions to the stream of services will in these cases be distributed in time in the way depicted by diagrams (b) and (c) respectively of the same figure. The essential point, however, is that in all these cases we cannot establish a marginal dependence on particular units of input for individual units of service maturing at all points during the life of the good, but only for those units of service which are due to the marginal addition total input. This means that while we cannot derive a complete input function from the given output func tion on a marginal productivity basis, and while in consequence it cannot be said that a particular input function is uniquely correlated with the given output function, there does exist a definite connection between the marginal investment in the durable good and the additions to the stream of services which that good renders.

The conclusion is that two factors now have to be taken into account in deciding how much it will be profitable to invest in any particular kind of durable good. The Combined ellect of marginal product of the input invested in varying quantity and anyone such good must be equal to the varying durability of durable goods value of a unit of that input, and the discounted value of all the services yielded by the good must be equal to the total value of the input invested in it. Two kinds of variation are possible in order to reach this equality: the durability of the individual good can be varied, and the number of goods of the same kind that are produced can be changed. Both these possibilities of variation will have to be used, until not only the discounted value of the variable physical marginal product of the last unit of inpu~ invested in a single good is equal to the value of that unit of input, but the discounted value of all the services of any good is equal to the value of all the input invested in it.

This case of durable goods of variable durability is the most important of the cases that are needed to supplement CH. XVI Marginal Value Product of Investment 215 the earlier discussion of the connection between the pro ductivity of investment and the rate of interest. For the " ideal" durable good (the "point input - continuous output" case), what has already been said is sufficient to indicate the complete solution. It would of course be possible to go on and apply the same kind of argument to the more complicated cases mentioned before. But though a casuistic elaboration of this kind may be neces sary for certain purposes, the general principle has prob ably been sufficiently well illustrated, and the gain we should derive from further refinements would hardly compensate for the effort which the more detailed analysis of these highly complicated cases would require.