Principles of Economics

Marshall, Alfred
(1842-1924)
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1890
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London: Macmillan and Co., Ltd.
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1920
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8th edition
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BOOK V, CHAPTER VI

JOINT AND COMPOSITE DEMAND. JOINT AND COMPOSITE SUPPLY.

V.VI.1

§ 1. Bread satisfies man's wants directly: and the demand for it is said to be direct. But a flour mill and an oven satisfy wants only indirectly, by helping to make bread, etc., and the demand for them is said to be indirect. More generally:—

V.VI.2

The demand for raw materials and other means of production is indirect and is derived from the direct demand for those directly serviceable products which they help to produce.

V.VI.3

The services of the flour mill and the oven are joined together in the ultimate product, bread: the demand for them is therefore called a joint demand. Again, hops and malt are complementary to one another; and are joined together in the common destination of ale: and so on. Thus the demand for each of several complementary things is derived from the services which they jointly render in the production of some ultimate product, as for instance a loaf of bread, a cask of ale. In other words there is a joint demand for the services which any of these things render in helping to produce a thing which satisfies wants directly and for which there is therefore a direct demand: the direct demand for the finished product is in effect split up into many derived demands for the things used in producing it*46.

V.VI.4

To take another illustration, the direct demand for houses gives rise to a joint demand for the labour of all the various building trades, and for bricks, stone, wood, etc. which are factors of production of building work of all kinds, or as we may say for shortness, of new houses. The demand for any one of these, as for instance the labour of plasterers, is only an indirect or derived demand.

V.VI.5

Let us pursue this last illustration with reference to a class of events that are of frequent occurrence in the labour market; the period over which the disturbance extends being short, and the causes of which we have to take account as readjusting demand and supply being only such as are able to operate within that short period.

V.VI.6

This case has important practical bearings, which give it a special claim on our attention; but we should notice that, referring as it does to short periods, it is an exception to our general rule of selecting illustrations in this and the neighbouring chapters from cases in which there is time enough for the full long-period action of the forces of supply to be developed.

V.VI.7

Let us then suppose that the supply and demand for building being in equilibrium, there is a strike on the part of one group of workers, say the plasterers, or that there is some other disturbance to the supply of plasterers' labour. In order to isolate and make a separate study of the demand for that factor, we suppose firstly that the general conditions of the demand for new houses remain unchanged (that is, that the demand schedule for new houses remains valid); and secondly we assume that there is no change in the general conditions of supply of the other factors, two of which are of course the business faculties and the business organizations of the master builders; (that is, we assume that their lists of supply prices also remain valid). Then a temporary check to the supply of plasterers' labour will cause a proportionate check to the amount of building: the demand price for the diminished number of houses will be a little higher than before; and the supply prices for the other factors of production will not be greater than before*47. Thus new houses can now be sold at prices which exceed by a good margin the sum of the prices at which these other requisites for the production of houses can be bought; and that margin gives the limit to the possible rise of the price that will be offered for plasterers' labour, on the supposition that plasterers' labour is indispensable. The different amounts of this margin, corresponding to different checks to the supply of plasterers' labour, are governed by the general rule that:— The price that will be offered for any thing used in producing a commodity is, for each separate amount for the commodity, limited by the excess of the price at which that amount of the commodity can find purchasers, over the sum of the prices at which the corresponding supplies of the other things needed for making it will be forthcoming.

V.VI.8

To use technical terms, the demand schedule for any factor of production of a commodity can be derived from that for the commodity by subtracting from the demand price of each separate amount of the commodity the sum of the supply prices for corresponding amounts of the other factors*48.

V.VI.9

§ 2. When however we come to apply this theory to the actual conditions of life, it will be important to remember that if the supply of one factor is disturbed, the supply of others is likely to be disturbed also. In particular, when the factor of which the supply is disturbed is one class of labour, as that of the plasterers, the employers' earnings generally act as a buffer. That is to say, the loss falls in the first instance on them; but by discharging some of their workmen and lowering the wages of others, they ultimately distribute a great part of it among the other factors of production. The details of the process by which this is effected are various, and depend on the action of trade combinations, on the higgling and bargaining of the market, and on other causes with which we are not just at present concerned.

V.VI.10

Let us inquire what are the conditions, under which a check to the supply of a thing that is wanted not for direct use, but as a factor of production of some commodity, may cause a very great rise in its price. The first condition is that the factor itself should be essential, or nearly essential to the production of the commodity, no good substitute being available at a moderate price.

V.VI.11

The second condition is that the commodity in the production of which it is a necessary factor, should be one for which the demand is stiff and inelastic; so that a check to its supply will cause consumers to offer a much increased price for it rather than go without it; and this of course includes the condition that no good substitutes for the commodity are available at a price but little higher than its equilibrium price. If the check to house building raises the price of houses very much, builders, anxious to secure the exceptional profits, will bid against one another for such plasterers' labour as there is in the market*49.

V.VI.12

The third condition is that only a small part of the expenses of production of the commodity should consist of the price of this factor. Since the plasterer's wages are but a small part of the total expenses of building a house, a rise of even 50 per cent. in them would add but a very small percentage to the expenses of production of a house and would check demand but little*50.

V.VI.13

The fourth condition is that even a small check to the amount demanded should cause a considerable fall in the supply prices of other factors of production; as that will increase the margin available for paying a high price for this one*51. If, for instance, bricklayers and other classes of workmen, or the employers themselves cannot easily find other things to do, and cannot afford to remain idle, they may be willing to work for much lower earnings than before, and this will increase the margin available for paying higher wages to plasterers. These four conditions are independent, and the effects of the last three are cumulative.

V.VI.14

The rise in plasterers' wages would be checked if it were possible either to avoid the use of plaster, or to get the work done tolerably well and at a moderate price by people outside the plasterers' trade: the tyranny, which one factor of production of a commodity might in some cases exercise over the other factors through the action of derived demand, is tempered by the principle of substitution*52.

V.VI.15

Again, an increased difficulty in obtaining one of the factors of a finished commodity can often be met by modifying the character of the finished product. Some plasterers' labour may be indispensable; but people are often in doubt how much plaster work it is worth while to have in their houses, and if there is a rise in its price they will have less of it. The intensity of the satisfaction of which they would be deprived if they had a little less of it, is its marginal utility; the price which they are just willing to pay in order to have it, is the true demand price for plasterers' work up to the amount which is being used.

V.VI.16

So again there is a joint demand for malt and hops in ale. But their proportions can be varied. A higher price can be got for an ale which differs from others only in containing more hops; and this excess price represents the demand for hops*53.

V.VI.17

The relations between plasterers, bricklayers, etc., are representative of much that is both instructive and romantic in the history of alliances and conflicts between trades-unions in allied trades. But the most numerous instances of joint demand are those of the demand for a raw material and the operatives who work it up; as for instance cotton or jute or iron or copper, and those who work up these several materials. Again, the relative prices of different articles of food vary a good deal with the supply of skilled cooks' labour: thus for instance many kinds of meat and many parts of vegetables which are almost valueless in America, where skilled cooks are rare and expensive, have a good value in France, where the art of cooking is widely diffused.

V.VI.18

§ 3. We have already*54 discussed the way in which the aggregate demand for any commodity is compounded of the demands of the different groups of people who may need it. But we now may extend this notion of composite demand to requisites of production which are needed by several groups of producers.

V.VI.19

Nearly every raw material and nearly every kind of labour is applied in many different branches of industry, and contributes to the production of a great variety of commodities. Each of these commodities has its own direct demand; and from that the derived demand for any of the things used in making it can be found, and the thing is "distributed between its various uses" in the manner which we have already discussed*55. The various uses are rivals, or competitors with one another; and the corresponding derived demands are rival or competitive demands relatively to one another. But in relation to the supply of the product, they co-operate with one another; being "compounded" into the total demand that carries off the supply: in just the same way as the partial demands of several classes of society for a finished commodity are aggregated, or compounded together into the total demand for it*56.

V.VI.20

§ 4. We may now pass to consider the case of joint products: i.e. of things which cannot easily be produced separately; but are joined in a common origin, and may therefore be said to have a joint supply, such as beef and hides, or wheat and straw*57. This case corresponds to that of things which have a joint demand, and it may be discussed almost in the same words, by merely substituting "demand" for "supply," and vice versâ. As there is a joint demand for things joined in a common destination: so there is a joint supply of things which have a common origin. The single supply of the common origin is split up into so many derived supplies of the things that proceed from it*58.

V.VI.21

For instance, since the repeal of the Corn Laws much of the wheat consumed in England has been imported, of course without any straw. This has caused a scarcity and a consequent rise in the price of straw, and the farmer who grows wheat looks to the straw for a great part of the value of the crop. The value of straw then is high in countries which import wheat, and low in those which export wheat. In the same way the price of mutton in the wool-producing districts of Australia was at one time very low. The wool was exported, the meat had to be consumed at home; and as there was no great demand for it, the price of the wool had to defray almost the whole of the joint expenses of production of the wool and the meat. Afterwards the low price of meat gave a stimulus to the industries of preserving meat for exportation, and now its price in Australia is higher.

V.VI.22

There are very few cases of joint products the cost of production of both of which together is exactly the same as that of one of them alone. So long as any product of a business has a market value, it is almost sure to have devoted to it some special care and expense, which would be diminished, or dispensed with if the demand for that product were to fall very much. Thus, for instance, if straw were valueless, farmers would exert themselves more than they do to make the ear bear as large a proportion as possible to the stalk. Again, the importation of foreign wool has caused English sheep to be adapted by judicious crossing and selection so as to develop heavy weights of good meat at an early age, even at the expense of some deterioration of their wool. It is only when one of two things produced by the same process is valueless, unsaleable, and yet does not involve any expense for its removal, that there is no inducement to attempt to alter its amount; and, it is only in these exceptional cases that we have no means of assigning its separate supply price to each of the joint products. For when it is possible to modify the proportions of these products, we can ascertain what part of the whole expense of the process of production would be saved, by so modifying these proportions as slightly to diminish the amount of one of the joint products without affecting the amounts of the others. That part of the expense is the expense of production of the marginal element of that product; it is the supply price of which we are in search*59.

V.VI.23

But these are exceptional cases. It more frequently happens that a business, or even an industry finds its advantage in using a good deal of the same plant, technical skill, and business organization for several classes of products. In such cases the cost of anything used for several purposes has to be defrayed by its fruits in all of them: but there is seldom any rule of nature to determine either the relative importance of these uses, or the proportions in which the total cost should be distributed among them: much depends on the changing features of markets*60.

V.VI.24

§ 5. We may pass to the problem of composite supply which is analogous to that of composite demand. A demand can often be satisfied by any one of several routes, according to the principle of substitution. These various routes are rivals or competitors with one another; and the corresponding supplies of commodities are rival, or competitive supplies relatively to one another. But in relation to the demand they co-operate with one another; being "compounded" into the total supply that meets the demand*61.

V.VI.25

If the causes which govern their production are nearly the same, they may for many purposes be treated as one commodity*62. For instance, beef and mutton may be treated as varieties of one commodity for many purposes; but they must be treated as separate for others, as for instance for those in which the question of the supply of wool enters. Rival things are however often not finished commodities, but factors of production: for instance, there are many rival fibres which are used in making ordinary printing paper. We have just noticed how the fierce action of derived demand for one of several complementary supplies, as e.g. for the supply of plasterers' labour, was liable to be moderated, when the demand was met by the competitive supply of a rival thing, which could be substituted for it*63.

V.VI.26

§ 6. All the four chief problems which have been discussed in this chapter have some bearing on the causes that govern the value of almost every commodity: and many of the most important cross connections between the values of different commodities are not obvious at first sight.

V.VI.27

Thus when charcoal was generally used in making iron, the price of leather depended in some measure on that of iron; and the tanners petitioned for the exclusion of foreign iron in order that the demand on the part of English iron smelters for oak charcoal might cause the production of English oak to be kept up, and thus prevent oak bark from becoming dear*64. This instance may serve to remind us of the way in which an excessive demand for a thing may cause its sources of supply to be destroyed, and thus render scarce any joint products that it may have: for the demand for wood on the part of the ironmakers led to a relentless destruction of many forests in England. Again, an excessive demand for lamb was assigned as a cause of the prevailing scarcity of sheep some years ago; while some argued on the contrary that the better the price to be got for spring lamb sold to the rich, the more profitable would be the production of sheep, and the cheaper would mutton be for the people. The fact is that an increase of demand may have opposite effects according as it does or does not act so suddenly as to prevent producers from adapting their action to it.

V.VI.28

Again, the development of railways and other means of communication for the benefit of one trade, as for instance wheat growing in some parts of America and silver mining in others, greatly lowers some of the chief expenses of production of nearly every other product of those districts. Again, the prices of soda, and bleaching materials and other products of industries, the chief raw material of which is salt, move up and down relatively to one another with almost every improvement in the various processes which are used in those industries; and every change in those prices affects the prices of many other goods, for the various products of the salt industries are more or less important factors in many branches of manufacture.

V.VI.29

Again, cotton and cotton-seed oil are joint products, and the recent fall in the price of cotton is largely due to the improved manufacture and uses of cotton-seed oil: and further, as the history of the cotton famine shows, the price of cotton largely affects that of wool, linen and other things of its own class; while cotton-seed oil is ever opening up new rivalries with things of its own class. Again, many new uses have been found for straw in manufacture; and these inventions are giving value to straw that used to be burnt in the West of America, and tend to hinder the rise in the marginal cost of producing wheat*65.


Notes for this chapter


46.
Compare III. III. 6. It will be recollected that the things in a form ready for immediate use have been called goods of the first order, or consumers' goods; and that things used as factors of production of other goods have been called producers' goods, or goods of the second and higher orders or intermediate goods: also that it is difficult to say when goods are really finished; that many things are commonly treated as finished consumers' goods before they are really ready for consumption, e.g. flour. See II. III. 1. The vagueness of the notion of instrumental goods, regarded as things the value of which is derived from that of their products, is indicated in II. IV. 13.
47.
This is at any rate true under all ordinary conditions: there will be less extra charges for overtime; and the price of the labour of carpenters, bricklayers and others is likely rather to go down than to go up, and the same is true of bricks and other building materials.
48.
The broad account given in the text may suffice for most purposes; and the general reader should perhaps omit the remaining footnotes to this chapter.

It must be remembered that this Derived schedule has no validity except on the suppositions that we are isolating this one factor for separate study; that its own conditions of supply are disturbed; that there is at the time no independent disturbance affecting any other element in the problem; and that therefore in the case of each of the other factors of production the selling price may be taken to coincide always with the supply price.

In illustrating this by a diagram, it will be well, for the sake of shortness of wording, to divide the expenses of production of a commodity into the supply prices of two things of which it is made; let us then regard the supply price of a knife as the sum of the supply prices of its blade and handle, and neglect the expense of putting the two together. Figure 20.  Click to enlarge in new window.Let ss' be the supply curve for handles and SS' that for knives; so that M being any point on Ox, and MqQ being drawn vertically to cut ss' in q and SS' in Q, Mq is the supply price for OM handles, qQ is the supply price for OM blades and MQ the supply price for OM knives. Let DD' the demand curve for knives cut SS' in A, and AaB be drawn vertically as in the figure. Then in equilibrium OB knives are sold at a price BA of which Ba goes for the handle and aA for the blade.

(In this illustration we may suppose that sufficient time is allowed to enable the forces which govern supply price to work themselves out fully; and we are at liberty therefore to make our supply curves inclined negatively. This change will not affect the argument; but on the whole it is best to take our typical instance with the supply curve inclined positively.)

Now let us suppose that we want to isolate for separate study the demand for knife handles. Accordingly we suppose that the demand for knives and the supply of blades conform to the laws indicated by their respective curves: also that the supply curve for handles still remains in force and represents the circumstances of normal supply of handles, although the supply of handles is temporarily disturbed. Let MQ cut DD' in P, then MP is the demand price for OM knives and Qq is the supply price for OM blades. Take a point p in MP such that Pp is equal to Qq, and therefore Mp is the excess of MP over Qq; then Mp is the demand price for OM handles. Let dd' be the locus of p obtained by giving M successive positions along Ox and finding the corresponding positions of p; then dd' is the derived demand curve for handles. Of course it passes through a. We may now neglect all the rest of the figure except the curves dd', ss'; and regard them as representing the relations of demand for and supply of handles, other things being equal, that is to say, in the absence of any disturbing cause which affects the law of supply of blades and the law of demand for knives. Ba is then the equilibrium price of handles, about which the market price oscillates, in the manner investigated in the preceding chapter, under the influence of demand and supply, of which the schedules are represented by dd' and ss'. It has already been remarked that the ordinary demand and supply curves have no practical value except in the immediate neighbourhood of the point of equilibrium. And the same remark applies with even greater force to the equation of derived demand.

[Since Mp - Mq = MP - MQ; therefore A being a point of stable equilibrium, the equilibrium at a also is stable. But this statement needs to be somewhat qualified if the supply curves are negatively inclined: see Appendix H.]

In the illustration that has just been worked out the unit of each of the factors remains unchanged whatever be the amount of the commodity produced; for one blade and one handle are always required for each knife; but when a change in the amount of the commodity produced occasions a change in the amount of each factor that is required for the production of a unit of the commodity, the demand and supply curves for the factor got by the above process are not expressed in terms of fixed units of the factor. They must be translated back into fixed units before they are available for general use. (See Mathematical Note XIV bis.)

49.
We have to inquire under what conditions the ratio pM to aB will be the greatest, pM being the demand price for the factor in question corresponding to a supply reduced from OB to OM, that is reduced by the given amount BM. The second condition is that PM should be large; and since the elasticity of demand is measured by the ratio which BM bears to the excess of PM over AB, the greater PM is, the smaller other things being equal, is the elasticity of demand.
50.
The third condition is that when PM exceeds AB in a given ratio, pM shall be caused to exceed Ba in a large ratio: and other things being equal, that requires Ba to be but a small part of BA.
51.
That is, if Qq had been smaller than it is, Pp would have been smaller and Mp would have been larger. See also Mathematical Note XV.
52.
It is shown in Böhm-Bawerk's excellent Grundzüge der Theorie des wirtschaftlichen Güterwerts (Jahrbuch für Nationalökonomie und Statistik, vol. XIII. p. 59) that if all but one of the factors of production of a commodity have available substitutes in unlimited supply, by which their own price is rigidly fixed, the derived demand price for the remaining factor will be the excess of the demand price for the finished product over the sum of the supply prices thus fixed for the remaining factors. This is an interesting special case of the law given in the text.
53.
See Mathematical Note XVI.
54.
See above, III. IV. 2, 4.
55.
See III. V.
56.
Figure 21.  Click to enlarge in new window.Thus, let a factor of production have three uses. Let d1d1' be the demand curve for it in its first use. From N any point on Oy draw Np1 horizontally to cut d1d1' in p1; then Np1 is the amount that is demanded for the first use at price ON. Produce Np1 to p2, and further on to P making p1p2 and p2P of such lengths as to represent the amounts of the factor demanded at price ON for the second and third uses respectively. As N moves along Oy let p2 trace out the curve d2d2' and let P trace out the curve DD'. Thus d2d2' would be the demand curve for the factor if it had only its first and second uses. DD' is its demand curve for all three uses. It is immaterial in what order we take the several uses. In the case represented, the demand for the second use begins at a lower price and that for the third use begins at a higher price than does the demand for the first use. (See Mathematical Note XVII.)
57.
Professor Dewsnup (American Economic Review, Supplement 1914, p. 89) suggests that things should be described as joint products, when their "total costs of production by a single plant are less than the sum of the costs of their production by separate plants." This definition is less general than that reached at the end of this section; but it is convenient for some special uses.
58.
If it is desired to isolate the relations of demand and supply for a joint product, the derived supply price is found in just the same way as the derived demand price for a factor of production was found in the parallel case of demand. Other things must be assumed to be equal (that is, the supply schedule for the whole process of production must be assumed to remain in force and so must the demand schedule for each of the joint products except that to be isolated). The derived supply price is then found by the rule that it must equal the excess of the supply price for the whole process of production over the sum of the demand prices of all the other joint products; the prices being taken throughout with reference to corresponding amounts.

Figure 22.  Click to enlarge in new window.We may again illustrate by a simple example in which it is assumed that the relative amounts of the two joint products are unalterable. Let SS' be the supply curve for bullocks which yield meat and leather in fixed quantities; dd' the demand curve for their carcases, that is, for the meat derived from them. M being any point on Ox draw Mp vertically to cut dd' in p, and produce it to P so that pP represents the demand price for OM hides. Then MP is the demand price for OM bullocks, and DD' the locus of P is the demand curve for bullocks: it may be called the total demand curve. Let DD' cut SS' in A; and draw AaB as in the figure. Then in equilibrium OB bullocks are produced and sold at the price BA of which Ba goes for the carcase and aA for the hide.

Let MP cut SS' in Q. From QM cut off Qq equal to Pp; then q is a point on the derived supply curve for carcases. For if we assume that the selling price of OM hides is always equal to the corresponding demand price Pp, it follows that since it costs QM to produce each of OM bullocks there remains a price QM - Pp, that is qM, to be borne by each of the OM carcases. Then ss' the locus of q, and dd' are the supply and demand curves for carcases. (See Mathematical Note XVIII.)

59.
See Mathematical Note XIX.
60.
A little more is said on this subject in the next chapter: it is discussed fully in the forthcoming work on Industry and Trade.
61.
The latter phrase "competing commodities" is used by Prof. Fisher in his brilliant Mathematical Investigations in the theory of value and prices, which throw much light on the subjects discussed in the present chapter.
62.
Comp. Jevons, l. c. pp. 145 6. See also above, footnotes on pp. 100, 105.
63.
The want which all the rivals tend to satisfy is met by a composite supply, the total supply at any price being the sum of the partial supplies at that price.

Figure 23.  Click to enlarge in new window.Thus, for instance, N being any point on Oy draw Nq1q2Q parallel to Ox such that Nq1, q1q2 and q2Q are respectively the amounts of the first, second and third of those rivals which can be supplied at the price ON. Then NQ is the total composite supply at that price, and the locus of Q is the total supply curve of the means of satisfying the want in question. Of course the units of the several things which are rivals must be so taken that each of them satisfies the same amount of the want. In the case represented in the figure small quantities of the first rival can be put on the market at a price too low to call forth any supply of the other two, and small quantities of the second at a price too low to call forth any of the third. (See Mathematical Note XX.)

Continued rivalry is as a rule possible only when none of the rivals has its supply governed by the law of increasing return. The equilibrium is stable only when none of them is able to drive the others out; and this is the case when all of them conform to the law of diminishing return; because then if one did obtain a temporary advantage and its use increased, its supply price would rise, and then the others would begin to undersell it. But if one of them conformed to the law of increasing return, the rivalry would soon cease; for whenever it happened to gain a temporary advantage over its rivals its increased use would lower its supply price and therefore increase its sale—its supply price would then be further lowered, and so on: thus its advantage over its rivals would be continually increased until it had driven them out of the field. It is true that there are apparent exceptions to this rule; and things which conform to the law of increasing return do sometimes seem to remain for a long time in the field as rivals: such is the case perhaps with different kinds of sewing machines and of electric lights. But in these cases the things do not really satisfy the same wants, they appeal to slightly different needs or tastes; there is still some difference of opinion as to their relative merits; or else perhaps some of them are patented or in some other way have become the monopoly of particular firms. In such cases custom and the force of advertising may keep many rivals in the field for a long time; particularly if the producers of those things which are really the best in proportion to their expenses of production are not able effectively to advertise and push their wares by travellers and other agencies.

64.
Toynbee (Industrial Revolution, p. 80).
65.
Again, since sheep and oxen compete for the use of land, leather and cloth compete in indirect demand for the use of a factor of production. But also in the upholsterer's shop they compete as supplying means for meeting the same want. There is thus a composite demand on the part of upholsterer and shoemaker for leather; and also for cloth when the upper part of a shoe is made of cloth: the shoe offers a joint demand for cloth and leather, they offering complementary supplies: and so on, in endless complications. See Mathematical Note XXI. The Austrian doctrine of "imputed value" has something in common with that of derived value given in this chapter. Whichever phrase be used, it is important that we should recognize the continuity between the old doctrine of value and the new; and that we should treat imputed or derived values merely as elements which take their place with many others in the broad problem of distribution and exchange. The new phrases merely give the means of applying to the ordinary affairs of life, some of that precision of expression which is the special property of mathematical language. Producers have always to consider how the demand for any raw material in which they are interested is dependent on the demand for the things in making which it is used, and how it is influenced by every change that affects them; and this is really a special case of the problem of ascertaining the efficient strength of any one of the forces, which contribute to a common result. In mathematical language this common result is called a function of the various forces: and the (marginal) contribution, which any of them is making to it, is represented by the (small) change in the result which would result from a (small) change in that force; that is by the differential coefficient of the result with regard to that force. In other words, the imputed value, or the derived value of a factor of production, if used for only one product, is the differential coefficient of that product with regard to that factor; and so on in successive complications, as indicated in Notes XIV.—XXI. of the Mathematical Appendix. (Some objections to parts of Prof. Wieser's doctrine of imputed values are well urged by Prof. Edgeworth, Economic Journal, Vol. V. pp. 279-85.)

End of Notes


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