The Economics of Welfare

Pigou, Arthur C.
(1877-1959)
CEE
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First Pub. Date
1920
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London: Macmillan and Co.
Pub. Date
1932
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4th edition.
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APPENDIX I
UNCERTAINTY-BEARING AS A FACTOR OF PRODUCTION

App.I.1

§ 1. IT is customary in economic discussion to class together as factors of production, along with the services of Nature, waiting and various sorts of mental and manual labour. In a world in which all future events were perfectly foreseen this catalogue would be substantially adequate. But in the actual world some future events are not perfectly foreseen. On the contrary, in the vast majority of enterprises, in the conduct of which resources are waited for, they are also exposed to uncertainty; they are turned, that is to say, into a use, the result of which cannot be certainly predicted. In these circumstances it is proper that there should be added to the list of factors of production enumerated above a further group comprising various sorts of uncertainty-bearing.

App.I.2

§ 2. The principal reason why this arrangement is not usually adopted seems to be that, in practice, uncertainty-bearing is bound up in such intimate association with waiting that the possibility of separating the two in analysis is not immediately apparent. Reflection, however, makes it plain that the connection between them is not a necessary or inherent connection,—that they are, in fact, two things generally found together, and not a single thing. Thus let us imagine a man in possession of a vase, which, as a vase, is worth £100, but, if broken, would be worth nothing; and let us suppose the owner to know that this vase contains something, whose value is equally likely to be anything between nothing and £250. If the owner breaks the vase, he is, then, equally likely to lose any sum up to £100 or to gain any sum up to £150. The actuarial value of his chance is, therefore, £25, and, if there were a million people in his position, and they all elected to break their vases, the aggregate wealth of them all would probably be increased by about £25,000,000. In other words, the services of these million people, in bearing the uncertainty of placing £100 each in a position where it is equally likely to become anything between nothing and £250, are responsible for an addition of £25,000,000 to national wealth. This example shows that uncertainty-bearing, though generally associated with waiting, is analytically quite distinct from it. Nor was it really necessary to seek an illustration so far removed from actual life. If a man contracts to deliver 100 bushels of wheat six months hence, with the intention of buying them for that purpose on the day of delivery at a price which he hopes will be lower than his contract price, that man, no less than the breaker of the vase, provides uncertainty-bearing without providing any waiting. Uncertainty-bearing is thus seen to be an independent and elementary factor of production standing on the same level as any of the better-known factors.

App.I.3

§ 3. In the way of this general conception there are two serious difficulties. The first of them can be set out as follows. It is well known that the ordinary factors of production are two-dimensional, in the sense that a unit of any of them can only be expressed as a quantity of stuff multiplied by a quantity of time. Waiting consists in the provision of a given quantity of resources, and labour in the provision of a given quantity of labour, during a given period. Thus the unit of waiting is said to be a year-pound, and the unit of labour a year-labourer.*38 It would seem, therefore, that, if uncertainty-bearing, as a factor of production, is to stand on a level with waiting and labour, it must somehow bear a relation to time analogous to that which they bear. But uncertainty-bearing, unlike waiting and labour, is in its essence independent of time, and, so far as pure theory goes, capable of instantaneous consummation. Consequently, the provision of a given quantity of uncertainty-bearing of any sort for a given period seems at first sight a mere phrase without substantial meaning. The difficulty thus suggested is, however, obviated by the fact that, as a matter of practice, the consummation of any act of uncertainty-bearing is not instantaneous, but involves a process in time. The uncertainty-bearing, for example, which a company promoter undertakes, is not completed until the public has come in and allowed him to unload, and this, of course, will not happen till a considerable interval has elapsed. This circumstance enables us to fashion a unit of uncertainty-bearing on the same plan as the units of waiting and of labour. This unit is the exposure of a £ to a given scheme of uncertainty, in an act the consummation of which occupies a year. The exposure of a £ to a succession of like schemes of uncertainty during a year, in acts the consummation of which occupies on the average, say, ten days, will thus embrace 365/10 of these units. We have in this way obtained a two-dimensional unit of uncertainty-bearing analogous to the units of waiting and of labour, and the difficulty, which this section was designed to discuss, has been overcome.

App.I.4

§ 4. The second difficulty is in this wise. Labour and waiting are objective services, the aversion to providing which may vary with different people, but which, in themselves, are the same for everybody. Uncertainty-bearing, however, it may be said, is in its essence a subjective state, invoked, indeed, by external conditions but bearing a quite different relation to these conditions for people of different temperaments and with different information. It would seem, therefore, at first sight that the amount of uncertainty-bearing involved in carrying through any operation must depend, not only on the nature of the operation, but also on the temperament and knowledge of the people who bear the uncertainty. Such a conception, however, is fatal to the symmetry of our analysis. If there is to be any real parallel with labour and waiting, we must define uncertainty-bearing objectively. Thus the uncertainty-bearing involved in the investment of any given amount of resources means for us the uncertainty-bearing which that investment would involve if it were made by a man of representative temperament and with representative knowledge. If the investment is actually made by a man who never feels subjective uncertainty, whatever the evidence, or by a man who possesses information adequate to destroy subjective uncertainty, we shall say, not that less uncertainty-bearing has been taken up, but that a given amount has been taken up by a person, who, from temperament or information, is an exceptionally ready bearer of uncertainty. There is, it must be admitted, an arbitrary and artificial appearance about this method of defining our key term; but there appears to be no way in which this can be avoided.

App.I.5

§ 5. Figure.  Click to enlarge in new window.Up to this point we have taken no account of the fact that uncertainty-bearing, like labour, is a term embracing a large group of factors of production, rather than a single factor. It must now be observed, however, that, just as there are many different sorts of labour, so there are many different sorts of uncertainty, embodied in many different schemes of prospective returns, to which, in the course of industry, resources may be exposed. A scheme of prospective returns can be represented diagrammatically in the following manner. Along a base-line OX mark off all possible yields that may result from the exposure of a £ to the scheme in question; and, through each point on OX, draw an ordinate proportionate to the probability, on the evidence, of the corresponding return. Join the tops of all these ordinates, as in the figure on the next page. Evidently any scheme of prospective returns can be represented by a curve formed upon this plan. Furthermore, the principal species of schemes that are liable to occur can be distinguished into certain broad groups. Find on OX a point B, such that OB represents the actuarial value of the chances of the returns indicated on the curve, or, in other words, such that OB is equal to the sum of the products of each several ordinate multiplied by the corresponding abscissa, divided by the sum of the ordinates; and let the ordinate through B cut the curve in H. In like manner, find on OX a point M, such that OM represents the most probable, or most "frequent," return relevant to the scheme of prospective returns under review; and let the ordinate through M cut the curve in K. On this basis we may distinguish, in the first place, between curves which are symmetrical, in such wise that BH and MK coincide, and curves which are asymmetrical. The symmetrical group includes schemes of such a sort that, if r is the actuarial value of a pound exposed to any scheme, the chance of obtaining a return (r - h) is equal to the chance of obtaining a return (r + h), for all values of h. The asymmetrical group includes all other schemes. The symmetrical type is only possible when the conditions are such that the exposure of a pound to uncertainty cannot yield a gain greater than a pound, since, from the nature of things, it cannot yield a loss greater than this. Secondly, within the symmetrical group we may distinguish curves which are spread out, like open umbrellas, and curves which are narrow, like closed umbrellas. The former sort represent schemes in which a wide divergence, the latter schemes in which only a small divergence, of the actual from the most probable return is probable. Thirdly, within the asymmetrical group we may distinguish curves in which MK lies respectively to the right and to the left of BH. The former sort represent schemes in which the most probable outcome is a moderate return on the money invested, but a small return is more probable than a large one. A scheme of this kind would be embodied in a lottery offering a great number of small prizes and one or two blanks. Again, a £ might be lent to somebody: and there might be 96 chances of its return in full, 1 chance that 10s. would be returned, 1 chance that 5s. would be returned, and 2 chances that none would be returned. The actuarial value of this scheme of prospect is £ 968/100: the most probable return is £1. The latter sort of curves represent schemes in which the most probable outcome is a small return, but large returns are possible. A lottery of the ordinary kind, containing a few large prizes and many blanks, affords an example of this sort of scheme. Within each of the groups thus distinguished an indefinite number of further subdivisions could be made. Of course a great many schemes of prospective returns are not represented by continuous curves, but by a few isolated points with gaps between them, these gaps corresponding to returns which are not possible.

App.I.6

§ 6. The existence of the great variety of schemes of prospective returns, each representing different sorts of uncertainty, might seem at first sight to vitiate the attempt, which was made in an earlier section, to treat "the factor uncertainty-bearing" and "the factor waiting" on the same footing. For waiting is a single thing, while uncertainty-bearing is a group of different things. The meaning of a change in the supply of waiting is, therefore, clear; but how are we to conceive of a change in the supply of uncertainty-bearing? This difficulty, though it is a natural one to raise, is easily overcome. For, after all, uncertainty-bearing in this regard stands in exactly the same position as labour. Labour in general includes an immense variety of different sorts and qualities of labour. This circumstance does not prevent us from making use of the general concept labour alongside of the concept waiting. In order to render this procedure legitimate, all that we need do is to select in an arbitrary manner some particular sort of labour as our fundamental unit, and to express quantities of other sorts of labour in terms of this unit on the basis of their comparative values in the market. In this way all the various sorts of labour supplied or demanded at any time can be expressed in a single figure, as the equivalent of so much labour of a particular arbitrarily chosen grade. Exactly the same device is available for uncertainty-bearing. The uncertainty involved in exposing a pound to a particular arbitrarily chosen scheme of prospective returns can be selected as a fundamental unit, and the uncertainty involved in other exposures can be reduced, on the basis of comparative market values, to its equivalent in terms of this unit. So soon as this is understood, an apparently formidable obstacle in the way of assimilating uncertainty-bearing to the other factors of production can be successfully overcome.

App.I.7

§ 7. When the assimilation is accomplished, and all the various sorts of uncertainty, to which, in different industries, people submit resources, are translated into terms of the uncertainty-bearing involved in some representative scheme of prospective returns, there will be a supply schedule and a demand schedule for pounds to be exposed to this scheme, just as there are a supply and a demand schedule for pounds to be exposed to "waiting." The demand price or the supply price for the exposure of any given quantity of pounds is the excess of money offered or asked above the actuarial value of a £ so exposed. For different quantities of uncertainty-bearing the demand price and the supply price will, of course, both be different. For some quantities the supply price will be negative. Up to a point, people will gamble because they like the excitement, even though they know that, on the whole, they are likely to lose money. But, though some amount of uncertainty-bearing, like some amount of labour, would be forthcoming for industry, even if there were no expectation of reward, in present conditions more is wanted than can be obtained on those terms. The main reason is that an uncertain prospect actuarially worth £100 of money is much less satisfactory than a certain prospect also worth £100 of money. This follows from the law of diminishing utility. One income of £90 plus one of £110 carry less satisfaction, other things being equal, than two incomes each of £100. Thus, in respect of such quantities of uncertainty-bearing as are actually made use of in modern industry, the supply price, like the supply price of the other factors of production, is positive; and the general conditions determining the value, or price, of uncertainty-bearing are similar to those determining the price of those factors.

App.I.8

§ 8. It must be clearly understood that the payment thus asked and offered for uncertainty-bearing is by no means the same thing as the exceptional profits obtained by persons who have succeeded in risky businesses. An uncertain undertaking is a risky undertaking. But the term risk is generally used to mean the chance of obtaining a smaller return than the actuarially probable return. This must be compensated by a corresponding chance of obtaining a larger return than this. Even though no payment whatever is made for uncertainty-bearing, the successful undertakings in a risky business would still need to make exceptional profits as an offset to the exceptional losses of those which fail. Otherwise the whole body of investors in the business, taken collectively, would be obtaining less than normal returns from investment in it. The payment for uncertainty-bearing, therefore, consists, not in the whole of the excess above normal profits earned by these successful undertakers, but only in that (generally small) part of this excess which is not cancelled by the corresponding losses of other undertakers who have fallen out of the race.

App.I.9

§ 9. It has next to be observed—and here we follow the line of though indicated in § 4—that the supply of uncertainty-bearing, as defined in the objective manner there set out, will be increased by anything that enables people with more knowledge to undertake risky enterprises in lieu of people with less knowledge. Every form of organisation that enables risks to be shifted on to the shoulders of specialists—the resort of farmers to speculators in grain prices through hedging on the produce exchange, the resort of bankers to specialist bill-brokers in negotiating the discount of bills, the resort of manufacturers for the foreign market to specialist export houses, and so on, has this effect. It may be added that a similar effect is produced when risky undertakings are taken over by rich persons instead of by poor persons. For, if a man possesses (x + 100)£, to expose £100 to a 5 per cent range of uncertainty is to accept an even chance of having (x + 105)£ or £(x + 95). But there is reason to believe that, not merely the desire for an extra unit of resources in general, but also the rate of diminution of this desire, diminishes as the number of units in our possession grows. It follows that the probable loss of satisfaction involved in accepting the above even chance instead of a certain (x + 100)£ is smaller the larger is the value of x.

App.I.10

§ 10. Like any other factor of production, uncertainty-bearing may improve in technical efficiency. The central fact, upon which the improvements in it that have actually taken place depend, is that forecasts based upon existing knowledge are, in general, more certain when they are made about collections than when they are made about individual members of collections. If all the individual members were so linked together that they necessarily always acted in the same way, this, of course, would not be so. But in many collections there are some individual members that are complementary to one another. Thus, on a holiday, it is uncertain whether indoor entertainers will make a great deal of money or very little money, because it is uncertain whether the weather will be wet or fine. In like manner and for the same reason it is uncertain whether outdoor entertainers will make very little money or a great deal of money. But the amount that the two sorts of entertainers together will make may be susceptible of nearly accurate forecast.*39 The case is similar with exporters and importers between countries the mutual exchange rate of whose moneys is varying; the exporters and importers will be affected in opposite senses if the exchange moves up or down between the making and the completion of foreign trade contracts. In circumstances such as these, so soon as an organisation is set up that combines the two complementary uncertainties under a single head, they neutralise or destroy one another. Nor is it only when uncertainties are complementary that combination reduces them. The same result follows, though in a less marked degree, when they are simply independent. The measure of reduction to be expected from combination in these circumstances is indicated in the familiar corollary to the normal law of error, which asserts that the "precision of an average is proportional to the square root of the number of terms it contains."*40 This implies that, if there is an even chance that the investment of £100 in one assigned venture will yield a return greater than £95 and less than £115, there is an even chance that £100 scattered among a hundred similar investments will, if all the causes affecting the different investments are independent, yield a return lying between £104 and £106. If only some of the causes are independent and some common, the range within which it is more probable than not that the return will lie will be greater than that enclosed between £104 and £106, but it will still be smaller than that enclosed between £95 and £115. It follows, that, if out of a hundred people, each of whom has £100 to invest, every one divides his investment among a hundred enterprises, the aggregate amount of uncertainty-bearing undertaken by the group is smaller than it would have been had every investor concentrated on a single enterprise. The physical results of the investments taken together must, however, be the same. Therefore, whenever more or less independent uncertainties are combined together, a given result can be attained by a smaller amount of uncertainty-bearing, or, to put the matter otherwise, the factor uncertainty-bearing has been made technically more efficient.*41 The principle thus explained is fully recognised by business men, and has long lain at the root both of insurance and of much speculative dealing on 'Change. Thus the segregation of the speculative element in certain forms of business and its concentration upon a relatively small number of speculators have not only changed the distribution, but have reduced the aggregate amount, of uncertainty-bearing required in industry. In modern times the range over which this principle can be applied has been greatly extended by three important developments. Of these the first is a legal change, namely, the concession to joint-stock companies of the privilege of limited liability; the second an economic change, namely, the development of organised speculative markets; the third also an economic change, namely, the development of the means of transport and communication. The ways in which these three changes have facilitated the application of the above principle will now be examined.

App.I.11

§ 11. So long as liability was unlimited, it was often against a man's interest to spread his investments; for, if he did so, he multiplied the points from which an unlimited call on his resources might be made. The English Limited Liability Act of 1862 and its foreign counterparts enabled investments to be spread without evoking this danger. Furthermore, intermediary organisations, themselves fortified by limited liability, have been developed, capable of spreading investments on behalf of persons whose resources are too small to allow of their spreading them for themselves. Since the minimum share in industrial enterprises is seldom less than £1, the small investor's capacity for direct spreading is narrowly restricted. Savings banks, friendly societies, trade unions, building societies, co-operative societies, trust companies and so forth—all of them limited liability associations—are able, however, to put him in a position as favourable in this respect as is occupied by the large capitalist. Nor is it only the spreading of investments that the system of limited liability has facilitated. It has also made possible the spreading, or combination, of risks in a wider sense. For, in general, each business deals directly or indirectly with many businesses. If one of them fails for a million pounds, under unlimited liability the whole of the loss falls on the shareholders or partners—provided, of course, that their total resources are adequate to meet it—but under limited liability a part of it is scattered among the shareholders or partners of a great number of businesses. Hence any shareholder in one business combines with the uncertainty proper to his own business some of that proper to other businesses also. It follows that the range of uncertainty, to which a normal £100 invested in industry is subjected by reason of failures, is still further diminished in amount. This advantage is additional to, and quite distinct from, any direct national gain which limited liability may give to a country by throwing a part of the real cost of its unsuccessful enterprises upon foreigners.

App.I.12

§ 12. The development of organised speculative markets enables the producing classes to shift uncertainty-bearing on to speculators, in whose hands they in great part cancel out. Thus the miller, who is contracting to deliver flour at a fixed price some months hence, can protect himself by buying "a future" in wheat at the same time that he makes his contract, and afterwards selling the "future" pari passu with purchases of "spot" wheat of various grades as he needs them for his milling. In like manner, the farmer can protect himself by selling a future at an early stage, and afterwards buying to cover it in the speculative market at the same time that he sells his actual wheat in the spot market. Plainly these processes involve a large reduction in the amount of uncertainty-bearing that has to be undergone to accomplish a given result. The chief conditions needed to render any class of products suitable to be handled in the type of organised market that permits of their use have been succinctly stated by Marshall as follows: "(1) That the product is not quickly perishable; (2) that the quantity of each thing can be expressed by number, weight, or measure; (3) that the quality can be determined by tests that yield almost identical results when applied by different officials, assumed to be expert and honest; and (4) that the class is important enough to occupy large bodies of buyers and sellers."*42

App.I.13

§ 13. There remains the development of the means of communication. This facilitates the combination of uncertainties in one very simple way. It puts investors into contact with a greater number of different openings than were formerly available. This effect, though of great importance, is so obvious and direct that no comment upon it is required. There is, however, a more subtle way in which the development in the means of communication works. Dr. Cassel has observed that industrial firms have, in recent times, been lessening the quantity of stock that they carry in store waiting to be worked up, relatively to their total business. The improvement in this respect applies all round. As regards production, "there is, in the best-organised industries, very little in the way of material lying idle between two different acts of production, even if these acts have to be carried out in different factories, perhaps at great distances from each other. A modern iron-works has no large stock either of raw materials or of their product, yet there is a continuous stream of ore and coal entering, and of iron being turned out of it."*43 In like manner, factories are coming to keep a smaller amount of capital locked up in the form of reserve machines not ordinarily in use. The same tendency is apparent in retail trading. The ratio of the average amount of stock kept to the aggregate annual turn-over is smaller than it used to be. "Under modern conditions the trade of the country is conducted on a retail system which is growing year by year. The practice of keeping large stocks has almost ceased, and goods are ordered in quantities only sufficient to meet the current demands."*44 One reason for this is the improvement in the means of communication. "The trunk lines of America, with their wide-spreading branches, enable merchants in the cities and the larger towns to replenish their counters and shelves every day. Stocks, therefore, need not be so large as of old, when, let us say, a whole winter's goods were laid in by October.... The inter-urban roads are extending these advantages to the village storekeeper, who, in the morning, telephones his wants to Toledo, Cleveland, or Detroit, and, in the afternoon, disposes the ordered wares on his shelves."*45 Now, prima facie, this change of custom would seem to be of little significance. After all, a reduction in the amount of finished goods held by retailers, of reserve machinery held by manufacturers, and so on, does not necessarily imply a reduction in the aggregate amount of these things held by the whole body of industrialists. On the contrary, we are naturally inclined to suggest that the wholesaler and the machine-maker must increase their stocks pari passu with the decrease in the stocks of their clients. As a matter of fact, however, this suggestion is incorrect. The reason is that the wholesaler and the machine-maker represent points at which uncertainties can be combined. The development of the means of communication, therefore, in so far as it directly transfers to them the task of bearing uncertainty, indirectly lessens the amount of uncertainty that needs to be borne. Uncertainty-bearing, in short, is rendered more efficient. The same result as before can be achieved with a smaller quantity of it, or, what comes to be the same thing, with a smaller quantity of waiting designed to obviate the need for employing it.


Notes for this chapter


38.
Cf. ante, pp. 161-2, footnote.
39.
Cf. Marshall, Industry and Trude, p. 255.
40.
Bowley, Elements of Statistics, p. 305.
41.
This circumstance, of course, permits the release, partly for immediate consumption and partly for investment, of resources which must otherwise have been stored. For example, the combination of the community's gold reserves in a central bank lowers the amount of aggregate gold reserve necessary, increases the capital available for investment, and pro tanto lowers the rate of interest. (Cf. H. Y. Brown, Quarterly Journal of Economics, 1910, pp. 743 et seq.)
42.
Industry and Trade, p. 256.
43.
The Nature and Necessity of Interest, p. 126.
44.
Inglis, Report of the Board of Trade Railway Conference, 1909, p. 33.
45.
Iles, Inventors at Work, p. 483.

Appendix II

End of Notes


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