APPENDIX II
THE MEASUREMENT OF ELASTICITIES OF DEMAND
§ 1. WITH the information at present available it is not possible to lay down any propositions about the elasticity of demand for different commodities beyond those general propositions that are set out in Part II. Chapter XIV. As has been pointed out by Marshall, attempts to determine the elasticity of demand for any commodity in any market by a direct comparison of the prices and the quantities consumed at different times are exposed to very great difficulties. If it could be presumed that the reactions exercised by price-changes upon quantity demanded came about immediately, if the association of actual price-changes with people's expectation of connected future price-changes in the same or the opposite direction could be eliminated, and if allowance could be made for those upward and downward shiftings of demand schedules, for which movements of confidence and alterations in the supply of monetary purchasing power are responsible, a comparison of the percentage changes of price between successive years with the percentage changes in consumption between the same years might, for commodities about which adequate statistics exist, yield a rough numerical measure of elasticity for amounts of consumption in the neighbourhood of the average actual consumption. It seems that for certain commodities the above presumption can reasonably be made. On the basis of it Professor Lehfeldt calculated, immediately before the war, that the elasticity of the aggregate demand for wheat in the United Kingdom was about -0.6. But there is little hope that many elasticities will lend themselves to calculation in this direct way. It is, therefore, important to inquire whether any indirect method of calculation is available for overcoming difficulties due to the slowness with which reactions work themselves out.
§ 2. Some years ago I devised a method, the basis of which is a comparison of the amounts of a commodity consumed by persons of different incomes at a given price, instead of a comparison of the amounts consumed by persons of given incomes at different prices. Statistical data needed for this method are found in family budgets. Considerable attention has been paid both by State Departments and by private persons to the study of these budgets; and a number of tables have been printed to show the proportion of their income which families in different income groups expend upon the various principal sorts of commodities. It is possible so to manipulate these data as to derive from them information about certain elasticities of demand.
§ 3. Let us suppose that the data are better than they are, and that our tables give the expenditure of the group of workpeople whose wages lie between 30s. and 31s., of the group whose wages lie between 31s. and 32s., and so on continually for all wage levels. With this close grouping we may fairly assume that the tastes and temperament of the people in any two adjacent groups are approximately the same. That is to say, the desire for the xth unit of any commodity (or group of commodities), the demand for which is not markedly correlated with the demand for other commodities, is equal for typical men in the 30s. to 31s. group and in the 31s. to 32s. group. Let the quantity of desire for the xth unit of the commodity be φ(x): or, in other words, y being the desire for the xth unit, let the desire curve for the commodity be represented by y = φ(x). We are entitled to assume further, in the absence of special knowledge as to the existence of correlation, that the desire curve of both groups for the commodity is independent of the quantity of other commodities consumed and, therefore, of the marginal desiredness of money. Let this marginal desiredness to the lower and higher income groups respectively be µ_{1} and µ_{2}, and the quantities of the commodity consumed by these groups x_{1} and x_{2}. Then, since the price paid for the commodity must be the same for both groups, we know that this price p is equal both to and to . These two expressions are, therefore, equal to one another. But, if, as it is reasonable to suppose when the incomes of the two groups are close together, x_{2} differs only slightly from x_{1}, φ(x_{2}) may in general be written φ(x_{1}) + (x_{2}—x_{1})φ'(x_{1});
But the elasticity of the desire curve in respect of any consumption x_{1} is known to be equal to . Let this elasticity be written η_{x1}. It follows that
But, since a small change in the consumption of any ordinary commodity, on which a small proportion of a man's total income is spent, cannot involve any appreciable change in the marginal desiredness of money to him, the elasticity of the desire curve in respect of any consumption x_{1} is equal to the elasticity of the demand curve in respect of that consumption. Therefore the elasticity of demand, as well as the elasticity of desire, of the lower income group, in respect of its consumption of x_{1} units, may be represented by η_{x1}, when:
.
§ 4. If we knew the relative values of µ_{1} and µ_{2}, this equation would enable us to determine the elasticity of demand of the lowest income group for any commodity, the demand for which is not markedly correlated with the demand for other commodities, in respect of such quantity of the commodity as that group is consuming. Similar equations would enable us to determine the corresponding elasticities of each of the other income groups. If it is objected that our result would in practice be impaired by the fact that the higher income groups are apt to consume a better quality of commodity, and not merely a greater quantity, than the lower income groups, the difficulty is easily overcome by substituting in our formula for the quantities of the commodity that are consumed by the different groups figures representing their aggregate expenditures upon it. This device escapes the suggested objection by treating improved quality as another form of increased quantity. In order to obtain the elasticity of demand for the commodity as a whole, it would be necessary to calculate the separate elasticities for all income groups and to combine them on the basis of the quantity of purchases to which they respectively refer.
§ 5. Unfortunately we do not know, and cannot ascertain, the relative values of µ_{1} and µ_{2}. Consequently we are estopped from using the above analysis to determine the elasticity of the demand for any commodity in absolute terms. But this does not block our investigation. For, by the process indicated above, the elasticities of demand in any income group can be determined, for all the things consumed in that income group, in expressions into which µ_{1} and µ_{2} enter in exactly the same way, namely, as the term . If, then, the several elasticities be η_{x}, η_{y}, η_{z}, and so on, any one of them can be expressed in terms of any other without reference to µ_{1} and µ_{2}. These unknowns are eliminated, and we obtain the formula
This result, it should be observed, only follows directly from the preceding argument, provided that the commodities concerned are both such that only a small part of a typical man's income is normally spent upon them. In general, however, though the absolute formula for elasticities, from which the result is derived, is only valid on this assumption, the above comparative formula is approximately valid also for two commodities on which a large part of a typical man's income is spent, so long as the part spent on the one does not differ greatly from that spent on the other. The reason for this is that the errors in the two formulae for absolute elasticities, which have to be combined, will tend to balance one another. Our comparative formula is seriously suspect only when it is used to obtain the relative elasticities of the demands of a group for two things, on one of which that group spends a large proportion, and on the other a small proportion, of its income. Apart from this, the formula, when applied to the statistics of quantities of, or expenditures upon, different commodities by neighbouring income groups, enables us to determine numerically the ratio of the elasticity of demand of any income group for any one commodity (in respect of the quantity of the commodity actually consumed by it) to the elasticity of demand of the group for any other commodity. This information will often be valuable in itself. It is important to know whether the demand of workers with 35s. a week for clothes is about twice, or about ten times, as elastic as their demand for food. But the information is also valuable indirectly. For, if we can in some other way—through the examination of shopkeepers' books or otherwise—determine the elasticity of demand of any income group, or collection of income groups, for one thing, we have here a bridge along which we may proceed to determine the elasticity of their demand for all other things.
§ 6. In explaining the above method I have, as indicated at the outset, assumed that our data are better than they are. This, I think, is legitimate, because there is no reason in the nature of things why these data should not be improved; and, indeed, there is little doubt that they will be improved. Even then, of course, any one attempting a detailed application of the method is certain to encounter serious difficulties, among which, perhaps, not the least will be that of deciding how far to treat different commodities separately and how far to group them together according to the purpose which they jointly serve. When put to the test, these difficulties may, no doubt, in some applications, prove insurmountable. From the results of an experiment made upon figures given in the second Fiscal Blue-book (pp. 215 and 217), I am, however, tempted to hope for better things. The figures refer to the expenditure upon "food" and "clothing" of groups of workpeople whose wages were respectively under 20s., between 20s. and 25s., between 25s. and 30s., between 30s. and 35s., and between 35s. and 40s. My method gave the ratio of the elasticity of demand for clothes to that for food for the several groups as follows:
Workmen under 20s.... |
1.16 |
From 20s. to 25s.... |
1.31 |
From 25s. to 30s.... |
1.62 |
From 30s. to 35s.... |
1.25 |
From 35s. to 40s.... |
2.46 |
Apart from the drop in the ratio for workpeople earning from 30s. to 35s.—and it may be remarked in passing that the instances from which the average in this group is made up are only half as numerous as those in the two adjacent groups—these figures are continuous and in no wise incompatible with what we should expect from general observation. It is natural that among the very poor the demand for clothes should be nearly as inelastic as the demand for food, and that, as we proceed to groups of greater wealth, its relative elasticity should grow. This small experiment, therefore, is not discouraging, and it is much to be desired that some economist should undertake a more extended study along similar lines.
Principles of Economics, pp. 109 et seq.
Professor Moore, in his Economic Cycles (chapters iv. and v.), makes calculations of the "elasticity" of demand for certain commodities without resort to the allowances stipulated for in the text. But, as he himself fully recognises, the elasticity, which his method enables him to measure, is not the same thing as, and is not, in general, equal to, the elasticity of demand as defined by Marshall and employed here. Marshall's elasticity, if known, would make it possible to predict how far the introduction of a new cause modifying supply in a given manner would affect prices; Professor Moore's to predict with what price-changes changes in supply coming about naturally, in company with such various other changes as have hitherto been found to accompany them, are likely to be associated. That this distinction is of great practical importance is shown by the fact that, whereas the elasticity of the demand for pig-iron, in Marshall's sense, is, of course, negative—that is to say, an increase in supply involves a fall in price—the elasticity in Professor Moore's sense, as calculated from his statistics, is positive. The reason for this is that the principal changes in the price of pig-iron that have in fact occurred are mainly caused by expansions of demand (general uplifts in the demand schedule), and not by changes in supply taking place while the demand schedule is unaltered. In certain conditions it might be possible to derive Marshall's elasticity from Professor Moore's elasticity, provided that the reactions exercised by supply changes upon prices could be presumed to take place very rapidly. Apart from this presumption derivation would be impossible, however ample the statistical material.
Economic Journal, 1914, pp. 212 et seq.
The direct method and any possible indirect method are seriously hampered by the fact that the elasticity of demand for a thing may be different in respect of different amounts. Thus suppose we start with a consumption A at a price P: that the price rises by
p per cent, and that this rise is the direct and sole cause of a fall in consumption of
a per cent. We cannot infer that the elasticity of demand either for consumption A or for consumption
is equal to
unless
p is small—strictly unless it is infinitesimal. If
p is not small, some assumption as to the relation of neighbouring elasticities must be made before any inference can be drawn. One possible assumption is that the demand curve is a straight line. On this assumption the elasticity of demand in respect of consumption A will be
: and in respect of consumption
it will be
. Another possible assumption is that the elasticity of demand is constant for all amounts of consumption from A to
. On this assumption it can be proved, as Dr. H. Dalton has pointed out to me, that the said elasticity is not
but
This must lie between and : and is probably not far from
Strictly, of course, such a change must involve some alteration in the marginal desiredness of money, unless the demand for the commodity in question has an elasticity equal to unity. If the elasticity is anything other than this, a change in the consumption of the commodity will be accompanied by a transference of money from expenditure upon it to expenditure upon other things, or vice versa. This must affect the marginal desiredness of money spent on these things, and its marginal desiredness, if affected in one field, is, since it must be the same in all, affected in all.
Professor Vinci, in his very interesting monograph L' elasticità dei consumi, suggests that the method described above can be extended to yield an absolute measure of elasticity by reference to the distinction between nominal and real prices. The money price paid by the higher income group is the same as that paid by the lower income group. But the real price is, he holds, less than this, in the proportion in which the income of the higher income group exceeds that of the lower. Thus, if the higher income group has 10 per cent more income, an equal money price paid by it implies a real price 10/11ths as great; and the elasticity of demand is obtained by dividing a virtual price difference of 1/11th into whatever fraction represents the associated consumption difference (loc. cit. p. 22). This procedure is, however, illegitimate, because, on the assumptions taken, the virtual price of all commodities to the higher income group is 10/11ths of what it is to the lower income group. Consequently, the difference in the consumption of any particular commodity is not due solely to the difference in price of that commodity, and cannot, therefore, in general, be inserted in the formula for elasticity of demand. Professor Vinci has, in fact, tacitly assumed that the marginal desiredness of money is equal for the two groups—an assumption which would only be warranted if the demand of both for the sum of commodities other than the particular one under investigation had an elasticity equal to unity.
Cf. my article "A Method of Determining the Numerical Value of Elasticities of Demand,"
Economic Journal of December 1910.
Appendix III