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 (x): or, in other words, y being the desire for the xth unit, let the desire curve for the commodity be represented by y = (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 1 and 2, and the quantities of the commodity consumed by these groups x1 and x2. 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, x2 differs only slightly from x1, (x2) may in general be written (x1) + (x2—x1) '(x1); But the elasticity of the desire curve in respect of any consumption x1 is known to be equal to . Let this elasticity be written 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 x1 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 x1 units, may be represented by x1, when: .

§ 4. If we knew the relative values of 1 and 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 1 and 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 1 and 2 enter in exactly the same way, namely, as the term . If, then, the several elasticities be x, y, z, and so on, any one of them can be expressed in terms of any other without reference to 1 and 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:

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.