A Hypothesis about the Consumption Function. * Unpublished typescript, 8 June 1951.

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1 A Hypothesis about the onsumption Function. * Unpublished typescript, 8 June onsumption expenditures in any year by a particular family can be regarded as the sum of a number of components, in the simplest case of two components, the permanent component and the transitory component. The permanent component is to be regarded as the level of consumption that would prevail on the average if no random or transitory factors had impinged on consumption in this year. t is analogous to the expected value of a probability distribution. The transitory component reflects the influence of such transitory factors as unusual sickness, special purchases of durable or semi-durable goods if these are treated as consumption expenditures, and the like. n symbols, 1 t where = consumption expenditures in the year in question, p = permanent component and t = transitory component. t will be taken to have an expected value equal to zero and to be uncorrelated with p. A more complex version would distinguish a larger number of components according to the time duration of the factors responsible for them. (See treatment of income in rofessional ncomes.). ncome in any year of a particular family is to be regarded as the sum of a number of components, in the simplest form, of two components, the permanent component and the transitory component, conceived of in the same way as the two components of consumption. Thus, t where = income in any year, p = permanent component. t = transitory component. t will be taken to have an expected value equal to zero and to be uncorrelated with p. (Again, see rofessional ncomes for more sophisticated and extended treatment.) t is clear that p is equivalent, economically, to a measure of wealth. 1

2 3. The relation between the permanent component of income and the permanent component of expenditure is given by K, 3 where K may itself be a function of other variables. n particular it is conjectured that K does depend on (a) the importance of the transitory components of income, (b) the relative importance of property and nonproperty income in p, (c) family size and perhaps age. For the time being, the tentative hypothesis will be that no other factors affect K except as they operate through these three, e.g., that occupation, size of city, level of prices, color and nativity do not. Both the form given in (3) and the list of factors said to affect it can be rationalized readily by the pure theory of consumer choice. 4. The transitory component of consumption is uncorrelated with the transitory component of income..e., E 0. 4 t t 5. Suppose observations are available on and for a group of families for which K is the same. Suppose a regression were computed from these observations giving as a linear function of, i.e., 5 a b, 6 where a b 7 and b, and where c and are the mean values of and respectively. t follows from the preceding assumptions that b as given by (7) is an estimate of K, 8

3 where K is as defined by (3), and is the fraction of the total variance in, attributable to the permanent component of income, i.e.,, and a as given by (6) is an estimate of K 1. 9 Since, under these assumptions, K, it follows that the computed elasticity of consumption with respect to income from (5) is given by: d c 10 N 6. Similarly, if a regression were computed giving as a linear function of, say 11 a bc, the computed value of b would be an estimate of K, 1 where c is the fraction of the total variance of attributable to the permanent component of consumption, and there are similar counterparts to the other equations in paragraph The theory so far described could be tested by budget data for single years alone only by computing K from, from (8) and seeing whether the computed K s were in fact more closely related to the variables listed under (a), (b), and (c) of paragraph 3 than to other variables. But this would be a rather weak and unsatisfactory test, partly because 3(a) would itself be computed from the same data as K, partly because this procedure could be justified 3

4 much more simply. A more powerful and significant test would be provided if or c could be computed independently. 8. The theory and techniques developed in rofessional ncomes do provide a method of computing or independently from data on income (or consumption) in two consecutive years. ndeed, if one had income in two consecutive years and consumption in at least one of them, they would do even more. They would permit the calculation of the average permanent income component of each income class. The average consumption of each income class is, under the assumptions of paragraphs 1 and 3, an estimate of the average permanent consumption component of that income class. f the hypothesis is valid, the average permanent income and consumption components so computed would satisfy an equation like (3). 9. Suppose an estimate of and were available for some group. Then, an alternative method of estimating K would be to compute a mutual regression (i.e., a regression in which a weighted sum of vertical and horizontal deviations is minimized) using as weights (1 ) σ and (1 ) σ, where σ and σ are the variances of income and consumption respectively. The closeness of the K so computed to the K computed from the group means would be a test of the hypothesis. 10. So far, we have stated the hypothesis in its simplest form. t can be complicated in various ways. (a) By extending the number of components of consumption and income (i) as in rofessional ncomes according to the temporal duration of an influence (ii) to distinguish between individual and group effects so that there would be a group transitory as well as individual transitory. The latter can also be considered equivalent to altering the assumption that transitory components are uncorrelated with permanent components. (b) By altering assumptions, of which the most important might be to introduce a correlation between the transitory components of consumption and income. Addendum to oints 5, 6, and 7 4

5 An additional way of testing the hypothesis from data for a single year is suggested by equation (1). onsider groups of families classified by a characteristic that may be expected to be related to size of transitory component of income but not to size of transitory component of consumption, e.g., occupation. From regressions of income on consumption, could compute estimates of size of transitory component of consumption; from regressions of consumption on income, size of transitory component of income, on assumption, of course, that hypothesis is correct. ould now compare estimated size of transitory component of consumption and of income for different occupational groups. Would expect estimates for consumption to vary considerably less than for income. f so, would tend to confirm hypothesis, if not, to contradict it. Notes *This document contains the earliest written version of the consumption function hypothesis. 5