# Adding-Up Problem

**DOI:**https://doi.org/10.1057/978-1-349-95121-5_507-1

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## Abstract

In any theory of income distribution in which one type of return is determined residually, it will be tautologically true that the various different incomes, as determined by the theory, will add up so as to exhaust the total product. By contrast, any theory which provides a ‘positive’ explanation for every category or return, treating none as a residual, must show that the various returns so explained do indeed exhaust the product. In practice, it has been with reference to the marginal productivity theory that this consistency requirement has received considerable attention. By the early 1890s a number of authors had sought to extend the ‘principle of rent’ into a completely general theory of distribution but it was P.H. Wicksteed, in his Co-ordination of the Laws of Distribution (1894) who first clearly stated, and attempted to resolve, the resulting adding-up problem.

## Keywords

Marginal Productivity Demand Curve Product Price Economic Journal Constant ReturnIn any theory of income distribution in which one type of return is determined residually, it will be tautologically true that the various different incomes, as determined by the theory, will add up so as to exhaust the total product. By contrast, any theory which provides a ‘positive’ explanation for every category or return, treating none as a residual, must show that the various returns so explained do indeed exhaust the product. In practice, it has been with reference to the marginal productivity theory that this consistency requirement has received considerable attention. By the early 1890s a number of authors had sought to extend the ‘principle of rent’ into a completely general theory of distribution but it was P.H. Wicksteed, in his Co-ordination of the Laws of Distribution (1894) who first clearly stated, and attempted to resolve, the resulting adding-up problem.

Consider first the simplest case, in which all markets are perfectly competitive, there is no uncertainty and ‘entrepreneurs’ are seen as mere hiring agents. If it is supposed also that all productive processes exhibit constant returns to scale, then the adding-up problem is shown by Euler’s theorem on homogeneous functions to be a quite trivial problem, as Flux (1894) pointed out in his *Economic Journal* review of Wicksteed’s book. When assuming constant returns one should, of course, be mindful of Samuelson’s warning that ‘Any function whatever in *n* variables may be regarded as a subset of a larger function in more than *n* variables which is homogeneous of the first order’ (1983, p. 84, n. 13). Attention can also be drawn to the indeterminacy of the sizes of firms in the constant returns case, and thus to the question of how the perfect competition assumption can be underpinned, but these (perfectly proper) questions are not specific to the adding-up problem. It is, however, vital to appreciate that linear homogeneity of production relations does *not*, by itself, dispose of the adding-up problem; it is linear homogeneity in production, combined with perfectly competitive market conditions, which does that. This was forcefully demonstrated by Wicksteed himself in 1894. Whilst he upheld the assumption of constant returns to scale in production, he also held that a proportional increase of *all* inputs – both those used in production and those used in selling activities – would not result in an equal proportional increase in the quantity sold, at a given price. Thus there is not a ‘constant returns’ relationship between total outlays and total revenue. Wicksteed examined the consequences for ‘adding-up’, first in the case of monopoly and then with an ever-increasing number of firms in the industry, and was able to show that, as the number of firms became very large, marginal productivity pricing would approximately exhaust the product. Adding-up, or otherwise, is thus intimately related to market conditions.

Wicksteed’s assumption of linear homogeneity in production, together with what was taken by Walras, at least, to be his implicit slighting of the work of others, resulted in his work receiving a hostile response from Pareto, Edgeworth and Walras. In the third edition of his *Eléments* Walras inserted an Appendix III, dated October 1895, which ended with the words ‘Mr. Wicksteed … would have been better inspired if he had not made such efforts to appear ignorant of the work of his predecessors’. (This appendix was, however, dropped from subsequent editions; Stigler and Schumpeter have disagreed over the precise import of, and degree of justification for, Walras’ displeasure.) More constructively, the second half of Walras’ appendix outlined a proof of the adding-up theorem under competitive conditions (see below), a proof based on work by Barone. (It seems that Barone had submitted a review of Wicksteed’s book to the *Economic Journal* and that Edgeworth had first accepted the review for publication but then subsequently withdrew his acceptance.) In his *Economic Journal* (1906) review of Pareto’s *Manuale di Economia Politica* (1906), Wicksteed acknowledged the justice of the criticisms which Edgeworth and Pareto had made of his 1894 *Co*-*ordination* argument; and in the *Common Sense* (1910) he again referred to Edgeworth and Pareto and stated that paragraph 6 of the *Co*-*ordination* ‘must be regarded as formally withdrawn’ (p. 373, n. 1). (It is to be noted that Wicksteed does *not* refer to Walras in either of these acknowledgements of justified criticism.) In Volume I of his *Lectures on Political Economy* (1901), Wicksell expressed surprise that Wicksteed had ‘declared – for reasons difficult to understand – that he desired to withdraw this work [the *Co*-*ordination*]’ (1934, p. 101, n. 4). It must be noted clearly, first that Wicksteed did *not* withdraw the work as a whole, but only its paragraph 6, and secondly that Wicksteed’s proof of the adding-up theorem under linear homogeneity and perfect competition is contained in paragraph 5. Paragraph 6, which he did declare to be withdrawn, concerns the extension of the result of paragraph 5 to the cases of imperfect product markets and of more than two inputs. This, together with Wicksteed’s continued use of marginal productivity theory in his *Common Sense*, supports the view of Hutchison, Robbins and Stigler that Wicksteed’s ‘recantation’ was ‘merely verbal’, and not a rejection of the substance of his earlier argument.

The solution to the adding-up problem which can be associated with the names of Barone, Walras and Wicksell dispenses with the linear homogeneity assumption but is still concerned with long run perfectly competitive equilibrium; it is centred not on the industry but on the individual firm. Any cost minimizing firm, which faces diminishing marginal products and given input prices, will so arrange its production that \( {w}_i = \left(\mathrm{m}\mathrm{c}\right)\left(\partial q/\partial {x}_i\right) \), for *each i*, where *w* _{ i } is the price of the *i*th variable input, (∂*q*/∂*x* _{i} ) its marginal product, and (mc) the marginal cost of the output in question. Multiplying both sides by *x* _{ i } and then summing over *i*, one finds that \( \left(\mathrm{a}\mathrm{v}\mathrm{c}\right)q=\left(\mathrm{m}\mathrm{c}\right)\;{\displaystyle \sum {x}_i\left(\partial q/\partial {x}_i\right)} \), where (avc) is average variable cost and *q* is output. For the cost minimizing firm, then, \( {\displaystyle \sum {x}_i\left(\partial q/\partial {x}_i\right)}\frac{\ge }{<}q \) according as \( \left(\mathrm{m}\mathrm{c}\right)\frac{\ge }{<}\left(\mathrm{a}\mathrm{v}\mathrm{c}\right) \), that is according as average variable cost is falling, constant, or rising. If the average variable cost curve has a minimum point then, at that point, it will be as *if* there are constant returns to scale and ‘adding-up’ will obtain. Now introduce the assumption of profit maximization; the perfectly competitive firm will obey the rule \( p=\left(\mathrm{m}\mathrm{c}\right)\ge \left(\mathrm{a}\mathrm{v}\mathrm{c}\right) \), where *p* is the product price. Hence \( {\displaystyle \sum {x}_i\left(\partial q/\partial {x}_i\right)\ \le\ q} \) for such a firm – and equality will hold in, and only in, the long-run equilibrium position (with *p* = (mc) = (avc) = minimum average total cost).

Consider now the long-run equilibrium position under imperfect competition. The results given above for the cost minimizing firm will still hold, of course, but now (mc) is equal to marginal revenue rather than to product price. The consequence is that, in an ‘imperfect’ long run equilibrium, \( {\displaystyle \sum {x}_i\left(\partial q/\partial {x}_i\right) = \left(e/e\hbox{--} 1\right)q} \), where ‘*e*’ is the (absolute) elasticity of the demand curve at the equilibrium point. (This result naturally tends to the corresponding perfectly competitive result as ‘*e*’ tends to infinity.) Analogous but inevitably more complex results can, of course, be obtained when both product and input markets are imperfect.

In the subsequently withdrawn paragraph 6 of his *Co*-*ordination*, Wicksteed noted that ‘In practical cases there is usually a speculator who … buys the other factors, speculatively, at their *estimated* values’ (p. 41, emphasis added) and that the speculator may make a gain or a loss, depending on how those anticipated values compare with the actual, realized values. He continued: ‘But these gains and losses may be resolved into (1st) compensation for risk, and (2nd) the share that falls to this special speculating ability, regarded as a factor of production, and receiving its share of the production in accordance with the general formula [of marginal productivity]’ (p. 42). Can entrepreneurship properly be regarded as simply ‘another factor’? If not – and Edgeworth and Wicksell, for example, appear to have thought not – if entrepreneurship is related to true uncertainty (as opposed to risk) and if uncertainty leads to the existence of *residual* ‘pure profits’ then, as observed above, there is no ‘adding-up problem’ to be solved. For that problem arises, within the marginal productivity context, only when *every* form of income is related to the marginal product of some input.

## See Also

## References

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