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The Kaldor–Hicks Potential Compensation Principle and the Constant Marginal Utility of Income

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Abstract

The Kaldor–Hicks potential compensation principle underlies partial equilibrium welfare analysis in imperfectly competitive markets. It depends on the assumptions that changes in consumer and producer surplus are weighted equally and that the marginal utility of income is constant. I show that if the first assumption is followed but there is decreasing marginal utility of income, the potential compensation principle does not give satisfactory indications of market performance.

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Notes

  1. For a concise account, see Arrow (1983, pp. 16–19).

  2. For another account, see Mishan (1968, pp. 504–506).

  3. In an only-recently published lecture delivered (one may infer) in 1954, Hicks changes his position (Kanari 2006, pp. 84–85):

    Some fifteen years ago, I published in the Economic Journal a paper entitled “The Foundations of Welfare Economics”. In this paper, developing a suggestion of Mr. Kaldor’s, I endeavoured to formulate a basis, on which a Welfare Economics something like Pigou’s could be constructed, while it avoided those particular assumptions about inter-personal comparability of satisfactions which Pigou’s critics had been unable to stomach. It was my hope that this endeavour would make the economics of welfare, in one form or the other, more widely and smoothly acceptable.

    But, as you will know, my hope was not fulfilled. All that my paper did was to create a turmoil; since its time there have been half-a-dozen books, and heaven knows how many articles, directed to showing, in one way or another, where it was that we (that is to say, Kaldor and I) had gone wrong.

  4. Marshall (1920, p. 125) defines consumers’ surplus as “The excess of the price which [consumers] would be willing to pay rather than go without the thing”. He defines worker’s surplus and saver’s surplus analogously (1920, p. 830).

  5. The first two postulates are a) “the competitive demand price for a given unit measures the value of that unit to the demander”, and b) “the competitive supply price for a given unit measures the value of that unit to the supplier”. I do not propose to discuss them—beyond remarking that the adjective “competitive” sows confusion if one is engaged in applied welfare analysis (for example, predicting the impact of a merger on market performance) for an imperfectly competitive market.

  6. In the same vein, see Samuelson (1990) and Slesnick (1998).

  7. But see Friedman (1942, fn. 45), who writes “[C]onstancy of the purchasing power of money was a standard assumption of economic theory long before Marshall’s day. It was made by Ricardo in his price theory, and Marshall refers to Cournot’s discussion of the reasons for making this assumption...”

  8. Discussing the first-order conditions for an individual budget-constrained utility maximization problem, Marshall writes (1920, Note II, p. 838) “Every increase in his means [income] diminishes the marginal degree of money to him...”. See, generally, entries in the index to Principles under “Money, changes in marginal utility of”.

  9. Vives (1987) develops conditions under which this rationale is valid.

  10. This is quoted by Friedman (1949).

  11. Pigou (1910, pp. 637–638) and Bowley (1924, p. 12) make the “small part of spending” argument. In an accompanying footnote, Pigou recognizes that the assumption cannot hold “strictly”.

  12. This is the kind of deus ex machina price increase that is considered by Williamson (1968). Alternatively, one might suppose a shift from a noncooperative oligopoly equilibrium price to a full or partial tacit collusion price. \(\pi\) would then depend on the specific oligopoly model that is chosen.

  13. In Martin (2019), I consider the alternative specification that firms are owned by workers, who receive dividends as part of disposable income. The results are qualitatively similar to those that are reported here.

  14. Georgescu-Roegen (1968, fn. 7) attributes this result to Pareto.

  15. Recall the assumption that income y is at least as large as the maximum reservation price, so all individuals can purchase one unit of the good, if they should care to do so. For the indicated parameter values, this assumption holds for \(\alpha \ge 0.8\). For smaller values of \(\alpha\), market demand must be recalculated to recognize that at high prices some individuals are priced out of the market for good 1.

  16. By implicit differentiation of \(\frac{\rho }{N}Q=\rho -y^{\alpha }+\left( y-p\right) ^{\alpha }\),

    $$\begin{aligned} \left. \frac{\partial p}{\partial \alpha }\right| _{Q=\overline{Q} }=-\left( \frac{y}{y-p}\right) ^{\alpha }\frac{\ln y}{\ln \left( y-p\right) } . \end{aligned}$$
    (27)

    The expression on the right is positive for \(y>p+1\) (so \(\ln y>0\), \(\ln \left( y-p\right) >0\)) and for \(y<1\) (so \(\ln y<0\), \(\ln \left( y-p\right) <0\)).

  17. See also the surveys by Frey and Stutzer (2002, especially Section 2) and Stevenson and Wolfers (2008, pp. 4–8). Maridal et al. (2018) survey alternative empirical measures of well-being; see also the discussion of Stevenson and Wolfers (2008, Appendix A). Bond and Lang (2019) despair of parametric analysis of the underlying survey data, and argue that conditions for the use of nonparametric techniques are not likely to be satisfied.

  18. See their Figure 1 and Table 1. See Easterlin (2005, p. 250) for similar results with U.S. data. Layard et al. (2008) use a constant elasticity of marginal utility of income specification and find decreasing marginal utility of income. Sacks et al. (2012) regard a linear relation between subjective well-being and the log of income as a stylized fact, and note that this implies diminishing marginal utility of income (2012, p. 1183): “[T]he logarithmic relationship implies that each percent increase in income raises measured well-being by a similar amount, and hence each extra dollar raises well-being by less than the previous”. (They make this comment in discussion of cross-country comparisons; the specification of their within-country tests, and the general nature of the results, are similar.)

  19. See Aguirregabiria and Slade (2017) for a survey.

  20. See their equation (2.3) and the associated text.

  21. One of BLP’s estimates of \(\alpha\) is 43.501 (1995, Table IV); marginal utility of income (\(\frac{43.501}{y_{i}-p_{j}}\)) then declines as income rises.

  22. It will be recalled (Sect. 2.2.2) that one of Alfred Marshall’s justifications for assuming a constant marginal utility of income was the case that spending on a good constitute a small part of income.

  23. In 1982–1984 dollars, \(\overline{y}_{1}=11129.33\), \(\overline{y}_{2}=27017.28\) . Mean income was $23,255.

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    Stephen Martin

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I am grateful for comments received at the 2nd Asia-Pacific Industrial Organization Conference, Auckland, December 2017, at the 16th International Industrial Organization Conference, Indianapolis, April 2018, from Javier Elizalde at the 33rd Jornadas de Economia Industrial, Barcelona, September 2018, and from the editor. I thank Tingminke Lu for excellent research assistance. Responsibility for errors is my own.

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Martin, S. The Kaldor–Hicks Potential Compensation Principle and the Constant Marginal Utility of Income. Rev Ind Organ 55, 493–513 (2019). https://doi.org/10.1007/s11151-019-09716-3

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