An Axiomatic Derivation of the Logarithmic Function as a Cardinal Utility Function on Money Income Levels

Author(s)
Mitsunobu Miyake

ABSTRACT

This note elaborates
Suppes’ (1977, *Erkenntnis* Vol. 11,
No. 1, pp 233-250) derivation of the logarithmic function as a consumer’s
cardinal utility function on money income levels, in which the consumer’s
preferences are specified by a level comparison relation and a difference
comparison relation. Without assuming Suppes’ hypothesis (Bernoulli’s
hypothesis or Weber-Fechner law), which asserts that the utility values are
proportional to the logarithmic values of income levels, it is shown that the
representability of the two relations by logarithmic utility function can be
characterized only by the three (mutually independent) axioms on the relations.

Cite this paper

M. Miyake, "An Axiomatic Derivation of the Logarithmic Function as a Cardinal Utility Function on Money Income Levels,"*Theoretical Economics Letters*, Vol. 4 No. 1, 2014, pp. 7-11. doi: 10.4236/tel.2014.41002.

M. Miyake, "An Axiomatic Derivation of the Logarithmic Function as a Cardinal Utility Function on Money Income Levels,"

References

[1] H. Watts, “An Economic Definition of Poverty,” In: D. P. Moynihan, Ed., On Understanding Poverty, Basic Books, New York, Chapter 11, 1968, pp. 316-329.

[2] S. R. Chakravarty, “A New Index of Poverty,” Mathematical Social Sciences, Vol. 6, No. 3, 1983, pp. 307-313.

http://dx.doi.org/10.1016/0165-4896(83)90064-1

[3] B. Zheng, “An Axiomatic Characterization of the Watts Poverty Index,” Economics Letters, Vol. 42, No. 1, 1993, pp. 81-88.

http://dx.doi.org/10.1016/0165-1765(93)90177-E

[4] E. Farhi and I. Werning, “Inequality and Social Discounting,” Journal of Political Economy, Vol. 115, No. 3, 2007, pp. 365-402. http://dx.doi.org/10.1086/518741

[5] D. Acemoglu, “Introduction to Modern Economic Growth,” Princeton University Press, Princeton, 2008.

[6] O. Blanchard, “Macroeconomics,” 5th Edition, Prentice Hall, Upper Saddle River, 2009.

[7] P. Suppes, “The Distributive Justice of Income Inequality,” Erkenntnis, Vol. 11, No. 1, 1977, pp. 233-250.

http://dx.doi.org/10.1007/BF00169854

[8] D. J. Murray, “A Perspective for Viewing the History of Psychophysics,” Behavioral and Brain Sciences, Vol. 16, No. 1, 1993, pp. 115-115.

http://dx.doi.org/10.1017/S0140525X00029277

[9] H. Dalton, “The Measurement of the Inequality of Incomes,” Economic Journal, Vol. 30, No. 119, 1920, pp. 348-361. http://dx.doi.org/10.2307/2223525

[10] P. A. Samuelson, “St. Petersburg Paradoxes: Defanged, Dissected, and Historically Described,” Journal of Economic Literature, Vol. 15, No. 1, 1977, pp. 24-55.

[11] F. Alt, “über die Mebarkeit des Nutzens,” Zeitschrift für Nationalökonomie, Vol. 7, No. 2, 1936, pp. 161-169.

http://dx.doi.org/10.1007/BF01316465

[12] L. S. Shapley, “Cardinal Utility Comparisons from Intensity Comparisons,” Report R-1683-PR, The Rand Corporation, Santa Monica, 1975.

[13] V. Köberling, “Strength of Preference and Cardinal Utility,” Economic Theory, Vol. 27, No. 2, 2006, pp. 375-391.

http://dx.doi.org/10.1007/s00199-005-0598-5

[14] H. L. Royden, “Real Analysis,” 3rd Edition, Macmillan, New York, 1988.

[1] H. Watts, “An Economic Definition of Poverty,” In: D. P. Moynihan, Ed., On Understanding Poverty, Basic Books, New York, Chapter 11, 1968, pp. 316-329.

[2] S. R. Chakravarty, “A New Index of Poverty,” Mathematical Social Sciences, Vol. 6, No. 3, 1983, pp. 307-313.

http://dx.doi.org/10.1016/0165-4896(83)90064-1

[3] B. Zheng, “An Axiomatic Characterization of the Watts Poverty Index,” Economics Letters, Vol. 42, No. 1, 1993, pp. 81-88.

http://dx.doi.org/10.1016/0165-1765(93)90177-E

[4] E. Farhi and I. Werning, “Inequality and Social Discounting,” Journal of Political Economy, Vol. 115, No. 3, 2007, pp. 365-402. http://dx.doi.org/10.1086/518741

[5] D. Acemoglu, “Introduction to Modern Economic Growth,” Princeton University Press, Princeton, 2008.

[6] O. Blanchard, “Macroeconomics,” 5th Edition, Prentice Hall, Upper Saddle River, 2009.

[7] P. Suppes, “The Distributive Justice of Income Inequality,” Erkenntnis, Vol. 11, No. 1, 1977, pp. 233-250.

http://dx.doi.org/10.1007/BF00169854

[8] D. J. Murray, “A Perspective for Viewing the History of Psychophysics,” Behavioral and Brain Sciences, Vol. 16, No. 1, 1993, pp. 115-115.

http://dx.doi.org/10.1017/S0140525X00029277

[9] H. Dalton, “The Measurement of the Inequality of Incomes,” Economic Journal, Vol. 30, No. 119, 1920, pp. 348-361. http://dx.doi.org/10.2307/2223525

[10] P. A. Samuelson, “St. Petersburg Paradoxes: Defanged, Dissected, and Historically Described,” Journal of Economic Literature, Vol. 15, No. 1, 1977, pp. 24-55.

[11] F. Alt, “über die Mebarkeit des Nutzens,” Zeitschrift für Nationalökonomie, Vol. 7, No. 2, 1936, pp. 161-169.

http://dx.doi.org/10.1007/BF01316465

[12] L. S. Shapley, “Cardinal Utility Comparisons from Intensity Comparisons,” Report R-1683-PR, The Rand Corporation, Santa Monica, 1975.

[13] V. Köberling, “Strength of Preference and Cardinal Utility,” Economic Theory, Vol. 27, No. 2, 2006, pp. 375-391.

http://dx.doi.org/10.1007/s00199-005-0598-5

[14] H. L. Royden, “Real Analysis,” 3rd Edition, Macmillan, New York, 1988.