Is Product of Two Convex Functions Necessarily Convex? A Case of the MRP Curve


The curvature of the marginal revenue product curve plays an important role in most theoretic microeconomic models since it determines the size of profit contribution to an employer and optimality conditions of solutions. There are many well established introductory and intermediate microeconomic textbooks portray marginal revenue product curves as linear or concave to the origin. In nearly all cases, the MRP cannot be linear, nor can it be concave. In this analysis, most of the well-known production functions generate convex MRP curves.

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C. Wei Yang, H. Wen Cheng, K. Hung, P. R. Woodburne and P. Kim, "Is Product of Two Convex Functions Necessarily Convex? A Case of the MRP Curve," Theoretical Economics Letters, Vol. 2 No. 5, 2012, pp. 511-516. doi: 10.4236/tel.2012.25094.

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The authors declare no conflicts of interest.


[1] C. W. Yang, D. B. Means and G. E. Moody, “Tax Rates and Total Tax Revenues from Local Property Taxes,” Public Finance Quarterly, Vol. 21, No. 4, 1993, pp. 355-377. doi:10.1177/109114219302100401
[2] A. Takayama, “Behavior of the Firm under Regulatory Constraint,” The American Economic Review, Vol. 59, No. 3, 1969, pp. 255-260.
[3] W. J. Baumol and A. K. Klevorick, “Input Choices and Rate-Of-Return Regulation: An Overview of the Discussion,” The Bell Journal of Economics and Management Science, Vol. 1, No. 2, 1970, pp. 162-190. doi:10.2307/3003179
[4] C. W. Cobb and P. H. Douglas, “A Theory of Production,” American Economic Review, Vol. 18, No. 1, 1928, pp. 139-165.
[5] K. J. Arrow, H. B. Chenery, B. S. Minhas and R. M. Solow, “Capital-labor Substitution and Economic Efficiency,” The Review of Economics and Statistics, Vol. 43, No. 3, 1961, pp. 225-250. doi:10.2307/1927286
[6] V. Mukerji, “A Generalized S.M.A.C. Function with Constant Ratios of Elasticity of Substitution,” Review of Economic Studies, Vol. 30, No. 3, 1963, pp. 233-236. doi:10.2307/2296324
[7] N. S. Revankar, “A Class of Variable Elasticity of Substitution Production Functions,” Econometrica, Vol. 39, No. 1, 1971, pp. 61-71. doi:10.2307/1909140
[8] L. R. Christensen, D. W. Jorgenson and L. J. Lau, “Transcendental Logarithmic Production Frontiers,” Review of Economics and Statistics, Vol. 55, No. 1, 1973, pp. 8-45. doi:10.2307/1927992
[9] P. H. Douglas, “Are There Laws of Production?” American Economic Review, Vol. 38, No. 1, 1948, pp. 1-42.
[10] C. A. K. Lovell, “Estimation and Prediction with CES and VES Production Functions,” International Economic Review, Vol. 14, No. 3, 1973, pp. 676-692. doi:10.2307/2525980
[11] D. B. Humphrey and J. R. Moroney, “Substitution among Capital, Labor, and Natural Resource Products in American Manufacturing,” Journal of Political Economy, Vol. 83, No. 1, 1975, pp. 57-82. doi:10.1086/260306
[12] M. Friedman, “Price Theory,” 2nd Edition, Aldine Publishing Company, New York, 1976.
[13] P. A. Samuelson and W. D. Nordhaus, “Economics,” 17th Edition, McGraw-Hill, New York, 2005.
[14] D. N. McCloskey, “The Rhetoric of Economics,” 2nd Edition, University of WI Press, Madison, 1998.

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