Monotone Comparative Statics: Geometric Approach



We consider the comparative statics of solutions to parameterized optimization problems. A geometric method is developed for finding a vector field that, at each point in the parameter space, indicates a direction in which monotone comparative statics obtains. Given such a vector field, we provide sufficient conditions under which the problem can be reparameterized on the parameter space (or a subset thereof) in a way that guarantees monotone comparative statics. A key feature of our method is that it does not require the feasible set to be a lattice and works in the absence of the standard quasi-supermodularity and single-crossing assumptions on the objective function. We illustrate our approach with a variety of applications.


Change of parameters Parameterized optimization problems Single-crossing Supermodularity 


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  1. 1.
    Samuelson, P.A.: The stability of equilibrium: comparative statics and dynamics. Econometrica 9(2), 97–120 (1941) CrossRefGoogle Scholar
  2. 2.
    Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000) MATHGoogle Scholar
  3. 3.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation I. Springer, New York (2006) Google Scholar
  4. 4.
    Topkis, D.M.: Ordered optimal solutions. Doctoral Dissertation, Stanford University, Stanford, CA (1968) Google Scholar
  5. 5.
    Topkis, D.M.: Supermodularity and Complementarity. Princeton University Press, Princeton (1998) Google Scholar
  6. 6.
    Milgrom, P., Shannon, C.: Monotone comparative statics. Econometrica 62(1), 157–180 (1994) CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Baumol, W.J., Quandt, R.E.: Rules of thumb and optimally imperfect decisions. Am. Econ. Rev. 54(2), 23–46 (1964) Google Scholar
  8. 8.
    Granot, F., Veinott, A.F.: Substitutes, complements and ripples in network flows. Math. Oper. Res. 10(3), 471–497 (1985) MathSciNetMATHGoogle Scholar
  9. 9.
    Bertsekas, D.P.: Nonlinear Programming. Athena Scientific, Belmont (1995) MATHGoogle Scholar
  10. 10.
    Zorich, V.A.: Mathematical Analysis, vols. I, II. Springer, New York (2004) Google Scholar
  11. 11.
    Sard, A.: The measure of the critical points of differentiable maps. Bull. Am. Math. Soc. 48, 883–890 (1942) CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    Sard, A.: Images of critical sets. Ann. Math. 68(2), 247–259 (1958) CrossRefMathSciNetGoogle Scholar
  13. 13.
    Milgrom, P.: Comparing optima: do simplifying assumptions affect conclusions? J. Political Econ. 102(3), 607–615 (1994) CrossRefGoogle Scholar
  14. 14.
    Berge, C.: Topological Spaces. Oliver and Boyd, Edinburgh (1963). Reprinted by Dover Publications, Mineola, in 1997 MATHGoogle Scholar
  15. 15.
    Khalil, H.K.: Nonlinear Systems. Macmillan, New York (1992) MATHGoogle Scholar
  16. 16.
    Kakutani, S.: A generalization of Brouwer’s fixed point theorem. Duke Math. J. 8, 457–459 (1941) CrossRefMathSciNetMATHGoogle Scholar
  17. 17.
    Arnold, V.I.: Ordinary Differential Equations. MIT Press, Cambridge (1973) MATHGoogle Scholar
  18. 18.
    Lee, J.M.: Introduction to Smooth Manifolds. Springer, New York (2003) Google Scholar
  19. 19.
    Veblen, T.: The Theory of the Leisure Class (1899). Reprinted by Penguin Books, New York, in 1994 Google Scholar
  20. 20.
    Hirsch, F.: Social Limits to Growth. Harvard University Press, Cambridge (1976) Google Scholar
  21. 21.
    Roberts, K.W.S.: Welfare considerations of nonlinear pricing. Econ. J. 89(353), 66–83 (1979) CrossRefGoogle Scholar
  22. 22.
    Mirman, L.J., Sibley, D.S.: Optimal nonlinear prices for multiproduct monopolies. Bell J. Econ. 11(2), 659–670 (1980) CrossRefGoogle Scholar
  23. 23.
    Matthews, S., Moore, J.: Monopoly provision of quality and warranties: an exploration of the theory of multidimensional screening. Econometrica 55(2), 441–467 (1987) CrossRefMATHGoogle Scholar
  24. 24.
    Rochet, J.-C., Stole, L.A.: The economics of multidimensional screening. In: Dewatripont, M., Hansen, L.-P., Turnovsky, S.J. (eds.) Advances in Economics and Econometrics: Theory and Applications: Eighth World Congress, vol. I, pp. 150–197. Cambridge University Press, New York (2003) Google Scholar
  25. 25.
    Mussa, M., Rosen, S.: Monopoly and product quality. J. Econ. Theory 18(2), 301–317 (1978) CrossRefMathSciNetMATHGoogle Scholar
  26. 26.
    Mirrlees, J.A.: An exploration in the theory of optimal income taxation. Rev. Econ. Stud. 38(2), 175–208 (1971) CrossRefMATHGoogle Scholar
  27. 27.
    Athey, S.: Monotone comparative statics under uncertainty. Q. J. Econ. 117(1), 187–223 (2002) CrossRefMathSciNetMATHGoogle Scholar
  28. 28.
    Edgeworth, F.Y.: The Pure Theory of Monopoly (1897). Reprinted in: Edgeworth, F.Y. Papers Relating to Political Economy, Macmillan, London, 1925 Google Scholar
  29. 29.
    Samuelson, P.A.: Complementarity: an essay on the 40th anniversary of the Hicks-Allen revolution in demand theory. J. Econ. Lit. 12(4), 1255–1289 (1974) Google Scholar
  30. 30.
    Milgrom, P., Roberts, J.: The economics of modern manufacturing: technology, strategy, and organization. Am. Econ. Rev. 80(3), 511–528 (1990) Google Scholar
  31. 31.
    Milgrom, P., Roberts, J.: Comparing equilibria. Am. Econ. Rev. 84(3), 441–459 (1994) Google Scholar

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© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Nuffield CollegeOxford UniversityOxfordUK
  2. 2.Department of Management Science and Engineering, 442 Terman Engineering CenterStanford UniversityStanfordUSA

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