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A Complete Theory Of Comparative Statics For Differentiable Optimization Problems

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  • M. Hossein Partovi
  • Michael R. Caputo

Abstract

A new comparative statics formalism using generalized compensated derivatives is presented that, in contrast to existing methodologies, directly yields constraint‐free semidefiniteness results for any differentiable, constrained optimization problem. The formalism provides a natural and powerful method of constructing comparative statics results, free of constraints and unrestricted in scope. New results on envelope relations, invariance conditions, rank inequalities and non‐uniqueness are derived that greatly extend their utility and reach. The methodology is illustrated by deriving the comparative statics of multiple linear constraint utility maximization models and the principal‐agent problem with hidden actions, both highly nontrivial and hitherto unsolved problems.

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  • M. Hossein Partovi & Michael R. Caputo, 2006. "A Complete Theory Of Comparative Statics For Differentiable Optimization Problems," Metroeconomica, Wiley Blackwell, vol. 57(1), pages 31-67, February.
    • Handle: RePEc:bla:metroe:v:57:y:2006:i:1:p:31-67
    DOI: 10.1111/j.1467-999X.2006.00232.x
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    1. Chichilnisky, Graciela & Kalman, Peter J, 1978. "Comparative Statics of Less Neoclassical Agents," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 19(1), pages 141-148, February.
    2. M. Partovi & Michael Caputo, 2007. "Existence of a Universal Comparative Statics Matrix for Differentiable Optimization Problems," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 30(2), pages 385-394, February.
    3. Michael R. Caputo & M. Hossein Partovi, 2008. "Intrinsic comparative statics of a general class of profit-maximizing rate-of-return regulated firms," Oxford Economic Papers, Oxford University Press, vol. 60(2), pages 369-382, April.
    4. Chichilnisky, Graciela & Kalman, P.J., 1977. "Properties of critical points and operators in economics," MPRA Paper 7976, University Library of Munich, Germany.
    5. Caputo, Michael R, 1999. "The Relationship between Two Dual Methods of Comparative Statics," Journal of Economic Theory, Elsevier, vol. 84(2), pages 243-250, February.
    6. Blomquist, N Soren, 1989. "Comparative Statics for Utility Maximization Models with Nonlinear Budget Constraints," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 30(2), pages 275-296, May.
    7. Kalman, Peter J & Intriligator, Michael D, 1973. "Generalized Comparative Statics with Applications to Consumer Theory and Producer Theory," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 14(2), pages 473-486, June.
    8. Silberberg, Eugene, 1974. "A revision of comparative statics methodology in economics, or, how to do comparative statics on the back of an envelope," Journal of Economic Theory, Elsevier, vol. 7(2), pages 159-172, February.
    9. Tatsuo Hatta, 1980. "Structure of the Correspondence Principle at an Extremum Point," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 47(5), pages 987-997.
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    Cited by:

    1. Manuel Besada & Javier García & Miguel Mirás & Carmen Vázquez, 2011. "Generalized marginal rate of substitution in multiconstraint consumer’s problems and their reciprocal expenditure problems," SERIEs: Journal of the Spanish Economic Association, Springer;Spanish Economic Association, vol. 2(3), pages 401-421, September.
    2. Caputo, Michael R., 2014. "Comparative statics of a monopolistic firm facing price-cap and command-and-control environmental regulations," Energy Economics, Elsevier, vol. 46(C), pages 464-471.
    3. Michael R. Caputo & Dmitriy Popov, 2014. "Comparative Statics Of A Monopolistic Firm Facing Rate-Of-Return And Command-And-Control Pollution Constraints," Bulletin of Economic Research, Wiley Blackwell, vol. 66(S1), pages 17-35, December.
    4. Caputo, Michael R. & Ling, Chen, 2013. "The intrinsic comparative dynamics of locally differentiable feedback Nash equilibria of autonomous and exponentially discounted infinite horizon differential games," Journal of Economic Dynamics and Control, Elsevier, vol. 37(10), pages 1982-1994.
    5. Caputo, Michael R., 2013. "The intrinsic comparative dynamics of infinite horizon optimal control problems with a time-varying discount rate and time-distance discounting," Journal of Economic Dynamics and Control, Elsevier, vol. 37(4), pages 810-820.
    6. Michael R. Caputo, 2016. "Intrinsic Comparative Statics of a Nash Bargaining Solution," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 18(04), pages 1-11, December.
    7. Caputo, Michael R. & Paris, Quirino, 2008. "Comparative statics of the generalized maximum entropy estimator of the general linear model," European Journal of Operational Research, Elsevier, vol. 185(1), pages 195-203, February.
    8. Michael Caputo & Chen Ling, 2015. "Intrinsic Comparative Dynamics of Locally Differentiable Feedback Stackelberg Equilibria," Dynamic Games and Applications, Springer, vol. 5(1), pages 1-25, March.
    9. Caputo Michael R, 2018. "Uncovering the Behavioral Implications of the Rational Addiction Assertion," The B.E. Journal of Theoretical Economics, De Gruyter, vol. 18(1), pages 1-13, January.
    10. Larson, Douglas M. & Shaikh, Sabina L., 1999. "Empirical Specification Requirements For Two-Constraint Models Of Recreation Demand," 1999 Annual meeting, August 8-11, Nashville, TN 21629, American Agricultural Economics Association (New Name 2008: Agricultural and Applied Economics Association).

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