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Comparative Dynamics via Envelope Methods in Variational Calculus

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  • Michael R. Caputo

Abstract

The primal-dual methodology of Samuelson (1965) is extended and applied to a nonautonomous variational calculus problem with a fixed vector of initial stocks, fixed initial and terminal time values, a free vector of terminal stocks, and a time-independent vector of parameters. It is shown that if the solution of the variational problem is smooth enough, the qualitative effects of parameter perturbations on the entire optimal arcs can be represented by a generalized Slutsky-type matrix, which holds in integral form and is symmetric negative semidefinite. Sufficient conditions for the optimal value function to be convex in the parameters are also given.

Suggested Citation

  • Michael R. Caputo, 1990. "Comparative Dynamics via Envelope Methods in Variational Calculus," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 57(4), pages 689-697.
    • Handle: RePEc:oup:restud:v:57:y:1990:i:4:p:689-697.
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    Cited by:

    1. Kane, Sally & Reilly, John & Tobey, James, 1991. "Climate Change: Economic Implications for World Agriculture," Agricultural Economic Reports 308153, United States Department of Agriculture, Economic Research Service.
    2. D. Dragone & H. Strulik, 2017. "Human Health and Aging over an Infinite Time Horizon," Working Papers wp1104, Dipartimento Scienze Economiche, Universita' di Bologna.
    3. Judd, Kenneth L., 1996. "Approximation, perturbation, and projection methods in economic analysis," Handbook of Computational Economics, in: H. M. Amman & D. A. Kendrick & J. Rust (ed.), Handbook of Computational Economics, edition 1, volume 1, chapter 12, pages 509-585, Elsevier.
    4. YIlmaz, Ensar, 2011. "Income distribution, efficiency and rationing," Economic Modelling, Elsevier, vol. 28(3), pages 1247-1255, May.
    5. Alfons Oude Lansink & Spiro Stefanou, 2001. "Dynamic Area Allocation and Economies of Scale and Scope," Journal of Agricultural Economics, Wiley Blackwell, vol. 52(3), pages 38-52, September.
    6. Krysiak, Frank C., 2006. "Stochastic intertemporal duality: An application to investment under uncertainty," Journal of Economic Dynamics and Control, Elsevier, vol. 30(8), pages 1363-1387, August.
    7. LaFrance, Jeffrey T., 1992. "Do Increased Commodity Prices Lead To More Or Less Soil Degradation?," Australian Journal of Agricultural Economics, Australian Agricultural and Resource Economics Society, vol. 36(1), pages 1-26, April.
    8. Ben White, 2000. "A Review of the Economics of Biological Natural Resources," Journal of Agricultural Economics, Wiley Blackwell, vol. 51(3), pages 419-462, September.
    9. Davide Dragone & Paolo Vanin, 2022. "Substitution Effects in Intertemporal Problems," American Economic Journal: Microeconomics, American Economic Association, vol. 14(3), pages 791-809, August.
    10. Krasachat, W. & Coelli, T. J., 1995. "Dynamics of Agricultural Production on Thailand," 1995 Conference (39th), February 14-16, 1995, Perth, Australia 170898, Australian Agricultural and Resource Economics Society.
    11. Caputo, Michael R., 2003. "The comparative dynamics of closed-loop controls for discounted infinite horizon optimal control problems," Journal of Economic Dynamics and Control, Elsevier, vol. 27(8), pages 1335-1365, June.
    12. D. Dragone & P. Vanin, 2015. "Price Effect in the Short and in the Long Run," Working Papers wp1040, Dipartimento Scienze Economiche, Universita' di Bologna.
    13. Favard, Pascal & Karp, Larry, 2004. "How many cake-eaters? Chouette, on a du monde a diner !," Journal of Economic Dynamics and Control, Elsevier, vol. 28(4), pages 801-815, January.
    14. 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.
    15. Barney, L. Jr., 1997. "Uncertainty and the comparative dynamics of stock price," International Review of Economics & Finance, Elsevier, vol. 6(4), pages 405-419.
    16. Ngo Long & Horst Siebert, 1992. "A model of the socialist firm in transition to a market economy," Journal of Economics, Springer, vol. 56(1), pages 1-21, February.
    17. Löfgren, Karl-Gustaf, 2011. "On Envelope Theorems in Economics: Inspired by a Revival of a Forgotten Lecture," Umeå Economic Studies 830, Umeå University, Department of Economics.
    18. Lichtenberg, Erik, 2001. "Adoption Of Soil Conservation Practices: A Revealed Preference Approach," Working Papers 28609, University of Maryland, Department of Agricultural and Resource Economics.
    19. Yilmaz, Ensar & Ünveren, Burak, 2011. "Income distribution and exchange in a dynamic search model," International Review of Economics & Finance, Elsevier, vol. 20(4), pages 665-678, October.
    20. Dragone, Davide & Strulik, Holger, 2020. "Negligible senescence: An economic life cycle model for the future," Journal of Economic Behavior & Organization, Elsevier, vol. 171(C), pages 264-285.
    21. Favard, Pascal & Karp, Larry, 1999. "How many cake-eaters? Chouette, on a du monde a diner!," Department of Agricultural & Resource Economics, UC Berkeley, Working Paper Series qt61d3d33p, Department of Agricultural & Resource Economics, UC Berkeley.
    22. Lichtenberg, Erik, 2004. "Cost-Responsiveness of Conservation Practice Adoption: A Revealed Preference Approach," Journal of Agricultural and Resource Economics, Western Agricultural Economics Association, vol. 29(3), pages 1-16, December.
    23. Caputo, Michael R. & Wilen, James E., 1995. "Optimality conditions and comparative statics for horizon and endpoint choices in optimal control theory," Journal of Economic Dynamics and Control, Elsevier, vol. 19(1-2), pages 351-369.

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