# Directional monotone comparative statics

- 237 Downloads
- 4 Citations

## Abstract

Many questions of interest in economics can be stated in terms of monotone comparative statics: If a parameter of a constrained optimization problem increases, when does its solution increase as well. We characterize monotone comparative statics in different directions in finite-dimensional Euclidean space by extending the monotonicity theorem of Milgrom and Shannon (Econometrica 62(1):157–180, 1994) to constraint sets ordered in Quah (Econometrica 75(2):401–431, 2007)’s set order. Our characterizations are ordinal and retain the same flavor as their counterparts in the standard theory, showing new connections to the standard theory and presenting new results. The results are highlighted with several applications (in consumer theory, producer theory, and game theory) which were previously outside the scope of the standard theory of monotone comparative statics.

## Keywords

Monotone comparative statics*i*-Directional single crossing property

*i*-Directional set order Quasisupermodular function

## JEL Classification

C61 C70 D00## References

- Aaberge, R., Gagsvik, J.K., Strøm, S.: Labor supply responses and welfare effects of tax reforms. Scand. J. Econ.
**97**(4), 635–659 (1995)CrossRefGoogle Scholar - Amir, R.: Cournot oligopoly and the theory of supermodular games. Games Econ. Behav.
**15**, 132–148 (1996)CrossRefGoogle Scholar - Amir, R., Lambson, V.E.: On the effects of entry in Cournot markets. Rev. Econ. Stud.
**67**(2), 235–254 (2000)CrossRefGoogle Scholar - Amir, R., Lazzati, N.: Endgoenous information acquisition in Bayesian games with strategic complementarities. J. Econ. Theory
**163**, 684–698 (2016)CrossRefGoogle Scholar - Antoniadu, E.: Comparative statics for the consumer problem. Econ. Theory
**31**(1), 189–203 (2007)CrossRefGoogle Scholar - Balbus, Ł., Reffett, K., Woźny, Ł.: A constructive study of Markov equilibria in stochastic games with strategic complementarities. J. Econ. Theory
**150**, 815–840 (2014)CrossRefGoogle Scholar - Bruneau, J.F.: A note on permits, standards, and technological innovation. J. Environ. Econ. Manag.
**48**(3), 1192–1199 (2004)CrossRefGoogle Scholar - Bulow, J.I., Geanakoplos, J.D., Klemperer, P.D.: Multimarket oligopoly: strategic substitutes and complements. J. Polit. Econ.
**93**(3), 488–511 (1985)CrossRefGoogle Scholar - Cosandier, C., Garcia, F., Knauff, M.: Price competition with differentiated goods and incomplete product awareness. Econ. Theory (2017). doi: 10.1007/s00199-017-1050-3
- Dobzinski, S., Lavi, R., Nisan, N.: Multi-unit auctions with budget limits. Games Econ. Behav.
**74**(2), 486–503 (2012)CrossRefGoogle Scholar - Echenique, F.: Comparative statics by adaptive dynamics and the correspondence principle. Econometrica
**70**(2), 257–289 (2002)CrossRefGoogle Scholar - Echenique, F.: A characterization of strategic complementarities. Games Econ. Behav.
**46**(2), 325–347 (2004)CrossRefGoogle Scholar - Harbaugh, W.T.: The prestige motive for making charitable transfers. Am. Econ. Rev.
**88**(2), 277–282 (1998)Google Scholar - Heikkilä, S., Reffett, K.: Fixed point theorems and their applications to theory of Nash equilibria. Nonlinear Anal.
**64**, 1415–1436 (2006)CrossRefGoogle Scholar - Hoynes, H.H.: Welfare transfers in two-parent families: labor supply and welfare participation under AFDC-UP. Econometrica
**64**(2), 295–332 (1996)CrossRefGoogle Scholar - Jensen, M.K.: Aggregative games and best-reply potentials. Econ. Theory
**43**(1), 45–66 (2010)CrossRefGoogle Scholar - LiCalzi, M., Veinott, A.F., Jr.: Subextremal Functions and Lattice Programming, Working paper (1992)Google Scholar
- Milgrom, P., Roberts, J.: Rationalizability, learning, and equilibrium in games with strategic complementarities. Econometrica
**58**(6), 1255–1277 (1990)CrossRefGoogle Scholar - Milgrom, P., Shannon, C.: Monotone comparative statics. Econometrica
**62**(1), 157–180 (1994)CrossRefGoogle Scholar - Mirman, L.J., Ruble, R.: Lattice theory and the consumer’s problem. Math. Oper. Res.
**33**(2), 301–314 (2008)CrossRefGoogle Scholar - Monaco, A.J., Sabarwal, T.: Games with strategic complements and substitutes. Econ. Theory
**62**(1), 65–91 (2016)CrossRefGoogle Scholar - Montero, J.-P.: Permits, standards, and technology innovation. J. Environ. Econ. Manag.
**44**(1), 23–44 (2002)CrossRefGoogle Scholar - Palfrey, T.R.: Multiple-object, discriminatory auctions with bidding constraints: a game-theoretic analysis. Manag. Sci.
**26**(9), 935–946 (1980)CrossRefGoogle Scholar - Quah, J.K.-H.: The comparative statics of constrained optimization problems. Econometrica
**75**(2), 401–431 (2007)CrossRefGoogle Scholar - Quah, J.K.-H., Strulovici, B.: Comparative statics, informativeness, and the interval dominance order. Econometrica
**77**(6), 1949–1992 (2009)CrossRefGoogle Scholar - Reynolds, S.S., Rietzke, D.: Price caps, oligopoly, and entry. Econ. Theory (2017). doi: 10.1007/s00199-016-0963-6
- Rothkopf, M.H.: Bidding in simultaneous auctions with a constraint on exposure. Oper. Res.
**25**(4), 620–629 (1977)CrossRefGoogle Scholar - Roy, S., Sabarwal, T.: On the (non-)lattice structure of the equilibrium set in games with strategic substitutes. Econ. Theory
**37**(1), 161–169 (2008)CrossRefGoogle Scholar - Roy, S., Sabarwal, T.: Monotone comparative statics for games with strategic substitutes. J. Math. Econ.
**46**(5), 793–806 (2010)CrossRefGoogle Scholar - Roy, S., Sabarwal, T.: Characterizing stability properties in games with strategic substitutes. Games Econ. Behav.
**75**(1), 337–353 (2012)CrossRefGoogle Scholar - Smithson, R.E.: Fixed points of order preserving multifunctions. Proc. Am. Math. Soc.
**28**(1), 304–310 (1971)CrossRefGoogle Scholar - Topkis, D.: Minimizing a submodular function on a lattice. Oper. Res.
**26**, 305–321 (1978)CrossRefGoogle Scholar - Topkis, D.: Equilibrium points in nonzero-sum n-person submodular games. SIAM J. Control Optim.
**17**(6), 773–787 (1979)CrossRefGoogle Scholar - Topkis, D.: Supermodularity and Complementarity. Princeton University Press, Princeton (1998)Google Scholar
- van Soest, A.: Structural models of family labor supply: a discrete choice approach. J. Hum. Resour.
**30**(1), 63–88 (1995)CrossRefGoogle Scholar - Veinott, A.F., Jr: Lattice Programming: Qualitative Optimization and Equilibria, Working paper (1992)Google Scholar
- Vives, X.: Nash equilibrium with strategic complementarities. J. Math. Econ.
**19**(3), 305–321 (1990)CrossRefGoogle Scholar - Zhou, L.: The set of Nash equilibria of a supermodular game is a complete lattice. Games Econ. Behav.
**7**(2), 295–300 (1994)CrossRefGoogle Scholar - Zimper, A.: A fixed point characterization of the dominance-solvability of lattice games with strategic substitutes. Int. J. Game Theory
**36**(1), 107–117 (2007)CrossRefGoogle Scholar