generalization of the above method allows the optimization problem to include a set of constraints. This leads to the general envelope theorem. Applications include determining changes in Marshallian demand in response to changes in price or wage. Limitations and extensions One limitation of comparative statics using the implicit function theorem is that results are valid only in a (potentially very small) neighborhood of the optimum—that is, only for very small changes in the exogenous variables. Another limitation is the potentially overly restrictive nature of the assumptions conventionally used to justify comparative statics procedures. For example, John Nachbar discovered in one of his case studies that using comparative statics in general equilibrium analysis works best with very small, individual level of data rather than at an aggregate level. Paul Milgrom and Chris Shannon pointed out in 1994 that the assumptions conventionally used to justify the use of comparative statics on optimization problems are not actually necessary—specifically, the assumptions of convexity of preferred sets or constraint sets, smoothness of their boundaries, first and second derivative conditions, and linearity of budget sets or objective functions. In fact, sometimes a problem meeting these conditions can be monotonically transformed to give a problem with identical comparative statics but violating some or all of these conditions; hence these conditions are not necessary to justify the comparative statics. Stemming from the article by Milgrom and Shannon as well as the results obtained by Veinott and Topkis an important strand of operational research was developed called monotone comparative statics. In particular, this theory concentrates on the comparative statics analysis using only conditions that are independent of order-preserving transformations. The method uses lattice theory and introduces the notions of quasi-supermodularity and the single-crossing condition. The wide application of monotone comparative statics to economics includes production theory, consumer theory, game theory with complete and incomplete information, auction theory, and others. See also Model (economics) Qualitative economics Notes References John Eatwell et al., ed. (1987). "Comparative dynamics," The New Palgrave: A Dictionary of Economics, v. 1, p. 517. John R. Hicks (1939). Value and Capital. Timothy J. Kehoe, 1987. "Comparative statics," The New Palgrave: A Dictionary of Economics, v. 1, pp. 517–20. Andreu Mas-Colell, Michael D. Whinston, and Jerry R. Green, 1995. Microeconomic Theory. Paul A. Samuelson (1947). Foundations of Economic Analysis. Eugene Silberberg and Wing Suen, 2000. The Structure of Economics: A Mathematical Analysis, 3rd edition. External links San Jose