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Van Zandt - CompStatics

# Van Zandt - CompStatics - An Introduction to Monotone...

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An Introduction to Monotone Comparative Statics Timothy Van Zandt INSEAD 14 November 2002 1. Overview Comparative statics (called sensitivity analysis in engineering felds) means character- izing how the endogenous outcomes oF a model change with its exogenous parame- ters. ±e “statics” in the name is because one is simply comparing two versions oF the model, For two di²erent values oF exogenous parameters, rather than tracing out a dy- namic process oF change. One can do comparative statics in a dynamic model. ³or example, one can characterize how the time-paths oF investments di²er For di²erent market conditions, For di²erent investment costs, or For di²erent initial values oF R&D. ±ere are many di²erent techniques For comparative statics. ±ere are some tech- niques oF general use, but each comparative statics exercise can involve also some ad hoc techniques. Two oF the “general-purpose” tools are monotone comparative statics and the implicit function theorem . ±e purpose oF these notes is to convey the main ideas oF monotone comparative statics. A common setting in which one may do comparative statics is in optimization problems. How does the solution to an optimization problem depend on parameters? How does demand depend on prices? How does a frm’s optimal price depend on the prices oF competing frms? What type oF monopolist would choose a higher level oF investment in cost reduction: a proft-maximizer or a social-welFare maximizer? ±e parameters that a²ect decisions in optimization problems may do so either because they a²ect preFerences (e.g., this is the case oF all the examples above except For the demand example) or because they a²ect the Feasible set (e.g., prices a²ect de- mand not because the person cares directly about prices but because prices determine the budget set). Monotone comparative statics are methods For characterizing how pa- rameters that a²ect preFerences thereby a²ect choices, keeping fxed the constraint set. ±ey are methods For unconstrained maximization. ±e term “monotone” in “monotone comparative statics” is For two reasons. ³irst, these are methods For characterizing whether an increase in a parameter causes the de- cision to increase or decrease. Second, historically the implicit Function theorem was used For this purpose and the implicit Function theorem not only tells you whether the decision increases or decreases but also the rate oF change. In contrast, monotone com- parative statics tells you only “up” or “down”, i.e., it gives an ordinal rather than cardinal answer. However, compared to the implicit Function theorem, monotone comparative statics is more direct, more widely applicable, and requires Fewer assumptions.

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Van Zandt • Monotone Comparative Statics 2 2. Onechoicevariableandoneparameter 2.1. Increasing diferences Let X R and Y R .Let u : X × Y R .
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Van Zandt - CompStatics - An Introduction to Monotone...

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