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 timepaths 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 “generalpurpose” 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 proftmaximizer or a socialwelFare 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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View Full DocumentVan 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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 '10
 AARON
 Order theory, Monotonic function, Convex function, Monotone comparative statics

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