Monotone comparative statics is a sub-field of comparative statics that focuses on the conditions under which endogenous variables undergo monotone changes (that is, either increasing or decreasing) when there is a change in the exogenous parameters.
Traditionally, comparative results in economics are obtained using the Implicit Function Theorem, an approach that requires the concavity and differentiability of the objective function as well as the interiority and uniqueness of the optimal solution.
The methods of monotone comparative statics typically dispense with these assumptions.
It focuses on the main property underpinning monotone comparative statics, which is a form of complementarity between the endogenous variable and exogenous parameter.
Roughly speaking, a maximization problem displays complementarity if a higher value of the exogenous parameter increases the marginal return of the endogenous variable.
This guarantees that the set of solutions to the optimization problem is increasing with respect to the exogenous parameter.
Standard comparative statics approach: Assume that set
The notion of complementarity between exogenous and endogenous variables is formally captured by single crossing differences.
Definition (single crossing differences):[2] The family of functions
Application (monopoly output and changes in costs): A monopolist chooses
Single crossing differences is not a necessary condition for the optimal solution to be increasing with respect to a parameter.
Like single crossing differences, the interval dominance order (IDO) is an ordinal property.
The next result gives useful sufficient conditions for single crossing differences and IDO.
Application (Optimal stopping problem):[8] At each moment in time, agent gains profit of
obey IDO and, by Theorem 2, the set of optimal stopping times is decreasing.
Examples of the strong set order in higher dimensions.
As in the case of single crossing differences, and unlike supermodularity, quasisupermodularity is an ordinal property.
In some important economic applications, the relevant change in the constraint set cannot be easily understood as an increase with respect to the strong set order and so Theorem 3 cannot be easily applied.
Theorem 3 cannot be straightforwardly applied to obtain conditions for normality, because
that obey single crossing differences or the interval dominance order.
is realized; then it seems reasonable that the optimal action should increase with the likelihood of higher states.
obey single crossing differences or the interval dominance order.
In the following theorem, X can be either ``single crossing differences" or ``the interval dominance order".
is ordered with respect to the monotone likelihood ratio property.
The monotone likelihood ratio condition in this theorem cannot be weakened, as the next result demonstrates.
, obeys single crossing (though not necessarily increasing) differences.
is ordered with respect to the monotone likelihood ratio property.
is required to be a single crossing function for any finite measure
is the Bernoulli utility function representing the firm’s attitude towards uncertainty.
, and obeys decreasing absolute risk aversion (DARA).