Monotone comparative statics


Monotone comparative statics is a sub-field of comparative statics that focuses on the conditions under which endogenous variables undergo monotone changes 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.

Basic results

Motivation

Let and let be a family of functions parameterized by, where is a partially ordered set. How does the correspondence vary with ?
Standard comparative statics approach: Assume that set is a compact interval and is a continuously differentiable, strictly quasiconcave function of. If is the unique maximizer of, it suffices to show that for any, which guarantees that is increasing in. This guarantees that the optimum has shifted to the right, i.e.,. This approach makes various assumptions, most notably the quasiconcavity of.

One-dimensional optimization problems

While it is clear what it means for a unique optimal solution to be increasing, it is not immediately clear what it means for the correspondence to be increasing in. The standard definition adopted by the literature is the following.
Definition : Let and be subsets of. Set dominates in the strong set order if for any in and in, we have in and in.
In particular, if and, then if and only if. The correspondence is said to be increasing if whenever.
The notion of complementarity between exogenous and endogenous variables is formally captured by single crossing differences.
Definition : Let. Then is a single crossing function if for any we have.
Definition : The family of functions,, obey single crossing differences if for all, function is a single crossing function.
Obviously, an increasing function is a single crossing function and, if is increasing in , we say that obey increasing differences. Unlike increasing differences, single crossing differences is an ordinal property, i.e., if obey single crossing differences, then so do, where for some function that is strictly increasing in.
Theorem 1: Define. The family obey single crossing differences if and only if for all, we have for any.
Application : A monopolist chooses to maximise its profit, where is the inverse demand function and is the constant marginal cost. Note that obey single crossing differences. Indeed, take any and assume that ; for any such that , we obtain. By Theorem 1, the profit-maximizing output decreases as the marginal cost of output increases, i.e., as decreases.

Interval dominance order

Single crossing differences is not a necessary condition for the optimal solution to be increasing with respect to a parameter. In fact, the condition is necessary only for to be increasing in for any. Once the sets are restricted to a narrower class of subsets of, the single crossing differences condition is no longer necessary.
Definition : Let. A set is an interval of if, whenever and are in, then any such that is also in.
For example, if, then is an interval of but not. Denote.
Definition : The family obey the interval dominance order if for any and, such that, for all, we have.
Like single crossing differences, the interval dominance order is an ordinal property. An example of an IDO family is a family of quasiconcave functions where increasing in. Such a family need not obey single crossing differences.
A function is regular if is non-empty for any, where denotes the interval.
Theorem 2: Denote. A family of regular functions obeys the interval dominance order if and only if is increasing in for all intervals.
The next result gives useful sufficient conditions for single crossing differences and IDO.
Proposition 1: Let be an interval of and be a family of continuously differentiable functions. If, for any, there exists a number such that for all, then obey single crossing differences. If, for any, there exists a nondecreasing, strictly positive function such that for all, then obey IDO.
Application : At each moment in time, agent gains profit of, which can be positive or negative. If agent decides to stop at time, the present value of his accumulated profit is
where is the discount rate. Since, the function has many turning points and they do not vary with the discount rate. We claim that the optimal stopping time is decreasing in, i.e., if then. Take any. Then, Since is positive and increasing, Proposition 1 says that obey IDO and, by Theorem 2, the set of optimal stopping times is decreasing.

Multi-dimensional optimization problems

The above results can be extended to a multi-dimensional setting. Let be a lattice. For any two, in, we denote their supremum by and their infimum by.
Definition : Let be a lattice and, be subsets of. We say that dominates in the strong set order if for any in and in, we have in and in.
Examples of the strong set order in higher dimensions.
Definition : Let be a lattice. The function is quasisupermodular if
The function is said to be a supermodular function if Every supermodular function is quasisupermodular. As in the case of single crossing differences, and unlike supermodularity, quasisupermodularity is an ordinal property. That is, if function is quasisupermodular, then so is function, where is some strictly increasing function.
Theorem 3: Let is a lattice, a partially ordered set, and, subsets of. Given , we denote by. Then for any and
Application : Let denote the vector of inputs of a profit-maximizing firm, be the vector of input prices, and the revenue function mapping input vector to revenue. The firm's profit is. For any,,, is increasing in. Hence, has increasing differences. Moreover, if is supermodular, then so is. Therefore, it is quasisupermodular and by Theorem 3, for.

Constrained optimization problems

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. For example, consider a consumer who maximizes a utility function subject to a budget constraint. At price in and wealth, his budget set is and his demand set at is . A basic property of consumer demand is normality, which means that the demand of each good is increasing in wealth. Theorem 3 cannot be straightforwardly applied to obtain conditions for normality, because if . In this case, the following result holds.
Theorem 4: Suppose is supermodular and concave. Then the demand correspondence is normal in the following sense: suppose, and ; then there is and such that and.
The supermodularity of alone guarantees that, for any and,. Note that the four points,,, and form a rectangle in Euclidean space. On the other hand, supermodularity and concavity together guarantee that
for any, where. In this case, crucially, the four points,,, and form a backward-leaning parallelogram in Euclidean space.

Monotone comparative statics under uncertainty

Let, and be a family of real-valued functions defined on that obey single crossing differences or the interval dominance order. Theorem 1 and 3 tell us that is increasing in. Interpreting to be the state of the world, this says that the optimal action is increasing in the state if the state is known. Suppose, however, that the action is taken before is realized; then it seems reasonable that the optimal action should increase with the likelihood of higher states. To capture this notion formally, let be a family of density functions parameterized by in the poset, where higher is associated with a higher likelihood of higher states, either in the sense of first order stochastic dominance or the monotone likelihood ratio property. Choosing under uncertainty, the agent maximizes
For to be increasing in, it suffices that family obey single crossing differences or the interval dominance order. The results in this section give condition under which this holds.
Theorem 5: Suppose obeys increasing differences. If is ordered with respect to first order stochastic dominance, then obeys increasing differences.
In the following theorem, X can be either ``single crossing differences" or ``the interval dominance order".
Theorem 6: Suppose obeys X. Then the family obeys X if 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.
Proposition 2: Let and be two probability mass functions defined on and suppose is does not dominate with respect to the monotone likelihood ratio property. Then there is a family of functions, defined on, that obey single crossing differences, such that, where .
Application : An agent maximizes expected utility with the strictly increasing Bernoulli utility function. The wealth of the agent,, can be invested in a safe or risky asset. The prices of the two assets are normalized at 1. The safe asset gives a constant return, while the return of the risky asset is governed by the
probability distribution. Let denote the agent's investment in the risky asset. Then the wealth of the agent in state is. The agent chooses to maximize
Note that, where, obeys single crossing differences. By Theorem 6, obeys single crossing differences, and hence is increasing in, if is ordered with
respect to the monotone likelihood ratio property.

Aggregation of the single crossing property

While the sum of increasing functions is also increasing, it is clear that the single crossing property need not be preserved by aggregation. For the sum of single crossing functions to have the same property requires that the functions be related to each other in a particular manner.
Definition : Let be a poset. Two functions obey signedratio monotonicity if, for any, the following holds:
Proposition 3: Let and be two single crossing functions. Then is a single crossing function for any nonnegative scalars and if and only if and obey signed-ratio monotonicity.
This result can be generalized to infinite sums in the following sense.
Theorem 7: Let be a finite measure space and suppose that, for each, is a bounded and measurable function of. Then is a single crossing function if, for all,, the pair of functions and of satisfy signed-ratio monotonicity. This condition is also necessary if contains all singleton sets and is required to be a single crossing function for any finite measure.
Application : A firm faces uncertainty over the demand for its output and the profit at state is given
by, where is the marginal cost and is the inverse demand function in state. The firm maximizes
where is the probability of state and is the Bernoulli utility function representing the firm’s attitude towards uncertainty. By Theorem 1, is increasing in if the family obeys single crossing differences. By definition, the latter says that, for any, the function
is a single crossing function. For each, is s single crossing function of. However, unless is linear, will not, in general, be increasing in. Applying Theorem 6, is a single crossing function if, for any, the functions and obey signed-ratio monotonicity. This is guaranteed when is decreasing in and increasing in and obeys increasing differences; and is twice differentiable, with, and obeys decreasing absolute risk aversion.

Selected literature on monotone comparative statics and its applications