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.
Basic results
Motivation
Let and let be a family of functions parameterized by , where is a partially ordered set (or poset, for short). 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 (strong set order):See Veinott (1992): Lattice programming: qualitative optimization and equilibria. MS Stanford. 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 (single crossing function): Let . Then is a single crossing function if for any we have .
Definition (single crossing differences):See Milgrom, P., and C. Shannon (1994): “Monotone Comparative Statics,” Econometrica, 62(1), 157–180; or Quah, J. K.-H., and B. Strulovici (2012): “Aggregating the Single Crossing Property,” Econometrica, 80(5), 2333–2348. The family of functions , , obey single crossing differences (or satisfy the single crossing property) if for all , function is a single crossing function.
Obviously, an increasing function is a single crossing function and, if is increasing in (in the above definition, for any ), 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:Milgrom, P., and C. Shannon (1994): “Monotone Comparative Statics,” Econometrica, 62(1), 157–180. Define . The family obey single crossing differences if and only if for all , we have for any .
Proof: Assume and , and . We have to show that and . We only need to consider the case where . Since , we obtain , which guarantees that . Furthermore, so that . If not, which implies (by single crossing differences) that , contradicting the optimality of at . To show the necessity of single crossing differences, set , where . Then for any guarantees that, if , then . Q.E.D.
Application (monopoly output and changes in costs): 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 (Interval):Quah, J. K.-H., and B. Strulovici (2009): “Comparative Statics, Informativeness, and the Interval Dominance Order,” Econometrica, 77(6), 1949–1992. 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 (Interval Dominance Order):Quah, J. K.-H., and B. Strulovici (2009): “Comparative Statics, Informativeness, and the Interval Dominance Order,” Econometrica, 77(6), 1949–1992. The family obey the interval dominance order (IDO) if for any and , such that , for all , we have .
Like single crossing differences, the interval dominance order (IDO) 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:Quah, J. K.-H., and B. Strulovici (2009): “Comparative Statics, Informativeness, and the Interval Dominance Order,” Econometrica, 77(6), 1949–1992. Denote . A family of regular functions obeys the interval dominance order if and only if is increasing in for all intervals .
Proof: To show the sufficiency of IDO, take any two , and assume that and . We only need to consider the case where . By definition , for all . Moreover, by IDO we have . Therefore, . Furthermore, it must be that . Otherwise, i.e., if , then by IDO we have , which contradicts that . To show the necessity of IDO, assume that there is an interval such that for all . This means that . There are two possible violations of IDO. One possibility is that . In this case, by the regularity of , the set is non-empty but does not contain which is impossible since increases in . Another possible violation of IDO occurs if but . In this case, the set either contains , which is not possible since increases in (note that in this case ) or it does not contain , which also violates monotonicity of . Q.E.D.
The next result gives useful sufficient conditions for single crossing differences and IDO.
Proposition 1:Quah, J. K.-H., and B. Strulovici (2009): “Comparative Statics, Informativeness, and the Interval Dominance Order,” Econometrica, 77(6), 1949–1992. Let be an interval of and be a family of continuously differentiable functions. (i) If, for any , there exists a number such that for all , then obey single crossing differences. (ii) If, for any , there exists a nondecreasing, strictly positive function such that for all , then obey IDO.
Application (Optimal stopping problem):Quah, J. K.-H., and B. Strulovici (2009): “Comparative Statics, Informativeness, and the Interval Dominance Order,” Econometrica, 77(6), 1949–1992; and Quah, J. K.-H., and B. Strulovici (2013): “Discounting, Values, and Decisions,” Journal of Political Economy, 121(5), 896-939. 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 (or least upper bound, or join) by and their infimum (or greatest lower bound, or meet) by .
Definition (Strong Set Order):See Veinott (1992): Lattice programming: qualitative optimization and equilibria. MS Stanford. 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. Let and , be some closed intervals in . Clearly , where is the standard ordering on , is a lattice. Therefore, as it was shown in the previous section if and only if and ; Let and , be some hyperrectangles. That is, there exist some vectors , , , in such that and , where is the natural, coordinate-wise ordering on . Note that is a lattice. Moreover, if and only if and ; Let be a space of all probability distributions with support being a subset of , endowed with the first order stochastic dominance order . Note that is a lattice. Let , denote sets of probability distributions with support and respectively. Then, with respect to if and only if and .
Definition (Quasisupermodular function):Milgrom, P., and C. Shannon (1994): “Monotone Comparative Statics,” Econometrica, 62(1), 157–180. Let be a lattice. The function is quasisupermodular (QSM) 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:Milgrom, P., and C. Shannon (1994): “Monotone Comparative Statics,” Econometrica, 62(1), 157–180. Let is a lattice, a partially ordered set, and , subsets of . Given , we denote by . Then for any and Proof: . Let , , and , . Since and , then . By quasisupermodularity, , and by the single crossing differences, . Hence . Now assume that . Then . By quasisupermodularity, , and by single crossing differences . But this contradicts that . Hence, .
. Set and . Then, and thus , which guarantees that, if , then . To show that single crossing differences also hold, set , where . Then for any guarantees that, if , then . Q.E.D.
Application (Production with multiple goods):See Milgrom, P., and J. Roberts (1990a): “The Economics of Modern Manufacturing: Technology, Strategy, and Organization,” American Economic Review, 80(3), 511–528; or Topkis, D. M. (1979): “Equilibrium Points in Nonzero-Sum n-Person Submodular Games,” SIAM Journal of Control and Optimization, 17, 773–787. Let denote the vector of inputs (drawn from a sublattice of ) of a profit-maximizing firm, be the vector of input prices, and the revenue function mapping input vector to revenue (in ). The firm's profit is . For any , , , is increasing in . Hence, has increasing differences (and so it obeys single crossing 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 (by definition) . A basic property of consumer demand is normality, which means (in the case where demand is unique) that the demand of each good is increasing in wealth. Theorem 3 cannot be straightforwardly applied to obtain conditions for normality, because if (when is derived from the Euclidean order). In this case, the following result holds.