The Milgrom-Shannon single crossing property is essential for monotone comparative statics of optimization problems and noncooperative games. This paper formulates conditions for an additively separable objective function to satisfy the single crossing property. One component of the objective function is assumed to allow a monotone concave transformation with increasing differences, and to be nondecreasing in the parameter variable. The other component is assumed to exhibit increasing differences, and to be nonincreasing in the choice variable. As an application, I prove existence of an isotone pure strategy Nash equilibrium in a Cournot duopoly with logconcave demand, affiliated types, and nondecreasing costs

Ewerhart, Christian

ZORA

Monotone comparative statics with separable objective functions

The sum of a supermodular function, assumed nondecreasing in the choice variable, and of a 'concavely supermodularizable' function, assumed nonincreasing in the parameter variable, satisfies the Milgrom-Shannon (1994, Monotone comparative statics, Econometrica 62, 157-180) single crossing condition. As an application, I prove existence of a pure strategy Nash equilibrium in a Cournot duopoly with logconcave demand, affiliated types, and nondecreasing costs

     Institute for Empirical Research in Economics University of Zurich  Working Paper Series ISSN 1424-0459       Working Paper No. 472 Monotone comparative statics with separable objective functions Christian Ewerhart January 2010   Monotone comparative statics with separable objective functionsChristian EwerhartaaUniversity of Zurich; postal address: Winterthurerstrasse 30, 8006 Zurich,Switzerland; e-mail: christian.ewerhart@iew.uzh.ch; phone: 41-44-6343733;fax: 41-44-6344978.Abstract. The sum of a supermodular function, assumed nondecreas-ing in the choice variable, and of a concavely supermodularizable func-tion, assumed nonincreasing in the parameter variable, satises the Milgrom-Shannon (1994, Monotone comparative statics, Econometrica 62, 157-180)single crossing condition. As an application, I prove existence of a purestrategy Nash equilibrium in a Cournot duopoly with logconcave demand,a¢ liated types, and nondecreasing costs.Keywords. Supermodularity; separable objective function; single crossingcondition; quantity competition.JEL-Codes. C02, C72, C65.11. IntroductionMonotone comparative statics has proven to be a useful tool in numerous eco-nomic applications (Topkis, 1978; Milgrom and Roberts, 1990). Over time,cardinal notions of supermodularity have been rened into more general, or-dinal variants (Milgrom and Shannon, 1994). However, ordinal techniquesare less straightforward to apply when the problem is characterized by a sep-arable objective function. For instance, while the sum of two supermodularfunctions is again supermodular, the sum of two logsupermodular functionsneed not be logsupermodular.1In this paper, I show that the sum of a supermodular function, assumedto be monotone increasing in the choice variable, and a concavely supermod-ularizable function, assumed to be monotone decreasing in the parametervariable, satises the Milgrom-Shannon single crossing condition. As an ap-plication, I prove existence of an isotone equilibrium in a Cournot duopolywith logconcave demand, a¢ liated types, and nondecreasing costs. This isthe rst result of this sort that both makes assumptions solely on primitivesof the model and works without negative prices.The rest of the paper is structured as follows. Section 2 denes andcharacterizes the notion of concavely increasing di¤erences. The key result ofthe paper is stated and proved in Section 3. Section 4 contains the applicationto the Cournot model.2. Concavely increasing di¤erencesThis section introduces the notion of concavely increasing di¤erences and1Cf. Athey (2002).2o¤ers a characterization that will be useful to prove the key result of thispaper. I will make use of the standard terminology and notation introducedin Milgrom and Shannon (1994). Denitions and clarications of the basicconcepts should be sought there.Denition 1. Let X be a lattice, and T be a partially ordered set. Afunction g : X  T ! R has [strict] concavely increasing di¤erencesin (x; t) if for any x0 > x00 and t0 > t00,(g(x0; t0))  (g(x00; t0)) [>](g(x0; t00))  (g(x00; t00)) (1)for some strictly increasing, concave transformation .2Obviously, any function with increasing di¤erences has concavely increasingdi¤erences. Moreover, any function that is logsupermodular on the productspace XT has concavely increasing di¤erences.3 Note, however, that thereare functions that fulll neither property and still satisfy Denition 1. In- Example1 heredeed, in Example 1, the inequality (19)   (29)  (18)   (27) fails fortransformations (y) = y and (y) = ln y, yet holds for (y) =  1=y.Denition 1 allows formulating su¢ cient conditions on a separable ob-jective function to satisfy the Milgrom-Shannon single crossing property. Toprepare the key result, I state the following characterization of concavelyincreasing di¤erences.Lemma 1. A function g has [strict] concavely increasing di¤erences in (x; t)if and only if for any x0 > x00 and t0 > t00 such that minfg(x00; t0); g(x0; t00)g 2The transformation may, but need not depend on the vector (x0; x00; t0; t00). Moreover,the domain of the transformation must be some interval.3Another example are rootsupermodular functions, as dened by Eeckhout and Kircher(forthcoming).3minfg(x00; t00); g(x0; t0)g, the inequalityg(x0; t0)  g(x00; t0) [>]g(x0; t00)  g(x00; t00) (2)holds.Proof. The proof is given for the case of nonstrict di¤erences only. The othercase is completely analogous. Only if. Assume g has concavely increasingdi¤erences in (x; t), and x x0 > x00 and t0 > t00. Write a = g(x00; t00), b =g(x00; t0), c = g(x0; t00), and d = g(x0; t0). Then there is a strictly increasing,concave transformation  such that (d)   (b)  (c)   (a). Considerminfb; cg  minfa; dg, say a  b  c. Then necessarily c  d, because  isstrictly increasing and (a) + (d)  (b) + (c). To prove (2), assume tothe contrary that d b < c a, so that with a concave and strictly increasing, (d) (b) < (c) (a), a contradiction. If. Using the same notation,assume g satises d  b  c  a provided that minfb; cg  minfa; dg. I needto nd a strictly increasing, concave  such that (b) + (c)  (a) + (d).Consider rst minfb; cg < minfa; dg. If b = c, the claim follows for anystrictly increasing . Therefore, without loss of generality, b < a; c; d. Inthis case, dene (a) = a, (c) = c, (d) = d, and (b) su¢ ciently negativeso that (b) + (c)  (a) + (d). Clearly,  can be extended to a strictlyincreasing, concave function on R. Consider now minfb; cg  minfa; dg.Then, by assumption, b + c  a + d. Clearly, in this case,  can be chosenlinear. Thus, concavely increasing di¤erences requires increasing di¤erences onlywhen o¤-diagonal entries do not undercut the minimum diagonal entry.For illustration, note that Lemma 1 implies that any monotone function,4either increasing or decreasing, that exhibits concavely increasing di¤erencesmust have increasing di¤erences.3. Separable objective functionsThe key result of the paper is the following.Theorem 1. Let X be a lattice, and T be a partially ordered set. Considerfunctions g; h : X  T ! R. Assume that g has concavely increasing di¤er-ences in (x; t) and is nondecreasing [nonincreasing] in t. Assume also thath has increasing di¤erences in (x; t) and is nonincreasing [nondecreasing] inx. Then g + h satises the single crossing condition in (x; t). If, in addi-tion, g has strict concavely increasing di¤erences and is strictly increasing[decreasing] in t, then g + h satises the strict single crossing property in(x; t).Proof. According to the denition, f = g + h satises the single crossingproperty in (x; t) if for x0 > x00 and t0 > t00, f(x0; t00)  f(x00; t00) implies thatf(x0; t0)  f(x00; t0) and f(x0; t00) > f(x00; t00) implies that f(x0; t0) > f(x00; t0).So take arbitrary x0 > x00 and t0 > t00. Imposef(x0; t00)  f(x00; t00). (3)Since h is nonincreasing in x, inequality (3) implies g(x0; t00)  g(x00; t00).Moreover, g is nondecreasing in t, so g(x00; t0)  g(x00; t00). By assumption, ghas concavely increasing di¤erences. Thus, by Lemma 1,g(x0; t0)  g(x00; t0)  g(x0; t00)  g(x00; t00). (4)But h has increasing di¤erences in (x; t), so thath(x0; t0)  h(x00; t0)  h(x0; t00)  h(x00; t00). (5)5Adding (4) and (5) term by term yieldsf(x0; t0)  f(x00; t0)  f(x0; t00)  f(x00; t00). (6)Combining this with (3), one obtainsf(x0; t0)  f(x00; t0)  0. (7)as desired. Moreover, if inequality (3) holds strictly, so does (7). This provesthe claim for nonstrict di¤erences. To prove the claim also for strict di¤er-ences, note that inequality (4) is then strict, so that inequality (3) impliesthe strict version of (7), as required by the strict single crossing property. For intuition, focus on T and X being two-element subsets of R, and being the logarithm. Clearly, the conclusion is obvious when g has actuallyincreasing di¤erences. So assume that the slope g(x0;t) g(x00;t)x0 x00 , regarded as afunction of t, strictly decreases, while the ratio g(x0;t)g(x00;t) weakly increases in t,as illustrated in Figure 1. Since g is nondecreasing in t, a moments reectionshows that this is possible only when g is strictly downward-sloping at t00.4But then, adding a function h that is nonincreasing in x implies the singlecrossing condition for the sum. Figure1 hereTo extend Theorem 1, re-order T (or, equivalently, X). E.g., assume thatg has concavely decreasing di¤erences in (x; t), which is dened in analogy toDenition 1, and that g is nonincreasing [nondecreasing] in t. Then, with hhaving decreasing di¤erences in (x; t) and being nonincreasing [nondecreas-ing] in x, it follows that g + h satises the dual single crossing condition in(x; t).4Indeed, if g were upwards sloping or at at t00, then the strictly lower slope at t0 wouldmake the ratio g(x0; t)=g(x00; t) decline strictly in t.6Another extension assumes that g has convexly increasing or decreasingdi¤erences in (x; t), where again, the notions are dened in analogy to De-nition 1. For instance, when g has convexly increasing di¤erences in (x; t)and is nondecreasing [nonincreasing] in t, and h has increasing di¤erences in(x; t) and is nondecreasing [nonincreasing] in x, then g+h satises the singlecrossing property.5Theorem 1 is readily applied to the comparative statics of optimizationproblems and noncooperative games using the results in Milgrom and Shan-non (1994).6 For instance, Amirs (1996) main result follows directly fromTheorem 1. But Theorem 1 can also be fruitfully applied to Bayesian games,as illustrated in the next section.4. ApplicationThis section deals with equilibrium existence in the Cournot duopoly witha¢ liated types. A pure strategy Nash equilibrium is known to exist re-gardless of distributional assumptions provided certainty payo¤s are sub-modular in rms actions (Vives, 1990). Under additional complementaritiesbetween actions and types, even an isotone equilibrium exists for a¢ liatedtypes (Athey, 2001, McAdams, 2003, Van Zandt and Vives, 2007). However,cardinal submodularity in actions is not a completely innocuous assumptionin the Bayesian Cournot model because uncertainty then tends to generatenegative prices (Einy et al., 2010).75Indeed, it is not di¢ cult to check that g has convexly increasing di¤erences in (x; t)if and only if  g has concavely decreasing di¤erences in (x; t), so that the claim followsfrom the rst extension.6Only when the choice set is not a chain, each term needs to be supermodular in thechoice variable to ensure the sum is quasisupermodular in x.7Indeed, Cournot prots that are submodular in actions imply Novsheks (1985) mar-7Addressing this issue, I will show now that a logconcave inverse demandand nondecreasing costs are su¢ cient for the existence of an isotone equilib-rium in a duopoly with a¢ liated costs. In contrast to prior results, negativeprices do not obtain even though all assumptions are on the primitives of themodel.Inverse demand is given by a nonincreasing function p, assumed to benonnegative and logconcave. There are two rms i = 1; 2, each receiving aprivate signal ti, referred to as the rms type, and drawn from a compactinterval Ti  R. Types are inversely a¢ liated, i.e., jointly distributed ac-cording to some a.e. logsubmodular density.8 Each rm i produces outputxi  0 at costs Ci(xi; t), where t = (ti; t i). Costs are assumed nondecreas-ing and continuous in output. Moreover, marginal costs are nonincreasingin own type, while (potentially) nondecreasing in the other rms type. It isassumed that there is an output level x above which no rm has an incentiveto operate.9 Denote rm is strategy by i = i(ti). Expected prots of arm i of type ti producing output xi readfi(xi; ti) =Zxip(xi +  i(t i))  Ci(xi; t)di(t ijti), (8)where i denotes the conditional distribution function of t i given ti. It isclaimed that fi has the single crossing property in (xi; ti) provided  i isginal revenue condition on inverse demand. The marginal revenue condition, in turn, canbe seen to be equivalent to inverse demand being a concave function of log-output. Hence,if inverse demand is declining somewhere, it must eventually cause negative prices.8The density is assumed bounded and atomless, to ensure that expected revenues andcosts exist and are nite for all type subintervals and for all nondecreasing strategies ofthe rival.9For instance, assume that xip(xi)  Ci(xi; t) <  Ci(0; t) for all i = 1; 2, t 2 Ti  T i,and xi > x.8monotone increasing. For this, write fi = gi + hi, wheregi(xi; ti) =Zxip(xi +  i(t i))di(t ijti) and (9)hi(xi; ti) =  ZCi(xi; t)di(t ijti), (10)respectively, denote expected revenues and (negatively signed) expected costs.To apply Theorem 1, note that ex post revenues xip(xi+x i) are logsubmodu-lar in (xi; x i). Hence, because  i is monotone increasing, xip(xi+ i(t i)) islogsubmodular in (xi; t i). Therefore, with inversely a¢ liated types, gi is log-supermodular in (xi; ti).10 Moreover, as p is nonincreasing,  i is monotoneincreasing, and types are inversely a¢ liated, it follows that gi is nondecreas-ing in ti. Consider now the cost term. By assumption, Ci(xi; t) is submodularin (xi; ti) and supermodular in (xi; t i). As types are inversely a¢ liated, itfollows that hi is supermodular in (xi; ti).11 Furthermore, since costs are non-decreasing in output, hi is nonincreasing in xi. Thus, fi satises the singlecrossing condition in (xi; ti) for any nondecreasing  i. It follows now fromCorollary 2.1 in Athey (2001) that an isotone pure strategy Nash equilibriumexists.ReferencesAmir, R., 1996, Cournot equilibrium and the theory of supermodular games,Games and Economic Behavior 15, 132-148.Athey, S., 2001, Single crossing properties and the existence of pure strategyequilibria in games of incomplete information, Econometrica 69, 861-889.10See the discussion following Lemma 2 in Athey (2002).11Cf. Fact (v) in Athey (2001, p. 872).9Athey, S., 2002, Monotone comparative statics under uncertainty, QuarterlyJournal of Economics 117, 187-222.Eeckhout, J., Kircher, P., forthcoming, Sorting and decentralized price com-petition, Econometrica.Einy, E., Haimanko, O., Moreno, D., Shitovitz, B., 2010, On the existence ofBayesian Cournot equilibrium, Games and Economic Behavior 68, 77-94.McAdams, D., 2003, Isotone equilibrium in games of incomplete information,Econometrica 71, 11911214.Milgrom, P., Roberts, J., 1990, Rationalizability, learning, and equilibriumin games with strategic complementarities, Econometrica 58, 1255-1277.Milgrom, P., Shannon, D., 1994, Monotone comparative statics, Economet-rica 62, 157-180.Novshek, W., 1985, On the existence of Cournot equilibrium, Review of Eco-nomic Studies 52, 85-98.Topkis, D., 1978, Minimizing a function on a lattice, Operations Research26, 305-321.Van Zandt, T., Vives, X., 2007, Monotone equilibria in Bayesian games ofstrategic complementarities, Journal of Economic Theory 134, 339360.Vives, X., 1990, Nash equilibrium with strategic complementarities, Journalof Mathematical Economics 19, 305-321.10xt0 10127 2918 19Example 1 − The tabulated function g fails to satisfy increasing diff i ( t) d l d l it b t h lerences n x; an  ogsupermo u ar y, u  as concave y increasing differences in (x;t).g(x‘‘,t‘)t‘g(x‘‘ t‘‘) t‘‘g(x‘,t‘‘)g(x‘,t‘),x‘‘ x‘Figure 1 − If the slope of g with respect to x strictly decreases in t, th ti ( ‘ t)/ ( ‘‘ t) kl i i t d i te ra o g x , g x ,  wea y ncreases n , an  g s mono one increasing in t, then g must be downwards sloping at t‘‘.

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Monotone comparative statics with separable objective functions

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