158:
63:
22:
1057:
868:
1173:
403:
This example will show how using Topkis's theorem gives the same result as using more standard tools. The advantage of using Topkis's theorem is that it can be applied to a wider class of problems than can be studied with standard economics tools.
857:
1327:
The problem with the above approach is that it relies on the differentiability of the objective function and on concavity. We could get at the same answer using Topkis's theorem in the following way. We want to show that
1518:
1458:
1052:{\displaystyle {\frac {\partial s^{\ast }(p)}{\partial p}}={\underset {{\text{negative since we assumed }}U(.){\text{ was concave in }}s}{\underbrace {\frac {-U_{sp}(s^{\ast }(p),p)}{U_{ss}(s^{\ast }(p),p)}} }}.}
1577:
1319:
385:
528:
1254:
691:
1068:
1397:
466:
623:
583:
284:. The theorem allows researchers to understand how the optimal value for a choice variable changes when a feature of the environment changes. The result states that if
703:
1361:
391:. The result is especially helpful for establishing comparative static results when the objective function is not differentiable. The result is named after
1722:
629:
is defined. Note that the optimal speed is a function of the amount of potholes. Taking the first order condition, we know that at the optimum,
73:
1712:
1673:
1463:
1406:
35:
1702:
1523:
1265:
311:
259:
241:
49:
477:
222:
131:
1717:
194:
103:
179:
172:
88:
1727:
1629:
201:
110:
1692:
1697:
208:
117:
1665:
1598:
1583:
1192:
534:
190:
99:
41:
1168:{\displaystyle {\frac {\partial s^{\ast }(p)}{\partial p}}{\overset {\text{sign}}{=}}U_{sp}(s^{\ast }(p),p).}
632:
471:
We would like to understand how the driver's speed (a choice variable) changes with the amount of potholes:
1324:
and more potholes causes less speeding. Clearly it is more reasonable to assume that they are substitutes.
411:. Going faster is desirable, but is more likely to result in a crash. There is some prevalence of potholes,
168:
1638:
273:
423:
is a parameter of the environment that is fixed from the perspective of the driver. The driver seeks to
1707:
1366:
392:
426:
1643:
852:{\displaystyle U_{ss}(s^{\ast }(p),p)(\partial s^{\ast }(p)/(\partial p))+U_{sp}(s^{\ast }(p),p)=0}
281:
592:
552:
1615:
215:
124:
1669:
1596:
Amir, Rabah (2005). "Supermodularity and
Complementarity in Economics: An Elementary Survey".
1648:
1607:
1331:
1586:
and Topkis's theorem gives the same result, but the latter does so with fewer assumptions.
305:
1686:
289:
157:
80:
62:
415:. The presence of potholes increases the probability of crashing. Note that
1627:
Topkis, Donald M. (1978). "Minimizing a
Submodular Function on a Lattice".
1652:
1619:
1399:. Note that the choice set is clearly a lattice. The cross partial of
1513:{\displaystyle {\frac {\partial ^{2}U}{\partial s\,\partial p}}<0,}
1611:
1453:{\displaystyle {\frac {\partial ^{2}U}{\partial s\,\partial p}}<0}
533:
If one wanted to solve the problem with standard tools such as the
1572:{\displaystyle {\frac {\partial s^{\ast }(p)}{\partial p}}<0}
1314:{\displaystyle {\frac {\partial s^{\ast }(p)}{\partial p}}<0}
693:. Differentiating the first order condition, with respect to
549:
is defined is convex, and that it there is a unique maximizer
380:{\displaystyle x^{*}(\theta )=\arg \max _{x\in D}f(x,\theta )}
151:
56:
15:
537:, one would have to assume that the problem is well behaved:
523:{\displaystyle {\frac {\partial s^{\ast }(p)}{\partial p}}.}
407:
A driver is driving down a highway and must choose a speed,
84:
697:
and using the implicit function theorem, we find that
1526:
1466:
1409:
1369:
1334:
1268:
1195:
1071:
871:
706:
635:
595:
555:
541:(.) is twice continuously differentiable, concave in
480:
429:
314:
1571:
1512:
1452:
1391:
1355:
1313:
1248:
1167:
1051:
851:
685:
617:
577:
522:
460:
379:
1363:is submodular (the opposite of supermodular) in
431:
344:
8:
280:is a result that is useful for establishing
89:introducing citations to additional sources
1249:{\displaystyle U_{sp}(s^{\ast }(p),p)<0}
50:Learn how and when to remove these messages
1642:
1537:
1527:
1525:
1491:
1474:
1467:
1465:
1434:
1417:
1410:
1408:
1368:
1333:
1279:
1269:
1267:
1216:
1200:
1194:
1138:
1122:
1108:
1082:
1072:
1070:
1035:
1018:
986:
970:
940:
924:
913:
911:
882:
872:
870:
819:
803:
776:
761:
727:
711:
705:
653:
640:
634:
625:is in the interior of the set over which
600:
594:
560:
554:
491:
481:
479:
434:
428:
347:
319:
313:
260:Learn how and when to remove this message
242:Learn how and when to remove this message
79:Relevant discussion may be found on the
1460:, is a sufficient condition. Hence if
686:{\displaystyle U_{s}(s^{\ast }(p),p)=0}
178:Please improve this article by adding
7:
1662:Supermodularity and Complementarity
1554:
1530:
1492:
1485:
1471:
1435:
1428:
1414:
1296:
1272:
1099:
1075:
899:
875:
784:
754:
508:
484:
14:
1723:Eponymous theorems of mathematics
31:This article has multiple issues.
1392:{\displaystyle \left(s,p\right)}
156:
72:relies largely or entirely on a
61:
20:
1020:negative since we assumed
461:{\displaystyle \max _{s}U(s,p)}
39:or discuss these issues on the
1549:
1543:
1350:
1338:
1291:
1285:
1237:
1228:
1222:
1209:
1159:
1150:
1144:
1131:
1094:
1088:
1032:
1026:
1007:
998:
992:
979:
961:
952:
946:
933:
894:
888:
840:
831:
825:
812:
793:
790:
781:
773:
767:
751:
748:
739:
733:
720:
674:
665:
659:
646:
612:
606:
572:
566:
503:
497:
455:
443:
374:
362:
331:
325:
1:
545:, that the domain over which
180:secondary or tertiary sources
1713:Optimization of ordered sets
618:{\displaystyle s^{\ast }(p)}
578:{\displaystyle s^{\ast }(p)}
1744:
1703:Theorems in lattice theory
1666:Princeton University Press
1660:Topkis, Donald M. (1998).
1037: was concave in
1599:Southern Economic Journal
1584:implicit function theorem
535:implicit function theorem
419:is a choice variable and
1718:Supermodular functions
1573:
1514:
1454:
1393:
1357:
1356:{\displaystyle U(s,p)}
1315:
1250:
1169:
1053:
853:
687:
619:
579:
524:
462:
381:
274:mathematical economics
167:relies excessively on
1653:10.1287/opre.26.2.305
1574:
1515:
1455:
1394:
1358:
1316:
1251:
1170:
1054:
854:
688:
620:
580:
525:
463:
382:
1728:Eponyms in economics
1590:Notes and references
1524:
1464:
1407:
1367:
1332:
1266:
1193:
1069:
869:
704:
633:
593:
553:
478:
427:
387:is nondecreasing in
312:
85:improve this article
1693:Comparative statics
1630:Operations Research
585:for every value of
282:comparative statics
1698:Economics theorems
1569:
1510:
1450:
1389:
1353:
1311:
1246:
1186:are substitutes,
1165:
1049:
1044:
1015:
849:
683:
615:
575:
520:
458:
439:
377:
358:
191:"Topkis's theorem"
100:"Topkis's theorem"
1675:978-0-691-03244-3
1561:
1499:
1442:
1303:
1116:
1115:
1106:
1038:
1021:
1011:
914:
912:
906:
515:
430:
343:
270:
269:
262:
252:
251:
244:
226:
150:
149:
135:
54:
1735:
1679:
1656:
1646:
1623:
1612:10.2307/20062066
1582:Hence using the
1578:
1576:
1575:
1570:
1562:
1560:
1552:
1542:
1541:
1528:
1519:
1517:
1516:
1511:
1500:
1498:
1483:
1479:
1478:
1468:
1459:
1457:
1456:
1451:
1443:
1441:
1426:
1422:
1421:
1411:
1403:being negative,
1398:
1396:
1395:
1390:
1388:
1384:
1362:
1360:
1359:
1354:
1320:
1318:
1317:
1312:
1304:
1302:
1294:
1284:
1283:
1270:
1255:
1253:
1252:
1247:
1221:
1220:
1208:
1207:
1174:
1172:
1171:
1166:
1143:
1142:
1130:
1129:
1117:
1113:
1109:
1107:
1105:
1097:
1087:
1086:
1073:
1058:
1056:
1055:
1050:
1045:
1043:
1039:
1036:
1022:
1019:
1016:
1010:
991:
990:
978:
977:
964:
945:
944:
932:
931:
915:
907:
905:
897:
887:
886:
873:
858:
856:
855:
850:
824:
823:
811:
810:
780:
766:
765:
732:
731:
719:
718:
692:
690:
689:
684:
658:
657:
645:
644:
624:
622:
621:
616:
605:
604:
584:
582:
581:
576:
565:
564:
529:
527:
526:
521:
516:
514:
506:
496:
495:
482:
467:
465:
464:
459:
438:
393:Donald M. Topkis
386:
384:
383:
378:
357:
324:
323:
278:Topkis's theorem
265:
258:
247:
240:
236:
233:
227:
225:
184:
160:
152:
145:
142:
136:
134:
93:
65:
57:
46:
24:
23:
16:
1743:
1742:
1738:
1737:
1736:
1734:
1733:
1732:
1683:
1682:
1676:
1659:
1644:10.1.1.557.5908
1626:
1595:
1592:
1553:
1533:
1529:
1522:
1521:
1484:
1470:
1469:
1462:
1461:
1427:
1413:
1412:
1405:
1404:
1374:
1370:
1365:
1364:
1330:
1329:
1295:
1275:
1271:
1264:
1263:
1212:
1196:
1191:
1190:
1134:
1118:
1098:
1078:
1074:
1067:
1066:
1017:
982:
966:
965:
936:
920:
916:
898:
878:
874:
867:
866:
815:
799:
757:
723:
707:
702:
701:
649:
636:
631:
630:
596:
591:
590:
556:
551:
550:
507:
487:
483:
476:
475:
425:
424:
401:
315:
310:
309:
266:
255:
254:
253:
248:
237:
231:
228:
185:
183:
177:
173:primary sources
161:
146:
140:
137:
94:
92:
78:
66:
25:
21:
12:
11:
5:
1741:
1739:
1731:
1730:
1725:
1720:
1715:
1710:
1705:
1700:
1695:
1685:
1684:
1681:
1680:
1674:
1657:
1637:(2): 305–321.
1624:
1606:(3): 636–660.
1591:
1588:
1568:
1565:
1559:
1556:
1551:
1548:
1545:
1540:
1536:
1532:
1509:
1506:
1503:
1497:
1494:
1490:
1487:
1482:
1477:
1473:
1449:
1446:
1440:
1437:
1433:
1430:
1425:
1420:
1416:
1387:
1383:
1380:
1377:
1373:
1352:
1349:
1346:
1343:
1340:
1337:
1322:
1321:
1310:
1307:
1301:
1298:
1293:
1290:
1287:
1282:
1278:
1274:
1257:
1256:
1245:
1242:
1239:
1236:
1233:
1230:
1227:
1224:
1219:
1215:
1211:
1206:
1203:
1199:
1176:
1175:
1164:
1161:
1158:
1155:
1152:
1149:
1146:
1141:
1137:
1133:
1128:
1125:
1121:
1112:
1104:
1101:
1096:
1093:
1090:
1085:
1081:
1077:
1060:
1059:
1048:
1042:
1034:
1031:
1028:
1025:
1014:
1009:
1006:
1003:
1000:
997:
994:
989:
985:
981:
976:
973:
969:
963:
960:
957:
954:
951:
948:
943:
939:
935:
930:
927:
923:
919:
910:
904:
901:
896:
893:
890:
885:
881:
877:
860:
859:
848:
845:
842:
839:
836:
833:
830:
827:
822:
818:
814:
809:
806:
802:
798:
795:
792:
789:
786:
783:
779:
775:
772:
769:
764:
760:
756:
753:
750:
747:
744:
741:
738:
735:
730:
726:
722:
717:
714:
710:
682:
679:
676:
673:
670:
667:
664:
661:
656:
652:
648:
643:
639:
614:
611:
608:
603:
599:
574:
571:
568:
563:
559:
531:
530:
519:
513:
510:
505:
502:
499:
494:
490:
486:
457:
454:
451:
448:
445:
442:
437:
433:
400:
397:
376:
373:
370:
367:
364:
361:
356:
353:
350:
346:
342:
339:
336:
333:
330:
327:
322:
318:
268:
267:
250:
249:
164:
162:
155:
148:
147:
83:. Please help
69:
67:
60:
55:
29:
28:
26:
19:
13:
10:
9:
6:
4:
3:
2:
1740:
1729:
1726:
1724:
1721:
1719:
1716:
1714:
1711:
1709:
1706:
1704:
1701:
1699:
1696:
1694:
1691:
1690:
1688:
1677:
1671:
1667:
1663:
1658:
1654:
1650:
1645:
1640:
1636:
1632:
1631:
1625:
1621:
1617:
1613:
1609:
1605:
1601:
1600:
1594:
1593:
1589:
1587:
1585:
1580:
1566:
1563:
1557:
1546:
1538:
1534:
1520:we know that
1507:
1504:
1501:
1495:
1488:
1480:
1475:
1447:
1444:
1438:
1431:
1423:
1418:
1402:
1385:
1381:
1378:
1375:
1371:
1347:
1344:
1341:
1335:
1325:
1308:
1305:
1299:
1288:
1280:
1276:
1262:
1261:
1260:
1243:
1240:
1234:
1231:
1225:
1217:
1213:
1204:
1201:
1197:
1189:
1188:
1187:
1185:
1181:
1162:
1156:
1153:
1147:
1139:
1135:
1126:
1123:
1119:
1110:
1102:
1091:
1083:
1079:
1065:
1064:
1063:
1046:
1040:
1029:
1023:
1012:
1004:
1001:
995:
987:
983:
974:
971:
967:
958:
955:
949:
941:
937:
928:
925:
921:
917:
908:
902:
891:
883:
879:
865:
864:
863:
846:
843:
837:
834:
828:
820:
816:
807:
804:
800:
796:
787:
777:
770:
762:
758:
745:
742:
736:
728:
724:
715:
712:
708:
700:
699:
698:
696:
680:
677:
671:
668:
662:
654:
650:
641:
637:
628:
609:
601:
597:
588:
569:
561:
557:
548:
544:
540:
536:
517:
511:
500:
492:
488:
474:
473:
472:
469:
452:
449:
446:
440:
435:
422:
418:
414:
410:
405:
398:
396:
394:
390:
371:
368:
365:
359:
354:
351:
348:
340:
337:
334:
328:
320:
316:
307:
303:
299:
295:
291:
287:
283:
279:
275:
264:
261:
246:
243:
235:
224:
221:
217:
214:
210:
207:
203:
200:
196:
193: –
192:
188:
187:Find sources:
181:
175:
174:
170:
165:This article
163:
159:
154:
153:
144:
133:
130:
126:
123:
119:
116:
112:
109:
105:
102: –
101:
97:
96:Find sources:
90:
86:
82:
76:
75:
74:single source
70:This article
68:
64:
59:
58:
53:
51:
44:
43:
38:
37:
32:
27:
18:
17:
1708:Order theory
1661:
1634:
1628:
1603:
1597:
1581:
1400:
1326:
1323:
1258:
1183:
1179:
1177:
1061:
861:
694:
626:
586:
546:
542:
538:
532:
470:
420:
416:
412:
408:
406:
402:
388:
301:
297:
293:
290:supermodular
285:
277:
271:
256:
238:
229:
219:
212:
205:
198:
186:
166:
138:
128:
121:
114:
107:
95:
71:
47:
40:
34:
33:Please help
30:
1259:and hence
1687:Categories
399:An example
202:newspapers
169:references
111:newspapers
36:improve it
1639:CiteSeerX
1555:∂
1539:∗
1531:∂
1493:∂
1486:∂
1472:∂
1436:∂
1429:∂
1415:∂
1297:∂
1281:∗
1273:∂
1218:∗
1140:∗
1100:∂
1084:∗
1076:∂
1013:⏟
988:∗
942:∗
918:−
900:∂
884:∗
876:∂
862:or that
821:∗
785:∂
763:∗
755:∂
729:∗
655:∗
602:∗
589:and that
562:∗
509:∂
493:∗
485:∂
372:θ
352:∈
341:
329:θ
321:∗
81:talk page
42:talk page
1620:20062066
232:May 2014
141:May 2014
308:, then
306:lattice
300:), and
216:scholar
125:scholar
1672:
1641:
1618:
218:
211:
204:
197:
189:
127:
120:
113:
106:
98:
1616:JSTOR
1062:So,
304:is a
223:JSTOR
209:books
132:JSTOR
118:books
1670:ISBN
1564:<
1502:<
1445:<
1306:<
1241:<
1182:and
1114:sign
292:in (
195:news
104:news
1649:doi
1608:doi
1178:If
432:max
345:max
338:arg
288:is
272:In
171:to
87:by
1689::
1668:.
1664:.
1647:.
1635:26
1633:.
1614:.
1604:71
1602:.
1579:.
468:.
395:.
276:,
182:.
45:.
1678:.
1655:.
1651::
1622:.
1610::
1567:0
1558:p
1550:)
1547:p
1544:(
1535:s
1508:,
1505:0
1496:p
1489:s
1481:U
1476:2
1448:0
1439:p
1432:s
1424:U
1419:2
1401:U
1386:)
1382:p
1379:,
1376:s
1372:(
1351:)
1348:p
1345:,
1342:s
1339:(
1336:U
1309:0
1300:p
1292:)
1289:p
1286:(
1277:s
1244:0
1238:)
1235:p
1232:,
1229:)
1226:p
1223:(
1214:s
1210:(
1205:p
1202:s
1198:U
1184:p
1180:s
1163:.
1160:)
1157:p
1154:,
1151:)
1148:p
1145:(
1136:s
1132:(
1127:p
1124:s
1120:U
1111:=
1103:p
1095:)
1092:p
1089:(
1080:s
1047:.
1041:s
1033:)
1030:.
1027:(
1024:U
1008:)
1005:p
1002:,
999:)
996:p
993:(
984:s
980:(
975:s
972:s
968:U
962:)
959:p
956:,
953:)
950:p
947:(
938:s
934:(
929:p
926:s
922:U
909:=
903:p
895:)
892:p
889:(
880:s
847:0
844:=
841:)
838:p
835:,
832:)
829:p
826:(
817:s
813:(
808:p
805:s
801:U
797:+
794:)
791:)
788:p
782:(
778:/
774:)
771:p
768:(
759:s
752:(
749:)
746:p
743:,
740:)
737:p
734:(
725:s
721:(
716:s
713:s
709:U
695:p
681:0
678:=
675:)
672:p
669:,
666:)
663:p
660:(
651:s
647:(
642:s
638:U
627:s
613:)
610:p
607:(
598:s
587:p
573:)
570:p
567:(
558:s
547:s
543:s
539:U
518:.
512:p
504:)
501:p
498:(
489:s
456:)
453:p
450:,
447:s
444:(
441:U
436:s
421:p
417:s
413:p
409:s
389:θ
375:)
369:,
366:x
363:(
360:f
355:D
349:x
335:=
332:)
326:(
317:x
302:D
298:θ
296:,
294:x
286:f
263:)
257:(
245:)
239:(
234:)
230:(
220:·
213:·
206:·
199:·
176:.
143:)
139:(
129:·
122:·
115:·
108:·
91:.
77:.
52:)
48:(
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
