1052:
758:
1047:{\displaystyle b_{11}>0,{\begin{vmatrix}b_{11}&b_{12}\\b_{21}&b_{22}\end{vmatrix}}>0,\ldots ,{\begin{vmatrix}b_{11}&b_{12}&\dots &b_{1n}\\b_{21}&b_{22}&\dots &b_{2n}\\\vdots &\vdots &\ddots &\vdots \\b_{n1}&b_{n2}&\dots &b_{nn}\end{vmatrix}}>0}
380:
722:
136:
174:
650:
686:
576:
202:
530:
751:
250:
224:
602:
413:
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448:
1076:
81:
270:
1342:
691:
1168:
89:
144:
1239:
1101:
607:
28:
663:
535:
179:
1096:
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491:
734:
233:
207:
1248:
728:
581:
40:
24:
1321:
1268:
1222:
1205:
1188:
388:
457:
426:
36:
8:
16:
Result in mathematical economics on existence of a non-negative equilibrium output vector
1126:
Hawkins, David; Simon, Herbert A. (1949). "Some
Conditions of Macroeconomic Stability".
1157:
1137:
1072:
264:, who empirically estimated it in the 1940s. Together, they describe a system in which
50:
1308:
1164:
44:
1080:
1064:
1060:
261:
32:
1288:
227:
35:, that guarantees the existence of a non-negative output vector that solves the
1300:
1336:
1296:
1292:
1312:
485:. Rearranged and written in vector notation, this gives the first equation.
1128:
375:{\displaystyle \sum _{j=1}^{n}a_{ij}x_{j}+d_{i}=x_{i}\quad i=1,2,\ldots ,n}
1328:(Second ed.). New York: Cambridge University Press. pp. 359–409.
1141:
1291:(1960). "Matrices with Dominant Diagonals and Economic Theory". In
1322:"Frobenius Theorems, Dominant Diagonal Matrices, and Applications"
1242:[On Some Properties of Matrices with Positive Elements]
1224:
Mathematics for
Stability and Optimization of Economic Systems
1207:
Equilibrium, Stability, and Growth: A Multi-sectoral
Analysis
1240:"О некоторых свойствах матриц с положительными элементами"
656:
states that the following two conditions are equivalent
865:
786:
1067:(1968), or Murata (1977). Condition (ii) is known as
761:
737:
694:
666:
610:
584:
538:
494:
460:
429:
391:
273:
236:
210:
182:
147:
92:
53:
1163:(2nd ed.). New York: Oxford University Press.
717:{\displaystyle \mathbf {B} \cdot \mathbf {x} >0}
1210:. London: Oxford University Press. pp. 15–17.
1156:
1046:
745:
716:
680:
644:
596:
570:
524:
473:
442:
407:
374:
244:
218:
196:
168:
130:
75:
1273:. Vol. 2. New York: Chelsea. pp. 71–73.
154:
1334:
131:{\displaystyle \cdot \mathbf {x} =\mathbf {d} }
1307:. Stanford University Press. pp. 47–62.
1237:
1125:
47:. More precisely, it states a condition for
1305:Mathematical Methods in the Social Sciences
1227:. New York: Academic Press. pp. 52–53.
1266:
1203:
1182:
1180:
169:{\displaystyle \mathbf {\hat {x}} \geq 0}
1319:
1287:
1154:
419:th good used to produce one unit of the
1186:
481:is the amount of final demand for good
1335:
1220:
1177:
1190:Convex Structures and Economic Theory
645:{\displaystyle b_{ij}\leq 0,i\neq j}
83:under which the input–output system
13:
1281:
681:{\displaystyle \mathbf {x} \geq 0}
571:{\displaystyle \mathbf {B} =\left}
197:{\displaystyle \mathbf {d} \geq 0}
14:
1354:
1193:. Academic Press. pp. 90–92.
739:
704:
696:
668:
540:
518:
507:
499:
238:
212:
184:
151:
124:
116:
105:
97:
66:
58:
344:
1260:
1231:
1214:
1197:
1148:
1119:
511:
495:
109:
93:
70:
54:
1:
1112:
525:{\displaystyle =\mathbf {B} }
1238:Kotelyanskiĭ, D. M. (1952).
746:{\displaystyle \mathbf {B} }
245:{\displaystyle \mathbf {A} }
219:{\displaystyle \mathbf {I} }
7:
1090:
10:
1359:
1343:Theorems in linear algebra
1267:Gantmacher, Felix (1959).
1204:Morishima, Michio (1964).
1187:Nikaido, Hukukane (1968).
1155:Leontief, Wassily (1986).
1097:Diagonally dominant matrix
1079:, as it is referred to by
597:{\displaystyle n\times n}
1320:Takayama, Akira (1985).
1102:Perron–Frobenius theorem
1073:independently discovered
729:leading principal minors
727:(ii) All the successive
1069:Hawkins–Simon condition
21:Hawkins–Simon condition
1326:Mathematical Economics
1270:The Theory of Matrices
1221:Murata, Yasuo (1977).
1159:Input-Output Economics
1048:
753:are positive, that is
747:
718:
682:
646:
598:
572:
526:
475:
454:th good produced, and
444:
409:
408:{\displaystyle a_{ij}}
376:
294:
246:
220:
198:
170:
132:
77:
25:mathematical economics
23:refers to a result in
1107:Sylvester's criterion
1049:
748:
719:
683:
654:Hawkins–Simon theorem
647:
599:
573:
527:
476:
474:{\displaystyle d_{i}}
450:is the amount of the
445:
443:{\displaystyle x_{j}}
415:is the amount of the
410:
377:
274:
247:
221:
199:
171:
133:
78:
759:
735:
692:
664:
660:(i) There exists an
608:
582:
536:
492:
458:
427:
389:
271:
234:
208:
180:
145:
90:
51:
45:demand equals supply
1071:. This theorem was
254:input–output matrix
1085:Kotelyanskiĭ lemma
1077:David Kotelyanskiĭ
1044:
1032:
839:
743:
714:
678:
642:
594:
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522:
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440:
405:
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242:
216:
194:
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128:
73:
41:input–output model
1293:Arrow, Kenneth J.
1059:For a proof, see
157:
1350:
1329:
1316:
1289:McKenzie, Lionel
1275:
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1184:
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1152:
1146:
1145:
1136:(3/4): 245–248.
1123:
1081:Felix Gantmacher
1053:
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262:Wassily Leontief
251:
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119:
108:
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82:
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79:
76:{\displaystyle }
74:
69:
61:
39:relation in the
33:Herbert A. Simon
27:, attributed to
1358:
1357:
1353:
1352:
1351:
1349:
1348:
1347:
1333:
1332:
1301:Suppes, Patrick
1284:
1282:Further reading
1279:
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278:
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269:
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258:Leontief matrix
237:
235:
232:
231:
228:identity matrix
211:
209:
206:
205:
183:
181:
178:
177:
149:
148:
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143:
142:
141:has a solution
123:
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96:
91:
88:
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65:
57:
52:
49:
48:
17:
12:
11:
5:
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1317:
1297:Karlin, Samuel
1283:
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1117:
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341:
337:
333:
328:
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252:is called the
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139:
138:
126:
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118:
114:
111:
107:
103:
99:
95:
72:
68:
64:
60:
56:
15:
9:
6:
4:
3:
2:
1355:
1344:
1341:
1340:
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1323:
1318:
1314:
1310:
1306:
1302:
1298:
1294:
1290:
1286:
1285:
1272:
1271:
1263:
1256:(3): 497–506.
1255:
1251:
1250:
1241:
1234:
1226:
1225:
1217:
1209:
1208:
1200:
1192:
1191:
1183:
1181:
1172:
1170:0-19-503525-9
1166:
1161:
1160:
1151:
1143:
1139:
1135:
1131:
1130:
1122:
1118:
1108:
1105:
1103:
1100:
1098:
1095:
1094:
1088:
1086:
1082:
1078:
1074:
1070:
1066:
1062:
1041:
1038:
1033:
1025:
1022:
1018:
1012:
1005:
1002:
998:
990:
987:
983:
975:
970:
965:
960:
951:
948:
944:
938:
931:
927:
919:
915:
905:
902:
898:
892:
885:
881:
873:
869:
862:
857:
854:
851:
848:
845:
840:
832:
828:
820:
816:
806:
802:
794:
790:
783:
778:
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767:
763:
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730:
726:
711:
708:
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675:
672:
659:
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624:
619:
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612:
591:
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564:
559:
556:
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548:
544:
514:
503:
486:
484:
466:
462:
453:
435:
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418:
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369:
366:
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360:
357:
354:
351:
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313:
309:
303:
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296:
290:
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267:
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259:
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229:
191:
188:
163:
160:
120:
112:
101:
86:
85:
84:
62:
46:
42:
38:
34:
30:
29:David Hawkins
26:
22:
1325:
1304:
1269:
1262:
1253:
1247:
1233:
1223:
1216:
1206:
1199:
1189:
1158:
1150:
1133:
1129:Econometrica
1127:
1121:
1084:
1068:
1058:
653:
604:matrix with
487:
482:
451:
420:
416:
384:
257:
253:
140:
20:
18:
652:. Then the
37:equilibrium
1113:References
688:such that
1061:Morishima
1013:…
976:⋮
971:⋱
966:⋮
961:⋮
939:…
893:…
855:…
701:⋅
673:≥
637:≠
625:≤
589:×
504:−
423:th good,
364:…
276:∑
189:≥
161:≥
155:^
113:⋅
102:−
63:−
1337:Category
1313:25792438
1303:(eds.).
1249:Mat. Sb.
1091:See also
1063:(1964),
532:, where
176:for any
1142:1905526
1065:Nikaido
488:Define
226:is the
204:. Here
1311:
1167:
1140:
578:is an
385:where
260:after
43:where
1252:N.S.
1244:(PDF)
1138:JSTOR
1309:OCLC
1165:ISBN
1039:>
846:>
773:>
709:>
230:and
31:and
19:The
1083:as
1075:by
731:of
256:or
1339::
1324:.
1299:;
1295:;
1254:31
1246:.
1179:^
1134:17
1132:.
1087:.
932:22
920:21
886:12
874:11
833:22
821:21
807:12
795:11
768:11
1315:.
1173:.
1144:.
1042:0
1034:|
1026:n
1023:n
1019:b
1006:2
1003:n
999:b
991:1
988:n
984:b
952:n
949:2
945:b
928:b
916:b
906:n
903:1
899:b
882:b
870:b
863:|
858:,
852:,
849:0
841:|
829:b
817:b
803:b
791:b
784:|
779:,
776:0
764:b
740:B
724:.
712:0
705:x
697:B
676:0
669:x
640:j
634:i
631:,
628:0
620:j
617:i
613:b
592:n
586:n
565:]
560:j
557:i
553:b
549:[
545:=
541:B
519:B
515:=
512:]
508:A
500:I
496:[
483:i
467:i
463:d
452:j
436:j
432:x
421:j
417:i
401:j
398:i
394:a
370:n
367:,
361:,
358:2
355:,
352:1
349:=
346:i
340:i
336:x
332:=
327:i
323:d
319:+
314:j
310:x
304:j
301:i
297:a
291:n
286:1
283:=
280:j
239:A
213:I
192:0
185:d
164:0
152:x
125:d
121:=
117:x
110:]
106:A
98:I
94:[
71:]
67:A
59:I
55:[
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