77:
this would result in a price less than average cost, and the firm could not survive without subsidy. The Ramsey problem is to decide exactly how much to raise each product's price above its marginal cost so the firm's revenue equals its total cost. If there is just one product, the problem is simple: raise the price to where it equals average cost. If there are two products, there is leeway to raise one product's price more and the other's less, so long as the firm can break even overall.
1099:
93:
in order to maximize overall social welfare. Customers sometimes object to it on that basis, since they care about their own individual welfare, not social welfare. Customers who are charged more may consider unfair, especially they, with less elastic demand, would say they "need" the good more. In such situations regulators may further limit an operator’s ability to adopt Ramsey prices.
843:
87:
In practice, government regulators are concerned with more than maximizing the sum of producer and consumer surplus. They may wish to put more weight on the surplus of politically powerful consumers, or they may wish to help the poor by putting more weight on their surplus. Moreover, many people will
92:
because the two products with different elasticities of demand are one physically identical product sold to two different groups of customers, e.g., electricity to residential customers and to commercial customers. Ramsey pricing says to charge whichever group has less elastic demand a higher price
76:
In a first-best world, without the need to earn enough revenue to cover fixed costs, the optimal solution would be to set the price for each product equal to its marginal cost. If the average cost curve is declining where the demand curve crosses it however, as happens when the fixed cost is large,
1094:{\displaystyle {\begin{aligned}p_{i}-C_{i}\left(\mathbf {q} \right)&=-\lambda \left({\frac {\partial R}{\partial q_{i}}}-C_{i}\left(\mathbf {q} \right)\right)\\&=-\lambda \left(p_{i}\left(1-{\frac {1}{\mathrm {Elasticity} _{i}}}\right)-C_{i}\left(\mathbf {q} \right)\right)\end{aligned}}}
67:
earns negative profits if it sets price equals to marginal cost, so it must set prices for some or all of the products it sells to above marginal cost if it is to be viable without government subsidies. Ramsey pricing says to mark up most the goods with the least elastic (that is, least
635:
1449:
1276:
483:
493:
806:
243:
738:. Typically, the fixed value is zero, which is to say that the regulator wants to maximize welfare subject to the constraint that the firm not lose money. The constraint can be stated generally as:
1499:
848:
155:
677:
353:
1336:
1161:
1614:
835:
1575:
392:
1122:
736:
709:
1328:
301:
270:
1549:
1299:
1519:
80:
The principle is applicable to pricing of goods that the government is the sole supplier of (public utilities) or regulation of natural monopolies, such as
88:
see Ramsey pricing as unfair, especially if they do not understand why it maximizes total surplus. In some contexts, Ramsey pricing is a form of
1628:, one must increase prices to rigid and elastic demands/supplies in the same proportion, in relation to the prices that would be charged at the
1521:
is again inversely proportional to the elasticity of demand. Note that the Ramsey mark-up is smaller than the ordinary monopoly markup of the
397:
84:
firms, where it is efficient for only one firm to operate but the government regulates its prices so it does not earn above-market profits.
630:{\displaystyle W\left(\mathbf {p,q} \right)=\sum _{i}\left(\int \limits _{0}^{q_{i}(p_{i})}p_{i}(q)dq\right)-C\left(\mathbf {q} \right).}
1616:
is non-binding). The Ramsey-price setting monopoly is in a second-best equilibrium, between ordinary monopoly and perfect competition.
1700:
744:
160:
1764:
1641:
1457:
1759:
1624:
An easier way to solve this problem in a two-output context is the Ramsey condition. According to Ramsey, as to minimize
104:
303:
is the price. Suppose that the products are sold in separate markets so demands are independent, and demand for good
48:
44:
643:
1629:
29:
815:
technique to yield the optimal output values, and backing out the optimal prices. The first order conditions on
1754:
310:
1444:{\displaystyle {\frac {p_{i}-C_{i}\left(\mathbf {q} \right)}{p_{i}}}={\frac {k}{\mathrm {Elasticity} _{i}}}}
1271:{\displaystyle \mathrm {Elasticity} _{i}=-{\frac {\partial q_{i}}{\partial p_{i}}}{\frac {p_{i}}{q_{i}}}}
89:
59:: the more elastic the demand or supply, the smaller the optimal tax. The rule was later applied by
1703:“Tariff Design: Economics of Tariff Design – Deviations from Marginal Cost Pricing: Ramsey Pricing”
812:
1580:
818:
40:(the sum of producer and consumer surplus) while earning enough revenue to cover its fixed costs.
1730:
1683:
1554:
358:
1107:
714:
682:
81:
37:
1722:
1675:
64:
1306:
279:
248:
1625:
52:
1528:
1281:
1504:
60:
1748:
1769:
1646:
1522:
56:
43:
Under Ramsey pricing, the price markup over marginal cost is inverse to the
63:(1956) to natural monopolies (industries with decreasing average cost). A
51:: the more elastic the product's demand or supply, the smaller the markup.
33:
478:{\displaystyle R\left(\mathbf {p,q} \right)=\sum _{i}p_{i}q_{i}(p_{i}).}
1734:
1687:
1713:
Ramsey, Frank P. (1927). "A Contribution to the Theory of
Taxation".
1666:
Ramsey, Frank P. (1927). "A Contribution to the Theory of
Taxation".
36:
should charge for the various products it sells in order to maximize
1726:
1679:
1501:. That is, the price margin compared to marginal cost for good
801:{\displaystyle R(\mathbf {p,q} )-C(\mathbf {q} )\geq \Pi ^{*}}
238:{\displaystyle C(q_{1},q_{2},\ldots ,q_{N})=C(\mathbf {q} ),}
101:
Consider the problem of a regulator seeking to set prices
679:
by choice of the subject to the requirement that profit
1494:{\displaystyle k={\frac {\lambda }{1+\lambda }}<1.}
1583:
1557:
1531:
1507:
1460:
1339:
1309:
1284:
1164:
1110:
846:
821:
747:
717:
685:
646:
496:
400:
361:
313:
282:
251:
163:
107:
1608:
1569:
1543:
1513:
1493:
1443:
1322:
1293:
1270:
1116:
1093:
829:
800:
730:
703:
671:
629:
477:
386:
347:
295:
264:
237:
149:
150:{\displaystyle \left(p_{1},\ldots ,p_{N}\right)}
32:policy problem concerning what prices a public
1701:Body of Knowledge on Infrastructure Regulation
8:
672:{\displaystyle W\left(\mathbf {p,q} \right)}
1632:solution (price equal to marginal cost).
1588:
1582:
1556:
1530:
1506:
1467:
1459:
1433:
1401:
1395:
1384:
1370:
1360:
1347:
1340:
1338:
1314:
1308:
1283:
1260:
1250:
1244:
1235:
1220:
1210:
1198:
1166:
1163:
1109:
1073:
1063:
1043:
1011:
1005:
988:
949:
939:
923:
905:
878:
868:
855:
847:
845:
822:
820:
792:
777:
754:
746:
722:
716:
684:
654:
645:
615:
579:
564:
551:
546:
541:
526:
504:
495:
463:
450:
440:
430:
408:
399:
366:
360:
332:
318:
312:
287:
281:
256:
250:
224:
206:
187:
174:
162:
157:for a multiproduct monopolist with costs
136:
117:
106:
348:{\displaystyle q_{i}\left(p_{i}\right),}
1658:
1278:is the elasticity of demand for good
811:This problem may be solved using the
7:
68:price-sensitive) demand or supply.
1585:
1429:
1426:
1423:
1420:
1417:
1414:
1411:
1408:
1405:
1402:
1228:
1213:
1194:
1191:
1188:
1185:
1182:
1179:
1176:
1173:
1170:
1167:
1039:
1036:
1033:
1030:
1027:
1024:
1021:
1018:
1015:
1012:
916:
908:
789:
719:
686:
55:found this 1927 in the context of
14:
1371:
1074:
950:
879:
823:
778:
761:
758:
755:
661:
658:
655:
616:
511:
508:
505:
415:
412:
409:
225:
97:Formal presentation and solution
1577:(the fixed-profit requirement,
1137:) is the partial derivative of
782:
774:
765:
751:
591:
585:
570:
557:
469:
456:
378:
372:
229:
221:
212:
167:
1:
355:with inverse demand function
1609:{\displaystyle \Pi ^{*}=R-C}
830:{\displaystyle \mathbf {q} }
640:The problem is to maximize
1786:
1570:{\displaystyle \lambda =1}
1124:is a Lagrange multiplier,
487:Total welfare is given by
49:Price elasticity of supply
45:price elasticity of demand
1642:Amoroso–Robinson relation
387:{\displaystyle p_{i}(q).}
1117:{\displaystyle \lambda }
731:{\displaystyle \Pi ^{*}}
704:{\displaystyle \Pi =R-C}
1330:and rearranging yields
711:equal some fixed value
1765:Mathematical economics
1610:
1571:
1545:
1515:
1495:
1445:
1324:
1295:
1272:
1118:
1095:
831:
802:
732:
705:
673:
631:
574:
479:
388:
349:
297:
272:is the output of good
266:
239:
151:
26:Ramsey–Boiteux pricing
1611:
1572:
1546:
1516:
1496:
1446:
1325:
1323:{\displaystyle p_{i}}
1296:
1273:
1119:
1096:
832:
803:
733:
706:
674:
632:
537:
480:
389:
350:
298:
296:{\displaystyle p_{i}}
267:
265:{\displaystyle q_{i}}
240:
152:
1760:Monopoly (economics)
1715:The Economic Journal
1668:The Economic Journal
1581:
1555:
1529:
1505:
1458:
1337:
1307:
1282:
1162:
1108:
844:
819:
745:
715:
683:
644:
494:
398:
359:
311:
280:
249:
161:
105:
90:price discrimination
1544:{\displaystyle k=1}
813:Lagrange multiplier
1606:
1567:
1541:
1511:
1491:
1441:
1320:
1294:{\displaystyle i.}
1291:
1268:
1145:) with respect to
1114:
1091:
1089:
827:
798:
728:
701:
669:
627:
531:
475:
435:
384:
345:
293:
262:
235:
147:
82:telecommunications
1626:deadweight losses
1514:{\displaystyle i}
1483:
1439:
1390:
1266:
1242:
1049:
930:
522:
426:
394:Total revenue is
1777:
1739:
1738:
1710:
1704:
1698:
1692:
1691:
1663:
1620:Ramsey condition
1615:
1613:
1612:
1607:
1593:
1592:
1576:
1574:
1573:
1568:
1550:
1548:
1547:
1542:
1520:
1518:
1517:
1512:
1500:
1498:
1497:
1492:
1484:
1482:
1468:
1450:
1448:
1447:
1442:
1440:
1438:
1437:
1432:
1396:
1391:
1389:
1388:
1379:
1378:
1374:
1365:
1364:
1352:
1351:
1341:
1329:
1327:
1326:
1321:
1319:
1318:
1300:
1298:
1297:
1292:
1277:
1275:
1274:
1269:
1267:
1265:
1264:
1255:
1254:
1245:
1243:
1241:
1240:
1239:
1226:
1225:
1224:
1211:
1203:
1202:
1197:
1123:
1121:
1120:
1115:
1100:
1098:
1097:
1092:
1090:
1086:
1082:
1081:
1077:
1068:
1067:
1055:
1051:
1050:
1048:
1047:
1042:
1006:
993:
992:
966:
962:
958:
957:
953:
944:
943:
931:
929:
928:
927:
914:
906:
886:
882:
873:
872:
860:
859:
836:
834:
833:
828:
826:
807:
805:
804:
799:
797:
796:
781:
764:
737:
735:
734:
729:
727:
726:
710:
708:
707:
702:
678:
676:
675:
670:
668:
664:
636:
634:
633:
628:
623:
619:
604:
600:
584:
583:
573:
569:
568:
556:
555:
545:
530:
518:
514:
484:
482:
481:
476:
468:
467:
455:
454:
445:
444:
434:
422:
418:
393:
391:
390:
385:
371:
370:
354:
352:
351:
346:
341:
337:
336:
323:
322:
302:
300:
299:
294:
292:
291:
271:
269:
268:
263:
261:
260:
244:
242:
241:
236:
228:
211:
210:
192:
191:
179:
178:
156:
154:
153:
148:
146:
142:
141:
140:
122:
121:
65:natural monopoly
57:Optimal taxation
1785:
1784:
1780:
1779:
1778:
1776:
1775:
1774:
1755:Economic policy
1745:
1744:
1743:
1742:
1727:10.2307/2222721
1712:
1711:
1707:
1699:
1695:
1680:10.2307/2222721
1665:
1664:
1660:
1655:
1638:
1622:
1584:
1579:
1578:
1553:
1552:
1527:
1526:
1503:
1502:
1472:
1456:
1455:
1400:
1380:
1366:
1356:
1343:
1342:
1335:
1334:
1310:
1305:
1304:
1280:
1279:
1256:
1246:
1231:
1227:
1216:
1212:
1165:
1160:
1159:
1154:, evaluated at
1153:
1132:
1106:
1105:
1088:
1087:
1069:
1059:
1010:
998:
994:
984:
983:
979:
964:
963:
945:
935:
919:
915:
907:
904:
900:
887:
874:
864:
851:
842:
841:
817:
816:
788:
743:
742:
718:
713:
712:
681:
680:
650:
642:
641:
611:
575:
560:
547:
536:
532:
500:
492:
491:
459:
446:
436:
404:
396:
395:
362:
357:
356:
328:
324:
314:
309:
308:
283:
278:
277:
252:
247:
246:
202:
183:
170:
159:
158:
132:
113:
112:
108:
103:
102:
99:
74:
53:Frank P. Ramsey
12:
11:
5:
1783:
1781:
1773:
1772:
1767:
1762:
1757:
1747:
1746:
1741:
1740:
1721:(145): 47–61.
1705:
1693:
1674:(145): 47–61.
1657:
1656:
1654:
1651:
1650:
1649:
1644:
1637:
1634:
1621:
1618:
1605:
1602:
1599:
1596:
1591:
1587:
1566:
1563:
1560:
1540:
1537:
1534:
1510:
1490:
1487:
1481:
1478:
1475:
1471:
1466:
1463:
1452:
1451:
1436:
1431:
1428:
1425:
1422:
1419:
1416:
1413:
1410:
1407:
1404:
1399:
1394:
1387:
1383:
1377:
1373:
1369:
1363:
1359:
1355:
1350:
1346:
1317:
1313:
1290:
1287:
1263:
1259:
1253:
1249:
1238:
1234:
1230:
1223:
1219:
1215:
1209:
1206:
1201:
1196:
1193:
1190:
1187:
1184:
1181:
1178:
1175:
1172:
1169:
1149:
1128:
1113:
1102:
1101:
1085:
1080:
1076:
1072:
1066:
1062:
1058:
1054:
1046:
1041:
1038:
1035:
1032:
1029:
1026:
1023:
1020:
1017:
1014:
1009:
1004:
1001:
997:
991:
987:
982:
978:
975:
972:
969:
967:
965:
961:
956:
952:
948:
942:
938:
934:
926:
922:
918:
913:
910:
903:
899:
896:
893:
890:
888:
885:
881:
877:
871:
867:
863:
858:
854:
850:
849:
825:
809:
808:
795:
791:
787:
784:
780:
776:
773:
770:
767:
763:
760:
757:
753:
750:
725:
721:
700:
697:
694:
691:
688:
667:
663:
660:
657:
653:
649:
638:
637:
626:
622:
618:
614:
610:
607:
603:
599:
596:
593:
590:
587:
582:
578:
572:
567:
563:
559:
554:
550:
544:
540:
535:
529:
525:
521:
517:
513:
510:
507:
503:
499:
474:
471:
466:
462:
458:
453:
449:
443:
439:
433:
429:
425:
421:
417:
414:
411:
407:
403:
383:
380:
377:
374:
369:
365:
344:
340:
335:
331:
327:
321:
317:
290:
286:
259:
255:
234:
231:
227:
223:
220:
217:
214:
209:
205:
201:
198:
195:
190:
186:
182:
177:
173:
169:
166:
145:
139:
135:
131:
128:
125:
120:
116:
111:
98:
95:
73:
70:
61:Marcel Boiteux
38:social welfare
22:Ramsey pricing
18:Ramsey problem
13:
10:
9:
6:
4:
3:
2:
1782:
1771:
1768:
1766:
1763:
1761:
1758:
1756:
1753:
1752:
1750:
1736:
1732:
1728:
1724:
1720:
1716:
1709:
1706:
1702:
1697:
1694:
1689:
1685:
1681:
1677:
1673:
1669:
1662:
1659:
1652:
1648:
1645:
1643:
1640:
1639:
1635:
1633:
1631:
1627:
1619:
1617:
1603:
1600:
1597:
1594:
1589:
1564:
1561:
1558:
1538:
1535:
1532:
1524:
1508:
1488:
1485:
1479:
1476:
1473:
1469:
1464:
1461:
1434:
1397:
1392:
1385:
1381:
1375:
1367:
1361:
1357:
1353:
1348:
1344:
1333:
1332:
1331:
1315:
1311:
1301:
1288:
1285:
1261:
1257:
1251:
1247:
1236:
1232:
1221:
1217:
1207:
1204:
1199:
1157:
1152:
1148:
1144:
1140:
1136:
1131:
1127:
1111:
1083:
1078:
1070:
1064:
1060:
1056:
1052:
1044:
1007:
1002:
999:
995:
989:
985:
980:
976:
973:
970:
968:
959:
954:
946:
940:
936:
932:
924:
920:
911:
901:
897:
894:
891:
889:
883:
875:
869:
865:
861:
856:
852:
840:
839:
838:
814:
793:
785:
771:
768:
748:
741:
740:
739:
723:
698:
695:
692:
689:
665:
651:
647:
624:
620:
612:
608:
605:
601:
597:
594:
588:
580:
576:
565:
561:
552:
548:
542:
538:
533:
527:
523:
519:
515:
501:
497:
490:
489:
488:
485:
472:
464:
460:
451:
447:
441:
437:
431:
427:
423:
419:
405:
401:
381:
375:
367:
363:
342:
338:
333:
329:
325:
319:
315:
306:
288:
284:
275:
257:
253:
232:
218:
215:
207:
203:
199:
196:
193:
188:
184:
180:
175:
171:
164:
143:
137:
133:
129:
126:
123:
118:
114:
109:
96:
94:
91:
85:
83:
78:
71:
69:
66:
62:
58:
54:
50:
46:
41:
39:
35:
31:
27:
23:
19:
1718:
1714:
1708:
1696:
1671:
1667:
1661:
1647:Lerner index
1623:
1453:
1303:Dividing by
1302:
1155:
1150:
1146:
1142:
1138:
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1129:
1125:
1103:
810:
639:
486:
304:
273:
100:
86:
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1523:Lerner Rule
72:Description
30:second-best
1749:Categories
1653:References
1630:first-best
1525:which has
1601:−
1590:∗
1586:Π
1559:λ
1480:λ
1470:λ
1354:−
1229:∂
1214:∂
1208:−
1112:λ
1057:−
1003:−
977:λ
974:−
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909:∂
898:λ
895:−
862:−
794:∗
790:Π
786:≥
769:−
724:∗
720:Π
696:−
687:Π
606:−
539:∫
524:∑
428:∑
197:…
127:…
1636:See also
1551:, since
47:and the
34:monopoly
1735:2222721
1688:2222721
1158:, and
28:, is a
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1454:where
1104:where
245:where
1731:JSTOR
1684:JSTOR
24:, or
20:, or
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276:and
16:The
1770:Tax
1723:doi
1676:doi
307:is
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