1523:
then the curve is inelastic at the point. If the tangent line is extended to the horizontal axis the problem is simply a matter of comparing angles created by the lines and the horizontal axis. If the marginal angle is greater than the average angle then the function is elastic at the point; if the marginal angle is less than the average angle then the function is inelastic at that point. If, however, one follows the convention adopted by economists and plots the independent variable
1518:(dQ/dP) to the value of the average function (Q(P)/P). This relationship provides an easy way of determining whether a demand curve is elastic or inelastic at a particular point. First, suppose one follows the usual convention in mathematics of plotting the independent variable (P) horizontally and the dependent variable (Q) vertically. Then the slope of a line tangent to the curve at that point is the value of the marginal function at that point. The slope of a
418:
119:
1522:
drawn from the origin through the point is the value of the average function. If the absolute value of the slope of the tangent is greater than the slope of the ray then the function is elastic at the point; if the slope of the secant is greater than the absolute value of the slope of the tangent
1858:
598:. Equivalently, it is the ratio of the infinitesimal change of the logarithm of a function with respect to the infinitesimal change of the logarithm of the argument. Generalizations to multi-input–multi-output cases also exist in the literature.
413:{\displaystyle =\lim _{x\to a}{\frac {f(x)-f(a)}{x-a}}{\frac {a}{f(a)}}=\lim _{x\to a}{\frac {f(x)-f(a)}{f(a)}}{\frac {a}{x-a}}=\lim _{x\to a}{\frac {{\frac {f(x)}{f(a)}}-1}{{\frac {x}{a}}-1}}\approx {\frac {\%\Delta f(a)}{\%\Delta a}}}
1243:
1076:
1674:
2023:
The elasticity can also be defined if the input and/or output is consistently negative, or simply away from any points where the input or output is zero, but in practice the elasticity is used for positive
1972:
503:
1323:
909:
113:
820:
1730:
1438:
1722:
664:
1488:
1375:
596:
619:
690:
535:
1870:
The opposite definition is sometimes used in the literature. That is, the term "semi-elasticity" is also sometimes used for the change (not percentage-wise) in
555:
1082:
915:
1568:
1884:
429:
1254:
826:
2231:
2206:
736:
45:
2300:
2295:
2178:
2057:
1853:{\displaystyle \ln {f}=\ln {C\alpha ^{x}}=\ln {C}+x\ln {\alpha }\implies {\frac {d\ln {f}}{dx}}=\ln {\alpha }.}
1381:
726:
Rules for finding the elasticity of products and quotients are simpler than those for derivatives. Let
1682:
707:
624:
2099:"A scale elasticity measure for directional distance function and its dual: Theory and DEA estimation"
1445:
2098:
29:
2004:
1993:
1988:
715:
711:
2049:
2042:
1342:
560:
2074:
604:
2130:"Elasticities in Metabolic Control Analysis: algebraic derivation of simplified expressions"
669:
1999:
1500:
511:
8:
2194:
540:
2146:
2129:
2227:
2202:
2174:
2151:
2053:
1864:
1515:
1238:{\displaystyle E(f(x)-g(x))={\frac {f(x)\cdot E(f(x))-g(x)\cdot E(g(x))}{f(x)-g(x)}}}
1071:{\displaystyle E(f(x)+g(x))={\frac {f(x)\cdot E(f(x))+g(x)\cdot E(g(x))}{f(x)+g(x)}}}
2274:
2141:
2110:
2037:
508:
It is thus the ratio of the relative (percentage) change in the function's output
1535:
1504:
2114:
1983:
1519:
699:
between two points as the separation between those two points approaches zero.
696:
2289:
1514:), and can be expressed as (dQ/dP)/(Q(P)/P) or the ratio of the value of the
1679:
The semi-elasticity will be constant for exponential functions of the form,
2155:
2247:
17:
1546:
A semi-elasticity (or semielasticity) gives the percentage change in
703:
2278:
1669:{\displaystyle Sf(x)={\frac {1}{f(x)}}f'(x)={\frac {d\ln f(x)}{dx}}}
2265:
Nievergelt, Yves (1983). "The
Concept of Elasticity in Economics".
35:
of a positive variable (positive input, positive output) at point
1531:
on the horizontal axis, then the opposite rules would apply.
2010:
1967:{\displaystyle {\frac {df(x)}{d\ln(x)}}={\frac {df(x)}{dx}}x}
1248:
The derivative can be expressed in terms of elasticity as
2173:(4th ed.). Boston: McGraw-Hill. pp. 192–193.
2075:"A Note on Equivalences in Measuring Returns to Scale"
1534:
The same graphical procedure can also be applied to a
1887:
1733:
1685:
1571:
1554:. Algebraically, the semi-elasticity S of a function
1448:
1384:
1345:
1257:
1085:
918:
829:
739:
672:
627:
607:
563:
543:
514:
432:
122:
48:
498:{\displaystyle Ef(x)={\frac {d\log f(x)}{d\log x}}.}
2041:
1966:
1852:
1716:
1668:
1482:
1432:
1369:
1318:{\displaystyle Df(x)={\frac {Ef(x)\cdot f(x)}{x}}}
1317:
1237:
1070:
903:
814:
684:
658:
613:
590:
549:
529:
497:
412:
107:
904:{\displaystyle E{\frac {f(x)}{g(x)}}=Ef(x)-Eg(x)}
537:with respect to the relative change in its input
2287:
2168:
2048:. Englewood Cliffs, NJ: Prentice Hall. pp.
2036:
1527:on the vertical axis and the dependent variable
1494:
295:
211:
127:
2079:International Journal of Business and Economics
1550:in terms of a change (not percentage-wise) in
695:The elasticity at a point is the limit of the
2171:Fundamental Methods of Mathematical Economics
815:{\displaystyle E(f(x)\cdot g(x))=Ef(x)+Eg(x)}
2201:(2nd ed.). South-Western. p. 656.
2199:Introductory Econometrics: A Modern Approach
702:The concept of elasticity is widely used in
108:{\displaystyle Ef(a)={\frac {a}{f(a)}}f'(a)}
2032:
2030:
601:The elasticity of a function is a constant
2264:
2193:
2127:
2011:=References==\frac{\Delta y}{\Delta x/x_1}
1804:
1800:
2145:
621:if and only if the function has the form
557:, for infinitesimal changes from a point
2134:Computer Applications in the Biosciences
2103:European Journal of Operational Research
2096:
2072:
2027:
2288:
2221:
1433:{\displaystyle E(a\cdot f(x))=Ef(x)}
2224:The theory of monetary institutions
2128:Woods, JH; Sauro, HM (April 1997).
1874:in terms of a percentage change in
13:
2258:
2226:. Malden: Blackwell. p. 148.
1541:
401:
398:
381:
378:
14:
2312:
2044:Mathematics for Economic Analysis
1863:An example of semi-elasticity is
1717:{\displaystyle f(x)=C\alpha ^{x}}
659:{\displaystyle f(x)=Cx^{\alpha }}
2147:10.1093/bioinformatics/13.2.123
2240:
2222:White, Lawrence Henry (1999).
2215:
2187:
2162:
2121:
2090:
2066:
2017:
1947:
1941:
1923:
1917:
1903:
1897:
1801:
1695:
1689:
1652:
1646:
1625:
1619:
1605:
1599:
1584:
1578:
1503:refers to the elasticity of a
1468:
1452:
1427:
1421:
1409:
1406:
1400:
1388:
1355:
1349:
1306:
1300:
1291:
1285:
1270:
1264:
1229:
1223:
1214:
1208:
1200:
1197:
1191:
1185:
1176:
1170:
1161:
1158:
1152:
1146:
1137:
1131:
1119:
1116:
1110:
1101:
1095:
1089:
1062:
1056:
1047:
1041:
1033:
1030:
1024:
1018:
1009:
1003:
994:
991:
985:
979:
970:
964:
952:
949:
943:
934:
928:
922:
898:
892:
880:
874:
859:
853:
845:
839:
809:
803:
791:
785:
773:
770:
764:
755:
749:
743:
637:
631:
585:
582:
576:
564:
524:
518:
472:
466:
445:
439:
393:
387:
339:
333:
325:
319:
302:
267:
261:
253:
247:
238:
232:
218:
201:
195:
169:
163:
154:
148:
134:
102:
96:
82:
76:
61:
55:
1:
2087:, and see references therein.
1495:Estimating point elasticities
1483:{\displaystyle E(bx^{a})=a\ }
7:
2248:"Stata 17 help for margins"
2169:Chiang; Wainwright (2005).
2097:Zelenyuk, Valentin (2013).
2073:Zelenyuk, Valentin (2013).
1977:
10:
2317:
2115:10.1016/j.ejor.2013.01.012
1501:price elasticity of demand
718:respectively for details.
708:metabolic control analysis
2040:; Hammond, Peter (1995).
730:be differentiable. Then
1370:{\displaystyle E(a)=0\ }
721:
591:{\displaystyle (a,f(a))}
614:{\displaystyle \alpha }
30:differentiable function
2301:Mathematical economics
2296:Functions and mappings
2005:Logarithmic derivative
1994:Elasticity coefficient
1989:Elasticity (economics)
1968:
1854:
1718:
1670:
1484:
1434:
1371:
1319:
1239:
1072:
905:
816:
716:elasticity coefficient
712:elasticity (economics)
686:
685:{\displaystyle C>0}
660:
615:
592:
551:
531:
499:
414:
109:
1969:
1855:
1719:
1671:
1485:
1435:
1372:
1320:
1240:
1073:
906:
817:
687:
661:
616:
593:
552:
532:
500:
415:
110:
2000:Homogeneous function
1885:
1731:
1683:
1569:
1538:or other functions.
1446:
1382:
1343:
1336:be constants. Then
1255:
1083:
916:
827:
737:
670:
625:
605:
561:
541:
530:{\displaystyle f(x)}
512:
430:
120:
46:
2195:Wooldridge, Jeffrey
1964:
1850:
1714:
1666:
1499:In economics, the
1480:
1430:
1367:
1315:
1235:
1068:
901:
812:
682:
656:
611:
588:
547:
527:
495:
410:
309:
225:
141:
105:
1959:
1927:
1867:in bond trading.
1865:modified duration
1831:
1664:
1609:
1516:marginal function
1479:
1366:
1313:
1233:
1066:
863:
550:{\displaystyle x}
490:
408:
370:
361:
343:
294:
289:
271:
210:
205:
184:
126:
86:
2308:
2282:
2252:
2251:
2244:
2238:
2237:
2219:
2213:
2212:
2191:
2185:
2184:
2166:
2160:
2159:
2149:
2125:
2119:
2118:
2094:
2088:
2086:
2070:
2064:
2063:
2047:
2034:
2025:
2021:
1973:
1971:
1970:
1965:
1960:
1958:
1950:
1933:
1928:
1926:
1906:
1889:
1859:
1857:
1856:
1851:
1846:
1832:
1830:
1822:
1821:
1806:
1799:
1782:
1768:
1767:
1766:
1744:
1723:
1721:
1720:
1715:
1713:
1712:
1675:
1673:
1672:
1667:
1665:
1663:
1655:
1632:
1618:
1610:
1608:
1591:
1489:
1487:
1486:
1481:
1477:
1467:
1466:
1439:
1437:
1436:
1431:
1376:
1374:
1373:
1368:
1364:
1324:
1322:
1321:
1316:
1314:
1309:
1277:
1244:
1242:
1241:
1236:
1234:
1232:
1203:
1126:
1077:
1075:
1074:
1069:
1067:
1065:
1036:
959:
910:
908:
907:
902:
864:
862:
848:
834:
821:
819:
818:
813:
691:
689:
688:
683:
665:
663:
662:
657:
655:
654:
620:
618:
617:
612:
597:
595:
594:
589:
556:
554:
553:
548:
536:
534:
533:
528:
504:
502:
501:
496:
491:
489:
475:
452:
423:or equivalently
419:
417:
416:
411:
409:
407:
396:
376:
371:
369:
362:
354:
351:
344:
342:
328:
314:
311:
308:
290:
288:
274:
272:
270:
256:
227:
224:
206:
204:
187:
185:
183:
172:
143:
140:
114:
112:
111:
106:
95:
87:
85:
68:
26:point elasticity
2316:
2315:
2311:
2310:
2309:
2307:
2306:
2305:
2286:
2285:
2279:10.1137/1025049
2261:
2259:Further reading
2256:
2255:
2246:
2245:
2241:
2234:
2220:
2216:
2209:
2192:
2188:
2181:
2167:
2163:
2126:
2122:
2095:
2091:
2071:
2067:
2060:
2038:Sydsaeter, Knut
2035:
2028:
2022:
2018:
2013:
1980:
1951:
1934:
1932:
1907:
1890:
1888:
1886:
1883:
1882:
1878:which would be
1842:
1823:
1817:
1807:
1805:
1795:
1778:
1762:
1758:
1754:
1740:
1732:
1729:
1728:
1708:
1704:
1684:
1681:
1680:
1656:
1633:
1631:
1611:
1595:
1590:
1570:
1567:
1566:
1544:
1542:Semi-elasticity
1536:supply function
1505:demand function
1497:
1462:
1458:
1447:
1444:
1443:
1383:
1380:
1379:
1344:
1341:
1340:
1278:
1276:
1256:
1253:
1252:
1204:
1127:
1125:
1084:
1081:
1080:
1037:
960:
958:
917:
914:
913:
849:
835:
833:
828:
825:
824:
738:
735:
734:
724:
671:
668:
667:
666:for a constant
650:
646:
626:
623:
622:
606:
603:
602:
562:
559:
558:
542:
539:
538:
513:
510:
509:
476:
453:
451:
431:
428:
427:
397:
377:
375:
353:
352:
329:
315:
313:
312:
310:
298:
278:
273:
257:
228:
226:
214:
191:
186:
173:
144:
142:
130:
121:
118:
117:
88:
72:
67:
47:
44:
43:
12:
11:
5:
2314:
2304:
2303:
2298:
2284:
2283:
2273:(2): 261–265.
2260:
2257:
2254:
2253:
2239:
2232:
2214:
2207:
2186:
2179:
2161:
2120:
2109:(3): 592–600.
2089:
2065:
2058:
2026:
2015:
2014:
2012:
2009:
2008:
2007:
2002:
1997:
1996:(biochemistry)
1991:
1986:
1984:Arc elasticity
1979:
1976:
1975:
1974:
1963:
1957:
1954:
1949:
1946:
1943:
1940:
1937:
1931:
1925:
1922:
1919:
1916:
1913:
1910:
1905:
1902:
1899:
1896:
1893:
1861:
1860:
1849:
1845:
1841:
1838:
1835:
1829:
1826:
1820:
1816:
1813:
1810:
1803:
1798:
1794:
1791:
1788:
1785:
1781:
1777:
1774:
1771:
1765:
1761:
1757:
1753:
1750:
1747:
1743:
1739:
1736:
1711:
1707:
1703:
1700:
1697:
1694:
1691:
1688:
1677:
1676:
1662:
1659:
1654:
1651:
1648:
1645:
1642:
1639:
1636:
1630:
1627:
1624:
1621:
1617:
1614:
1607:
1604:
1601:
1598:
1594:
1589:
1586:
1583:
1580:
1577:
1574:
1543:
1540:
1496:
1493:
1492:
1491:
1476:
1473:
1470:
1465:
1461:
1457:
1454:
1451:
1441:
1429:
1426:
1423:
1420:
1417:
1414:
1411:
1408:
1405:
1402:
1399:
1396:
1393:
1390:
1387:
1377:
1363:
1360:
1357:
1354:
1351:
1348:
1326:
1325:
1312:
1308:
1305:
1302:
1299:
1296:
1293:
1290:
1287:
1284:
1281:
1275:
1272:
1269:
1266:
1263:
1260:
1246:
1245:
1231:
1228:
1225:
1222:
1219:
1216:
1213:
1210:
1207:
1202:
1199:
1196:
1193:
1190:
1187:
1184:
1181:
1178:
1175:
1172:
1169:
1166:
1163:
1160:
1157:
1154:
1151:
1148:
1145:
1142:
1139:
1136:
1133:
1130:
1124:
1121:
1118:
1115:
1112:
1109:
1106:
1103:
1100:
1097:
1094:
1091:
1088:
1078:
1064:
1061:
1058:
1055:
1052:
1049:
1046:
1043:
1040:
1035:
1032:
1029:
1026:
1023:
1020:
1017:
1014:
1011:
1008:
1005:
1002:
999:
996:
993:
990:
987:
984:
981:
978:
975:
972:
969:
966:
963:
957:
954:
951:
948:
945:
942:
939:
936:
933:
930:
927:
924:
921:
911:
900:
897:
894:
891:
888:
885:
882:
879:
876:
873:
870:
867:
861:
858:
855:
852:
847:
844:
841:
838:
832:
822:
811:
808:
805:
802:
799:
796:
793:
790:
787:
784:
781:
778:
775:
772:
769:
766:
763:
760:
757:
754:
751:
748:
745:
742:
723:
720:
697:arc elasticity
681:
678:
675:
653:
649:
645:
642:
639:
636:
633:
630:
610:
587:
584:
581:
578:
575:
572:
569:
566:
546:
526:
523:
520:
517:
506:
505:
494:
488:
485:
482:
479:
474:
471:
468:
465:
462:
459:
456:
450:
447:
444:
441:
438:
435:
421:
420:
406:
403:
400:
395:
392:
389:
386:
383:
380:
374:
368:
365:
360:
357:
350:
347:
341:
338:
335:
332:
327:
324:
321:
318:
307:
304:
301:
297:
293:
287:
284:
281:
277:
269:
266:
263:
260:
255:
252:
249:
246:
243:
240:
237:
234:
231:
223:
220:
217:
213:
209:
203:
200:
197:
194:
190:
182:
179:
176:
171:
168:
165:
162:
159:
156:
153:
150:
147:
139:
136:
133:
129:
125:
115:
104:
101:
98:
94:
91:
84:
81:
78:
75:
71:
66:
63:
60:
57:
54:
51:
39:is defined as
28:of a positive
9:
6:
4:
3:
2:
2313:
2302:
2299:
2297:
2294:
2293:
2291:
2280:
2276:
2272:
2268:
2263:
2262:
2249:
2243:
2235:
2233:0-631-21214-0
2229:
2225:
2218:
2210:
2208:0-324-11364-1
2204:
2200:
2196:
2190:
2182:
2176:
2172:
2165:
2157:
2153:
2148:
2143:
2140:(2): 123–30.
2139:
2135:
2131:
2124:
2116:
2112:
2108:
2104:
2100:
2093:
2084:
2080:
2076:
2069:
2061:
2055:
2051:
2046:
2045:
2039:
2033:
2031:
2020:
2016:
2006:
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2267:SIAM Review
2024:quantities.
710:(MCA); see
18:mathematics
2290:Categories
2180:0070109109
2085:(1): 85–89
2059:013583600X
22:elasticity
1915:
1844:α
1840:
1815:
1802:⟹
1797:α
1793:
1776:
1760:α
1752:
1738:
1706:α
1641:
1558:at point
1395:⋅
1295:⋅
1218:−
1180:⋅
1165:−
1141:⋅
1105:−
1013:⋅
974:⋅
884:−
759:⋅
704:economics
652:α
609:α
484:
461:
402:Δ
399:%
382:Δ
379:%
373:≈
364:−
346:−
303:→
283:−
242:−
219:→
178:−
158:−
135:→
2197:(2003).
1978:See also
1616:′
93:′
2156:9146958
2050:173–175
1724:since,
2230:
2205:
2177:
2154:
2056:
1478:
1365:
20:, the
722:Rules
2228:ISBN
2203:ISBN
2175:ISBN
2152:PMID
2054:ISBN
1872:f(x)
1562:is
1548:f(x)
1332:and
1328:Let
728:f, g
714:and
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677:>
2275:doi
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2269:.
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2136:.
2132:.
2105:.
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2081:.
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