2388:
flexible functional forms (FFFs) because they do not impose any restrictions a priori on the degree of substitutability among inputs. These FFFs can provide a second-order approximation to any twice-differentiable function that meets the necessary regulatory conditions, including basic technological conditions and those consistent with cost minimization. Widely used examples of FFFs are the
2445:. This oversimplifies the reality where technological changes entail significant investments in plant and equipment, thus requiring time, often occurring over years rather than instantaneously. One way to address this issue will be to resort to a variable cost function that explicitly takes into account differences in the speed of adjustments among inputs.
817:
2362:
Furthermore, in the areas of trade, homothetic and monolithic functional models do not accurately predict results. One example is in the gravity equation for trade, or how much will two countries trade with each other based on GDP and distance. This led researchers to explore non-homothetic models of
2401:
A drawback of the GL function is its inability to be globally concave without sacrificing flexibility in the price space. This limitation also applies to the GO function, as it is a non-homothetic extension of the GL. In a subsequent study, Nakamura attempted to address this issue by employing the
2387:
for further details on this topic, including the potential for accommodating diverse elasticities of substitution among inputs, although this capability is somewhat constrained). To address this limitation, flexible functional forms have been developed. These general functional forms are called
1848:
In essence, under general conditions, a specific technology can be equally effectively represented by both cost and production functions. One advantage of using a cost function rather than a production function is that the demand functions for inputs can be easily derived from the former using
2464:
Up until 1990, the predominant user of this functional form was Iwao Ozaki, a
Japanese economist, which explains its namesake. Although much of Ozaki's work remains in Japanese and isn't readily accessible to the general public, there is an exception found in the paper "Economies of Scale and
28:
of production proposed by
Shinichiro Nakamura. The GO cost function is notable for explicitly considering nonhomothetic technology, where the proportions of inputs can vary as the output changes. This stands in contrast to the standard production model, which assumes homothetic technology.
384:
2343:
2382:
is a special case of CES) typically involve only two inputs, such as capital and labor. While they can be extended to include more than two inputs, assuming the same degree of substitutability for all inputs may seem overly restrictive (refer to
836:
The GO cost function is flexible in the price space, and treats scale effects and technical change in a highly general manner. The concavity condition which ensures that a constant function aligns with cost minimization for a specific set of
601:
1829:
158:
2465:
Input-Output
Coefficients" within the book "Applications of Input-Output Analysis," edited by A. Carter and A. Brody. This publication is available from North-Holland Publishers, dated 1969, spanning pages 280-302."
1675:
The duality theorems of cost and production functions state that once a well-behaved cost function is established, one can derive the corresponding production function, and vice versa. For a given cost function
2392:
and the
Generalized Leontief (GL) function. The translog function extends the Cobb-Douglas function to the second order, while the GL function performs a similar extension to the Leontief production function.
2348:
which implies that for a homothetic technology, the ratio of inputs depends solely on prices and not on the scale of output. However, empirical studies on the cross-section of establishments show that the
2359:) effectively explains the data, particularly for heavy industries such as steel mills, paper mills, basic chemical sectors, and power stations, indicating that homotheticity may not be applicable.
1655:
2020:
1362:
2230:
1968:
492:
2141:
2046:
446:
2550:
Charles
Blackorby, Daniel Primont, R. Robert Russell |title=Duality, Separability, and Functional Structure: Theory and Economic Applications, Elsevier Science Ltd, 1978,
1234:
963:
539:
812:{\displaystyle x_{i}={\partial c \over \partial p_{i}}=b_{ii}y^{b_{yi}}\exp ^{b_{it}t}+\textstyle \sum _{i\neq j}^{m}b_{ij}{\sqrt {p_{i}/p_{j}}}y^{b_{y}}\exp ^{b_{t}t}}
1117:
2074:
1070:
1709:
1559:
1484:
2575:
Nakamura, Shinichiro. "A non-homothetic globally concave flexible cost function and its application to panel data." The
Japanese Economic Review 52 (2001): 208-223.
1293:
1164:
1010:
909:
882:
590:
122:
148:
1264:) holds, the GO function reduces to the Generalized Leontief function of Diewert, A well-known flexible functional form for cost and production functions. When (
2443:
2423:
2223:
2203:
2183:
2163:
1907:
1887:
1729:
1524:
1504:
1449:
1429:
1409:
1254:
1184:
1137:
1030:
983:
855:
563:
95:
75:
55:
1740:
2367:
analysis of producer behavior, for example, when producers would begin to minimize costs by switching inputs or investing in increased production.
379:{\displaystyle C(p,y,t)=\sum _{i}b_{ii}\left(y^{b_{yi}}e^{b_{ti}t}p_{i}+\sum _{j\,:\,j\neq i}b_{ij}{\sqrt {p_{i}p_{j}}}y^{b_{y}}e^{b_{t}t}\right)}
2597:
Morrison, Catherine. "Quasi-fixed inputs in US and
Japanese manufacturing: a generalized Leontief restricted cost function approach."
2566:
Melvyn Fuss and Daniel McFadden, eds., Production
Economics: A Dual Approach to Theory and Applications, Volume 1, North Holland, 1978
2405:
Moreover, both the GO function and the underlying GL function presume immediate adjustments of inputs in response to changes in
1570:
2379:
2049:
1862:
1731:
can be obtained as (a more rigorous derivation involves using a distance function instead of a production function) :
2584:
Ryan, David L., and
Terence J. Wales. "Imposing local concavity in the translog and generalized Leontief cost functions."
2623:
2384:
2375:
2053:
1866:
2555:
2389:
2185:
is termed a unit cost function. From
Shephard's lemma, we obtain the following expression for the ratio of inputs
2650:
2534:
Diewert, W. Erwin. "An application of the Shephard duality theorem: A generalized Leontief production function."
1268:) hods, it reduces to a non-linear version of Leontief's model, which explains the cross-sectional variation of
1975:
2655:
1304:
2618:
2338:{\displaystyle {\frac {x_{i}}{x_{j}}}={\frac {\partial c(p)/\partial p_{i}}{\partial c(p)/\partial p_{j}}}}
1914:
2491:
Shinichiro Nakamura (1990). "A Nonhomothetic Generalized Leontief Cost Function Based on Pooled Data".
451:
2084:
2025:
405:
1196:
925:
2402:
Generalized McFadden function. For further advancements in this area, refer to Ryan and Wales.
497:
1086:
2059:
1042:
1679:
1529:
1454:
2364:
1271:
1142:
988:
887:
860:
568:
100:
1889:
can be represented as a positive monotone transformation of a linear-homogeneous function
857:, necessitates that its Hessian (the matrix of second partial derivatives with respect to
8:
2628:
2613:
1850:
542:
127:
1869:(CES) functions exhibit homothticity. This property means that the production function
1391:
In economics, production technology is typically represented by the production function
2508:
2428:
2408:
2208:
2188:
2168:
2148:
1892:
1872:
1714:
1509:
1489:
1434:
1414:
1394:
1239:
1169:
1122:
1015:
968:
840:
548:
80:
60:
40:
2551:
2633:
2500:
1824:{\displaystyle f(x)=\max(y|C(p,y)\leq p^{\top }x,{\text{ for all possible }}p)}
2644:
1853:, whereas this process can become cumbersome with the production function.
2512:
2056:
function for which the elasticity of substitution between the inputs,
2079:
For a homothetic technology, the cost function can be represented as
17:
2504:
1486:. When considering cost minimization for a given set of prices
1856:
25:
1650:{\displaystyle C(p,y)=\min _{x}(p^{\top }x|f(x)\geq y)}
1012:) scale proportionally with the overall output level (
707:
2431:
2411:
2233:
2211:
2191:
2171:
2151:
2087:
2062:
2028:
1978:
1917:
1895:
1875:
1861:
Commonly used forms of production functions, such as
1743:
1717:
1682:
1573:
1532:
1512:
1492:
1457:
1437:
1417:
1397:
1307:
1274:
1242:
1199:
1172:
1145:
1125:
1089:
1045:
1018:
991:
971:
928:
890:
863:
843:
604:
571:
551:
500:
454:
408:
161:
130:
103:
83:
63:
43:
2490:
2437:
2417:
2337:
2217:
2197:
2177:
2157:
2135:
2068:
2040:
2014:
1962:
1901:
1881:
1823:
1723:
1703:
1649:
1553:
1518:
1498:
1478:
1443:
1423:
1403:
1356:
1287:
1248:
1228:
1178:
1158:
1131:
1111:
1064:
1024:
1004:
977:
957:
903:
876:
849:
811:
584:
557:
533:
486:
440:
378:
142:
116:
89:
69:
49:
1386:
1295:when variations in input prices were negligible:
914:Several notable special cases can be identified:
2642:
1759:
1596:
2390:transcendental logarithmic (translog) function
2370:
124:, the generalized-Ozaki (GO) cost function
2546:
2544:
1035:Homogeneity of (of degree one) in output (
545:, we derive the demand function for input
2530:
2528:
2526:
2524:
2522:
2015:{\displaystyle h(\lambda x)=\lambda h(x)}
285:
281:
2541:
1857:Homothetic- and Nonhomothetic Technology
1711:, the corresponding production function
1411:, which, in the case of a single output
2165:is a monotone increasing function, and
2643:
2599:The Review of Economics and Statistics
2519:
2493:The Review of Economics and Statistics
1357:{\displaystyle x_{i}=b_{ii}y^{b_{yi}}}
2486:
2484:
2482:
1733:
1563:
1297:
594:
152:
22:generalized-Ozaki (GO) cost function
2624:Constant elasticity of substitution
1867:Constant Elasticity of Substitution
13:
2319:
2299:
2284:
2264:
2052:function is a special case of the
1963:{\displaystyle y=f(x)=\phi (h(x))}
1799:
1613:
1526:, the corresponding cost function
629:
621:
32:
14:
2667:
2479:
911:) being negative semidefinite.
487:{\displaystyle \sum _{i}b_{ij}=1}
24:is a general description of the
2136:{\displaystyle C(p,y)=c(p)d(y)}
2591:
2578:
2569:
2560:
2458:
2396:
2311:
2305:
2276:
2270:
2130:
2124:
2118:
2112:
2103:
2091:
2009:
2003:
1991:
1982:
1957:
1954:
1948:
1942:
1933:
1927:
1818:
1788:
1776:
1769:
1762:
1753:
1747:
1698:
1686:
1644:
1635:
1629:
1622:
1605:
1589:
1577:
1548:
1536:
1473:
1467:
1387:Cost- and production funcitons
183:
165:
137:
134:
1:
2499:(4). The MIT Press: 649?656.
2472:
2041:{\displaystyle \lambda >0}
1381:
441:{\displaystyle b_{ij}=b_{ji}}
2619:List of production functions
2536:Journal of political economy
1812: for all possible
1229:{\displaystyle b_{ti}=b_{t}}
1139:. None of the input levels (
958:{\displaystyle b_{yi}=b_{y}}
7:
2608:
2355:
1838:
1664:
1371:
826:
392:
10:
2672:
2363:production, to fit with a
1189:Neutral technicla change (
534:{\displaystyle i,j=1,..,m}
2371:Flexible Functional Forms
2451:
1112:{\displaystyle b_{yi}=0}
1079:Factor limitationality (
2069:{\displaystyle \sigma }
1065:{\displaystyle b_{y}=0}
2651:Functions and mappings
2439:
2419:
2339:
2219:
2199:
2179:
2159:
2137:
2070:
2042:
2016:
1964:
1903:
1883:
1825:
1725:
1705:
1704:{\displaystyle C(p,y)}
1651:
1555:
1554:{\displaystyle C(p,y)}
1520:
1500:
1480:
1479:{\displaystyle y=f(x)}
1451:inputs, is written as
1445:
1425:
1405:
1358:
1289:
1250:
1230:
1180:
1160:
1133:
1113:
1066:
1026:
1006:
979:
959:
905:
878:
851:
813:
728:
586:
559:
535:
488:
442:
380:
144:
118:
91:
71:
51:
2588:67.3 (2000): 253-260.
2538:79.3 (1971): 481-507.
2440:
2420:
2378:functions (note that
2340:
2220:
2200:
2180:
2160:
2138:
2071:
2043:
2017:
1965:
1904:
1884:
1826:
1726:
1706:
1652:
1561:can be expressed as:
1556:
1521:
1501:
1481:
1446:
1426:
1406:
1359:
1290:
1288:{\displaystyle x_{i}}
1251:
1231:
1181:
1161:
1159:{\displaystyle x_{i}}
1134:
1114:
1067:
1027:
1007:
1005:{\displaystyle x_{i}}
980:
960:
906:
904:{\displaystyle p_{j}}
879:
877:{\displaystyle p_{i}}
852:
814:
708:
587:
585:{\displaystyle x_{i}}
560:
536:
489:
443:
381:
145:
119:
117:{\displaystyle p_{i}}
92:
72:
52:
2656:Production economics
2429:
2409:
2231:
2209:
2189:
2169:
2149:
2085:
2060:
2026:
1976:
1915:
1893:
1873:
1741:
1715:
1680:
1571:
1530:
1510:
1490:
1455:
1435:
1415:
1395:
1305:
1272:
1240:
1197:
1170:
1143:
1123:
1087:
1043:
1016:
989:
985:. All input levels (
969:
926:
888:
861:
841:
602:
569:
549:
498:
452:
406:
159:
128:
101:
81:
61:
41:
2614:Production function
143:{\displaystyle C()}
37:For a given output
2435:
2415:
2335:
2215:
2195:
2175:
2155:
2133:
2066:
2038:
2012:
1960:
1899:
1879:
1821:
1721:
1701:
1647:
1604:
1551:
1516:
1496:
1476:
1441:
1421:
1401:
1354:
1285:
1246:
1226:
1176:
1156:
1129:
1109:
1062:
1022:
1002:
975:
955:
901:
874:
847:
809:
808:
582:
555:
541:. By applying the
531:
484:
464:
438:
376:
296:
198:
140:
114:
87:
67:
47:
2586:Economics Letters
2438:{\displaystyle y}
2418:{\displaystyle p}
2333:
2256:
2218:{\displaystyle j}
2198:{\displaystyle i}
2178:{\displaystyle c}
2158:{\displaystyle d}
1902:{\displaystyle h}
1882:{\displaystyle f}
1846:
1845:
1813:
1724:{\displaystyle f}
1672:
1671:
1595:
1519:{\displaystyle y}
1499:{\displaystyle p}
1444:{\displaystyle m}
1424:{\displaystyle y}
1404:{\displaystyle f}
1379:
1378:
1249:{\displaystyle i}
1179:{\displaystyle p}
1132:{\displaystyle i}
1025:{\displaystyle y}
978:{\displaystyle i}
850:{\displaystyle p}
834:
833:
769:
643:
558:{\displaystyle i}
455:
400:
399:
332:
273:
189:
90:{\displaystyle m}
70:{\displaystyle t}
50:{\displaystyle y}
2663:
2634:Returns to scale
2629:Shephard's lemma
2602:
2601:(1988): 275-287.
2595:
2589:
2582:
2576:
2573:
2567:
2564:
2558:
2548:
2539:
2532:
2517:
2516:
2488:
2466:
2462:
2444:
2442:
2441:
2436:
2424:
2422:
2421:
2416:
2344:
2342:
2341:
2336:
2334:
2332:
2331:
2330:
2318:
2297:
2296:
2295:
2283:
2262:
2257:
2255:
2254:
2245:
2244:
2235:
2224:
2222:
2221:
2216:
2204:
2202:
2201:
2196:
2184:
2182:
2181:
2176:
2164:
2162:
2161:
2156:
2142:
2140:
2139:
2134:
2075:
2073:
2072:
2067:
2047:
2045:
2044:
2039:
2021:
2019:
2018:
2013:
1969:
1967:
1966:
1961:
1908:
1906:
1905:
1900:
1888:
1886:
1885:
1880:
1851:Shephard's lemma
1840:
1830:
1828:
1827:
1822:
1814:
1811:
1803:
1802:
1772:
1734:
1730:
1728:
1727:
1722:
1710:
1708:
1707:
1702:
1666:
1656:
1654:
1653:
1648:
1625:
1617:
1616:
1603:
1564:
1560:
1558:
1557:
1552:
1525:
1523:
1522:
1517:
1505:
1503:
1502:
1497:
1485:
1483:
1482:
1477:
1450:
1448:
1447:
1442:
1430:
1428:
1427:
1422:
1410:
1408:
1407:
1402:
1373:
1363:
1361:
1360:
1355:
1353:
1352:
1351:
1350:
1333:
1332:
1317:
1316:
1298:
1294:
1292:
1291:
1286:
1284:
1283:
1255:
1253:
1252:
1247:
1235:
1233:
1232:
1227:
1225:
1224:
1212:
1211:
1185:
1183:
1182:
1177:
1165:
1163:
1162:
1157:
1155:
1154:
1138:
1136:
1135:
1130:
1118:
1116:
1115:
1110:
1102:
1101:
1071:
1069:
1068:
1063:
1055:
1054:
1031:
1029:
1028:
1023:
1011:
1009:
1008:
1003:
1001:
1000:
984:
982:
981:
976:
964:
962:
961:
956:
954:
953:
941:
940:
910:
908:
907:
902:
900:
899:
883:
881:
880:
875:
873:
872:
856:
854:
853:
848:
828:
818:
816:
815:
810:
807:
806:
802:
801:
787:
786:
785:
784:
770:
768:
767:
758:
753:
752:
743:
741:
740:
727:
722:
703:
702:
698:
697:
680:
679:
678:
677:
660:
659:
644:
642:
641:
640:
627:
619:
614:
613:
595:
591:
589:
588:
583:
581:
580:
564:
562:
561:
556:
543:Shephard's lemma
540:
538:
537:
532:
493:
491:
490:
485:
477:
476:
463:
447:
445:
444:
439:
437:
436:
421:
420:
394:
385:
383:
382:
377:
375:
371:
370:
369:
365:
364:
350:
349:
348:
347:
333:
331:
330:
321:
320:
311:
309:
308:
295:
269:
268:
259:
258:
254:
253:
236:
235:
234:
233:
211:
210:
197:
153:
150:is expressed as
149:
147:
146:
141:
123:
121:
120:
115:
113:
112:
96:
94:
93:
88:
77:and a vector of
76:
74:
73:
68:
56:
54:
53:
48:
2671:
2670:
2666:
2665:
2664:
2662:
2661:
2660:
2641:
2640:
2638:
2611:
2606:
2605:
2596:
2592:
2583:
2579:
2574:
2570:
2565:
2561:
2549:
2542:
2533:
2520:
2505:10.2307/2109605
2489:
2480:
2475:
2470:
2469:
2463:
2459:
2454:
2448:
2430:
2427:
2426:
2410:
2407:
2406:
2399:
2373:
2326:
2322:
2314:
2298:
2291:
2287:
2279:
2263:
2261:
2250:
2246:
2240:
2236:
2234:
2232:
2229:
2228:
2210:
2207:
2206:
2190:
2187:
2186:
2170:
2167:
2166:
2150:
2147:
2146:
2086:
2083:
2082:
2061:
2058:
2057:
2027:
2024:
2023:
1977:
1974:
1973:
1916:
1913:
1912:
1894:
1891:
1890:
1874:
1871:
1870:
1859:
1810:
1798:
1794:
1768:
1742:
1739:
1738:
1716:
1713:
1712:
1681:
1678:
1677:
1674:
1621:
1612:
1608:
1599:
1572:
1569:
1568:
1531:
1528:
1527:
1511:
1508:
1507:
1491:
1488:
1487:
1456:
1453:
1452:
1436:
1433:
1432:
1416:
1413:
1412:
1396:
1393:
1392:
1389:
1384:
1343:
1339:
1338:
1334:
1325:
1321:
1312:
1308:
1306:
1303:
1302:
1279:
1275:
1273:
1270:
1269:
1241:
1238:
1237:
1220:
1216:
1204:
1200:
1198:
1195:
1194:
1171:
1168:
1167:
1150:
1146:
1144:
1141:
1140:
1124:
1121:
1120:
1094:
1090:
1088:
1085:
1084:
1072:in addition to
1050:
1046:
1044:
1041:
1040:
1017:
1014:
1013:
996:
992:
990:
987:
986:
970:
967:
966:
949:
945:
933:
929:
927:
924:
923:
895:
891:
889:
886:
885:
868:
864:
862:
859:
858:
842:
839:
838:
797:
793:
792:
788:
780:
776:
775:
771:
763:
759:
754:
748:
744:
742:
733:
729:
723:
712:
690:
686:
685:
681:
670:
666:
665:
661:
652:
648:
636:
632:
628:
620:
618:
609:
605:
603:
600:
599:
576:
572:
570:
567:
566:
550:
547:
546:
499:
496:
495:
469:
465:
459:
453:
450:
449:
429:
425:
413:
409:
407:
404:
403:
360:
356:
355:
351:
343:
339:
338:
334:
326:
322:
316:
312:
310:
301:
297:
277:
264:
260:
246:
242:
241:
237:
226:
222:
221:
217:
216:
212:
203:
199:
193:
160:
157:
156:
129:
126:
125:
108:
104:
102:
99:
98:
82:
79:
78:
62:
59:
58:
42:
39:
38:
35:
33:The GO function
12:
11:
5:
2669:
2659:
2658:
2653:
2610:
2607:
2604:
2603:
2590:
2577:
2568:
2559:
2540:
2518:
2477:
2476:
2474:
2471:
2468:
2467:
2456:
2455:
2453:
2450:
2434:
2414:
2398:
2395:
2372:
2369:
2329:
2325:
2321:
2317:
2313:
2310:
2307:
2304:
2301:
2294:
2290:
2286:
2282:
2278:
2275:
2272:
2269:
2266:
2260:
2253:
2249:
2243:
2239:
2214:
2194:
2174:
2154:
2132:
2129:
2126:
2123:
2120:
2117:
2114:
2111:
2108:
2105:
2102:
2099:
2096:
2093:
2090:
2065:
2037:
2034:
2031:
2011:
2008:
2005:
2002:
1999:
1996:
1993:
1990:
1987:
1984:
1981:
1959:
1956:
1953:
1950:
1947:
1944:
1941:
1938:
1935:
1932:
1929:
1926:
1923:
1920:
1898:
1878:
1858:
1855:
1844:
1843:
1834:
1832:
1820:
1817:
1809:
1806:
1801:
1797:
1793:
1790:
1787:
1784:
1781:
1778:
1775:
1771:
1767:
1764:
1761:
1758:
1755:
1752:
1749:
1746:
1720:
1700:
1697:
1694:
1691:
1688:
1685:
1670:
1669:
1660:
1658:
1646:
1643:
1640:
1637:
1634:
1631:
1628:
1624:
1620:
1615:
1611:
1607:
1602:
1598:
1594:
1591:
1588:
1585:
1582:
1579:
1576:
1550:
1547:
1544:
1541:
1538:
1535:
1515:
1495:
1475:
1472:
1469:
1466:
1463:
1460:
1440:
1420:
1400:
1388:
1385:
1383:
1380:
1377:
1376:
1367:
1365:
1349:
1346:
1342:
1337:
1331:
1328:
1324:
1320:
1315:
1311:
1282:
1278:
1258:
1257:
1245:
1223:
1219:
1215:
1210:
1207:
1203:
1187:
1175:
1153:
1149:
1128:
1108:
1105:
1100:
1097:
1093:
1077:
1061:
1058:
1053:
1049:
1033:
1021:
999:
995:
974:
952:
948:
944:
939:
936:
932:
918:Homothticity (
898:
894:
871:
867:
846:
832:
831:
822:
820:
805:
800:
796:
791:
783:
779:
774:
766:
762:
757:
751:
747:
739:
736:
732:
726:
721:
718:
715:
711:
706:
701:
696:
693:
689:
684:
676:
673:
669:
664:
658:
655:
651:
647:
639:
635:
631:
626:
623:
617:
612:
608:
579:
575:
554:
530:
527:
524:
521:
518:
515:
512:
509:
506:
503:
483:
480:
475:
472:
468:
462:
458:
435:
432:
428:
424:
419:
416:
412:
398:
397:
388:
386:
374:
368:
363:
359:
354:
346:
342:
337:
329:
325:
319:
315:
307:
304:
300:
294:
291:
288:
284:
280:
276:
272:
267:
263:
257:
252:
249:
245:
240:
232:
229:
225:
220:
215:
209:
206:
202:
196:
192:
188:
185:
182:
179:
176:
173:
170:
167:
164:
139:
136:
133:
111:
107:
86:
66:
46:
34:
31:
9:
6:
4:
3:
2:
2668:
2657:
2654:
2652:
2649:
2648:
2646:
2639:
2636:
2635:
2631:
2630:
2626:
2625:
2621:
2620:
2616:
2615:
2600:
2594:
2587:
2581:
2572:
2563:
2557:
2556:0-444-00235-9
2553:
2547:
2545:
2537:
2531:
2529:
2527:
2525:
2523:
2514:
2510:
2506:
2502:
2498:
2494:
2487:
2485:
2483:
2478:
2461:
2457:
2449:
2446:
2432:
2412:
2403:
2394:
2391:
2386:
2381:
2377:
2368:
2366:
2365:cross section
2360:
2358:
2357:
2352:
2346:
2327:
2323:
2315:
2308:
2302:
2292:
2288:
2280:
2273:
2267:
2258:
2251:
2247:
2241:
2237:
2226:
2212:
2192:
2172:
2152:
2143:
2127:
2121:
2115:
2109:
2106:
2100:
2097:
2094:
2088:
2080:
2077:
2063:
2055:
2051:
2035:
2032:
2029:
2006:
2000:
1997:
1994:
1988:
1985:
1979:
1970:
1951:
1945:
1939:
1936:
1930:
1924:
1921:
1918:
1910:
1896:
1876:
1868:
1864:
1854:
1852:
1842:
1835:
1833:
1831:
1815:
1807:
1804:
1795:
1791:
1785:
1782:
1779:
1773:
1765:
1756:
1750:
1744:
1736:
1735:
1732:
1718:
1695:
1692:
1689:
1683:
1668:
1661:
1659:
1657:
1641:
1638:
1632:
1626:
1618:
1609:
1600:
1592:
1586:
1583:
1580:
1574:
1566:
1565:
1562:
1545:
1542:
1539:
1533:
1513:
1493:
1470:
1464:
1461:
1458:
1438:
1418:
1398:
1375:
1368:
1366:
1364:
1347:
1344:
1340:
1335:
1329:
1326:
1322:
1318:
1313:
1309:
1300:
1299:
1296:
1280:
1276:
1267:
1263:
1243:
1221:
1217:
1213:
1208:
1205:
1201:
1192:
1188:
1173:
1151:
1147:
1126:
1106:
1103:
1098:
1095:
1091:
1082:
1078:
1075:
1059:
1056:
1051:
1047:
1038:
1034:
1019:
997:
993:
972:
950:
946:
942:
937:
934:
930:
921:
917:
916:
915:
912:
896:
892:
869:
865:
844:
830:
823:
821:
819:
803:
798:
794:
789:
781:
777:
772:
764:
760:
755:
749:
745:
737:
734:
730:
724:
719:
716:
713:
709:
704:
699:
694:
691:
687:
682:
674:
671:
667:
662:
656:
653:
649:
645:
637:
633:
624:
615:
610:
606:
597:
596:
593:
577:
573:
552:
544:
528:
525:
522:
519:
516:
513:
510:
507:
504:
501:
481:
478:
473:
470:
466:
460:
456:
433:
430:
426:
422:
417:
414:
410:
396:
389:
387:
372:
366:
361:
357:
352:
344:
340:
335:
327:
323:
317:
313:
305:
302:
298:
292:
289:
286:
282:
278:
274:
270:
265:
261:
255:
250:
247:
243:
238:
230:
227:
223:
218:
213:
207:
204:
200:
194:
190:
186:
180:
177:
174:
171:
168:
162:
155:
154:
151:
131:
109:
105:
97:input prices
84:
64:
44:
30:
27:
23:
19:
2637:
2632:
2627:
2622:
2617:
2612:
2598:
2593:
2585:
2580:
2571:
2562:
2535:
2496:
2492:
2460:
2447:
2404:
2400:
2380:Cobb-Douglas
2374:
2361:
2354:
2350:
2347:
2227:
2144:
2081:
2078:
2050:Cobb-Douglas
1971:
1911:
1863:Cobb-Douglas
1860:
1847:
1836:
1737:
1673:
1662:
1567:
1390:
1369:
1301:
1265:
1261:
1259:
1190:
1166:) depend on
1080:
1073:
1036:
919:
913:
835:
824:
598:
401:
390:
36:
21:
15:
2397:Limitations
2076:, is one.
2645:Categories
2473:References
1382:Background
57:, at time
2320:∂
2300:∂
2285:∂
2265:∂
2064:σ
2030:λ
1998:λ
1986:λ
1940:ϕ
1800:⊤
1792:≤
1639:≥
1614:⊤
717:≠
710:∑
630:∂
622:∂
457:∑
290:≠
275:∑
191:∑
18:economics
2609:See also
2353:model (
2022:for any
1236:for all
1119:for all
965:for all
592: :
2513:2109605
2048:. The
402:Here,
2554:
2511:
2145:where
1972:where
1260:When (
2509:JSTOR
2452:Notes
2425:and
2552:ISBN
2205:and
2033:>
1865:and
1506:and
1431:and
884:and
448:and
26:cost
20:the
2501:doi
2385:CES
2376:CES
2225::
2054:CES
1760:max
1597:min
1193:):
1083:):
1039:):
920:HT)
790:exp
683:exp
16:In
2647::
2543:^
2521:^
2507:.
2497:72
2495:.
2481:^
2351:FL
2345:,
1909::
1266:FL
1262:HT
1191:NT
1081:FL
1074:HT
1037:HG
1032:).
922::
565:,
494:,
2515:.
2503::
2433:y
2413:p
2356:3
2328:j
2324:p
2316:/
2312:)
2309:p
2306:(
2303:c
2293:i
2289:p
2281:/
2277:)
2274:p
2271:(
2268:c
2259:=
2252:j
2248:x
2242:i
2238:x
2213:j
2193:i
2173:c
2153:d
2131:)
2128:y
2125:(
2122:d
2119:)
2116:p
2113:(
2110:c
2107:=
2104:)
2101:y
2098:,
2095:p
2092:(
2089:C
2036:0
2010:)
2007:x
2004:(
2001:h
1995:=
1992:)
1989:x
1983:(
1980:h
1958:)
1955:)
1952:x
1949:(
1946:h
1943:(
1937:=
1934:)
1931:x
1928:(
1925:f
1922:=
1919:y
1897:h
1877:f
1841:)
1839:5
1837:(
1819:)
1816:p
1808:,
1805:x
1796:p
1789:)
1786:y
1783:,
1780:p
1777:(
1774:C
1770:|
1766:y
1763:(
1757:=
1754:)
1751:x
1748:(
1745:f
1719:f
1699:)
1696:y
1693:,
1690:p
1687:(
1684:C
1667:)
1665:4
1663:(
1645:)
1642:y
1636:)
1633:x
1630:(
1627:f
1623:|
1619:x
1610:p
1606:(
1601:x
1593:=
1590:)
1587:y
1584:,
1581:p
1578:(
1575:C
1549:)
1546:y
1543:,
1540:p
1537:(
1534:C
1514:y
1494:p
1474:)
1471:x
1468:(
1465:f
1462:=
1459:y
1439:m
1419:y
1399:f
1374:)
1372:3
1370:(
1348:i
1345:y
1341:b
1336:y
1330:i
1327:i
1323:b
1319:=
1314:i
1310:x
1281:i
1277:x
1256:.
1244:i
1222:t
1218:b
1214:=
1209:i
1206:t
1202:b
1186:.
1174:p
1152:i
1148:x
1127:i
1107:0
1104:=
1099:i
1096:y
1092:b
1076:.
1060:0
1057:=
1052:y
1048:b
1020:y
998:i
994:x
973:i
951:y
947:b
943:=
938:i
935:y
931:b
897:j
893:p
870:i
866:p
845:p
829:)
827:2
825:(
804:t
799:t
795:b
782:y
778:b
773:y
765:j
761:p
756:/
750:i
746:p
738:j
735:i
731:b
725:m
720:j
714:i
705:+
700:t
695:t
692:i
688:b
675:i
672:y
668:b
663:y
657:i
654:i
650:b
646:=
638:i
634:p
625:c
616:=
611:i
607:x
578:i
574:x
553:i
529:m
526:,
523:.
520:.
517:,
514:1
511:=
508:j
505:,
502:i
482:1
479:=
474:j
471:i
467:b
461:i
434:i
431:j
427:b
423:=
418:j
415:i
411:b
395:)
393:1
391:(
373:)
367:t
362:t
358:b
353:e
345:y
341:b
336:y
328:j
324:p
318:i
314:p
306:j
303:i
299:b
293:i
287:j
283::
279:j
271:+
266:i
262:p
256:t
251:i
248:t
244:b
239:e
231:i
228:y
224:b
219:y
214:(
208:i
205:i
201:b
195:i
187:=
184:)
181:t
178:,
175:y
172:,
169:p
166:(
163:C
138:)
135:(
132:C
110:i
106:p
85:m
65:t
45:y
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