2539:
5030:
5596:
1953:
4520:
5102:
2534:{\displaystyle S(\mathbf {p} ,w)=\left({\begin{array}{ccc}{\frac {\partial x_{1}(\mathbf {p} ,w)}{\partial p_{1}}}+x_{1}(\mathbf {p} ,w){\frac {\partial x_{1}(\mathbf {p} ,w)}{\partial w}}&\cdots &{\frac {\partial x_{1}(\mathbf {p} ,w)}{\partial p_{n}}}+x_{n}(\mathbf {p} ,w){\frac {\partial x_{1}(\mathbf {p} ,w)}{\partial w}}\\\vdots &\ddots &\vdots \\{\frac {\partial x_{n}(\mathbf {p} ,w)}{\partial p_{1}}}+x_{1}(\mathbf {p} ,w){\frac {\partial x_{n}(\mathbf {p} ,w)}{\partial w}}&\cdots &{\frac {\partial x_{n}(\mathbf {p} ,w)}{\partial p_{n}}}+x_{n}(\mathbf {p} ,w){\frac {\partial x_{n}(\mathbf {p} ,w)}{\partial w}}\end{array}}\right)}
5025:{\displaystyle {\begin{bmatrix}{\frac {\partial x_{1}}{\partial p_{1}}}&{\frac {\partial x_{1}}{\partial p_{2}}}\\{\frac {\partial x_{2}}{\partial p_{1}}}&{\frac {\partial x_{2}}{\partial p_{2}}}\\\end{bmatrix}}={\begin{bmatrix}{\frac {\partial h_{1}}{\partial p_{1}}}&{\frac {\partial h_{1}}{\partial p_{2}}}\\{\frac {\partial h_{2}}{\partial p_{1}}}&{\frac {\partial h_{2}}{\partial p_{2}}}\\\end{bmatrix}}-{\begin{bmatrix}{\frac {\partial x_{1}}{\partial w}}x_{1}&{\frac {\partial x_{1}}{\partial w}}x_{2}\\{\frac {\partial x_{2}}{\partial w}}x_{1}&{\frac {\partial x_{2}}{\partial w}}x_{2}\\\end{bmatrix}}}
5591:{\displaystyle {\begin{bmatrix}\Delta x_{1}\\\Delta x_{2}\end{bmatrix}}\approx {\begin{bmatrix}{\frac {\partial h_{1}}{\partial p_{1}}}&{\frac {\partial h_{1}}{\partial p_{2}}}\\{\frac {\partial h_{2}}{\partial p_{1}}}&{\frac {\partial h_{2}}{\partial p_{2}}}\\\end{bmatrix}}\cdot {\begin{bmatrix}\Delta p_{1}\\\Delta p_{2}\end{bmatrix}}-{\begin{bmatrix}{\frac {\partial x_{1}}{\partial w}}x_{1}&{\frac {\partial x_{1}}{\partial w}}x_{2}\\{\frac {\partial x_{2}}{\partial w}}x_{1}&{\frac {\partial x_{2}}{\partial w}}x_{2}\\\end{bmatrix}}\cdot {\begin{bmatrix}\Delta p_{1}\\\Delta p_{2}\end{bmatrix}}}
1286:
5839:
3148:
1024:
4389:
307:
3345:
5607:
2910:
4018:
1938:
769:
For example, consumers who are running low of money for food purchase instant noodles, however, the product is not generally held as something people would normally consume on a daily basis. This is due to the constrains in terms of money; as wealth increases, consumption decreases. In this case, the
1720:
58:
can be negative for consumers as it can limit choices. He designed this formula to explore a consumer's response as the price changes. When the price increases, the budget set moves inward, which also causes the quantity demanded to decrease. In contrast, when the price decreases, the budget set
507:
The first term on the right-hand side represents the substitution effect, and the second term represents the income effect. Note that since utility is not observable, the substitution effect is not directly observable, but it can be calculated by reference to the other two terms in the
Slutsky
5856:
is a product that is in greater demand when the price increases, which are also special cases of inferior goods. In the extreme case of income inferiority, the size of income effect overpowers the size of the substitution effect, leading to a positive overall change in demand responding to an
1281:{\displaystyle {\frac {\partial h_{i}(\mathbf {p} ,u)}{\partial p_{j}}}={\frac {\partial x_{i}(\mathbf {p} ,e(\mathbf {p} ,u))}{\partial p_{j}}}+{\frac {\partial x_{i}(\mathbf {p} ,e(\mathbf {p} ,u))}{\partial e(\mathbf {p} ,u)}}\cdot {\frac {\partial e(\mathbf {p} ,u)}{\partial p_{j}}}}
4178:
112:
3159:
765:
Generally, not all goods are "normal". While in an economic sense, some are inferior. However, that does not equate quality-wise that they are poor rather that it sets a negative income profile - as income increases, consumers consumption of the good decreases.
5834:{\displaystyle \Delta x_{1}\approx \left({\frac {\partial h_{1}}{\partial p_{1}}}\Delta p_{1}+{\frac {\partial h_{1}}{\partial p_{2}}}\Delta p_{2}\right)-\left({\frac {\partial x_{1}}{\partial w}}\right)\left(x_{1}\Delta p_{1}+x_{2}\Delta p_{2}\right)}
1512:
3143:{\displaystyle {\frac {\partial h_{1}}{\partial p_{1}}}={\frac {\partial x_{1}}{\partial p_{1}}}+{\frac {\partial x_{1}}{\partial w_{1}}}x_{1}=-{\frac {.7w}{p_{1}^{2}}}+{\frac {.7}{p_{1}}}{\frac {.7w}{p_{1}}}=-{\frac {.21w}{p_{1}^{2}}}}
1381:
3804:
1801:
3702:
1573:
608:
956:
75:: when the price of good changes, as it becomes relatively cheaper, if hypothetically consumer's consumption remains same, income would be freed up which could be spent on a combination of each or more of the goods.
4384:{\displaystyle {\frac {\partial h_{1}}{\partial p_{2}}}={\frac {\partial x_{1}}{\partial p_{2}}}+{\frac {\partial x_{1}}{\partial w}}x_{2}=0+{\frac {.7}{p_{1}}}{\frac {.3w}{p_{2}}}=-{\frac {.21w}{p_{1}p_{2}}}}
660:
Overall, in simple words, the
Slutsky equation states the total change in demand consists of an income effect and a substitution effect and both effects collectively must equal the total change in demand.
3577:
734:
5035:
Although strictly speaking the
Slutsky equation only applies to infinitesimal changes in prices, it is standardly used a linear approximation for finite changes. If the prices of the two goods change by
4505:
has a negative income effect on good 1's demand, an opposite effect of the exact same size as the substitution effect, so the net effect is zero. This is a special property of the Cobb-Douglas function.
302:{\displaystyle {\partial x_{i}(\mathbf {p} ,w) \over \partial p_{j}}={\partial h_{i}(\mathbf {p} ,u) \over \partial p_{j}}-{\partial x_{i}(\mathbf {p} ,w) \over \partial w}x_{j}(\mathbf {p} ,w),\,}
5844:
The first term is the substitution effect. The second term is the income effect, composed of the consumer's response to income loss times the size of the income loss from each price's increase.
3340:{\displaystyle {\frac {\partial x_{1}}{\partial p_{1}}}={\frac {\partial h_{1}}{\partial p_{1}}}-{\frac {\partial x_{1}}{\partial w}}x_{1}=-{\frac {.21w}{p_{1}^{2}}}-{\frac {.49w}{p_{1}^{2}}}}
1789:
4170:
3770:
5857:
increase in the price. Slutsky's decomposition of the change in demand into a pure substitution effect and income effect explains why the law of demand doesn't hold for Giffen goods.
2721:
4116:
86:
of a consumer increases as a result of a price decrease, so the consumer can now afford better products or more of the same products, depending on whether the product itself is a
2647:
4476:
2761:
2684:
1549:
993:
486:
384:
347:
2902:
2848:
3464:
3391:
67:
is due to the effect of income being freed up. The equation demonstrates that the change in the demand for a good, caused by a price change, is the result of two effects:
5094:
5064:
2584:
406:
1393:
4503:
4419:
4051:
3418:
3796:
3153:
Going back to the original
Slutsky equation shows how the substitution and income effects add up to give the total effect of the price rise on quantity demanded:
4013:{\displaystyle {\frac {\partial h_{1}}{\partial p_{1}}}=-.21p_{1}^{-1.3}p_{2}^{.3}(wp_{1}^{-.7}p_{2}^{-.3})=-.21wp_{1}^{-2}p_{2}^{0}=-{\frac {.21w}{p_{1}^{2}}},}
1933:{\displaystyle \sigma (\mathbf {p} ,u):=\mathbf {D_{p}h} (\mathbf {p} ,u)={\begin{bmatrix}{\frac {\partial h_{i}(\mathbf {p} ,u)}{\partial p_{j}}}\end{bmatrix}}}
3600:
2797:
2604:
2562:
1295:
446:
426:
1715:{\displaystyle \mathbf {D_{p}x} (\mathbf {p} ,w)=\mathbf {D_{p}h} (\mathbf {p} ,u)-\mathbf {D_{w}x} (\mathbf {p} ,w)\mathbf {x} (\mathbf {p} ,w)^{\top },\,}
3608:
521:
3469:
One can check that the answer from the
Slutsky equation is the same as from directly differentiating the Hicksian demand function, which here is
870:
851:
However, whether the total effect will always be negative is impossible to tell if inferior complementary goods are mentioned. For instance, the
6009:
Varian, Hal R. “Chapter 8: Slutsky
Equation.” Essay. In Intermediate Microeconomics with Calculus, 1st ed., 137. New York, NY: W W Norton, 2014.
3420:
rises, 21/70 is from the substitution effect and 49/70 from the income effect. Good 1 is the good this consumer spends most of his income on (
4026:
The
Slutsky equation also can be applied to compute the cross-price substitution effect. One might think it was zero here because when
867:
While there are several ways to derive the
Slutsky equation, the following method is likely the simplest. Begin by noting the identity
3475:
667:
5886:
2776:
6053:
6038:
739:
The equation above is helpful as it represents the fluctuation in demand are indicative of different types of good. The
1745:
5881:
743:
will always turn out negative as indifference curves are always downward sloping. However, the same does not apply to
639:
4121:
2904:
Rearrange the
Slutsky equation to put the Hicksian derivative on the left-hand-side yields the substitution effect:
6021:
Varian, H. R. (2020). Intermediate microeconomics : a modern approach (Ninth edition.). W.W. Norton & Company.
621:
1552:
449:
3710:
5876:
508:
equation, which are observable. This process is sometimes known as the Hicks decomposition of a demand change.
2689:
6048:
4056:
512:
2609:
4424:
2729:
2652:
1517:
961:
454:
352:
315:
4172:), but that is wrong. Again rearranging the Slutsky equation, the cross-price substitution effect is:
2853:
5871:
2802:
2724:
996:
3423:
5916:
3353:
1384:
859:
pull in opposite directions. The total effect will depend on which effect is ultimately stronger.
852:
838:
833:
782:
771:
740:
72:
60:
55:
47:
5069:
5039:
2567:
1507:{\displaystyle h_{j}(\mathbf {p} ,u)=h_{j}(\mathbf {p} ,v(\mathbf {p} ,w))=x_{j}(\mathbf {p} ,w),}
389:
6043:
36:
5906:
5896:
818:
813:
759:
755:
83:
4481:
4397:
4029:
3396:
6033:
5866:
2764:
1376:{\displaystyle {\frac {\partial e(\mathbf {p} ,u)}{\partial p_{j}}}=h_{j}(\mathbf {p} ,u)}
40:
3775:
1985:
5992:
5972:
5953:
5911:
3585:
2782:
2589:
2547:
823:
431:
411:
20:
43:, which is known as such since it compensates to maintain a fixed level of utility.
6027:
3697:{\displaystyle {\frac {\partial h_{1}}{\partial p_{1}}}=-.21p_{1}^{-1.3}p_{2}^{.3}u,}
856:
786:
775:
744:
91:
79:
64:
51:
32:
448:
given by maximizing utility at the original price and income, formally given by the
5891:
843:
751:
603:{\displaystyle \varepsilon _{p,ij}=\varepsilon _{p,ij}^{h}-\varepsilon _{w,i}b_{j}}
2686:
is exactly equal to the corresponding element of the
Hicksian substitution matrix
488:. The right-hand side of the equation is equal to the change in demand for good
5901:
5853:
87:
1559:
one can substitute and rewrite the derivation above as the Slutsky equation.
789:
are positive or negative when prices increase depends on the type of goods:
5601:
Multiplying out the matrices, the effect on good 1, for example, would be
951:{\displaystyle h_{i}(\mathbf {p} ,u)=x_{i}(\mathbf {p} ,e(\mathbf {p} ,u))}
3798:
when the consumer uses the specified demand functions, the derivative is:
59:
moves outward, which leads to an increase in the quantity demanded. The
2649:, the Slutsky equation implies that each element of the Slutsky matrix
4514:
When there are two goods, the Slutsky equation in matrix form is:
747:
as it depends on how consumption of a good changes with income.
3572:{\displaystyle h_{1}(p_{1},p_{2},u)=.7p_{1}^{-.3}p_{2}^{.3}u,}
758:
or income goes up. The reverse holds when price increases and
729:{\displaystyle \Delta x_{1}=\Delta x_{1}^{s}+\Delta x_{1}^{l}}
98:
The Slutsky equation decomposes the change in demand for good
63:
is due to the effect of the relative price change while the
386:
is the Marshallian demand, at the vector of price levels
46:
There are two parts of the Slutsky equation, namely the
5940:(10th ed.). Mason, Ohio: Thomson Higher Education.
3707:
so since the Cobb-Douglas indirect utility function is
2564:
is the maximum utility the consumer achieves at prices
5547:
5372:
5323:
5160:
5111:
4855:
4692:
4529:
2723:. The Slutsky matrix is symmetric, and given that the
1868:
1732:
is the derivative operator with respect to prices and
1567:
The Slutsky equation can be rewritten in matrix form:
754:
is negative, and if the price decreases, consequently
500:
demanded, multiplied by the change in demand for good
5610:
5105:
5072:
5042:
4523:
4484:
4427:
4400:
4181:
4124:
4059:
4053:
rises, the Marshallian quantity demanded of good 1,
4032:
3807:
3778:
3713:
3611:
3588:
3478:
3426:
3399:
3356:
3162:
2913:
2856:
2805:
2785:
2732:
2692:
2655:
2612:
2592:
2570:
2550:
1956:
1804:
1748:
1576:
1520:
1396:
1298:
1027:
964:
873:
762:
or income decreases, as a result of, so does demand.
670:
524:
457:
434:
414:
392:
355:
318:
115:
1003:
is the utility obtained by maximizing utility given
5096:, the effect on the demands for the two goods are:
2799:generates Marshallian demand for goods 1 and 2 of
1739:is the derivative operator with respect to wealth.
5991:
5971:
5952:
5833:
5590:
5088:
5058:
5024:
4497:
4470:
4413:
4383:
4164:
4110:
4045:
4012:
3790:
3764:
3696:
3594:
3571:
3458:
3412:
3385:
3339:
3142:
2896:
2842:
2791:
2755:
2715:
2678:
2641:
2598:
2578:
2556:
2533:
1932:
1783:
1714:
1543:
1506:
1375:
1280:
987:
950:
728:
602:
480:
440:
420:
400:
378:
341:
301:
1784:{\displaystyle \mathbf {D_{p}h} (\mathbf {p} ,u)}
408:, wealth level (or, alternatively, income level)
3466:), which is why the income effect is so large.
4165:{\displaystyle \partial x_{1}/\partial p_{2}=0}
4023:which is indeed the Slutsky equation's answer.
4478:towards good 1. At the same time, the rise in
102:in response to a change in the price of good
8:
4421:rises, there is a substitution effect of
3765:{\displaystyle v=wp_{1}^{-.7}p_{2}^{-.3},}
1011:. Totally differentiating with respect to
511:The equation can be rewritten in terms of
5820:
5807:
5794:
5781:
5751:
5741:
5723:
5707:
5692:
5682:
5673:
5657:
5642:
5632:
5618:
5609:
5574:
5557:
5542:
5525:
5504:
5494:
5486:
5465:
5455:
5445:
5424:
5414:
5406:
5385:
5375:
5367:
5350:
5333:
5318:
5298:
5283:
5273:
5262:
5247:
5237:
5224:
5209:
5199:
5188:
5173:
5163:
5155:
5138:
5121:
5106:
5104:
5080:
5071:
5050:
5041:
5008:
4987:
4977:
4969:
4948:
4938:
4928:
4907:
4897:
4889:
4868:
4858:
4850:
4830:
4815:
4805:
4794:
4779:
4769:
4756:
4741:
4731:
4720:
4705:
4695:
4687:
4667:
4652:
4642:
4631:
4616:
4606:
4593:
4578:
4568:
4557:
4542:
4532:
4524:
4522:
4489:
4483:
4459:
4449:
4437:
4426:
4405:
4399:
4372:
4362:
4347:
4333:
4319:
4311:
4302:
4287:
4266:
4256:
4244:
4229:
4219:
4207:
4192:
4182:
4180:
4150:
4138:
4132:
4123:
4090:
4077:
4064:
4058:
4037:
4031:
3999:
3994:
3980:
3968:
3963:
3950:
3945:
3917:
3912:
3899:
3894:
3878:
3873:
3860:
3855:
3833:
3818:
3808:
3806:
3777:
3750:
3745:
3732:
3727:
3712:
3682:
3677:
3664:
3659:
3637:
3622:
3612:
3610:
3587:
3557:
3552:
3539:
3534:
3509:
3496:
3483:
3477:
3441:
3431:
3425:
3404:
3398:
3377:
3372:
3363:
3355:
3329:
3324:
3310:
3299:
3294:
3280:
3268:
3247:
3237:
3225:
3210:
3200:
3188:
3173:
3163:
3161:
3132:
3127:
3113:
3099:
3085:
3077:
3068:
3057:
3052:
3038:
3026:
3013:
2998:
2988:
2976:
2961:
2951:
2939:
2924:
2914:
2912:
2885:
2876:
2861:
2855:
2834:
2825:
2810:
2804:
2784:
2739:
2731:
2699:
2691:
2662:
2654:
2625:
2611:
2591:
2571:
2569:
2549:
2498:
2489:
2479:
2465:
2456:
2440:
2417:
2408:
2398:
2366:
2357:
2347:
2333:
2324:
2308:
2285:
2276:
2266:
2220:
2211:
2201:
2187:
2178:
2162:
2139:
2130:
2120:
2088:
2079:
2069:
2055:
2046:
2030:
2007:
1998:
1988:
1984:
1963:
1955:
1913:
1890:
1881:
1871:
1863:
1846:
1833:
1828:
1811:
1803:
1767:
1754:
1749:
1747:
1711:
1702:
1687:
1679:
1665:
1652:
1647:
1630:
1617:
1612:
1595:
1582:
1577:
1575:
1527:
1519:
1487:
1478:
1454:
1440:
1431:
1410:
1401:
1395:
1359:
1350:
1334:
1311:
1299:
1297:
1269:
1246:
1234:
1214:
1186:
1172:
1163:
1153:
1141:
1115:
1101:
1092:
1082:
1070:
1047:
1038:
1028:
1026:
971:
963:
931:
917:
908:
887:
878:
872:
720:
715:
699:
694:
678:
669:
594:
578:
565:
551:
529:
523:
464:
456:
433:
413:
393:
391:
362:
354:
325:
317:
298:
281:
272:
243:
234:
224:
212:
189:
180:
170:
158:
135:
126:
116:
114:
791:
5998:(3rd ed.). New York: W. W. Norton.
5978:(3rd ed.). New York: W. W. Norton.
5959:(3rd ed.). New York: W. W. Norton.
5928:
2763:is concave, the Slutsky matrix is also
2716:{\displaystyle \sigma (\mathbf {p} ,u)}
631:is the compensated price elasticity,
4111:{\displaystyle x_{1}(p_{1},p_{2},w),}
2775:A Cobb-Douglas utility function (see
7:
2642:{\displaystyle u=v(\mathbf {p} ,w)}
5813:
5787:
5759:
5744:
5716:
5700:
5685:
5666:
5650:
5635:
5611:
5567:
5550:
5512:
5497:
5473:
5458:
5432:
5417:
5393:
5378:
5343:
5326:
5291:
5276:
5255:
5240:
5217:
5202:
5181:
5166:
5131:
5114:
5073:
5043:
4995:
4980:
4956:
4941:
4915:
4900:
4876:
4861:
4823:
4808:
4787:
4772:
4749:
4734:
4713:
4698:
4660:
4645:
4624:
4609:
4586:
4571:
4550:
4535:
4510:Changes in multiple prices at once
4471:{\displaystyle -.21w/(p_{1}p_{2})}
4274:
4259:
4237:
4222:
4200:
4185:
4143:
4125:
3826:
3811:
3630:
3615:
3255:
3240:
3218:
3203:
3181:
3166:
3006:
2991:
2969:
2954:
2932:
2917:
2514:
2482:
2433:
2401:
2382:
2350:
2301:
2269:
2236:
2204:
2155:
2123:
2104:
2072:
2023:
1991:
1906:
1874:
1703:
1327:
1302:
1262:
1237:
1205:
1156:
1134:
1085:
1063:
1031:
708:
687:
671:
259:
227:
205:
173:
151:
119:
37:Marshallian (uncompensated) demand
14:
2756:{\displaystyle e(\mathbf {p} ,u)}
2679:{\displaystyle S(\mathbf {p} ,w)}
1544:{\displaystyle v(\mathbf {p} ,w)}
988:{\displaystyle e(\mathbf {p} ,u)}
481:{\displaystyle v(\mathbf {p} ,w)}
379:{\displaystyle x(\mathbf {p} ,w)}
342:{\displaystyle h(\mathbf {p} ,u)}
5887:Cobb-Douglas production function
2897:{\displaystyle x_{2}=.3w/p_{2}.}
2777:Cobb-Douglas production function
2740:
2700:
2663:
2626:
2572:
2499:
2466:
2418:
2367:
2334:
2286:
2221:
2188:
2140:
2089:
2056:
2008:
1964:
1891:
1847:
1839:
1834:
1830:
1812:
1768:
1760:
1755:
1751:
1688:
1680:
1666:
1658:
1653:
1649:
1631:
1623:
1618:
1614:
1596:
1588:
1583:
1579:
1528:
1488:
1455:
1441:
1411:
1360:
1312:
1247:
1215:
1187:
1173:
1116:
1102:
1048:
972:
932:
918:
888:
465:
394:
363:
326:
282:
244:
190:
136:
2843:{\displaystyle x_{1}=.7w/p_{1}}
4465:
4442:
4102:
4070:
3926:
3884:
3602:is utility. The derivative is
3521:
3489:
3459:{\displaystyle p_{1}q_{1}=.7w}
3350:Thus, of the total decline of
2750:
2736:
2710:
2696:
2673:
2659:
2636:
2622:
2509:
2495:
2476:
2462:
2428:
2414:
2377:
2363:
2344:
2330:
2296:
2282:
2231:
2217:
2198:
2184:
2150:
2136:
2099:
2085:
2066:
2052:
2018:
2004:
1974:
1960:
1901:
1887:
1857:
1843:
1822:
1808:
1778:
1764:
1699:
1684:
1676:
1662:
1641:
1627:
1606:
1592:
1538:
1524:
1498:
1484:
1468:
1465:
1451:
1437:
1421:
1407:
1370:
1356:
1322:
1308:
1257:
1243:
1225:
1211:
1200:
1197:
1183:
1169:
1129:
1126:
1112:
1098:
1058:
1044:
982:
968:
945:
942:
928:
914:
898:
884:
475:
461:
373:
359:
336:
322:
292:
278:
254:
240:
200:
186:
146:
132:
1:
3386:{\displaystyle .7w/p_{1}^{2}}
41:Hicksian (compensated) demand
5089:{\displaystyle \Delta p_{2}}
5059:{\displaystyle \Delta p_{1}}
2779:) with two goods and income
2579:{\displaystyle \mathbf {p} }
1795:and is formally defined as:
1793:Hicksian substitution matrix
1292:Making use of the fact that
401:{\displaystyle \mathbf {p} }
5882:Marshallian demand function
496:minus the quantity of good
349:is the Hicksian demand and
6070:
3393:in quantity demanded when
428:, and fixed utility level
1553:indirect utility function
1018:yields as the following:
653:the budget share of good
492:holding utility fixed at
450:indirect utility function
5877:Hicksian demand function
750:The income effect on a
620:is the (uncompensated)
6054:Mathematical economics
5994:Microeconomic Analysis
5974:Microeconomic Analysis
5955:Microeconomic Analysis
5936:Nicholson, W. (2005).
5835:
5592:
5090:
5060:
5026:
4499:
4472:
4415:
4385:
4166:
4112:
4047:
4014:
3792:
3766:
3698:
3596:
3573:
3460:
3414:
3387:
3341:
3144:
2898:
2844:
2793:
2765:negative semi-definite
2757:
2717:
2680:
2643:
2600:
2580:
2558:
2535:
1934:
1785:
1716:
1545:
1508:
1377:
1282:
989:
952:
730:
604:
482:
442:
422:
402:
380:
343:
303:
5836:
5593:
5091:
5061:
5027:
4500:
4498:{\displaystyle p_{2}}
4473:
4416:
4414:{\displaystyle p_{2}}
4386:
4167:
4113:
4048:
4046:{\displaystyle p_{2}}
4015:
3793:
3767:
3699:
3597:
3574:
3461:
3415:
3413:{\displaystyle p_{1}}
3388:
3342:
3145:
2899:
2845:
2794:
2758:
2718:
2681:
2644:
2601:
2581:
2559:
2536:
1935:
1786:
1717:
1546:
1509:
1387:and that at optimum,
1378:
1283:
990:
953:
774:is negative, but the
731:
605:
504:when wealth changes.
483:
443:
423:
403:
381:
344:
304:
35:, relates changes in
16:Equation in economics
6039:Eponyms in economics
5938:Microeconomic Theory
5608:
5103:
5070:
5040:
4521:
4482:
4425:
4398:
4394:This says that when
4179:
4122:
4057:
4030:
3805:
3776:
3711:
3609:
3586:
3476:
3424:
3397:
3354:
3160:
2911:
2854:
2803:
2783:
2730:
2725:expenditure function
2690:
2653:
2610:
2590:
2568:
2548:
1954:
1802:
1746:
1574:
1518:
1394:
1296:
1025:
997:expenditure function
962:
871:
802:Substitution Effect
668:
522:
455:
432:
412:
390:
353:
316:
113:
5990:Varian, H. (1992).
5970:Varian, H. (1992).
5951:Varian, H. (1992).
5917:Complementary goods
4004:
3973:
3958:
3925:
3907:
3883:
3868:
3791:{\displaystyle u=v}
3758:
3740:
3687:
3672:
3562:
3547:
3382:
3334:
3304:
3137:
3062:
853:substitution effect
839:Complementary goods
834:Complementary goods
793:
783:substitution effect
772:substitution effect
741:substitution effect
725:
704:
570:
73:substitution effect
61:substitution effect
56:substitution effect
48:substitution effect
5831:
5588:
5582:
5533:
5358:
5309:
5146:
5086:
5056:
5022:
5016:
4841:
4678:
4495:
4468:
4411:
4381:
4162:
4108:
4043:
4010:
3990:
3959:
3941:
3908:
3890:
3869:
3851:
3788:
3762:
3741:
3723:
3694:
3673:
3655:
3592:
3569:
3548:
3530:
3456:
3410:
3383:
3368:
3337:
3320:
3290:
3140:
3123:
3048:
2894:
2840:
2789:
2753:
2713:
2676:
2639:
2596:
2576:
2554:
2531:
2525:
1930:
1924:
1781:
1712:
1563:The Slutsky matrix
1541:
1504:
1373:
1278:
985:
948:
792:
778:is also negative.
726:
711:
690:
600:
547:
478:
438:
418:
398:
376:
339:
299:
54:. In general, the
5872:Hotelling's lemma
5766:
5714:
5664:
5519:
5480:
5439:
5400:
5305:
5269:
5231:
5195:
5002:
4963:
4922:
4883:
4837:
4801:
4763:
4727:
4674:
4638:
4600:
4564:
4379:
4339:
4317:
4281:
4251:
4214:
4005:
3840:
3644:
3595:{\displaystyle u}
3335:
3305:
3262:
3232:
3195:
3138:
3105:
3083:
3063:
3020:
2983:
2946:
2792:{\displaystyle w}
2599:{\displaystyle w}
2557:{\displaystyle u}
2521:
2447:
2389:
2315:
2243:
2169:
2111:
2037:
1920:
1341:
1276:
1229:
1148:
1077:
849:
848:
640:income elasticity
441:{\displaystyle u}
421:{\displaystyle w}
266:
219:
165:
6061:
6010:
6007:
6001:
5999:
5997:
5987:
5981:
5979:
5977:
5967:
5961:
5960:
5958:
5948:
5942:
5941:
5933:
5907:Substitute goods
5897:Purchasing power
5840:
5838:
5837:
5832:
5830:
5826:
5825:
5824:
5812:
5811:
5799:
5798:
5786:
5785:
5771:
5767:
5765:
5757:
5756:
5755:
5742:
5733:
5729:
5728:
5727:
5715:
5713:
5712:
5711:
5698:
5697:
5696:
5683:
5678:
5677:
5665:
5663:
5662:
5661:
5648:
5647:
5646:
5633:
5623:
5622:
5597:
5595:
5594:
5589:
5587:
5586:
5579:
5578:
5562:
5561:
5538:
5537:
5530:
5529:
5520:
5518:
5510:
5509:
5508:
5495:
5491:
5490:
5481:
5479:
5471:
5470:
5469:
5456:
5450:
5449:
5440:
5438:
5430:
5429:
5428:
5415:
5411:
5410:
5401:
5399:
5391:
5390:
5389:
5376:
5363:
5362:
5355:
5354:
5338:
5337:
5314:
5313:
5306:
5304:
5303:
5302:
5289:
5288:
5287:
5274:
5270:
5268:
5267:
5266:
5253:
5252:
5251:
5238:
5232:
5230:
5229:
5228:
5215:
5214:
5213:
5200:
5196:
5194:
5193:
5192:
5179:
5178:
5177:
5164:
5151:
5150:
5143:
5142:
5126:
5125:
5095:
5093:
5092:
5087:
5085:
5084:
5065:
5063:
5062:
5057:
5055:
5054:
5031:
5029:
5028:
5023:
5021:
5020:
5013:
5012:
5003:
5001:
4993:
4992:
4991:
4978:
4974:
4973:
4964:
4962:
4954:
4953:
4952:
4939:
4933:
4932:
4923:
4921:
4913:
4912:
4911:
4898:
4894:
4893:
4884:
4882:
4874:
4873:
4872:
4859:
4846:
4845:
4838:
4836:
4835:
4834:
4821:
4820:
4819:
4806:
4802:
4800:
4799:
4798:
4785:
4784:
4783:
4770:
4764:
4762:
4761:
4760:
4747:
4746:
4745:
4732:
4728:
4726:
4725:
4724:
4711:
4710:
4709:
4696:
4683:
4682:
4675:
4673:
4672:
4671:
4658:
4657:
4656:
4643:
4639:
4637:
4636:
4635:
4622:
4621:
4620:
4607:
4601:
4599:
4598:
4597:
4584:
4583:
4582:
4569:
4565:
4563:
4562:
4561:
4548:
4547:
4546:
4533:
4504:
4502:
4501:
4496:
4494:
4493:
4477:
4475:
4474:
4469:
4464:
4463:
4454:
4453:
4441:
4420:
4418:
4417:
4412:
4410:
4409:
4390:
4388:
4387:
4382:
4380:
4378:
4377:
4376:
4367:
4366:
4356:
4348:
4340:
4338:
4337:
4328:
4320:
4318:
4316:
4315:
4303:
4292:
4291:
4282:
4280:
4272:
4271:
4270:
4257:
4252:
4250:
4249:
4248:
4235:
4234:
4233:
4220:
4215:
4213:
4212:
4211:
4198:
4197:
4196:
4183:
4171:
4169:
4168:
4163:
4155:
4154:
4142:
4137:
4136:
4118:is unaffected (
4117:
4115:
4114:
4109:
4095:
4094:
4082:
4081:
4069:
4068:
4052:
4050:
4049:
4044:
4042:
4041:
4019:
4017:
4016:
4011:
4006:
4003:
3998:
3989:
3981:
3972:
3967:
3957:
3949:
3924:
3916:
3906:
3898:
3882:
3877:
3867:
3859:
3841:
3839:
3838:
3837:
3824:
3823:
3822:
3809:
3797:
3795:
3794:
3789:
3771:
3769:
3768:
3763:
3757:
3749:
3739:
3731:
3703:
3701:
3700:
3695:
3686:
3681:
3671:
3663:
3645:
3643:
3642:
3641:
3628:
3627:
3626:
3613:
3601:
3599:
3598:
3593:
3578:
3576:
3575:
3570:
3561:
3556:
3546:
3538:
3514:
3513:
3501:
3500:
3488:
3487:
3465:
3463:
3462:
3457:
3446:
3445:
3436:
3435:
3419:
3417:
3416:
3411:
3409:
3408:
3392:
3390:
3389:
3384:
3381:
3376:
3367:
3346:
3344:
3343:
3338:
3336:
3333:
3328:
3319:
3311:
3306:
3303:
3298:
3289:
3281:
3273:
3272:
3263:
3261:
3253:
3252:
3251:
3238:
3233:
3231:
3230:
3229:
3216:
3215:
3214:
3201:
3196:
3194:
3193:
3192:
3179:
3178:
3177:
3164:
3149:
3147:
3146:
3141:
3139:
3136:
3131:
3122:
3114:
3106:
3104:
3103:
3094:
3086:
3084:
3082:
3081:
3069:
3064:
3061:
3056:
3047:
3039:
3031:
3030:
3021:
3019:
3018:
3017:
3004:
3003:
3002:
2989:
2984:
2982:
2981:
2980:
2967:
2966:
2965:
2952:
2947:
2945:
2944:
2943:
2930:
2929:
2928:
2915:
2903:
2901:
2900:
2895:
2890:
2889:
2880:
2866:
2865:
2849:
2847:
2846:
2841:
2839:
2838:
2829:
2815:
2814:
2798:
2796:
2795:
2790:
2762:
2760:
2759:
2754:
2743:
2722:
2720:
2719:
2714:
2703:
2685:
2683:
2682:
2677:
2666:
2648:
2646:
2645:
2640:
2629:
2605:
2603:
2602:
2597:
2585:
2583:
2582:
2577:
2575:
2563:
2561:
2560:
2555:
2540:
2538:
2537:
2532:
2530:
2526:
2522:
2520:
2512:
2502:
2494:
2493:
2480:
2469:
2461:
2460:
2448:
2446:
2445:
2444:
2431:
2421:
2413:
2412:
2399:
2390:
2388:
2380:
2370:
2362:
2361:
2348:
2337:
2329:
2328:
2316:
2314:
2313:
2312:
2299:
2289:
2281:
2280:
2267:
2244:
2242:
2234:
2224:
2216:
2215:
2202:
2191:
2183:
2182:
2170:
2168:
2167:
2166:
2153:
2143:
2135:
2134:
2121:
2112:
2110:
2102:
2092:
2084:
2083:
2070:
2059:
2051:
2050:
2038:
2036:
2035:
2034:
2021:
2011:
2003:
2002:
1989:
1967:
1939:
1937:
1936:
1931:
1929:
1928:
1921:
1919:
1918:
1917:
1904:
1894:
1886:
1885:
1872:
1850:
1842:
1838:
1837:
1815:
1791:is known as the
1790:
1788:
1787:
1782:
1771:
1763:
1759:
1758:
1721:
1719:
1718:
1713:
1707:
1706:
1691:
1683:
1669:
1661:
1657:
1656:
1634:
1626:
1622:
1621:
1599:
1591:
1587:
1586:
1550:
1548:
1547:
1542:
1531:
1513:
1511:
1510:
1505:
1491:
1483:
1482:
1458:
1444:
1436:
1435:
1414:
1406:
1405:
1385:Shephard's lemma
1382:
1380:
1379:
1374:
1363:
1355:
1354:
1342:
1340:
1339:
1338:
1325:
1315:
1300:
1287:
1285:
1284:
1279:
1277:
1275:
1274:
1273:
1260:
1250:
1235:
1230:
1228:
1218:
1203:
1190:
1176:
1168:
1167:
1154:
1149:
1147:
1146:
1145:
1132:
1119:
1105:
1097:
1096:
1083:
1078:
1076:
1075:
1074:
1061:
1051:
1043:
1042:
1029:
994:
992:
991:
986:
975:
957:
955:
954:
949:
935:
921:
913:
912:
891:
883:
882:
819:Substitute goods
814:Substitute goods
794:
781:In any case the
760:purchasing power
756:purchasing power
735:
733:
732:
727:
724:
719:
703:
698:
683:
682:
622:price elasticity
609:
607:
606:
601:
599:
598:
589:
588:
569:
564:
543:
542:
487:
485:
484:
479:
468:
447:
445:
444:
439:
427:
425:
424:
419:
407:
405:
404:
399:
397:
385:
383:
382:
377:
366:
348:
346:
345:
340:
329:
308:
306:
305:
300:
285:
277:
276:
267:
265:
257:
247:
239:
238:
225:
220:
218:
217:
216:
203:
193:
185:
184:
171:
166:
164:
163:
162:
149:
139:
131:
130:
117:
84:purchasing power
29:Slutsky identity
25:Slutsky equation
6069:
6068:
6064:
6063:
6062:
6060:
6059:
6058:
6024:
6023:
6019:
6014:
6013:
6008:
6004:
6000:, pp. 120-121.
5989:
5988:
5984:
5969:
5968:
5964:
5950:
5949:
5945:
5935:
5934:
5930:
5925:
5867:Consumer choice
5863:
5850:
5816:
5803:
5790:
5777:
5776:
5772:
5758:
5747:
5743:
5737:
5719:
5703:
5699:
5688:
5684:
5669:
5653:
5649:
5638:
5634:
5631:
5627:
5614:
5606:
5605:
5581:
5580:
5570:
5564:
5563:
5553:
5543:
5532:
5531:
5521:
5511:
5500:
5496:
5492:
5482:
5472:
5461:
5457:
5452:
5451:
5441:
5431:
5420:
5416:
5412:
5402:
5392:
5381:
5377:
5368:
5357:
5356:
5346:
5340:
5339:
5329:
5319:
5308:
5307:
5294:
5290:
5279:
5275:
5271:
5258:
5254:
5243:
5239:
5234:
5233:
5220:
5216:
5205:
5201:
5197:
5184:
5180:
5169:
5165:
5156:
5145:
5144:
5134:
5128:
5127:
5117:
5107:
5101:
5100:
5076:
5068:
5067:
5046:
5038:
5037:
5015:
5014:
5004:
4994:
4983:
4979:
4975:
4965:
4955:
4944:
4940:
4935:
4934:
4924:
4914:
4903:
4899:
4895:
4885:
4875:
4864:
4860:
4851:
4840:
4839:
4826:
4822:
4811:
4807:
4803:
4790:
4786:
4775:
4771:
4766:
4765:
4752:
4748:
4737:
4733:
4729:
4716:
4712:
4701:
4697:
4688:
4677:
4676:
4663:
4659:
4648:
4644:
4640:
4627:
4623:
4612:
4608:
4603:
4602:
4589:
4585:
4574:
4570:
4566:
4553:
4549:
4538:
4534:
4525:
4519:
4518:
4512:
4485:
4480:
4479:
4455:
4445:
4423:
4422:
4401:
4396:
4395:
4368:
4358:
4357:
4349:
4329:
4321:
4307:
4283:
4273:
4262:
4258:
4240:
4236:
4225:
4221:
4203:
4199:
4188:
4184:
4177:
4176:
4146:
4128:
4120:
4119:
4086:
4073:
4060:
4055:
4054:
4033:
4028:
4027:
3982:
3829:
3825:
3814:
3810:
3803:
3802:
3774:
3773:
3709:
3708:
3633:
3629:
3618:
3614:
3607:
3606:
3584:
3583:
3505:
3492:
3479:
3474:
3473:
3437:
3427:
3422:
3421:
3400:
3395:
3394:
3352:
3351:
3312:
3282:
3264:
3254:
3243:
3239:
3221:
3217:
3206:
3202:
3184:
3180:
3169:
3165:
3158:
3157:
3115:
3095:
3087:
3073:
3040:
3022:
3009:
3005:
2994:
2990:
2972:
2968:
2957:
2953:
2935:
2931:
2920:
2916:
2909:
2908:
2881:
2857:
2852:
2851:
2830:
2806:
2801:
2800:
2781:
2780:
2773:
2728:
2727:
2688:
2687:
2651:
2650:
2608:
2607:
2588:
2587:
2566:
2565:
2546:
2545:
2524:
2523:
2513:
2485:
2481:
2452:
2436:
2432:
2404:
2400:
2396:
2391:
2381:
2353:
2349:
2320:
2304:
2300:
2272:
2268:
2263:
2262:
2257:
2252:
2246:
2245:
2235:
2207:
2203:
2174:
2158:
2154:
2126:
2122:
2118:
2113:
2103:
2075:
2071:
2042:
2026:
2022:
1994:
1990:
1980:
1952:
1951:
1923:
1922:
1909:
1905:
1877:
1873:
1864:
1829:
1800:
1799:
1750:
1744:
1743:
1737:
1730:
1698:
1648:
1613:
1578:
1572:
1571:
1565:
1516:
1515:
1474:
1427:
1397:
1392:
1391:
1346:
1330:
1326:
1301:
1294:
1293:
1265:
1261:
1236:
1204:
1159:
1155:
1137:
1133:
1088:
1084:
1066:
1062:
1034:
1030:
1023:
1022:
1016:
960:
959:
904:
874:
869:
868:
865:
674:
666:
665:
651:
636:
629:
618:
590:
574:
525:
520:
519:
453:
452:
430:
429:
410:
409:
388:
387:
351:
350:
314:
313:
268:
258:
230:
226:
208:
204:
176:
172:
154:
150:
122:
118:
111:
110:
31:), named after
17:
12:
11:
5:
6067:
6065:
6057:
6056:
6051:
6049:Microeconomics
6046:
6041:
6036:
6026:
6025:
6018:
6015:
6012:
6011:
6002:
5982:
5962:
5943:
5927:
5926:
5924:
5921:
5920:
5919:
5914:
5912:Inferior goods
5909:
5904:
5899:
5894:
5889:
5884:
5879:
5874:
5869:
5862:
5859:
5849:
5846:
5842:
5841:
5829:
5823:
5819:
5815:
5810:
5806:
5802:
5797:
5793:
5789:
5784:
5780:
5775:
5770:
5764:
5761:
5754:
5750:
5746:
5740:
5736:
5732:
5726:
5722:
5718:
5710:
5706:
5702:
5695:
5691:
5687:
5681:
5676:
5672:
5668:
5660:
5656:
5652:
5645:
5641:
5637:
5630:
5626:
5621:
5617:
5613:
5599:
5598:
5585:
5577:
5573:
5569:
5566:
5565:
5560:
5556:
5552:
5549:
5548:
5546:
5541:
5536:
5528:
5524:
5517:
5514:
5507:
5503:
5499:
5493:
5489:
5485:
5478:
5475:
5468:
5464:
5460:
5454:
5453:
5448:
5444:
5437:
5434:
5427:
5423:
5419:
5413:
5409:
5405:
5398:
5395:
5388:
5384:
5380:
5374:
5373:
5371:
5366:
5361:
5353:
5349:
5345:
5342:
5341:
5336:
5332:
5328:
5325:
5324:
5322:
5317:
5312:
5301:
5297:
5293:
5286:
5282:
5278:
5272:
5265:
5261:
5257:
5250:
5246:
5242:
5236:
5235:
5227:
5223:
5219:
5212:
5208:
5204:
5198:
5191:
5187:
5183:
5176:
5172:
5168:
5162:
5161:
5159:
5154:
5149:
5141:
5137:
5133:
5130:
5129:
5124:
5120:
5116:
5113:
5112:
5110:
5083:
5079:
5075:
5053:
5049:
5045:
5033:
5032:
5019:
5011:
5007:
5000:
4997:
4990:
4986:
4982:
4976:
4972:
4968:
4961:
4958:
4951:
4947:
4943:
4937:
4936:
4931:
4927:
4920:
4917:
4910:
4906:
4902:
4896:
4892:
4888:
4881:
4878:
4871:
4867:
4863:
4857:
4856:
4854:
4849:
4844:
4833:
4829:
4825:
4818:
4814:
4810:
4804:
4797:
4793:
4789:
4782:
4778:
4774:
4768:
4767:
4759:
4755:
4751:
4744:
4740:
4736:
4730:
4723:
4719:
4715:
4708:
4704:
4700:
4694:
4693:
4691:
4686:
4681:
4670:
4666:
4662:
4655:
4651:
4647:
4641:
4634:
4630:
4626:
4619:
4615:
4611:
4605:
4604:
4596:
4592:
4588:
4581:
4577:
4573:
4567:
4560:
4556:
4552:
4545:
4541:
4537:
4531:
4530:
4528:
4511:
4508:
4492:
4488:
4467:
4462:
4458:
4452:
4448:
4444:
4440:
4436:
4433:
4430:
4408:
4404:
4392:
4391:
4375:
4371:
4365:
4361:
4355:
4352:
4346:
4343:
4336:
4332:
4327:
4324:
4314:
4310:
4306:
4301:
4298:
4295:
4290:
4286:
4279:
4276:
4269:
4265:
4261:
4255:
4247:
4243:
4239:
4232:
4228:
4224:
4218:
4210:
4206:
4202:
4195:
4191:
4187:
4161:
4158:
4153:
4149:
4145:
4141:
4135:
4131:
4127:
4107:
4104:
4101:
4098:
4093:
4089:
4085:
4080:
4076:
4072:
4067:
4063:
4040:
4036:
4021:
4020:
4009:
4002:
3997:
3993:
3988:
3985:
3979:
3976:
3971:
3966:
3962:
3956:
3953:
3948:
3944:
3940:
3937:
3934:
3931:
3928:
3923:
3920:
3915:
3911:
3905:
3902:
3897:
3893:
3889:
3886:
3881:
3876:
3872:
3866:
3863:
3858:
3854:
3850:
3847:
3844:
3836:
3832:
3828:
3821:
3817:
3813:
3787:
3784:
3781:
3761:
3756:
3753:
3748:
3744:
3738:
3735:
3730:
3726:
3722:
3719:
3716:
3705:
3704:
3693:
3690:
3685:
3680:
3676:
3670:
3667:
3662:
3658:
3654:
3651:
3648:
3640:
3636:
3632:
3625:
3621:
3617:
3591:
3580:
3579:
3568:
3565:
3560:
3555:
3551:
3545:
3542:
3537:
3533:
3529:
3526:
3523:
3520:
3517:
3512:
3508:
3504:
3499:
3495:
3491:
3486:
3482:
3455:
3452:
3449:
3444:
3440:
3434:
3430:
3407:
3403:
3380:
3375:
3371:
3366:
3362:
3359:
3348:
3347:
3332:
3327:
3323:
3318:
3315:
3309:
3302:
3297:
3293:
3288:
3285:
3279:
3276:
3271:
3267:
3260:
3257:
3250:
3246:
3242:
3236:
3228:
3224:
3220:
3213:
3209:
3205:
3199:
3191:
3187:
3183:
3176:
3172:
3168:
3151:
3150:
3135:
3130:
3126:
3121:
3118:
3112:
3109:
3102:
3098:
3093:
3090:
3080:
3076:
3072:
3067:
3060:
3055:
3051:
3046:
3043:
3037:
3034:
3029:
3025:
3016:
3012:
3008:
3001:
2997:
2993:
2987:
2979:
2975:
2971:
2964:
2960:
2956:
2950:
2942:
2938:
2934:
2927:
2923:
2919:
2893:
2888:
2884:
2879:
2875:
2872:
2869:
2864:
2860:
2837:
2833:
2828:
2824:
2821:
2818:
2813:
2809:
2788:
2772:
2769:
2752:
2749:
2746:
2742:
2738:
2735:
2712:
2709:
2706:
2702:
2698:
2695:
2675:
2672:
2669:
2665:
2661:
2658:
2638:
2635:
2632:
2628:
2624:
2621:
2618:
2615:
2595:
2574:
2553:
2542:
2541:
2529:
2519:
2516:
2511:
2508:
2505:
2501:
2497:
2492:
2488:
2484:
2478:
2475:
2472:
2468:
2464:
2459:
2455:
2451:
2443:
2439:
2435:
2430:
2427:
2424:
2420:
2416:
2411:
2407:
2403:
2397:
2395:
2392:
2387:
2384:
2379:
2376:
2373:
2369:
2365:
2360:
2356:
2352:
2346:
2343:
2340:
2336:
2332:
2327:
2323:
2319:
2311:
2307:
2303:
2298:
2295:
2292:
2288:
2284:
2279:
2275:
2271:
2265:
2264:
2261:
2258:
2256:
2253:
2251:
2248:
2247:
2241:
2238:
2233:
2230:
2227:
2223:
2219:
2214:
2210:
2206:
2200:
2197:
2194:
2190:
2186:
2181:
2177:
2173:
2165:
2161:
2157:
2152:
2149:
2146:
2142:
2138:
2133:
2129:
2125:
2119:
2117:
2114:
2109:
2106:
2101:
2098:
2095:
2091:
2087:
2082:
2078:
2074:
2068:
2065:
2062:
2058:
2054:
2049:
2045:
2041:
2033:
2029:
2025:
2020:
2017:
2014:
2010:
2006:
2001:
1997:
1993:
1987:
1986:
1983:
1979:
1976:
1973:
1970:
1966:
1962:
1959:
1945:Slutsky matrix
1941:
1940:
1927:
1916:
1912:
1908:
1903:
1900:
1897:
1893:
1889:
1884:
1880:
1876:
1870:
1869:
1867:
1862:
1859:
1856:
1853:
1849:
1845:
1841:
1836:
1832:
1827:
1824:
1821:
1818:
1814:
1810:
1807:
1780:
1777:
1774:
1770:
1766:
1762:
1757:
1753:
1735:
1728:
1723:
1722:
1710:
1705:
1701:
1697:
1694:
1690:
1686:
1682:
1678:
1675:
1672:
1668:
1664:
1660:
1655:
1651:
1646:
1643:
1640:
1637:
1633:
1629:
1625:
1620:
1616:
1611:
1608:
1605:
1602:
1598:
1594:
1590:
1585:
1581:
1564:
1561:
1557:
1556:
1540:
1537:
1534:
1530:
1526:
1523:
1503:
1500:
1497:
1494:
1490:
1486:
1481:
1477:
1473:
1470:
1467:
1464:
1461:
1457:
1453:
1450:
1447:
1443:
1439:
1434:
1430:
1426:
1423:
1420:
1417:
1413:
1409:
1404:
1400:
1372:
1369:
1366:
1362:
1358:
1353:
1349:
1345:
1337:
1333:
1329:
1324:
1321:
1318:
1314:
1310:
1307:
1304:
1290:
1289:
1272:
1268:
1264:
1259:
1256:
1253:
1249:
1245:
1242:
1239:
1233:
1227:
1224:
1221:
1217:
1213:
1210:
1207:
1202:
1199:
1196:
1193:
1189:
1185:
1182:
1179:
1175:
1171:
1166:
1162:
1158:
1152:
1144:
1140:
1136:
1131:
1128:
1125:
1122:
1118:
1114:
1111:
1108:
1104:
1100:
1095:
1091:
1087:
1081:
1073:
1069:
1065:
1060:
1057:
1054:
1050:
1046:
1041:
1037:
1033:
1014:
984:
981:
978:
974:
970:
967:
947:
944:
941:
938:
934:
930:
927:
924:
920:
916:
911:
907:
903:
900:
897:
894:
890:
886:
881:
877:
864:
861:
847:
846:
841:
836:
831:
827:
826:
824:Inferior goods
821:
816:
811:
807:
806:
805:Income Effect
803:
800:
797:
737:
736:
723:
718:
714:
710:
707:
702:
697:
693:
689:
686:
681:
677:
673:
649:
634:
627:
616:
611:
610:
597:
593:
587:
584:
581:
577:
573:
568:
563:
560:
557:
554:
550:
546:
541:
538:
535:
532:
528:
477:
474:
471:
467:
463:
460:
437:
417:
396:
375:
372:
369:
365:
361:
358:
338:
335:
332:
328:
324:
321:
310:
309:
297:
294:
291:
288:
284:
280:
275:
271:
264:
261:
256:
253:
250:
246:
242:
237:
233:
229:
223:
215:
211:
207:
202:
199:
196:
192:
188:
183:
179:
175:
169:
161:
157:
153:
148:
145:
142:
138:
134:
129:
125:
121:
96:
95:
76:
39:to changes in
21:microeconomics
15:
13:
10:
9:
6:
4:
3:
2:
6066:
6055:
6052:
6050:
6047:
6045:
6042:
6040:
6037:
6035:
6032:
6031:
6029:
6022:
6016:
6006:
6003:
5996:
5995:
5986:
5983:
5976:
5975:
5966:
5963:
5957:
5956:
5947:
5944:
5939:
5932:
5929:
5922:
5918:
5915:
5913:
5910:
5908:
5905:
5903:
5900:
5898:
5895:
5893:
5890:
5888:
5885:
5883:
5880:
5878:
5875:
5873:
5870:
5868:
5865:
5864:
5860:
5858:
5855:
5847:
5845:
5827:
5821:
5817:
5808:
5804:
5800:
5795:
5791:
5782:
5778:
5773:
5768:
5762:
5752:
5748:
5738:
5734:
5730:
5724:
5720:
5708:
5704:
5693:
5689:
5679:
5674:
5670:
5658:
5654:
5643:
5639:
5628:
5624:
5619:
5615:
5604:
5603:
5602:
5583:
5575:
5571:
5558:
5554:
5544:
5539:
5534:
5526:
5522:
5515:
5505:
5501:
5487:
5483:
5476:
5466:
5462:
5446:
5442:
5435:
5425:
5421:
5407:
5403:
5396:
5386:
5382:
5369:
5364:
5359:
5351:
5347:
5334:
5330:
5320:
5315:
5310:
5299:
5295:
5284:
5280:
5263:
5259:
5248:
5244:
5225:
5221:
5210:
5206:
5189:
5185:
5174:
5170:
5157:
5152:
5147:
5139:
5135:
5122:
5118:
5108:
5099:
5098:
5097:
5081:
5077:
5051:
5047:
5017:
5009:
5005:
4998:
4988:
4984:
4970:
4966:
4959:
4949:
4945:
4929:
4925:
4918:
4908:
4904:
4890:
4886:
4879:
4869:
4865:
4852:
4847:
4842:
4831:
4827:
4816:
4812:
4795:
4791:
4780:
4776:
4757:
4753:
4742:
4738:
4721:
4717:
4706:
4702:
4689:
4684:
4679:
4668:
4664:
4653:
4649:
4632:
4628:
4617:
4613:
4594:
4590:
4579:
4575:
4558:
4554:
4543:
4539:
4526:
4517:
4516:
4515:
4509:
4507:
4490:
4486:
4460:
4456:
4450:
4446:
4438:
4434:
4431:
4428:
4406:
4402:
4373:
4369:
4363:
4359:
4353:
4350:
4344:
4341:
4334:
4330:
4325:
4322:
4312:
4308:
4304:
4299:
4296:
4293:
4288:
4284:
4277:
4267:
4263:
4253:
4245:
4241:
4230:
4226:
4216:
4208:
4204:
4193:
4189:
4175:
4174:
4173:
4159:
4156:
4151:
4147:
4139:
4133:
4129:
4105:
4099:
4096:
4091:
4087:
4083:
4078:
4074:
4065:
4061:
4038:
4034:
4024:
4007:
4000:
3995:
3991:
3986:
3983:
3977:
3974:
3969:
3964:
3960:
3954:
3951:
3946:
3942:
3938:
3935:
3932:
3929:
3921:
3918:
3913:
3909:
3903:
3900:
3895:
3891:
3887:
3879:
3874:
3870:
3864:
3861:
3856:
3852:
3848:
3845:
3842:
3834:
3830:
3819:
3815:
3801:
3800:
3799:
3785:
3782:
3779:
3759:
3754:
3751:
3746:
3742:
3736:
3733:
3728:
3724:
3720:
3717:
3714:
3691:
3688:
3683:
3678:
3674:
3668:
3665:
3660:
3656:
3652:
3649:
3646:
3638:
3634:
3623:
3619:
3605:
3604:
3603:
3589:
3566:
3563:
3558:
3553:
3549:
3543:
3540:
3535:
3531:
3527:
3524:
3518:
3515:
3510:
3506:
3502:
3497:
3493:
3484:
3480:
3472:
3471:
3470:
3467:
3453:
3450:
3447:
3442:
3438:
3432:
3428:
3405:
3401:
3378:
3373:
3369:
3364:
3360:
3357:
3330:
3325:
3321:
3316:
3313:
3307:
3300:
3295:
3291:
3286:
3283:
3277:
3274:
3269:
3265:
3258:
3248:
3244:
3234:
3226:
3222:
3211:
3207:
3197:
3189:
3185:
3174:
3170:
3156:
3155:
3154:
3133:
3128:
3124:
3119:
3116:
3110:
3107:
3100:
3096:
3091:
3088:
3078:
3074:
3070:
3065:
3058:
3053:
3049:
3044:
3041:
3035:
3032:
3027:
3023:
3014:
3010:
2999:
2995:
2985:
2977:
2973:
2962:
2958:
2948:
2940:
2936:
2925:
2921:
2907:
2906:
2905:
2891:
2886:
2882:
2877:
2873:
2870:
2867:
2862:
2858:
2835:
2831:
2826:
2822:
2819:
2816:
2811:
2807:
2786:
2778:
2770:
2768:
2766:
2747:
2744:
2733:
2726:
2707:
2704:
2693:
2670:
2667:
2656:
2633:
2630:
2619:
2616:
2613:
2593:
2551:
2527:
2517:
2506:
2503:
2490:
2486:
2473:
2470:
2457:
2453:
2449:
2441:
2437:
2425:
2422:
2409:
2405:
2393:
2385:
2374:
2371:
2358:
2354:
2341:
2338:
2325:
2321:
2317:
2309:
2305:
2293:
2290:
2277:
2273:
2259:
2254:
2249:
2239:
2228:
2225:
2212:
2208:
2195:
2192:
2179:
2175:
2171:
2163:
2159:
2147:
2144:
2131:
2127:
2115:
2107:
2096:
2093:
2080:
2076:
2063:
2060:
2047:
2043:
2039:
2031:
2027:
2015:
2012:
1999:
1995:
1981:
1977:
1971:
1968:
1957:
1950:
1949:
1948:
1947:is given by:
1946:
1925:
1914:
1910:
1898:
1895:
1882:
1878:
1865:
1860:
1854:
1851:
1825:
1819:
1816:
1805:
1798:
1797:
1796:
1794:
1775:
1772:
1740:
1738:
1731:
1708:
1695:
1692:
1673:
1670:
1644:
1638:
1635:
1609:
1603:
1600:
1570:
1569:
1568:
1562:
1560:
1554:
1535:
1532:
1521:
1501:
1495:
1492:
1479:
1475:
1471:
1462:
1459:
1448:
1445:
1432:
1428:
1424:
1418:
1415:
1402:
1398:
1390:
1389:
1388:
1386:
1367:
1364:
1351:
1347:
1343:
1335:
1331:
1319:
1316:
1305:
1270:
1266:
1254:
1251:
1240:
1231:
1222:
1219:
1208:
1194:
1191:
1180:
1177:
1164:
1160:
1150:
1142:
1138:
1123:
1120:
1109:
1106:
1093:
1089:
1079:
1071:
1067:
1055:
1052:
1039:
1035:
1021:
1020:
1019:
1017:
1010:
1006:
1002:
998:
979:
976:
965:
939:
936:
925:
922:
909:
905:
901:
895:
892:
879:
875:
862:
860:
858:
857:income effect
854:
845:
842:
840:
837:
835:
832:
829:
828:
825:
822:
820:
817:
815:
812:
809:
808:
804:
801:
799:Total Effect
798:
796:
795:
790:
788:
787:income effect
784:
779:
777:
776:income effect
773:
767:
763:
761:
757:
753:
748:
746:
745:income effect
742:
721:
716:
712:
705:
700:
695:
691:
684:
679:
675:
664:
663:
662:
658:
656:
652:
645:
641:
637:
630:
623:
619:
595:
591:
585:
582:
579:
575:
571:
566:
561:
558:
555:
552:
548:
544:
539:
536:
533:
530:
526:
518:
517:
516:
514:
509:
505:
503:
499:
495:
491:
472:
469:
458:
451:
435:
415:
370:
367:
356:
333:
330:
319:
295:
289:
286:
273:
269:
262:
251:
248:
235:
231:
221:
213:
209:
197:
194:
181:
177:
167:
159:
155:
143:
140:
127:
123:
109:
108:
107:
105:
101:
93:
92:inferior good
89:
85:
81:
80:income effect
77:
74:
70:
69:
68:
66:
65:income effect
62:
57:
53:
52:income effect
49:
44:
42:
38:
34:
33:Eugen Slutsky
30:
26:
22:
6020:
6005:
5993:
5985:
5973:
5965:
5954:
5946:
5937:
5931:
5892:Giffen Goods
5851:
5848:Giffen goods
5843:
5600:
5034:
4513:
4393:
4025:
4022:
3706:
3581:
3468:
3349:
3152:
2774:
2543:
1944:
1942:
1792:
1741:
1733:
1726:
1724:
1566:
1558:
1291:
1012:
1008:
1004:
1000:
866:
850:
844:Normal goods
780:
768:
764:
752:normal goods
749:
738:
659:
654:
647:
643:
632:
625:
614:
612:
510:
506:
501:
497:
493:
489:
311:
103:
99:
97:
88:normal good
45:
28:
24:
18:
5902:Normal good
5854:Giffen good
2606:, that is,
2586:and income
1742:The matrix
6028:Categories
6017:References
5923:References
863:Derivation
513:elasticity
6044:Equations
5980:, p. 121.
5814:Δ
5788:Δ
5760:∂
5745:∂
5735:−
5717:Δ
5701:∂
5686:∂
5667:Δ
5651:∂
5636:∂
5625:≈
5612:Δ
5568:Δ
5551:Δ
5540:⋅
5513:∂
5498:∂
5474:∂
5459:∂
5433:∂
5418:∂
5394:∂
5379:∂
5365:−
5344:Δ
5327:Δ
5316:⋅
5292:∂
5277:∂
5256:∂
5241:∂
5218:∂
5203:∂
5182:∂
5167:∂
5153:≈
5132:Δ
5115:Δ
5074:Δ
5044:Δ
4996:∂
4981:∂
4957:∂
4942:∂
4916:∂
4901:∂
4877:∂
4862:∂
4848:−
4824:∂
4809:∂
4788:∂
4773:∂
4750:∂
4735:∂
4714:∂
4699:∂
4661:∂
4646:∂
4625:∂
4610:∂
4587:∂
4572:∂
4551:∂
4536:∂
4429:−
4345:−
4275:∂
4260:∂
4238:∂
4223:∂
4201:∂
4186:∂
4144:∂
4126:∂
3978:−
3952:−
3933:−
3919:−
3901:−
3862:−
3846:−
3827:∂
3812:∂
3752:−
3734:−
3666:−
3650:−
3631:∂
3616:∂
3541:−
3308:−
3278:−
3256:∂
3241:∂
3235:−
3219:∂
3204:∂
3182:∂
3167:∂
3111:−
3036:−
3007:∂
2992:∂
2970:∂
2955:∂
2933:∂
2918:∂
2694:σ
2515:∂
2483:∂
2434:∂
2402:∂
2394:⋯
2383:∂
2351:∂
2302:∂
2270:∂
2260:⋮
2255:⋱
2250:⋮
2237:∂
2205:∂
2156:∂
2124:∂
2116:⋯
2105:∂
2073:∂
2024:∂
1992:∂
1907:∂
1875:∂
1806:σ
1704:⊤
1645:−
1328:∂
1303:∂
1263:∂
1238:∂
1232:⋅
1206:∂
1157:∂
1135:∂
1086:∂
1064:∂
1032:∂
709:Δ
688:Δ
672:Δ
576:ε
572:−
549:ε
527:ε
260:∂
228:∂
222:−
206:∂
174:∂
152:∂
120:∂
5861:See also
855:and the
642:of good
2771:Example
1551:is the
995:is the
6034:Demand
3582:where
2850:and
1725:where
1514:where
999:, and
958:where
646:, and
613:where
312:where
90:or an
82:: the
50:, and
23:, the
2544:When
5066:and
3772:and
1943:The
1007:and
638:the
27:(or
4432:.21
4351:.21
3984:.21
3936:.21
3865:1.3
3849:.21
3669:1.3
3653:.21
3314:.49
3284:.21
3117:.21
1383:by
785:or
635:w,i
78:an
19:In
6030::
5852:A
4323:.3
4305:.7
3922:.3
3904:.7
3880:.3
3755:.3
3737:.7
3684:.3
3559:.3
3544:.3
3528:.7
3451:.7
3358:.7
3089:.7
3071:.7
3042:.7
2871:.3
2820:.7
2767:.
1826::=
830:-
810:+
657:.
624:,
515::
106::
71:a
5828:)
5822:2
5818:p
5809:2
5805:x
5801:+
5796:1
5792:p
5783:1
5779:x
5774:(
5769:)
5763:w
5753:1
5749:x
5739:(
5731:)
5725:2
5721:p
5709:2
5705:p
5694:1
5690:h
5680:+
5675:1
5671:p
5659:1
5655:p
5644:1
5640:h
5629:(
5620:1
5616:x
5584:]
5576:2
5572:p
5559:1
5555:p
5545:[
5535:]
5527:2
5523:x
5516:w
5506:2
5502:x
5488:1
5484:x
5477:w
5467:2
5463:x
5447:2
5443:x
5436:w
5426:1
5422:x
5408:1
5404:x
5397:w
5387:1
5383:x
5370:[
5360:]
5352:2
5348:p
5335:1
5331:p
5321:[
5311:]
5300:2
5296:p
5285:2
5281:h
5264:1
5260:p
5249:2
5245:h
5226:2
5222:p
5211:1
5207:h
5190:1
5186:p
5175:1
5171:h
5158:[
5148:]
5140:2
5136:x
5123:1
5119:x
5109:[
5082:2
5078:p
5052:1
5048:p
5018:]
5010:2
5006:x
4999:w
4989:2
4985:x
4971:1
4967:x
4960:w
4950:2
4946:x
4930:2
4926:x
4919:w
4909:1
4905:x
4891:1
4887:x
4880:w
4870:1
4866:x
4853:[
4843:]
4832:2
4828:p
4817:2
4813:h
4796:1
4792:p
4781:2
4777:h
4758:2
4754:p
4743:1
4739:h
4722:1
4718:p
4707:1
4703:h
4690:[
4685:=
4680:]
4669:2
4665:p
4654:2
4650:x
4633:1
4629:p
4618:2
4614:x
4595:2
4591:p
4580:1
4576:x
4559:1
4555:p
4544:1
4540:x
4527:[
4491:2
4487:p
4466:)
4461:2
4457:p
4451:1
4447:p
4443:(
4439:/
4435:w
4407:2
4403:p
4374:2
4370:p
4364:1
4360:p
4354:w
4342:=
4335:2
4331:p
4326:w
4313:1
4309:p
4300:+
4297:0
4294:=
4289:2
4285:x
4278:w
4268:1
4264:x
4254:+
4246:2
4242:p
4231:1
4227:x
4217:=
4209:2
4205:p
4194:1
4190:h
4160:0
4157:=
4152:2
4148:p
4140:/
4134:1
4130:x
4106:,
4103:)
4100:w
4097:,
4092:2
4088:p
4084:,
4079:1
4075:p
4071:(
4066:1
4062:x
4039:2
4035:p
4008:,
4001:2
3996:1
3992:p
3987:w
3975:=
3970:0
3965:2
3961:p
3955:2
3947:1
3943:p
3939:w
3930:=
3927:)
3914:2
3910:p
3896:1
3892:p
3888:w
3885:(
3875:2
3871:p
3857:1
3853:p
3843:=
3835:1
3831:p
3820:1
3816:h
3786:v
3783:=
3780:u
3760:,
3747:2
3743:p
3729:1
3725:p
3721:w
3718:=
3715:v
3692:,
3689:u
3679:2
3675:p
3661:1
3657:p
3647:=
3639:1
3635:p
3624:1
3620:h
3590:u
3567:,
3564:u
3554:2
3550:p
3536:1
3532:p
3525:=
3522:)
3519:u
3516:,
3511:2
3507:p
3503:,
3498:1
3494:p
3490:(
3485:1
3481:h
3454:w
3448:=
3443:1
3439:q
3433:1
3429:p
3406:1
3402:p
3379:2
3374:1
3370:p
3365:/
3361:w
3331:2
3326:1
3322:p
3317:w
3301:2
3296:1
3292:p
3287:w
3275:=
3270:1
3266:x
3259:w
3249:1
3245:x
3227:1
3223:p
3212:1
3208:h
3198:=
3190:1
3186:p
3175:1
3171:x
3134:2
3129:1
3125:p
3120:w
3108:=
3101:1
3097:p
3092:w
3079:1
3075:p
3066:+
3059:2
3054:1
3050:p
3045:w
3033:=
3028:1
3024:x
3015:1
3011:w
3000:1
2996:x
2986:+
2978:1
2974:p
2963:1
2959:x
2949:=
2941:1
2937:p
2926:1
2922:h
2892:.
2887:2
2883:p
2878:/
2874:w
2868:=
2863:2
2859:x
2836:1
2832:p
2827:/
2823:w
2817:=
2812:1
2808:x
2787:w
2751:)
2748:u
2745:,
2741:p
2737:(
2734:e
2711:)
2708:u
2705:,
2701:p
2697:(
2674:)
2671:w
2668:,
2664:p
2660:(
2657:S
2637:)
2634:w
2631:,
2627:p
2623:(
2620:v
2617:=
2614:u
2594:w
2573:p
2552:u
2528:)
2518:w
2510:)
2507:w
2504:,
2500:p
2496:(
2491:n
2487:x
2477:)
2474:w
2471:,
2467:p
2463:(
2458:n
2454:x
2450:+
2442:n
2438:p
2429:)
2426:w
2423:,
2419:p
2415:(
2410:n
2406:x
2386:w
2378:)
2375:w
2372:,
2368:p
2364:(
2359:n
2355:x
2345:)
2342:w
2339:,
2335:p
2331:(
2326:1
2322:x
2318:+
2310:1
2306:p
2297:)
2294:w
2291:,
2287:p
2283:(
2278:n
2274:x
2240:w
2232:)
2229:w
2226:,
2222:p
2218:(
2213:1
2209:x
2199:)
2196:w
2193:,
2189:p
2185:(
2180:n
2176:x
2172:+
2164:n
2160:p
2151:)
2148:w
2145:,
2141:p
2137:(
2132:1
2128:x
2108:w
2100:)
2097:w
2094:,
2090:p
2086:(
2081:1
2077:x
2067:)
2064:w
2061:,
2057:p
2053:(
2048:1
2044:x
2040:+
2032:1
2028:p
2019:)
2016:w
2013:,
2009:p
2005:(
2000:1
1996:x
1982:(
1978:=
1975:)
1972:w
1969:,
1965:p
1961:(
1958:S
1926:]
1915:j
1911:p
1902:)
1899:u
1896:,
1892:p
1888:(
1883:i
1879:h
1866:[
1861:=
1858:)
1855:u
1852:,
1848:p
1844:(
1840:h
1835:p
1831:D
1823:)
1820:u
1817:,
1813:p
1809:(
1779:)
1776:u
1773:,
1769:p
1765:(
1761:h
1756:p
1752:D
1736:w
1734:D
1729:p
1727:D
1709:,
1700:)
1696:w
1693:,
1689:p
1685:(
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1677:)
1674:w
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1667:p
1663:(
1659:x
1654:w
1650:D
1642:)
1639:u
1636:,
1632:p
1628:(
1624:h
1619:p
1615:D
1610:=
1607:)
1604:w
1601:,
1597:p
1593:(
1589:x
1584:p
1580:D
1555:,
1539:)
1536:w
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1525:(
1522:v
1502:,
1499:)
1496:w
1493:,
1489:p
1485:(
1480:j
1476:x
1472:=
1469:)
1466:)
1463:w
1460:,
1456:p
1452:(
1449:v
1446:,
1442:p
1438:(
1433:j
1429:h
1425:=
1422:)
1419:u
1416:,
1412:p
1408:(
1403:j
1399:h
1371:)
1368:u
1365:,
1361:p
1357:(
1352:j
1348:h
1344:=
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1332:p
1323:)
1320:u
1317:,
1313:p
1309:(
1306:e
1288:.
1271:j
1267:p
1258:)
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1201:)
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1170:(
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1127:)
1124:u
1121:,
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1113:(
1110:e
1107:,
1103:p
1099:(
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1080:=
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1068:p
1059:)
1056:u
1053:,
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1045:(
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1015:j
1013:p
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983:)
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940:u
937:,
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929:(
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915:(
910:i
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902:=
899:)
896:u
893:,
889:p
885:(
880:i
876:h
722:l
717:1
713:x
706:+
701:s
696:1
692:x
685:=
680:1
676:x
655:j
650:j
648:b
644:i
633:ε
628:p
626:ε
617:p
615:ε
596:j
592:b
586:i
583:,
580:w
567:h
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559:i
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545:=
540:j
537:i
534:,
531:p
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494:u
490:i
476:)
473:w
470:,
466:p
462:(
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241:(
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201:)
198:u
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187:(
182:i
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168:=
160:j
156:p
147:)
144:w
141:,
137:p
133:(
128:i
124:x
104:j
100:i
94:.
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