723:
743:
691:
the household, the household will sell it; if the market values a good less than the household, the household will buy it. The process then continues until the market's and household's marginal rates of substitution are equal. Now, if the price of carrots were to change, and the price of all other goods were to remain constant, the gradient of the budget line would also change, leading to a different point of tangency and a different quantity demanded. These price / quantity combinations can then be used to deduce a full demand curve. A line connecting all points of tangency between the indifference curve and the
711:
668:
201:
indifference curve would imply that a consumer is indifferent between a bundle A and another bundle B because they lie on the same indifference curve, even in the case in which the quantity of both goods in bundle B is higher. Because of monotonicity of preferences and non-satiation, a bundle with more of both goods must be preferred to one with less of both, thus the first bundle must yield a higher utility, and lie on a different indifference curve at a higher utility level. The negative slope of the indifference curve implies that the
31:
139:
5262:, the indifference curve is a model for how animals 'decide' whether to perform a particular behavior, based on changes in two variables which can increase in intensity, one along the x-axis and the other along the y-axis. For example, the x-axis may measure the quantity of food available while the y-axis measures the risk involved in obtaining it. The indifference curve is drawn to predict the animal's behavior at various levels of risk and food availability.
656:: Consumers are willing to give up or trade-off some of one good to get more of another. The fundamental assertion is that there is a maximum amount that "a consumer will give up, of one commodity, to get one unit of another good, in that amount which will leave the consumer indifferent between the new and old situations" The negative slope of the indifference curves represents the willingness of the consumer to make a trade off.
154:. Points yielding different utility levels are each associated with distinct indifference curves and these indifference curves on the indifference map are like contour lines on a topographical graph. Each point on the curve represents the same elevation. If you move "off" an indifference curve traveling in a northeast direction (assuming positive
211:, such that all points on an indifference curve are ranked equally preferred and ranked either more or less preferred than every other point not on the curve. So, with (2), no two curves can intersect (otherwise non-satiation would be violated since the point(s) of intersection would have equal utility).
971:
then the indifference curves will be L-shaped. Examples of perfect complements include left shoes compared to right shoes: the consumer is no better off having several right shoes if she has only one left shoe - additional right shoes have zero marginal utility without more left shoes, so bundles of
904:
then the indifference curves will have a constant slope since the consumer would be willing to switch between at a fixed ratio. The marginal rate of substitution between perfect substitutes is likewise constant. An example of a utility function that is associated with indifference curves like these
690:
give a straight line on the indifference map showing all the possible distributions between the two goods; the point of maximum utility is then the point at which an indifference curve is tangent to the budget line (illustrated). This follows from common sense: if the market values a good more than
270:
imply that the indifference curves cannot be concave to the origin, i.e. they will either be straight lines or bulge toward the origin of the indifference curve. If the latter is the case, then as a consumer decreases consumption of one good in successive units, successively larger doses of the
200:
values.) The negative slope of the indifference curve reflects the assumption of the monotonicity of consumer's preferences, which generates monotonically increasing utility functions, and the assumption of non-satiation (marginal utility for all goods is always positive); an upward sloping
3762:
972:
goods differing only in the number of right shoes they include - however many - are equally preferred. The marginal rate of substitution is either zero or infinite. An example of the type of utility function that has an indifference map like that above is the
Leontief function:
5140:
722:
85:
of an indifference curve is called the MRS (marginal rate of substitution), and it indicates how much of good y must be sacrificed to keep the utility constant if good x is increased by one unit. Given a utility function u(x,y), to calculate the MRS, one takes the
4973:
3101:
2918:
5241:
5296:. He argues that the presence of an endowment effect indicates that a person has no indifference curve (see however Hanemann, 1991) rendering the neoclassical tools of welfare analysis useless, concluding that courts should instead use
895:
4607:
4726:
4488:
4375:
1038:
3576:
2577:
962:
259:. A negative slope and transitivity exclude indifference curves crossing, since straight lines from the origin on both sides of where they crossed would give opposite and intransitive preference rankings.
4141:
4268:
3557:
742:
176:
Negatively sloped. That is, as quantity consumed of one good (X) increases, total satisfaction would increase if not offset by a decrease in the quantity consumed of the other good (Y). Equivalently,
2343:
2043:
4984:
158:
for the goods) you are essentially climbing a mound of utility. The higher you go the greater the level of utility. The non-satiation requirement means that you will never reach the "top," or a "
3936:
542:"Continuous" means infinitely divisible - just like there are infinitely many numbers between 1 and 2 all bundles are infinitely divisible. This assumption makes indifference curves continuous.
5430:
Indifference curves can be used to derive the individual demand curve. However, the assumptions of consumer preference theory do not guarantee that the demand curve will have a negative slope.
4832:
4009:
2421:
645:
This assumption also set the stage for using techniques of constrained optimization because the shape of the curve assures that the first derivative is negative and the second is positive.
4082:
2659:
2503:
106:. If the marginal rate of substitution is diminishing along an indifference curve, that is the magnitude of the slope is decreasing or becoming less steep, then the preference is convex.
3820:
3380:
3330:
2712:
2929:
3277:
3153:
4824:
363:
This axiom precludes the possibility that the consumer cannot decide, It assumes that a consumer is able to make this comparison with respect to every conceivable bundle of goods.
77:
There are infinitely many indifference curves: one passes through each combination. A collection of (selected) indifference curves, illustrated graphically, is referred to as an
1939:
54:
for one combination or bundle of goods over a different combination on the same curve. One can also refer to each point on the indifference curve as rendering the same level of
3192:
2763:
2755:
2242:
4767:
3228:
2613:
2457:
2282:
2203:
5919:
805:. As price rises for a fixed money income, the consumer seeks the less expensive substitute at a lower indifference curve. The substitution effect is reinforced through the
1851:
710:
4514:
1878:
1821:
1321:
686:. For a single consumer, this is a relatively simple process. First, let one good be an example market e.g., carrots, and let the other be a composite of all other goods.
5151:
4793:
5300:
as a measure of value. Fischel (1995) however, raises the counterpoint that using WTA as a measure of value would deter the development of a nation's infrastructure and
1607:
1441:
1908:
1759:
1263:
1790:
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2141:
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1960:
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1507:
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1345:
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1240:
1219:
1198:
1173:
1152:
1131:
1108:
812:
4530:
1980:
801:
goods the marginal rate of substitution is not constant so their indifference curves are curved. The curves are convex to the origin, describing the negative
5311:
criticised the indifference curve as "never by definition exhibited in action, in actual exchanges, and is therefore unknowable and objectively meaningless."
4627:
671:
To maximise utility, a household should consume at (Qx, Qy). Assuming it does, a full demand schedule can be deduced as the price of one good fluctuates.
809:
of lower real income (Beattie-LaFrance). An example of a utility function that generates indifference curves of this kind is the Cobb–Douglas function
793:, but does not care where he/she is on a given indifference curve. The slope of an indifference curve (in absolute value), known by economists as the
50:. That is, any combinations of two products indicated by the curve will provide the consumer with equal levels of utility, and the consumer has no
1065:
in Fig. 3 would have no effect on equilibrium quantities demanded, since the budget line would rotate around the corner of the indifference curve.
4401:
4294:
975:
62:
of various points showing different combinations of two goods providing equal utility to the consumer. Utility is then a device to represent
3757:{\displaystyle {\frac {dU\left(x_{0},y_{0}\right)}{dx}}=0\Leftrightarrow {\frac {dy}{dx}}=-{\frac {U_{1}(x_{0},y_{0})}{U_{2}(x_{0},y_{0})}}}
2357:. The actual values of the function have no importance. Only the ranking of those values has content for the theory. More precisely, if
1047:. The results will only be stated here. A price-budget-line change that kept a consumer in equilibrium on the same indifference curve:
908:
5889:
4090:
5467:
4205:
3486:
2508:
289:. The consumer has ranked all available alternative combinations of commodities in terms of the satisfaction they provide him.
5618:
5135:{\displaystyle U_{2}(x,y)=(1-\alpha )\left(\alpha x^{\rho }+(1-\alpha )y^{\rho }\right)^{\left(1/\rho \right)-1}y^{\rho -1}.}
2287:
1987:
4196:
A class of utility functions known as Cobb-Douglas utility functions are very commonly used in economics for two reasons:
4618:
4202:
2. They are very flexible and can be adjusted to fit real-world data very easily. If the utility function is of the form
189:
4968:{\displaystyle U_{1}(x,y)=\alpha \left(\alpha x^{\rho }+(1-\alpha )y^{\rho }\right)^{\left(1/\rho \right)-1}x^{\rho -1}}
3884:
1078:, and use this representation to derive indifference curves showing combinations of equal preference to the consumer.
5979:
5861:
5733:
5708:
5683:
5648:
5587:
5366:
3962:
1043:
The different shapes of the curves imply different responses to a change in price as shown from demand analysis in
4035:
639:
The marginal rate of substitution tells how much 'y' a person is willing to sacrifice to get one more unit of 'x'.
3096:{\displaystyle {\frac {dU\left(x_{0},y_{0}\right)}{dx}}=U_{1}(x_{0},y_{0}).1+U_{2}(x_{0},y_{0}){\frac {dy}{dx}}}
897:. The negative slope of the indifference curve incorporates the willingness of the consumer to make trade offs.
797:, shows the rate at which consumers are willing to give up one good in exchange for more of the other good. For
736:
are perfect substitutes. The gray line perpendicular to all curves indicates the curves are mutually parallel.
5881:
5784:
Hanemann, W. Michael (1991). "Willingness To Pay and
Willingness To Accept: How Much Can They Differ? Reply".
3775:
3335:
3285:
2667:
46:
connects points on a graph representing different quantities of two goods, points between which a consumer is
4521:
3823:
794:
202:
17:
3233:
5326:
3112:
5483:
4802:
2913:{\displaystyle dU\left(x_{0},y_{0}\right)=U_{1}\left(x_{0},y_{0}\right)dx+U_{2}\left(x_{0},y_{0}\right)dy}
2360:
170:
2618:
2462:
1913:
180:, such that more of either good (or both) is equally preferred to no increase, is excluded. (If utility
3158:
2721:
2208:
4734:
3197:
2582:
2426:
2251:
2172:
130:
theory, which posits that individuals can always rank any consumption bundles by order of preference.
5969:
5964:
122:
was the first author to actually draw these curves, in his 1906 book. The theory can be derived from
5236:{\displaystyle {\frac {dx}{dy}}=-{\frac {1-\alpha }{\alpha }}\left({\frac {x}{y}}\right)^{1-\rho }.}
384:
are identical in all respects the consumer will recognize this fact and be indifferent in comparing
173:
of commodity quantities (i.e. the possibility of having negative quantities of any good is ignored).
5675:
3382:. That is, when preferences are represented by a utility function, the indifference curves are the
1826:
1051:
in Fig. 1 would reduce quantity demanded of a good smoothly as price rose relatively for that good.
115:
4493:
1856:
1799:
1299:
214:
4772:
4199:
1. They represent ‘well-behaved’ preferences, such as more is better and preference for variety.
159:
150:
A graph of indifference curves for several utility levels of an individual consumer is called an
66:
rather than something from which preferences come. The main use of indifference curves is in the
5821:"The offer/ask disparity and just compensation for takings: A constitutional choice perspective"
1585:
1576:, meaning not that they are unwanted but that they are equally good in satisfying preferences.
1419:
123:
51:
1886:
1744:
1248:
5297:
1054:
in Fig. 2 would have either no effect on quantity demanded of either good (at one end of the
310:. A consumer can unambiguously determine that one and only one of the following is the case:
5386:
The transitivity of weak preferences is sufficient for most indifference-curve analyses: If
1768:
890:{\displaystyle \scriptstyle U\left(x,y\right)=x^{\alpha }y^{1-\alpha },0\leq \alpha \leq 1}
67:
5271:
4602:{\displaystyle {\frac {dx}{dy}}=-{\frac {1-\alpha }{\alpha }}\left({\frac {x}{y}}\right).}
3455:
3410:
2122:
2098:
667:
8:
5500:
4380:
4273:
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4149:
4014:
3941:
3850:
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3434:
3389:
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2077:
2056:
1944:
1793:
1720:
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1157:
1136:
1115:
1092:
802:
625:
commodities can be disease, pollution etc. because we always desire less of such things.
59:
5949:
5906:
5801:
5766:
5606:
5509:]. Piccola Biblioteca Scientifica. Vol. 13. Milano: Societa Editrice Libraria.
5485:
Mathematical
Psychics: An Essay on the Application of Mathematics to the Moral Sciences
5336:
1965:
267:
177:
87:
5944:
5540:
716:
Figure 1: An example of an indifference map with three indifference curves represented
5910:
5902:
5857:
5837:
5820:
5770:
5729:
5704:
5679:
5644:
5614:
5583:
5321:
5293:
5277:
5247:
2154:
1059:
1055:
692:
687:
683:
679:
642:
This assumption assures that indifference curves are smooth and convex to the origin.
118:, who explained in his 1881 book the mathematics needed for their drawing; later on,
5974:
5898:
5832:
5793:
5758:
5341:
5281:
4796:
4721:{\displaystyle U(x,y)=\left(\alpha x^{\rho }+(1-\alpha )y^{\rho }\right)^{1/\rho }}
2715:
2346:
2150:
155:
3431:, without moving off the indifference curve, one must also change the quantity of
217:
with respect to points on distinct indifference curves. That is, if each point on
5361:
5331:
5308:
5301:
1266:
1044:
968:
901:
675:
653:
263:
127:
119:
5520:
5797:
5667:
5351:
696:
272:
5502:
Manuale di
Economia Politica — con una Introduzione alla Scienza Sociale
1110:
be a set of mutually exclusive alternatives among which a consumer can choose.
58:(satisfaction) for the consumer. In other words, an indifference curve is the
5958:
806:
138:
30:
34:
An example of an indifference map with three indifference curves represented
4483:{\displaystyle U_{2}\left(x,y\right)=(1-\alpha )\left(x/y\right)^{\alpha }}
2354:
143:
3332:
must deliver at each bundle on the curve the same utility level as bundle
5356:
5346:
4370:{\displaystyle U_{1}\left(x,y\right)=\alpha \left(x/y\right)^{\alpha -1}}
3383:
208:
3386:
of the utility function. Therefore, if one is to change the quantity of
5805:
5450:
Geanakoplos, John (1987). "Arrow-Debreu model of general equilibrium".
1033:{\displaystyle \scriptstyle U\left(x,y\right)=\min\{\alpha x,\beta y\}}
63:
3768:
Thus, the ratio of marginal utilities gives the absolute value of the
5289:
757:
39:
5749:
Hovenkamp, Herbert (1991). "Legal Policy and the
Endowment Effect".
5762:
586:
This assumption is commonly called the "more is better" assumption.
5259:
2245:
2146:
55:
5701:
Schaum's
Outline of Theory and Problems of Managerial Economics
957:{\displaystyle \scriptstyle U\left(x,y\right)=\alpha x+\beta y}
71:
5458:
4520:
of the indifference curve, and therefore the negative of the
4517:
3769:
82:
4136:{\displaystyle {\frac {dx}{dy}}=-{\frac {\beta }{\alpha }}.}
279:
5728:(2nd ed.). Reading: Addison-Wesley. pp. 141–143.
5662:
5660:
5582:(2nd ed.). Reading: Addison-Wesley. pp. 109–117.
1221:
is one such combination, such as 1 apple and 4 bananas and
589:
An alternative version of this assumption requires that if
4263:{\displaystyle U\left(x,y\right)=x^{\alpha }y^{1-\alpha }}
3552:{\displaystyle {\frac {dU\left(x_{0},y_{0}\right)}{dx}}=0}
1200:
is made of combinations of apples and bananas. The symbol
142:
An example of how indifference curves are obtained as the
74:
patterns for individual consumers over commodity bundles.
5443:
5285:
162:," a consumption bundle that is preferred to all others.
5657:
5481:
2572:{\displaystyle U\left(x,y\right)>U\left(x',y'\right)}
1058:) or would change quantity demanded from one end of the
98:
and divide it by the partial derivative of the function
5882:"The Law of Demand versus Diminishing Marginal Utility"
5611:
Microeconomics: Theory & Applications with
Calculus
1242:
is another combination such as 2 apples and 2 bananas.
2095:
is made of two numbers: The number of apples, call it
979:
912:
816:
748:
Figure 3: Indifference curves for perfect complements
5601:
5599:
5154:
4987:
4835:
4805:
4775:
4737:
4630:
4533:
4496:
4404:
4383:
4297:
4276:
4208:
4173:
4152:
4093:
4084:. The slope of the indifference curve is, therefore,
4038:
4017:
3965:
3944:
3887:
3853:
3832:
3778:
3579:
3489:
3458:
3437:
3413:
3392:
3338:
3288:
3236:
3200:
3161:
3115:
2932:
2766:
2724:
2670:
2621:
2585:
2511:
2465:
2429:
2363:
2338:{\displaystyle \left(x,y\right)\to U\left(x,y\right)}
2290:
2254:
2211:
2175:
2165:) such that any set of three or more bundles forms a
2161:
pairs of consumption bundles by order of preference (
2125:
2101:
2080:
2059:
2038:{\displaystyle {\mathcal {C}}_{a}=\{b\in A:b\sim a\}}
1990:
1968:
1947:
1916:
1889:
1859:
1829:
1802:
1771:
1747:
1723:
1702:
1681:
1660:
1639:
1618:
1588:
1561:
1540:
1515:
1494:
1473:
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1422:
1395:
1374:
1353:
1332:
1302:
1275:
1251:
1227:
1206:
1185:
1160:
1139:
1118:
1095:
978:
911:
815:
165:
Indifference curves are typically represented to be:
5945:
Anatomy of Cobb–Douglas Type
Utility Functions in 3D
5465:
Böhm, Volker; Haller, Hans (1987). "Demand theory".
5641:
The
Structure of Economics: A Mathematical Analysis
1069:
114:The theory of indifference curves was developed by
5596:
5235:
5134:
4967:
4818:
4787:
4761:
4720:
4601:
4508:
4482:
4390:
4369:
4283:
4262:
4180:
4159:
4135:
4076:
4024:
4003:
3951:
3931:{\displaystyle U\left(x,y\right)=\alpha x+\beta y}
3930:
3860:
3839:
3814:
3756:
3551:
3468:
3444:
3423:
3399:
3374:
3324:
3271:
3222:
3186:
3147:
3095:
2912:
2749:
2706:
2653:
2607:
2571:
2497:
2451:
2415:
2337:
2276:
2236:
2197:
2135:
2111:
2087:
2066:
2037:
1974:
1954:
1933:
1902:
1872:
1845:
1815:
1784:
1753:
1730:
1709:
1688:
1667:
1646:
1625:
1601:
1568:
1547:
1522:
1501:
1480:
1459:
1435:
1402:
1381:
1360:
1339:
1315:
1282:
1257:
1234:
1213:
1192:
1167:
1146:
1125:
1102:
1032:
956:
889:
702:
3194:with respect to its first argument, evaluated at
2615:is described as strictly preferred to the bundle
1074:Choice theory formally represents consumers by a
5956:
5880:Beattie, Bruce R.; LaFrance, Jeffrey T. (2006).
2048:
1005:
728:Figure 2: Three indifference curves where Goods
5879:
5498:
2459:is described as at least as good as the bundle
5638:
4188:: the indifference curves are straight lines.
1179:In the language of the example above, the set
294:Assume that there are two consumption bundles
5723:
5634:
5632:
5630:
5577:
4004:{\displaystyle U_{1}\left(x,y\right)=\alpha }
3476:such that, in the end, there is no change in
765:In Figure 1, the consumer would rather be on
4077:{\displaystyle U_{2}\left(x,y\right)=\beta }
2032:
2008:
1026:
1008:
275:are required to keep satisfaction unchanged.
5950:Anatomy of CES Type Utility Functions in 3D
5927:Australasian Journal of Economics Education
5468:The New Palgrave: A Dictionary of Economics
5452:The New Palgrave: A Dictionary of Economics
5449:
4799:is a special case of the CES utility, with
5856:. New York University Press. p. 242.
5627:
5464:
5280:(1991) has argued that the presence of an
4146:Observe that the slope does not depend on
2132:
2108:
2084:
2063:
1951:
1899:
1869:
1842:
1812:
1781:
1727:
1706:
1685:
1664:
1643:
1622:
1598:
1565:
1544:
1519:
1498:
1477:
1456:
1432:
1399:
1378:
1357:
1336:
1312:
1279:
1231:
1210:
1189:
1164:
1143:
1122:
1099:
631:diminishing marginal rates of substitution
188:, in the third dimension, does not have a
5836:
5825:International Review of Law and Economics
5748:
5742:
5698:
5692:
4815:
4387:
4280:
4177:
4156:
4021:
3948:
3857:
3836:
3465:
3441:
3420:
3396:
652:. It is the most critical assumption of
613:It also implies that the commodities are
280:Assumptions of consumer preference theory
224:is (strictly) preferred to each point on
5890:Applied Economic Perspectives and Policy
5851:
5783:
5777:
5573:
5571:
5569:
5567:
5565:
5563:
5561:
5145:Therefore, along an indifference curve,
3815:{\displaystyle \left(x_{0},y_{0}\right)}
3375:{\displaystyle \left(x_{0},y_{0}\right)}
3325:{\displaystyle \left(x_{0},y_{0}\right)}
2707:{\displaystyle \left(x_{0},y_{0}\right)}
2248:(satisfaction) relation associated with
1910:, the corresponding indifference curve,
666:
648:Another name for this assumption is the
597:have the same quantity of one good, but
137:
29:
5818:
5812:
5605:
5541:"William Stanley Jevons - Policonomics"
4826:.) The marginal utilities are given by
4191:
3881:If the utility function is of the form
1081:
14:
5957:
5917:
5666:
5613:. Boston: Addison-Wesley. p. 62.
3272:{\displaystyle U_{2}\left(x,y\right).}
5672:An Introduction to Positive Economics
5643:(3rd ed.). Boston: McGraw-Hill.
5558:
3148:{\displaystyle U_{1}\left(x,y\right)}
133:
5521:"Indifference curves | Policonomics"
4819:{\displaystyle \rho \rightarrow 0\,}
5246:These examples might be useful for
4619:Constant Elasticity of Substitution
3772:of the indifference curve at point
2416:{\displaystyle U(x,y)\geq U(x',y')}
2119:and the number of bananas, call it
24:
5873:
5394:, meaning that the consumer likes
2654:{\displaystyle \left(x',y'\right)}
2498:{\displaystyle \left(x',y'\right)}
2169:. This means that for each bundle
1994:
1934:{\displaystyle {\mathcal {C}}_{a}}
1920:
25:
5991:
5938:
5482:Francis Ysidro Edgeworth (1881).
5471:. Vol. 1. pp. 785–792 .
5454:. Vol. 1. pp. 116–124 .
5284:has significant implications for
3876:
3187:{\displaystyle U\left(x,y\right)}
2750:{\displaystyle U\left(x,y\right)}
2237:{\displaystyle U\left(x,y\right)}
2053:In the example above, an element
1792:can be ranked. The relation is a
474:This is a consistency assumption.
169:Defined only in the non-negative
5920:"Behavioral Indifference Curves"
5903:10.1111/j.1467-9353.2006.00286.x
5270:Indifference curves inherit the
5250:individual or aggregate demand.
4762:{\displaystyle \alpha \in (0,1)}
3223:{\displaystyle \left(x,y\right)}
2923:or, without loss of generality,
2608:{\displaystyle \left(x,y\right)}
2452:{\displaystyle \left(x,y\right)}
2277:{\displaystyle \left(x,y\right)}
2198:{\displaystyle \left(x,y\right)}
1070:Preference relations and utility
741:
721:
709:
302:each containing two commodities
5845:
5717:
5488:. London: C. Kegan Paul and Co.
5424:
3282:The indifference curve through
1245:A preference relation, denoted
756:. The elbows of the curves are
703:Examples of indifference curves
5533:
5513:
5492:
5475:
5380:
5272:criticisms directed at utility
5065:
5053:
5028:
5016:
5010:
4998:
4901:
4889:
4858:
4846:
4809:
4756:
4744:
4686:
4674:
4646:
4634:
4612:
4449:
4437:
3748:
3722:
3707:
3681:
3639:
3070:
3044:
3025:
2999:
2410:
2388:
2379:
2367:
2310:
1941:is made up of all elements of
1410:(in preference satisfaction).
662:
252:is preferred to each point on
238:is preferred to each point on
13:
1:
5437:
5265:
4522:marginal rate of substitution
3938:then the marginal utility of
3824:marginal rate of substitution
3155:is the partial derivative of
2664:Consider a particular bundle
2049:Formal link to utility theory
1846:{\displaystyle b\succeq c,\;}
795:marginal rate of substitution
678:uses indifference curves and
203:marginal rate of substitution
5838:10.1016/0144-8188(94)00005-F
5819:Fischel, William A. (1995).
5751:The Journal of Legal Studies
5726:Microeconomics with Calculus
5699:Salvatore, Dominick (1989).
5580:Microeconomics with Calculus
5367:Utility–possibility frontier
5327:Community indifference curve
5292:, particularly in regard to
4509:{\displaystyle \alpha <1}
4377:and the marginal utility of
4011:and the marginal utility of
2353:of the function is a set of
2205:there is a unique relation,
1873:{\displaystyle a\succeq c\;}
1816:{\displaystyle a\succeq b\;}
1316:{\displaystyle a\succeq b\;}
629:Indifference curves exhibit
601:has more of the other, then
7:
5507:Manual of Political Economy
5314:
4788:{\displaystyle \rho \leq 1}
3871:
3822:. This ratio is called the
1675:is not weakly preferred to
10:
5996:
5798:10.1257/000282803321455449
5253:
1602:{\displaystyle a\succ b\;}
109:
70:of potentially observable
5852:Rothbard, Murray (1998).
5676:Weidenfeld & Nicolson
5639:Silberberg; Suen (2000).
2157:is a function that ranks
1962:which are indifferent to
1717:is strictly preferred to
1436:{\displaystyle a\sim b\;}
779:, and would rather be on
501:is sufficiently close to
5980:1881 in economic history
5786:American Economic Review
5724:Binger; Hoffman (1998).
5578:Binger; Hoffman (1998).
5499:Vilfredo Pareto (1919).
5373:
4270:the marginal utility of
1903:{\displaystyle a\in A\;}
1754:{\displaystyle \succeq }
1741:The preference relation
1258:{\displaystyle \succeq }
116:Francis Ysidro Edgeworth
5417:is weakly preferred to
5409:is weakly preferred to
5390:is weakly preferred to
1633:is weakly preferred to
1509:is weakly preferred to
1467:is weakly preferred to
1389:is at least as good as
1347:is weakly preferred to
1154:be generic elements of
650:substitution assumption
5237:
5136:
4969:
4820:
4789:
4763:
4722:
4603:
4510:
4484:
4392:
4371:
4285:
4264:
4182:
4161:
4137:
4078:
4026:
4005:
3953:
3932:
3862:
3841:
3816:
3758:
3553:
3470:
3446:
3425:
3401:
3376:
3326:
3273:
3224:
3188:
3149:
3097:
2914:
2751:
2708:
2655:
2609:
2573:
2499:
2453:
2417:
2339:
2278:
2238:
2199:
2137:
2113:
2089:
2068:
2039:
1976:
1956:
1935:
1904:
1874:
1847:
1817:
1786:
1755:
1732:
1711:
1690:
1669:
1648:
1627:
1603:
1570:
1549:
1524:
1503:
1482:
1461:
1437:
1404:
1383:
1362:
1341:
1317:
1284:
1259:
1236:
1215:
1194:
1169:
1148:
1127:
1104:
1034:
958:
891:
684:consumer demand curves
672:
352:, formally written as
336:, formally written as
320:, formally written as
147:
124:William Stanley Jevons
35:
5854:The Ethics of Liberty
5238:
5137:
4970:
4821:
4790:
4764:
4723:
4604:
4511:
4485:
4393:
4372:
4286:
4265:
4183:
4162:
4138:
4079:
4027:
4006:
3954:
3933:
3863:
3842:
3817:
3759:
3554:
3471:
3447:
3426:
3402:
3377:
3327:
3274:
3225:
3189:
3150:
3098:
2915:
2752:
2709:
2656:
2610:
2574:
2500:
2454:
2418:
2340:
2279:
2239:
2200:
2138:
2114:
2090:
2069:
2040:
1977:
1957:
1936:
1905:
1875:
1848:
1818:
1787:
1785:{\displaystyle a,b\;}
1756:
1733:
1712:
1691:
1670:
1649:
1628:
1604:
1571:
1550:
1525:
1504:
1483:
1462:
1438:
1405:
1384:
1363:
1342:
1318:
1285:
1260:
1237:
1216:
1195:
1170:
1149:
1128:
1105:
1035:
959:
892:
670:
146:of a utility function
141:
102:with respect to good
94:with respect to good
33:
5678:. pp. 182–186.
5545:www.policonomics.com
5152:
4985:
4833:
4803:
4773:
4735:
4628:
4531:
4494:
4402:
4381:
4295:
4274:
4206:
4192:Cobb–Douglas utility
4171:
4150:
4091:
4036:
4015:
3963:
3942:
3885:
3851:
3830:
3776:
3577:
3487:
3469:{\displaystyle dy\,}
3456:
3435:
3424:{\displaystyle dx\,}
3411:
3390:
3336:
3286:
3234:
3198:
3159:
3113:
2930:
2764:
2722:
2668:
2619:
2583:
2509:
2463:
2427:
2361:
2288:
2252:
2209:
2173:
2136:{\displaystyle y.\;}
2123:
2112:{\displaystyle x,\;}
2099:
2078:
2057:
1988:
1966:
1945:
1914:
1887:
1857:
1827:
1800:
1769:
1745:
1721:
1700:
1679:
1658:
1637:
1616:
1586:
1559:
1538:
1513:
1492:
1471:
1450:
1420:
1393:
1372:
1351:
1330:
1300:
1273:
1249:
1225:
1204:
1183:
1158:
1137:
1116:
1093:
1082:Preference relations
976:
909:
813:
547:Preferences exhibit
231:, and each point on
27:Concept in economics
5674:(Fourth ed.).
5607:Perloff, Jeffrey M.
5307:Austrian economist
4391:{\displaystyle y\,}
4284:{\displaystyle x\,}
4181:{\displaystyle y\,}
4160:{\displaystyle x\,}
4025:{\displaystyle y\,}
3952:{\displaystyle x\,}
3861:{\displaystyle y\,}
3840:{\displaystyle x\,}
3567:above to solve for
3559:, or, substituting
3445:{\displaystyle y\,}
3400:{\displaystyle x\,}
2244:, representing the
2167:transitive relation
2088:{\displaystyle A\;}
2067:{\displaystyle a\;}
1955:{\displaystyle A\;}
1794:transitive relation
1731:{\displaystyle b\;}
1710:{\displaystyle a\;}
1689:{\displaystyle a\;}
1668:{\displaystyle b\;}
1647:{\displaystyle b\;}
1626:{\displaystyle a\;}
1569:{\displaystyle b\;}
1548:{\displaystyle a\;}
1530:.' That is, one is
1523:{\displaystyle a\;}
1502:{\displaystyle b\;}
1481:{\displaystyle b\;}
1460:{\displaystyle a\;}
1403:{\displaystyle b\;}
1382:{\displaystyle a\;}
1361:{\displaystyle b\;}
1340:{\displaystyle a\;}
1283:{\displaystyle A\;}
1235:{\displaystyle b\;}
1214:{\displaystyle a\;}
1193:{\displaystyle A\;}
1168:{\displaystyle A\;}
1147:{\displaystyle b\;}
1126:{\displaystyle a\;}
1103:{\displaystyle A\;}
1076:preference relation
969:perfect complements
902:perfect substitutes
803:substitution effect
549:strong monotonicity
376:This means that if
205:is always positive;
5918:Komlos, J (2015).
5668:Lipsey, Richard G.
5337:Convex preferences
5233:
5132:
4965:
4816:
4785:
4759:
4718:
4599:
4506:
4480:
4388:
4367:
4281:
4260:
4178:
4157:
4133:
4074:
4022:
4001:
3949:
3928:
3858:
3837:
3812:
3754:
3549:
3466:
3442:
3421:
3397:
3372:
3322:
3269:
3220:
3184:
3145:
3093:
2910:
2757:about this point:
2747:
2704:
2651:
2605:
2569:
2495:
2449:
2423:, then the bundle
2413:
2335:
2274:
2234:
2195:
2133:
2109:
2085:
2064:
2035:
1972:
1952:
1931:
1900:
1870:
1843:
1813:
1782:
1751:
1728:
1707:
1696:.' One says that '
1686:
1665:
1644:
1623:
1599:
1566:
1545:
1520:
1499:
1478:
1457:
1433:
1400:
1379:
1358:
1337:
1313:
1280:
1269:define on the set
1255:
1232:
1211:
1190:
1165:
1144:
1123:
1100:
1030:
1029:
954:
953:
887:
886:
688:Budget constraints
680:budget constraints
673:
348:is indifferent to
268:convex preferences
148:
134:Map and properties
88:partial derivative
44:indifference curve
36:
5620:978-0-321-27794-7
5322:Budget constraint
5294:welfare economics
5278:Herbert Hovenkamp
5212:
5197:
5173:
4590:
4576:
4552:
4128:
4112:
3752:
3660:
3631:
3541:
3091:
2984:
1975:{\displaystyle a}
1612:is described as '
1534:to the choice of
1446:is described as '
1326:is described as '
1060:budget constraint
1056:budget constraint
967:If two goods are
900:If two goods are
693:budget constraint
561:has more of both
16:(Redirected from
5987:
5970:Economics curves
5965:Family economics
5934:
5924:
5914:
5886:
5868:
5867:
5849:
5843:
5842:
5840:
5816:
5810:
5809:
5781:
5775:
5774:
5746:
5740:
5739:
5721:
5715:
5714:
5696:
5690:
5689:
5664:
5655:
5654:
5636:
5625:
5624:
5603:
5594:
5593:
5575:
5556:
5555:
5553:
5551:
5537:
5531:
5530:
5528:
5527:
5517:
5511:
5510:
5496:
5490:
5489:
5479:
5473:
5472:
5462:
5456:
5455:
5447:
5431:
5428:
5422:
5399:at least as much
5384:
5342:Endowment effect
5282:endowment effect
5274:more generally.
5242:
5240:
5239:
5234:
5229:
5228:
5217:
5213:
5205:
5198:
5193:
5182:
5174:
5172:
5164:
5156:
5141:
5139:
5138:
5133:
5128:
5127:
5112:
5111:
5104:
5100:
5096:
5082:
5078:
5077:
5076:
5049:
5048:
4997:
4996:
4974:
4972:
4971:
4966:
4964:
4963:
4948:
4947:
4940:
4936:
4932:
4918:
4914:
4913:
4912:
4885:
4884:
4845:
4844:
4825:
4823:
4822:
4817:
4794:
4792:
4791:
4786:
4768:
4766:
4765:
4760:
4727:
4725:
4724:
4719:
4717:
4716:
4712:
4703:
4699:
4698:
4697:
4670:
4669:
4608:
4606:
4605:
4600:
4595:
4591:
4583:
4577:
4572:
4561:
4553:
4551:
4543:
4535:
4515:
4513:
4512:
4507:
4489:
4487:
4486:
4481:
4479:
4478:
4473:
4469:
4465:
4433:
4429:
4414:
4413:
4397:
4395:
4394:
4389:
4376:
4374:
4373:
4368:
4366:
4365:
4354:
4350:
4346:
4326:
4322:
4307:
4306:
4290:
4288:
4287:
4282:
4269:
4267:
4266:
4261:
4259:
4258:
4243:
4242:
4230:
4226:
4187:
4185:
4184:
4179:
4166:
4164:
4163:
4158:
4142:
4140:
4139:
4134:
4129:
4121:
4113:
4111:
4103:
4095:
4083:
4081:
4080:
4075:
4067:
4063:
4048:
4047:
4031:
4029:
4028:
4023:
4010:
4008:
4007:
4002:
3994:
3990:
3975:
3974:
3958:
3956:
3955:
3950:
3937:
3935:
3934:
3929:
3909:
3905:
3867:
3865:
3864:
3859:
3846:
3844:
3843:
3838:
3821:
3819:
3818:
3813:
3811:
3807:
3806:
3805:
3793:
3792:
3763:
3761:
3760:
3755:
3753:
3751:
3747:
3746:
3734:
3733:
3721:
3720:
3710:
3706:
3705:
3693:
3692:
3680:
3679:
3669:
3661:
3659:
3651:
3643:
3632:
3630:
3622:
3621:
3617:
3616:
3615:
3603:
3602:
3581:
3558:
3556:
3555:
3550:
3542:
3540:
3532:
3531:
3527:
3526:
3525:
3513:
3512:
3491:
3475:
3473:
3472:
3467:
3451:
3449:
3448:
3443:
3430:
3428:
3427:
3422:
3406:
3404:
3403:
3398:
3381:
3379:
3378:
3373:
3371:
3367:
3366:
3365:
3353:
3352:
3331:
3329:
3328:
3323:
3321:
3317:
3316:
3315:
3303:
3302:
3278:
3276:
3275:
3270:
3265:
3261:
3246:
3245:
3230:. (Likewise for
3229:
3227:
3226:
3221:
3219:
3215:
3193:
3191:
3190:
3185:
3183:
3179:
3154:
3152:
3151:
3146:
3144:
3140:
3125:
3124:
3102:
3100:
3099:
3094:
3092:
3090:
3082:
3074:
3069:
3068:
3056:
3055:
3043:
3042:
3024:
3023:
3011:
3010:
2998:
2997:
2985:
2983:
2975:
2974:
2970:
2969:
2968:
2956:
2955:
2934:
2919:
2917:
2916:
2911:
2903:
2899:
2898:
2897:
2885:
2884:
2870:
2869:
2851:
2847:
2846:
2845:
2833:
2832:
2818:
2817:
2805:
2801:
2800:
2799:
2787:
2786:
2756:
2754:
2753:
2748:
2746:
2742:
2716:total derivative
2713:
2711:
2710:
2705:
2703:
2699:
2698:
2697:
2685:
2684:
2660:
2658:
2657:
2652:
2650:
2646:
2645:
2634:
2614:
2612:
2611:
2606:
2604:
2600:
2578:
2576:
2575:
2570:
2568:
2564:
2563:
2552:
2533:
2529:
2504:
2502:
2501:
2496:
2494:
2490:
2489:
2478:
2458:
2456:
2455:
2450:
2448:
2444:
2422:
2420:
2419:
2414:
2409:
2398:
2347:utility function
2344:
2342:
2341:
2336:
2334:
2330:
2309:
2305:
2283:
2281:
2280:
2275:
2273:
2269:
2243:
2241:
2240:
2235:
2233:
2229:
2204:
2202:
2201:
2196:
2194:
2190:
2151:utility function
2142:
2140:
2139:
2134:
2118:
2116:
2115:
2110:
2094:
2092:
2091:
2086:
2073:
2071:
2070:
2065:
2044:
2042:
2041:
2036:
2004:
2003:
1998:
1997:
1981:
1979:
1978:
1973:
1961:
1959:
1958:
1953:
1940:
1938:
1937:
1932:
1930:
1929:
1924:
1923:
1909:
1907:
1906:
1901:
1883:For any element
1879:
1877:
1876:
1871:
1852:
1850:
1849:
1844:
1822:
1820:
1819:
1814:
1791:
1789:
1788:
1783:
1760:
1758:
1757:
1752:
1737:
1735:
1734:
1729:
1716:
1714:
1713:
1708:
1695:
1693:
1692:
1687:
1674:
1672:
1671:
1666:
1653:
1651:
1650:
1645:
1632:
1630:
1629:
1624:
1608:
1606:
1605:
1600:
1575:
1573:
1572:
1567:
1554:
1552:
1551:
1546:
1529:
1527:
1526:
1521:
1508:
1506:
1505:
1500:
1487:
1485:
1484:
1479:
1466:
1464:
1463:
1458:
1442:
1440:
1439:
1434:
1409:
1407:
1406:
1401:
1388:
1386:
1385:
1380:
1367:
1365:
1364:
1359:
1346:
1344:
1343:
1338:
1322:
1320:
1319:
1314:
1289:
1287:
1286:
1281:
1264:
1262:
1261:
1256:
1241:
1239:
1238:
1233:
1220:
1218:
1217:
1212:
1199:
1197:
1196:
1191:
1174:
1172:
1171:
1166:
1153:
1151:
1150:
1145:
1132:
1130:
1129:
1124:
1109:
1107:
1106:
1101:
1039:
1037:
1036:
1031:
1001:
997:
963:
961:
960:
955:
934:
930:
896:
894:
893:
888:
867:
866:
851:
850:
838:
834:
745:
725:
713:
605:is preferred to
577:is preferred to
509:is preferred to
493:is preferred to
479:Preferences are
414:Preferences are
368:Preferences are
332:is preferred to
316:is preferred to
285:Preferences are
245:, each point on
156:marginal utility
152:indifference map
90:of the function
79:indifference map
21:
5995:
5994:
5990:
5989:
5988:
5986:
5985:
5984:
5955:
5954:
5941:
5922:
5884:
5876:
5874:Further reading
5871:
5864:
5850:
5846:
5817:
5813:
5782:
5778:
5747:
5743:
5736:
5722:
5718:
5711:
5703:. McGraw-Hill.
5697:
5693:
5686:
5665:
5658:
5651:
5637:
5628:
5621:
5604:
5597:
5590:
5576:
5559:
5549:
5547:
5539:
5538:
5534:
5525:
5523:
5519:
5518:
5514:
5497:
5493:
5480:
5476:
5463:
5459:
5448:
5444:
5440:
5435:
5434:
5429:
5425:
5385:
5381:
5376:
5371:
5362:Substitute good
5332:Consumer theory
5317:
5309:Murray Rothbard
5302:economic growth
5268:
5256:
5218:
5204:
5200:
5199:
5183:
5181:
5165:
5157:
5155:
5153:
5150:
5149:
5117:
5113:
5092:
5088:
5084:
5083:
5072:
5068:
5044:
5040:
5036:
5032:
5031:
4992:
4988:
4986:
4983:
4982:
4953:
4949:
4928:
4924:
4920:
4919:
4908:
4904:
4880:
4876:
4872:
4868:
4867:
4840:
4836:
4834:
4831:
4830:
4804:
4801:
4800:
4774:
4771:
4770:
4736:
4733:
4732:
4708:
4704:
4693:
4689:
4665:
4661:
4657:
4653:
4652:
4629:
4626:
4625:
4617:A general CES (
4615:
4582:
4578:
4562:
4560:
4544:
4536:
4534:
4532:
4529:
4528:
4495:
4492:
4491:
4474:
4461:
4457:
4453:
4452:
4419:
4415:
4409:
4405:
4403:
4400:
4399:
4382:
4379:
4378:
4355:
4342:
4338:
4334:
4333:
4312:
4308:
4302:
4298:
4296:
4293:
4292:
4275:
4272:
4271:
4248:
4244:
4238:
4234:
4216:
4212:
4207:
4204:
4203:
4194:
4172:
4169:
4168:
4151:
4148:
4147:
4120:
4104:
4096:
4094:
4092:
4089:
4088:
4053:
4049:
4043:
4039:
4037:
4034:
4033:
4016:
4013:
4012:
3980:
3976:
3970:
3966:
3964:
3961:
3960:
3943:
3940:
3939:
3895:
3891:
3886:
3883:
3882:
3879:
3874:
3852:
3849:
3848:
3831:
3828:
3827:
3801:
3797:
3788:
3784:
3783:
3779:
3777:
3774:
3773:
3742:
3738:
3729:
3725:
3716:
3712:
3711:
3701:
3697:
3688:
3684:
3675:
3671:
3670:
3668:
3652:
3644:
3642:
3623:
3611:
3607:
3598:
3594:
3593:
3589:
3582:
3580:
3578:
3575:
3574:
3533:
3521:
3517:
3508:
3504:
3503:
3499:
3492:
3490:
3488:
3485:
3484:
3457:
3454:
3453:
3436:
3433:
3432:
3412:
3409:
3408:
3391:
3388:
3387:
3361:
3357:
3348:
3344:
3343:
3339:
3337:
3334:
3333:
3311:
3307:
3298:
3294:
3293:
3289:
3287:
3284:
3283:
3251:
3247:
3241:
3237:
3235:
3232:
3231:
3205:
3201:
3199:
3196:
3195:
3169:
3165:
3160:
3157:
3156:
3130:
3126:
3120:
3116:
3114:
3111:
3110:
3083:
3075:
3073:
3064:
3060:
3051:
3047:
3038:
3034:
3019:
3015:
3006:
3002:
2993:
2989:
2976:
2964:
2960:
2951:
2947:
2946:
2942:
2935:
2933:
2931:
2928:
2927:
2893:
2889:
2880:
2876:
2875:
2871:
2865:
2861:
2841:
2837:
2828:
2824:
2823:
2819:
2813:
2809:
2795:
2791:
2782:
2778:
2777:
2773:
2765:
2762:
2761:
2732:
2728:
2723:
2720:
2719:
2693:
2689:
2680:
2676:
2675:
2671:
2669:
2666:
2665:
2638:
2627:
2626:
2622:
2620:
2617:
2616:
2590:
2586:
2584:
2581:
2580:
2556:
2545:
2544:
2540:
2519:
2515:
2510:
2507:
2506:
2482:
2471:
2470:
2466:
2464:
2461:
2460:
2434:
2430:
2428:
2425:
2424:
2402:
2391:
2362:
2359:
2358:
2320:
2316:
2295:
2291:
2289:
2286:
2285:
2284:. The relation
2259:
2255:
2253:
2250:
2249:
2219:
2215:
2210:
2207:
2206:
2180:
2176:
2174:
2171:
2170:
2124:
2121:
2120:
2100:
2097:
2096:
2079:
2076:
2075:
2058:
2055:
2054:
2051:
1999:
1993:
1992:
1991:
1989:
1986:
1985:
1967:
1964:
1963:
1946:
1943:
1942:
1925:
1919:
1918:
1917:
1915:
1912:
1911:
1888:
1885:
1884:
1858:
1855:
1854:
1828:
1825:
1824:
1801:
1798:
1797:
1770:
1767:
1766:
1746:
1743:
1742:
1722:
1719:
1718:
1701:
1698:
1697:
1680:
1677:
1676:
1659:
1656:
1655:
1638:
1635:
1634:
1617:
1614:
1613:
1587:
1584:
1583:
1560:
1557:
1556:
1539:
1536:
1535:
1514:
1511:
1510:
1493:
1490:
1489:
1472:
1469:
1468:
1451:
1448:
1447:
1421:
1418:
1417:
1394:
1391:
1390:
1373:
1370:
1369:
1352:
1349:
1348:
1331:
1328:
1327:
1301:
1298:
1297:
1274:
1271:
1270:
1267:binary relation
1250:
1247:
1246:
1226:
1223:
1222:
1205:
1202:
1201:
1184:
1181:
1180:
1159:
1156:
1155:
1138:
1135:
1134:
1117:
1114:
1113:
1094:
1091:
1090:
1084:
1072:
1045:consumer theory
987:
983:
977:
974:
973:
920:
916:
910:
907:
906:
856:
852:
846:
842:
824:
820:
814:
811:
810:
791:
784:
777:
770:
761:
746:
737:
726:
717:
714:
705:
676:Consumer theory
665:
654:consumer theory
282:
257:
250:
243:
236:
229:
222:
136:
128:ordinal utility
120:Vilfredo Pareto
112:
28:
23:
22:
15:
12:
11:
5:
5993:
5983:
5982:
5977:
5972:
5967:
5953:
5952:
5947:
5940:
5939:External links
5937:
5936:
5935:
5915:
5897:(2): 263–271.
5875:
5872:
5870:
5869:
5862:
5844:
5831:(2): 187–203.
5811:
5792:(3): 635–647.
5776:
5763:10.1086/467886
5741:
5734:
5716:
5709:
5691:
5684:
5656:
5649:
5626:
5619:
5595:
5588:
5557:
5532:
5512:
5491:
5474:
5457:
5441:
5439:
5436:
5433:
5432:
5423:
5378:
5377:
5375:
5372:
5370:
5369:
5364:
5359:
5354:
5352:Microeconomics
5349:
5344:
5339:
5334:
5329:
5324:
5318:
5316:
5313:
5267:
5264:
5255:
5252:
5244:
5243:
5232:
5227:
5224:
5221:
5216:
5211:
5208:
5203:
5196:
5192:
5189:
5186:
5180:
5177:
5171:
5168:
5163:
5160:
5143:
5142:
5131:
5126:
5123:
5120:
5116:
5110:
5107:
5103:
5099:
5095:
5091:
5087:
5081:
5075:
5071:
5067:
5064:
5061:
5058:
5055:
5052:
5047:
5043:
5039:
5035:
5030:
5027:
5024:
5021:
5018:
5015:
5012:
5009:
5006:
5003:
5000:
4995:
4991:
4976:
4975:
4962:
4959:
4956:
4952:
4946:
4943:
4939:
4935:
4931:
4927:
4923:
4917:
4911:
4907:
4903:
4900:
4897:
4894:
4891:
4888:
4883:
4879:
4875:
4871:
4866:
4863:
4860:
4857:
4854:
4851:
4848:
4843:
4839:
4814:
4811:
4808:
4784:
4781:
4778:
4758:
4755:
4752:
4749:
4746:
4743:
4740:
4729:
4728:
4715:
4711:
4707:
4702:
4696:
4692:
4688:
4685:
4682:
4679:
4676:
4673:
4668:
4664:
4660:
4656:
4651:
4648:
4645:
4642:
4639:
4636:
4633:
4614:
4611:
4610:
4609:
4598:
4594:
4589:
4586:
4581:
4575:
4571:
4568:
4565:
4559:
4556:
4550:
4547:
4542:
4539:
4505:
4502:
4499:
4477:
4472:
4468:
4464:
4460:
4456:
4451:
4448:
4445:
4442:
4439:
4436:
4432:
4428:
4425:
4422:
4418:
4412:
4408:
4386:
4364:
4361:
4358:
4353:
4349:
4345:
4341:
4337:
4332:
4329:
4325:
4321:
4318:
4315:
4311:
4305:
4301:
4279:
4257:
4254:
4251:
4247:
4241:
4237:
4233:
4229:
4225:
4222:
4219:
4215:
4211:
4193:
4190:
4176:
4155:
4144:
4143:
4132:
4127:
4124:
4119:
4116:
4110:
4107:
4102:
4099:
4073:
4070:
4066:
4062:
4059:
4056:
4052:
4046:
4042:
4020:
4000:
3997:
3993:
3989:
3986:
3983:
3979:
3973:
3969:
3947:
3927:
3924:
3921:
3918:
3915:
3912:
3908:
3904:
3901:
3898:
3894:
3890:
3878:
3877:Linear utility
3875:
3873:
3870:
3856:
3835:
3810:
3804:
3800:
3796:
3791:
3787:
3782:
3766:
3765:
3750:
3745:
3741:
3737:
3732:
3728:
3724:
3719:
3715:
3709:
3704:
3700:
3696:
3691:
3687:
3683:
3678:
3674:
3667:
3664:
3658:
3655:
3650:
3647:
3641:
3638:
3635:
3629:
3626:
3620:
3614:
3610:
3606:
3601:
3597:
3592:
3588:
3585:
3572:
3548:
3545:
3539:
3536:
3530:
3524:
3520:
3516:
3511:
3507:
3502:
3498:
3495:
3464:
3461:
3440:
3419:
3416:
3395:
3370:
3364:
3360:
3356:
3351:
3347:
3342:
3320:
3314:
3310:
3306:
3301:
3297:
3292:
3268:
3264:
3260:
3257:
3254:
3250:
3244:
3240:
3218:
3214:
3211:
3208:
3204:
3182:
3178:
3175:
3172:
3168:
3164:
3143:
3139:
3136:
3133:
3129:
3123:
3119:
3107:
3106:
3089:
3086:
3081:
3078:
3072:
3067:
3063:
3059:
3054:
3050:
3046:
3041:
3037:
3033:
3030:
3027:
3022:
3018:
3014:
3009:
3005:
3001:
2996:
2992:
2988:
2982:
2979:
2973:
2967:
2963:
2959:
2954:
2950:
2945:
2941:
2938:
2921:
2920:
2909:
2906:
2902:
2896:
2892:
2888:
2883:
2879:
2874:
2868:
2864:
2860:
2857:
2854:
2850:
2844:
2840:
2836:
2831:
2827:
2822:
2816:
2812:
2808:
2804:
2798:
2794:
2790:
2785:
2781:
2776:
2772:
2769:
2745:
2741:
2738:
2735:
2731:
2727:
2702:
2696:
2692:
2688:
2683:
2679:
2674:
2649:
2644:
2641:
2637:
2633:
2630:
2625:
2603:
2599:
2596:
2593:
2589:
2567:
2562:
2559:
2555:
2551:
2548:
2543:
2539:
2536:
2532:
2528:
2525:
2522:
2518:
2514:
2493:
2488:
2485:
2481:
2477:
2474:
2469:
2447:
2443:
2440:
2437:
2433:
2412:
2408:
2405:
2401:
2397:
2394:
2390:
2387:
2384:
2381:
2378:
2375:
2372:
2369:
2366:
2345:is called the
2333:
2329:
2326:
2323:
2319:
2315:
2312:
2308:
2304:
2301:
2298:
2294:
2272:
2268:
2265:
2262:
2258:
2232:
2228:
2225:
2222:
2218:
2214:
2193:
2189:
2186:
2183:
2179:
2131:
2128:
2107:
2104:
2083:
2062:
2050:
2047:
2034:
2031:
2028:
2025:
2022:
2019:
2016:
2013:
2010:
2007:
2002:
1996:
1971:
1950:
1928:
1922:
1898:
1895:
1892:
1868:
1865:
1862:
1841:
1838:
1835:
1832:
1811:
1808:
1805:
1780:
1777:
1774:
1750:
1726:
1705:
1684:
1663:
1642:
1621:
1610:
1609:
1597:
1594:
1591:
1579:The statement
1564:
1543:
1518:
1497:
1476:
1455:
1444:
1443:
1431:
1428:
1425:
1413:The statement
1398:
1377:
1356:
1335:
1324:
1323:
1311:
1308:
1305:
1293:The statement
1278:
1254:
1230:
1209:
1188:
1177:
1176:
1163:
1142:
1121:
1111:
1098:
1083:
1080:
1071:
1068:
1067:
1066:
1063:
1052:
1028:
1025:
1022:
1019:
1016:
1013:
1010:
1007:
1004:
1000:
996:
993:
990:
986:
982:
952:
949:
946:
943:
940:
937:
933:
929:
926:
923:
919:
915:
885:
882:
879:
876:
873:
870:
865:
862:
859:
855:
849:
845:
841:
837:
833:
830:
827:
823:
819:
789:
782:
775:
768:
763:
762:
747:
740:
738:
727:
720:
718:
715:
708:
704:
701:
697:expansion path
695:is called the
664:
661:
660:
659:
658:
657:
646:
643:
640:
634:
633:
621:. Examples of
611:
610:
587:
584:
583:
582:
552:
551:
544:
543:
540:
539:
538:
514:
484:
483:
476:
475:
472:
471:
470:
446:
419:
418:
411:
410:
409:
408:
373:
372:
365:
364:
361:
360:
359:
343:
327:
291:
290:
281:
278:
277:
276:
260:
255:
248:
241:
234:
227:
220:
212:
206:
174:
135:
132:
111:
108:
68:representation
26:
9:
6:
4:
3:
2:
5992:
5981:
5978:
5976:
5973:
5971:
5968:
5966:
5963:
5962:
5960:
5951:
5948:
5946:
5943:
5942:
5932:
5928:
5921:
5916:
5912:
5908:
5904:
5900:
5896:
5892:
5891:
5883:
5878:
5877:
5865:
5863:9780814775592
5859:
5855:
5848:
5839:
5834:
5830:
5826:
5822:
5815:
5807:
5803:
5799:
5795:
5791:
5787:
5780:
5772:
5768:
5764:
5760:
5756:
5752:
5745:
5737:
5735:0-321-01225-9
5731:
5727:
5720:
5712:
5710:0-07-054513-8
5706:
5702:
5695:
5687:
5685:0-297-76899-9
5681:
5677:
5673:
5669:
5663:
5661:
5652:
5650:0-07-118136-9
5646:
5642:
5635:
5633:
5631:
5622:
5616:
5612:
5608:
5602:
5600:
5591:
5589:0-321-01225-9
5585:
5581:
5574:
5572:
5570:
5568:
5566:
5564:
5562:
5546:
5542:
5536:
5522:
5516:
5508:
5504:
5503:
5495:
5487:
5486:
5478:
5470:
5469:
5461:
5453:
5446:
5442:
5427:
5420:
5416:
5412:
5408:
5404:
5400:
5397:
5393:
5389:
5383:
5379:
5368:
5365:
5363:
5360:
5358:
5355:
5353:
5350:
5348:
5345:
5343:
5340:
5338:
5335:
5333:
5330:
5328:
5325:
5323:
5320:
5319:
5312:
5310:
5305:
5303:
5299:
5295:
5291:
5287:
5283:
5279:
5275:
5273:
5263:
5261:
5251:
5249:
5230:
5225:
5222:
5219:
5214:
5209:
5206:
5201:
5194:
5190:
5187:
5184:
5178:
5175:
5169:
5166:
5161:
5158:
5148:
5147:
5146:
5129:
5124:
5121:
5118:
5114:
5108:
5105:
5101:
5097:
5093:
5089:
5085:
5079:
5073:
5069:
5062:
5059:
5056:
5050:
5045:
5041:
5037:
5033:
5025:
5022:
5019:
5013:
5007:
5004:
5001:
4993:
4989:
4981:
4980:
4979:
4960:
4957:
4954:
4950:
4944:
4941:
4937:
4933:
4929:
4925:
4921:
4915:
4909:
4905:
4898:
4895:
4892:
4886:
4881:
4877:
4873:
4869:
4864:
4861:
4855:
4852:
4849:
4841:
4837:
4829:
4828:
4827:
4812:
4806:
4798:
4782:
4779:
4776:
4753:
4750:
4747:
4741:
4738:
4713:
4709:
4705:
4700:
4694:
4690:
4683:
4680:
4677:
4671:
4666:
4662:
4658:
4654:
4649:
4643:
4640:
4637:
4631:
4624:
4623:
4622:
4620:
4596:
4592:
4587:
4584:
4579:
4573:
4569:
4566:
4563:
4557:
4554:
4548:
4545:
4540:
4537:
4527:
4526:
4525:
4523:
4519:
4503:
4500:
4497:
4475:
4470:
4466:
4462:
4458:
4454:
4446:
4443:
4440:
4434:
4430:
4426:
4423:
4420:
4416:
4410:
4406:
4384:
4362:
4359:
4356:
4351:
4347:
4343:
4339:
4335:
4330:
4327:
4323:
4319:
4316:
4313:
4309:
4303:
4299:
4277:
4255:
4252:
4249:
4245:
4239:
4235:
4231:
4227:
4223:
4220:
4217:
4213:
4209:
4200:
4197:
4189:
4174:
4153:
4130:
4125:
4122:
4117:
4114:
4108:
4105:
4100:
4097:
4087:
4086:
4085:
4071:
4068:
4064:
4060:
4057:
4054:
4050:
4044:
4040:
4018:
3998:
3995:
3991:
3987:
3984:
3981:
3977:
3971:
3967:
3945:
3925:
3922:
3919:
3916:
3913:
3910:
3906:
3902:
3899:
3896:
3892:
3888:
3869:
3854:
3833:
3825:
3808:
3802:
3798:
3794:
3789:
3785:
3780:
3771:
3743:
3739:
3735:
3730:
3726:
3717:
3713:
3702:
3698:
3694:
3689:
3685:
3676:
3672:
3665:
3662:
3656:
3653:
3648:
3645:
3636:
3633:
3627:
3624:
3618:
3612:
3608:
3604:
3599:
3595:
3590:
3586:
3583:
3573:
3570:
3566:
3562:
3546:
3543:
3537:
3534:
3528:
3522:
3518:
3514:
3509:
3505:
3500:
3496:
3493:
3483:
3482:
3481:
3479:
3462:
3459:
3452:by an amount
3438:
3417:
3414:
3393:
3385:
3368:
3362:
3358:
3354:
3349:
3345:
3340:
3318:
3312:
3308:
3304:
3299:
3295:
3290:
3280:
3266:
3262:
3258:
3255:
3252:
3248:
3242:
3238:
3216:
3212:
3209:
3206:
3202:
3180:
3176:
3173:
3170:
3166:
3162:
3141:
3137:
3134:
3131:
3127:
3121:
3117:
3105:
3087:
3084:
3079:
3076:
3065:
3061:
3057:
3052:
3048:
3039:
3035:
3031:
3028:
3020:
3016:
3012:
3007:
3003:
2994:
2990:
2986:
2980:
2977:
2971:
2965:
2961:
2957:
2952:
2948:
2943:
2939:
2936:
2926:
2925:
2924:
2907:
2904:
2900:
2894:
2890:
2886:
2881:
2877:
2872:
2866:
2862:
2858:
2855:
2852:
2848:
2842:
2838:
2834:
2829:
2825:
2820:
2814:
2810:
2806:
2802:
2796:
2792:
2788:
2783:
2779:
2774:
2770:
2767:
2760:
2759:
2758:
2743:
2739:
2736:
2733:
2729:
2725:
2717:
2714:and take the
2700:
2694:
2690:
2686:
2681:
2677:
2672:
2662:
2647:
2642:
2639:
2635:
2631:
2628:
2623:
2601:
2597:
2594:
2591:
2587:
2579:, the bundle
2565:
2560:
2557:
2553:
2549:
2546:
2541:
2537:
2534:
2530:
2526:
2523:
2520:
2516:
2512:
2491:
2486:
2483:
2479:
2475:
2472:
2467:
2445:
2441:
2438:
2435:
2431:
2406:
2403:
2399:
2395:
2392:
2385:
2382:
2376:
2373:
2370:
2364:
2356:
2352:
2348:
2331:
2327:
2324:
2321:
2317:
2313:
2306:
2302:
2299:
2296:
2292:
2270:
2266:
2263:
2260:
2256:
2247:
2230:
2226:
2223:
2220:
2216:
2212:
2191:
2187:
2184:
2181:
2177:
2168:
2164:
2160:
2156:
2152:
2148:
2143:
2129:
2126:
2105:
2102:
2081:
2060:
2046:
2029:
2026:
2023:
2020:
2017:
2014:
2011:
2005:
2000:
1983:
1982:. Formally,
1969:
1948:
1926:
1896:
1893:
1890:
1881:
1866:
1863:
1860:
1839:
1836:
1833:
1830:
1809:
1806:
1803:
1795:
1778:
1775:
1772:
1765:if all pairs
1764:
1748:
1739:
1724:
1703:
1682:
1661:
1640:
1619:
1595:
1592:
1589:
1582:
1581:
1580:
1577:
1562:
1541:
1533:
1516:
1495:
1474:
1453:
1429:
1426:
1423:
1416:
1415:
1414:
1411:
1396:
1375:
1354:
1333:
1309:
1306:
1303:
1296:
1295:
1294:
1291:
1276:
1268:
1252:
1243:
1228:
1207:
1186:
1161:
1140:
1119:
1112:
1096:
1089:
1088:
1087:
1079:
1077:
1064:
1062:to the other.
1061:
1057:
1053:
1050:
1049:
1048:
1046:
1041:
1023:
1020:
1017:
1014:
1011:
1002:
998:
994:
991:
988:
984:
980:
970:
965:
950:
947:
944:
941:
938:
935:
931:
927:
924:
921:
917:
913:
903:
898:
883:
880:
877:
874:
871:
868:
863:
860:
857:
853:
847:
843:
839:
835:
831:
828:
825:
821:
817:
808:
807:income effect
804:
800:
796:
792:
785:
778:
771:
759:
755:
751:
744:
739:
735:
731:
724:
719:
712:
707:
706:
700:
698:
694:
689:
685:
681:
677:
669:
655:
651:
647:
644:
641:
638:
637:
636:
635:
632:
628:
627:
626:
624:
620:
616:
608:
604:
600:
596:
592:
588:
585:
580:
576:
572:
568:
564:
560:
556:
555:
554:
553:
550:
546:
545:
541:
536:
533:
529:
525:
521:
518:
515:
512:
508:
504:
500:
496:
492:
488:
487:
486:
485:
482:
478:
477:
473:
468:
465:
461:
458:
454:
451:
447:
444:
441:
437:
434:
430:
427:
423:
422:
421:
420:
417:
413:
412:
407:
404:
400:
396:
393:
392:
391:
387:
383:
379:
375:
374:
371:
367:
366:
362:
358:
355:
351:
347:
344:
342:
339:
335:
331:
328:
326:
323:
319:
315:
312:
311:
309:
305:
301:
297:
293:
292:
288:
284:
283:
274:
269:
265:
261:
258:
251:
244:
237:
230:
223:
216:
213:
210:
207:
204:
199:
195:
191:
190:local maximum
187:
183:
179:
175:
172:
168:
167:
166:
163:
161:
157:
153:
145:
140:
131:
129:
125:
121:
117:
107:
105:
101:
97:
93:
89:
84:
80:
75:
73:
69:
65:
61:
57:
53:
49:
45:
41:
32:
19:
18:Utility curve
5930:
5926:
5894:
5888:
5853:
5847:
5828:
5824:
5814:
5789:
5785:
5779:
5754:
5750:
5744:
5725:
5719:
5700:
5694:
5671:
5640:
5610:
5579:
5548:. Retrieved
5544:
5535:
5524:. Retrieved
5515:
5506:
5501:
5494:
5484:
5477:
5466:
5460:
5451:
5445:
5426:
5418:
5414:
5410:
5406:
5402:
5398:
5395:
5391:
5387:
5382:
5306:
5276:
5269:
5257:
5245:
5144:
4977:
4797:Cobb–Douglas
4730:
4616:
4201:
4198:
4195:
4145:
3880:
3767:
3568:
3564:
3560:
3477:
3384:level curves
3281:
3108:
3103:
2922:
2663:
2355:real numbers
2163:completeness
2162:
2158:
2149:theory, the
2144:
2052:
1984:
1882:
1796:if whenever
1762:
1740:
1611:
1578:
1531:
1445:
1412:
1368:.' That is,
1325:
1292:
1244:
1178:
1085:
1075:
1073:
1042:
966:
899:
798:
787:
780:
773:
766:
764:
753:
749:
733:
729:
682:to generate
674:
649:
630:
622:
618:
617:rather than
614:
612:
606:
602:
598:
594:
590:
578:
574:
570:
566:
562:
558:
548:
534:
531:
527:
523:
519:
516:
510:
506:
502:
498:
494:
490:
480:
466:
463:
459:
456:
452:
449:
442:
439:
435:
432:
428:
425:
415:
405:
402:
398:
394:
389:
385:
381:
377:
369:
356:
353:
349:
345:
340:
337:
333:
329:
324:
321:
317:
313:
307:
303:
299:
295:
286:
266:. With (2),
253:
246:
239:
232:
225:
218:
197:
193:
185:
181:
164:
151:
149:
144:level curves
113:
103:
99:
95:
91:
78:
76:
47:
43:
37:
5357:Rationality
5347:Level curve
5258:As used in
4613:CES utility
2074:of the set
1532:indifferent
663:Application
262:(Strictly)
182:U = f(x, y)
160:bliss point
64:preferences
48:indifferent
5959:Categories
5757:(2): 225.
5526:2018-12-08
5438:References
5266:Criticisms
4621:) form is
4524:, is then
481:continuous
416:transitive
273:other good
215:Transitive
52:preference
5911:154152189
5771:155051169
5290:economics
5248:modelling
5226:ρ
5223:−
5195:α
5191:α
5188:−
5179:−
5122:−
5119:ρ
5106:−
5098:ρ
5074:ρ
5063:α
5060:−
5046:ρ
5038:α
5026:α
5023:−
4958:−
4955:ρ
4942:−
4934:ρ
4910:ρ
4899:α
4896:−
4882:ρ
4874:α
4865:α
4810:→
4807:ρ
4780:≤
4777:ρ
4742:∈
4739:α
4714:ρ
4695:ρ
4684:α
4681:−
4667:ρ
4659:α
4574:α
4570:α
4567:−
4558:−
4498:α
4476:α
4447:α
4444:−
4360:−
4357:α
4331:α
4256:α
4253:−
4240:α
4126:α
4123:β
4118:−
4072:β
3999:α
3923:β
3914:α
3666:−
3640:⇔
2383:≥
2311:→
2027:∼
2015:∈
1894:∈
1864:⪰
1834:⪰
1807:⪰
1749:⪰
1593:≻
1427:∼
1307:⪰
1253:⪰
1021:β
1012:α
948:β
939:α
905:would be
881:≤
878:α
875:≤
864:α
861:−
848:α
758:collinear
370:reflexive
178:satiation
40:economics
5670:(1975).
5609:(2008).
5550:23 March
5315:See also
3872:Examples
3826:between
2643:′
2632:′
2561:′
2550:′
2487:′
2476:′
2407:′
2396:′
1763:complete
448:Also if
287:complete
209:Complete
192:for any
171:quadrant
5975:Utility
5933:: 1–11.
5806:2006525
5413:, then
5260:biology
5254:Biology
4795:. (The
4490:.Where
3565:(Eq. 1)
3104:(Eq. 1)
2246:utility
2147:utility
1265:, is a
573:, then
462:, then
438:, then
110:History
56:utility
5909:
5860:
5804:
5769:
5732:
5707:
5682:
5647:
5617:
5586:
5405:, and
4731:where
4516:. The
3109:where
2349:. The
2153:of an
1654:, but
1488:, and
264:convex
81:. The
72:demand
5923:(PDF)
5907:S2CID
5885:(PDF)
5802:JSTOR
5767:S2CID
5505:[
5374:Notes
4518:slope
3770:slope
3569:dy/dx
3563:into
2505:. If
2351:range
2155:agent
1853:then
786:than
772:than
569:than
505:then
83:slope
60:locus
42:, an
5858:ISBN
5730:ISBN
5705:ISBN
5680:ISBN
5645:ISBN
5615:ISBN
5584:ISBN
5552:2018
5288:and
4978:and
4769:and
4501:<
3847:and
2535:>
1823:and
1133:and
1086:Let
799:most
752:and
732:and
615:good
593:and
565:and
522:and
497:and
455:and
431:and
388:and
380:and
306:and
298:and
196:and
5899:doi
5833:doi
5794:doi
5759:doi
5401:as
5298:WTA
5286:law
4398:is
4291:is
4167:or
4032:is
3959:is
3407:by
2718:of
2159:all
2145:In
1761:is
1738:.'
1555:or
1006:min
623:bad
619:bad
557:If
489:If
424:If
38:In
5961::
5929:.
5925:.
5905:.
5895:28
5893:.
5887:.
5829:15
5827:.
5823:.
5800:.
5790:81
5788:.
5765:.
5755:20
5753:.
5659:^
5629:^
5598:^
5560:^
5543:.
5304:.
3868:.
3480::
3279:)
3029:.1
2661:.
2045:.
1880:.
1290:.
1040:.
964:.
699:.
530:⇒
526:→
401:⇒
397:=
184:,
126:'
5931:2
5913:.
5901::
5866:.
5841:.
5835::
5808:.
5796::
5773:.
5761::
5738:.
5713:.
5688:.
5653:.
5623:.
5592:.
5554:.
5529:.
5421:.
5419:C
5415:A
5411:C
5407:B
5403:B
5396:A
5392:B
5388:A
5231:.
5220:1
5215:)
5210:y
5207:x
5202:(
5185:1
5176:=
5170:y
5167:d
5162:x
5159:d
5130:.
5125:1
5115:y
5109:1
5102:)
5094:/
5090:1
5086:(
5080:)
5070:y
5066:)
5057:1
5054:(
5051:+
5042:x
5034:(
5029:)
5020:1
5017:(
5014:=
5011:)
5008:y
5005:,
5002:x
4999:(
4994:2
4990:U
4961:1
4951:x
4945:1
4938:)
4930:/
4926:1
4922:(
4916:)
4906:y
4902:)
4893:1
4890:(
4887:+
4878:x
4870:(
4862:=
4859:)
4856:y
4853:,
4850:x
4847:(
4842:1
4838:U
4813:0
4783:1
4757:)
4754:1
4751:,
4748:0
4745:(
4710:/
4706:1
4701:)
4691:y
4687:)
4678:1
4675:(
4672:+
4663:x
4655:(
4650:=
4647:)
4644:y
4641:,
4638:x
4635:(
4632:U
4597:.
4593:)
4588:y
4585:x
4580:(
4564:1
4555:=
4549:y
4546:d
4541:x
4538:d
4504:1
4471:)
4467:y
4463:/
4459:x
4455:(
4450:)
4441:1
4438:(
4435:=
4431:)
4427:y
4424:,
4421:x
4417:(
4411:2
4407:U
4385:y
4363:1
4352:)
4348:y
4344:/
4340:x
4336:(
4328:=
4324:)
4320:y
4317:,
4314:x
4310:(
4304:1
4300:U
4278:x
4250:1
4246:y
4236:x
4232:=
4228:)
4224:y
4221:,
4218:x
4214:(
4210:U
4175:y
4154:x
4131:.
4115:=
4109:y
4106:d
4101:x
4098:d
4069:=
4065:)
4061:y
4058:,
4055:x
4051:(
4045:2
4041:U
4019:y
3996:=
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3988:y
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3978:(
3972:1
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3663:=
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3634:=
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3625:d
3619:)
3613:0
3609:y
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3591:(
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3571::
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3544:=
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3000:(
2995:1
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2987:=
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2807:=
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2192:)
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2178:(
2130:.
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2106:,
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2082:A
2061:a
2033:}
2030:a
2024:b
2021::
2018:A
2012:b
2009:{
2006:=
2001:a
1995:C
1970:a
1949:A
1927:a
1921:C
1897:A
1891:a
1867:c
1861:a
1840:,
1837:c
1831:b
1810:b
1804:a
1779:b
1776:,
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1454:a
1430:b
1424:a
1397:b
1376:a
1355:b
1334:a
1310:b
1304:a
1277:A
1229:b
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1187:A
1175:.
1162:A
1141:b
1120:a
1097:A
1027:}
1024:y
1018:,
1015:x
1009:{
1003:=
999:)
995:y
992:,
989:x
985:(
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951:y
945:+
942:x
936:=
932:)
928:y
925:,
922:x
918:(
914:U
884:1
872:0
869:,
858:1
854:y
844:x
840:=
836:)
832:y
829:,
826:x
822:(
818:U
790:1
788:I
783:2
781:I
776:2
774:I
769:3
767:I
760:.
754:Y
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734:Y
730:X
609:.
607:B
603:A
599:A
595:B
591:A
581:.
579:B
575:A
571:B
567:y
563:x
559:A
537:.
535:C
532:A
528:B
524:C
520:B
517:A
513:.
511:C
507:A
503:B
499:C
495:B
491:A
469:.
467:C
464:A
460:C
457:B
453:B
450:A
445:.
443:C
440:A
436:C
433:B
429:B
426:A
406:B
403:A
399:B
395:A
390:B
386:A
382:B
378:A
357:B
354:A
350:B
346:A
341:A
338:B
334:A
330:B
325:B
322:A
318:B
314:A
308:y
304:x
300:B
296:A
256:1
254:I
249:3
247:I
242:2
240:I
235:3
233:I
228:1
226:I
221:2
219:I
198:y
194:x
186:U
104:y
100:u
96:x
92:u
20:)
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