Editing Indifference curve (section)

== Assumptions of consumer preference theory ==
*Preferences are '''complete'''. The consumer has ranked all available alternative combinations of commodities in terms of the satisfaction they provide him.
:Assume that there are two consumption bundles ''A'' and ''B'' each containing two commodities ''x'' and ''y''. A consumer can unambiguously determine that one and only one of the following is the case:
:*''A'' is preferred to ''B'', formally written as ''A'' <sup>p</sup> ''B''<ref name="Binger">{{cite book |last1=Binger |last2=Hoffman |year=1998 |title=Microeconomics with Calculus |edition=2nd |publisher=Addison-Wesley |location=Reading |isbn=0-321-01225-9 |pages=109–117 }}</ref>
:*''B'' is preferred to ''A'', formally written as ''B'' <sup>p</sup> ''A''<ref name="Binger"/>
:*''A'' is indifferent to ''B'', formally written as ''A'' <sup>I</sup> ''B''<ref name="Binger"/>
:This axiom precludes the possibility that the consumer cannot decide,<ref name="Perloff 2008. p. 62">{{cite book |last=Perloff |first=Jeffrey M. |author-link=Jeffrey M. Perloff|year=2008 |title=Microeconomics: Theory & Applications with Calculus |publisher=Addison-Wesley |location=Boston |isbn=978-0-321-27794-7 |page=62 }}</ref> It assumes that a consumer is able to make this comparison with respect to every conceivable bundle of goods.<ref name="Binger"/>
*Preferences are '''reflexive'''
:This means that if ''A'' and ''B'' are identical in all respects the consumer will recognize this fact and be indifferent in comparing ''A'' and ''B''
:*''A'' = ''B'' ⇒ ''A'' <sup>I</sup> ''B''<ref name="Binger"/>
*Preferences are '''transitive'''{{refn|The transitivity of weak preferences is sufficient for most indifference-curve analyses: If ''A'' is weakly preferred to ''B'', meaning that the consumer likes ''A'' ''at least as much'' as ''B'', and ''B'' is weakly preferred to ''C'', then ''A'' is weakly preferred to ''C''.<ref name="Perloff 2008. p. 62"/>|group=nb}}
:*If ''A'' <sup>p</sup> ''B'' and ''B'' <sup>p</sup> ''C'', then ''A'' <sup>p</sup> ''C''.<ref name="Binger"/>
:*Also if ''A'' <sup>I</sup> ''B'' and ''B'' <sup>I</sup> ''C'', then ''A'' <sup>I</sup> ''C''.<ref name="Binger"/>
:This is a consistency assumption.
*Preferences are '''continuous'''
:*If ''A'' is preferred to ''B'' and ''C'' is sufficiently close to ''B'' then ''A'' is preferred to ''C''.
:*''A'' <sup>p</sup> ''B'' and ''C'' → ''B'' ⇒ ''A'' <sup>p</sup> ''C''.
:"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.
*Preferences exhibit '''strong monotonicity'''
:*If ''A'' has more of both ''x'' and ''y'' than ''B'', then ''A'' is preferred to ''B''.
:This assumption is commonly called the "more is better" assumption.
:An alternative version of this assumption requires that if ''A'' and ''B'' have the same quantity of one good, but ''A'' has more of the other, then ''A'' is preferred to ''B''.
It also implies that the commodities are '''good''' rather than '''bad'''. Examples of '''bad''' commodities can be disease, pollution etc. because we always desire less of such things.
*Indifference curves exhibit '''diminishing marginal rates of substitution'''
:*The marginal rate of substitution tells how much 'y' a person is willing to sacrifice to get one more unit of 'x'.{{clarify|reason=After having explained 'marginal rate of substitution', explain when they are called 'diminishing.|date=December 2015}}
:*This assumption assures that indifference curves are smooth and convex to the origin.
:*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.
:*Another name for this assumption is the '''substitution assumption'''.  It is the most critical assumption of [[consumer theory]]: 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"<ref name="Silberberg">{{cite book |last1=Silberberg |last2=Suen |year=2000 |title=The Structure of Economics: A Mathematical Analysis |edition=3rd |publisher=McGraw-Hill |location=Boston |isbn=0-07-118136-9 }}</ref> The negative slope of  the indifference curves represents the willingness of the consumer to make a trade off.<ref name="Silberberg"/>
<!-- If reflexivity is needed, it seems rather odd to assume also irreflexivity. And the definition of negative transitivity seems not to be correct.
*There are also many sub-assumptions:{{clarify|reason=Why are they called so? Are they inferior? Do they follow from the above assumptions? Are they debated in the literature?|date=December 2015}}
:*Irreflexivity - for no ''x'' is ''x'' <sup>p</sup> ''x''
:*Negative transitivity - if not ''x'' <sup>p</sup> ''y'', then for any third commodity{{clarify|reason=Should be 'bundle'?|date=December 2015}} ''z'', either not ''x'' <sup>p</sup> ''z'', or not ''z'' <sup>p</sup> ''y'', or neither of them.
-->

=== Application ===
[[File:Indifference curves showing budget line.svg|thumb|right|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.]]

[[Consumer theory]] uses indifference curves and [[budget constraint|budget constraints]] to generate [[Supply and demand|consumer demand curves]]. 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. [[budget constraint|Budget constraints]] 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 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.<ref name="Lipsey 1975. pp. 182-186">{{Cite book|title=An Introduction to Positive Economics |edition=Fourth |first=Richard G. |last=Lipsey |author-link=Richard Lipsey|year=1975 |publisher=[[Weidenfeld & Nicolson]] | isbn=0-297-76899-9 |pages=182–186 }}</ref> 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.<ref name="Lipsey 1975. pp. 182-186"/> Stated precisely, a set of indifference curve for representative of different price ratios between two goods are used to generate the [[Price-consumption curve]] in good-good vector space, which is equivalent to the [[demand curve]] in good-price vector space. The line connecting all points of tangency between the indifference curve and the [[budget constraint]] as the budget constraint changes is called the [[expansion path]],<ref name="Salvatore">{{cite book |last=Salvatore |first=Dominick |year=1989 |title=Schaum's Outline of Theory and Problems of Managerial Economics |publisher=McGraw-Hill |isbn=0-07-054513-8 }}</ref> and correlates to shifts in demand.  The line connecting all points of tangency between the indifference curve and budget constraint as the price of either good changes is the price-consumption curve, and correlates to movement along the demand curve.
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=== Examples of indifference curves ===
<gallery class="center">
File:Simple-indifference-curves.svg|Figure 1: An example of an indifference map with three indifference curves represented
File:Indifference-curves-perfect-substitutes.svg|Figure 2: Three indifference curves where Goods ''X'' and ''Y'' are perfect substitutes. The gray line perpendicular to all curves indicates the curves are mutually parallel.  
File:Indifference-curves-perfect-complements.svg|Figure 3: Indifference curves for perfect complements ''X'' and ''Y''. The elbows of the curves are [[collinear]]. The grey line shows the [[Income–consumption curve]] (the consumer theory equivalent to the [[Expansion path]]) of a series of [[Leontief utilities|Leontief utility curves]].
</gallery>

In Figure 1, the consumer would rather be on ''I<sub>3</sub>'' than ''I<sub>2</sub>'', and would rather be on ''I<sub>2</sub>'' than ''I<sub>1</sub>'', 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 [[marginal rate of substitution]], shows the rate at which consumers are willing to give up one good in exchange for more of the other good. For ''most'' 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 [[substitution effect]]. 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 [[income effect]] of lower real income (Beattie-LaFrance). An example of a utility function that generates indifference curves of this kind is the Cobb–Douglas function <math>\scriptstyle U\left(x,y\right)=x^\alpha y^{1-\alpha }, 0 \leq \alpha \leq 1</math>. The negative slope of the indifference curve incorporates the willingness of the consumer to make trade offs.<ref name="Silberberg"/>

If two goods are [[substitute good|perfect substitutes]] 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 would be <math>\scriptstyle U\left(x,y\right)=\alpha x + \beta y</math>.

If two goods are [[complement good|perfect complements]] 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 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: <math>\scriptstyle U\left(x,y\right)= \min \{ \alpha x, \beta y \}</math>.

The different shapes of the curves imply different responses to a change in price as shown from demand analysis in [[consumer theory]]. The results will only be stated here. A price-budget-line change that kept a consumer in  equilibrium on the same indifference curve:
:in Fig. 1 would reduce quantity demanded of a good smoothly as price rose relatively for that good.
:in Fig. 2 would have either no effect on quantity demanded of either good (at one end of the [[budget constraint]]) or would change quantity demanded from one end of the [[budget constraint]] to the other.
:in Fig. 3 would have no effect on equilibrium quantities demanded, since the budget line would rotate around the corner of the indifference curve.{{refn|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.<ref>{{cite book |last1=Binger |last2=Hoffman |year=1998 |title=Microeconomics with Calculus |edition=2nd |publisher=Addison-Wesley |location=Reading |isbn=0-321-01225-9 |pages=141–143 }}</ref>|group=nb}}