Editing Indifference curve (section)

== Preference relations and utility ==
Choice theory formally represents consumers by a '''preference relation''', and use this representation to derive indifference curves showing combinations of equal preference to the consumer.

=== Preference relations ===
Let
:<math>A\;</math> be a set of mutually exclusive alternatives among which a consumer can choose.
:<math>a\;</math> and <math>b\;</math> be generic elements of <math>A\;</math>.
In the language of the example above, the set <math>A\;</math> is made of combinations of apples and bananas. The symbol <math>a\;</math> is one such combination, such as 1 apple and 4 bananas and <math>b\;</math> is another combination such as 2 apples and 2 bananas.

A preference relation, denoted <math>\succeq</math>, is a [[binary relation]] define on the set <math>A\;</math>.

The statement
:<math>a\succeq b\;</math>
is described as '<math>a\;</math> is weakly preferred to <math>b\;</math>.' That is, <math>a\;</math> is at least as good as <math>b\;</math> (in preference satisfaction).

The statement
:<math>a\sim b\;</math>
is described as '<math>a\;</math> is weakly preferred to <math>b\;</math>, and <math>b\;</math> is weakly preferred to <math>a\;</math>.' That is, one is ''indifferent'' to the choice of <math>a\;</math> or <math>b\;</math>, meaning not that they are unwanted but that they are equally good in satisfying preferences.

The statement
:<math>a\succ b\;</math>
is described as '<math>a\;</math> is weakly preferred to <math>b\;</math>, but <math>b\;</math> is not weakly preferred to <math>a\;</math>.' One says that '<math>a\;</math> is strictly preferred to <math>b\;</math>.'

The preference relation <math>\succeq</math> is '''complete''' if all pairs <math>a,b\;</math> can be ranked. The relation is a [[transitive relation]] if whenever <math>a\succeq b\;</math> and <math>b\succeq c,\;</math> then <math>a\succeq c\;</math>.

For any element <math>a \in A\;</math>, the corresponding indifference curve, <math>\mathcal{C}_a</math> is made up of all elements of <math>A\;</math> which are indifferent to <math>a</math>.  Formally,

<math>\mathcal{C}_a=\{b \in A:b \sim a\}</math>.

=== Formal link to utility theory ===

In the example above, an element <math>a\;</math> of the set <math>A\;</math> is made of two numbers: The number of apples, call it <math>x,\;</math> and the number of bananas, call it <math>y.\;</math>

In [[utility]] theory, the [[utility function]] of an [[agent (economics)|agent]] is a function that ranks ''all'' pairs of consumption bundles by order of preference (''completeness'') such that any set of three or more bundles forms a [[transitive relation]]. This means that for each bundle <math>\left(x,y\right)</math> there is a unique relation, <math>U\left(x,y\right)</math>, representing the [[utility]] (satisfaction) relation associated with <math>\left(x,y\right)</math>. The relation <math>\left(x,y\right)\to U\left(x,y\right)</math> is called the [[utility function]]. The [[Range of a function|range]] of the function is a set of [[real numbers]]. The actual values of the function have no importance. Only the ranking of those values has content for the theory. More precisely, if <math>U(x,y)\geq U(x',y')</math>, then the bundle <math>\left(x,y\right)</math> is described as at least as good as the bundle <math>\left(x',y'\right)</math>. If <math>U\left(x,y\right)>U\left(x',y'\right)</math>, the bundle <math>\left(x,y\right)</math> is described as strictly preferred to the bundle <math>\left(x',y'\right)</math>.

Consider a particular bundle <math>\left(x_0,y_0\right)</math> and take the [[total derivative]] of <math>U\left(x,y\right)</math> about this point:

:<math>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 </math> 

or, without loss of generality,

:<math>\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}</math> '''(Eq. 1)'''

where <math>U_1\left(x,y\right)</math> is the partial derivative of <math>U\left(x,y\right)</math> with respect to its first argument, evaluated at <math>\left(x,y\right)</math>. (Likewise for <math>U_2\left(x,y\right).</math>)

The indifference curve through <math>\left(x_0,y_0\right)</math> must deliver at each bundle on the curve the same utility level as bundle <math>\left(x_0,y_0\right)</math>. That is, when preferences are represented by a utility function, the indifference curves are the [[level curve]]s of the utility function. Therefore, if one is to change the quantity of <math>x\,</math> by <math>dx\,</math>, without moving off the indifference curve, one must also change the quantity of <math>y\,</math> by an amount <math>dy\,</math> such that, in the end, there is no change in ''U'':
:<math>\frac{dU\left(x_0,y_0\right)}{dx}= 0</math>, or, substituting ''0'' into ''(Eq. 1)'' above to solve for ''dy/dx'':
:<math>\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)}</math>.
Thus, the ratio of marginal utilities gives the absolute value of the [[slope]] of the indifference curve at point <math>\left(x_0,y_0\right)</math>. This ratio is called the [[marginal rate of substitution]] between <math>x\,</math> and <math>y\,</math>.

=== Examples ===

==== Linear utility ====

If the utility function is of the form <math>U\left(x,y\right)=\alpha x+\beta y</math> then the marginal utility of <math>x\,</math> is <math>U_1\left(x,y\right)=\alpha</math> and the marginal utility of <math>y\,</math> is <math>U_2\left(x,y\right)=\beta</math>. The slope of the indifference curve is, therefore,
:<math>\frac{dx}{dy}=-\frac{\beta}{\alpha}.</math>
Observe that the slope does not depend on <math>x\,</math> or <math>y\,</math>: the indifference curves are straight lines.

==== Cobb–Douglas utility ====

A class of utility functions known as Cobb-Douglas utility functions are very commonly used in economics for two reasons:

1. They represent ‘well-behaved’ preferences, such as more is better and preference for variety.

2. They are very flexible and can be adjusted to fit real-world data very easily.
If the utility function is of the form <math>U\left(x,y\right)=x^\alpha y^{1-\alpha}</math> the marginal utility of <math>x\,</math> is <math>U_1\left(x,y\right)=\alpha \left(x/y\right)^{\alpha-1}</math> and the marginal utility of <math>y\,</math> is <math>U_2\left(x,y\right)=(1-\alpha) \left(x/y\right)^{\alpha}</math>.Where <math>\alpha<1</math>. The [[slope]] of the indifference curve, and therefore the negative of the [[marginal rate of substitution]], is then
:<math>\frac{dx}{dy}=-\frac{1-\alpha}{\alpha}\left(\frac{x}{y}\right).</math>

==== CES utility ====
A general CES ([[Constant Elasticity of Substitution]]) form is
:<math>U(x,y)=\left(\alpha x^\rho +(1-\alpha)y^\rho\right)^{1/\rho}</math>
where <math>\alpha\in(0,1)</math> and <math>\rho\leq 1</math>. (The [[Cobb–Douglas]] is a special case of the CES utility, with <math>\rho\rightarrow 0\,</math>.) The marginal utilities are given by
:<math>U_1(x,y)=\alpha \left(\alpha x^\rho +(1-\alpha)y^\rho\right)^{\left(1/\rho\right)-1} x^{\rho-1}</math>
and
:<math>U_2(x,y)=(1-\alpha)\left(\alpha x^\rho +(1-\alpha)y^\rho\right)^{\left(1/\rho\right)-1} y^{\rho-1}.</math>
Therefore, along an indifference curve,
:<math>\frac{dx}{dy}=-\frac{1-\alpha}{\alpha}\left(\frac{x}{y}\right)^{1-\rho}.</math>
These examples might be useful for [[model (economics)|modelling]] individual or aggregate demand.

==== Biology ====
As used in [[biology]], 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.