Editing Indifference curve (section)

=== 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.