Editing Indifference curve (section)

=== 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>.