Editing Marshallian demand function (section)

== Uniqueness ==
<math>x^*(p, I)</math> is called a ''correspondence'' because in general it may be set-valued - there may be several different bundles that attain the same maximum utility. In some cases, there is a ''unique'' utility-maximizing bundle for each price and income situation; then, <math>x^*(p, I)</math> is a function and it is called the '''Marshallian demand function'''. 

If the consumer has strictly [[convex preferences]] and the prices of all goods are strictly positive, then there is a unique utility-maximizing bundle.<ref name=Varian>{{Cite Varian Microeconomic Analysis 3}}</ref>{{rp|156}} To prove this, suppose, by contradiction, that there are two different bundles, <math>x_1</math> and <math>x_2</math>, that maximize the utility. Then <math>x_1</math> and <math>x_2</math> are equally preferred. By definition of strict convexity, the mixed bundle <math>0.5 x_1 + 0.5 x_2</math> is strictly better than <math>x_1 , x_2</math>. But this contradicts the optimality of <math>x_1 , x_2</math>.