Utility maximization problem: Difference between revisions

Revision as of 15:22, 6 September 2009

Revision as of 14:33, 24 August 2010

Line 2:

==Basic setup==

Suppose their [[consumption set]], or the enumeration of all possible consumption bundles that could be selected if there are no budget constraints, has ''L'' commodities and is limited to positive amounts of consumption of each commodity. Let ''x'' be the vector ''x''={''x_i'';''i''=1,...''L''} containing the ammounts of each commodity, then

:<math>x \in \textbf R^L_+ \ .</math>

Line 10:

:<math>p \in \textbf R^L_+ \ ,</math>

and that the consumer's wealth is ''w'', then the set of all affordable packages, the [[budget set]], is

:<math>B(p, w) = \{x \in \textbf R^L_+ : \langle p , x \rangle \leq w\} \ ,</math>

where <math>\langle p , x \rangle</math> is the [[Inner-product|inner product]] of ''p'' and ''x'', or the total cost of consuming ''x'' of the products at price level ''p''.:
:<math>\langle p , x \rangle=\sum_{i=1}^L p_i x_i .</math>

The consumer would like to buy the best package of commodities it can afford. Suppose that the consumer's utility function (''u'') is a real valued function with ___domain of the commodity bundles, or

:<math>u : \textbf R^L_+ \rightarrow \textbf R \ .</math>

Line 22 ⟶ 25:

:<math>x(p, w) = \operatorname{argmax}_{x^* \in B(p, w)} u(x^*)</math>.

Finding ''x''(''p'', ''w'') is the '''utility maximization problem'''. If ''u'' is continuous and no commodities are free of charge, then x(p, w) exists.{{fact}} If there is always a unique maximizer, then it is called the [[Marshallian demand function]]. The relationship between the [[utility function]] and [[Marshallian demand]] in the Utility Maximization Problem mirrors the relationship between the [[expenditure function]] and [[Hicksian demand]] in the [[Expenditure Minimization Problem]].

In practice, a consumer may not always pick an optimal package. For example, it may require too much thought. [[Bounded rationality]] is a theory that explains this behaviour with [[satisficing]] - picking packages that are suboptimal but good enough.

=== non unique solution ===

The solution ''x''(''p'', ''w'') need not be unique.{{fact}} If a consumer always picks an optimal package as defined above, then ''x''(''p'', ''w'') is called the [[Marshallian demand correspondence]].

==See also==