Utility maximization problem: Difference between revisions

Revision as of 14:33, 24 August 2010

Revision as of 23:27, 24 August 2010

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In [[microeconomics]], the '''utility maximization problem''' is the problem [[consumers]] face: "how should I spend my [[money]] in order to maximize my [[utility]]?" It is a type of [[Optimal_decisionOptimal decision|optimal decision problem]].

==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>

Suppose also that the prices (''p'') of the ''L'' commodities are positive

:<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>

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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>

Then the consumer's optimal choices ''x''(''p'', ''w'') are the utility maximizing bundle that is in the budget set, or

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:<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.{{factCitation needed|date=August 2010}} 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.{{factCitation needed|date=August 2010}} If a consumer always picks an optimal package as defined above, then ''x''(''p'', ''w'') is called the [[Marshallian demand correspondence]].

==See also==

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*[[Linear programming|Profit maximization formulae]]

{{No footnotes|date=August 2010}}

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==References==

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*[http://students.washington.edu/fuleky/anatomy/anatomy.html Anatomy of Cobb-Douglas Type Utility Functions in 3D]

{{DEFAULTSORT:Utility Maximization Problem}}

[[Category:Consumer theory]]

[[Category:Optimal decisions]]

[[Category:Utility]]

{{econ-stub}}