Utility maximization problem: Difference between revisions

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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]]?"

Suppose their [[consumption set]], or the enumeration of all possible consumption bundles that could be selected if there are no budget constraint has ''L'' commodities and is limited to positive amounts of consumption of each

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

If a consumer always picks an optimal package as defined above, thenFinding ''x''(''p'', ''w'') is called the [[Marshallian'''utility demandmaximization correspondence]]problem'''. If ''u'' is continuous and no commodities are free of charge, then x(p, w) exists. 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]].

Finding ''x''(''p'', ''w'') is the '''utility maximization 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.

The solution ''x''(''p'', ''w'') need not be unique. If ''u'' is continuous and no commodities are free of charge, then x(p, w) is nonempty.

=== non unique solution ===

If a consumer always picks an optimal package as defined above, then ''x''(''p'', ''w'') is called the [[Marshallian demand correspondence]]. 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]].

The solution ''x''(''p'', ''w'') need not be unique. If ''u''a isconsumer continuousalways andpicks noan commoditiesoptimal arepackage freeas ofdefined chargeabove, then ''x''(''p'', ''w'') is nonemptycalled the [[Marshallian demand correspondence]].

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.

==See also==