Utility maximization problem: Difference between revisions

Revision as of 06:12, 26 September 2004

Revision as of 08:17, 26 September 2004

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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. Proof: B(p, w) is a [[compact space]]. So if u is [[continuous]], then the [[Karl Weierstraß|Weierstrass]] theorem implies that u(B(p, w)) is a compact subset of <math>\textbf R</math>. By the [[Heine-Borel theorem]], every compact set contains its maximum, so we can conclude that u(B(p, w)) has a maximum and hence there must be a package in B(p, w) that maps to this maximum.

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 indirectthe utility[[expenditure function]] and [[Hicksian demand]] in the [[Expenditure Minimization Problem]].

In practise, 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.