Utility functions on divisible goods

This page compares the properties of several typical utility functions of divisible goods. These functions are commonly used as examples in consumer theory.

The functions are ordinal utility functions, which means that their properties are invariant under positive monotone transformation. For example, the Cobb–Douglas function could also be written as: . Such functions only become interesting when there are two or more goods (with a single good, all monotonically increasing functions are ordinally equivalent).

The utility functions are exemplified for two goods, and . and are their prices. and are constant positive parameters and is another constant parameter. is a utility function of a single commodity (). is the total income (wealth) of the consumer.

NameFunctionMarshallian Demand curveIndirect utilityIndifference curvesMonotonicityConvexityHomothetyGood typeExample
Leontiefhyperbolic:  ?L-shapesWeakWeakYesPerfect complementsLeft and right shoes
Cobb–Douglashyperbolic: hyperbolicStrongStrongYesIndependentApples and socks
Linear"Step function" correspondence: only goods with minimum are demanded ?Straight linesStrongWeakYesPerfect substitutesPotatoes of two different farms
QuasilinearDemand for is determined by: where v is a function of price onlyParallel curvesStrong, if is increasingStrong, if is quasiconcaveNoSubstitutes, if is quasiconcaveMoney () and another product ()
MaximumDiscontinuous step function: only one good with minimum is demanded ?ר-shapesWeakConcaveYesSubstitutes and interferingTwo simultaneous movies
CESSee Marshallian demand function#Example ?Leontief, Cobb–Douglas, Linear and Maximum are special cases
when , respectively.
Translog ? ?Cobb–Douglas is a special case when .
Isoelastic ? ? ? ? ? ? ? ?

References

  • Hal Varian (2006). Intermediate micro-economics. ISBN 0393927024. chapter 5.

Acknowledgements

This page has been greatly improved thanks to comments and answers in Economics StackExchange.

See also

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