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Utility functions on divisible goods

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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.

Name Function Marshallian Demand curve Indirect utility Indifference curves Monotonicity Convexity Homothety Good type Example
Leontief hyperbolic: ? L-shapes Weak Weak Yes Perfect complements Left and right shoes
Cobb–Douglas hyperbolic: hyperbolic Strong Strong Yes Independent Apples and socks
Linear "Step function" correspondence: only goods with minimum are demanded ? Straight lines Strong Weak Yes Perfect substitutes Potatoes of two different farms
Quasilinear Demand for is determined by: where v is a function of price only Parallel curves Strong, if is increasing Strong, if is quasiconcave No Substitutes, if is quasiconcave Money () and another product ()
Maximum Discontinuous step function: only one good with minimum is demanded ? ר-shapes Weak Concave Yes Substitutes and interfering Two simultaneous movies
CES See 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

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  • Hal Varian (2006). Intermediate micro-economics. W.W. Norton & Company. ISBN 0393927024. chapter 5.

Acknowledgements

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

See also

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