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: $w_{x}\log {x}+w_{y}\log {y}$. 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, $x$ and $y$. $p_{x}$ and $p_{y}$ are their prices. $w_{x}$ and $w_{y}$ are constant positive parameters. $u_{y}$ is a utility function of a single commodity ($y$).

Name	Function	Indifference curves	Demand curve	Monotonicity	Convexity	Good type	Example
Linear	${{x \over w_{x}}+{y \over w_{y}}}$	Straight lines	Step function: only goods with minimum ${w_{i}p_{i}}$ are demanded	Strong	Weak	Perfect substitutes	Potatoes of two different farms
Quasilinear	$x+u_{y}(y)$	Parallel curves	Demand for $y$ is determined by: $u_{y}'(y)=p_{y}/p_{x}$	Strong, if $u_{y}$ is increasing	Strong, if $u_{y}$ is concave	Substitutes, if $u_{y}$ is concave	Money ($x$) and another product ($y$)
Leontief	$\min({x \over w_{x}},{y \over w_{y}})$	L-shapes	hyperbolic: ${{\text{Income}} \over w_{x}p_{x}+w_{y}p_{y}}$	Weak	Weak	Perfect complements	Left and right shoes
Cobb–Douglas	$x^{w_{x}}y^{w_{y}}$	hyperbolic	hyperbolic: ${\frac {w_{x}}{w_{x}+w_{y}}}{{\text{Income}} \over p_{x}}$	Strong	Strong	Independent	Apples and socks
Maximum	$\max({x \over w_{x}},{y \over w_{y}})$	ר-shapes	Discontinuous step function: only one good with minimum ${w_{i}p_{i}}$ is demanded	Weak	Concave	Substitutes and interfering	Two simultaneous movies

References

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

Acknowledgements

This page has been greatly improved thanks to comments in the following Economics StackExchange thread.