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: ${\displaystyle w_{x}\log {x}+w_{y}\log {y}}$. 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, ${\displaystyle x}$ and ${\displaystyle y}$. ${\displaystyle p_{x}}$ and ${\displaystyle p_{y}}$ are their prices. ${\displaystyle w_{x}}$ and ${\displaystyle w_{y}}$ are constant positive parameters and ${\displaystyle r}$ is another constant parameter. ${\displaystyle u_{y}}$ is a utility function of a single commodity (${\displaystyle y}$). ${\displaystyle I}$ is the total income (wealth) of the consumer.

Name	Function	Marshallian Demand curve	Indirect utility	Indifference curves	Monotonicity	Convexity	Homothety	Good type	Example
Leontief	${\displaystyle \min \left({x \over w_{x}},{y \over w_{y}}\right)}$	hyperbolic: ${\displaystyle I \over w_{x}p_{x}+w_{y}p_{y}}$	?	L-shapes	Weak	Weak	Yes	Perfect complements	Left and right shoes
Cobb–Douglas	${\displaystyle x^{w_{x}}y^{w_{y}}}$	hyperbolic: ${\displaystyle {\frac {w_{x}}{w_{x}+w_{y}}}{I \over p_{x}}}$	${\displaystyle I \over p_{x}^{w_{x}}p_{y}^{w_{y}}}$	hyperbolic	Strong	Strong	Yes	Independent	Apples and socks
Linear	${\displaystyle {{x \over w_{x}}+{y \over w_{y}}}}$	"Step function" correspondence: only goods with minimum ${\displaystyle {w_{i}p_{i}}}$ are demanded	?	Straight lines	Strong	Weak	Yes	Perfect substitutes	Potatoes of two different farms
Quasilinear	${\displaystyle x+u_{y}(y)}$	Demand for ${\displaystyle y}$ is determined by: ${\displaystyle u_{y}'(y)=p_{y}/p_{x}}$	${\displaystyle v(p)+I}$ where v is a function of price only	Parallel curves	Strong, if ${\displaystyle u_{y}}$ is increasing	Strong, if ${\displaystyle u_{y}}$ is quasiconcave	No	Substitutes, if ${\displaystyle u_{y}}$ is quasiconcave	Money (${\displaystyle x}$) and another product (${\displaystyle y}$)
Maximum	${\displaystyle \left({x \over w_{x}},{y \over w_{y}}\right)}$	Discontinuous step function: only one good with minimum ${\displaystyle {w_{i}p_{i}}}$ is demanded	?	ר-shapes	Weak	Concave	Yes	Substitutes and interfering	Two simultaneous movies
CES	${\displaystyle \left(\left({x \over w_{x}}\right)^{r}+\left({y \over w_{y}}\right)^{r}\right)^{1/r}}$	See Marshallian demand function#Example	?	Leontief, Cobb–Douglas, Linear and Maximum are special cases when ${\displaystyle r=-\infty ,0,1,\infty }$, respectively.
Translog	${\displaystyle w_{x}\ln {x}+w_{y}\ln {y}+w_{xy}\ln {x}\ln {y}}$	?	?	Cobb–Douglas is a special case when ${\displaystyle w_{xy}=0}$.
Isoelastic	${\displaystyle x^{w_{x}}+y^{w_{y}}}$	?	?	?	?	?	?	?	?

References

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

Acknowledgements

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

See also

• Utility functions on indivisible goods