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 and $r$ is another constant parameter. $u_{y}$ is a utility function of a single commodity (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle y} ).

Name	Function	Indifference curves	Marshallian Demand curve	Monotonicity	Convexity	Homothety	Good type	Example
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	Yes	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	Yes	Independent	Apples and socks
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	Yes	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 quasiconcave	No	Substitutes, if $u_{y}$ is quasiconcave	Money ( $x$ ) and another product ( $y$ )
Maximum	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \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	Yes	Substitutes and interfering	Two simultaneous movies
CES	$(({x \over w_{x}})^{r}+({y \over w_{y}})^{r})^{1/r}$	Depends on $r$ . See Marshallian demand function#Example. Leontief, Cobb–Douglas, Linear and Maximum are special cases when $r=-\infty ,0,1,\infty$ , respectively.
Translog	$w_{x}\ln {x}+w_{y}\ln {y}+w_{xy}\ln {x}\ln {y}$	Cobb–Douglas is a special case when $w_{xy}=0$ .
Isoelastic	$x^{w_{x}}+y^{w_{y}}$	?	?	?	?	?	?	?