Utility functions on divisible goods
# 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 ($y$). $I$ is the total income (wealth) of the consumer.

| Name | Function | Marshallian Demand curve | Indirect utility | Indifference curves | Monotonicity | Convexity | Homothety | Good type | Example |
| - | - | - | - | - | - | - | - | - | - |
| Leontief | $\min \left({x \over w_{x}},{y \over w_{y}}\right)$ | hyperbolic: $I \over w_{x}p_{x}+w_{y}p_{y}$ | ? | L-shapes | Weak | Weak | Yes | Perfect complements | Left and right shoes |
| Cobb–Douglas | $x^{w_{x}}y^{w_{y}}$ | hyperbolic: ${\frac {w_{x}}{w_{x}+w_{y}}}{I \over p_{x}}$ | $I \over p_{x}^{w_{x}}p_{y}^{w_{y}}$ | hyperbolic | Strong | Strong | Yes | Independent | Apples and socks |
| Linear | ${{x \over w_{x}}+{y \over w_{y}}}$ | "Step function" correspondence: only goods with minimum ${w_{i}p_{i}}$ are demanded | ? | Straight lines | Strong | Weak | Yes | Perfect substitutes | Potatoes of two different farms |
| Quasilinear | $x+u_{y}(y)$ | Demand for $y$ is determined by: $u_{y}'(y)=p_{y}/p_{x}$ | $v(p)+I$ where v is a function of price only | Parallel curves | 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 | $\left({x \over w_{x}},{y \over w_{y}}\right)$ | Discontinuous step function: only one good with minimum ${w_{i}p_{i}}$ is demanded | ? | ר-shapes | Weak | Concave | Yes | Substitutes and interfering | Two simultaneous movies |
| CES | $\left(\left({x \over w_{x}}\right)^{r}+\left({y \over w_{y}}\right)^{r}\right)^{1/r}$ | See Marshallian demand function#Example | ? | Leontief, Cobb–Douglas, Linear and Maximum are special cases when $r=-\infty ,0,1,\infty$, respectively. | Leontief, Cobb–Douglas, Linear and Maximum are special cases when $r=-\infty ,0,1,\infty$, respectively. | Leontief, Cobb–Douglas, Linear and Maximum are special cases when $r=-\infty ,0,1,\infty$, respectively. | Leontief, Cobb–Douglas, Linear and Maximum are special cases when $r=-\infty ,0,1,\infty$, respectively. | Leontief, Cobb–Douglas, Linear and Maximum are special cases when $r=-\infty ,0,1,\infty$, respectively. | 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 $w_{xy}=0$. | Cobb–Douglas is a special case when $w_{xy}=0$. | Cobb–Douglas is a special case when $w_{xy}=0$. | Cobb–Douglas is a special case when $w_{xy}=0$. | Cobb–Douglas is a special case when $w_{xy}=0$. | Cobb–Douglas is a special case when $w_{xy}=0$. |
| Isoelastic | $x^{w_{x}}+y^{w_{y}}$ | ? | ? | ? | ? | ? | ? | ? | ? |
