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: w x log x + w y log y {\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, x {\displaystyle x} and y {\displaystyle y}. p x {\displaystyle p_{x}} and p y {\displaystyle p_{y}} are their prices. w x {\displaystyle w_{x}} and w y {\displaystyle w_{y}} are constant positive parameters and r {\displaystyle r} is another constant parameter. u y {\displaystyle u_{y}} is a utility function of a single commodity (y {\displaystyle y}). I {\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 | min ( x w x , y w y ) {\displaystyle \min \left({x \over w_{x}},{y \over w_{y}}\right)} | hyperbolic: I w x p x + w y p y {\displaystyle 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 {\displaystyle x^{w_{x}}y^{w_{y}}} | hyperbolic: w x w x + w y I p x {\displaystyle {\frac {w_{x}}{w_{x}+w_{y}}}{I \over p_{x}}} | I p x w x p y w y {\displaystyle I \over p_{x}^{w_{x}}p_{y}^{w_{y}}} | hyperbolic | Strong | Strong | Yes | Independent | Apples and socks |
| Linear | x w x + y w y {\displaystyle {{x \over w_{x}}+{y \over w_{y}}}} | "Step function" correspondence: only goods with minimum w i p i {\displaystyle {w_{i}p_{i}}} are demanded | ? | Straight lines | Strong | Weak | Yes | Perfect substitutes | Potatoes of two different farms |
| Quasilinear | x + u y ( y ) {\displaystyle x+u_{y}(y)} | Demand for y {\displaystyle y} is determined by: u y ′ ( y ) = p y / p x {\displaystyle u_{y}'(y)=p_{y}/p_{x}} | v ( p ) + I {\displaystyle v(p)+I} where v is a function of price only | Parallel curves | Strong, if u y {\displaystyle u_{y}} is increasing | Strong, if u y {\displaystyle u_{y}} is quasiconcave | No | Substitutes, if u y {\displaystyle u_{y}} is quasiconcave | Money (x {\displaystyle x}) and another product (y {\displaystyle y}) |
| Maximum | ( x w x , y w y ) {\displaystyle \left({x \over w_{x}},{y \over w_{y}}\right)} | Discontinuous step function: only one good with minimum w i p i {\displaystyle {w_{i}p_{i}}} is demanded | ? | ר-shapes | Weak | Concave | Yes | Substitutes and interfering | Two simultaneous movies |
| CES | ( ( x w x ) r + ( y w y ) r ) 1 / r {\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 r = − ∞ , 0 , 1 , ∞ {\displaystyle r=-\infty ,0,1,\infty }, respectively. | |||||
| Translog | w x ln x + w y ln y + w x y ln x ln y {\displaystyle w_{x}\ln {x}+w_{y}\ln {y}+w_{xy}\ln {x}\ln {y}} | ? | ? | Cobb–Douglas is a special case when w x y = 0 {\displaystyle w_{xy}=0}. | |||||
| Isoelastic | x w x + y w y {\displaystyle x^{w_{x}}+y^{w_{y}}} | ? | ? | ? | ? | ? | ? | ? | ? |
- 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 .