Finance: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: [math]\displaystyle{ w_x \log{x} + w_y\log{y} }[/math]. 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, [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math]. [math]\displaystyle{ p_x }[/math] and [math]\displaystyle{ p_y }[/math] are their prices. [math]\displaystyle{ w_x }[/math] and [math]\displaystyle{ w_y }[/math] are constant positive parameters and [math]\displaystyle{ r }[/math] is another constant parameter. [math]\displaystyle{ u_y }[/math] is a utility function of a single commodity ([math]\displaystyle{ y }[/math]). [math]\displaystyle{ I }[/math] is the total income (wealth) of the consumer.
| Name | Function | Marshallian Demand curve | Indirect utility | Indifference curves | Monotonicity | Convexity | Homothety | Good type | Example |
|---|---|---|---|---|---|---|---|---|---|
| Leontief | [math]\displaystyle{ \min\left({x\over w_x},{y\over w_y}\right) }[/math] | hyperbolic: [math]\displaystyle{ I \over w_x p_x+w_y p_y }[/math] | ? | L-shapes | Weak | Weak | Yes | Perfect complements | Left and right shoes |
| Cobb–Douglas | [math]\displaystyle{ x^{w_x} y^{w_y} }[/math] | hyperbolic: [math]\displaystyle{ \frac{w_x}{w_x+w_y} {I \over p_x} }[/math] | [math]\displaystyle{ I \over p_x^{w_x} p_y^{w_y} }[/math] | hyperbolic | Strong | Strong | Yes | Independent | Apples and socks |
| Linear | [math]\displaystyle{ {{x\over w_x}+{y\over w_y}} }[/math] | "Step function" correspondence: only goods with minimum [math]\displaystyle{ {w_i p_i} }[/math] are demanded | ? | Straight lines | Strong | Weak | Yes | Perfect substitutes | Potatoes of two different farms |
| Quasilinear | [math]\displaystyle{ x + u_y(y) }[/math] | Demand for [math]\displaystyle{ y }[/math] is determined by: [math]\displaystyle{ u_y'(y) = p_y/p_x }[/math] | [math]\displaystyle{ v(p) + I }[/math] where v is a function of price only | Parallel curves | Strong, if [math]\displaystyle{ u_y }[/math] is increasing | Strong, if [math]\displaystyle{ u_y }[/math] is quasiconcave | No | Substitutes, if [math]\displaystyle{ u_y }[/math] is quasiconcave | Money ([math]\displaystyle{ x }[/math]) and another product ([math]\displaystyle{ y }[/math]) |
| Maximum | [math]\displaystyle{ \left({x\over w_x},{y\over w_y}\right) }[/math] | Discontinuous step function: only one good with minimum [math]\displaystyle{ {w_i p_i} }[/math] is demanded | ? | ר-shapes | Weak | Concave | Yes | Substitutes and interfering | Two simultaneous movies |
| CES | [math]\displaystyle{ \left(\left({x\over w_x}\right)^r + \left({y\over w_y}\right)^r\right)^{1/r} }[/math] | See Marshallian demand function#Example | ? | Leontief, Cobb–Douglas, Linear and Maximum are special cases when [math]\displaystyle{ r=-\infty,0,1,\infty }[/math], respectively. | |||||
| Translog | [math]\displaystyle{ w_x \ln{x} + w_y \ln{y} + w_{xy} \ln{x} \ln{y} }[/math] | ? | ? | Cobb–Douglas is a special case when [math]\displaystyle{ w_{xy}=0 }[/math]. | |||||
| Isoelastic | [math]\displaystyle{ x^{w_x}+y^{w_y} }[/math] | ? | ? | ? | ? | ? | ? | ? | ? |
References
- Hal Varian (2006). Intermediate micro-economics. ISBN 0393927024. chapter 5.
Acknowledgements
This page has been greatly improved thanks to comments and answers in Economics StackExchange.
See also
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