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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: . 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, and . and are their prices. and are constant positive parameters and is another constant parameter. is a utility function of a single commodity (). is the total income (wealth) of the consumer.
Name
Function
Marshallian Demand curve
Indirect utility
Indifference curves
Monotonicity
Convexity
Homothety
Good type
Example
Leontief
hyperbolic:
?
L-shapes
Weak
Weak
Yes
Perfect complements
Left and right shoes
Cobb–Douglas
hyperbolic:
hyperbolic
Strong
Strong
Yes
Independent
Apples and socks
Linear
"Step function" correspondence: only goods with minimum are demanded
?
Straight lines
Strong
Weak
Yes
Perfect substitutes
Potatoes of two different farms
Quasilinear
Demand for is determined by:
where v is a function of price only
Parallel curves
Strong, if is increasing
Strong, if is quasiconcave
No
Substitutes, if is quasiconcave
Money () and another product ()
Maximum
Discontinuous step function: only one good with minimum is demanded
?
ר-shapes
Weak
Concave
Yes
Substitutes and interfering
Two simultaneous movies
CES
See Marshallian demand function#Example
?
Leontief, Cobb–Douglas, Linear and Maximum are special cases when , respectively.
Translog
?
?
Cobb–Douglas is a special case when .
Isoelastic
?
?
?
?
?
?
?
?
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