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 utility functions are exemplified for two commodity types, $x$ and $y$. $p_{x}$ and $p_{y}$ are their prices. $w_{x}$ and $w_{y}$ are constant parameters.

Name	Function	Indifference curves	Demand curve	Monotonicity	Convexity	Good type	Example
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	perfect substitutes	Potatoes of two different farms
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	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	independent	Apples and socks
Maximum	$\max({x \over w_{x}},{y \over w_{y}})$	ר-shapes	Discontinuous step function: only one good with minimum ${w_{i}p_{i}}$ is demanded	Weak	Concave	substitutes	?

References

• Hal Varian (2006). Intermediate micro-economics. ISBN 0393927024. chapter 5.