Leontief utilities

In economics, especially in consumer theory, a Leontief utility function is a function of the form: [math]\displaystyle{ u(x_1,\ldots,x_m)=\min\left\{\frac{x_1}{w_1},\ldots,\frac{x_m}{w_m}\right\} . }[/math] where:

• [math]\displaystyle{ m }[/math] is the number of different goods in the economy.
• [math]\displaystyle{ x_i }[/math] (for [math]\displaystyle{ i\in 1,\dots,m }[/math]) is the amount of good [math]\displaystyle{ i }[/math] in the bundle.
• [math]\displaystyle{ w_i }[/math] (for [math]\displaystyle{ i\in 1,\dots,m }[/math]) is the weight of good [math]\displaystyle{ i }[/math] for the consumer.

This form of utility function was first conceptualized by Wassily Leontief.

Examples

Leontief utility functions represent complementary goods. For example:

• Suppose [math]\displaystyle{ x_1 }[/math] is the number of left shoes and [math]\displaystyle{ x_2 }[/math] the number of right shoes. A consumer can only use pairs of shoes. Hence, his utility is [math]\displaystyle{ \min(x_1,x_2) }[/math].
• In a cloud computing environment, there is a large server that runs many different tasks. Suppose a certain type of a task requires 2 CPUs, 3 gigabytes of memory and 4 gigabytes of disk-space to complete. The utility of the user is equal to the number of completed tasks. Hence, it can be represented by: [math]\displaystyle{ \min({x_{\mathrm{CPU}}\over 2}, {x_{\mathrm{MEM}}\over 3}, {x_{\mathrm{DISK}}\over 4}) }[/math].

Properties

A consumer with a Leontief utility function has the following properties:

• The preferences are weakly monotone but not strongly monotone: having a larger quantity of a single good does not increase utility, but having a larger quantity of all goods does.
• The preferences are weakly convex, but not strictly convex: a mix of two equivalent bundles may be either equivalent to or better than the original bundles.
• The indifference curves are L-shaped and their corners are determined by the weights. E.g., for the function [math]\displaystyle{ \min(x_1/2, x_2/3) }[/math], the corners of the indifferent curves are at [math]\displaystyle{ (2t,3t) }[/math] where [math]\displaystyle{ t\in[0,\infty) }[/math].
• The consumer's demand is always to get the goods in constant ratios determined by the weights, i.e. the consumer demands a bundle [math]\displaystyle{ (w_1 t,\ldots,w_m t) }[/math] where [math]\displaystyle{ t }[/math] is determined by the income: [math]\displaystyle{ t = \text{Income} / (p_1 w_1 + \dots + p_m w_m) }[/math].^[1] Since the Marshallian demand function of every good is increasing in income, all goods are normal goods.^[2]

Competitive equilibrium

Since Leontief utilities are not strictly convex, they do not satisfy the requirements of the Arrow–Debreu model for existence of a competitive equilibrium. Indeed, a Leontief economy is not guaranteed to have a competitive equilibrium. There are restricted families of Leontief economies that do have a competitive equilibrium.

There is a reduction from the problem of finding a Nash equilibrium in a bimatrix game to the problem of finding a competitive equilibrium in a Leontief economy.^[3] This has several implications:

• It is NP-hard to say whether a particular family of Leontief exchange economies, that is guaranteed to have at least one equilibrium, has more than one equilibrium.
• It is NP-hard to decide whether a Leontief economy has an equilibrium.

Moreover, the Leontief market exchange problem does not have a fully polynomial-time approximation scheme, unless PPAD ⊆ P.^[4]

On the other hand, there are algorithms for finding an approximate equilibrium for some special Leontief economies.^[3][5]

Application

Dominant resource fairness is a common rule for resource allocation in cloud computing systems, which assums that users have Leontief preferences.

References

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