Leontief utilities

In economics and consumer theory, a Leontief utility function is a function of the form:

$u(x_{1},\ldots ,x_{m})=\min \left\{{\frac {x_{1}}{w_{1}}},\ldots ,{\frac {x_{m}}{w_{m}}}\right\}$.

where:

• $m$ is the number of different goods in the economy.
• $x_{i}$ (for $i\in 1,\dots ,m$) is the amount of good $i$ in the bundle..
• $w_{i}$ (for $i\in 1,\dots ,m$) is the weight of good $i$ for the consumer.

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

Examples

Leontief utility functions represent complementary goods. For example:

• Suppose $x_{1}$ is the number of left shoes and $x_{2}$ the number of right shoes. A consumer can only use pairs of shoes. Hence, his utility is $\min(x_{1},x_{2})$.
• 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: $\min({x_{CPU} \over 2},{x_{MEM} \over 3},{x_{DISK} \over 4})$.

Properties

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

• The preferences are weakly monotone but not strictly 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 $\min(x_{1}/2,x_{2}/3)$, the corners of the indifferent curves are at $(2t,3t)$ where $t\in [0,\infty )$.

Solving for Walrasian–Marshallian demand

Solving for Walrasian–Marshallian demand is a simple affair with utility functions representing Leontief preferences. One need only set equal the terms of the min function and solve then w.r.t the income constraint [Income = (p1)*(x1)+ … + (pn)(xn)]

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 two-player game to the problem of finding a competitive equilibrium in a Leontief economy. 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.

On the other hand, there are algorithms for finding an approximate equilibrium for some special Leontief economies.