Leontief utilities

In economics, especially in consumer theory, a Leontief utility function is a function of the form: $${\displaystyle u(x_{1},\ldots ,x_{m})=\min \left\{{\frac {x_{1}}{w_{1}}},\ldots ,{\frac {x_{m}}{w_{m}}}\right\}.}$$ where:

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

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

Leontief utility functions represent complementary goods. For example:

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

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 ${\displaystyle \min(x_{1}/2,x_{2}/3)}$, the corners of the indifferent curves are at ${\displaystyle (2t,3t)}$ where ${\displaystyle t\in [0,\infty )}$.
• The consumer's demand is always to get the goods in constant ratios determined by the weights, i.e. the consumer demands a bundle ${\displaystyle (w_{1}t,\ldots ,w_{m}t)}$ where ${\displaystyle t}$ is determined by the income: ${\displaystyle t={\text{Income}}/(p_{1}w_{1}+\dots +p_{m}w_{m})}$.^[1] Since the Marshallian demand function of every good is increasing in income, all goods are normal goods.^[2]

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]

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