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\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6765c338c11769b4e3923c72a8ac2ff55f2bf2e5)
where:
is the number of different goods in the economy.
(for
) is the amount of good
in the bundle.
(for
) is the weight of good
for the consumer.
This form of utility function was first conceptualized by Wassily Leontief.
Examples
Leontief utility functions represent complementary goods. For example:
- Suppose
is the number of left shoes and
the number of right shoes. A consumer can only use pairs of shoes. Hence, his utility is
.
- 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:
.
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
, the corners of the indifferent curves are at
where
.
- The consumer's demand is always to get the goods in constant ratios determined by the weights, i.e. the consumer demands a bundle
where
is determined by the income:
.[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.