Leontief utilities explained

In economics, especially in consumer theory, a Leontief utility function is a function of the form:$$u(x\_1,\backslash ldots,x\_m)=\backslash min\backslash left\backslash \; .$$where:

m is the number of different goods in the economy.x _{i} (for i\in 1 , ... ,m ) is the amount of good i in the bundle.w _{i} (for i\in 1 , ... ,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: $\backslash min$.

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

min (x _{1/2,} x _{2/3)} , the corners of the indifferent curves are at ( 2 t, 3 t) where t\in[ 0 , inf ty) .

• The consumer's demand is always to get the goods in constant ratios determined by the weights, i.e. the consumer demands a bundle

(w _{1} t,\ldots,w _{m} t) where t is determined by the income: t = Income / (p _{1} w _{1} + ... + p _{m} w _{m)} .^[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.

Notes and References

1. Web site: Intermediate Micro Lecture Notes. 21 October 2013. Yale University. 21 October 2013.
2. Web site: Perfect complements have to be normal goods . 2015-05-11 . 17 December 2015 . Greinecker, Michael.
3. Book: 10.1145/1109557.1109629. Leontief economies encode nonzero sum two-player games. Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm - SODA '06. 659. 2006. Codenotti. Bruno. Saberi. Amin. Varadarajan. Kasturi. Ye. Yinyu. 0898716055.
4. Book: 10.1007/978-3-540-73814-5_9 . On the Approximation and Smoothed Complexity of Leontief Market Equilibria . Frontiers in Algorithmics . 4613 . 96 . Lecture Notes in Computer Science . 2007 . Huang . Li-Sha . Teng . Shang-Hua . 978-3-540-73813-8 .
5. Book: 10.1007/978-3-540-27836-8_33. Efficient Computation of Equilibrium Prices for Markets with Leontief Utilities. Automata, Languages and Programming. 3142. 371. Lecture Notes in Computer Science. 2004. Codenotti. Bruno. Varadarajan. Kasturi. 978-3-540-22849-3.