Leontief Utility

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

.

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.

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: .

A consumer with a Leontief 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 , 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]}

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]}

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

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