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

References

↑ "Intermediate Micro Lecture Notes" (PDF). Yale University. 21 October 2013. Retrieved 21 October 2013.
↑ Greinecker, Michael (2015-05-11). "Perfect complements have to be normal goods". Retrieved 17 December 2015.
1 2 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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