Leontief utilities

In economics, especially in 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: $\min({x_{\mathrm{CPU}}\over 2}, {x_{\mathrm{MEM}}\over 3}, {x_{\mathrm{DISK}}\over 4})$ .

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 $(2t,3t)$ 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: $t = \text{Income} / (p_1 w_1 + \dots + p_m w_m)$ .{{cite web|url=http://dirkbergemann.commons.yale.edu/files/lecture_notes-vp-db.pdf|title=Intermediate Micro Lecture Notes|last=|first=|date=21 October 2013|work=Yale University|accessdate=21 October 2013}} Since the Marshallian demand function of every good is increasing in income, all goods are normal goods.{{cite web | url=http://economics.stackexchange.com/a/5618/385 | title=Perfect complements have to be normal goods | date=2015-05-11 | accessdate=17 December 2015 | author=Greinecker, Michael}}

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.{{Cite book|doi=10.1145/1109557.1109629|chapter=Leontief economies encode nonzero sum two-player games|title=Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm - SODA '06|pages=659|year=2006|last1=Codenotti|first1=Bruno|last2=Saberi|first2=Amin|last3=Varadarajan|first3=Kasturi|last4=Ye|first4=Yinyu|isbn=0898716055}} 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.{{Cite book|doi=10.1007/978-3-540-73814-5_9 |chapter=On the Approximation and Smoothed Complexity of Leontief Market Equilibria |title=Frontiers in Algorithmics |volume=4613 |pages=96 |series=Lecture Notes in Computer Science |year=2007 |last1=Huang |first1=Li-Sha |last2=Teng |first2=Shang-Hua |isbn=978-3-540-73813-8 }}

On the other hand, there are algorithms for finding an approximate equilibrium for some special Leontief economies.{{Cite book|doi=10.1007/978-3-540-27836-8_33|chapter=Efficient Computation of Equilibrium Prices for Markets with Leontief Utilities|title=Automata, Languages and Programming|volume=3142|pages=371|series=Lecture Notes in Computer Science|year=2004|last1=Codenotti|first1=Bruno|last2=Varadarajan|first2=Kasturi|isbn=978-3-540-22849-3}}

Application

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

References

Category:Utility function types