Leontief utilities

In economics, especially in consumer theory, a Leontief utility function is a function of the form:

${\displaystyle u(x_{1},\ldots ,x_{m})=\min \left\{{\frac {x_{1}}{w_{1}}},\ldots ,{\frac {x_{m}}{w_{m}}}\right\}}$.

where:

• ${\displaystyle m}$ is the number of different goods in the economy.
• ${\displaystyle x_{i}}$ (for ${\displaystyle i\in 1,\dots ,m}$) is the amount of good ${\displaystyle i}$ in the bundle..
• ${\displaystyle w_{i}}$ (for ${\displaystyle i\in 1,\dots ,m}$) is the weight of good ${\displaystyle i}$ for the consumer.

This form of utility function was first conceptualized by Wassily Leontief.

Examples

Leontief utility functions represent complementary goods. For example:

• Suppose ${\displaystyle x_{1}}$ is the number of left shoes and ${\displaystyle x_{2}}$ the number of right shoes. A consumer can only use pairs of shoes. Hence, his utility is ${\displaystyle \min(x_{1},x_{2})}$.
• 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: ${\displaystyle \min({x_{\mathrm {CPU} } \over 2},{x_{\mathrm {MEM} } \over 3},{x_{\mathrm {DISK} } \over 4})}$.

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

1. "Intermediate Micro Lecture Notes" (PDF). Yale University. 21 October 2013. Retrieved 21 October 2013.
2. Greinecker, Michael (2015-05-11). "Perfect complements have to be normal goods". Retrieved 17 December 2015.
3. 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.
4. 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.
5. 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.