Finance:Leontief utilities

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

[math]u(x_1,\ldots,x_m)=\min\left\{\frac{x_1}{w_1},\ldots,\frac{x_m}{w_m}\right\}[/math].

where:

  • [math]m[/math] is the number of different goods in the economy.
  • [math]x_i[/math] (for [math]i\in 1,\dots,m[/math]) is the amount of good [math]i[/math] in the bundle..
  • [math]w_i[/math] (for [math]i\in 1,\dots,m[/math]) is the weight of good [math]i[/math] for the consumer.

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

Examples[edit]

Leontief utility functions represent complementary goods. For example:

  • Suppose [math]x_1[/math] is the number of left shoes and [math]x_2[/math] the number of right shoes. A consumer can only use pairs of shoes. Hence, his utility is [math]\min(x_1,x_2)[/math].
  • 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: [math]\min({x_{\mathrm{CPU}}\over 2}, {x_{\mathrm{MEM}}\over 3}, {x_{\mathrm{DISK}}\over 4})[/math].

Properties[edit]

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 [math]\min(x_1/2, x_2/3)[/math], the corners of the indifferent curves are at [math](2t,3t)[/math] where [math]t\in[0,\infty)[/math].
  • The consumer's demand is always to get the goods in constant ratios determined by the weights, i.e. the consumer demands a bundle [math](w_1 t,\ldots,w_m t)[/math] where [math]t[/math] is determined by the income: [math]t = Income / (p_1 w_1 + \ldots + p_m w_m)[/math].[1] Since the Marshallian demand function of every good is increasing in income, all goods are normal goods.[2]

Competitive equilibrium[edit]

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[edit]

  1. "Intermediate Micro Lecture Notes". Yale University. 21 October 2013. http://dirkbergemann.commons.yale.edu/files/lecture_notes-vp-db.pdf. Retrieved 21 October 2013. 
  2. Greinecker, Michael (2015-05-11). "Perfect complements have to be normal goods". http://economics.stackexchange.com/a/5618/385. Retrieved 17 December 2015. 
  3. 3.0 3.1 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. pp. 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. pp. 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. pp. 371. doi:10.1007/978-3-540-27836-8_33. ISBN 978-3-540-22849-3. 

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