ⓘ Leontief utilities. In economics, especially in consumer theory, a Leontief utility function is a function of the form: u x 1, …, x m = min { x 1 w 1, …, x m w ..

ⓘ Leontief utilities

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

u x 1, …, x m = min { x 1 w 1, …, x m w m } {\displaystyle ux_{1},\ldots,x_{m}=\min \left\ for i ∈ 1, …, m {\displaystyle i\in 1,\dots,m} is the weight of good i {\displaystyle i} for the consumer.

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

1. Examples

Leontief utility functions represent complementary goods. For example:

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

2. Properties

A consumer with a Leontief utility function has the following properties:

• 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 {\displaystyle \minx_{1}/2,x_{2}/3}, the corners of the indifferent curves are at 2 t, 3 t {\displaystyle 2t,3t} where t ∈ "0, ∞) {\displaystyle t\in "0,\infty)}.
• 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 consumers demand is always to get the goods in constant ratios determined by the weights, i.e. the consumer demands a bundle w 1 t, …, w m t {\displaystyle w_{1}t,\ldots,w_{m}t} where t {\displaystyle t} is determined by the income: t = I n c o m e / p 1 w 1 + … + p m w m {\displaystyle t=Income/p_{1}w_{1}+\ldots +p_{m}w_{m}}. Since the Marshallian demand function of every good is increasing in income, all goods are normal goods.
• 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.

3. 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. This has several implications:

• It is NP-hard to decide whether a Leontief economy has an equilibrium.
• 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.

Moreover, the Leontief market exchange problem does not have a fully polynomial-time approximation scheme, unless PPAD ⊆ P.

On the other hand, there are algorithms for finding an approximate equilibrium for some special Leontief economies.

• the problem of finding a competitive equilibrium in an economy with Leontief utilities A zero - sum game is a special case of a bimatrix game in which A
• weakly monotonic but not strongly monotonic are those represented by a Leontief utility function. Monotonic function Monotonicity in calculus and analysis
• Sometimes cardinal utility is used to aggregate utilities across persons, to create a social welfare function. When ordinal utilities are used, differences
• based upon the respective marginal utilities of the goods that they have or desire with these marginal utilities being distinct for each potential trader
• exhibits constant elasticity of substitution between capital and labor. Leontief linear and Cobb Douglas functions are special cases of the CES production
• sectors of a national economy or different regional economies. Wassily Leontief 1906 1999 is credited with developing this type of analysis and earned
• a special case of the Gorman form. Particularly: linear, Leontief and Cobb - Douglas utilities are homothetic and thus have the Gorman form. To prove that
• illustrated by the figure. Such preferences can be represented by a Leontief utility function. Few goods behave as perfect complements. One example is a
• 1934 sequential - move version of Cournot duopoly. Other examples include Leontief s 1946 monopoly - union model and Rubenstein s bargaining model. Lastly
• Aversion of Individuals vs Risk Aversion of the Whole Economy The benefit of utilities a plausible explanation for small risky parts in the portfolio
• input - output models pioneered by Wassily Leontief but assign a more important role to prices. Thus, where Leontief assumed that, say, a fixed amount of labour