# ⓘ 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