ⓘ 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 diskspace 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 Lshaped 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 NPhard to decide whether a Leontief economy has an equilibrium.
 It is NPhard 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 polynomialtime 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
Dichotomous preferences 

Exponential utility 
Fractionally subadditive 
Gorman polar form 
Gross substitutes (indivisible items) 
Homothetic preferences 
Indirect utility function 

Isoelastic utility 
King–Plosser–Rebelo preferences 
Linear utility 
Responsive set extension 
Stone–Geary utility function 

Unit demand 
Utility functions on divisible goods 

Film 

Television show 

Game 

Sport 

Science 

Hobby 

Travel 

Technology 

Brand 

Outer space 

Cinematography 

Photography 

Music 

Literature 

Theatre 

History 

Transport 

Visual arts 

Recreation 

Politics 

Religion 

Nature 

Fashion 

Subculture 

Animation 

Award 

Interest 