• Images
• Videos
• Documents
• Books
• Articles

Formal definitions

Given a relation on pairs of elements of set X {\displaystyle X}

≽   ⊆   X 2 {\displaystyle \succcurlyeq ~\subseteq ~X^{2}}

and an element x {\displaystyle x} of X {\displaystyle X}

x ∈ X {\displaystyle x\in X}

The upper contour set of x {\displaystyle x} is the set of all y {\displaystyle y} that are related to x {\displaystyle x} :

{ y   ∍   y ≽ x } {\displaystyle \left\{y~\backepsilon ~y\succcurlyeq x\right\}}

The lower contour set of x {\displaystyle x} is the set of all y {\displaystyle y} such that x {\displaystyle x} is related to them:

{ y   ∍   x ≽ y } {\displaystyle \left\{y~\backepsilon ~x\succcurlyeq y\right\}}

The strict upper contour set of x {\displaystyle x} is the set of all y {\displaystyle y} that are related to x {\displaystyle x} without x {\displaystyle x} being in this way related to any of them:

{ y   ∍   ( y ≽ x ) ∧ ¬ ( x ≽ y ) } {\displaystyle \left\{y~\backepsilon ~(y\succcurlyeq x)\land \lnot (x\succcurlyeq y)\right\}}

The strict lower contour set of x {\displaystyle x} is the set of all y {\displaystyle y} such that x {\displaystyle x} is related to them without any of them being in this way related to x {\displaystyle x} :

{ y   ∍   ( x ≽ y ) ∧ ¬ ( y ≽ x ) } {\displaystyle \left\{y~\backepsilon ~(x\succcurlyeq y)\land \lnot (y\succcurlyeq x)\right\}}

The formal expressions of the last two may be simplified if we have defined

≻   =   { ( a , b )   ∍   ( a ≽ b ) ∧ ¬ ( b ≽ a ) } {\displaystyle \succ ~=~\left\{\left(a,b\right)~\backepsilon ~\left(a\succcurlyeq b\right)\land \lnot (b\succcurlyeq a)\right\}}

so that a {\displaystyle a} is related to b {\displaystyle b} but b {\displaystyle b} is not related to a {\displaystyle a} , in which case the strict upper contour set of x {\displaystyle x} is

{ y   ∍   y ≻ x } {\displaystyle \left\{y~\backepsilon ~y\succ x\right\}}

and the strict lower contour set of x {\displaystyle x} is

{ y   ∍   x ≻ y } {\displaystyle \left\{y~\backepsilon ~x\succ y\right\}}

Contour sets of a function

In the case of a function f ( ) {\displaystyle f()} considered in terms of relation ▹ {\displaystyle \triangleright } , reference to the contour sets of the function is implicitly to the contour sets of the implied relation

( a ≽ b )   ⇐   [ f ( a ) ▹ f ( b ) ] {\displaystyle (a\succcurlyeq b)~\Leftarrow ~[f(a)\triangleright f(b)]}

Examples

Arithmetic

Consider a real number x {\displaystyle x} , and the relation ≥ {\displaystyle \geq } . Then

• the upper contour set of x {\displaystyle x} would be the set of numbers that were greater than or equal to x {\displaystyle x} ,
• the strict upper contour set of x {\displaystyle x} would be the set of numbers that were greater than x {\displaystyle x} ,
• the lower contour set of x {\displaystyle x} would be the set of numbers that were less than or equal to x {\displaystyle x} , and
• the strict lower contour set of x {\displaystyle x} would be the set of numbers that were less than x {\displaystyle x} .

Consider, more generally, the relation

( a ≽ b )   ⇐   [ f ( a ) ≥ f ( b ) ] {\displaystyle (a\succcurlyeq b)~\Leftarrow ~[f(a)\geq f(b)]}

Then

• the upper contour set of x {\displaystyle x} would be the set of all y {\displaystyle y} such that f ( y ) ≥ f ( x ) {\displaystyle f(y)\geq f(x)} ,
• the strict upper contour set of x {\displaystyle x} would be the set of all y {\displaystyle y} such that f ( y ) > f ( x ) {\displaystyle f(y)>f(x)} ,
• the lower contour set of x {\displaystyle x} would be the set of all y {\displaystyle y} such that f ( x ) ≥ f ( y ) {\displaystyle f(x)\geq f(y)} , and
• the strict lower contour set of x {\displaystyle x} would be the set of all y {\displaystyle y} such that f ( x ) > f ( y ) {\displaystyle f(x)>f(y)} .

It would be technically possible to define contour sets in terms of the relation

( a ≽ b )   ⇐   [ f ( a ) ≤ f ( b ) ] {\displaystyle (a\succcurlyeq b)~\Leftarrow ~[f(a)\leq f(b)]}

though such definitions would tend to confound ready understanding.

In the case of a real-valued function f ( ) {\displaystyle f()} (whose arguments might or might not be themselves real numbers), reference to the contour sets of the function is implicitly to the contour sets of the relation

( a ≽ b )   ⇐   [ f ( a ) ≥ f ( b ) ] {\displaystyle (a\succcurlyeq b)~\Leftarrow ~[f(a)\geq f(b)]}

Note that the arguments to f ( ) {\displaystyle f()} might be vectors, and that the notation used might instead be

[ ( a 1 , a 2 , … ) ≽ ( b 1 , b 2 , … ) ]   ⇐   [ f ( a 1 , a 2 , … ) ≥ f ( b 1 , b 2 , … ) ] {\displaystyle [(a_{1},a_{2},\ldots )\succcurlyeq (b_{1},b_{2},\ldots )]~\Leftarrow ~[f(a_{1},a_{2},\ldots )\geq f(b_{1},b_{2},\ldots )]}

Economics

In economics, the set X {\displaystyle X} could be interpreted as a set of goods and services or of possible outcomes, the relation ≻ {\displaystyle \succ } as strict preference, and the relationship ≽ {\displaystyle \succcurlyeq } as weak preference. Then

• the upper contour set, or better set,¹ of x {\displaystyle x} would be the set of all goods, services, or outcomes that were at least as desired as x {\displaystyle x} ,
• the strict upper contour set of x {\displaystyle x} would be the set of all goods, services, or outcomes that were more desired than x {\displaystyle x} ,
• the lower contour set, or worse set,² of x {\displaystyle x} would be the set of all goods, services, or outcomes that were no more desired than x {\displaystyle x} , and
• the strict lower contour set of x {\displaystyle x} would be the set of all goods, services, or outcomes that were less desired than x {\displaystyle x} .

Such preferences might be captured by a utility function u ( ) {\displaystyle u()} , in which case

• the upper contour set of x {\displaystyle x} would be the set of all y {\displaystyle y} such that u ( y ) ≥ u ( x ) {\displaystyle u(y)\geq u(x)} ,
• the strict upper contour set of x {\displaystyle x} would be the set of all y {\displaystyle y} such that u ( y ) > u ( x ) {\displaystyle u(y)>u(x)} ,
• the lower contour set of x {\displaystyle x} would be the set of all y {\displaystyle y} such that u ( x ) ≥ u ( y ) {\displaystyle u(x)\geq u(y)} , and
• the strict lower contour set of x {\displaystyle x} would be the set of all y {\displaystyle y} such that u ( x ) > u ( y ) {\displaystyle u(x)>u(y)} .

Complementarity

On the assumption that ≽ {\displaystyle \succcurlyeq } is a total ordering of X {\displaystyle X} , the complement of the upper contour set is the strict lower contour set.

X 2 ∖ { y   ∍   y ≽ x } = { y   ∍   x ≻ y } {\displaystyle X^{2}\backslash \left\{y~\backepsilon ~y\succcurlyeq x\right\}=\left\{y~\backepsilon ~x\succ y\right\}} X 2 ∖ { y   ∍   x ≻ y } = { y   ∍   y ≽ x } {\displaystyle X^{2}\backslash \left\{y~\backepsilon ~x\succ y\right\}=\left\{y~\backepsilon ~y\succcurlyeq x\right\}}

and the complement of the strict upper contour set is the lower contour set.

X 2 ∖ { y   ∍   y ≻ x } = { y   ∍   x ≽ y } {\displaystyle X^{2}\backslash \left\{y~\backepsilon ~y\succ x\right\}=\left\{y~\backepsilon ~x\succcurlyeq y\right\}} X 2 ∖ { y   ∍   x ≽ y } = { y   ∍   y ≻ x } {\displaystyle X^{2}\backslash \left\{y~\backepsilon ~x\succcurlyeq y\right\}=\left\{y~\backepsilon ~y\succ x\right\}}

See also

• Epigraph
• Hypograph

Bibliography

• Andreu Mas-Colell, Michael D. Whinston, and Jerry R. Green, Microeconomic Theory (LCC HB172.M6247 1995), p43. ISBN 0-19-507340-1 (cloth) ISBN 0-19-510268-1 (paper)

References

1. Robert P. Gilles (1996). Economic Exchange and Social Organization: The Edgeworthian Foundations of General Equilibrium Theory. Springer. p. 35. ISBN 9780792342007. 9780792342007 ↩

2. Robert P. Gilles (1996). Economic Exchange and Social Organization: The Edgeworthian Foundations of General Equilibrium Theory. Springer. p. 35. ISBN 9780792342007. 9780792342007 ↩