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Concave functions_ Quasiconcave functions_ Convex sets_ and more

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					Concave functions, Quasiconcave functions, Convex sets, and more

Definition: A convex combination of two points, x and x , is any point λx + (1 − λ)x , where
0 ≤ λ ≤ 1.

Definition: A set is a convex set if for any choice of two points, x and x , in the set, any convex
combination of x and x is in the set.

Definition: A function, f (x), is concave if f (λx + (1 − λ)x ) ≥ λf (x ) + (1 − λ)f (x ), where
0 ≤ λ ≤ 1.

Definition: A function f (x) is a convex function if f (λx + (1 − λ)x ) ≤ λf (x ) + (1 − λ)f (x ),
where 0 ≤ λ ≤ 1.

Note: It is important not to confuse convex functions with convex sets.

Definition: A function f (x) is quasiconcave if we have f (λx + (1 − λ)x ) ≥ min[f (x ), f (x )],
where 0 ≤ λ ≤ 1.

More useful Definition: A function f (x) is quasiconcave if its upper contour sets are convex sets.
That is, if the set {x : f (x) ≥ K} is a convex set for any constant K.

Note: Concavity implies (but is not implied by) quasiconcavity.

Note: Quasiconcavity (but not concavity) is preserved by monotonic transformations.

Definition: A function f (x) is quasiconvex if we have f (λx +(1−λ)x ) ≤ max[f (x ), f (x )], where
0 ≤ λ ≤ 1.

More useful Definition: A function f (x) is quasiconvex if its lower contour sets are convex sets.
That is, if the set {x : f (x) ≤ K} is a convex set for any constant K.

Examples: A function which is concave (and therefore quasiconcave) is a Cobb-Douglas technology
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with DRS, such as f (x1 , x2 ) = x1 x2 . A function which is quasiconcave but not concave is a Cobb-
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Douglas technology with IRS, such as f (x1 , x2 ) = x1 x2 .

Definition: A function of the form f (x) = f (x1 , x2 , . . . , xn ) = a0 + a1 x1 + a2 x2 + · · · + an xn = a0 + ax
is an affine function. If a0 = 0, it is a linear function.


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Note: An affine function is both concave and convex.

Fact: A function f (x) is convex if and only if −f (x) is concave.

Fact: If the functions f1 (x), f2 (x), . . . , fn (x) are all concave, then min[f1 (x), f2 (x), . . . , fn (x)] is a
concave function. In particular the minimum over a collection of affine functions is concave.

Fact: If the functions f1 (x), f2 (x), . . . , fn (x) are all convex, then max[f1 (x), f2 (x), . . . , fn (x)] is a
convex function. In particular the maximum over a collection of affine functions is convex.

Definition: A consumer possessing a quasiconcave utility function is said to have convex prefer-
ences.

Definition: A firm possessing a concave production function is said to have access to a convex
technology.

Note: Convex preferences in consumer theory are not equivalent to convex technologies in producer
theory. (The former is a result of quasiconcavity, the latter a result of concavity.) Likewise, noncon-
vexity of preferences and nonconvexity in production are not equivalent.

Fact: A concave transformation of a concave function is itself concave.

Negative fact: An increasing transformation of a concave function is not necessarily concave. E.g.
f (x) = 2x is concave and g(f ) = f 3 is increasing, but g[f (x)] = 8x3 is not concave.




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