VIEWS: 30 PAGES: 2 POSTED ON: 8/13/2011 Public Domain
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 1/3 1/3 with DRS, such as f (x1 , x2 ) = x1 x2 . A function which is quasiconcave but not concave is a Cobb- 2/3 2/3 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. 1 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. 2