The Class o k -Convex Functions Martin Kaae Jensen May 5, 0 Abstract This brie note characterizes the class o k -convex and k -concave unctions which represent a strengthening o convexity and a weakening o concavity, respectively. The note is meant primarily as a mathematical support note or Acemoglu and Jensen (0) and Jensen (0), but may be o independent interest to some readers. Keywords: k -convexity, k -concavity, HARA unctions. JEL Classiication Codes: C6, D90, E. Department o Economics, University o Birmingham. (e-mail: m.k.jensen@bham.ac.uk)
The class o k -Convex Functions Throughout X is assumed to be a convex subset o an ordered vector space. Deinition Let k 0. A unction : X + is said to be k -convex i: When k, the unction k [ (x)] k is convex, or equivalently, i or all α [0, ]: [α( (x)) k + ( α)( (y )) k ] k (αx + ( α)y ) () When k =, the unction log (x ) is convex (i.e. α [0, ]: is log-convex), or equivalently, i or all ( (x)) α ( (y )) α (αx + ( α)y ) () k -concavity is deined similarly by requiring the unction k [ (x)] k to be concave (or reversing the inequalities in ()-()). Notice that the let-hand sides o ()-() are weighted power means (the exponent is k and the weights are (α, α)). So k -convexity can be seen as convexity except that the usual weighted mean is allowed to be any power mean. When k = 0 we thus recoup convexity in the usual sense since the power mean with exponent is just the usual (arithmetic) mean. Notice also that by the power mean inequality, the power mean is smaller than or equal to the usual arithmetic mean whenever k 0 : α[ (x)] + ( α) (y ) [α( (x)) k + ( α)( (y )) k ] k Combining with (), we immediately see that any k -convex unction is convex (in act k -convexity implies k -convexity or all k k by the general ormulation o the power mean inequality). By the same line o reasoning, one sees that k -concavity implies k -concavity or all k k. So while k -convexity is a stronger condition than convexity, k -concavity is weaker than concavity. These statements are also true in the log-convex and log-concave cases, i.e., when k = (and here they are well known). The ollowing lemma tells us how we can conclude that a composite unction is k -convex: Lemma Consider (x ) = h(g (x),..., g n (x)) where h : n is k -convex and g i : n. Then is k -convex i or each i one o the ollowing holds: h is non-increasing in the i th argument and g i is concave. h is non-decreasing in the i th argument and g i is convex. g i is aine. We also note that k -concavity implies quasi-concavity because the power mean on the let-hand side o () converges to min{ (x ), (y )} as k (thus () with the inequality reversed precisely becomes the condition or quasi-concavity in the limit k ).
Proo. The statement o the lemma is well-known or convex unctions ( is convex when h is convex and the lemma s conditions on g = (g,..., g n ) hold). Use this on the composition o h(y,...) = k [h(y,..., y n )] k (which is convex) and g to conclude that h g is convex, i.e., = h g is k - convex. For a given k, the set o k -convex unctions is a very well behaved class: It orms a closed and convex cone in the topological vector space o real-valued unctions with the pointwise topology ((i)-(ii)) o the ollowing lemma). And k -convexity is preserved by integration ((iii) o the lemma). All o this is o course well known or 0- and -convex unctions (i.e., convex and log-convex unctions). Lemma (i) I α,β 0, and and g are k -convex, the weighted sum α + β g is k -convex. (ii) The pointwise limit o a sequence o k -convex unctions is k -convex. (iii) k -convexity is preserved by integration: I (x, z ) is measurable in z and k -convex in x or almost every z Z then (x, z )µ(d z ) Z is k -convex. Proo. (i) We take α = β = (the case where these weights are non-trivial adds no extra diiculties to the proo below). Let F = k k and G = k g k and assume that these are convex. We wish to show that, k ( + g ) k = [(( k )F ) k + (( k )G ) k ] k, k is then a convex unction. But this ollows directly rom the act that H(y, y ) = k [(( k )y ) k + (( k )y ) k ] k is increasing and convex or k 0 since the above unction is the composition o H with F and G (the limit case k = where is log-convex is proved by the same argument except that one now uses that H(y, y ) = log(exp(y ) + exp(y )) is convex). (ii) Let (x) = lim n n (x) where each n is k -convex. Since the pointwise limit o a sequence o convex unctions is convex, k [ (x)] k = lim n k [ n(x)] k is convex. (iii) Follows rom (i)-(ii) by a standard argument. Since is not k -concave i is k -convex (in act k -concavity cannot even be deined or ), we cannot in general subtract a k -concave unction rom a k -convex unction and expect that anything nice comes out o it (in this respect k -convex/k -concave unctions behave more like quasiconvex/quasi-concave unctions than they behave like convex/concave unctions). Using that the sum o k -convex unctions is k -convex one can however prove the ollowing subtraction result. Lemma 3 Fix k 0, and let (x) = v (x) u (x) where v : X + is k -convex, and u : X + is k -concave. Then {x X : (x) r } is convex or all r 0, i.e., is quasi-convex when restricted to the domain {x X : (x) 0}. I instead v is k -concave and u is k -convex, then is quasi-concave on the restricted domain {x X : (x) 0}. That k -convexity is preserved by integration was irst proved by Carroll and Kimball (996) in the special case where is twice dierentiable and X + (Carroll and Kimball (996), Lemma ). That proo, however, uses dierentiability in an indispensable manner. The generalization presented here plays an important role or the results in Jensen (0).
Proo. Fix r 0 and consider the set L r = {x X : v (x ) u (x) r }. Clearly L r = {x X : [v (x) k r ] k k [u (x)] k 0} (in particular, v (x ) r is a well-deined positive k -convex unction by (i) o lemma ). But since k [v (x ) r ] k k [u (x )] k is convex, it is quasi-convex and thereore L r is a convex set. For the second claim, simply repeat the argument or (x ) = u (x) v (x) in order to conclude that {x X : u (x) v (x) r } is convex or all r 0, which is equivalent to {x X : (x) r } convex or all r 0. Since k -convexity is equivalent to convexity o k [ (x )] k, we can use the Hessian criteria or convexity o a unction to establish k -convexity/k -concavity when the unctions involved are suiciently smooth. Note that since the irst derivative o k [ (x)] k is [ (x)] k (x) (here taking X ) we must assume that (x ) > 0 or all x X in order to be able to use such dierentiability arguments. Lemma 4 Assume that : X + is twice dierentiable and that (x ) > 0 or all x X. I X, is k -convex (k -concave) i and only i: (x) (x) ( )k ( (x)) (3) When (x ) 0 or all x X, this is in turn equivalent to the condition: (x) (x ) ( (x )) ( ) k (4) I X, will be k -convex (k -concave) i and only i the ollowing matrix is positive semideinite (negative semi-deinite): x x k ( x ) x k x x k x x k ( ) The determinant o this matrix will be non-negative i, or example: (i) The Hessian matrix o has a non-negative determinant, and (ii) satisies (assuming here in addition that x, x 0): x x x [ x ( x ) + x ( ) ] Hence will be k -convex (k -concave) i (i)-(ii) hold and one o the diagonal entries o (5) are non-negative (non-positive). 3 Proo. This is just calculus and linear algebra. The determinant o the matrix in the lemma is k [ x x x x ( x ) x x ( x ) x x ] + ( ) [ x x x x ( x x ) ]. (5) The second o these terms 3 O course i one o the diagonal entries is non-negative (non-positive) both will be non-negative (non-positive) when the determinant is non-negative. The point is that we only need to veriy this or one o them when we are aced with a concrete application. 3
is the determinant o the Hessian o (multiplied by a positive constant). The irst term equals k ( x ) ( ) [ x x x ] when the irst derivatives are non-zero. That (i)-(ii) imply a x x ( x ) ( x ) non-negative determinant is thereore obvious. Remark It is clear that the diagonal criterion at the end o the previous lemma is a necessary condition or k -convexity or k -concavity. (i) is also a necessary condition in the k -convex case (any k -convex unction is convex), but it is not a necessary condition in the k -concave case (concavity implies k -concavity, but the converse is alse). (ii) is not a necessary condition in either case, but it provides a very convenient way to veriy k -convexity/k -concavity. 4 Remark (The HARA Class) A unction F : X, X, with F 0, is said to exhibit Hyperbolic Absolute Risk Aversion (or lie in the HARA class) i F F (F ) = k 0 (Carroll and Kimball (996)). From the previous lemma we see that this is equivalent to assuming that the derivative o F, F = is k -convex as well as k -concave. Reerences Acemoglu, D. and M. K. Jensen (0): Robust Comparative Statics in Large Dynamic Economies, Mimeo MIT. Carroll, C.D. and M.S. Kimball (996): On the Concavity o the Consumption Function, Econometrica 64, 98-99. Jensen, M.K. (0): Distributional Comparative Statics, Mimeo, University o Birmingham. 4 As may be veriied, (ii) is in act a necessary condition in the special case where s Hessian matrix has a zero determinant. This is a situation one requently encounters in applications. 4