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Microeconomics principles and analysis phần 9

Bây giờ hãy xem xét trạng thái cân bằng. Hãy để chúng tôi tập trung rẽ vào subgame sau từ một quyết định của các đương nhiệm đầu tư (đối với trường hợp các đương nhiệm không đầu tư xem Tập thể dục 10,11). Nếu các thách thức để sau này | 506 APPENDIX A. MATHEMATICS BACKGROUND Figure A.9 A strictly concave-contoured strictly quasiconcave function There are functions for which the contours look like those of a concave function but which are not themselves concave. An example here would be f x where f is a concave function and is an arbitrary monotonic transformation. These remarks lead us to the definition Definition A.23 A function f is strictly concave-contoured if all the sets B yo in A.31 are strictly convex. A synonym for strictly concave-contoured is strictly quasiconcave. Try not to let this unfortunately necessary jargon confuse you. Take for example a conventional looking utility function such as U x X1X2- A.32 According to definition A.23 this function is strictly quasiconcave if you draw the set of points B v xi x2 X1X2 v you will get a strictly convex set. Furthermore although U in A.32 is not a concave function it is a simple transformation of the strictly concave function U x log X1 log x2 A.33 and has the same shape of contour map as U. But when we draw those contours on a diagram with the usual axes we would colloquially describe their shape A.7. MAXIMISATION 507 as being convex to the origin There is nothing seriously wrong here the definition the terminology and our intuitive view are all correct it is just a matter of the way in which we visualise the function. Finally the following complementary property is sometimes useful Definition A.24 A function f is strictly quasiconvex if f is strictly quasiconcave. A.6.6 The Hessian property Consider a twice-differentiable function f from D c Rn to R. Let fij x denote O2 f x rp 1 . . 1. . the symmetric matrix OXiOXj J f11 x f12 x . f1n x f21 x f22 x . f2n x . . . . fn1 x fn2 x . fnn x is known as the Hessian matrix of f. Definition A.25 The Hessian matrix of f at x is negative semidefinite if for any vector w 2 Rn it is true that ỵỵ fij x 0. i 1j 1 A twice-differentiable function f from D to R is concave if and only if f is negative .