Social welfare function W: Rn-R gives social utility W(u1, u2,. un ). W is strictly increasing is socially optimal if it solves max Wu(a1), u2(a2),., un(n) st>Tis>w Proposition 1.29. If is SO, it is PO. I Proposition 1. 30. Suppose that preferences are continuous, strictly monotonic, and strictly convex. Then, for any PO allocation x* with >>0,v i, there exist ai
then there exists AE R\ such that (Kuhn-Tucker condition) G(s') =0 and 1. Lagrange Method for Constrained Optimization FOC: D.L(,\)=0. The following classical theorem is from Takayama(1993, p.114). Theorem A-4 (Sufficieney). Let f and, i= ,..m, be quasi-concave, where Theorem A-1. (Lagrange). For f: and G\\, consider the following G=(.8 ) Let r' satisfy the Kuhn-Tucker condition and the FOC for (A.2). Then, x' problem is a global maximum point if max f() (1)Df(x') =0, and f is locally twice continuously differentiable,or
Nobel Lecture, June 2, 1920 The genesis and present state of development of the quantum theory If i take it correctly that the duty imposed upon me today is to give a public lecture on my writings, then i believe that this task, the importance of which i am well aware through the gratitude felt towards the noble-minded founder of our Foundation, cannot be more suitably fulfilled than by my trying to give you the story of the origin of the quantum theory in broad outlines and to couple with this, a picture in a small frame of the development of this theory up