数学附录、偏好和效用、消费者决策

数学附录 1欧氏空间:欧氏空间R啪每一点有n个分量,它们都是实数; 两点x=(x1xn)和y=(y1…,yn)之间的距离为 d(x,y)=I(xry1)2+…+(xnyn)212。给出实数:>0,欧氏空间中一点 x的减B(,包含所有對这点距离小子的点 2序列和极限:假设X是欧氏空间R咱的一个子集,{x}=是X 中的一个无穷)序列。称x为这个序列的一个聚点,如果x的任 意一个邻城中都包含有这个序列中的点。注意:x不必在X中; 又一个序列可以有多个聚点 特别地,如果给出x的每一个邻城B(x*),这个序列从某一项开 始,这一项及其后所有的项都包含于B(x*),那么x就是这个序 列的唯一聚点,称为这个序列的极限。这时又称这个序列收敛于 是 3开集和闭集:欧氏空间中的一个子集F叫做闭的,如果包含于 的X每一个序列的所有聚点都属于F本身。欧氏空间中一个子集G 叫做开的,如果它的余集G=R叫G是闭的

数学附录 • １ 欧氏空间：欧氏空间Rn的每一点有n个分量，它们都是实数； 两点x=(x1 ,…xn )和y=(y1 ,…,yn )之间的距离为 d(x,y) = [(x1 -y1 ) 2+…+(xn -yn ) 2 ] 1/2 。给出实数>0，欧氏空间中一点 x的-邻域B(x,)包含所有到这点距离小于的点。 • ２ 序列和极限：假设X是欧氏空间Rn的一个子集，{xk }k=1  是X 中的一个(无穷)序列。称x*为这个序列的一个聚点，如果x*的任 意一个邻域中都包含有这个序列中的点。注意：x*不必在X中； 又一个序列可以有多个聚点。 • 特别地，如果给出x*的每一个邻域B(x*,)，这个序列从某一项开 始，这一项及其后所有的项都包含于B(x*,)，那么x*就是这个序 列的唯一聚点，称为这个序列的极限。这时又称这个序列收敛于 是x*。 • ３ 开集和闭集：欧氏空间中的一个子集F叫做闭的，如果包含于 的X每一个序列的所有聚点都属于F本身。欧氏空间中一个子集G 叫做开的，如果它的余集Gc=Rn \G是闭的

数学附录 4内点和界点:欧氏空间中一个子集X的点x叫做一个 内点,如果它的某一个邻域B(x:)整个被包含在X内; 称x为X的一个界点,如果x的每一个邻域既包含X的点 也包含余集X的点。X的全体内点之集记为In(X),X 的全体界点之集记为Bdy(X) 个开集G的每个点都是内点,一个闭集F包含它所有 界点。 5紧致性:欧氏空间中一个子集X叫做紧致的(序列 紧),如果X中每一个序列都有子序列收敛于X本身某 个点。可以证明,欧氏空间中的每个有界闭集都是紧 致集,反之亦然

数学附录 • ４ 内点和界点：欧氏空间中一个子集X的点x叫做一个 内点，如果它的某一个邻域B(x,)整个被包含在X内； 称x为X的一个界点，如果x的每一个邻域既包含X的点 也包含余集Xc的点。X的全体内点之集记为Int(X)，X 的全体界点之集记为Bdy(X)。 • 一个开集G的每个点都是内点，一个闭集F包含它所有 界点。 • ５ 紧致性：欧氏空间中一个子集X叫做紧致的（序列 紧），如果X中每一个序列都有子序列收敛于X本身某 个点。可以证明，欧氏空间中的每个有界闭集都是紧 致集，反之亦然

数学附录 6连续函数:定义在欧氏空间子集X上的实值函数叫 做在某点x∈X连续的,如果对于每一个收敛于x的序列 {x}kX,相应的函数值序列{(x}=都收敛于f(x) 如果函数在X上每一个点都连续,就称在X上连续。 定义在紧致集X上的每一个连续函数都在X中取得最大 值和最小值

数学附录 • ６ 连续函数：定义在欧氏空间子集X上的实值函数f叫 做在某点xX连续的，如果对于每一个收敛于x的序列 {xk }k=1 X，相应的函数值序列{f(xk )}k=1  都收敛于f(x)。 如果函数f在X上每一个点都连续，就称f在X上连续。 • 定义在紧致集X上的每一个连续函数都在X中取得最大 值和最小值

数学附录 7凸集: a set Xc界 n is said to be a convex set if for any two points xand x"in X and any real numberλ∈[0,1 the point x2=(1-x2+x”is contained in X. The intersection of any number of convex set is also convex 8. For any m points x),..., x m in Rn, and any 1min[0,1] such that∑入=1,x=∑入 x: is said to be a convex combination of x),.X

数学附录 • ７ 凸集：A set Xn is said to be a convex set if for any two points x’ and x” in X and any real number [0, 1], the point x =(1-)x’+x” is contained in X. The intersection of any number of convex set is also convex. • ８. For any m points x1 ,…,xm in n , and any 1 ,…,m in [0, 1] such that ii=1, x=iixi is said to be a convex combination of x1 ,…,xm

数学附录 9凹函数: Assume that a real- valued function f is defined on a convex set Xc f is said to be concave if for any xand x in X, and any real number nE(0, 1), it holds that (1-)f(x)+^f(x”)≤f(1-)x+x”) f is said to be strictly concave if the sign "<"in the above inequality is replaced by“<

数学附录 • 9 凹函数： Assume that a real-valued function f is defined on a convex set Xn . f is said to be concave if for any x’ and x” in X, and any real number (0, 1), it holds that (1-)f(x’)+f(x”)  f((1-)x’+x”) • f is said to be strictly concave if the sign “” in the above inequality is replaced by “<

数学附录 10凸函数: assume that a real-valued function f is defined on a convex set XoRn f is said to be convex if for any xand x?"in X, and any real number nE(0, 1), it holds that (1-λ)f(x3)+λf(x”)≥f(1)x,+x”) f is said to be strictly concave if the sign“≥” in the above inequality is replaced by">

数学附录 • 10 凸函数： Assume that a real-valued function f is defined on a convex set Xn . f is said to be convex if for any x’ and x” in X, and any real number (0, 1), it holds that (1-)f(x’)+f(x”)  f((1-)x’+x”) • f is said to be strictly concave if the sign “” in the above inequality is replaced by “>

数学附录 l1拟凹感数: A function f defined on a convex set Xcr" is said to be quasi- concave,. if for any X,andx” in X and anyλ∈[0,1 f(1-x+x”)≥min{f(x3),f(x”) f is said to be strictly quasi- convex, if the sign“≥” in the above inequality is replaced with“>” 7. Any concave(strictly concave) function is quasi-concave(strictly quasi-concave), but the converse is not true

数学附录 • 11 拟凹函数：A function f defined on a convex set Xn is said to be quasi-concave, if for any x’ and x” in X and any [0, 1]: f((1-)x’+x”)  min {f(x’),f(x”)} • f is said to be strictly quasi-convex, if the sign “” in the above inequality is replaced with “>”. • 7. Any concave (strictly concave) function is quasi-concave (strictly quasi-concave),but the converse is not true

数学附录 12拟凸函数: A function f defined on a convex set XcRn is said to be quasi-- convX, if for any xand x” in X and anyλ∈[0,1: f(1-)x,+x”)≤min{f(x),f(x”)} f is said to be strictly quasi- convex, if the sign“≤” in the a bove inequality is replaced with“<” Any convex(strictly convex) function is quasi-convex(strictly quasi-convex), but the converse is not true

数学附录 • 12 拟凸函数：A function f defined on a convex set Xn is said to be quasi-convx, if for any x’ and x” in X and any [0, 1]: f((1-)x’+x”)  min {f(x’),f(x”)} • f is said to be strictly quasi-convex, if the sign “ ” in the above inequality is replaced with “<”. . Any convex (strictly convex) function is quasi-convex (strictly quasi-convex),but the converse is not true

数学附录 14等值集,上值集,下值集: assume that f be defined on XCR, XEX, and f(x)=y. The level set, the superior set, and the inferior set for level y are respectively, the sets (y)={x∈X:f(x)=y};S(y)={x∈X:f(x)≥y;Iy {x∈x:f(x)y"}

数学附录 • 14 等值集，上值集，下值集： Assume that f be defined on Xn , x 0X, and f(x0 ) = y 0 . The level set, the superior set, and the inferior set for level y 0 are, respectively, the sets: L(y0 ) = {xX: f(x)=y0 }; S(y0 ) = {xX: f(x)y 0 }; I(y0 ) = {xX: f(x)y 0 }

数学附录 A necessary and sufficient condition for a function f defined on a convex set Crn to be quasi-concave(resp quas-convex)is that all the superior sets s(y)(resp all the inferior sets l(y))are convex; f is strictly quasi- concave(resp. strictly quas-convex) if all s(y(resp. I(y)) are convex, and for any two points xand x? "in any s(y), (resp. I(y)), the points on the line segment x=(1 A)x2+Ax”:λ∈(0,1} expect possibly the two end points are all contained in Int(s(y))(resp. l(y))

数学附录 • A necessary and sufficient condition for a function f defined on a convex set Xn to be quasi-concave (resp. quas-convex) is that all the superior sets S(y) (resp. all the inferior sets I(y)) are convex; f is strictly quasi￾concave (resp. strictly quas-convex) if all S(y) (resp. I(y)) are convex, and for any two points x’ and x” in any S(y), (resp. I(y)), the points on the line segment {x=(1- )x’+x”: [0, 1]} expect possibly the two endpoints are all contained in Int(S(y)) (resp. I(y))