数学附录、偏好和效用、消费者决策

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))