高级微观经济学(上海财经大学 陶佶)note01
高级微观经济课件
——愈来愈深化的问题,愈来愈能启发
新问题的问题。‛
5. 关于教材
教材:范里安:《微观经济学:现代观 点》(第八版),上海三联书店、上海人
民出版社2011年1月版
6. 主要参考书: 平狄克、鲁宾费尔德:《微观经济学》(
第七版),中国人民大学出版社2009年
7. 参考文献 ①图书:《经济学方法》,复旦大学出版社 2006年版 《青年经济学家指南》,上海财经大学出版 社2001年版 《应用经济学研究方法论》,经济科学出版 社1998年版 ②报刊杂志: 中国人民大学复印报刊资料经济类各专题、 CSSCI来源期刊
经济学中常用的数学理论
经济学是选择的科学,应用数学的目的——最优 化(优化理论) 数学分析、高等代数、微分方程、概率论、实变 函数、集合论、拓扑学、泛函分析——经济学语 言 经济学帝国主义——实证研究工具 社会科学研究现实的模式,数学研究逻辑可能的 模式。 理论研究:数理经济学(逻辑演绎) 经验研究:计量经济学(统计归纳)
数学(大海)与经济学(陆地)
人总希望脚踏实地。当被带离海岸线很远 时,会因失去对陆地的知觉而产生恐惧感 ,这是就初入海者而言的。渔民和航海家 则不同,他们会如鱼得水,如果把他们留 在岸边,他们会无所事事。但毕竟大多数 人都不是渔民和航海家,他们在海中游玩 时希望时刻看到岸边,并能随时上岸。岸 上的世界七彩斑斓,海中的世界单调乏味 ,但生命的本源却来自海洋。因此,我们 要培养自己在海中的生存能力。
know-what—知其然 显性知识 know-why—知其所以然 know-how—技巧、诀窍 隐性知识 know-who( 隔行如隔山
拥有:信息<知识<智慧<素质<觉悟
解决问题:?→。发现问题:?→? 波普尔《猜想与反驳》:‚科学和 知识的增长永远始于问题,终于问题
高级微观经济学(上海财经大学 陶佶)note06
消费者理论专题
Lecture Note 6 – Topics in Consumer Theory
1. The Money Metric Utility Functions The money metric utility functions are useful in integrability theory and welfare economics. The money metric direct utility function gives the minimum expenditure at prices p necessary to purchase a bundle at least as good as x. Let the consumption bundle x be given. Question: how much money would a consumer need at the price vector p to be as well off as he could be by consuming the bundle x ? Graphically, it asks how much money consumer would need to reach the indifference cure passing through x .
A Numerical Example In previous numerical example, the direct utility is a CES function,
ρ u ( x1 , x2 ) = ( x1ρ + x2 )
1/ ρ
, where 0 ≠ ρ < 1.
《高级微观经济学》课件
考试安排
课程结束后将进行一次期末考试,考察学生对微观经济学理论和实践的理解和运用能力。
结语
通过学习高级微观经济学,您将拥有深入洞察经济问题的能力,成为经济学 的专家,并能运用所学知识解决实际经济问题。
研究消费者的偏好和选择行为,分析消费者 的需求曲线和边际效用。
市场结构和竞争
了解不同市场结构的特点,包括完全竞争、 垄断、寡头垄断和垄断竞争。
学习方法
1
课堂学习
通过听课和参与讨论,加深对微观经济学的理解和思考。
2
案例分析
通过分析实际经济问题和案例,将理论知识应用到实际情境中。
3
小组讨论
与同学一起合作讨论,分享思考和观点,促进深度学习和交流。
《高级微观经济学》 课件
让我们一起探索高级微观经济学的奥秘吧!本课程将帮助您深入了解微观经 济学的核心概念和分析方法,让您成为经济学的专家。
课程简介
通过本课程,您将了解微观经济学的基本原理和理论框架,掌握市场经济中个体和企业的行为分析方法, 以及了解市场失灵和政府干预等相关问题。
教材介绍
我们将使用《高级微观经济学》教材,该教材包含了丰富的案例研究和实际 问题分析,帮助学生将理论知识应用到实际经济问题中。
课程目标
本课程的目标是帮助学生深入理解微观经济学的核心概念,掌握经济学的思维方式和分析工具,以及培 养学生独立思考和问题解决的能力。
主要内容
供求关系分析
通过供求关系曲线的分析,了解市场价格和 数量的决定因素。
生产者行为分析
研究生产者的成本和利润最大化行为,分析 生产者的供给曲线和边际成本。
消费者行为分析
高级微观经济学(上海财经大学 陶佶)note02
Let x1 , x2 and x3 be any three consumption bundles in X .
Axiom 2.1 - Complete. Either x1 \ x2 or x2 \ x1 .
上海财大经济学院
2
作者:陶佶
2005 年秋季
高等微观经济学 I
Axiom 2.2 - Reflexive. For all x in X , x \ x . Axiom 2.3 - Transitive. If x1 \ x2 and x2 \ x3 , then x1 \ x3 .
Let X be a consumption set, a collection of all alternatives or complete consumption plans. The consumption set is also called as the choice set. Let xi ∈ be the number of units of ith
good, and x = ( x1, x2 , , xn ) be a vector containing different quantities of n commodities,
called as a consumption bundle or a consumption plan.
Properties of the Consumption Set, X : The minimal requirements are
Terminology 1. Let x0 be any points in the consumption set X . Relative to any such point,
高级微观经济学(上海财经大学 陶佶)note01
d x , y ≡ ( x1 − y1 ) 2 + ( x2 − y2 ) 2 ≡ x − y
for x and y in . It is obvious to see that the space with the metric d above is a metric space. The metric d called as Euclidean metric or Euclidean norm (欧几里德范数) can be generalized to an n-dimensional Euclidean space. Definition 8. Open and Closed ε -Balls (开球和闭球): Let ε be a real positive number.
11nn10and0nnnn?????????????
2005 年秋季
高等微观经济学 I
实分析简介
Lecture Note I
1. Logic Consider two statements, A and B. Suppose B ⇒ A is true. 1. A is necessary (必要条件) for B. 2. B is sufficient (充分条件) for A. Contra-positive (逆否) form of B ⇒ A: ~A ⇒ ~B. If both A ⇒ B and B ⇒ A are true, then A and B are equivalent: A ⇔ B. 2. Set Theory We begin with a few definitions. A set (集合) is a collection of objects called elements (元素). Usually, sets are denoted by the capital letters A, B,
高级微观经济学-课件4-chapter-1
三、The Consumer ’s Problem本部分考察消费者选择理论的其他组成要素:Consumption Set 、Feasible Set 、Behavior Assumption ,然后构建消费者选择理论的正式表述。
Assumption1.2消费者偏好:消费者偏好关系具有完备性、传递性、严格单调性和严格凸的,那由定理1.1和1.3可知,该偏好关系可以有一个连续的、严格增加的、严格拟凹的实值效用函数u 代表。
(考察两种商品情形) 消费者的行为假设:假设消费者根据其偏好关系在可行集中选择最偏好的消费束,即消费者选择能够支付得起的最优商品组合,即:*B ∈x ,使得对于所有的B ∈x ,有*x x f % (1.4) “支付得起”——预算集“最优”——偏好关系 预算集:{}R ,,0,0nB y y +=∈≤>>≥x x px p⏹ 消费者从预算集中选择最偏好的商品组合(点)*x : *B ∈x ,且对于所有的B ∈x ,有*x x f %。
⏹ 消费者从预算集中选择最大化效用函数的点*x : ()()()**arg max ,..u u u s t y ⇓≥=≤x x x x px 144444424444443给定假设1.2,并给定对消费者可行集的限制, 消费者问题(1.4)⇔受到约束的效用函数最大化问题; 即消费者问题转化为下面的优化问题:()1max ,..ni i i u B s t y p x y=∈≤⇒≤∑x x px (1.5) 接着需要考虑的问题是:此最大化问题是否有解? 是否有唯一解?定理A1.10:极值的存在性定理(解决了解的存在性问题) 设R nS +∈是非空紧集,:R f S →是连续的实值映射,则存在向量*S ∈x 和向量S ∈x %,对于所有的S ∈x ,有()()()*f f f ≤≤x x x % 该定理在(1.5)问题的应用:()u x 连续 {}R ,,0,0n B y y +=∈≤>>≥x x px p :非空、闭集、有界集 (其中,闭集+有界集⇒紧集)定理A2.14:目标函数严格凹(解决了解的唯一性问题)如果*x 最大化严格凹函数f ,那么*x 就是该函数唯一的全局最大值点; 如果*x 最小化严格凸函数f ,那么*x 就是该函数唯一的全局最大值点; 定理1.4:消费者效用最大化问题一阶条件的充分性假设()u x 是R n +上的连续拟凹函数,而且(p,y)0>>,如果u 在*x 处可微,而且**(x ,)0λ>>满足效用最大化问题的一阶条件(1.10),那么*x 就是使得消费者在价格p 和收入y 处达到效用最大化的解。
高级微观经济学第二讲
• 但是,进一步仔细观察就会发现,商品的 买卖活动不见得要在某个固定或特定地方 或场所进行,它可以分散在许多不同的地 方。由此可见,交易活动的场所并不是市 场的特征。
• 但是,任何市场都离不开买者、卖者以及 他们之间的交易活动,缺少其中任何一个, 市场都无法形成。三者并存才能使交易活 动得以进行,从而才能形成市场。因此, 买卖双方及其交易才是构成市场的基本要 素。经济学最关心的是交易活动如何影响 和决定商品的价格。
• 所谓完全的信息,是指买卖双方完全了解市场现在和未来 的情况,信息不需付出任何代价即可得到。因此,完全市 场中没有不确定性问题,一切都是事先可知的,至少客观 概率存在。信息的畅通,使得买卖双方对市场上的任何变 化都了如指掌。 • 第二个条件是说买卖双方听从价格召唤,完全依据价格行 事。某个销售者抬高物价,他就会失去顾客;反之,降价 则会招引顾客。为了获利,卖者必然希望高价售出他的商 品,而买者必然希望低价买进他所需的商品。在这种频繁 的买卖交易过程中,价格在不同买者与卖者之间的差别便 会趋于消失。
• (一)经济人是经济活动的主体 经济学中所说的人是经济人(agent),是 发生经济活动的社会基本单位。他可以是 一个个人、一个家庭 、一个集团或者一个 组织,具有独立的决策机构或中心,这个 机构或中心决定和指挥着它的一切经济活 动。因此,经济人是经济活动的主体。
• 经济人概念的关键,在于经济人能够独立 决策。例如一个企业是一个经济人,企业 具有领导核心机构(如董事会、董事长、总 经理),这个机构决定和指挥着企业的一切 经济活动。企业雇用的工人不是企业经济 人,因为他决定不了企业的经济活动。但 工人是劳动活动的主体,他有权决定自己 是否向企业提供劳动,因而工人作为劳动 者一方是劳动经济人。
高级微观经济学上财经济学院课件(1)
2. Utility function is a convenient way to describe a preference relation. For example, if I tell you that my preferences over apples and bananas is u (a, b) = a0.5 + b, then you would know how I would choose between any combinations of apples and bananas. 3. Theorem. A preference relation can be represented by a utility function only if it
∀x, y ∈ X, y ≫ x =⇒ y (b) Strict monotonicity: The prtone if x
∀x, y ∈ X, y ≥ x =⇒ y while y ≫ x =⇒ y ≻ x. (c) Strong monotonicity: The preference
notation ∼ to represent the indifference relation. If x
strictly prefers x to x′ . We use ≻ to represent the strict preference relation. 5. The transitivity of implies the transitivity of ∼ and the transitivity of ≻, and vice
1.2
Utility function
1. A utility function defined over X assigns a real number to each member of X . We say a utility function u : Rn → R represents a preference relation and x′ , x x′ iff u(x) ≥ u(x′ ). if for any objects x
高级微观经济学讲义(清华 白重恩) Notes1-04
⇒ u(x) ≥ u(y), u(y) ≥ u(z) ⇒ u(x) ≥ u(z) ⇒x z ⇒ 传递性
7、Question: if
is rational preference relation, can we find a U function representation?
答案:不一定能找到一个效用函数与之对应。 一个反例: X = ( x1 , x2 ) : x1 ∈ [ o, ∞ ) , x2 ∈ [ o, ∞ ) ,定义偏好关系为字典顺序,即,
∀x1,
( x1 ,2)
( x1 ,1)
u ( x1 ,2) > u ( x1 ,1) ⇒ ∃有理数ϕ ( x1 ),u ( x1 ,2) > ϕ ( x1 ) > u ( x1 ,1)
可以证明 x1 → ϕ ( x1 ) 是一个一一对应
设x1 < x 2
⇒ u ( x1 ,1) < u ( x1 ,2) < u ( x 2 ,1) < u ( x 2 ,2) ⇒ ϕ ( x1 ) < ϕ ( x 2 )
y.
∀x ∈ R n . define: e ≡ (1,1,...,1) ∃λ1 , λ2 ∈ R, s.t. λ1e ≥ x ≥ λ2 e ⇒ λ1e
define: set A,B.
x
λ2 e; (单调性的定义)
A ≡ {λ1 ∈ R : λ1e B ≡ {λ2 ∈ R : x
x};
λ2 e}.
Î
( x1 , x2 ) ( y1 , y2 ), if x1 > y1 , or x1 = y1 andx2 ≥ y2
{
}
可以证明字典序是理性的。 问题:是否存在一个函数 u, s.t , ( x1 , x 2 ) ( y1 , y 2 ) iff u ( x1 , x 2 ) ≥ ( y1 , y 2 ) ? Answer: No! 证明:反设存在这样一个 u 满足上述条件。
高级宏微观经济学---第1章
13
四、效用函数的凹性和拟凹性
1.效用函数的凹性。 1.效用函数的凹性。是指效用函数 u 定义在凸消费集 X = R n 中 效用函数的凹性 +
x 的实值函数。 对于其中两个任意消费束 也可认定为消费品) 0 , x1 ∈ X (也可认定为消费品) 的实值函数。
而言, 而言,当 0 ≤ θ ≤1 时,存在
λx1 + (1 − λ )x0 ≻ x 0 。
凸性是经济学中专用的一个核心假定, 凸性是经济学中专用的一个核心假定,相关概念有凸集和凸函 数。关于凸集可这样理解: 关于凸集可这样理解: ①设 x , y, z ∈ R n ,对于任意的 0 ≤ λ ≤ 1 ,若存在 z = λx + (1 − λ ) y , 一个凸组合, 则称 z 是 x 和 y 一个凸组合,请参阅图 1—1。 ②设 X ⊂ R n ,若 X 中任意两点的凸组合都在 X 中,则称 X 为凸 为凸集。 集。即 x, y ∈ X ,若 [λx + (1− λ ) y] ∈ X , 0 ≤ λ ≤ 1 ,则 X 为凸集。参阅图 1—2。 ③凸性偏好假定的经济意义有两点。其一、消费者更喜欢商品的 凸性偏好假定的经济意义有两点。其一、 多样化选择;其二、消费者消费商品的边际替代率递减。 多样化选择;其二、消费者消费商品的边际替代率递减。
Chapter 1Introduction(高级微观经济学-上海财经大学,沈凌)
Chapter 1: Introductionz How to build an economic model? (Hal R.Varian)1. An economic model: an idealization of the reality, but not the reality.2. Why do we need an economic model?3. How to build an economic model? z Getting ideas from reality: An interesting one? Is the idea worth pursuing? z Don’t look at the literature too soon z Simplifying and Generalizing your model z Making mistakes: team work z Searching the literature z Giving a seminar1z Mathematics 1. Set theoryA Set (A) is a collection of objects called elements (a): a ∈ A The empty set is Φ , and the universal set is U .Binary operations on set: 1. 2. 3. 4. the union of A and B is the set A ∪ B = {x : x ∈ A or x ∈ B} the intersection of A and B is A ∩ B = {x : x ∈ A and x ∈ B} the difference of A and B isA \ B = {x : x ∈ A and x ∉ B}the symmetric difference of A and B is A∆B = ( A ∪ B ) \ ( A ∩ B )The complement of A is Ac = U \ ATheorem 1Let A, B and C be sets,1. 2.A \ (B ∪ C ) = ( A \ B ) ∩ ( A \ C ) A \ (B ∩ C ) = ( A \ B ) ∪ ( A \ C )ACBCorollary 2 (DeMorgan’s Law)( A ∪ B )c = Ac ∩ B cand ( A ∩ B ) = Ac ∪ B cc2Generalizing theorem 1 to theorem 3: A\⎛ ⎜ ∪ Si ⎞ ⎟ = ∩( A \ S i ) and A \ ⎛ ⎜ ∩ Si ⎞ ⎟ = ∪( A \ S i ) ⎝ i∈I ={1, 2,3...} ⎠ i∈I ⎝ i∈I ={1, 2,3...} ⎠ i∈IGiven any set A, the power set of A, written by Ρ( A) is the set consisting of all subsets of A, i.e., Ρ( A) = {B | B ⊂ A}Question : If a set A has n elements, how many elements are there in Ρ( A) ?The Cartesian Product of two sets A and B (also called the product set or cross product) is defined to be the set of all points (a, b ) where a ∈ A and b ∈ B . It is denoted A × B .Example:R2 ≡ R × RR n ≡ R × R × R × ...R = {( x1 , x2 ,..., xn ) | xi ∈ R, i = 1,2,...n} ,wheretheelement(x1 , x2 ,...xn ) ofR n is an n-dimensional ordered vector. We denote: xS ⊂ R n is a convex set if ∀x, y ∈ S , we have tx + (1 − t ) y ∈ S for all t ∈ [0,1]The intersection of convex sets is convex, but the union of them is not.32. TopologyA metric space is a set S with a global distance function (the metric d ) that, for every pointsx and y in S , gives the distance between them as a nonnegative real number d (x, y ) . Ametric space must satisfy: 1. d (x, y ) = 0 iff x = y 2. d (x, y ) = d ( y, x ) 3. d ( x, y ) + d ( x, z ) ≥ d (x, z ) Example: Euclidean metric in R 2 : d (x, y ) =(x1 − y1 )2 + (x2 − y2 )2Open and Closed ε − Balls: let ε be a real positive number, then 1. The open ε − ball with center x 0 and radius ε > 0 isBε x 0 = x ∈ R n | d x 0 , x < ε( ) {() }) }2. The closed ε − ball with center x 0 and radius ε > 0 isBε x 0 = x ∈ R n | d x 0 , x ≤ ε( ) {(Open and Closed sets in R n : A set S ⊂ R n is open if ∀x ∈ S , ∃ε > 0,Bε ( x ) ⊂ S .A set S ⊂ R n is closed if its complement, S c , is open.Some important properties of open and closed sets: 1. The union of open sets is open. 2. The intersection of any finite number of open sets is open. 3. The union of any finite number of closed sets is closed.44. The intersection of closed sets is closed.Question: what if the collection is infinite for 2. and 3.?Theorem 4: Every open set is a collection of open balls.Bounded sets in R n : A set S ⊂ R n is bounded if ∃ε > 0 and x ∈ R n , S ⊂ Bε ( x ) .Let S ⊂ R be a nonempty set of real numbers: 1. Any real number l is a lower bound if ∀ x ∈ S , 2. Any real number u is an upper bound if ∀ x ∈ S ,x ≥ l . The set is bounded from below. x ≤ u . The set is bounded from above.3. The largest number among lower bounds is called the greatest lower bound of S. 4. The smallest number among upper bounds is called the least upper bound of S.A bounded set is bounded both from below and above. We can show that for any bounded subsets of the real line, there always exists a g.l.b. and l.u.b.Let S ⊂ R be a bounded set and let a be the g.l.b of S and b be the l.u.b. of S, then we have: 1. If S is open, then a ∉ S and b ∉ S 2. if S is closed, then a ∈ S and b ∈ SCompact sets : A set is compact if it is closed and bounded.53. Relations and functionsConsider an ordered pair (s, t ) that associated an element s ∈ S to another element t ∈ T . Any collection of such ordered pairs is said to constitute a binary relation between the sets S andT . Note that a binary relation R is a subset of the cross product S × T .Some properties of relations: 1. The relation is complete if either xRy or yRx. 2. 3. 4. transitive if xRy and yRz implies xRz. reflexive if xRx. symmetric if xRy ⇔ yRxExamples: The preference relation ( ≿ ) is complete, transitive and reflexive.The function is a mapping from one set D (domain) to another set R (range) denoted as:f :D→ RThe image of f : I ≡ {y | y = f ( x)} ⊂ R The inverse image of a set of points S ⊂ R is: f −1 (S ) ≡ {x | x ∈ D, f ( x ) ∈ S } The graph of f : G ≡ {(x, y ) | x ∈ D, y = f ( x ) ∈ S } A function is a surjective function if the range ran( f ) = R A function is an injective function (or one to one) if f (a) = f (b) implies a = b A function is bijective if it is both surjective and injective. In a sense, the domain and the range must have the same number of elements.6Homogeneous function A function f ( x1 ,...x N ) is homogeneous of degree r ( for r = ...,−1,0,1,... ) if ∀t > 0 we have:f (tx1 ,...tx N ) = t r f ( x1 ,...x N )Theorem 5 (Euler’s Formula) suppose that f ( x1 ,...x N ) is homogeneous of degree r ( forr = ...,−1,0,1,... ) and differentiable. Then at any (x1 ,..., x N ) we have∑∂f ( x1 ,...x N ) xn = rf ( x1 ,..., x N ) ∂xn n =1NProof: by definition, we have f (tx1 ,...tx N ) − t r f (x1 ,...x N ) = 0 Differentiation this expression with respect to t gives∑∂f (tx1 ,...tx N ) xn − rt r −1 f ( x1 ,..., x N ) = 0 ∂ (txn ) n =1NEvaluating at t = 1 , we obtain Euler’s Formula.□74. ContinuityA function f : R → R is continuous at a point x 0 if ∀ε > 0 , ∃δ > 0 such that d (x, x 0 ) < δ implies that d ( f ( x), f ( x 0 ) ) < ε . Cauchy definition: the function is continuous at the point x 0 ∈ D if ∀ε > 0 , ∃δ > 0 such that f (Bδ ( x 0 ) ∩ D ) ⊂ Bε ( f (x 0 )) A function is continuous function if it is continuous at every point in its domain.Open and closed set in D: A subset S ⊂ D is open in D if for every x ∈ S , there is an ε > 0 such that Bε ( x) ∩ D ⊂ S . A subset is closed in D if its complement S in D, is open in D.Theorem 6The following statements are equivalent: 1.f : D → R n is continuous;2. for every open ball B in R n , f −1 (B ) is open in D; 3. for every open set S in R n , f −1 (S ) is open in D. Remark: the continuous inverse image of an open set is an open set. In short, the inverse mapping of a continuous function can preserve the openness of sets. For what kind of sets the continuous image can preserve its properties? It turns out to be a compact set. The continuous image of a compact set is a compact set.85. Some existence theoremsTheorem 7 (Weierstrass) Existence of Extreme ValuesLet f : S → R be a continuous real-values mapping, where S is a nonempty compact subset of R n . Then there exists two vectors xmax,xmin∈ S such that for all x ∈ S ,f x( ) ≤ f (x ) ≤ f (x )min maxTheorem 8 (Brouwer) Fixed-pointLet S ⊂ R n be a nonempty compact set and f : S → S be a continuous real-values mapping, then there exists at least one fixed point x of f in S . I.e., f ( x ) = x .∗ ∗ ∗96. Real valued functionsSets related to a real valued function f : D → R :L( y 0 ) is a Level set if L( y 0 ) = {x | x ∈ D, f ( x ) = y 0 }, where y 0 ∈ R .Superior and inferior sets: 1. S ( y 0 ) = {x | x ∈ D, f ( x ) ≥ y 0 } is called the superior set (or upper contour set) for levely 0 ∈ R . S ' ( y 0 ) = {x | x ∈ D, f ( x ) > y 0 } is called the strictly superior set for level y 0 ∈ R2. I ( y 0 ) = {x | x ∈ D, f ( x ) ≤ y 0 } is called the inferior set (or lower contour set) for levely 0 ∈ R . I ' ( y 0 ) = {x | x ∈ D, f ( x ) < y 0 } is called the strictly inferior set for level y 0 ∈ RConcave function: ∀x1 , x 2 ∈ D, t ∈ [0,1] , f (tx1 + (1 − t )x 2 ) ≥ tf (x1 ) + (1 − t ) f (x 2 ) Strictly concave function: ∀x1 , x 2 ∈ D, t ∈ (0,1) , f (tx1 + (1 − t )x 2 ) > tf (x1 ) + (1 − t ) f (x 2 ) If the function is differentiable, then it is concave iff f ' ' ( x ) ≤ 0 , and it is strictly concave iff ' ' ( x ) < 0 but the reverse is not true.In the multi-dimensional case, the condition for concavity is equivalent to that the matrix (known as Hessian matrix) of second order derivative is negative semidefinite at every point. If the Hessian matrix of a function is negative definite at every point, then the function must be strictly concave. But the reverse is not true.⎛ ∂ 2 f (x ) ∂ 2 f (x ) ⎞ ⎜ ⎟ ... ⎜ ∂x1∂x1 ∂x1∂xn ⎟ ⎟ D 2 f (x ) = ⎜ ..... ⎜ ⎟ ⎜ ∂ 2 f (x ) ∂ 2 f (x ) ⎟ ... ⎜ ⎟ ⎝ ∂xn ∂x1 ∂xn ∂xn ⎠1011 Convex function: []1,0,,21∈∈∀t D x x , ()()()()()212111x f t x tf x t tx f −+≤−+ Strictly convex function: ()1,0,,21∈∈∀t D x x , ()()()()()212111x f t x tf x t tx f −+<−+ If the function is differentiable, then it is concave iff ()0''≥x f , and it is strictly concave if ()0''>x f but the reverse is not true.In the multi-dimensional case, the condition for concavity is equivalent to that the matrix (known as Hessian matrix) of second order derivative is positive semidefinite at every point. If the Hessian matrix of a function is positive definite at every point, then the function must be strictly concave. But the reverse is not true.()x f is a (strictly) concave iff ()x f − is (strictly) convex.The following statements are equivalent:1. R D f →: is Quasiconcave Function2. []1,0,,21∈∈∀t D x x , ()()()()[]2121,min 1x f x f x t tx f ≥−+3. ()x f − is quasiconvex4. ()y S is convex set for all R y ∈ Quasiconvex Function ?• A function R A f →: is quasiconcave if its upper contour sets (){}t x f A x ≥∈: are convex sets; i.e., if ()t x f ≥ and ()t x f ≥′, then ()()t x x f ≥′−+αα1 for any R t ∈, A x x ∈′, and []1,0∈α.• A function R A f →: is strictly quasiconcave if ()()t x x f >′−+αα1 for x x ′≠ and()1,0∈α.。
夏纪军-高级微观经济学讲义1
微观经济理论I上海财经大学经济学院1微观经济理论•参考书:–G. A. Jehle & P. Reny•Advanced Microeconomic Theory,–A. Mas-Colell, M. D. Whinston & J. R. Green •Microeconomic TheoryA. Rubinstein–A Rubinstein•Microeconomics Lecture noteVarian H R Microeconomic Analysis–Varian, H. R., Microeconomic Analysis2第一讲第讲偏好与选择31. 偏好与选择问题界定:选择函数(choice function)•(choice function)•偏好关系(preference relation)•显示偏好弱公理(weak Axiom of revealed preference)•理性选择与WA:选择函数的可合理化定理选择函数的可合理化定理41.1 选择函数择项•选择项:x–根据问题分析需要,设定个体可以选择的变量•比如–例1:利率或税收政策对消费会产生怎样的影响?•选项:x=(c1, c2)–例2:要素投入决策•选项:x=(k, l)–例3:地方政府基础设施投资激励•选项:x=(I i,C i )51.1 选择函数•选择项:x•选择集:X给定经济环境下,个体可以–想象自己可能选择的各种选择项的集合。
–根据经济环境设定每个变量可能的取值范围•例:•消费者选择集:={(0X {(c 1, c 2):c 1 0, c 2 0}•生产者选择集:X ={(k , l ):k 0, l 0}X 0•地方政府选择集:={(I i , C i ):I i 0, C i 0}61.1 选择函数•选择项:x•选择集:X•可行集:A X⊂–决策者可以选择的选项的集合–反映各种约束对选择的影响•例, c2):(1+r)c1+ c2≤(1+r)y1+ y2}消费者可行集{((1+)+(1+)+–消费者可行集:B={(c1–生产者可行集:B={(x1, x2):y≤f(k, l) }, C i):I i +C i ≤S+tF i }地方政府的可行集{(–地方政府的可行集:B={(Ii71.1 选择函数择项•选择项:x•选择集:X•可行集:A⊂X•D:可能的可行集的集合(X部分子集的集合)–例:–消费者可行集:B(r, y1,y2)–生产者可行集:B(ρ)–地方政府可行集: B(t, F,S)81.1 选择函数•例:大学的选择–可以申请的所有学校:X={x1,x2,..,x N}–A:接受申请的所有学校,可以从中选择一所接受申请的所有学校可以从中选择所–D:2N -1 种可能的选择情形(不考虑∅)•如果学校根据排名,从高到低进行了排序,而且排名高的学校录取了该学生,排名较低的学校都会录取。
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∪ i∈I Si = {x: x ∈ Si for some i ∈ I};
b) ∩ i∈I Si = {x: x ∈ Si for all i ∈ I}. DeMorgan’s Law can be generalized to indexed collections. Theorem 3. Let A be a set and {Si}i∈I be an indexed collection of sets, then a) A \ ∪ i∈I Si = ∩ i∈I (A \ Si); b) A \ ∩ i∈I Si = ∪ i∈I (A \ Si). Problem 2. Prove Theorem 3. Definition 4. Given any set A, the power set (幂集) of A, written P(A) is the set consisting of all subsets of A; i.e., P(A) = {B: B ⊂ A}. Problem 3. If a set S has n elements, how many elements are there in P(S)? Definition 5. The Cartesian product (笛卡尔乘集) of two sets A and B (also called the product set or cross product) is defined to be the set of all points (a, b) where a ∈ A and b ∈ B. It is denoted
, S,
, Z . A set can consist of any type of element. Even sets can be
elements of some set. A consumption set is a collection of consumption plans. The typical sets we deal with have real numbers as their elements. If a is an element of A, we write a ∈ A. If a is not an element of A, we write a ∉ A. If all the elements of A are also elements of B, then A is a subset of B. We write either A ⊂ B or
with boldface or underscored type. This note uses x ≡ ( x1 , Vector Relation: for any two vectors x and y in and x y if xi > yi , i = 1,
n
, xn ) for convenie two-
. An n-dimensional space is defined as the set product
≡
× ×
.xn ) of
×
n
≡ {( x1 , x2 ,
, xn ) | xi ∈ , i = 1, 2,
, n} .
The element ( x1 , x2 ,
is an n-dimensional ordered tuple, or vector, usually denoted
n
, we say that x ≥ y if xi ≥ yi , i = 1,
, n;
, n.
Definition 6. S ⊂
is a convex set (凸集) if for all x and y ∈ S , we have tx + (1 − t ) y ∈ S
for all t in the interval [0, 1]. Intuitively, a set is convex iff we can connect any two points in the set by a straight line that lies entirely within the set. Note that convex sets play a fundamental role in microeconomic theory. In theoretical analysis, convexity is assumed by economists to get well-behaved analytical results. Remark 7. The intersection of convex sets is convex, but the union of them is not. 3. A Little Topology We begin with a rigorous definition of metric space. A metric space (测度空间) is a set S with a global distance function (the metric d) that, for every two points x and y in S, gives the distance between them as a nonnegative real number d(x, y). A metric space must satisfy: 1. d(x, y) = 0 iff x = y; 2. d(x, y) = d(y, x); 3. The triangle inequality d(x, y) + d(y, z) ≥ d(x, z). A natural example is the Cartesian plane . Define the distance function
上海财大经济学院
1
作者:陶佶
2005 年秋季
高等微观经济学 I
实分析简介
2. The intersection (交集) of A and B is the set A ∩ B = {x: x ∈ A and x ∈ B}. 3. The difference of A and B is the set A \ B = {x: x ∈ A and x ∉ B}. 4. The symmetric difference of A and B is the set A Δ B = (A ∪ B)\(A ∩ B). It can be easily seen that A Δ B = (A \ B) ∪ (B \ A). Another common set operation is complementation (补集). Let U be a well-defined universal set that contains all the elements in the question. Then the complementation of a set A ⊂ U is Ac = U \ A . Theorem 1. Let A, B, and C be sets. a) A \ (B ∪ C) = (A \ B) ∩ (A \ C); b) A \ (B ∩ C) = (A \ B) ∪ (A \ C). Proof: Theorem 1 can be proved as a sequence of equivalences. Problem 1. Prove Theorem 1. The familiar DeMorgan’s Law is an obvious consequence of Theorem 1 when there is a universal set to make the complementation well-defined. Corollary 2. (DeMorgan’s Law) Let A, and B be sets. a) (A ∪ B)c = Ac ∩ Bc; b) (A ∩ B)c = Ac ∪ Bc. To deal with large collections of sets, we use index set (索引集) I = {1, 2, 3,…} and denote the collection of sets as {Si}i∈I. Union and Intersection can be extended to work with indexed collections. In particular, we define a)
Figure 1. Venn Diagrams The Venn diagrams above show four standard binary operations on sets. 1. The union (并集) of A and B is the set A ∪ B = {x: x ∈ A or x ∈ B}.
B ⊃ A . If A ⊂ B and B ⊂ A , we say that A and B are equal: A = B.
A set S is empty (空集) if it contains no elements at all. An empty set denoted as ∅ is a subset of any set.
d x , y ≡ ( x1 − y1 ) 2 + ( x2 − y2 ) 2 ≡ x − y
for x and y in . It is obvious to see that the space with the metric d above is a metric space. The metric d called as Euclidean metric or Euclidean norm (欧几里德范数) can be generalized to an n-dimensional Euclidean space. Definition 8. Open and Closed ε -Balls (开球和闭球): Let ε be a real positive number.