高宏Chapter 2A
高宏2014年第1次习题课
]] σ／(σ-1)
] σ／(σ-1)
所以生产函数为规模报酬不变的。 b) 对生产函数两边均除以 AL：Y／AL=[ (K／AL)(σ-1)／σ +1] σ／(σ-1) 取 k=K／AL,y=Y／AL=f(k), 则可以得到生产函数的密集形式 ： a) f(k)= [k(σ-1)／σ +1] σ／(σ-1) c）对(1)式两边对 k 求导数： f′(k)= [σ／(σ-1)] [k(σ-1)／σ +1] σ／(σ-1)-1[(σ-1)／σ] k[(σ-1)／
2014 年高级宏观经济学第 1 次习题课
1.1 考虑一处于平衡增长路径上的索洛经济，为了简单，假定无技术进步。现在假定 人口增长率下降。 a) 处于平衡增长路径上的每工人平均资本、每工人平均产量和每工人平均消费 将发生什么变化？画出经济向其新平衡增长路径移动的过程中这些变量的 路径。 b) 说明人口增长率下降对产量路径(总产量，而非每工人平均产量)的影响。 答： a) k˙=sf(k)-(n+σ)k 假 设 经 济 在 初 始 时 处 于 平 衡 增 长 路 径 上 ， 即 满 足 k*˙=sf(k*)-(n+σ)k*=0.此时，经济中的实际投资等于持平投资，两条线相交于 (k*,y*) 。在 t0 时刻，当人口增长率由 n 下降到 nnew 后，实际投资线不变， 持平投资线发生偏转。此时，k*˙=sf(k*)-(nnew+σ)k*>0,实际投资超过持平投资， 每个人平均资本开始增加。在 t1 时刻之后，经济重新达到平衡增长路径后。 k*new˙=sf(k* new)-(nnew+σ)k* new =0, 单位有效劳动（由于没有技术进步，相当于 每人）实际投资等处持平投资 。在 t0 时刻到 t1 时刻之间，由于 k˙>0，所以 每个人平均资本逐步增长。而由于生产函数固定，所以每个人平均产出伴随 每个人平均资本的增加而增加，每个人平均消费也将随之逐步增加 （c=(1-s)f(k) ） 。
《高等代数2》教学大纲
《高等代数Ⅱ》教学大纲
一、课程基本信息
二、课程教学目标
通过本课程的学习，使学生较系统地掌握多项式、线性空间、线性变换、欧几里得空间等数学科
学的基础理论知识和基本计算技巧，学会严密的逻辑推理方法，大力加强学生的归纳、演绎、类比、抽象等能力，为学生学习后继有关课程如近世代数、离散数学、数论等奠定坚实的基础。
三、理论教学内容与要求
四、考核方式
本课程为考试课。
采用期末考试、平时考核相结合的考核方式。
总成绩为100分，其中期末考试成绩占总成绩的70%，平时成绩（包括作业、出勤、课堂表现等）占总成绩的30%。
罗默《高级宏观经济学》课件
罗默的《高级宏观经济学》课件一、引言罗默的《高级宏观经济学》是一本在学术界广受推崇的经济学教材，深入浅出地介绍了宏观经济学的基本理论和分析方法。
本书共分为两个部分，第一部分主要讨论宏观经济学的微观基础，包括消费者行为、生产者行为、市场均衡和一般均衡等；第二部分则重点介绍宏观经济学的核心议题，如经济增长、失业、通货膨胀、经济周期等。
本课件旨在帮助读者更好地理解和掌握罗默的《高级宏观经济学》的内容，为宏观经济学的学习和研究提供有益的参考。
二、微观基础1.消费者行为罗默认为，消费者行为是宏观经济学微观基础的重要组成部分。
消费者的目标是实现效用最大化，即在预算约束下，选择一组商品和服务，使得总效用达到最大。
为此，罗默介绍了无差异曲线、预算线等概念，以及如何通过求解拉格朗日函数来找到消费者最优消费组合的方法。
2.生产者行为生产者行为是宏观经济学微观基础的另一个重要组成部分。
罗默认为，生产者的目标是实现利润最大化，即在生产技术、要素价格和市场价格的约束下，选择一组生产要素和产量，使得总利润达到最大。
罗默详细介绍了生产函数、成本函数、利润函数等概念，以及如何通过求解拉格朗日函数来找到生产者最优生产组合的方法。
3.市场均衡和一般均衡罗默认为，市场均衡是宏观经济学的核心概念之一。
市场均衡指的是在某一市场价格下，市场需求等于市场供给。
罗默介绍了如何通过供求曲线来分析市场均衡，以及市场均衡的稳定性条件。
罗默还介绍了一般均衡理论，即在一个包含多个市场和多种商品的体系中，各个市场之间相互影响，形成一个稳定的均衡状态。
三、核心议题1.经济增长罗默认为，经济增长是宏观经济学研究的核心议题之一。
经济增长指的是一个国家或地区在一定时期内，实际产出水平持续提高的过程。
罗默详细介绍了经济增长的源泉，包括技术进步、资本积累、劳动力增长等，并分析了不同增长模型的特点和适用条件。
2.失业罗默认为，失业是宏观经济学的另一个核心议题。
失业指的是劳动力市场上，愿意工作但未能找到工作的人口。
上财罗大庆高宏课件 (9)
• Since all periods are identical, the competitive equilibrium prices and consumption allocation are
∞ {r t }∞ t=1 = {0.01}t=1 , ∞ {c 1 t , c 2 t }∞ t=1 = {2.2, 2}t=1 .
• Optimization problem solved by an individual born in period t is to maximize u(c1t , c2t+1 ) subject to the Present Value budget constraint. • Logarithmic utility example: – Maximize u(c1t , c2t+1 ) = ln c1t + β ln c2t+1 subject to c 1t + c2t+1 y2t+1 = y 1t + . 1 + rt 1 + rt
(II) Heterogeneity within Cohorts
• Heterogeneity creates incentives for trade. • Assume there are two types of agents born in each period. • Type 1 (group 1) – Nt individuals of type 1 are born in period t – they have utility function u(c1t , c2t+1 ) – and an endowment stream {y1t , y2t+1 }. • Type 2 (group 2) – Nt∗ individuals of type 2 are born in period t
高等代数第三章2
§3.3 向量组的秩三个辅助概念定义 设12,,,ns K ααα∈"，12,,,r i i i αα"αi 是其一个部分组。
若12,,,r i i αα"α线性无关，且对任意(,j k j i α≠1,2,,;1)k r j =≤"s ≤均有12,,,,r j i i i αααα"线性相关，则称12,,,r i i i αα"α是一个极大线性无关组，简称极大无关组。
显然，若12,,,s ααα"线性无关，则极大无关组就是其自身。
例 向量组的任何一个线性无关的部分组均可扩充为一个极大无关组。
定义 设 12,,,s ααα"与 12,,,t ββ"β)是两组n 元 向量，若每个均可由(1,2,,i i s α="12,,,t ββ"β线性 12,,,s ααα"12,,,t 可由向量组βββ" 表出，则称向量组线性表出。
若向量组12,,,s αα"α与向量组12,,,t βββ"可相互线性表出，则称向量组12,,,s ααα"12,,,t 与向量组βββ"12,,,等价，记为{s ααα"}{12,,,t βββ"}≅例 讨论下列向量之间的关系：(1) )1,0(),0,1(21==εε与 )0,0(),0,1(21==αα(2) )1,0(),0,1(21==εε与 )1,2(),2,1(21==ββ性质 向量组的等价具有(1)自反性：{m ααα,...,,21}≅{m ααα,...,,21}；(2)对称性：{s ααα,...,,21}≅{t βββ,...,,21}⇒{t βββ,...,,21}≅{s ααα,...,,21}；(3)传递性：12121212{,,...,}{,,...,}{,,...,}{,,...,}s t t r αααββββββγγγ≅⎫⎬≅⎭⇒{12,,...,s ααα}≅{12,,...,m γγγ}。
罗默《高级宏观经济学》(第3版)课后习题详解(第2章 无限期界与世代交叠模型)
罗默《高级宏观经济学》（第3版）第2章 无限期界与世代交叠模型跨考网独家整理最全经济学考研真题，经济学考研课后习题解析资料库，您可以在这里查阅历年经济学考研真题，经济学考研课后习题，经济学考研参考书等内容，更有跨考考研历年辅导的经济学学哥学姐的经济学考研经验，从前辈中获得的经验对初学者来说是宝贵的财富，这或许能帮你少走弯路，躲开一些陷阱。
以下内容为跨考网独家整理，如您还需更多考研资料，可选择经济学一对一在线咨询进行咨询。
2.1 考虑N 个厂商，每个厂商具有规模报酬不变的生产函数()Y F K AL =，，或者（利用密集形式）()Y ALf k =。
设()·0f '>，()()***1c s f k =-。
设所有厂商以工资wA 雇用工人，以成本r 租借资本，并且拥有相同的A 值。
（a ）考虑一位厂商试图以最小成本生产Y 单位产出的问题。
证明k 的成本最小化水平()()()**1001t t t f c c k cs f k n g k L n L αδ*+⎛⎫"==-=++=+ ⎪⎝⎭＜唯一地被确定并独立于Y ，所有厂商因此选择相同的k 值。
（b ）证明N 个成本最小化厂商的总产出等于具有相同生产函数的一个单个厂商利用N 个厂商所拥有的全部劳动与资本所生产的产出。
证明：（a ）题目的要求是厂商选择资本K 和有效劳动AL 以最小化成本rK wAL +，同时厂商受到生产函数()Y ALf k =的约束。
这是一个典型的最优化问题。
().mi . n s t w Y ALf k AL rK = +本题使用拉格朗日方法求解，构造拉格朗日函数： 求一阶条件：用第一个结果除以第二个结果：上式潜在地决定了最佳资本k 的选择。
很明显，k 的选择独立于Y 。
上式表明，资本和有效劳动的边际产品之比必须等于两种要素的价格之比，这便是成本最小化条件。
（b ）因为每个厂商拥有同样的k 和A ，下面是N 个成本最小化厂商的总产量关系式：单一厂商拥有同样的A 并且选择相同数量的k ，k 的决定独立于Y 的选择。
高等代数教案
高等代数教案 The pony was revised in January 2021
高等代数
教案
秦文钊
一、章（节、目）授课计划第页
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一、章（节、目）授课计划第页
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a的代数余子式.称为元素
ij
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二、课时教学内容第页。
高宏第三章 最优控制
第三章 最优控制（上）-变分法第一节 动态优化简介一、静态优化问题如果一个企业要确定一个最优产出水平x *以最大利润()F x ：0max ()x F x ≥ （1）这样的问题的解通常将是一数，即确定选择变量的单个最优值。
最优值常可由一阶条件()0F x *'=确定。
动态问题是多期（multiperiod ）的，但是并不是有多期的时间就是动态问题．．．．．．．．．．．．．．．．．。
考虑企业的多期决策问题：1max (,)Tt t F t x =∑ （2）(0,1)t x t T = 描述的是每阶段的产出组成的序列，即给出了一个产出的时间路径。
显而易见，总利润不是由单期的产出决定，而是由整个的产出的时间路径确定，所以要使利润最大化，实质上是要找到一条最优的路径（而不是单个期的t x ）。
但由于t 期利润只与t 期的产出有关，所以要在整个时间序列内最大化利润，就只要分别在每一期最大化利润即可，即这一个问题的解是一个有T 个数的集合，1{,}T x x ** 。
所以由于任一产量只影响该期利润，问题（2）实际上是一系列的．．．．静态问题，即在每一期选择当前产量使该期利润最大化。
问题（2）有类似的T 个一阶条件，各期的一阶条件之间没有联系。
在Ramsey 模型的竞争性均衡结构中，生产者问题就具有这样的性质。
二、动态问题具有动态性质的问题是，当前的产出不但影响到当前的利润，还影响到未．．．．．来．的利润。
更为一般地来说，当前决策影响未来决策。
11max (,,).. 0,1Tt t t t F t x x s t x t T-=≥=∑0x 给定或0(0)x x = （3）在问题（3）中，每一期的利润不但取决于当前产量，还与过去的产量有关；换句话说，t 期选择的产量t x 不但影响t 期的利润，还会影响到以后的利润。
注意，上述问题中已指定了0x 。
0x 影响到了以后各期的利润（从而也影响到总利润）。
问题（3）与问题（2）不同，它的最优解的T 个一阶条件不能分别确定，而是要同时确定，也就是我们实际上要“一次性”确定一条最优路径。
罗默《高级宏观经济学》课件
y * s K (k*) s y 1 K (k *)
k k *
在多数国家， 支付给资本 的收入份额 约为1/3。则 产出的储蓄 率弹性约为 0.5。
Y K K (k*) K Y
k * f (k*) f (k*)
产出的资本弹性等于资本收入在总收入 中所占的份额。
(t ) 0 k (t ) 0 k
k增加 k下降 k不变
9
(t ) 0 k
2018/12/21
2、稳态
单位有效劳动资 本存量的变化： f(k)
(t ) sf (k (t )) (n g )k (t ) k
(n＋g+δ)k f(k) 在A点，实际投资与 持平相等，资本存量 不变，经济达到稳态： 单位有效劳动的 k*,y*,c*固定不变。 消费： c*＝f(k*)-sf(k*) =f(k*)-(n+g+δ）k*
2018/12/21 6
假设的含义
根据规模报酬不变假设可以得到 生产函数的紧凑形式：
f(k) c
f(k)
y=f(k)
y=Y/AL单位有效劳动的产出 k=K/AL 单位有效劳动的资本 稻田条件意味着，在资本存量充 分小时资本的边际产出十分大， 当资本存量很大时，其会变得很 小。其作用在于确保经济的路径 不发散。 根据假设1得到： f (0) 0
资本增长率
y* Y / AL Y y * AL
ln Y ln y * ln A ln L
L A Y n g Y L A
k* K / AL K k * AL
K n g K
消费C=（1-s）Y，因此，消费和产出具有相同 的增长率，等于n+g。 同样方法可以计算出每个工人的产出Y/L、人均 资本K/L和人均消费C/L具有相同的增长率：g。
高宏：内生增长理论
干中学 learning by doing
• 干中学理论的核心思想：个人在制造产品时，他 会考虑生产过程的改进方法。因此，有些知识的 积累是传统经济活动的副产品。
在该模型中，所有资源 都用于产品生产。学习 是生产新资本的副产品， 因此，知识存量是资本 存量的函数。 在该模型中，只有资本 是内生变量。
ln Y ln(1 aK ) ln K (1 ) ln A (1 ) ln(1 aL ) (1 ) ln L
等式两边对时间求导得到产出的增长率：
gY g K (1 )(n g A)
每个工人产出的增 长率：
g y gY n
Y (t ) K (t ) B K (t ) L(t ) gY (1 )g K (1 ቤተ መጻሕፍቲ ባይዱ )n
1
(1 )
1
(t ) sY (t ) sK (t ) B1 K (t ) (1 ) L(t )1 K
2 g K (1 ) 1 g K (1 )ngK
整个经济
研发部门
生产部门
劳动力
资本 知识
aL
aK 1
1- aL
1-aK 1
12
生产函数
生产部门
Y (t ) (1 aK ) K (t )
技术部门：B为转 移参数，θ反映了 现有知识存量对研 发成败的影响。
A(t )(1 aL ) L(t )
1
,0 1
(t ) Ba K (t ) a L(t ) A(t ) A K L B 0, 0, 0
高宏1
1. Euler’s Theorem292. First Welfare Theorem II 165二、简答题1. The two important standard assumptions on the aggregate production function:Y(t) = F(K(t), L(t), A(t)). 29、332. Uzawa’s Theorem I 603. Gorman’s Aggregation Theorem 1514. Representative Firm Theorem 158-159三、证明题1. Let x(t), a, b ∈ R. If |a| < 1, then the unique steady state of the linear difference equation x(t + 1) = ax(t) + b is globally (asymptotically) stable, in the sense that x(t)→x∗ = b/(1− a). 452.the Cobb-Douglas production function has an elasticity of substitution between capital and labor equal to 1. 523.(Existence of a Normative Representative Household) Consider an economy with a finite number N <∞ of commodities, a set H of households, and a convex aggregate production possibilities set Y . Suppose that the preferences of each household h ∈H can be represented by Gorman form vh(p, wh) = ah(p) + b(p)wh, where p = (p1, . . . , pN) is the price vector, and that each household h ∈H has a positive demand for each commodity.1. Then any feasible allocation that maximizes the utility of the representative household,v(p, w) =_h∈H ah(p) + b(p)w, with w ≡_h∈H wh, is Pareto optimal.2. Moreover, if ah(p) = ah for all p and all h ∈ H, then any Pareto optimal allocation maximizes the utility of the representative household. 1544.(Contraction Mapping Theorem) Let (S, d) be a complete metric space and suppose that T : S→S is a contraction. Then T has a unique fixed point, ˆz; that is, there exists a unique ˆz ∈S such thatT ˆz = ˆz. 1911. Inada Conditions 332. Contraction Mapping and fixed point 191二、简答题1. Types of Neutral Technological Progress 58-592. Uzawa’s Theorem II 633. Debreu-Mantel-Sonnenschein Theorem 1504. Second Welfare Theorem 167三、证明题1. (Euler’s Theorem) Suppose that g :R K+2→R is differentiable in x ∈ R and y ∈ R, with partial derivatives denoted by gx and gy, and is homogeneous of degree m inx and y. Then mg(x, y, z) = gx(x, y, z)x + gy(x, y, z)y for all x ∈ R, y ∈ R, and z ∈ R K.Moreover, gx(x, y, z) and gy(x, y, z) are themselves homogeneous of degree m − 1in x and y. 292.Let g :R→R be differentiable in the neighborhood of the steady state x∗, defined by g(x∗) =x∗,and suppose that__g(x∗)__< 1. Then the steady state x∗ of the nonlinear difference equation x(t + 1) = g(x(t)) is locally (asymptotically) stable. Moreover, if g is continuously differentiable and satisfies __g(x)__< 1 for all x ∈ R, then x∗ is globally(asymptotically) stable. 453 (Uzawa’s Theorem I) Consider a growth model with aggregate production functionY(t) = ˜ F(K(t), L(t), ˜ A(t)),where ˜ F :R2+ × A→R+ and ˜ A(t) ∈ A represents technology at time t (where A is anarbitrary set, e.g., a subset of R N for some natural numberN). Suppose that ˜ F exhibits constant returns to scale in K and L. The aggregate resource constraint is˙ K(t) = Y(t) −C(t) −δK(t ).Suppose that there is a constant growth rate of population, L(t) = exp(nt)L(0), and that there exists T <∞ such that for all t ≥ T , ˙ Y (t)/Y (t) = gY > 0, ˙K (t)/K(t) = gK > 0,and˙C(t)/C(t) = gC > 0. Then1. gY= gK= gC; and2. for any t ≥ T , there exists a functionF :R2+→R+ homogeneous of degree 1 in its two arguments, such that the aggregate production function can be represented asY(t) = F(K(t), A(t)L(t)),where A(t) ∈ R+ and˙ A(t)A(t)= g = gY−n.604. Construct a continuous-time version of the model with finite lives and random deaths (recall(5.12)in the text). In particular suppose that an individual faces a constant (Poisson) flow rate of death equal to ν >0 and has a true discount factor equal to ρ. Show that this individual behaves as if she were infinitely lived with an effective discount factor of ρ + ν. 1791. First Welfare Theorem I 1642. Locally asymptotically stable and globally asymptotically stable 44二、简答题1. The two reasonable microfoundations for the Infinite Planning Horizon 156-1572. Existence of a Normative Representative Household 1543. Why is there such a difference between the representative household and representative firm assumptions? 1594 The five assumptions on Stationary Dynamic Programming Theorems 187-188三、证明题1. Suppose that Assumptions 1 and 2 hold. Then the steady-state equilibrium of the Solow growth model described by the difference equation (2.17) is globally asymptotically stable, and starting from any k(0) > 0, k(t) monotonically converges to k∗. 452. (Uzawa’s Theorem II) Suppose that all hypotheses in Theorem 2.6 are satisfied, so that ˜F :R2+ × A→R+ has a representation of the form F(K(t), A(t)L(t)) withA(t) ∈ R+ and ˙ A(t)/A(t) = g = gY−n (for t ≥ T ). In addition, suppose that factor markets are competitive and that for all t ≥ T , the rental rate satisfies R(t) = R∗(or equivalently,αK(t) = α∗K). Then, denoting the partial derivatives of ˜ F and F with respect to their first two arguments by ˜ FK, ˜ FL, FK, and FL, we have˜ FK(K(t ), L(t), ˜ A(t)) = FK(K(t ), A(t)L(t)) and˜ FL(K(t ), L(t), ˜ A(t)) = A(t)FL(K(t ), A(t)L(t)). (2.43)Moreover, if (2.43) holds and factor markets are competitive, then R(t) = R∗(and αK(t) = α∗K) for all t ≥ T . 633. (FirstWelfare Theorem I) Suppose that (x∗, y∗, p∗) is a competitive equilibrium of economy E ≡ (H, F, U, ω, Y, X, θ) with H finite. Assume that all households arelocally nonsatiated. Then (x∗, y∗) is Pareto optimal. 1644. (Representative Firm Theorem) Consider a competitive production economy with N ∈ N ∪{+∞} commodities and a countable set F of firms, each with a production possibilities set Y f ⊂R N. Let p ∈ R N+ be the price vector in this economy and denote the set of profit-maximizing net supplies of firm f ∈ F by ˆ Y f (p) ⊂Y f (so that for any ˆy f ∈ ˆ Y f (p), we have p . ˆy f ≥ p . yf for all yf ∈Y f ). Then there exists a representative firm with production possibilities set Y ⊂ R N and a set of profit-maximizing net supplies ˆ Y(p) such that for any p ∈ R N+, ˆy ∈ ˆ Y(p) if and only if ˆy =_f ∈Fˆyf for some ˆy f ∈ ˆ Y f (p) for each f ∈ F. 158-159。
上财罗大庆高宏作业1
Advanced Macroeconomics IInstructor:Luo DaqingAssignment1Note:You may use either English or Chinese to answer the questions.Question1(3points,1point for each part)This question seeks to clarify the concept of expectations,conditional and unconditional.(a)Consider a two-period model where consumption in period t is a random variable denoted C t,for t=0,1.For simplicity,assume that C t can take on two values,c t and c t, for t=0,1.Show that when C t is identically and independently distributed over timeE0(C1)=E[C1].(b)Consider a two-period model where consumption in period t is a random variable denoted C t,for t=0,1.For simplicity,assume that C t can take on two values,c t and c t,for t=0,1.Show thatE[E0(C1)]=E[C1].(c)Consider a three-period model where a consumer seeks to maximize her lifetime expected utilityE02t=0βt u(C t)=u(c0)+β[u(c1)·P r(C1=c1|c0)+u(c1)·P r(C1=c1|c0)] +β2[u(c2)·P r(C2=c2|c0)+u(c2)·P r(C2=c2|c0)]where C1and C2are random variables that can take on only two values,c t and c t,for t=1,2.How would you calculate the conditional probabilities P r(C2=c2|c0)and P r(C2=c2|c0)?Question2(3.5points)Consider an extension of the two-period endowment economy studied in class where agents live for four periods.A representative agent has lifetime expected utilityE1U=ln(c1)+0.96E1ln(c2)+0.962E1ln(c3)+0.963E1ln(c4)where c t denotes consumption in period t.Each agent receives an endowment y t in period t.denote the price of a one-period bond purchased in period t and (a)(0.5pts)Let q Sbtpaying one unit of consumption in period t+1.State(no deviations nor explanations required)an expression relating the price q Sto y1and y2.b1(b)(0.5pts)Let q Mdenote the price of a two-period bond purchased in period t and btpaying one unit of consumption in period t+2.State(no deviations nor explanationsto y1and y3.required)an expression relating the price q Mb1(c)(0.5pts)Let q Ldenote the price of a three-period bond purchased in period t and btpaying one unit of consumption in period t+3.State(no deviations nor explanations required)an expression relating the price q Lto y1and y4.b1(d)(1pts)Consider the following contract.In period1,buyer and seller agree on the following:(1)in period3,the buyer has to pay f units of consumption to the seller;(2)in period4,the seller has to pay1unit of consumption to the buyer.Explain what are the utility costs and benefits to the buyer of signing the above described contract. What is the equation relating the price f to endowments?and (e)(1pts)Using your answer from part(d),derive an expression relating f,q Lb1q Mand briefly explain why this equation has to hold in equilibrium.b1Question3(3.5points)Consider a three-period economy with a large number of identical consumers,firms and banks.Assume the number offirms coincides with the number of consumers and that eachfirm is entirely owned by a consumer with the following utility functionU=ln(c1)+0.95E1ln(c2)+0.952E1ln(c3).A representativefirm considers investing into a new machine.The machine would be bought at price P and installed in period1and would be productive in periods2and3 only.The payoffs generated by the machine in periods2and3are denoted py2and py3, respectively.Assume that consumption is contingent on the state of the economy.More specifically, consumption is equal to1.2when the economy is in a good state and to0.4when the economy is in a bad state.Assume that the economy is in a good state in period1(i.e. c1=1.2).The payoffs generated by the machine are also state contingent.The payoffis0.2in the good state and0.1in the bad state.Finally,the state of the economy evolves independently over time and both states have probability0.5.(a)(0.5pts)Suppose thefirm has the resources necessary to completely cover the purchasing price of the machine in period1.State the Euler equation linking the price of the machine,its payoffs and current and future consumption.Clearly explain your answer in terms of costs and benefits related to such an investment.(b)(0.5pts)Using your answer to part(a),show that the equilibrium price thefirm is willing to pay for the machine is0.4631.From now on,we assume that the market price of the machine is P=0.4631and that thefirm has to borrow to be able to acquire the machine.(c)(0.5pts)Suppose the bank requires thefirm to make a down payment D in period 1and payments of equal amount m in periods2and3.Write the Euler equation linking D,m,the machine’s payoffs and current and future consumption.Clearly explain your answer in terms of costs and benefits related to such an investment.(d)(0.5pts)Using your answer to part(c)and assuming D=0.23,solve for the equilibrium amount m thefirm is willing to pay in periods2and3.(e)(0.5pts)Suppose the bank charges a periodic real interest rate r=0.03and uses the following present-value formulaP=D+m1+r+m(1+r)2to determine m.Again,assuming D=0.23,solve for the amount m the bank will charge afirm borrowing to buy a machine.(f)(1pts)In light of your results,are the market for machines and loans in equilibrium? If not,discuss the adjustments that will take place in the economy to bring the markets to equilibrium.。
高宏数学预备知识总结
第一章 数学预备知识本章讲述若干数学预备知识，包括导数及其应用、静态优化、积分、微分方程、差分方程以及相位图分析等内容。
这些预备性的数学知识对于学习高级宏观经济学是必须的，但是在微观经济学、数理经济学、时间序列分析、高等数学等课程中有详细的讨论，在这里我们只是将与我们后面的学习有关的知识要点罗列在一起并在必要时做出一定的经济解释。
这里的数学知识只是与动态优化相关的部分，对于学习高级宏观经济学必须的其他数学知识并未涉及，特别是时间序列、概率论等知识。
第一节 导数及其应用一、导数有函数()f q π=，导数就是111()()limlim q q q f q q f q d dqq qππ∆→∞∆→∞+∆-∆==∆∆。
导数的经济含义是：边际量、q 变动一单位时π变动的大小、q 对π的变动速率。
二、常用求导公式（1）f b =为常数，0df dbdx dx ==； （2）b 为常数，(())d bf x dfb bf dx dx'==； （3）b 为常数，1bb dx bx dx-=； （4）1(ln )x x'=； （5）()ln x x a a a '=； （6）()x x e e '=； （7）()f g f g '''+=+；（8）()fg f gfg '''=+； （9）2()f f gfg g g ''-'=；（10）链式法则：(),()y f x x g z dy dy dxdz dx dz===【例题1-1】：求下面各题的导数。
（1）32 3y x y x '=⇒= （2）34 3y x y x --'=⇒=-（3）23 25621212(25)z y y x dz d z dy y y x dx dy dx ==+'=⋅=⨯==+（4）()()ax ax ax de de d ax e a dx d ax dx =⋅=⋅练习：求导数[]ln()d ax dx、[]ln ()d x t dt、2(ln )d x dx三、二阶导数二阶导数表示边际量的变化速率，可用如下方式表示：22(),,()d y d dy f x dx dx dx''四、微分22(),[]()y f x dy f dx d f dx d dy d y dx f dx dx f dxdx'=='''''==== 导数是微商。
Generation of higher order gauss-laguerre modes in single-pass 2nd harmonic generation
Preben Buchhave and Peter Tidemand-Lichtenberg
DTU Physics, Department of Physics, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark *Corresponding author: pbu@fysik.dtu.dk
©2008 Optical Society of America
OCIS codes: (190.2620) Harmonic generation and mixing; (190.4410) Nonlinear optics, parametric; (190.4420) Nonlinear optics, transverse effects.
difference frequency generation. However, nonlinear effects such as depletion and phase shift were not included, and the method only applies to very low power beams. In this paper an alternative formulation is used, where the beams are expanded in higher order Gauss-Hermite (G-H) or Gauss-Laguerre (G-L) modes after passage of each thin slab. The choice of mode expansion depends on the symmetry of the problem. Since both G-H and G-L modes form complete orthogonal sets of functions, an expansion of the beams in these modes fully describes the result of the interaction, and diffraction due to the distortion of the wave front is included by the mix of the higher order modes. Besides forming an alternative way of performing a wave propagation calculation, this method has the advantage that it allows to follow the generation of higher order modes both spatially and temporally through the crystal. Furthermore it is possible to calculate the power generated in each of these modes. It is also possible to input higher order pump modes or superposition of modes and calculate the power conversion efficiency into the generated modes, a method that has recently been used in generation of squeezed light in higher order G-H modes [6]. In the calculations the phase matching may be adjusted at will; it is for example possible to adjust the phase matching to compensate at least partially for the Gouy phase shift in a particular mode and thus selectively generate light in one mode by pumping in another mode [7]. In the following, single pass second harmonic generation (SHG) in an efficient nonlinear crystal pumped by a pump beam of known mode composition is considered. As an example we shall look at a single mode Q-switched Nd:YAG laser pulse frequency doubled through a PPKTP crystal and investigate the formation of higher order G-L modes as the fields propagate through the crystal, as well as illustrate the temporal development of the pulse shape as a function of time. 2. Wave propagation with G-L expansion The calculations are performed in the following way: 1. The crystal is divided into thin slabs normal to the optical axis. 2. The three interacting beams that make up the total field incident on a slab interact on their way through the slab. The slab is assumed so thin that the nonlinear differential equations governing the interaction (the coupled wave equations) can be approximated by difference equations, and diffraction effects on the way through the slab are unimportant (plane wave approximation within a slab). Losses can be included in each slab. The beams may be focused anywhere inside or outside the crystal. Collinear beams are assumed, although the simulations can easily be expanded to include non-collinear beams and walk-off. 3. After passage of a slab, the field incident on the slab has been distorted due to the nonlinear interaction. The fields leaving the slab are expanded in G-L modes. 4. The beam waist radius and phase of the individual modes of the expansion are recalculated for the position of the next slab. The different phase shifts of the different order G-L modes implicitly take care of diffraction. 5. The G-L modes with the new phases and beam waists are added for each beam, and these new beams are used as input for the next slab; the calculation is repeated from 2. 6. The output beams are found by summing the G-L modes leaving the last slab. 2.1. Nonlinear interaction The coupled wave equations (CWE) for the 2nd order nonlinear interaction between three beams, described by the normalized electric fields a j ( x, y, z ) , where j identifies the field, are in the difference equation form given by [8]
高宏第3次习题课
不变相对风险回避系数效用函数的替代弹性。
考虑一个人，他只存活两期，且其效用函数由（2.46）给出。
令12,P P 代表消费品在这两期的价格，W 代表他一生收入的价值；因此他的预算约束为1122PC P C W +=。
a) 若12,P P 和W 给定，则使他效用最大化的1C 和2C 是多少？b) 两期消费之间的替代弹性为()()1212ln //ln /C C P P -∂∂。
证明：若效用函数由（2.46）式给出，则1C 和2C 之间的替代弹性为1/θ。
答：a) 该效用最大化问题为：st 1122PC P C W += 容易求得其FOC 条件为：()()1/1/12121/C P P C θθρ=+将其代入到约束条件中得到：()()()221/1/21/11/W P C P Pθθθρ-=⎡⎤++⎣⎦()()()()()()1/1/21211/1/211//11/P P W P C P P θθθθθρρ-+=⎡⎤++⎣⎦b) 证明：根据FOC 条件()()1/1/12121/C P P C θθρ=+()()()()()1221ln /1/ln 11/ln /C C P P θρθ⇒=++所以消费的替代弹性为()()1212ln /1ln /C C P P θ∂-=∂。
生产率增长放慢和储蓄。
考虑一处于平衡增长路径上的拉姆齐-卡斯-库普曼斯经济，并假定g 有一个永久性的下降。
a) 这如何影响0k= 曲线？ b) 这如何影响0c= 曲线？ c) 在这种变化发生之时，c 如何变动？d) 用一个式子表示g 的一个边际变化对平衡增长路径上的储蓄率的影响。
可否判定这一表达式为正还是为负？e) 若生产函数为柯布-道格拉斯生产函数()f k k α=。
请用,,,,n g ρθα重写上一问的答案。
答：a) 关于资本的欧拉方程为：()()()()()()k t f k t c t n g k t =--+ 。
所以g 的永久性下降会导致0k= 曲线向上移动。
高等代数课件 5.2 标准形
为标准型, 转化为 求可逆阵C , 使得 C T AC 为对角阵.
d1 d2 T C AC = B = 为对角阵, 化二次型 d n
注 此时
2
二次型的标准形
一、用配方法化二次型成标准形
例 化二次型 2 2 f ( x1 , x2 , x3 ) = x12 + 2 x2 + 5 x3 + 2 x1 x2 + 2 x1 x3 + 6 x2 x3 为标准形, 并求所用的变换矩阵. 含x1的平方项 含有x1的项配方 解 去掉配方后多出的项
8
二次型的标准形
注 (1)正交变换法化的标准型系数是A 的特征值, 而 配方法则与它无关. (2)使用不同的方法, 所得到的标准形可能不相同 (标准型不唯一). (3)标准形中含有的项数必定相同, 项数等于所给 二次型的秩. 且其中所含的正项的个数(负项的个数) 是固定的, 称为二次型的正(负)惯性指数.
9
二次型的标准形
注 对对称阵A, 求可逆阵 C 使 CT AC = Λ 为对角阵. 设 C = P1 P2 Ps 为初等矩阵之积.而C T = PsT P2T P1T ,
T (i , j ) = E (i , j ) E C AC = P P P AP1 P2 Ps = Λ . 1 0 0 1 0 0 注 三种初等矩阵: 0 0 1 A 0 0 1 1 0 1 0 0 1 0 ← 第 i 行 0 1 E (i , j ) = ← 第 j 行 1 0
= ( x1 + 2 x2 )2 − 3( x2 + x3 )2 y1 = x1 + 2 x2 若令 y2 = x2 + x3 , y =x 3 3
北大经济学院高宏张延课件 (4)
有大的影响，如图3.12中的(a)所示。另外，资本的收
入份额不大意味着：f(k*)对k* 不是很敏感。最终结果
是，储蓄率的变化对产量的影响不是很大。
2013-10-8 高宏（4）
《高宏》讲义，张延著。版权所有
16
• y* = (k*)a ，ln y* = aln k*
• 均衡时，存在：
• k˙ = s f(k) - (n+g)k = 0， • s f(k*) = (n+g)k* ， sy* = (n+g)k* • 两边取ln ，得到：
《高宏》讲义，张延著。版权所有
23
•
如果a → 1 ──→ s f(k*) 越接近于线性 ──→ s
的变化对y* 的影响越大。
•
而事实上，实证分析的结果是：a =1／3 ，如
何扩展a ？
•
扩展资本的内涵：
•
资本包括：实物资本K 和人力资本H，从L 的
份额中剥离出一部分，作为人力资本，扩充进资本的
份额，进而扩充a 。
2013-10-8 高宏（4）
《高宏》讲义，张延著。版权所有
2
• 第2章 研究资本和知识作用的增长理论
• 上篇 研究资本作用的增长理论
•
2.1 人力资本模型
•
3.6节的讨论指出，以知识积累为基础的理论
难以解释收入的国家间差别。因此，本章的这一部
分，讨论新增长理论的另一条思路：强调人力资本
积累的模型。
雷曾(1990年) Azariadis, Costas, and Drazen, Allan.
(1990), “Threshold Externalities in Economic
Development.” Q. J. E. 105(May): 501-526.；
Carl Walsh高宏经济学教学大纲
Economics 205C Spring 2010 UCSC C. WalshAdvanced Macroeconomic TheoryEconomics 205CThis course is the third of the three-quarter sequence in macroeconomics for students in the Ph.D. Program in International Economics. The main emphasis will be on new Keynesian models for closed and open economies, optimal monetary policy, and financial frictions.The basic text for the course will be C. E. Walsh, Monetary Theory and Policy (3nd ed., MIT Press, 2010). Required readings are denoted by *. Other readings not listed as required are useful references if you are interested in pursuing a particular topic It is important that you do the required readings before coming to lecture!Your evaluation in this course will be based on an in-class midterm (tentatively scheduled for Monday, May 3rd), a referee report, a three hour final exam, and your participation in class discussions. The final exam is scheduled for Wednesday, June 9 7:30-10:30pm. Exercises and many of the readings will be posted at /~walshc/205Csp10/.My office hours will be Tuesday 9-10am and Wednesday 3:30-4:30 (or by appointment) in 467 E2. My office phone is 9-4082 and my email address is walshc@.I. Nominal frictions, time dependent and state dependent models of price adjustment*Walsh, C. E., Monetary Theory and Policy, 3rd ed. 2009, chapter 6.Nakamura, E. and J. Steinsson, “Five Facts about Prices: An Evaluation of Menu Cost Models,” Quarterly Journal of Economics, 123(4), Nov. 2008, 1415-1464.Klenow, Peter J. and Oleksiy Kryvtsov, “State-Dependent or Time-Dependent Pricing: Does it Matter for Recent U.S. Inflation?” Quarterly Journal of Economics, 123(3), Aug. 2008, 863-904.II. The basic new Keynesian frameworkII. 1 *Walsh, C. E., Monetary Theory and Policy, 3rd ed. 2009, chapter 8, sections 1-3.Woodford, M., Interest and Prices, Princeton University Press, 2003, Chapters 3 and 4.II.2 Linear quadratic toolsLjungqvist and Sargent, Sections 5.1-5.5.Svensson, L. E. O., “Optimization under Commitment and Discretion, the Recursive Saddlepoint Method, and Targeting Rules and Instrument Rules: Lecture Notes,” January 2010, http://people.su.se/~leosven/papers/CommDiscTRIR.pdfII.3 Policy issues*Walsh, C. E., Monetary Theory and Policy, 3rd ed. 2009, chapter 8, sections 4-5.Clarida, R., J. Galí, and M. Gertler, “The Science of Monetary Policy: A New Keynesian Perspective,” Journal of Economic Perspectives, 37(4), 1999, 1661-1707.Chari, V. V., P. J. Kehoe and E. R. McGrattan. 2009. “New Keynesian Models: Not Yet Useful for Policy Analysis,” American Economic Journal Macroeconomics, 1(1): 242-266.II.4 Fiscal policy*Walsh, Monetary Theory and Policy, 3rd ed., 4.1-4.5, 11.1-11.4.*Gali, J., J. Valles, and D. Lopez-Salido, Understanding the effects of government spending on consumption, Journal of the European economic Association March 2007, 5(1): 227-270.Woodford, M., Simple Analytics of the Government Expenditure Multiplier, NBER Working paper 15714, Jan. 2010.III. The open economy*Walsh, Ch. 9, sections 9.4-9.5.*Galí, J. and T. Monacelli, Monetary Policy and Exchange Rate Volatility in a Small Open Economy, Review of Economic Studies, 707-734, 2005.Galí, J. and T. Monacelli, Optimal Monetary and Fiscal Policy in a Currency Union, Journal of International Economics, December 76 (2008), 116-132.IV. Financial factorsIV.1. Interest rates*Walsh, Ch. 10, sections 10.1-10.4Eggertsson, G. B. and M. Woodford. 2003. “The Zero Bound on Interest Rates and Optimal Monetary Policy,” Brookings Papers on Economic Activity, 1, 139-211.Rudebusch, G. D., and T. Wu, "A Macro-Finance Model of the Term Structure, Monetary Policy, and the Economy,” Economic Journal 118, July 2008, 906-926.Ang, A. and M. Piazzesi, “A No-Arbitrage Vector Autoregression of Term Structure Dynamics with Macroeconomic and Latent Variables,” Journal of Monetary Economics, 50,2003, 745-787.Piazzi, M.,IV.2. Market segmentation*Walsh, Ch. 5, section 5.3.Alvarez, F., A. Atkeson, and P. J., Kehoe, “Money, Interest Rates, and Exchange Rates with Endogenously Segmented Markets,” Journal of Political Economy 2002, vol. 110 (1), 73-112.Carlstrom, C. T. and T. S. Fuerst, "Interest Rate Rules vs. Money Growth Rules: A Welfare Comparion in a Cash-in-Advance Economy," Journal of Monetary Economics, 36(2), Nov. 1995, 247-267.IV.3. Imperfect and asymmetric information*Walsh, Ch. 10, sections 10.5—10.6*Akerlof, George A., "The market for `lemons': Quality uncertainty and the market mechanism," Quarterly Journal of Economics, 84(3), Aug. 1970, 488-500.*Gertler, M. and N. Kiyotaki, “Financial Intermediation and Credit Policy in Business Cycle Analysis,” October 2009Diamond, Douglas W., "Banks and liquidity creation: A simple exposition of the Diamond-Dybvig model," Federal Reserve Bank of Richmond Economic Quarterly, 93(2), Spring 2007.Bernanke, B. S. and M. Gertler, "Agency Costs, Net Worth, and Business Fluctuations,"American Economic Review, 79(1), Mar. 1989, 14-31.Carlstrom, C. T., and T. S. Fuerst, "Agency Costs, Net Worth, and Business Fluctuations: A Computable General Equilibrium Analysis," American Economic Review, 87(5), Dec. 1997, 893-910.IV.4. And optimal policyC. T. Carlstrom, T. S. Fuerst and M. Paustian, "Optimal Monetary Policy in a Model with Agency Costs," Aug. 2009.Cúrdia, V. and M. Woodford. 2008. "Credit Frictions and Optimal Monetary Policy," Discussion Papers 0809-02, Columbia University.De Fiore, F. and O. Tristani. 2009. "Optimal Monetary Policy in a Model of the Credit Channel." March.Faia, E. and T. Monacelli. 2007. "Optimal Interest rate Rules, Asset Prices, and Credit Frictions," Journal of Economic Dynamics and Control. 31: 3228-3254.Iacoviello, M. 2005."Housing Prices, Borrowing Constraints, and Monetary Policy in the Business Cycle," American Economic Review 95(3): 739-764.V. Real FrictionsV.1. Informational frictions*Walsh, Ch. 5, section 5.2.Mankiw, N. G. and R. Reis, “Sticky Information versus Sticky Prices: A Proposal to Replace the New Keynesian Phillips Curve,” Quarterly Journal of Economics, 2002, 1295-1328.Mankiw, N. G. and R. Reis,”Imperfect Information and Aggregate Supply," NBER Working Papers 15773, 2010.Mankiw, N. G. and R. Reis,”Sticky Information in General Equilibrium," Journal of the European Economic Association, MIT Press, 2007, vol. 5(2-3), pages 603-613, 04-05.V.2. Variable capital utilization, habit persistence in empirical DSGE modelsChristiano, L. J., M. Eichenbaum, and C. Evans, “Nominal Rigidities and the Dynamic Effects of a Shock to Monetary Policy,” Journal of Political Economy 113(1), Feb. 2005, 1-45.Smets, F. and R. Wouters, “An Estimated Dynamic Stochastic General Equilibrium Model of the Euro Area,” Journal of the European Economic Association, 1(5), 2003, 1123-1175.Smets, F. and R. Wouter, “Shocks and Frictions in US Business Cycles: A Bayesian Approach,” American Economic Review, 97(3), June 2007, 586-606.Adolfsen, M., S. Laséen, J. Lindé, and M. Villani, “Bayesian Estimation of an Open Economy DSGE Model with Incomplete Pass-Through,” Journal of International Economics,72(2), July 2007, 481-511.V.3. UnemploymentSearchMortensen, D. T. and C. A. Pissarides, “Job creation and job destruction in the theory of unemployment,” Review of Economic Studies, 61 (3), July 1994, 397-416.Shimer, Robert, “The Cyclical Behavior of Equilibrium Unemployment and Vacancies,” American Economic Review, 2005, 95(1), 25-49.Walsh, C. E., “Labor market search, sticky prices, and interest rate policies,” Review of Economic Dynamics, 8(4), Oct. 2005, 829-849.Ravenna, F. and C. Walsh, “Welfare-based optimal monetary policy with unemployment and sticky prices: A linear-quadratic framework,” March 2010.。
高级宏观经济学课件(厦门大学,龚敏)
Lecture 1 The Solow Growth Model Romer(2001), Advanced Macroeconomics, Chapter 1 Blanchard, Olivier J., 2nd edition, Macroeconomics, Prentice Hall曼昆，《宏观经济学》，人民大学，2000年巴罗，萨拉伊马丁，《经济增长》，中国社会科学出版社，2000年Some Basic Facts about Economic Growth Assumptions of One-Sector Growth Models The Dynamics of the ModelThe Impact of a Change in the Saving Rate The Speed of ConvergenceThe Solow Model and the Central Questions of Growth TheoryEmpirical Applications一、Some Basic Facts about EconomicGrowth: the stylized facts of growth 1. Economic growth through deep time(F2)2. The stylized facts✓ 0,>>y L Y （F3）✓ o k L K >> ,(F4)✓ tcons Y K Y K tan ,==(F5)✓ tcons YP YWt cons YP t cons KP tan 1,tan ,tan =-====ρ3. international differences in the standard of living (F6)4. The Solow growth modelThe ultimate objective of research on economic growth is to determine whether there are possibilities for raising overall growth or bringing standards of living in poor countries closer to those in the worldleaders.The Solow model is the starting point for almost all analyses of growth.The principal conclusion of the Solow model is that the accumulation ofphysical capital cannot account for either the vast growth over time in output per person or the cast geographic differences in output per person. The model treats other potential sources of differences in real incomes as either exogenous and thus not explained by the model.( in the case of technological progress) or absent altogether(in the case of positive externalities from capital).二、 Basic assumptions of one-sectorgrowth models1． Input and Output),(t t t t L A K F Y1) Labor-augmenting or Harrod-neutral),(t t t t L K A F Y = capitalaugmenting),(t t t t L K F A Y = Hicks-ueutral2) The ratio of capital to output, K/Y, eventually settles down 3) Constant returns to scale (first-degree homogeneity)✓ The economy is big enough that the gains from specialization have been exhausted.✓ Inputs other than capital, labor, and knowledge are relatively unimportant. 4) intensive form :)()1,(t t k f ALK F ALY y ===✓ The amount of output per unit of effective labor depends only on the quantity of capital per unit of effective labor, and not on the overall size of theeconomy.✓ output per worker)(k Af ALY AL Y ==5) Assumptions 0)0(=f 0,0<''>'f f)(lim )(lim 0='∞='∞→→k f k f k k ( Inada conditions)✓ The path of the economy does not diverge✓ Cobb-Douglas functionαα==kALK k f )()(2． the evolution of the input into production1) Labor and knowledge grow at constant rates:nt L L nL L eL L t tt ntt +=⇔=⇔=00ln )ln(gtL A gA A eA A t tt gtt +=⇔=⇔=00ln )ln(2) Investment and capital formationtt t t t sY S I dtdK dtdK I S G I T S =====+=+t t tK sY Kδ-=三、 The simple dynamics of growth model1． The Harrod-Domar model1) The fixed-coefficients productionfunction⎥⎦⎤⎢⎣⎡αυ=t t t L K Y ,min2) Saving, investment, and the warranted rate of growthυ==s I Y3) Labor force growth:g n Y +=4) The Harrod-Domar condition:υ=+s g n2． The Solow model1) Cobb-Douglas function(F7)kk y ALK AL YK Y k k MPk kMPkkALY y AL K Y )(1)1()(0)(211===υ<-αα=∂∂>α====-α-ααα-α2) The dynamics of kg yLY k f LeK f LeY EY y E K F Y eL E gtgtt t t tg n t +=======+ )()()(),()(0O ne way: actual and break-even investmentk g n k sf k dtdk tt)()(δ++-==0)()(>⇒δ++>kk g n k sf t , k is rising0)()(>⇒δ++<k k g n k sf t , k is falling 0)()(>⇒δ++=k k g n k sf t, k is constant*)(*)(k g n k sf δ++=T he second way: speedk g n k sf k dtdk tt)()(δ++-==)()()ln(δ++-=g n kk sf dt k d t t)(**)(δ++=g n k k sf)()ln(1δ++-=-αg n skdtk d tt)]([)ln()()ln(1δ++-⨯α=α==-ααg n skdtk d dtk dlin dty d tt t t3) the balanced growth path*************)(*k A L K y A L Y y k Y K k L A K Y L A Y k f y t tt t t t t t t t t t t t ======**)(k sgn k f ++δ=nLL g n II CC KK Y Y =>+==== 0)()()()(====ALIAL C AL K AL Yg LIL C L K L Y ====)()()()()]()([)()()),((t t t t t tt t t t t t t k f k k f A k f L K k f A dL L A K F d w '-='-==)()),((t t t t t k f dKL A K F d r '==t t t t t Y K r L w =+tt tt t t L K L k f k k f A K k f L w K r S S ⋅'-⋅'==*)](**)([**)(*)](**)([**)(k f k k f k k f '-⋅'=(constant)The Solow model implies that, regardless of its starting point, the economy converges to a balanced growth path --- a situation where each variable of the model is growing at a constant rate. On the balanced growth path, the growth rate of output per worker is determined solely by the rate of technological progress.四、 The impact of a change in the savingrateThe division of the government ’spurchanses between consumption and investment goods, the division of its revenues between taxed and borrowing, and its tax treatments of saving andinvestment are all likely to affect the fraction of output that is invested.1. the impact on output(F24,F25)A change in the saving rate has a level effect but not a growth effec t: it changes the economy’s balanced growth path, and thus the level of output per worker at any point in time, but it does not affect the growth rate of output per worker in the balanced growth path.In the Solow model only changes in the rate of technological progress have growth effects; all other changes have only level effects .2. saving and consumption (welfare) )()()()1(t t t t k sf k f k f s c -=-=*)(*)(*)(*)(*)()1(*);(**k g n k f k sf k f k f s c s k k δ++-=-=-== sg n s k g n g n s k f s c ∂δ∂δ++-δ'=∂∂),,,(*)](),,,(*([*if )(),,,(*(δ++>δ'g n g n s k f , 0*>∂∂s c if )(),,,(*(δ++<δ'g n g n s k f , 0*<∂∂s c if )(),,,(*(δ++=δ'g n g n s k f ,0*=∂∂s c the golden-rule level of the capital stock: Consumption is at its maximum possible level among balanced growth paths . δ++=δ'=g n g n s k f MPK )),,,(*(Quantitative Implications1． The effect of s on output in the long run s g n s k k f s y ∂δ∂'=∂∂),,,(**)(*),,,(*)()),,,(*(δδ++=δg n s k g n g n s k sf0*)()(*)(*>'-δ++=∂∂k f s g n k f s k ?*)()(*)(*)(*k f s g n k f k f sy '-δ++'=∂∂*)](/*)(*[1*)(/*)(***k f k f k k f k f k s y y s'-'=∂∂⋅ *)(/*)(*k f k f k 'is the elasticity ofoutput with respect to capital at *k k =. Denoting this by *)(k K α*)(1*)(**k k s y y sK K α-α=∂∂⋅If markets are competitive and there are no externalities, *)(/*)(*k f k f k ' is the share of total income that goes to capital on the balanced growth path. 31*)(≈αk K 21**=∂∂⋅⇒s y y sA 10% increase in the saving rate (from20% of output to 22%) raises output per worker in the long run by about 5% relative to the path it would have followed.Significant changes in saving have only moderate effects on the level of output on the balanced growth path.2． The speed of convergence1) One way)()(δ++-==γg n k k sf k k k k k f )(资本的平均产品。
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U
t 0
e
t
Lt u C t dt H
The lifetime utility function
U
t 0
e
t
Lt u C t dt H
暨 南 大 学 经 济 学 院 周 泳 宏
C(t): consumption of each member of the household at time t L(t): total population of the economy L(t)/H:the number of members of the household u(·): each member’s utility at a given date uL/H: household’s total utility
暨 南 大 学 经 济 学 院 周 泳 宏
There is no uncertainty in this model, risk aversion is irrelevant θ determines the househould’s willingness to shift consumption between different periods
Chapter 2 Infinite-Horizon and Overlapping-Generations Model
周泳宏 tzhouyhjnu@
Introduction
This chapter investigates two models
Ramsey-Cass-Koopmans Model Diamond Model
暨 南 大 学 经 济 学 院 周 泳 宏
Each household supplies one unit of labor at each point in time. Capital does not depreciate
K 0
The initial stock of capital
K 0 H
☆ prove: Discount factor
U
暨 南 大 学 经 济 学 院 周 泳 宏
t 0
e
t
Lt u C t dt H
ρ:discount rate The e t acts to discount the future The greater is the discount rate, the less the household values future consumption relative to current consumption.
If this condition does not hold, the household can attain infinite lifetime utility, and its maximization problem does not have a well-defined solution
模型假设
完全竞争市场结构 长生不老的不断扩展的家庭 家庭和个人完全同质 忽略资本的折旧
2.1 Assumptions
1）Firms
暨 南 大 学 经 济 学 院 周 泳 宏
The economy is viewed as consisting of a large number of identical firms. With perfectly competitive markets, factor returns are equal to the marginal products Firms take A as given
each household initially owns and rents to firms
暨 南 大 学 经 济 学 院 周 泳 宏
The household chooses the consumption/saving plan that maximises their discounted ‘lifetime’ utility. The lifetime utility function for the household is
instantaneous utility function
C t u C t ， n (1 ) g 0 1
1
暨 南 大 学 经 济 学 院 周 泳 宏
This function form is needed for the economy to convergence to BGP. The utility function is known as constant relative risk aversion (CRRA) utility.
暨 南 大 学 经 济 学 院 周 泳 宏
The production side (firm) of the economy is identical to that used in the Solow model. The household side consists of a ‘representative agent’ who owns the capital, supplies the labor used by firms, and consumes or saves the resulting output.
Given the parameters,the representative agent chooses the consumption/saving plan that maximises their (expected) discounted lifetime utility.
暨 南 大 学 经 济 学 院 周 泳 宏
暨 南 大 学 经 济 学 院 周 泳 宏
Both the models continue to treat the growth rates of labor and knowledge as exogenous But derive the evolution of the capital stock from the interaction of maximizing households and firms in competitive markets
2.2 The Behavior of Households and Firms
1）Firms
Markets are competitive, with no depreciation, the real interest rate (real rate of return) on capital is
The smaller isθ, the more slowly marginal utility falls as consumption rises, and so the more willing the household is to allow its consumption to vary over time to take advantage of differences between its discount rate and the rate of return it gets on its saving
暨 南 大 学 经 济 学 院 周 泳 宏
Ramsey: there is a fixed number of infinitely lived households Diamond:there is turnover in the population
New individuals are continually being born, and old individuals are continually dying
A grows exogenously at rate g
Y t F K t , At Lt
2）Household
暨 南 大 学 经 济 学 院 周 泳 宏
There is a large and fixed number of infinitely lived identical households. The size of each household grows at rate n The number of households, H, remains fixed.
Part A
Ramsey-Cass-Koopmans model
Introduction
The Ramsey model is a general equilibrium model in which households and firms interact in competitive markets.
The Ramsey-Cass-Koopmans model The Overlapping Generations or Diamond model
The central difference between the Diamond model and the Ramsey-CassKoopmans model is that
暨 南 大 学 经 济 学 院 周 泳 宏
Firms pay rent and wages to the households who decide how much to spend on consumption (revenue for the firms) and how much to save (invest in new capital). The heart of the model is the consumption/saving decision and this is based on micro-optimising behaviour.