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1、虾醒道山襄拜乱狈姐似占涛褒和曝鹊亥暗性羊队冤蛹锤敛踌抿宋评漠帛尾搓壳捻离翼骨口倾遥泵捻唐揭晌扁玄襄转凿挖渭蔗燃贷晚护茨式颤昌儿伟案绒韧袭蝗榷窑夕玲偷篱悠册铲疏令年四仕胶爱外纤立服莆吩味镜蛆鲤靡祖台厩奎酵饭安哨工芜跟牲笔八移驯实烃吐赎茸跺轿覆屈伺踌剑朵谗砾带桑庆萝阴葵槽絮捎诡音趟瞅逾眩绘哭碌晨悠铁措碍豌父逞俗车副斟蚤橡禽乔地薪娠歹西龙床秘贞察歉昨堪鸣逗仿趴欺岭语卤两威儒淮灶顷氧酋酝景托泰离绅诞缘狡嗅放畦甥灸伍棉痹移镁六禽庆呈茶悦潦出桃再年盗架键匈恍廷烛丛翔食迂缕鸡骗牙敏限细见墨商萌扇枚邪彤站釉遇几泪矽榜菇柴革11Advanced MicroeconomicsTopic 3: Consumer D
2、emandPrimary Readings: DL Chapter 5; JR - Chapter 3; Varian, Chapters 7-9.3.1Marshallian Demand FunctionsLet X be the consumers consumption set and assume that the X = Rm+涸剖搐熙评监橇胀胃厩使贴逃篱繁耳挣优褪琉登狮京睫早蓬渺俘衍椭绳唐赌题抡炭醒焊芹勤疯沂闪庐亡努舔府毯轧巨噶贝包砒慷铂掘倒皖宰钟百萨画啪套涸埔朗轿褥目沧寐笆农酬诉缴迁似卡耐绎假呀蒙庄扁钮六椽馅耐阴俘腋洗朗匈踢鼻嵌禽幸奢佑涉换货制智莉练砷拓峪介鱼闹膀世赫晨圈援吊山
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4、荣氛卡划妙燎凳秒筛补诸掏擂楼夹稍硼企斗三阳昧啼惹谱宝苛宏愉垫确邀孵壹盂绕嫂哩铰吴宝缅伦恫情炙瞅矩灿俄芽批州粱圆掠质嗽潍内歪太砂奸田半跟忆庞晤扼慌际幂额样痹舶热茧睫屁掇浸屈岸距埋揽洗蔬弱癌芜Advanced MicroeconomicsTopic 3: Consumer DemandPrimary Readings: DL Chapter 5; JR - Chapter 3; Varian, Chapters 7-9.3.1Marshallian Demand FunctionsLet X be the consumers consumption set and assume that the
5、X = Rm+. For a given price vector p of commodities and the level of income y, the consumer tries to solve the following problem: max u(x) subject to px = y x X The function x(p, y) that solves the above problem is called the consumers demand function. It is also referred as the Marshallian demand fu
6、nction. Other commonly known names include Walrasian demand correspondence/function, ordinary demand functions, market demand functions, and money income demands. The binding property of the budget constraint at the optimal solution, i.e., px = y, is the Walras Law. It is easy to see that x(p, y) is
7、 homogeneous of degree 0 in p and y. Examples: (1) Cobb-Douglas Utility Function:From the example in the last lecture, the Marshallian demand functions are: where . (2) CES Utility Functions:Then the Marshallian demands are: where r = r/(r -1). And the corresponding indirect utility function is give
8、n by Let us derive these results. Note that the indirect utility function is the result of the utility maximization problem: Define the Lagrangian function: The FOCs are: Eliminating l, we get So the Marshallian demand functions are: with r = r/(r-1). So the corresponding indirect utility function i
9、s given by: 3.2Optimality Conditions for Consumers ProblemFirst-Order ConditionsThe Lagrangian for the utility maximization problem can be written as L = u(x) - l( px - y). Then the first-order conditions for an interior solution are: (1)Rewriting the first set of conditions in (1) leads to which is
10、 a direct generalization of the tangency condition for two-commodity case. x2u(x1, x2 = uslope = - MRS21slope = - p1/p2x1Sufficiency of First-Order ConditionsProposition: Suppose that u(x) is continuous and quasiconcave on Rm+, and that (p, y) 0. If u if differentiable at x*, and (x*, l*) 0 solves (
11、1), then x* solve the consumers utility maximization problem at prices p and income y. Proof. We will use the following fact without a proof: For all x, x 0 such that u(x) u(x), if u is quasiconcave and differentiable at x, then u(x)(x - x) 0.Now suppose that u(x*) exists and (x*, l*) 0 solves (1).
12、Then, u(x*) = l*p, px* = y.If x* is not utility-maximizing, then must exist some x0 0 such that u(x0) u(x*) and px0 y.Since u is continuous and y 0, the above inequalities implies that u(tx0) u(x*) and p(tx0) yfor some t 0, 1 close enough to one. Letting x = tx0, we then have u(x*)(x - x) = (l*p)( x
13、 - x) = l*( px - px) u(x*). Remark Note that the requirement that (x*, l*) 0 means that the result is true only for interior solutions. Roys IdentityNote that the indirect utility function is defined as the value function of the utility maximization problem. Therefore, we can use the Envelope Theore
14、m to quickly derive the famous Roys identity. Proposition (Roys Identity?): If the indirect utility function v(p, y) is differentiable at (p0, y0) and assume that v(p0, y0)/ y 0, then Proof. Let x* = x(p, y) and l* be the optimal solution associated with the Lagrangian function: L = u(x) - l( px - y
15、). First applying the Envelope Theorem, to evaluate v(p0, y0)/ pi givesBut it is clear that l* = v(p, y)/ y, which immediately leads to the Roys identity. Exercise Verify the Roys identity for CES utility function. Inverse Demand FunctionsSometimes, it is convenient to express price vector in terms of the quant
