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1、Chapter 2Consumer Behavior and Demand Theory(3)1學院Chapter 2 includes:n 2.1 Preference and Utilityn 2.2 Utilityization and Choicen 2.3 Comparative Static Analysis in Optimal Choicen 2.4 Income and Substitution Effectsn 2.5 Market Demand and Elasticity2學院Overview of Last Classn Budget Constraint(預算約束)

2、n Application of Budget Constraintn Utilityization (ConsumersOptimal Choice)n Types of Optimal Solution3學院2.3 Comparative Analysis in Optimal Choicen 2.3.1 From Utility Functionization Mto Demandn 2.3.2 From P.C.C to Demand Curven 2.3.3 Judging ed from P.C.Cn 2.3.4 From I.C.C to Engle Curven 2.3.5 H

3、omothetic(同位性) andEngel Curvesn 2.3.6 Judging Types of Goods from I.C.Cn 2.3.7 Judging eI from Engle Curve4學院A Survey(概述)This section studies how people change their choices when conditions such as income or changes in the prices of goods affect the amount that people choose to consume.This section

4、then compares the new choices with those that were made before conditions changedThe main result of this approach is toconstruct an individuals demand curve5學院2.3 Comparative Analysis in Optimal Choicen 2.3.1 From Utility Functionization Mto Demandn 2.3.2 From P.C.C to Demand Curven 2.3.3 Judging ed

5、 from P.C.Cn 2.3.4 From I.C.C to Engle Curven 2.3.5 Homothetic(同位性) andEngel Curvesn 2.3.6 Judging Types of Goods from I.C.Cn 2.3.7 Judging eI from Engle Curve6學院2.3.1 From Utility Functionization Mto Demandn Defining Individual Demand Functionn An individual demand Function shows the relationship b

6、etween the price of a good and the quantity of that good purchased by an individual assuming that all other determinants of demand are held constant.= dqd(P , P , I ; preferences)xXYx7學院2.3.1From Utility Functionization Mto Demandn Max : U=U(X1,X2)n S.t. P1X1+P2X2=I8學院2.3.1 From Utility Demand Funct

7、ionization MtoL( x1, x2 , l ) = u( x1 , x2 ) - l p1x1Take first order necessary conditions for+ p2 x2um- I ¶u(x , x)- lp = 0121¶x¶u(x ,1x ) 12- lp2= 0¶x2p1x1 + p2 x2 - I = 0學院92.3.1 From Utility Functionization Mto DemandFrom 3 first-order conditions, we can get three equations,

8、and then solve them to get individual demand functions for two goods.x1*=x1(p1,p2,I) x2*=x2(p1,p2,I)xi*= xi(p1,p2 I) ( i=1,2) is called ordinary demand function(普通需求函數), is also called Marshallian Demand Function（ 爾需求函數）10學院Cobb-Douglas Example： Ordinary Demandsn Given p1, p2 and I, how to calculate

9、 optimal ordinary demands (x1*,x2*)?n Assume Cobb-Douglas preferences:bU (x ,12= ¶U= axa-1 xbMU1¶ x121= ¶U= bxa xb-1MU2¶ x12211學院Cobb-Douglas Example： Ordinary Demandsn So the MRS is¶U /¶ xaxa-1 xbdxaxMRS = 2 = -1 = -12 = -2 .bx¶U /¶ xbxa xb-1dx12121n At (x1*,

10、x2*), MRS = -p1/p2*soaxpbp-= -Þx2 =* 2 1 1 x1(1)bx *pap122n Also, at (x1*,x2*), the budget is exhausted, sop x* + p x*= I(2)學院112212Cobb-Douglas Example： Ordinary Demandsn Substitute for x2* from (1) into (2) to getbp1p x* + p= Ix*1121ap2n Solve for x1* to getn Substitute x1* into (2) and solve

11、 for x2* to get13學院x* =bI2(a + b) p2x* =aI1(a + b) p1Cobb-Douglas Example： Ordinary Demandsx2b2U (x ,1bI=x*2(a + b) p2x1=aIx*1(a +14學院Cobb-Douglas Example： Ordinary Demandsöa+bøæèaa)a+b1-a2U (x , x ) =a+b112p x*aIa= ap1 x1 =Þ=* 1 1 I (a + b)(a + b)bIx*pba= 1- a.p2 x2 =*Þ

12、;= 1- 22 I (a + b)(a + b)(a + b)n Note that for Cobb-Douglas utility functionn 1.Demands are linear in incomen 2.Expenditure shares are constantn 3.Expenditure shares sum to one15學院2.3 Comparative Analysis in Optimal Choicen 2.3.1 From Utility Functionization Mto Demandn 2.3.2 From P.C.C to Demand C

13、urven 2.3.3 Judging ed from P.C.Cn 2.3.4 From I.C.C to Engle Curven 2.3.5 Homothetic(同位性) andEngel Curvesn 2.3.6 Judging Types of Goods from I.C.Cn 2.3.7 Judging eI from Engle Curve16學院2.3.2 From P.C.C to Demand Curven By the way of comparative static analysis, if we let the price of a good change h

14、olding other factors constant, we get ordinary demand curven Example: the price of a good increasen Suppose p1 increases, from p1 to p1, and to p1n Hold p2 and I constant學院172.3.2 From P.C.C to Demand Curvethe price of a good Changesx2p1x1+ p2x2= Ip1=p1x118學院2.3.2 From P.C.C to Demand Curvethe price

15、 of a good Increasex2p1x1+ p2x2= Ip1= p1p1= p1x119學院2.3.2 From P.C.C to Demand Curvep1x1+ p2x2= Ix2p1= p1p1= p1p1= p1x120學院2.3.2 From P.C.C to Demand Curvex2p1= p1x1x1*(p1)21學院From P.C.C to Demand Curvep1x2p= p 11p1x1*x1*(p1)xx1*(p1)22學院2.3.2 From P.C.C to Demand Curvep1x2p1= p1p1x1*x1*(p1)x1*(p1)x1

16、*(p1)23學院2.3.2 From P.C.C to Demand Curvep1x2p1p1x1*x1*(p1)x1*(p1)x1*(p1)x1*(p1)24學院2.3.2 From P.C.C to Demand Curvep1x2p1= p1p1p1x1*x1*(p1)x1*(p1)x1*(p1)xx1*(p1)x1*(p1)25學院2.3.2From P.C.C to Demand Curvep1p1x2p1p1x1*x1*(p1)x1*(p1)x1*(p1)x1*(p1) x *(p )x1*(p1)1126學院2.3.2 From P.C.C to Demand Curvep1

17、p1x2demand curvep1p1x1*x1*(p1)x1*(p1)x1*(p1)x1*(p1) x *(p ) x1*(p1)1127學院2.3.2 From P.C.C to Demand Curvep1p1x2demand curvep1p1x1*x1*(p1)x1*(p1)x1*(p1) x *(p ) x1*(p1)1128學院x1*(p1)2.3.2 From P.C.C to Demand Curvep1p1demand curvex2p1p1P.C.Cx1*x1*(p1)x1*(p1)x1*(p1) x *(p ) x1*(p1)1129學院x1*(p1)Summary:

18、 the price of a good Changesn Price consumption curve（P.C.C）:n contains all utility-izing bundles traced out as p1changes holding p2 and I constantn (在收入和其他商品價格不變條件下，某一商品價格變化所引起的消費者效用最大化的消費組合的點的軌跡)n Ordinary demand curve for commodity 1:n Plot of x1-coordinate of p1- price offer curve for each value

19、 of p1n Reflects optimal consumption of x1 at each p130學院Example: Cobb-Douglas Utilityn Assume Cobb-Douglas Utility functionbU (x ,12n Ordinary demand functions areaIbIx*( p , p , I ) =x*( p , p , I ) =,112212a + b pa + b p12n x2* is constant (flat) not f(p1)n x1* demand is rectangular hyperbola(直角雙

20、曲線) (px=k)學院31Example: Cobb-Douglas Utility=aIx *x1+ b )pa(21bI=x *2+ ba()p2P.C.Cx1*(p1)x1*(p1)x1x1*(p1)32學院Example: Cobb-Douglas Utilityp1demand curvex2aI=x *1+ b )a(p1x1*x1*(p1) x *(p ) x1*(p1)11x33學院P.C.C2.3 Comparative Analysis in Optimal Choicen 2.3.1 From Utility Functionization Mto Demandn 2.

21、3.2 From P.C.C to Demand Curven 2.3.3 Judging ed from P.C.Cn 2.3.4 From I.C.C to Engle Curven 2.3.5 Homothetic(同位性) andEngel Curvesn 2.3.6 Judging Types of Goods from I.C.Cn 2.3.7 Judging eI from Engle Curve34學院2.3.3 Judging ed from P.C.Cn 1. Defining edDqe= - q = - Dq · pDpdDpqpæp öD

22、qdqpor = lim ç-Dp®0è÷ = -··Dpq ødpq35學院2.3.3 Judging ed from P.C.Cn 2. ed and Consumer's ExpenditureTE = p ´ q,( p)dTE = d p ´ q( p) =·(1- e)ddpdpèe< 1 Þ dTE > 0 Û p ­ (¯) Þ TE ­ (¯)ddpe> 1 Þ

23、dTE < 0 Û p ­ (¯) Þ TE ¯ (­)ddpe= 1 Þ dTE = 0 Û p ­ (¯) Þ TEddp36學院2.3.3 Judging ed from P.C.Cn 3. Judging edfrom P.C.C37學院2.3.3 Judging edfrom P.C.C（p 1¯Þ）¯Û商x品的 < 1（ep 2¯Þ）­Û商x品的 > 1eTETEdd38學院2

24、.3.3 Judging edfrom P.C.C（ p4¯Þ）¯Û商x 品的 <e0（ p 3 ¯）ÞÛ商x品的= 1eqTEdd39學院402.3 Comparative Analysis in Optimal Choicen 2.3.1 From Utility Functionization Mto Demandn 2.3.2 From P.C.C to Demand Curven 2.3.3 Judging ed from P.C.Cn 2.3.4 From I.C.C to Engle Curven 2.3

25、.5 Homothetic(同位性) andEngel Curvesn 2.3.6 Judging Types of Goods from I.C.Cn 2.3.7 Judging eI from Engle Curve41學院2.3.4 From I.C.C to Engle Curven Change income I, holding prices constantn Income consumption curve(I.C.C)（收入消）:在商品價格不變條件下，由于消費者收入水平變化所引起的消費者最優(yōu)消費組合的軌跡。n Engle curve: graph of quantity de

26、manded and income42學院2.3.4 From I.C.C to Engle CurveIncome changes, holding prices constantx2I < I < Ix143學院2.3.4 From I.C.C to Engle Curvex2x144學院2.3.4 From I.C.C to Engle Curvex2x2 x2x2x1x1學院x1x1452.3.4 From I.C.C to Engle Curvex2Income consumption curvex2 x2x2x1x1學院x1x1462.3.4 From I.C.C to

27、 Engle CurveIEngel Curvex2II II.C.Cx1*xxx2 x2x211x 1x1x1x1x147學院2.3.4 From I.C.C to Engle Curvex1 Engel CurveII Ix2I.C.Cx1 x1 x1x2Engel Curvex2 x2x2I IIx1x1x1x x2x2x1248學院Example: Cobb-Douglas Utilityn Cobb-Douglas Engel curves:U(x1, x2) = x1an Ordinary demands:x2bx* =aI;=bI.x*12(a + b)p1(a + b)p249

28、學院Example: Cobb-Douglas Utilityn Rearranging to solve for I:n Engel curve for good 1:+(ab)p*1=Ix1an Engel curve for good 2:+(ab)p*2=Ix2b50學院Example: Cobb-Douglas Utility+(ab)p(ab)p*II=1*=2IxIx12abx1*x2*51學院2.3 Comparative Analysis in Optimal Choicen 2.3.1 From Utility Functionization Mto Demandn 2.3

29、.2 From P.C.C to Demand Curven 2.3.3 Judging ed from P.C.Cn 2.3.4 From I.C.C to Engle Curven 2.3.5 Homothetic(同位性) andEngel Curvesn 2.3.6 Judging Types of Goods from I.C.Cn 2.3.7 Judging eI from Engle Curve52學院2.3.5 Homothetic(同位性) andEngel CurvesEngel curves are straight lines if consumers preferen

30、ces are homothetic (all examples thus far)A consumers preferences are homothetic if and only if, for every k>0,Û(x1,x2) < (y1,y2)(kx1,kx2)<(ky1,ky2)n That is, the consumers MRS is the same anywhere on a straight line drawn from the originn Often assumed in dynamic optimization problems

31、53學院2.3.5 Homothetic(同位性) andEngel CurvesIncome Effects: Nonhomothetic Examplen Quasilinear preferences are not homotheticU(x1, x2) = f(x1) + x2n For example,1 + x 2 .U54學院2.3.5 Homothetic(同位性) andEngel Curvesx2Quasi-linear Indifference CurvesEach curve is vertically shifted copy of otherEach curve

32、intersects both axes.x155學院2.3.5 Homothetic(同位性) and CurvesEngelx2Income Changes: Quasi-linear Utilityx1x156學院Income Changes: Quasi-linear UtilityEngel CurveIx2x1*x1x1x157學院Income Changes: Quasi-linear UtilityEngel CurveIx2x2*x1x158學院Income Changes: Quasi-linear Utilityx1 Engel CurveIx2x1*x1Ix2 Enge

33、l Curvex1x2*x159學院2.3 Comparative Analysis in Optimal Choicen 2.3.1 From Utility Functionization Mto Demandn 2.3.2 From P.C.C to Demand Curven 2.3.3 Judging ed from P.C.Cn 2.3.4 From I.C.C to Engle Curven 2.3.5 Homothetic(同位性) andEngel Curvesn 2.3.6 Judging Types of Goods from I.C.Cn 2.3.7 Judging e

34、I from Engle Curve60學院2.3.6 Judging Types of Goods from I.C.Cn Normal good(正常品):n quantity demanded rises with incomen Engel curve is positively slopedn Inferior good（低檔品）:n quantity demanded falls with incomen Engel curve is negatively sloped61學院1.Income Changes: x1, x2Normal Goodsx1Engel CurveII Ix2I.C.Cx1 x1 x1x2 x2x2x2Engel CurveI II學院x1x1x1x2 x2x2x1622.Income Change