Advanced Microeconomics.ppt

1、Microeconomics,Advanced Microeconomics(lecture 1: production theory I),lecture 1,Prof. Zhou Qin,Summary,textbook: Varian, Hal R., 1992, Microeconomics Analysis, 3rd ed. Mas-Colell, A., M. Whinston, and J. Green, 1995, Microeconomics Theory. assignments: Once two weeks; Team work; Deliver on the clas

2、s. examinations: Final-term: 80% of the questions coming from the assignments. Score Final-term:50%() Assignments:30%(Same Score in the team) Homework: 20%,lecture 1,Prof. Zhou Qin,The Basic Framework of Microeconomics,Technology,Preference,Choice (production),Choice (purchase),Supply,Demand,Market

3、Equilibrium,lecture 1,Prof. Zhou Qin,Technology,Contain: “production (possibilities) set” (PS) and “production function”; Properties of the “PS”; Technical rate of substitution; Returns to scale.,lecture 1,Prof. Zhou Qin,1. Production set,Production plan (production vector, or input-output vector):

4、Production set Y: all technological feasible y. Restricted production set Y (z): Some of yi in y are restricted on z . Short-run production set.,lecture 1,Prof. Zhou Qin,1. Production set,Input requirement set: All yi in y are negative, let them be x (then x is positive), and the rest yj to be q . S

5、o, and , and the input requirement set is :,lecture 1,Prof. Zhou Qin,1. Production set,Transformation function: and satisfied if and only if y is technologically efficient, then,y2,y1,lecture 1,Prof. Zhou Qin,1.Production set,Production function: If O = 1,and I = n - 1, then: means: Isoquant: Questi

6、on1: calculate the PS, RS, TF, PF, Isoquant of Cobb-Douglas technology and Leontief technology.,lecture 1,Prof. Zhou Qin,2.Properties of PS.,Y is nonempty: we have something to do. Y is close: Y contain its boundary, No free lunch: See the fig. Free disposal: See the fig.,lecture 1,Prof. Zhou Qin,2.

7、Properties of PS.,Additive (free entrance) : Convexity: See the fig. Proposition1: if Y is convex, so is V(q). Proposition2: if V(q) is convex, f(x) is quasi-concave.,lecture 1,Prof. Zhou Qin,3.Technical rate of substitution,Marginal rate of transformation (MRT) of good j for good i at y and T(y)=0.

8、 T(y)=0, we got: See the fig.,lecture 1,Prof. Zhou Qin,3.Technical rate of substitution,one production: the marginal rate technical of substitution at x* . As x changes , we got technical rate of substitution,lecture 1,Prof. Zhou Qin,3.Technical rate of substitution,The elasticity of substitution: t

9、he curvature of the isoquant. Question2: calculate the “TRS” and the “elasticity of substitution” of Cobb-Douglas technology and CES technology.,lecture 1,Prof. Zhou Qin,4.Returns to scale,Nonincreasing returns to scale: Nondecreasing returns to scale: Constant return to scale:,lecture 1,Prof. Zhou

10、Qin,4.Returns to scale,Proposition 3: Y is constant returns to scale if Y is both “additive” and “convexity”. Proposition 4: single production, if and only if f(.) is homogenous of degree 1, Y is constant returns to scale.,lecture 1,Prof. Zhou Qin,4.Returns to scale,Why we assume that Y is non-incre

11、asing (or decreasing for usual) returns to scale? Suppose Y is decreasing returns to scale, and its PF f(x), now we introduce new input z, and difine a new PF: F(.) is homogenous of degree 1.,lecture 1,Prof. Zhou Qin,4.Returns to scale,Homogeneous and homothetic tech. Homogeneous of degree k: Homoth

12、etic function: Elasticity of scale:,lecture 1,Prof. Zhou Qin,Assignment,Textbook: ex.1.1, ex.1.3, ex.1.7. ex.1.9,lecture 1,Prof. Zhou Qin,Properties of PS.,y2,y1,Free lunch,lecture 1,Prof. Zhou Qin,Properties of PS.,lecture 1,Prof. Zhou Qin,Properties of PS.,y2,y1,lecture 1,Prof. Zhou Qin,Technical

13、rate of substitution,lecture 1,Prof. Zhou Qin,THANK YOU!,Advanced Microeconomics(lecture 2: production theory II),Prof. Zhou Qin,2007-7-12,Prof. Zhou Qin,profit maximization,Contain: Some definition Isoprofit curve Demand function and its properties supply function and its properties Profit function

14、 and its properties,2007-7-12,Prof. Zhou Qin,1.Some definition,Basic assumption: Y is changeless, no technical improvement. The firm is the price taking. Profit function: When it is one production:,2007-7-12,Prof. Zhou Qin,2. Isoprofit curve,The profit : We then got the isoprpfit curve . Then and is

15、 negative semidifinite.,q,x,x*,2007-7-12,Prof. Zhou Qin,2.Isoprofit curve,Weak Axiom of Profit Maximization (WAPM) if ys, yt are in Y, and choice by firm under price ps and pt. then . we can get: or,2007-7-12,Prof. Zhou Qin,2. Isoprofit curve,2007-7-12,Prof. Zhou Qin,2. Isoprofit curve,y1,y2,x,q,y1,

16、y2,x,q,YI,YO,2007-7-12,Prof. Zhou Qin,3.Demand function,factor demand function (set): proposition5: its homogenous of degree zero.,2007-7-12,Prof. Zhou Qin,3.Demand function,One production: regular p as 1. let x(,w) be the profit maximization choice (function means x is single point under w) of inpu

17、t factor vector under factor price w. then must hold:,2007-7-12,Prof. Zhou Qin,3.Demand function,When , differentiate with respect to w, we get: or Substitution matrix is a symmetric negative matrix. 1. 2.,2007-7-12,Prof. Zhou Qin,3.Demand function,Are the profit maximization give us the sufficient

18、information about the production behavior? Such as (1) factor demand function; and (2)yeild (supply) function? Proposition6:demand for factor i is: or,2007-7-12,Prof. Zhou Qin,4.supply function,Supply set: Proposition7: y(.) is homogenous of degree 1. Proposition8: If Y is convex for any p, then is

19、y(p), if Y is strictly convex, y(p) is a single point.,2007-7-12,Prof. Zhou Qin,4.Supply function,Proposition9: (Hotellings lemma) y(p) is single point, then (recall prop.6) The supply substitution matrix is symmetric positive matrix. And (why?) see the fig.,2007-7-12,Prof. Zhou Qin,5.Profit functio

20、n,Proposition10: is homogenous of degree 1. Proposition11: is convex.,2007-7-12,Prof. Zhou Qin,2.5profit function,LeChatelier Principle: something is limited in short-term, but not long-term, definite as z. and , let: if ,2007-7-12,Prof. Zhou Qin,Assignment,Textbook: ex.1.7, ex.2.2, ex.2.5, ex.3.4.,

21、2007-7-12,Prof. Zhou Qin,Supply function,y1,y2,y(p),2007-7-12,Prof. Zhou Qin,Advanced Microeconomics(LECTURE 3:production theory III),PROF. ZHOU QIN,lecture3,School of Economics and Management Southeast University,Cost Minimization,Content: Definitions Properties of cost function WACM Some forms of

22、cost functions,lecture3,School of Economics and Management Southeast University,1.Definitions,One production, cost function is The optimal solution x(w,q), is the conditional factor demand function. Question1:calculate the conditional factor demand function of C-D tech. and CES tech.,lecture3,School

23、 of Economics and Management Southeast University,1.Definitions,Recall the cost minimization condition, let x0, then set Lagrangian: We got: see the fig. What is?,lecture3,School of Economics and Management Southeast University,2.Properties of cost function,Proposition1: c(w,q) is homogeneous of deg

24、ree 1 in w, and non-decreasing in q. Proposition2: c(w,q) is concave function of w. Proposition3: x(w,q) is homogeneous of degree 0 in w. Proposition4:if V(q) is convex, then is x(.) if V(q) is strictly convex, x(.) is single point.,lecture3,School of Economics and Management Southeast University,2.

25、Properties of cost function,Proposition5: (Shephards lemma) if x(w,q) is single point, then Proposition6: is symmetric negative semi-definite, and see the fig. Proposition7: if f(.) is HD1, c(.) and x(.) is HD1 too, if f(.) is concave, c(.) is convex in q.,lecture3,School of Economics and Management

26、 Southeast University,3.WACM,Weak Axiom of Cost Minimization (WACM): if xs, xt are in Y, and choice by firm under price ws and wt. then . we can get:,lecture3,School of Economics and Management Southeast University,3.WACM,x2,xA,x1,xB,lecture3,School of Economics and Management Southeast University,3

27、.WACM,x2,xA,x1,xB,VI,VO,lecture3,School of Economics and Management Southeast University,4.Some forms of cost functions,If , corresponding price , and is the limited factor (fixed assets), while is the variable factor. The total cost is,lecture3,School of Economics and Management Southeast Universit

28、y,4.Some forms of cost functions,Envelope theorem: an objective function depend on some parameter. Then: see the prove.,lecture3,School of Economics and Management Southeast University,4.Some forms of cost functions,LAC and SAC: So See the fig.,lecture3,School of Economics and Management Southeast U

29、niversity,4. Duality,Given the technology, we can obtain the cost function, are the cost function contains the same information of the technology (production function)? If the answer is “yes”, then the cost minimization behavior will indicate the technology of the firm.,lecture3,School of Economics

30、and Management Southeast University,Assignment,Textbook: ex.5.3, ex.5.8, ex.5.11,ex.5.12,ex.5.17,lecture3,School of Economics and Management Southeast University,Definitions,x2,x1,x(w,q),w,lecture3,School of Economics and Management Southeast University,Properties of cost function,x2,x(w,q),w,lectur

31、e3,School of Economics and Management Southeast University,Prove of envelope theorem,Let be the solution of the problem, then Differentiating both sides, we have: And x* maximize f, so Then,lecture3,School of Economics and Management Southeast University,LAC and SAC,q,AC,AC,q,q*,lecture3,School of E

32、conomics and Management Southeast University,Advanced Economics(lecture 4: Duality ),Zhou Qin,lecture4,63,Duality,Given the technology, we can obtain the cost function, are the cost function contains the same information of the technology (production function)? If the answer is “yes”, then the cost

33、minimization behavior will indicate the technology of the firm.,lecture4,64,Duality,Content: Duality in mathematics Sufficient condition for cost function Factor demand function Geometry of duality,lecture4,65,1.Duality in mathematics,Some concepts and properties: Half-spaces: Normal vector: Hyperpl

34、ane: K is convex closure: K is concave, K* is the closed convex hull:,lecture4,66,1.Duality in mathematics,Support function: (infimum) given an alternative description for K. Proposition8: is HD1 and concave. See the fig.,lecture4,67,1.Duality in mathematics,Duality theorem: K is nonempty closure, a

35、nd its support function is differentiable at , then there is only one , that See the fig.,lecture4,68,2.S.C. for cost function,Differentiable function satisfied: HD1 of w Concave of w Non-decreasing of w Nonnegative for . Then is the cost function definite by the tech. of,lecture4,69,3.what about de

36、mand function,Duality indicate that HD1 and concave is what the convex tech need for the cost. What about the other function, such as factor demand function. is HD0 and is symmetric negative semidefinite. Then it is the conditional factor demand function of a certain tech.,lecture4,70,3.what about d

37、emand function,Example: given a cost function , then whats the corresponded tech. and factor demand function? Proposition9: elasticity of scale,lecture4,71,4.Geometry of duality,lecture4,72,4.Geometry of duality,lecture4,73,4.Geometry of duality,lecture4,74,4.Geometry of duality,lecture4,75,Support

38、function,p,K,lecture4,76,Duality theorem,K,x1,x2,p2 is decreasing,lecture4,77,Duality theorem,K,p2,lecture4,78,Advanced Economics(lecture 5: consumption theory I),Zhou Qin,Preference and Choice,Content: Preference and rationality The rule of choice Budget Demand function WA and demanded law,1.Prefer

39、ence,An (acceptable) alternatives set (or a consumption set ) A (weak) preference relation“ ”: a binary relationship on X, for any , means” x is at least as good as y”. A strict preference “ ”: An indifference relation “ ”:,1.Preference,Definition: preference on X is rational if: Its complete: Its r

40、eflexive: Its transitive:,1.Preference,Propersition1: if is rational, then: is irreflexive and transitive is reflexive and transitive and symmetric If,2.The rule of choice,We can only observe what a man choice under a certain constrain. Is the mans preference rational when we observed several choice

41、 of him? Or if his preference was rational, what rules will his choice take?,2.The rule of choice,Choice construct: a choice under certain budget: Budget set: and , is a budget family, represented a limited list of all possible choice under certain system, nature, technology and other social environ

42、ments that the decision-maker must take. Choice rule: , maybe contain more than one elements.,2.The rule of choice,Weak Axiom of Revealed Preference (WARP or WA) : if for certain , and for and , then must Revealed Preference “ ”:,2.The rule of choice,Choice set Proposition2: if are rational, then ch

43、oice construct satisfied WA. Prop.2 indicate that if the decision-maker have rational preference, than his choice satisfied WA. Is his preference rational if his choice is observed satisfy WA?,2.The rule of choice,Definition: for , we say “rational preference rationalization Propersition3: if satisf

44、ied WA and contain all subsets of X that is ternary and less than ternary. example: X=x,y,z, B =x,y,y,z,x,z, C(x,y)=x,C(y,z)=y,C(x,z)=z,(B , C(.) satisfied WARP, but there is no rationalization preference relation over it.,3.Competive budget,Consumption set X and its price with the wealth w0. Defini

45、tion: Walras (competive) budget set Proposition4: if X is concave, then is,4.Demand function,For any p and w are strictly positive, the corresponding demand set x(p,w) are nonempty, if x(p,w) is single point, we call it the Walrasian( Marshallian) demand function. x(p,w) is homogenous of degree zero

46、 and satisfied Walras law. That is px=w for all (we will prove them next lecture),4.Demand function,Wealth effects: Engel function: given price ,function over wealth Wealth expansion path: Wealth effects:,4.Demand function,Wealth effects: Commodity l is normal goods if Commodity l is inferior goods

47、if The demand is normal if See the fig,4.Demand function,Price effects: Supply curve: Price effects: Giffen good: See the fig.,4.Demand function,Cournot Aggregation: if Walrasian demand function x(p,w) is HD0 and satisfied Walras Law , then: Engel Aggregation: if Walrasian demand function x(p,w) sat

48、isfied Walras Law, then:,5.WA and demand law,Walrasian demand function x(p,w) satisfied WA if for any we have: See the fig.,5.WA and demand law,Changing in price will change wealth too. But how can we tell the demand changing by price changing from wealth changing? Given a changing from ,and people

49、will not get worse that is here wealth changing (compensation ) was called “Slutsky wealth compensation” and “(Slutsky) compensated price changing”.,5.WA and demand law,Proposition5: x(p,w) satisfied WA if and only if: and when , Prop.5 indicates, , or thats called “demand law ”, or “compensation de

50、mand law ”.,5.WA and demand law,Slutsky matrix (substitution matrix) Substitution effects is n.s.d Giffen good is necessary inferior good.,Assignment,If Then For then If Laspeyres quantity index is less than 1, then ,and if Paasche quantity index is large than1, then,Consumption set,Consumption set,

51、Consumption set,expansion path,x2,x1,Wealth effects,luxury goods,Supply curve:,x2,x1,Giffen good,x2,x1,5.WA and demand law,x2,x1,x2,x1,5.WA and demand law,x2,x1,x2,x1,Thank You!,Advanced Microeconomicslecture 5:consumption theory II,Zhou Qin,111,Utility maximization,Content: Utility function Utility

52、 maximization Expenditure minimization Relationship between UMP and EMP Integrability,112,1.From preferences to utility,Definition: is a utility function of preference , if Can we always find a utility function of ? Maybe If X is finite, there always exist utility function. Proposition1: only ration

53、al can be represented by a utility function. (N.C not S.C) Lexicographic preference: rational but no utility function exist.,113,1.From preferences to utility,Continuity: then ,or x upper contour sets and lower contour sets are closure. Proposition2:If is continuous, then exist a continuous utility

54、function representing .,114,1.From preferences to utility,Desirability: preference is desirable if is monotone: is local non-satiation: there is a proposition3: is strong monotone, then its monotone; is monotone, its local non-satiation.,115,1.From preferences to utility,Convexity: x upper contour s

55、ets are convex. Decreasing in marginal rate of substitution. people like variety. Preferences are convex means utility function is quasi-concave. Note: quasi-concave does not contain concave.,116,1.From preferences to utility,If , then are homothetic. Proposition4: The are homothetic if and only if

56、its utility function is HD1. If then are quasi-linear of commodity 1(its called standard commodity) Proposition5: are quasi-linear if and only if its utility function is Gorman form,117,1.From preferences to utility,Preference,Utility function,rationality,exist,X finite,Desirability,increasing,Conti

57、nuity,continuity,Convexity,Quasi-concave,Homothetic,HD1,Quasi-linear,Gorman form,118,2.Utility maximization,Consumers problem (UMP): The solutions are called Walras Demand Correspondence, or function if its single point.,119,2.Utility maximization,Properties of HD0 Satisfied Walras Law If are concav

58、e, are concave too, if are strictly concave, is single point.,120,2.Utility maximization,If the solution , then Here is the shadow price: the marginal utility of optimal consumption . The value function is called “indirect utility function”:,121,3.Expenditure minimization,EMP: The solution was called “Hicks (or compensation ) demand correspondence”. single point-function. Its HD0 of p. Its convex, as is . and single point when are strictly convex.,122,3.E