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1、Advanced Microeconomics2023/2/2【可编辑】Advanced Microeconomics(lecture 1:production theory I)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 class.examinations:Final-te

2、rm:80%of the questions coming from the assignments.ScoreFinal-term:50%()Assignments:30%(Same Score in the team)Homework:20%lecture 1lecture 1Prof.Zhou QinProf.Zhou QinThe Basic Framework of MicroeconomicsTechnology Preference Choice(production)Choice(purchase)Supply Demand MarketEquilibriumlecture 1

3、lecture 1Prof.Zhou QinProf.Zhou QinTechnologyContain:“production(possibilities)set”(PS)and“production function”;Properties of the“PS”;Technical rate of substitution;Returns to scale.lecture 1lecture 1Prof.Zhou QinProf.Zhou Qin1.Production setTransformation function:and satisfied if and only if y is

4、technologically efficient,then y2y1lecture 1lecture 1Prof.Zhou QinProf.Zhou Qin1.Production setProduction function:If O=1,and I=n-1,then:means:Isoquant:Question1:calculate the PS,RS,TF,PF,Isoquant of Cobb-Douglas technology and Leontief technology.lecture 1lecture 1Prof.Zhou QinProf.Zhou Qin2.Proper

5、ties 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 1lecture 1Prof.Zhou QinProf.Zhou Qin2.Properties of PS.Additive(free entrance):Convexity:See the fig.Proposition1:if Y is convex,so is V(q).Proposition2:if V

6、(q)is convex,f(x)is quasi-concave.lecture 1lecture 1Prof.Zhou QinProf.Zhou Qin3.Technical rate of substitutionMarginal rate of transformation(MRT)of good j for good i at y and T(y)=0.T(y)=0,we got:See the fig.lecture 1lecture 1Prof.Zhou QinProf.Zhou Qin3.Technical rate of substitutionone production:

7、the marginal rate technical of substitution at x*.As x changes,we got technical rate of substitution lecture 1lecture 1Prof.Zhou QinProf.Zhou Qin3.Technical rate of substitutionThe elasticity of substitution:the curvature of the isoquant.Question2:calculate the“TRS”and the“elasticity of substitution

8、”of Cobb-Douglas technology and CES technology.lecture 1lecture 1Prof.Zhou QinProf.Zhou Qin4.Returns to scaleNonincreasing returns to scale:Nondecreasing returns to scale:Constant return to scale:lecture 1lecture 1Prof.Zhou QinProf.Zhou Qin4.Returns to scaleProposition 3:Y is constant returns to sca

9、le 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 1lecture 1Prof.Zhou QinProf.Zhou Qin4.Returns to scaleWhy we assume that Y is non-increasing(or decreasing for usual)returns to scale?Suppose Y

10、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 1lecture 1Prof.Zhou QinProf.Zhou Qin4.Returns to scaleHomogeneous and homothetic tech.Homogeneous of degree k:Homothetic function:Elasticity of scale:lecture 1lecture

11、 1Prof.Zhou QinProf.Zhou QinAssignmentTextbook:ex.1.1,ex.1.3,ex.1.7.ex.1.9 lecture 1lecture 1Prof.Zhou QinProf.Zhou QinProperties of PS.y2y1y2y1Free lunchlecture 1lecture 1Prof.Zhou QinProf.Zhou QinProperties of PS.y2y1lecture 1lecture 1Prof.Zhou QinProf.Zhou QinProperties of PS.y2y1lecture 1lecture

12、 1Prof.Zhou QinProf.Zhou QinTechnical rate of substitutiony2y1lecture 1lecture 1Prof.Zhou QinProf.Zhou QinTHANK YOU!lecture 1lecture 1Prof.Zhou QinProf.Zhou QinAdvanced Microeconomics(lecture 2:production theory II)Prof.Zhou Qinprofit maximizationContain:Some definitionIsoprofit curveDemand function

13、 and its propertiessupply function and its propertiesProfit function and its properties2007-7-122007-7-12Prof.Zhou QinProf.Zhou Qin1.Some definitionBasic assumption:Y is changeless,no technical improvement.The firm is the price taking.Profit function:When it is one production:2007-7-122007-7-12Prof.

14、Zhou QinProf.Zhou Qin2.Isoprofit curveThe profit:We then got the isoprpfit curve.Then and is negative semidifinite.qxx*2007-7-122007-7-12Prof.Zhou QinProf.Zhou Qin2.Isoprofit curveWeak Axiom of Profit Maximization(WAPM)if ys,y yt t are in Y,and choice by firm under price ps and pt.then .we can get:o

15、r 2007-7-122007-7-12Prof.Zhou QinProf.Zhou Qin2.Isoprofit curvey1y2xqy1y2xq2007-7-122007-7-12Prof.Zhou QinProf.Zhou Qin2.Isoprofit curvey1y2xqy1y2xqYIYO2007-7-122007-7-12Prof.Zhou QinProf.Zhou Qin3.Demand functionfactor demand function(set):proposition5:its homogenous of degree zero.2007-7-122007-7-

16、12Prof.Zhou QinProf.Zhou Qin3.Demand functionOne production:regular p as 1.let x(,w)be the profit maximization choice(function means x is single point under w)of input factor vector under factor price w.then must hold:2007-7-122007-7-12Prof.Zhou QinProf.Zhou Qin3.Demand functionWhen ,differentiate w

17、ith respect to w,we get:orSubstitution matrix is a symmetric negative matrix.1.2.2007-7-122007-7-12Prof.Zhou QinProf.Zhou Qin3.Demand functionAre the profit maximization give us the sufficient information about the production behavior?Such as(1)factor demand function;and(2)yeild(supply)function?Prop

18、osition6:demand for factor i is:or2007-7-122007-7-12Prof.Zhou QinProf.Zhou Qin4.supply functionSupply set:Proposition7:y(.)is homogenous of degree 1.Proposition8:If Y is convex for any p,then is y(p),if Y is strictly convex,y(p)is a single point.2007-7-122007-7-12Prof.Zhou QinProf.Zhou Qin4.Supply f

19、unctionProposition9:(Hotellings lemma)y(p)is single point,then (recall prop.6)The supply substitution matrix is symmetric positive matrix.And (why?)see the fig.2007-7-122007-7-12Prof.Zhou QinProf.Zhou Qin5.Profit functionProposition10:is homogenous of degree 1.Proposition11:is convex.2007-7-122007-7

20、-12Prof.Zhou QinProf.Zhou Qin2.5 profit functionLeChatelier Principle:something is limited in short-term,but not long-term,definite as z.and ,let:if ,y(p)pyL(p)yS(p)p*2007-7-122007-7-12Prof.Zhou QinProf.Zhou QinAssignmentTextbook:ex.1.7,ex.2.2,ex.2.5,ex.3.4.2007-7-122007-7-12Prof.Zhou QinProf.Zhou Q

21、inSupply functiony1y2y(p)2007-7-122007-7-12Prof.Zhou QinProf.Zhou Qin2007-7-122007-7-12Prof.Zhou QinProf.Zhou QinAdvanced Microeconomics(LECTURE 3:production theory III)PROF.ZHOU QINCost MinimizationContent:DefinitionsProperties of cost functionWACMSome forms of cost functions lecture3lecture3School

22、 of Economics and School of Economics and Management Southeast UniversityManagement Southeast University1.DefinitionsOne 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 te

23、ch.lecture3lecture3School of Economics and School of Economics and Management Southeast UniversityManagement Southeast University1.DefinitionsRecall the cost minimization condition,let x0,then set Lagrangian:We got:see the fig.What is?lecture3lecture3School of Economics and School of Economics and M

24、anagement Southeast UniversityManagement Southeast University2.Properties of cost functionProposition1:c(w,q)is homogeneous of degree 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(

25、.)if V(q)is strictly convex,x(.)is single point.lecture3lecture3School of Economics and School of Economics and Management Southeast UniversityManagement Southeast University2.Properties of cost functionProposition5:(Shephards lemma)if x(w,q)is single point,thenProposition6:is symmetric negative sem

26、i-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.lecture3lecture3School of Economics and School of Economics and Management Southeast UniversityManagement Southeast University3.WACMWeak Axiom of Cost Minimization(WACM):if xs,x xt t are

27、 in Y,and choice by firm under price ws and wt.then .we can get:lecture3lecture3School of Economics and School of Economics and Management Southeast UniversityManagement Southeast University3.WACMx2xAx1xBx2xAx1xBlecture3lecture3School of Economics and School of Economics and Management Southeast Uni

28、versityManagement Southeast University3.WACMx2xAx1xBx2xAx1xBVIVOlecture3lecture3School of Economics and School of Economics and Management Southeast UniversityManagement Southeast University4.Some forms of cost functionsIf ,corresponding price ,and is the limited factor(fixed assets),while is the va

29、riable factor.The total cost islecture3lecture3School of Economics and School of Economics and Management Southeast UniversityManagement Southeast University4.Some forms of cost functionsEnvelope theorem:an objective function depend on some parameter.Then:see the prove.lecture3lecture3School of Econ

30、omics and School of Economics and Management Southeast UniversityManagement Southeast University4.Some forms of cost functionsLAC and SAC:So See the fig.lecture3lecture3School of Economics and School of Economics and Management Southeast UniversityManagement Southeast University4.Duality Given the t

31、echnology,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.lecture3lecture3School of Economics and School of Economics and Mana

32、gement Southeast UniversityManagement Southeast UniversityAssignment Textbook:ex.5.3,ex.5.8,ex.5.11,ex.5.12,ex.5.17lecture3lecture3School of Economics and School of Economics and Management Southeast UniversityManagement Southeast UniversityDefinitionsx2x1x(w,q)wlecture3lecture3School of Economics a

33、nd School of Economics and Management Southeast UniversityManagement Southeast UniversityProperties of cost functionx2x(w,q)wlecture3lecture3School of Economics and School of Economics and Management Southeast UniversityManagement Southeast UniversityProve of envelope theoremLet be the solution of t

34、he problem,thenDifferentiating both sides,we have:And x*maximize f,soThen lecture3lecture3School of Economics and School of Economics and Management Southeast UniversityManagement Southeast UniversityLAC and SACqACACqq*lecture3lecture3School of Economics and School of Economics and Management Southe

35、ast UniversityManagement Southeast Universitylecture3lecture3School of Economics and School of Economics and Management Southeast UniversityManagement Southeast UniversityAdvanced Economics(lecture 4:Duality)Zhou QinDuality Given the technology,we can obtain the cost function,are the cost function c

36、ontains 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.lecture4lecture46363DualityContent:Duality in mathematicsSufficient condition for cost functionFactor demand functionGeometry of dual

37、itylecture4lecture464641.Duality in mathematicsSome concepts and properties:Half-spaces:Normal vector:Hyperplane:K is convex closure:K is concave,K*is the closed convex hull:lecture4lecture465651.Duality in mathematicsSupport function:(infimum)given an alternative description for K.Proposition8:is H

38、D1 and concave.See the fig.lecture4lecture466661.Duality in mathematicsDuality theorem:K is nonempty closure,and its support function is differentiable at ,then there is only one ,that See the fig.lecture4lecture467672.S.C.for cost functionDifferentiable function satisfied:HD1 of wConcave of wNon-de

39、creasing of wNonnegative for .Then is the cost function definite by the tech.of lecture4lecture468683.what about demand functionDuality 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 negativ

40、e semidefinite.Then it is the conditional factor demand function of a certain tech.lecture4lecture469693.what about demand functionExample:given a cost function ,then whats the corresponded tech.and factor demand function?Proposition9:elasticity of scale lecture4lecture470704.Geometry of dualityx2x1

41、isoquantw1w2isocostx1w1/w2factor demandlecture4lecture471714.Geometry of dualityx2x1isoquantw1w2isocostx1w1/w2factor demandlecture4lecture472724.Geometry of dualityx1x2isoquantw2w1isocostx1w1/w2factor demandlecture4lecture473734.Geometry of dualityw1w2isocostx2x1isoquantx1w1/w2factor demandlecture4l

42、ecture47474Support functionpKxlecture4lecture47575Duality theoremKx1x2p2 is decreasinglecture4lecture47676Duality theoremKp2lecture4lecture47777lecture4lecture47878Advanced Economics(lecture 5:consumption theory I)Zhou QinPreference and ChoiceContent:Preference and rationalityThe rule of choiceBudge

43、t Demand functionWA and demanded law1.PreferenceAn(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.PreferenceDefinition:preference on X is rational i

44、f:Its complete:Its reflexive:Its transitive:1.PreferencePropersition1:if is rational,then:is irreflexive and transitive is reflexive and transitive and symmetricIf 2.The rule of choiceWe can only observe what a man choice under a certain constrain.Is the mans preference rational when we observed sev

45、eral choice of him?Or if his preference was rational,what rules will his choice take?2.The rule of choiceChoice 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

46、ments that the decision-maker must take.Choice rule:,maybe contain more than one elements.2.The rule of choiceWeak Axiom of Revealed Preference(WARP or WA):if for certain ,and for and ,then must Revealed Preference“”:2.The rule of choiceChoice setProposition2:if are rational,then choice construct sa

47、tisfied 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 choiceDefinition:for ,we say“rational preference rationalizationPropersition3:if satisfied WA and contain all subs

48、ets 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 budgetConsumption set X and its price with the wealth w0.Definition:Walras(competive)budget setPropos

49、ition4:if X is concave,then is 4.Demand functionFor 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 and satisfied Walras law.That is px=w for all (we wil

50、l prove them next lecture)4.Demand functionWealth effects:Engel function:given price ,function over wealth Wealth expansion path:Wealth effects:4.Demand functionWealth effects:Commodity l is normal goods ifCommodity l is inferior goods ifThe demand is normal ifSee the fig 4.Demand functionPrice effe