1、The University of AucklandDepartment of EconomicsFirst Semester, 1999616.301 FC Advanced,MicroeconomicsJohn Hillas,AbstractContentsHome PageGo BackCloseQuit,3,4,8,10,20,28,30,42,54,62,66,ContentsChapter 1. General Equilibrium Theory1. The basic model of a competitive economy2. Walrasian equilibrium3
2、. Edgeworth Boxes4. The First and Second Fundamental Theorems of,Welfare Economics5. ExercisesChapter 2. Noncooperative Game Theory1. Normal Form Games2. Extensive Form Games3. Existence of Equilibrium,17,616.301 FC AdvancedMicroeconomicsJohn Hillas,Chapter 3. Auctions1. Introduction2. Types of Auct
3、ion3. Analysis of the Auctions I: Private Values4. Bibliography,596078,Title PageContentsGo BackCloseQuitPage 2 of 78,Contents,Go Back,Close,Quit,CHAPTER 1General Equilibrium TheoryThere are two central multi-agent models used in economics: the gen-,eral equilibrium model and the strategic or game t
4、heoretic model. In thestrategic model we say what each actor in the economy (or in the part ofthe economy under consideration) can do. Each agent acts taking into con-sideration the plans of each other agent in the economy. There is a certaincoherence to this. It is clearly specied what each person
5、knows and howknowledge ows from one to another. It becomes dicult to specify in acompletely satisfying way all the relevant details of the economy.In the general equilibrium model on the other hand each actor does nottake explicitly into account the actions of each other. Rather we assumethat each r
6、eacts optimally to a market aggregate, the price vector. Incomparison with the strategic model there is a certain lack of coherence.It is not specied exactly how the consumers interact or how informationows from one consumer to another. On the other hand the very lack ofdetail can be seen as a stren
7、gth of the model. Since the details of how theactors interact is not specied we do not get bogged down in the somewhatunnatural details of a particular mode of interaction.,616.301 FC AdvancedMicroeconomicsJohn HillasTitle PagePage 3 of 78,L,Contents,Go Back,Close,Quit,L,L,L,L,L,1. The basic model o
8、f a competitive economyWe summarise the basic ingredients of the model. All the followingitems except the last are part of the exogenous description of the economy.The price vector is endogenous. That is it will be specied as part of thesolution of the model., L goods N consumers a typical consumer
9、is indexed consumer n. Theset of all consumers is (abusively) denoted N . (That is, the samesymbol N stands for both the number of consumers and the set of allconsumers. This will typically not cause any confusion and is suchcommon practice that you should become used to it.) the consumption set for
10、 each consumer is R+, the set of all L-,616.301 FC AdvancedMicroeconomicsJohn HillasTitle Page,dimensional vectors of nonnegative real numbers. n the rational preference relation of consumer n on R+ or un autility function for consumer n mapping R+ to R the set of realnumbers. That is, for any consu
11、mption bundle x = (x1, . . . , xL) R+ un tells us the utility that consumer n associates to that bundle. n = (1n, 2n, . . . , Ln) in R+ the endowment of consumer n p in R+ a strictly positive price vector; p = (p1, . . . , p , . . . , pL)where p 0 is the price of the th good.Page 4 of 78,L,Contents,
12、Quit,L,Definition 1.1. An allocationx = (x11, x21, . . . , xL1), . . . , (x1N , x2N , . . . , xLN )in (R+)N species a consumption bundle for each consumer. A feasibleallocation is an allocation such that,nN,xn ,nN,n,616.301 FC AdvancedMicroeconomics,or equivalently that, for each,x n , n.,John Hilla
13、s,nN,nN,Title Page,That is, for each good, the amount that the consumers together consumeis no more than the amount that together they have. (Note that we areimplicitly assuming that the goods are freely disposable. That is, we donot assume that all the good is necessarily consumed. If there is some
14、 left,over it is costlessly disposed of.),Go Back,CloseDefinition 1.2. Consumer ns budget set is,B(p, n) = x R+ | p x p n,Page 5 of 78,Go Back,Quit,Thus the budget set tells us all the consumption bundles that the con-sumer could aord to buy at prices p = (p1, p2, . . . , p , . . . , pL) if she rsts
15、old all of her endowment at those prices and funded her purchases with thereceipts. Since we assume that the consumer faces the same prices whenshe sells as when she buys it does not make any dierence whether we thinkof her as rst selling all of her endowment and then buying what she wantsor selling
16、 only part of what she has and buying a dierent incremental,bundle to adjust her overall consumption bundle.Definition 1.3. Consumer ns demand correspondence is,616.301 FC AdvancedMicroeconomicsJohn Hillas,xn(p, n) = x B(p, n) | there is no y B(p, n) with yor, in terms of the utility function,n,x,Ti
17、tle Page,Contentsxn(p, n) = x B(p, n) | there is no y B(p, n) with un(y) un(x).In words we say that the demand correspondence for consumer n is a rulethat associates to any price vector the set of all aordable consumptionbundles for consumer n for which there is no aordable consumption bundle,that c
18、onsumer n would rather have.Let us now make some fairly strong assumptions about the ns, orequivalently, the utility functions un. For the most part the full strengthof these assumptions is unnecessary. Most of the results that we give are,ClosePage 6 of 78,Contents,L L,true with weaker assumptions.
19、 However these assumptions will imply thatthe demand correspondences are, in fact, functions, which will somewhatsimplify the presentation.We assume that for each n the preference relation n is (a) continuous(this is technical and we wont say anything further about it), (b) strictlyincreasing (if x
20、y and x = y then x n y), and (c) strictly convex (ifx n y, x = y, and (0, 1) then x + (1 )y n y).,If we speak instead of the utility functions then we assume that theutility function un is (a) continuous (this is again technical, but you shouldknow what a continuous function is), (b) strictly increa
21、sing (if x y andx = y then un(x) un(y), and (c) strictly quasi-convex (if un(x) un(y),x = y, and (0, 1) then un(x + (1 )y) un(y).Proposition 1.1. If n is continuous, strictly increasing, and strictly,616.301 FC AdvancedMicroeconomicsJohn HillasTitle Page,convex (un is continuous, strictly increasing
22、, and strictly quasi-convex)then1. xn(p, n) = for any n in R+ and any p in R+,2. xn(p, n) is a singleton so xn(, n) is a function, and3. xn(, n) is a continuous function.Go BackCloseQuitPage 7 of 78,+,L,2. Walrasian equilibriumWe come now to the central solution concept of general equilibriumtheory,
23、 the concept of competitive or Walrasian equilibrium. Very briey aWalrasian equilibrium is a situation in which total demand does not exceedtotal supply. Indeed, is all goods are desired in the economy, as we assumethey are, then it is a situation in which total demand exactly equals totalsupply. We
24、 state this more formally in the following denition.616.301 FC AdvancedMicroeconomics,Definition 1.4. The price vector p is a Walrasian (or competitive) equi-librium price if,John Hillas,nN,xn(p, n) ,nN,n.,Title Page,ContentsIf we do not assume that the demand functions are single valued thewe need
25、a slightly more general form of the denition.Go Back,Definition 1.4 . The pair (p, x) in,RL,(R+)N,is a Walrasian equilib-,Close,rium if x is a feasible allocation (that is,each n in N,nN xn ,nN n) and, for,Quit,Page 8 of 78,xn,n,y for all y in B(p, n).,Since we assume that n is strictly increasing (
26、in fact local nonsatiationis enough) it is fairly easy to see that the only feasible allocations that willbe involved in any equilibria are those for which,(1),nN,xn =,nN,n.,616.301 FC AdvancedMicroeconomicsJohn HillasTitle PageContentsGo BackCloseQuitPage 9 of 78,4,Contents,Go Back,Close,Quit,3. Ed
27、geworth BoxesWe shall now examine graphically the case L = N = 2. An allocationin this case is a vector in R+. However, since we have the two equations ofthe vector equation 1 we can eliminate two of the variables and illustratethe allocations in two dimensions. A particularly meaningful way of doin
28、gthis is by what is known as the Edgeworth box.,Let us rst draw the consumption set and the budget set for eachconsumer, as we usually do for the two good case in consumer theory. Weshow this in Figure 1 and Figure 2. The only new feature of this graph isthat rather than having a xed amount of wealt
29、h each consumer starts owith an initial endowment bundle n. The boundary of their budget set(that is, the budget line) is then given by a line through n perpendicularto the price vector p.What we want to do is to draw Figure 1 and Figure 2 in the samediagram. We do this by rotating Figure 2 through
30、180 and then liningthe gures up so that 1 and 2 coincide. We do this in Figure 3. Anypoint x in the diagram now represents (x11, x21) if viewed from 01 lookingup with the normal perspective and simultaneously represents (x12, x22) ifviewed from 02 looking down. Notice that while all the feasible all
31、ocationsare within the “box” part of each consumers budget set goes outside the“box.” One of the central ideas of general equilibrium theory is that thedecision making can be decentralized by the price mechanism. Thus neitherconsumer is required to take into account when making their choices what,61
32、6.301 FC AdvancedMicroeconomicsJohn HillasTitle PagePage 10 of 78,d,r,x21Tdd616.301 FC Advanced,dd,d,MicroeconomicsJohn Hillas,ddd,d,d,Title Page,21,d,d,dd1 d,p,Contents,dd,01,11,d,Ex11,Go Back,Close,Figure 1,QuitPage 11 of 78,r,Go Back,Close,Quit,x22Tdd,22,d,dd2 d,p,d,d,616.301 FC Advanced,d,d,dd,d
33、,d,MicroeconomicsJohn Hillas,d,Title Page,02,12,d,Ex12,Contents,Figure 2is globally feasible for the economy. Thus we really do want to draw thediagrams as I have and not leave out the parts “outside the box.”We can represent preferences in the usual manner by indierence curves.,I shall not again dr
34、aw separate pictures for consumers 1 and 2, but rathergo straight to drawing them in the Edgeworth box, as in Figure 4.,Page 12 of 78,d,r,x,x21Tdd,x12,d,d,12,02,dd616.301 FC Advanced,dd,d,MicroeconomicsJohn Hillas,dd,21,d,d,d d,pd,22,Title PageContents,d,01,11,d,d,Ex11,c22,Figure 3Let us look at the
35、 denition of a Walrasian equilibrium. If some allo-cation feasible x (= ) is to be an equilibrium allocation then it must be,Go BackCloseQuitPage 13 of 78,x,Quit,x21T,x12,01,02c22,Ex11,616.301 FC AdvancedMicroeconomicsJohn HillasTitle PageContents,Figure 4,Go Back,Closein the budget sets of both con
36、sumers. (Such an allocation is shown in Fig-,ure 5.) Thus the boundary of the budget sets must be the line through xand (and the equilibrium price vector will be perpendicular to this line).,Page 14 of 78,Contents,Also x must be, for each consumer, at least as good as any other bundlein their budget
37、 set. Now any feasible allocation y that makes Consumer 1better o than he is at allocation x must not be in Consumer 1s budgetset. (Otherwise he would have chosen it.) Thus the allocation must bestrictly above the budget line through and x. But then there are pointsin Consumer 2s budget set which gi
38、ve her strictly more of both goods thanshe gets in the allocation y. So, since her preferences are strictly increasing,there is a point in her budget set that she strictly prefers to what she getsin the allocation y. But since the allocation x is a competitive equilibriumwith the given budget sets t
39、hen what she gets in the allocation x must beat least as good any other point in her budget set, and thus strictly betterthan what she gets at y.What have we shown? We have shown that if x is a competitive al-location from the endowments then any feasible allocation that makes,616.301 FC AdvancedMic
40、roeconomicsJohn HillasTitle Page,Consumer 1 better o makes Consumer 2 worse o. We can similarlyshow that any feasible allocation that makes Consumer 2 better o makesConsumer 1 worse o. In other words x is Pareto optimal.Go BackCloseQuitPage 15 of 78,d,r,r ,x,x21Tdd,x12,d,d,02,dd616.301 FC Advanced,d
41、d,dr yx d,MicroeconomicsJohn Hillas,d,d,d,d,p,Title Page,d,d,d,Contents,01,d,d,Ex11,c22,Figure 5,Go BackCloseQuitPage 16 of 78,Go Back,Close,Quit,4. The First and Second Fundamental Theorems of WelfareEconomicsWe shall now generalise this intuition into the relationship betweenequilibrium and ecienc
42、y to the more general model. We rst dene moreformally our idea of eciency.,Definition 1.5. A feasible allocation x is Pareto optimal (or Pareto e-cient) if there is no other feasible allocation y such that yn n xn for all nin N and yn n xn for at least one n in N .In words we say that a feasible all
43、ocation is Pareto optimal if thereis no other feasible allocation that makes at least one consumer strictlybetter o without making any consumer worse o. The following result,616.301 FC AdvancedMicroeconomicsJohn HillasTitle Page,generalises our observation about the Edgeworth box.ContentsTheorem 1.1
44、 (The First Fundamental Theorem of Welfare Economics).Suppose that for each n the preferences n are strictly increasing and that(p, x) is a Walrasian equilibrium. Then x is Pareto optimal.In fact, we can say something in the other direction as well. It clearly isnot the case that any Pareto optimal
45、allocation is a Walrasian equilibrium.A Pareto optimal allocation may well redistribute the goods, giving moreto some consumers and less to others. However, if we are permitted to,make such transfers then any Pareto optimal allocation is a Walrasianequilibrium from some redistributed initial endowme
46、nt. Suppose that in,Page 17 of 78,Contents,the Edgeworth box there is some point such as x in Figure 6 that is Paretooptimal. Since x is Pareto optimal Consumer 2s indierence curve throughx must lie everywhere below Consumer 1s indierence curve through x.Thus the indierence curves must be tangent to each other. Lets draw thecommon tangent. Now, if we redistribute the initial endowments to somepoint on this tangent line then with the new endowments the allocationx is a competitive equilibrium. This result is true with some generality, as,
