3/1/2016. Intermediate Microeconomics W3211. Lecture 7: The Endowment Economy. Today s Aims. The Story So Far. An Endowment Economy.

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1 1 Intermedite Microeconomics W3211 Lecture 7: The Endowment Economy Introduction Columbi University, Spring 2016 Mrk Den: 2 The Story So Fr. 3 Tody s Aims 4 Remember: the course hd two bsic ims: 1. Introduce you to models of how people mke choices Constrined optimiztion 2. Introduce you to models of wht hppens when people interct We hve now done firly thorough job of modelling the behvior of one type of economic gent the consumer Set up the consumer s problem Solved the consumer s problem Derived demnd functions Thought bout how demnd chnges with income nd prices We re now going to tke our first stb in thinking bout wht hppens when economic gents interct In prticulr, we re going to think bout how prices nd income re determined in n economy So fr we hve treted these s exogenous prmeters But they hve to come from somewhere! We will ssume tht they come bout through the interction of economic gents buying nd selling They move in order to blnce supply nd demnd This is the study of equilibrium We will strt with the simplest possible setting Two consumers Two goods No firms This is the study of n endowment economy Vrin Ch. 9, 15, 16 Feldmn nd Serrno Ch 15 An Endowment Economy 6 We re going to think bout the simplest possible exmple of n economy There re two people: 1 nd 2 A Simple Endowment Economy 1: Revisiting the consumer s problem There re two goods: pples nd bnns Ech person strts off with n endowment of pples nd bnns, : mount of pples nd bnns tht person 1 hs, : mount of pples nd bnns tht person 2 hs People re llowed to buy nd sell t mrket prices nd 5 Questions we wnt to nswer: Wht should prices be in this economy? Who ends up with wht stuff? 1

2 An Endowment Economy 7 8 Note: Apples nd bnns only exist in the endowment of the consumers They re not produced There re no firms This is simplifiction, mkes it esier to see wht is going on Don t worry, firms will pper soon enough First order of business: think wht the consumer s problem looks like in n endowment economy Remember the stndrd consumer problem 1. CHOOSE consumption bundle 2. IN ORDER TO MAXIMIZE preferences 3. SUBJECT TO the budget constrint The budget constrint (for consumer 1) ws given by Now, rther thn getting mysterious income I, the income the consumer gets comes from their endowment Or, equivlently 0 is the net demnd for good I Budget constrint sys tht the cost of net demnd hs to be less thn zero 9 10 Wht hppens to the budget constrint if the price of good chnges? Wht hppens to the budget constrint if the price of good chnges? implies Could buy units of good b (incresing in ) Could buy units of good (decresing in ) w, w b Alwys fesible to consume, ) Budget constrint pivots round the endowment Increse in p Another question: wht hppens when the prices of both goods chnge in proportion? A quick exmple Budget constrint is Now proportionlly increse prices of both nd b A quick exmple Budget constrint is

3 13 14 The only thing tht mtters for the consumers problem is the rtio of the price of good to tht of good b This is obvious from the budget constrint This mens tht we cn normlize the price of good b to 1 i.e. ssume tht the price of good b is 1 Think only bout chnges in the price of good Wht we relly men by the price of good is This will mke our life lot esier We hve now chnged the prmeters of the consumer s problem Rther thn the prmeters being,, they re now,, We cn similrly think of the demnd function in terms of these prmeters:,, ) is the demnd of person 1 for good, given prices nd endowment, ),, ) =,, ) - is the net demnd of person 1 for good But is quite subtle point. Mke sure you understnd it! A worked exmple: Choose, to Mximize, =,, Subject to First set up the Lgrnge Function,, Then tke derivtives Then tke derivtives First two equtions gives Lst eqution gives the 17 Substituting in gives And so which implies,, 1 2,, 1 2,,,, A Simple Endowment Economy 2: The Edgeworth Box 18 3

4 The Edgeworth Box 19 The Edgeworth Box 20 For the consumer s problem we found it hndy to drw grphs which llowed us to see wht is going on It will be eqully hndy for us to do so for our simple 2 person, 2 good economy However, the grph we need to drw is bit more complicted First step: how cn we represent ll the stuff tht there is in the economy? With box! Width of the box is totl mount of good Height of the box is the totl mount of good b The Edgeworth Box! Fig 2: An Edgeworth Box 21 The Edgeworth Box 22 Second step, how do we represent fesible lloction in the Edgeworth Box? A fesible lloction is n mount of ech good for person 1 nd 2 tht uses up the totl lloction,,, such tht + = + + = + Any point in the box is fesible lloction A Fesible Alloction 23 The Edgeworth Box 24 x 2 x 2 b Third step, we cn represent the consumer s problem for person 1 x 1 x 1 4

5 Consumer 1 in the Edgeworth Box 25 Fig 5: Consumer 1 s Optiml Bundle 26 Budget Line Slope: -p Consumer 1 s Indifference Curves Alloction x 1 b Optiml Consumption Bundle for Consumer 1 Alloction x 1 The Edgeworth Box 27 Consumer 2 Budget Set in the Edgeworth Box 28 Fourth step (nd this is the clever bit), we cn represent the consumer s problem for person 2 Consumer 2 s Budget Set Consumer 2 s Preferences in the Edgeworth Box 29 Consumer 2 s Optiml Bundle x 2 30 x 2 b Better Bundles Optiml Consumption Bundle for Consumer 2 5

6 32 We re now redy to clculte the equilibrium for our nice, simple, two person, two good exchnge economy A Simple Endowment Economy 3: An importnt first step: we better define wht we men by n equilibrium An equilibrium is description of wht we think will hppen in n economy We cn think of it s prediction of where we might expect the economy to end up Consists of two types of thing Prices of the vrious goods Quntities of ech good tht ech person receives. So in our simple, two person two good economy, n equilibrium consists of A price of good : (remember we don t need price for good b) The mount of stuff tht person 1 gets: nd The mount of stuff tht person 2 gets: nd A hndy tip: If I sk you, in n exm or homework, to clculte n equilibrium, you hve to tell me wht the prices nd quntities re If you hven t, you hven t nswered the question! 35 The Definition of n 36 Wht properties do you think n equilibrium should hve? i.e., if I told you tht,,, nd were the equilibrium for the economy, wht would you expect to be true? It my be useful to think of the equilibrium s the resting point of the economy We define n equilibrium in the following wy:,,, nd form n equilibrium if two things re true 1. Optimlity: The mount tht ech person gets is wht they wnt, given the prices nd their endowments,, ),, ),, ),, ) 2. Mrket Clering: Supply equls demnd for ech good

7 Finding n So now we know wht n equilibrium is How do we find one? Luckily we hve nother recipe! 1. Clculte the demnd for ech consumer nd ech good s function of prices i.e. clculte,, ),,, ),,, ),,, ) 2. Find the price t which mrkets cler i.e. find the price such tht,, )+,, )= +,, )+,, )= + 3. Use this price to clculte wht ech person gets Let s work through n exmple First, we need to describe the economy 1. The endowment of ech gent =3 =2 =1 =5 2. The preferences of ech gent, =, = Stge 1: Clculte the demnd for ech consumer nd ech good s function of prices Luckily we hve lredy done this for these preferences (erlier in the lecture),, ,, ,, ,, Stge 2: Find the price t which mrkets cler Let s first think bout the mrket for pples,, )+,, ) Gret! We hve the price tht clers the mrket for pples! The sme price clers the mrket for pples nd bnns! But there is lso the mrket for bnns! A mircle! And we only hve one price (it s not mircle. I will explin why in minute) This sounds like trouble First, is it the cse tht is n equilibrium? NO! Let s see wht hppens in the mrket for bnns,, )+,, ) An equilibrium consists of,,, nd 7

8 43 Wlrs Lw 44 Stge 3: Use this price to clculte wht ech person gets,, ,, ,, 1 2,, The equilibrium of this economy is these lloctions, plus the price So let s go bck to the mircle We hd two mrkets (pples nd bnns) nd only one price ( ) How did the one price cler the two mrkets? It turns out this is lwys true Wlrs lw: The price tht clers the mrket for good will lso cler the mrket for good b Why (strp in for some lgebr) Wlrs Lw 45 Both gents obey their budget constrint So If the mrket for good clers then A Simple Endowment Economy 4: in Pictures Implies Mrket for good b clers Hving introduced the Edgeworth box, it might be useful to think bout wht equilibrium looks like in this digrm. First lets think wht hppens to the demnd of consumer 1 s we chnge the price Offer curve for consumer 1 Increse in p w 1 1+w 2 1 8

9 49 50 Hving introduced the Edgeworth box, it might be useful to think bout wht equilibrium looks like in this digrm. First lets think wht hppens to the demnd of consumer 1 s we chnge the price Offer curve of consumer trces out their optiml bundle s chnges Similrly, we cn trce out the offer curve of consumer 2 Offer curve for consumer 2 Increse in p x 2 Hving introduced the Edgeworth box, it might be useful to think bout wht equilibrium looks like in this digrm. x 2 b First lets think wht hppens to the demnd of consumer 1 s we chnge the price Offer curve of consumer trces out their optiml bundle s chnges Similrly, we cn trce out the offer curve of consumer 2 x 1 b Where is equilibrium going to occur? x x 2 In the Edgeworth box, equilibrium occurs where the offer curves intersect x 2 b x 1 b x 1 9

10 55 56 Chnge In In the Edgeworth box, equilibrium occurs where the offer curves intersect This is lso the point t which three things re tngent The price line The indifference curve of person 1 The indifference curve of person 2 w 1 2+w 2 2 Good 2 This will be importnt for the next lecture! Notice: different endowments will give rise to different equilibri Good 1 Chnge In Endowment w 1 1+w 2 1, Supply nd Demnd 57, Supply nd Demnd 58 Here is finl wy of thinking bout equilibrium Remember the mrket clering condition:,, )+,, )= + Left hnd side is the totl (or mrket) demnd for good Typiclly (though not lwys) downwrd sloping Right hnd side is the totl supply of good Good 1 w 1 1+w 2 1 Demnd Curve= x ( p, w, w ) x ( p, w, w ) b Supply curve b occurs when prices mke supply equl to demnd Price P 1 Summry 60 Tody we hve done the following 1. Defined simple endowment economy Rther thn income, consumers hve endowments of ech good We hve defined the consumer s problem in this cse Also defined the demnd function,, ) Summry 2. Introduced the Edgeworth Box s wy of representing simple two person, two good endowment economy Shown how it represents fesible lloctions Shown how it cn represent the consumer s problem for both consumer 1 nd 2 Defined the concept of n equilibrium Shown how to solve for the equilibrium of simple endowment economy 59 10