ECON 251 – Lecture 3 – Computing Equilibrium | Open Yale Courses

Financial Theory

ECON 251 – Lecture 3 – Computing Equilibrium

Chapter 1. Introduction [00:00:00]

Professor John Geanakoplos: now, the course, precisely to summarize again, the course is the standard fiscal theory course that was made popular over the last ten-spot years in a bunch of business schools, and those guys who developed this material basically thought that markets were bang-up and finance was about a discriminate part–could be walled off from much of economics. so here at Yale we ’ ve never taught finance that manner .
We ’ ve always taught it as a function of economics and the crisis recently, I think, has made it clearly that that ’ randomness probably the means one should truly think about the trouble. So it ’ s become very fashionable now to say that fiscal theorists had everything wholly incorrectly and to ask how it is that they got everything all wrong. Why didn ’ thyroxine they anticipate the crash ? And the two standard critiques of criterion fiscal economics are a ) it didn ’ t allow for psychology, and you ’ ll hear about that from Shiller next semester, and bacillus ) it didn ’ t take into report collateral. And it was all done in a very special case, a knife-edge case .
If you looked at it a little more broadly you would realize that the kind of crisis we ’ ve had now is not such an unfathomable matter. In fact it ’ sulfur happened many times earlier. So that ’ s the position I ’ m going to take in this class. so to put it a different room, Krugman, identical recently in the New York Times, you may have read his magazine article, wrote precisely that, that there are two problems. The fiscal hypothesis has failed us. Why has it failed us ? well, because it didn ’ t have psychology and it didn ’ t have collateral.

And he didn ’ deoxythymidine monophosphate speak much about the collateral which is obviously something he ’ s not thought very much about earlier. But in concert with the collateral he sort of said it ’ s excessively much–how did he put it ? He said, “ Too much seduction by mathematics. The fiscal economists were seduced by their own mathematics into believing stuff that a sensible person who didn ’ t pay then much attention to mathematics wouldn ’ thymine do. ”
well, although the review in this course is going to be partially based on collateral the rest of what Krugman said I wholly disagree with. I regard that as a kind of Taliban access to economics. The more engineering and firepower you use the more you ’ re going to be misled. That ’ s what the Taliban believe, and they want to get rid of modern engineering and return to beginning principles. sol I think, in fact, the problem with modern finance was not besides much mathematics, but besides little mathematics, and they made these identical limited simplifying assumptions and didn ’ metric ton realize how important the assumptions were to the conclusions. So we ’ re going to reexamine all that and that ’ s what we ’ rhenium starting at nowadays .

Chapter 2. Welfare and Utility in Free Markets [00:02:48]

We ’ re going to consider, first gear of all, the argumentation that free markets work good. So we started with a small example. Oh, by the way, the first trouble stage set, if I didn ’ triiodothyronine mention it, is due Tuesday. So you need to bring it to class and there will be a box with each of your section leaders ’ name calling on it. so purportedly you ’ ve been able to sign up for sections by now. Is that right ? anyhow, it ’ mho on the web, so you pick a segment and sign up for it. If you wait excessively long your incision will fill up the time .
sol there are eight sections. You ought to be able to find one of them that fits your schedule. And you need to turn it in on Tuesday in class, by the end of classify. possibly you can scribble something during the class, but by the end of the class we ’ rhenium going to take the problem sets and after that it ’ sulfur besides late to hand them in. So all of you are going to have problems, you ’ re going to have midnight sessions, all night things that you ’ re going to have to do for some early course, or grandmothers are going to die. All sorts of things are going to happen, but we don ’ t take the problem sets recently. So there will be ten problem sets .
We ’ re lone going to count nine of the grades so you ’ ll have one loose pass, and that ’ s what life is. And it ’ s just besides complicated to keep track of people handing them late all the prison term. The answers are going to go on the vane right after the class, and then it ’ s just in the past we negotiated with every person who was late and it ’ s good excessively complicated. And when you make a bare rule grandmothers don ’ thyroxine die anymore. So anyhow, that ’ s how we ’ re going to work it in the class. You just have to turn it in. And there are ten of them. only nine count. If you miss one raw it ’ s actually not going to change your grade anyhow. If you miss half of them that ’ sulfur going to have some consequence on your grade, and then I don ’ t promote you to do that, but I ’ meter sure you won ’ metric ton do that. so I think that ’ s all the preliminaries. There are two midterms, one in the middle of the class and one correct at the end .
anyhow, the motion we want to spend the whole of the class on today is whether the free marketplace is in truth such a great idea. And the quintessential example in which it is a great theme is the one we did in class on the very first day. We had a bunch of football tickets and there were buyers each of whom knew his own valuation and sellers each of whom knew her own valuation and we threw everybody together and equitable very briefly explained the rules. By the way, the only important rule was–there were two crucial rules. You had to announce publicly and obstreperously what price you were offering. That ’ randomness identical crucial. You don ’ t have any clandestine deals. That would have screwed everything up .
And second, we had a rule about once you make a deal what happens. How does the thing actually get transferred ? so one of the TAs digest by and wrote it down, and the two people exchanged the footballs and agreed to it and walked off stage. So the actual mechanics of the transaction to make sure that the person turning over the money actually gets the football ticket, that of naturally is fabulously important and that ’ s the thing that gets left out much in finance. That ’ s the collateral business that we ’ rhenium going to come to by and by. How do you know that the guy ’ s actually going to pay you what he promises ? well, he ’ mho got to put up collateral so that you can trust him. so without that there would have been a boastfully problem, and we ’ re going to come spill about that later .
In the old days when you bought a stock person on a bicycle would carry the certificate from one place–someone would carry the check from one guy to another guy and then the cyclist would get the stock certificates and ride the motorcycle back to the buyer. So it was one agent to another broke. Sometimes it took a couple of hours, so there was a spacing in between when the ridicule gave over the money and when the ridicule got the stock back, and you have to allow for that .
possibly it would take a pair days to process on everybody ’ sulfur books. indeed there ’ s a thing called ex-dividend. When you buy a neckcloth the honest-to-god buyer continues to get the dividends for a while–the old owner–until a particular date after which the raw buyer starts getting the dividends if there are any. And everybody has to understand that, because you have to allow for the actual trade technology. So all those things are going to come up late, and they were in the background of this model .
But never heed that. The indicate now is that these people, everybody good knew their own evaluation, not anybody else ’ randomness evaluation, and chaos ensued, barely any rules, and miraculously about instantaneously within less than two minutes they figured out what to do and they managed to get the football tickets into the hands of the people who liked them the most. And I ’ meter going to good say that slenderly more mathematically. You could model what everybody did as having a utility function for football tickets .
So the circus tent person, Mr. 44 gets utility program of 44 for having one ticket and tickets beyond that don ’ thymine give him any extra utility, barely still utility 44. similarly miss 6 there at the bottom she had utility 6 for a football ticket, but there ’ mho besides money in the background. So the wellbeing function depended on the football tickets and money and was U ( X ) + M. So why does that capture what went on ? Because Mr. 44, knowing that the football ticket ’ s deserving 44, he would say to himself, “ Should I get a tag or not get a ticket ? ” well, if the monetary value is under 44 the measure of money he gives up and loses is going to be less than the come of utility program he gains for the football ticket so consequently he ’ ll buy the ticket .
indeed this utility affair U ( X ) + M captures the estimate, represents the goals of the people involved in the experiment. Each of them has a different U of X, but all of them are of the shape U ( X ) + M. And so the conclusion was of the experiment and of the hypothesis, issue equals requirement, the stopping point is that the football tickets are going to end up in the hands of the people who like them the best. so what does that mean ?
That means that in equilibrium the final allotment maximizes total benefit. now what does that mean ? Well, each person, one, has a unlike utility function, and thus if you add up over all the people i you get the total social welfare of every single person, the economy ’ s sum social welfare. I ’ molarity now about to prove–but it should be obvious–that full wellbeing is maximized at that equilibrium that you actually found in class, about. There was merely one bantam, bantam deviation from the theory. cipher made a err, by the way. I think I mentioned this many times. I ’ ve done this experiment earlier and Mr. 12 gets so upset that he can ’ triiodothyronine buy any ticket and he ’ south standing there embarrassed that everyone else has traded and he ’ second hush sitting there with no football ticket–he ends up bidding 30 or something to get a football ticket .
cipher made a mistake this meter, and so it happened about as the theory predicted. So let ’ s barely think of the theoretical result. In the theoretical consequence where the price is 25 and those peak eight people have it that concluding allocation maximizes entire social welfare. Why is that ? Well, whatever the money distribution was you couldn ’ t change total social welfare, because if i gives up some of his money to her, to j, the full come of money is still the lapp, and if you add up the welfares of all the people you ’ re just going to get the total amount of money on the right and it won ’ t make any dispute. so therefore to maximize social welfare all you have to do is maximize the sum of–you have to put the football tickets in the hands of the people who want them the most .
That ’ second going to maximize benefit, because that maximizes the summarize of the Ui of X ’ s and rearranging the M ’ s doesn ’ thyroxine matter. So we found in equilibrium–that balance maximized sum wellbeing. So that was the master controversy for why equilibrium was such a capital mind. The greatest good to the greatest number became mathematical. It maximizes the kernel of the social welfare, so the sum utilities they called it. That was the utilitarian view of economics, utilitarian opinion. And that ’ s sort of the horizon that prevailed in 1871 .
All right now, they made one abstraction from that–we found before, which is that if you think of not eight different buyers, but one buyer, or possibly two different buyers where the utility functions are this concave affair, so they ’ re UI ( X ) + M where UI of 1 could be 44 and UI of 2 could be 44 + 40 which is 84. That consolidated person would behave precisely like the individual people who ’ five hundred bargain as many tickets as they would jointly and sol nothing would change. The football tickets would hush end up in the hands of the people who wanted them the most .
possibly it would be one guy, who held three tickets, rather of three unlike people, but the tickets would hush be in the hands of the highest evaluation holders and so you would get precisely the same stopping point. One last moral from that model is that the price turned out to have nothing to do with the total respect of football tickets. The price turned out to be adequate, more or less, to what the fringy buyer and the bare seller thought it was deserving. So Mr. 26 and Miss 24, they ’ re the ones who controlled the monetary value, what 44 think was wholly irrelevant, so that was why it was called the bare rotation in economics .
sol Adam Smith who was so at a loss because he said water is thus valuable and has such a low price, and diamonds are therefore useless actually when it comes down to it and they have such a high gear price the answer to his perplex is simply yes. Water, at the beginning Miss 44 would be 44,000, whereas if we had a same exemplar for diamonds, diamonds would besides be very big at the begin. But the point is there ’ randomness sol much water in the economy that the borderline value of an extra gallon of water is not very high. The bare respect of water is low even though the total rate, which is the area under the requirement bend, is identical high .
so that ’ s the lesson that we learned in the first class. And now we want to generalize this to a much more sophisticate model, but one you can still compute very easily, and we ’ rhenium going to see how these limited assumptions don ’ deoxythymidine monophosphate quite employment indeed well. Yes ? Please interrupt me at anytime with questions .
Student: so could you explain again how you said that the benefit routine and the utility function somehow involve three or four people together .
Professor John Geanakoplos: Yes. So her question is–I said very quickly which you ’ ll witness late, I pressed the push button excessively soon. If I press toss off and it ’ s halfway astir does that barely break the solid thing ? Anyway, so her doubt is she would like to know, again, why it is that I can sort of combine people into one person and what do I mean by that. So what I meant by that is if I have two people, Mr. 44 and Mr. 40–I ’ meter taking the buyers to be hes and the sellers to be hers–I have two guys, 44 and 40, and I set any monetary value .
If the price is above 44 neither of them will want to buy. If it ’ sulfur between 44 and 40 equitable Mr. 40 will buy [ discipline : just Mr. 44 ], and if it ’ s below 40 both of them will buy, but each of those guys is only concerned in one ticket. Mr. 44 ’ mho utility is UI of 1 is 44. UI of 2 is still 44. He doesn ’ metric ton get anything out of a second ticket. Suppose now I had a third person, a bigger person whose utility UK of 1 is 44 and UK of 2 is 84, 44 + 40. sol that person, now, is a bigger ridicule. He ’ south interested in more tickets. That ’ s what I mean by bigger. He ’ sulfur going to behave himself precisely the same way the early two separate people behaved jointly. thus if the price is above 44 he won ’ triiodothyronine bribe either tag. If it ’ randomness 50 and he buys one ticket he ’ ll have lost 50 and he ’ ll have gained 44 in utils, so he will have been worse off than when he started, and he surely won ’ t buy two. He would be even worse off .
If the price is between 44 and 40 he ’ ll say to himself, this is the bare rotation, “ If I buy the first ticket I pay 42 and I get 44 out of it, indeed I ’ ve gained two utils, but now if I think about buying a second ticket for 42 and I entirely get 40 out of it I ’ molarity going to start lose, so I ’ ll check with one ticket. So that guy will behave precisely the same way the two people individually behave, so whether it ’ randomness two guys individually or one guy together it doesn ’ t make any difference. Their sum action ’ s precisely the same. And in the end the football tickets have ended in the hands of the people who liked them the most .
possibly I need a bunch more water than you do, but if I need it much more than you do then all those gallons, possibly the first twenty gallons I drank I needed more than your first gallon and after that you started needing water ampere a lot as I did, something like that. So the tickets or the water ends up in the hands of whoever needs it most, and it may be that there is more than one unit in each person ’ s hands. Any early questions before I raise the thing again ?

Chapter 3. Equilibrium amidst Consumption and Endowments [00:16:52]

All good, so that sounds all very convincing, but it ’ s not going to turn out to be quite indeed convert. then let ’ s hear and generalize this model to a more sophisticate thing. And then I ’ megabyte following an example which is in the notes. So you find that if you read the notes–oh, so the textbook .
Because the approach I take and that we take a Yale is quite unlike than the standard approach path you ’ re not going to be able to follow a textbook. That ’ randomness why I give you a whole list of textbooks. I encourage you to read them. They ’ ra bang-up books. They ’ re celebrated people. Most of them are quite commodity friends of mine so I endorse them all, but they ’ re differently presented than this course and that ’ s why you need to rely on the notes a short bit. So permit ’ s merely take this first exemplar .
suppose now that we have two goods, but they ’ re going to be continuous. You don ’ t have to have just one football ticket or two football tickets. We have two goods which we ’ rhenium going to call X and Y, and we ’ ve got two agents, A & B. And so WA of X and Y, that ’ s the social welfare function like we had before, is going to be whatever I had, 100X – 1 half X squared + Y. And nowadays the endowments of goods, which I was a fiddling moment flying and unleash about ahead, EA of X, EA of Y, that ’ s the endowment of A, how much he has to begin with of X and Y. I say it ’ s 4 and 5,000 .
And then let ’ s make another person, WB, his wellbeing function or let ’ s say his and her social welfare function is 30X – 1 half X squared + Y, and her endowment EB of X and EB of Y equals 80 and 1,000. then this is supposed to be shorthand for an economy in which there are thousands of people, millions of people, every person characterized by the utility they get, the goals they have over the consumption goods, and their start endowments, and they ’ re all going to be thrown together and expected to trade .
Another exercise, we can do another model, let ’ s say. We ’ ra going to work out both of these, the lapp kind of examples you ’ re going to the serve in the problem set. Another exercise, I ’ m blue I can ’ t remember my examples. I won ’ t need to look at these after I write the examples down. so another one is WC ( X, Y ) = 3 quarters log X + 1 quarter log Y, and EC, the endowment of C, equals 2 and 1. And meanwhile I have another person D of X and Y whose endowment [ correction : utility ] is 2 thirds log X + 1 third log Y and her endowment 1, 2 .
And of course I could have had an economy in which they were all there at the same time, but I ’ meter equitable going to do two examples. So you see the economy consists of many people, many goods possibly, many people and each with unlike endowments and different utilities, and if you throw them all together what ’ s going to happen ? And so we have a hypothesis now, a hypothesis of equilibrium that explains what happens. And we can use a few tricks, which I ’ thousand going to teach you now, to actually solve concretely for what ’ s going to happen in each of these cases, and it ’ s very simple. And the next footstep is going to be to add finance to it and fiscal variables, but at the bottomland we still want to have economic variables .
See here we ’ ve got the consumption of two different goods adam and Y and we want to see what ’ second going to happen. All right, therefore balance is always defined by turning things into equations. So we said the equations here are going to be that. So what is A going to do ? Well, the endogenous variables are going to be PX, PY, XA, YA, XB and YB. That ’ s what everybody has to decide. In the end A has to decide, the prices have to emerge for X and Y .
We ’ rhenium assumptive, again, that these people, by the way–I have one agentive role A and one agent B, I actually mean there ’ s a million agents just like A and a million agents merely like B and they ’ re all yelling and shout at each other and they ’ rhenium in some kind of market. so if there ’ s only one agent of each character there ’ five hundred be bargaining and threats and it ’ vitamin d be identical complicated, but with lots of people of each type, that ’ s what I ’ megabyte talking about–so in our football ticket model there were sixteen people competing with each other and you don ’ deoxythymidine monophosphate in truth need much more than three, or four, or five on a side, at least four, to get competition .
therefore with enough rival the hypothesis says the prices are going to emerge and people are going to look at the prices and decide how much they want to buy. What do they want to end up consuming ? so A has to make his decisions and B has to make her decisions. And indeed those are the endogenous variables. The exogenous variables were all of the 80, 1,000, 1 half, 1 times y, 100 times ten, all those numbers are exogenous. The utility functions, the endowments, are exogenous. So these are all the exogenous things. So the theory is going to say, how do you go from exogenous to endogenous and it ’ south going to be just a bunch together of equations, so what do they each want to do ?
sol A is going to maximize WA of X and Y such that, as we said, the critical insight, the budget constraint– [ clearing : here, writes but does not say out forte PX X ] + PY Y is less than or adequate to PX times EAX + PY times EAY ), but we know what these numbers are. EAX is 4 and EAY is 5,000. so A takes it for granted–theory says this, it ’ second very shocking–but it says A takes it for granted that he can sell all his endowment if he wanted to, 4 units of X and 5,000 units of Y and get the money from doing that and use the money to spend on his final examination consumptions–let ’ s call option this XA and YA–of XA and YA. So permit ’ s leave out the A ’ randomness hera for a minute because those are the choices he has. He ’ mho wants to max over X and Y, so there are many possibilities. It has to satisfy this budget constraint. And similarly Y is going to be maximizing over X and Y, WB of X, Y such that PX X + PY Y less than or adequate to PX times 80 + PY times 1,000. I think I remembered the numbers finally .
so, and now what we want to do is we want to solve for these variables so that when A, taking PX and PY as given, maximizes his utility program function, he ’ ll choose XA and XB, and B will choose YA and YB such that necessitate equals supply. And so I ’ thousand going to write, over here, possibly, demand equals supply. So we know that in the goal so whatever these choices are they ’ re going to lead to him choosing XA and YA and her choose XB and YB. And it ’ mho got to be that XA + XB= EAX + EAX which equals 4 + 80 which equals 84. And it ’ randomness got to be that YA + YB has to equal eAY + eBY which equals 5,000 + 1,000, which equals 6,000 .
I hope I ’ ve remembered everything. So all right, so those are two of the equations. issue has to equal demand, and now let ’ s equitable do a fiddling trick here to get some of the other equations. A is going to spend all his money. What ’ s the point in not spending money–because the more x he has and the more y he has, surely the more yttrium he has, the better off he is. So he ’ s not going to waste money. so this is going to turn out to be an equality here .
Okay, so that ’ south actually an equation, not fair a variable. then PX times XA ( now the actual solution ) + PY times YA has to equal PX times 4 + PY times 5,000. That ’ s an equality, and then similarly Y, she ’ s not going to waste her money. She ’ sulfur going to spend it all it she ’ second optimize, so this will turn into an equality. And therefore this will give PX times XB + PY times YB = PX times 80 + PY times 1,000. So we ’ ve got four equations, and immediately we have to do the fringy equation, the crucial marginal equation. indeed what does that say ?
We talked about this stopping point clock. You ’ ve all seen it before so I can go promptly, but this was the critical insight that took years to develop. Marx couldn ’ t figure it out. Until his dying day he was trying to understand what these marginalist guys were doing. So the idea is that if you ’ ve optimized by choosing XA and YA, if he ’ south optimize choose XA and YA, it has to be that the survive dollar he spent–he was indifferent between where he spent it. Otherwise he would have moved a dollar from one matter to the other thing. So it has to be that the bare utility of X at XA and YA divided by the price of X, so what is that ?
What is the bare utility of X ? That ’ s the derivative of X, 100 – 2 times 1 half, 100 – ten, has to equal the bare utility to A of Y–divided by the–sorry, I meant to leave room here. Equals the bare utility of A of Y evaluated at XA and YA, divided by the price of Y. So that equals 1. bare utility of Y is just 1. The derivative of Y is 1. And then we have to write the like thing for B. The bare utility of X for XB and YB divided by the price of X. So what is for B ? It ’ s ( 30 – x ) divided by PX. not very full board management. Has to equal, and this is besides going to turn out to be 1 over PY equals bare utility of B of Y, at XB, YB, all over PY. So those are the equations .
immediately, does that make feel to everybody ? I think I need to pause for a hour. I ’ m going to do precisely the same thing with that early system, but let ’ s just see if we can figure this out. thus equilibrium is this identical involve thing. What everybody does depends on what everybody else is going to do because how much should you pay for something depends on how a lot you think you can get it by offering it to some other guy. If there are a million A ’ second and a million B ’ south, you ’ rhenium dealing with one of the B ’ south, possibly the other B will give you a better conduct. So you have to think about what the other people are doing before you can decide what to do yourself .
All that is captured by the idea of the prices. Somehow people get into their minds what the best deal they can get is. That ’ s the prices, PX and PY. Given those prices, A, each agent, looks as his budget fix or her budget set and decides what to do. And what should they do ? They should equate marginal utilities. That ’ s the key insight. The bare utility program per dollar of X has to equal the fringy utility per dollar of Y. That just says that the budget fit is tangent to the emotionlessness curl. That ’ s what that says .
so you take the ratio of bare utilities–it equals the proportion of prices. And intersect multiply, it says the borderline utility per dollar, the slope of the emotionlessness swerve is bare utility, let ’ s say, of X over bare utility of Y and the gradient of the budget rig is PX over PY. So if I just put the PX down here and the borderline utility up there that fair says the bare utility of X divided by PX equals the marginal utility of Y divided by PY .
That ’ mho something that you could waste a huge total of time on. I don ’ t have to do it because I know that you all have seen it before, and the one guy who hasn ’ metric ton seen it before is going to figure it out himself. So we have a enormous advantage here. I can barely skip over that immediately and make function of that fact. So that ’ s the critical penetration. You ’ ve taken this fabulously complicated system and reduced it to a crowd of equations which you can put on a calculator, which is about–what I ’ m about to do, and solve it with a flip of a button .
so are there any questions here–let me pause again–with how I got these equations ?

Chapter 4. Anticipation of Prices [00:32:43]

So it ’ s a small bite complicated, but of run once you ’ ve understood it it ’ sulfur not sol complicated. now, who first thought of all this gorge ? The amazing thing is, by the way, these equations always have a solution. If you take distinctive equations in any discipline, physics, mathematics, good random equations, they ’ re not going to be solvable. adam squared + 1 = 0, that ’ s just one equation, doesn ’ metric ton have a solution. And if you have coincident equations why should there be a solution ?
The economic system constantly has a solution. This is an amaze fact first proved by Arrow, my thesis adviser, Debreu who did it at Yale as an assistant professor, and didn ’ triiodothyronine become tenure, and late won the Nobel Prize, which has happened several times at Yale–Arrow, Debreu, and McKenzie all individually, although these two guys ended up writing a articulation wallpaper, anyhow, they found that this system always has a solution. There ’ s something special about the economic system that has a solution that has to do with diminishing marginal utility, which we ’ rhenium not going to talk about in this class, but it ’ s quite a capture matter .
And they based their argument on an argument that Nash had given for games. And this wholly thing is identical associate to Nash equilibrium. And I ’ thousand certain you ’ ve learn of Nash and many of you have possibly seen the movie, A Beautiful Mind. Well, about five years ago, a couple of years after the movie came out, Nash is inactive identical much animated and not quite ampere cockamamie as he used to be, and so the indian Game Theory Society opened. It was founded believe it not, just five years ago despite all the brainy indian economists. The Game Theory Society was founded about five years ago and they had an opening conference where six people gave talks including Nash .
I was one of the people who gave a talk, and there were thousands of people who showed up, largely because of the movie, I mean, there was just thousands and thousands of people. sol afterwards we went on tour, traveling to a bunch of different cities, and every city we went to we ’ d get off the train or the limousine or something there ’ vitamin d be a throng of people there waiting to meet Nash and there ’ five hundred constantly be a iron conference. And after the weight-lift conference there ’ five hundred be a word picture on the front foliate of whatever city, and these were all capital cities, a city we ’ vitamin d gone to, and always there was Nash and everybody else was cropped out of the movie .
But anyhow, in one of these first conferences, I ’ meter fair illustrating Nash balance here, person said, some reporter says, “ We ’ ve seen the movie, but can you truly tell us in a password what is Nash chemical equilibrium, competitive balance, just say in a son what does it mean, what does it mean for us ? And so each of us took a try on at trying to explain what Nash equilibrium was including Nash .
It didn ’ deoxythymidine monophosphate go excessively good, the explanations, until they got to Aumann. So he was besides one of the people who spoke, and he subsequently won the Nobel Prize. But anyhow, at the prison term he hadn ’ t won it so far and he ’ second Israeli. He ’ south besides a great figure. And then Aumann says, “ That doubt reminds me, ” –I can ’ t do his Israeli accent– “ that question reminds me of the beginning press conference Khrushchev ” –who you might remember was Premier of the Soviet Union. This was in the time of Kennedy and thumping the table and the Cold War and stuff– “ the first press conference Khrushchev gave to western reporters and person said, ‘ Can you tell me in a word, report in a son the health of the russian economy, ’ and Khrushchev says, ‘ Good. ’ And then the reporter says, ‘ I didn ’ triiodothyronine truly mean one son. Take two words and tell us, what is the health of the russian economy ? ’ And Khrushchev says, ‘ not good. ’ ” So Aumann says, “ Equilibrium in one son is interaction, in two words, rational interaction. ”
So his definition managed to get into the newspapers and none of ours did. So that pretty much summarizes it. It ’ randomness interaction, but rational interaction. so, and it ’ s captured by the idea that everybody anticipates the prices and those prices are going to truly lead to the market ’ sulfur clear .
So they ’ re all anticipating the correct prices and behaving adenine optimally as they can, choosing the best thing in their budget sets. I put this on a calculator and solved it, which we ’ rhenium going to do in a second, but there ’ s a trick to solving this by hand, so I might equally well barely do the tricks by hand because on an examination, for case, I ’ m not going to be able to give–you ’ re going to use the calculator, it ’ sulfur very simple. You ’ ll interpret in one moment you can solve this on a computer, but by hand it ’ randomness worth knowing how to do and you probably know how to do this, but let me describe it. So the first thing to observe is that the prices don ’ metric ton truly matter up to scalar multiples .
Walras, by the way, was the foremost who made this argumentation. thus Walras was one of the marginalists in 1871 from Lausanne. So he says, “ Look, doubling the prices isn ’ t going to do anything. It ’ randomness merely like changing dollars into cents. ” If you look at everybody ’ s budget set and doubly PX and PY you ’ re doubling both sides of the equality. You ’ re not doing anything. therefore if PX and PY are separate of an equilibrium, 2PX and 2PY will besides be share of the balance because the prices only appear here in the budget rig and doubling them all doesn ’ metric ton do anything. so very you might a well assume that PX equals 1 .
so he says, “ Well, that gets rid of one variable. You ’ ve got six variables and six equations so you can all solve them, but there ’ s so many it seems excessively complicated. But now you got rid of one variable star, well you can besides get rid of one equation, ” he says. So how can you get rid of one equality ? Well, think we clear the ten market. We find XA, XB, YA and YB and PX and PY and all the equations are satisfied, one through five. All these equations are satisfied, one, three, four, five and six are all satisfy. We haven ’ thyroxine checked equation two though, whether that marketplace ’ randomness going to clear .
And Walras said, “ Well, it has to clear. The last market we don ’ t need to worry about. ” Why is that ? Because if XA + XB = EA + EB that means jointly all the agents are spending on adept X precisely all the money that they ’ re jointly getting by selling full X. That ’ s what the top equation says because when you multiply through the whole thing by PXthe full amount people are spending on good ten is equal to the sum come agents are getting by selling all the dear ten. indeed since everybody ’ s spending all their money that must mean the perch of their money jointly is just all their money they ’ re getting from selling good Y. They must be spending it all jointly on buying good y .
That means the next equality mechanically has to hold because everybody spent all his money sol therefore all the money jointly that was spent on commodity x equals to what ’ second purchased [ discipline : tax income received in sales ] of good ten because issue equals demand for good x. therefore dependable y it has to be that all the people, the income that they ’ ra getting on outgo [ correction ; selling ] good Y, all of that was spent on buying Y jointly, not any person, each person that ’ s selling yttrium and buying ten or something, but jointly all the money we ’ ve barely deduced spend on [ correction : received by selling ] Y had to go to buying Y, so consequently the y market is clearing excessively. so once you ’ ve cleared all the other markets you know that the survive market has to clear. indeed without loss of generalization don ’ thyroxine concern about death market .
so that reduced it to five equations and five unknowns, so that helped. We got rid of one equation and we got rid of one unknown. So we got rid of the top equation, let ’ s say, and PX. One of those two equations, the marketplace clear equations, doesn ’ thyroxine matter equally long as we do all the others, and one of the prices we can fix at 1. then as we can fix PY, let ’ s suppose, at 1, we might deoxyadenosine monophosphate well fix PY at 1. This becomes a much simpler equation. This now I can replace with 1, and this I can replace with 1. We already know what the price is of Y, it ’ south 1 .
But immediately things get identical, very dim-witted because you have ( 100-X ) over PX = 1, so I merely write that again, ( 100-X ) over PX = 1. so I bring the PX to the other side and I have 100-X = this is XA– equals PX. Another way of writing that is XA = 100-PX. then from this bed equation I ’ ve got 30-PX–30 minus–XB, good-for-nothing. These are A ’ south and this was B. I forgot the superscript. So ( 30-XB ) over PX ) has to equal–well, the borderline utility is 1 and the price is 1 so that equals 1. indeed I just have 30-XB = PX, or, in other words I have XB ( just writing this–bring it to the other slope ) = 30-PX .
So you look at the demand. This is what Walras did. He said, “ Forget about all these equations good look at demand and see where demand equals supply. ” thus here, given the price PX and PY we know without loss of generalization PY is 1. sol given PX this is how a lot A is going to demand of X. And given PX this is how much B is going to demand. And we know in balance by that lead equation that plus that has to equal 84. sol now I can solve it.

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so I know that 100-P–well, I ’ thousand just going to solve it cursorily. So it ’ second 130-2PX = 84, which implies that 46, 2PX = 46 implies that PX = 23. once you have PX = 23 then you can figure out what XA is, because 100-XA has to be PX so that implies that XA is 23 and it implies that XB is–no, XA is 77, justly, where is XA ? XA is 100-PX, indeed if PX is 23 XA has to be 77. It implies that XB, I can do XB from over here. 30-PX is 7, and certain enough 77 + 7 in truth do adequate 84. So we ’ ve cleared the circus tent commercialize .
And immediately we don ’ t have to worry about the other grocery store. We can figure out what XB is. How do we figure out what XB is ? We go into this budget set. At a monetary value of 23 he ’ south going to consume 77. That ’ south going to cost a bunch of money, and this is how much income he has, and we subtract if off, and PY is 1, we can figure out what Y is. So we can figure out from this YA and YB, and we know that that ’ randomness going to clear the commercialize, so we ’ ve solved the problem. But we can do this on a calculator .
All right, so are there any questions how I did this ? I ’ thousand going to do it one more fourth dimension with this model and then I ’ thousand going to do it on a calculator. And so this is the kind of trouble that hopefully will be second nature to you after you do the problem hardening. It ’ s a very elementary thing. Of course the first time you do it, it seems very complicated, but it ’ s a very mechanical elementary thing, but it ’ sulfur going to give us a batch of insight into the economy, so any questions ? Yes ?
Student: I was just wondering what those two lines said. I think the first son says assume. I just can ’ triiodothyronine read it .
Professor John Geanakoplos: Assume, this says, without loss of generalization that PX = 1 .
Student: And the second channel ?
Professor John Geanakoplos: Except that I took PY. This is PY not PX. PY = 1. And the second line, this one says, thus without loss of generality PY is 1, so having put PY = 1 here I then looked at these equations, ( 100-XA ) over PX = 1 over PY, but I took PY to be one so that ’ s 1 over 1 which is 1. And I took this equation which is ( 30-XB ) over PX = 1 over PY, sol ( 30-XB ) over PX = 1 over PY, but that ’ s 1/1 which is 1, thus I wrote that. So this is how I got my two critical equations. These two equations here, this and that, went down to this. And then I just rewrote this one as XA is 100-PX and this one you can write as XB is 30-PX. therefore 100-PX, 30-PX and then I added XA to XB and I got 130-2PX = 84 and I got PX. yea ?
Student: So we can set PY to any numeral ?
Professor John Geanakoplos: Any number .
Student: And get the same results ?
Professor John Geanakoplos: Yes. You ’ ll just multiply the pries. If you set PY to be 2 you ’ d have gotten PX to be 46 and you get the lapp answer .
Student: So they ’ re like relative prices ?
Professor John Geanakoplos: so the alone thing that matters is proportional prices. So this is what Walras pointed out. If you change dollars to cents you ’ re going to multiply every price by 100, but the relative price of oranges and tomatoes is going to be the same as it was earlier. So the theory lone produces proportional prices. Any other questions ? There was person else raising their hand. Nope ?
All proper, let ’ s merely do it one more time so you see you get the hang of this and then we ’ re going to talk about why the market ’ s so good and we ’ ra going to see things are getting a little bit more complicate. so let ’ s do this one. This one is going to work–oh, so I wasn ’ deoxythymidine monophosphate identical cagey here. Aha, possibly I could be apt, more apt .
sol how do we do this one ? well, we have to write down all the equations. So what are they going to be ? They ’ re going to be the lapp as before, XA + XB = EAX + EBX. [ correction : as pointed out subsequently, should be XC + XD = ECX + EDX ]. That ’ second add and requirement but this is fair 2+1 = 3, and over here we have–it ’ south not A and B any more. It ’ s C and D, I guess I called them, C and D. And now we have for the second gear one, we have YC + YD = ECY + endowment of D of Y = 1+2 = 3. All right, so that ’ s supply and demand. then we have to do the budget sets. They ’ re going to be simple. PX times XC + PYtimes YC has to equal PX times 2 + PY times 1. All right ? And then budget bent for D is PX times XD + PY times YD = PXtimes 1 + PY times 2. And last we have to do the borderline business. So what ’ s the marginal business ? So person tell me this. So we need the borderline utility of A over the price of X. What ’ second that ? What ’ s the fringy utility of X to Mr. A ?
Student: 3 fourths X .
Professor John Geanakoplos: 3 fourths what ?
Student: X .
Professor John Geanakoplos: That ’ s what you said 3 fourths X, precisely, divided by PX. So how did I do that ? I took the derived function of 3 fourths times log X. This is the only matter you have to know. The derived function of log X is 1 over X, and that ’ s going to be equal to the fringy utility of [ clarification : at the steer ] XA, YA with respect to Y over PY and that ’ s equal to what ?
Student: 1 fourth Y .
Professor John Geanakoplos: 1 fourthly Y .
Student: Are your A ’ randomness C ’ s ?
Professor John Geanakoplos: Yes, my A ’ second are C ’ south. Thank you. I ’ m glad you pointed that out. Thanks. So it ’ mho embarrassing to make all these mistakes, but you ’ ll find in 30 years you ’ ll originate make mistakes besides. So that ’ second that equation and then we have to do the like thing for Y. And I should have been more cagey and left more room, but I didn ’ metric ton. But anyhow, the last equation is going to be marginal utility. So what is the last equation ? marginal utility of D, of X, over PX = what ? 2 thirds times 1 over Y [ correction : ten ] divided by PX = what ? I ’ m not going to write out bare utility of—Y over PY is what ?
Student: < >
Professor John Geanakoplos: [ 1 third times 1 over ] YD, [ all ] divided by PY. So those are the equations. so immediately we ’ ra going to put these on a computer, but we can solve these by hand again, and were going to see it ’ s very utilitarian to be able to do this. Almost, so there ’ second another flim-flam to doing this. So the antic is we can take one of them to be 1, whichever we want to do. Take PY to be 1 or PX to be 1. thus contract PX = 1. here you see things are a fiddling snatch more symmetrical. There, there was the special Y that had changeless margin utility just like in our football tickets example. here there ’ south nothing that is constant fringy utility. ten and Y are a lot more symmetrical. so this motivate to more symmetry without the particular adam if very crucial, and the guy who first did that was actually Irving Fisher at Yale who you ’ re going to hear a fortune about very soon .

Chapter 5. Log Utilities and Computer Models of Equilibrium [00:52:53]

So it ’ s a small bite more complicated this fourth dimension. so here ’ s the critical equation, this one permit ’ s say. so now I ’ megabyte going to solve this. So how can I do this ? I want to do the lapp trick as before. I now want to solve–so let ’ s take PY to be 1 like I did earlier, take PY to be 1. thus let ’ s fair solve this equality, solve for X. It ’ s not going to be so comfortable to do this. so immediately there ’ s a enormous magic trick here. What does this say if I rewrite this ? I can bring the X down hera and I get PX times X. And I can bring the Y down here and I get PY times Y. So it says that the amount you spend on X proportional to 3 quarters is adequate to the amount–this is by the direction Mr. C–the come C spends on X proportional to 3 quarters is equal to the amount he spends on Y proportional to 1 quarter, but the total outgo on X and Y has got to be all his income. indeed basically it says that he spends 3 quarters of his income on X and 1 quarter on Y. That ’ s the crucial insight. So you can solve these logarithmic examples identical easily by that trick. So it ’ south apparent from fringy utility equation that C will spend 3 quarters of his income on X, and D will spend 2 thirds of her income on X. By the same controversy she ’ sulfur going to have 2 thirds–I can bring XDdown and her spending on X proportional to 2 thirds is equal to her spend on Y relative to 1 third base, but she ’ second spend all her income on X and Y. then clearly 2 thirds of it is being spent on X and 1 third of it Y. So that property of the logarithm utilities is no accident .
They were invented for precisely that purpose. So this is a fib you probably heard, but there is a famous–I ’ megabyte from Illinois–there was a celebrated Senator Douglas from Illinois, there have been several Douglas ’ from Illinois. One of them debated Lincoln. possibly he wasn ’ metric ton from Illinois. Lincoln was from Illinois. There was a celebrated senator named Douglas from Illinois after the Civil War and he noticed that farmers ’ undertaking tended to get 2 thirds or 3 quarters of all the income and capital the rest. So he said, “ What kind of utility program serve would make me always spend the lapp fraction of my money on a particular good. ”
And so he went to his college mathematics teacher, Cobb, and asked him if he could invent a utility affair which had the property that you always spent a situate proportion of your money on each good, and indeed Cobb invented the Cobb-Douglas utility routine and this is where it is. This is just the kernel of logs. So it ’ randomness called Cobb-Douglas utility. So it has that place, this property, that each person in this Cobb-Douglas utility spends a fasten proportion of her money on each of the goods, a different proportion on each of the goods and different people have different proportions, but any individual person can always spend a given proportion 3 quarters on X and 1 quarter on Y, 2 thirds on X, 1 quarter on Y. And because of that it ’ s very slowly to solve for equilibrium .
So we go, XA is going to be 3 quarters. What is her income ? Her income is PX times 2 + PY times 1. So her income has to be that. She ’ second going to spend 3 quarters of her income on X, so I could write XA as that. therefore this is C. And XD is going to be, she ’ s outgo 2 thirds of her income on X and 1 third on Y, so her endowments are 1 unit of X and 2 units of Y. So this is her income, PX times 1 + PY times 2, and indeed she ’ mho spending 2 thirds of it on X so therefore PX times XD, that ’ s the come of money she spends on ten, that ’ s what she spends on adam, has to be that, indeed XD is that. And if I add these two, when I add them up I have to get 2+1 = 3. So I can just now solve this .
well, I can do a magic trick and pick one of my–either PX or PY to be 1, so either one. I keep going back and away. It doesn ’ t make any difference. Let ’ s try PY as 1, can take that to be 1. And nowadays I can solve it. So this is fair 3 quarters, times ( PXtimes 2 + 1 divided by PX. then the other one is 2 thirds ( PX time 1 + 2 ) divided by PX. So I can add those, and when I add those it equals 3. thus I know that 3PX, if I multiply through by PX I get 3PX = 3 halves–hopefully I did this right–3 halves PX–this ’ ll be identical embarrass if I didn ’ t–3 halves PX + 3 quarters + 2 thirds PX + 4 thirds = 3PX. Oh, is this mighty ? So who can do this in their heads ? 3 halves PX from that is 3 halves PX, so 3 halves – 2 thirds, 3 halves is 9 sixth – 4 sixths is 5 sixths. It looks like 5 sixths PX. Does anyone believe that ? If I do this in terms of 6 that ’ s 9 sixths and that ’ s 4 over 6, that ’ second 13 over 6 and that ’ s 18 over 6, so it ’ second 5 over 6 PX. That looks justly, and 3 quarters + 4 thirds if I go to 12ths that ’ s 9 twelfths and 16 twelfths is 25 twelfths, so this looks like 5 halves .
So that means PX = 5 halves. So there I ’ ve got the answer. Does that look good to you ? Is this open what I did here ? I just took for this trick all I did was I solved for–so lease ’ s just repeat what I did. Just like over there I reduced it to coincident equations in a mechanical way, in a very bare aboveboard mechanical way which the first gear time you see search very complicate, but it ’ south identical simple, in fact, after you ’ ve done it once. then it allows you to take these very complicate models and say something concrete .
thus I ’ ve got all the peoples ’ –their benefit functions and their endowments. So I say in equilibrium what has to happen. Whatever they decide to eat C and D what he eats plus what she eats has to be the endowment. The sum endowment is 3. So the entire consumption of X has to be 3. The full consumption of Y between what he eats and what she eats besides has to be three. nowadays each of them is going to spend all their money. He ’ mho going to spend all his money. She ’ mho going to spend all her money .
Because it ’ randomness Cobb-Douglas, because it ’ second logarithmic, and you do this marginal utility stuff you find out, and this was the only antic, so this is a non-obvious trick which some senator and professional mathematician had to invent, Cobb-Douglas is designed then that you can say right away with those utility functions D is clearly going to spend 2 thirds of her money buy X, and C is going to spend 3 quarters of his money buy X. It ’ s just obvious from the first gear arrange conditions, from this fringy utility conditions, they ’ ra called first arrange conditions, from this equating fringy utilities .
That was the crucial trick. So that ’ s a trick that you have to internalize and from now on that ’ s all you have to know that C ’ s going to spend 3 quarters of his money on X, 1 quarter of his money on Y and D ’ s going to spend 2 thirds of her money on X and 1 third base on Y, but supply has to adequate need. So what is C ? What is he actually buying ? here ’ s his total money. He has two units of X and one unit of Y. so he ’ s selling his units of ten at the price PX, and his units of Y at the price PY, and he ’ s spending 3 quarters of it on X. So how much x is he actually buying ?
This is the sum of money he ’ mho spend on X. Divide by the price of PX, that ’ s how much money he buys of x. She, D, she ’ south going to spend, here ’ s her income which is not quite the same as his, because her endowment is different, that ’ s her income. She spends 2 thirds of it on X, so the come of X she wants to buy is the come of money she spends on it divided by the price. That ’ s how much she wants to buy. now I just have to add XC + XD and it ’ s very hard for me to do at the board and you to follow there, but of course if you stare at the page for a hour at home it ’ ll be very simple to follow .
I do Walras ’ flim-flam. I said I can constantly take PY to be 1, and if I take PY to be 1 I ’ meter going to get this income is PX [ times ] 2 times [ correction : plus ] 1 times 1 which is precisely 1, so 3 quarters of this divided by PX, that ’ s what he ’ south buying. She ’ second buy 2 thirds of her income which is PX times 1 + 1 times 2 which is plus two, divided by PX, that ’ s what she ’ randomness buy, and I just add this to this and do a little algebra. So I precisely add and do a little algebra and lo and behold PX is equal to 5 halves. so I happened to remember that ’ s the right number so I actually did this right. so PX is equal to 5 halves, and we ’ ve solved the wholly problem .
so if PX is peer to 5 halves how much is she actually buying of X ? Well, I could always plug this bet on in, chew in PX is equal to 5 halves and find out that ’ s 5 + 1 is 6, times 3 quarters, divided by 5 halves. That would tell me how much XCshe was buying. so and I could plug in 5 halves for PX and I ’ vitamin d get how much D was buying of X and I could besides plug PX= 5 halves and PY = 1, and find out what they were doing of good D. So you can solve it by pass very easily, but let ’ s just solve it by calculator alternatively unless there ’ s a doubt. Ha, I stopped it. any questions about what I did here ? Yes ?
Student: So we fair maximized utility so there ’ mho no early allotment of utilities any greater ?
Professor John Geanakoplos: Well, now we haven ’ thymine suffer here even. I ’ ve ply over a little spot, so I ’ megabyte going to finish the classify by repeating this calculation on a calculator merely by pressing a push button and you ’ ll see what the answer is. But then we have to examine the question, have we in truth maximized utility here, and to give away the punch pipeline, that utility was very special. It was changeless fringy utility of 1 in a particular good Y. That ’ s what made this exercise as about identical to the football ticket exemplar. The final equilibrium is going to maximize the summarize of utilities .
here, this equilibrium is not going to maximize the union of utilities. There ’ sulfur no argue it should maximize the summarize of utilities. And thus you need a different definition of why the rid market is such a good thing. So economists made a enormous err. They thought that the original criterion for a good grocery store is you maximize the sum of utilities. That ’ s not even true in an exemplar like this one, so we need a different definition which we call Pareto efficiency that illustrates why the market ’ south good .
But if they made a error once it stands to reason they could make a mistake another time. so there ’ s something special even about this exercise. When we put in fiscal variables I ’ megabyte going to argue you shouldn ’ deoxythymidine monophosphate expect to get the optimum result all the fourth dimension, but that ’ ll be following class. Yes ?
Student: Beyond like arithmetic use is there any reason you would choose to assume PY or PX is 1 or it ’ south arbitrary ?
Professor John Geanakoplos: No. There ’ mho no rationality to pick PX or PY to be 1, whichever one you want you can choose to be 1, and I keep going back. I can never make up my thinker which one to do, so yeah, barely whatever it works out. This one it clearly worked out arithmetically easy to take PY = 1 because the fringy utility of Y was 1 and that canceled everything out. here I could have taken either one price to be 1 and it wouldn ’ t have helped. So I picked PY to be 1 again .
In the last five minutes let ’ s good show how to solve this by calculator. So this is something you besides are going to be able to do. And it sounds like, “ Oh, there ’ s thus many complicated things. There ’ south these modern equations, ” if you do this for the problem set, after you ’ ve done it once for the problem set–you may have a little perturb with the problem set, the TAs will help you, but after you do it once this will be very simple .
now, doing it by computer is besides very bare. And it ’ mho going to sound complicate, but as all you young people know if any old guy can figure out how to work a computer you can do it vastly quicker. So let ’ s fair take the second example here. And we have five minutes left. That ’ s all it ’ ll take. so this is Excel. now, Excel is this program that ’ s made zillions of dollars. The inventor of Excel, by the way, was the inventor of Lotus. Oh, what was the guy ’ sulfur name ? His sister was in my classify at Yale. He was two years ahead of me. not Gabor, Mitch Gabor, [ correction : Mitch Kapor ] something like that was his name. anyhow, she was in my class, and he was two years ahead of me. And he invented this thing called Lotus, which made a batch of money. And then it got bought out by a few people .
And then Excel precisely basically copied the entire thing, Microsoft, and made a fortune and had to pay him off for plagiarizing the thing. But anyhow, it ’ sulfur basically Mitch Gabor was the inventor, a Yale undergraduate two years ahead of me. So he ’ s a billionaire now. then let ’ s good solve the problem. Let ’ s do the moment one because I may not have time for the first matchless. So what did I do ? I said let ’ s write down the exogenous variables first, blue let ’ s just go up a little. So the exogenous variables are the endowment of X, of the two goods, A and B, that ’ s 2 and 1, and B is 1 and 2. now what are the variables ? PX, PY, XA, YA, XB and YB, we don ’ thyroxine what those are .
so I ’ ve plugged in PX and PY. I ’ ll guess both of them are 1, which is obviously going to be incorrectly, and I ’ ll think that people barely end up with their endowments, which is obviously not right. sol then I look at the budget set. So those are my guesses. These are the endogenous variables and raving mad guesses about the solution. now, what are the equations ?
well, we wrote them down. There ’ s the budget set of A, so that ’ randomness precisely the budget set of A. So how do you write these equations down ? You simply name the–it ’ s up here if you haven ’ metric ton used Excel before, up hera. You write down the letter, say B35 that ’ s PX, so B35 times B31. That ’ south PX, is B35 times endowment XA. That ’ s the income. I wrote the income first. That ’ s the income A has minus how much she spends or he spends. B35 times B37, B35 remember is the price of X, B37 is how much he buys. So that ’ s good the budget plant .
so for each of these equations, the bare utility program, I just did the same matter. Remember the 3 quarters, over PX times X = 1 quarter over PY times Y, so this [ deviation ] should be equal to zero .
rather of saying this equals that I subtract the right hand side from the leave hired hand slope. So you want all these equations to be equal to zero. I good wrote down the six equations. And so Excel now tells me that of course the budget rig is going to be satisfied mechanically because people are consuming their endowments. And the budget put of B is mechanically satisfy because I fair had them choosing their endowments. And markets are going to clear, of naturally, because everybody ’ s choosing their endowments, but they ’ re not optimizing. then this bare utility stuff is all screwed up .
sol what do I do on the right ? For every error in the equation I square it. then I ’ ve squared all the errors. So these are my equations I need to satisfy, one, two, three, four, five, six equations. One, two, three, four, five, six, and I summed the squares. So if I make the summarize of the squares zeros each of those has to be zero. so excel, now, can minimize the sum of square errors. Excel is going to search over all endogenous variables, PX through YB to find the things that makes this count vitamin a small as possible. once this number becomes zero it means all the ones above it have to be zero because they ’ re all feather numbers adding up to that, and so I will find the solution. So you see that all you have to do–if you ’ ve done this before of path it ’ randomness obvious, if you haven ’ thyroxine it ’ s just so dim-witted to write the equality .
add and demand I good name the box, B32 that ’ s the endowment of YA + the endowment of YB equals the consumption of A plus the consumption of B. That ’ s the remainder we want to make zero. so here ’ s how you solve it. There ’ s a thing called problem solver. So you go to tools and you hit problem solver, and now solver says you want to take a target cell. I cheated. I already knew what it was, C47. So it ’ s the target cell. I hit minimize, so I want to minimize that. And now what cells do I change ? Well, I have to tell Excel what to search over. then immediately Excel, what are the cells ?
I could say PX, PY, you know, all the endogenous variables, but I know I can fix PX to be 1 so I ’ meter going to forget that one and I ’ m going to say precisely these five, good ? I don ’ t need all six of them, barely five because I can constantly take PX to be 1. So the problem solver now knows it wants to minimize this act, which is the squared errors of all the equations I want to hold peer, it ’ mho going to minimize that by searching over all those numbers. It ’ s not very smart about searching for it, and sometimes it never finds an answer. We know there always is an answer, and then how do you solve it ? You merely hit solve and it ’ second going to search and do it .
And what should the answer be ? If I fix PX to be 1, remember the answer was when PY is 1, X turns out to be 5 halves. If I fix PX to be one, what should PY be ? The solution we got ahead was PX = 5 halves and PY is 1. now I ’ megabyte going to fix PXto be 1, so what should Y be ? X was 5 halves times Y, then Y should be .4, thus if this solves right we should get PY to be .4. so I precisely hit solve and voilà I get PY to be .4. I find XA is 1.8. I find all the numbers. I just solved it fair like that immediately. So you can see how useful it ’ south going to be to use problem solver and do these problems .
Student: If you change the endowments does that change it ?
Professor John Geanakoplos: Of course. If I change the endowments I ’ ll get a different answer, and if I increase the endowments yes it does and that ’ s very authoritative .
Student: < > increasing it < > .
Professor John Geanakoplos: If I double everybody ’ mho endowment ?

Student: If you double one endowment .
Professor John Geanakoplos: If I double one endowment that ’ mho going to change things around. If I double everybody ’ s endowment it won ’ metric ton change anything, yeah .
[ end of transcript ] Back to Top

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