WEBVTT 00:00:00.000 --> 00:00:01.968 align:middle line:90% [SQUEAKING] 00:00:01.968 --> 00:00:03.936 align:middle line:90% [RUSTLING] 00:00:03.936 --> 00:00:05.904 align:middle line:90% [CLICKING] 00:00:05.904 --> 00:00:11.830 align:middle line:90% 00:00:11.830 --> 00:00:14.250 align:middle line:84% ROBERT TOWNSEND: All right, so let's get started. 00:00:14.250 --> 00:00:16.355 align:middle line:84% Are there any questions from last time? 00:00:16.355 --> 00:00:20.860 align:middle line:90% 00:00:20.860 --> 00:00:24.900 align:middle line:84% So as you are no doubt well aware, 00:00:24.900 --> 00:00:28.470 align:middle line:84% this is the last lecture I'm going to give. 00:00:28.470 --> 00:00:34.330 align:middle line:84% The exam is a week from yesterday, on Monday. 00:00:34.330 --> 00:00:39.240 align:middle line:84% So what I want to do today is to review the material 00:00:39.240 --> 00:00:41.980 align:middle line:90% and see if there are questions. 00:00:41.980 --> 00:00:47.490 align:middle line:84% So today, we're covering the material from lectures 16 00:00:47.490 --> 00:00:50.220 align:middle line:90% on, 16 through 21. 00:00:50.220 --> 00:00:54.000 align:middle line:84% And normally, the exam would cover 16 through 21 00:00:54.000 --> 00:00:55.920 align:middle line:84% and I could have written that in there. 00:00:55.920 --> 00:00:58.470 align:middle line:84% I am just trying to avoid confusion though 00:00:58.470 --> 00:01:01.560 align:middle line:84% because the course is cumulative and one thing 00:01:01.560 --> 00:01:03.150 align:middle line:90% builds on another. 00:01:03.150 --> 00:01:08.340 align:middle line:84% Essentially, it will focus on 16 onwards, 00:01:08.340 --> 00:01:11.700 align:middle line:84% but you should be, by now, very familiar 00:01:11.700 --> 00:01:13.500 align:middle line:90% with the early material. 00:01:13.500 --> 00:01:15.430 align:middle line:90% And some of it may come up. 00:01:15.430 --> 00:01:18.690 align:middle line:84% So this is just to prevent me from misleading you. 00:01:18.690 --> 00:01:23.250 align:middle line:84% If I said it was only on the last one third of the class, 00:01:23.250 --> 00:01:27.090 align:middle line:84% then you may get lulled into thinking 00:01:27.090 --> 00:01:30.030 align:middle line:84% that it's quite narrowly about only those things, 00:01:30.030 --> 00:01:33.630 align:middle line:84% as opposed to the things on which it is building. 00:01:33.630 --> 00:01:36.450 align:middle line:84% Anyway, just one word of clarification. 00:01:36.450 --> 00:01:39.460 align:middle line:84% And again, it's only the third midterm. 00:01:39.460 --> 00:01:43.150 align:middle line:84% It doesn't count anything special in terms of your grade. 00:01:43.150 --> 00:01:43.950 align:middle line:90% Let's see. 00:01:43.950 --> 00:01:47.670 align:middle line:84% So last time, we talked about money. 00:01:47.670 --> 00:01:50.010 align:middle line:90% Bitcoin, in particular. 00:01:50.010 --> 00:01:53.790 align:middle line:84% And I singled out two questions, one of which 00:01:53.790 --> 00:01:56.610 align:middle line:84% I kind of mentioned in class, but let's 00:01:56.610 --> 00:01:58.620 align:middle line:90% get you guys engaged in this. 00:01:58.620 --> 00:02:03.480 align:middle line:84% A central banker, namely, Dudley at New York Fed 00:02:03.480 --> 00:02:06.180 align:middle line:84% is heard to say that unlike Fiat money, 00:02:06.180 --> 00:02:08.310 align:middle line:84% Bitcoin's positive value in markets 00:02:08.310 --> 00:02:11.950 align:middle line:84% is just a bubble and with wild fluctuations. 00:02:11.950 --> 00:02:16.200 align:middle line:84% So Bitcoin should be regulated, if not banned altogether. 00:02:16.200 --> 00:02:20.520 align:middle line:84% And then it says discuss this statement using the tools 00:02:20.520 --> 00:02:22.570 align:middle line:90% you have learned in the class. 00:02:22.570 --> 00:02:25.725 align:middle line:84% So could I get some volunteers to answer that question? 00:02:25.725 --> 00:02:37.662 align:middle line:90% 00:02:37.662 --> 00:02:39.370 align:middle line:84% STUDENT: To be honest, I'm pretty unsure, 00:02:39.370 --> 00:02:41.530 align:middle line:84% but I do remember we talked about how you can 00:02:41.530 --> 00:02:44.440 align:middle line:90% have equilibrium with bubbles. 00:02:44.440 --> 00:02:47.440 align:middle line:84% So they're not inherently something 00:02:47.440 --> 00:02:49.150 align:middle line:90% that you have to regulate. 00:02:49.150 --> 00:02:52.540 align:middle line:84% It's possible to still have an equilibrium, despite Bitcoin 00:02:52.540 --> 00:02:55.350 align:middle line:90% being overvalued. 00:02:55.350 --> 00:02:56.430 align:middle line:90% ROBERT TOWNSEND: Right. 00:02:56.430 --> 00:02:57.240 align:middle line:90% That's true. 00:02:57.240 --> 00:03:01.690 align:middle line:90% 00:03:01.690 --> 00:03:04.690 align:middle line:84% The question actually asks about Fiat money, 00:03:04.690 --> 00:03:07.430 align:middle line:90% I kind of slipped that in there. 00:03:07.430 --> 00:03:12.080 align:middle line:84% That unlike Fiat money, Bitcoin is just a bubble. 00:03:12.080 --> 00:03:14.360 align:middle line:84% Can you maybe respond to that part? 00:03:14.360 --> 00:03:18.740 align:middle line:90% 00:03:18.740 --> 00:03:20.870 align:middle line:84% Maria, or anyone else for that matter. 00:03:20.870 --> 00:03:24.890 align:middle line:90% 00:03:24.890 --> 00:03:27.370 align:middle line:84% STUDENT: I mean, I guess that is somewhat true, 00:03:27.370 --> 00:03:34.490 align:middle line:84% as in it's not tied to a tangible asset underlying it. 00:03:34.490 --> 00:03:37.580 align:middle line:84% I'm not sure what exactly you're looking for here though. 00:03:37.580 --> 00:03:38.510 align:middle line:90% ROBERT TOWNSEND: OK. 00:03:38.510 --> 00:03:41.170 align:middle line:90% Fiat money is a bubble. 00:03:41.170 --> 00:03:44.800 align:middle line:84% It's a bubble the way that Bitcoin is potentially 00:03:44.800 --> 00:03:47.150 align:middle line:90% a bubble. 00:03:47.150 --> 00:03:50.410 align:middle line:84% Fiat money used to be backed by gold. 00:03:50.410 --> 00:03:52.740 align:middle line:84% So there, if you didn't want to hold the notes, 00:03:52.740 --> 00:03:55.200 align:middle line:84% you could go to the Federal Reserve 00:03:55.200 --> 00:04:01.140 align:middle line:84% and demand a real commodity, so then it would not potentially 00:04:01.140 --> 00:04:06.700 align:middle line:84% be a bubble because it was just a certificate for a commodity 00:04:06.700 --> 00:04:09.040 align:middle line:90% that would have real value. 00:04:09.040 --> 00:04:11.140 align:middle line:84% But when Nixon broke the link to gold, 00:04:11.140 --> 00:04:15.070 align:middle line:84% then Fiat money is just pieces of paper 00:04:15.070 --> 00:04:17.980 align:middle line:90% and nevertheless, it has value. 00:04:17.980 --> 00:04:21.790 align:middle line:84% So the point of the lecture was that Fiat money is a bubble. 00:04:21.790 --> 00:04:26.560 align:middle line:84% Now we use the word "bubble" is almost deliberately provocative 00:04:26.560 --> 00:04:28.135 align:middle line:90% because it sounds bad. 00:04:28.135 --> 00:04:31.930 align:middle line:90% 00:04:31.930 --> 00:04:36.750 align:middle line:84% But Dudley, who made the statement seem to have not 00:04:36.750 --> 00:04:40.980 align:middle line:84% let's say, forgotten-- seems to have forgotten that his 00:04:40.980 --> 00:04:41.520 align:middle line:90% domain-- 00:04:41.520 --> 00:04:45.160 align:middle line:84% Fiat money-- is potentially fragile, as well. 00:04:45.160 --> 00:04:47.250 align:middle line:84% Now, the difference is the Federal Reserve 00:04:47.250 --> 00:04:50.670 align:middle line:90% behaves itself, by and large-- 00:04:50.670 --> 00:04:54.810 align:middle line:84% or not to jump in with some parenthetical comments on that, 00:04:54.810 --> 00:04:57.160 align:middle line:84% but let's argue that that's the case. 00:04:57.160 --> 00:05:00.810 align:middle line:84% So then with forward guidance and so on, 00:05:00.810 --> 00:05:03.160 align:middle line:84% no one is expecting high inflation, 00:05:03.160 --> 00:05:07.185 align:middle line:84% which would lead to speculations about how much 00:05:07.185 --> 00:05:09.370 align:middle line:84% Fiat money is really worth, and so on. 00:05:09.370 --> 00:05:14.110 align:middle line:84% So whereas with Bitcoin, we don't have a long history. 00:05:14.110 --> 00:05:16.480 align:middle line:84% We haven't set expectations, and so on. 00:05:16.480 --> 00:05:19.530 align:middle line:84% So there are differences between the two. 00:05:19.530 --> 00:05:23.040 align:middle line:84% But in a model, it's pretty hard to make a distinction 00:05:23.040 --> 00:05:27.190 align:middle line:84% because they're both, essentially, worthless 00:05:27.190 --> 00:05:28.360 align:middle line:90% even though they have value. 00:05:28.360 --> 00:05:30.870 align:middle line:90% 00:05:30.870 --> 00:05:34.500 align:middle line:84% So that's what I was looking for. 00:05:34.500 --> 00:05:38.280 align:middle line:84% The second question, should currency pay interest? 00:05:38.280 --> 00:05:40.890 align:middle line:90% 00:05:40.890 --> 00:05:44.170 align:middle line:84% Or more specifically, what did we 00:05:44.170 --> 00:05:45.820 align:middle line:90% learn in the lecture about that? 00:05:45.820 --> 00:05:50.640 align:middle line:90% 00:05:50.640 --> 00:05:56.040 align:middle line:84% STUDENT: Is it related to the non-zero price of a currency 00:05:56.040 --> 00:06:00.540 align:middle line:84% because I remember we say that cash 00:06:00.540 --> 00:06:04.270 align:middle line:84% has positive value because it can be used to pay taxes. 00:06:04.270 --> 00:06:06.230 align:middle line:90% And that's a cash advance. 00:06:06.230 --> 00:06:10.740 align:middle line:90% 00:06:10.740 --> 00:06:20.080 align:middle line:84% So if the price is equal to 0, then nobody 00:06:20.080 --> 00:06:23.850 align:middle line:90% will demand the currency, right? 00:06:23.850 --> 00:06:27.790 align:middle line:84% ROBERT TOWNSEND: Yeah, you're speaking to the levels. 00:06:27.790 --> 00:06:31.270 align:middle line:84% So the point of having cash in advance or taxes and so on 00:06:31.270 --> 00:06:33.190 align:middle line:84% is that money would always have value 00:06:33.190 --> 00:06:36.250 align:middle line:84% because there's something you need to do with it, 00:06:36.250 --> 00:06:39.430 align:middle line:84% either buy stuff or pay taxes in it. 00:06:39.430 --> 00:06:43.120 align:middle line:84% This question interest implies something 00:06:43.120 --> 00:06:49.510 align:middle line:84% about how the price of money is changing over time, 00:06:49.510 --> 00:06:50.710 align:middle line:90% or that currency-- 00:06:50.710 --> 00:06:55.740 align:middle line:90% 00:06:55.740 --> 00:06:57.000 align:middle line:90% how can I put this-- 00:06:57.000 --> 00:06:59.880 align:middle line:84% the mechanical way to bear interest, 00:06:59.880 --> 00:07:03.600 align:middle line:84% even though you can't take it necessarily to a bank 00:07:03.600 --> 00:07:07.620 align:middle line:84% and exchange it for more money is 00:07:07.620 --> 00:07:14.300 align:middle line:84% to have the thing that money exchanges against be 00:07:14.300 --> 00:07:16.260 align:middle line:90% lowering in price. 00:07:16.260 --> 00:07:21.660 align:middle line:84% In other words, a deflation where the nominal price level 00:07:21.660 --> 00:07:25.320 align:middle line:84% is going down means that you can buy 00:07:25.320 --> 00:07:28.770 align:middle line:84% more goods with money tomorrow than you can buy with it today. 00:07:28.770 --> 00:07:33.620 align:middle line:84% And that's effectively like paying interest. 00:07:33.620 --> 00:07:35.450 align:middle line:84% The way to remember that is inflation 00:07:35.450 --> 00:07:38.240 align:middle line:84% is bad, obviously, because the purchasing power of money 00:07:38.240 --> 00:07:39.710 align:middle line:90% is going down. 00:07:39.710 --> 00:07:41.720 align:middle line:84% It takes more and more money to buy goods. 00:07:41.720 --> 00:07:46.220 align:middle line:84% With deflation, the purchasing power of money is going up. 00:07:46.220 --> 00:07:49.490 align:middle line:84% So that's effectively a way to pay interest. 00:07:49.490 --> 00:07:51.980 align:middle line:84% And what did we learn in class about 00:07:51.980 --> 00:07:56.390 align:middle line:84% whether or not it's optimal to engineer that? 00:07:56.390 --> 00:08:03.180 align:middle line:90% 00:08:03.180 --> 00:08:04.290 align:middle line:90% Any volunteers? 00:08:04.290 --> 00:08:09.690 align:middle line:90% 00:08:09.690 --> 00:08:10.560 align:middle line:90% Yeah, sorry. 00:08:10.560 --> 00:08:16.180 align:middle line:84% I know it's the end of the term and you're all very, very busy. 00:08:16.180 --> 00:08:19.230 align:middle line:84% So the answer was, in that model of money 00:08:19.230 --> 00:08:23.820 align:middle line:84% was spatially separated agents, there 00:08:23.820 --> 00:08:28.970 align:middle line:84% was a monetary equilibrium with a constant price level, 00:08:28.970 --> 00:08:32.740 align:middle line:90% but it was not Pareto Optimal. 00:08:32.740 --> 00:08:36.789 align:middle line:84% And to get to one of the Pareto Optimal allocations, 00:08:36.789 --> 00:08:40.179 align:middle line:84% it was necessary to engineer this kind of deflation. 00:08:40.179 --> 00:08:43.409 align:middle line:90% 00:08:43.409 --> 00:08:47.580 align:middle line:84% So that's an example of paying interest on money 00:08:47.580 --> 00:08:51.610 align:middle line:84% is actually something the Federal Reserve should 00:08:51.610 --> 00:08:52.110 align:middle line:90% be doing. 00:08:52.110 --> 00:08:55.110 align:middle line:90% 00:08:55.110 --> 00:08:56.655 align:middle line:84% I hope it's coming back to you now. 00:08:56.655 --> 00:09:00.120 align:middle line:90% 00:09:00.120 --> 00:09:02.460 align:middle line:84% Anyway, those were the two main ingredients 00:09:02.460 --> 00:09:06.180 align:middle line:84% of this very last lecture on bubbles. 00:09:06.180 --> 00:09:09.140 align:middle line:84% So I probably won't, if history is any guide, 00:09:09.140 --> 00:09:14.090 align:middle line:84% I won't have time to open up lecture, this last lecture 21 00:09:14.090 --> 00:09:18.920 align:middle line:84% and review it, but those were the two principal components. 00:09:18.920 --> 00:09:25.265 align:middle line:84% Namely, money as a bubble and this policy aspect. 00:09:25.265 --> 00:09:33.170 align:middle line:90% 00:09:33.170 --> 00:09:38.630 align:middle line:84% So again, the study guide should help you prepare for the exam. 00:09:38.630 --> 00:09:43.910 align:middle line:84% And now we're ready to review the lectures. 00:09:43.910 --> 00:09:45.480 align:middle line:90% There's only six of them. 00:09:45.480 --> 00:09:50.330 align:middle line:84% So this material is probably pretty fresh in your minds. 00:09:50.330 --> 00:09:53.720 align:middle line:84% I think what I'll do rather than obviously I wouldn't have time 00:09:53.720 --> 00:09:57.290 align:middle line:84% anyway to go over all the individual slides 00:09:57.290 --> 00:09:59.690 align:middle line:84% of each lecture, I will try to remind you 00:09:59.690 --> 00:10:03.770 align:middle line:84% what we did and maybe draw some connections 00:10:03.770 --> 00:10:06.830 align:middle line:84% to the other lectures and how things are kind of fitting 00:10:06.830 --> 00:10:07.650 align:middle line:90% together. 00:10:07.650 --> 00:10:11.720 align:middle line:84% So this was the lecture on the fundamental welfare theorems, 00:10:11.720 --> 00:10:14.480 align:middle line:84% namely, that competitive equilibria 00:10:14.480 --> 00:10:16.850 align:middle line:84% under certain assumptions are Paretp Optimal. 00:10:16.850 --> 00:10:20.240 align:middle line:84% And any optimum can be supported with transfers. 00:10:20.240 --> 00:10:23.120 align:middle line:84% So this lecture was about sufficient assumptions 00:10:23.120 --> 00:10:25.520 align:middle line:90% and the corresponding proofs. 00:10:25.520 --> 00:10:30.350 align:middle line:84% And it ended with extensions to more general commodity spaces. 00:10:30.350 --> 00:10:33.680 align:middle line:84% So you may remember that I did this. 00:10:33.680 --> 00:10:36.320 align:middle line:84% I showed this-- back to the earlier lectures, 00:10:36.320 --> 00:10:38.330 align:middle line:84% the first part of the course on risk sharing, 00:10:38.330 --> 00:10:41.660 align:middle line:84% I showed the income movements of households 00:10:41.660 --> 00:10:44.570 align:middle line:84% over time and compared to each other 00:10:44.570 --> 00:10:48.470 align:middle line:84% and including the consumption, we noted this gradient here 00:10:48.470 --> 00:10:51.170 align:middle line:84% that the average level of consumption 00:10:51.170 --> 00:10:54.380 align:middle line:84% is increasing as one moves from landless households 00:10:54.380 --> 00:10:56.810 align:middle line:90% to larger landowners. 00:10:56.810 --> 00:10:58.640 align:middle line:84% In the risk sharing framework, that 00:10:58.640 --> 00:11:02.000 align:middle line:84% was related to lambda, the Pareto weight. 00:11:02.000 --> 00:11:04.400 align:middle line:84% And very clearly, for certain functional forms, 00:11:04.400 --> 00:11:05.900 align:middle line:90% it was exactly that. 00:11:05.900 --> 00:11:09.020 align:middle line:84% Pareto weight of one household relative to the others. 00:11:09.020 --> 00:11:12.800 align:middle line:84% But the motivation here was to introduce the welfare theorems. 00:11:12.800 --> 00:11:16.310 align:middle line:84% So then the issue was whether the lambdas that 00:11:16.310 --> 00:11:18.770 align:middle line:84% are implicit in that diagram are related 00:11:18.770 --> 00:11:23.570 align:middle line:84% to wealth, like landholdings and the number of plow animals, 00:11:23.570 --> 00:11:24.810 align:middle line:90% and so on. 00:11:24.810 --> 00:11:29.780 align:middle line:84% So I was anticipating a connection between lambdas 00:11:29.780 --> 00:11:34.190 align:middle line:84% as in the Pareto problem of Pareto Optimal allocations 00:11:34.190 --> 00:11:37.310 align:middle line:84% versus the wealth that may be needed 00:11:37.310 --> 00:11:39.050 align:middle line:90% in a competitive equilibrium. 00:11:39.050 --> 00:11:42.510 align:middle line:84% You'll see that again, or be reminded of it momentarily. 00:11:42.510 --> 00:11:47.960 align:middle line:84% So the first welfare theorem is that any price equilibrium-- 00:11:47.960 --> 00:11:50.600 align:middle line:84% competitive equilibrium, including a price equilibrium 00:11:50.600 --> 00:11:54.950 align:middle line:84% with transfers is Pareto Optimal under minimal sufficient 00:11:54.950 --> 00:11:58.250 align:middle line:84% assumptions, namely rational preferences-- 00:11:58.250 --> 00:12:00.890 align:middle line:84% which we haven't reviewed for a while but it was in lecture 00:12:00.890 --> 00:12:02.150 align:middle line:90% two-- 00:12:02.150 --> 00:12:07.250 align:middle line:84% complete transitive and continuous preferences 00:12:07.250 --> 00:12:09.350 align:middle line:90% and local non-satiation. 00:12:09.350 --> 00:12:12.770 align:middle line:84% With those two assumptions, we get this sufficient 00:12:12.770 --> 00:12:14.430 align:middle line:90% for this theorem. 00:12:14.430 --> 00:12:17.420 align:middle line:84% So this was a picture in our familiar Edgeworth 00:12:17.420 --> 00:12:23.240 align:middle line:84% box of a competitive equilibrium being Pareto Optimal. 00:12:23.240 --> 00:12:27.590 align:middle line:84% And the proof, you may remember, relied 00:12:27.590 --> 00:12:32.720 align:middle line:84% on starting with a given allocation, the star allocation 00:12:32.720 --> 00:12:36.110 align:middle line:84% and then conjuring up, by contradiction, that there might 00:12:36.110 --> 00:12:38.510 align:middle line:90% be something Pareto improving. 00:12:38.510 --> 00:12:42.620 align:middle line:84% We prove this little Lemma, that anything which is weekly better 00:12:42.620 --> 00:12:45.810 align:middle line:90% has to cost us at least as much. 00:12:45.810 --> 00:12:47.840 align:middle line:84% And then we just went through the math, 00:12:47.840 --> 00:12:49.970 align:middle line:84% summing up over all the households. 00:12:49.970 --> 00:12:53.270 align:middle line:84% Obviously, if it's strictly better for some households 00:12:53.270 --> 00:12:55.580 align:middle line:84% i prime, then it must have costed more, 00:12:55.580 --> 00:12:57.260 align:middle line:84% or otherwise, they would have chose it. 00:12:57.260 --> 00:13:00.170 align:middle line:84% So we get, in equation one, the statement 00:13:00.170 --> 00:13:04.340 align:middle line:84% about total expenditures, the value of total expenditures 00:13:04.340 --> 00:13:05.840 align:middle line:90% summing up. 00:13:05.840 --> 00:13:08.720 align:middle line:84% All the households-- I'm going quickly now-- 00:13:08.720 --> 00:13:10.970 align:middle line:90% we used profit maximization. 00:13:10.970 --> 00:13:14.840 align:middle line:84% We coupled equations one and two to get a statement 00:13:14.840 --> 00:13:16.550 align:middle line:90% about resource constraints. 00:13:16.550 --> 00:13:19.370 align:middle line:84% And at the bottom line, concluded 00:13:19.370 --> 00:13:22.700 align:middle line:84% that this alternative Pareto dominating allocation would not 00:13:22.700 --> 00:13:26.660 align:middle line:84% be feasible in contradiction to what we assumed. 00:13:26.660 --> 00:13:27.980 align:middle line:90% So that was QED. 00:13:27.980 --> 00:13:31.650 align:middle line:84% That was the end of the proof of the first welfare theorem. 00:13:31.650 --> 00:13:34.670 align:middle line:84% Now later, even today, we'll review it. 00:13:34.670 --> 00:13:38.690 align:middle line:84% This thing fails with if these sums are infinite sums 00:13:38.690 --> 00:13:41.720 align:middle line:90% and don't have finite value. 00:13:41.720 --> 00:13:46.520 align:middle line:84% Then the second welfare theorem was illustrated here 00:13:46.520 --> 00:13:50.270 align:middle line:84% that we have Pareto Optimum characterized by tangency 00:13:50.270 --> 00:13:51.800 align:middle line:90% of indifference curves. 00:13:51.800 --> 00:13:55.440 align:middle line:84% But that nevertheless, it's not reachable 00:13:55.440 --> 00:13:58.100 align:middle line:84% if we start from the endowment and start 00:13:58.100 --> 00:14:02.180 align:middle line:84% drawing potential budget lines through the endowment point, 00:14:02.180 --> 00:14:04.430 align:middle line:84% as you would for candidate budget lines 00:14:04.430 --> 00:14:05.910 align:middle line:90% in a competitive equilibrium. 00:14:05.910 --> 00:14:09.230 align:middle line:84% So this was meant to illustrate that to get 00:14:09.230 --> 00:14:11.270 align:middle line:84% to this particular Pareto Optimum that's 00:14:11.270 --> 00:14:13.460 align:middle line:84% in the second welfare theorem, we're 00:14:13.460 --> 00:14:19.010 align:middle line:84% going to have to take some value away from agent A over here 00:14:19.010 --> 00:14:23.150 align:middle line:84% and give it to Agent B. And that's what this said. 00:14:23.150 --> 00:14:28.820 align:middle line:84% The tax on A is equal to the subsidy on B. T being transfer, 00:14:28.820 --> 00:14:31.410 align:middle line:90% so negative is a tax. 00:14:31.410 --> 00:14:34.100 align:middle line:84% So then we could get, as in the second welfare theorem, 00:14:34.100 --> 00:14:38.330 align:middle line:84% to any allocation along that contract curve where 00:14:38.330 --> 00:14:42.230 align:middle line:84% the indifference curves are tangent by reallocating wealth. 00:14:42.230 --> 00:14:46.490 align:middle line:84% Now to prove it, we went through this constructive argument. 00:14:46.490 --> 00:14:48.080 align:middle line:90% We defined an economy. 00:14:48.080 --> 00:14:50.030 align:middle line:84% And this slide, or a version of it, 00:14:50.030 --> 00:14:53.660 align:middle line:84% shows up at least three times in these six lectures. 00:14:53.660 --> 00:14:56.240 align:middle line:84% So an economy consisting of preferences 00:14:56.240 --> 00:15:00.290 align:middle line:84% endowments production sets and profit shares here, 00:15:00.290 --> 00:15:01.670 align:middle line:90% assuming a lot more. 00:15:01.670 --> 00:15:06.350 align:middle line:84% Concavity and convexity in preferences, production sets, 00:15:06.350 --> 00:15:07.160 align:middle line:90% et cetera. 00:15:07.160 --> 00:15:09.350 align:middle line:84% And then we get the second welfare theorem, 00:15:09.350 --> 00:15:11.750 align:middle line:84% that any Pareto Optimum can be supported. 00:15:11.750 --> 00:15:15.620 align:middle line:84% For some wealth, any of optimum associated with some lambda 00:15:15.620 --> 00:15:17.900 align:middle line:84% vector of maximized weighted utilities 00:15:17.900 --> 00:15:22.250 align:middle line:84% can be supported by a suitable redistribution of wealth. 00:15:22.250 --> 00:15:25.460 align:middle line:84% The way we prove this was to go back to what 00:15:25.460 --> 00:15:28.160 align:middle line:90% you know about Lagrangians. 00:15:28.160 --> 00:15:31.430 align:middle line:84% So this was, I think, also in the second lecture 00:15:31.430 --> 00:15:32.690 align:middle line:90% of the class. 00:15:32.690 --> 00:15:35.300 align:middle line:84% And you've used it over and over again. 00:15:35.300 --> 00:15:38.270 align:middle line:84% Maximizing concave functions on convex 00:15:38.270 --> 00:15:41.640 align:middle line:84% sets, generating all the shadow prices. 00:15:41.640 --> 00:15:43.940 align:middle line:90% So we used it here three times. 00:15:43.940 --> 00:15:46.950 align:middle line:84% First we use it for the Pareto problem. 00:15:46.950 --> 00:15:50.120 align:middle line:84% So we're starting with a Pareto Optimal allocation. 00:15:50.120 --> 00:15:54.800 align:middle line:84% So we know, again, from things you did earlier in the class, 00:15:54.800 --> 00:15:57.440 align:middle line:84% when we introduced the notion of Pareto optimality-- 00:15:57.440 --> 00:15:59.690 align:middle line:84% I don't remember the lecture number-- 00:15:59.690 --> 00:16:02.720 align:middle line:84% we maximize weighted sums of utilities, 00:16:02.720 --> 00:16:05.930 align:middle line:84% lambda's being these weights, subject to resource constraints 00:16:05.930 --> 00:16:08.000 align:middle line:90% and production constraints. 00:16:08.000 --> 00:16:09.950 align:middle line:90% That's all written down here. 00:16:09.950 --> 00:16:12.380 align:middle line:84% The solution to any one of these Pareto problems 00:16:12.380 --> 00:16:15.320 align:middle line:84% is Pareto Optimal, and all the Optima 00:16:15.320 --> 00:16:18.500 align:middle line:84% can be described as solutions for some Lambdas. 00:16:18.500 --> 00:16:21.290 align:middle line:84% So we start thinking about these so-called Pareto 00:16:21.290 --> 00:16:24.800 align:middle line:84% problems as equivalent with Pareto Optimal allocations. 00:16:24.800 --> 00:16:28.310 align:middle line:84% Use the Lagrangian, set it up this way 00:16:28.310 --> 00:16:31.880 align:middle line:84% and get the first order conditions, including 00:16:31.880 --> 00:16:35.150 align:middle line:84% the primal, duel feasibility on the shadow 00:16:35.150 --> 00:16:38.420 align:middle line:84% prices, complementary, slackness and all of that. 00:16:38.420 --> 00:16:41.810 align:middle line:84% The point is that with all that concavity and convexity as 00:16:41.810 --> 00:16:45.830 align:middle line:84% appropriate, these first order conditions from differentiating 00:16:45.830 --> 00:16:50.010 align:middle line:84% the Lagrangian are not only necessary, they're sufficient. 00:16:50.010 --> 00:16:55.010 align:middle line:84% So any solution to these first order conditions, allocations 00:16:55.010 --> 00:16:59.090 align:middle line:84% and shadow prices that satisfy the first order conditions will 00:16:59.090 --> 00:17:03.720 align:middle line:84% be Pareto Optimal, will be a solution to the Pareto problem. 00:17:03.720 --> 00:17:05.760 align:middle line:90% So we did that. 00:17:05.760 --> 00:17:08.690 align:middle line:84% Then we went to the second part, reminded ourselves 00:17:08.690 --> 00:17:10.910 align:middle line:84% of the definition of a competitive equilibrium 00:17:10.910 --> 00:17:12.050 align:middle line:90% with transfers. 00:17:12.050 --> 00:17:16.790 align:middle line:84% Namely, consumer optimization, firm maximization of profits, 00:17:16.790 --> 00:17:21.200 align:middle line:84% and so on, along with a feasible distribution of wealth, as I 00:17:21.200 --> 00:17:22.550 align:middle line:90% was just saying. 00:17:22.550 --> 00:17:24.710 align:middle line:84% So here's the second and third time. 00:17:24.710 --> 00:17:27.170 align:middle line:84% We're going to use the Lagrangian for the consumer 00:17:27.170 --> 00:17:31.070 align:middle line:84% optimization problem and use it for the firm profit 00:17:31.070 --> 00:17:31.880 align:middle line:90% maximization. 00:17:31.880 --> 00:17:35.120 align:middle line:84% Here is the consumer maximization problem. 00:17:35.120 --> 00:17:37.550 align:middle line:84% Maximizing utility subject to a budget, 00:17:37.550 --> 00:17:39.210 align:middle line:84% we get this first order condition. 00:17:39.210 --> 00:17:41.570 align:middle line:84% And again, with a concavity in the constraint 00:17:41.570 --> 00:17:44.390 align:middle line:84% setting in the objective function and convexity 00:17:44.390 --> 00:17:47.120 align:middle line:84% in the constraint set, these are not only necessary, 00:17:47.120 --> 00:17:49.490 align:middle line:84% but they're sufficient condition for an optimum. 00:17:49.490 --> 00:17:54.170 align:middle line:84% All we need to do is find the mu i at given prices, p, 00:17:54.170 --> 00:17:58.175 align:middle line:84% and in Xi*, and it will be maximizing for the consumer. 00:17:58.175 --> 00:18:01.040 align:middle line:90% 00:18:01.040 --> 00:18:04.190 align:middle line:84% Likewise, for the firm maximization, 00:18:04.190 --> 00:18:07.550 align:middle line:84% we have necessary and sufficient conditions 00:18:07.550 --> 00:18:10.820 align:middle line:84% to find candidates for the decentralized competitive 00:18:10.820 --> 00:18:12.380 align:middle line:90% equilibrium. 00:18:12.380 --> 00:18:15.080 align:middle line:84% And then all we did was match them up. 00:18:15.080 --> 00:18:17.180 align:middle line:84% Now I think the tricky thing here 00:18:17.180 --> 00:18:19.280 align:middle line:84% is to remember what the given is and what 00:18:19.280 --> 00:18:20.330 align:middle line:90% the objective function. 00:18:20.330 --> 00:18:23.570 align:middle line:84% The given was a Pareto Optimal allocation. 00:18:23.570 --> 00:18:25.460 align:middle line:84% So for the consumer, we started out 00:18:25.460 --> 00:18:28.550 align:middle line:84% with these conditions, which are the necessary and sufficient 00:18:28.550 --> 00:18:31.740 align:middle line:84% conditions from the Pareto problem being satisfied. 00:18:31.740 --> 00:18:36.650 align:middle line:84% So we have the gamma sub L for commodity L as a given. 00:18:36.650 --> 00:18:38.120 align:middle line:84% The lambdas were given because we 00:18:38.120 --> 00:18:41.120 align:middle line:84% had a given certain candidate, Pareto Optimum. 00:18:41.120 --> 00:18:42.380 align:middle line:90% These were all satisfied. 00:18:42.380 --> 00:18:43.910 align:middle line:90% We're looking for this thing. 00:18:43.910 --> 00:18:47.600 align:middle line:84% We're trying to see if consumers are going to be maximizing. 00:18:47.600 --> 00:18:49.640 align:middle line:84% So the candidate for the price would 00:18:49.640 --> 00:18:51.380 align:middle line:90% be the Lagrange multiplier. 00:18:51.380 --> 00:18:54.560 align:middle line:84% And the candidate for the marginal utility 00:18:54.560 --> 00:18:57.050 align:middle line:84% of income, the shadow price and the budget constraint 00:18:57.050 --> 00:18:59.330 align:middle line:90% would be one over lambda. 00:18:59.330 --> 00:19:03.650 align:middle line:84% So here, I go back to the slides of the Rocky 00:19:03.650 --> 00:19:04.880 align:middle line:90% Mountain and Kansas. 00:19:04.880 --> 00:19:08.960 align:middle line:84% This establishes a direct relationship between the Pareto 00:19:08.960 --> 00:19:11.660 align:middle line:84% weights and the Pareto problem and the marginal utility 00:19:11.660 --> 00:19:12.200 align:middle line:90% of wealth. 00:19:12.200 --> 00:19:13.820 align:middle line:90% And hence, the level of wealth. 00:19:13.820 --> 00:19:15.410 align:middle line:90% It's quite general. 00:19:15.410 --> 00:19:17.390 align:middle line:90% It's super robust. 00:19:17.390 --> 00:19:21.200 align:middle line:84% And so then the rest of this was just the tedious part 00:19:21.200 --> 00:19:24.770 align:middle line:84% of constructing the wealth, has the evaluation of expenditure 00:19:24.770 --> 00:19:27.470 align:middle line:84% and making sure that they all add up. 00:19:27.470 --> 00:19:29.930 align:middle line:84% We refer to this way of finding the prices 00:19:29.930 --> 00:19:34.910 align:middle line:84% from the second welfare theorem as marginal utility pricing. 00:19:34.910 --> 00:19:39.200 align:middle line:84% I mean, more accurately, it's the lambda weighted margin 00:19:39.200 --> 00:19:42.590 align:middle line:84% utilities which are equated all over the households, not 00:19:42.590 --> 00:19:44.310 align:middle line:90% the levels. 00:19:44.310 --> 00:19:48.380 align:middle line:84% But when you take ratios across different goods, L and K, 00:19:48.380 --> 00:19:49.820 align:middle line:90% the marginal rates of-- 00:19:49.820 --> 00:19:51.270 align:middle line:90% the lambdas cancel out. 00:19:51.270 --> 00:19:53.840 align:middle line:84% So the marginal rates of substitution 00:19:53.840 --> 00:19:56.210 align:middle line:84% are equated over all the households, 00:19:56.210 --> 00:20:01.130 align:middle line:84% as they are in this picture, along this orange, 00:20:01.130 --> 00:20:04.040 align:middle line:90% green, orange, green? 00:20:04.040 --> 00:20:06.850 align:middle line:90% A yellow, orange line. 00:20:06.850 --> 00:20:09.310 align:middle line:84% And this made the wealth's add up. 00:20:09.310 --> 00:20:13.030 align:middle line:84% And we did a second proof of the welfare theorem, which 00:20:13.030 --> 00:20:16.570 align:middle line:84% is namely one nuanced thing, which is instead of just 00:20:16.570 --> 00:20:18.850 align:middle line:84% maximizing utility subject to budget, 00:20:18.850 --> 00:20:21.790 align:middle line:84% we minimize expenditures subject to achieving 00:20:21.790 --> 00:20:24.010 align:middle line:90% a certain level of utility. 00:20:24.010 --> 00:20:28.570 align:middle line:90% That conjured up this. 00:20:28.570 --> 00:20:33.340 align:middle line:84% So this is the target and different curve, 00:20:33.340 --> 00:20:37.690 align:middle line:84% as it were, as if it were coming from one consumer-- 00:20:37.690 --> 00:20:39.490 align:middle line:90% interesting comment. 00:20:39.490 --> 00:20:42.400 align:middle line:84% But it's just the aggregated better 00:20:42.400 --> 00:20:48.880 align:middle line:84% than set associated with the Pareto Optimal allocation. 00:20:48.880 --> 00:20:52.520 align:middle line:84% The other part of this diagram is two production functions, 00:20:52.520 --> 00:20:54.520 align:middle line:84% different from each other, but aggregated up 00:20:54.520 --> 00:20:56.800 align:middle line:84% to an aggregated production function. 00:20:56.800 --> 00:20:59.590 align:middle line:84% And then you can see, as if it were a Robinson Crusoe 00:20:59.590 --> 00:21:04.180 align:middle line:84% representative consumer, the optimum is characterized 00:21:04.180 --> 00:21:06.820 align:middle line:84% by a maximal utility level subject 00:21:06.820 --> 00:21:08.980 align:middle line:90% to the single production set. 00:21:08.980 --> 00:21:15.550 align:middle line:84% And then my point is that we get this separating hyper-plane. 00:21:15.550 --> 00:21:19.450 align:middle line:84% So all these definitions supporting hyperplane, 00:21:19.450 --> 00:21:23.800 align:middle line:84% et cetera, that's really where given all the convexity, 00:21:23.800 --> 00:21:25.510 align:middle line:90% that's all you need to do. 00:21:25.510 --> 00:21:28.280 align:middle line:84% These convex sets are separated from each other. 00:21:28.280 --> 00:21:30.220 align:middle line:84% So this is a quite general proof and it 00:21:30.220 --> 00:21:34.480 align:middle line:84% works in many spaces, not just finite dimensional Euclidean 00:21:34.480 --> 00:21:35.530 align:middle line:90% spaces. 00:21:35.530 --> 00:21:39.760 align:middle line:84% And we're about to review the representative consumer, 00:21:39.760 --> 00:21:41.590 align:middle line:84% as I said, this looks all the world 00:21:41.590 --> 00:21:46.450 align:middle line:84% like a representative consumer because I've seemingly added up 00:21:46.450 --> 00:21:48.160 align:middle line:90% the indifference curves. 00:21:48.160 --> 00:21:51.010 align:middle line:84% But one should be aware that it is not the Gorman 00:21:51.010 --> 00:21:53.320 align:middle line:90% consumer that's going on here. 00:21:53.320 --> 00:21:56.590 align:middle line:84% And this particular Pareto Optimum 00:21:56.590 --> 00:21:59.380 align:middle line:84% is sensitive to the distribution of wealth. 00:21:59.380 --> 00:22:00.790 align:middle line:90% You just can't see that. 00:22:00.790 --> 00:22:05.290 align:middle line:84% I did not go through that when I gave this lecture because we 00:22:05.290 --> 00:22:08.820 align:middle line:84% hadn't gotten to Gorman and it wouldn't have made any sense. 00:22:08.820 --> 00:22:12.640 align:middle line:84% But I might as well say it now, if you don't have a Gorman 00:22:12.640 --> 00:22:17.170 align:middle line:84% representative consumer, then as you increase this guy's budget 00:22:17.170 --> 00:22:20.770 align:middle line:84% line and decreases this one, you may 00:22:20.770 --> 00:22:24.482 align:middle line:84% change the shape of this seemingly aggregated 00:22:24.482 --> 00:22:26.440 align:middle line:84% up in indifference curve and, therefore, change 00:22:26.440 --> 00:22:28.540 align:middle line:90% the whole optimal allocation. 00:22:28.540 --> 00:22:30.490 align:middle line:84% So hopefully that helps you think 00:22:30.490 --> 00:22:33.790 align:middle line:84% about the hidden way in which heterogeneity 00:22:33.790 --> 00:22:36.690 align:middle line:90% is entering into this picture. 00:22:36.690 --> 00:22:40.570 align:middle line:84% And this lecture, I finished it off in an abstract way 00:22:40.570 --> 00:22:43.360 align:middle line:84% by saying the proofs of the welfare theorems 00:22:43.360 --> 00:22:46.050 align:middle line:84% are quite general, that does not require 00:22:46.050 --> 00:22:49.720 align:middle line:84% a finite dimensional commodity spaces. 00:22:49.720 --> 00:22:53.680 align:middle line:84% I gave you the sufficient assumptions for the two welfare 00:22:53.680 --> 00:22:57.400 align:middle line:90% theorems and kind of quit there. 00:22:57.400 --> 00:23:01.840 align:middle line:84% If we'd had more time, I could have gone into greater detail. 00:23:01.840 --> 00:23:05.320 align:middle line:84% I only asserted without proving it 00:23:05.320 --> 00:23:07.060 align:middle line:84% that if we took what we did earlier 00:23:07.060 --> 00:23:10.360 align:middle line:84% in the course about Pareto Optima 00:23:10.360 --> 00:23:13.330 align:middle line:84% with obstacles to trade with incentive 00:23:13.330 --> 00:23:16.390 align:middle line:84% constraints and potentially non-convex consumption 00:23:16.390 --> 00:23:19.420 align:middle line:84% sets because of discrete choices, 00:23:19.420 --> 00:23:22.480 align:middle line:84% we could nevertheless introduce lotteries. 00:23:22.480 --> 00:23:24.820 align:middle line:84% And that would have turned the space 00:23:24.820 --> 00:23:29.110 align:middle line:84% into a convex set and the theorems, as in Debreu, 00:23:29.110 --> 00:23:30.590 align:middle line:90% would apply to that. 00:23:30.590 --> 00:23:33.850 align:middle line:84% So although this lecture just proves the welfare theorems 00:23:33.850 --> 00:23:36.880 align:middle line:84% carefully for finite dimensional commodity spaces 00:23:36.880 --> 00:23:40.510 align:middle line:84% and only asserted that the theorems extend, 00:23:40.510 --> 00:23:43.810 align:middle line:84% it would cover competitive equilibria 00:23:43.810 --> 00:23:46.480 align:middle line:84% in contract space where households 00:23:46.480 --> 00:23:48.940 align:middle line:84% can buy whatever contracts they want, 00:23:48.940 --> 00:23:50.650 align:middle line:90% granted at certain prices. 00:23:50.650 --> 00:23:54.610 align:middle line:84% And firms as intermediaries will create contracts 00:23:54.610 --> 00:23:56.800 align:middle line:90% that maximize profits. 00:23:56.800 --> 00:23:59.110 align:middle line:84% Tying some of the loose ends together, 00:23:59.110 --> 00:24:01.940 align:middle line:84% but without actually going through much of the details 00:24:01.940 --> 00:24:02.440 align:middle line:90% there. 00:24:02.440 --> 00:24:05.410 align:middle line:90% So that was lecture 16. 00:24:05.410 --> 00:24:09.637 align:middle line:90% Questions about 16? 00:24:09.637 --> 00:24:11.470 align:middle line:84% So we'll come back to it again because we're 00:24:11.470 --> 00:24:13.060 align:middle line:84% going to go through the failures, 00:24:13.060 --> 00:24:14.980 align:middle line:90% or review them quickly. 00:24:14.980 --> 00:24:18.430 align:middle line:84% Then this was the existence lecture. 00:24:18.430 --> 00:24:21.110 align:middle line:84% The point being that, for example, 00:24:21.110 --> 00:24:23.770 align:middle line:84% the first welfare theorem assumes existence 00:24:23.770 --> 00:24:26.290 align:middle line:84% of an equilibrium and then assures everybody 00:24:26.290 --> 00:24:28.850 align:middle line:84% that it would be optimal under sufficient conditions. 00:24:28.850 --> 00:24:31.540 align:middle line:84% But if the model could be vacuous if there 00:24:31.540 --> 00:24:33.730 align:middle line:90% is no competitive equilibrium. 00:24:33.730 --> 00:24:35.590 align:middle line:84% So we're dealing with things like that 00:24:35.590 --> 00:24:37.210 align:middle line:90% directly in this lecture. 00:24:37.210 --> 00:24:40.480 align:middle line:84% And this was a listing of the things covered. 00:24:40.480 --> 00:24:44.470 align:middle line:84% So the main mathematical object tool really 00:24:44.470 --> 00:24:48.550 align:middle line:84% was fixed point theorem, this was an example 00:24:48.550 --> 00:24:51.220 align:middle line:84% that some function has across the 45 degree 00:24:51.220 --> 00:24:54.970 align:middle line:84% line, in which case whatever you put in, you get back out. 00:24:54.970 --> 00:24:57.790 align:middle line:84% The argument in the range is the argument 00:24:57.790 --> 00:25:00.370 align:middle line:84% in the domain under the function, F. 00:25:00.370 --> 00:25:02.830 align:middle line:84% And it was sufficient for this to have 00:25:02.830 --> 00:25:06.490 align:middle line:84% the set of bases of values in the domain and the range 00:25:06.490 --> 00:25:10.390 align:middle line:84% being non-empty convex and compact and for the function F 00:25:10.390 --> 00:25:11.770 align:middle line:90% to be continuous. 00:25:11.770 --> 00:25:14.620 align:middle line:84% And that's Brouwer's Fixed Point Theorem. 00:25:14.620 --> 00:25:19.030 align:middle line:84% I should say though, well, we're about to get there, that this 00:25:19.030 --> 00:25:20.650 align:middle line:90% is all about existence. 00:25:20.650 --> 00:25:22.840 align:middle line:90% It's not about how to find it. 00:25:22.840 --> 00:25:26.500 align:middle line:84% And at least one of you has asked me questions about that. 00:25:26.500 --> 00:25:29.170 align:middle line:84% Anyway, this is Brouwer's Fixed Point Theorem. 00:25:29.170 --> 00:25:32.590 align:middle line:84% It's paired with Kakutani's Fixed Point Theorem. 00:25:32.590 --> 00:25:34.450 align:middle line:84% And the way in which they differed, 00:25:34.450 --> 00:25:37.360 align:middle line:84% primarily, was not the set A, which is still 00:25:37.360 --> 00:25:41.080 align:middle line:84% compact and convex and non-empty, but the function 00:25:41.080 --> 00:25:45.050 align:middle line:84% F, which is no longer single, necessarily single value. 00:25:45.050 --> 00:25:46.630 align:middle line:90% It could be multiple values. 00:25:46.630 --> 00:25:48.920 align:middle line:84% Or there are multiple values in the range 00:25:48.920 --> 00:25:52.940 align:middle line:84% for any particular input x in the domain. 00:25:52.940 --> 00:25:55.060 align:middle line:84% And of course, not anything goes and had 00:25:55.060 --> 00:25:58.640 align:middle line:84% to be convex value and continuous in a certain sense. 00:25:58.640 --> 00:26:03.060 align:middle line:84% And then with Kakutani, again, we have a fixed point. 00:26:03.060 --> 00:26:05.730 align:middle line:84% So these are very powerful tools and they're 00:26:05.730 --> 00:26:10.050 align:middle line:84% used for competitive equilibria in one way or another 00:26:10.050 --> 00:26:13.020 align:middle line:90% and used in games. 00:26:13.020 --> 00:26:18.090 align:middle line:84% And we actually combined those at the end of this lecture 00:26:18.090 --> 00:26:21.030 align:middle line:84% to have a game that would be achieved under a Nash 00:26:21.030 --> 00:26:25.440 align:middle line:84% equilibrium as a Walrasian equilibrium. 00:26:25.440 --> 00:26:29.820 align:middle line:84% So here are what happens if you don't 00:26:29.820 --> 00:26:33.000 align:middle line:84% assume continuity of the function, you get these jumps. 00:26:33.000 --> 00:26:36.120 align:middle line:84% So if the aggregate demand for an economy 00:26:36.120 --> 00:26:38.430 align:middle line:84% somehow had jumps like this, then you 00:26:38.430 --> 00:26:40.680 align:middle line:84% could not be assured that there would exist 00:26:40.680 --> 00:26:43.110 align:middle line:90% a competitive equilibrium. 00:26:43.110 --> 00:26:48.180 align:middle line:84% And we're looking for the existence of prices, such 00:26:48.180 --> 00:26:52.680 align:middle line:84% that you put prices in to each of the agents maximization 00:26:52.680 --> 00:26:56.940 align:middle line:84% problems, look at their demands, see 00:26:56.940 --> 00:26:58.930 align:middle line:90% if demands add up to supply. 00:26:58.930 --> 00:27:02.940 align:middle line:84% And if we find the price that makes that happen 00:27:02.940 --> 00:27:06.120 align:middle line:84% and all of this can be formulated as a fixed point. 00:27:06.120 --> 00:27:09.240 align:middle line:84% Now I did not go through the math of this 00:27:09.240 --> 00:27:12.760 align:middle line:84% because it's surprisingly tedious. 00:27:12.760 --> 00:27:16.410 align:middle line:84% However, let me say that the existence proof is 00:27:16.410 --> 00:27:19.740 align:middle line:84% the existence of a competitive equilibrium as in at least one 00:27:19.740 --> 00:27:20.910 align:middle line:90% fixed point. 00:27:20.910 --> 00:27:24.390 align:middle line:84% Herbert Scarfe did a lot of work on this subject. 00:27:24.390 --> 00:27:27.780 align:middle line:84% And in his proof of the Walrasian equilibrium, 00:27:27.780 --> 00:27:32.640 align:middle line:84% he actually did more, which was to go back to Brouwer 00:27:32.640 --> 00:27:35.790 align:middle line:84% and actually show how to find the fixed point. 00:27:35.790 --> 00:27:39.150 align:middle line:84% That is to say, in an algorithm, take a guess, 00:27:39.150 --> 00:27:43.350 align:middle line:84% look at directional derivatives, formulate a new guess 00:27:43.350 --> 00:27:49.470 align:middle line:84% and iterate until you get to the end or approximately the end, 00:27:49.470 --> 00:27:51.540 align:middle line:84% however close the computer can make it. 00:27:51.540 --> 00:27:57.670 align:middle line:84% And he would have constructively found the Brouwer's fix point, 00:27:57.670 --> 00:28:00.510 align:middle line:84% which works much more generally than just finding the existence 00:28:00.510 --> 00:28:02.550 align:middle line:90% of a competitive equilibrium. 00:28:02.550 --> 00:28:06.060 align:middle line:84% This comes up later when we talked about restrictions 00:28:06.060 --> 00:28:08.550 align:middle line:84% on data having to do with algorithms 00:28:08.550 --> 00:28:10.920 align:middle line:84% and what the algorithms actually look 00:28:10.920 --> 00:28:13.830 align:middle line:84% like when computing solutions to individual problems 00:28:13.830 --> 00:28:16.110 align:middle line:84% and defining a competitive equilibrium. 00:28:16.110 --> 00:28:17.970 align:middle line:84% Anyway, I just wanted to draw your attention 00:28:17.970 --> 00:28:20.460 align:middle line:84% to the distinction as an overview-- 00:28:20.460 --> 00:28:21.840 align:middle line:90% that's my job today-- 00:28:21.840 --> 00:28:24.330 align:middle line:84% between existence of a competitive equilibrium 00:28:24.330 --> 00:28:29.880 align:middle line:84% and how to find the competitive equilibrium, algorithmically. 00:28:29.880 --> 00:28:33.150 align:middle line:84% What we did do in this lecture was 00:28:33.150 --> 00:28:36.990 align:middle line:84% to prove the existence of a competitive equilibrium using 00:28:36.990 --> 00:28:39.480 align:middle line:84% the second welfare theorem, which we just 00:28:39.480 --> 00:28:41.400 align:middle line:90% reviewed in lecture 16. 00:28:41.400 --> 00:28:45.240 align:middle line:84% So the idea is instead of iterating on candidate prices 00:28:45.240 --> 00:28:47.790 align:middle line:84% to find where supply is equal to demand, 00:28:47.790 --> 00:28:50.490 align:middle line:84% we iterate on these lambda weights. 00:28:50.490 --> 00:28:53.250 align:middle line:84% And the idea is very much, here's 00:28:53.250 --> 00:28:56.620 align:middle line:84% that economy again, I promised you it would keep coming back. 00:28:56.620 --> 00:28:58.770 align:middle line:84% The statement of the economy with the convexity 00:28:58.770 --> 00:29:00.660 align:middle line:90% and concavity. 00:29:00.660 --> 00:29:05.500 align:middle line:84% The idea is to go back to the Pareto problems, 00:29:05.500 --> 00:29:07.200 align:middle line:84% this is now the second time today 00:29:07.200 --> 00:29:09.750 align:middle line:84% that the Pareto problem has come up. 00:29:09.750 --> 00:29:13.500 align:middle line:84% We just used it in 16 for the second welfare theorem. 00:29:13.500 --> 00:29:16.740 align:middle line:84% Here it is again, stated rather succinctly 00:29:16.740 --> 00:29:21.210 align:middle line:84% to remind us of these lambda i weights, such that when we max, 00:29:21.210 --> 00:29:22.720 align:middle line:90% we get Pareto Optimal. 00:29:22.720 --> 00:29:25.860 align:middle line:84% So we start with those lambdas and then 00:29:25.860 --> 00:29:30.660 align:middle line:84% we essentially look at the solution 00:29:30.660 --> 00:29:33.390 align:middle line:84% to the Pareto problem, these x hats, 00:29:33.390 --> 00:29:35.130 align:middle line:90% for those particular lambdas. 00:29:35.130 --> 00:29:38.370 align:middle line:84% Look at the prices, which you now know and remember 00:29:38.370 --> 00:29:40.950 align:middle line:84% are the shadow prices on the resource constraints 00:29:40.950 --> 00:29:43.500 align:middle line:84% and we generate the wealth associating 00:29:43.500 --> 00:29:45.580 align:middle line:90% with this purchasing power. 00:29:45.580 --> 00:29:49.320 align:middle line:84% So what we're going to look for to get a standard Walrasian 00:29:49.320 --> 00:29:52.440 align:middle line:84% equilibrium his particular lambdas, 00:29:52.440 --> 00:29:57.480 align:middle line:84% such that the valuation of privately owned resources 00:29:57.480 --> 00:30:01.230 align:middle line:84% for a given household, I, is equal to the wealth 00:30:01.230 --> 00:30:03.210 align:middle line:90% coming from the lambdas. 00:30:03.210 --> 00:30:05.460 align:middle line:84% Here this is a better statement of it. 00:30:05.460 --> 00:30:08.040 align:middle line:84% A particular lambda star, the fixed point-- 00:30:08.040 --> 00:30:12.150 align:middle line:84% find me, please-- the fixed point would be such that when 00:30:12.150 --> 00:30:14.880 align:middle line:84% we substitute lambda into the Pareto problem and generate 00:30:14.880 --> 00:30:19.050 align:middle line:84% the optimal allocations, we'll have some valuation at those 00:30:19.050 --> 00:30:21.760 align:middle line:84% shadow prices of the expenditures, 00:30:21.760 --> 00:30:23.550 align:middle line:90% which when we sum up over-- 00:30:23.550 --> 00:30:26.130 align:middle line:84% and summed up over all the commodities, L, 00:30:26.130 --> 00:30:28.650 align:middle line:84% would be exactly equal to the right hand 00:30:28.650 --> 00:30:32.220 align:middle line:84% side of the budget, the valuation of purchasing power 00:30:32.220 --> 00:30:35.190 align:middle line:84% that the consumer has as a result of owning the endowment 00:30:35.190 --> 00:30:38.040 align:middle line:84% and getting shares in the profits of the firms 00:30:38.040 --> 00:30:41.800 align:middle line:84% when the firms are maximizing at that same lambda star. 00:30:41.800 --> 00:30:44.100 align:middle line:90% So this is the fixed point. 00:30:44.100 --> 00:30:47.460 align:middle line:84% I think I mentioned in a question and answer 00:30:47.460 --> 00:30:51.210 align:middle line:84% at the beginning of lecture 18 when we reviewed 17, 00:30:51.210 --> 00:30:54.570 align:middle line:84% that I felt I hadn't done a good enough job explaining 00:30:54.570 --> 00:30:57.300 align:middle line:90% the intuition for this mapping. 00:30:57.300 --> 00:31:00.990 align:middle line:84% But it does come from the second welfare theorem. 00:31:00.990 --> 00:31:05.310 align:middle line:84% Start with a given guess about the lambda that pins down 00:31:05.310 --> 00:31:09.570 align:middle line:84% which particular Pareto Optimum were targeting and suppose that 00:31:09.570 --> 00:31:12.420 align:middle line:84% were the case that the valuation of expenditure 00:31:12.420 --> 00:31:15.750 align:middle line:84% at that optimum for consumer I were strictly greater 00:31:15.750 --> 00:31:22.200 align:middle line:84% than the valuation of his or her privately owned resources. 00:31:22.200 --> 00:31:25.480 align:middle line:84% Well in this case, we gave this consumer too much. 00:31:25.480 --> 00:31:26.640 align:middle line:90% They can't afford it. 00:31:26.640 --> 00:31:29.250 align:middle line:90% So the obvious way to iterate-- 00:31:29.250 --> 00:31:31.800 align:middle line:84% if indeed you were even searching-- 00:31:31.800 --> 00:31:35.040 align:middle line:84% is to reduce the lambda for this household 00:31:35.040 --> 00:31:37.000 align:middle line:84% and increase the lambdas for the other. 00:31:37.000 --> 00:31:40.350 align:middle line:84% So this is very much like that Edgeworth Box diagram 00:31:40.350 --> 00:31:44.280 align:middle line:84% for the second welfare theorem, where I said household A has 00:31:44.280 --> 00:31:45.360 align:middle line:90% to give up stuff. 00:31:45.360 --> 00:31:47.910 align:middle line:84% This would be household A if the inequality 00:31:47.910 --> 00:31:49.230 align:middle line:90% is greater than or equal to. 00:31:49.230 --> 00:31:51.480 align:middle line:84% And household B would be the recipient 00:31:51.480 --> 00:31:53.100 align:middle line:90% and be getting the transfers. 00:31:53.100 --> 00:31:56.970 align:middle line:84% For household B, you'd be under valuing evaluation 00:31:56.970 --> 00:31:59.730 align:middle line:84% of the resources that the assigned optimum could 00:31:59.730 --> 00:32:03.150 align:middle line:84% be less than the valuation of their privately 00:32:03.150 --> 00:32:04.800 align:middle line:90% owned resources. 00:32:04.800 --> 00:32:07.860 align:middle line:84% Now again, Negishi didn't actually 00:32:07.860 --> 00:32:11.160 align:middle line:84% show that one could find the fixed 00:32:11.160 --> 00:32:13.950 align:middle line:84% point by iterating on the lambdas, 00:32:13.950 --> 00:32:16.960 align:middle line:84% but in practice, it typically works. 00:32:16.960 --> 00:32:19.260 align:middle line:84% So as a numerical algorithm, you could just 00:32:19.260 --> 00:32:22.800 align:middle line:84% be trying out candidates, lambda checking out 00:32:22.800 --> 00:32:26.950 align:middle line:84% these kinds of budget equations and so on. 00:32:26.950 --> 00:32:30.900 align:middle line:84% So here again, this links up to lambda equal marginal utility 00:32:30.900 --> 00:32:31.680 align:middle line:90% of income. 00:32:31.680 --> 00:32:35.130 align:middle line:84% That's the Kansas Rocky Mountain pictures at the beginning 00:32:35.130 --> 00:32:37.800 align:middle line:84% and I reinforce that when we went to the second welfare 00:32:37.800 --> 00:32:38.610 align:middle line:90% theorem. 00:32:38.610 --> 00:32:40.980 align:middle line:84% So this proof of the existence is working off 00:32:40.980 --> 00:32:43.770 align:middle line:84% the same equivalence between the lambda 00:32:43.770 --> 00:32:46.560 align:middle line:84% and the inverse marginal utility of wealth. 00:32:46.560 --> 00:32:50.610 align:middle line:84% This is the computer science part 00:32:50.610 --> 00:32:54.870 align:middle line:84% that it looks like it might be hard to find, 00:32:54.870 --> 00:32:56.250 align:middle line:90% actually constructively. 00:32:56.250 --> 00:32:58.680 align:middle line:84% So these guys algorithmically want 00:32:58.680 --> 00:33:01.140 align:middle line:84% to find the Walrasian equilibrium, 00:33:01.140 --> 00:33:04.440 align:middle line:84% not just be happy with the notion 00:33:04.440 --> 00:33:06.990 align:middle line:84% that there's a fixed point so that it exists. 00:33:06.990 --> 00:33:11.010 align:middle line:84% And this literature works on the same, they kind of discovered, 00:33:11.010 --> 00:33:12.780 align:middle line:90% again, Negishi. 00:33:12.780 --> 00:33:15.360 align:middle line:84% So that's why I told you about it. 00:33:15.360 --> 00:33:18.090 align:middle line:84% But you get a better sense here and I gave you 00:33:18.090 --> 00:33:21.750 align:middle line:84% a hint of that, with respect to Herb Scarf, that the way 00:33:21.750 --> 00:33:26.850 align:middle line:84% to find it is to periodically look at the aggregate demand 00:33:26.850 --> 00:33:30.760 align:middle line:84% oracle, as it were, and take a local derivative of it 00:33:30.760 --> 00:33:33.528 align:middle line:84% to see which way you ought to move. 00:33:33.528 --> 00:33:35.070 align:middle line:84% So it's a little bit harder than what 00:33:35.070 --> 00:33:38.620 align:middle line:84% I said about just iterating on Negishi's algorithm. 00:33:38.620 --> 00:33:41.800 align:middle line:84% But their point is that they designed an algorithm that 00:33:41.800 --> 00:33:45.220 align:middle line:84% finds the competitive equilibrium using Negishi 00:33:45.220 --> 00:33:47.830 align:middle line:84% and it's not that hard in the sense 00:33:47.830 --> 00:33:50.980 align:middle line:84% that it's polynomial of degree three 00:33:50.980 --> 00:33:53.410 align:middle line:84% and the number of goods, which is N. 00:33:53.410 --> 00:33:58.090 align:middle line:84% So it's polynomial, rather than exponentially hard. 00:33:58.090 --> 00:34:01.180 align:middle line:84% So then we went to the second part of the lecture, which 00:34:01.180 --> 00:34:04.000 align:middle line:90% was Strategic Form Games. 00:34:04.000 --> 00:34:05.395 align:middle line:90% This looks quite different. 00:34:05.395 --> 00:34:07.660 align:middle line:84% You have utility functions again, 00:34:07.660 --> 00:34:10.670 align:middle line:84% but they can be a function of other people's strategies, 00:34:10.670 --> 00:34:12.630 align:middle line:90% not just your own. 00:34:12.630 --> 00:34:15.460 align:middle line:84% Finite set of players, that's kind of similar. 00:34:15.460 --> 00:34:18.179 align:middle line:84% And we talked about strategy profiles. 00:34:18.179 --> 00:34:21.510 align:middle line:84% What household i is doing, or trader i, 00:34:21.510 --> 00:34:24.030 align:middle line:84% versus what all the others are doing. 00:34:24.030 --> 00:34:28.650 align:middle line:84% And we defined mixed strategies, where the players 00:34:28.650 --> 00:34:33.389 align:middle line:84% are randomizing, potentially, over a finite set 00:34:33.389 --> 00:34:35.409 align:middle line:90% of discrete actions. 00:34:35.409 --> 00:34:38.409 align:middle line:84% So this is one place in the course where, 00:34:38.409 --> 00:34:41.159 align:middle line:84% again, these lotteries come up to convexity 00:34:41.159 --> 00:34:42.210 align:middle line:90% things in this case. 00:34:42.210 --> 00:34:45.030 align:middle line:84% And indeed, the convexification is 00:34:45.030 --> 00:34:49.380 align:middle line:84% what allows us to prove the existence of this mixed 00:34:49.380 --> 00:34:51.210 align:middle line:90% strategy equilibrium. 00:34:51.210 --> 00:34:53.699 align:middle line:84% A mixed strategy equilibrium is a set 00:34:53.699 --> 00:34:57.600 align:middle line:84% of randomization routines for each player stacked up 00:34:57.600 --> 00:35:00.330 align:middle line:84% in a vector, such that given what 00:35:00.330 --> 00:35:02.910 align:middle line:84% all the other players "I" are doing, 00:35:02.910 --> 00:35:05.250 align:middle line:90% minus "I", all the others. 00:35:05.250 --> 00:35:09.840 align:middle line:84% But player "I" wants to do is the I-star, signa-star vector 00:35:09.840 --> 00:35:14.350 align:middle line:84% component of that vector, better than any other component. 00:35:14.350 --> 00:35:18.410 align:middle line:84% So this is the definition of a mixed strategy equilibrium. 00:35:18.410 --> 00:35:22.080 align:middle line:84% And then we had this interesting side note here. 00:35:22.080 --> 00:35:26.240 align:middle line:84% And this is a third related, but not identical 00:35:26.240 --> 00:35:30.680 align:middle line:84% statement, which is you have a candidate for an equilibrium, 00:35:30.680 --> 00:35:34.870 align:middle line:84% how do it really is an equilibrium? 00:35:34.870 --> 00:35:37.210 align:middle line:84% It's related to the idea of the fixed point, 00:35:37.210 --> 00:35:39.770 align:middle line:90% momentarily I'll show you that. 00:35:39.770 --> 00:35:43.030 align:middle line:84% But here, the question is, how many things do you 00:35:43.030 --> 00:35:46.810 align:middle line:84% have to check at the candidate mixed equilibrium Nash, mixed 00:35:46.810 --> 00:35:48.410 align:middle line:90% strategy Nash equilibrium. 00:35:48.410 --> 00:35:52.450 align:middle line:84% And it turns out you only need to check pure strategy 00:35:52.450 --> 00:35:58.760 align:middle line:84% deviations here, rather than all possible mixed strategies. 00:35:58.760 --> 00:36:00.910 align:middle line:84% Thank goodness because that would be hard. 00:36:00.910 --> 00:36:02.980 align:middle line:84% And the reason for that is, if you 00:36:02.980 --> 00:36:07.330 align:middle line:84% can find a degenerate action which 00:36:07.330 --> 00:36:10.090 align:middle line:84% dominates the supposed mixed strategy 00:36:10.090 --> 00:36:11.530 align:middle line:90% maximum, that's enough. 00:36:11.530 --> 00:36:14.230 align:middle line:84% That's a particular degenerate mixed strategy 00:36:14.230 --> 00:36:16.040 align:middle line:84% to show that you could do better. 00:36:16.040 --> 00:36:19.780 align:middle line:84% And if there were multiple ones that were doing better, 00:36:19.780 --> 00:36:22.360 align:middle line:84% you pick the best one that dominates 00:36:22.360 --> 00:36:24.170 align:middle line:90% to break up the solution. 00:36:24.170 --> 00:36:27.490 align:middle line:84% And if there were two or more that 00:36:27.490 --> 00:36:31.180 align:middle line:84% generated the same utility, you could randomize over them, 00:36:31.180 --> 00:36:34.040 align:middle line:84% but there's no need because they all generate the same utility 00:36:34.040 --> 00:36:35.290 align:middle line:90% so you might as well pick one. 00:36:35.290 --> 00:36:38.710 align:middle line:84% So that's the intuition for why it's 00:36:38.710 --> 00:36:41.860 align:middle line:84% enough to look to check a given candidate 00:36:41.860 --> 00:36:48.460 align:middle line:84% profile against these degenerate actions, not mix strategies. 00:36:48.460 --> 00:36:55.330 align:middle line:84% The rest of it is remembering exactly these kinds 00:36:55.330 --> 00:36:56.290 align:middle line:90% of responses. 00:36:56.290 --> 00:37:04.850 align:middle line:84% The better-than sets, or best response sets for each player. 00:37:04.850 --> 00:37:07.120 align:middle line:84% So given the others are doing, you 00:37:07.120 --> 00:37:09.550 align:middle line:84% want a best response for player I, obviously 00:37:09.550 --> 00:37:11.020 align:middle line:90% is going to maximize. 00:37:11.020 --> 00:37:13.750 align:middle line:84% Hopefully, generating the same mixed strategy as a candidate, 00:37:13.750 --> 00:37:18.130 align:middle line:84% but maybe not, if it's not a mixed strategy equilibrium. 00:37:18.130 --> 00:37:20.060 align:middle line:84% But all the other guys are doing the same. 00:37:20.060 --> 00:37:23.380 align:middle line:84% So we're going to look for a fixed point in the space 00:37:23.380 --> 00:37:25.960 align:middle line:90% of best response strategies. 00:37:25.960 --> 00:37:29.440 align:middle line:84% And the proof of the existence of a strategy Nash equilibrium 00:37:29.440 --> 00:37:34.060 align:middle line:84% had to do with the convexity of these best responses, 00:37:34.060 --> 00:37:36.920 align:middle line:90% as in Kakutani, for example. 00:37:36.920 --> 00:37:38.917 align:middle line:90% In particular, all this stuff. 00:37:38.917 --> 00:37:40.750 align:middle line:84% I'm a little worried I'm going to get bogged 00:37:40.750 --> 00:37:45.280 align:middle line:84% down in the weeds, but I did want you to, on the other hand, 00:37:45.280 --> 00:37:47.090 align:middle line:90% see the big picture. 00:37:47.090 --> 00:37:50.140 align:middle line:90% So now we have two parts. 00:37:50.140 --> 00:37:52.420 align:middle line:84% The Nash equilibrium for strategic games 00:37:52.420 --> 00:37:57.670 align:middle line:84% and the Walrasian equilibria, they're all using fixed points, 00:37:57.670 --> 00:38:00.580 align:middle line:84% but the economic problem looks kind of different. 00:38:00.580 --> 00:38:04.660 align:middle line:84% So I finished this lecture off by drawing the connection, 00:38:04.660 --> 00:38:08.770 align:middle line:84% like how do you actually set up a market to find a Walrasian 00:38:08.770 --> 00:38:09.940 align:middle line:90% equilibria. 00:38:09.940 --> 00:38:14.950 align:middle line:84% And we did that with respect to these limit order strategies 00:38:14.950 --> 00:38:18.760 align:middle line:84% on the supply and demand side and then wrote it down 00:38:18.760 --> 00:38:19.790 align:middle line:90% as the game. 00:38:19.790 --> 00:38:22.270 align:middle line:84% So once we wrote this down as the game, 00:38:22.270 --> 00:38:24.970 align:middle line:84% where players decide what to bid, 00:38:24.970 --> 00:38:29.380 align:middle line:84% given what others are bidding it is a strategic form game 00:38:29.380 --> 00:38:32.290 align:middle line:84% and we characterize the equilibrium. 00:38:32.290 --> 00:38:36.010 align:middle line:84% Turned out, you needed some penalties, otherwise, they 00:38:36.010 --> 00:38:37.900 align:middle line:84% would spend more than they have and that 00:38:37.900 --> 00:38:41.650 align:middle line:84% can't be consistent with maximization-- well, 00:38:41.650 --> 00:38:44.940 align:middle line:84% it would be maximization, but there'd be no limit. 00:38:44.940 --> 00:38:48.080 align:middle line:84% So it would not be consistent with finding the Walrasian 00:38:48.080 --> 00:38:49.700 align:middle line:90% equilibrium. 00:38:49.700 --> 00:38:53.350 align:middle line:84% So we did have to find the optimal penalty structure 00:38:53.350 --> 00:38:56.440 align:middle line:84% in order to prove the theorem that any Walrasian 00:38:56.440 --> 00:39:00.190 align:middle line:84% equilibrium could be supported as a mixed strategy equilibrium 00:39:00.190 --> 00:39:01.840 align:middle line:90% in this market game. 00:39:01.840 --> 00:39:04.520 align:middle line:84% And the other direction is also true. 00:39:04.520 --> 00:39:07.270 align:middle line:90% So that was lecture 17. 00:39:07.270 --> 00:39:08.830 align:middle line:84% Are there any questions about what 00:39:08.830 --> 00:39:10.180 align:middle line:90% we were doing in lecture 17? 00:39:10.180 --> 00:39:16.450 align:middle line:90% 00:39:16.450 --> 00:39:19.060 align:middle line:90% 18 was aggregation. 00:39:19.060 --> 00:39:21.040 align:middle line:84% And again, this is getting more recent. 00:39:21.040 --> 00:39:25.400 align:middle line:84% So hopefully, a little more fresh in your minds. 00:39:25.400 --> 00:39:28.640 align:middle line:84% We want to do this positive and normative. 00:39:28.640 --> 00:39:30.980 align:middle line:84% And I would say the content of this lecture 00:39:30.980 --> 00:39:34.520 align:middle line:84% is, what do we have to assume, like it or not, 00:39:34.520 --> 00:39:36.800 align:middle line:84% in order to be in the land of macro, 00:39:36.800 --> 00:39:39.590 align:middle line:84% where you have a representative consumer? 00:39:39.590 --> 00:39:43.190 align:middle line:84% And is usable in the sense that what the representative 00:39:43.190 --> 00:39:46.940 align:middle line:84% consumer would do, which from a positive economic standpoint, 00:39:46.940 --> 00:39:49.310 align:middle line:84% generate aggregate demand as if you 00:39:49.310 --> 00:39:50.750 align:middle line:84% had all the underlying households, 00:39:50.750 --> 00:39:51.917 align:middle line:90% but you don't need to do it. 00:39:51.917 --> 00:39:53.850 align:middle line:84% You just use the representative consumer. 00:39:53.850 --> 00:39:55.940 align:middle line:90% So it's a very powerful tool. 00:39:55.940 --> 00:40:01.010 align:middle line:84% But again, for you to decide how strong or potentially 00:40:01.010 --> 00:40:03.350 align:middle line:84% unrealistic are the underlying assumptions 00:40:03.350 --> 00:40:05.610 align:middle line:90% that allows us to do this? 00:40:05.610 --> 00:40:09.420 align:middle line:84% So we reviewed the notion of the indirect utility function, 00:40:09.420 --> 00:40:12.410 align:middle line:84% which is the maximized utility after solving 00:40:12.410 --> 00:40:16.310 align:middle line:84% max utility subject to budget, given the parameters, the price 00:40:16.310 --> 00:40:17.840 align:middle line:90% and the wealth of a household. 00:40:17.840 --> 00:40:22.280 align:middle line:84% And we quickly reviewed some properties 00:40:22.280 --> 00:40:24.560 align:middle line:84% of the indirect utility functions. 00:40:24.560 --> 00:40:28.340 align:middle line:84% In particular, Roy's identity, which miraculously 00:40:28.340 --> 00:40:31.040 align:middle line:84% says that if you start differentiating 00:40:31.040 --> 00:40:34.310 align:middle line:84% the indirect utility function with respect to prices 00:40:34.310 --> 00:40:37.310 align:middle line:84% and respect to wealth and start taking ratios, 00:40:37.310 --> 00:40:40.190 align:middle line:84% you'll get the underlying demands. 00:40:40.190 --> 00:40:41.630 align:middle line:90% And this was a tool. 00:40:41.630 --> 00:40:43.850 align:middle line:90% And it's heavily used. 00:40:43.850 --> 00:40:46.640 align:middle line:84% Again, now for the third time, it just 00:40:46.640 --> 00:40:51.170 align:middle line:84% came from the Lagrangian problem for the individual household. 00:40:51.170 --> 00:40:54.540 align:middle line:84% And Roy's identity came from that. 00:40:54.540 --> 00:40:58.340 align:middle line:84% And then we got to Gorman, which is a particular representation 00:40:58.340 --> 00:41:00.500 align:middle line:84% of these indirect utility functions 00:41:00.500 --> 00:41:02.240 align:middle line:84% as a function of prices and wealth, 00:41:02.240 --> 00:41:05.060 align:middle line:90% namely, this thing in six. 00:41:05.060 --> 00:41:08.990 align:middle line:84% So it's separable, it has an intercept of the function of p, 00:41:08.990 --> 00:41:12.170 align:middle line:84% slope term on wealth as a function of p. 00:41:12.170 --> 00:41:14.990 align:middle line:84% I noted that the Cobb Douglas would satisfy this. 00:41:14.990 --> 00:41:16.700 align:middle line:90% We actually did the math. 00:41:16.700 --> 00:41:18.890 align:middle line:84% This math is never very hard to do, 00:41:18.890 --> 00:41:21.410 align:middle line:84% as long as you remember that when you have these Cobb 00:41:21.410 --> 00:41:24.200 align:middle line:84% Douglas utility function, these coefficients represent 00:41:24.200 --> 00:41:25.580 align:middle line:90% expenditure shares. 00:41:25.580 --> 00:41:31.910 align:middle line:84% So P1 times x1 is alpha share of wealth, w-- 00:41:31.910 --> 00:41:36.920 align:middle line:84% likewise, p2x2 is 1 minus alpha share of wealth, W, 00:41:36.920 --> 00:41:38.600 align:middle line:90% for the second good. 00:41:38.600 --> 00:41:40.580 align:middle line:84% So you can plop them down right away 00:41:40.580 --> 00:41:44.540 align:middle line:84% and just substitute into the given utility function 00:41:44.540 --> 00:41:46.770 align:middle line:84% and you end up with this functional form. 00:41:46.770 --> 00:41:49.640 align:middle line:84% I also noted that although this is 00:41:49.640 --> 00:41:52.400 align:middle line:84% special in that the intercepts are 0, 00:41:52.400 --> 00:41:57.170 align:middle line:84% it also seemingly assumed that the coefficients, B, are 00:41:57.170 --> 00:41:57.950 align:middle line:90% the same. 00:41:57.950 --> 00:42:02.780 align:middle line:84% But no, this was a particular indirect utility function 00:42:02.780 --> 00:42:05.210 align:middle line:84% for a household-- a given household. 00:42:05.210 --> 00:42:09.170 align:middle line:84% We have to start putting "I" in the utility function to decide. 00:42:09.170 --> 00:42:11.510 align:middle line:84% And if these alphas were different 00:42:11.510 --> 00:42:14.870 align:middle line:84% across different households, we would need an alpha I on them 00:42:14.870 --> 00:42:18.960 align:middle line:84% and then this thing, the Gorman form, would be violated. 00:42:18.960 --> 00:42:21.680 align:middle line:84% So the first part of the positive representative 00:42:21.680 --> 00:42:25.400 align:middle line:84% consumer was just saying a definition of aggregate demand, 00:42:25.400 --> 00:42:27.320 align:middle line:84% namely, it's the sum of the demands 00:42:27.320 --> 00:42:30.680 align:middle line:84% over all the individual households at price vector, p, 00:42:30.680 --> 00:42:32.690 align:middle line:90% and Wealth's WI. 00:42:32.690 --> 00:42:39.590 align:middle line:84% And the Gorman part is to find that demand-- 00:42:39.590 --> 00:42:43.160 align:middle line:84% this guy here-- depends on the vector of wealth's 00:42:43.160 --> 00:42:44.480 align:middle line:90% in a particular way. 00:42:44.480 --> 00:42:47.540 align:middle line:84% Namely, it's just the sum of them. 00:42:47.540 --> 00:42:49.850 align:middle line:90% Or in other words, it's enough. 00:42:49.850 --> 00:42:53.810 align:middle line:84% We should be able to find a utility function 00:42:53.810 --> 00:42:57.740 align:middle line:84% and give this mythical representative household all 00:42:57.740 --> 00:43:02.180 align:middle line:84% the wealth in the economy and have that household maximize 00:43:02.180 --> 00:43:05.810 align:middle line:84% utility, generating aggregate demand, which 00:43:05.810 --> 00:43:07.670 align:middle line:84% would be the sum of the individual demands 00:43:07.670 --> 00:43:10.580 align:middle line:84% if we'd bothered to keep track of the micro economy 00:43:10.580 --> 00:43:13.210 align:middle line:84% and the individual wealth levels. 00:43:13.210 --> 00:43:17.520 align:middle line:84% So this statement was necessary, but it's 00:43:17.520 --> 00:43:21.450 align:middle line:84% about to be sufficient if the individual households have 00:43:21.450 --> 00:43:24.450 align:middle line:84% this particular indirect utility function of the Gorman 00:43:24.450 --> 00:43:29.790 align:middle line:84% form with the I only on the A intercept piece, 00:43:29.790 --> 00:43:32.250 align:middle line:84% although w depends on I, but that's because it's 00:43:32.250 --> 00:43:34.440 align:middle line:90% individual households wealth. 00:43:34.440 --> 00:43:36.600 align:middle line:84% Then the utility function we're looking 00:43:36.600 --> 00:43:39.690 align:middle line:84% for is just going to be an aggregated version, 00:43:39.690 --> 00:43:43.140 align:middle line:84% aggregating up the intercept terms 00:43:43.140 --> 00:43:47.940 align:middle line:84% and imposing the same coefficient on the wealth 00:43:47.940 --> 00:43:50.850 align:middle line:90% across all the households. 00:43:50.850 --> 00:43:53.680 align:middle line:90% Now do you like it or not? 00:43:53.680 --> 00:43:57.640 align:middle line:84% It is going to require that we have these linear expansion 00:43:57.640 --> 00:44:00.190 align:middle line:84% paths, if we go back to the micro economy 00:44:00.190 --> 00:44:01.840 align:middle line:90% and see how they're behaving. 00:44:01.840 --> 00:44:04.120 align:middle line:84% But giving wealth to one guy would 00:44:04.120 --> 00:44:08.620 align:middle line:84% increase utility and consumption in exactly the same way 00:44:08.620 --> 00:44:11.330 align:middle line:84% that consumption would be decreased for the other one. 00:44:11.330 --> 00:44:14.470 align:middle line:84% So moving wealth around because the slopes of these lines 00:44:14.470 --> 00:44:16.810 align:middle line:84% are the same, they're parallel lines, 00:44:16.810 --> 00:44:20.130 align:middle line:84% moving wealth's around has no effect on aggregate demand. 00:44:20.130 --> 00:44:23.050 align:middle line:84% It's a restatement of the positive part 00:44:23.050 --> 00:44:25.690 align:middle line:90% of the representative consumer. 00:44:25.690 --> 00:44:28.420 align:middle line:84% If you're listening to macro economists, 00:44:28.420 --> 00:44:30.670 align:middle line:84% worried a lot about marginal propensity 00:44:30.670 --> 00:44:34.630 align:middle line:84% to consume out of wealth and out of transfers. 00:44:34.630 --> 00:44:37.780 align:middle line:84% They're trying to find transfers to particular constrained 00:44:37.780 --> 00:44:39.430 align:middle line:84% households in a way that they would 00:44:39.430 --> 00:44:43.060 align:middle line:84% eat a lot as if there was a special virtue of pumping up 00:44:43.060 --> 00:44:44.290 align:middle line:90% aggregate demand. 00:44:44.290 --> 00:44:46.720 align:middle line:84% Well they don't have Gorman in mind 00:44:46.720 --> 00:44:50.650 align:middle line:84% because aggregate demand is not determined by the distribution 00:44:50.650 --> 00:44:51.820 align:middle line:90% of wealth under Gorman. 00:44:51.820 --> 00:44:54.340 align:middle line:84% It's determined by the aggregate wealth. 00:44:54.340 --> 00:44:56.830 align:middle line:84% So it wouldn't matter who you gave the money to, 00:44:56.830 --> 00:44:59.080 align:middle line:90% if it were Gorman world. 00:44:59.080 --> 00:45:02.080 align:middle line:84% Positive representative consumer. 00:45:02.080 --> 00:45:04.660 align:middle line:84% The proof of it I'm not going to go through. 00:45:04.660 --> 00:45:08.260 align:middle line:84% It's basically using Roy's identity over and over again 00:45:08.260 --> 00:45:10.450 align:middle line:90% and exploiting the linearity. 00:45:10.450 --> 00:45:14.770 align:middle line:84% Adding up and then taking it back down. 00:45:14.770 --> 00:45:18.580 align:middle line:84% There's examples of Gorman aggregate preferences, 00:45:18.580 --> 00:45:21.080 align:middle line:84% allowing variation in the intercepts. 00:45:21.080 --> 00:45:24.430 align:middle line:84% So households do not have to be all alike. 00:45:24.430 --> 00:45:27.250 align:middle line:84% And the normative household, normative 00:45:27.250 --> 00:45:31.700 align:middle line:84% in addition to being positive, there's another definition, 00:45:31.700 --> 00:45:33.940 align:middle line:90% namely we're looking-- 00:45:33.940 --> 00:45:36.880 align:middle line:84% we have the underlying micro economy with the heterogeneity 00:45:36.880 --> 00:45:38.650 align:middle line:90% across all the households. 00:45:38.650 --> 00:45:41.650 align:middle line:84% We're looking for this fictitious utility function, 00:45:41.650 --> 00:45:46.570 align:middle line:84% U*, which we label the utility function of the normative 00:45:46.570 --> 00:45:50.530 align:middle line:84% representative household to satisfy two properties. 00:45:50.530 --> 00:45:55.180 align:middle line:84% Namely, if we want to see if something x and y is Pareto 00:45:55.180 --> 00:45:58.960 align:middle line:84% Optimal, then we can search for alternative allocations 00:45:58.960 --> 00:46:00.880 align:middle line:90% that might Pareto dominate. 00:46:00.880 --> 00:46:04.360 align:middle line:84% In the normal micro land of all these different households, 00:46:04.360 --> 00:46:06.040 align:middle line:84% you have to check out each and every one 00:46:06.040 --> 00:46:10.150 align:middle line:84% to see if they're better off under the proposed alternative. 00:46:10.150 --> 00:46:13.660 align:middle line:84% With Gorman normative, you don't have to do that. 00:46:13.660 --> 00:46:17.410 align:middle line:84% You just sum up all the demands in the baseline 00:46:17.410 --> 00:46:22.450 align:middle line:84% and in the alternative and evaluate that sum under this 00:46:22.450 --> 00:46:25.360 align:middle line:90% mythical utility function, U*. 00:46:25.360 --> 00:46:29.290 align:middle line:84% So if something predominates, then it has to have a higher 00:46:29.290 --> 00:46:31.700 align:middle line:90% return under U*. 00:46:31.700 --> 00:46:34.700 align:middle line:84% So that makes the search a whole lot easier. 00:46:34.700 --> 00:46:39.550 align:middle line:84% And the flip-side is the same with an important caveat. 00:46:39.550 --> 00:46:43.760 align:middle line:84% We start with a baseline and you find a prime allocation, 00:46:43.760 --> 00:46:47.230 align:middle line:84% which is dominant under U*, after you sum up. 00:46:47.230 --> 00:46:49.780 align:middle line:84% Then we should be able to do better. 00:46:49.780 --> 00:46:53.770 align:middle line:84% So there is a way in the micro space 00:46:53.770 --> 00:46:57.430 align:middle line:84% to assign allocations that add up 00:46:57.430 --> 00:47:00.490 align:middle line:84% to the prime bundle that are not necessarily 00:47:00.490 --> 00:47:02.500 align:middle line:90% the actual prime allocation. 00:47:02.500 --> 00:47:06.580 align:middle line:84% A way to do that in such a way that everyone is at least as 00:47:06.580 --> 00:47:09.740 align:middle line:84% well off and some people are better off. 00:47:09.740 --> 00:47:14.420 align:middle line:84% So the point of this is, it's very strong when it works, 00:47:14.420 --> 00:47:16.720 align:middle line:84% it's a way of evaluating allocations. 00:47:16.720 --> 00:47:19.960 align:middle line:84% But is it is not to say that you don't have to worry 00:47:19.960 --> 00:47:21.640 align:middle line:90% about the distribution. 00:47:21.640 --> 00:47:23.770 align:middle line:84% You could make people better off, 00:47:23.770 --> 00:47:25.660 align:middle line:84% as I've said before, I think when 00:47:25.660 --> 00:47:29.960 align:middle line:84% I gave this lecture, free trade could make everyone better off. 00:47:29.960 --> 00:47:33.790 align:middle line:84% But you shouldn't stop there because even under Gorman, you 00:47:33.790 --> 00:47:38.470 align:middle line:84% have to redistribute wealth in such a way as to let it 00:47:38.470 --> 00:47:42.360 align:middle line:84% Pareto dominate the tariff equilibrium. 00:47:42.360 --> 00:47:45.000 align:middle line:84% Maybe that's a less that trade theorists kind of forgot 00:47:45.000 --> 00:47:46.470 align:middle line:90% to emphasize 00:47:46.470 --> 00:47:48.420 align:middle line:90% So we did this in-- 00:47:48.420 --> 00:47:51.510 align:middle line:84% we actually did the proof in the space of indirect utilities. 00:47:51.510 --> 00:47:56.560 align:middle line:84% And again, I don't want to go through that too much. 00:47:56.560 --> 00:47:59.130 align:middle line:84% So at the end, we have the application 00:47:59.130 --> 00:48:02.670 align:middle line:84% of those positive and normative representative consumers 00:48:02.670 --> 00:48:04.090 align:middle line:90% to this slide. 00:48:04.090 --> 00:48:07.980 align:middle line:84% This is the third time that I've shown you this economy. 00:48:07.980 --> 00:48:09.840 align:middle line:84% And at least maybe the third time, 00:48:09.840 --> 00:48:12.600 align:middle line:84% we talked about the Pareto problem in this economy. 00:48:12.600 --> 00:48:15.570 align:middle line:84% This Pareto problem, of course, has the lambda weights, 00:48:15.570 --> 00:48:16.470 align:middle line:90% as well it should. 00:48:16.470 --> 00:48:19.530 align:middle line:84% But if we have a positive and normative representative 00:48:19.530 --> 00:48:23.520 align:middle line:84% consumer, then amazingly, we can drop the Pareto weights 00:48:23.520 --> 00:48:26.740 align:middle line:90% and just maximize U*. 00:48:26.740 --> 00:48:28.800 align:middle line:84% So it's as if there's a single representative 00:48:28.800 --> 00:48:31.710 align:middle line:84% consumer with an aggregated production technology, 00:48:31.710 --> 00:48:33.210 align:middle line:90% like Robinson Crusoe. 00:48:33.210 --> 00:48:36.090 align:middle line:84% And this is a much, much easier problem to solve. 00:48:36.090 --> 00:48:39.450 align:middle line:84% Hence the power of it for macro because we're solving it 00:48:39.450 --> 00:48:40.800 align:middle line:90% in the space of macros. 00:48:40.800 --> 00:48:43.320 align:middle line:84% And again, typically, people forget 00:48:43.320 --> 00:48:46.230 align:middle line:84% to think about the underlying distribution and the underlying 00:48:46.230 --> 00:48:48.915 align:middle line:84% wealth distribution of welfare gains 00:48:48.915 --> 00:48:52.890 align:middle line:84% and maybe even losses so stick with the aggregates. 00:48:52.890 --> 00:48:55.140 align:middle line:84% So are there any questions about lecture 18? 00:48:55.140 --> 00:48:59.860 align:middle line:90% 00:48:59.860 --> 00:49:02.030 align:middle line:90% 19 went the other direction. 00:49:02.030 --> 00:49:04.840 align:middle line:90% And I paired them deliberately. 00:49:04.840 --> 00:49:06.910 align:middle line:84% Instead of saying, look at all the special things 00:49:06.910 --> 00:49:09.100 align:middle line:84% you can do if you assume special structure, 00:49:09.100 --> 00:49:11.990 align:middle line:84% like linear income expansion paths, 00:49:11.990 --> 00:49:15.790 align:middle line:84% let's try to avoid any restrictions whatsoever 00:49:15.790 --> 00:49:19.630 align:middle line:84% on utility and production sets and so on and still see 00:49:19.630 --> 00:49:23.345 align:middle line:84% whether we can say anything at all. 00:49:23.345 --> 00:49:24.970 align:middle line:84% So the spirit of this is very different 00:49:24.970 --> 00:49:26.140 align:middle line:90% from the previous lecture. 00:49:26.140 --> 00:49:29.470 align:middle line:84% And this is about not assuming Gorman aggregation, 00:49:29.470 --> 00:49:35.260 align:middle line:84% or not even assuming anything much at all about expansion. 00:49:35.260 --> 00:49:37.420 align:middle line:90% Could have given goods. 00:49:37.420 --> 00:49:40.540 align:middle line:84% What, if anything, can we say or given data, 00:49:40.540 --> 00:49:43.660 align:middle line:84% can we just always accept that there's an economy that 00:49:43.660 --> 00:49:45.730 align:middle line:90% would have generated the data. 00:49:45.730 --> 00:49:48.460 align:middle line:84% And we've talked about this at the beginning 00:49:48.460 --> 00:49:50.450 align:middle line:90% of the following class. 00:49:50.450 --> 00:49:53.510 align:middle line:84% So we want to put as little structure as possible. 00:49:53.510 --> 00:49:56.680 align:middle line:84% And we did this a couple of ways with an infinite amount 00:49:56.680 --> 00:49:58.870 align:middle line:84% of data, with a finite amount of data. 00:49:58.870 --> 00:50:01.690 align:middle line:84% We did it in partial equilibrium under consumer choice 00:50:01.690 --> 00:50:03.820 align:middle line:84% and then we made some connections 00:50:03.820 --> 00:50:07.360 align:middle line:90% to general equilibrium theory. 00:50:07.360 --> 00:50:10.240 align:middle line:84% The Slutsky thing came from a property, 00:50:10.240 --> 00:50:13.060 align:middle line:84% namely, that we minimized the expenditure 00:50:13.060 --> 00:50:16.180 align:middle line:84% necessary to achieve a certain level of utility. 00:50:16.180 --> 00:50:19.270 align:middle line:84% That expenditure function is concave. 00:50:19.270 --> 00:50:22.990 align:middle line:84% And then the derivative of the expenditure function-- 00:50:22.990 --> 00:50:26.071 align:middle line:84% a bit like Roy's identity, not quite-- 00:50:26.071 --> 00:50:29.030 align:middle line:90% is the hexane demands. 00:50:29.030 --> 00:50:32.080 align:middle line:84% So the second derivative of the expenditure function, a matrix, 00:50:32.080 --> 00:50:34.120 align:middle line:84% really of cross partial derivatives 00:50:34.120 --> 00:50:38.440 align:middle line:84% has to be symmetric and negative semi-definite. 00:50:38.440 --> 00:50:42.280 align:middle line:84% This statement is what led to the Slutsky equation. 00:50:42.280 --> 00:50:45.430 align:middle line:84% This is a version of that equation. 00:50:45.430 --> 00:50:47.770 align:middle line:84% I guess for the purposes of the lecture in terms 00:50:47.770 --> 00:50:51.970 align:middle line:84% of what's the point of it all, one is we get restrictions 00:50:51.970 --> 00:50:53.620 align:middle line:90% from the data. 00:50:53.620 --> 00:50:56.830 align:middle line:84% Assuming preferences are regular, 00:50:56.830 --> 00:51:01.060 align:middle line:84% we get restrictions on something we never see. 00:51:01.060 --> 00:51:04.630 align:middle line:84% Namely, these compensated Hicksian demand functions, 00:51:04.630 --> 00:51:07.390 align:middle line:84% as I think we went through in lecture 3. 00:51:07.390 --> 00:51:11.140 align:middle line:84% But fortunately, we're able to write out the left hand 00:51:11.140 --> 00:51:15.280 align:middle line:84% side, these Slutsky derivative, cross partial derivatives, 00:51:15.280 --> 00:51:17.500 align:middle line:84% in terms of observables on the right hand 00:51:17.500 --> 00:51:20.530 align:middle line:84% side, which are directly from the consumers optimization 00:51:20.530 --> 00:51:21.670 align:middle line:90% problem. 00:51:21.670 --> 00:51:25.930 align:middle line:84% How demand changes with prices, how demands change with wealth. 00:51:25.930 --> 00:51:27.820 align:middle line:84% And of course, we would be observing 00:51:27.820 --> 00:51:29.740 align:middle line:90% the actual allocations. 00:51:29.740 --> 00:51:31.160 align:middle line:90% This is with infinite data. 00:51:31.160 --> 00:51:33.760 align:middle line:84% So again, we're allowed conceptually 00:51:33.760 --> 00:51:36.820 align:middle line:84% to think about having all these derivatives 00:51:36.820 --> 00:51:40.690 align:middle line:84% for infinitesimally small changes available to us. 00:51:40.690 --> 00:51:44.530 align:middle line:84% And again, as I went through in class from lecture 3, 00:51:44.530 --> 00:51:47.290 align:middle line:84% when we did the income and substitution effects 00:51:47.290 --> 00:51:51.070 align:middle line:84% we had written with the S on the right hand side 00:51:51.070 --> 00:51:54.380 align:middle line:90% and DXDP on the left hand side. 00:51:54.380 --> 00:51:55.730 align:middle line:90% So there was a negative sign. 00:51:55.730 --> 00:51:57.110 align:middle line:90% But it's all the same. 00:51:57.110 --> 00:51:59.350 align:middle line:90% It's all equivalent. 00:51:59.350 --> 00:52:03.580 align:middle line:84% So then we're saying Slutsky would be satisfied 00:52:03.580 --> 00:52:06.010 align:middle line:84% and a couple of other things are satisfied 00:52:06.010 --> 00:52:10.810 align:middle line:84% if we have the premise of an underlying rational consumer 00:52:10.810 --> 00:52:12.460 align:middle line:90% maximizing utility. 00:52:12.460 --> 00:52:15.460 align:middle line:84% And we reviewed this in the discussion 00:52:15.460 --> 00:52:16.600 align:middle line:90% of the subsequence class. 00:52:16.600 --> 00:52:20.920 align:middle line:84% We have homogeneity of degree 0, and Walras law. 00:52:20.920 --> 00:52:24.550 align:middle line:84% Spending your budget, no price solution and spending 00:52:24.550 --> 00:52:28.060 align:middle line:84% and being on the budget line that plus Slutsky are 00:52:28.060 --> 00:52:34.030 align:middle line:84% the restrictions implied by having a utility maximizing 00:52:34.030 --> 00:52:35.710 align:middle line:90% rational household. 00:52:35.710 --> 00:52:39.790 align:middle line:84% And it turned out that it goes the other way, as well. 00:52:39.790 --> 00:52:43.390 align:middle line:84% That is to say, if you had data that satisfied these three 00:52:43.390 --> 00:52:47.500 align:middle line:84% properties, then there would be an underlying utility function 00:52:47.500 --> 00:52:51.130 align:middle line:84% that's quasi-concave increasing and so on that 00:52:51.130 --> 00:52:53.750 align:middle line:90% would have generated the data. 00:52:53.750 --> 00:52:57.370 align:middle line:84% So this seemingly bears some similarity to Gorman 00:52:57.370 --> 00:53:00.520 align:middle line:84% in the sense of could we generate the aggregate demand 00:53:00.520 --> 00:53:03.250 align:middle line:84% by maximizing a utility function? 00:53:03.250 --> 00:53:06.010 align:middle line:84% Could we find the utility function, U*, 00:53:06.010 --> 00:53:08.290 align:middle line:84% such that given the household all the wealth, 00:53:08.290 --> 00:53:11.230 align:middle line:84% the solution to that problem would be the underlying 00:53:11.230 --> 00:53:12.100 align:middle line:90% demands. 00:53:12.100 --> 00:53:15.670 align:middle line:84% Here though, we're giving the data, including demands, 00:53:15.670 --> 00:53:20.140 align:middle line:84% for a given household and trying to find whether or not 00:53:20.140 --> 00:53:23.800 align:middle line:84% there exists a utility function that will generate demands 00:53:23.800 --> 00:53:25.090 align:middle line:90% with all these properties. 00:53:25.090 --> 00:53:27.430 align:middle line:84% And the answer is, yes, that it's 00:53:27.430 --> 00:53:30.500 align:middle line:84% sufficient and necessary to have these properties. 00:53:30.500 --> 00:53:32.860 align:middle line:84% So the theory has content, you just 00:53:32.860 --> 00:53:35.530 align:middle line:84% need to scroll through this list in the data 00:53:35.530 --> 00:53:38.680 align:middle line:84% and see if you can violate one of them. 00:53:38.680 --> 00:53:43.240 align:middle line:84% And it could not have come from a maximizing household. 00:53:43.240 --> 00:53:48.790 align:middle line:84% Convexity was not testable, not with this demand theory 00:53:48.790 --> 00:53:50.440 align:middle line:84% because you generate the same data 00:53:50.440 --> 00:53:54.740 align:middle line:84% and we actually made a connection to computer science. 00:53:54.740 --> 00:53:58.630 align:middle line:84% So this part, this is testing with finite data, 00:53:58.630 --> 00:54:00.700 align:middle line:90% not an infinite amount of data. 00:54:00.700 --> 00:54:04.460 align:middle line:84% And the key here was the beginning, at least, 00:54:04.460 --> 00:54:07.850 align:middle line:84% this Weak Axiom of revealed preference, which 00:54:07.850 --> 00:54:09.260 align:middle line:90% you should all know-- 00:54:09.260 --> 00:54:11.900 align:middle line:84% and again, we reviewed it in the subsequence lecture 00:54:11.900 --> 00:54:13.040 align:middle line:90% at the beginning-- 00:54:13.040 --> 00:54:19.370 align:middle line:84% that pairwise, if we have two observations prices and demands 00:54:19.370 --> 00:54:23.330 align:middle line:84% and situation one and situation two, then a price P2, 00:54:23.330 --> 00:54:26.810 align:middle line:84% if they're observed choosing X2, it better be the case. 00:54:26.810 --> 00:54:30.110 align:middle line:84% And X1 was available and not chosen, 00:54:30.110 --> 00:54:31.880 align:middle line:84% it better be the case that it price 00:54:31.880 --> 00:54:34.100 align:middle line:84% is P1, where they're choosing X1, 00:54:34.100 --> 00:54:36.260 align:middle line:90% they don't have X2 available. 00:54:36.260 --> 00:54:38.810 align:middle line:84% Otherwise, they were supposed to have chosen it. 00:54:38.810 --> 00:54:40.430 align:middle line:84% If it had been interior in the budget, 00:54:40.430 --> 00:54:42.020 align:middle line:84% according to the first statement, 00:54:42.020 --> 00:54:45.170 align:middle line:90% it would be utility maximizing. 00:54:45.170 --> 00:54:48.510 align:middle line:84% So this is putting restrictions on data. 00:54:48.510 --> 00:54:50.615 align:middle line:84% The Weak Axiom Revealed Preference 00:54:50.615 --> 00:54:53.480 align:middle line:84% is looking at pairwise observations 00:54:53.480 --> 00:54:57.620 align:middle line:84% as we move prices and demands around to see if it's quote, 00:54:57.620 --> 00:54:59.190 align:middle line:90% "rational". 00:54:59.190 --> 00:55:02.370 align:middle line:84% Weak Axiom of Revealed Preference. 00:55:02.370 --> 00:55:05.010 align:middle line:90% And then we generalize that. 00:55:05.010 --> 00:55:09.630 align:middle line:84% We call those Directly Revealed Preferred pairwise comparisons. 00:55:09.630 --> 00:55:14.670 align:middle line:84% And then we went on to talk about Indirectly Revealed 00:55:14.670 --> 00:55:17.280 align:middle line:84% Preferred, or just Revealed Preferred. 00:55:17.280 --> 00:55:21.690 align:middle line:84% And the content of this was, we never see 00:55:21.690 --> 00:55:24.270 align:middle line:90% all the pairwise comparisons. 00:55:24.270 --> 00:55:30.300 align:middle line:84% We never see XT directly compared to XS, 00:55:30.300 --> 00:55:33.510 align:middle line:84% but we do see it indirectly because we 00:55:33.510 --> 00:55:38.970 align:middle line:84% have XT, in this case, Reveal Directly Preferred to XR1, 00:55:38.970 --> 00:55:41.800 align:middle line:90% XR1 reveal preferred to XR2. 00:55:41.800 --> 00:55:43.540 align:middle line:90% And so on down the chain. 00:55:43.540 --> 00:55:48.960 align:middle line:84% So under one of those properties of a rational consumer, 00:55:48.960 --> 00:55:53.370 align:middle line:84% namely transitivity, if we had a sequence of Reveal Directly 00:55:53.370 --> 00:55:56.730 align:middle line:84% Preferred and under transitivity, XT 00:55:56.730 --> 00:56:01.060 align:middle line:84% should be revealed preferred, directly preferred to XS too. 00:56:01.060 --> 00:56:05.520 align:middle line:84% So when you can do the pairwise comparison, a TNS, 00:56:05.520 --> 00:56:08.310 align:middle line:84% it better be the case they don't violate 00:56:08.310 --> 00:56:10.110 align:middle line:90% that XT is preferred to-- 00:56:10.110 --> 00:56:12.380 align:middle line:90% weakly preferred XS. 00:56:12.380 --> 00:56:19.220 align:middle line:84% So this second part is, assuming underlying consumer rationality 00:56:19.220 --> 00:56:22.100 align:middle line:84% in order to set up a situation where we can actually 00:56:22.100 --> 00:56:24.170 align:middle line:90% impose the test. 00:56:24.170 --> 00:56:28.640 align:middle line:84% Then we did that in computational economics, 00:56:28.640 --> 00:56:32.950 align:middle line:84% which I mentioned I elaborated on in the subsequence lecture. 00:56:32.950 --> 00:56:35.300 align:middle line:84% And we took it to general equilibrium. 00:56:35.300 --> 00:56:38.680 align:middle line:84% So are there any questions about risk sharing? 00:56:38.680 --> 00:56:41.260 align:middle line:90% Any questions about lecture 19? 00:56:41.260 --> 00:56:44.790 align:middle line:90% 00:56:44.790 --> 00:56:46.950 align:middle line:84% These reviews get faster and faster 00:56:46.950 --> 00:56:49.170 align:middle line:84% as we approach the end of the hour. 00:56:49.170 --> 00:56:52.920 align:middle line:84% But on the other hand, it's more and more recent, so 00:56:52.920 --> 00:56:54.420 align:middle line:90% not so worried. 00:56:54.420 --> 00:56:55.770 align:middle line:90% 21, we already did. 00:56:55.770 --> 00:56:58.290 align:middle line:90% So that just leaves us with 20. 00:56:58.290 --> 00:57:02.830 align:middle line:84% And 20 is pretty familiar, which was the failure of the welfare 00:57:02.830 --> 00:57:03.330 align:middle line:90% theorem. 00:57:03.330 --> 00:57:07.380 align:middle line:84% So we kind of covered sufficient assumptions already. 00:57:07.380 --> 00:57:09.540 align:middle line:90% And then for the fourth-- 00:57:09.540 --> 00:57:10.560 align:middle line:90% third or fourth time-- 00:57:10.560 --> 00:57:14.760 align:middle line:84% I'm showing you this economy again in all of its glory, 00:57:14.760 --> 00:57:20.100 align:middle line:84% with the convexity and concavity in the appropriate spots. 00:57:20.100 --> 00:57:22.080 align:middle line:84% Statement of the second welfare theorem, 00:57:22.080 --> 00:57:25.290 align:middle line:84% you actually saw this slide before in the review today. 00:57:25.290 --> 00:57:28.290 align:middle line:84% And the second welfare theorem says 00:57:28.290 --> 00:57:32.160 align:middle line:84% that any Pareto Optimum can be supported 00:57:32.160 --> 00:57:34.690 align:middle line:90% as of Walrasian equilibrium. 00:57:34.690 --> 00:57:38.190 align:middle line:84% So here's a case where everything's working out fine. 00:57:38.190 --> 00:57:40.950 align:middle line:84% There is a Pareto Optimum, obviously. 00:57:40.950 --> 00:57:44.160 align:middle line:84% This is the supporting hyperplane, again. 00:57:44.160 --> 00:57:47.200 align:middle line:84% So it's going to be a competitive equilibrium. 00:57:47.200 --> 00:57:51.910 align:middle line:84% But the second welfare theorem has sufficient assumptions 00:57:51.910 --> 00:57:53.660 align:middle line:90% to convexity. 00:57:53.660 --> 00:57:56.750 align:middle line:84% So as I said, I think last time-- 00:57:56.750 --> 00:58:01.150 align:middle line:84% or in answer to this second to last time-- 00:58:01.150 --> 00:58:03.340 align:middle line:84% the starting point here is something 00:58:03.340 --> 00:58:04.630 align:middle line:90% that's Pareto Optimal. 00:58:04.630 --> 00:58:07.990 align:middle line:84% And it's easy to lose track of the direction of the argument. 00:58:07.990 --> 00:58:09.250 align:middle line:90% Is this Pareto Optimal? 00:58:09.250 --> 00:58:12.940 align:middle line:84% Yes, it certainly is, or at least you can imagine so. 00:58:12.940 --> 00:58:16.060 align:middle line:84% This is the indifference curve that's tangent, locally. 00:58:16.060 --> 00:58:19.270 align:middle line:84% And you could try to move away from there, globally. 00:58:19.270 --> 00:58:21.880 align:middle line:84% But whatever you do, it has to be 00:58:21.880 --> 00:58:24.910 align:middle line:84% on this production set to be feasible 00:58:24.910 --> 00:58:28.090 align:middle line:84% and that's going to be associated with a lower 00:58:28.090 --> 00:58:30.530 align:middle line:90% level of an indifference curve. 00:58:30.530 --> 00:58:33.460 align:middle line:84% So this is the best level of utility 00:58:33.460 --> 00:58:36.700 align:middle line:84% achievable within and on the production set, and likewise 00:58:36.700 --> 00:58:37.600 align:middle line:90% over here. 00:58:37.600 --> 00:58:39.820 align:middle line:84% Oh, but then the second, I'm forgetting 00:58:39.820 --> 00:58:42.880 align:middle line:84% the second part of the theorem is, can it be supported 00:58:42.880 --> 00:58:44.150 align:middle line:90% as a competitive equilibrium? 00:58:44.150 --> 00:58:46.360 align:middle line:84% And the answer is no because given 00:58:46.360 --> 00:58:49.360 align:middle line:84% this imagine supporting hyperplane, 00:58:49.360 --> 00:58:51.040 align:middle line:84% the household would be better off 00:58:51.040 --> 00:58:53.110 align:middle line:90% moving away to some extreme. 00:58:53.110 --> 00:58:54.040 align:middle line:90% Have a higher utility. 00:58:54.040 --> 00:58:56.170 align:middle line:84% So it's not going to be consistent with utility 00:58:56.170 --> 00:58:57.070 align:middle line:90% maximization. 00:58:57.070 --> 00:59:00.280 align:middle line:84% And this one is not consistent with profit maximization, 00:59:00.280 --> 00:59:04.450 align:middle line:84% even though it is still a Pareto Optimal point. 00:59:04.450 --> 00:59:08.190 align:middle line:84% As you could construct, graphically. 00:59:08.190 --> 00:59:10.310 align:middle line:84% So I showed you this picture earlier. 00:59:10.310 --> 00:59:13.500 align:middle line:84% This is kind of an aggregated version of it. 00:59:13.500 --> 00:59:15.960 align:middle line:84% It says those assumptions are sufficient, 00:59:15.960 --> 00:59:18.720 align:middle line:84% they're not necessarily necessary. 00:59:18.720 --> 00:59:22.020 align:middle line:84% The first welfare theorem as a competitive equilibrium 00:59:22.020 --> 00:59:22.770 align:middle line:90% is optimal. 00:59:22.770 --> 00:59:26.730 align:middle line:84% This one assumes the existence of a competitive equilibrium. 00:59:26.730 --> 00:59:30.090 align:middle line:84% All we need is rationality and non-satiation. 00:59:30.090 --> 00:59:32.460 align:middle line:84% But this was a picture showing that if we 00:59:32.460 --> 00:59:35.340 align:middle line:84% don't have non-satiation locally, 00:59:35.340 --> 00:59:38.700 align:middle line:84% we're not necessarily going to show 00:59:38.700 --> 00:59:42.360 align:middle line:84% that the competitive equilibrium is Pareto Optimal because we're 00:59:42.360 --> 00:59:44.430 align:middle line:84% given the competitive equilibrium we can find 00:59:44.430 --> 00:59:46.980 align:middle line:90% something Pareto dominating. 00:59:46.980 --> 00:59:49.570 align:middle line:84% Then we went through this pollution example, 00:59:49.570 --> 00:59:53.260 align:middle line:84% which is not allowing trade in all the goods. 00:59:53.260 --> 00:59:57.510 align:middle line:84% And again, the intuition for this miraculous price system 00:59:57.510 --> 01:00:00.210 align:middle line:84% when it works means that households only 01:00:00.210 --> 01:00:03.780 align:middle line:84% have to evaluate things on the margin at those prices. 01:00:03.780 --> 01:00:07.230 align:middle line:84% They internalize through the prices 01:00:07.230 --> 01:00:09.390 align:middle line:90% and make the right decision. 01:00:09.390 --> 01:00:12.510 align:middle line:84% The prices reflect the social trade off 01:00:12.510 --> 01:00:16.510 align:middle line:84% of increasing or decreasing demands. 01:00:16.510 --> 01:00:18.310 align:middle line:84% And of course, they're much aware 01:00:18.310 --> 01:00:20.860 align:middle line:84% of their utility consequences, their marginal rate 01:00:20.860 --> 01:00:22.340 align:middle line:90% of substitution. 01:00:22.340 --> 01:00:24.670 align:middle line:84% So if we get it right in a competitive equilibrium 01:00:24.670 --> 01:00:27.730 align:middle line:84% in complete markets, and the price system conveys 01:00:27.730 --> 01:00:29.470 align:middle line:84% all the right things on the margin 01:00:29.470 --> 01:00:34.120 align:middle line:84% and all the people have to do is take from God this price system 01:00:34.120 --> 01:00:39.000 align:middle line:84% from Adam Smith, you might say, as if by an invisible hand, 01:00:39.000 --> 01:00:43.560 align:middle line:84% namely, the price system be guided to Pareto Optimal 01:00:43.560 --> 01:00:44.880 align:middle line:90% allocations. 01:00:44.880 --> 01:00:48.330 align:middle line:84% But likewise, if we screw up the markets, 01:00:48.330 --> 01:00:50.910 align:middle line:84% then the whole thing falls apart. 01:00:50.910 --> 01:00:53.580 align:middle line:84% And the solution example was where 01:00:53.580 --> 01:00:55.690 align:middle line:84% we didn't have a price for all the goods, 01:00:55.690 --> 01:00:58.410 align:middle line:84% so we don't have a rate of substitution of one 01:00:58.410 --> 01:01:02.940 align:middle line:84% good for the other in the market and households and firms 01:01:02.940 --> 01:01:04.530 align:middle line:90% make efficient choices. 01:01:04.530 --> 01:01:08.130 align:middle line:84% So we did that with the pollution as a leading example. 01:01:08.130 --> 01:01:11.160 align:middle line:84% We went to arrow to generalize it, 01:01:11.160 --> 01:01:14.130 align:middle line:84% where households care about other's consumption-- 01:01:14.130 --> 01:01:16.230 align:middle line:84% either they're altruistic or envious. 01:01:16.230 --> 01:01:17.790 align:middle line:84% And the same thing kind of works. 01:01:17.790 --> 01:01:20.160 align:middle line:84% This is the fourth or fifth time today 01:01:20.160 --> 01:01:22.560 align:middle line:84% that we talk about a Pareto problem 01:01:22.560 --> 01:01:25.050 align:middle line:84% and associated first order condition generating 01:01:25.050 --> 01:01:25.860 align:middle line:90% shadow prices. 01:01:25.860 --> 01:01:28.710 align:middle line:84% So we're following exactly the same steps as 01:01:28.710 --> 01:01:31.560 align:middle line:84% in the second welfare theorem and finding 01:01:31.560 --> 01:01:35.880 align:middle line:84% a way, despite the externalities and utility functions, 01:01:35.880 --> 01:01:38.520 align:middle line:84% to support the Pareto Optimal allocation 01:01:38.520 --> 01:01:41.520 align:middle line:90% as a competitive equilibrium. 01:01:41.520 --> 01:01:44.160 align:middle line:84% We already talked about failure when we have 01:01:44.160 --> 01:01:45.630 align:middle line:90% an infinite number of goods. 01:01:45.630 --> 01:01:51.420 align:middle line:84% This was an example of how to Pareto dominate an allocation 01:01:51.420 --> 01:01:55.960 align:middle line:84% by, essentially, giving the first household some goods 01:01:55.960 --> 01:01:58.980 align:middle line:84% the second had and then the third giving to the second, 01:01:58.980 --> 01:02:00.060 align:middle line:90% and so on. 01:02:00.060 --> 01:02:03.870 align:middle line:84% This is exactly what's going on in the overlapping generations, 01:02:03.870 --> 01:02:07.200 align:middle line:90% monetary economy of lecture 21. 01:02:07.200 --> 01:02:09.450 align:middle line:90% Money is worthless. 01:02:09.450 --> 01:02:11.310 align:middle line:90% It's just pieces of paper. 01:02:11.310 --> 01:02:16.050 align:middle line:84% There's no intrinsic value, but if household two here came up 01:02:16.050 --> 01:02:19.860 align:middle line:84% with money and gave it to household-- took money, sorry, 01:02:19.860 --> 01:02:24.750 align:middle line:84% from household one and exchanged it for goods, 01:02:24.750 --> 01:02:28.860 align:middle line:84% then one gets the goods and two gets the money and then two 01:02:28.860 --> 01:02:32.400 align:middle line:84% can pass the money along to household three. 01:02:32.400 --> 01:02:36.460 align:middle line:84% So there is a connection between this artificial example 01:02:36.460 --> 01:02:42.740 align:middle line:84% and the fact that we get a non-optimal autarky equilibrium 01:02:42.740 --> 01:02:47.360 align:middle line:84% dominated by a monetary equilibrium in the overlapping 01:02:47.360 --> 01:02:51.010 align:middle line:84% generations model, which we talked about at the beginning 01:02:51.010 --> 01:02:51.510 align:middle line:90% of class. 01:02:51.510 --> 01:02:52.230 align:middle line:90% OK. 01:02:52.230 --> 01:02:58.780 align:middle line:84% So essentially, that was it for lecture 23. 01:02:58.780 --> 01:03:00.910 align:middle line:84% And we already reviewed lecture 21. 01:03:00.910 --> 01:03:03.660 align:middle line:90% So questions? 01:03:03.660 --> 01:03:09.820 align:middle line:90% 01:03:09.820 --> 01:03:12.100 align:middle line:90% I enjoyed teaching the class. 01:03:12.100 --> 01:03:15.190 align:middle line:84% I really enjoyed the interaction with you guys. 01:03:15.190 --> 01:03:18.190 align:middle line:90% You're really doing well. 01:03:18.190 --> 01:03:19.510 align:middle line:90% It's a great joy to me. 01:03:19.510 --> 01:03:23.110 align:middle line:84% I hope you like this way of learning economics 01:03:23.110 --> 01:03:25.950 align:middle line:84% and you're really quite well trained now. 01:03:25.950 --> 01:03:28.040 align:middle line:90% Thank you very much. 01:03:28.040 --> 01:03:35.480 align:middle line:90%