WEBVTT 00:00:00.000 --> 00:00:02.475 align:middle line:90% [SQUEAKING] 00:00:02.475 --> 00:00:03.465 align:middle line:90% [RUSTLING] 00:00:03.465 --> 00:00:05.940 align:middle line:90% [CLICKING] 00:00:05.940 --> 00:00:11.748 align:middle line:90% 00:00:11.748 --> 00:00:13.540 align:middle line:84% ROBERT M. TOWNSEND: Let's think about where 00:00:13.540 --> 00:00:16.100 align:middle line:90% we are in the class. 00:00:16.100 --> 00:00:18.250 align:middle line:84% You should have the calendar in front of you. 00:00:18.250 --> 00:00:21.790 align:middle line:90% So we're at lecture 7 today. 00:00:21.790 --> 00:00:25.180 align:middle line:84% This is kind of a big moment because we're fully 00:00:25.180 --> 00:00:27.820 align:middle line:84% embracing general equilibrium although you've 00:00:27.820 --> 00:00:30.410 align:middle line:90% seen versions of it before. 00:00:30.410 --> 00:00:33.520 align:middle line:84% The other thing would be the reading list. 00:00:33.520 --> 00:00:38.440 align:middle line:84% For today, there is not a lot of new material on the reading is, 00:00:38.440 --> 00:00:40.150 align:middle line:90% not a lot of starred materials. 00:00:40.150 --> 00:00:42.220 align:middle line:84% We're going to do Pareto optimality. 00:00:42.220 --> 00:00:46.150 align:middle line:84% And the assigned starred reading is 00:00:46.150 --> 00:00:48.190 align:middle line:84% some of the sections in Kreps, which 00:00:48.190 --> 00:00:52.360 align:middle line:84% is available on the Canvas website, so not much 00:00:52.360 --> 00:00:53.890 align:middle line:90% additional stuff to read. 00:00:53.890 --> 00:00:55.900 align:middle line:84% We're going to do risk sharing at the end of it 00:00:55.900 --> 00:00:57.250 align:middle line:90% and then look-- 00:00:57.250 --> 00:00:59.136 align:middle line:84% get in more into the applications 00:00:59.136 --> 00:01:02.800 align:middle line:90% in lectures 8 and 9. 00:01:02.800 --> 00:01:05.420 align:middle line:90% In terms of the study guide-- 00:01:05.420 --> 00:01:09.490 align:middle line:84% so, again, let me ask you a few questions. 00:01:09.490 --> 00:01:13.330 align:middle line:84% We did dynamic optimization last time. 00:01:13.330 --> 00:01:16.660 align:middle line:84% And the first question is really the motivation, but not 00:01:16.660 --> 00:01:17.860 align:middle line:90% everything-- 00:01:17.860 --> 00:01:20.680 align:middle line:84% the motivation and the economic problems we're trying to solve 00:01:20.680 --> 00:01:24.190 align:middle line:84% are as important to the class as the tools and techniques 00:01:24.190 --> 00:01:25.040 align:middle line:90% that we're learning. 00:01:25.040 --> 00:01:29.380 align:middle line:84% So let me ask you, what are the observations on storage 00:01:29.380 --> 00:01:33.670 align:middle line:84% on carryover of last year's crop to this year's harvest that 00:01:33.670 --> 00:01:38.320 align:middle line:84% motivated that lecture that we did last time? 00:01:38.320 --> 00:01:39.925 align:middle line:90% Wang Peng, can you answer that? 00:01:39.925 --> 00:01:44.600 align:middle line:90% 00:01:44.600 --> 00:01:47.750 align:middle line:90% AUDIENCE: [INAUDIBLE] sorry. 00:01:47.750 --> 00:01:49.610 align:middle line:84% ROBERT M. TOWNSEND: Guanpeng can you 00:01:49.610 --> 00:01:51.020 align:middle line:90% take a stab at this question? 00:01:51.020 --> 00:01:55.700 align:middle line:84% What are the observations on storage carryover 00:01:55.700 --> 00:01:58.400 align:middle line:84% of last year's crop to this year's crop 00:01:58.400 --> 00:02:01.805 align:middle line:84% that motivated the lecture from last time? 00:02:01.805 --> 00:02:04.860 align:middle line:90% 00:02:04.860 --> 00:02:13.700 align:middle line:84% AUDIENCE: I believe the observation was that it wasn't 00:02:13.700 --> 00:02:15.243 align:middle line:90% necessarily that common for-- 00:02:15.243 --> 00:02:18.330 align:middle line:90% 00:02:18.330 --> 00:02:20.180 align:middle line:84% so I think it was like the crops that 00:02:20.180 --> 00:02:24.050 align:middle line:84% were carried over generally there's two ways to use both. 00:02:24.050 --> 00:02:28.070 align:middle line:84% One is carried over to be planted for next year 00:02:28.070 --> 00:02:30.620 align:middle line:90% and carry over for consumption. 00:02:30.620 --> 00:02:35.847 align:middle line:90% 00:02:35.847 --> 00:02:37.180 align:middle line:90% ROBERT M. TOWNSEND: That's true. 00:02:37.180 --> 00:02:42.670 align:middle line:84% Those were the two ways to quote, "store the harvest." 00:02:42.670 --> 00:02:49.048 align:middle line:84% Did they carry it over for consumption very frequently? 00:02:49.048 --> 00:02:50.590 align:middle line:84% AUDIENCE: I don't think so especially 00:02:50.590 --> 00:02:54.530 align:middle line:84% because the spoilage rate was relatively high. 00:02:54.530 --> 00:02:56.330 align:middle line:90% ROBERT M. TOWNSEND: OK, yeah. 00:02:56.330 --> 00:02:57.740 align:middle line:90% So it was quite rare. 00:02:57.740 --> 00:03:00.140 align:middle line:90% Carryover was quite rare. 00:03:00.140 --> 00:03:03.290 align:middle line:84% And the purpose of the model and its parameters 00:03:03.290 --> 00:03:08.307 align:middle line:84% was to explain why that would be the case. 00:03:08.307 --> 00:03:08.974 align:middle line:90% AUDIENCE: Right. 00:03:08.974 --> 00:03:10.967 align:middle line:90% 00:03:10.967 --> 00:03:12.800 align:middle line:84% ROBERT M. TOWNSEND: The title of the lecture 00:03:12.800 --> 00:03:16.170 align:middle line:90% was storage seed and starvation. 00:03:16.170 --> 00:03:20.880 align:middle line:84% So do you remember how often were they starving to death? 00:03:20.880 --> 00:03:22.917 align:middle line:84% AUDIENCE: It was once every decade or 12 years. 00:03:22.917 --> 00:03:24.500 align:middle line:84% ROBERT M. TOWNSEND: Yes, that's right. 00:03:24.500 --> 00:03:25.820 align:middle line:90% Thank you. 00:03:25.820 --> 00:03:28.820 align:middle line:90% Good, all right. 00:03:28.820 --> 00:03:32.970 align:middle line:84% All right, so that was the motivation for that lecture. 00:03:32.970 --> 00:03:38.520 align:middle line:90% And then this question-- 00:03:38.520 --> 00:03:41.920 align:middle line:84% although I don't-- you won't be able to write it down for me, 00:03:41.920 --> 00:03:44.520 align:middle line:84% but maybe you can say it in words. 00:03:44.520 --> 00:03:49.050 align:middle line:84% What was the notation we used for the return to seed? 00:03:49.050 --> 00:03:54.920 align:middle line:84% And in particular, in what way was it linear or at least up 00:03:54.920 --> 00:03:55.550 align:middle line:90% to a limit? 00:03:55.550 --> 00:04:00.080 align:middle line:84% And in which way did it depend on uncertainty? 00:04:00.080 --> 00:04:06.820 align:middle line:84% And let me see, Armando, can you take a stab at that? 00:04:06.820 --> 00:04:16.060 align:middle line:90% 00:04:16.060 --> 00:04:17.399 align:middle line:90% Are you there, Armando? 00:04:17.399 --> 00:04:18.070 align:middle line:90% Armando? 00:04:18.070 --> 00:04:18.820 align:middle line:90% AUDIENCE: Yes. 00:04:18.820 --> 00:04:21.649 align:middle line:90% Sorry, my microphone was muted. 00:04:21.649 --> 00:04:23.170 align:middle line:90% Yeah, so there was-- 00:04:23.170 --> 00:04:27.910 align:middle line:84% I believe it was there was three multipliers, right. 00:04:27.910 --> 00:04:31.930 align:middle line:84% It was they each had a random chance of occurring. 00:04:31.930 --> 00:04:34.870 align:middle line:84% And it was like a poor harvest, medium harvest, 00:04:34.870 --> 00:04:36.280 align:middle line:90% and good harvest. 00:04:36.280 --> 00:04:38.260 align:middle line:90% And it was just linear. 00:04:38.260 --> 00:04:40.540 align:middle line:84% The harvest, the amount harvested 00:04:40.540 --> 00:04:44.640 align:middle line:84% is linear with how much they planted with either a 1, 00:04:44.640 --> 00:04:49.160 align:middle line:84% 2, or 4 multiplier, I believe, was it? 00:04:49.160 --> 00:04:51.470 align:middle line:84% ROBERT M. TOWNSEND: That's right. 00:04:51.470 --> 00:04:56.370 align:middle line:84% Yeah, it was epsilon times alpha times k, 00:04:56.370 --> 00:04:59.420 align:middle line:84% where k was the amount of land planted, 00:04:59.420 --> 00:05:04.860 align:middle line:84% alpha was the linear constant, and epsilon 00:05:04.860 --> 00:05:07.390 align:middle line:84% capture those shocks you just mentioned low, medium, 00:05:07.390 --> 00:05:07.890 align:middle line:90% and high. 00:05:07.890 --> 00:05:09.960 align:middle line:90% So that's a perfect answer. 00:05:09.960 --> 00:05:10.560 align:middle line:90% Thank you. 00:05:10.560 --> 00:05:13.880 align:middle line:90% 00:05:13.880 --> 00:05:17.420 align:middle line:84% So it's blending-- the point of why I'm asking 00:05:17.420 --> 00:05:21.290 align:middle line:84% is just to remind you that, oh, because it's linear, 00:05:21.290 --> 00:05:25.360 align:middle line:90% its constant returns to scale. 00:05:25.360 --> 00:05:28.760 align:middle line:84% And yet it also incorporated the uncertainty. 00:05:28.760 --> 00:05:31.600 align:middle line:84% So we're going to come back to that technology 00:05:31.600 --> 00:05:35.380 align:middle line:84% when we look at applications of risk sharing 00:05:35.380 --> 00:05:37.950 align:middle line:90% in a couple of lectures. 00:05:37.950 --> 00:05:42.710 align:middle line:84% All right, and this next one, write down 00:05:42.710 --> 00:05:47.270 align:middle line:84% a flow equation for consumption in terms of the above notation, 00:05:47.270 --> 00:05:50.730 align:middle line:84% but I can't expect you to write it down. 00:05:50.730 --> 00:05:53.310 align:middle line:84% So could someone describe in words 00:05:53.310 --> 00:05:59.185 align:middle line:84% what the constraint is onto the problem in terms of this flow 00:05:59.185 --> 00:05:59.685 align:middle line:90% equation? 00:05:59.685 --> 00:06:02.470 align:middle line:90% 00:06:02.470 --> 00:06:03.850 align:middle line:90% I'll take a volunteer. 00:06:03.850 --> 00:06:06.970 align:middle line:90% 00:06:06.970 --> 00:06:09.520 align:middle line:84% AUDIENCE: I remember the consumption 00:06:09.520 --> 00:06:15.880 align:middle line:84% is equal to the output of the last year 00:06:15.880 --> 00:06:27.045 align:middle line:84% input plus the output of the inventory and a minus 00:06:27.045 --> 00:06:32.730 align:middle line:84% how much you want to leave for next year as the seed 00:06:32.730 --> 00:06:37.230 align:middle line:84% and how much you leave for next year as the inventory. 00:06:37.230 --> 00:06:39.012 align:middle line:90% That was consumption. 00:06:39.012 --> 00:06:40.470 align:middle line:84% ROBERT M. TOWNSEND: That's perfect. 00:06:40.470 --> 00:06:46.470 align:middle line:84% AUDIENCE: And the consumption should be larger than 0, right? 00:06:46.470 --> 00:06:48.150 align:middle line:90% ROBERT M. TOWNSEND: Yeah. 00:06:48.150 --> 00:06:50.370 align:middle line:90% AUDIENCE: For each period? 00:06:50.370 --> 00:06:52.590 align:middle line:84% ROBERT M. TOWNSEND: That's another constraint, yeah. 00:06:52.590 --> 00:06:54.930 align:middle line:84% This flow equation you just described so well. 00:06:54.930 --> 00:06:58.650 align:middle line:84% It's exactly what I was looking for. 00:06:58.650 --> 00:07:03.530 align:middle line:84% So to elaborate, we talked about resources available today, 00:07:03.530 --> 00:07:06.320 align:middle line:84% and that's the stuff coming from current harvest 00:07:06.320 --> 00:07:10.490 align:middle line:90% plus depreciated storage. 00:07:10.490 --> 00:07:15.500 align:middle line:84% And then also, how much would it be available tomorrow? 00:07:15.500 --> 00:07:18.560 align:middle line:84% Well, that is co-determined with consumption. 00:07:18.560 --> 00:07:21.900 align:middle line:84% Because you have available resources, 00:07:21.900 --> 00:07:24.880 align:middle line:84% but you can use it for consumption or you can save it. 00:07:24.880 --> 00:07:28.490 align:middle line:84% And there were two different ways to save. 00:07:28.490 --> 00:07:31.850 align:middle line:84% So your answer was very good, and I'm just repeating 00:07:31.850 --> 00:07:35.170 align:middle line:90% in slightly different words. 00:07:35.170 --> 00:07:35.670 align:middle line:90% OK. 00:07:35.670 --> 00:07:40.260 align:middle line:90% 00:07:40.260 --> 00:07:47.830 align:middle line:84% I guess last one is principal of dynamic programming. 00:07:47.830 --> 00:07:50.830 align:middle line:90% This is a bit more technical. 00:07:50.830 --> 00:07:53.440 align:middle line:84% But, again, I don't want to hide anything from you, especially 00:07:53.440 --> 00:07:56.530 align:middle line:84% when we subsequently jump back and forth 00:07:56.530 --> 00:08:00.010 align:middle line:84% between finite horizon and infinite horizon problems. 00:08:00.010 --> 00:08:04.330 align:middle line:84% Does anyone remember how we converted 00:08:04.330 --> 00:08:07.980 align:middle line:90% into two period problems? 00:08:07.980 --> 00:08:11.090 align:middle line:84% AUDIENCE: So we picked the last date 00:08:11.090 --> 00:08:13.580 align:middle line:84% and solved the maximization problem using that last date, 00:08:13.580 --> 00:08:15.560 align:middle line:84% and then iterated to the next last date 00:08:15.560 --> 00:08:18.210 align:middle line:84% until we got back to the first date. 00:08:18.210 --> 00:08:19.460 align:middle line:90% ROBERT M. TOWNSEND: Excellent. 00:08:19.460 --> 00:08:20.085 align:middle line:90% That's perfect. 00:08:20.085 --> 00:08:24.470 align:middle line:90% 00:08:24.470 --> 00:08:27.320 align:middle line:84% And then the next part, the infinite horizon part, 00:08:27.320 --> 00:08:29.450 align:middle line:90% is we just keep going like that. 00:08:29.450 --> 00:08:33.860 align:middle line:84% We just keep iterating, so that the final terminal date 00:08:33.860 --> 00:08:37.940 align:middle line:84% is so far away that effectively they're 00:08:37.940 --> 00:08:43.390 align:middle line:84% acting the way they would in an infinite horizon problem. 00:08:43.390 --> 00:08:48.870 align:middle line:84% All right, so that's probably enough time spent 00:08:48.870 --> 00:08:53.080 align:middle line:90% on those review questions. 00:08:53.080 --> 00:09:01.370 align:middle line:84% Let me then turn to the lecture today, lecture 7. 00:09:01.370 --> 00:09:03.990 align:middle line:90% 00:09:03.990 --> 00:09:09.830 align:middle line:84% All right, so a brief outline of what we're going to do-- 00:09:09.830 --> 00:09:14.780 align:middle line:84% we're going to define this concept of various concepts, 00:09:14.780 --> 00:09:19.250 align:middle line:84% Pareto optimality, then Pareto frontier. 00:09:19.250 --> 00:09:23.210 align:middle line:84% Then I'm going to show you a programming problem, a math 00:09:23.210 --> 00:09:26.210 align:middle line:84% problem so to speak, that allows us to determine, 00:09:26.210 --> 00:09:30.860 align:middle line:84% as its solution, all possible Pareto optimal allocations. 00:09:30.860 --> 00:09:33.830 align:middle line:84% And then we're going to go and apply it 00:09:33.830 --> 00:09:37.910 align:middle line:84% in the case of uncertainty and call the solution 00:09:37.910 --> 00:09:40.080 align:middle line:90% the optimal allocation of risk. 00:09:40.080 --> 00:09:44.540 align:middle line:84% So although this slide says utility possibilities set, 00:09:44.540 --> 00:09:47.600 align:middle line:84% and that's going to be the bottom line of the slide, 00:09:47.600 --> 00:09:52.160 align:middle line:84% this is also a hugely important slide for the whole course 00:09:52.160 --> 00:09:55.790 align:middle line:84% because it's the first time, I think, that we've introduced, 00:09:55.790 --> 00:10:00.290 align:middle line:84% in notation, all the ingredients we've been covering so far 00:10:00.290 --> 00:10:03.860 align:middle line:84% and, in the notation, describing an economy. 00:10:03.860 --> 00:10:05.570 align:middle line:84% I went back and looked at lecture 1. 00:10:05.570 --> 00:10:09.380 align:middle line:84% It does say an economy is preferences, endowments, 00:10:09.380 --> 00:10:10.280 align:middle line:90% and technology. 00:10:10.280 --> 00:10:13.580 align:middle line:84% We called it a PET economy as a way to remember. 00:10:13.580 --> 00:10:15.300 align:middle line:90% But here it is in notation. 00:10:15.300 --> 00:10:19.580 align:middle line:84% So an economy, script epsilon or economy, 00:10:19.580 --> 00:10:22.070 align:middle line:84% is the consumption set and preferences 00:10:22.070 --> 00:10:24.390 align:middle line:84% of each of finite number of agents, 00:10:24.390 --> 00:10:28.070 align:middle line:84% the production set of each of the finite number of firms, 00:10:28.070 --> 00:10:31.610 align:middle line:84% and some aggregated social endowment. 00:10:31.610 --> 00:10:36.080 align:middle line:84% So it's a very succinct list, preferences, endowments, 00:10:36.080 --> 00:10:37.310 align:middle line:90% and technology. 00:10:37.310 --> 00:10:40.850 align:middle line:84% Then an allocation for the whole economy 00:10:40.850 --> 00:10:47.660 align:middle line:84% now is an allocation x for each of the capital I consumers, 00:10:47.660 --> 00:10:51.230 align:middle line:84% an allocation in the production set for each of the capital 00:10:51.230 --> 00:10:52.820 align:middle line:90% J producers. 00:10:52.820 --> 00:10:56.210 align:middle line:84% If we're in finite dimensional Euclidean space 00:10:56.210 --> 00:11:01.730 align:middle line:84% and there are capital L goods, then we have L times I plus J 00:11:01.730 --> 00:11:06.320 align:middle line:84% dimensional vectors here belied by this relatively brief 00:11:06.320 --> 00:11:07.120 align:middle line:90% notation. 00:11:07.120 --> 00:11:09.620 align:middle line:84% In other words, we're going to have to specify a consumption 00:11:09.620 --> 00:11:12.890 align:middle line:84% vector Xi in the consumption set for each consumer 00:11:12.890 --> 00:11:16.775 align:middle line:84% i, a production plan Yj in the production set. 00:11:16.775 --> 00:11:20.930 align:middle line:84% It's capital Yj for each firm j of the capital J. 00:11:20.930 --> 00:11:24.380 align:middle line:84% And we say an allocation is feasible if and only 00:11:24.380 --> 00:11:29.070 align:middle line:84% if the consumptions are feasible given resources available, 00:11:29.070 --> 00:11:32.880 align:middle line:84% which would be coming from the endowment and from production. 00:11:32.880 --> 00:11:35.540 align:middle line:84% So a couple of comments-- sometimes 00:11:35.540 --> 00:11:38.060 align:middle line:84% people will write consumption less than 00:11:38.060 --> 00:11:41.670 align:middle line:84% or equal to as if you could throw something away. 00:11:41.670 --> 00:11:42.560 align:middle line:90% That's fine. 00:11:42.560 --> 00:11:46.850 align:middle line:84% It's just more useful today to talk about it as an equality. 00:11:46.850 --> 00:11:50.390 align:middle line:84% The second thing is this has buried behind it-- 00:11:50.390 --> 00:11:51.830 align:middle line:90% and it's easy to write-- 00:11:51.830 --> 00:11:55.040 align:middle line:84% all of the business we've been working with respect 00:11:55.040 --> 00:11:57.090 align:middle line:84% to negative and positive elements. 00:11:57.090 --> 00:12:00.200 align:middle line:84% So for example, some of these Yj's are negative. 00:12:00.200 --> 00:12:03.440 align:middle line:84% I refer to it as production as if it were all positive, 00:12:03.440 --> 00:12:04.940 align:middle line:90% like grain or something. 00:12:04.940 --> 00:12:07.880 align:middle line:84% But if an input is negative over here, 00:12:07.880 --> 00:12:11.120 align:middle line:84% then there has to be some offsetting element. 00:12:11.120 --> 00:12:16.370 align:middle line:84% Namely, consumers supply it, or it could be more evident 00:12:16.370 --> 00:12:19.190 align:middle line:90% coming from their endowment. 00:12:19.190 --> 00:12:21.380 align:middle line:84% So this is where the accounting quote 00:12:21.380 --> 00:12:24.170 align:middle line:84% of all the positive and negative stuff is kind of buried. 00:12:24.170 --> 00:12:27.300 align:middle line:84% But it is easier to remember this way, 00:12:27.300 --> 00:12:29.690 align:middle line:84% because it looks like you just have resources 00:12:29.690 --> 00:12:32.450 align:middle line:84% available from endowments and production 00:12:32.450 --> 00:12:36.320 align:middle line:84% meeting up with resources used in consumption. 00:12:36.320 --> 00:12:40.130 align:middle line:84% OK, now given the same economy, we 00:12:40.130 --> 00:12:43.430 align:middle line:84% can define the set of all possible utilities, 00:12:43.430 --> 00:12:46.910 align:middle line:84% utility possibilities set, which is this script U 00:12:46.910 --> 00:12:50.510 align:middle line:84% and which is a specification of utility 00:12:50.510 --> 00:12:52.140 align:middle line:90% for each and every household. 00:12:52.140 --> 00:12:54.830 align:middle line:84% So this has the dimension capital I, which 00:12:54.830 --> 00:12:57.260 align:middle line:90% is the number of households. 00:12:57.260 --> 00:13:00.740 align:middle line:84% And it's a vector of that dimension. 00:13:00.740 --> 00:13:05.180 align:middle line:84% And we're going to say that a vector U bar is in this set. 00:13:05.180 --> 00:13:09.500 align:middle line:84% And if two things are true, first of all, 00:13:09.500 --> 00:13:11.870 align:middle line:84% there has to be a feasible allocation 00:13:11.870 --> 00:13:15.560 align:middle line:84% generating it or generating something with at least 00:13:15.560 --> 00:13:17.030 align:middle line:90% as much utility. 00:13:17.030 --> 00:13:22.580 align:middle line:84% So a vector over here, say, the i-th component of this vector, 00:13:22.580 --> 00:13:27.920 align:middle line:84% has to be able to be generated by part of a feasible 00:13:27.920 --> 00:13:31.490 align:middle line:84% allocation, which is specifying Xi for household 00:13:31.490 --> 00:13:34.820 align:middle line:84% i, which, when substituted in the utility function, 00:13:34.820 --> 00:13:38.030 align:middle line:84% generates a utility level at least as great 00:13:38.030 --> 00:13:43.250 align:middle line:84% as the target, namely U bar sub i. 00:13:43.250 --> 00:13:45.980 align:middle line:84% Again, it would be a little more intuitive just 00:13:45.980 --> 00:13:47.700 align:middle line:90% to put equality here. 00:13:47.700 --> 00:13:50.030 align:middle line:84% But what this allows us to say is, 00:13:50.030 --> 00:13:53.720 align:middle line:84% if you can generate a utility point through a feasible 00:13:53.720 --> 00:13:57.680 align:middle line:84% allocation, then all the utility points which are southwest 00:13:57.680 --> 00:14:00.330 align:middle line:90% of it are also feasible. 00:14:00.330 --> 00:14:03.000 align:middle line:84% So it's as if you could throw things away. 00:14:03.000 --> 00:14:06.590 align:middle line:84% So in words, this utility possibility set is a set up all 00:14:06.590 --> 00:14:10.340 align:middle line:84% utility profiles for households in the economy that can be 00:14:10.340 --> 00:14:15.060 align:middle line:84% attained by feasible allocation, a feasible allocation with 00:14:15.060 --> 00:14:17.440 align:middle line:84% greater than or equal-- small typo-- 00:14:17.440 --> 00:14:19.990 align:middle line:84% so you can generate v. In other words, 00:14:19.990 --> 00:14:24.456 align:middle line:84% if you can generate U and v is less than U, 00:14:24.456 --> 00:14:28.620 align:middle line:84% you can have v in the production possibility set as well. 00:14:28.620 --> 00:14:30.510 align:middle line:84% Because whatever that generated U 00:14:30.510 --> 00:14:34.230 align:middle line:84% or something generating utility greater than or equal to U 00:14:34.230 --> 00:14:37.500 align:middle line:84% will generate utility greater than or equal to v, all right? 00:14:37.500 --> 00:14:40.800 align:middle line:84% So given a utility possibility set, script U, 00:14:40.800 --> 00:14:47.490 align:middle line:84% we define the Pareto frontier, P of U, as the boundary. 00:14:47.490 --> 00:14:52.380 align:middle line:84% So it's just the set of subset of utility vectors 00:14:52.380 --> 00:14:54.960 align:middle line:84% such that there is nothing northeast. 00:14:54.960 --> 00:14:58.050 align:middle line:84% If U is in the utility possibility set, 00:14:58.050 --> 00:15:00.060 align:middle line:84% there's no U prime in there, which 00:15:00.060 --> 00:15:03.780 align:middle line:84% would generate at least as much utility for all households 00:15:03.780 --> 00:15:08.490 align:middle line:84% i and strictly greater utility for some households j. 00:15:08.490 --> 00:15:11.050 align:middle line:84% So if the right hand side were true, 00:15:11.050 --> 00:15:14.670 align:middle line:84% then you could quote, "Pareto dominate" the vector U. 00:15:14.670 --> 00:15:17.400 align:middle line:84% And that would eliminate it from being in what we want 00:15:17.400 --> 00:15:20.040 align:middle line:90% to call the Pareto frontier. 00:15:20.040 --> 00:15:22.630 align:middle line:84% So here's an example and then a picture. 00:15:22.630 --> 00:15:26.490 align:middle line:84% Let's simplify our economy and get rid of production, 00:15:26.490 --> 00:15:28.510 align:middle line:84% although we will still have endowments. 00:15:28.510 --> 00:15:31.050 align:middle line:84% So this is a pure exchange economy. 00:15:31.050 --> 00:15:32.430 align:middle line:90% There are only two goods. 00:15:32.430 --> 00:15:34.500 align:middle line:90% And there are two consumers. 00:15:34.500 --> 00:15:36.630 align:middle line:90% L and I are equal to 2. 00:15:36.630 --> 00:15:40.410 align:middle line:84% And we're going to say, for simplicity of notation, 00:15:40.410 --> 00:15:44.160 align:middle line:84% first good is denoted X. The second good is denoted Z. 00:15:44.160 --> 00:15:47.520 align:middle line:84% So the first household has the utility function over X and Z, 00:15:47.520 --> 00:15:50.490 align:middle line:84% which is just linearly additive in the 2. 00:15:50.490 --> 00:15:53.980 align:middle line:84% And the second household is very similar, 00:15:53.980 --> 00:15:58.440 align:middle line:84% except it is multiplied by 2, but it's still linear, 00:15:58.440 --> 00:16:00.550 align:middle line:90% additive, and so on. 00:16:00.550 --> 00:16:04.410 align:middle line:84% Now, each has equal endowment of both goods. 00:16:04.410 --> 00:16:06.210 align:middle line:84% And the endowment of household one 00:16:06.210 --> 00:16:09.570 align:middle line:84% is equal to the endowment of household two. 00:16:09.570 --> 00:16:14.250 align:middle line:84% So 1/2, 1/2 everywhere means there's 1 unit of good 1 00:16:14.250 --> 00:16:17.820 align:middle line:84% in total, although it's split initially equally 00:16:17.820 --> 00:16:22.350 align:middle line:84% across the two households, and 1 unit of good 2 available, 00:16:22.350 --> 00:16:24.610 align:middle line:90% also split equally. 00:16:24.610 --> 00:16:27.760 align:middle line:84% So a feasible allocation over these two goods 00:16:27.760 --> 00:16:30.880 align:middle line:90% X and Z has to sum up properly. 00:16:30.880 --> 00:16:33.750 align:middle line:84% So if we take the first good and add up 00:16:33.750 --> 00:16:37.470 align:middle line:84% what household 1 is getting, what household 2 is getting, 00:16:37.470 --> 00:16:40.740 align:middle line:84% it's got to equal the endowment, which is 1, 00:16:40.740 --> 00:16:45.540 align:middle line:84% likewise for the second good, Z. What the first guy gets 00:16:45.540 --> 00:16:47.685 align:middle line:84% and the second guy gets in allocation-- 00:16:47.685 --> 00:16:50.555 align:middle line:84% it's not utility-- has to add up to 1, 00:16:50.555 --> 00:16:53.040 align:middle line:90% the sum of the endowments, OK. 00:16:53.040 --> 00:16:55.200 align:middle line:90% So it can be shown-- 00:16:55.200 --> 00:16:56.970 align:middle line:84% and I will show you the picture-- 00:16:56.970 --> 00:16:59.640 align:middle line:84% that the utility possibility set consists 00:16:59.640 --> 00:17:04.920 align:middle line:84% of all points U2, which are less than 4 minus 2U1. 00:17:04.920 --> 00:17:10.500 align:middle line:84% And the frontier is just the outer portion of that. 00:17:10.500 --> 00:17:12.720 align:middle line:84% Without doing all the algebra right now, 00:17:12.720 --> 00:17:15.160 align:middle line:84% you can just imagine, for example, 00:17:15.160 --> 00:17:19.200 align:middle line:84% what would happen if we gave the entire social endowment 00:17:19.200 --> 00:17:20.880 align:middle line:90% to agent 2. 00:17:20.880 --> 00:17:23.670 align:middle line:84% Agent 2 would then get 1 here, 1 here. 00:17:23.670 --> 00:17:25.980 align:middle line:90% And you get 2 plus 2 equal 4. 00:17:25.980 --> 00:17:29.460 align:middle line:84% And you see this 4 down here, which is the maximum utility 00:17:29.460 --> 00:17:31.050 align:middle line:90% that agent 2 could get. 00:17:31.050 --> 00:17:34.920 align:middle line:84% And the fact that it's minus means moving away from there. 00:17:34.920 --> 00:17:38.100 align:middle line:84% We're giving agent 1 some of the social endowment 00:17:38.100 --> 00:17:40.470 align:middle line:84% and taking away from household 2. 00:17:40.470 --> 00:17:45.150 align:middle line:84% And since 2's utility level's twice that of household 1, 00:17:45.150 --> 00:17:49.980 align:middle line:84% then every unit utility that 2 gives up 00:17:49.980 --> 00:17:52.660 align:middle line:90% counts twice for household 1. 00:17:52.660 --> 00:17:53.160 align:middle line:90% OK. 00:17:53.160 --> 00:17:57.600 align:middle line:84% So again, that's just to be motivational without actually 00:17:57.600 --> 00:17:59.130 align:middle line:90% going through all the algebra. 00:17:59.130 --> 00:18:02.580 align:middle line:84% Here is what the utility of possibilities frontier 00:18:02.580 --> 00:18:03.330 align:middle line:90% looks like. 00:18:03.330 --> 00:18:08.040 align:middle line:84% It's linear, again, with 4 over here as the maximal point. 00:18:08.040 --> 00:18:11.160 align:middle line:90% This is a slope of minus 2. 00:18:11.160 --> 00:18:14.280 align:middle line:84% It's negative because you're giving up the social endowment 00:18:14.280 --> 00:18:19.560 align:middle line:84% away from 2 toward 1 as you move along this line. 00:18:19.560 --> 00:18:23.940 align:middle line:84% What's not drawn in here very carefully are the boundaries 00:18:23.940 --> 00:18:29.250 align:middle line:84% of consumption and so on, which if X1 and X2, Z1, 00:18:29.250 --> 00:18:32.940 align:middle line:84% Z2 cannot go negative, then we shouldn't be drawing in this 00:18:32.940 --> 00:18:36.000 align:middle line:84% thing that extends beyond the first orthon. 00:18:36.000 --> 00:18:39.600 align:middle line:84% Pareto optimality, this is the big one, 00:18:39.600 --> 00:18:42.360 align:middle line:90% but it's almost obvious already. 00:18:42.360 --> 00:18:47.190 align:middle line:84% An allocation x and y for consumption and production 00:18:47.190 --> 00:18:50.670 align:middle line:84% is Pareto optimal if it's feasible 00:18:50.670 --> 00:18:53.820 align:middle line:84% and there's no other feasible allocation x prime, 00:18:53.820 --> 00:18:57.840 align:middle line:84% y prime such that the prime is at least 00:18:57.840 --> 00:19:00.970 align:middle line:84% as good from every agent i's point of view 00:19:00.970 --> 00:19:04.980 align:middle line:84% and there's at least one i prime household agent 00:19:04.980 --> 00:19:08.700 align:middle line:90% that really likes prime better. 00:19:08.700 --> 00:19:12.450 align:middle line:84% And we call these allocations the Pareto set, the set 00:19:12.450 --> 00:19:14.490 align:middle line:90% of Pareto optimal allocations. 00:19:14.490 --> 00:19:16.560 align:middle line:84% Well, at one level, it's kind of intuitive 00:19:16.560 --> 00:19:18.930 align:middle line:84% that those should be called optimal 00:19:18.930 --> 00:19:21.780 align:middle line:84% because you can't make anyone better off without making 00:19:21.780 --> 00:19:23.640 align:middle line:90% someone worse off. 00:19:23.640 --> 00:19:25.560 align:middle line:84% If you could make everyone better off, 00:19:25.560 --> 00:19:28.230 align:middle line:84% then it wouldn't be Pareto optimal to begin with. 00:19:28.230 --> 00:19:31.170 align:middle line:84% A little more telling, this definition 00:19:31.170 --> 00:19:34.090 align:middle line:84% doesn't give a darn about the distribution of income. 00:19:34.090 --> 00:19:38.010 align:middle line:84% So there's no normative statement here about equality 00:19:38.010 --> 00:19:40.870 align:middle line:84% and the income distribution or anything like that. 00:19:40.870 --> 00:19:42.570 align:middle line:90% So that's a little bit jarring. 00:19:42.570 --> 00:19:46.440 align:middle line:84% We could give all the allocation to household 2 and none 00:19:46.440 --> 00:19:49.950 align:middle line:84% to household 1, and that would still be Pareto optimal. 00:19:49.950 --> 00:19:52.170 align:middle line:84% We can look at the distribution of income, 00:19:52.170 --> 00:19:55.710 align:middle line:84% and we surely will often in the subsequence lectures 00:19:55.710 --> 00:19:58.500 align:middle line:84% and in some of the pictures about to follow. 00:19:58.500 --> 00:20:01.890 align:middle line:84% That means, if you want to talk about the distribution 00:20:01.890 --> 00:20:05.160 align:middle line:84% of income and put some normative criteria for that, 00:20:05.160 --> 00:20:07.680 align:middle line:84% you're going to need more than this definition of Pareto 00:20:07.680 --> 00:20:10.600 align:middle line:90% optimality. 00:20:10.600 --> 00:20:15.790 align:middle line:84% OK, so let's do a two-person economy. 00:20:15.790 --> 00:20:20.140 align:middle line:84% This is literally putting the pieces together. 00:20:20.140 --> 00:20:23.950 align:middle line:84% We did for households, utility functions 00:20:23.950 --> 00:20:25.150 align:middle line:90% and preference orderings. 00:20:25.150 --> 00:20:29.020 align:middle line:84% And this puts consumer 1 with two goods, changing the names 00:20:29.020 --> 00:20:33.190 align:middle line:84% to a and b now, nice, smooth, regular decreasing 00:20:33.190 --> 00:20:34.880 align:middle line:90% marginal rates of substitution. 00:20:34.880 --> 00:20:38.170 align:middle line:84% Oh, and household 1's endowment is 00:20:38.170 --> 00:20:41.890 align:middle line:84% here, mark e, which means a certain amount of good b 00:20:41.890 --> 00:20:43.600 align:middle line:90% and a certain amount of good a. 00:20:43.600 --> 00:20:45.370 align:middle line:84% So there are two households, but we're 00:20:45.370 --> 00:20:47.720 align:middle line:84% going to make this guy stand on his head. 00:20:47.720 --> 00:20:51.400 align:middle line:84% So the 0 point for household b is here. 00:20:51.400 --> 00:20:54.490 align:middle line:84% And more of good b moves downward. 00:20:54.490 --> 00:20:57.400 align:middle line:90% More of good a moves leftward. 00:20:57.400 --> 00:21:03.520 align:middle line:84% So these are regular, nice, smooth, concave, quasi-concave 00:21:03.520 --> 00:21:05.680 align:middle line:84% indifference curves preference sets 00:21:05.680 --> 00:21:07.000 align:middle line:90% with more preferred to less. 00:21:07.000 --> 00:21:11.440 align:middle line:84% And here's the endowment of agent 2, boom, boom-- 00:21:11.440 --> 00:21:14.350 align:middle line:84% certain amount of good 2, a certain amount of be, 00:21:14.350 --> 00:21:15.820 align:middle line:90% a certain amount of good a. 00:21:15.820 --> 00:21:17.900 align:middle line:90% You with me so far? 00:21:17.900 --> 00:21:20.580 align:middle line:84% I think this came from Kreps, but, you know, 00:21:20.580 --> 00:21:24.790 align:middle line:84% the Edgeworth box is a big part of economics. 00:21:24.790 --> 00:21:28.300 align:middle line:84% Now, I'm going to take this point e 00:21:28.300 --> 00:21:33.020 align:middle line:84% and start moving this whole figure for consumer 2 00:21:33.020 --> 00:21:37.930 align:middle line:84% to the point that these two e's correspond and are identically 00:21:37.930 --> 00:21:40.010 align:middle line:90% lying on top of one another. 00:21:40.010 --> 00:21:42.040 align:middle line:84% So that's like moving this edge of what's 00:21:42.040 --> 00:21:45.260 align:middle line:84% about to become the box over in this direction. 00:21:45.260 --> 00:21:48.190 align:middle line:84% And here, although e should have been ideally put 00:21:48.190 --> 00:21:51.920 align:middle line:84% on this picture, this would be the Edgeworth box. 00:21:51.920 --> 00:21:54.850 align:middle line:84% So now, we've got household 1 down here, 00:21:54.850 --> 00:21:58.660 align:middle line:84% consumer 2 here, and the indifference curves of both. 00:21:58.660 --> 00:22:05.740 align:middle line:84% The endowment, say, here, is denoting the good a assigned 00:22:05.740 --> 00:22:08.650 align:middle line:84% initially as an endowment to consumer 1, 00:22:08.650 --> 00:22:12.010 align:middle line:84% but the other direction here is good 00:22:12.010 --> 00:22:15.790 align:middle line:90% b being assigned to household 2. 00:22:15.790 --> 00:22:17.350 align:middle line:90% So things add up. 00:22:17.350 --> 00:22:19.630 align:middle line:84% The total adds up except you're moving-- 00:22:19.630 --> 00:22:23.170 align:middle line:84% the amounts are signifying different directions 00:22:23.170 --> 00:22:26.770 align:middle line:84% for the two consumers because consumer 2 is upside down. 00:22:26.770 --> 00:22:30.550 align:middle line:84% So are there questions about the construction of the box? 00:22:30.550 --> 00:22:34.870 align:middle line:84% All right, so without putting the endowment in there, 00:22:34.870 --> 00:22:40.990 align:middle line:84% this would be red indifference curves for household agent 1, 00:22:40.990 --> 00:22:44.440 align:middle line:84% bluish indifference curves for consumer 2. 00:22:44.440 --> 00:22:46.900 align:middle line:84% I call them consumers, households, and agents. 00:22:46.900 --> 00:22:48.610 align:middle line:90% I'm not being consistent. 00:22:48.610 --> 00:22:51.280 align:middle line:84% They're all called consumers in this diagram. 00:22:51.280 --> 00:22:54.100 align:middle line:84% And you can see these indifference curves cross. 00:22:54.100 --> 00:22:56.830 align:middle line:84% And I want to claim, going back to the definition, 00:22:56.830 --> 00:22:59.140 align:middle line:84% that this cannot be Pareto optimal. 00:22:59.140 --> 00:22:59.920 align:middle line:90% Why? 00:22:59.920 --> 00:23:03.220 align:middle line:84% Because there are plenty of other feasible allocations 00:23:03.220 --> 00:23:05.860 align:middle line:84% that make at least one household better off 00:23:05.860 --> 00:23:07.180 align:middle line:90% and the other no worse off. 00:23:07.180 --> 00:23:09.520 align:middle line:84% And in fact, there are many allocations 00:23:09.520 --> 00:23:11.050 align:middle line:90% that make them both better off. 00:23:11.050 --> 00:23:14.920 align:middle line:84% This whole region here lies above the red indifference 00:23:14.920 --> 00:23:21.040 align:middle line:84% curve for household consumer 1 and above the blue indifference 00:23:21.040 --> 00:23:22.690 align:middle line:90% curve for consumer 2. 00:23:22.690 --> 00:23:25.300 align:middle line:84% Here, it's a different picture where 00:23:25.300 --> 00:23:29.150 align:middle line:84% the two baseline indifference curves are tangent. 00:23:29.150 --> 00:23:32.020 align:middle line:84% And in contrast, this is Pareto optimal 00:23:32.020 --> 00:23:34.390 align:middle line:84% because I challenge you to find an allocation that 00:23:34.390 --> 00:23:37.390 align:middle line:84% makes one better off without making the other one worse off. 00:23:37.390 --> 00:23:39.760 align:middle line:84% If you're going to make consumer one better off, 00:23:39.760 --> 00:23:42.380 align:middle line:84% you have to keep track of these red indifference curves. 00:23:42.380 --> 00:23:46.052 align:middle line:84% So we would be on or above the red indifference curve 00:23:46.052 --> 00:23:48.010 align:middle line:84% depending on whether we're making them no worse 00:23:48.010 --> 00:23:49.840 align:middle line:90% off or strictly better off. 00:23:49.840 --> 00:23:53.020 align:middle line:84% And any of those points other than the starting point 00:23:53.020 --> 00:23:56.480 align:middle line:84% actually makes consumer 2 worse off. 00:23:56.480 --> 00:23:59.350 align:middle line:84% So now, we can talk-- this is still 00:23:59.350 --> 00:24:01.870 align:middle line:84% called Edgeworth Box, although it's starting 00:24:01.870 --> 00:24:03.520 align:middle line:90% to look more like a rectangle. 00:24:03.520 --> 00:24:07.240 align:middle line:84% That's because there's more of good 1 than there is of good 2. 00:24:07.240 --> 00:24:09.310 align:middle line:90% It's wider than it is high. 00:24:09.310 --> 00:24:11.590 align:middle line:84% And what's being traced out here, 00:24:11.590 --> 00:24:14.020 align:middle line:84% like this is from the previous slide, 00:24:14.020 --> 00:24:17.650 align:middle line:84% that there are many other allocations which are also 00:24:17.650 --> 00:24:20.260 align:middle line:84% Pareto optimal, where those indifference curves are 00:24:20.260 --> 00:24:21.500 align:middle line:90% tangent. 00:24:21.500 --> 00:24:25.360 align:middle line:84% So this dotted orange yellow brick road, so to speak, 00:24:25.360 --> 00:24:28.300 align:middle line:84% dotted line, is a set of all Pareto 00:24:28.300 --> 00:24:31.030 align:middle line:84% optimal allocations in the whole economy. 00:24:31.030 --> 00:24:34.390 align:middle line:84% On the one hand, it's a lot less, a lot fewer, 00:24:34.390 --> 00:24:37.570 align:middle line:84% allocations than any other feasible allocation. 00:24:37.570 --> 00:24:40.240 align:middle line:84% Any allocation in this box is feasible 00:24:40.240 --> 00:24:43.190 align:middle line:84% because resource allocations add up to the total endowment. 00:24:43.190 --> 00:24:44.680 align:middle line:90% So [INAUDIBLE] worse off. 00:24:44.680 --> 00:24:48.250 align:middle line:84% And again, you can see these extreme distributions, 00:24:48.250 --> 00:24:53.410 align:middle line:84% where, if we could go that far to 0,0, consumer 1 over here 00:24:53.410 --> 00:24:55.600 align:middle line:90% is in a terrible situation. 00:24:55.600 --> 00:24:58.660 align:middle line:84% And consumer 2 just got everything. 00:24:58.660 --> 00:25:01.480 align:middle line:84% And that's feasible, as is at this point-- 00:25:01.480 --> 00:25:03.850 align:middle line:90% feasible, also optimal. 00:25:03.850 --> 00:25:05.210 align:middle line:90% AUDIENCE: I have a question. 00:25:05.210 --> 00:25:06.550 align:middle line:90% ROBERT M. TOWNSEND: Yes. 00:25:06.550 --> 00:25:12.090 align:middle line:84% AUDIENCE: Does this [INAUDIBLE] have some special properties? 00:25:12.090 --> 00:25:13.830 align:middle line:84% ROBERT M. TOWNSEND: Yeah, it's where, 00:25:13.830 --> 00:25:17.340 align:middle line:84% in this sense with these all smooth indifferent curves, 00:25:17.340 --> 00:25:19.500 align:middle line:84% the marginal rates of substitution 00:25:19.500 --> 00:25:24.840 align:middle line:84% that we learned about are equal across the two households. 00:25:24.840 --> 00:25:27.630 align:middle line:90% 00:25:27.630 --> 00:25:28.633 align:middle line:90% AUDIENCE: OK. 00:25:28.633 --> 00:25:30.300 align:middle line:84% ROBERT M. TOWNSEND: You know, the reason 00:25:30.300 --> 00:25:34.640 align:middle line:84% this isn't optimal is because the marginal rates 00:25:34.640 --> 00:25:37.360 align:middle line:90% of substitution are not equal. 00:25:37.360 --> 00:25:40.510 align:middle line:84% So the amount that consumer 2 requires 00:25:40.510 --> 00:25:44.440 align:middle line:84% to maintain indifference, giving up good a and getting good b, 00:25:44.440 --> 00:25:47.920 align:middle line:84% is greater than what consumer 1 requires 00:25:47.920 --> 00:25:50.680 align:middle line:84% to maintain indifference in terms of giving up good a 00:25:50.680 --> 00:25:51.910 align:middle line:90% and getting good b. 00:25:51.910 --> 00:25:55.150 align:middle line:84% So there's plenty of quote, "gains to trade." 00:25:55.150 --> 00:26:00.255 align:middle line:90% Both can be made better off, OK? 00:26:00.255 --> 00:26:05.690 align:middle line:84% All right, now, I mean, one thing we could have done 00:26:05.690 --> 00:26:08.780 align:middle line:84% is put the endowment point here and then 00:26:08.780 --> 00:26:11.180 align:middle line:84% looked at the set of-- well, maybe 00:26:11.180 --> 00:26:13.100 align:middle line:84% imagine this is the endowment point. 00:26:13.100 --> 00:26:15.870 align:middle line:84% Then we could look at the set of allocations, 00:26:15.870 --> 00:26:17.180 align:middle line:90% which are Pareto optimal. 00:26:17.180 --> 00:26:20.610 align:middle line:84% And that yellow brick road would run like that. 00:26:20.610 --> 00:26:23.450 align:middle line:84% And so these allocations in the lens shaped region 00:26:23.450 --> 00:26:26.630 align:middle line:84% are a subset of the Pareto optimal allocations, which 00:26:26.630 --> 00:26:29.420 align:middle line:84% make one or both agents strictly better off 00:26:29.420 --> 00:26:30.900 align:middle line:90% than at the endowment. 00:26:30.900 --> 00:26:35.420 align:middle line:84% So that's a more refined notion of optimality 00:26:35.420 --> 00:26:40.280 align:middle line:84% called the core, which I didn't create a slide to show you. 00:26:40.280 --> 00:26:43.520 align:middle line:84% But that's sort of beginning to take into account that you 00:26:43.520 --> 00:26:46.460 align:middle line:84% can't make someone worse off than their endowment, 00:26:46.460 --> 00:26:50.690 align:middle line:84% whereas Pareto optimality, as I keep saying, does not do that. 00:26:50.690 --> 00:26:51.890 align:middle line:90% It's much more general. 00:26:51.890 --> 00:26:54.890 align:middle line:84% OK, Pareto optimality and the utility possibility 00:26:54.890 --> 00:26:58.970 align:middle line:84% set, let's define, if it's not obvious already. 00:26:58.970 --> 00:27:03.020 align:middle line:84% An allocation of consumption and production is Pareto optimal-- 00:27:03.020 --> 00:27:04.280 align:middle line:90% actually, it's a theorem. 00:27:04.280 --> 00:27:06.740 align:middle line:84% We defined Pareto optimality already. 00:27:06.740 --> 00:27:09.530 align:middle line:84% And we defined the Pareto set already. 00:27:09.530 --> 00:27:13.550 align:middle line:84% So this puts them together in a rather obvious way, namely 00:27:13.550 --> 00:27:17.120 align:middle line:84% an allocation x, y is Pareto optimal if and only 00:27:17.120 --> 00:27:20.330 align:middle line:84% if the utilities that are generated under x 00:27:20.330 --> 00:27:23.570 align:middle line:84% are on the frontier of the Pareto set, 00:27:23.570 --> 00:27:27.945 align:middle line:84% which we denoted P script U. So you can go both ways here. 00:27:27.945 --> 00:27:32.000 align:middle line:84% It's if and only if the direction from Pareto 00:27:32.000 --> 00:27:36.050 align:middle line:84% optimal to being on the frontier is left as an exercise. 00:27:36.050 --> 00:27:39.740 align:middle line:84% It's pretty much as trivial as the one we're going to do, 00:27:39.740 --> 00:27:42.270 align:middle line:84% but we'll do one of the many way. 00:27:42.270 --> 00:27:44.450 align:middle line:84% So let's go in the opposite direction. 00:27:44.450 --> 00:27:48.170 align:middle line:84% That is we start with an allocation on the Pareto 00:27:48.170 --> 00:27:51.950 align:middle line:84% frontier, then we want to show that it's a Pareto optimum. 00:27:51.950 --> 00:27:53.960 align:middle line:84% So this is really almost just a test 00:27:53.960 --> 00:27:55.640 align:middle line:90% of remembering the definitions. 00:27:55.640 --> 00:27:58.670 align:middle line:84% Suppose that it were not Pareto optimal, 00:27:58.670 --> 00:28:01.170 align:middle line:84% then we'll do a proof by contradiction. 00:28:01.170 --> 00:28:04.610 align:middle line:84% So if it's not Pareto optimal, then there has to exist, 00:28:04.610 --> 00:28:07.580 align:middle line:84% by the definition, some other feasible allocation 00:28:07.580 --> 00:28:11.060 align:middle line:84% denoted with a tilde such that utility 00:28:11.060 --> 00:28:15.290 align:middle line:84% is at least as high under the tilde allocation 00:28:15.290 --> 00:28:17.210 align:middle line:84% as it was under the original and strictly 00:28:17.210 --> 00:28:20.540 align:middle line:84% greater in utility for some household j. 00:28:20.540 --> 00:28:24.350 align:middle line:84% But then these vectors of allocations Pareto 00:28:24.350 --> 00:28:28.160 align:middle line:84% dominate the original one and, hence, the original one 00:28:28.160 --> 00:28:30.300 align:middle line:84% could not have been in the Pareto frontier. 00:28:30.300 --> 00:28:31.820 align:middle line:84% That's one way to remember that is 00:28:31.820 --> 00:28:33.440 align:middle line:84% that, if it's on the Pareto frontier, 00:28:33.440 --> 00:28:35.090 align:middle line:90% there's nothing northeast of it. 00:28:35.090 --> 00:28:37.280 align:middle line:84% We just constructed something northeast of it. 00:28:37.280 --> 00:28:39.090 align:middle line:90% That's the contradiction. 00:28:39.090 --> 00:28:41.720 align:middle line:90% So this direction is now proved. 00:28:41.720 --> 00:28:44.930 align:middle line:84% And separately after class, you can take a stab 00:28:44.930 --> 00:28:46.280 align:middle line:90% at the other direction. 00:28:46.280 --> 00:28:48.320 align:middle line:84% OK, so the moral of the story here 00:28:48.320 --> 00:28:51.830 align:middle line:84% is we can go back and forth between the Pareto frontier 00:28:51.830 --> 00:28:54.380 align:middle line:90% and Pareto optimal allocations. 00:28:54.380 --> 00:28:58.220 align:middle line:84% And even I have to be careful not to intermingle them 00:28:58.220 --> 00:29:00.320 align:middle line:84% because I know it principle they're the same. 00:29:00.320 --> 00:29:02.840 align:middle line:84% And here's the proof that they are. 00:29:02.840 --> 00:29:05.720 align:middle line:84% Next step, we defined a welfare function 00:29:05.720 --> 00:29:09.270 align:middle line:84% for the entire economy, and we make it linear. 00:29:09.270 --> 00:29:13.190 align:middle line:84% So this big welfare comes from the utility vector 00:29:13.190 --> 00:29:15.320 align:middle line:90% across the individual agents. 00:29:15.320 --> 00:29:18.290 align:middle line:84% And this welfare function is linear 00:29:18.290 --> 00:29:22.010 align:middle line:84% when it's just a weighted average of the underlying 00:29:22.010 --> 00:29:25.190 align:middle line:84% utility components with weights lambda i. 00:29:25.190 --> 00:29:29.300 align:middle line:84% Weights good in principle be 0, somebody doesn't count at all. 00:29:29.300 --> 00:29:32.580 align:middle line:84% Otherwise, the weights are positive. 00:29:32.580 --> 00:29:35.870 align:middle line:84% And when I say we take a weighted average, 00:29:35.870 --> 00:29:38.360 align:middle line:84% each of the weights might as well sum up to 1. 00:29:38.360 --> 00:29:40.370 align:middle line:84% Because if they didn't, we could add up 00:29:40.370 --> 00:29:42.500 align:middle line:84% whatever they do total to and divide 00:29:42.500 --> 00:29:44.450 align:middle line:84% through the same objective function. 00:29:44.450 --> 00:29:48.380 align:middle line:84% And if you're going to maximize one welfare function, 00:29:48.380 --> 00:29:50.970 align:middle line:84% you maximize the scalar multiple of it. 00:29:50.970 --> 00:29:55.370 align:middle line:84% So the idea of this welfare function is a way I don't like, 00:29:55.370 --> 00:29:57.590 align:middle line:84% but then I'll tell you what I do like. 00:29:57.590 --> 00:30:01.430 align:middle line:84% It's a way of aggregating utilities across individuals 00:30:01.430 --> 00:30:06.050 align:middle line:84% in society as if there were some social planner who determines 00:30:06.050 --> 00:30:08.060 align:middle line:90% who counts more than others. 00:30:08.060 --> 00:30:10.640 align:middle line:84% And particularly under the linear welfare function, 00:30:10.640 --> 00:30:14.780 align:middle line:84% the lambda weights are the weights on the individuals. 00:30:14.780 --> 00:30:19.210 align:middle line:84% Now, what I do like about this is this math problem, as it 00:30:19.210 --> 00:30:22.510 align:middle line:84% were, which, if there were a social planner, 00:30:22.510 --> 00:30:26.230 align:middle line:84% we would be maximizing the weighted sum of utilities 00:30:26.230 --> 00:30:32.950 align:middle line:84% subject to allocations being in the utility possibilities set. 00:30:32.950 --> 00:30:36.740 align:middle line:84% So it looks like we've ignored feasibility and everything, 00:30:36.740 --> 00:30:37.540 align:middle line:90% but no. 00:30:37.540 --> 00:30:39.760 align:middle line:84% The feasibility of an allocation is 00:30:39.760 --> 00:30:42.910 align:middle line:84% loaded into the definition of the utility possibilities set 00:30:42.910 --> 00:30:47.710 align:middle line:84% script U, and we're constrained to pick utility vectors which 00:30:47.710 --> 00:30:48.900 align:middle line:90% are in that set. 00:30:48.900 --> 00:30:50.650 align:middle line:84% And we're going to pick them in such a way 00:30:50.650 --> 00:30:54.880 align:middle line:84% as to maximize this lambda weighted sum of utilities. 00:30:54.880 --> 00:30:58.710 align:middle line:84% And we can call that the planner's problem. 00:30:58.710 --> 00:31:03.290 align:middle line:84% So the big result is going to be-- let me say it in words-- 00:31:03.290 --> 00:31:07.100 align:middle line:84% that, with certain relatively weak assumptions, 00:31:07.100 --> 00:31:10.550 align:middle line:84% any Pareto optimal allocation that we pick 00:31:10.550 --> 00:31:13.370 align:middle line:84% will be a solution to the planning problem 00:31:13.370 --> 00:31:15.290 align:middle line:90% for some weights lambda. 00:31:15.290 --> 00:31:18.590 align:middle line:84% And any solution to the planning problem for given weights 00:31:18.590 --> 00:31:21.620 align:middle line:90% lambda will be Pareto optimal. 00:31:21.620 --> 00:31:23.810 align:middle line:84% So the conceptualization thing here, 00:31:23.810 --> 00:31:26.120 align:middle line:84% before we get lost in the notation, 00:31:26.120 --> 00:31:28.400 align:middle line:84% is that we can now go back and forth 00:31:28.400 --> 00:31:31.640 align:middle line:84% between the notions of Pareto optimality 00:31:31.640 --> 00:31:36.190 align:middle line:84% and the idea of maximizing the social welfare function, 00:31:36.190 --> 00:31:39.100 align:middle line:90% although the weights can vary. 00:31:39.100 --> 00:31:40.900 align:middle line:90% So let's look at the math. 00:31:40.900 --> 00:31:44.860 align:middle line:84% Suppose we add a solution to the planner's problem seven. 00:31:44.860 --> 00:31:48.160 align:middle line:84% And just to make it easier, suppose the lambda weights 00:31:48.160 --> 00:31:50.140 align:middle line:90% are all strictly positive. 00:31:50.140 --> 00:31:54.250 align:middle line:84% Then the first claim is that solution to seven 00:31:54.250 --> 00:31:55.840 align:middle line:90% must be in the Pareto set. 00:31:55.840 --> 00:31:58.720 align:middle line:84% And if it's on the Pareto frontier, sorry, 00:31:58.720 --> 00:32:00.700 align:middle line:90% then it must be Pareto optimal. 00:32:00.700 --> 00:32:03.970 align:middle line:84% Going the other way, if the utility possibility set 00:32:03.970 --> 00:32:06.970 align:middle line:84% is convex and closed, then picking 00:32:06.970 --> 00:32:11.200 align:middle line:84% any utility on the Pareto frontier 00:32:11.200 --> 00:32:15.040 align:middle line:84% and, hence, Pareto optimal, we can find particular lambda 00:32:15.040 --> 00:32:18.970 align:middle line:84% weights that makes that utility vector a solution 00:32:18.970 --> 00:32:20.560 align:middle line:90% to the planner's problem. 00:32:20.560 --> 00:32:23.230 align:middle line:84% All right, so let's do the two parts a bit. 00:32:23.230 --> 00:32:25.420 align:middle line:84% For the first part, suppose we're 00:32:25.420 --> 00:32:27.820 align:middle line:84% given some U star that solves the planning 00:32:27.820 --> 00:32:32.080 align:middle line:84% problem at positive weights, but claim by contradiction 00:32:32.080 --> 00:32:34.030 align:middle line:84% that it's not on the Pareto frontier. 00:32:34.030 --> 00:32:37.010 align:middle line:84% Well, if it's on the Pareto frontier, we can dominate. 00:32:37.010 --> 00:32:41.110 align:middle line:84% There's a U prime vector, which is no worse for all i 00:32:41.110 --> 00:32:42.880 align:middle line:90% and strictly better for j. 00:32:42.880 --> 00:32:45.220 align:middle line:84% But since the lambda weights are strictly positive, 00:32:45.220 --> 00:32:48.610 align:middle line:84% we've now found a new vector, the prime vector, 00:32:48.610 --> 00:32:51.880 align:middle line:84% such that the weighted average of the prime utilities 00:32:51.880 --> 00:32:53.920 align:middle line:84% is strictly greater than the weighted average 00:32:53.920 --> 00:32:55.510 align:middle line:90% of the starting point. 00:32:55.510 --> 00:32:58.570 align:middle line:84% Hence, in more cryptic notation, the dot product 00:32:58.570 --> 00:33:01.600 align:middle line:84% lambda U is strictly greater than lambda U star, 00:33:01.600 --> 00:33:04.030 align:middle line:84% but we started with the claim that we've 00:33:04.030 --> 00:33:05.380 align:middle line:90% got a solution to seven. 00:33:05.380 --> 00:33:07.510 align:middle line:84% And we just found a feasible solution 00:33:07.510 --> 00:33:10.690 align:middle line:84% to seven, which gives a higher welfare number. 00:33:10.690 --> 00:33:12.310 align:middle line:90% So that's a contradiction. 00:33:12.310 --> 00:33:14.050 align:middle line:90% You with me so far? 00:33:14.050 --> 00:33:15.260 align:middle line:90% OK. 00:33:15.260 --> 00:33:18.730 align:middle line:84% So suppose that the utility possibility 00:33:18.730 --> 00:33:23.050 align:middle line:84% set is convex and closed, and take some solution U 00:33:23.050 --> 00:33:25.810 align:middle line:84% star, which is on the Pareto frontier and, hence, Pareto 00:33:25.810 --> 00:33:26.590 align:middle line:90% optimal. 00:33:26.590 --> 00:33:29.890 align:middle line:84% Take a vector U star, which is Pareto optimal. 00:33:29.890 --> 00:33:31.820 align:middle line:90% It's got to be on the boundary. 00:33:31.820 --> 00:33:35.170 align:middle line:84% And because it's on the boundary of the utility possibility set, 00:33:35.170 --> 00:33:38.590 align:middle line:84% in turn is convex, we can apply something called-- 00:33:38.590 --> 00:33:40.480 align:middle line:84% and I'll show you this momentarily-- 00:33:40.480 --> 00:33:42.940 align:middle line:84% the supporting hyperplane theorem, which 00:33:42.940 --> 00:33:46.060 align:middle line:84% is going to give us, guarantee the existence of, a lambda 00:33:46.060 --> 00:33:50.440 align:middle line:84% vector such that at that lambda, lambda dot U star, 00:33:50.440 --> 00:33:53.860 align:middle line:84% is strictly greater than lambda U for any other U 00:33:53.860 --> 00:33:55.510 align:middle line:90% in the utility possibility set. 00:33:55.510 --> 00:33:59.440 align:middle line:84% In other words, lambda star at lambda, the particular lambda, 00:33:59.440 --> 00:34:02.230 align:middle line:84% lambda star U, is the highest possible, 00:34:02.230 --> 00:34:05.320 align:middle line:84% cannot be dominated by in some senses weakly, 00:34:05.320 --> 00:34:08.650 align:middle line:84% the highest possible lambda weighted utility that you can 00:34:08.650 --> 00:34:12.610 align:middle line:84% get running over all possible other feasible utilities 00:34:12.610 --> 00:34:14.739 align:middle line:90% in the utility possibility set. 00:34:14.739 --> 00:34:18.100 align:middle line:84% That means that U star is a solution to the planning 00:34:18.100 --> 00:34:20.090 align:middle line:90% problem by definition. 00:34:20.090 --> 00:34:22.600 align:middle line:84% So that completes the proof of going both ways, 00:34:22.600 --> 00:34:24.280 align:middle line:84% but it kind of begs the issue of what 00:34:24.280 --> 00:34:26.650 align:middle line:84% the heck is this supporting hyperplane theorem. 00:34:26.650 --> 00:34:28.960 align:middle line:84% So let me show you some pictures. 00:34:28.960 --> 00:34:32.540 align:middle line:84% And this is, along the way, an extremely useful tool. 00:34:32.540 --> 00:34:35.840 align:middle line:84% It's just the first time that we've needed it. 00:34:35.840 --> 00:34:39.320 align:middle line:84% OK, so here's the utility possibility set. 00:34:39.320 --> 00:34:41.659 align:middle line:84% Here's the U star that we picked. 00:34:41.659 --> 00:34:45.650 align:middle line:84% It's on the Pareto frontier, so it's Pareto optimal. 00:34:45.650 --> 00:34:47.330 align:middle line:84% And that theorem is true, so I should 00:34:47.330 --> 00:34:49.850 align:middle line:84% be able to find some lambda weight such that, when 00:34:49.850 --> 00:34:53.120 align:middle line:84% I maximize the lambda weighted sum of utilities, 00:34:53.120 --> 00:34:55.550 align:middle line:84% I would be at a maximum at U star, 00:34:55.550 --> 00:34:58.550 align:middle line:84% which visually you can see is true. 00:34:58.550 --> 00:35:01.340 align:middle line:84% There are other points that are Pareto optimal, 00:35:01.340 --> 00:35:05.840 align:middle line:84% but they would not be maximal under the lambdas, which 00:35:05.840 --> 00:35:09.080 align:middle line:84% are implicitly the slope of this red line. 00:35:09.080 --> 00:35:10.400 align:middle line:90% These are level curves. 00:35:10.400 --> 00:35:12.960 align:middle line:84% We could call them iso-welfare curves. 00:35:12.960 --> 00:35:17.600 align:middle line:84% These are level curves of the social planner 00:35:17.600 --> 00:35:21.290 align:middle line:84% at particular lambda weights lambda. 00:35:21.290 --> 00:35:25.520 align:middle line:84% And it's kind of tradition to denote the slope 00:35:25.520 --> 00:35:30.740 align:middle line:84% by a line, which is right angled to the line in question. 00:35:30.740 --> 00:35:33.140 align:middle line:84% So this lambda going up this way is just 00:35:33.140 --> 00:35:38.210 align:middle line:84% a way to say that the slope of both the Pareto 00:35:38.210 --> 00:35:41.690 align:middle line:84% frontier and the iso-welfare curve 00:35:41.690 --> 00:35:46.910 align:middle line:84% are the same and equal to lambda 1 over lambda 2. 00:35:46.910 --> 00:35:50.450 align:middle line:84% All right, so that's a picture of why it's true 00:35:50.450 --> 00:35:53.180 align:middle line:84% that, when you pick a point on the Pareto frontier, 00:35:53.180 --> 00:35:55.220 align:middle line:84% you can find lambdas which makes it 00:35:55.220 --> 00:35:57.780 align:middle line:84% a solution to the programming problem. 00:35:57.780 --> 00:35:59.390 align:middle line:90% What is this hyperplane? 00:35:59.390 --> 00:36:02.090 align:middle line:90% So we'll do everything in R2. 00:36:02.090 --> 00:36:03.920 align:middle line:90% Here's a line in R2. 00:36:03.920 --> 00:36:06.650 align:middle line:84% And in higher dimensions, this is called 00:36:06.650 --> 00:36:09.270 align:middle line:90% a hyperplane, fancy language. 00:36:09.270 --> 00:36:11.630 align:middle line:84% You could probably draw it as a plane in R3. 00:36:11.630 --> 00:36:13.580 align:middle line:84% It gets harder to draw, unless you're better 00:36:13.580 --> 00:36:16.495 align:middle line:90% than I am, in R4 and so on. 00:36:16.495 --> 00:36:17.780 align:middle line:90% But we'll stick to R2. 00:36:17.780 --> 00:36:18.590 align:middle line:90% So here's a line. 00:36:18.590 --> 00:36:19.850 align:middle line:90% It's called a hyperplane. 00:36:19.850 --> 00:36:22.190 align:middle line:84% And we can obviously talk about the half space 00:36:22.190 --> 00:36:26.700 align:middle line:84% above the line and the half space below the line. 00:36:26.700 --> 00:36:31.430 align:middle line:84% So above the line means, if we specify the intercept c 00:36:31.430 --> 00:36:35.360 align:middle line:84% and the slope p then the sets above the line, 00:36:35.360 --> 00:36:39.800 align:middle line:84% the half space above, is a set of all values z 00:36:39.800 --> 00:36:45.380 align:middle line:84% which gives no lower value than c at slopes p and vice versa 00:36:45.380 --> 00:36:47.550 align:middle line:90% for the lower half space. 00:36:47.550 --> 00:36:51.410 align:middle line:84% Here's another line that separates these two sets. 00:36:51.410 --> 00:36:53.810 align:middle line:84% We could call one of these set A and the other one 00:36:53.810 --> 00:36:57.080 align:middle line:84% set B. Note that they are both convex sets. 00:36:57.080 --> 00:36:57.800 align:middle line:90% Pick a set. 00:36:57.800 --> 00:37:01.100 align:middle line:84% Pick the pink one, pick any two points, 00:37:01.100 --> 00:37:03.200 align:middle line:84% then weighted average of those points 00:37:03.200 --> 00:37:05.570 align:middle line:90% are, in turn, in the set. 00:37:05.570 --> 00:37:07.550 align:middle line:84% This is a separating line in the sense 00:37:07.550 --> 00:37:10.040 align:middle line:84% that the half space above this line 00:37:10.040 --> 00:37:13.250 align:middle line:84% contains the pink convex set is contained in that. 00:37:13.250 --> 00:37:15.440 align:middle line:84% It's within the half space above. 00:37:15.440 --> 00:37:19.790 align:middle line:84% And the blue convex set is part of the half space below. 00:37:19.790 --> 00:37:21.920 align:middle line:84% And that's what all this stuff up here says. 00:37:21.920 --> 00:37:25.190 align:middle line:84% And the culmination of this is called 00:37:25.190 --> 00:37:26.900 align:middle line:90% the supporting hyperplane. 00:37:26.900 --> 00:37:29.720 align:middle line:84% So again, we run from hyperplanes 00:37:29.720 --> 00:37:34.250 align:middle line:84% to separating hyperplane theorem having to do with convex sets 00:37:34.250 --> 00:37:36.770 align:middle line:84% to supporting hyperplane theorem. 00:37:36.770 --> 00:37:40.280 align:middle line:84% So this pick this set B. And let's require 00:37:40.280 --> 00:37:42.500 align:middle line:90% that it be a convex set. 00:37:42.500 --> 00:37:46.160 align:middle line:84% Pick this point x in the set and on the boundary. 00:37:46.160 --> 00:37:50.330 align:middle line:84% Then the claim is we can find a hyperplane, 00:37:50.330 --> 00:37:54.140 align:middle line:84% in particular a supporting hyperplane, which 00:37:54.140 --> 00:37:57.890 align:middle line:84% just touches the set B at x and is 00:37:57.890 --> 00:38:01.820 align:middle line:90% tangent to the boundary at x. 00:38:01.820 --> 00:38:03.410 align:middle line:90% Are you with me? 00:38:03.410 --> 00:38:07.970 align:middle line:84% So the slopes of this hyperplane are determined by p. 00:38:07.970 --> 00:38:11.210 align:middle line:84% And we don't need a constant c anymore. 00:38:11.210 --> 00:38:15.290 align:middle line:84% This is said to be a supporting hyperplane in the sense 00:38:15.290 --> 00:38:20.390 align:middle line:84% that, if you picked any other x in the interior other 00:38:20.390 --> 00:38:24.170 align:middle line:84% than the one on the boundary, any other y in this case 00:38:24.170 --> 00:38:26.450 align:middle line:84% in terms of the notation, it will have 00:38:26.450 --> 00:38:28.670 align:middle line:90% a return which is less than-- 00:38:28.670 --> 00:38:31.550 align:middle line:84% p dot y will be less than p dot x. 00:38:31.550 --> 00:38:34.160 align:middle line:84% So this is what we were using a moment ago, this 00:38:34.160 --> 00:38:35.960 align:middle line:90% separating hyperplane theorem. 00:38:35.960 --> 00:38:38.660 align:middle line:84% Because I kind of asserted we could draw this picture, 00:38:38.660 --> 00:38:41.330 align:middle line:84% and it seems really natural that we ought to be able to. 00:38:41.330 --> 00:38:47.030 align:middle line:84% It is using the convexity of the utility possibility set 00:38:47.030 --> 00:38:51.230 align:middle line:84% and picking a point, like x was three slides away, 00:38:51.230 --> 00:38:54.590 align:middle line:84% and then drawing this hyperplane which is tangent. 00:38:54.590 --> 00:38:56.300 align:middle line:84% And the separating hyperplane theorem 00:38:56.300 --> 00:38:59.210 align:middle line:84% says we can always do that under these weak assumptions 00:38:59.210 --> 00:39:00.650 align:middle line:90% like convexity. 00:39:00.650 --> 00:39:03.110 align:middle line:84% That also means we can pick another one, right? 00:39:03.110 --> 00:39:05.330 align:middle line:84% If we pick another point on the frontier, 00:39:05.330 --> 00:39:07.610 align:middle line:84% we can find a separating hyperplane. 00:39:07.610 --> 00:39:10.520 align:middle line:84% But it's clearly, in this diagram at least, 00:39:10.520 --> 00:39:12.810 align:middle line:90% going to have a different slope. 00:39:12.810 --> 00:39:14.150 align:middle line:90% So we can support it. 00:39:14.150 --> 00:39:16.460 align:middle line:84% We're just going to have to change the lambdas. 00:39:16.460 --> 00:39:18.450 align:middle line:90% And that's what's going on here. 00:39:18.450 --> 00:39:19.910 align:middle line:90% This was the initial point. 00:39:19.910 --> 00:39:21.780 align:middle line:84% We could have picked another point. 00:39:21.780 --> 00:39:24.350 align:middle line:84% If we did that, we'd have to find new slopes 00:39:24.350 --> 00:39:28.280 align:middle line:84% for the supporting hyperplane, so that it would kind of pivot 00:39:28.280 --> 00:39:32.570 align:middle line:84% and allow it to be tangent at this new predetermined point. 00:39:32.570 --> 00:39:35.090 align:middle line:84% So again, not to get lost, this is 00:39:35.090 --> 00:39:37.700 align:middle line:84% how we're going back and forth between Pareto 00:39:37.700 --> 00:39:40.790 align:middle line:84% optimal allocations, which are on the Pareto frontier, 00:39:40.790 --> 00:39:44.270 align:middle line:84% and finding a lambda such that at those lambdas 00:39:44.270 --> 00:39:47.150 align:middle line:84% we would be maximizing a linear welfare function. 00:39:47.150 --> 00:39:51.230 align:middle line:84% And any Pareto optimal point which is on the boundary 00:39:51.230 --> 00:39:52.910 align:middle line:90% has to have that property. 00:39:52.910 --> 00:39:54.860 align:middle line:84% And likewise, the first part, which 00:39:54.860 --> 00:39:58.160 align:middle line:84% was anything that is maximal under the welfare function, 00:39:58.160 --> 00:40:00.320 align:middle line:90% must also be Pareto optimal. 00:40:00.320 --> 00:40:04.530 align:middle line:84% And the proof there that we did was by contradiction. 00:40:04.530 --> 00:40:09.120 align:middle line:84% All right, so in notation, we can 00:40:09.120 --> 00:40:12.270 align:middle line:84% parameterize Pareto optimal allocations 00:40:12.270 --> 00:40:14.640 align:middle line:90% by choosing Pareto weights. 00:40:14.640 --> 00:40:17.940 align:middle line:84% And this lambda and delta sort of means there in the simplex, 00:40:17.940 --> 00:40:20.160 align:middle line:84% which, again, without loss of generality, 00:40:20.160 --> 00:40:23.100 align:middle line:84% the lambdas are non-negative and they add up to 1. 00:40:23.100 --> 00:40:25.470 align:middle line:84% We used simplex earlier for lotteries. 00:40:25.470 --> 00:40:28.230 align:middle line:84% Here's the second time that we're using it. 00:40:28.230 --> 00:40:32.160 align:middle line:84% OK, so if we want to maximize a particular lambda weighted sum 00:40:32.160 --> 00:40:35.580 align:middle line:84% of utilities, lambda i weighting Ui, 00:40:35.580 --> 00:40:38.520 align:middle line:84% we can do that and find the maximum to it 00:40:38.520 --> 00:40:42.150 align:middle line:84% and call the solution X star and Y star. 00:40:42.150 --> 00:40:43.800 align:middle line:84% But here, we're going to earmark them 00:40:43.800 --> 00:40:47.310 align:middle line:84% by lambda because it was a solution at particular lambdas. 00:40:47.310 --> 00:40:49.830 align:middle line:84% This is just a repeat of consumption and production 00:40:49.830 --> 00:40:51.280 align:middle line:90% sets and feasibility. 00:40:51.280 --> 00:40:55.770 align:middle line:84% So any solution that maximizes we earmark by lambda, this 00:40:55.770 --> 00:40:58.080 align:middle line:90% is a particular Pareto optimum. 00:40:58.080 --> 00:41:01.410 align:middle line:84% And also, given any Pareto optimal allocation, 00:41:01.410 --> 00:41:04.660 align:middle line:84% we can find a lambda which makes it a solution. 00:41:04.660 --> 00:41:06.790 align:middle line:84% So this is just restating in words 00:41:06.790 --> 00:41:08.920 align:middle line:90% what we already determined. 00:41:08.920 --> 00:41:12.000 align:middle line:84% So the task of finding Pareto optimal allocations 00:41:12.000 --> 00:41:16.810 align:middle line:84% in the Edgeworth box is simply solving this math problem. 00:41:16.810 --> 00:41:18.330 align:middle line:90% So let's apply it. 00:41:18.330 --> 00:41:22.020 align:middle line:84% Suppose we had this date space with uncertainty 00:41:22.020 --> 00:41:25.650 align:middle line:84% and this sort of, Debreu tree about the set 00:41:25.650 --> 00:41:29.280 align:middle line:84% of possible outcomes at date 1, date 2, and date 3. 00:41:29.280 --> 00:41:31.410 align:middle line:84% And we're going to index consumption 00:41:31.410 --> 00:41:33.780 align:middle line:84% by the history of all shocks leading up 00:41:33.780 --> 00:41:35.730 align:middle line:90% to that particular date. 00:41:35.730 --> 00:41:37.570 align:middle line:90% And this was expected utility. 00:41:37.570 --> 00:41:38.730 align:middle line:90% And this is a review slide. 00:41:38.730 --> 00:41:42.150 align:middle line:84% We went over this last time or at least earlier-- 00:41:42.150 --> 00:41:44.710 align:middle line:84% not last time, but the time before that. 00:41:44.710 --> 00:41:48.270 align:middle line:84% So let's determine an optimal allocation of risk 00:41:48.270 --> 00:41:52.630 align:middle line:84% in this economy that has states determined at random. 00:41:52.630 --> 00:41:55.710 align:middle line:84% So we're going to want to-- let me say it in words. 00:41:55.710 --> 00:41:59.130 align:middle line:84% We're going to maximize a lambda weighted sum of utility subject 00:41:59.130 --> 00:42:00.840 align:middle line:90% to resource constraints. 00:42:00.840 --> 00:42:03.892 align:middle line:84% Note we're not going to do this in the space of utilities. 00:42:03.892 --> 00:42:05.850 align:middle line:84% We're going to do it in the space of underlying 00:42:05.850 --> 00:42:08.290 align:middle line:84% allocations, but it's the same thing. 00:42:08.290 --> 00:42:10.020 align:middle line:84% Instead of going from allocations 00:42:10.020 --> 00:42:13.397 align:middle line:84% to utilities and from utilities to maximizing 00:42:13.397 --> 00:42:15.480 align:middle line:84% a lambda weighted sum, we'll just kind of directly 00:42:15.480 --> 00:42:18.690 align:middle line:84% search over the underlying allocations. 00:42:18.690 --> 00:42:20.160 align:middle line:90% It's small typeset. 00:42:20.160 --> 00:42:22.620 align:middle line:84% You may be squinting to try to see this. 00:42:22.620 --> 00:42:26.820 align:middle line:84% Here was a repeat from the previous slide, the utility 00:42:26.820 --> 00:42:29.620 align:middle line:84% function, with a little more notation. 00:42:29.620 --> 00:42:32.610 align:middle line:84% So this is expected utility as we 00:42:32.610 --> 00:42:36.510 align:middle line:84% studied in the second lecture on preferences and utility. 00:42:36.510 --> 00:42:41.040 align:middle line:84% This counted by beta, as we kind of did in the storage problem, 00:42:41.040 --> 00:42:45.690 align:middle line:84% and weighting outcomes by probabilities. 00:42:45.690 --> 00:42:49.200 align:middle line:84% The summation here is over all dates and all states, 00:42:49.200 --> 00:42:50.760 align:middle line:90% as in this tree. 00:42:50.760 --> 00:42:55.140 align:middle line:84% And then we take those expected utility outcomes 00:42:55.140 --> 00:42:57.660 align:middle line:84% and pre-multiply by the lambda weights, 00:42:57.660 --> 00:42:59.460 align:middle line:90% as we've now been trained to do. 00:42:59.460 --> 00:43:01.380 align:middle line:84% And because we're not in utility space 00:43:01.380 --> 00:43:03.970 align:middle line:84% and we're in space of underlying allocations, 00:43:03.970 --> 00:43:06.270 align:middle line:84% we have to make sure that consumptions add up 00:43:06.270 --> 00:43:07.200 align:middle line:90% to the total. 00:43:07.200 --> 00:43:10.470 align:middle line:84% Now, here's, again, a bit of a switch of notation. 00:43:10.470 --> 00:43:12.000 align:middle line:90% These Y's are now-- 00:43:12.000 --> 00:43:16.950 align:middle line:84% these right-hand sides are using the Y for endowment, not 00:43:16.950 --> 00:43:18.180 align:middle line:90% for production. 00:43:18.180 --> 00:43:20.650 align:middle line:90% But it's not that misleading. 00:43:20.650 --> 00:43:24.480 align:middle line:84% So we have an exogenously given set of endowments 00:43:24.480 --> 00:43:27.660 align:middle line:84% depending on the data and the state. 00:43:27.660 --> 00:43:30.000 align:middle line:84% And then consumption has to add up to that 00:43:30.000 --> 00:43:32.670 align:middle line:84% or certainly could not be greater than that because it 00:43:32.670 --> 00:43:34.480 align:middle line:90% would not be feasible. 00:43:34.480 --> 00:43:36.900 align:middle line:84% So we call this the Pareto problem 00:43:36.900 --> 00:43:40.830 align:middle line:84% or the planner's problem for, here, particular weights 00:43:40.830 --> 00:43:41.640 align:middle line:90% lambda. 00:43:41.640 --> 00:43:45.100 align:middle line:84% What does the solution look like? 00:43:45.100 --> 00:43:48.670 align:middle line:84% First of all, let me tell you intuitively what's going on, 00:43:48.670 --> 00:43:50.400 align:middle line:90% and then we can generalize it. 00:43:50.400 --> 00:43:52.590 align:middle line:84% We wanted to maximize the lambda weighted sum 00:43:52.590 --> 00:43:54.150 align:middle line:90% of expected utilities. 00:43:54.150 --> 00:43:57.990 align:middle line:84% All I've done here is pull out in front of those expressions 00:43:57.990 --> 00:44:01.440 align:middle line:84% the thing that is common across all the agents i, 00:44:01.440 --> 00:44:03.690 align:middle line:84% namely the beta and the probabilities. 00:44:03.690 --> 00:44:05.740 align:middle line:84% Otherwise, it's the same expression. 00:44:05.740 --> 00:44:09.420 align:middle line:84% But maximizing one is equivalent to maximizing the other 00:44:09.420 --> 00:44:12.130 align:middle line:84% because this is just some scalar constant. 00:44:12.130 --> 00:44:14.010 align:middle line:84% We also know that consumptions are 00:44:14.010 --> 00:44:17.700 align:middle line:84% going to add up to the total social endowment, which 00:44:17.700 --> 00:44:20.160 align:middle line:84% is the sum of individual incomes, which 00:44:20.160 --> 00:44:24.250 align:middle line:84% was I think I forgot to mention at the bottom of this slide. 00:44:24.250 --> 00:44:25.170 align:middle line:90% This is the aggregate. 00:44:25.170 --> 00:44:28.600 align:middle line:84% Social endowment is just the sum of the individual endowments. 00:44:28.600 --> 00:44:31.080 align:middle line:84% So there's lots of dates and lots of states, 00:44:31.080 --> 00:44:32.820 align:middle line:84% but for every date and state there's 00:44:32.820 --> 00:44:34.560 align:middle line:90% a particular social endowment. 00:44:34.560 --> 00:44:38.310 align:middle line:84% So we can simplify this so that it looks like a static problem. 00:44:38.310 --> 00:44:41.190 align:middle line:84% We can maximize each component one at a time 00:44:41.190 --> 00:44:43.410 align:middle line:84% and just respect the resource constraint. 00:44:43.410 --> 00:44:46.410 align:middle line:84% In the objective function, a component one at a time 00:44:46.410 --> 00:44:48.720 align:middle line:84% is just the lambda weighted sum of utilities. 00:44:48.720 --> 00:44:50.790 align:middle line:84% And this would be the resource constraint 00:44:50.790 --> 00:44:53.800 align:middle line:84% that consumptions have to add up appropriately. 00:44:53.800 --> 00:44:58.140 align:middle line:84% So amazingly, we've sort of been able to abstract away 00:44:58.140 --> 00:45:03.810 align:middle line:84% from dates and states and about to determine an optimal risk 00:45:03.810 --> 00:45:06.780 align:middle line:84% sharing rule as long as we keep track 00:45:06.780 --> 00:45:09.480 align:middle line:84% of the aggregate social endowment. 00:45:09.480 --> 00:45:11.340 align:middle line:90% Questions? 00:45:11.340 --> 00:45:14.880 align:middle line:84% All right, so another tool we can use that you already know 00:45:14.880 --> 00:45:16.260 align:middle line:90% is the Lagrangian. 00:45:16.260 --> 00:45:20.250 align:middle line:84% So we'll maximize this lambda weighted sum of expected 00:45:20.250 --> 00:45:23.100 align:middle line:84% utilities, and we'll throw in the constraint 00:45:23.100 --> 00:45:27.720 align:middle line:84% sets with Lagrange multipliers where y is greater than 00:45:27.720 --> 00:45:29.970 align:middle line:84% and is going to end up being equal to the sum 00:45:29.970 --> 00:45:31.410 align:middle line:90% of consumptions. 00:45:31.410 --> 00:45:35.190 align:middle line:84% So these thetas are the quote, "Lagrange multipliers." 00:45:35.190 --> 00:45:37.770 align:middle line:84% And you can think of them as shadow prices, 00:45:37.770 --> 00:45:39.930 align:middle line:84% like the marginal utility of income. 00:45:39.930 --> 00:45:41.940 align:middle line:84% But there are lots of them because we 00:45:41.940 --> 00:45:46.380 align:middle line:84% have a theta for every possible date and all possible histories 00:45:46.380 --> 00:45:48.610 align:middle line:84% of states leading up to that date. 00:45:48.610 --> 00:45:50.070 align:middle line:90% So there's a large number. 00:45:50.070 --> 00:45:54.150 align:middle line:84% Anyway, max away, we get, for a particular date 00:45:54.150 --> 00:45:56.700 align:middle line:84% and a particular set of states, st. 00:45:56.700 --> 00:45:59.730 align:middle line:84% We have the derivative of the utility function with respect 00:45:59.730 --> 00:46:03.520 align:middle line:84% to its argument preceded by beta and lambda. 00:46:03.520 --> 00:46:05.730 align:middle line:84% So we get this thing, theta to the t 00:46:05.730 --> 00:46:08.520 align:middle line:84% because it's at beta t on the left-hand side. 00:46:08.520 --> 00:46:11.820 align:middle line:84% And on the right-hand side, we get-- where else does it enter? 00:46:11.820 --> 00:46:12.370 align:middle line:90% Here. 00:46:12.370 --> 00:46:14.910 align:middle line:84% So we pick up the Lagrange multiplier. 00:46:14.910 --> 00:46:18.090 align:middle line:84% And then this is just a repeat of the constraints. 00:46:18.090 --> 00:46:20.760 align:middle line:84% You could write it out more formally 00:46:20.760 --> 00:46:24.670 align:middle line:84% with theta times the sum being 0 and so on, 00:46:24.670 --> 00:46:27.090 align:middle line:84% but the problem is implicitly assuming 00:46:27.090 --> 00:46:30.030 align:middle line:84% that more is preferred to less, resources are scarce. 00:46:30.030 --> 00:46:32.430 align:middle line:84% So you're not going to throw anything away. 00:46:32.430 --> 00:46:35.490 align:middle line:84% These first order conditions look like lambda times 00:46:35.490 --> 00:46:38.400 align:middle line:84% the marginal utility of consumption for household 1 00:46:38.400 --> 00:46:41.760 align:middle line:84% is equal to lambda i times the marginal utility of consumption 00:46:41.760 --> 00:46:44.370 align:middle line:90% for all other households i. 00:46:44.370 --> 00:46:47.640 align:middle line:84% And this is true overall, for every i. 00:46:47.640 --> 00:46:52.410 align:middle line:84% And this characterizes solution for all possible dates 00:46:52.410 --> 00:46:55.680 align:middle line:84% and states where the aggregate social endowment is 00:46:55.680 --> 00:46:56.740 align:middle line:90% a parameter. 00:46:56.740 --> 00:46:59.580 align:middle line:84% So there are how many households? 00:46:59.580 --> 00:47:02.580 align:middle line:84% Capital I. So we have I minus 1 equations here. 00:47:02.580 --> 00:47:04.470 align:middle line:84% We pick up one more equation here. 00:47:04.470 --> 00:47:07.680 align:middle line:84% We can solve this thing and denote the consumption 00:47:07.680 --> 00:47:09.480 align:middle line:84% of household I to be some function 00:47:09.480 --> 00:47:12.990 align:middle line:84% g of the aggregate social endowment. 00:47:12.990 --> 00:47:16.470 align:middle line:84% This is a Pareto optimal allocation. 00:47:16.470 --> 00:47:18.450 align:middle line:84% It's certainly a subset of the set 00:47:18.450 --> 00:47:20.790 align:middle line:84% of all feasible allocations, and it 00:47:20.790 --> 00:47:23.730 align:middle line:84% has to do with these weighted marginal utilities being 00:47:23.730 --> 00:47:27.060 align:middle line:84% equated at particular lambda weights. 00:47:27.060 --> 00:47:29.700 align:middle line:84% Because utility functions are concave, 00:47:29.700 --> 00:47:33.820 align:middle line:84% strictly concave, if they were strictly concave, 00:47:33.820 --> 00:47:36.930 align:middle line:84% then this g is going to be positive 00:47:36.930 --> 00:47:38.980 align:middle line:90% with a positive derivative. 00:47:38.980 --> 00:47:41.640 align:middle line:84% So there's a monotonicity argument here 00:47:41.640 --> 00:47:44.790 align:middle line:84% that, if the aggregate social endowment is going up, 00:47:44.790 --> 00:47:47.710 align:middle line:84% then the consumptions of all the agents are going up. 00:47:47.710 --> 00:47:50.880 align:middle line:84% How fast they go up as the social endowment goes up 00:47:50.880 --> 00:47:52.410 align:middle line:90% is a function of the g. 00:47:52.410 --> 00:47:54.120 align:middle line:90% And the g has an i on it. 00:47:54.120 --> 00:47:58.320 align:middle line:84% That i reflects potential heterogeneity in the underlying 00:47:58.320 --> 00:47:59.220 align:middle line:90% utility functions. 00:47:59.220 --> 00:48:02.580 align:middle line:84% We did not require that everyone has the same utility 00:48:02.580 --> 00:48:06.390 align:middle line:84% levels or even the same utility functions. 00:48:06.390 --> 00:48:09.490 align:middle line:84% And I'll show you some solutions momentarily. 00:48:09.490 --> 00:48:13.680 align:middle line:84% So we'll eventually get to an application, risk and insurance 00:48:13.680 --> 00:48:16.440 align:middle line:84% in village India and then one with production. 00:48:16.440 --> 00:48:19.230 align:middle line:84% We will rederive this equation and use it. 00:48:19.230 --> 00:48:24.310 align:middle line:84% But the restrictions on the data come from equations like 17. 00:48:24.310 --> 00:48:27.580 align:middle line:84% Remember, economic science is about making predictions. 00:48:27.580 --> 00:48:29.070 align:middle line:90% So we have a whole economy. 00:48:29.070 --> 00:48:32.310 align:middle line:84% Feasibility is one way to restrict the prediction. 00:48:32.310 --> 00:48:36.060 align:middle line:84% Optimality is a further restriction on the prediction. 00:48:36.060 --> 00:48:40.200 align:middle line:84% It's a criterion function, a metric we're going to use. 00:48:40.200 --> 00:48:42.300 align:middle line:84% We're now in the position to make predictions 00:48:42.300 --> 00:48:45.120 align:middle line:84% about what allocations ought to look like at least 00:48:45.120 --> 00:48:47.070 align:middle line:90% in this risk sharing setting. 00:48:47.070 --> 00:48:49.950 align:middle line:84% And they have to satisfy equation seven. 00:48:49.950 --> 00:48:52.680 align:middle line:84% So there are lots of things that are inconsistent with equation 00:48:52.680 --> 00:48:53.430 align:middle line:90% seven. 00:48:53.430 --> 00:48:55.530 align:middle line:90% And that creates a test. 00:48:55.530 --> 00:48:59.380 align:middle line:84% We can go to an actual application and start testing. 00:48:59.380 --> 00:49:02.430 align:middle line:84% So let me just give you two examples. 00:49:02.430 --> 00:49:06.210 align:middle line:84% If we started with constant absolute risk aversion-- 00:49:06.210 --> 00:49:08.400 align:middle line:84% and that concept was introduced when 00:49:08.400 --> 00:49:12.520 align:middle line:84% we talked about choice under uncertainty under the consumer 00:49:12.520 --> 00:49:13.020 align:middle line:90% preferences. 00:49:13.020 --> 00:49:15.960 align:middle line:84% It's sort of negative exponential, which makes 00:49:15.960 --> 00:49:17.910 align:middle line:90% the margin utility positive. 00:49:17.910 --> 00:49:20.670 align:middle line:84% This gamma j, which is a weighting term, 00:49:20.670 --> 00:49:23.250 align:middle line:84% is close to absolute risk aversion. 00:49:23.250 --> 00:49:26.160 align:middle line:84% Actually, 1 over gamma j is risk aversion. 00:49:26.160 --> 00:49:30.030 align:middle line:84% The higher this gamma j here, lower this thing is. 00:49:30.030 --> 00:49:33.450 align:middle line:84% So gamma j itself is the inverse of risk aversion. 00:49:33.450 --> 00:49:35.130 align:middle line:90% It's like risk tolerance. 00:49:35.130 --> 00:49:37.260 align:middle line:90% And I'll spare you the algebra. 00:49:37.260 --> 00:49:40.950 align:middle line:84% But if you maximized a lambda weighted sum of utilities 00:49:40.950 --> 00:49:44.280 align:middle line:84% subject to the aggregate social endowment being e, 00:49:44.280 --> 00:49:46.120 align:middle line:90% you would get this solution. 00:49:46.120 --> 00:49:48.990 align:middle line:84% So this is saying consumption of household 1 00:49:48.990 --> 00:49:51.390 align:middle line:84% is a linear function of the endowment. 00:49:51.390 --> 00:49:53.820 align:middle line:84% This is the intercept, and this is the slope. 00:49:53.820 --> 00:49:56.700 align:middle line:84% The intercept has to do with those lambda weights. 00:49:56.700 --> 00:50:00.600 align:middle line:84% So very naturally, in this case, the higher is lambda 1, 00:50:00.600 --> 00:50:02.760 align:middle line:84% then the higher is the log of this ratio. 00:50:02.760 --> 00:50:06.900 align:middle line:84% And the higher will be the level of consumption of household 1. 00:50:06.900 --> 00:50:10.230 align:middle line:84% And the only thing left is how those consumptions 00:50:10.230 --> 00:50:13.410 align:middle line:84% will change as we change the aggregate social endowment 00:50:13.410 --> 00:50:16.080 align:middle line:84% determined over time or randomly or whatever. 00:50:16.080 --> 00:50:18.900 align:middle line:84% And it's saying that consumption of household 1 00:50:18.900 --> 00:50:21.960 align:middle line:84% will go up with e at a faster rate. 00:50:21.960 --> 00:50:26.550 align:middle line:84% And for household 2, the higher is gamma 1 relative to gamma 2. 00:50:26.550 --> 00:50:29.250 align:middle line:84% Gamma 1, as I said, is like risk tolerance. 00:50:29.250 --> 00:50:31.270 align:middle line:84% It's the inverse of risk aversion. 00:50:31.270 --> 00:50:33.930 align:middle line:84% So now, we're introducing some randomness 00:50:33.930 --> 00:50:36.150 align:middle line:90% in the economy's GDP. 00:50:36.150 --> 00:50:38.820 align:middle line:84% And the issue is, who's going to bear that risk? 00:50:38.820 --> 00:50:40.740 align:middle line:84% Who's going to suffer in recessions? 00:50:40.740 --> 00:50:44.550 align:middle line:84% Who's going to bear the bonus of being in a boom? 00:50:44.550 --> 00:50:46.030 align:middle line:90% We have risk aversion. 00:50:46.030 --> 00:50:50.670 align:middle line:84% So the guy that's most risk averse with the 1 over gamma 00:50:50.670 --> 00:50:53.310 align:middle line:84% would be the one where this is close to 0, 00:50:53.310 --> 00:50:56.370 align:middle line:84% meaning that his concerns over consumption 00:50:56.370 --> 00:50:59.730 align:middle line:84% is pretty stable against the aggregate social endowment. 00:50:59.730 --> 00:51:01.830 align:middle line:84% And the other guy is bearing all the risk, 00:51:01.830 --> 00:51:04.530 align:middle line:84% but that's a Pareto optimal allocation. 00:51:04.530 --> 00:51:06.660 align:middle line:84% The riskiest person should bear the brunt 00:51:06.660 --> 00:51:08.730 align:middle line:90% of the aggregate fluctuations. 00:51:08.730 --> 00:51:09.840 align:middle line:90% Here's another example. 00:51:09.840 --> 00:51:13.650 align:middle line:84% This is constant relative risk aversion, a version 00:51:13.650 --> 00:51:14.910 align:middle line:90% of which you saw before. 00:51:14.910 --> 00:51:18.240 align:middle line:84% We're going to raise consumption to the power d. 00:51:18.240 --> 00:51:22.020 align:middle line:84% This is actually consumption over and above subsistence, 00:51:22.020 --> 00:51:23.080 align:middle line:90% as here. 00:51:23.080 --> 00:51:27.810 align:middle line:84% So the consumption set has a positive lower bound. 00:51:27.810 --> 00:51:32.700 align:middle line:84% And this is the distance from that subsistence consumption 00:51:32.700 --> 00:51:34.890 align:middle line:90% raised to the power d. 00:51:34.890 --> 00:51:38.670 align:middle line:84% And when you solve these equations, they're linear. 00:51:38.670 --> 00:51:41.490 align:middle line:84% And again, you get an intercept and a slope. 00:51:41.490 --> 00:51:44.190 align:middle line:84% However, it's a bit different from before 00:51:44.190 --> 00:51:48.180 align:middle line:84% because the intercept contains not only the lambda weights, 00:51:48.180 --> 00:51:51.960 align:middle line:84% but it contains the a thing, which is related to the b, 00:51:51.960 --> 00:51:55.330 align:middle line:84% related to the power in the utility function. 00:51:55.330 --> 00:51:58.507 align:middle line:84% It looks like higher lambda is going to lower consumption, 00:51:58.507 --> 00:52:00.090 align:middle line:84% but then you've got to remember lambda 00:52:00.090 --> 00:52:02.020 align:middle line:90% is a number between 0 and 1. 00:52:02.020 --> 00:52:05.850 align:middle line:84% We're taking it to a power so you can make it smaller instead 00:52:05.850 --> 00:52:07.110 align:middle line:90% of bigger, et cetera. 00:52:07.110 --> 00:52:09.300 align:middle line:84% So when you do it right and think about it, 00:52:09.300 --> 00:52:12.540 align:middle line:84% it does mean that higher lambda 1 is associated 00:52:12.540 --> 00:52:14.370 align:middle line:84% with higher levels of consumption, 00:52:14.370 --> 00:52:17.970 align:middle line:84% but it's weighted, effectively, by risk aversion 00:52:17.970 --> 00:52:21.330 align:middle line:84% and by the subsistence level of consumption. 00:52:21.330 --> 00:52:24.600 align:middle line:84% And now, if we vary the aggregate social endowment, 00:52:24.600 --> 00:52:27.000 align:middle line:84% the allocations will change, but they're not 00:52:27.000 --> 00:52:30.150 align:middle line:84% going to go up 1 to 1 the way they did before. 00:52:30.150 --> 00:52:33.930 align:middle line:84% That's going to be determined by both the lambda weights as well 00:52:33.930 --> 00:52:35.670 align:middle line:90% as risk aversion. 00:52:35.670 --> 00:52:38.250 align:middle line:84% Both those pictures, constant absolute risk aversion 00:52:38.250 --> 00:52:40.320 align:middle line:84% and constant relative risk aversion, 00:52:40.320 --> 00:52:42.540 align:middle line:84% look like this with different slopes 00:52:42.540 --> 00:52:44.730 align:middle line:90% and different intercepts. 00:52:44.730 --> 00:52:46.740 align:middle line:90% And then this is-- 00:52:46.740 --> 00:52:50.310 align:middle line:84% I'm not and I probably should have written out the underlying 00:52:50.310 --> 00:52:51.420 align:middle line:90% utility functions. 00:52:51.420 --> 00:52:54.390 align:middle line:84% But you can think of this as one person having 00:52:54.390 --> 00:52:56.670 align:middle line:84% constant relative risk aversion, the other one having 00:52:56.670 --> 00:52:59.100 align:middle line:90% constant absolute risk aversion. 00:52:59.100 --> 00:53:01.470 align:middle line:84% And then you get non-linear schedules, 00:53:01.470 --> 00:53:04.680 align:middle line:84% but they are both monotone increasing as predicted. 00:53:04.680 --> 00:53:08.160 align:middle line:84% But now, the slopes vary and the levels are clearly different 00:53:08.160 --> 00:53:11.400 align:middle line:84% depending on the aggregate social endowment e. 00:53:11.400 --> 00:53:13.630 align:middle line:84% But we can still put restrictions on the data. 00:53:13.630 --> 00:53:14.910 align:middle line:90% So those are three pictures. 00:53:14.910 --> 00:53:17.250 align:middle line:84% And we're going to come back to these pictures 00:53:17.250 --> 00:53:21.150 align:middle line:84% when we do one of the applications, which is dividing 00:53:21.150 --> 00:53:23.040 align:middle line:90% up land in medieval England. 00:53:23.040 --> 00:53:24.660 align:middle line:84% We've referred to the several times. 00:53:24.660 --> 00:53:27.120 align:middle line:84% We're actually now empowered with the tools 00:53:27.120 --> 00:53:31.060 align:middle line:84% to be able to analyze that application. 00:53:31.060 --> 00:53:34.300 align:middle line:90% All right, are there questions? 00:53:34.300 --> 00:53:35.140 align:middle line:90% OK. 00:53:35.140 --> 00:53:38.770 align:middle line:84% So again, next Thursday we're going to not 00:53:38.770 --> 00:53:40.130 align:middle line:90% have another new lecture. 00:53:40.130 --> 00:53:43.660 align:middle line:84% We're going to have a review session over the first seven 00:53:43.660 --> 00:53:45.370 align:middle line:90% of them, including this one. 00:53:45.370 --> 00:53:49.420 align:middle line:84% So review the material, take advantage of the opportunity. 00:53:49.420 --> 00:53:53.590 align:middle line:84% Feel free to ask me questions, and I 00:53:53.590 --> 00:53:55.300 align:middle line:84% will try to provide a summary of what 00:53:55.300 --> 00:53:59.130 align:middle line:90% we've done so far in the class. 00:53:59.130 --> 00:54:02.060 align:middle line:90% OK, thank you very much. 00:54:02.060 --> 00:54:09.308 align:middle line:90%