WEBVTT 00:00:00.000 --> 00:00:11.440 align:middle line:90% 00:00:11.440 --> 00:00:13.750 align:middle line:84% ROBERT TOWNSEND: Welcome, everybody, today, 00:00:13.750 --> 00:00:16.810 align:middle line:84% for the next segment of the class. 00:00:16.810 --> 00:00:20.840 align:middle line:84% Actually, this is literally a continuation. 00:00:20.840 --> 00:00:23.740 align:middle line:84% It's not a sharp break between the last lecture and today. 00:00:23.740 --> 00:00:26.230 align:middle line:84% Fortunately, the lecture today has 00:00:26.230 --> 00:00:29.680 align:middle line:84% some repeat of the equations, or at least a version 00:00:29.680 --> 00:00:31.330 align:middle line:90% of those equations. 00:00:31.330 --> 00:00:35.230 align:middle line:84% Plus, I know you're all totally fresh from the exam, 00:00:35.230 --> 00:00:38.980 align:middle line:84% so you know what the last lecture was anyway. 00:00:38.980 --> 00:00:40.480 align:middle line:90% Just teasing you a little bit. 00:00:40.480 --> 00:00:41.515 align:middle line:90% Sorry. 00:00:41.515 --> 00:00:45.040 align:middle line:84% I'll just say a few words about the calendar. 00:00:45.040 --> 00:00:51.130 align:middle line:84% So we are right here, today, Thursday, 10/1. 00:00:51.130 --> 00:00:51.880 align:middle line:90% Oh, my goodness. 00:00:51.880 --> 00:00:53.620 align:middle line:90% It is October. 00:00:53.620 --> 00:00:57.730 align:middle line:84% And we're going to do risk-sharing applications today 00:00:57.730 --> 00:00:59.860 align:middle line:90% and then also 9. 00:00:59.860 --> 00:01:02.440 align:middle line:84% Although this says risk-sharing applications, 00:01:02.440 --> 00:01:06.020 align:middle line:84% I want to feature that we'll combine it with a production 00:01:06.020 --> 00:01:08.770 align:middle line:84% even in a more direct way than we will today. 00:01:08.770 --> 00:01:13.930 align:middle line:84% So two lectures today, Thursday and Tuesday, on risk-sharing. 00:01:13.930 --> 00:01:16.600 align:middle line:90% So that's the calendar. 00:01:16.600 --> 00:01:19.960 align:middle line:84% The second thing would be the reading list. 00:01:19.960 --> 00:01:22.060 align:middle line:90% And we are here at lecture 8. 00:01:22.060 --> 00:01:25.060 align:middle line:84% So the two readings today are risk and insurance 00:01:25.060 --> 00:01:28.120 align:middle line:84% in Village India, my paper in Econometrica. 00:01:28.120 --> 00:01:30.760 align:middle line:84% And we will cover that in a fair amount of detail 00:01:30.760 --> 00:01:33.760 align:middle line:84% today, although not every word, obviously. 00:01:33.760 --> 00:01:37.880 align:middle line:84% And also, chapter 2 in Medieval Village Economy. 00:01:37.880 --> 00:01:40.710 align:middle line:90% So that's the reading list. 00:01:40.710 --> 00:01:47.080 align:middle line:84% So today is lecture 8, and that is here. 00:01:47.080 --> 00:01:49.620 align:middle line:84% So risk-sharing applications-- we're 00:01:49.620 --> 00:01:51.900 align:middle line:84% going to do two different economies using 00:01:51.900 --> 00:01:54.270 align:middle line:84% the same theory, although the questions to be 00:01:54.270 --> 00:01:56.730 align:middle line:84% asked, though related, are a little bit different. 00:01:56.730 --> 00:02:00.930 align:middle line:84% One is Village India using consumption and income 00:02:00.930 --> 00:02:04.920 align:middle line:84% data, which I refer to as an ex-post version 00:02:04.920 --> 00:02:07.500 align:middle line:84% of the implications of the risk-sharing model, 00:02:07.500 --> 00:02:11.760 align:middle line:84% and also, as anticipated, the medieval village economy, 00:02:11.760 --> 00:02:14.850 align:middle line:84% where we're going to talk about dividing up the land. 00:02:14.850 --> 00:02:17.680 align:middle line:84% And you've seen maps of this before, 00:02:17.680 --> 00:02:19.710 align:middle line:84% but we didn't have all of the tools we 00:02:19.710 --> 00:02:21.750 align:middle line:90% needed to really do a good job. 00:02:21.750 --> 00:02:23.350 align:middle line:90% But we do now. 00:02:23.350 --> 00:02:23.850 align:middle line:90% OK. 00:02:23.850 --> 00:02:28.280 align:middle line:84% So for India, here's the motivation from my paper. 00:02:28.280 --> 00:02:32.220 align:middle line:84% A large part of the population in developing countries 00:02:32.220 --> 00:02:34.800 align:middle line:90% live in high-risk environments. 00:02:34.800 --> 00:02:38.890 align:middle line:84% That said, there are numerous ways to cope with it. 00:02:38.890 --> 00:02:43.410 align:middle line:84% One is to diversify the activities you're undertaking-- 00:02:43.410 --> 00:02:46.860 align:middle line:84% trying to have a portfolio, as it were-- 00:02:46.860 --> 00:02:51.060 align:middle line:84% or ex-post, after incomes get realized, 00:02:51.060 --> 00:02:53.290 align:middle line:84% to engage in financial transactions, 00:02:53.290 --> 00:02:56.640 align:middle line:84% including gifts and transfers among family and family 00:02:56.640 --> 00:02:57.840 align:middle line:90% networks. 00:02:57.840 --> 00:03:01.560 align:middle line:84% How well do any one of these ex-post or ex-ante mechanisms 00:03:01.560 --> 00:03:02.730 align:middle line:90% work? 00:03:02.730 --> 00:03:05.880 align:middle line:84% Well, the main, quote, "insight" of the paper 00:03:05.880 --> 00:03:08.160 align:middle line:84% was that we really don't necessarily 00:03:08.160 --> 00:03:10.200 align:middle line:90% have to enumerate the mechanism. 00:03:10.200 --> 00:03:12.560 align:middle line:90% We just look at the outcome. 00:03:12.560 --> 00:03:14.300 align:middle line:84% I'll qualify that a little bit, and we 00:03:14.300 --> 00:03:16.250 align:middle line:90% can keep coming back to this. 00:03:16.250 --> 00:03:20.510 align:middle line:84% If they diversified perfectly and everyone had a balanced 00:03:20.510 --> 00:03:24.450 align:middle line:84% portfolio, then they would all have the same outcome, 00:03:24.450 --> 00:03:27.390 align:middle line:84% and there would be no further risk to share. 00:03:27.390 --> 00:03:29.000 align:middle line:84% So in some sense, what we're going 00:03:29.000 --> 00:03:33.320 align:middle line:84% to go through today would not have much content in the sense 00:03:33.320 --> 00:03:37.220 align:middle line:84% that today, we're referring to risk sharing, 00:03:37.220 --> 00:03:41.640 align:middle line:84% but it's not done, per se, by sharing risk after the fact. 00:03:41.640 --> 00:03:44.540 align:middle line:84% It's done by ex-ante diversification. 00:03:44.540 --> 00:03:46.400 align:middle line:84% Anyway, those two strands to repeat 00:03:46.400 --> 00:03:48.570 align:middle line:84% are going to be in today's lecture. 00:03:48.570 --> 00:03:51.510 align:middle line:84% We're going to do the ex-post part first, 00:03:51.510 --> 00:03:55.070 align:middle line:84% which assumes partial but not complete diversification, 00:03:55.070 --> 00:03:56.960 align:middle line:84% and then look at how they could do 00:03:56.960 --> 00:04:01.100 align:middle line:84% it ex-ante with partial but potentially not complete 00:04:01.100 --> 00:04:03.330 align:middle line:90% diversification. 00:04:03.330 --> 00:04:04.010 align:middle line:90% All right. 00:04:04.010 --> 00:04:09.580 align:middle line:84% So characteristics of village economies as in India-- 00:04:09.580 --> 00:04:15.150 align:middle line:84% very risky place, in large part because agriculture does not 00:04:15.150 --> 00:04:18.390 align:middle line:84% provide a stable source of income. 00:04:18.390 --> 00:04:21.149 align:middle line:84% There's a lot of risk, idiosyncratic 00:04:21.149 --> 00:04:22.530 align:middle line:90% and aggregate risk. 00:04:22.530 --> 00:04:25.330 align:middle line:84% Households diversify in many ways. 00:04:25.330 --> 00:04:29.430 align:middle line:84% One-- not planting all of the same crops. 00:04:29.430 --> 00:04:32.190 align:middle line:84% And we are going to use this benchmark which 00:04:32.190 --> 00:04:35.610 align:middle line:84% assumes there's no information problem, no moral hazard 00:04:35.610 --> 00:04:39.630 align:middle line:84% problem, no contract enforcement problem. 00:04:39.630 --> 00:04:42.500 align:middle line:84% All of those words at this point in the class 00:04:42.500 --> 00:04:44.250 align:middle line:84% don't make a whole lot of sense because we 00:04:44.250 --> 00:04:49.720 align:middle line:84% haven't studied anything but these frictionless economies. 00:04:49.720 --> 00:04:51.150 align:middle line:90% But that was the premise. 00:04:51.150 --> 00:04:53.940 align:middle line:84% And again, to anticipate subsequent lectures, 00:04:53.940 --> 00:04:56.130 align:middle line:84% today we'll see how well this theory does 00:04:56.130 --> 00:04:59.100 align:middle line:90% without any of these frictions. 00:04:59.100 --> 00:05:01.500 align:middle line:84% And then subsequent literature has 00:05:01.500 --> 00:05:04.860 align:middle line:84% started to assume things like contract enforcement 00:05:04.860 --> 00:05:06.960 align:middle line:84% and information asymmetries, but we are not 00:05:06.960 --> 00:05:09.470 align:middle line:90% going to do that today. 00:05:09.470 --> 00:05:11.470 align:middle line:90% So how do they diversify? 00:05:11.470 --> 00:05:14.410 align:middle line:84% Well, in a village Aurepalle, which 00:05:14.410 --> 00:05:18.450 align:middle line:84% was under the auspices of the Institute for Crops 00:05:18.450 --> 00:05:22.810 align:middle line:84% in the Semi-Arid Tropics, ICRISAT, farmers 00:05:22.810 --> 00:05:24.520 align:middle line:90% are a poly-growth sorghum. 00:05:24.520 --> 00:05:25.780 align:middle line:90% They grow castor. 00:05:25.780 --> 00:05:28.120 align:middle line:90% Sorghum is a grain like wheat. 00:05:28.120 --> 00:05:32.230 align:middle line:84% Castor you may have heard of as in castor oil. 00:05:32.230 --> 00:05:34.840 align:middle line:84% It's both used in diet, but also, it 00:05:34.840 --> 00:05:38.410 align:middle line:84% turns out to be a lubricant for jet engines. 00:05:38.410 --> 00:05:44.020 align:middle line:84% Anyway, the coefficients of variation are 0.5 and about 1. 00:05:44.020 --> 00:05:47.230 align:middle line:84% Now, you may not remember that in medieval villages, 00:05:47.230 --> 00:05:49.540 align:middle line:84% we talked about a very risky environment 00:05:49.540 --> 00:05:51.970 align:middle line:84% with the coefficient of variation of yields 00:05:51.970 --> 00:05:56.560 align:middle line:84% of about 0.35 or something like that, which delivered 00:05:56.560 --> 00:05:58.640 align:middle line:90% a disaster every 12 years. 00:05:58.640 --> 00:06:01.990 align:middle line:84% So it's more risky than that taking any one crop 00:06:01.990 --> 00:06:02.950 align:middle line:90% one at a time. 00:06:02.950 --> 00:06:05.740 align:middle line:84% But they can diversify over crops. 00:06:05.740 --> 00:06:08.860 align:middle line:84% So looking at the crop correlation, 00:06:08.860 --> 00:06:13.285 align:middle line:84% those numbers range from 0.09 to 0.81. 00:06:13.285 --> 00:06:17.200 align:middle line:84% At .09, it's almost the case that the yields 00:06:17.200 --> 00:06:20.290 align:middle line:84% on one type of crop are independent of the yields 00:06:20.290 --> 00:06:21.880 align:middle line:90% on the other one. 00:06:21.880 --> 00:06:25.390 align:middle line:84% Again, you can go back and look at the lecture under Consumer 00:06:25.390 --> 00:06:27.940 align:middle line:84% Preferences where we were introducing risk 00:06:27.940 --> 00:06:30.820 align:middle line:84% and derived these kinds of statistics. 00:06:30.820 --> 00:06:34.000 align:middle line:84% Soil isn't uniform either, so even taking 00:06:34.000 --> 00:06:36.190 align:middle line:84% as given that they're going to do castor, 00:06:36.190 --> 00:06:39.290 align:middle line:84% they can plant it in one type of soil or another. 00:06:39.290 --> 00:06:42.610 align:middle line:84% Each type is risky with a coefficient of variation 00:06:42.610 --> 00:06:45.640 align:middle line:84% from 0.7 to, again, a little over 1. 00:06:45.640 --> 00:06:49.270 align:middle line:84% But you can diversify over soil as well 00:06:49.270 --> 00:06:53.800 align:middle line:84% with a correlation of only 0.37, so pretty good diversification 00:06:53.800 --> 00:06:55.570 align:middle line:90% possibilities. 00:06:55.570 --> 00:07:00.250 align:middle line:84% That said, households don't hold completely balanced portfolios 00:07:00.250 --> 00:07:01.900 align:middle line:90% across soils and crops. 00:07:01.900 --> 00:07:05.050 align:middle line:84% One household will plant more of one crop than the other, 00:07:05.050 --> 00:07:09.670 align:middle line:84% or for castor, have more of one type of soil than the other 00:07:09.670 --> 00:07:12.610 align:middle line:84% despite all the diversification possibilities. 00:07:12.610 --> 00:07:18.310 align:middle line:84% And going outwards from crops to other income-generating 00:07:18.310 --> 00:07:22.690 align:middle line:84% activities, they can engage in wage labor. 00:07:22.690 --> 00:07:27.880 align:middle line:84% They can do trade in handicrafts or animal husbandry. 00:07:27.880 --> 00:07:31.510 align:middle line:84% So for example, trade in handicrafts in Aurepalle 00:07:31.510 --> 00:07:35.920 align:middle line:84% consists of climbing palm trees to get 00:07:35.920 --> 00:07:40.000 align:middle line:84% the fruit from which they will make palm liquor, for example. 00:07:40.000 --> 00:07:42.430 align:middle line:84% And it's a pretty specialized activity. 00:07:42.430 --> 00:07:46.300 align:middle line:84% It looks like telephone repairmen climbing poles 00:07:46.300 --> 00:07:50.080 align:middle line:84% with pretty heavy and fancy equipment. 00:07:50.080 --> 00:07:54.290 align:middle line:84% Animal husbandry consists of grazing animals and so on. 00:07:54.290 --> 00:07:57.250 align:middle line:84% Anyway, if you look at how much land they hold, 00:07:57.250 --> 00:08:02.140 align:middle line:84% it varies from no land at all to being a small holder or medium 00:08:02.140 --> 00:08:03.710 align:middle line:90% or large holder. 00:08:03.710 --> 00:08:07.400 align:middle line:84% And so for example, not surprisingly, 00:08:07.400 --> 00:08:09.520 align:middle line:84% the more land they have, the more 00:08:09.520 --> 00:08:14.560 align:middle line:84% is their income from crops, going from about 2% to 56% 00:08:14.560 --> 00:08:15.940 align:middle line:90% of the total income. 00:08:15.940 --> 00:08:19.630 align:middle line:84% Likewise, when they hold very little land, most of the income 00:08:19.630 --> 00:08:22.300 align:middle line:84% is from labor, wage labor, and that 00:08:22.300 --> 00:08:26.830 align:middle line:84% drops to 4% for the rich guys, large landholders 00:08:26.830 --> 00:08:29.500 align:middle line:90% in the village, and so on. 00:08:29.500 --> 00:08:32.770 align:middle line:84% Those patterns are pretty much repeated in the other two 00:08:32.770 --> 00:08:35.650 align:middle line:84% villages, and Kanzara especially. 00:08:35.650 --> 00:08:38.559 align:middle line:84% Shirapur is much flatter in the sense 00:08:38.559 --> 00:08:41.980 align:middle line:84% that we don't see very sharp gradients in terms 00:08:41.980 --> 00:08:45.790 align:middle line:84% of percentage of income from various sources 00:08:45.790 --> 00:08:47.290 align:middle line:90% as we move from small. 00:08:47.290 --> 00:08:50.440 align:middle line:84% So villages are kind of a metaphor. 00:08:50.440 --> 00:08:52.638 align:middle line:90% Villages are not all alike. 00:08:52.638 --> 00:08:54.430 align:middle line:84% But the techniques we're going to use today 00:08:54.430 --> 00:08:56.920 align:middle line:84% do take into account the differences. 00:08:56.920 --> 00:08:58.690 align:middle line:84% Now let's look at diversification 00:08:58.690 --> 00:09:03.260 align:middle line:84% over income sources the way we did for crops and soil types. 00:09:03.260 --> 00:09:07.810 align:middle line:84% This is a little bit difficult to read, and I'll tell you why. 00:09:07.810 --> 00:09:12.760 align:middle line:84% Going down the diagonal here is the coefficient of variation 00:09:12.760 --> 00:09:14.210 align:middle line:90% of each income source. 00:09:14.210 --> 00:09:17.680 align:middle line:84% So for example, crop production is 00:09:17.680 --> 00:09:21.730 align:middle line:84% quite risky with a coefficient of 0.42, which I almost 00:09:21.730 --> 00:09:24.220 align:middle line:90% said a minute ago. 00:09:24.220 --> 00:09:29.530 align:middle line:84% Of livestock income, the CV is 0.21, et cetera. 00:09:29.530 --> 00:09:32.050 align:middle line:84% Now, these other off diagonal elements 00:09:32.050 --> 00:09:35.200 align:middle line:90% here represent the covariation. 00:09:35.200 --> 00:09:38.650 align:middle line:84% So what would have been better would 00:09:38.650 --> 00:09:42.100 align:middle line:84% have been to list profits, livestock, income, 00:09:42.100 --> 00:09:44.560 align:middle line:84% earned wages, and trade and handicrafts 00:09:44.560 --> 00:09:47.320 align:middle line:84% going down the rows, and then picking 00:09:47.320 --> 00:09:51.860 align:middle line:84% a row, move across the column to find the covariance. 00:09:51.860 --> 00:09:55.090 align:middle line:84% So for example, this guy here is the covariance 00:09:55.090 --> 00:09:58.700 align:middle line:84% between profits from crop production and livestock 00:09:58.700 --> 00:09:59.200 align:middle line:90% income. 00:09:59.200 --> 00:10:01.130 align:middle line:90% And it's actually negative. 00:10:01.130 --> 00:10:05.350 align:middle line:84% So huge diversification possibilities for a household 00:10:05.350 --> 00:10:08.800 align:middle line:84% if they were doing both crops and livestock. 00:10:08.800 --> 00:10:12.040 align:middle line:84% These lower off-diagonal elements 00:10:12.040 --> 00:10:15.370 align:middle line:84% are not filled in because the matrix is symmetric. 00:10:15.370 --> 00:10:19.300 align:middle line:84% The covariation of activity i with activity j 00:10:19.300 --> 00:10:22.840 align:middle line:84% is the same as the variation of i with i. 00:10:22.840 --> 00:10:25.450 align:middle line:84% So it would be redundant to fill them all in, 00:10:25.450 --> 00:10:29.370 align:middle line:84% and they were deleted to make it easier to read. 00:10:29.370 --> 00:10:32.990 align:middle line:84% So taking into account different soils, different crops, 00:10:32.990 --> 00:10:35.720 align:middle line:84% different income sources, piling them all together, 00:10:35.720 --> 00:10:39.870 align:middle line:84% and plotting income over the 10 years for which we 00:10:39.870 --> 00:10:44.730 align:middle line:84% have the data, we look what happens over time 00:10:44.730 --> 00:10:50.200 align:middle line:90% from 1976 to 1985, essentially. 00:10:50.200 --> 00:10:52.990 align:middle line:84% And what's getting plotted is not just 00:10:52.990 --> 00:10:56.130 align:middle line:84% the time series of one household. 00:10:56.130 --> 00:10:57.860 align:middle line:84% All of them are getting plotted here. 00:10:57.860 --> 00:11:02.030 align:middle line:84% In fact, the way the data were coded, 00:11:02.030 --> 00:11:05.000 align:middle line:84% low numbers correspond to little or no land, 00:11:05.000 --> 00:11:08.630 align:middle line:84% and high numbers correspond to lots of land. 00:11:08.630 --> 00:11:12.650 align:middle line:84% So as we move backwards along the household number, 00:11:12.650 --> 00:11:15.320 align:middle line:84% the households are getting more and more land 00:11:15.320 --> 00:11:17.480 align:middle line:90% and are actually-- 00:11:17.480 --> 00:11:20.460 align:middle line:90% net worth is going up. 00:11:20.460 --> 00:11:22.560 align:middle line:84% So that 1% of the population that 00:11:22.560 --> 00:11:26.520 align:middle line:84% owns 50% of the wealth, that's this guy back here. 00:11:26.520 --> 00:11:31.920 align:middle line:84% Now, I refer to this graph as the Rocky Mountains 00:11:31.920 --> 00:11:34.140 align:middle line:84% because you can see all the peaks-- 00:11:34.140 --> 00:11:34.980 align:middle line:90% well, not quite. 00:11:34.980 --> 00:11:38.480 align:middle line:84% I was going to say you can see all the peaks and valleys. 00:11:38.480 --> 00:11:39.800 align:middle line:90% That's wrong. 00:11:39.800 --> 00:11:44.430 align:middle line:84% When you have a peak, you can't see the valley behind it. 00:11:44.430 --> 00:11:48.420 align:middle line:84% If all these incomes communed with each other, 00:11:48.420 --> 00:11:51.160 align:middle line:84% then everyone's peak would happen at the same time, 00:11:51.160 --> 00:11:53.220 align:middle line:90% and it would look like waves. 00:11:53.220 --> 00:11:56.730 align:middle line:84% Instead, we see this rather jagged, sharp picture. 00:11:56.730 --> 00:12:00.120 align:middle line:84% And again, when you're able to see the valley behind, 00:12:00.120 --> 00:12:02.520 align:middle line:84% it's because there's not a peak in front of it. 00:12:02.520 --> 00:12:05.310 align:middle line:84% Now, the other thing is moving along this gradient-- 00:12:05.310 --> 00:12:08.730 align:middle line:84% I should have pointed this out earlier-- there is a 0 here. 00:12:08.730 --> 00:12:13.080 align:middle line:84% This 0 corresponds to the overall average of income 00:12:13.080 --> 00:12:15.970 align:middle line:84% both over households and over time. 00:12:15.970 --> 00:12:19.230 align:middle line:84% So these guys are below-average income. 00:12:19.230 --> 00:12:22.020 align:middle line:84% These guys behind it are higher-average income. 00:12:22.020 --> 00:12:25.020 align:middle line:84% And again, you can see that that's quite a salient 00:12:25.020 --> 00:12:26.980 align:middle line:90% gradient. 00:12:26.980 --> 00:12:30.880 align:middle line:84% Here is the corresponding picture for grain. 00:12:30.880 --> 00:12:33.850 align:middle line:84% And I could have plotted all consumption items, 00:12:33.850 --> 00:12:35.410 align:middle line:84% but it would have looked similar. 00:12:35.410 --> 00:12:37.690 align:middle line:90% The paper actually does that. 00:12:37.690 --> 00:12:42.370 align:middle line:84% When I say corresponding, I mean the scale hasn't changed. 00:12:42.370 --> 00:12:47.200 align:middle line:84% The scale that was used to plot income, 4,000 at the peak, 00:12:47.200 --> 00:12:51.670 align:middle line:84% is the same as the scale that's being used to plot consumption. 00:12:51.670 --> 00:12:55.200 align:middle line:84% So you can see everybody's consumption is compressed, 00:12:55.200 --> 00:12:57.310 align:middle line:90% so they're smoothing. 00:12:57.310 --> 00:12:59.560 align:middle line:90% They're smoothing over time. 00:12:59.560 --> 00:13:02.380 align:middle line:84% And to a degree there, a large degree, 00:13:02.380 --> 00:13:06.130 align:middle line:84% they're actually smoothing over households as well. 00:13:06.130 --> 00:13:08.770 align:middle line:84% Those very wealthy guys are consuming 00:13:08.770 --> 00:13:11.470 align:middle line:84% more than the poor guys but not nearly as much 00:13:11.470 --> 00:13:17.270 align:middle line:84% as income is higher for them relative to income of low guys. 00:13:17.270 --> 00:13:19.620 align:middle line:84% The main point of the picture, however, 00:13:19.620 --> 00:13:23.390 align:middle line:84% is that here, you can almost see the waves. 00:13:23.390 --> 00:13:26.780 align:middle line:84% Grain consumption is kind of co-moving. 00:13:26.780 --> 00:13:29.420 align:middle line:84% That would be easier to see if we blew it up, 00:13:29.420 --> 00:13:31.310 align:middle line:84% but we will look at this statistically 00:13:31.310 --> 00:13:34.820 align:middle line:84% as well in a few subsequent slides. 00:13:34.820 --> 00:13:37.100 align:middle line:90% And I refer to this as Kansas. 00:13:37.100 --> 00:13:40.580 align:middle line:84% I don't if any of you have ever been in Kansas. 00:13:40.580 --> 00:13:44.510 align:middle line:84% I ask this every time I teach this diagram. 00:13:44.510 --> 00:13:47.000 align:middle line:84% At one point, some student in the class 00:13:47.000 --> 00:13:48.290 align:middle line:90% said, I'm from Kansas. 00:13:48.290 --> 00:13:49.430 align:middle line:90% And I said, oh, I'm sorry. 00:13:49.430 --> 00:13:51.057 align:middle line:90% I hope I haven't insulted you. 00:13:51.057 --> 00:13:51.890 align:middle line:90% And he said, no, no. 00:13:51.890 --> 00:13:54.680 align:middle line:90% It is really boring there. 00:13:54.680 --> 00:13:55.790 align:middle line:90% So I like Kansas. 00:13:55.790 --> 00:13:56.780 align:middle line:90% I'm not saying that. 00:13:56.780 --> 00:13:59.990 align:middle line:84% But anyway, amber waves of grain, 00:13:59.990 --> 00:14:03.330 align:middle line:90% so consumption versus income. 00:14:03.330 --> 00:14:05.990 align:middle line:84% So let's think about a theory to try 00:14:05.990 --> 00:14:09.810 align:middle line:84% to explain the difference between those two diagrams. 00:14:09.810 --> 00:14:12.570 align:middle line:84% And this is the bit that's a little bit of a review, 00:14:12.570 --> 00:14:14.020 align:middle line:90% but I think a useful one. 00:14:14.020 --> 00:14:17.250 align:middle line:84% So we're going to talk about individual k 00:14:17.250 --> 00:14:20.310 align:middle line:84% alive at date t that has experienced 00:14:20.310 --> 00:14:23.830 align:middle line:90% a history of shocks ht. 00:14:23.830 --> 00:14:27.150 align:middle line:84% This history is the shock at date 1, the shock at date 2, 00:14:27.150 --> 00:14:29.340 align:middle line:90% and the shock at date 3. 00:14:29.340 --> 00:14:34.170 align:middle line:84% So again, this is the state space representation 00:14:34.170 --> 00:14:37.620 align:middle line:84% of the tree that you saw in Debreu, 00:14:37.620 --> 00:14:40.290 align:middle line:84% and I featured at least twice already. 00:14:40.290 --> 00:14:42.010 align:middle line:90% What is the objective function? 00:14:42.010 --> 00:14:44.460 align:middle line:84% We're going to solve that programming problem 00:14:44.460 --> 00:14:47.640 align:middle line:84% to go back and forth between solutions to that problem 00:14:47.640 --> 00:14:49.770 align:middle line:90% in Pareto optimal allocations. 00:14:49.770 --> 00:14:53.160 align:middle line:84% So we're going to maximize a lambda-weighted sum 00:14:53.160 --> 00:14:55.710 align:middle line:84% of discounted expected utilities, 00:14:55.710 --> 00:14:59.790 align:middle line:84% weighted lambda k for weight k over all the individuals 00:14:59.790 --> 00:15:01.890 align:middle line:90% and their capital M of them. 00:15:01.890 --> 00:15:06.450 align:middle line:84% The contemporary utility functions have as arguments 00:15:06.450 --> 00:15:10.320 align:middle line:84% the consumption of individual k at date thq, which we just did, 00:15:10.320 --> 00:15:13.950 align:middle line:90% and then this age-gender index. 00:15:13.950 --> 00:15:16.440 align:middle line:90% So not everyone is the same age. 00:15:16.440 --> 00:15:18.960 align:middle line:90% Not everyone is the same gender. 00:15:18.960 --> 00:15:21.810 align:middle line:84% 18-year-old males eat twice as much 00:15:21.810 --> 00:15:23.610 align:middle line:90% as anybody else, essentially. 00:15:23.610 --> 00:15:26.130 align:middle line:84% So we want to adjust the utility for 00:15:26.130 --> 00:15:28.440 align:middle line:90% these metabolic requirements. 00:15:28.440 --> 00:15:33.510 align:middle line:84% And they may not be treating the women very equitably either. 00:15:33.510 --> 00:15:36.900 align:middle line:84% But we are going to take these as metabolic weights that were 00:15:36.900 --> 00:15:40.060 align:middle line:90% measured in a dietary survey. 00:15:40.060 --> 00:15:44.670 align:middle line:84% So the other objects here are the probability 00:15:44.670 --> 00:15:47.190 align:middle line:84% of that history actually happening. 00:15:47.190 --> 00:15:52.320 align:middle line:84% Note that for any day t, we sum over all possible histories 00:15:52.320 --> 00:15:56.980 align:middle line:84% that could have generated that node at date t. 00:15:56.980 --> 00:15:59.450 align:middle line:84% And then in addition, there are multiple dates, 00:15:59.450 --> 00:16:03.460 align:middle line:84% so we sum over those dates and discount them by beta. 00:16:03.460 --> 00:16:06.310 align:middle line:84% And again, the lambdas are the Pareto weights 00:16:06.310 --> 00:16:09.220 align:middle line:84% that vary between 0 and 1, and they sum to 1, 00:16:09.220 --> 00:16:11.170 align:middle line:90% and we take the weighted sum. 00:16:11.170 --> 00:16:13.720 align:middle line:84% There are many, many resource constraints. 00:16:13.720 --> 00:16:16.840 align:middle line:84% There is a particular constraint for every date t 00:16:16.840 --> 00:16:18.700 align:middle line:84% and all possible histories that could 00:16:18.700 --> 00:16:22.390 align:middle line:84% have led to t that simply says that the way you hand out 00:16:22.390 --> 00:16:26.020 align:middle line:84% consumption must not sum to something greater 00:16:26.020 --> 00:16:28.450 align:middle line:90% than the total that's available. 00:16:28.450 --> 00:16:30.370 align:middle line:84% So usually, we talk about income. 00:16:30.370 --> 00:16:33.250 align:middle line:84% Now we just went to total expenditure. 00:16:33.250 --> 00:16:37.480 align:middle line:84% But obviously, whatever income is, if it were spent, 00:16:37.480 --> 00:16:39.520 align:middle line:84% it would lead to expenditure, and then you 00:16:39.520 --> 00:16:40.690 align:middle line:90% could reallocate it. 00:16:40.690 --> 00:16:44.080 align:middle line:84% And some people might be consuming more than others. 00:16:44.080 --> 00:16:47.980 align:middle line:84% But here we take the total aggregate consumption as fixed. 00:16:47.980 --> 00:16:49.870 align:middle line:84% We're not going to put labor in here. 00:16:49.870 --> 00:16:51.850 align:middle line:90% There's just consumption. 00:16:51.850 --> 00:16:54.470 align:middle line:84% And we get first-order conditions. 00:16:54.470 --> 00:16:58.150 align:middle line:84% So if you pick a particular consumption for a household 00:16:58.150 --> 00:17:02.650 align:middle line:84% k at day t as a function of a particular history ht, 00:17:02.650 --> 00:17:06.010 align:middle line:84% it would appear in the w contemporary utility 00:17:06.010 --> 00:17:11.290 align:middle line:84% function with the prime for the derivative weighted by lambda k 00:17:11.290 --> 00:17:14.260 align:middle line:84% And that beta to the t probability of ht 00:17:14.260 --> 00:17:17.150 align:middle line:84% would also be over there, but we've multiplied through. 00:17:17.150 --> 00:17:20.599 align:middle line:84% So that now becomes a term in the denominator. 00:17:20.599 --> 00:17:23.109 align:middle line:84% And the numerator term here is mu. 00:17:23.109 --> 00:17:24.530 align:middle line:90% So let me go back. 00:17:24.530 --> 00:17:28.150 align:middle line:84% So the mu is the Lagrange multiplier 00:17:28.150 --> 00:17:30.560 align:middle line:90% on this resource constraint. 00:17:30.560 --> 00:17:35.320 align:middle line:84% And it really should have a mu indexed by t and the history 00:17:35.320 --> 00:17:38.230 align:middle line:84% h of t because there is a constraint like this 00:17:38.230 --> 00:17:41.540 align:middle line:84% for every single possible history end date. 00:17:41.540 --> 00:17:43.570 align:middle line:84% So there are many, many constraints. 00:17:43.570 --> 00:17:45.100 align:middle line:84% That's why I caught myself when I 00:17:45.100 --> 00:17:46.570 align:middle line:90% said subject to one constraint. 00:17:46.570 --> 00:17:47.590 align:middle line:90% That's wrong. 00:17:47.590 --> 00:17:49.480 align:middle line:84% There's only one good, but there's 00:17:49.480 --> 00:17:52.060 align:middle line:90% many dates and many states. 00:17:52.060 --> 00:17:53.800 align:middle line:90% The mu that's appearing here-- 00:17:53.800 --> 00:17:57.370 align:middle line:84% I didn't write an additional little t which 00:17:57.370 --> 00:17:58.720 align:middle line:90% ought to be there probably. 00:17:58.720 --> 00:18:02.650 align:middle line:84% But anyway, mu tilde is the normalized version 00:18:02.650 --> 00:18:04.570 align:middle line:84% after dividing through by the discount 00:18:04.570 --> 00:18:06.500 align:middle line:90% rate and the probability. 00:18:06.500 --> 00:18:09.850 align:middle line:84% And if you started with particular utility functions 00:18:09.850 --> 00:18:13.990 align:middle line:84% like one of our favorites, constant exponential utility, 00:18:13.990 --> 00:18:16.970 align:middle line:84% where sigma is the degree of risk aversion, 00:18:16.970 --> 00:18:18.550 align:middle line:90% it is of this form. 00:18:18.550 --> 00:18:20.840 align:middle line:84% Now, there is a step that's happening here 00:18:20.840 --> 00:18:24.730 align:middle line:84% which is imposing the way that these age and gender 00:18:24.730 --> 00:18:27.580 align:middle line:84% weights are entering into the utility function. 00:18:27.580 --> 00:18:31.810 align:middle line:84% Namely, it's consumption per unit index. 00:18:31.810 --> 00:18:35.950 align:middle line:84% So pick a 30-year-old male and then look at the other age 00:18:35.950 --> 00:18:37.510 align:middle line:90% groups and the other genders. 00:18:37.510 --> 00:18:39.010 align:middle line:84% And the question is whether they're 00:18:39.010 --> 00:18:43.480 align:middle line:84% consuming more or consuming less than that baseline case. 00:18:43.480 --> 00:18:47.410 align:middle line:84% So we, for short, can talk about this consumption per unit age. 00:18:47.410 --> 00:18:51.400 align:middle line:84% That's really the object that's entering into individual K's 00:18:51.400 --> 00:18:52.870 align:middle line:90% utility function. 00:18:52.870 --> 00:18:56.890 align:middle line:84% And c tilde then replaces that normalized consumption c 00:18:56.890 --> 00:18:58.000 align:middle line:90% over a. 00:18:58.000 --> 00:19:01.990 align:middle line:84% And when you do the math, you'll get the consumption formula, 00:19:01.990 --> 00:19:04.340 align:middle line:84% the risk-sharing formula, as it were. 00:19:04.340 --> 00:19:07.600 align:middle line:84% So this is the per-unit consumption 00:19:07.600 --> 00:19:10.600 align:middle line:84% of household individual K at date t. 00:19:10.600 --> 00:19:13.660 align:middle line:84% It has an intercept term, a term that 00:19:13.660 --> 00:19:15.940 align:middle line:84% depends on the demographics, and a term 00:19:15.940 --> 00:19:20.110 align:middle line:84% that depends on this Pareto, this shadow price mu. 00:19:20.110 --> 00:19:22.840 align:middle line:84% You've kind of seen this before, but let me remind you 00:19:22.840 --> 00:19:25.450 align:middle line:84% this intercept term, quite logically, has 00:19:25.450 --> 00:19:26.780 align:middle line:90% to do with the Pareto weight. 00:19:26.780 --> 00:19:29.410 align:middle line:84% The higher is the Pareto weight of individual K, 00:19:29.410 --> 00:19:31.540 align:middle line:84% the higher is the log of the Pareto weight, 00:19:31.540 --> 00:19:35.290 align:middle line:84% it's normalized by this degree of risk aversion. 00:19:35.290 --> 00:19:39.370 align:middle line:84% If we move over here to the mu term, 00:19:39.370 --> 00:19:42.460 align:middle line:84% mu is the shadow price of consumption. 00:19:42.460 --> 00:19:44.380 align:middle line:84% It's how much the objective function 00:19:44.380 --> 00:19:46.840 align:middle line:84% would increase if you increase consumption 00:19:46.840 --> 00:19:50.330 align:middle line:84% by an infinitesimally small amount in the aggregate. 00:19:50.330 --> 00:19:54.860 align:middle line:84% So as mu goes up, consumption is going down. 00:19:54.860 --> 00:19:55.750 align:middle line:90% Why is that? 00:19:55.750 --> 00:19:57.430 align:middle line:90% Mu is the shadow price. 00:19:57.430 --> 00:19:59.890 align:middle line:84% So the lower is aggregate consumption. 00:19:59.890 --> 00:20:01.880 align:middle line:84% The higher will be the shadow price. 00:20:01.880 --> 00:20:04.660 align:middle line:84% So as mu goes up, there's less consumption to go around, 00:20:04.660 --> 00:20:07.690 align:middle line:84% so individual consumption is also dropping. 00:20:07.690 --> 00:20:10.180 align:middle line:84% And we will replace this by aggregate consumption 00:20:10.180 --> 00:20:13.390 align:middle line:84% and get a positive sign momentarily. 00:20:13.390 --> 00:20:15.550 align:middle line:84% The other interesting thing about this equation 00:20:15.550 --> 00:20:18.940 align:middle line:84% is, again, this 1 over sigma which I feature in this term, 00:20:18.940 --> 00:20:21.190 align:middle line:84% although it's on all the other ones as well. 00:20:21.190 --> 00:20:25.000 align:middle line:84% And that is if individual risk aversion goes up, 00:20:25.000 --> 00:20:27.400 align:middle line:90% this coefficient is going down. 00:20:27.400 --> 00:20:31.480 align:middle line:84% So as risk aversion goes up, this individual K 00:20:31.480 --> 00:20:34.000 align:middle line:84% should not experience too much variation 00:20:34.000 --> 00:20:37.540 align:middle line:84% in his or her consumption when the aggregate is moving 00:20:37.540 --> 00:20:40.630 align:middle line:84% because other people would be in a better position 00:20:40.630 --> 00:20:43.600 align:middle line:84% to absorb the aggregate shock if they were 00:20:43.600 --> 00:20:46.330 align:middle line:84% less-risk averse than individual K. 00:20:46.330 --> 00:20:49.810 align:middle line:84% And again, you'll see that momentarily. 00:20:49.810 --> 00:20:51.660 align:middle line:90% Questions? 00:20:51.660 --> 00:20:54.260 align:middle line:84% STUDENT: My question is why there 00:20:54.260 --> 00:21:01.430 align:middle line:84% was a negative sign before the A, log A, the age and sex 00:21:01.430 --> 00:21:02.810 align:middle line:90% index. 00:21:02.810 --> 00:21:07.810 align:middle line:84% It looks like in the utility function when A goes up, 00:21:07.810 --> 00:21:09.520 align:middle line:90% the utility will go up. 00:21:09.520 --> 00:21:13.960 align:middle line:90% 00:21:13.960 --> 00:21:15.460 align:middle line:84% ROBERT TOWNSEND: When A is going up, 00:21:15.460 --> 00:21:21.810 align:middle line:84% though, consumption per unit age is going down. 00:21:21.810 --> 00:21:24.120 align:middle line:84% STUDENT: But in the utility function-- 00:21:24.120 --> 00:21:26.530 align:middle line:84% ROBERT TOWNSEND: And they're becoming more urgent, 00:21:26.530 --> 00:21:29.680 align:middle line:90% so to speak, other things equal. 00:21:29.680 --> 00:21:34.980 align:middle line:90% So I'm sorry. 00:21:34.980 --> 00:21:38.690 align:middle line:84% I think I said it right the first time. 00:21:38.690 --> 00:21:42.740 align:middle line:84% Holding aggregate consumption fixed, for example, 00:21:42.740 --> 00:21:47.330 align:middle line:84% as their index is going up, the consumption per unit index 00:21:47.330 --> 00:21:48.410 align:middle line:90% is going down. 00:21:48.410 --> 00:21:49.812 align:middle line:90% And that's the negative sign. 00:21:49.812 --> 00:21:51.020 align:middle line:90% It's really just that simple. 00:21:51.020 --> 00:21:57.400 align:middle line:90% 00:21:57.400 --> 00:21:59.090 align:middle line:90% All right. 00:21:59.090 --> 00:22:02.660 align:middle line:84% But thank you for asking the question. 00:22:02.660 --> 00:22:03.160 align:middle line:90% All right. 00:22:03.160 --> 00:22:05.860 align:middle line:84% Now, that was just one individual. 00:22:05.860 --> 00:22:10.270 align:middle line:84% We can put all the individuals together and allow explicitly 00:22:10.270 --> 00:22:12.220 align:middle line:90% for different risk aversion. 00:22:12.220 --> 00:22:14.080 align:middle line:84% And this formula appears in the paper, 00:22:14.080 --> 00:22:16.060 align:middle line:84% and I'm not going to try to derive it. 00:22:16.060 --> 00:22:19.520 align:middle line:90% It's somewhat tedious algebra. 00:22:19.520 --> 00:22:21.340 align:middle line:90% But the interpretation is easy. 00:22:21.340 --> 00:22:23.950 align:middle line:84% Namely, there's an intercept, something 00:22:23.950 --> 00:22:26.440 align:middle line:84% to do with demographics, and something 00:22:26.440 --> 00:22:29.590 align:middle line:84% to do with aggregate consumption. 00:22:29.590 --> 00:22:32.860 align:middle line:84% In a little bit more detail, this is no longer 00:22:32.860 --> 00:22:34.720 align:middle line:90% individual I consumption. 00:22:34.720 --> 00:22:38.170 align:middle line:84% It's all the individuals in a household J. 00:22:38.170 --> 00:22:40.240 align:middle line:84% And unfortunately, that's all we see in the data. 00:22:40.240 --> 00:22:44.080 align:middle line:84% We don't see individual eating apart from that dietary survey. 00:22:44.080 --> 00:22:47.170 align:middle line:84% We just see aggregate household consumption. 00:22:47.170 --> 00:22:48.970 align:middle line:84% But that functional form allowed us 00:22:48.970 --> 00:22:51.370 align:middle line:84% to sum up over all the consumptions, 00:22:51.370 --> 00:22:52.780 align:middle line:90% and that's what we see. 00:22:52.780 --> 00:22:56.990 align:middle line:84% And we also have the age and sex of all the members. 00:22:56.990 --> 00:22:59.780 align:middle line:90% So we also have the denominator. 00:22:59.780 --> 00:23:03.820 align:middle line:84% So this is household-level consumption per unit age. 00:23:03.820 --> 00:23:05.390 align:middle line:90% And that should be higher. 00:23:05.390 --> 00:23:08.320 align:middle line:84% The higher is household J's lambda weight 00:23:08.320 --> 00:23:10.960 align:middle line:84% relative to the lambda weights of the other people 00:23:10.960 --> 00:23:13.210 align:middle line:90% in the village. 00:23:13.210 --> 00:23:15.970 align:middle line:84% It's a bit more complicated because these lambda weights 00:23:15.970 --> 00:23:18.920 align:middle line:84% are weighted by the degree of risk aversion. 00:23:18.920 --> 00:23:22.630 align:middle line:84% So it's a risk-aversion-weighted version of the Lambda 00:23:22.630 --> 00:23:25.360 align:middle line:84% that we care about, comparing it to lambda 00:23:25.360 --> 00:23:28.960 align:middle line:90% K. This is the demographic. 00:23:28.960 --> 00:23:30.260 align:middle line:90% I may come back to that. 00:23:30.260 --> 00:23:33.320 align:middle line:84% This is the formula now in terms of aggregate consumption. 00:23:33.320 --> 00:23:37.840 align:middle line:84% This is the consumption of individual K and household J-- 00:23:37.840 --> 00:23:41.350 align:middle line:84% I, actually-- but then summing over all households I. 00:23:41.350 --> 00:23:44.410 align:middle line:84% So this is aggregate consumption per unit age 00:23:44.410 --> 00:23:45.460 align:middle line:90% in the whole village. 00:23:45.460 --> 00:23:49.330 align:middle line:84% And again, you can see quite explicitly that J's consumption 00:23:49.330 --> 00:23:52.390 align:middle line:84% will move with that when sigma J is going down, 00:23:52.390 --> 00:23:55.270 align:middle line:84% or the other way around-- when household J is very risk 00:23:55.270 --> 00:23:57.880 align:middle line:84% averse, this coefficient is going to 0, 00:23:57.880 --> 00:23:59.800 align:middle line:84% and individual consumption would not 00:23:59.800 --> 00:24:01.630 align:middle line:90% very much with the aggregate. 00:24:01.630 --> 00:24:05.530 align:middle line:84% That's the aggregate allocation of risk bearing. 00:24:05.530 --> 00:24:09.100 align:middle line:84% Because the theory has those age and gender weights in it, 00:24:09.100 --> 00:24:10.750 align:middle line:90% you can treat those as shocks. 00:24:10.750 --> 00:24:13.660 align:middle line:84% Even though they're quite not moving in a surprise way, 00:24:13.660 --> 00:24:15.042 align:middle line:90% they are moving over time. 00:24:15.042 --> 00:24:17.500 align:middle line:84% They're moving intertemporally, and maybe the whole village 00:24:17.500 --> 00:24:18.730 align:middle line:90% is getting older. 00:24:18.730 --> 00:24:23.230 align:middle line:84% Maybe one household is moving across metabolic categories 00:24:23.230 --> 00:24:25.460 align:middle line:84% faster than the other households are. 00:24:25.460 --> 00:24:29.110 align:middle line:84% And so the formula allows for that adjustment, 00:24:29.110 --> 00:24:32.420 align:middle line:84% comparing household j to the other ones. 00:24:32.420 --> 00:24:37.330 align:middle line:84% So let's look at an empirical implication, a linear equation. 00:24:37.330 --> 00:24:41.440 align:middle line:84% The aggregate consumption, age weighted, of household 00:24:41.440 --> 00:24:44.410 align:middle line:90% J has an intercept. 00:24:44.410 --> 00:24:46.840 align:middle line:84% It depends on aggregate consumption. 00:24:46.840 --> 00:24:50.890 align:middle line:84% It depends on the index of household J 00:24:50.890 --> 00:24:53.390 align:middle line:90% and maybe on other things. 00:24:53.390 --> 00:24:56.410 align:middle line:90% This is just a version of this. 00:24:56.410 --> 00:24:58.570 align:middle line:84% Household J's consumption depends 00:24:58.570 --> 00:25:02.440 align:middle line:84% on an intercept, its age, and aggregate consumption. 00:25:02.440 --> 00:25:04.960 align:middle line:84% It's just much easier to write this way. 00:25:04.960 --> 00:25:07.300 align:middle line:90% And the theory has implications. 00:25:07.300 --> 00:25:10.510 align:middle line:84% Namely, if everyone had the same risk aversion, 00:25:10.510 --> 00:25:14.980 align:middle line:84% then this beta J on aggregate consumption would be 1. 00:25:14.980 --> 00:25:16.600 align:middle line:84% You can convince yourself this would 00:25:16.600 --> 00:25:21.550 align:middle line:84% be 1 because the sigmas are all alike, for example. 00:25:21.550 --> 00:25:25.720 align:middle line:84% Likewise, this demographic thing would have a coefficient of 1 00:25:25.720 --> 00:25:27.730 align:middle line:90% over sigma, the common sigma. 00:25:27.730 --> 00:25:29.320 align:middle line:90% And this other stuff-- 00:25:29.320 --> 00:25:31.610 align:middle line:90% I saved the best for last. 00:25:31.610 --> 00:25:35.140 align:middle line:84% This is any other variable, including household income. 00:25:35.140 --> 00:25:39.790 align:middle line:84% And the theory is saying that this coefficient should be 0. 00:25:39.790 --> 00:25:42.520 align:middle line:84% Now, that's very counterintuitive. 00:25:42.520 --> 00:25:46.270 align:middle line:84% How could I possibly be claiming that household consumption 00:25:46.270 --> 00:25:49.580 align:middle line:84% doesn't depend on household income? 00:25:49.580 --> 00:25:52.950 align:middle line:84% What about marginal propensity to consume and all of that? 00:25:52.950 --> 00:25:56.900 align:middle line:84% Well, the answer is they're pooling their incomes together 00:25:56.900 --> 00:26:00.320 align:middle line:84% as if in a mutual fund and then redistributing it 00:26:00.320 --> 00:26:02.600 align:middle line:84% as a function of the Pareto weights. 00:26:02.600 --> 00:26:06.270 align:middle line:84% So household J's income is kind of in there, 00:26:06.270 --> 00:26:08.630 align:middle line:84% but it's in the pooled version, and it shows up 00:26:08.630 --> 00:26:09.890 align:middle line:90% in aggregate consumption. 00:26:09.890 --> 00:26:13.860 align:middle line:84% And if you control for that, then there's nothing leftover. 00:26:13.860 --> 00:26:18.440 align:middle line:84% There's no idiosyncratic risk for household J to bear. 00:26:18.440 --> 00:26:20.360 align:middle line:84% Another way to say that is we're going 00:26:20.360 --> 00:26:22.670 align:middle line:84% to get perfect consumption insurance 00:26:22.670 --> 00:26:25.160 align:middle line:84% because, like a large mutual fund, 00:26:25.160 --> 00:26:28.690 align:middle line:84% all the individual shocks go to 0. 00:26:28.690 --> 00:26:32.980 align:middle line:84% A version of this in China that I'm becoming aware of-- 00:26:32.980 --> 00:26:36.310 align:middle line:84% Alibaba runs a mutual fund over health shocks. 00:26:36.310 --> 00:26:38.650 align:middle line:84% There's 100 million households joining, 00:26:38.650 --> 00:26:40.930 align:middle line:90% and they pay ex-post premia. 00:26:40.930 --> 00:26:45.820 align:middle line:84% Typical medical expense must be as high as $20,000. 00:26:45.820 --> 00:26:50.750 align:middle line:84% How much do they pay in? $9 in yuan equivalent. 00:26:50.750 --> 00:26:54.290 align:middle line:84% So when you spread that damage of an individual household, 00:26:54.290 --> 00:26:57.470 align:middle line:84% which is high, over all the hundreds 00:26:57.470 --> 00:27:00.980 align:middle line:84% of thousands of households, it almost goes to 0. 00:27:00.980 --> 00:27:04.130 align:middle line:84% And this is the extreme version of perfect pooling. 00:27:04.130 --> 00:27:09.440 align:middle line:84% That's contemporary China, not a village in India in 1985. 00:27:09.440 --> 00:27:12.440 align:middle line:84% So we can test the theory by looking at each household one 00:27:12.440 --> 00:27:15.380 align:middle line:84% at a time over the 10 years, and we can test the theory 00:27:15.380 --> 00:27:17.870 align:middle line:84% by pooling households together into one 00:27:17.870 --> 00:27:21.650 align:middle line:84% big cross-sectional time-dependent regression. 00:27:21.650 --> 00:27:25.400 align:middle line:84% This is a little hard to see, but what's going on here 00:27:25.400 --> 00:27:29.990 align:middle line:84% statistically is testing these coefficients xe, which 00:27:29.990 --> 00:27:33.710 align:middle line:84% should be 0, to see whether they actually are in practice. 00:27:33.710 --> 00:27:37.010 align:middle line:84% Under the null hypothesis that it's 0, 00:27:37.010 --> 00:27:40.190 align:middle line:84% can you reject in a statistically significant way 00:27:40.190 --> 00:27:43.010 align:middle line:84% that it's greater than 0 or less than 0? 00:27:43.010 --> 00:27:47.390 align:middle line:84% And for most households, the bulk of the households, 00:27:47.390 --> 00:27:50.330 align:middle line:84% you can't reject much one way or the other. 00:27:50.330 --> 00:27:52.340 align:middle line:84% There's just a handful of households 00:27:52.340 --> 00:27:54.590 align:middle line:90% on either side of the null. 00:27:54.590 --> 00:27:56.540 align:middle line:90% This test has power. 00:27:56.540 --> 00:27:59.300 align:middle line:84% This thing is, oh, well, maybe it's just noisy data, 00:27:59.300 --> 00:28:01.490 align:middle line:84% and we can't infer much of anything. 00:28:01.490 --> 00:28:03.410 align:middle line:84% Maybe they're literally hand-to-mouth, 00:28:03.410 --> 00:28:06.060 align:middle line:84% and they're eating entirely their income, in which case 00:28:06.060 --> 00:28:08.210 align:middle line:90% the coefficient would be 1. 00:28:08.210 --> 00:28:12.990 align:middle line:84% And we can't reject that for half of them. 00:28:12.990 --> 00:28:15.770 align:middle line:84% But for the other half, we can statistically 00:28:15.770 --> 00:28:18.320 align:middle line:90% reject that it's 1. 00:28:18.320 --> 00:28:20.990 align:middle line:84% So there is a lot of smoothing going on. 00:28:20.990 --> 00:28:24.950 align:middle line:84% And this is for the other two villages down here. 00:28:24.950 --> 00:28:28.970 align:middle line:84% Now let's talk about pooling it together. 00:28:28.970 --> 00:28:32.040 align:middle line:84% Actually, let's just jump here, and then I'll go back. 00:28:32.040 --> 00:28:33.290 align:middle line:90% So here's the villages-- 00:28:33.290 --> 00:28:36.050 align:middle line:90% Aurepalle, Shirapur, Kandara. 00:28:36.050 --> 00:28:39.830 align:middle line:84% And we're going to somehow take each household over time 00:28:39.830 --> 00:28:42.920 align:middle line:84% and take households in the cross-section 00:28:42.920 --> 00:28:45.320 align:middle line:84% and come up with a coefficient on how much 00:28:45.320 --> 00:28:50.840 align:middle line:84% consumption is moving on average with household income. 00:28:50.840 --> 00:28:52.750 align:middle line:90% And the answer is roughly-- 00:28:52.750 --> 00:28:54.040 align:middle line:90% should be in rupees-- 00:28:54.040 --> 00:28:56.810 align:middle line:90% $0.07 for the dollar. 00:28:56.810 --> 00:28:59.860 align:middle line:84% So for every dollar change in income, 00:28:59.860 --> 00:29:04.060 align:middle line:84% consumption is moving by only $0.07. 00:29:04.060 --> 00:29:05.050 align:middle line:90% It's positive. 00:29:05.050 --> 00:29:07.060 align:middle line:90% It's statistically significant. 00:29:07.060 --> 00:29:09.380 align:middle line:90% We now reject the model. 00:29:09.380 --> 00:29:11.650 align:middle line:84% But as an approximation, we're doing 00:29:11.650 --> 00:29:14.920 align:middle line:84% quite well because these coefficients are small. 00:29:14.920 --> 00:29:18.160 align:middle line:84% So most, although not all, of the 00:29:18.160 --> 00:29:22.150 align:middle line:84% household-specific idiosyncratic risk is being pooled. 00:29:22.150 --> 00:29:25.660 align:middle line:84% And you can peruse this table and see-- 00:29:25.660 --> 00:29:30.820 align:middle line:84% this is profits as opposed to all income, labor income, 00:29:30.820 --> 00:29:33.230 align:middle line:84% profits from trade and handicraft. 00:29:33.230 --> 00:29:35.420 align:middle line:84% You can look for the high numbers. 00:29:35.420 --> 00:29:37.990 align:middle line:84% So there are certain activities in certain villages 00:29:37.990 --> 00:29:40.480 align:middle line:84% that look relatively under-insured. 00:29:40.480 --> 00:29:43.440 align:middle line:84% This is who's the most vulnerable. 00:29:43.440 --> 00:29:45.870 align:middle line:84% If we had the ability to come in externally 00:29:45.870 --> 00:29:48.270 align:middle line:84% and introduce a better insurance product, 00:29:48.270 --> 00:29:51.330 align:middle line:84% we would want to offer it to households that need it. 00:29:51.330 --> 00:29:54.720 align:middle line:84% Or if no one needed it, we would expect take-up to be zero. 00:29:54.720 --> 00:30:00.090 align:middle line:84% So labor incomes had a pretty high coefficient in Shirapur 00:30:00.090 --> 00:30:00.820 align:middle line:90% and so on. 00:30:00.820 --> 00:30:03.400 align:middle line:90% But a lot of these are 0-- 00:30:03.400 --> 00:30:06.430 align:middle line:84% that is to say, not statistically different from 0. 00:30:06.430 --> 00:30:09.910 align:middle line:84% So in that sense, it's a good approximation. 00:30:09.910 --> 00:30:14.470 align:middle line:84% Remember, according to Lucas, models are abstractions. 00:30:14.470 --> 00:30:18.040 align:middle line:84% They're meant to be stylized versions of reality. 00:30:18.040 --> 00:30:20.860 align:middle line:84% Because we have a model, we have an exact prediction 00:30:20.860 --> 00:30:22.810 align:middle line:90% of what we should expect to see. 00:30:22.810 --> 00:30:24.430 align:middle line:84% And through the lens of the model, 00:30:24.430 --> 00:30:28.420 align:middle line:84% we can decide how reality is doing relative to the model. 00:30:28.420 --> 00:30:30.520 align:middle line:84% So I'm not overly alarmed that we're going 00:30:30.520 --> 00:30:32.920 align:middle line:90% to get statistical rejections. 00:30:32.920 --> 00:30:35.770 align:middle line:84% The question is, is it a useful starting point 00:30:35.770 --> 00:30:37.610 align:middle line:90% as an approximation? 00:30:37.610 --> 00:30:40.330 align:middle line:90% And the answer seems to be yes. 00:30:40.330 --> 00:30:43.630 align:middle line:84% I will share with you that when I wrote this paper, 00:30:43.630 --> 00:30:46.540 align:middle line:84% it kind of created a firestorm of alarm 00:30:46.540 --> 00:30:49.900 align:middle line:84% among policymakers who were astounded 00:30:49.900 --> 00:30:54.400 align:middle line:84% that poor households in villages in the caste system, and so on, 00:30:54.400 --> 00:30:59.110 align:middle line:84% could ever possibly be achieving anything close to an optimum. 00:30:59.110 --> 00:31:03.340 align:middle line:84% It just went against the grain of their prior presumptions. 00:31:03.340 --> 00:31:05.540 align:middle line:90% But it doesn't fit perfectly. 00:31:05.540 --> 00:31:08.440 align:middle line:84% So this slide which I deliberately skipped over just 00:31:08.440 --> 00:31:11.680 align:middle line:84% now is to say, look, maybe I'm getting 00:31:11.680 --> 00:31:15.790 align:middle line:84% into statistical stuff a bit, and we can follow up later. 00:31:15.790 --> 00:31:18.760 align:middle line:84% One reason why this coefficient might be low 00:31:18.760 --> 00:31:22.960 align:middle line:84% is because this variable x is very poorly measured. 00:31:22.960 --> 00:31:24.960 align:middle line:84% Say it just got a lot of measurement error 00:31:24.960 --> 00:31:26.200 align:middle line:90% that is not real. 00:31:26.200 --> 00:31:29.940 align:middle line:84% Then it's moving around a lot for spurious reasons, 00:31:29.940 --> 00:31:32.530 align:middle line:84% and you would not expect that to influence consumption. 00:31:32.530 --> 00:31:36.090 align:middle line:84% So that biases this coefficient to something close to 0. 00:31:36.090 --> 00:31:40.050 align:middle line:84% Fortunately, we can correct for it with the panel. 00:31:40.050 --> 00:31:43.260 align:middle line:84% And it's putting explicitly this measurement error 00:31:43.260 --> 00:31:47.430 align:middle line:84% into, in this case, idiosyncratic income 00:31:47.430 --> 00:31:51.330 align:middle line:84% and then substituting that into those equations. 00:31:51.330 --> 00:31:54.870 align:middle line:84% We now see a correlation between the measurement error 00:31:54.870 --> 00:31:57.730 align:middle line:84% and the measured version of income, 00:31:57.730 --> 00:32:00.660 align:middle line:84% which means you can't run an OLS regression. 00:32:00.660 --> 00:32:03.000 align:middle line:84% But on the other hand, another way to do this 00:32:03.000 --> 00:32:05.100 align:middle line:84% is just to take first differences. 00:32:05.100 --> 00:32:07.530 align:middle line:84% Differences over time over households 00:32:07.530 --> 00:32:08.940 align:middle line:90% gives you another version. 00:32:08.940 --> 00:32:12.520 align:middle line:84% And it turns out this within coefficient 00:32:12.520 --> 00:32:17.880 align:middle line:84% and this across-time coefficient allow you to essentially get 00:32:17.880 --> 00:32:19.150 align:middle line:90% rid of the measurement error. 00:32:19.150 --> 00:32:23.580 align:middle line:84% So that's this IV Griliches-Hausman thing, 00:32:23.580 --> 00:32:26.340 align:middle line:90% which I'm not going to go over. 00:32:26.340 --> 00:32:30.210 align:middle line:84% I still don't know what the best strategy is for me. 00:32:30.210 --> 00:32:33.870 align:middle line:84% If I skip stuff, then it looks like it came out of nowhere, 00:32:33.870 --> 00:32:35.460 align:middle line:84% and you don't really understand it. 00:32:35.460 --> 00:32:38.790 align:middle line:84% Hence, I try in all these slides to show you 00:32:38.790 --> 00:32:40.480 align:middle line:90% the steps that are being taken. 00:32:40.480 --> 00:32:42.990 align:middle line:84% I'm deliberately not concealing something from you. 00:32:42.990 --> 00:32:45.870 align:middle line:84% But on the other hand, this particular sequence 00:32:45.870 --> 00:32:49.230 align:middle line:84% involves statistical material you may not have seen 00:32:49.230 --> 00:32:51.040 align:middle line:90% something similar to before. 00:32:51.040 --> 00:32:54.400 align:middle line:84% So if it's still quite opaque, don't worry too much about it. 00:32:54.400 --> 00:32:56.580 align:middle line:90% I'm just trying to be clearer. 00:32:56.580 --> 00:32:57.210 align:middle line:90% OK. 00:32:57.210 --> 00:33:00.960 align:middle line:84% So that's summary, risk and insurance and village India. 00:33:00.960 --> 00:33:03.010 align:middle line:84% These villages are doing reasonably well, 00:33:03.010 --> 00:33:04.260 align:middle line:90% although not perfectly. 00:33:04.260 --> 00:33:06.340 align:middle line:84% They're sharing a lot of the risk. 00:33:06.340 --> 00:33:08.490 align:middle line:84% The reason that consumption looks so flat 00:33:08.490 --> 00:33:12.600 align:middle line:84% relative to income is because they're actively, somehow 00:33:12.600 --> 00:33:15.420 align:middle line:84% or another, engaged in borrowing and lending 00:33:15.420 --> 00:33:19.900 align:middle line:84% or transfers to bridge the gap between consumption and income. 00:33:19.900 --> 00:33:22.990 align:middle line:84% Now, another way they could handle the situation-- 00:33:22.990 --> 00:33:26.370 align:middle line:84% so for this, we're going to jump to medieval villages-- 00:33:26.370 --> 00:33:30.060 align:middle line:84% is to potentially divide up the land. 00:33:30.060 --> 00:33:32.550 align:middle line:84% So this is the diversification part, 00:33:32.550 --> 00:33:35.040 align:middle line:84% whether land holdings of a typical village 00:33:35.040 --> 00:33:38.250 align:middle line:84% would be consistent with the optimal allocation of risk 00:33:38.250 --> 00:33:41.940 align:middle line:84% predicted in that model just now on the premise 00:33:41.940 --> 00:33:43.980 align:middle line:84% that what you get is what you eat. 00:33:43.980 --> 00:33:45.330 align:middle line:90% You have these strips. 00:33:45.330 --> 00:33:47.040 align:middle line:90% The strips are subject to risk. 00:33:47.040 --> 00:33:49.300 align:middle line:84% You add it all up across all your land holdings, 00:33:49.300 --> 00:33:50.100 align:middle line:90% then you eat it. 00:33:50.100 --> 00:33:51.840 align:middle line:90% So this is the other extreme-- 00:33:51.840 --> 00:33:55.590 align:middle line:84% no expo smoothing, no credit markets. 00:33:55.590 --> 00:33:59.580 align:middle line:84% Even going back to lecture 1 or something where I presented 00:33:59.580 --> 00:34:01.260 align:middle line:84% all the different economies-- we talked 00:34:01.260 --> 00:34:03.150 align:middle line:84% about the medieval village economy. 00:34:03.150 --> 00:34:06.870 align:middle line:84% We're about to do it now in terms of dividing up the land. 00:34:06.870 --> 00:34:09.120 align:middle line:84% We also talked about Thai village economy 00:34:09.120 --> 00:34:12.840 align:middle line:84% with that temple where households did not diversify. 00:34:12.840 --> 00:34:13.860 align:middle line:90% Some had low land. 00:34:13.860 --> 00:34:15.360 align:middle line:90% Some had high land. 00:34:15.360 --> 00:34:17.250 align:middle line:84% They made donations to the temple. 00:34:17.250 --> 00:34:20.909 align:middle line:84% And people that had a bad year, God gives back. 00:34:20.909 --> 00:34:26.909 align:middle line:84% So the second thing is what's going on in the first, 00:34:26.909 --> 00:34:29.880 align:middle line:84% some mechanism for risk sharing ex-post. 00:34:29.880 --> 00:34:32.429 align:middle line:84% This is the mechanism in the second, which 00:34:32.429 --> 00:34:35.310 align:middle line:90% is ex-ante division of land. 00:34:35.310 --> 00:34:37.409 align:middle line:84% I said that in reverse order, but anyway. 00:34:37.409 --> 00:34:40.380 align:middle line:84% So you've seen this picture twice before. 00:34:40.380 --> 00:34:43.020 align:middle line:90% This is Elford Stratfordshire. 00:34:43.020 --> 00:34:46.320 align:middle line:84% Mr. Darlaston's land is shaded in black, 00:34:46.320 --> 00:34:52.969 align:middle line:84% and he's got something like 50 strips throughout the village. 00:34:52.969 --> 00:34:55.250 align:middle line:90% So why were they doing that-- 00:34:55.250 --> 00:34:58.980 align:middle line:84% not just Mr. Darlaston, but all of them to some degree? 00:34:58.980 --> 00:35:03.870 align:middle line:84% So if you think about this map, for example, 00:35:03.870 --> 00:35:08.200 align:middle line:84% are these types of land differ in some way 00:35:08.200 --> 00:35:12.340 align:middle line:84% from these types of land in terms of soil or slope 00:35:12.340 --> 00:35:13.750 align:middle line:90% or elevation? 00:35:13.750 --> 00:35:16.720 align:middle line:84% Or alternatively, land is all alike. 00:35:16.720 --> 00:35:20.740 align:middle line:84% And then the question is, when the storm comes and destroys 00:35:20.740 --> 00:35:24.760 align:middle line:84% crops, it could destroy these crops and not these. 00:35:24.760 --> 00:35:26.770 align:middle line:84% So there's really two different models, 00:35:26.770 --> 00:35:28.930 align:middle line:84% and we're only going to have time to do one, 00:35:28.930 --> 00:35:30.310 align:middle line:84% although the other one is covered 00:35:30.310 --> 00:35:33.080 align:middle line:84% in that missing medieval village chapter. 00:35:33.080 --> 00:35:36.490 align:middle line:84% We're going to do uniform shocks on non-uniform land, 00:35:36.490 --> 00:35:41.720 align:middle line:84% and the non-uniform shocks we'll save for another time. 00:35:41.720 --> 00:35:43.820 align:middle line:90% So there are two types of land. 00:35:43.820 --> 00:35:45.570 align:middle line:90% Start off keeping this simple. 00:35:45.570 --> 00:35:49.400 align:middle line:90% Say high-low or clay-sandy. 00:35:49.400 --> 00:35:53.450 align:middle line:84% There's only two households, another abstraction. 00:35:53.450 --> 00:35:55.130 align:middle line:90% And there's only one date. 00:35:55.130 --> 00:35:59.810 align:middle line:84% And land type k has a yield vector which 00:35:59.810 --> 00:36:02.180 align:middle line:90% varies with the shock epsilon. 00:36:02.180 --> 00:36:05.280 align:middle line:84% And epsilon can take on capital S values. 00:36:05.280 --> 00:36:08.820 align:middle line:84% So this epsilon is a measure of rainfall, temperature, 00:36:08.820 --> 00:36:12.260 align:middle line:84% humidity, or other events which have 00:36:12.260 --> 00:36:15.410 align:middle line:84% beneficial or adverse consequences for the yield 00:36:15.410 --> 00:36:19.700 align:middle line:84% as captured by the way which the yield of type k land 00:36:19.700 --> 00:36:21.680 align:middle line:90% varies with those shocks. 00:36:21.680 --> 00:36:24.320 align:middle line:84% You've, again, seen this before a bit. 00:36:24.320 --> 00:36:28.640 align:middle line:84% When we did the dynamics, we planted land with seed 00:36:28.640 --> 00:36:33.150 align:middle line:84% and talked about how the yield could vary with low, medium, 00:36:33.150 --> 00:36:33.910 align:middle line:90% and high shock. 00:36:33.910 --> 00:36:37.500 align:middle line:84% So this is very similar notation. 00:36:37.500 --> 00:36:40.340 align:middle line:84% Now, let's be serious even though we're 00:36:40.340 --> 00:36:41.810 align:middle line:90% trying to keep this simple. 00:36:41.810 --> 00:36:46.180 align:middle line:84% What numbers do we use for this vector? 00:36:46.180 --> 00:36:49.990 align:middle line:84% One thing we want to capture is how much 00:36:49.990 --> 00:36:53.710 align:middle line:84% that yield is varying as measured by the coefficient 00:36:53.710 --> 00:36:54.910 align:middle line:90% of variation. 00:36:54.910 --> 00:36:58.745 align:middle line:84% And in medieval villages, it was about 0.35, 00:36:58.745 --> 00:37:00.370 align:middle line:84% which we've covered before, and I'm not 00:37:00.370 --> 00:37:03.410 align:middle line:84% going to go back and dig up those slides for you. 00:37:03.410 --> 00:37:05.140 align:middle line:84% But that was the number we used before. 00:37:05.140 --> 00:37:07.120 align:middle line:90% The other one is-- 00:37:07.120 --> 00:37:11.260 align:middle line:84% there's two types of land here, k equal 1 or 2. 00:37:11.260 --> 00:37:15.310 align:middle line:84% And we want to choose vectors of returns for both. 00:37:15.310 --> 00:37:17.440 align:middle line:84% So we want to capture the covariance 00:37:17.440 --> 00:37:19.900 align:middle line:84% in the two types of land, how different are they 00:37:19.900 --> 00:37:21.560 align:middle line:90% from one another. 00:37:21.560 --> 00:37:24.820 align:middle line:84% And again, going back in the earlier lecture, the estates 00:37:24.820 --> 00:37:30.040 align:middle line:84% of the Bishop of Winchester, we had a cross-space correlation 00:37:30.040 --> 00:37:31.280 align:middle line:90% of 0.6. 00:37:31.280 --> 00:37:34.840 align:middle line:84% So we're going to take those numbers, .6 and .347, 00:37:34.840 --> 00:37:36.640 align:middle line:90% as givens. 00:37:36.640 --> 00:37:40.570 align:middle line:84% But here's a simple representation. 00:37:40.570 --> 00:37:42.340 align:middle line:84% Let's take, again, only two types 00:37:42.340 --> 00:37:48.460 align:middle line:84% of land, 1 or 2, only three events, epsilon equal 1, 2, 3. 00:37:48.460 --> 00:37:53.800 align:middle line:84% Type 1 land has the highest return in event 1, then medium, 00:37:53.800 --> 00:37:54.880 align:middle line:90% then low. 00:37:54.880 --> 00:37:59.140 align:middle line:84% Type 2 land has the highest return 00:37:59.140 --> 00:38:03.940 align:middle line:84% at event 2 surrounded by the lower numbers, 3 and 2. 00:38:03.940 --> 00:38:09.400 align:middle line:84% So if you aggregated up the total yield over all types, 00:38:09.400 --> 00:38:11.500 align:middle line:84% it would just be the sum of those columns. 00:38:11.500 --> 00:38:14.690 align:middle line:90% And it would go 12, 10, 7. 00:38:14.690 --> 00:38:18.680 align:middle line:84% Now suppose we divided up the types of land 00:38:18.680 --> 00:38:20.480 align:middle line:90% in a very extreme way. 00:38:20.480 --> 00:38:24.140 align:middle line:84% We let household 1 have all of type 1 land 00:38:24.140 --> 00:38:27.170 align:middle line:84% and household two all of type 2 land. 00:38:27.170 --> 00:38:29.840 align:middle line:84% And the question for you is whether that would be 00:38:29.840 --> 00:38:32.180 align:middle line:90% an optimal allocation of land. 00:38:32.180 --> 00:38:35.030 align:middle line:84% If we did it that way and they were forced 00:38:35.030 --> 00:38:37.220 align:middle line:84% to eat the grain off their strips, 00:38:37.220 --> 00:38:39.350 align:middle line:84% would the consumption that we see 00:38:39.350 --> 00:38:42.960 align:middle line:84% correspond to the implications of the model? 00:38:42.960 --> 00:38:47.090 align:middle line:84% And the answer is going to be no because the aggregate is going 00:38:47.090 --> 00:38:50.280 align:middle line:90% down as epsilon is going up. 00:38:50.280 --> 00:38:52.340 align:middle line:84% And that's true of household 1's consumption, 00:38:52.340 --> 00:38:55.130 align:middle line:84% but it is not true of household 2's consumption. 00:38:55.130 --> 00:38:59.060 align:middle line:84% This proposal fails the monotonicity requirement 00:38:59.060 --> 00:39:02.300 align:middle line:84% of the theory that as aggregate consumption 00:39:02.300 --> 00:39:04.430 align:middle line:84% goes down everybody's consumption ought 00:39:04.430 --> 00:39:06.950 align:middle line:84% to go down, although some people's consumption go 00:39:06.950 --> 00:39:08.940 align:middle line:90% down more than others. 00:39:08.940 --> 00:39:12.800 align:middle line:84% So we reject this proposal for the allocation of land 00:39:12.800 --> 00:39:16.010 align:middle line:84% if the goal is to achieve an optimal allocation of risk 00:39:16.010 --> 00:39:19.250 align:middle line:90% bearing by land division alone. 00:39:19.250 --> 00:39:22.650 align:middle line:84% Well, there are other ways to do it. 00:39:22.650 --> 00:39:27.080 align:middle line:84% Let's take the land type and give household J 00:39:27.080 --> 00:39:30.500 align:middle line:84% a proportional number of shares in the output 00:39:30.500 --> 00:39:32.870 align:middle line:90% of each type of land. 00:39:32.870 --> 00:39:35.750 align:middle line:84% So that means household J would get 00:39:35.750 --> 00:39:39.980 align:middle line:84% fraction alpha J of the output over each type of land. 00:39:39.980 --> 00:39:43.050 align:middle line:84% So we're kind of re-dividing things here. 00:39:43.050 --> 00:39:45.800 align:middle line:84% We're going to have household 1 getting 00:39:45.800 --> 00:39:50.060 align:middle line:84% some proportion of this column but also some proportion, 00:39:50.060 --> 00:39:53.630 align:middle line:84% not necessarily-- yes, in this case, the same 00:39:53.630 --> 00:39:56.190 align:middle line:90% of the other type of land. 00:39:56.190 --> 00:39:59.600 align:middle line:84% So this alpha J doesn't have a K on it. 00:39:59.600 --> 00:40:01.850 align:middle line:84% The shares depend only on the household, 00:40:01.850 --> 00:40:03.740 align:middle line:90% not on the type of land. 00:40:03.740 --> 00:40:08.350 align:middle line:84% The shares can depend on the household. 00:40:08.350 --> 00:40:08.850 align:middle line:90% All right. 00:40:08.850 --> 00:40:12.410 align:middle line:90% So it's linear in the aggregate. 00:40:12.410 --> 00:40:14.980 align:middle line:84% If we were over at school, you'd come up to my office, 00:40:14.980 --> 00:40:16.450 align:middle line:84% and you'll see my bookshelf filled 00:40:16.450 --> 00:40:21.130 align:middle line:84% with stacks of books devoted to medieval English villages. 00:40:21.130 --> 00:40:24.400 align:middle line:84% I had a happy time doing a lot of research. 00:40:24.400 --> 00:40:27.520 align:middle line:84% And you'd discover these amazing things. 00:40:27.520 --> 00:40:29.380 align:middle line:84% Holdings were scattered over fields 00:40:29.380 --> 00:40:33.010 align:middle line:84% so that each household partook in equal proportion 00:40:33.010 --> 00:40:36.130 align:middle line:90% of fertilizer and poor soils. 00:40:36.130 --> 00:40:37.780 align:middle line:84% Dividing up the land in accordance 00:40:37.780 --> 00:40:39.970 align:middle line:84% with drainage and exposure, resources 00:40:39.970 --> 00:40:41.920 align:middle line:90% were divided into shares. 00:40:41.920 --> 00:40:44.170 align:middle line:84% One man might have more shares than another, 00:40:44.170 --> 00:40:48.150 align:middle line:84% but the shares themselves were equal. 00:40:48.150 --> 00:40:51.090 align:middle line:84% Now, there are potential problems. 00:40:51.090 --> 00:40:52.870 align:middle line:84% For the theory to fit exactly, we 00:40:52.870 --> 00:40:55.500 align:middle line:84% got to worry a little bit about the intercepts 00:40:55.500 --> 00:40:59.580 align:middle line:84% because those divisions did not allow the intercepts to be 00:40:59.580 --> 00:41:00.810 align:middle line:90% other than 0. 00:41:00.810 --> 00:41:05.070 align:middle line:84% If you go back to where we first introduced risk sharing 00:41:05.070 --> 00:41:08.310 align:middle line:84% and looked at particular utility functions, 00:41:08.310 --> 00:41:11.730 align:middle line:84% I showed you, as example of the risk-sharing rules, 00:41:11.730 --> 00:41:14.400 align:middle line:84% two equations-- constant relative risk 00:41:14.400 --> 00:41:17.460 align:middle line:84% aversion, constant absolute risk aversion. 00:41:17.460 --> 00:41:22.200 align:middle line:84% And the intercept was 0 under these various types 00:41:22.200 --> 00:41:22.960 align:middle line:90% of conditions. 00:41:22.960 --> 00:41:25.650 align:middle line:84% So it's not like they can't be 0, 00:41:25.650 --> 00:41:28.440 align:middle line:84% but it does require certain parameters 00:41:28.440 --> 00:41:31.240 align:middle line:84% to hold in order for that to be the case. 00:41:31.240 --> 00:41:34.280 align:middle line:84% So a little bit of a qualification. 00:41:34.280 --> 00:41:36.870 align:middle line:90% But all is not lost. 00:41:36.870 --> 00:41:39.210 align:middle line:90% We can even do more. 00:41:39.210 --> 00:41:43.260 align:middle line:84% And namely, we don't have to have a given household having 00:41:43.260 --> 00:41:47.740 align:middle line:84% the same number of shares in every type of land. 00:41:47.740 --> 00:41:51.100 align:middle line:84% And that's going to allow various kinds of intercepts 00:41:51.100 --> 00:41:53.330 align:middle line:90% and so on. 00:41:53.330 --> 00:41:56.470 align:middle line:90% So here's the idea finally. 00:41:56.470 --> 00:42:00.250 align:middle line:84% Suppose there are at least as many land types 00:42:00.250 --> 00:42:04.930 align:middle line:84% with independent return vectors as there are states of nature. 00:42:04.930 --> 00:42:07.380 align:middle line:84% So what I showed you just now violated that. 00:42:07.380 --> 00:42:11.430 align:middle line:84% There were three states of nature and two types of land. 00:42:11.430 --> 00:42:14.480 align:middle line:84% But if we had three states of nature, 00:42:14.480 --> 00:42:17.150 align:middle line:84% we'd have to have three types of land 00:42:17.150 --> 00:42:18.920 align:middle line:90% with independent return streams. 00:42:18.920 --> 00:42:22.435 align:middle line:84% And likewise, s types of land, we need 00:42:22.435 --> 00:42:26.160 align:middle line:90% s dimensional return vectors. 00:42:26.160 --> 00:42:30.000 align:middle line:84% So one column cannot be derived as a linear combination 00:42:30.000 --> 00:42:31.830 align:middle line:90% of the other columns. 00:42:31.830 --> 00:42:34.530 align:middle line:84% Now let's solve the risk-sharing problem 00:42:34.530 --> 00:42:37.830 align:middle line:84% and take this consumption to be the target, this consumption 00:42:37.830 --> 00:42:40.230 align:middle line:90% of household J in state epsilon. 00:42:40.230 --> 00:42:43.320 align:middle line:84% And we're going to do that, obviously, over all states 00:42:43.320 --> 00:42:48.090 align:middle line:84% epsilon and all households j, their little n of them. 00:42:48.090 --> 00:42:50.760 align:middle line:90% We get an equation of this form. 00:42:50.760 --> 00:42:52.410 align:middle line:84% So don't get lost in the notation. 00:42:52.410 --> 00:42:53.730 align:middle line:90% I'll help. 00:42:53.730 --> 00:42:57.840 align:middle line:84% This is consumption of household J in state 1, consumption 00:42:57.840 --> 00:43:03.830 align:middle line:84% of household J in state 2, likewise in state capital S. 00:43:03.830 --> 00:43:05.150 align:middle line:90% That's a target. 00:43:05.150 --> 00:43:08.310 align:middle line:84% And just imagine it solved the Pareto problem. 00:43:08.310 --> 00:43:10.100 align:middle line:90% How can we achieve it? 00:43:10.100 --> 00:43:14.750 align:middle line:84% We have as controls the fraction of type k land held 00:43:14.750 --> 00:43:18.860 align:middle line:84% by household J. And now those fractions, alpha j, 00:43:18.860 --> 00:43:20.480 align:middle line:90% vary with k. 00:43:20.480 --> 00:43:24.590 align:middle line:84% So we're allowing land-type-specific allocation 00:43:24.590 --> 00:43:26.090 align:middle line:90% of shares. 00:43:26.090 --> 00:43:28.400 align:middle line:84% Anyway, if you do the linear algebra here, 00:43:28.400 --> 00:43:31.940 align:middle line:84% here's the yield of type 1 land in state 00:43:31.940 --> 00:43:36.440 align:middle line:84% 1, the yield of type 2 land in state 1, et cetera. 00:43:36.440 --> 00:43:39.080 align:middle line:84% These alphas would then multiply-- 00:43:39.080 --> 00:43:43.500 align:middle line:84% e1, alpha 1, e2, alpha 2, et cetera. 00:43:43.500 --> 00:43:45.990 align:middle line:84% So adding it up like the dot product, 00:43:45.990 --> 00:43:48.320 align:middle line:90% we would get consumption. 00:43:48.320 --> 00:43:51.470 align:middle line:84% Consumption is coming from a share-weighted version 00:43:51.470 --> 00:43:54.560 align:middle line:90% of the yield. 00:43:54.560 --> 00:43:57.230 align:middle line:90% Hopefully, the algebra is clear. 00:43:57.230 --> 00:43:59.330 align:middle line:84% Writing it in matrix notation, we just 00:43:59.330 --> 00:44:02.150 align:middle line:84% want to get this quote endowment matrix 00:44:02.150 --> 00:44:07.680 align:middle line:84% e times the share vector for household J varying over type k 00:44:07.680 --> 00:44:08.180 align:middle line:90% land. 00:44:08.180 --> 00:44:12.530 align:middle line:84% It's a vector equal to the state-dependent consumptions. 00:44:12.530 --> 00:44:16.220 align:middle line:90% So e times alpha equals cj. 00:44:16.220 --> 00:44:19.010 align:middle line:84% So then the question is, to achieve this target, 00:44:19.010 --> 00:44:20.690 align:middle line:90% can we find alphas that do it? 00:44:20.690 --> 00:44:22.620 align:middle line:84% And that's like solving this equation. 00:44:22.620 --> 00:44:26.590 align:middle line:84% So essentially, take an inverse of something. 00:44:26.590 --> 00:44:31.030 align:middle line:84% If there are as many return vectors over types of land 00:44:31.030 --> 00:44:34.300 align:middle line:84% that are independent of each other as there are states, 00:44:34.300 --> 00:44:37.720 align:middle line:90% then this matrix has an inverse. 00:44:37.720 --> 00:44:41.990 align:middle line:84% It's non-singular, so you can solve this equation. 00:44:41.990 --> 00:44:45.130 align:middle line:84% So now we'll have solutions for the alphas. 00:44:45.130 --> 00:44:48.220 align:middle line:84% And it turns out if you do that not just for household j 00:44:48.220 --> 00:44:51.610 align:middle line:84% but over all of them, you'll get an implication 00:44:51.610 --> 00:44:56.550 align:middle line:84% that the sum of the alphas will add up to 1. 00:44:56.550 --> 00:44:58.680 align:middle line:90% So they do look like shares. 00:44:58.680 --> 00:45:04.920 align:middle line:84% Type k land held in fraction alpha j by the j-th household 00:45:04.920 --> 00:45:08.160 align:middle line:84% when summed over-- there should be a sum here-- 00:45:08.160 --> 00:45:09.720 align:middle line:90% there was one there-- 00:45:09.720 --> 00:45:11.940 align:middle line:84% when summed over j would equal 1. 00:45:11.940 --> 00:45:14.490 align:middle line:90% So all the land is divided. 00:45:14.490 --> 00:45:17.130 align:middle line:84% All the land of each type k is totally divided 00:45:17.130 --> 00:45:19.680 align:middle line:90% among all the households. 00:45:19.680 --> 00:45:21.780 align:middle line:84% You could run into a little bit of problem 00:45:21.780 --> 00:45:24.480 align:middle line:84% if the alphas went negative because then 00:45:24.480 --> 00:45:26.500 align:middle line:90% that's like trying to go short. 00:45:26.500 --> 00:45:29.470 align:middle line:84% We're not going to allow that to happen. 00:45:29.470 --> 00:45:31.980 align:middle line:84% So there's a little bit of a fly in the ointment. 00:45:31.980 --> 00:45:33.630 align:middle line:84% But roughly speaking, there's a way 00:45:33.630 --> 00:45:39.070 align:middle line:84% to achieve an optimal allocation of risk sharing just by shares. 00:45:39.070 --> 00:45:43.320 align:middle line:84% So they don't need to transfer risk around ex post. 00:45:43.320 --> 00:45:45.990 align:middle line:90% They can do it all ex ante. 00:45:45.990 --> 00:45:48.180 align:middle line:84% Now, in order to achieve that, we 00:45:48.180 --> 00:45:50.280 align:middle line:84% had to ignore the negative alphas. 00:45:50.280 --> 00:45:52.620 align:middle line:84% But more to the point, we had to assume 00:45:52.620 --> 00:45:54.780 align:middle line:84% there are as many types of land as there 00:45:54.780 --> 00:45:56.580 align:middle line:90% are states of the world. 00:45:56.580 --> 00:45:57.808 align:middle line:90% Maybe that's not true. 00:45:57.808 --> 00:45:58.350 align:middle line:90% I don't know. 00:45:58.350 --> 00:45:59.910 align:middle line:90% It could go both ways. 00:45:59.910 --> 00:46:02.190 align:middle line:84% How many different types of land are there, 00:46:02.190 --> 00:46:05.760 align:middle line:84% and how many different states can a given land type take on? 00:46:05.760 --> 00:46:08.490 align:middle line:84% If there's incredible heterogeneity over land, 00:46:08.490 --> 00:46:10.680 align:middle line:84% it's entirely possible that the number of types 00:46:10.680 --> 00:46:14.370 align:middle line:84% was equal if not greater than the number of states. 00:46:14.370 --> 00:46:17.040 align:middle line:84% But I could also imagine the other way around. 00:46:17.040 --> 00:46:19.740 align:middle line:84% Like the Thai village, you've got low land and high land. 00:46:19.740 --> 00:46:21.810 align:middle line:84% You put them in two categories, and then you 00:46:21.810 --> 00:46:22.860 align:middle line:90% have multiple states. 00:46:22.860 --> 00:46:25.980 align:middle line:84% You'd have an incomplete version. 00:46:25.980 --> 00:46:27.660 align:middle line:90% Let's see how bad it is, though. 00:46:27.660 --> 00:46:31.530 align:middle line:84% Maybe as an approximation, they were doing pretty well 00:46:31.530 --> 00:46:33.360 align:middle line:84% by dividing up the land even if they 00:46:33.360 --> 00:46:37.930 align:middle line:84% couldn't achieve the optimal allocation perfectly. 00:46:37.930 --> 00:46:41.680 align:middle line:84% So let's put in two different utility functions. 00:46:41.680 --> 00:46:44.950 align:middle line:84% For household 1, we have constant relative risk 00:46:44.950 --> 00:46:49.240 align:middle line:84% aversion, and for household 2, it's a quadratic function. 00:46:49.240 --> 00:46:53.170 align:middle line:84% This quadratic thing is a pain and always scary 00:46:53.170 --> 00:46:53.960 align:middle line:90% when you see it. 00:46:53.960 --> 00:46:55.660 align:middle line:90% It's like there's a bliss point. 00:46:55.660 --> 00:46:56.740 align:middle line:90% We've talked about this. 00:46:56.740 --> 00:46:59.200 align:middle line:84% Consumption is less than the bliss point 00:46:59.200 --> 00:47:02.500 align:middle line:84% assumed so that the marginal utility of consumption 00:47:02.500 --> 00:47:04.398 align:middle line:90% is always positive. 00:47:04.398 --> 00:47:06.190 align:middle line:84% Otherwise, you see a lot of negative signs, 00:47:06.190 --> 00:47:07.930 align:middle line:90% and it's kind of disturbing. 00:47:07.930 --> 00:47:10.870 align:middle line:84% Can this consumption be higher than the bliss point? 00:47:10.870 --> 00:47:14.860 align:middle line:84% No, because you'll get marginal utility being negative, 00:47:14.860 --> 00:47:16.760 align:middle line:84% moving negatively with consumption. 00:47:16.760 --> 00:47:20.110 align:middle line:84% So anyway, just so you feel a bit more comfortable, 00:47:20.110 --> 00:47:24.100 align:middle line:84% let's put in alpha equal to 0.5, k equal 0.002, 00:47:24.100 --> 00:47:27.040 align:middle line:90% and be equal to 80. 00:47:27.040 --> 00:47:30.010 align:middle line:84% So we have completely parameterized this family 00:47:30.010 --> 00:47:31.270 align:middle line:90% of utility functions. 00:47:31.270 --> 00:47:33.910 align:middle line:84% And again, we have different preferences here 00:47:33.910 --> 00:47:37.250 align:middle line:84% but nice concave utility, strictly, for both. 00:47:37.250 --> 00:47:40.870 align:middle line:84% We've got two types of land, and we're back to, say, 00:47:40.870 --> 00:47:43.880 align:middle line:84% having three or more states of the world. 00:47:43.880 --> 00:47:48.580 align:middle line:84% So ex ante division into shares cannot work perfectly. 00:47:48.580 --> 00:47:51.760 align:middle line:84% To see how far we get, we constrain the programming 00:47:51.760 --> 00:47:57.730 align:middle line:84% problem to nevertheless search over these shares, 00:47:57.730 --> 00:48:02.230 align:middle line:84% alpha jk of household J share of type k land. 00:48:02.230 --> 00:48:05.290 align:middle line:84% And we maximize a lambda-weighted sum 00:48:05.290 --> 00:48:07.870 align:middle line:90% of ex ante expected utilities. 00:48:07.870 --> 00:48:13.390 align:middle line:84% Oh, well there's only one date here, so ignore the dynamics. 00:48:13.390 --> 00:48:15.640 align:middle line:90% But there are multiple states. 00:48:15.640 --> 00:48:17.800 align:middle line:84% And we're going to max this thing subject to shares 00:48:17.800 --> 00:48:20.740 align:middle line:84% can't go negative, and they have to add up to 1. 00:48:20.740 --> 00:48:23.770 align:middle line:84% So it's a constrained programming problem. 00:48:23.770 --> 00:48:27.040 align:middle line:84% We could call the solution to this constrained optimal. 00:48:27.040 --> 00:48:30.280 align:middle line:84% It's constrained by the premise that the mechanism only 00:48:30.280 --> 00:48:32.980 align:middle line:84% allows ex-ante division into shares 00:48:32.980 --> 00:48:38.570 align:middle line:84% and not ex-post consumption transfers, not gifts, not 00:48:38.570 --> 00:48:40.970 align:middle line:90% borrowing and lending, nothing. 00:48:40.970 --> 00:48:43.790 align:middle line:84% Now, you know from our work on Pareto 00:48:43.790 --> 00:48:47.720 align:middle line:84% optimal allocations there's going to be a Pareto frontier. 00:48:47.720 --> 00:48:50.390 align:middle line:84% So it's not like there's one solution. 00:48:50.390 --> 00:48:53.510 align:middle line:84% It's going to vary with the lambdas. 00:48:53.510 --> 00:48:56.930 align:middle line:84% So we'll do something special as an example, which 00:48:56.930 --> 00:49:00.050 align:middle line:84% is to maximize the utility of household 1 00:49:00.050 --> 00:49:02.360 align:middle line:84% subject to the utility of household 00:49:02.360 --> 00:49:05.730 align:middle line:90% 2 at an arbitrary constant. 00:49:05.730 --> 00:49:10.160 align:middle line:84% So let me show you a picture, and then I'll go back. 00:49:10.160 --> 00:49:12.710 align:middle line:84% We're in utility space, so we have the utilities 00:49:12.710 --> 00:49:15.050 align:middle line:90% of household 1 and household 2. 00:49:15.050 --> 00:49:18.470 align:middle line:84% And we have-- if we didn't have a constrained problem, 00:49:18.470 --> 00:49:21.200 align:middle line:84% we'd have this outer frontier of the set 00:49:21.200 --> 00:49:24.300 align:middle line:84% of all possible utilities on the boundary. 00:49:24.300 --> 00:49:28.160 align:middle line:84% And we'll call that the utility at the full risk sharing. 00:49:28.160 --> 00:49:30.080 align:middle line:90% There are many solutions. 00:49:30.080 --> 00:49:33.710 align:middle line:84% We got these through hyperplanes previously. 00:49:33.710 --> 00:49:36.810 align:middle line:84% There is an equivalent way to do it, 00:49:36.810 --> 00:49:40.910 align:middle line:84% which is to fix the utility of one household, namely 2, 00:49:40.910 --> 00:49:43.920 align:middle line:84% and maximize the utility of the other one. 00:49:43.920 --> 00:49:46.070 align:middle line:90% So we would end up right here. 00:49:46.070 --> 00:49:48.260 align:middle line:84% But in the constrained problem, we're 00:49:48.260 --> 00:49:51.290 align:middle line:84% limited to dividing up land by shares. 00:49:51.290 --> 00:49:55.850 align:middle line:84% That constrained Pareto frontier is interior, still 00:49:55.850 --> 00:49:57.770 align:middle line:90% a bit concave. 00:49:57.770 --> 00:50:00.140 align:middle line:84% And we would move, therefore, from where 00:50:00.140 --> 00:50:02.600 align:middle line:84% that horizontal utility level for household 2 00:50:02.600 --> 00:50:05.700 align:middle line:84% hits the outer frontier versus the inner frontier. 00:50:05.700 --> 00:50:07.430 align:middle line:84% So let's focus on the inner frontier, 00:50:07.430 --> 00:50:09.530 align:middle line:84% the constrained problem, and then see 00:50:09.530 --> 00:50:13.120 align:middle line:84% how close they're going to get to the outer frontier. 00:50:13.120 --> 00:50:15.010 align:middle line:90% I solved that thing numerically. 00:50:15.010 --> 00:50:17.740 align:middle line:84% I was trying to give you all the parameters for utility 00:50:17.740 --> 00:50:18.910 align:middle line:90% functions. 00:50:18.910 --> 00:50:21.970 align:middle line:84% Some of it's missing here-- covariance, 00:50:21.970 --> 00:50:23.950 align:middle line:90% coefficient of variation. 00:50:23.950 --> 00:50:29.470 align:middle line:84% And you get a solution that household 1 has about 3/4, 00:50:29.470 --> 00:50:35.530 align:middle line:84% 72%, of land type I 2 and about a quarter of land type 2. 00:50:35.530 --> 00:50:37.690 align:middle line:84% And household 2 would be on the other side 00:50:37.690 --> 00:50:41.420 align:middle line:90% of this at roughly 1/4, 3/4. 00:50:41.420 --> 00:50:44.830 align:middle line:84% So one thing to note is that that 00:50:44.830 --> 00:50:51.040 align:middle line:84% generates a non-linear sharing rule but non-optimal rule. 00:50:51.040 --> 00:50:52.970 align:middle line:90% I think I missed-- 00:50:52.970 --> 00:50:54.980 align:middle line:84% when I presented this constraint problem, 00:50:54.980 --> 00:50:58.850 align:middle line:84% I neglected the little paragraph at the bottom. 00:50:58.850 --> 00:51:01.150 align:middle line:90% No, it's not there. 00:51:01.150 --> 00:51:03.910 align:middle line:90% Well, let me remind you. 00:51:03.910 --> 00:51:07.300 align:middle line:84% Again, back to the preference lecture where we did risk, 00:51:07.300 --> 00:51:11.260 align:middle line:84% and I showed you two particular utility functions, 00:51:11.260 --> 00:51:15.490 align:middle line:84% constant exponential and constant relative risk aversion 00:51:15.490 --> 00:51:16.930 align:middle line:90% with linear schedules. 00:51:16.930 --> 00:51:21.160 align:middle line:84% But then I showed you another figure which was non-linear-- 00:51:21.160 --> 00:51:24.700 align:middle line:84% monotone increasing for both households, but nonlinear. 00:51:24.700 --> 00:51:29.020 align:middle line:84% That came from this specification of utilities. 00:51:29.020 --> 00:51:32.020 align:middle line:84% So that's what I was looking for, seen in previous lecture 00:51:32.020 --> 00:51:33.130 align:middle line:90% 7. 00:51:33.130 --> 00:51:37.990 align:middle line:84% So we know the optimal sharing rules are nonlinear. 00:51:37.990 --> 00:51:41.590 align:middle line:84% Dividing up into shares suggests we're going 00:51:41.590 --> 00:51:44.300 align:middle line:90% to end up with linear rules. 00:51:44.300 --> 00:51:47.050 align:middle line:84% So we're not going to end up with the optimum. 00:51:47.050 --> 00:51:49.150 align:middle line:84% But we can still try to figure out 00:51:49.150 --> 00:51:54.320 align:middle line:84% how close the inner frontier is to the outer frontier. 00:51:54.320 --> 00:51:57.560 align:middle line:90% So let's look at the transfers. 00:51:57.560 --> 00:51:59.330 align:middle line:90% Let me state this clearly. 00:51:59.330 --> 00:52:03.860 align:middle line:84% We're going to try to do it ex-ante by division of shares, 00:52:03.860 --> 00:52:07.520 align:middle line:84% get the constrained optimum, and then look 00:52:07.520 --> 00:52:11.570 align:middle line:84% at transfers across the two households, who's 00:52:11.570 --> 00:52:14.300 align:middle line:84% paying who depending on the state, and then 00:52:14.300 --> 00:52:17.270 align:middle line:84% finally, how those transfers would need potentially 00:52:17.270 --> 00:52:20.760 align:middle line:90% to be augmented. 00:52:20.760 --> 00:52:22.350 align:middle line:84% So here's the aggregate endowment. 00:52:22.350 --> 00:52:26.300 align:middle line:84% This is the sum of the yields over the two types of land. 00:52:26.300 --> 00:52:28.220 align:middle line:84% That's also aggregate consumption 00:52:28.220 --> 00:52:30.650 align:middle line:84% because this is a static problem. 00:52:30.650 --> 00:52:33.830 align:middle line:84% Here is consumption of the individual agents, 00:52:33.830 --> 00:52:35.300 align:middle line:90% both of them. 00:52:35.300 --> 00:52:39.830 align:middle line:84% This is the consumption of person 1, constrained, 00:52:39.830 --> 00:52:42.350 align:middle line:90% linear dotted line. 00:52:42.350 --> 00:52:44.960 align:middle line:84% Here's the consumption of person 2. 00:52:44.960 --> 00:52:49.250 align:middle line:84% Constrained is this dot dashed line for person 2. 00:52:49.250 --> 00:52:53.090 align:middle line:84% Constrained is the dot-dot-dot line for person 1. 00:52:53.090 --> 00:52:55.160 align:middle line:90% They're both linear schedules. 00:52:55.160 --> 00:52:57.800 align:middle line:84% Now, remember, they're not doing this with transfers-- 00:52:57.800 --> 00:52:59.360 align:middle line:84% I kind of misspoke a minute ago-- 00:52:59.360 --> 00:53:02.390 align:middle line:84% because they're doing it with ex-ante division into shares. 00:53:02.390 --> 00:53:05.210 align:middle line:84% But nevertheless, it's not the full-blown optimum. 00:53:05.210 --> 00:53:08.540 align:middle line:84% It's not the outer frontier in which you can not only 00:53:08.540 --> 00:53:11.750 align:middle line:84% have division but also ex-post transfers 00:53:11.750 --> 00:53:13.130 align:middle line:90% to get to the full optimum. 00:53:13.130 --> 00:53:16.310 align:middle line:84% And those schedules are listed here as unconstrained. 00:53:16.310 --> 00:53:20.160 align:middle line:84% And they are non-linear, as I was trying to say earlier. 00:53:20.160 --> 00:53:25.520 align:middle line:84% So if we look at these states, say five states, 00:53:25.520 --> 00:53:28.880 align:middle line:84% and look at consumption under ex-ante division of land 00:53:28.880 --> 00:53:33.680 align:middle line:84% versus consumption under the full optimum, for household 1, 00:53:33.680 --> 00:53:36.980 align:middle line:84% you can see household 1 is monotonic. 00:53:36.980 --> 00:53:39.020 align:middle line:84% Household consumption is logically 00:53:39.020 --> 00:53:41.420 align:middle line:84% increasing with the aggregate, but it's 00:53:41.420 --> 00:53:43.430 align:middle line:84% a bit different, although monotonic, 00:53:43.430 --> 00:53:45.380 align:middle line:84% from what it would be under the full optimum. 00:53:45.380 --> 00:53:48.560 align:middle line:84% And here you can see at the lowest state, 00:53:48.560 --> 00:53:52.190 align:middle line:84% household 1 would be consuming more under the full optimum 00:53:52.190 --> 00:53:56.120 align:middle line:84% but then less under the full optimum relative 00:53:56.120 --> 00:54:00.500 align:middle line:84% to the division, less, and then back to being more. 00:54:00.500 --> 00:54:02.640 align:middle line:90% So here it is over here. 00:54:02.640 --> 00:54:04.680 align:middle line:90% Why am I struggling so hard? 00:54:04.680 --> 00:54:07.610 align:middle line:84% So this is the amount of the transfer 00:54:07.610 --> 00:54:11.480 align:middle line:84% from household 2 to household 1 that 00:54:11.480 --> 00:54:15.860 align:middle line:84% is necessary to add on to the solution 00:54:15.860 --> 00:54:19.900 align:middle line:84% from the ex-ante division of land shares only. 00:54:19.900 --> 00:54:22.640 align:middle line:84% If you wanted to look at it in a picture, 00:54:22.640 --> 00:54:27.400 align:middle line:84% the picture is literally the difference 00:54:27.400 --> 00:54:29.780 align:middle line:90% between these two lines. 00:54:29.780 --> 00:54:32.560 align:middle line:84% So here, the amount that household 1 00:54:32.560 --> 00:54:37.840 align:middle line:84% ought to be getting under the full optimum would be higher, 00:54:37.840 --> 00:54:40.990 align:middle line:84% and therefore, ought to get a positive transfer. 00:54:40.990 --> 00:54:44.570 align:middle line:84% Here, it would be lower, hence the negative transfer. 00:54:44.570 --> 00:54:47.620 align:middle line:84% And finally, at the end, positive again. 00:54:47.620 --> 00:54:50.170 align:middle line:84% So that's consistent with these numbers 00:54:50.170 --> 00:54:53.260 align:middle line:84% where these states, going from 1 to 5, 00:54:53.260 --> 00:54:55.660 align:middle line:84% is like moving across the different values 00:54:55.660 --> 00:54:57.550 align:middle line:90% of the aggregate endowment. 00:54:57.550 --> 00:55:01.330 align:middle line:84% So in summary, I guess there is a couple 00:55:01.330 --> 00:55:04.450 align:middle line:84% of takeaways from the second half of this lecture. 00:55:04.450 --> 00:55:07.960 align:middle line:84% One, we looked at special mechanisms-- in this case, 00:55:07.960 --> 00:55:11.410 align:middle line:84% ex-ante division of land and whether or not we 00:55:11.410 --> 00:55:13.300 align:middle line:84% could get to an optimal allocation 00:55:13.300 --> 00:55:15.430 align:middle line:90% of risk bearing that way. 00:55:15.430 --> 00:55:18.960 align:middle line:84% And we saw some special cases where could. 00:55:18.960 --> 00:55:22.210 align:middle line:84% We had to make the shares different across the different 00:55:22.210 --> 00:55:25.390 align:middle line:84% households and maybe different across the different land 00:55:25.390 --> 00:55:28.990 align:middle line:84% types, especially so if we accommodated more and more 00:55:28.990 --> 00:55:30.920 align:middle line:90% diversity and preferences. 00:55:30.920 --> 00:55:33.670 align:middle line:84% The second takeaway from the second part 00:55:33.670 --> 00:55:37.330 align:middle line:84% is we might not achieve a full optimum, 00:55:37.330 --> 00:55:39.940 align:middle line:84% so we would have expected some kind of borrowing and lending 00:55:39.940 --> 00:55:42.310 align:middle line:84% and gift-giving in that medieval village economy 00:55:42.310 --> 00:55:46.390 align:middle line:84% even if we can't find it in the historical records. 00:55:46.390 --> 00:55:49.990 align:middle line:84% Or we could just say, evidently, for institutional reasons 00:55:49.990 --> 00:55:52.780 align:middle line:84% we don't understand, they just weren't doing it. 00:55:52.780 --> 00:55:54.560 align:middle line:90% But we don't lose our tools. 00:55:54.560 --> 00:55:55.660 align:middle line:90% Not all is lost. 00:55:55.660 --> 00:56:00.040 align:middle line:84% We are still able to solve for a constrained optimal allocation 00:56:00.040 --> 00:56:03.220 align:middle line:84% solving the so-called Pareto problem or programming 00:56:03.220 --> 00:56:06.890 align:middle line:84% problem, maximizing lambda-weighted sums of utility 00:56:06.890 --> 00:56:10.690 align:middle line:84% subject to constraints with the appropriate control variables, 00:56:10.690 --> 00:56:12.860 align:middle line:90% in this case shares. 00:56:12.860 --> 00:56:14.530 align:middle line:84% And then the big picture of this lecture 00:56:14.530 --> 00:56:17.260 align:middle line:84% is we're looking at the application 00:56:17.260 --> 00:56:20.770 align:middle line:84% of the theory of the optimal allocation of risk bearing. 00:56:20.770 --> 00:56:22.900 align:middle line:84% We did that in villages in India, 00:56:22.900 --> 00:56:25.100 align:middle line:84% looking at consumption and income data. 00:56:25.100 --> 00:56:27.940 align:middle line:84% So from a science point of view, we have a model. 00:56:27.940 --> 00:56:28.840 align:middle line:90% We have data. 00:56:28.840 --> 00:56:31.390 align:middle line:84% The data can vary across the applications. 00:56:31.390 --> 00:56:33.100 align:middle line:84% You can still ask, even if we don't 00:56:33.100 --> 00:56:35.440 align:middle line:84% have the data on everything, whether we can 00:56:35.440 --> 00:56:37.270 align:middle line:90% see how well the theory fits. 00:56:37.270 --> 00:56:39.610 align:middle line:90% And we could do that in-- 00:56:39.610 --> 00:56:41.500 align:middle line:84% in the medieval villages, we don't even 00:56:41.500 --> 00:56:44.260 align:middle line:84% have data on consumption of individual households, 00:56:44.260 --> 00:56:46.060 align:middle line:84% and all we see are pictures of land. 00:56:46.060 --> 00:56:49.510 align:middle line:84% But we do have the covariance and variance statistics. 00:56:49.510 --> 00:56:51.250 align:middle line:84% So we're kind of able to calibrate 00:56:51.250 --> 00:56:54.730 align:middle line:84% the model a bit the way we were calibrating the dynamic storage 00:56:54.730 --> 00:56:56.890 align:middle line:84% model, but in this case, not to try 00:56:56.890 --> 00:57:00.130 align:middle line:84% to explain why they didn't store very often, but rather, 00:57:00.130 --> 00:57:03.730 align:middle line:84% to explain this very salient pattern of dividing land 00:57:03.730 --> 00:57:05.320 align:middle line:90% into different strips. 00:57:05.320 --> 00:57:05.830 align:middle line:90% OK. 00:57:05.830 --> 00:57:08.860 align:middle line:90% So that's all I have for today. 00:57:08.860 --> 00:57:10.410 align:middle line:90% All right. 00:57:10.410 --> 00:57:11.470 align:middle line:90% See you soon. 00:57:11.470 --> 00:57:13.430 align:middle line:90% Thank you so much. 00:57:13.430 --> 00:57:21.000 align:middle line:90%