WEBVTT 00:00:00.000 --> 00:00:01.482 align:middle line:90% [SQUEAKING] 00:00:01.482 --> 00:00:02.964 align:middle line:90% [RUSTLING] 00:00:02.964 --> 00:00:04.940 align:middle line:90% [CLICKING] 00:00:04.940 --> 00:00:11.370 align:middle line:90% 00:00:11.370 --> 00:00:14.940 align:middle line:84% ROBERT TOWNSEND: OK, we'll get started on time. 00:00:14.940 --> 00:00:17.640 align:middle line:84% If you look at the reading list and scroll 00:00:17.640 --> 00:00:22.560 align:middle line:84% down to where we are, which is Dynamics and Programming, 00:00:22.560 --> 00:00:25.200 align:middle line:90% you'll see Lecture 6 there. 00:00:25.200 --> 00:00:30.270 align:middle line:84% There's a chapter on Varians, which is starred, called Time. 00:00:30.270 --> 00:00:32.070 align:middle line:84% That's because the application today 00:00:32.070 --> 00:00:35.070 align:middle line:84% will be featuring dynamics and what 00:00:35.070 --> 00:00:36.910 align:middle line:90% happens over various dates. 00:00:36.910 --> 00:00:40.320 align:middle line:84% And we're going to do it in the context of a storage 00:00:40.320 --> 00:00:43.230 align:middle line:84% problem, where people were close to starving to death, as 00:00:43.230 --> 00:00:45.400 align:middle line:90% in the medieval villages. 00:00:45.400 --> 00:00:49.410 align:middle line:84% And that is Chapter 3 of "Medville." 00:00:49.410 --> 00:00:53.640 align:middle line:84% It's also starred, which fills out even more 00:00:53.640 --> 00:00:57.000 align:middle line:84% than I'm able to in one CLASS the material for the lecture 00:00:57.000 --> 00:00:57.880 align:middle line:90% today. 00:00:57.880 --> 00:01:01.200 align:middle line:90% So that's the reading list. 00:01:01.200 --> 00:01:03.630 align:middle line:90% Then there's study guide. 00:01:03.630 --> 00:01:06.150 align:middle line:84% So I went back and earmarked what 00:01:06.150 --> 00:01:08.010 align:middle line:84% we did last time, which was Decision 00:01:08.010 --> 00:01:10.380 align:middle line:84% Making under Uncertainty, Linear Programming. 00:01:10.380 --> 00:01:12.030 align:middle line:84% Some of these questions are worded 00:01:12.030 --> 00:01:14.430 align:middle line:90% as if you were doing it at home. 00:01:14.430 --> 00:01:18.850 align:middle line:84% But nevertheless, the question asks about important concepts. 00:01:18.850 --> 00:01:20.340 align:middle line:90% So let me read the first one. 00:01:20.340 --> 00:01:22.560 align:middle line:84% Draw a consumption set which consists 00:01:22.560 --> 00:01:27.210 align:middle line:84% of discrete number of points argue that th set is not convex 00:01:27.210 --> 00:01:30.720 align:middle line:84% and then show how lotteries which put mass 00:01:30.720 --> 00:01:34.620 align:middle line:84% on these discrete points, transforms the consumption set 00:01:34.620 --> 00:01:36.780 align:middle line:90% into something convex. 00:01:36.780 --> 00:01:41.160 align:middle line:84% So there's two sentences there, effectively two questions. 00:01:41.160 --> 00:01:42.890 align:middle line:90% Could I get a volunteer, please? 00:01:42.890 --> 00:01:46.770 align:middle line:90% 00:01:46.770 --> 00:01:48.190 align:middle line:90% So you can't actually draw it. 00:01:48.190 --> 00:01:52.260 align:middle line:84% Can you describe a consumption set? 00:01:52.260 --> 00:01:55.770 align:middle line:90% Maybe these words suffice. 00:01:55.770 --> 00:01:58.590 align:middle line:84% Try to describe, in words, a consumption set which 00:01:58.590 --> 00:02:00.390 align:middle line:84% contains a discrete number of points 00:02:00.390 --> 00:02:04.790 align:middle line:90% and tell me why it's not convex. 00:02:04.790 --> 00:02:05.980 align:middle line:90% AUDIENCE: Yeah. 00:02:05.980 --> 00:02:10.190 align:middle line:84% So I think we discussed this earlier. 00:02:10.190 --> 00:02:14.590 align:middle line:84% But one consumption set was consuming lunch 00:02:14.590 --> 00:02:17.860 align:middle line:84% in New York at noon versus consuming lunch 00:02:17.860 --> 00:02:19.930 align:middle line:90% in Washington, DC at noon. 00:02:19.930 --> 00:02:21.190 align:middle line:90% ROBERT TOWNSEND: That's true. 00:02:21.190 --> 00:02:23.440 align:middle line:84% AUDIENCE: So that would be a discrete consumption set. 00:02:23.440 --> 00:02:26.050 align:middle line:84% And since you can't consume lunch 00:02:26.050 --> 00:02:28.540 align:middle line:84% in a mixture of those two cities, 00:02:28.540 --> 00:02:31.000 align:middle line:84% then the consumption is not convex. 00:02:31.000 --> 00:02:33.880 align:middle line:84% But if you had a sum probability of eating lunch 00:02:33.880 --> 00:02:37.390 align:middle line:84% in either location, then you could have a probability 00:02:37.390 --> 00:02:40.240 align:middle line:90% between zero and one. 00:02:40.240 --> 00:02:43.480 align:middle line:84% ROBERT TOWNSEND: And how would you draw it? 00:02:43.480 --> 00:02:48.038 align:middle line:90% 00:02:48.038 --> 00:02:50.330 align:middle line:84% AUDIENCE: So for the original discrete consumption set, 00:02:50.330 --> 00:02:52.360 align:middle line:90% you'd just be two points. 00:02:52.360 --> 00:02:54.160 align:middle line:84% I think one of those points would be one, 00:02:54.160 --> 00:02:57.160 align:middle line:84% zero and one of those points will be zero, one. 00:02:57.160 --> 00:02:59.680 align:middle line:90% ROBERT TOWNSEND: Good. 00:02:59.680 --> 00:03:01.300 align:middle line:90% And can you imagine how-- 00:03:01.300 --> 00:03:05.320 align:middle line:84% I'll help in a second if it's not clear-- 00:03:05.320 --> 00:03:07.600 align:middle line:84% can you imagine how you would be able to convert 00:03:07.600 --> 00:03:12.760 align:middle line:84% that set into something that is convex, using the lottery? 00:03:12.760 --> 00:03:15.972 align:middle line:84% Is there something you could draw to depict it? 00:03:15.972 --> 00:03:18.430 align:middle line:84% AUDIENCE: If you just sort of drew a line between those two 00:03:18.430 --> 00:03:20.890 align:middle line:84% points, like your location, instead 00:03:20.890 --> 00:03:25.030 align:middle line:84% of doing how many lunches you eat at DC and New York, 00:03:25.030 --> 00:03:28.960 align:middle line:84% you could do the probability that you eat at DC or New York. 00:03:28.960 --> 00:03:30.700 align:middle line:90% ROBERT TOWNSEND: Exactly. 00:03:30.700 --> 00:03:34.870 align:middle line:84% So now we have a line running on the diagonal 00:03:34.870 --> 00:03:37.510 align:middle line:90% from one, zero to zero one. 00:03:37.510 --> 00:03:39.410 align:middle line:90% The end points are still there. 00:03:39.410 --> 00:03:41.350 align:middle line:84% Those are lotteries that are degenerate. 00:03:41.350 --> 00:03:45.580 align:middle line:84% They're putting probability one on each of the endpoints. 00:03:45.580 --> 00:03:49.360 align:middle line:84% An intermediate lottery, a non-trivial lottery, say, 00:03:49.360 --> 00:03:54.410 align:middle line:84% halfway between, would put probability 1/2 on-- 00:03:54.410 --> 00:03:58.880 align:middle line:84% would be distance 1/2 to one, zero and distance 1/4 00:03:58.880 --> 00:04:00.710 align:middle line:90% to zero, one. 00:04:00.710 --> 00:04:03.630 align:middle line:84% And that represents an equal weighted probability. 00:04:03.630 --> 00:04:07.810 align:middle line:84% And you can inch closer to one endpoint or the other. 00:04:07.810 --> 00:04:10.990 align:middle line:84% Those are still non-degenerate lotteries. 00:04:10.990 --> 00:04:13.330 align:middle line:84% So a lottery, when you put a point in there, 00:04:13.330 --> 00:04:17.589 align:middle line:90% represents the weights. 00:04:17.589 --> 00:04:19.339 align:middle line:84% But yes, your answer was perfect. 00:04:19.339 --> 00:04:21.310 align:middle line:90% Thank you. 00:04:21.310 --> 00:04:22.315 align:middle line:90% Questions about that? 00:04:22.315 --> 00:04:25.900 align:middle line:90% 00:04:25.900 --> 00:04:28.170 align:middle line:90% OK. 00:04:28.170 --> 00:04:32.340 align:middle line:84% What does risk aversion imply about the consumer's utility 00:04:32.340 --> 00:04:34.800 align:middle line:90% function? 00:04:34.800 --> 00:04:37.470 align:middle line:90% Again, I'll take volunteers. 00:04:37.470 --> 00:04:39.690 align:middle line:84% AUDIENCE: If you're risk averse, then you 00:04:39.690 --> 00:04:42.610 align:middle line:84% have diminishing marginal utility of income. 00:04:42.610 --> 00:04:47.220 align:middle line:84% So the utility function should be concave. 00:04:47.220 --> 00:04:48.490 align:middle line:90% ROBERT TOWNSEND: Good. 00:04:48.490 --> 00:04:49.555 align:middle line:90% Strictly or weakly? 00:04:49.555 --> 00:04:52.810 align:middle line:90% 00:04:52.810 --> 00:04:55.300 align:middle line:90% AUDIENCE: Convex? 00:04:55.300 --> 00:04:56.710 align:middle line:90% ROBERT TOWNSEND: Let me repeat. 00:04:56.710 --> 00:04:59.890 align:middle line:84% The question says what does risk aversion imply 00:04:59.890 --> 00:05:03.080 align:middle line:84% about the consumer's utility function. 00:05:03.080 --> 00:05:04.810 align:middle line:90% So your answer, concave, yes. 00:05:04.810 --> 00:05:08.290 align:middle line:84% And I'm asking back, you mean strictly concave, 00:05:08.290 --> 00:05:11.450 align:middle line:90% weakly concave. 00:05:11.450 --> 00:05:14.390 align:middle line:84% AUDIENCE: I mean, I think it would be weakly concave, right? 00:05:14.390 --> 00:05:17.510 align:middle line:84% Because if you have a straight line-- 00:05:17.510 --> 00:05:20.120 align:middle line:84% or actually would that still be considered risk averse, 00:05:20.120 --> 00:05:24.170 align:middle line:84% if your utility from getting a certain value 00:05:24.170 --> 00:05:29.270 align:middle line:84% is like completely equal to the utility of having 00:05:29.270 --> 00:05:30.920 align:middle line:90% that as your expected value? 00:05:30.920 --> 00:05:31.340 align:middle line:90% ROBERT TOWNSEND: All right. 00:05:31.340 --> 00:05:32.000 align:middle line:90% That's perfect. 00:05:32.000 --> 00:05:33.030 align:middle line:90% You figured it out. 00:05:33.030 --> 00:05:36.320 align:middle line:84% So the answer is we typically think of risk aversion 00:05:36.320 --> 00:05:39.860 align:middle line:84% to mean strictly averse to the lottery. 00:05:39.860 --> 00:05:46.010 align:middle line:84% And hence, the linear case of the linear utility function, 00:05:46.010 --> 00:05:48.170 align:middle line:84% or weekly concave function, but not 00:05:48.170 --> 00:05:51.830 align:middle line:84% strictly concave, we call them risk neutral 00:05:51.830 --> 00:05:53.780 align:middle line:90% and not risk averse. 00:05:53.780 --> 00:05:55.380 align:middle line:90% But I never said that in class. 00:05:55.380 --> 00:05:59.200 align:middle line:90% So I'm amplifying. 00:05:59.200 --> 00:06:02.260 align:middle line:84% But yes, risk aversion is being risk averse. 00:06:02.260 --> 00:06:07.340 align:middle line:84% Strictly speaking implies strictly concave utility. 00:06:07.340 --> 00:06:08.750 align:middle line:90% So perfect, thank you. 00:06:08.750 --> 00:06:14.490 align:middle line:90% 00:06:14.490 --> 00:06:17.660 align:middle line:90% So one more here. 00:06:17.660 --> 00:06:21.290 align:middle line:84% This, again, is going to require you to describe something. 00:06:21.290 --> 00:06:26.480 align:middle line:84% Combining states and time of the world, time and states 00:06:26.480 --> 00:06:30.140 align:middle line:84% of the world, what is the complete commodity space 00:06:30.140 --> 00:06:33.590 align:middle line:90% for a consumer? 00:06:33.590 --> 00:06:35.060 align:middle line:84% And what I'm really looking for is 00:06:35.060 --> 00:06:39.680 align:middle line:84% for you to describe to me the tree of possibilities that we 00:06:39.680 --> 00:06:41.900 align:middle line:90% had on the slide last time. 00:06:41.900 --> 00:06:44.810 align:middle line:90% 00:06:44.810 --> 00:06:46.610 align:middle line:84% And, again, I'll take volunteers. 00:06:46.610 --> 00:06:53.750 align:middle line:90% 00:06:53.750 --> 00:06:56.940 align:middle line:84% AUDIENCE: So this is a tree that's oriented upwards. 00:06:56.940 --> 00:07:01.730 align:middle line:84% And so at where time increases as you go up in this tree. 00:07:01.730 --> 00:07:04.280 align:middle line:84% As you make a specific choice at any given time, 00:07:04.280 --> 00:07:08.675 align:middle line:84% you limit your future options to some specific set, 00:07:08.675 --> 00:07:11.300 align:middle line:84% and you exclude some others that are no longer possible for you 00:07:11.300 --> 00:07:11.800 align:middle line:90% to consume. 00:07:11.800 --> 00:07:15.510 align:middle line:90% 00:07:15.510 --> 00:07:17.040 align:middle line:90% ROBERT TOWNSEND: That's right. 00:07:17.040 --> 00:07:22.390 align:middle line:84% So the tricky part of it is to remember 00:07:22.390 --> 00:07:26.170 align:middle line:84% that the state of the world at any date 00:07:26.170 --> 00:07:29.230 align:middle line:84% is not just a set of possible realizations 00:07:29.230 --> 00:07:31.330 align:middle line:90% of the contemporary shock. 00:07:31.330 --> 00:07:35.890 align:middle line:84% It also includes the history of shocks, up to and including 00:07:35.890 --> 00:07:38.550 align:middle line:90% the contemporary date. 00:07:38.550 --> 00:07:41.180 align:middle line:84% So that's what's going on with that tree. 00:07:41.180 --> 00:07:44.930 align:middle line:90% You pick a node in the tree. 00:07:44.930 --> 00:07:48.740 align:middle line:84% It's described not just by the set of all things that 00:07:48.740 --> 00:07:51.730 align:middle line:84% could have happened on the current branch, 00:07:51.730 --> 00:07:56.410 align:middle line:84% but the set of all things that could possibly have happened up 00:07:56.410 --> 00:07:59.260 align:middle line:90% to and including that point. 00:07:59.260 --> 00:08:02.970 align:middle line:84% And actually, I made the comment in class 00:08:02.970 --> 00:08:05.460 align:middle line:84% that one has to consider whether you're already 00:08:05.460 --> 00:08:09.510 align:middle line:84% at the current date, or whether you're at date zero, 00:08:09.510 --> 00:08:12.870 align:middle line:84% imagining what could happen in the future. 00:08:12.870 --> 00:08:14.582 align:middle line:84% Because if you're at zero and imagining 00:08:14.582 --> 00:08:16.290 align:middle line:84% what could happen in the future, then you 00:08:16.290 --> 00:08:20.100 align:middle line:84% have to delineate all possible histories 00:08:20.100 --> 00:08:23.890 align:middle line:84% and all possible states that can happen in any given date, 00:08:23.890 --> 00:08:27.945 align:middle line:84% including up through and including the date in question. 00:08:27.945 --> 00:08:30.860 align:middle line:90% 00:08:30.860 --> 00:08:31.500 align:middle line:90% Great. 00:08:31.500 --> 00:08:32.000 align:middle line:90% OK. 00:08:32.000 --> 00:08:35.169 align:middle line:90% 00:08:35.169 --> 00:08:37.240 align:middle line:84% So, again, I earmarked some of these. 00:08:37.240 --> 00:08:40.720 align:middle line:84% I only went through three of the four. 00:08:40.720 --> 00:08:41.750 align:middle line:90% That was very helpful. 00:08:41.750 --> 00:08:43.570 align:middle line:90% Thank you. 00:08:43.570 --> 00:08:51.730 align:middle line:90% OK, so the lecture today is-- 00:08:51.730 --> 00:08:57.520 align:middle line:84% so this is on dynamics and dynamic programming. 00:08:57.520 --> 00:09:00.430 align:middle line:84% It's going to focus on this application 00:09:00.430 --> 00:09:03.520 align:middle line:84% for medieval villages on storage, 00:09:03.520 --> 00:09:08.050 align:middle line:84% or call it inventory, or carryover, as well as the seed 00:09:08.050 --> 00:09:10.190 align:middle line:90% that they put in the ground. 00:09:10.190 --> 00:09:12.440 align:middle line:84% And I'll come back to that momentarily. 00:09:12.440 --> 00:09:16.670 align:middle line:84% And if you go back to the economic science lecture, 00:09:16.670 --> 00:09:22.490 align:middle line:84% this lecture is illustrative of something we call calibration. 00:09:22.490 --> 00:09:26.290 align:middle line:84% So you have a model of the way the world works and you 00:09:26.290 --> 00:09:28.840 align:middle line:84% find, somehow, parameter values that 00:09:28.840 --> 00:09:30.850 align:middle line:90% are appropriate for the model. 00:09:30.850 --> 00:09:34.270 align:middle line:84% And then you simulate the model and figure out 00:09:34.270 --> 00:09:36.290 align:middle line:90% how well it fits the data. 00:09:36.290 --> 00:09:38.710 align:middle line:84% And if it fits reasonably well, then it's 00:09:38.710 --> 00:09:41.980 align:middle line:84% a success, that you have a logical explanation 00:09:41.980 --> 00:09:45.790 align:middle line:84% of in an artificial economy of how the world actually 00:09:45.790 --> 00:09:49.840 align:middle line:84% works, or in this case, past tense, was working. 00:09:49.840 --> 00:09:52.120 align:middle line:84% There should be another bullet point here. 00:09:52.120 --> 00:09:56.950 align:middle line:84% Because I like to feature and earmark that when it comes up, 00:09:56.950 --> 00:09:58.990 align:middle line:84% we go through various techniques. 00:09:58.990 --> 00:10:01.750 align:middle line:84% And the technique that's going to become apparent 00:10:01.750 --> 00:10:03.500 align:middle line:90% is dynamic programming. 00:10:03.500 --> 00:10:07.420 align:middle line:84% So most of the lectures up to this point, not all of them 00:10:07.420 --> 00:10:09.340 align:middle line:90% were static. 00:10:09.340 --> 00:10:12.430 align:middle line:84% We made an exception with that interest rate and income 00:10:12.430 --> 00:10:16.000 align:middle line:84% and substitution effects and a couple of the consumption 00:10:16.000 --> 00:10:17.230 align:middle line:90% set diagrams. 00:10:17.230 --> 00:10:20.030 align:middle line:84% So time is playing a big role here. 00:10:20.030 --> 00:10:22.510 align:middle line:84% And so if we're going to solve a problem or program, 00:10:22.510 --> 00:10:24.640 align:middle line:84% it's going to be a dynamic program. 00:10:24.640 --> 00:10:26.590 align:middle line:84% And there's some useful techniques 00:10:26.590 --> 00:10:30.320 align:middle line:84% that fall right out of the end of this lecture. 00:10:30.320 --> 00:10:33.760 align:middle line:84% So we're going to talk about actual and potential use 00:10:33.760 --> 00:10:39.130 align:middle line:84% of risk reduction arrangements, namely carrying over grain 00:10:39.130 --> 00:10:40.970 align:middle line:90% from one year to the next. 00:10:40.970 --> 00:10:43.660 align:middle line:84% This first line says alternative, 00:10:43.660 --> 00:10:46.690 align:middle line:84% begging the issue of what else have we looked at. 00:10:46.690 --> 00:10:50.500 align:middle line:84% Well, we looked at an individual putting his strips of land 00:10:50.500 --> 00:10:52.390 align:middle line:90% all over the village. 00:10:52.390 --> 00:10:56.500 align:middle line:84% And I referred to that as portfolio diversification 00:10:56.500 --> 00:10:57.790 align:middle line:90% at one point. 00:10:57.790 --> 00:11:00.530 align:middle line:84% We looked at, and we are about to see, again, 00:11:00.530 --> 00:11:03.830 align:middle line:84% the picture of the estates of the Bishop of Winchester, 00:11:03.830 --> 00:11:05.710 align:middle line:84% which were spread out over space, 00:11:05.710 --> 00:11:08.350 align:middle line:90% arguably, again, to diversify. 00:11:08.350 --> 00:11:10.930 align:middle line:84% And now we're going to be stuck in one village, 00:11:10.930 --> 00:11:13.570 align:middle line:84% and effectively, with one person in one village, 00:11:13.570 --> 00:11:19.030 align:middle line:84% on his or her own, looking for alternative ways of reducing 00:11:19.030 --> 00:11:20.780 align:middle line:90% exposure to risk. 00:11:20.780 --> 00:11:22.870 align:middle line:84% Now the stylized facts, if we go back 00:11:22.870 --> 00:11:25.480 align:middle line:84% to the same Bishop of Winchester accounts, 00:11:25.480 --> 00:11:28.240 align:middle line:84% we don't see a whole lot of carryover. 00:11:28.240 --> 00:11:33.310 align:middle line:84% There are one estate at the time, rare spikes where 00:11:33.310 --> 00:11:35.410 align:middle line:90% a carryover jumps up. 00:11:35.410 --> 00:11:37.330 align:middle line:84% And then the next year, it's down. 00:11:37.330 --> 00:11:39.520 align:middle line:84% I should say, or this comes up later, 00:11:39.520 --> 00:11:44.770 align:middle line:84% "carry over" means carried over from one harvest to the next. 00:11:44.770 --> 00:11:48.280 align:middle line:84% Obviously, time passes by the day and by the month. 00:11:48.280 --> 00:11:50.080 align:middle line:84% They only produce one crop a year. 00:11:50.080 --> 00:11:53.320 align:middle line:84% They have to have something to eat during the interim. 00:11:53.320 --> 00:11:55.810 align:middle line:84% So there is some storage going on. 00:11:55.810 --> 00:11:57.700 align:middle line:84% The issue is whether, by the time 00:11:57.700 --> 00:11:59.410 align:middle line:84% they get to the next harvest, there's 00:11:59.410 --> 00:12:00.910 align:middle line:90% any storage left at all. 00:12:00.910 --> 00:12:02.920 align:middle line:84% And if there is, we call it carryover. 00:12:02.920 --> 00:12:06.190 align:middle line:84% Well, likewise, if you look at the same estate accounts, 00:12:06.190 --> 00:12:08.950 align:middle line:84% at a point in time, you can see that some of them 00:12:08.950 --> 00:12:10.270 align:middle line:90% have carryover. 00:12:10.270 --> 00:12:11.960 align:middle line:90% But most of them don't. 00:12:11.960 --> 00:12:16.157 align:middle line:84% So this model is going to be similar to a macro model. 00:12:16.157 --> 00:12:17.740 align:middle line:84% That probably doesn't mean much to you 00:12:17.740 --> 00:12:22.240 align:middle line:84% right now, a neoclassical growth stabilization model. 00:12:22.240 --> 00:12:25.840 align:middle line:84% We're going to put in two kinds of storage technologies, 00:12:25.840 --> 00:12:29.590 align:middle line:84% grain in the bin for storage, but also seed in the ground, 00:12:29.590 --> 00:12:32.530 align:middle line:84% because that's another way to carry over the crop, 00:12:32.530 --> 00:12:34.073 align:middle line:90% by planting it. 00:12:34.073 --> 00:12:35.740 align:middle line:84% And we're going to choose the parameters 00:12:35.740 --> 00:12:38.020 align:middle line:84% to be consistent with the available data. 00:12:38.020 --> 00:12:42.040 align:middle line:84% And we're going to, again, look at the numerical predictions 00:12:42.040 --> 00:12:45.460 align:middle line:84% of the model and see if we can match the fact that carryover 00:12:45.460 --> 00:12:47.140 align:middle line:90% was not common. 00:12:47.140 --> 00:12:49.960 align:middle line:84% This goes more into how uncommon it 00:12:49.960 --> 00:12:54.010 align:middle line:84% was where, if they're not carrying anything over, 00:12:54.010 --> 00:12:56.380 align:middle line:84% and there's only one person in a village, 00:12:56.380 --> 00:12:59.150 align:middle line:84% like Robinson Crusoe as an abstraction, 00:12:59.150 --> 00:13:01.150 align:middle line:84% then they're eating all their output-- 00:13:01.150 --> 00:13:03.980 align:middle line:84% well, maybe-- over a 12-month time period. 00:13:03.980 --> 00:13:07.030 align:middle line:84% So variability in consumption is one to one 00:13:07.030 --> 00:13:09.440 align:middle line:90% with variability in output. 00:13:09.440 --> 00:13:13.330 align:middle line:84% And if output varies a lot, that's a lot of risk to bear. 00:13:13.330 --> 00:13:15.790 align:middle line:84% We have data from that same lord. 00:13:15.790 --> 00:13:19.300 align:middle line:84% Treating him as the single villager, 00:13:19.300 --> 00:13:22.360 align:middle line:84% we can look at output as representative output 00:13:22.360 --> 00:13:28.840 align:middle line:84% in the population and see, in the cross section in 1236, 00:13:28.840 --> 00:13:32.320 align:middle line:84% only three of the 18 states had carryover. 00:13:32.320 --> 00:13:35.560 align:middle line:84% In a good year, there was carryover. 00:13:35.560 --> 00:13:38.230 align:middle line:84% 1223 was good, for example, for one 00:13:38.230 --> 00:13:42.370 align:middle line:84% of those estates with 57% of the crop being 00:13:42.370 --> 00:13:44.140 align:middle line:90% stored till next year. 00:13:44.140 --> 00:13:48.710 align:middle line:84% And the following year for that same estate, 1224, 00:13:48.710 --> 00:13:50.450 align:middle line:90% zero carryover. 00:13:50.450 --> 00:13:51.960 align:middle line:90% This is a review picture. 00:13:51.960 --> 00:13:55.460 align:middle line:84% These are the estates of the Bishop of Winchester. 00:13:55.460 --> 00:13:57.710 align:middle line:84% And as I said last time, it's hard to figure out 00:13:57.710 --> 00:14:00.740 align:middle line:84% what country you're in, let alone where you are in it. 00:14:00.740 --> 00:14:03.870 align:middle line:84% But this is the English Channel down here. 00:14:03.870 --> 00:14:08.840 align:middle line:84% So all these dots are the Bishop's of Winchester estates. 00:14:08.840 --> 00:14:12.140 align:middle line:84% So that's the context that's generating the data. 00:14:12.140 --> 00:14:15.800 align:middle line:84% OK, two ways to store, well, actually, four. 00:14:15.800 --> 00:14:18.710 align:middle line:84% We're going to adopt two of them and rule out the two others. 00:14:18.710 --> 00:14:21.350 align:middle line:84% Not model the two others, I should say. 00:14:21.350 --> 00:14:24.500 align:middle line:84% In the artificial economy, first, grain 00:14:24.500 --> 00:14:28.280 align:middle line:84% can be stored after harvest, as I've been saying. 00:14:28.280 --> 00:14:31.460 align:middle line:84% Second, they could take some of that harvest 00:14:31.460 --> 00:14:34.730 align:middle line:84% and put the grain in the ground as seed 00:14:34.730 --> 00:14:37.160 align:middle line:90% planted for next year's crop. 00:14:37.160 --> 00:14:40.100 align:middle line:84% The other things we're not going to consider is taking the grain 00:14:40.100 --> 00:14:41.840 align:middle line:90% and feeding it to the animals. 00:14:41.840 --> 00:14:44.390 align:middle line:84% And eventually, you might kill the animal. 00:14:44.390 --> 00:14:47.390 align:middle line:84% So in some respects, you're carrying over the calories 00:14:47.390 --> 00:14:48.050 align:middle line:90% from the grain. 00:14:48.050 --> 00:14:51.920 align:middle line:84% But it's not very efficient, since the animals are also 00:14:51.920 --> 00:14:55.290 align:middle line:90% surviving on the grain. 00:14:55.290 --> 00:14:58.460 align:middle line:84% So we don't have animals in the model. 00:14:58.460 --> 00:15:01.910 align:middle line:84% Another more humorous possibility is beer. 00:15:01.910 --> 00:15:05.930 align:middle line:84% So they would take the grain and ferment it and make 00:15:05.930 --> 00:15:09.200 align:middle line:84% barley malt, for example, which English people still 00:15:09.200 --> 00:15:12.747 align:middle line:84% like to drink quite a bit, no doubt due to this history. 00:15:12.747 --> 00:15:14.330 align:middle line:84% But we're not going to do that either, 00:15:14.330 --> 00:15:16.050 align:middle line:90% no drinking in the model. 00:15:16.050 --> 00:15:19.130 align:middle line:84% So let's take this storage possibility. 00:15:19.130 --> 00:15:21.530 align:middle line:84% The most obvious one is you take the grain 00:15:21.530 --> 00:15:24.860 align:middle line:84% and put it in the bin, and potentially have some 00:15:24.860 --> 00:15:27.230 align:middle line:90% left or not by next harvest. 00:15:27.230 --> 00:15:32.750 align:middle line:84% How the depreciation rate being delta, why does grain go bad? 00:15:32.750 --> 00:15:36.320 align:middle line:84% Well, in England, it's a very wet climate. 00:15:36.320 --> 00:15:39.950 align:middle line:84% If it gets wet, and the grain gets wet if it's not secured, 00:15:39.950 --> 00:15:41.060 align:middle line:90% it spoils. 00:15:41.060 --> 00:15:44.030 align:middle line:84% Mildew, even if it doesn't directly rain on the grain, 00:15:44.030 --> 00:15:45.800 align:middle line:90% is not good for the grain. 00:15:45.800 --> 00:15:48.740 align:middle line:84% And grain gets eaten by rats and other rodents. 00:15:48.740 --> 00:15:50.900 align:middle line:84% And you may think that's humorous. 00:15:50.900 --> 00:15:53.090 align:middle line:90% But it is real. 00:15:53.090 --> 00:15:55.730 align:middle line:84% I know in Thailand, we would go and look 00:15:55.730 --> 00:15:58.520 align:middle line:84% at what they call the rice bank, which was a shed where 00:15:58.520 --> 00:16:00.200 align:middle line:90% they had stored the rice. 00:16:00.200 --> 00:16:02.030 align:middle line:84% And they were showing us the shed. 00:16:02.030 --> 00:16:04.970 align:middle line:84% And as they opened the door, all these mice ran out. 00:16:04.970 --> 00:16:07.070 align:middle line:90% So we're eating the grain. 00:16:07.070 --> 00:16:08.250 align:middle line:90% How big is delta? 00:16:08.250 --> 00:16:10.160 align:middle line:90% We have to calibrate this model. 00:16:10.160 --> 00:16:15.560 align:middle line:84% So drawing from McCloskey, you can see prices increasing month 00:16:15.560 --> 00:16:18.110 align:middle line:84% after month after month, from harvest 00:16:18.110 --> 00:16:20.060 align:middle line:90% up through the next month. 00:16:20.060 --> 00:16:25.500 align:middle line:84% And from this fact, McCloskey gets a depreciation rate of 30% 00:16:25.500 --> 00:16:28.430 align:middle line:84% So here's that table that he's using. 00:16:28.430 --> 00:16:31.550 align:middle line:84% Between September and October, grain prices 00:16:31.550 --> 00:16:36.620 align:middle line:84% inflated by 3 and 1/2%, between October and November, 2 and. 00:16:36.620 --> 00:16:37.520 align:middle line:90% 1/4%. 00:16:37.520 --> 00:16:40.490 align:middle line:84% So these are kind of grain price inflation rates 00:16:40.490 --> 00:16:41.690 align:middle line:90% through the year. 00:16:41.690 --> 00:16:46.550 align:middle line:84% And roughly, if you add them up and do a little compounding, 00:16:46.550 --> 00:16:48.230 align:middle line:90% you'll get 30%. 00:16:48.230 --> 00:16:49.730 align:middle line:90% Now who cares? 00:16:49.730 --> 00:16:54.830 align:middle line:84% Well, let's imagine that whatever you store at one date, 00:16:54.830 --> 00:16:59.930 align:middle line:84% you get back 1 minus delta that number, 00:16:59.930 --> 00:17:02.030 align:middle line:90% say, the following month. 00:17:02.030 --> 00:17:03.270 align:middle line:90% It's linear. 00:17:03.270 --> 00:17:05.930 align:middle line:84% You don't get out what you put in 1 to 1. 00:17:05.930 --> 00:17:12.109 align:middle line:84% If the depreciation were 30%, you put in 1, you get back 0.7. 00:17:12.109 --> 00:17:13.910 align:middle line:90% You put in 10, you get back 7. 00:17:13.910 --> 00:17:14.900 align:middle line:90% It's linear. 00:17:14.900 --> 00:17:19.280 align:middle line:84% The storage technology displays constant returns to scale. 00:17:19.280 --> 00:17:22.640 align:middle line:84% If you were a firm maximizing profits 00:17:22.640 --> 00:17:25.940 align:middle line:84% by the choice of how much to store over time, 00:17:25.940 --> 00:17:30.080 align:middle line:84% you would be thinking about this constant returns 00:17:30.080 --> 00:17:34.970 align:middle line:84% to scale technology and asking, how much would they be storing, 00:17:34.970 --> 00:17:37.790 align:middle line:84% depending on how much the price next month is 00:17:37.790 --> 00:17:39.600 align:middle line:90% relative to the price today. 00:17:39.600 --> 00:17:42.170 align:middle line:90% So it says exercise. 00:17:42.170 --> 00:17:44.330 align:middle line:84% So I urge you to write that down. 00:17:44.330 --> 00:17:46.700 align:middle line:84% I did it just to make sure this morning. 00:17:46.700 --> 00:17:50.960 align:middle line:84% It's, like, maximize P1 times y minus i, 00:17:50.960 --> 00:17:57.170 align:middle line:84% where i is investment, plus P2 times Y2 plus i multiplied 00:17:57.170 --> 00:17:58.460 align:middle line:90% by 1 minus delta. 00:17:58.460 --> 00:18:00.770 align:middle line:84% It's kind of silly to be describing this in words. 00:18:00.770 --> 00:18:03.800 align:middle line:84% Now what do you know about constant returns to scale? 00:18:03.800 --> 00:18:05.930 align:middle line:84% You already know, with constant returns 00:18:05.930 --> 00:18:08.630 align:middle line:84% to scale when we did the Production Lecture 4, 00:18:08.630 --> 00:18:11.870 align:middle line:84% that the solution is either infinite or indeterminate 00:18:11.870 --> 00:18:12.740 align:middle line:90% or zero. 00:18:12.740 --> 00:18:14.750 align:middle line:90% Well, it's not indeterminate. 00:18:14.750 --> 00:18:17.840 align:middle line:84% And it's not zero, either, because they were storing 00:18:17.840 --> 00:18:19.400 align:middle line:90% from month to month, at least. 00:18:19.400 --> 00:18:23.750 align:middle line:84% So the premise is that, in order to compensate the economics is, 00:18:23.750 --> 00:18:26.960 align:middle line:84% in order to compensate for the loss of the crop, 00:18:26.960 --> 00:18:30.050 align:middle line:84% rather than selling it today at a certain price, 00:18:30.050 --> 00:18:31.850 align:middle line:84% they can sell it tomorrow they're going 00:18:31.850 --> 00:18:35.100 align:middle line:90% to have 70% left, a 30% loss. 00:18:35.100 --> 00:18:38.810 align:middle line:84% So the price has to accumulate at 30% over the year. 00:18:38.810 --> 00:18:41.090 align:middle line:84% I'm going back and forth between month and years. 00:18:41.090 --> 00:18:41.840 align:middle line:90% Sorry. 00:18:41.840 --> 00:18:43.800 align:middle line:84% So that's the economic intuition. 00:18:43.800 --> 00:18:45.740 align:middle line:90% And you already know the math. 00:18:45.740 --> 00:18:47.870 align:middle line:84% So that's one advantage for us having 00:18:47.870 --> 00:18:50.600 align:middle line:84% spent some time on production and, in particular, 00:18:50.600 --> 00:18:52.410 align:middle line:90% on profit maximization. 00:18:52.410 --> 00:18:58.580 align:middle line:84% So you know how McCloskey is calibrating parameter delta. 00:18:58.580 --> 00:19:03.890 align:middle line:84% Second possibility is putting seed in the ground. 00:19:03.890 --> 00:19:07.970 align:middle line:84% Now it turns out that's a pretty healthy return. 00:19:07.970 --> 00:19:10.820 align:middle line:90% We have some data on that. 00:19:10.820 --> 00:19:14.810 align:middle line:84% And the yield to seed ratio is about 2.6 00:19:14.810 --> 00:19:18.870 align:middle line:84% for one of these manners and 1.67 for the other one. 00:19:18.870 --> 00:19:25.430 align:middle line:84% So that's a yield-to-seed ratio, meaning you get roughly, 00:19:25.430 --> 00:19:28.820 align:middle line:84% taking the average, two units of output 00:19:28.820 --> 00:19:30.810 align:middle line:90% for every units of input. 00:19:30.810 --> 00:19:33.400 align:middle line:90% So that's pretty productive. 00:19:33.400 --> 00:19:35.280 align:middle line:84% So if we stopped right there, you'd 00:19:35.280 --> 00:19:37.290 align:middle line:84% say, well, putting seed in the ground 00:19:37.290 --> 00:19:39.750 align:middle line:84% is a lot more productive than depreciation, 00:19:39.750 --> 00:19:42.750 align:middle line:90% even setting aside the risk. 00:19:42.750 --> 00:19:44.980 align:middle line:84% Well, so there's a couple of things. 00:19:44.980 --> 00:19:48.150 align:middle line:84% One, there's a limit to how much seed you can put in the ground. 00:19:48.150 --> 00:19:50.860 align:middle line:84% And two, we're going to be considering the risk. 00:19:50.860 --> 00:19:53.280 align:middle line:90% So here's the technology. 00:19:53.280 --> 00:19:55.890 align:middle line:90% If you planted Kt units-- 00:19:55.890 --> 00:20:00.120 align:middle line:84% put Kt seeds in the ground, at t minus 1, 00:20:00.120 --> 00:20:03.630 align:middle line:84% the output would be a function f of that amount 00:20:03.630 --> 00:20:05.190 align:middle line:90% of seed, the harvest. 00:20:05.190 --> 00:20:07.230 align:middle line:90% And there's some shock epsilon. 00:20:07.230 --> 00:20:10.320 align:middle line:84% In fact, let's assume this is a linear technology. 00:20:10.320 --> 00:20:12.720 align:middle line:84% So whatever seed got put in the ground, 00:20:12.720 --> 00:20:14.490 align:middle line:90% alpha times it is the output. 00:20:14.490 --> 00:20:15.900 align:middle line:90% Alpha could be 2. 00:20:15.900 --> 00:20:19.710 align:middle line:84% And epsilon is a shock that's going to make it random 00:20:19.710 --> 00:20:20.970 align:middle line:90% and make it move around. 00:20:20.970 --> 00:20:26.220 align:middle line:84% OK, first thing, you can't plant an unbounded amount of seed. 00:20:26.220 --> 00:20:29.850 align:middle line:84% In fact, the best way to think about this is the seed per unit 00:20:29.850 --> 00:20:31.110 align:middle line:90% land is constant. 00:20:31.110 --> 00:20:33.960 align:middle line:84% You have so much land, there's no point 00:20:33.960 --> 00:20:35.370 align:middle line:90% in putting down more seed. 00:20:35.370 --> 00:20:39.180 align:middle line:84% So it's really a decision about how much land to plant. 00:20:39.180 --> 00:20:40.770 align:middle line:84% And we'll put an upper bound on that 00:20:40.770 --> 00:20:42.540 align:middle line:84% because there are finite land holdings. 00:20:42.540 --> 00:20:46.740 align:middle line:84% So K bar is going to be the most land that you can plant. 00:20:46.740 --> 00:20:48.390 align:middle line:90% You plant all of your land. 00:20:48.390 --> 00:20:51.300 align:middle line:84% And if you were to try to put down more seed, 00:20:51.300 --> 00:20:52.890 align:middle line:90% it's like throwing it away. 00:20:52.890 --> 00:20:55.830 align:middle line:84% You're still going to only have the maximum amount, 00:20:55.830 --> 00:20:59.860 align:middle line:84% K bar in this case times epsilon alpha. 00:20:59.860 --> 00:21:05.335 align:middle line:84% So we're about ready to get to a key equation here. 00:21:05.335 --> 00:21:06.210 align:middle line:90% Let's do it this way. 00:21:06.210 --> 00:21:09.000 align:middle line:84% If you focus on equation 1, you've 00:21:09.000 --> 00:21:15.420 align:middle line:84% got output f, harvest, which is a function of the seed planted 00:21:15.420 --> 00:21:19.480 align:middle line:84% last year plus an influence of epsilon. 00:21:19.480 --> 00:21:22.720 align:middle line:84% Now you could have had something in storage as well. 00:21:22.720 --> 00:21:24.360 align:middle line:84% Imagine they had done both things. 00:21:24.360 --> 00:21:25.860 align:middle line:84% So they put some seed in the ground. 00:21:25.860 --> 00:21:27.480 align:middle line:90% And they had some storage. 00:21:27.480 --> 00:21:29.310 align:middle line:84% You're going to get 1 minus delta times 00:21:29.310 --> 00:21:31.200 align:middle line:90% that amount of storage back. 00:21:31.200 --> 00:21:33.660 align:middle line:84% And that would be the end of the story for consumption, 00:21:33.660 --> 00:21:35.850 align:middle line:84% except they have to plan for next year. 00:21:35.850 --> 00:21:39.270 align:middle line:84% So they still have these two options, storage planted now 00:21:39.270 --> 00:21:43.200 align:middle line:84% at t that will be available at t plus 1, 00:21:43.200 --> 00:21:46.360 align:middle line:84% and seed, which they can squirrel away now, 00:21:46.360 --> 00:21:49.380 align:middle line:84% which will be planted and influence 00:21:49.380 --> 00:21:53.290 align:middle line:90% the harvest at Kt plus 1. 00:21:53.290 --> 00:21:55.540 align:middle line:90% It's kind of weird dating. 00:21:55.540 --> 00:21:57.670 align:middle line:90% Please be alert to that. 00:21:57.670 --> 00:22:02.380 align:middle line:84% In fact, the last slide for today will change the dating. 00:22:02.380 --> 00:22:05.020 align:middle line:84% Different people use different conventions. 00:22:05.020 --> 00:22:07.780 align:middle line:84% What I'm complaining about, or alerting you, 00:22:07.780 --> 00:22:10.690 align:middle line:84% is this is consumption at date t. 00:22:10.690 --> 00:22:14.680 align:middle line:84% And these things, seed in the ground and inventory, 00:22:14.680 --> 00:22:16.270 align:middle line:90% have a date t on them. 00:22:16.270 --> 00:22:19.900 align:middle line:84% But they are predetermined from t minus 1. 00:22:19.900 --> 00:22:22.120 align:middle line:84% They are not decision variables at t. 00:22:22.120 --> 00:22:27.040 align:middle line:84% The decision variables at t are labeled t plus 1. 00:22:27.040 --> 00:22:29.560 align:middle line:84% The pro case for doing it this way 00:22:29.560 --> 00:22:33.580 align:middle line:84% is that we care about the state variable at a given day t, 00:22:33.580 --> 00:22:37.630 align:middle line:84% and that's determined by the seed that they planted before 00:22:37.630 --> 00:22:40.870 align:middle line:84% and the inventory that they have from before. 00:22:40.870 --> 00:22:42.490 align:middle line:90% Anyway, just to alert you. 00:22:42.490 --> 00:22:46.100 align:middle line:84% I trust you will interrupt me when you have questions. 00:22:46.100 --> 00:22:49.060 align:middle line:84% So this is the amount that's available. 00:22:49.060 --> 00:22:52.810 align:middle line:84% This is the amount that you squirrel away for next year. 00:22:52.810 --> 00:22:55.430 align:middle line:84% And the difference between what's available 00:22:55.430 --> 00:22:58.720 align:middle line:84% and what you store in one way or another is consumption. 00:22:58.720 --> 00:23:01.780 align:middle line:84% Now the risk averse, we discussed that a minute 00:23:01.780 --> 00:23:05.290 align:middle line:84% ago, so this is a strictly concave utility 00:23:05.290 --> 00:23:10.190 align:middle line:84% function over consumption over a finite number of dates t. 00:23:10.190 --> 00:23:14.810 align:middle line:84% So this runs from 1 to capital T. Hopefully, 00:23:14.810 --> 00:23:17.810 align:middle line:84% capital T is really a long time away. 00:23:17.810 --> 00:23:20.060 align:middle line:90% And I'll come back to that. 00:23:20.060 --> 00:23:21.770 align:middle line:84% Alternatively, I could have written 00:23:21.770 --> 00:23:23.450 align:middle line:90% cap T equal to infinity. 00:23:23.450 --> 00:23:25.100 align:middle line:90% They live forever. 00:23:25.100 --> 00:23:26.850 align:middle line:84% But we're not quite ready to do that. 00:23:26.850 --> 00:23:30.140 align:middle line:84% So I'm just going to show you finite time horizon first 00:23:30.140 --> 00:23:33.290 align:middle line:84% and then argue how to extend it at the end of the class 00:23:33.290 --> 00:23:35.600 align:middle line:84% when we get to dynamic programming. 00:23:35.600 --> 00:23:37.760 align:middle line:90% There's a discounted utility. 00:23:37.760 --> 00:23:41.630 align:middle line:84% Beta is a discount rate, beta some number less than 1. 00:23:41.630 --> 00:23:43.610 align:middle line:84% So you care about the present more 00:23:43.610 --> 00:23:46.220 align:middle line:84% than you care about the future, present utility 00:23:46.220 --> 00:23:47.840 align:middle line:90% more than future utility. 00:23:47.840 --> 00:23:50.540 align:middle line:84% This is, with beta less than 1, geometric. 00:23:50.540 --> 00:23:54.470 align:middle line:84% You can see, as the horizon gets further and further away, 00:23:54.470 --> 00:23:58.560 align:middle line:84% you care less and less relative to today's consumption. 00:23:58.560 --> 00:24:02.510 align:middle line:84% But these are strictly concave, so you don't like variability. 00:24:02.510 --> 00:24:05.270 align:middle line:84% We talked about concave utility when we 00:24:05.270 --> 00:24:07.100 align:middle line:90% introduced the concept of risk. 00:24:07.100 --> 00:24:11.030 align:middle line:84% Here, even if there were no risks, 00:24:11.030 --> 00:24:13.940 align:middle line:84% they would try to avoid variability in consumption. 00:24:13.940 --> 00:24:17.000 align:middle line:84% They'd rather have the same consumption every day, 00:24:17.000 --> 00:24:19.970 align:middle line:84% rather than having it low one date and high another. 00:24:19.970 --> 00:24:22.100 align:middle line:84% Because the utility consequence of that 00:24:22.100 --> 00:24:24.390 align:middle line:84% would be less than having the same amount. 00:24:24.390 --> 00:24:28.280 align:middle line:84% So that's where the context drift concavity is coming in. 00:24:28.280 --> 00:24:33.140 align:middle line:84% So we already calibrated the depreciation rate at 30%. 00:24:33.140 --> 00:24:37.380 align:middle line:84% The discount rate we're going to put at 0.95. 00:24:37.380 --> 00:24:40.520 align:middle line:84% Which is kind of standard, although we could vary it, 00:24:40.520 --> 00:24:41.990 align:middle line:90% we could make it lower. 00:24:41.990 --> 00:24:43.730 align:middle line:84% Once we have the program written, 00:24:43.730 --> 00:24:45.710 align:middle line:90% we can plug in whatever we want. 00:24:45.710 --> 00:24:47.840 align:middle line:84% Ideally, though, the spirit of this 00:24:47.840 --> 00:24:51.110 align:middle line:84% should be that we're somewhat confident about these numbers 00:24:51.110 --> 00:24:52.430 align:middle line:90% we're about to choose. 00:24:52.430 --> 00:24:56.150 align:middle line:84% Constant relative risk aversion, I defined that utility function 00:24:56.150 --> 00:24:57.380 align:middle line:90% last time. 00:24:57.380 --> 00:25:01.820 align:middle line:84% That's seed to the gamma divided by sometimes normalized. 00:25:01.820 --> 00:25:03.530 align:middle line:84% And we're going to set that at 0.5. 00:25:03.530 --> 00:25:05.730 align:middle line:90% This is kind of a low number. 00:25:05.730 --> 00:25:07.820 align:middle line:84% It's definitely a strictly concave utility, 00:25:07.820 --> 00:25:10.950 align:middle line:84% but they may have been even more risk averse than that. 00:25:10.950 --> 00:25:16.670 align:middle line:84% However, you'll see this ex post justification that all three 00:25:16.670 --> 00:25:18.560 align:middle line:84% of these numbers kind of work in terms 00:25:18.560 --> 00:25:21.260 align:middle line:90% of explaining the paradox. 00:25:21.260 --> 00:25:24.680 align:middle line:84% Namely, they're about to starve to death every 12 years. 00:25:24.680 --> 00:25:26.570 align:middle line:84% And they weren't storing anything. 00:25:26.570 --> 00:25:28.970 align:middle line:90% That's the paradox. 00:25:28.970 --> 00:25:30.920 align:middle line:84% Back to that epsilon, this is the way 00:25:30.920 --> 00:25:32.240 align:middle line:90% you put the uncertainty in. 00:25:32.240 --> 00:25:37.440 align:middle line:84% One unit of land planted today yields two tomorrow. 00:25:37.440 --> 00:25:40.010 align:middle line:84% So that's the yield-to-seed ratio. 00:25:40.010 --> 00:25:42.140 align:middle line:84% But there is a little uncertainty. 00:25:42.140 --> 00:25:44.900 align:middle line:84% 10 out of 12 times, you get 2 tomorrow. 00:25:44.900 --> 00:25:48.860 align:middle line:84% But 1/12 of the time, you get 1/2 of it. 00:25:48.860 --> 00:25:53.370 align:middle line:84% And 1/12 of the time, you get 4, double it. 00:25:53.370 --> 00:25:57.650 align:middle line:84% So the expected yield is a little bit above 2, 00:25:57.650 --> 00:26:01.040 align:middle line:84% since the weights, 1/12 are same on the low and the high end. 00:26:01.040 --> 00:26:04.010 align:middle line:90% But 4 is 2 times greater than 2. 00:26:04.010 --> 00:26:06.320 align:middle line:84% And the difference between 2 and 1 is just 1. 00:26:06.320 --> 00:26:10.280 align:middle line:84% So it's just the expected value is slightly higher than 2. 00:26:10.280 --> 00:26:13.760 align:middle line:84% We're going to set the upper bound on storage at 1. 00:26:13.760 --> 00:26:18.590 align:middle line:84% And this number, 1/12, I actually just set y. 00:26:18.590 --> 00:26:23.300 align:middle line:84% And you may be reminded of that when you go back 00:26:23.300 --> 00:26:26.300 align:middle line:84% and look at, in the data, the coefficient of variation 00:26:26.300 --> 00:26:28.610 align:middle line:84% that we introduced last time of yields. 00:26:28.610 --> 00:26:33.410 align:middle line:84% It was 0.35, which meant roughly every 12 years, 00:26:33.410 --> 00:26:38.070 align:middle line:84% output fell below less than half of the mean value. 00:26:38.070 --> 00:26:43.080 align:middle line:84% So we called that disaster last time for these scattered plots. 00:26:43.080 --> 00:26:48.192 align:middle line:84% So this mass over here under the solid density is 1/12. 00:26:48.192 --> 00:26:50.150 align:middle line:84% [INAUDIBLE] it's also true they could get lucky 00:26:50.150 --> 00:26:51.810 align:middle line:90% and have a high yield. 00:26:51.810 --> 00:26:54.020 align:middle line:84% So you saw this picture last time. 00:26:54.020 --> 00:26:55.880 align:middle line:84% I'm really just using it to remind you 00:26:55.880 --> 00:26:59.180 align:middle line:84% of the context of where the risk is coming from. 00:26:59.180 --> 00:27:01.730 align:middle line:84% We wanted-- we just did calibrate 00:27:01.730 --> 00:27:04.760 align:middle line:84% the yield-to-seed ratio in such a way 00:27:04.760 --> 00:27:09.220 align:middle line:84% as to deliver disaster every 12 years, on average. 00:27:09.220 --> 00:27:13.610 align:middle line:84% OK, so let's look at the dynamic decision problem. 00:27:13.610 --> 00:27:16.910 align:middle line:84% Let's call this the total available this year 00:27:16.910 --> 00:27:20.060 align:middle line:84% and denote, on the y-axis so to speak, 00:27:20.060 --> 00:27:23.660 align:middle line:84% the amount of each of the possible savings decisions, 00:27:23.660 --> 00:27:28.440 align:middle line:84% either of storage or planting, back to the key equation. 00:27:28.440 --> 00:27:32.030 align:middle line:84% The amount available is this, output plus storage. 00:27:32.030 --> 00:27:36.978 align:middle line:84% And the savings decisions are inventory or seed. 00:27:36.978 --> 00:27:38.270 align:middle line:90% So that's what's being plotted. 00:27:38.270 --> 00:27:41.260 align:middle line:90% 00:27:41.260 --> 00:27:44.770 align:middle line:84% But we are varying the total amount available 00:27:44.770 --> 00:27:47.500 align:middle line:90% going from 0 to 6. 00:27:47.500 --> 00:27:49.580 align:middle line:90% And let's look at-- 00:27:49.580 --> 00:27:54.530 align:middle line:90% oh, where is this coming from? 00:27:54.530 --> 00:27:57.400 align:middle line:84% So we actually solved this problem. 00:27:57.400 --> 00:28:01.960 align:middle line:84% We maximize 2 subject to 1, putting 00:28:01.960 --> 00:28:04.620 align:middle line:90% in all these parameters. 00:28:04.620 --> 00:28:07.566 align:middle line:90% And I didn't show you the code. 00:28:07.566 --> 00:28:09.880 align:middle line:84% You can solve it on Matlab or something. 00:28:09.880 --> 00:28:12.730 align:middle line:90% I'm not asking you to do that. 00:28:12.730 --> 00:28:17.140 align:middle line:84% But anyway, it's, like, this picture could be anything. 00:28:17.140 --> 00:28:21.690 align:middle line:84% It's pinned down, because it's the solution to a max problem 00:28:21.690 --> 00:28:24.580 align:middle line:90% at the calibrated parameters. 00:28:24.580 --> 00:28:27.960 align:middle line:84% So this is our model version of what they were doing, in fact. 00:28:27.960 --> 00:28:34.890 align:middle line:84% So if the total amount available was a little less than 1, 00:28:34.890 --> 00:28:38.940 align:middle line:84% then they were not planting all the land, 00:28:38.940 --> 00:28:40.870 align:middle line:90% despite that high yield. 00:28:40.870 --> 00:28:44.700 align:middle line:84% However, for anything else, they hit the upper bound, 00:28:44.700 --> 00:28:47.070 align:middle line:84% and they plant everything they can 00:28:47.070 --> 00:28:51.130 align:middle line:90% at the upper bound, which was 1. 00:28:51.130 --> 00:28:53.080 align:middle line:84% Let's look at the storage decision. 00:28:53.080 --> 00:28:55.400 align:middle line:84% As a function of the total available, 00:28:55.400 --> 00:28:59.080 align:middle line:84% it's only when the total available is 3 or higher 00:28:59.080 --> 00:29:01.630 align:middle line:90% that positive storage kicks in. 00:29:01.630 --> 00:29:05.260 align:middle line:84% And when it does, it's moving linearly 00:29:05.260 --> 00:29:07.870 align:middle line:84% with the amount available this year. 00:29:07.870 --> 00:29:09.730 align:middle line:90% It's actually not moving 1 to 1. 00:29:09.730 --> 00:29:14.230 align:middle line:84% If it were 1 to 1, higher available means higher storage, 00:29:14.230 --> 00:29:16.750 align:middle line:84% it would be parallel with this 45 degree line. 00:29:16.750 --> 00:29:18.580 align:middle line:90% But it's not. 00:29:18.580 --> 00:29:20.170 align:middle line:90% So they never store 1 to 1. 00:29:20.170 --> 00:29:23.390 align:middle line:84% They may store less than the harvest. 00:29:23.390 --> 00:29:28.240 align:middle line:84% But they are storing some, something here. 00:29:28.240 --> 00:29:28.740 align:middle line:90% All right. 00:29:28.740 --> 00:29:31.860 align:middle line:90% So those are the savings plans. 00:29:31.860 --> 00:29:34.230 align:middle line:84% Things to keep in the back of your mind 00:29:34.230 --> 00:29:36.630 align:middle line:84% when we go through some of the other slides 00:29:36.630 --> 00:29:39.900 align:middle line:84% is the fact that mostly they put all the seed in the ground, 00:29:39.900 --> 00:29:43.800 align:middle line:84% but not when the amount available is less 00:29:43.800 --> 00:29:47.970 align:middle line:84% and the amount of storage is zero, 00:29:47.970 --> 00:29:51.600 align:middle line:84% unless the amount available is above 3. 00:29:51.600 --> 00:29:54.300 align:middle line:90% Now how much is consumption? 00:29:54.300 --> 00:29:58.920 align:middle line:84% Well, back to this thing, so consumption 00:29:58.920 --> 00:30:01.560 align:middle line:84% is the difference between the amount available and those two 00:30:01.560 --> 00:30:03.130 align:middle line:90% savings decisions. 00:30:03.130 --> 00:30:05.700 align:middle line:84% So if you look at this diagram, consumption 00:30:05.700 --> 00:30:08.320 align:middle line:84% is the difference between the amount available 00:30:08.320 --> 00:30:10.290 align:middle line:90% and the sum of these decisions. 00:30:10.290 --> 00:30:12.930 align:middle line:84% This 45-degree line is now playing a role. 00:30:12.930 --> 00:30:17.070 align:middle line:84% Because the amount available, say, 2, 00:30:17.070 --> 00:30:19.270 align:middle line:90% go up to the 45-degree line. 00:30:19.270 --> 00:30:22.080 align:middle line:90% Then this amount is 2. 00:30:22.080 --> 00:30:24.430 align:middle line:84% That's the definition of the 45-degree line. 00:30:24.430 --> 00:30:24.930 align:middle line:90% Right? 00:30:24.930 --> 00:30:27.960 align:middle line:84% It's a way to convert the units on the x-axis 00:30:27.960 --> 00:30:29.880 align:middle line:84% to the same amount on the y-axis. 00:30:29.880 --> 00:30:31.080 align:middle line:90% That's convenient here. 00:30:31.080 --> 00:30:32.880 align:middle line:84% Because we can look at, OK, here's 00:30:32.880 --> 00:30:36.480 align:middle line:84% the amount that would be available next year 00:30:36.480 --> 00:30:37.950 align:middle line:90% if they didn't save anything. 00:30:37.950 --> 00:30:40.200 align:middle line:84% But they are storing in this case. 00:30:40.200 --> 00:30:44.040 align:middle line:84% So consumption is the difference between this 45-degree line 00:30:44.040 --> 00:30:47.190 align:middle line:84% and the sum of these two other lines. 00:30:47.190 --> 00:30:50.220 align:middle line:84% That means that as we get over here, 00:30:50.220 --> 00:30:52.980 align:middle line:84% the 45-degree line is really close to the only thing they're 00:30:52.980 --> 00:30:56.620 align:middle line:84% doing, which is planting seed, and very close. 00:30:56.620 --> 00:31:01.440 align:middle line:84% So when you have less than 1.1 available this year, 00:31:01.440 --> 00:31:02.580 align:middle line:90% they're still eating. 00:31:02.580 --> 00:31:04.410 align:middle line:84% And they're not planting all the land, 00:31:04.410 --> 00:31:06.060 align:middle line:90% but they're not eating much. 00:31:06.060 --> 00:31:07.650 align:middle line:90% It's a really hard problem. 00:31:07.650 --> 00:31:09.780 align:middle line:84% Do we starve now or starve later? 00:31:09.780 --> 00:31:14.310 align:middle line:84% I mean, they kind of gamble on getting a reasonable return. 00:31:14.310 --> 00:31:18.480 align:middle line:84% If they ate the seed, they have nothing tomorrow. 00:31:18.480 --> 00:31:20.230 align:middle line:84% They're starving to death for sure. 00:31:20.230 --> 00:31:22.950 align:middle line:84% In fact, I don't if you've heard this expression, 00:31:22.950 --> 00:31:24.960 align:middle line:90% don't eat the seed corn. 00:31:24.960 --> 00:31:27.600 align:middle line:84% Yeah, that came right out of this episode. 00:31:27.600 --> 00:31:30.120 align:middle line:84% Eating the seed corn is the worst thing you could do. 00:31:30.120 --> 00:31:33.330 align:middle line:84% Because you're kind of guaranteeing zero for tomorrow. 00:31:33.330 --> 00:31:35.100 align:middle line:90% But they actually do cut back. 00:31:35.100 --> 00:31:36.180 align:middle line:90% They eat some of it. 00:31:36.180 --> 00:31:41.310 align:middle line:84% OK, so this is a related way to think about the dynamics. 00:31:41.310 --> 00:31:43.920 align:middle line:84% Let's take total available this year, 00:31:43.920 --> 00:31:46.260 align:middle line:84% implement those two savings decisions 00:31:46.260 --> 00:31:49.380 align:middle line:84% on the previous diagram, and then ask 00:31:49.380 --> 00:31:51.870 align:middle line:84% how much are you going to have next year. 00:31:51.870 --> 00:31:53.535 align:middle line:90% Sorry if I'm belaboring this. 00:31:53.535 --> 00:31:56.040 align:middle line:84% This is the amount available this year. 00:31:56.040 --> 00:31:58.110 align:middle line:90% You can save it or eat it. 00:31:58.110 --> 00:32:01.860 align:middle line:84% If you save it in one or the other of these two devices, 00:32:01.860 --> 00:32:05.910 align:middle line:84% we're now talking about the first part of this equation 00:32:05.910 --> 00:32:06.750 align:middle line:90% with the two terms. 00:32:06.750 --> 00:32:09.630 align:middle line:90% But the date would be t plus 1. 00:32:09.630 --> 00:32:13.820 align:middle line:84% So this would be the amount available next year, 00:32:13.820 --> 00:32:16.970 align:middle line:84% given the amount available this year and the endogenously 00:32:16.970 --> 00:32:20.000 align:middle line:84% determined-- oh, I used that word for the first time-- 00:32:20.000 --> 00:32:24.500 align:middle line:84% endogenously determined savings decisions. 00:32:24.500 --> 00:32:27.740 align:middle line:84% So here, we can't plot savings and the total amount 00:32:27.740 --> 00:32:31.890 align:middle line:84% available unless we're in some 3-dimensional space. 00:32:31.890 --> 00:32:33.890 align:middle line:84% So the savings decisions are suppressed, 00:32:33.890 --> 00:32:35.420 align:middle line:90% though they're still active. 00:32:35.420 --> 00:32:38.690 align:middle line:84% Well, first of all, output is stochastic. 00:32:38.690 --> 00:32:43.250 align:middle line:84% Epsilon could be and likely to be medium, but it could be high 00:32:43.250 --> 00:32:45.170 align:middle line:90% or it could be low. 00:32:45.170 --> 00:32:48.880 align:middle line:84% So the amount that's available next year, even given 00:32:48.880 --> 00:32:53.660 align:middle line:84% the savings decisions, will vary with the epsilon. 00:32:53.660 --> 00:32:57.190 align:middle line:84% So if you had three units available this year, which 00:32:57.190 --> 00:32:59.500 align:middle line:84% is right on the margin of storing, 00:32:59.500 --> 00:33:04.450 align:middle line:84% say zero, the most likely outcome next year is medium. 00:33:04.450 --> 00:33:07.820 align:middle line:84% And hence, there would be two units available next year. 00:33:07.820 --> 00:33:10.880 align:middle line:84% And so there would be no storage next year, either. 00:33:10.880 --> 00:33:13.360 align:middle line:84% So what do those words correspond to? 00:33:13.360 --> 00:33:16.330 align:middle line:84% You're at three units available now. 00:33:16.330 --> 00:33:19.330 align:middle line:84% The most likely outcome next year is medium. 00:33:19.330 --> 00:33:21.910 align:middle line:84% The amount available next year is two. 00:33:21.910 --> 00:33:24.340 align:middle line:84% And, again, from the savings decisions, 00:33:24.340 --> 00:33:28.720 align:middle line:84% storage at three and at two is zero, so no storage. 00:33:28.720 --> 00:33:31.490 align:middle line:84% So this is like what would happen over time. 00:33:31.490 --> 00:33:33.010 align:middle line:90% But let's just come back. 00:33:33.010 --> 00:33:35.200 align:middle line:84% Suppose we redid this experiment, 00:33:35.200 --> 00:33:37.510 align:middle line:84% and you had two units available this year, 00:33:37.510 --> 00:33:39.760 align:middle line:84% and then you got the medium output. 00:33:39.760 --> 00:33:41.180 align:middle line:90% You would have two units. 00:33:41.180 --> 00:33:43.210 align:middle line:90% So what does that referring to? 00:33:43.210 --> 00:33:46.100 align:middle line:84% If we have two units available this year, 00:33:46.100 --> 00:33:49.510 align:middle line:84% and you had the medium output, which is the most likely thing, 00:33:49.510 --> 00:33:52.790 align:middle line:84% you get 2 units available for the following year. 00:33:52.790 --> 00:33:55.660 align:middle line:84% So two in and two out is the most likely outcome 00:33:55.660 --> 00:33:56.830 align:middle line:90% if you start with two. 00:33:56.830 --> 00:33:59.740 align:middle line:84% So you could refer to that as a kind of steady state, 00:33:59.740 --> 00:34:02.020 align:middle line:90% meaning things aren't moving. 00:34:02.020 --> 00:34:03.200 align:middle line:90% Actually, they are moving. 00:34:03.200 --> 00:34:05.260 align:middle line:84% So technically, it's not a steady state, 00:34:05.260 --> 00:34:07.120 align:middle line:84% because we picked a particular shock. 00:34:07.120 --> 00:34:09.070 align:middle line:90% But it is the most likely shock. 00:34:09.070 --> 00:34:12.400 align:middle line:84% So it's approximately true that if you have two today, 00:34:12.400 --> 00:34:14.139 align:middle line:84% you're going to have two tomorrow. 00:34:14.139 --> 00:34:17.230 align:middle line:84% But if you had two units available this year, 00:34:17.230 --> 00:34:22.300 align:middle line:84% and you got the low output, epsilon 1/12 being probability, 00:34:22.300 --> 00:34:24.909 align:middle line:84% the output available next year is about one. 00:34:24.909 --> 00:34:27.909 align:middle line:84% And at that point, you're not planting all the land. 00:34:27.909 --> 00:34:29.139 align:middle line:90% So that's over here. 00:34:29.139 --> 00:34:31.690 align:middle line:84% You could have two today and get really unlucky 00:34:31.690 --> 00:34:35.620 align:middle line:84% and have less than two available, about one available 00:34:35.620 --> 00:34:36.500 align:middle line:90% next year. 00:34:36.500 --> 00:34:39.520 align:middle line:84% And, again, if you memorized a bit, those savings diagrams, 00:34:39.520 --> 00:34:43.239 align:middle line:84% with one available, you're not storing. 00:34:43.239 --> 00:34:46.480 align:middle line:84% And you're eating some of the seed. 00:34:46.480 --> 00:34:48.000 align:middle line:90% So let's go the other way. 00:34:48.000 --> 00:34:50.949 align:middle line:84% Suppose you had two units available, which is, roughly 00:34:50.949 --> 00:34:54.219 align:middle line:84% speaking, a reasonable starting point, and you got lucky 00:34:54.219 --> 00:34:55.820 align:middle line:90% and got the high output. 00:34:55.820 --> 00:34:57.580 align:middle line:84% Then you would finally be storing. 00:34:57.580 --> 00:35:00.880 align:middle line:84% Suppose you had two, and instead of getting the medium yield, 00:35:00.880 --> 00:35:02.020 align:middle line:90% you got the high yield. 00:35:02.020 --> 00:35:04.940 align:middle line:84% Then you'd have four units available next year. 00:35:04.940 --> 00:35:06.850 align:middle line:84% And at four units available, there's 00:35:06.850 --> 00:35:08.900 align:middle line:90% definitely storage going on. 00:35:08.900 --> 00:35:10.180 align:middle line:90% So that's carryover. 00:35:10.180 --> 00:35:11.920 align:middle line:90% Rare, but it can happen. 00:35:11.920 --> 00:35:13.150 align:middle line:90% Does it persist? 00:35:13.150 --> 00:35:16.030 align:middle line:84% Well, suppose, just to run this experiment, 00:35:16.030 --> 00:35:18.550 align:middle line:84% that you get lucky and lucky and lucky. 00:35:18.550 --> 00:35:21.400 align:middle line:90% So where would you end up? 00:35:21.400 --> 00:35:24.490 align:middle line:84% You'd end up at roughly a little above four. 00:35:24.490 --> 00:35:27.760 align:middle line:84% Four replicates itself, four in, four out. 00:35:27.760 --> 00:35:29.270 align:middle line:90% It's on the 45-degree line. 00:35:29.270 --> 00:35:31.120 align:middle line:84% So that's the best case scenario, 00:35:31.120 --> 00:35:32.690 align:middle line:90% that you would converge to that. 00:35:32.690 --> 00:35:34.180 align:middle line:90% And you would be storing. 00:35:34.180 --> 00:35:36.520 align:middle line:84% But I'm sure you can see this coming. 00:35:36.520 --> 00:35:40.030 align:middle line:84% If you were at that fortunate situation with storage, 00:35:40.030 --> 00:35:42.620 align:middle line:90% you got a medium or low output. 00:35:42.620 --> 00:35:45.880 align:middle line:84% Then you would climb back below three. 00:35:45.880 --> 00:35:47.590 align:middle line:84% And you wouldn't be storing at all. 00:35:47.590 --> 00:35:51.550 align:middle line:84% So they had carryover once in a while, but not often. 00:35:51.550 --> 00:35:52.450 align:middle line:90% Questions? 00:35:52.450 --> 00:35:53.680 align:middle line:90% AUDIENCE: I have a question. 00:35:53.680 --> 00:35:54.555 align:middle line:90% ROBERT TOWNSEND: Yes? 00:35:54.555 --> 00:35:57.800 align:middle line:84% AUDIENCE: If the depreciation rate goes down, 00:35:57.800 --> 00:36:00.665 align:middle line:84% then the inventory line will move, right? 00:36:00.665 --> 00:36:02.290 align:middle line:84% ROBERT TOWNSEND: It will move up, yeah. 00:36:02.290 --> 00:36:03.670 align:middle line:90% It would shift up. 00:36:03.670 --> 00:36:08.860 align:middle line:84% Well, it would not only have a higher slope, 00:36:08.860 --> 00:36:11.387 align:middle line:90% but it would kick in earlier. 00:36:11.387 --> 00:36:11.970 align:middle line:90% AUDIENCE: Yes. 00:36:11.970 --> 00:36:17.280 align:middle line:84% So it is also implied that in the medieval England maybe 00:36:17.280 --> 00:36:19.500 align:middle line:84% the depreciation rate was so high, 00:36:19.500 --> 00:36:25.620 align:middle line:84% so the carryover was so low, or we cannot observe 00:36:25.620 --> 00:36:27.557 align:middle line:90% the carryover. 00:36:27.557 --> 00:36:28.890 align:middle line:90% ROBERT TOWNSEND: Say that again? 00:36:28.890 --> 00:36:34.050 align:middle line:84% AUDIENCE: I mean, maybe it implies that in the medieval 00:36:34.050 --> 00:36:37.500 align:middle line:84% England, the depreciation rate is so high, 00:36:37.500 --> 00:36:42.690 align:middle line:84% that is why we cannot observe the carryover. 00:36:42.690 --> 00:36:43.960 align:middle line:90% ROBERT TOWNSEND: Yeah. 00:36:43.960 --> 00:36:45.550 align:middle line:90% Well, if-- 00:36:45.550 --> 00:36:48.030 align:middle line:90% AUDIENCE: Yeah. 00:36:48.030 --> 00:36:49.620 align:middle line:84% ROBERT TOWNSEND: I mean, we picked 00:36:49.620 --> 00:36:52.080 align:middle line:90% it to try to match the data. 00:36:52.080 --> 00:36:56.150 align:middle line:84% You could run the model with a higher or lower number. 00:36:56.150 --> 00:36:59.130 align:middle line:84% If we make the depreciation rate less severe, 00:36:59.130 --> 00:37:03.920 align:middle line:84% so it's like 20% or 30%, they would store more often. 00:37:03.920 --> 00:37:06.110 align:middle line:84% And then we would have to look at that diagram 00:37:06.110 --> 00:37:09.290 align:middle line:84% and see whether carryover is still rare. 00:37:09.290 --> 00:37:13.610 align:middle line:84% I mean, I imagine if you go from 30% to 28% 00:37:13.610 --> 00:37:17.360 align:middle line:84% or even 25%, not too much might change. 00:37:17.360 --> 00:37:22.880 align:middle line:84% But it's clear if you ran the depreciation to zero, 00:37:22.880 --> 00:37:26.990 align:middle line:84% then they're going to be storing quite a lot. 00:37:26.990 --> 00:37:28.430 align:middle line:84% They would have a way of carrying 00:37:28.430 --> 00:37:31.190 align:middle line:90% the crop over one to one. 00:37:31.190 --> 00:37:34.640 align:middle line:84% It would still be uncertain, but that 00:37:34.640 --> 00:37:36.470 align:middle line:84% would be a big step toward keeping 00:37:36.470 --> 00:37:40.040 align:middle line:90% their consumption steady. 00:37:40.040 --> 00:37:40.953 align:middle line:90% AUDIENCE: Thank you. 00:37:40.953 --> 00:37:41.870 align:middle line:90% ROBERT TOWNSEND: Yeah. 00:37:41.870 --> 00:37:43.430 align:middle line:90% So no, it's a great question. 00:37:43.430 --> 00:37:47.540 align:middle line:84% Because that parameter and all the other ones are key. 00:37:47.540 --> 00:37:50.600 align:middle line:84% We could vary each, one at a time. 00:37:50.600 --> 00:37:52.745 align:middle line:84% We could make them more risk averse, for example. 00:37:52.745 --> 00:37:53.370 align:middle line:90% AUDIENCE: Yeah. 00:37:53.370 --> 00:37:57.470 align:middle line:84% ROBERT TOWNSEND: That would be an interesting experiment. 00:37:57.470 --> 00:38:03.590 align:middle line:84% Then they're going to be more worried about that low output. 00:38:03.590 --> 00:38:05.660 align:middle line:84% And that probably means they're going 00:38:05.660 --> 00:38:11.350 align:middle line:84% to be trying to maintain more seed and more inventory, 00:38:11.350 --> 00:38:12.930 align:middle line:90% which one I'm not quite sure. 00:38:12.930 --> 00:38:14.680 align:middle line:84% Well, there's an upper bound on the amount 00:38:14.680 --> 00:38:16.540 align:middle line:84% of seed, the amount of land they can plant. 00:38:16.540 --> 00:38:21.370 align:middle line:84% So that's probably going to push inventory up. 00:38:21.370 --> 00:38:27.870 align:middle line:84% That's a version of what we saw last time in a static context. 00:38:27.870 --> 00:38:32.290 align:middle line:84% Namely, that you're willing to accept something 00:38:32.290 --> 00:38:36.590 align:middle line:84% less than the mean return in order to guarantee, 00:38:36.590 --> 00:38:37.625 align:middle line:90% to get rid of the risk. 00:38:37.625 --> 00:38:40.620 align:middle line:90% 00:38:40.620 --> 00:38:47.780 align:middle line:84% So in this case, even with storage being somewhat costly, 00:38:47.780 --> 00:38:49.415 align:middle line:84% when they're risk averse, then we 00:38:49.415 --> 00:38:51.240 align:middle line:84% could make them even more risk averse. 00:38:51.240 --> 00:38:54.050 align:middle line:84% They're going to be more inclined to give up 00:38:54.050 --> 00:38:56.330 align:middle line:84% some of the consumption today, even 00:38:56.330 --> 00:39:00.920 align:middle line:84% though it's going to not survive the rats into tomorrow. 00:39:00.920 --> 00:39:05.030 align:middle line:84% Because when they have it, the consumption today 00:39:05.030 --> 00:39:07.610 align:middle line:84% and consumption tomorrow are more likely to be the same. 00:39:07.610 --> 00:39:10.230 align:middle line:90% 00:39:10.230 --> 00:39:12.720 align:middle line:90% AUDIENCE: Thanks. 00:39:12.720 --> 00:39:14.310 align:middle line:90% ROBERT TOWNSEND: Beta matters. 00:39:14.310 --> 00:39:17.290 align:middle line:84% If beta goes to zero, you don't care about the future. 00:39:17.290 --> 00:39:20.790 align:middle line:84% So obviously, these guys would be very myopic. 00:39:20.790 --> 00:39:23.440 align:middle line:84% And they would be behaving very differently, and so on. 00:39:23.440 --> 00:39:24.930 align:middle line:90% So all these parameters matter. 00:39:24.930 --> 00:39:27.040 align:middle line:90% I guess I got lucky. 00:39:27.040 --> 00:39:31.840 align:middle line:84% I mean, some of these parameters were picked. 00:39:31.840 --> 00:39:34.030 align:middle line:84% And I was complaining they seem a little bit off, 00:39:34.030 --> 00:39:36.400 align:middle line:90% like beta and risk aversion. 00:39:36.400 --> 00:39:39.850 align:middle line:84% But that, along with the depreciation rate 00:39:39.850 --> 00:39:44.260 align:middle line:84% and the very carefully calibrated technology, 00:39:44.260 --> 00:39:51.280 align:middle line:84% did succeed in producing the data. 00:39:51.280 --> 00:39:54.330 align:middle line:90% So I want to come back to this. 00:39:54.330 --> 00:39:56.250 align:middle line:90% We cheated a bit. 00:39:56.250 --> 00:39:58.470 align:middle line:84% Because we had a finite horizon and then 00:39:58.470 --> 00:40:00.870 align:middle line:90% I kind of told you to ignore it. 00:40:00.870 --> 00:40:04.710 align:middle line:84% That diagram was produced as if the horizon was infinite. 00:40:04.710 --> 00:40:06.360 align:middle line:90% And how would I know that? 00:40:06.360 --> 00:40:09.150 align:middle line:84% Because if the horizon is finite, then 00:40:09.150 --> 00:40:11.760 align:middle line:84% unlike those figures, the amount stored 00:40:11.760 --> 00:40:15.390 align:middle line:84% is going to depend on how many years are left. 00:40:15.390 --> 00:40:18.520 align:middle line:84% Intuitively, if you're at the terminal date-- 00:40:18.520 --> 00:40:20.760 align:middle line:84% it sounds kind of foreboding, no? 00:40:20.760 --> 00:40:21.840 align:middle line:90% The terminal date? 00:40:21.840 --> 00:40:24.030 align:middle line:84% Well, you've got one more year to go. 00:40:24.030 --> 00:40:24.780 align:middle line:90% That's it. 00:40:24.780 --> 00:40:26.100 align:middle line:90% Eat now. 00:40:26.100 --> 00:40:27.780 align:middle line:84% You're not going to be around tomorrow. 00:40:27.780 --> 00:40:30.630 align:middle line:84% So there's not any savings in seed. 00:40:30.630 --> 00:40:32.490 align:middle line:90% There's not any storage. 00:40:32.490 --> 00:40:35.710 align:middle line:84% Now you could move back from there to the next 00:40:35.710 --> 00:40:38.220 align:middle line:84% to the last date, knowing that you're not 00:40:38.220 --> 00:40:39.330 align:middle line:90% going to save tomorrow. 00:40:39.330 --> 00:40:42.660 align:middle line:84% You're still facing a choice between today and tomorrow, 00:40:42.660 --> 00:40:44.560 align:middle line:90% or this year versus next year. 00:40:44.560 --> 00:40:46.890 align:middle line:84% So you could solve that two period problem. 00:40:46.890 --> 00:40:49.200 align:middle line:84% We call this working back from the end, right? 00:40:49.200 --> 00:40:51.360 align:middle line:90% We start at the last date. 00:40:51.360 --> 00:40:53.220 align:middle line:84% We move to the next to the last date 00:40:53.220 --> 00:40:55.500 align:middle line:84% and solve the maximization problem. 00:40:55.500 --> 00:40:58.530 align:middle line:84% Substitute in the values, and then we 00:40:58.530 --> 00:41:02.550 align:middle line:84% have the maximized utility at the next to last date 00:41:02.550 --> 00:41:04.810 align:middle line:84% because we've already solved that problem. 00:41:04.810 --> 00:41:07.260 align:middle line:84% And then you move to the third to last date. 00:41:07.260 --> 00:41:09.510 align:middle line:84% But it's still going to look like a two period problem 00:41:09.510 --> 00:41:12.000 align:middle line:84% because we've collapsed the last date and the next 00:41:12.000 --> 00:41:16.380 align:middle line:84% to the last date into one because we already solved 00:41:16.380 --> 00:41:18.840 align:middle line:90% that last two period problem. 00:41:18.840 --> 00:41:21.630 align:middle line:84% I'm going to show you some notation. 00:41:21.630 --> 00:41:24.870 align:middle line:84% I'm not sure you could possibly be following me. 00:41:24.870 --> 00:41:26.520 align:middle line:84% But I'll say it in English first. 00:41:26.520 --> 00:41:29.190 align:middle line:84% And then we'll do it in the notation. 00:41:29.190 --> 00:41:33.900 align:middle line:84% Anyway, there's a sequence of two period problems. 00:41:33.900 --> 00:41:38.490 align:middle line:84% As you let the horizon t get larger and larger, 00:41:38.490 --> 00:41:41.730 align:middle line:84% you're kind of farther and farther away as you go back 00:41:41.730 --> 00:41:43.830 align:middle line:90% toward t equal 1. 00:41:43.830 --> 00:41:46.500 align:middle line:84% So the larger is that distance between t 00:41:46.500 --> 00:41:51.060 align:middle line:84% equal 1 and cap T. Especially when beta is close to zero, 00:41:51.060 --> 00:41:55.230 align:middle line:84% it's as if you're solving an infinite horizon problem. 00:41:55.230 --> 00:41:58.650 align:middle line:84% And that is the way numerically to solve it, literally. 00:41:58.650 --> 00:42:00.070 align:middle line:90% Start with the last day. 00:42:00.070 --> 00:42:02.320 align:middle line:90% Do the two period optimization. 00:42:02.320 --> 00:42:05.160 align:middle line:84% So the next to the last to the last, plug in the solution, 00:42:05.160 --> 00:42:07.420 align:middle line:84% move to the third to last, and so on. 00:42:07.420 --> 00:42:09.240 align:middle line:90% And then keep iterating forever. 00:42:09.240 --> 00:42:12.600 align:middle line:84% Well, keep iterating until the solution isn't 00:42:12.600 --> 00:42:13.990 align:middle line:90% changing anymore. 00:42:13.990 --> 00:42:16.350 align:middle line:84% And then, amazingly, you will be guaranteed 00:42:16.350 --> 00:42:18.850 align:middle line:84% to have solved the infinite horizon problem. 00:42:18.850 --> 00:42:22.380 align:middle line:84% So let me show you the math of that. 00:42:22.380 --> 00:42:25.140 align:middle line:84% And I apologize the notation changes a bit. 00:42:25.140 --> 00:42:27.690 align:middle line:84% This is from engineering or economics. 00:42:27.690 --> 00:42:32.170 align:middle line:84% It's called a control problem, the Bellman problem. 00:42:32.170 --> 00:42:36.330 align:middle line:84% So x is the current state, which in our application 00:42:36.330 --> 00:42:40.650 align:middle line:84% was the current harvest plus the storage. 00:42:40.650 --> 00:42:43.800 align:middle line:84% And y are the control variables, namely, 00:42:43.800 --> 00:42:47.160 align:middle line:84% how much you can store and save for tomorrow. 00:42:47.160 --> 00:42:48.450 align:middle line:90% That you get to choose. 00:42:48.450 --> 00:42:50.670 align:middle line:90% Also, you choose consumption. 00:42:50.670 --> 00:42:54.210 align:middle line:84% You can choose two out of three, that determines the third. 00:42:54.210 --> 00:42:57.360 align:middle line:84% And then you'll get beta times the utility 00:42:57.360 --> 00:43:00.630 align:middle line:84% for next year, which in this control problem language 00:43:00.630 --> 00:43:03.960 align:middle line:84% is this so-called value function which has in it 00:43:03.960 --> 00:43:07.140 align:middle line:84% why the endogenous variable chosen today 00:43:07.140 --> 00:43:10.860 align:middle line:84% becomes the state of the world for next period. 00:43:10.860 --> 00:43:13.130 align:middle line:84% So this is the two period problem. 00:43:13.130 --> 00:43:18.910 align:middle line:84% Utility contemporary, utility today, and utility 00:43:18.910 --> 00:43:23.860 align:middle line:84% next period, it's really already maximized utility from tomorrow 00:43:23.860 --> 00:43:25.100 align:middle line:90% on. 00:43:25.100 --> 00:43:26.785 align:middle line:84% But think of it as a two period problem. 00:43:26.785 --> 00:43:28.660 align:middle line:84% That's the way I was describing [INAUDIBLE].. 00:43:28.660 --> 00:43:30.910 align:middle line:84% And what is the overall solution? 00:43:30.910 --> 00:43:36.580 align:middle line:84% It has to do with J, the guess about the next to last date, 00:43:36.580 --> 00:43:37.940 align:middle line:90% and so on. 00:43:37.940 --> 00:43:41.500 align:middle line:90% But T now, here's my apology. 00:43:41.500 --> 00:43:43.390 align:middle line:90% T is not time. 00:43:43.390 --> 00:43:45.760 align:middle line:84% T is the mapping in this language of Bellman. 00:43:45.760 --> 00:43:48.100 align:middle line:84% And I realized only this morning that it 00:43:48.100 --> 00:43:51.700 align:middle line:84% might have been better to change the notation to be consistent. 00:43:51.700 --> 00:43:54.970 align:middle line:84% But T does not refer to the time horizon. 00:43:54.970 --> 00:43:59.770 align:middle line:84% T refers to the mapping from the utility from next period 00:43:59.770 --> 00:44:01.690 align:middle line:90% on to the utility today. 00:44:01.690 --> 00:44:06.123 align:middle line:84% This is the maximized utility, where J is a given. 00:44:06.123 --> 00:44:08.290 align:middle line:84% You're surely going to have questions about this, so 00:44:08.290 --> 00:44:09.100 align:middle line:90% interrupt me. 00:44:09.100 --> 00:44:11.770 align:middle line:84% Now there's something called a contraction mapping. 00:44:11.770 --> 00:44:13.990 align:middle line:84% And don't worry about the details here. 00:44:13.990 --> 00:44:17.920 align:middle line:84% But it turns out we have this mapping T here, 00:44:17.920 --> 00:44:22.210 align:middle line:84% and we have the underlying space being a metric m 00:44:22.210 --> 00:44:25.030 align:middle line:84% and we have some distance metric, 00:44:25.030 --> 00:44:28.330 align:middle line:84% some way to measure distance in that metric space. 00:44:28.330 --> 00:44:34.120 align:middle line:84% Then the difference between two guesses about v times beta 00:44:34.120 --> 00:44:40.700 align:middle line:84% is an upper bound for the distance between the result. 00:44:40.700 --> 00:44:47.140 align:middle line:84% So, again, here we should think about v1 and v2 as guesses 00:44:47.140 --> 00:44:48.530 align:middle line:90% for the right-hand side. 00:44:48.530 --> 00:44:51.310 align:middle line:90% So this would be J1 and J2. 00:44:51.310 --> 00:44:56.860 align:middle line:84% And then we apply T to it, and this would be TJ instead of Tv. 00:44:56.860 --> 00:44:58.900 align:middle line:84% So that's probably the second thing to correct, 00:44:58.900 --> 00:45:00.790 align:middle line:90% is to change the v's to Js. 00:45:00.790 --> 00:45:03.760 align:middle line:84% Anyway, the point is the distance over here 00:45:03.760 --> 00:45:07.240 align:middle line:84% on the left-hand side is not greater, typically less 00:45:07.240 --> 00:45:09.860 align:middle line:84% than the distance on the right-hand side. 00:45:09.860 --> 00:45:12.520 align:middle line:84% And so intuitively what that means is things 00:45:12.520 --> 00:45:13.600 align:middle line:90% are going to contract. 00:45:13.600 --> 00:45:15.790 align:middle line:84% Hence this is called a contraction mapping. 00:45:15.790 --> 00:45:19.630 align:middle line:84% Or to put it more formally, if you start with an arbitrary 00:45:19.630 --> 00:45:24.790 align:middle line:84% J or an arbitrary v, and keep iterating, start here, 00:45:24.790 --> 00:45:27.020 align:middle line:90% get this, put it in. 00:45:27.020 --> 00:45:27.790 align:middle line:90% Do it again. 00:45:27.790 --> 00:45:28.660 align:middle line:90% Get this. 00:45:28.660 --> 00:45:29.860 align:middle line:90% Do it again. 00:45:29.860 --> 00:45:36.070 align:middle line:84% As N gets large, and we keep successively applying that 00:45:36.070 --> 00:45:39.610 align:middle line:84% mapping to an initial guess value of v, 00:45:39.610 --> 00:45:42.790 align:middle line:84% we're going to converge to something called v*. 00:45:42.790 --> 00:45:47.770 align:middle line:84% And when T of v* is equal to v*, we are done. 00:45:47.770 --> 00:45:50.050 align:middle line:84% That's the infinite horizon solution. 00:45:50.050 --> 00:45:55.450 align:middle line:84% That's like saying the J up here and the TJ is also J. 00:45:55.450 --> 00:45:57.878 align:middle line:84% So the time horizon has disappeared. 00:45:57.878 --> 00:45:59.170 align:middle line:90% So this is kind of tough stuff. 00:45:59.170 --> 00:46:01.212 align:middle line:84% I thought it would be better to show you the math 00:46:01.212 --> 00:46:03.110 align:middle line:84% rather than be mysterious about it. 00:46:03.110 --> 00:46:06.110 align:middle line:90% But let me pause for questions. 00:46:06.110 --> 00:46:06.610 align:middle line:90% All right. 00:46:06.610 --> 00:46:09.670 align:middle line:84% We have everything we needed in that application 00:46:09.670 --> 00:46:11.920 align:middle line:84% to apply the contraction mapping. 00:46:11.920 --> 00:46:15.130 align:middle line:84% Our guess for v0 is just the solution 00:46:15.130 --> 00:46:16.810 align:middle line:90% to the last period problem. 00:46:16.810 --> 00:46:20.710 align:middle line:84% We get the optimized value, as if you were at the last date. 00:46:20.710 --> 00:46:23.860 align:middle line:84% Substitute in the maximized utility number. 00:46:23.860 --> 00:46:26.770 align:middle line:84% And then go to the next to the last period 00:46:26.770 --> 00:46:30.400 align:middle line:84% and iterate again with that new guess for the second period 00:46:30.400 --> 00:46:31.240 align:middle line:90% utility. 00:46:31.240 --> 00:46:33.670 align:middle line:84% And just keep iterating, and we'll 00:46:33.670 --> 00:46:35.660 align:middle line:84% get to the infinite horizon solution. 00:46:35.660 --> 00:46:40.690 align:middle line:84% So we can think about applying this technique more generally. 00:46:40.690 --> 00:46:44.830 align:middle line:84% So this is a related, but somewhat distinct problem. 00:46:44.830 --> 00:46:49.150 align:middle line:84% It's in a contemporary setting, a household 00:46:49.150 --> 00:46:52.720 align:middle line:84% trying to maximize the utility of consumption, 00:46:52.720 --> 00:46:56.500 align:middle line:84% but facing stochastic earnings, say, from year to year. 00:46:56.500 --> 00:46:59.500 align:middle line:84% And they're limited in the way that they can save. 00:46:59.500 --> 00:47:03.220 align:middle line:84% But there is a savings account in the bank at some interest 00:47:03.220 --> 00:47:06.580 align:middle line:84% rate, or even a riskless bond, like treasuries. 00:47:06.580 --> 00:47:08.350 align:middle line:84% So it's like putting stuff in the ground 00:47:08.350 --> 00:47:10.480 align:middle line:84% today and getting stuff out tomorrow. 00:47:10.480 --> 00:47:11.780 align:middle line:90% That's like storage. 00:47:11.780 --> 00:47:14.170 align:middle line:84% In fact, in the first lecture, I said something 00:47:14.170 --> 00:47:16.810 align:middle line:90% about these agrarian metaphors. 00:47:16.810 --> 00:47:18.130 align:middle line:90% And we just jump like that. 00:47:18.130 --> 00:47:22.240 align:middle line:84% We went from grain in the bin as kind of a storage technology 00:47:22.240 --> 00:47:27.430 align:middle line:84% to riskless bonds as another way to store money over time, 00:47:27.430 --> 00:47:29.230 align:middle line:90% hopefully with a positive yield. 00:47:29.230 --> 00:47:31.910 align:middle line:84% Actually, the real rates are negative right now. 00:47:31.910 --> 00:47:35.270 align:middle line:84% So anyway, that's a bit of a problem. 00:47:35.270 --> 00:47:38.380 align:middle line:90% So we maximized utility. 00:47:38.380 --> 00:47:41.980 align:middle line:84% And now I have the courage to put infinite horizon here. 00:47:41.980 --> 00:47:45.640 align:middle line:84% So we're going to maximize discounted expected utility 00:47:45.640 --> 00:47:47.200 align:middle line:90% over the infinite horizon. 00:47:47.200 --> 00:47:49.180 align:middle line:90% Beta is still less than one. 00:47:49.180 --> 00:47:50.600 align:middle line:90% This thing would converge. 00:47:50.600 --> 00:47:52.750 align:middle line:90% So this term is finite. 00:47:52.750 --> 00:47:55.270 align:middle line:84% And the choice will be over how much to eat 00:47:55.270 --> 00:47:56.590 align:middle line:90% and how much to save. 00:47:56.590 --> 00:47:58.420 align:middle line:84% So the way to think about this is 00:47:58.420 --> 00:48:01.300 align:middle line:84% you're coming in with a realization 00:48:01.300 --> 00:48:03.040 align:middle line:90% of your stochastic income. 00:48:03.040 --> 00:48:06.070 align:middle line:84% You're having the amount of the bond 00:48:06.070 --> 00:48:08.080 align:middle line:90% that you bought last period. 00:48:08.080 --> 00:48:10.430 align:middle line:84% You get that back plus the interest rate. 00:48:10.430 --> 00:48:13.840 align:middle line:84% That's the resources available today at T. 00:48:13.840 --> 00:48:17.290 align:middle line:84% And you can either eat that, or buy a bond 00:48:17.290 --> 00:48:19.340 align:middle line:84% and carry it over to the next period. 00:48:19.340 --> 00:48:22.690 align:middle line:84% So this looks really similar to the medieval storage problem. 00:48:22.690 --> 00:48:27.220 align:middle line:84% A couple of comments about it, as I tried to alert you. 00:48:27.220 --> 00:48:30.250 align:middle line:90% The timing here is the date. 00:48:30.250 --> 00:48:31.990 align:middle line:90% The dating is different. 00:48:31.990 --> 00:48:34.780 align:middle line:84% So now, at t, this is no longer a state variable. 00:48:34.780 --> 00:48:37.240 align:middle line:84% This is a decision variable, bonds 00:48:37.240 --> 00:48:39.650 align:middle line:90% to becoming due next period. 00:48:39.650 --> 00:48:42.820 align:middle line:84% Likewise, on the right-hand side, this doesn't have a t. 00:48:42.820 --> 00:48:44.140 align:middle line:90% It had a t minus 1. 00:48:44.140 --> 00:48:47.860 align:middle line:84% Because the savings decisions are dated at the time t 00:48:47.860 --> 00:48:50.770 align:middle line:84% that you make the decision, rather than the time 00:48:50.770 --> 00:48:52.870 align:middle line:90% t where the yield comes in. 00:48:52.870 --> 00:48:54.430 align:middle line:90% And this is also standard. 00:48:54.430 --> 00:48:56.500 align:middle line:84% You just have to keep an eye out in terms of what 00:48:56.500 --> 00:48:58.330 align:middle line:90% the authors have in mind. 00:48:58.330 --> 00:49:01.060 align:middle line:90% And this starts at date zero. 00:49:01.060 --> 00:49:03.710 align:middle line:84% So you need some initial conditions, 00:49:03.710 --> 00:49:09.010 align:middle line:84% which should have been k at t equals 0 minus 1 or k1. 00:49:09.010 --> 00:49:10.660 align:middle line:90% This is also a typo. 00:49:10.660 --> 00:49:14.380 align:middle line:84% This should have been k minus 1, not k0. 00:49:14.380 --> 00:49:15.880 align:middle line:90% And the y0 is fine. 00:49:15.880 --> 00:49:17.690 align:middle line:90% That's the contemporary yield. 00:49:17.690 --> 00:49:19.660 align:middle line:84% So I apologize about these typos. 00:49:19.660 --> 00:49:21.670 align:middle line:84% Although it is kind of an interesting story. 00:49:21.670 --> 00:49:23.560 align:middle line:90% They're not mine. 00:49:23.560 --> 00:49:25.670 align:middle line:84% I took this slide from somebody else, 00:49:25.670 --> 00:49:27.460 align:middle line:84% and they had actually made the mistake, 00:49:27.460 --> 00:49:29.660 align:middle line:84% and it wasn't corrected in the publication. 00:49:29.660 --> 00:49:31.040 align:middle line:90% But we'll fix it. 00:49:31.040 --> 00:49:32.905 align:middle line:84% The other thing about the typos-- 00:49:32.905 --> 00:49:35.770 align:middle line:84% I know you hate them, I do, too-- when 00:49:35.770 --> 00:49:39.400 align:middle line:84% we did that input-output matrix for Leontief last, 00:49:39.400 --> 00:49:41.800 align:middle line:84% when we did the last part of lecture 00:49:41.800 --> 00:49:44.710 align:middle line:84% before the first part of last lecture, the beginning 00:49:44.710 --> 00:49:47.830 align:middle line:84% of Lecture 5, we went back over Lecture 4, 00:49:47.830 --> 00:49:51.790 align:middle line:84% and I was saying how much of the service sector 00:49:51.790 --> 00:49:53.530 align:middle line:90% you need as an input. 00:49:53.530 --> 00:49:55.480 align:middle line:84% Produce one unit of output of services. 00:49:55.480 --> 00:49:57.670 align:middle line:84% How much do you need from manufacturing 00:49:57.670 --> 00:49:58.840 align:middle line:90% and raw materials? 00:49:58.840 --> 00:50:00.400 align:middle line:90% It was right in the slide. 00:50:00.400 --> 00:50:01.930 align:middle line:90% The typo had been corrected. 00:50:01.930 --> 00:50:04.270 align:middle line:84% But that typo exists in Wikipedia. 00:50:04.270 --> 00:50:05.720 align:middle line:90% And it's still wrong there. 00:50:05.720 --> 00:50:07.900 align:middle line:84% So what I'm saying is, I work very 00:50:07.900 --> 00:50:09.310 align:middle line:90% hard to eliminate the typos. 00:50:09.310 --> 00:50:11.140 align:middle line:84% But sometimes, they still persist. 00:50:11.140 --> 00:50:14.410 align:middle line:84% If you're reading carefully, though, you can spot them. 00:50:14.410 --> 00:50:17.020 align:middle line:84% And please don't hesitate, if you find other things 00:50:17.020 --> 00:50:18.160 align:middle line:90% that you find questionable. 00:50:18.160 --> 00:50:19.390 align:middle line:90% That's the bad part. 00:50:19.390 --> 00:50:22.880 align:middle line:84% Then it becomes confusing about whether there's a typo or not. 00:50:22.880 --> 00:50:24.520 align:middle line:90% So if in doubt, ask us. 00:50:24.520 --> 00:50:28.690 align:middle line:84% And we'll try to answer questions. 00:50:28.690 --> 00:50:33.340 align:middle line:84% AUDIENCE: Here, the kt could be [INAUDIBLE],, right? 00:50:33.340 --> 00:50:38.470 align:middle line:84% in the latter constraint, and say that a kt just 00:50:38.470 --> 00:50:41.782 align:middle line:90% are larger than minus k? 00:50:41.782 --> 00:50:43.240 align:middle line:84% ROBERT TOWNSEND: Oh, good question. 00:50:43.240 --> 00:50:47.560 align:middle line:90% 00:50:47.560 --> 00:50:50.240 align:middle line:90% So borrowing is allowed. 00:50:50.240 --> 00:50:53.600 align:middle line:84% So I referred to this as a riskless bond, 00:50:53.600 --> 00:50:56.300 align:middle line:90% as if you could save. 00:50:56.300 --> 00:50:59.930 align:middle line:84% But in fact, if savings is negative, you're borrowing. 00:50:59.930 --> 00:51:02.940 align:middle line:90% 00:51:02.940 --> 00:51:08.420 align:middle line:84% So if you multiplied through by the negative sign, 00:51:08.420 --> 00:51:13.360 align:middle line:84% you'd have a minus kt less than or equal to k bar. 00:51:13.360 --> 00:51:16.330 align:middle line:84% So kt positive for savings minus kt 00:51:16.330 --> 00:51:20.110 align:middle line:90% would be now denoting borrowing. 00:51:20.110 --> 00:51:25.600 align:middle line:84% And borrowing has an upper bound, a lower bar. 00:51:25.600 --> 00:51:29.850 align:middle line:84% So this formulation of the problem allows borrowing, 00:51:29.850 --> 00:51:30.675 align:middle line:90% but it's limited. 00:51:30.675 --> 00:51:33.815 align:middle line:90% 00:51:33.815 --> 00:51:34.940 align:middle line:90% So that's a great question. 00:51:34.940 --> 00:51:35.815 align:middle line:90% Thank you for asking. 00:51:35.815 --> 00:51:36.880 align:middle line:90% Is it clear? 00:51:36.880 --> 00:51:38.630 align:middle line:90% AUDIENCE: Yes. 00:51:38.630 --> 00:51:42.580 align:middle line:84% Another question is, what is the difference 00:51:42.580 --> 00:51:46.180 align:middle line:84% between the self-insurance and full insurance? 00:51:46.180 --> 00:51:49.760 align:middle line:84% Is the self-insurance-- that means we cannot borrow, 00:51:49.760 --> 00:51:53.560 align:middle line:84% or we can borrow, but there were some constraints like this. 00:51:53.560 --> 00:51:58.490 align:middle line:84% We can borrow at a certain range, or-- 00:51:58.490 --> 00:52:01.005 align:middle line:90% 00:52:01.005 --> 00:52:02.880 align:middle line:84% ROBERT TOWNSEND: Yeah, so I actually took out 00:52:02.880 --> 00:52:04.305 align:middle line:90% slides that I used to have. 00:52:04.305 --> 00:52:08.230 align:middle line:90% 00:52:08.230 --> 00:52:10.000 align:middle line:84% Self-insurance is what we've been 00:52:10.000 --> 00:52:11.140 align:middle line:90% studying the whole lecture. 00:52:11.140 --> 00:52:14.470 align:middle line:84% Because it's, like, one village, or one person. 00:52:14.470 --> 00:52:17.260 align:middle line:84% And they're pretty much in isolation. 00:52:17.260 --> 00:52:20.800 align:middle line:84% Here, it's still self-insurance, although there 00:52:20.800 --> 00:52:24.730 align:middle line:84% is this bond, or this borrowing possibility at a fixed interest 00:52:24.730 --> 00:52:27.250 align:middle line:84% rate, which I didn't tell you where that came from. 00:52:27.250 --> 00:52:30.680 align:middle line:84% But this is, like, partial equilibrium problem 00:52:30.680 --> 00:52:33.650 align:middle line:84% in which the agent can borrow or lend 00:52:33.650 --> 00:52:37.370 align:middle line:90% with the rest of the economy. 00:52:37.370 --> 00:52:40.520 align:middle line:84% And that's still referred to as self-insurance, although it's 00:52:40.520 --> 00:52:43.910 align:middle line:90% a bit misleading. 00:52:43.910 --> 00:52:48.440 align:middle line:84% Because one person's saving is another person's borrowing. 00:52:48.440 --> 00:52:51.120 align:middle line:84% But anyway, we're not doing the general equilibrium here. 00:52:51.120 --> 00:52:55.280 align:middle line:84% We're taking-- the interest rate is given. 00:52:55.280 --> 00:52:58.100 align:middle line:84% But the other thing would be, what 00:52:58.100 --> 00:53:02.110 align:middle line:84% if there were multiple people like this, 00:53:02.110 --> 00:53:06.880 align:middle line:84% and some people had high income at a given date, 00:53:06.880 --> 00:53:11.650 align:middle line:84% and other people had low income at a given date? 00:53:11.650 --> 00:53:14.590 align:middle line:84% Then we're back to thinking about 00:53:14.590 --> 00:53:16.945 align:middle line:84% whether they could make deals with each other. 00:53:16.945 --> 00:53:20.860 align:middle line:90% 00:53:20.860 --> 00:53:24.490 align:middle line:84% So we'll get there actually next time. 00:53:24.490 --> 00:53:28.770 align:middle line:84% But the risk-sharing problem would be-- 00:53:28.770 --> 00:53:31.560 align:middle line:84% an extreme version of that is get rid 00:53:31.560 --> 00:53:34.020 align:middle line:84% of the storage altogether and the seed. 00:53:34.020 --> 00:53:34.698 align:middle line:90% Get rid of them. 00:53:34.698 --> 00:53:36.240 align:middle line:84% Not the seed, get rid of the storage. 00:53:36.240 --> 00:53:38.130 align:middle line:90% Get rid of the bond. 00:53:38.130 --> 00:53:41.130 align:middle line:84% But imagine these guys can contract with each other, 00:53:41.130 --> 00:53:45.030 align:middle line:84% write financial contracts, or informal risk-sharing 00:53:45.030 --> 00:53:47.970 align:middle line:84% arrangements with each other so as 00:53:47.970 --> 00:53:50.400 align:middle line:84% to pool the risk that they're facing. 00:53:50.400 --> 00:53:54.080 align:middle line:90% 00:53:54.080 --> 00:53:58.110 align:middle line:84% Like in the village, they had scattered those strips. 00:53:58.110 --> 00:54:02.450 align:middle line:84% So they're trying to stabilize their starting point. 00:54:02.450 --> 00:54:05.330 align:middle line:84% But on top of it, they could be borrowing and lending. 00:54:05.330 --> 00:54:08.780 align:middle line:90% They could be making a deal. 00:54:08.780 --> 00:54:12.140 align:middle line:84% If I'm high and you're low, I'll give you money, and likewise, 00:54:12.140 --> 00:54:14.430 align:middle line:90% vice versa. 00:54:14.430 --> 00:54:17.120 align:middle line:90% And that's not self-insurance. 00:54:17.120 --> 00:54:21.710 align:middle line:84% That's kind of dynamics of explicit risk-sharing 00:54:21.710 --> 00:54:22.580 align:middle line:90% insurance. 00:54:22.580 --> 00:54:26.350 align:middle line:90% 00:54:26.350 --> 00:54:27.340 align:middle line:90% AUDIENCE: Thank you. 00:54:27.340 --> 00:54:27.790 align:middle line:90% ROBERT TOWNSEND: Yeah. 00:54:27.790 --> 00:54:29.090 align:middle line:90% No, it's a great question. 00:54:29.090 --> 00:54:32.530 align:middle line:84% As I said, I used to have more slides here about that. 00:54:32.530 --> 00:54:36.400 align:middle line:84% And I thought it was a little too early in the class. 00:54:36.400 --> 00:54:39.025 align:middle line:84% But there are clearly trade offs about-- 00:54:39.025 --> 00:54:42.950 align:middle line:90% 00:54:42.950 --> 00:54:45.170 align:middle line:84% how can I put this-- whether or not 00:54:45.170 --> 00:54:48.590 align:middle line:84% want to do self-insurance as in a dynamic problem 00:54:48.590 --> 00:54:55.550 align:middle line:84% versus whether or not want to do risk sharing, 00:54:55.550 --> 00:54:59.800 align:middle line:84% but do it repeatedly over many dates. 00:54:59.800 --> 00:55:05.850 align:middle line:84% In fact, the very first lecture, after we did economic science, 00:55:05.850 --> 00:55:10.190 align:middle line:84% we went through some example economies. 00:55:10.190 --> 00:55:12.100 align:middle line:90% And I gave you five of them. 00:55:12.100 --> 00:55:15.790 align:middle line:84% And the very first two were one, medieval villages 00:55:15.790 --> 00:55:17.740 align:middle line:84% with these scattered strips that we've 00:55:17.740 --> 00:55:27.170 align:middle line:84% been studying now, last time and today, implicitly, or the Thai 00:55:27.170 --> 00:55:27.920 align:middle line:90% temple scheme. 00:55:27.920 --> 00:55:29.720 align:middle line:84% You remember the Thai temple scheme? 00:55:29.720 --> 00:55:32.630 align:middle line:90% They had low land and high land. 00:55:32.630 --> 00:55:35.210 align:middle line:84% If the monsoon rains were really abundant, 00:55:35.210 --> 00:55:37.910 align:middle line:84% the low land flooded too much, and they only 00:55:37.910 --> 00:55:40.130 align:middle line:90% got crops off the high land. 00:55:40.130 --> 00:55:42.810 align:middle line:84% And when the monsoon rains were adequate or low, 00:55:42.810 --> 00:55:44.810 align:middle line:90% then the highlands failed. 00:55:44.810 --> 00:55:49.070 align:middle line:84% And what they did was contribute their crops to the monks. 00:55:49.070 --> 00:55:51.620 align:middle line:84% Not for eating, that really threw me. 00:55:51.620 --> 00:55:54.920 align:middle line:90% Because I know monks don't-- 00:55:54.920 --> 00:55:57.840 align:middle line:84% I mean, you can feed the monks, but the monks don't feed you. 00:55:57.840 --> 00:56:01.700 align:middle line:84% But nevertheless, they were putting their crops 00:56:01.700 --> 00:56:05.390 align:middle line:84% under the auspices of the temple with the monks 00:56:05.390 --> 00:56:07.580 align:middle line:84% and then handing it out to people that 00:56:07.580 --> 00:56:11.800 align:middle line:90% had either drought or flood. 00:56:11.800 --> 00:56:15.200 align:middle line:84% So that's an active risk-sharing scheme. 00:56:15.200 --> 00:56:18.090 align:middle line:84% And your question comes back to that, 00:56:18.090 --> 00:56:21.290 align:middle line:84% which is comparing and contrasting these two 00:56:21.290 --> 00:56:22.985 align:middle line:90% different ways to handle risk. 00:56:22.985 --> 00:56:25.960 align:middle line:90% 00:56:25.960 --> 00:56:30.525 align:middle line:90% OK, so this is Lecture 6. 00:56:30.525 --> 00:56:32.150 align:middle line:84% There aren't quite as many slides here. 00:56:32.150 --> 00:56:35.240 align:middle line:84% But you can see there are some very important concepts. 00:56:35.240 --> 00:56:38.060 align:middle line:84% And we've now succeeded in having both time 00:56:38.060 --> 00:56:43.070 align:middle line:84% and uncertainty and dynamic programming techniques 00:56:43.070 --> 00:56:43.950 align:middle line:90% for solving that. 00:56:43.950 --> 00:56:48.680 align:middle line:84% So we'll come back and use those techniques 00:56:48.680 --> 00:56:50.390 align:middle line:90% in subsequent lectures. 00:56:50.390 --> 00:56:55.850 align:middle line:84% Next Tuesday is a lecture on Pareto optimality, which 00:56:55.850 --> 00:56:58.580 align:middle line:84% is finally the full blown general equilibrium, 00:56:58.580 --> 00:57:00.920 align:middle line:84% although we've kind of been going back and forth 00:57:00.920 --> 00:57:03.540 align:middle line:84% and getting very close to that in the last couple of lectures. 00:57:03.540 --> 00:57:06.350 align:middle line:84% But in the lecture on Tuesday, we 00:57:06.350 --> 00:57:09.980 align:middle line:84% will fully embrace it and talk about efficiency. 00:57:09.980 --> 00:57:12.730 align:middle line:90% OK, that's all for today. 00:57:12.730 --> 00:57:19.000 align:middle line:90%