WEBVTT 00:00:00.000 --> 00:00:01.473 align:middle line:90% [SQUEAKING] 00:00:01.473 --> 00:00:03.928 align:middle line:90% [RUSTLING] 00:00:03.928 --> 00:00:05.401 align:middle line:90% [CLICKING] 00:00:05.401 --> 00:00:13.270 align:middle line:90% 00:00:13.270 --> 00:00:16.940 align:middle line:84% ROBERT TOWNSEND: Well, greetings everyone. 00:00:16.940 --> 00:00:22.120 align:middle line:84% First of all, in terms of the reading list, 00:00:22.120 --> 00:00:27.220 align:middle line:84% today we're going to finish the two, three 00:00:27.220 --> 00:00:31.060 align:middle line:84% things I deliberately didn't do at the end of the lecture 00:00:31.060 --> 00:00:35.620 align:middle line:84% on firms and production sets, which was largely Kreps, 00:00:35.620 --> 00:00:38.020 align:middle line:90% but with plenty of stuff added. 00:00:38.020 --> 00:00:42.130 align:middle line:84% I should say the stuff that's added is largely anticipating 00:00:42.130 --> 00:00:44.650 align:middle line:84% an application we got to at the end, which 00:00:44.650 --> 00:00:49.870 align:middle line:84% is the impact of tariffs and so on, on economic welfare. 00:00:49.870 --> 00:00:53.680 align:middle line:84% So you'll have the tools at the time we get there, 00:00:53.680 --> 00:00:56.590 align:middle line:84% although we will also provide some reminders. 00:00:56.590 --> 00:00:59.950 align:middle line:84% The second thing is, although it's not starred, 00:00:59.950 --> 00:01:01.832 align:middle line:90% there are these things about-- 00:01:01.832 --> 00:01:03.790 align:middle line:84% other things on the reading list, which are not 00:01:03.790 --> 00:01:08.380 align:middle line:84% starred in this case, about the great Japanese earthquake. 00:01:08.380 --> 00:01:11.590 align:middle line:84% And that's-- I'm going to talk about that momentarily, 00:01:11.590 --> 00:01:14.710 align:middle line:84% but I know everyone's quite busy, 00:01:14.710 --> 00:01:17.930 align:middle line:84% and there's plenty of required things to do. 00:01:17.930 --> 00:01:21.310 align:middle line:84% But if you get a chance, I'm sure you actually would enjoy 00:01:21.310 --> 00:01:25.120 align:middle line:90% taking a look at that article. 00:01:25.120 --> 00:01:29.280 align:middle line:84% Today, the main thing I want to draw your attention to 00:01:29.280 --> 00:01:32.790 align:middle line:84% is decision making under uncertainty and also 00:01:32.790 --> 00:01:34.560 align:middle line:90% linear programs. 00:01:34.560 --> 00:01:36.990 align:middle line:84% And that material is largely coming 00:01:36.990 --> 00:01:41.140 align:middle line:84% from Nicholson and Schneider, although it 00:01:41.140 --> 00:01:44.860 align:middle line:84% does have some material that was moved relative 00:01:44.860 --> 00:01:46.540 align:middle line:90% to earlier years. 00:01:46.540 --> 00:01:49.240 align:middle line:84% As you will see, this Medville, which 00:01:49.240 --> 00:01:51.880 align:middle line:84% is short for the medieval village economy-- 00:01:51.880 --> 00:01:54.970 align:middle line:84% several of the slides today have material 00:01:54.970 --> 00:01:58.240 align:middle line:84% on that from that book, which you therefore 00:01:58.240 --> 00:01:59.500 align:middle line:90% might find helpful. 00:01:59.500 --> 00:02:05.350 align:middle line:84% In terms of the study guide, we have some questions here. 00:02:05.350 --> 00:02:09.550 align:middle line:84% And let me try something a bit new-- 00:02:09.550 --> 00:02:11.200 align:middle line:90% but nevertheless, let me try. 00:02:11.200 --> 00:02:14.810 align:middle line:84% Armando, could you take a crack at this one-- 00:02:14.810 --> 00:02:18.250 align:middle line:84% define increasing, decreasing, and constant returns 00:02:18.250 --> 00:02:20.695 align:middle line:90% to scale and production? 00:02:20.695 --> 00:02:24.460 align:middle line:84% AUDIENCE: Yeah, so first, constant return to scale 00:02:24.460 --> 00:02:27.340 align:middle line:84% means that if you double the inputs, 00:02:27.340 --> 00:02:29.050 align:middle line:90% you get double the outputs. 00:02:29.050 --> 00:02:33.680 align:middle line:84% Increasing means that efficiency increases as we scale it up, 00:02:33.680 --> 00:02:35.770 align:middle line:84% so if you double the inputs, you get 00:02:35.770 --> 00:02:37.600 align:middle line:90% more than double the outputs. 00:02:37.600 --> 00:02:40.390 align:middle line:84% And decreasing is that if you double the inputs, 00:02:40.390 --> 00:02:43.140 align:middle line:90% you would get less than double. 00:02:43.140 --> 00:02:44.670 align:middle line:84% ROBERT TOWNSEND: Yeah, that's fair. 00:02:44.670 --> 00:02:45.885 align:middle line:90% That's a pretty good answer. 00:02:45.885 --> 00:02:49.130 align:middle line:90% 00:02:49.130 --> 00:02:51.770 align:middle line:84% We can define these things in various ways. 00:02:51.770 --> 00:02:54.200 align:middle line:84% They're all equivalent with one another. 00:02:54.200 --> 00:02:55.415 align:middle line:90% The way you did it is fine. 00:02:55.415 --> 00:02:58.380 align:middle line:90% 00:02:58.380 --> 00:03:00.830 align:middle line:84% And in some respects, the definitions 00:03:00.830 --> 00:03:03.950 align:middle line:84% are the reverse of what you would expect. 00:03:03.950 --> 00:03:09.470 align:middle line:84% Decreasing returns to scales means that if you pick any-- 00:03:09.470 --> 00:03:12.140 align:middle line:84% if you pick zero and any other point in the production 00:03:12.140 --> 00:03:17.480 align:middle line:84% possibilities set, and then pick alpha between 0 and 1, 00:03:17.480 --> 00:03:21.620 align:middle line:84% that that point is also in the production set. 00:03:21.620 --> 00:03:24.440 align:middle line:84% I say it's weird, because decreasing-- 00:03:24.440 --> 00:03:27.110 align:middle line:84% you're actually scaling back as opposed 00:03:27.110 --> 00:03:30.530 align:middle line:84% to doubling and moving forward, but it's the same definition 00:03:30.530 --> 00:03:33.560 align:middle line:84% because you do get that curvature 00:03:33.560 --> 00:03:39.880 align:middle line:84% in the production set, and likewise for increasing 00:03:39.880 --> 00:03:40.850 align:middle line:90% returns. 00:03:40.850 --> 00:03:43.430 align:middle line:90% So thank you, that's good. 00:03:43.430 --> 00:03:46.670 align:middle line:84% All right, I marked a few of these. 00:03:46.670 --> 00:03:49.070 align:middle line:90% We don't have to do all of them. 00:03:49.070 --> 00:03:52.400 align:middle line:84% How would you prove that maximized profits 00:03:52.400 --> 00:03:57.640 align:middle line:84% are homogeneous of degree 1 in prices p? 00:03:57.640 --> 00:04:00.470 align:middle line:84% Caleb, if I'm saying your name correctly-- 00:04:00.470 --> 00:04:00.970 align:middle line:90% Aulins? 00:04:00.970 --> 00:04:04.720 align:middle line:90% 00:04:04.720 --> 00:04:07.315 align:middle line:84% AUDIENCE: Yeah, sorry, what was the question? 00:04:07.315 --> 00:04:09.190 align:middle line:84% ROBERT TOWNSEND: Prove that maximized profits 00:04:09.190 --> 00:04:12.250 align:middle line:84% are homogeneous of degree 1 and prices p. 00:04:12.250 --> 00:04:14.870 align:middle line:90% 00:04:14.870 --> 00:04:17.880 align:middle line:84% How would you go about proving that? 00:04:17.880 --> 00:04:22.360 align:middle line:90% AUDIENCE: I'm not sure. 00:04:22.360 --> 00:04:24.880 align:middle line:84% ROBERT TOWNSEND: Do you recall what homogeneous of degree 1 00:04:24.880 --> 00:04:27.020 align:middle line:90% means? 00:04:27.020 --> 00:04:28.100 align:middle line:90% AUDIENCE: No. 00:04:28.100 --> 00:04:31.040 align:middle line:84% ROBERT TOWNSEND: OK, so homogeneous of degree 1 00:04:31.040 --> 00:04:41.600 align:middle line:84% means that if you increase every element of the price factor 00:04:41.600 --> 00:04:45.770 align:middle line:84% by a constant that maximized profits, 00:04:45.770 --> 00:04:49.033 align:middle line:90% increased by that same constant. 00:04:49.033 --> 00:04:49.575 align:middle line:90% AUDIENCE: OK. 00:04:49.575 --> 00:04:51.873 align:middle line:90% 00:04:51.873 --> 00:04:53.540 align:middle line:84% ROBERT TOWNSEND: Do you feel comfortable 00:04:53.540 --> 00:04:57.405 align:middle line:84% taking a crack at the rest of this? 00:04:57.405 --> 00:04:59.370 align:middle line:84% AUDIENCE: I'm not sure what the answer is. 00:04:59.370 --> 00:05:02.110 align:middle line:84% ROBERT TOWNSEND: OK, all right, that's fine. 00:05:02.110 --> 00:05:05.750 align:middle line:90% 00:05:05.750 --> 00:05:09.350 align:middle line:84% Wang Ping-- and correct me if I'm saying your names wrong. 00:05:09.350 --> 00:05:11.360 align:middle line:90% I'm sure I am. 00:05:11.360 --> 00:05:14.130 align:middle line:84% AUDIENCE: In the future, you can just call me Andy. 00:05:14.130 --> 00:05:15.330 align:middle line:90% ROBERT TOWNSEND: Andy, OK. 00:05:15.330 --> 00:05:16.462 align:middle line:90% AUDIENCE: Yeah. 00:05:16.462 --> 00:05:18.670 align:middle line:84% ROBERT TOWNSEND: That's the A in your middle initial. 00:05:18.670 --> 00:05:19.295 align:middle line:90% AUDIENCE: Yeah. 00:05:19.295 --> 00:05:22.760 align:middle line:84% ROBERT TOWNSEND: OK, so let's see. 00:05:22.760 --> 00:05:26.290 align:middle line:84% Do you want to try the same question? 00:05:26.290 --> 00:05:30.650 align:middle line:84% AUDIENCE: Sure, yeah, so homogeneity 00:05:30.650 --> 00:05:35.920 align:middle line:84% means basically, if we double the prices, 00:05:35.920 --> 00:05:38.420 align:middle line:90% then we also double the profit. 00:05:38.420 --> 00:05:39.670 align:middle line:90% ROBERT TOWNSEND: Right. 00:05:39.670 --> 00:05:44.320 align:middle line:84% AUDIENCE: Right, so I believe that profit function is 00:05:44.320 --> 00:05:48.970 align:middle line:84% equal to-- is p.z, where z is the production vector. 00:05:48.970 --> 00:05:50.840 align:middle line:90% ROBERT TOWNSEND: Yes. 00:05:50.840 --> 00:05:56.870 align:middle line:84% AUDIENCE: And so if we double the price p, if we double the-- 00:05:56.870 --> 00:05:59.360 align:middle line:84% if we double the production, so if we 00:05:59.360 --> 00:06:00.860 align:middle line:84% maintain the same production vector, 00:06:00.860 --> 00:06:05.540 align:middle line:84% z, then we can guarantee that we at least double the profits. 00:06:05.540 --> 00:06:07.280 align:middle line:84% But if we were able to do better, 00:06:07.280 --> 00:06:11.670 align:middle line:84% then we could then halve the prices again 00:06:11.670 --> 00:06:14.240 align:middle line:90% and obtain a better-- 00:06:14.240 --> 00:06:19.120 align:middle line:84% basically, a better production for the original price level. 00:06:19.120 --> 00:06:25.470 align:middle line:84% And so basically, the optimal profits at the doubled prices 00:06:25.470 --> 00:06:28.980 align:middle line:84% must exactly equal twice the value at the regular prices. 00:06:28.980 --> 00:06:32.070 align:middle line:90% 00:06:32.070 --> 00:06:36.180 align:middle line:84% ROBERT TOWNSEND: Right, I think that's the correct answer. 00:06:36.180 --> 00:06:40.020 align:middle line:84% The way I like to think about it is if we-- 00:06:40.020 --> 00:06:45.960 align:middle line:84% graphically, if we were doubling the price of every component 00:06:45.960 --> 00:06:47.910 align:middle line:84% of the price vector, p, then we're 00:06:47.910 --> 00:06:53.770 align:middle line:84% not changing the slope of those isoprofit lines. 00:06:53.770 --> 00:06:57.900 align:middle line:84% And so the tangency with the production possibilities 00:06:57.900 --> 00:07:01.260 align:middle line:84% frontier would be the same point. 00:07:01.260 --> 00:07:06.330 align:middle line:84% And that's why the z, as in p.z is not moving. 00:07:06.330 --> 00:07:09.050 align:middle line:90% 00:07:09.050 --> 00:07:10.610 align:middle line:90% Great, thank you. 00:07:10.610 --> 00:07:13.770 align:middle line:90% And let's try just one more. 00:07:13.770 --> 00:07:17.620 align:middle line:84% What is the definition of an isoquant? 00:07:17.620 --> 00:07:19.330 align:middle line:90% So Ke Qiu-- 00:07:19.330 --> 00:07:22.190 align:middle line:90% 00:07:22.190 --> 00:07:24.590 align:middle line:84% AUDIENCE: I think the definition of isoquant 00:07:24.590 --> 00:07:30.890 align:middle line:84% is like, there was a set of input boundaries 00:07:30.890 --> 00:07:38.180 align:middle line:84% which, given the productivity, they 00:07:38.180 --> 00:07:42.725 align:middle line:84% will output the same level of output y-- 00:07:42.725 --> 00:07:48.320 align:middle line:90% 00:07:48.320 --> 00:07:53.810 align:middle line:84% all input bounders which produce the same level of output. 00:07:53.810 --> 00:07:55.310 align:middle line:84% ROBERT TOWNSEND: Yes that's correct. 00:07:55.310 --> 00:07:58.620 align:middle line:90% Great, thank you. 00:07:58.620 --> 00:08:04.740 align:middle line:84% OK, so I was too ambitious and I marked off too many of these. 00:08:04.740 --> 00:08:07.500 align:middle line:84% As I said, I sort of choose at random. 00:08:07.500 --> 00:08:10.870 align:middle line:84% In this case, this morning, I just marked a few of them. 00:08:10.870 --> 00:08:12.510 align:middle line:90% Take a look at the other ones. 00:08:12.510 --> 00:08:16.240 align:middle line:84% As always, these review questions are for your benefit, 00:08:16.240 --> 00:08:17.970 align:middle line:90% for the class. 00:08:17.970 --> 00:08:22.260 align:middle line:84% I'm only calling on you now instead of just taking 00:08:22.260 --> 00:08:26.310 align:middle line:84% volunteers because I like class participation, 00:08:26.310 --> 00:08:31.769 align:middle line:84% and it's just a reminder to try to keep up rather than having 00:08:31.769 --> 00:08:34.260 align:middle line:84% to make up lost ground later, when you're studying 00:08:34.260 --> 00:08:35.789 align:middle line:90% for the midterm or something. 00:08:35.789 --> 00:08:41.850 align:middle line:84% If you keep up with the class, you'll be fine. 00:08:41.850 --> 00:08:51.480 align:middle line:84% OK, so let me go to lecture 4 and talk a little bit 00:08:51.480 --> 00:08:56.760 align:middle line:84% about three related examples of production 00:08:56.760 --> 00:09:01.530 align:middle line:84% I featured in the lecture, the example of trade. 00:09:01.530 --> 00:09:03.210 align:middle line:84% But here are some others that actually 00:09:03.210 --> 00:09:07.680 align:middle line:84% do relate to what we will get to at the end of lecture 5 00:09:07.680 --> 00:09:08.880 align:middle line:90% as well. 00:09:08.880 --> 00:09:13.900 align:middle line:84% First off, I want to talk about Leontief input-output matrices. 00:09:13.900 --> 00:09:19.940 align:middle line:84% So Leontief's view of the economy's production 00:09:19.940 --> 00:09:25.370 align:middle line:84% possibilities was interesting, albeit a bit simple. 00:09:25.370 --> 00:09:31.130 align:middle line:84% Suppose there are three sectors, for example, as in this matrix. 00:09:31.130 --> 00:09:35.390 align:middle line:84% You have a sector representing raw materials, which 00:09:35.390 --> 00:09:38.630 align:middle line:84% could be agriculture, or mining, and so on, 00:09:38.630 --> 00:09:41.540 align:middle line:84% and a sector representing services, 00:09:41.540 --> 00:09:44.960 align:middle line:84% which would be retailing, advertising, transportation, 00:09:44.960 --> 00:09:49.400 align:middle line:84% lawyers, real estate agents, and financial services-- 00:09:49.400 --> 00:09:51.020 align:middle line:90% could go on and on. 00:09:51.020 --> 00:09:56.640 align:middle line:84% And manufacturing, which I guess is pretty self explanatory-- 00:09:56.640 --> 00:10:00.140 align:middle line:84% this entire page of lecture is just 00:10:00.140 --> 00:10:01.760 align:middle line:90% meant to be representative. 00:10:01.760 --> 00:10:06.950 align:middle line:84% Obviously, you can have many particular rows, 00:10:06.950 --> 00:10:09.380 align:middle line:84% depending on your willingness to disaggregate 00:10:09.380 --> 00:10:12.540 align:middle line:84% further and further into in more and more detail. 00:10:12.540 --> 00:10:16.520 align:middle line:84% So this matrix, what it's saying is the motivation 00:10:16.520 --> 00:10:19.100 align:middle line:84% for this is you have some final demand, 00:10:19.100 --> 00:10:21.230 align:middle line:84% and how much would you need to produce 00:10:21.230 --> 00:10:23.720 align:middle line:90% to satisfy the final demand? 00:10:23.720 --> 00:10:26.360 align:middle line:84% You could have final demand for services, 00:10:26.360 --> 00:10:30.860 align:middle line:84% but the problem with that is that's the final demand, not 00:10:30.860 --> 00:10:33.080 align:middle line:90% all the intermediate demands. 00:10:33.080 --> 00:10:37.880 align:middle line:84% Services are in turn used in other sectors. 00:10:37.880 --> 00:10:40.820 align:middle line:84% And in fact, the column here represents 00:10:40.820 --> 00:10:45.050 align:middle line:84% that to produce $1 worth of services-- 00:10:45.050 --> 00:10:48.800 align:middle line:84% services are used in other sectors, as in this row. 00:10:48.800 --> 00:10:52.880 align:middle line:84% Services show up in producing raw materials, 00:10:52.880 --> 00:10:56.090 align:middle line:84% services show up in producing services 00:10:56.090 --> 00:10:57.980 align:middle line:84% because of that aggregation problem, 00:10:57.980 --> 00:11:00.560 align:middle line:90% and get used in manufacturing. 00:11:00.560 --> 00:11:03.890 align:middle line:84% Likewise, if you go down the column, 00:11:03.890 --> 00:11:07.400 align:middle line:84% to produce $1 worth of services, you 00:11:07.400 --> 00:11:10.190 align:middle line:84% need $0.04 worth of raw materials, 00:11:10.190 --> 00:11:16.440 align:middle line:84% $0.03 worth of services, and $0.01 worth of manufacturing. 00:11:16.440 --> 00:11:20.300 align:middle line:84% So you can see, its input-output-- 00:11:20.300 --> 00:11:23.060 align:middle line:84% these are both in some sense direct outputs, 00:11:23.060 --> 00:11:28.280 align:middle line:84% but also indirect inputs into other sectors. 00:11:28.280 --> 00:11:32.170 align:middle line:84% This is a version, by the way, of the question 00:11:32.170 --> 00:11:34.770 align:middle line:90% having to do with aggregation. 00:11:34.770 --> 00:11:39.990 align:middle line:84% Last time, we had a slide showing two goods, x and y, 00:11:39.990 --> 00:11:40.650 align:middle line:90% I think-- 00:11:40.650 --> 00:11:43.170 align:middle line:90% or was it Y1 and Y2. 00:11:43.170 --> 00:11:47.160 align:middle line:84% And one used Y1 as an input and used Y2, 00:11:47.160 --> 00:11:52.440 align:middle line:84% and the other industry used Y2 as an input to produce Y1. 00:11:52.440 --> 00:11:54.720 align:middle line:90% That was decreasing returns. 00:11:54.720 --> 00:11:58.420 align:middle line:84% This is linear constant returns, because everything here 00:11:58.420 --> 00:12:04.550 align:middle line:84% is linear, but it is the same idea of inputs and outputs. 00:12:04.550 --> 00:12:10.000 align:middle line:84% So we take this information in the input-output matrix 00:12:10.000 --> 00:12:12.880 align:middle line:84% and summarize it in a matrix A, and here we're 00:12:12.880 --> 00:12:16.780 align:middle line:84% just reproducing these row column elements exactly. 00:12:16.780 --> 00:12:20.080 align:middle line:90% 00:12:20.080 --> 00:12:25.080 align:middle line:84% OK, so the problem is we have final demands, which are given, 00:12:25.080 --> 00:12:31.990 align:middle line:84% which is a three dimensional column vector. 00:12:31.990 --> 00:12:36.460 align:middle line:84% And it tells us how much we're going 00:12:36.460 --> 00:12:40.210 align:middle line:84% to need to end up delivering either to consumers 00:12:40.210 --> 00:12:45.780 align:middle line:84% or for exports, of raw material services 00:12:45.780 --> 00:12:50.080 align:middle line:84% and manufactured goods, in that order. 00:12:50.080 --> 00:12:54.030 align:middle line:84% So if we think about these equations here, 00:12:54.030 --> 00:12:57.240 align:middle line:90% we want to produce x. 00:12:57.240 --> 00:13:01.440 align:middle line:84% And then we want to figure out how much 00:13:01.440 --> 00:13:06.780 align:middle line:84% to produce if we want to meet the final demands of 400, 200, 00:13:06.780 --> 00:13:07.500 align:middle line:90% and 600. 00:13:07.500 --> 00:13:11.220 align:middle line:84% But in addition to that, some of that output 00:13:11.220 --> 00:13:14.490 align:middle line:84% gets used in turn as inputs, and is not 00:13:14.490 --> 00:13:20.860 align:middle line:84% available for consumer and export demands. 00:13:20.860 --> 00:13:27.930 align:middle line:84% So the key equation here would be, if we need to produce x, 00:13:27.930 --> 00:13:32.610 align:middle line:84% but we lose a times x because some of that output 00:13:32.610 --> 00:13:36.690 align:middle line:84% gets used as inputs, then we have 00:13:36.690 --> 00:13:44.300 align:middle line:84% to produce i minus a times X. I is the identity matrix here. 00:13:44.300 --> 00:13:47.440 align:middle line:90% So it just picks up the same-- 00:13:47.440 --> 00:13:52.680 align:middle line:84% it is 1 on the diagonal and 0 on the off diagonal. 00:13:52.680 --> 00:14:01.140 align:middle line:84% So for example, ia would just give you x back, 00:14:01.140 --> 00:14:04.590 align:middle line:84% as if you didn't have to use things as inputs. 00:14:04.590 --> 00:14:08.520 align:middle line:84% Ax, that we've been going through here, gives you-- 00:14:08.520 --> 00:14:10.620 align:middle line:84% if you're going to produce x, how much of that 00:14:10.620 --> 00:14:15.870 align:middle line:84% x gets used up in producing services, raw materials, 00:14:15.870 --> 00:14:17.550 align:middle line:90% and manufacturing? 00:14:17.550 --> 00:14:25.140 align:middle line:84% And what's left, I minus a times x, should just equal demand. 00:14:25.140 --> 00:14:31.550 align:middle line:84% So this is a linear equation in the control variables, namely 00:14:31.550 --> 00:14:33.090 align:middle line:90% x. 00:14:33.090 --> 00:14:38.230 align:middle line:90% And also, demands are given. 00:14:38.230 --> 00:14:41.230 align:middle line:84% Although to anticipate a bit, you 00:14:41.230 --> 00:14:45.220 align:middle line:84% might think there's some trade off in demands, 00:14:45.220 --> 00:14:49.090 align:middle line:84% or maybe some linear combination of demands is fine, 00:14:49.090 --> 00:14:52.030 align:middle line:90% not just specific numbers. 00:14:52.030 --> 00:14:56.990 align:middle line:84% And then you could imagine maximizing demand, 00:14:56.990 --> 00:15:00.260 align:middle line:84% but in any event subject to this constraint. 00:15:00.260 --> 00:15:01.130 align:middle line:90% Here, it's simpler. 00:15:01.130 --> 00:15:02.630 align:middle line:84% We're just given the demands, and we 00:15:02.630 --> 00:15:07.050 align:middle line:84% have to produce x in such a way as to meet the demand. 00:15:07.050 --> 00:15:12.790 align:middle line:84% So this is a linear system of linear equations. 00:15:12.790 --> 00:15:17.120 align:middle line:84% And if you remember a little matrix algebra, 00:15:17.120 --> 00:15:21.860 align:middle line:84% all we need to do is to invert this matrix, in minus a, 00:15:21.860 --> 00:15:23.940 align:middle line:84% which eliminates it on both sides-- 00:15:23.940 --> 00:15:27.630 align:middle line:84% on the left hand side, and puts it on to the right hand side. 00:15:27.630 --> 00:15:33.300 align:middle line:84% So the x solution we want is i minus a times dx. 00:15:33.300 --> 00:15:36.710 align:middle line:84% So I going to say in this case, you can go back up 00:15:36.710 --> 00:15:39.590 align:middle line:84% and realize that this was 0.2, so i-- 00:15:39.590 --> 00:15:43.910 align:middle line:90% 1 minus 0.2 is 0.98, and so on. 00:15:43.910 --> 00:15:46.280 align:middle line:84% Nothing gets subtracted on the off diagonal, 00:15:46.280 --> 00:15:49.770 align:middle line:90% because i is 0 there. 00:15:49.770 --> 00:15:53.270 align:middle line:84% So as long as this is what we would call a non singular 00:15:53.270 --> 00:15:55.730 align:middle line:84% matrix, that the rows and columns are 00:15:55.730 --> 00:16:00.740 align:middle line:84% independent of each other, one is not a multiplicative factor 00:16:00.740 --> 00:16:04.370 align:middle line:84% of any of the other two, then this inverse exists 00:16:04.370 --> 00:16:06.120 align:middle line:90% and we can solve the problem. 00:16:06.120 --> 00:16:11.000 align:middle line:84% And you can see although it should be intuitive by now 00:16:11.000 --> 00:16:17.030 align:middle line:84% that they need to produce more than the 400 of raw materials. 00:16:17.030 --> 00:16:20.000 align:middle line:90% They need to produce 449. 00:16:20.000 --> 00:16:25.910 align:middle line:84% Actually, these were denoted in billions, and likewise, 00:16:25.910 --> 00:16:31.400 align:middle line:84% how much more of each relative to the exogenously given demand 00:16:31.400 --> 00:16:40.410 align:middle line:84% actually depends in a pretty complicated way on that matrix 00:16:40.410 --> 00:16:43.890 align:middle line:90% A, in the following sense. 00:16:43.890 --> 00:16:49.870 align:middle line:84% To produce a unit of services, one 00:16:49.870 --> 00:16:55.080 align:middle line:84% has to take into account the inputs of manufacturing 00:16:55.080 --> 00:16:56.250 align:middle line:90% and raw materials. 00:16:56.250 --> 00:17:00.330 align:middle line:84% But to produce manufacturing and raw materials, 00:17:00.330 --> 00:17:06.960 align:middle line:84% one has to take into account that they too utilize services, 00:17:06.960 --> 00:17:07.883 align:middle line:90% and so on. 00:17:07.883 --> 00:17:09.300 align:middle line:84% And actually, you can kind of keep 00:17:09.300 --> 00:17:12.730 align:middle line:90% going back into the matrix. 00:17:12.730 --> 00:17:15.069 align:middle line:84% Another way to write the system of equations, 00:17:15.069 --> 00:17:20.589 align:middle line:84% actually, to get the inverse, is to successively 00:17:20.589 --> 00:17:24.190 align:middle line:84% plug-in first approximation, second approximation, 00:17:24.190 --> 00:17:27.819 align:middle line:84% third approximation as you take into account 00:17:27.819 --> 00:17:33.940 align:middle line:84% more and more of the layers of the interaction. 00:17:33.940 --> 00:17:39.210 align:middle line:84% So you could have a sector which doesn't contribute 00:17:39.210 --> 00:17:43.680 align:middle line:84% much to final demand, but is crucial as an input 00:17:43.680 --> 00:17:45.400 align:middle line:90% into the other two sectors. 00:17:45.400 --> 00:17:48.330 align:middle line:84% So that sector could have a high importance, 00:17:48.330 --> 00:17:51.090 align:middle line:84% even though it's not directly satisfying 00:17:51.090 --> 00:17:53.680 align:middle line:90% the demand of itself. 00:17:53.680 --> 00:17:58.920 align:middle line:84% So then we come to Google, and in particular their algorithm. 00:17:58.920 --> 00:18:03.060 align:middle line:84% When you type in a keyword, it pops up in a rank order 00:18:03.060 --> 00:18:05.580 align:middle line:84% the most likely things that are related 00:18:05.580 --> 00:18:07.290 align:middle line:90% to what you just searched for. 00:18:07.290 --> 00:18:11.910 align:middle line:84% This was invented by Sergey Brin and Larry Page. 00:18:11.910 --> 00:18:15.330 align:middle line:84% I sometimes wonder if people's last names are not 00:18:15.330 --> 00:18:17.730 align:middle line:84% predictors of their eventually careers, 00:18:17.730 --> 00:18:21.930 align:middle line:84% Larry Page being one of the fathers of PageRank, 00:18:21.930 --> 00:18:26.550 align:middle line:84% but anyway, that algorithm is really based on the same ideas 00:18:26.550 --> 00:18:31.950 align:middle line:84% that underlie Leontief's ranking of industries, which is you 00:18:31.950 --> 00:18:37.560 align:middle line:84% Google up something, and it may be 00:18:37.560 --> 00:18:41.460 align:middle line:84% that there's an article that you might not 00:18:41.460 --> 00:18:44.940 align:middle line:84% have thought to be the most important one, 00:18:44.940 --> 00:18:47.940 align:middle line:84% because it may not seem to answer the question directly. 00:18:47.940 --> 00:18:50.280 align:middle line:84% But in fact, it's the key article 00:18:50.280 --> 00:18:53.220 align:middle line:84% that started it all, that has been cited and used 00:18:53.220 --> 00:18:57.210 align:middle line:84% by other things that are more directly linked to the Google 00:18:57.210 --> 00:18:58.980 align:middle line:90% item that you're searching for. 00:18:58.980 --> 00:19:02.910 align:middle line:84% And this display is not by sectors of industries, 00:19:02.910 --> 00:19:06.060 align:middle line:90% but just by a network diagram. 00:19:06.060 --> 00:19:10.020 align:middle line:84% You can see these things directly related 00:19:10.020 --> 00:19:14.700 align:middle line:84% via pathways to things in the center, these outliers 00:19:14.700 --> 00:19:15.900 align:middle line:90% or clusters. 00:19:15.900 --> 00:19:18.720 align:middle line:84% And then it gets very thin, but they could end up 00:19:18.720 --> 00:19:20.880 align:middle line:84% being important because they're related 00:19:20.880 --> 00:19:22.900 align:middle line:90% to things in the interior. 00:19:22.900 --> 00:19:25.710 align:middle line:84% These things are ranked by distance, actually, 00:19:25.710 --> 00:19:29.160 align:middle line:84% reordered in order for the diagram 00:19:29.160 --> 00:19:32.720 align:middle line:84% to display the degree of closeness. 00:19:32.720 --> 00:19:35.360 align:middle line:84% And they-- Page and those guys actually 00:19:35.360 --> 00:19:38.990 align:middle line:84% cited on Leontief and output matrix. 00:19:38.990 --> 00:19:44.700 align:middle line:84% And then as a final example, the Great East 00:19:44.700 --> 00:19:47.150 align:middle line:90% Japanese earthquake-- 00:19:47.150 --> 00:19:52.760 align:middle line:84% to remind you, there was an earthquake, a big one, and then 00:19:52.760 --> 00:19:54.920 align:middle line:90% a subsequence tidal wave. 00:19:54.920 --> 00:19:58.580 align:middle line:84% And actually, the coastline I think 00:19:58.580 --> 00:20:03.480 align:middle line:84% subsided by a meter, or some outrageous amount. 00:20:03.480 --> 00:20:08.780 align:middle line:84% And so in came the tidal wave, doing a lot of damage. 00:20:08.780 --> 00:20:14.070 align:middle line:84% And this is highlighting the damaged structures. 00:20:14.070 --> 00:20:19.340 align:middle line:84% And this is highlighting the headquarters 00:20:19.340 --> 00:20:23.690 align:middle line:84% of firms whose headquarters' location was destroyed 00:20:23.690 --> 00:20:27.500 align:middle line:84% in the immediate aftermath of the earthquake and tsunami. 00:20:27.500 --> 00:20:32.000 align:middle line:84% By the way, the Fukushima nuclear power plant 00:20:32.000 --> 00:20:34.430 align:middle line:84% was right there, adding to the mix. 00:20:34.430 --> 00:20:36.050 align:middle line:90% So then we move over here. 00:20:36.050 --> 00:20:38.780 align:middle line:84% This is all of the main island of Japan. 00:20:38.780 --> 00:20:43.580 align:middle line:84% And this is the area that's highlighted, the small area 00:20:43.580 --> 00:20:44.900 align:middle line:90% here. 00:20:44.900 --> 00:20:48.380 align:middle line:84% And then we ask, as an input-output sense, 00:20:48.380 --> 00:20:54.290 align:middle line:84% which industries had been supplying to headquarters which 00:20:54.290 --> 00:20:58.070 align:middle line:84% is not per se just where all the management is, but like 00:20:58.070 --> 00:21:00.140 align:middle line:90% the primary production facility. 00:21:00.140 --> 00:21:03.810 align:middle line:84% You have inputs supplying to General Motors, for example, 00:21:03.810 --> 00:21:04.370 align:middle line:90% in Detroit. 00:21:04.370 --> 00:21:07.650 align:middle line:84% You have tire manufacturers that are linked-- this is Japan. 00:21:07.650 --> 00:21:11.090 align:middle line:84% So we have input suppliers linked to headquarters, 00:21:11.090 --> 00:21:12.650 align:middle line:90% or intermediate. 00:21:12.650 --> 00:21:17.780 align:middle line:84% And likewise, in principle, it's not just inputs from upstream, 00:21:17.780 --> 00:21:19.580 align:middle line:90% but outputs downstream. 00:21:19.580 --> 00:21:22.220 align:middle line:84% The disruption here means you can't produce something 00:21:22.220 --> 00:21:25.880 align:middle line:84% which is in turn supplied downstream, quote unquote, 00:21:25.880 --> 00:21:26.670 align:middle line:90% to others. 00:21:26.670 --> 00:21:30.920 align:middle line:84% So the first order connection, as in the PageRank, 00:21:30.920 --> 00:21:32.390 align:middle line:90% are these areas in red. 00:21:32.390 --> 00:21:34.370 align:middle line:84% And now, we've covered all of Japan, 00:21:34.370 --> 00:21:37.730 align:middle line:84% because there's either an immediate input supplier 00:21:37.730 --> 00:21:41.210 align:middle line:84% or output purchaser indirectly damaged 00:21:41.210 --> 00:21:44.005 align:middle line:84% in the sense of the first degree, again, 00:21:44.005 --> 00:21:45.840 align:middle line:84% that's the primary area of damage. 00:21:45.840 --> 00:21:49.130 align:middle line:84% This is second degree damage, A linked to B, 00:21:49.130 --> 00:21:51.960 align:middle line:90% who was hit with C, and so on. 00:21:51.960 --> 00:21:55.820 align:middle line:84% So eventually, as you go in this case down to the fifth order, 00:21:55.820 --> 00:21:57.590 align:middle line:90% most of Japan was covered. 00:21:57.590 --> 00:22:00.470 align:middle line:84% And in fact, this Japanese earthquake 00:22:00.470 --> 00:22:05.150 align:middle line:84% caused a drop in national level GDP of about 1.5%. 00:22:05.150 --> 00:22:10.490 align:middle line:84% It is, though, just another illustration 00:22:10.490 --> 00:22:12.350 align:middle line:90% of the same mathematics. 00:22:12.350 --> 00:22:14.690 align:middle line:84% So now, with some trepidation, let 00:22:14.690 --> 00:22:17.300 align:middle line:84% me go to lecture 5 and verify that you 00:22:17.300 --> 00:22:22.590 align:middle line:84% can see this on your screens as I move through it. 00:22:22.590 --> 00:22:24.560 align:middle line:90% Yes, you can see it right? 00:22:24.560 --> 00:22:28.430 align:middle line:84% So this is decision making under uncertainty, 00:22:28.430 --> 00:22:31.820 align:middle line:84% and then we'll get back to some of our techniques, 00:22:31.820 --> 00:22:34.410 align:middle line:84% in particular programming, linear programming. 00:22:34.410 --> 00:22:37.460 align:middle line:84% So let's suppose we have a lottery, 00:22:37.460 --> 00:22:42.260 align:middle line:84% and there are little n distinct prizes, possibly zero, possibly 00:22:42.260 --> 00:22:45.120 align:middle line:90% negative if it takes money away. 00:22:45.120 --> 00:22:51.140 align:middle line:84% So these X's are the amount of the winnings or losings, 00:22:51.140 --> 00:22:54.770 align:middle line:84% if that even turns up in the lottery. 00:22:54.770 --> 00:22:56.780 align:middle line:84% There are n possible things that can happen, 00:22:56.780 --> 00:22:59.150 align:middle line:84% and they happen with these probabilities 00:22:59.150 --> 00:23:04.190 align:middle line:84% pi, pi1 being the probability that the gains or losses is 00:23:04.190 --> 00:23:08.420 align:middle line:84% in the order of x1, pi2 for x2, and so on. 00:23:08.420 --> 00:23:11.360 align:middle line:84% Because it's a lottery, the probabilities 00:23:11.360 --> 00:23:13.550 align:middle line:90% themselves are not negative. 00:23:13.550 --> 00:23:15.860 align:middle line:90% They're 0 or positive. 00:23:15.860 --> 00:23:18.950 align:middle line:84% And it's a comprehensive lottery, 00:23:18.950 --> 00:23:22.710 align:middle line:84% choosing over this vector of possible outcomes, 00:23:22.710 --> 00:23:25.460 align:middle line:84% so the probabilities add up to 1. 00:23:25.460 --> 00:23:30.140 align:middle line:84% In other words, the lottery is a well-defined random variable, 00:23:30.140 --> 00:23:33.500 align:middle line:84% with these n distinct realizations. 00:23:33.500 --> 00:23:36.920 align:middle line:84% And we can call the expectation of the lottery, 00:23:36.920 --> 00:23:39.710 align:middle line:84% the expected outcome of the lottery 00:23:39.710 --> 00:23:42.470 align:middle line:84% of the random variable capital X, 00:23:42.470 --> 00:23:46.430 align:middle line:84% just to be the weighted average of the outcomes 00:23:46.430 --> 00:23:48.900 align:middle line:84% where the weights are these probabilities. 00:23:48.900 --> 00:23:51.260 align:middle line:84% So part of this-- these next few slides 00:23:51.260 --> 00:23:55.640 align:middle line:84% have to do with the ideas of lotteries over discrete items. 00:23:55.640 --> 00:24:00.020 align:middle line:84% Part of the next few slides are quick reviews 00:24:00.020 --> 00:24:02.640 align:middle line:84% of some basic ideas and statistics. 00:24:02.640 --> 00:24:05.630 align:middle line:84% Here's a basic idea from statistics. 00:24:05.630 --> 00:24:08.690 align:middle line:84% If we have a random variable, capital X, 00:24:08.690 --> 00:24:12.660 align:middle line:84% when we want to talk about its variability or its variance, 00:24:12.660 --> 00:24:17.900 align:middle line:84% denoted is sigma squared X, that is just the dispersion of X 00:24:17.900 --> 00:24:22.010 align:middle line:84% around the mean, mu X. Mu X was defined here 00:24:22.010 --> 00:24:26.140 align:middle line:84% in the previous slide as the mean squared. 00:24:26.140 --> 00:24:27.820 align:middle line:90% So we square the differences. 00:24:27.820 --> 00:24:31.800 align:middle line:84% If X were equal to its mean, this would be 0. 00:24:31.800 --> 00:24:34.290 align:middle line:84% The further away it is, the higher 00:24:34.290 --> 00:24:36.090 align:middle line:90% the squared value would be. 00:24:36.090 --> 00:24:39.030 align:middle line:84% Expectation again means that we're just 00:24:39.030 --> 00:24:43.080 align:middle line:84% taking weighted averages, with the weights being these pi I 00:24:43.080 --> 00:24:44.290 align:middle line:90% probabilities. 00:24:44.290 --> 00:24:46.500 align:middle line:84% So this would be the formal expression. 00:24:46.500 --> 00:24:50.820 align:middle line:84% The variance is the proxy for risk. 00:24:50.820 --> 00:24:52.710 align:middle line:84% The square root of the variance is 00:24:52.710 --> 00:24:55.420 align:middle line:90% called the standard deviation. 00:24:55.420 --> 00:24:58.260 align:middle line:84% And if we want to normalize it so we can interpret 00:24:58.260 --> 00:25:01.440 align:middle line:84% the units of it without knowing the magnitudes 00:25:01.440 --> 00:25:05.890 align:middle line:84% of the underlying variable, we divide by its mean. 00:25:05.890 --> 00:25:10.260 align:middle line:84% So the standard deviation divided by the mean 00:25:10.260 --> 00:25:13.980 align:middle line:84% is called the coefficient of variation. 00:25:13.980 --> 00:25:17.850 align:middle line:84% What I mean by normalizing isn't just that we're dividing, 00:25:17.850 --> 00:25:19.800 align:middle line:84% but the reason for dividing-- you 00:25:19.800 --> 00:25:22.830 align:middle line:84% could well imagine if we have two random variables 00:25:22.830 --> 00:25:26.580 align:middle line:84% and one is twice the other one, then all the values 00:25:26.580 --> 00:25:31.140 align:middle line:84% are doubled, and the measure of the dispersion will go up too. 00:25:31.140 --> 00:25:33.150 align:middle line:84% So if we wanted to measure variability, 00:25:33.150 --> 00:25:35.880 align:middle line:84% that's kind of independent of the scale of the underlying 00:25:35.880 --> 00:25:37.830 align:middle line:90% random variable itself. 00:25:37.830 --> 00:25:41.070 align:middle line:90% One more slide on statistics-- 00:25:41.070 --> 00:25:47.220 align:middle line:84% so suppose the prize has two dimensions, X and Y, not just 00:25:47.220 --> 00:25:50.760 align:middle line:84% the winning and losings of dollars, but maybe 00:25:50.760 --> 00:25:52.590 align:middle line:90% two different commodities. 00:25:52.590 --> 00:25:57.390 align:middle line:84% So a particular realization or outcome 00:25:57.390 --> 00:26:00.840 align:middle line:84% would be a specification of both the value of X 00:26:00.840 --> 00:26:04.710 align:middle line:84% and the value of Y Xi, Yi for realization i. 00:26:04.710 --> 00:26:08.430 align:middle line:84% Realization i happens with probability pi i, 00:26:08.430 --> 00:26:10.870 align:middle line:90% which is the same as before. 00:26:10.870 --> 00:26:13.140 align:middle line:84% Now, though, we can define the covariance 00:26:13.140 --> 00:26:16.590 align:middle line:84% between two random variables, x and y, 00:26:16.590 --> 00:26:22.170 align:middle line:84% to be the expectation of the product subtracting off 00:26:22.170 --> 00:26:23.400 align:middle line:90% the means. 00:26:23.400 --> 00:26:28.320 align:middle line:84% So X plus its mean uX, times Y minus its mean uY. 00:26:28.320 --> 00:26:30.750 align:middle line:84% And I didn't write it out, but you could just sum it up 00:26:30.750 --> 00:26:36.150 align:middle line:84% over the same pi is instead of writing the expectation 00:26:36.150 --> 00:26:36.960 align:middle line:90% operator. 00:26:36.960 --> 00:26:40.920 align:middle line:84% It tells you whether the variables x and y 00:26:40.920 --> 00:26:44.740 align:middle line:84% are moving together or moving in opposite directions. 00:26:44.740 --> 00:26:49.860 align:middle line:84% So for example, when a high value of X 00:26:49.860 --> 00:26:52.530 align:middle line:84% corresponds with a high value of Y, 00:26:52.530 --> 00:26:57.190 align:middle line:84% thinking collectively about these realizations Xi or Yi, 00:26:57.190 --> 00:27:00.030 align:middle line:84% then this covariance is positive. 00:27:00.030 --> 00:27:04.290 align:middle line:84% And likewise, if a high x means a low y, 00:27:04.290 --> 00:27:06.700 align:middle line:84% the covariance would be negative. 00:27:06.700 --> 00:27:11.610 align:middle line:84% You could think about plotting the Xi, Yi variables 00:27:11.610 --> 00:27:14.460 align:middle line:84% as in a cloud and trying to figure out 00:27:14.460 --> 00:27:18.000 align:middle line:84% if the cloud is running on the 45 degree line 00:27:18.000 --> 00:27:20.490 align:middle line:84% or running in the reverse direction, 00:27:20.490 --> 00:27:24.600 align:middle line:90% from northwest to southeast. 00:27:24.600 --> 00:27:29.700 align:middle line:84% Again, it's useful to normalize, so we take that covariance 00:27:29.700 --> 00:27:34.170 align:middle line:84% and just divide it by the product of the variance, called 00:27:34.170 --> 00:27:35.490 align:middle line:90% the correlation. 00:27:35.490 --> 00:27:38.910 align:middle line:84% And correlations are by construction going 00:27:38.910 --> 00:27:41.610 align:middle line:90% to be between minus 1 and 1. 00:27:41.610 --> 00:27:46.140 align:middle line:84% So again, 0 means no correlation, 1 means perfect 00:27:46.140 --> 00:27:50.160 align:middle line:84% correlation, positive direction, n minus 1, 00:27:50.160 --> 00:27:52.990 align:middle line:90% means negative correlation. 00:27:52.990 --> 00:27:56.460 align:middle line:84% So those are the three statistical review slides, 00:27:56.460 --> 00:27:58.800 align:middle line:84% and we're going to come back to them momentarily. 00:27:58.800 --> 00:28:00.910 align:middle line:84% They're all defined for random variables, 00:28:00.910 --> 00:28:04.200 align:middle line:84% but we defined a random variable by first thinking 00:28:04.200 --> 00:28:05.130 align:middle line:90% about a lottery. 00:28:05.130 --> 00:28:08.280 align:middle line:84% So let's come back to the lottery. 00:28:08.280 --> 00:28:14.700 align:middle line:84% Here, in fact, the underlying ingredients are x1 and x2, 00:28:14.700 --> 00:28:17.940 align:middle line:84% but they can only take on a finite number of values. 00:28:17.940 --> 00:28:23.850 align:middle line:84% These dots represent a discrete point, and this commodity space 00:28:23.850 --> 00:28:27.900 align:middle line:84% here, the values of x1 and x2, which are feasible, 00:28:27.900 --> 00:28:31.410 align:middle line:84% this is not a convex space, because you can easily 00:28:31.410 --> 00:28:35.700 align:middle line:84% find lines between any two of these dots. 00:28:35.700 --> 00:28:39.420 align:middle line:84% And intermediate points on the line other than the end point 00:28:39.420 --> 00:28:45.250 align:middle line:90% are not in this consumption set. 00:28:45.250 --> 00:28:47.500 align:middle line:90% So this is a non-convex set. 00:28:47.500 --> 00:28:49.810 align:middle line:84% Back to the lottery interpretation, 00:28:49.810 --> 00:28:55.840 align:middle line:84% lets call, say, Cs a particular realization or state or point 00:28:55.840 --> 00:28:57.400 align:middle line:90% in the commodity space-- 00:28:57.400 --> 00:29:03.310 align:middle line:84% and capital C. S denotes the set of all possible states. 00:29:03.310 --> 00:29:05.230 align:middle line:84% It's just counting up the number of dots 00:29:05.230 --> 00:29:07.460 align:middle line:90% you're seeing on this picture. 00:29:07.460 --> 00:29:11.710 align:middle line:84% So for example, if there are capital S dots or states, 00:29:11.710 --> 00:29:15.130 align:middle line:84% each state corresponds to a particular consumption 00:29:15.130 --> 00:29:20.710 align:middle line:84% bundle in R2, and we can call pi S the probability 00:29:20.710 --> 00:29:27.670 align:middle line:84% of being in state S. So we could have a lottery over this space. 00:29:27.670 --> 00:29:31.690 align:middle line:84% If we define those lotteries, a particular lottery 00:29:31.690 --> 00:29:36.880 align:middle line:84% would put a non-negative probability, perhaps 0, 00:29:36.880 --> 00:29:39.040 align:middle line:90% on any particular state. 00:29:39.040 --> 00:29:42.190 align:middle line:84% And the sum of the probabilities over all the states 00:29:42.190 --> 00:29:44.440 align:middle line:90% would have to add up to 1. 00:29:44.440 --> 00:29:47.500 align:middle line:90% So that's a particular lottery. 00:29:47.500 --> 00:29:51.070 align:middle line:84% The set of all possible lottery is the set 00:29:51.070 --> 00:29:55.960 align:middle line:84% of all probabilities pi that satisfy these two characters 00:29:55.960 --> 00:29:58.210 align:middle line:90% called, call it capital Pi. 00:29:58.210 --> 00:30:00.220 align:middle line:90% It's also called a simplex. 00:30:00.220 --> 00:30:02.620 align:middle line:84% By definition, a simplex is a set 00:30:02.620 --> 00:30:05.170 align:middle line:84% where each element is non-negative, 00:30:05.170 --> 00:30:07.930 align:middle line:84% and all the elements add up to 1. 00:30:07.930 --> 00:30:11.110 align:middle line:84% So a lottery satisfies the definition of a simplex. 00:30:11.110 --> 00:30:12.790 align:middle line:90% Other things do, too. 00:30:12.790 --> 00:30:17.260 align:middle line:84% OK, so this becomes the new commodity space, not 00:30:17.260 --> 00:30:19.210 align:middle line:84% the set of underlying points, but the set 00:30:19.210 --> 00:30:21.370 align:middle line:90% of lotteries on those points. 00:30:21.370 --> 00:30:25.270 align:middle line:84% And likewise, we now can define preferences 00:30:25.270 --> 00:30:28.780 align:middle line:84% with this squiggly script inequality 00:30:28.780 --> 00:30:31.300 align:middle line:84% over the set of possible lotteries. 00:30:31.300 --> 00:30:34.810 align:middle line:84% And what we're going to use is von Neumann Morgenstern 00:30:34.810 --> 00:30:37.960 align:middle line:84% expected utility, which is quite natural. 00:30:37.960 --> 00:30:41.290 align:middle line:84% And it's saying, take a particular point 00:30:41.290 --> 00:30:46.370 align:middle line:84% Cs in that space, take the underlying utility of it-- 00:30:46.370 --> 00:30:48.070 align:middle line:90% which we've been using before. 00:30:48.070 --> 00:30:49.930 align:middle line:84% Take the probability of that state 00:30:49.930 --> 00:30:53.620 align:middle line:84% and then sum it up over all possible states 00:30:53.620 --> 00:30:57.940 align:middle line:84% and define this expected utility. 00:30:57.940 --> 00:31:01.600 align:middle line:84% So these are the underlying real value numbers 00:31:01.600 --> 00:31:03.460 align:middle line:90% of deterministic utility. 00:31:03.460 --> 00:31:05.110 align:middle line:84% This object over here on the left 00:31:05.110 --> 00:31:10.030 align:middle line:84% is the expected utility for a particular lottery pi. 00:31:10.030 --> 00:31:13.120 align:middle line:84% And the idea now is that all possible lotteries 00:31:13.120 --> 00:31:17.590 align:middle line:84% can be ranked by this expected utility criterion. 00:31:17.590 --> 00:31:21.370 align:middle line:84% Let's just note in passing that this expected utility 00:31:21.370 --> 00:31:25.660 align:middle line:84% criterion is linear in the choice objects, which 00:31:25.660 --> 00:31:27.310 align:middle line:90% are the lotteries. 00:31:27.310 --> 00:31:31.990 align:middle line:84% These u of Cs things become coefficients, real numbers. 00:31:31.990 --> 00:31:34.730 align:middle line:84% But we're not varying the underlying commodity points. 00:31:34.730 --> 00:31:37.060 align:middle line:90% So these become fixed numbers. 00:31:37.060 --> 00:31:39.790 align:middle line:84% And the thing that we're interested in varying 00:31:39.790 --> 00:31:43.610 align:middle line:84% would be the probabilities that constitute the lottery. 00:31:43.610 --> 00:31:47.980 align:middle line:84% So the objective function is linear in the probabilities. 00:31:47.980 --> 00:31:51.820 align:middle line:84% There are some criteria we need to satisfy in order 00:31:51.820 --> 00:31:56.470 align:middle line:90% to rank using expected utility. 00:31:56.470 --> 00:31:59.110 align:middle line:84% What we want to get at the end of the day 00:31:59.110 --> 00:32:03.760 align:middle line:84% is that a lottery pi is strictly preferred to a lottery pi 00:32:03.760 --> 00:32:07.300 align:middle line:84% prime, if the expected utility under pi 00:32:07.300 --> 00:32:10.150 align:middle line:84% is strictly greater than the expected utility under pi 00:32:10.150 --> 00:32:11.690 align:middle line:90% prime. 00:32:11.690 --> 00:32:17.300 align:middle line:84% In order to do that, we need three axioms, these three. 00:32:17.300 --> 00:32:19.160 align:middle line:90% The first combines two-- 00:32:19.160 --> 00:32:20.780 align:middle line:90% cheating a bit. 00:32:20.780 --> 00:32:24.800 align:middle line:84% Completeness says that all lotteries can 00:32:24.800 --> 00:32:28.630 align:middle line:90% be ranked and give us a number. 00:32:28.630 --> 00:32:32.470 align:middle line:84% And transitivity, I think you can see, 00:32:32.470 --> 00:32:37.630 align:middle line:84% is going to be satisfied, because if A is preferred to B, 00:32:37.630 --> 00:32:40.360 align:middle line:84% lottery pi is preferred to lottery pi prime, 00:32:40.360 --> 00:32:46.090 align:middle line:84% and pi prime is preferred to pi double prime. 00:32:46.090 --> 00:32:48.070 align:middle line:84% Under the preference ordering, we 00:32:48.070 --> 00:32:50.230 align:middle line:84% can map those preference orderings 00:32:50.230 --> 00:32:54.420 align:middle line:84% into these expected utility objects, 00:32:54.420 --> 00:32:59.030 align:middle line:84% which are just real numbers that satisfy transitivity. 00:32:59.030 --> 00:33:03.950 align:middle line:84% So those are intuitive properties, completeness 00:33:03.950 --> 00:33:05.990 align:middle line:90% and transitivity. 00:33:05.990 --> 00:33:09.920 align:middle line:84% Here's something called independence. 00:33:09.920 --> 00:33:15.350 align:middle line:84% Take two lotteries of focus, denoted pi and pi prime, 00:33:15.350 --> 00:33:19.460 align:middle line:84% and say that pi is weakly preferred under that expected 00:33:19.460 --> 00:33:22.050 align:middle line:90% utility ordering to pi prime. 00:33:22.050 --> 00:33:30.600 align:middle line:84% Then it has to be true that some alpha weighted lottery-- 00:33:30.600 --> 00:33:33.750 align:middle line:84% this is a lottery, pi is a lottery. 00:33:33.750 --> 00:33:36.300 align:middle line:90% And pi prime is a lottery. 00:33:36.300 --> 00:33:39.630 align:middle line:84% And pi triple prime is also a lottery. 00:33:39.630 --> 00:33:44.230 align:middle line:84% So now we're taking compound probabilities, if you like, 00:33:44.230 --> 00:33:48.660 align:middle line:84% when we take the alpha weighted sum of the lottery pi and pi 00:33:48.660 --> 00:33:51.440 align:middle line:90% prime. 00:33:51.440 --> 00:33:55.880 align:middle line:84% If alpha were, say, equal to 1, then we're 00:33:55.880 --> 00:33:59.692 align:middle line:84% already given that pi is weakly preferred to pi prime, 00:33:59.692 --> 00:34:02.150 align:middle line:84% and that would be an equivalent statement on the right hand 00:34:02.150 --> 00:34:03.020 align:middle line:90% side. 00:34:03.020 --> 00:34:06.230 align:middle line:84% If alpha is less than 1, however, 00:34:06.230 --> 00:34:09.320 align:middle line:84% then we're putting some weight on this extraneous third 00:34:09.320 --> 00:34:13.580 align:middle line:84% lottery, pi double prime, and the notion of independence 00:34:13.580 --> 00:34:18.199 align:middle line:84% means that whatever this other lottery looks like, 00:34:18.199 --> 00:34:21.199 align:middle line:84% it doesn't matter for fixed alpha 00:34:21.199 --> 00:34:26.780 align:middle line:84% when it comes to weighting pi and pi double-- 00:34:26.780 --> 00:34:28.790 align:middle line:90% weighting the composite lottery. 00:34:28.790 --> 00:34:32.840 align:middle line:84% Because each one of these, The second term 00:34:32.840 --> 00:34:36.949 align:middle line:84% in each one of these expressions is 1 minus alpha times 00:34:36.949 --> 00:34:40.370 align:middle line:84% pi double prime, which is the same on the left hand side 00:34:40.370 --> 00:34:42.719 align:middle line:90% as it is on the right hand side. 00:34:42.719 --> 00:34:47.423 align:middle line:84% So the ranking is independent of whatever that term is. 00:34:47.423 --> 00:34:48.840 align:middle line:84% It seems like it ought to be true, 00:34:48.840 --> 00:34:50.880 align:middle line:90% but it's actually an axiom. 00:34:50.880 --> 00:34:55.810 align:middle line:84% Continuity is similar, although not identical. 00:34:55.810 --> 00:35:00.120 align:middle line:84% It says if we have three lotteries pi, pi prime, 00:35:00.120 --> 00:35:03.780 align:middle line:84% pi triple prime, and they're rank ordered-- 00:35:03.780 --> 00:35:08.470 align:middle line:84% pi, preferred to pi prime, preferred to pi double prime, 00:35:08.470 --> 00:35:14.060 align:middle line:84% then there exist numbers alpha and beta such 00:35:14.060 --> 00:35:19.550 align:middle line:84% that this weighted sum of pi and pi 00:35:19.550 --> 00:35:22.760 align:middle line:84% double prime, weighted by alpha, is strictly 00:35:22.760 --> 00:35:24.980 align:middle line:84% preferred to pi prime, which is strictly 00:35:24.980 --> 00:35:29.180 align:middle line:84% preferred by the beta weighted sum of pi and pi double prime. 00:35:29.180 --> 00:35:34.630 align:middle line:84% Now the way to see this is that it's not 00:35:34.630 --> 00:35:37.090 align:middle line:84% required to be true for all alpha and beta, 00:35:37.090 --> 00:35:39.350 align:middle line:84% it's only required to be true for some of them. 00:35:39.350 --> 00:35:42.610 align:middle line:84% So if we were allowed to set alpha equal to 1, 00:35:42.610 --> 00:35:47.610 align:middle line:84% we would have alpha pi on the left hand side, 00:35:47.610 --> 00:35:51.690 align:middle line:90% and pi prime would be over here. 00:35:51.690 --> 00:35:53.760 align:middle line:84% We'd have pi preferred to pi prime, 00:35:53.760 --> 00:35:58.630 align:middle line:84% but that was a given to begin with, so no new information. 00:35:58.630 --> 00:36:00.960 align:middle line:84% This is saying if alpha is arbitrarily 00:36:00.960 --> 00:36:03.270 align:middle line:84% close to 1, then that ought to be true, 00:36:03.270 --> 00:36:05.970 align:middle line:84% and hence the notion of continuity. 00:36:05.970 --> 00:36:08.340 align:middle line:84% Likewise, on the right hand side, 00:36:08.340 --> 00:36:13.650 align:middle line:84% if beta were zero, which it's not allowed to be, 00:36:13.650 --> 00:36:16.620 align:middle line:84% then we'd be just getting pi double prime here. 00:36:16.620 --> 00:36:20.520 align:middle line:84% And we're already given that pi prime is 00:36:20.520 --> 00:36:23.270 align:middle line:90% preferred to pi double prime. 00:36:23.270 --> 00:36:28.490 align:middle line:84% The content of it is if beta is not 0 but very close to 0, 00:36:28.490 --> 00:36:31.850 align:middle line:84% then we should preserve the rank order. 00:36:31.850 --> 00:36:34.280 align:middle line:90% These are intuitive properties. 00:36:34.280 --> 00:36:37.730 align:middle line:84% They're here only to say that expected utility may not 00:36:37.730 --> 00:36:41.840 align:middle line:84% be, quote unquote, as general as you might think when facing 00:36:41.840 --> 00:36:43.860 align:middle line:84% problems of choice under uncertainty, 00:36:43.860 --> 00:36:46.490 align:middle line:90% although it is very natural. 00:36:46.490 --> 00:36:50.180 align:middle line:84% And these properties are similar in some respects 00:36:50.180 --> 00:36:52.880 align:middle line:84% to what we did before with ordinary preference orders. 00:36:52.880 --> 00:36:55.700 align:middle line:84% And with regard to what we did before, 00:36:55.700 --> 00:36:57.680 align:middle line:84% there was already a question from one of you 00:36:57.680 --> 00:37:01.460 align:middle line:84% about this, what about choice under uncertainty? 00:37:01.460 --> 00:37:05.570 align:middle line:84% Now, our measure of utility becomes cardinal, not 00:37:05.570 --> 00:37:07.290 align:middle line:90% just ordinal. 00:37:07.290 --> 00:37:11.750 align:middle line:84% We are really going to care about the degree of curvature 00:37:11.750 --> 00:37:16.730 align:middle line:84% in this orange line, which is mapping the expected utility 00:37:16.730 --> 00:37:20.150 align:middle line:90% outcome as we vary the input x. 00:37:20.150 --> 00:37:23.060 align:middle line:84% So what's going on in this setup here? 00:37:23.060 --> 00:37:26.030 align:middle line:84% We have income, say, moving around, 00:37:26.030 --> 00:37:30.680 align:middle line:84% and it has a mean expected value of x, 00:37:30.680 --> 00:37:34.130 align:middle line:84% and it has a couple of endpoints-- the lowest value 00:37:34.130 --> 00:37:36.530 align:middle line:90% x0, the highest value x1. 00:37:36.530 --> 00:37:40.280 align:middle line:84% Let's say for simplicity that's all that's going on, 00:37:40.280 --> 00:37:42.920 align:middle line:84% that the lottery that creates the random variable 00:37:42.920 --> 00:37:47.990 align:middle line:84% is nothing other than some weighted average of x0 and x1. 00:37:47.990 --> 00:37:51.920 align:middle line:84% And those weights are such as to produce exactly this expected 00:37:51.920 --> 00:37:53.810 align:middle line:90% value. 00:37:53.810 --> 00:37:58.630 align:middle line:84% And that's what's being drawn with this line, 00:37:58.630 --> 00:38:04.750 align:middle line:84% namely expectation of x is a weighted combination 00:38:04.750 --> 00:38:06.640 align:middle line:90% of x0 and x1. 00:38:06.640 --> 00:38:08.890 align:middle line:84% Given the expectation of x, we could 00:38:08.890 --> 00:38:14.170 align:middle line:84% ask what the utility would be for that deterministic value, 00:38:14.170 --> 00:38:15.430 align:middle line:90% and it would be-- 00:38:15.430 --> 00:38:19.360 align:middle line:84% we would have to go up here if they were getting it for sure, 00:38:19.360 --> 00:38:21.640 align:middle line:84% and over here, and have the utility 00:38:21.640 --> 00:38:24.640 align:middle line:90% of the expectation of x. 00:38:24.640 --> 00:38:26.110 align:middle line:84% That may be a funny place to start, 00:38:26.110 --> 00:38:28.510 align:middle line:84% because there's a genuine lottery here, 00:38:28.510 --> 00:38:31.480 align:middle line:84% and you can't get the mean for sure. 00:38:31.480 --> 00:38:33.910 align:middle line:84% You have to engage in the lottery 00:38:33.910 --> 00:38:37.520 align:middle line:84% and suffer the randomness around the mean. 00:38:37.520 --> 00:38:41.890 align:middle line:84% So what is the actual utility of the lottery? 00:38:41.890 --> 00:38:44.830 align:middle line:84% Well, that can be achieved in this diagram 00:38:44.830 --> 00:38:47.410 align:middle line:84% by taking a weighted average of the utility 00:38:47.410 --> 00:38:48.830 align:middle line:90% of the extreme points. 00:38:48.830 --> 00:38:52.540 align:middle line:84% Here's the utility you would get for the high value, x1. 00:38:52.540 --> 00:38:55.300 align:middle line:84% Here's the utility you get for the low value, x0. 00:38:55.300 --> 00:38:57.430 align:middle line:84% If you take a weighted average, you'll 00:38:57.430 --> 00:38:59.810 align:middle line:90% be moving along this line. 00:38:59.810 --> 00:39:04.510 align:middle line:84% You'll actually take this weighted average here 00:39:04.510 --> 00:39:08.080 align:middle line:84% that also determines the mean, and go over here and plot 00:39:08.080 --> 00:39:09.980 align:middle line:90% the utility of the outcome. 00:39:09.980 --> 00:39:13.780 align:middle line:84% So this is the expected utility under the actual lottery, 00:39:13.780 --> 00:39:16.420 align:middle line:84% and it's just the appropriately weighted average 00:39:16.420 --> 00:39:20.200 align:middle line:84% of the utilities of the two extremes. 00:39:20.200 --> 00:39:22.620 align:middle line:90% You're with me so far? 00:39:22.620 --> 00:39:28.460 align:middle line:84% So as you can see, if instead of suffering the lottery, 00:39:28.460 --> 00:39:32.810 align:middle line:84% you gave this decision maker a deterministic value 00:39:32.810 --> 00:39:35.870 align:middle line:84% with the same utility level as the utility has 00:39:35.870 --> 00:39:38.870 align:middle line:84% under the lottery, what would that value be? 00:39:38.870 --> 00:39:40.130 align:middle line:90% That would be right here. 00:39:40.130 --> 00:39:43.760 align:middle line:84% If you started with this value of x for sure, 00:39:43.760 --> 00:39:46.430 align:middle line:84% and went up to the orange line, you 00:39:46.430 --> 00:39:48.830 align:middle line:90% would get the utility there. 00:39:48.830 --> 00:39:53.330 align:middle line:84% But that value of x is obviously a lot lower than the mean. 00:39:53.330 --> 00:39:55.910 align:middle line:84% This is called the certainty equivalent 00:39:55.910 --> 00:39:58.350 align:middle line:90% value of the lottery. 00:39:58.350 --> 00:39:59.970 align:middle line:90% Let me put it a bit differently. 00:39:59.970 --> 00:40:03.240 align:middle line:84% If you told the household to choose 00:40:03.240 --> 00:40:06.690 align:middle line:84% between the lottery and a deterministic value 00:40:06.690 --> 00:40:10.020 align:middle line:84% with a mean less than the mean of the lottery, 00:40:10.020 --> 00:40:13.290 align:middle line:84% they might choose the deterministic value 00:40:13.290 --> 00:40:18.930 align:middle line:84% as long as that value is not less than the risk premium. 00:40:18.930 --> 00:40:20.850 align:middle line:84% Anything here, they would choose. 00:40:20.850 --> 00:40:22.950 align:middle line:84% Anything over here, they would not. 00:40:22.950 --> 00:40:26.020 align:middle line:84% Right here, they would be indifferent. 00:40:26.020 --> 00:40:31.090 align:middle line:84% Now, as I think you can imagine, if you increase 00:40:31.090 --> 00:40:34.330 align:middle line:84% the curvature of this function, you're 00:40:34.330 --> 00:40:37.600 align:middle line:84% going to actually be lowering the expected utility 00:40:37.600 --> 00:40:40.000 align:middle line:84% of the lottery, because there's more risk associated 00:40:40.000 --> 00:40:43.900 align:middle line:84% with it, in the sense that you care more about the variance 00:40:43.900 --> 00:40:46.990 align:middle line:84% because your utility is more curved. 00:40:46.990 --> 00:40:50.230 align:middle line:84% And that's going to change the certainty equivalent 00:40:50.230 --> 00:40:52.280 align:middle line:90% amount as well. 00:40:52.280 --> 00:40:54.250 align:middle line:84% So from now on, for much of the class, 00:40:54.250 --> 00:40:57.550 align:middle line:84% we're going to be considering cardinal rather than ordinal 00:40:57.550 --> 00:40:59.320 align:middle line:90% utility. 00:40:59.320 --> 00:41:02.140 align:middle line:84% What are standard measures of risk aversion? 00:41:02.140 --> 00:41:03.970 align:middle line:84% There's something called absolute risk 00:41:03.970 --> 00:41:09.140 align:middle line:84% aversion and another thing called relative risk aversion. 00:41:09.140 --> 00:41:11.840 align:middle line:84% Absolute risk aversion has to do with how 00:41:11.840 --> 00:41:15.590 align:middle line:84% your marginal utility is changing as you change 00:41:15.590 --> 00:41:18.500 align:middle line:90% the reward, the consumption. 00:41:18.500 --> 00:41:21.770 align:middle line:84% And relative risk aversion has to do with how much utility 00:41:21.770 --> 00:41:25.700 align:middle line:84% is changing as you change the percentage of wealth 00:41:25.700 --> 00:41:27.230 align:middle line:90% that you have. 00:41:27.230 --> 00:41:30.650 align:middle line:84% I'll spare you the derivations of all of this, 00:41:30.650 --> 00:41:34.190 align:middle line:84% but I will show you some example utility functions that 00:41:34.190 --> 00:41:36.860 align:middle line:84% display either absolute risk aversion 00:41:36.860 --> 00:41:42.150 align:middle line:84% constant or relative risk aversion on the next slide. 00:41:42.150 --> 00:41:45.830 align:middle line:84% So absolute risk aversion is your willingness 00:41:45.830 --> 00:41:50.090 align:middle line:84% or not to take up gambles, like, starting with $100, 00:41:50.090 --> 00:41:52.070 align:middle line:90% add plus or minus 10. 00:41:52.070 --> 00:41:55.340 align:middle line:90% The lottery is from 110 to 90. 00:41:55.340 --> 00:41:57.890 align:middle line:84% The relative risk aversion has to do 00:41:57.890 --> 00:41:59.750 align:middle line:90% with proportional gambles. 00:41:59.750 --> 00:42:04.820 align:middle line:84% You start with, say, 100 and then have 10% gains or losses. 00:42:04.820 --> 00:42:07.610 align:middle line:84% That would be the same, but you could start with 1,000 00:42:07.610 --> 00:42:10.940 align:middle line:84% and have 10% gains and losses, in which case 00:42:10.940 --> 00:42:17.490 align:middle line:84% you'd be facing a lottery that moves you from 1,100 to 900. 00:42:17.490 --> 00:42:21.270 align:middle line:84% Under constant relative risk aversion, all you care about 00:42:21.270 --> 00:42:24.720 align:middle line:84% are those percentage differences from your initial wealth. 00:42:24.720 --> 00:42:27.360 align:middle line:84% Under absolute risk aversion, you actually 00:42:27.360 --> 00:42:29.220 align:middle line:90% don't like that new-- 00:42:29.220 --> 00:42:32.940 align:middle line:84% if you started with $1,000 as opposed to $100, 00:42:32.940 --> 00:42:35.730 align:middle line:84% your absolute risk aversion would actually be going down. 00:42:35.730 --> 00:42:38.040 align:middle line:84% Because of the curved utility, you 00:42:38.040 --> 00:42:41.520 align:middle line:84% would be thinking about a lottery around a point that 00:42:41.520 --> 00:42:42.930 align:middle line:90% has less curvature. 00:42:42.930 --> 00:42:45.580 align:middle line:84% Attitudes toward risk do depend on these functions. 00:42:45.580 --> 00:42:47.020 align:middle line:90% So here are the examples. 00:42:47.020 --> 00:42:52.290 align:middle line:84% This one is constant absolute risk aversion. 00:42:52.290 --> 00:42:55.030 align:middle line:84% And it's an exponential function-- 00:42:55.030 --> 00:42:56.580 align:middle line:84% looks kind of weird at first because 00:42:56.580 --> 00:42:58.860 align:middle line:90% of all these negative signs. 00:42:58.860 --> 00:43:01.440 align:middle line:84% Say, well, exponential is positive-- yes, 00:43:01.440 --> 00:43:04.350 align:middle line:84% so as C goes up, utility is going down. 00:43:04.350 --> 00:43:06.450 align:middle line:90% What the heck's going on here? 00:43:06.450 --> 00:43:09.480 align:middle line:84% But we put a negative sign in front of all of this 00:43:09.480 --> 00:43:11.500 align:middle line:90% to correct for that. 00:43:11.500 --> 00:43:14.040 align:middle line:84% So you could take the derivative and the second derivative 00:43:14.040 --> 00:43:17.190 align:middle line:84% and verify that this function has constant absolute risk 00:43:17.190 --> 00:43:21.120 align:middle line:84% aversion, and the order of magnitude is actually a. 00:43:21.120 --> 00:43:22.560 align:middle line:84% Or if you started down here, we'd 00:43:22.560 --> 00:43:25.140 align:middle line:84% have constant relative risk aversion, 00:43:25.140 --> 00:43:27.330 align:middle line:84% and you can verify if you did the math-- 00:43:27.330 --> 00:43:32.730 align:middle line:84% u prime u double prime times C that you would get a number R. 00:43:32.730 --> 00:43:37.500 align:middle line:84% So R is the constant level of relative risk aversion 00:43:37.500 --> 00:43:39.400 align:middle line:90% under this function. 00:43:39.400 --> 00:43:42.960 align:middle line:84% And we often use these for different purposes. 00:43:42.960 --> 00:43:45.330 align:middle line:84% They'll show up in an application 00:43:45.330 --> 00:43:48.930 align:middle line:84% we'll get to in about, I guess about four lectures. 00:43:48.930 --> 00:43:52.320 align:middle line:84% Let's talk some more about lotteries and uncertainty. 00:43:52.320 --> 00:43:55.710 align:middle line:84% Actually, we'll now throw in not just uncertainty 00:43:55.710 --> 00:43:59.730 align:middle line:84% as in states of the world S, but also time. 00:43:59.730 --> 00:44:04.410 align:middle line:84% So time is evolving from day 1 to day 2 to day 3. 00:44:04.410 --> 00:44:09.300 align:middle line:84% So from the perspective of your initial starting point, i0, 00:44:09.300 --> 00:44:13.290 align:middle line:84% you could experience a low or a high state at date one. 00:44:13.290 --> 00:44:17.800 align:middle line:84% And if you're at date one, you're still facing risk, 00:44:17.800 --> 00:44:21.120 align:middle line:84% so you could go low or high from there. 00:44:21.120 --> 00:44:24.240 align:middle line:84% If you went to medium, then you could go and turn 00:44:24.240 --> 00:44:27.210 align:middle line:84% all over again from low, medium, to high. 00:44:27.210 --> 00:44:30.300 align:middle line:84% These branches of the trees with the nodes 00:44:30.300 --> 00:44:34.050 align:middle line:84% reflect every possible realization and the path 00:44:34.050 --> 00:44:35.250 align:middle line:90% to get there. 00:44:35.250 --> 00:44:38.640 align:middle line:84% From the standpoint of expected utility, 00:44:38.640 --> 00:44:44.130 align:middle line:84% we would start here, ex ante, before anything happens at all, 00:44:44.130 --> 00:44:48.300 align:middle line:84% and trace the probabilities at date one. 00:44:48.300 --> 00:44:52.440 align:middle line:84% You get this or this or this with certain probabilities, 00:44:52.440 --> 00:44:55.620 align:middle line:90% pi s1, given s0. 00:44:55.620 --> 00:44:59.250 align:middle line:84% But you'd also take into account day two, 00:44:59.250 --> 00:45:03.360 align:middle line:84% so you take the expectation of being here or here, 00:45:03.360 --> 00:45:07.330 align:middle line:84% or any of these intermediate branches. 00:45:07.330 --> 00:45:10.740 align:middle line:84% And finally, you'd have to add in the day three, 00:45:10.740 --> 00:45:15.840 align:middle line:84% where all these events at the thinnest branches of the tree 00:45:15.840 --> 00:45:17.580 align:middle line:90% could come to pass. 00:45:17.580 --> 00:45:19.140 align:middle line:84% If there's anything counterintuitive 00:45:19.140 --> 00:45:22.170 align:middle line:84% about this diagram, it is that when some things happen, 00:45:22.170 --> 00:45:23.710 align:middle line:90% other things don't happen. 00:45:23.710 --> 00:45:25.680 align:middle line:84% So when you're over here and move left, 00:45:25.680 --> 00:45:29.230 align:middle line:84% you've eliminated the other two branches of the tree. 00:45:29.230 --> 00:45:33.030 align:middle line:84% If you took expected utility from day one onward, 00:45:33.030 --> 00:45:35.310 align:middle line:84% you would just be taking into account 00:45:35.310 --> 00:45:38.610 align:middle line:84% what's left that could potentially happen at random. 00:45:38.610 --> 00:45:42.540 align:middle line:84% But this criterion down here is taking expected utility 00:45:42.540 --> 00:45:45.700 align:middle line:84% as if nothing yet has happened at all. 00:45:45.700 --> 00:45:48.630 align:middle line:84% So taking into account all the possible events at day 00:45:48.630 --> 00:45:51.390 align:middle line:90% 1, at day 2, and day 3. 00:45:51.390 --> 00:45:56.430 align:middle line:84% So we can immediately get to a building block 00:45:56.430 --> 00:46:01.110 align:middle line:84% that's crucial to understand finance and financial markets, 00:46:01.110 --> 00:46:04.770 align:middle line:84% and that's the notion of a state contingent commodity. 00:46:04.770 --> 00:46:06.420 align:middle line:90% We've already been doing it. 00:46:06.420 --> 00:46:09.780 align:middle line:84% The state contingent commodity with only one date 00:46:09.780 --> 00:46:13.620 align:middle line:84% would be the probability of getting a certain wealth payoff 00:46:13.620 --> 00:46:17.940 align:middle line:84% or certain xy combination of goods and so on. 00:46:17.940 --> 00:46:22.915 align:middle line:84% So what you get depends on the state, commodity bundle you get 00:46:22.915 --> 00:46:23.790 align:middle line:90% depends on the state. 00:46:23.790 --> 00:46:26.430 align:middle line:90% We call it state contingent. 00:46:26.430 --> 00:46:31.000 align:middle line:84% You get wheat tomorrow if it rains, but nothing otherwise. 00:46:31.000 --> 00:46:34.770 align:middle line:84% And we imagine you can actually buy these kinds of contingent 00:46:34.770 --> 00:46:38.640 align:middle line:84% pay off securities in the financial markets. 00:46:38.640 --> 00:46:41.010 align:middle line:84% However, a much better way to put 00:46:41.010 --> 00:46:44.460 align:middle line:84% that is any financial security is 00:46:44.460 --> 00:46:47.640 align:middle line:84% a weighted average, probabilistic weighted average 00:46:47.640 --> 00:46:50.340 align:middle line:90% over all possible events. 00:46:50.340 --> 00:46:53.490 align:middle line:84% For example, you're an investor and you bought 00:46:53.490 --> 00:46:55.080 align:middle line:90% in-- someone's lending to you. 00:46:55.080 --> 00:46:58.020 align:middle line:84% And let's imagine they are never going to default. 00:46:58.020 --> 00:47:00.180 align:middle line:84% Then, you're going to get that money back for sure, 00:47:00.180 --> 00:47:02.460 align:middle line:84% in which case, it's not risky anymore, 00:47:02.460 --> 00:47:05.700 align:middle line:84% but that's only because the event is the same. 00:47:05.700 --> 00:47:09.220 align:middle line:84% Or you could have something like a stock option, 00:47:09.220 --> 00:47:13.530 align:middle line:84% so that it doesn't come in to play unless the stock 00:47:13.530 --> 00:47:16.900 align:middle line:84% prices are particularly high and you execute the option. 00:47:16.900 --> 00:47:21.450 align:middle line:84% So all possible securities can be represented as linear 00:47:21.450 --> 00:47:26.430 align:middle line:84% combinations of these primitive Arrow-Debreu securities-- 00:47:26.430 --> 00:47:29.010 align:middle line:84% primitive in the sense that they, building blocks, 00:47:29.010 --> 00:47:33.690 align:middle line:84% only pay off in the event of one well delineated event, and 0 00:47:33.690 --> 00:47:34.750 align:middle line:90% otherwise. 00:47:34.750 --> 00:47:37.050 align:middle line:84% So when we get to competitive markets 00:47:37.050 --> 00:47:39.420 align:middle line:84% and how to handle uncertainty, we'll 00:47:39.420 --> 00:47:43.035 align:middle line:84% come back to these state contingent commodities. 00:47:43.035 --> 00:47:47.910 align:middle line:84% OK, so let's move to an application of various 00:47:47.910 --> 00:47:54.000 align:middle line:84% of these things, one lotteries, two, random variables, three, 00:47:54.000 --> 00:47:56.340 align:middle line:90% diversification of portfolios. 00:47:56.340 --> 00:47:58.170 align:middle line:84% Except in this case, they're not going 00:47:58.170 --> 00:48:00.420 align:middle line:84% to be trading financial assets, they're 00:48:00.420 --> 00:48:02.400 align:middle line:84% going to be holding plots of land. 00:48:02.400 --> 00:48:04.680 align:middle line:84% So the next map I'm going to show you 00:48:04.680 --> 00:48:08.580 align:middle line:84% is a village of Elford in Stratfordshire, England, 00:48:08.580 --> 00:48:12.030 align:middle line:90% in 1719, before enclosures. 00:48:12.030 --> 00:48:16.410 align:middle line:84% And shaded in black are the holdings of Mr. Darlaston. 00:48:16.410 --> 00:48:20.190 align:middle line:84% So here's that village, Elford, Stratfordshire. 00:48:20.190 --> 00:48:22.710 align:middle line:84% And you have to look a little close, 00:48:22.710 --> 00:48:26.920 align:middle line:84% but you see these darker areas, they are shaded in 00:48:26.920 --> 00:48:30.420 align:middle line:84% and represent the holdings of Mr. Darlaston. 00:48:30.420 --> 00:48:34.290 align:middle line:84% Mr. Darlaston has these long, narrow strips of land 00:48:34.290 --> 00:48:37.230 align:middle line:84% all over the village, in all the three fields. 00:48:37.230 --> 00:48:38.850 align:middle line:84% And you could well think about that 00:48:38.850 --> 00:48:42.600 align:middle line:84% as a portfolio of land holdings, and in some respects 00:48:42.600 --> 00:48:45.790 align:middle line:84% a portfolio of securities where in this case, 00:48:45.790 --> 00:48:50.280 align:middle line:84% the holding of land yields a dividend, which is the harvest. 00:48:50.280 --> 00:48:54.000 align:middle line:84% And you may well surmise there's some risk here, 00:48:54.000 --> 00:48:57.240 align:middle line:84% or they're weird people, because why else would they 00:48:57.240 --> 00:48:59.320 align:middle line:90% be doing something like this? 00:48:59.320 --> 00:49:03.840 align:middle line:84% Here's another view of the village of Laxton. 00:49:03.840 --> 00:49:06.870 align:middle line:84% I should at least tell you what enclosures are. 00:49:06.870 --> 00:49:08.850 align:middle line:84% Given the way this land was spread out, 00:49:08.850 --> 00:49:12.000 align:middle line:84% it was impractical to put fences up 00:49:12.000 --> 00:49:15.480 align:middle line:84% that would mark the division of the boundary of all 00:49:15.480 --> 00:49:16.800 align:middle line:90% these various strips. 00:49:16.800 --> 00:49:19.080 align:middle line:84% It would have been very costly to put up the fences 00:49:19.080 --> 00:49:20.730 align:middle line:90% and very difficult to plow. 00:49:20.730 --> 00:49:22.620 align:middle line:90% See the little plows here? 00:49:22.620 --> 00:49:26.220 align:middle line:84% Eventually, enclosures meant they put-- 00:49:26.220 --> 00:49:27.720 align:middle line:90% they divided it up. 00:49:27.720 --> 00:49:32.220 align:middle line:84% They got rid of the area and divided up 00:49:32.220 --> 00:49:34.950 align:middle line:84% the land in a different way, just 00:49:34.950 --> 00:49:38.640 align:middle line:84% put fences around the consolidated pieces of land. 00:49:38.640 --> 00:49:41.700 align:middle line:84% But they didn't do that for hundreds of years. 00:49:41.700 --> 00:49:45.780 align:middle line:84% They kept this, the scattered strips, for quite a long time. 00:49:45.780 --> 00:49:48.390 align:middle line:90% OK, so let's move to the data. 00:49:48.390 --> 00:49:51.270 align:middle line:84% The data we have from this period of time 00:49:51.270 --> 00:49:53.940 align:middle line:84% is from the Bishop of Winchester. 00:49:53.940 --> 00:49:58.170 align:middle line:84% The church was wealthy and they kept records. 00:49:58.170 --> 00:50:00.990 align:middle line:84% So this is what survives, the holdings 00:50:00.990 --> 00:50:05.940 align:middle line:84% of the Bishop of Winchester, all domains, the crops being 00:50:05.940 --> 00:50:08.790 align:middle line:84% wheat, barley, and oats, various types of grains. 00:50:08.790 --> 00:50:12.480 align:middle line:84% And these objects denote, say, the coefficient 00:50:12.480 --> 00:50:15.720 align:middle line:84% of variation, which I've alerted you is the variance divided 00:50:15.720 --> 00:50:19.360 align:middle line:84% by the mean, standard deviation divided by the mean. 00:50:19.360 --> 00:50:24.840 align:middle line:84% So this number, say 0.35, is in the middle of the two others. 00:50:24.840 --> 00:50:26.730 align:middle line:84% Actually, it's a bit on the lower end. 00:50:26.730 --> 00:50:31.020 align:middle line:84% Even at 0.35, the coefficient of variation of that magnitude 00:50:31.020 --> 00:50:32.380 align:middle line:90% is a lot of risk. 00:50:32.380 --> 00:50:34.950 align:middle line:90% So let's look at the picture. 00:50:34.950 --> 00:50:38.040 align:middle line:84% If we have those scattered plots, 00:50:38.040 --> 00:50:42.420 align:middle line:84% those strips, we could plot the entire distribution of outcomes 00:50:42.420 --> 00:50:44.220 align:middle line:90% with their probabilities. 00:50:44.220 --> 00:50:48.510 align:middle line:84% We could denote the mean, Us, for scattered plots. 00:50:48.510 --> 00:50:51.480 align:middle line:84% And then we have disastrous events, 00:50:51.480 --> 00:50:56.880 align:middle line:84% so d represents minimal, subsistence consumption. 00:50:56.880 --> 00:51:00.450 align:middle line:84% And if you have these scattered plots 00:51:00.450 --> 00:51:03.810 align:middle line:84% with that coefficient of variation of 0.35, 00:51:03.810 --> 00:51:06.930 align:middle line:84% then roughly one in every 12 years 00:51:06.930 --> 00:51:10.680 align:middle line:84% you get a very low yield, less than 2 times-- 00:51:10.680 --> 00:51:14.160 align:middle line:84% 2 standard deviations below the mean, and that's starvation. 00:51:14.160 --> 00:51:17.940 align:middle line:84% And they were starving, we know this, periodically, every 12 00:51:17.940 --> 00:51:18.960 align:middle line:90% years or so. 00:51:18.960 --> 00:51:23.103 align:middle line:84% Not like clockwork, but on average every 12 years. 00:51:23.103 --> 00:51:25.020 align:middle line:84% Now, could they have done something different? 00:51:25.020 --> 00:51:28.400 align:middle line:84% They could have consolidated the land 00:51:28.400 --> 00:51:32.750 align:middle line:84% and that would have gotten rid of certain costs and problems. 00:51:32.750 --> 00:51:35.360 align:middle line:84% That would have moved up the mean, 00:51:35.360 --> 00:51:40.640 align:middle line:84% but you'd be left with more mass in the tails, and hence 00:51:40.640 --> 00:51:44.780 align:middle line:84% a higher probability of having a yield that was disastrous. 00:51:44.780 --> 00:51:47.000 align:middle line:84% So the reason that they're spreading out 00:51:47.000 --> 00:51:49.730 align:middle line:84% the land like that despite costs has 00:51:49.730 --> 00:51:53.150 align:middle line:84% to do with trying to increase their expected utility 00:51:53.150 --> 00:51:56.260 align:middle line:84% by putting less mass in the tails. 00:51:56.260 --> 00:52:01.030 align:middle line:84% Also from the Bishop of Winchester estates, 00:52:01.030 --> 00:52:03.040 align:middle line:90% which you can hardly see here-- 00:52:03.040 --> 00:52:05.770 align:middle line:84% this, believe it or not, is Southern England. 00:52:05.770 --> 00:52:08.200 align:middle line:90% This is the water down here. 00:52:08.200 --> 00:52:10.090 align:middle line:84% And each one of these dots represents 00:52:10.090 --> 00:52:12.400 align:middle line:90% an ecclesiastical estate. 00:52:12.400 --> 00:52:17.020 align:middle line:84% And we have the data of the yields, the harvest yields. 00:52:17.020 --> 00:52:20.710 align:middle line:84% In this case, the covariate, the correlation-- 00:52:20.710 --> 00:52:24.070 align:middle line:84% the statistical concept we were chatting about earlier-- 00:52:24.070 --> 00:52:27.760 align:middle line:84% is what is being plotted here for villages 00:52:27.760 --> 00:52:30.520 align:middle line:84% that are relatively near to one another, 00:52:30.520 --> 00:52:33.220 align:middle line:84% and villages that are relatively far away. 00:52:33.220 --> 00:52:35.200 align:middle line:90% So D means near-- 00:52:35.200 --> 00:52:37.630 align:middle line:84% sorry, it's hard to read the legend. 00:52:37.630 --> 00:52:40.660 align:middle line:84% And so take two villages pairwise, 00:52:40.660 --> 00:52:43.690 align:middle line:84% look at the correlation across the two villages. 00:52:43.690 --> 00:52:47.950 align:middle line:84% It's 0.68 on average, picking villages that are near. 00:52:47.950 --> 00:52:51.970 align:middle line:84% And if you have villages that are far away one another, 00:52:51.970 --> 00:52:54.730 align:middle line:84% that correlation drops to something lower, 00:52:54.730 --> 00:52:58.160 align:middle line:84% so the yields are not moving as much together. 00:52:58.160 --> 00:53:01.690 align:middle line:84% So again, if you were the Bishop of Winchester, maybe 00:53:01.690 --> 00:53:04.270 align:middle line:84% you chose the location of your estates 00:53:04.270 --> 00:53:06.820 align:middle line:84% in order to try to ensure yourself 00:53:06.820 --> 00:53:10.570 align:middle line:84% of having somewhat regular grain for harvest, for eating, 00:53:10.570 --> 00:53:13.840 align:middle line:84% and to feed the horses, and so on. 00:53:13.840 --> 00:53:15.430 align:middle line:90% Horses are serious, by the way. 00:53:15.430 --> 00:53:16.300 align:middle line:90% That's the cavalry. 00:53:16.300 --> 00:53:20.980 align:middle line:84% That's the main defense, so they need it to keep them alive. 00:53:20.980 --> 00:53:27.550 align:middle line:84% OK, then we come to a technique which we've already been using, 00:53:27.550 --> 00:53:30.320 align:middle line:90% but let's do it from scratch. 00:53:30.320 --> 00:53:33.950 align:middle line:84% And that's called linear programming. 00:53:33.950 --> 00:53:36.040 align:middle line:84% So this is a linear programming problem. 00:53:36.040 --> 00:53:38.900 align:middle line:84% X is the control variable, a vector. 00:53:38.900 --> 00:53:42.250 align:middle line:90% This-- say, column vector-- 00:53:42.250 --> 00:53:48.310 align:middle line:84% these constants, coefficient C, represent a row vector, 00:53:48.310 --> 00:53:51.640 align:middle line:84% so these are weights on the various x values. 00:53:51.640 --> 00:53:56.560 align:middle line:84% This is a set of constraints, which is this matrix a times x, 00:53:56.560 --> 00:53:58.030 align:middle line:90% less than or equal to b. 00:53:58.030 --> 00:54:00.560 align:middle line:90% And the x's are non-negative. 00:54:00.560 --> 00:54:06.790 align:middle line:84% So this is notation for a standard linear program, namely 00:54:06.790 --> 00:54:09.160 align:middle line:90% the objective function. 00:54:09.160 --> 00:54:12.880 align:middle line:84% And the constraint sets just represent linear combinations 00:54:12.880 --> 00:54:15.130 align:middle line:90% of the control variables x. 00:54:15.130 --> 00:54:16.720 align:middle line:84% I had this in the back of my head 00:54:16.720 --> 00:54:20.140 align:middle line:84% when we were talking about the Leontief input-output matrix. 00:54:20.140 --> 00:54:24.820 align:middle line:84% And I said in words, instead of taking final demand D as given, 00:54:24.820 --> 00:54:27.760 align:middle line:84% maybe it's a weighted average of demands 00:54:27.760 --> 00:54:30.130 align:middle line:84% that you're trying to maximize, and you actually 00:54:30.130 --> 00:54:33.280 align:middle line:84% have a choice of various production modes 00:54:33.280 --> 00:54:35.390 align:middle line:84% to maximize that weighted average. 00:54:35.390 --> 00:54:39.580 align:middle line:84% So we actually saw an early version of a linear programming 00:54:39.580 --> 00:54:43.450 align:middle line:84% problem, but the objective function was not made explicit. 00:54:43.450 --> 00:54:47.500 align:middle line:84% This would be the consumer choice problem 00:54:47.500 --> 00:54:50.080 align:middle line:84% in a competitive environment, so where 00:54:50.080 --> 00:54:51.440 align:middle line:90% there's a budget constraint. 00:54:51.440 --> 00:54:54.880 align:middle line:84% So these discrete points, commodity points C, 00:54:54.880 --> 00:54:59.260 align:middle line:84% denote subscript S varying over error. 00:54:59.260 --> 00:55:04.840 align:middle line:84% S, states of the world was the expected utility criterion, 00:55:04.840 --> 00:55:07.060 align:middle line:84% so that's the objective function. 00:55:07.060 --> 00:55:10.240 align:middle line:84% By choice of lotteries, so the lotteries 00:55:10.240 --> 00:55:12.280 align:middle line:84% are non-negative and add up to 1. 00:55:12.280 --> 00:55:14.500 align:middle line:84% To qualify as a lottery, any given lottery 00:55:14.500 --> 00:55:17.590 align:middle line:84% has to satisfy those two criteria. 00:55:17.590 --> 00:55:21.640 align:middle line:84% The choice object, as I said-- to belabor it-- is not c of s. 00:55:21.640 --> 00:55:23.380 align:middle line:84% Those are fixed commodity points. 00:55:23.380 --> 00:55:26.020 align:middle line:90% The choice object are the pis. 00:55:26.020 --> 00:55:28.690 align:middle line:84% And we'll talk about a budget problem, which 00:55:28.690 --> 00:55:33.710 align:middle line:84% is there are prices p depending on s-- that is to say, 00:55:33.710 --> 00:55:36.100 align:middle line:84% depending on the underlying discrete bundle, quite 00:55:36.100 --> 00:55:37.930 align:middle line:90% naturally. 00:55:37.930 --> 00:55:40.330 align:middle line:84% And you've got a budget in dollars 00:55:40.330 --> 00:55:43.600 align:middle line:84% that you can use to buy a lottery, 00:55:43.600 --> 00:55:47.080 align:middle line:84% but if the probability is non-trivial, 00:55:47.080 --> 00:55:51.580 align:middle line:84% it's a number less than 1 and greater than 0, 00:55:51.580 --> 00:55:54.190 align:middle line:84% then you scale the price of getting 00:55:54.190 --> 00:55:56.800 align:middle line:90% that bundle s back by pi. 00:55:56.800 --> 00:55:59.440 align:middle line:84% In other words, you're kind of paying the expected price. 00:55:59.440 --> 00:56:04.240 align:middle line:84% If pi s were one, then you're buying cs for sure 00:56:04.240 --> 00:56:07.180 align:middle line:84% and the price would be ps, and that would look like 00:56:07.180 --> 00:56:10.240 align:middle line:84% the standard price times quantity cannot exceed 00:56:10.240 --> 00:56:10.900 align:middle line:90% the budget. 00:56:10.900 --> 00:56:12.650 align:middle line:84% But here, what you're buying is a lottery, 00:56:12.650 --> 00:56:15.970 align:middle line:84% and the lottery can have non degenerate elements. 00:56:15.970 --> 00:56:19.480 align:middle line:84% You have to decide how you would be paying for, 00:56:19.480 --> 00:56:22.300 align:middle line:84% what you would be paying if you were buying a lottery which 00:56:22.300 --> 00:56:23.320 align:middle line:90% were non degenerate. 00:56:23.320 --> 00:56:29.440 align:middle line:84% And here, you pay as per the actuarial value of the payoff. 00:56:29.440 --> 00:56:31.510 align:middle line:84% Well, this is another linear program. 00:56:31.510 --> 00:56:33.970 align:middle line:84% The objective function is linear in the pis. 00:56:33.970 --> 00:56:36.310 align:middle line:84% The constraint set is in the pis, 00:56:36.310 --> 00:56:39.790 align:middle line:84% including the budget set, which is linear in the pis. 00:56:39.790 --> 00:56:43.390 align:middle line:84% So we can solve these things conceptually and numerically, 00:56:43.390 --> 00:56:46.180 align:middle line:84% recognizing that it's just a version 00:56:46.180 --> 00:56:48.610 align:middle line:90% of the general notation. 00:56:48.610 --> 00:56:49.840 align:middle line:90% And here's the third. 00:56:49.840 --> 00:56:51.610 align:middle line:90% Suppose we have a firm-- 00:56:51.610 --> 00:56:53.150 align:middle line:90% back to production again. 00:56:53.150 --> 00:56:56.380 align:middle line:84% The firm can produce n possible goods. 00:56:56.380 --> 00:56:59.350 align:middle line:84% The jth good here has a price, pj. 00:56:59.350 --> 00:57:02.830 align:middle line:84% The firm has a fixed supply of inputs, which 00:57:02.830 --> 00:57:04.150 align:middle line:90% makes the problem special. 00:57:04.150 --> 00:57:07.450 align:middle line:84% They're not buying these, they already have them in inventory. 00:57:07.450 --> 00:57:11.860 align:middle line:84% And the technology tells us how much of input 00:57:11.860 --> 00:57:14.890 align:middle line:90% i is needed to produce good j. 00:57:14.890 --> 00:57:16.510 align:middle line:90% Those are the aijs. 00:57:16.510 --> 00:57:18.730 align:middle line:84% Now here, the natural objective function 00:57:18.730 --> 00:57:22.240 align:middle line:84% would be the revenue, price times quantity, 00:57:22.240 --> 00:57:24.850 align:middle line:90% summing over all the j goods. 00:57:24.850 --> 00:57:26.740 align:middle line:84% And you're going to choose the qjs, 00:57:26.740 --> 00:57:31.250 align:middle line:84% but you can't over utilize your inputs. 00:57:31.250 --> 00:57:37.310 align:middle line:84% So if the firm is endowed with input i, then you would say, 00:57:37.310 --> 00:57:40.180 align:middle line:84% how much of input i is needed to produce good j 00:57:40.180 --> 00:57:43.300 align:middle line:84% times the proposed quantity of good j, 00:57:43.300 --> 00:57:45.310 align:middle line:90% summing over all goods j. 00:57:45.310 --> 00:57:47.950 align:middle line:84% This would be the total input requirement 00:57:47.950 --> 00:57:53.380 align:middle line:84% for input i, associated with producing this vector of goods 00:57:53.380 --> 00:57:55.630 align:middle line:90% q, indexed by j. 00:57:55.630 --> 00:57:57.730 align:middle line:84% And so this is a linear constraint. 00:57:57.730 --> 00:57:59.350 align:middle line:90% The qs are non-negative. 00:57:59.350 --> 00:58:01.630 align:middle line:90% This is another linear program. 00:58:01.630 --> 00:58:04.810 align:middle line:84% And the last one is what's called a transportation 00:58:04.810 --> 00:58:06.010 align:middle line:90% problem-- 00:58:06.010 --> 00:58:08.440 align:middle line:90% maybe just say it in words. 00:58:08.440 --> 00:58:13.640 align:middle line:84% There are m warehouses, which are stocked with goods, 00:58:13.640 --> 00:58:17.550 align:middle line:84% and n retail outlets where goods are sold. 00:58:17.550 --> 00:58:20.450 align:middle line:90% So here, we have geography. 00:58:20.450 --> 00:58:24.200 align:middle line:84% And in particular, in order to ship the product 00:58:24.200 --> 00:58:28.950 align:middle line:84% from a warehouse to an outlet, you use up resources. 00:58:28.950 --> 00:58:34.410 align:middle line:84% So this cost, cij, is the amount of resources you used up, 00:58:34.410 --> 00:58:39.620 align:middle line:84% say, in dollars, to ship the good from warehouse i to outlet 00:58:39.620 --> 00:58:40.700 align:middle line:90% j. 00:58:40.700 --> 00:58:44.960 align:middle line:84% And then the decision problem is you're given demands 00:58:44.960 --> 00:58:48.350 align:middle line:84% at all the outlets, and you're given the amount, initially, 00:58:48.350 --> 00:58:52.580 align:middle line:84% in the warehouses, and you want to then determine the shipments 00:58:52.580 --> 00:58:54.350 align:middle line:84% from all the different warehouses 00:58:54.350 --> 00:58:56.390 align:middle line:84% to all the different potential outlets 00:58:56.390 --> 00:59:00.470 align:middle line:84% to solve the problem, which minimizes the total cost-- 00:59:00.470 --> 00:59:04.370 align:middle line:84% cost from i to j, warehouses i to outlets j, 00:59:04.370 --> 00:59:06.380 align:middle line:90% and these are the constraints. 00:59:06.380 --> 00:59:10.640 align:middle line:84% The constraint says that you've got so much in inventory 00:59:10.640 --> 00:59:17.510 align:middle line:84% at warehouse i, so whatever you're shipping out of i to all 00:59:17.510 --> 00:59:21.917 align:middle line:84% these outlets j, it cannot exceed that level of inventory, 00:59:21.917 --> 00:59:23.000 align:middle line:90% although it could be less. 00:59:23.000 --> 00:59:26.840 align:middle line:84% And likewise, you have to meet the demand at outlet j-- 00:59:26.840 --> 00:59:30.560 align:middle line:84% So the shipping that's coming from all possible outlets i 00:59:30.560 --> 00:59:36.230 align:middle line:84% to that particular j cannot be less than the demand. 00:59:36.230 --> 00:59:40.080 align:middle line:84% So this is another geographic programming problem. 00:59:40.080 --> 00:59:41.930 align:middle line:90% So we did ordinary production. 00:59:41.930 --> 00:59:44.480 align:middle line:90% We did choice under uncertainty. 00:59:44.480 --> 00:59:48.320 align:middle line:84% We did a spatial geographic problem. 00:59:48.320 --> 00:59:51.950 align:middle line:84% They're all well known prototypes in the literature, 00:59:51.950 --> 00:59:53.900 align:middle line:84% and they all use this common tool, 00:59:53.900 --> 00:59:56.434 align:middle line:90% which is linear programming. 00:59:56.434 --> 01:00:00.140 align:middle line:90% OK, that's all I have for today. 01:00:00.140 --> 01:00:08.000 align:middle line:90%