WEBVTT 00:00:00.000 --> 00:00:01.988 align:middle line:90% [SQUEAKING] 00:00:01.988 --> 00:00:03.479 align:middle line:90% [RUSTLING] 00:00:03.479 --> 00:00:05.467 align:middle line:90% [CLICKING] 00:00:05.467 --> 00:00:11.002 align:middle line:90% 00:00:11.002 --> 00:00:11.960 align:middle line:90% ROBERT M. TOWNSEND: OK. 00:00:11.960 --> 00:00:13.680 align:middle line:90% Greetings, everyone. 00:00:13.680 --> 00:00:16.122 align:middle line:90% Thank you for coming today. 00:00:16.122 --> 00:00:17.420 align:middle line:90% I'll say a few things. 00:00:17.420 --> 00:00:20.410 align:middle line:84% So in terms of the calendar, and we're 00:00:20.410 --> 00:00:23.770 align:middle line:84% finishing off this section right before Thanksgiving break, 00:00:23.770 --> 00:00:24.670 align:middle line:90% when I get-- 00:00:24.670 --> 00:00:27.400 align:middle line:84% excuse me, on identification today. 00:00:27.400 --> 00:00:29.500 align:middle line:90% We're on that momentarily. 00:00:29.500 --> 00:00:31.690 align:middle line:84% Then, we have two formal lectures 00:00:31.690 --> 00:00:34.600 align:middle line:84% after the break, on Tuesday, Thursday. 00:00:34.600 --> 00:00:37.270 align:middle line:90% So that's the calendar. 00:00:37.270 --> 00:00:40.780 align:middle line:84% Then, in terms of the readings today, it's 00:00:40.780 --> 00:00:44.980 align:middle line:84% important to point out, the section on identification 00:00:44.980 --> 00:00:48.530 align:middle line:84% and falsification we're about to do, 00:00:48.530 --> 00:00:50.740 align:middle line:84% there are three starred readings, 00:00:50.740 --> 00:00:54.700 align:middle line:84% and they all come from Varian, and they're all really, really 00:00:54.700 --> 00:00:55.430 align:middle line:90% good. 00:00:55.430 --> 00:00:58.480 align:middle line:84% In fact, I had to abbreviate in spots, in order 00:00:58.480 --> 00:01:00.070 align:middle line:90% to cover all the topics. 00:01:00.070 --> 00:01:03.040 align:middle line:84% So we are drawing on this material in part, 00:01:03.040 --> 00:01:07.480 align:middle line:90% and you will find it useful. 00:01:07.480 --> 00:01:08.800 align:middle line:90% Again, it's not the whole book. 00:01:08.800 --> 00:01:10.990 align:middle line:84% It's just Varian, in these sections, 00:01:10.990 --> 00:01:15.490 align:middle line:90% 8.1 through 8.3, 8.5, and 8.7. 00:01:15.490 --> 00:01:20.960 align:middle line:84% Then, from last time, let's have a little bit of a discussion. 00:01:20.960 --> 00:01:23.300 align:middle line:90% Let's see. 00:01:23.300 --> 00:01:26.810 align:middle line:84% Daniel, can you tell me, as best you 00:01:26.810 --> 00:01:30.860 align:middle line:84% can, what the definition of is the positive representative 00:01:30.860 --> 00:01:36.770 align:middle line:84% consumer and the normative representative consumer, 00:01:36.770 --> 00:01:39.950 align:middle line:84% explaining what they are Intuitively 00:01:39.950 --> 00:01:42.020 align:middle line:84% AUDIENCE: So like the idea is you're 00:01:42.020 --> 00:01:47.550 align:middle line:84% trying to treat the entire economy as a single person. 00:01:47.550 --> 00:01:50.150 align:middle line:84% So a positive representative consumer 00:01:50.150 --> 00:01:56.540 align:middle line:84% is, at any given level of wealth and input, 00:01:56.540 --> 00:02:05.030 align:middle line:84% the way to maximize utility is to spend your wealth the way 00:02:05.030 --> 00:02:11.000 align:middle line:84% that customer would and then distribute that 00:02:11.000 --> 00:02:14.240 align:middle line:90% to the people like in some way. 00:02:14.240 --> 00:02:17.850 align:middle line:84% We don't really know which way from the consumer, 00:02:17.850 --> 00:02:21.080 align:middle line:84% but like it'll just be like the best distribution. 00:02:21.080 --> 00:02:25.290 align:middle line:84% And then the normative is very similar, 00:02:25.290 --> 00:02:28.560 align:middle line:84% except for instead of the utility maximizing, 00:02:28.560 --> 00:02:32.900 align:middle line:84% it's the Pareto optimal way to distribute the wealth. 00:02:32.900 --> 00:02:36.253 align:middle line:90% 00:02:36.253 --> 00:02:37.295 align:middle line:90% ROBERT M. TOWNSEND: Yeah. 00:02:37.295 --> 00:02:40.290 align:middle line:90% 00:02:40.290 --> 00:02:43.050 align:middle line:84% I guess I would sharpen up a little bit the positive one. 00:02:43.050 --> 00:02:49.230 align:middle line:84% The idea is to get the excess demand curve for the economy, 00:02:49.230 --> 00:02:53.790 align:middle line:84% as if it were coming from this representative consumer 00:02:53.790 --> 00:02:58.170 align:middle line:84% maximizing utility subject to resource constraints. 00:02:58.170 --> 00:03:00.780 align:middle line:84% And then that representative consumer 00:03:00.780 --> 00:03:04.560 align:middle line:84% has all the resource constraints in the entire economy. 00:03:04.560 --> 00:03:07.950 align:middle line:84% In particular, the budget just says, don't spend more 00:03:07.950 --> 00:03:10.490 align:middle line:90% than the economy-wide wealth. 00:03:10.490 --> 00:03:13.460 align:middle line:84% And you said that, but maybe better 00:03:13.460 --> 00:03:18.330 align:middle line:84% for me to repeat it a little bit. 00:03:18.330 --> 00:03:20.510 align:middle line:84% Yeah, and then the normative one is a way 00:03:20.510 --> 00:03:23.930 align:middle line:84% to rank order allocations, according 00:03:23.930 --> 00:03:27.190 align:middle line:90% to the Pareto principal. 00:03:27.190 --> 00:03:28.900 align:middle line:84% But ironically, rather than keeping 00:03:28.900 --> 00:03:32.470 align:middle line:84% track of all the individuals, and whether they're 00:03:32.470 --> 00:03:36.460 align:middle line:84% made better off or worse off, we can collapse it to 00:03:36.460 --> 00:03:41.170 align:middle line:84% as if we were maximizing utility for the whole economy 00:03:41.170 --> 00:03:44.840 align:middle line:84% through the lens of this representative consumer. 00:03:44.840 --> 00:03:51.490 align:middle line:84% And as you said, that determines the overall allocation. 00:03:51.490 --> 00:03:57.190 align:middle line:84% It doesn't pin down the potential reallocation 00:03:57.190 --> 00:04:03.280 align:middle line:84% of wealth that is necessary, if you go from something Pareto 00:04:03.280 --> 00:04:08.590 align:middle line:84% inferior to something Pareto superior or Pareto optimal, 00:04:08.590 --> 00:04:12.880 align:middle line:84% you may have to compensate the losers 00:04:12.880 --> 00:04:17.410 align:middle line:90% and by taxing the winners. 00:04:17.410 --> 00:04:18.039 align:middle line:90% OK, great. 00:04:18.039 --> 00:04:22.380 align:middle line:90% 00:04:22.380 --> 00:04:26.520 align:middle line:84% So we made a lot of use of this Gorman form. 00:04:26.520 --> 00:04:31.980 align:middle line:84% Caleb, can you tell me in words, what is this Gorman polar form? 00:04:31.980 --> 00:04:37.265 align:middle line:90% 00:04:37.265 --> 00:04:37.890 align:middle line:90% AUDIENCE: Yeah. 00:04:37.890 --> 00:04:46.930 align:middle line:84% So like it breaks down the equilibrium 00:04:46.930 --> 00:04:48.220 align:middle line:90% into like a linear form. 00:04:48.220 --> 00:04:50.290 align:middle line:84% I guess, I don't know exactly how to explain it. 00:04:50.290 --> 00:04:54.570 align:middle line:90% 00:04:54.570 --> 00:04:57.990 align:middle line:84% ROBERT M. TOWNSEND: Do you remember any of the notation? 00:04:57.990 --> 00:05:00.330 align:middle line:84% What is it breaking down exactly? 00:05:00.330 --> 00:05:02.870 align:middle line:90% 00:05:02.870 --> 00:05:05.530 align:middle line:84% AUDIENCE: Like the equilibrium in the economy. 00:05:05.530 --> 00:05:09.187 align:middle line:90% 00:05:09.187 --> 00:05:11.270 align:middle line:84% ROBERT M. TOWNSEND: Does the word indirect utility 00:05:11.270 --> 00:05:11.990 align:middle line:90% come to mind? 00:05:11.990 --> 00:05:18.404 align:middle line:90% 00:05:18.404 --> 00:05:19.340 align:middle line:90% OK. 00:05:19.340 --> 00:05:23.660 align:middle line:84% It's the indirect utility function, 00:05:23.660 --> 00:05:28.990 align:middle line:84% which is the maximized utility subject to the given 00:05:28.990 --> 00:05:30.455 align:middle line:90% vector of prices and wealth. 00:05:30.455 --> 00:05:33.120 align:middle line:90% 00:05:33.120 --> 00:05:40.250 align:middle line:84% And the statement about Gorman form is this form in which 00:05:40.250 --> 00:05:42.950 align:middle line:84% that that indirect utility function 00:05:42.950 --> 00:05:48.230 align:middle line:84% must take in order to do this positive and normative thing. 00:05:48.230 --> 00:05:52.420 align:middle line:84% Can someone volunteer to tell me what, in words if you can, 00:05:52.420 --> 00:05:55.185 align:middle line:90% what the green form looks like? 00:05:55.185 --> 00:05:55.810 align:middle line:90% AUDIENCE: Sure. 00:05:55.810 --> 00:05:57.647 align:middle line:90% I can take this one. 00:05:57.647 --> 00:05:58.730 align:middle line:90% ROBERT M. TOWNSEND: Great. 00:05:58.730 --> 00:06:03.200 align:middle line:84% AUDIENCE: So the Gorman form is basically an indirect utility 00:06:03.200 --> 00:06:08.030 align:middle line:84% function that is linear in W, the overall wealth 00:06:08.030 --> 00:06:08.760 align:middle line:90% of the economy. 00:06:08.760 --> 00:06:11.540 align:middle line:84% So it's some function A, which depends on prices, 00:06:11.540 --> 00:06:15.768 align:middle line:84% plus sum function B, which depends on prices, plus W. 00:06:15.768 --> 00:06:16.810 align:middle line:90% ROBERT M. TOWNSEND: Yeah. 00:06:16.810 --> 00:06:18.790 align:middle line:90% That's exactly right. 00:06:18.790 --> 00:06:22.510 align:middle line:84% Well, it's at the level of the individuals. 00:06:22.510 --> 00:06:23.350 align:middle line:90% AUDIENCE: Yes. 00:06:23.350 --> 00:06:26.290 align:middle line:84% ROBERT M. TOWNSEND: So if we had an individual I, 00:06:26.290 --> 00:06:28.030 align:middle line:84% where do we put the I in that equation? 00:06:28.030 --> 00:06:30.590 align:middle line:90% 00:06:30.590 --> 00:06:35.350 align:middle line:84% AUDIENCE: So it is on both the wealth 00:06:35.350 --> 00:06:37.420 align:middle line:84% that an individual person faces, and I 00:06:37.420 --> 00:06:41.950 align:middle line:84% think as well on the functions A and B. 00:06:41.950 --> 00:06:46.518 align:middle line:84% Each individual can have their own specific form for that. 00:06:46.518 --> 00:06:48.060 align:middle line:84% ROBERT M. TOWNSEND: Two out of three. 00:06:48.060 --> 00:06:51.720 align:middle line:84% So it's on the wealth, because at the individual level, 00:06:51.720 --> 00:06:55.860 align:middle line:84% it's the indirect utility is a function of the individual's 00:06:55.860 --> 00:07:02.040 align:middle line:84% wealth, say W sub I. The intercept term, the A of P 00:07:02.040 --> 00:07:08.730 align:middle line:84% is the AI of P. So that's where the heterogeneity is loaded. 00:07:08.730 --> 00:07:13.050 align:middle line:84% But it's not on the B, and maybe you can remember why. 00:07:13.050 --> 00:07:15.930 align:middle line:84% What did we say about those linear expansion paths? 00:07:15.930 --> 00:07:18.510 align:middle line:84% AUDIENCE: Oh, those need to be parallel linear expansion 00:07:18.510 --> 00:07:21.093 align:middle line:84% paths, and so they need to be identical for each consumer. 00:07:21.093 --> 00:07:22.260 align:middle line:90% ROBERT M. TOWNSEND: Exactly. 00:07:22.260 --> 00:07:23.220 align:middle line:90% That's perfect. 00:07:23.220 --> 00:07:28.230 align:middle line:90% So there's no I for the B part. 00:07:28.230 --> 00:07:31.430 align:middle line:90% OK, and then finally-- 00:07:31.430 --> 00:07:36.270 align:middle line:84% oh, I almost answer this already, 00:07:36.270 --> 00:07:37.890 align:middle line:90% but we can discuss it some more. 00:07:37.890 --> 00:07:41.880 align:middle line:84% How realistic are the necessary and sufficient conditions 00:07:41.880 --> 00:07:44.370 align:middle line:84% for Gorman aggregation, based on what you've 00:07:44.370 --> 00:07:45.720 align:middle line:90% learned in previous lectures? 00:07:45.720 --> 00:07:48.420 align:middle line:90% 00:07:48.420 --> 00:07:50.460 align:middle line:84% In other words, do you believe it or not? 00:07:50.460 --> 00:07:55.830 align:middle line:90% 00:07:55.830 --> 00:07:56.330 align:middle line:90% Guanpeng? 00:07:56.330 --> 00:07:59.100 align:middle line:90% 00:07:59.100 --> 00:08:00.210 align:middle line:90% AUDIENCE: Yeah. 00:08:00.210 --> 00:08:01.920 align:middle line:84% I believe one of the necessary conditions 00:08:01.920 --> 00:08:05.700 align:middle line:84% was that the aggregate demand doesn't 00:08:05.700 --> 00:08:07.920 align:middle line:84% depend on the distribution of the wealth. 00:08:07.920 --> 00:08:11.443 align:middle line:84% It has to depend only on the total wealth. 00:08:11.443 --> 00:08:13.360 align:middle line:84% ROBERT M. TOWNSEND: That's true, and then what 00:08:13.360 --> 00:08:18.870 align:middle line:84% does that imply about the underlying 00:08:18.870 --> 00:08:20.580 align:middle line:90% households in the economy? 00:08:20.580 --> 00:08:25.370 align:middle line:90% 00:08:25.370 --> 00:08:29.260 align:middle line:84% AUDIENCE: I believe it implies some uniformity. 00:08:29.260 --> 00:08:32.387 align:middle line:90% 00:08:32.387 --> 00:08:33.429 align:middle line:90% ROBERT M. TOWNSEND: Yeah. 00:08:33.429 --> 00:08:37.909 align:middle line:84% It's uniform in the way that wealth 00:08:37.909 --> 00:08:40.070 align:middle line:90% impacts marginal changes. 00:08:40.070 --> 00:08:41.659 align:middle line:90% AUDIENCE: Right. 00:08:41.659 --> 00:08:43.929 align:middle line:90% ROBERT M. TOWNSEND: So yeah. 00:08:43.929 --> 00:08:47.280 align:middle line:84% Those linear expansion paths are a shorthand way 00:08:47.280 --> 00:08:51.330 align:middle line:84% of saying different people can consume 00:08:51.330 --> 00:08:55.000 align:middle line:84% different amounts, because they have different wealth. 00:08:55.000 --> 00:08:58.840 align:middle line:84% But if you start changing wealth distribution, 00:08:58.840 --> 00:09:01.150 align:middle line:84% by definition, holding the aggregate fixed, if you're 00:09:01.150 --> 00:09:03.250 align:middle line:84% going to change the distribution, 00:09:03.250 --> 00:09:05.590 align:middle line:84% you're increasing wealth for some households 00:09:05.590 --> 00:09:07.660 align:middle line:90% and decreasing it for others. 00:09:07.660 --> 00:09:09.620 align:middle line:90% And those marginal changes-- 00:09:09.620 --> 00:09:12.130 align:middle line:84% and not just local changes, global changes-- 00:09:12.130 --> 00:09:17.710 align:middle line:84% these could be big redistributions, offset 00:09:17.710 --> 00:09:21.940 align:middle line:90% each other in the aggregate. 00:09:21.940 --> 00:09:24.760 align:middle line:84% The amount that consumption of a given good 00:09:24.760 --> 00:09:26.950 align:middle line:84% is going down for the people who are taxed 00:09:26.950 --> 00:09:32.410 align:middle line:84% is exactly equal to the amount that consumption of that good 00:09:32.410 --> 00:09:35.650 align:middle line:84% is going up for the people who get the subsidies. 00:09:35.650 --> 00:09:38.690 align:middle line:90% 00:09:38.690 --> 00:09:42.230 align:middle line:84% And that's why the redistribution doesn't 00:09:42.230 --> 00:09:43.490 align:middle line:90% matter for the aggregate. 00:09:43.490 --> 00:09:45.890 align:middle line:84% That's why we can do this representative consumer 00:09:45.890 --> 00:09:50.500 align:middle line:84% thing to get the aggregates, because only 00:09:50.500 --> 00:09:54.780 align:middle line:84% aggregate wealth matters for aggregate demand, as you said. 00:09:54.780 --> 00:09:59.040 align:middle line:84% However, it matters a whole lot for the individuals. 00:09:59.040 --> 00:10:00.840 align:middle line:84% I mean if you're paying taxes, lump 00:10:00.840 --> 00:10:07.420 align:middle line:84% sum to the government for it to be redistributed, you get hurt. 00:10:07.420 --> 00:10:09.180 align:middle line:90% So it's not a welfare statement. 00:10:09.180 --> 00:10:13.310 align:middle line:90% 00:10:13.310 --> 00:10:16.360 align:middle line:84% And I'll just repeat, that's why, 00:10:16.360 --> 00:10:19.030 align:middle line:84% depending on what we're doing with free trade or something 00:10:19.030 --> 00:10:23.170 align:middle line:84% else, in principle, we may be able to make everyone 00:10:23.170 --> 00:10:27.960 align:middle line:84% better off, as we are told by situations 00:10:27.960 --> 00:10:30.840 align:middle line:84% where there is a representative in indifference curve 00:10:30.840 --> 00:10:35.610 align:middle line:84% and we're using implicitly Gorman's form 00:10:35.610 --> 00:10:36.870 align:middle line:90% for the indirect utilities. 00:10:36.870 --> 00:10:39.240 align:middle line:84% But that doesn't mean everyone is actually 00:10:39.240 --> 00:10:42.370 align:middle line:84% better off without the intervention, 00:10:42.370 --> 00:10:44.290 align:middle line:84% because the factor prices are moving, 00:10:44.290 --> 00:10:48.710 align:middle line:84% and some people own more labor than capital and vice versa. 00:10:48.710 --> 00:10:53.140 align:middle line:84% So we really do need to be actively redistributing wealth 00:10:53.140 --> 00:10:58.360 align:middle line:84% in order to reap the gains for everyone 00:10:58.360 --> 00:11:02.460 align:middle line:84% to benefit from the gains to trade. 00:11:02.460 --> 00:11:02.960 align:middle line:90% OK. 00:11:02.960 --> 00:11:05.480 align:middle line:90% 00:11:05.480 --> 00:11:12.330 align:middle line:84% So this lecture from 18 was about a tool, 00:11:12.330 --> 00:11:15.000 align:middle line:84% namely Gorman aggregation, which you 00:11:15.000 --> 00:11:18.840 align:middle line:84% saw in the form of the earlier lectures 00:11:18.840 --> 00:11:21.380 align:middle line:90% without me elaborating on it. 00:11:21.380 --> 00:11:23.890 align:middle line:84% So this was an opportunity to do that, but it 00:11:23.890 --> 00:11:25.480 align:middle line:84% does take underlying assumptions, 00:11:25.480 --> 00:11:29.850 align:middle line:84% and in part, the microdata seem to contradict. 00:11:29.850 --> 00:11:30.350 align:middle line:90% Yeah. 00:11:30.350 --> 00:11:32.380 align:middle line:84% I didn't push you guys very hard, 00:11:32.380 --> 00:11:34.030 align:middle line:90% but do you believe it or not? 00:11:34.030 --> 00:11:37.840 align:middle line:84% Then, it's do we see these differentiable wealth effects 00:11:37.840 --> 00:11:38.660 align:middle line:90% in practice? 00:11:38.660 --> 00:11:44.170 align:middle line:84% We have normal goods, necessary goods, and so on and even given 00:11:44.170 --> 00:11:45.040 align:middle line:90% goods, as I said. 00:11:45.040 --> 00:11:48.550 align:middle line:90% 00:11:48.550 --> 00:11:50.470 align:middle line:84% To know if it's a bad approximation, 00:11:50.470 --> 00:11:52.750 align:middle line:84% we'd have to take a stand on the underlying 00:11:52.750 --> 00:11:55.960 align:middle line:84% non-Gorman indirect utility functions 00:11:55.960 --> 00:11:59.230 align:middle line:84% and simulate the economy, and see if it looks almost linear, 00:11:59.230 --> 00:12:01.180 align:middle line:90% even though it's not. 00:12:01.180 --> 00:12:03.865 align:middle line:84% In which case, the redistribution might be small. 00:12:03.865 --> 00:12:06.490 align:middle line:90% 00:12:06.490 --> 00:12:07.900 align:middle line:90% Sorry. 00:12:07.900 --> 00:12:09.070 align:middle line:90% I said that wrong. 00:12:09.070 --> 00:12:14.290 align:middle line:84% The literal form of Gorman aggregation 00:12:14.290 --> 00:12:16.960 align:middle line:84% as an analytic technique might get 00:12:16.960 --> 00:12:20.230 align:middle line:84% us close to what we want as a benchmark, 00:12:20.230 --> 00:12:24.140 align:middle line:84% and since it's so tractable, it's very tempting to use it. 00:12:24.140 --> 00:12:26.770 align:middle line:84% But then again, for other utility functions, 00:12:26.770 --> 00:12:29.440 align:middle line:84% we could be really, really far off 00:12:29.440 --> 00:12:33.220 align:middle line:90% and be misleading policymakers. 00:12:33.220 --> 00:12:35.510 align:middle line:90% Hence the microdata. 00:12:35.510 --> 00:12:38.480 align:middle line:84% Where my pitch is, if you're going to do macro policy, 00:12:38.480 --> 00:12:41.570 align:middle line:84% you have to have micro data, period, 00:12:41.570 --> 00:12:46.160 align:middle line:84% but that's partly a belief that I've acquired over time. 00:12:46.160 --> 00:12:49.550 align:middle line:90% It's a matter of experience. 00:12:49.550 --> 00:12:52.760 align:middle line:90% Although, it's not always true. 00:12:52.760 --> 00:12:56.160 align:middle line:84% The approximations can be very helpful. 00:12:56.160 --> 00:12:56.660 align:middle line:90% OK. 00:12:56.660 --> 00:13:03.030 align:middle line:84% So the lecture today is kind of the reverse of that. 00:13:03.030 --> 00:13:08.820 align:middle line:84% Instead of putting structure on indirect utility functions, 00:13:08.820 --> 00:13:12.780 align:middle line:84% we're going to see if we can get by with almost no structure 00:13:12.780 --> 00:13:15.900 align:middle line:84% at all on preferences and still be 00:13:15.900 --> 00:13:21.300 align:middle line:84% able to make predictions which are testable or rejectable. 00:13:21.300 --> 00:13:25.440 align:middle line:84% So this part, this lecture is all about economic science. 00:13:25.440 --> 00:13:29.460 align:middle line:84% I featured the science of experiments in Lecture 1 00:13:29.460 --> 00:13:31.500 align:middle line:90% as a motivation for the class. 00:13:31.500 --> 00:13:37.380 align:middle line:84% We went through [Koopmans] and Lucas and Josh and Matzkin 00:13:37.380 --> 00:13:38.730 align:middle line:90% and so on. 00:13:38.730 --> 00:13:42.030 align:middle line:84% So this is very much the science of doing experiments, 00:13:42.030 --> 00:13:43.890 align:middle line:84% and I'll tell you exactly what we're 00:13:43.890 --> 00:13:46.620 align:middle line:84% going to do with micro data, depending 00:13:46.620 --> 00:13:50.730 align:middle line:84% on the amount of data available, and also macro data, 00:13:50.730 --> 00:13:54.460 align:middle line:84% trying to put as little structure as possible 00:13:54.460 --> 00:13:58.800 align:middle line:84% and still get rejectable restrictions. 00:13:58.800 --> 00:14:00.100 align:middle line:90% Said that. 00:14:00.100 --> 00:14:02.730 align:middle line:84% So we're going to go through consumer optimization again, 00:14:02.730 --> 00:14:06.180 align:middle line:84% which is, as always, a bit of a review. 00:14:06.180 --> 00:14:07.740 align:middle line:84% We're going to approach that as if we 00:14:07.740 --> 00:14:10.890 align:middle line:84% had an infinite amount of data, and that will take us 00:14:10.890 --> 00:14:13.080 align:middle line:84% back to the Slutsky matrix, which 00:14:13.080 --> 00:14:17.200 align:middle line:84% you may have forgotten by now, but I'll try to remind you. 00:14:17.200 --> 00:14:19.000 align:middle line:84% That's with an infinite amount of data, 00:14:19.000 --> 00:14:23.230 align:middle line:84% we get restrictions that we can reject consumer rationality, 00:14:23.230 --> 00:14:24.740 align:middle line:90% in principle. 00:14:24.740 --> 00:14:27.280 align:middle line:84% And then we'll do the same thing with finite data 00:14:27.280 --> 00:14:31.510 align:middle line:84% and still derive algorithms running on finite data 00:14:31.510 --> 00:14:36.010 align:middle line:84% sets that would allow us to not reject or alternatively 00:14:36.010 --> 00:14:40.360 align:middle line:84% reject the rationality of the household generating the data. 00:14:40.360 --> 00:14:42.430 align:middle line:84% Then, we'll come back to Lucas a bit 00:14:42.430 --> 00:14:45.040 align:middle line:84% in computational considerations, which 00:14:45.040 --> 00:14:47.410 align:middle line:84% I think has been clear throughout the lectures 00:14:47.410 --> 00:14:50.240 align:middle line:90% that we visit from time to time. 00:14:50.240 --> 00:14:53.740 align:middle line:84% And I have dug up some really interesting computer science 00:14:53.740 --> 00:14:56.980 align:middle line:84% material to share with you about how hard or easy 00:14:56.980 --> 00:15:00.760 align:middle line:84% it is to solve certain problems and whether it matters, 00:15:00.760 --> 00:15:02.260 align:middle line:90% be provocative. 00:15:02.260 --> 00:15:04.870 align:middle line:84% And then we'll go to general equilibrium theory 00:15:04.870 --> 00:15:07.850 align:middle line:84% and kind of do the same thing all over again. 00:15:07.850 --> 00:15:09.310 align:middle line:84% And that's going to again come back 00:15:09.310 --> 00:15:14.800 align:middle line:84% to this you need micro to do macro statement that I just 00:15:14.800 --> 00:15:16.090 align:middle line:90% mentioned. 00:15:16.090 --> 00:15:17.320 align:middle line:90% OK. 00:15:17.320 --> 00:15:20.260 align:middle line:84% To get in the content of it, this slide 00:15:20.260 --> 00:15:23.120 align:middle line:90% appears to come out of nowhere. 00:15:23.120 --> 00:15:26.840 align:middle line:84% It was actually almost part of the lecture 00:15:26.840 --> 00:15:28.990 align:middle line:84% we had on income and substitution effects, 00:15:28.990 --> 00:15:31.150 align:middle line:84% and I'll remind you of that in a minute. 00:15:31.150 --> 00:15:35.140 align:middle line:84% In particular, it has to do with the Hicksian demand 00:15:35.140 --> 00:15:36.950 align:middle line:90% and the expenditure function. 00:15:36.950 --> 00:15:39.060 align:middle line:90% Well, I'll remind you now. 00:15:39.060 --> 00:15:42.210 align:middle line:84% We did utility maximization subject to the budget, 00:15:42.210 --> 00:15:46.570 align:middle line:84% and we got the Walrasian or Marshallian demand. 00:15:46.570 --> 00:15:50.340 align:middle line:84% We also decomposed that into income and substitution 00:15:50.340 --> 00:15:53.160 align:middle line:84% effects, and the way we got the substitution effects 00:15:53.160 --> 00:15:55.170 align:middle line:90% was to do compensation. 00:15:55.170 --> 00:15:57.720 align:middle line:84% Prices, say if they decrease, are associated 00:15:57.720 --> 00:15:59.760 align:middle line:90% with increases in income. 00:15:59.760 --> 00:16:01.950 align:middle line:84% We take the income effects away, and just 00:16:01.950 --> 00:16:05.380 align:middle line:84% look at how demands change, as we change prices. 00:16:05.380 --> 00:16:08.520 align:middle line:84% In other words, we minimize the expenditure 00:16:08.520 --> 00:16:11.280 align:middle line:84% necessary to achieve a certain level of utility, 00:16:11.280 --> 00:16:13.720 align:middle line:90% and we got the Hicksian demand. 00:16:13.720 --> 00:16:18.180 align:middle line:84% So with the Hicksian demand and the expenditure function, 00:16:18.180 --> 00:16:21.630 align:middle line:84% we didn't actually go through these properties in Lecture 3, 00:16:21.630 --> 00:16:23.790 align:middle line:90% but they are intuitive. 00:16:23.790 --> 00:16:26.430 align:middle line:84% If the utility function is continuous 00:16:26.430 --> 00:16:29.750 align:middle line:84% represents locally non-satiated preferences. 00:16:29.750 --> 00:16:32.550 align:middle line:84% We might as well define the consumption 00:16:32.550 --> 00:16:35.760 align:middle line:84% set to be the non-negative L dimensional orthont. 00:16:35.760 --> 00:16:39.240 align:middle line:84% Then, we have the following four properties 00:16:39.240 --> 00:16:44.100 align:middle line:84% the expenditure function, this minimized expenditure at prices 00:16:44.100 --> 00:16:47.400 align:middle line:84% p to achieve a certain utility level u, 00:16:47.400 --> 00:16:51.150 align:middle line:90% is homogeneous of degree what? 00:16:51.150 --> 00:16:55.200 align:middle line:90% 1 in those prices p. 00:16:55.200 --> 00:16:57.750 align:middle line:84% I think I slipped up the other day, when I said 0, 00:16:57.750 --> 00:16:59.160 align:middle line:90% and then I switched back to 1. 00:16:59.160 --> 00:17:00.960 align:middle line:90% It depends on what object. 00:17:00.960 --> 00:17:03.630 align:middle line:84% When we do utility maximization to budgets, 00:17:03.630 --> 00:17:05.700 align:middle line:84% if we increase income and prices, 00:17:05.700 --> 00:17:09.900 align:middle line:84% things are homogeneous of degree 0, because nothing changes. 00:17:09.900 --> 00:17:12.430 align:middle line:84% But this is a property of the expenditure function. 00:17:12.430 --> 00:17:14.940 align:middle line:84% So logically, if prices go up, and you're 00:17:14.940 --> 00:17:16.619 align:middle line:84% hitting the same level of utility, 00:17:16.619 --> 00:17:18.119 align:middle line:90% it's going to cost you more. 00:17:18.119 --> 00:17:20.069 align:middle line:84% So the expenditure is just double, 00:17:20.069 --> 00:17:22.200 align:middle line:90% if your doubling prices. 00:17:22.200 --> 00:17:24.930 align:middle line:90% It's homogeneous of degree 1. 00:17:24.930 --> 00:17:29.880 align:middle line:84% The expenditure function for any given price vector p 00:17:29.880 --> 00:17:31.830 align:middle line:90% is increasing in utility. 00:17:31.830 --> 00:17:34.320 align:middle line:84% That makes sense, when you think about having 00:17:34.320 --> 00:17:36.840 align:middle line:84% nice concave indifference curves, or even 00:17:36.840 --> 00:17:38.100 align:middle line:90% linear indifference curves. 00:17:38.100 --> 00:17:41.253 align:middle line:84% If you do increase u by moving further and further north out, 00:17:41.253 --> 00:17:43.170 align:middle line:84% hitting higher and higher indifference curves, 00:17:43.170 --> 00:17:46.410 align:middle line:84% then to minimize the expenditure of achieving those points, 00:17:46.410 --> 00:17:49.090 align:middle line:90% the expenditure has to go up. 00:17:49.090 --> 00:17:52.050 align:middle line:84% Likewise, as prices go up, expenditures 00:17:52.050 --> 00:17:54.510 align:middle line:84% go up, which I kind of already said. 00:17:54.510 --> 00:17:57.300 align:middle line:90% And this is a key property. 00:17:57.300 --> 00:17:59.550 align:middle line:84% In addition, the expenditure function 00:17:59.550 --> 00:18:02.890 align:middle line:90% is concave in the prices. 00:18:02.890 --> 00:18:07.260 align:middle line:84% So we're given that the utility function represents 00:18:07.260 --> 00:18:09.480 align:middle line:90% local non-satiated preferences. 00:18:09.480 --> 00:18:11.080 align:middle line:90% We're not given much else. 00:18:11.080 --> 00:18:13.200 align:middle line:84% So this might be a bit surprising. 00:18:13.200 --> 00:18:16.290 align:middle line:84% This allows for it to be linear, however, and in a minute, 00:18:16.290 --> 00:18:18.540 align:middle line:90% we'll make it strictly concave. 00:18:18.540 --> 00:18:21.235 align:middle line:84% Anyway, you remember how these proofs go. 00:18:21.235 --> 00:18:24.720 align:middle line:84% Pick two prices, pick a linear combination, 00:18:24.720 --> 00:18:29.310 align:middle line:84% prove that an intermediate combination is also minimizing 00:18:29.310 --> 00:18:30.930 align:middle line:90% expenditure, et cetera. 00:18:30.930 --> 00:18:33.660 align:middle line:84% And then the utility expenditure function 00:18:33.660 --> 00:18:38.220 align:middle line:84% is continuous in both objects, but what I want to focus on 00:18:38.220 --> 00:18:40.290 align:middle line:84% is number three, that the expenditure 00:18:40.290 --> 00:18:44.370 align:middle line:84% function is concave in prices for fixed utility. 00:18:44.370 --> 00:18:45.000 align:middle line:90% OK. 00:18:45.000 --> 00:18:51.050 align:middle line:84% Relatedly, the Hicksian demand is, 00:18:51.050 --> 00:18:54.110 align:middle line:84% if you're going to achieve a target utility u, 00:18:54.110 --> 00:18:58.430 align:middle line:84% then you will achieve it exactly, essentially. 00:18:58.430 --> 00:19:02.030 align:middle line:84% That is to say, the x's, which are part of the solution 00:19:02.030 --> 00:19:04.460 align:middle line:84% to the minimizing expenditure problem 00:19:04.460 --> 00:19:07.190 align:middle line:84% when substituted into the utility function, 00:19:07.190 --> 00:19:09.170 align:middle line:90% generate u exactly. 00:19:09.170 --> 00:19:12.410 align:middle line:84% if preferences are, number two, convex, 00:19:12.410 --> 00:19:15.710 align:middle line:84% then these minimized Hicksian demands 00:19:15.710 --> 00:19:19.820 align:middle line:84% are convex valued, entertaining multiple solutions 00:19:19.820 --> 00:19:22.850 align:middle line:84% that minimize expenditure, for given p and u. 00:19:22.850 --> 00:19:25.400 align:middle line:84% But if preferences are strictly convex, 00:19:25.400 --> 00:19:29.430 align:middle line:84% then this h function is single-valued and continuous. 00:19:29.430 --> 00:19:32.960 align:middle line:84% So here, it's where the strict convexity of the preferences 00:19:32.960 --> 00:19:38.750 align:middle line:84% comes in, which we're going to get to again in 4. 00:19:38.750 --> 00:19:42.110 align:middle line:84% 3 says, this Hicksian demand is single value 00:19:42.110 --> 00:19:46.430 align:middle line:84% if it's single valued, which it would be with strictly convex 00:19:46.430 --> 00:19:47.360 align:middle line:90% preferences. 00:19:47.360 --> 00:19:49.370 align:middle line:84% Then, it's differentiable, and moreover, 00:19:49.370 --> 00:19:51.020 align:middle line:84% when you take the derivative of it, 00:19:51.020 --> 00:19:55.190 align:middle line:84% with respect to say a price p of L, for the Lth good, 00:19:55.190 --> 00:19:57.380 align:middle line:90% you get the minimized-- 00:19:57.380 --> 00:19:59.180 align:middle line:90% you get the quantities back. 00:19:59.180 --> 00:20:03.410 align:middle line:84% So the intuition for this is expenditure 00:20:03.410 --> 00:20:09.560 align:middle line:84% is just p1 x1 p2 x2 dot dot dot pL xL. 00:20:09.560 --> 00:20:11.450 align:middle line:84% You take the derivative with respect 00:20:11.450 --> 00:20:13.940 align:middle line:90% to p, you're going to get the x. 00:20:13.940 --> 00:20:17.750 align:middle line:84% That's a bit sloppy, and we did the envelope theorem earlier 00:20:17.750 --> 00:20:19.310 align:middle line:90% and reviewed it last time. 00:20:19.310 --> 00:20:23.510 align:middle line:84% But basically, the derivative at the optimizing minimized 00:20:23.510 --> 00:20:27.230 align:middle line:84% expenditures, at the parametric price, p sub L, 00:20:27.230 --> 00:20:33.860 align:middle line:84% is you get the solution back, the Hicksian demand. 00:20:33.860 --> 00:20:39.440 align:middle line:84% So these two pieces, 3 and 3, on each of the two 00:20:39.440 --> 00:20:43.190 align:middle line:90% slides generate this statement. 00:20:43.190 --> 00:20:45.530 align:middle line:84% That if you take the Hicksian demands 00:20:45.530 --> 00:20:49.610 align:middle line:84% and differentiate with respect to prices, then-- 00:20:49.610 --> 00:20:52.610 align:middle line:84% because level h is on the right-hand side in 8, 00:20:52.610 --> 00:20:55.650 align:middle line:84% but the derivative is already on the left hand side. 00:20:55.650 --> 00:20:58.280 align:middle line:84% So if you're differentiating the right all over 00:20:58.280 --> 00:21:01.110 align:middle line:84% again with respect to p, we're double differentiating, 00:21:01.110 --> 00:21:02.760 align:middle line:90% so to speak, the left-hand side. 00:21:02.760 --> 00:21:06.740 align:middle line:84% So we get these second-order partial derivatives, 00:21:06.740 --> 00:21:09.260 align:middle line:84% [? own ?] end cross partial derivatives, 00:21:09.260 --> 00:21:12.770 align:middle line:84% of the expenditure function, when we differentiate 00:21:12.770 --> 00:21:14.870 align:middle line:90% the Hicksian demands. 00:21:14.870 --> 00:21:16.670 align:middle line:90% This is really compact notation. 00:21:16.670 --> 00:21:18.050 align:middle line:84% Obviously, there's a whole vector 00:21:18.050 --> 00:21:21.320 align:middle line:84% of prices, p1 through pL, et cetera. 00:21:21.320 --> 00:21:24.350 align:middle line:84% We're writing it down almost as if it were single dimension, 00:21:24.350 --> 00:21:28.310 align:middle line:84% but it's meant to capture the entire L by L matrix. 00:21:28.310 --> 00:21:30.980 align:middle line:84% That is to say, derivative of the first Hicksian 00:21:30.980 --> 00:21:34.190 align:middle line:84% demand with respect to the first price, the first Hicksian 00:21:34.190 --> 00:21:37.700 align:middle line:84% demand good with respect to the second, price and so on, 00:21:37.700 --> 00:21:39.650 align:middle line:90% going across the row. 00:21:39.650 --> 00:21:41.130 align:middle line:90% Fill out all the rows. 00:21:41.130 --> 00:21:48.260 align:middle line:84% So it has all the own II, HIPI derivatives, as well as HIJ, 00:21:48.260 --> 00:21:51.170 align:middle line:84% Ith good with respect to the Jth price. 00:21:51.170 --> 00:21:54.140 align:middle line:84% It's all loaded into this compact notation. 00:21:54.140 --> 00:21:57.920 align:middle line:84% Now, I'm saving the best for last, which is this guy here 00:21:57.920 --> 00:22:02.300 align:middle line:84% is the matrix of second-order derivatives of the expenditure 00:22:02.300 --> 00:22:05.090 align:middle line:84% function, and three already told us 00:22:05.090 --> 00:22:07.970 align:middle line:84% the expenditure function is concave. 00:22:07.970 --> 00:22:11.750 align:middle line:84% So that means, if you think about first-order conditions 00:22:11.750 --> 00:22:15.110 align:middle line:84% and second-order conditions, that second-order conditions 00:22:15.110 --> 00:22:19.170 align:middle line:84% are going to be satisfied for a concave function. 00:22:19.170 --> 00:22:24.440 align:middle line:84% And in fact, if you look it up-- stuff does slip memory a bit-- 00:22:24.440 --> 00:22:28.970 align:middle line:84% it means that this matrix of own and cross-partial derivatives 00:22:28.970 --> 00:22:32.750 align:middle line:84% is symmetric and negative semidefinite. 00:22:32.750 --> 00:22:35.570 align:middle line:90% You may not remember that. 00:22:35.570 --> 00:22:37.520 align:middle line:84% You think about one good, then you just 00:22:37.520 --> 00:22:40.280 align:middle line:90% have first the ii derivative. 00:22:40.280 --> 00:22:44.730 align:middle line:84% You think about two, then you're going to have a 2 by 2 matrix. 00:22:44.730 --> 00:22:47.180 align:middle line:84% Then, you have the determinant of that matrix. 00:22:47.180 --> 00:22:49.640 align:middle line:84% If you have three goods, then it's a bit tricky. 00:22:49.640 --> 00:22:52.130 align:middle line:84% To factor the matrix, got to figure out 00:22:52.130 --> 00:22:55.140 align:middle line:84% the determinant of each of the subcomponents and so on. 00:22:55.140 --> 00:22:57.920 align:middle line:84% But an equivalent way to check and see 00:22:57.920 --> 00:23:01.700 align:middle line:84% whether all the conditions are satisfied for concavity is just 00:23:01.700 --> 00:23:04.580 align:middle line:84% to pre-multiply and post-multiply 00:23:04.580 --> 00:23:10.400 align:middle line:84% the matrix by a prime and a, as arbitrary vectors, 00:23:10.400 --> 00:23:14.143 align:middle line:84% and make sure the result is negative. 00:23:14.143 --> 00:23:15.310 align:middle line:90% So that's a bit of a review. 00:23:15.310 --> 00:23:17.410 align:middle line:84% Don't worry about it if you never learned it. 00:23:17.410 --> 00:23:22.000 align:middle line:84% It's the same thing as getting concavity of the expenditure 00:23:22.000 --> 00:23:23.170 align:middle line:90% function. 00:23:23.170 --> 00:23:24.070 align:middle line:90% OK. 00:23:24.070 --> 00:23:25.520 align:middle line:90% So what? 00:23:25.520 --> 00:23:26.930 align:middle line:90% OK. 00:23:26.930 --> 00:23:29.600 align:middle line:90% Well, it's a bit problematic. 00:23:29.600 --> 00:23:32.000 align:middle line:84% I said the motive today was to get restrictions 00:23:32.000 --> 00:23:34.790 align:middle line:84% on observables, but we don't see this guy. 00:23:34.790 --> 00:23:37.040 align:middle line:84% We don't see the Hicksian demand. 00:23:37.040 --> 00:23:41.300 align:middle line:84% We just see the Marshallian demand, so we're in trouble. 00:23:41.300 --> 00:23:44.420 align:middle line:84% Except this-- this is a statement 00:23:44.420 --> 00:23:46.310 align:middle line:90% about the Slutsky matrix. 00:23:46.310 --> 00:23:49.100 align:middle line:84% What we want to find, which is the derivative 00:23:49.100 --> 00:23:52.520 align:middle line:84% of that thing we just looked at, turns out, 00:23:52.520 --> 00:23:55.010 align:middle line:84% it is associated with observables. 00:23:55.010 --> 00:23:59.510 align:middle line:84% This is the Lth good changing with respect to the Kth price 00:23:59.510 --> 00:24:00.830 align:middle line:90% is equal to-- 00:24:00.830 --> 00:24:03.560 align:middle line:84% the way the whole demand is changing of the Lth good, 00:24:03.560 --> 00:24:05.930 align:middle line:90% as we change the price pk. 00:24:05.930 --> 00:24:08.520 align:middle line:84% But we adjust for the income effect, 00:24:08.520 --> 00:24:10.550 align:middle line:84% which is how the Lth grid is changing, 00:24:10.550 --> 00:24:14.030 align:middle line:84% as we change income or wealth, post multiplied 00:24:14.030 --> 00:24:17.270 align:middle line:84% by the quantity of the good price of which is changing, 00:24:17.270 --> 00:24:19.850 align:middle line:90% namely pxk for pk. 00:24:19.850 --> 00:24:20.660 align:middle line:90% OK? 00:24:20.660 --> 00:24:24.000 align:middle line:84% So then, you're probably still struggling. 00:24:24.000 --> 00:24:25.820 align:middle line:90% I went back to check. 00:24:25.820 --> 00:24:27.590 align:middle line:90% You may remember this. 00:24:27.590 --> 00:24:32.090 align:middle line:84% Remember this lecture where we did the income and substitution 00:24:32.090 --> 00:24:35.650 align:middle line:90% effects, these guys? 00:24:35.650 --> 00:24:40.990 align:middle line:84% And then we decomposed the total demand change, 00:24:40.990 --> 00:24:43.000 align:middle line:84% as a consequence of the price change 00:24:43.000 --> 00:24:47.860 align:middle line:84% into the substitution effect and the income effect. 00:24:47.860 --> 00:24:51.550 align:middle line:84% And we had to clarify that the income effect was negative, 00:24:51.550 --> 00:24:56.860 align:middle line:84% because in that diagram we had p on the wrong axis, given 00:24:56.860 --> 00:25:00.920 align:middle line:84% the strange way economists invented demand curves. 00:25:00.920 --> 00:25:03.700 align:middle line:84% So then, what we're trying to do today 00:25:03.700 --> 00:25:06.730 align:middle line:84% is to get this substitution effect matrix, 00:25:06.730 --> 00:25:08.860 align:middle line:84% and we're getting it from observables, 00:25:08.860 --> 00:25:13.510 align:middle line:84% how demand is changing as price changes plus how demand is 00:25:13.510 --> 00:25:15.980 align:middle line:90% changing as income is changing. 00:25:15.980 --> 00:25:19.720 align:middle line:84% So this simple example equation is elaborated 00:25:19.720 --> 00:25:21.490 align:middle line:90% in this Slutsky matrix. 00:25:21.490 --> 00:25:23.470 align:middle line:84% The sign changes, only because we 00:25:23.470 --> 00:25:25.450 align:middle line:84% changed what's on the left-hand side, 00:25:25.450 --> 00:25:28.150 align:middle line:84% from the total demand to the substitution effect. 00:25:28.150 --> 00:25:31.750 align:middle line:84% Substitute effect equals total demand plus income changes. 00:25:31.750 --> 00:25:34.060 align:middle line:84% Substitution effect equals change 00:25:34.060 --> 00:25:36.620 align:middle line:84% in total demand plus income changes. 00:25:36.620 --> 00:25:38.650 align:middle line:84% So in fact, the Slutsky thing is really 00:25:38.650 --> 00:25:41.620 align:middle line:84% familiar to you from that entire lecture 00:25:41.620 --> 00:25:44.710 align:middle line:84% that we did on income and substitution effects. 00:25:44.710 --> 00:25:48.340 align:middle line:84% And now back to the point, to reiterate. 00:25:48.340 --> 00:25:52.390 align:middle line:84% This thing under the assumptions on the previous slide 00:25:52.390 --> 00:25:57.040 align:middle line:84% has to be, if we enumerated it out for all l and k, 00:25:57.040 --> 00:26:01.120 align:middle line:84% a symmetric negative semidefinite matrix. 00:26:01.120 --> 00:26:04.240 align:middle line:84% So we can fill in the I, the row column 00:26:04.240 --> 00:26:06.880 align:middle line:84% elements from all these observables, 00:26:06.880 --> 00:26:11.090 align:middle line:84% and just check and see whether it's true. 00:26:11.090 --> 00:26:11.590 align:middle line:90% OK. 00:26:11.590 --> 00:26:16.210 align:middle line:84% So I should have said, this is with infinite data, 00:26:16.210 --> 00:26:20.500 align:middle line:84% because we're talking about having derivatives, which 00:26:20.500 --> 00:26:22.720 align:middle line:90% is infinitesimally small data. 00:26:22.720 --> 00:26:26.230 align:middle line:84% But it's still good to know that, in principle, there's 00:26:26.230 --> 00:26:27.220 align:middle line:90% a way to do it. 00:26:27.220 --> 00:26:30.280 align:middle line:84% If you can't identify something with infinite data, 00:26:30.280 --> 00:26:34.150 align:middle line:84% you're not going to identify something with finite data. 00:26:34.150 --> 00:26:36.840 align:middle line:90% So let's review where we are. 00:26:36.840 --> 00:26:41.130 align:middle line:84% The goal here is to see whether we have testable restrictions. 00:26:41.130 --> 00:26:44.580 align:middle line:84% In particular, ask the question whether what we see, 00:26:44.580 --> 00:26:49.530 align:middle line:84% in terms of the demand function, came from a utility maximizing 00:26:49.530 --> 00:26:52.120 align:middle line:90% rational consumer or not. 00:26:52.120 --> 00:26:54.600 align:middle line:84% So I'm just going to stick you with a big database, 00:26:54.600 --> 00:26:57.120 align:middle line:84% in this case an infinite dimensional database, 00:26:57.120 --> 00:27:00.360 align:middle line:84% and have you check and see whether it could have come 00:27:00.360 --> 00:27:04.230 align:middle line:84% from a rational maximizing-- maximizing some utility 00:27:04.230 --> 00:27:06.160 align:middle line:90% function, consumer. 00:27:06.160 --> 00:27:06.870 align:middle line:90% OK? 00:27:06.870 --> 00:27:12.450 align:middle line:84% So what are the three properties homogeneity of degree 0. 00:27:12.450 --> 00:27:16.530 align:middle line:84% Again, multiplying income and prices by a scalar, 00:27:16.530 --> 00:27:17.430 align:middle line:90% you just get-- 00:27:17.430 --> 00:27:19.770 align:middle line:90% nothing changes. 00:27:19.770 --> 00:27:25.080 align:middle line:84% That's a statement about not being naive about inflation, 00:27:25.080 --> 00:27:26.370 align:middle line:90% for example. 00:27:26.370 --> 00:27:29.460 align:middle line:84% Although, I guess arguably, it could be violated, 00:27:29.460 --> 00:27:33.720 align:middle line:84% but it seems pretty reasonable for a thoughtful consumer. 00:27:33.720 --> 00:27:36.480 align:middle line:84% In two different situations with prices and income 00:27:36.480 --> 00:27:38.070 align:middle line:84% just being multiples of one another, 00:27:38.070 --> 00:27:40.350 align:middle line:84% they shouldn't change what they do. 00:27:40.350 --> 00:27:43.170 align:middle line:84% The second property is called Walras Law, 00:27:43.170 --> 00:27:46.830 align:middle line:84% which is a fancy name for spending all your money. 00:27:46.830 --> 00:27:50.460 align:middle line:84% Namely, if you have wealth w, and you add up 00:27:50.460 --> 00:27:53.730 align:middle line:84% all the expenses observed to take place 00:27:53.730 --> 00:27:57.000 align:middle line:84% at prices p, that's also equal to your wealth. 00:27:57.000 --> 00:27:58.410 align:middle line:90% So you don't underspend. 00:27:58.410 --> 00:28:01.830 align:middle line:84% You can't overspend, because that would violate the budget. 00:28:01.830 --> 00:28:03.480 align:middle line:90% You don't have the money. 00:28:03.480 --> 00:28:05.310 align:middle line:84% But you could, in principle, underspend. 00:28:05.310 --> 00:28:07.860 align:middle line:84% However, a rational consumer with 00:28:07.860 --> 00:28:12.190 align:middle line:84% local non-satiated preferences would never do that. 00:28:12.190 --> 00:28:14.400 align:middle line:90% They go right out to the budget. 00:28:14.400 --> 00:28:18.960 align:middle line:84% Technically, Walras Law is used to mean you only 00:28:18.960 --> 00:28:20.670 align:middle line:84% need to solve for the equilibrium-- 00:28:20.670 --> 00:28:23.130 align:middle line:84% if there are capital L goods, you only 00:28:23.130 --> 00:28:26.160 align:middle line:84% need to solve for the equilibrium in L minus 1 00:28:26.160 --> 00:28:26.820 align:middle line:90% of them. 00:28:26.820 --> 00:28:32.800 align:middle line:84% Because the Lth is pinned down by this budget equation. 00:28:32.800 --> 00:28:35.910 align:middle line:84% So it's like no extra restrictions. 00:28:35.910 --> 00:28:38.820 align:middle line:84% And finally, the Slutsky matrix, if the demand function 00:28:38.820 --> 00:28:41.970 align:middle line:84% is differentiable and single-valued, 00:28:41.970 --> 00:28:46.200 align:middle line:84% then that matrix, s sub p and w, is symmetric and negative 00:28:46.200 --> 00:28:48.130 align:middle line:90% semidefinite. 00:28:48.130 --> 00:28:49.070 align:middle line:90% OK. 00:28:49.070 --> 00:28:50.450 align:middle line:90% Who cares? 00:28:50.450 --> 00:28:51.980 align:middle line:90% Here's the content. 00:28:51.980 --> 00:28:55.430 align:middle line:84% If the other two things are true-- 00:28:55.430 --> 00:28:58.160 align:middle line:84% which one would argue is likely to be 00:28:58.160 --> 00:29:00.860 align:middle line:90% true in any generated data set-- 00:29:00.860 --> 00:29:04.970 align:middle line:84% if you can find anything that's not symmetric or not 00:29:04.970 --> 00:29:08.030 align:middle line:84% negative semidefinite, then preferences 00:29:08.030 --> 00:29:11.720 align:middle line:84% could not have come from a locally-- 00:29:11.720 --> 00:29:14.360 align:middle line:84% from locally non-satiated preference, as a solution 00:29:14.360 --> 00:29:15.990 align:middle line:90% to the max problem. 00:29:15.990 --> 00:29:19.010 align:middle line:90% So this is the necessity part. 00:29:19.010 --> 00:29:24.260 align:middle line:84% If we have utility max, then all three properties have to hold. 00:29:24.260 --> 00:29:28.970 align:middle line:84% Otherwise, can't be true that it came from utility max problem. 00:29:28.970 --> 00:29:32.610 align:middle line:84% Now, one thing-- we'll get to convexity in a minute, 00:29:32.610 --> 00:29:36.410 align:middle line:84% but I'll just earmark that, that there is no statement here 00:29:36.410 --> 00:29:41.800 align:middle line:84% about convexity, just locally non-satiated preferences, 00:29:41.800 --> 00:29:43.850 align:middle line:90% and it's not quite fair. 00:29:43.850 --> 00:29:47.390 align:middle line:84% The Slutsky thing we derived did rely on-- 00:29:47.390 --> 00:29:48.980 align:middle line:84% at least the way I presented it, it 00:29:48.980 --> 00:29:51.410 align:middle line:90% relied on convex preferences. 00:29:51.410 --> 00:29:54.110 align:middle line:84% The question here is whether one can 00:29:54.110 --> 00:29:58.280 align:middle line:84% test for non-convex preferences with the data, which 00:29:58.280 --> 00:30:00.200 align:middle line:84% is a bit of a different question, 00:30:00.200 --> 00:30:02.580 align:middle line:90% and I'll come back to that. 00:30:02.580 --> 00:30:05.480 align:middle line:84% So those three properties not only 00:30:05.480 --> 00:30:08.420 align:middle line:84% have to be true, if we maximize utility, 00:30:08.420 --> 00:30:12.500 align:middle line:84% but if they're true in the data, then we can always 00:30:12.500 --> 00:30:16.400 align:middle line:84% find a utility function that generated the observations, 00:30:16.400 --> 00:30:18.110 align:middle line:90% and that's called integrability. 00:30:18.110 --> 00:30:20.610 align:middle line:84% Let me give you the formal statement of it. 00:30:20.610 --> 00:30:22.820 align:middle line:84% So this is like the sufficient part. 00:30:22.820 --> 00:30:27.470 align:middle line:84% If preferences of a household are strictly increasing, 00:30:27.470 --> 00:30:32.390 align:middle line:84% strictly quasiconcave, then the Walrasian demand function 00:30:32.390 --> 00:30:38.230 align:middle line:84% is homogeneous of degree 0, satisfies Walras Law, 00:30:38.230 --> 00:30:41.740 align:middle line:84% and it's Slutsky matrix is symmetric and negative 00:30:41.740 --> 00:30:43.540 align:middle line:90% semidefinite. 00:30:43.540 --> 00:30:45.380 align:middle line:90% That's the first part. 00:30:45.380 --> 00:30:49.690 align:middle line:84% The second part is, if the demand function is homogeneous 00:30:49.690 --> 00:30:53.290 align:middle line:84% of degree 0, satisfies Walras's Law, 00:30:53.290 --> 00:30:55.750 align:middle line:84% and the Slutsky matrix is symmetric and negative 00:30:55.750 --> 00:30:59.560 align:middle line:84% semidefinite, then there exists a utility function that 00:30:59.560 --> 00:31:04.840 align:middle line:84% is increasing quasiconcave that would generate the observed 00:31:04.840 --> 00:31:06.020 align:middle line:90% data. 00:31:06.020 --> 00:31:07.180 align:middle line:90% So let me backtrack. 00:31:07.180 --> 00:31:10.490 align:middle line:84% This first part A is what we did initially. 00:31:10.490 --> 00:31:13.690 align:middle line:84% The second part B is the integrability theorem. 00:31:13.690 --> 00:31:16.150 align:middle line:84% It says we can work backwards from the data set 00:31:16.150 --> 00:31:19.810 align:middle line:84% to an underlying utility function, which again has 00:31:19.810 --> 00:31:23.590 align:middle line:84% to be increasing quasiconcave and would 00:31:23.590 --> 00:31:27.230 align:middle line:84% generate the data as if a solution to the max problem. 00:31:27.230 --> 00:31:32.170 align:middle line:84% So this is the bit about not testability of convex 00:31:32.170 --> 00:31:36.280 align:middle line:84% preferences, and the idea is we've deliberately 00:31:36.280 --> 00:31:40.840 align:middle line:84% drawn underlying utility function, which is not concave 00:31:40.840 --> 00:31:44.740 align:middle line:84% or these upper contour sets are not strictly convex. 00:31:44.740 --> 00:31:47.060 align:middle line:90% You see the wiggle here. 00:31:47.060 --> 00:31:48.880 align:middle line:84% But if we start rotating a budget 00:31:48.880 --> 00:31:52.340 align:middle line:84% line or along the indifference curve 00:31:52.340 --> 00:31:54.830 align:middle line:84% and picking out the tangencies, we 00:31:54.830 --> 00:31:58.610 align:middle line:84% would generate the Hicksian demand, et cetera. 00:31:58.610 --> 00:32:02.060 align:middle line:84% Then, we get to this particular budget line, 00:32:02.060 --> 00:32:04.250 align:middle line:90% where we're no longer pivoting. 00:32:04.250 --> 00:32:05.810 align:middle line:90% We jump. 00:32:05.810 --> 00:32:08.570 align:middle line:84% For a moment, we would be indifferent between this point 00:32:08.570 --> 00:32:11.570 align:middle line:84% and this point, and as we keep rotating the line 00:32:11.570 --> 00:32:13.940 align:middle line:84% and making it have less and less slope, 00:32:13.940 --> 00:32:17.810 align:middle line:84% we will start finding pivoting along this curve indifference 00:32:17.810 --> 00:32:18.740 align:middle line:90% curve. 00:32:18.740 --> 00:32:21.380 align:middle line:84% That shouldn't be bending back up again, but anyway. 00:32:21.380 --> 00:32:25.410 align:middle line:84% So the point is we never end up in this non-concave portion. 00:32:25.410 --> 00:32:27.740 align:middle line:84% So we don't generate observables in there. 00:32:27.740 --> 00:32:33.350 align:middle line:84% What we would infer is they have strictly concave, or weakly 00:32:33.350 --> 00:32:38.720 align:middle line:84% linear, preferences over certain ranges of prices, which 00:32:38.720 --> 00:32:43.520 align:middle line:84% is consistent with an underlying weakly quasiconcave utility 00:32:43.520 --> 00:32:44.810 align:middle line:90% function. 00:32:44.810 --> 00:32:48.590 align:middle line:84% Now this doesn't mean that we can't test for convexity 00:32:48.590 --> 00:32:51.230 align:middle line:84% of preferences in other ways, but this whole lecture 00:32:51.230 --> 00:32:55.850 align:middle line:84% is about restrictions on data that come from market behavior, 00:32:55.850 --> 00:32:58.310 align:middle line:84% maximizing utility subject to budgets. 00:32:58.310 --> 00:33:00.230 align:middle line:84% And if that's all we have in the data, 00:33:00.230 --> 00:33:03.210 align:middle line:90% we cannot test for convexity. 00:33:03.210 --> 00:33:06.840 align:middle line:84% Again, the statement, there exists a utility function that 00:33:06.840 --> 00:33:09.630 align:middle line:90% is increasing in quasiconcave. 00:33:09.630 --> 00:33:11.280 align:middle line:84% That's probably confusing, because it 00:33:11.280 --> 00:33:13.650 align:middle line:84% looks like we just said, the utility function 00:33:13.650 --> 00:33:15.630 align:middle line:90% is quasiconcave. 00:33:15.630 --> 00:33:18.960 align:middle line:90% No, this one is not. 00:33:18.960 --> 00:33:20.790 align:middle line:84% That's the true underlying function, 00:33:20.790 --> 00:33:22.680 align:middle line:90% but we can't test for that. 00:33:22.680 --> 00:33:26.010 align:middle line:84% All we see in the observables are the one 00:33:26.010 --> 00:33:29.280 align:middle line:84% that as if generated the dashed line, 00:33:29.280 --> 00:33:32.790 align:middle line:84% and that utility function is quasiconcave. 00:33:32.790 --> 00:33:35.970 align:middle line:84% So that's the content of this statement. 00:33:35.970 --> 00:33:41.400 align:middle line:84% It's as if there were a utility function which is quasiconcave. 00:33:41.400 --> 00:33:44.060 align:middle line:90% Questions? 00:33:44.060 --> 00:33:45.410 align:middle line:90% All right. 00:33:45.410 --> 00:33:49.100 align:middle line:84% So operationally, we might as well assume convex preferences, 00:33:49.100 --> 00:33:52.440 align:middle line:84% because we're not going to be able to reject it anyway. 00:33:52.440 --> 00:33:56.720 align:middle line:84% That's an odd-sounding corollary to this whole first part 00:33:56.720 --> 00:33:59.900 align:middle line:84% of the lecture, which is the goal of placing restrictions 00:33:59.900 --> 00:34:04.080 align:middle line:84% on data, whether data place restrictions on theory or not. 00:34:04.080 --> 00:34:07.190 align:middle line:84% And finally, let me say again, why are we 00:34:07.190 --> 00:34:11.330 align:middle line:84% trying so hard to see what we could potentially reject? 00:34:11.330 --> 00:34:13.850 align:middle line:84% The answer is, if say it's not Slutsky, 00:34:13.850 --> 00:34:17.989 align:middle line:84% then it could not have come from a rational consumer maximizing 00:34:17.989 --> 00:34:20.840 align:middle line:90% utility, and that's good news. 00:34:20.840 --> 00:34:22.940 align:middle line:84% If for any data set, we can always 00:34:22.940 --> 00:34:25.730 align:middle line:84% find a consumer maximizing utility 00:34:25.730 --> 00:34:27.199 align:middle line:84% that could have generated the data, 00:34:27.199 --> 00:34:29.780 align:middle line:84% then there's no content to the theory. 00:34:29.780 --> 00:34:32.210 align:middle line:84% It might be fun to do the theory, 00:34:32.210 --> 00:34:34.710 align:middle line:84% but it's a vacuous empirical exercise. 00:34:34.710 --> 00:34:37.670 align:middle line:84% It has no content, if it's always true. 00:34:37.670 --> 00:34:39.600 align:middle line:84% Fortunately, it's not always true, 00:34:39.600 --> 00:34:42.469 align:middle line:84% and these are the conditions we can check. 00:34:42.469 --> 00:34:43.969 align:middle line:84% It's an odd thing, because then you 00:34:43.969 --> 00:34:46.820 align:middle line:84% could end up with, oh my god, where did this data come from? 00:34:46.820 --> 00:34:51.690 align:middle line:84% It couldn't be coming from a rational consumer, but anyway. 00:34:51.690 --> 00:34:56.610 align:middle line:84% We actually like it if we can reject things, strange science. 00:34:56.610 --> 00:34:59.910 align:middle line:90% So now, let's go to finite data. 00:34:59.910 --> 00:35:01.950 align:middle line:84% We had observed, a minute ago, we 00:35:01.950 --> 00:35:04.140 align:middle line:84% had the entire Walrasian demand function 00:35:04.140 --> 00:35:08.560 align:middle line:84% for arbitrarily small changes in prices and wealth and so on, 00:35:08.560 --> 00:35:10.410 align:middle line:84% but typically, we don't see that. 00:35:10.410 --> 00:35:12.940 align:middle line:84% We don't have an infinite data set. 00:35:12.940 --> 00:35:15.720 align:middle line:84% We may have a household, and if we're lucky 00:35:15.720 --> 00:35:18.900 align:middle line:84% we can observe that household over long periods of time 00:35:18.900 --> 00:35:22.470 align:middle line:84% and see what decisions are being made, 00:35:22.470 --> 00:35:26.310 align:middle line:84% if those things are recorded on Alipay, for example. 00:35:26.310 --> 00:35:29.520 align:middle line:84% And if it were true that everything that was bought 00:35:29.520 --> 00:35:32.940 align:middle line:84% was acquired on Alipay, then we would see-- 00:35:32.940 --> 00:35:35.550 align:middle line:84% and if we also saw the wealth, because they're 00:35:35.550 --> 00:35:37.590 align:middle line:84% spending everything on Alipay, then we 00:35:37.590 --> 00:35:40.740 align:middle line:84% would actually, potentially, have a very long data set, 00:35:40.740 --> 00:35:42.670 align:middle line:90% but not infinite. 00:35:42.670 --> 00:35:46.020 align:middle line:84% So we want to see whether we can, nevertheless, 00:35:46.020 --> 00:35:49.110 align:middle line:84% test the hypothesis of rationality, 00:35:49.110 --> 00:35:51.450 align:middle line:84% and the answer is going to be positive. 00:35:51.450 --> 00:35:56.010 align:middle line:84% There is a way to take the data and run an algorithm 00:35:56.010 --> 00:36:00.990 align:middle line:84% and see whether the results are consistent but potentially 00:36:00.990 --> 00:36:02.730 align:middle line:90% inconsistent. 00:36:02.730 --> 00:36:05.500 align:middle line:84% And in the latter case, we can reject. 00:36:05.500 --> 00:36:08.040 align:middle line:84% So this is the definition of the Weak 00:36:08.040 --> 00:36:10.230 align:middle line:90% Axiom of Revealed Preference. 00:36:10.230 --> 00:36:14.190 align:middle line:84% Let's start with just having two observations 00:36:14.190 --> 00:36:17.580 align:middle line:84% with different prices and different choices. 00:36:17.580 --> 00:36:21.390 align:middle line:84% (p1, x1) (p2, x2) is a pair of observations 00:36:21.390 --> 00:36:23.640 align:middle line:84% on prices and consumption bundles, 00:36:23.640 --> 00:36:26.550 align:middle line:84% implicitly for a given household. 00:36:26.550 --> 00:36:31.020 align:middle line:84% pt doesn't mean, necessarily, t over time, 00:36:31.020 --> 00:36:34.980 align:middle line:84% although you could imagine we observed 00:36:34.980 --> 00:36:38.100 align:middle line:84% at t equal 1 the first choice and a t equal 2 00:36:38.100 --> 00:36:39.510 align:middle line:90% the second choice. 00:36:39.510 --> 00:36:42.810 align:middle line:84% But that's confusing, because it raises the dynamic issues 00:36:42.810 --> 00:36:43.690 align:middle line:90% and everything else. 00:36:43.690 --> 00:36:49.530 align:middle line:84% So t is really only meant here to index the data point, 00:36:49.530 --> 00:36:53.370 align:middle line:84% and there's two data points and hence two values for t. 00:36:53.370 --> 00:36:56.890 align:middle line:84% And let's suppose it's not trivial, 00:36:56.890 --> 00:36:59.610 align:middle line:90% so these choices are different. 00:36:59.610 --> 00:37:02.970 align:middle line:84% And we say the individual choices, these choices, 00:37:02.970 --> 00:37:05.760 align:middle line:84% satisfy the weak axiom of real preference, 00:37:05.760 --> 00:37:13.260 align:middle line:84% if whenever at the second price is p2 with x2 chosen and x1 00:37:13.260 --> 00:37:16.470 align:middle line:84% in the interior of the budget, we must then have, 00:37:16.470 --> 00:37:22.200 align:middle line:84% at prices p1, spending (p1, x1), that x2 is not attainable. 00:37:22.200 --> 00:37:23.850 align:middle line:90% So let me say it again. 00:37:23.850 --> 00:37:25.650 align:middle line:90% It's an if-then thing. 00:37:25.650 --> 00:37:29.760 align:middle line:84% If this is true in the data that, if they had chosen x1, 00:37:29.760 --> 00:37:32.910 align:middle line:84% they would not have been spending all their money-- 00:37:32.910 --> 00:37:35.520 align:middle line:84% if that configuration is true in the data, 00:37:35.520 --> 00:37:39.600 align:middle line:84% then we better find that, at prices p1, 00:37:39.600 --> 00:37:44.640 align:middle line:84% the valuation of expenditures on x2 is strictly greater than x1, 00:37:44.640 --> 00:37:47.650 align:middle line:84% and again, the idea is revealed preference. 00:37:47.650 --> 00:37:54.420 align:middle line:84% So at prices p2, they are revealed to prefer x2 over x1, 00:37:54.420 --> 00:37:57.120 align:middle line:84% because x1 is available, and they didn't choose it. 00:37:57.120 --> 00:38:00.480 align:middle line:84% So it better not be the case that, in some other way 00:38:00.480 --> 00:38:03.960 align:middle line:84% of looking at the data, this inequality is violated. 00:38:03.960 --> 00:38:08.640 align:middle line:84% Because if at prices p1, x2 were available, 00:38:08.640 --> 00:38:11.790 align:middle line:84% because it's interior to the budget at prices p1, 00:38:11.790 --> 00:38:14.610 align:middle line:84% then to be consistent, they should have chosen it, 00:38:14.610 --> 00:38:18.720 align:middle line:84% because it's already shown to be weakly preferred to x1. 00:38:18.720 --> 00:38:20.310 align:middle line:90% So that's the idea. 00:38:20.310 --> 00:38:22.270 align:middle line:84% If this is true, then this is true. 00:38:22.270 --> 00:38:25.680 align:middle line:84% So it's like a statement of an algorithm running 00:38:25.680 --> 00:38:29.230 align:middle line:84% over the data, and we're going to generalize that. 00:38:29.230 --> 00:38:32.740 align:middle line:84% So what happens if this isn't true? 00:38:32.740 --> 00:38:34.840 align:middle line:90% This is true, but this is not. 00:38:34.840 --> 00:38:37.110 align:middle line:90% So let's look at a picture. 00:38:37.110 --> 00:38:44.140 align:middle line:84% At prices p2, they choose x2, so on the budget. 00:38:44.140 --> 00:38:51.680 align:middle line:84% And x1 is interior, so that's this condition. 00:38:51.680 --> 00:38:56.420 align:middle line:84% x1 is interior to the budget at prices p2. 00:38:56.420 --> 00:39:01.730 align:middle line:84% Then, we look at prices p2, p1, are they? 00:39:01.730 --> 00:39:03.710 align:middle line:90% Here. 00:39:03.710 --> 00:39:05.050 align:middle line:90% So here's x1. 00:39:05.050 --> 00:39:07.480 align:middle line:90% It's on the budget at price p1. 00:39:07.480 --> 00:39:09.970 align:middle line:90% The question is, where is x2? 00:39:09.970 --> 00:39:15.960 align:middle line:84% In this case, x2 is interior to the p1 budget line. 00:39:15.960 --> 00:39:17.400 align:middle line:90% That violates. 00:39:17.400 --> 00:39:18.735 align:middle line:90% We didn't want to find that. 00:39:18.735 --> 00:39:22.990 align:middle line:84% That violates what ought to be true, 00:39:22.990 --> 00:39:25.950 align:middle line:84% and so it violates the weak axiom. 00:39:25.950 --> 00:39:29.490 align:middle line:84% Not only that, try to draw indifference curves 00:39:29.490 --> 00:39:34.830 align:middle line:84% from a quasiconcave or linear or strictly convex, 00:39:34.830 --> 00:39:37.160 align:middle line:90% you can't do it. 00:39:37.160 --> 00:39:41.600 align:middle line:84% You'd get a tangency here, you'd get a tangency here, 00:39:41.600 --> 00:39:45.140 align:middle line:84% or looking at the bottom, the indifference curves 00:39:45.140 --> 00:39:46.335 align:middle line:90% would cross. 00:39:46.335 --> 00:39:48.710 align:middle line:84% There's no way you can draw this picture without crossing 00:39:48.710 --> 00:39:53.190 align:middle line:84% the indifference curves and have a tangent at the two choices. 00:39:53.190 --> 00:39:57.230 align:middle line:84% Well, we learned right in Lecture 2 00:39:57.230 --> 00:40:01.610 align:middle line:84% that indifference curves can cross for rational consumers. 00:40:01.610 --> 00:40:04.130 align:middle line:84% A is preferred to B. B is preferred to C . 00:40:04.130 --> 00:40:08.370 align:middle line:84% A is preferred to C. You may remember that diagram. 00:40:08.370 --> 00:40:12.110 align:middle line:84% So that's meant to be an intuitive proof of the picture 00:40:12.110 --> 00:40:13.950 align:middle line:90% of the proposition. 00:40:13.950 --> 00:40:16.700 align:middle line:84% So in conclusion, for the weak axiom 00:40:16.700 --> 00:40:19.370 align:middle line:84% if the agent has preferences which 00:40:19.370 --> 00:40:21.920 align:middle line:84% are locally non-satiated and rational, 00:40:21.920 --> 00:40:27.080 align:middle line:84% then given observations on prices, expenditures, 00:40:27.080 --> 00:40:31.790 align:middle line:84% and wealth, you must have that the pairs of data points 00:40:31.790 --> 00:40:36.200 align:middle line:84% satisfy this Weak Axiom of Revealed Preference, WARP. 00:40:36.200 --> 00:40:38.210 align:middle line:90% OK? 00:40:38.210 --> 00:40:40.530 align:middle line:90% So now, we want to generalize. 00:40:40.530 --> 00:40:43.310 align:middle line:84% We have here still two data points, 00:40:43.310 --> 00:40:46.470 align:middle line:84% just slightly different notation. 00:40:46.470 --> 00:40:49.581 align:middle line:84% The data points are indexed by t and s, 00:40:49.581 --> 00:40:54.060 align:middle line:84% and we write that the t data point of x 00:40:54.060 --> 00:41:00.285 align:middle line:84% is revealed directly preferred to the data point indexed by s, 00:41:00.285 --> 00:41:03.870 align:middle line:90% if x is available at prices pt. 00:41:03.870 --> 00:41:07.290 align:middle line:84% So again, at prices pt, they chose xt. 00:41:07.290 --> 00:41:09.420 align:middle line:84% That's the way the data are organized. 00:41:09.420 --> 00:41:12.390 align:middle line:84% They could have chosen xs, because it's in the budget, 00:41:12.390 --> 00:41:14.100 align:middle line:90% but they didn't do it. 00:41:14.100 --> 00:41:17.580 align:middle line:84% So we say directly reveal preferred. 00:41:17.580 --> 00:41:19.860 align:middle line:84% This might imply strict preference, 00:41:19.860 --> 00:41:21.720 align:middle line:90% but it doesn't really mean that. 00:41:21.720 --> 00:41:25.980 align:middle line:84% It just means, as with the axioms of reveal preference, 00:41:25.980 --> 00:41:27.600 align:middle line:90% what we see is what they want. 00:41:27.600 --> 00:41:29.370 align:middle line:90% That's all we're saying. 00:41:29.370 --> 00:41:33.930 align:middle line:84% They reveal to prefer it, then they continue to prefer it. 00:41:33.930 --> 00:41:36.840 align:middle line:84% In this case, they prefer it over other things they could 00:41:36.840 --> 00:41:38.300 align:middle line:90% have chosen and didn't. 00:41:38.300 --> 00:41:40.380 align:middle line:90% It's pretty simple-minded. 00:41:40.380 --> 00:41:44.500 align:middle line:84% Then, we come to I almost want to say indirectly revealed 00:41:44.500 --> 00:41:45.000 align:middle line:90% preferred. 00:41:45.000 --> 00:41:47.430 align:middle line:84% Actually, it's in parentheses here 00:41:47.430 --> 00:41:50.460 align:middle line:84% to contrast with the directly revealed preferred. 00:41:50.460 --> 00:41:52.488 align:middle line:84% So now we're going to have a bigger data set. 00:41:52.488 --> 00:41:54.030 align:middle line:84% We're going to have a whole sequence, 00:41:54.030 --> 00:41:58.095 align:middle line:84% and we say that xt R without the d 00:41:58.095 --> 00:42:02.040 align:middle line:84% is just revealed preferred not directly preferred. 00:42:02.040 --> 00:42:05.820 align:middle line:84% Or if you like, I for indirectly preferred, 00:42:05.820 --> 00:42:10.050 align:middle line:84% xt is revealed preferred to xs, if we 00:42:10.050 --> 00:42:14.820 align:middle line:84% have a sequence of data points, such that pairwise we 00:42:14.820 --> 00:42:16.980 align:middle line:90% have direct revelation. 00:42:16.980 --> 00:42:20.640 align:middle line:84% So we want xt to, at the beginning, 00:42:20.640 --> 00:42:24.130 align:middle line:90% to be compared to xs at the end. 00:42:24.130 --> 00:42:27.240 align:middle line:84% But we do that, we never see, say in the data, 00:42:27.240 --> 00:42:30.570 align:middle line:84% that they're directly comparable. 00:42:30.570 --> 00:42:34.170 align:middle line:84% Instead, we see xt in a situation 00:42:34.170 --> 00:42:38.600 align:middle line:84% where, within the budget, xt is revealed directly preferred 00:42:38.600 --> 00:42:40.200 align:middle line:90% to xr1. 00:42:40.200 --> 00:42:42.750 align:middle line:84% In another piece of the data, we see 00:42:42.750 --> 00:42:49.770 align:middle line:84% xr1 is revealed directly to xr2, and so on down the chain, 00:42:49.770 --> 00:42:54.430 align:middle line:84% with xrk being revealed directly preferred to xs. 00:42:54.430 --> 00:42:56.770 align:middle line:84% So this is like the transitivity axiom, 00:42:56.770 --> 00:43:01.130 align:middle line:84% which is part of the axioms of rational consumer. 00:43:01.130 --> 00:43:01.630 align:middle line:90% Right? 00:43:01.630 --> 00:43:06.790 align:middle line:84% We have a chain in which we have directly revealed preference. 00:43:06.790 --> 00:43:09.100 align:middle line:84% Hence, the one at the beginning ought 00:43:09.100 --> 00:43:11.830 align:middle line:84% to be indirectly revealed preferred to the one 00:43:11.830 --> 00:43:12.860 align:middle line:90% at the end. 00:43:12.860 --> 00:43:15.020 align:middle line:90% Next xt to xs. 00:43:15.020 --> 00:43:18.700 align:middle line:84% So then we come to the Generalized Axiom 00:43:18.700 --> 00:43:21.700 align:middle line:84% of Revealed Preference, called GARP 00:43:21.700 --> 00:43:24.070 align:middle line:84% so I don't know there was a famous movie "The World 00:43:24.070 --> 00:43:25.030 align:middle line:90% According to Garp." 00:43:25.030 --> 00:43:27.890 align:middle line:84% I don't know if you-- was a Robin Williams movie. 00:43:27.890 --> 00:43:28.870 align:middle line:90% They didn't mean this. 00:43:28.870 --> 00:43:32.380 align:middle line:84% So generalized revealed preference 00:43:32.380 --> 00:43:37.790 align:middle line:84% is building on this notion of indirectly revealed preference. 00:43:37.790 --> 00:43:40.600 align:middle line:84% If we have the data points of the dimensionality-- 00:43:40.600 --> 00:43:46.210 align:middle line:84% the data set is capital T, and we have prices and demands 00:43:46.210 --> 00:43:51.040 align:middle line:84% for each T, then the general axiom of revealed preference 00:43:51.040 --> 00:43:57.370 align:middle line:84% means that, if xt is indirectly revealed to xs, then 00:43:57.370 --> 00:43:59.830 align:middle line:84% if we're able to make the comparison, 00:43:59.830 --> 00:44:02.710 align:middle line:90% we have the strict inequality. 00:44:02.710 --> 00:44:06.130 align:middle line:84% Namely, xt is preferred to xs, it better 00:44:06.130 --> 00:44:11.860 align:middle line:84% be the case that, at prices ps, xt is outside the budget, 00:44:11.860 --> 00:44:14.560 align:middle line:84% relative to the expenditures on xs. 00:44:14.560 --> 00:44:18.400 align:middle line:84% And the intuition, as I wrote in red at the bottom of the slide, 00:44:18.400 --> 00:44:22.120 align:middle line:84% is simply imagine this inequality is reversed. 00:44:22.120 --> 00:44:27.010 align:middle line:84% Then, we would have, at prices ps, xt is within the budget, 00:44:27.010 --> 00:44:31.060 align:middle line:84% and xs ought to be at least weakly if not strictly 00:44:31.060 --> 00:44:35.320 align:middle line:84% preferred, because xs was chosen, and xt was available. 00:44:35.320 --> 00:44:41.200 align:middle line:84% So xs would be directly revealed to be preferred to xt, 00:44:41.200 --> 00:44:42.820 align:middle line:90% according to this definition. 00:44:42.820 --> 00:44:47.290 align:middle line:84% But it goes the other way here, where xt is revealed, 00:44:47.290 --> 00:44:50.260 align:middle line:84% granted indirectly, to be preferred to xs. 00:44:50.260 --> 00:44:54.310 align:middle line:84% So that if this were a strict inequality going the other way, 00:44:54.310 --> 00:44:56.740 align:middle line:90% we would intuitively violate. 00:44:56.740 --> 00:44:59.080 align:middle line:84% These two pieces of information would not 00:44:59.080 --> 00:45:01.240 align:middle line:90% be consistent with each other. 00:45:01.240 --> 00:45:05.110 align:middle line:84% Now, again, you should think about this as an algorithm. 00:45:05.110 --> 00:45:06.310 align:middle line:90% Some classes do this. 00:45:06.310 --> 00:45:08.560 align:middle line:90% I found one on the web today. 00:45:08.560 --> 00:45:10.720 align:middle line:84% They give students a very large data 00:45:10.720 --> 00:45:13.300 align:middle line:84% set of prices and quantities and just 00:45:13.300 --> 00:45:17.170 align:middle line:84% ask them is this consistent with a rational household? 00:45:17.170 --> 00:45:19.010 align:middle line:90% And how do you approach that? 00:45:19.010 --> 00:45:22.400 align:middle line:84% Well, you start looking for these chains. 00:45:22.400 --> 00:45:26.290 align:middle line:84% You have to construct them algorithmically and get, 00:45:26.290 --> 00:45:29.860 align:middle line:84% as a result when it's true, a statement about x and t 00:45:29.860 --> 00:45:31.660 align:middle line:90% indirectly reveal preferred. 00:45:31.660 --> 00:45:33.880 align:middle line:84% And then in the same data set, go 00:45:33.880 --> 00:45:37.420 align:middle line:84% looking for an instance where there was a direct comparison, 00:45:37.420 --> 00:45:41.110 align:middle line:84% and it better be that the inequality is weakly 00:45:41.110 --> 00:45:42.770 align:middle line:90% going this direction. 00:45:42.770 --> 00:45:45.777 align:middle line:84% So it's an algorithm, I'm trying to say. 00:45:45.777 --> 00:45:47.860 align:middle line:84% Because again, if it doesn't go in this direction, 00:45:47.860 --> 00:45:51.250 align:middle line:84% then it contradicts the whole notion of revealed preference 00:45:51.250 --> 00:45:53.050 align:middle line:90% and transitivity. 00:45:53.050 --> 00:45:56.770 align:middle line:90% So again, good news, ironically. 00:45:56.770 --> 00:46:00.280 align:middle line:84% We are potentially able to reject. 00:46:00.280 --> 00:46:03.580 align:middle line:84% There are comparisons in the data which, 00:46:03.580 --> 00:46:05.980 align:middle line:84% if this inequality goes the other way, 00:46:05.980 --> 00:46:08.830 align:middle line:84% would tell us there's no way that this data came 00:46:08.830 --> 00:46:11.620 align:middle line:84% from a rational maximizing consumer, 00:46:11.620 --> 00:46:13.000 align:middle line:90% and that's good for us. 00:46:13.000 --> 00:46:16.220 align:middle line:84% Now, it doesn't mean that, in any given data set, 00:46:16.220 --> 00:46:19.330 align:middle line:84% you will necessarily find a violation, 00:46:19.330 --> 00:46:21.490 align:middle line:84% but you end up with a weak statement 00:46:21.490 --> 00:46:25.600 align:middle line:84% that the data set look as if there was nothing inconsistent 00:46:25.600 --> 00:46:28.300 align:middle line:84% in this data, with respect to the statement that they 00:46:28.300 --> 00:46:32.090 align:middle line:84% came from a rational maximizing consumer. 00:46:32.090 --> 00:46:35.130 align:middle line:84% So we should check and make sure that that's true, 00:46:35.130 --> 00:46:36.970 align:middle line:90% and that's the spirit of it. 00:46:36.970 --> 00:46:39.290 align:middle line:90% Questions? 00:46:39.290 --> 00:46:40.340 align:middle line:90% OK. 00:46:40.340 --> 00:46:43.400 align:middle line:84% So let me get to the computational part. 00:46:43.400 --> 00:46:46.040 align:middle line:84% These guys at Caltech, consumer theory 00:46:46.040 --> 00:46:49.490 align:middle line:84% assumes consumers possess infinite computational 00:46:49.490 --> 00:46:50.360 align:middle line:90% abilities. 00:46:50.360 --> 00:46:54.380 align:middle line:84% Like they can solve max problems and potentially hard problems. 00:46:54.380 --> 00:46:56.780 align:middle line:84% Proponents of bounded rationality 00:46:56.780 --> 00:47:00.830 align:middle line:84% want to require that any reasonable model of consumer 00:47:00.830 --> 00:47:04.920 align:middle line:84% behavior incorporate computational constraints. 00:47:04.920 --> 00:47:08.660 align:middle line:84% In other words, they may not be able to solve a hard problem. 00:47:08.660 --> 00:47:10.760 align:middle line:84% They may choose something, but it 00:47:10.760 --> 00:47:13.190 align:middle line:84% wouldn't be the full solution, because they're not 00:47:13.190 --> 00:47:14.570 align:middle line:90% able to compute it. 00:47:14.570 --> 00:47:18.560 align:middle line:84% And Echenique and his coauthors are saying, that's false. 00:47:18.560 --> 00:47:21.110 align:middle line:84% Very much in the spirit of revealed preference, 00:47:21.110 --> 00:47:24.290 align:middle line:84% any consumption data set that is compatible 00:47:24.290 --> 00:47:27.650 align:middle line:90% with a rational consumer-- 00:47:27.650 --> 00:47:29.930 align:middle line:84% excuse me-- is also compatible with 00:47:29.930 --> 00:47:33.440 align:middle line:84% a rational and computationally bounded consumer. 00:47:33.440 --> 00:47:36.410 align:middle line:84% That is to say, the data set in hand 00:47:36.410 --> 00:47:40.280 align:middle line:84% can always be rationalized by a utility function that's 00:47:40.280 --> 00:47:45.610 align:middle line:84% pretty easy to solve, meaning solved in polynomial time. 00:47:45.610 --> 00:47:47.760 align:middle line:84% So this is very much like not being 00:47:47.760 --> 00:47:50.530 align:middle line:84% able to test the convexity of preferences. 00:47:50.530 --> 00:47:53.160 align:middle line:84% You start with something non-convex, 00:47:53.160 --> 00:47:55.530 align:middle line:84% but it generates a data set that could also be 00:47:55.530 --> 00:47:58.220 align:middle line:90% generated by something convex. 00:47:58.220 --> 00:48:00.620 align:middle line:84% We start with something that could 00:48:00.620 --> 00:48:04.340 align:middle line:84% have come from the household solving a hard problem, 00:48:04.340 --> 00:48:08.150 align:middle line:84% but it could also have come from a household solving a simpler 00:48:08.150 --> 00:48:09.140 align:middle line:90% problem. 00:48:09.140 --> 00:48:11.540 align:middle line:84% I'm not giving you any intuition on this right now. 00:48:11.540 --> 00:48:13.790 align:middle line:84% I'm telling you what these computer scientists 00:48:13.790 --> 00:48:14.600 align:middle line:90% have figured out. 00:48:14.600 --> 00:48:18.470 align:middle line:84% And again their metric is, how hard is it 00:48:18.470 --> 00:48:20.900 align:middle line:90% to solve the problem? 00:48:20.900 --> 00:48:22.640 align:middle line:84% Looking at derivatives and so on, 00:48:22.640 --> 00:48:24.620 align:middle line:84% how long does it take, as we, say, 00:48:24.620 --> 00:48:26.580 align:middle line:90% increase the number of goods? 00:48:26.580 --> 00:48:28.430 align:middle line:90% If it's exponential, it's bad. 00:48:28.430 --> 00:48:30.830 align:middle line:90% If it's polynomial, it's good. 00:48:30.830 --> 00:48:31.490 align:middle line:90% All right. 00:48:31.490 --> 00:48:34.730 align:middle line:84% Another version of this, instead of having a single household 00:48:34.730 --> 00:48:38.930 align:middle line:84% maximizing, or otherwise, we have multiple agents 00:48:38.930 --> 00:48:41.220 align:middle line:90% interacting in a market economy. 00:48:41.220 --> 00:48:45.500 align:middle line:84% We want to say, if possible, that the observed market 00:48:45.500 --> 00:48:50.570 align:middle line:84% outcomes are consistent not only with individual maximization 00:48:50.570 --> 00:48:53.550 align:middle line:90% but with Walrasian equilibrium. 00:48:53.550 --> 00:48:56.240 align:middle line:84% So what is the empirical content? 00:48:56.240 --> 00:48:59.180 align:middle line:84% Is it really hard to solve the general equilibrium 00:48:59.180 --> 00:49:00.290 align:middle line:90% computationally? 00:49:00.290 --> 00:49:02.930 align:middle line:84% In which case, it's not likely we're 00:49:02.930 --> 00:49:05.390 align:middle line:84% going to be able to accept in the data 00:49:05.390 --> 00:49:07.430 align:middle line:84% that it came from the underlying economy, 00:49:07.430 --> 00:49:11.090 align:middle line:84% or it's a version of the individual maximization 00:49:11.090 --> 00:49:12.260 align:middle line:90% problem. 00:49:12.260 --> 00:49:15.860 align:middle line:84% That there's another economy, which is actually easier 00:49:15.860 --> 00:49:19.800 align:middle line:84% to solve, that could have generated the same data. 00:49:19.800 --> 00:49:22.400 align:middle line:84% This is totally remarkable stuff. 00:49:22.400 --> 00:49:24.410 align:middle line:84% Because it goes against the grain 00:49:24.410 --> 00:49:28.760 align:middle line:84% of the way one would like to think about it, as rational 00:49:28.760 --> 00:49:31.580 align:middle line:84% as consumers can possibly be, it's 00:49:31.580 --> 00:49:34.760 align:middle line:84% unlikely they can solve in their minds problems that 00:49:34.760 --> 00:49:37.520 align:middle line:84% prove intractable for computer scientists equipped 00:49:37.520 --> 00:49:39.080 align:middle line:90% with the latest technology. 00:49:39.080 --> 00:49:40.190 align:middle line:90% It's a cool statement. 00:49:40.190 --> 00:49:42.620 align:middle line:84% These are really famous theorists, by the way. 00:49:42.620 --> 00:49:45.230 align:middle line:84% Gilboa, Schmaidler, and Postlewaite 00:49:45.230 --> 00:49:47.210 align:middle line:90% solve in their minds-- 00:49:47.210 --> 00:49:52.700 align:middle line:84% get it-- what you would need for a supercomputer. 00:49:52.700 --> 00:49:55.250 align:middle line:84% If an equilibrium is not efficiently computed-- 00:49:55.250 --> 00:49:57.230 align:middle line:90% this is the market version-- 00:49:57.230 --> 00:50:00.620 align:middle line:84% much of the credibility of using the predicate that we have 00:50:00.620 --> 00:50:04.370 align:middle line:84% rational agents is lost, and a shorthand version of that-- 00:50:04.370 --> 00:50:07.740 align:middle line:84% if your laptop can't find it, neither can the market. 00:50:07.740 --> 00:50:10.260 align:middle line:84% But these guys are saying, that's just wrong. 00:50:10.260 --> 00:50:12.470 align:middle line:90% It's just completely wrong. 00:50:12.470 --> 00:50:15.800 align:middle line:84% Because they misunderstand the role of models 00:50:15.800 --> 00:50:22.190 align:middle line:84% in economics, which is as if the model were a reality, then 00:50:22.190 --> 00:50:23.750 align:middle line:90% there are restrictions on data. 00:50:23.750 --> 00:50:26.210 align:middle line:84% It is not a statement that we have 00:50:26.210 --> 00:50:29.250 align:middle line:84% all of the reality captured in the model. 00:50:29.250 --> 00:50:33.890 align:middle line:84% So the reality behaves as if the theory were true. 00:50:33.890 --> 00:50:38.880 align:middle line:84% A consumption data set is either not rationalizable at all-- 00:50:38.880 --> 00:50:40.430 align:middle line:90% so that's the rejection part. 00:50:40.430 --> 00:50:42.440 align:middle line:84% They're not going to overturn that. 00:50:42.440 --> 00:50:43.910 align:middle line:90% There's still content. 00:50:43.910 --> 00:50:46.280 align:middle line:84% It's either not rationalizable at all, 00:50:46.280 --> 00:50:50.090 align:middle line:84% or it's rationalizable by utility function that's 00:50:50.090 --> 00:50:51.980 align:middle line:90% fairly easy to maximize. 00:50:51.980 --> 00:50:54.800 align:middle line:84% When I show you these slides, I want to show you more, 00:50:54.800 --> 00:50:56.480 align:middle line:84% but I guess we have limited time. 00:50:56.480 --> 00:50:59.630 align:middle line:84% So I probably should have put this paper on the reading list 00:50:59.630 --> 00:51:01.040 align:middle line:90% as optional. 00:51:01.040 --> 00:51:01.760 align:middle line:90% OK. 00:51:01.760 --> 00:51:06.080 align:middle line:84% So let's go to general equilibrium in a paper of Brown 00:51:06.080 --> 00:51:07.970 align:middle line:84% and Matzkin, which all things considered, 00:51:07.970 --> 00:51:09.890 align:middle line:90% is relatively recent. 00:51:09.890 --> 00:51:12.830 align:middle line:84% Suppose we're given data set on a number of households, 00:51:12.830 --> 00:51:16.340 align:middle line:84% capital I in the economy, and we see 00:51:16.340 --> 00:51:18.890 align:middle line:90% their individual endowments-- 00:51:18.890 --> 00:51:24.740 align:middle line:84% private ownership economy, omega I, across households capital I. 00:51:24.740 --> 00:51:28.730 align:middle line:84% Then, there's something that I'll elaborate in words now 00:51:28.730 --> 00:51:30.500 align:middle line:90% called Anything Goes. 00:51:30.500 --> 00:51:34.460 align:middle line:84% General equilibrium theory does not put restrictions on data. 00:51:34.460 --> 00:51:36.770 align:middle line:90% Anything goes. 00:51:36.770 --> 00:51:37.850 align:middle line:90% And that was a killer. 00:51:37.850 --> 00:51:41.000 align:middle line:84% That killed the whole subject for a long time, 00:51:41.000 --> 00:51:42.800 align:middle line:90% because of what I'm saying. 00:51:42.800 --> 00:51:47.570 align:middle line:84% Now, the object in question was our famous excess demand 00:51:47.570 --> 00:51:50.240 align:middle line:84% function that we have used before, the aggregate demand 00:51:50.240 --> 00:51:52.370 align:middle line:84% function that we've been focusing 00:51:52.370 --> 00:51:55.070 align:middle line:90% on with aggregation and so on. 00:51:55.070 --> 00:51:57.590 align:middle line:84% What shape does that aggregate demand function 00:51:57.590 --> 00:51:59.960 align:middle line:90% have, as we vary prices? 00:51:59.960 --> 00:52:02.990 align:middle line:84% And the answer was whatever wiggle and turn it has, 00:52:02.990 --> 00:52:06.750 align:middle line:84% including upsloping portions and so on. 00:52:06.750 --> 00:52:09.230 align:middle line:84% You can always find an underlying economy 00:52:09.230 --> 00:52:12.180 align:middle line:84% that would have generated that excess demand. 00:52:12.180 --> 00:52:15.800 align:middle line:84% So this part of it is the seeming answer to the question. 00:52:15.800 --> 00:52:18.290 align:middle line:84% What happens if we don't put any restrictions 00:52:18.290 --> 00:52:22.190 align:middle line:84% on utility functions, other than things like rationality? 00:52:22.190 --> 00:52:25.790 align:middle line:84% And the answer is anything could happen. 00:52:25.790 --> 00:52:27.890 align:middle line:84% That aggregate demand function could 00:52:27.890 --> 00:52:29.450 align:middle line:90% look any way you want it to. 00:52:29.450 --> 00:52:31.820 align:middle line:84% For any aggregate demand function, 00:52:31.820 --> 00:52:35.720 align:middle line:84% we can populate an economy with a finite number of households, 00:52:35.720 --> 00:52:40.640 align:middle line:84% give them certain preferences and ownership of resources 00:52:40.640 --> 00:52:42.950 align:middle line:84% in such a way that we can get that aggregate demand 00:52:42.950 --> 00:52:46.080 align:middle line:84% to shift and turn and do whatever we want. 00:52:46.080 --> 00:52:49.010 align:middle line:84% So again, the shorthand is anything goes, 00:52:49.010 --> 00:52:52.340 align:middle line:84% and because we could never reject the model, 00:52:52.340 --> 00:52:55.580 align:middle line:84% people thought that general equilibrium had no content, 00:52:55.580 --> 00:52:57.050 align:middle line:90% and they stopped doing it. 00:52:57.050 --> 00:52:59.340 align:middle line:90% It was discredited. 00:52:59.340 --> 00:53:00.900 align:middle line:90% This was not just one person. 00:53:00.900 --> 00:53:05.550 align:middle line:84% It was Hugo Sonnenschein, Debreu, and Mantel. 00:53:05.550 --> 00:53:06.840 align:middle line:90% OK. 00:53:06.840 --> 00:53:09.150 align:middle line:90% But let's just change this. 00:53:09.150 --> 00:53:12.270 align:middle line:84% First, what is this seemingly reasonable but actually crazy 00:53:12.270 --> 00:53:14.520 align:middle line:84% thing that we actually see the excess demand? 00:53:14.520 --> 00:53:16.500 align:middle line:90% We never see that. 00:53:16.500 --> 00:53:18.450 align:middle line:84% If we see market economy, we only 00:53:18.450 --> 00:53:19.860 align:middle line:90% see market clearing prices. 00:53:19.860 --> 00:53:23.380 align:middle line:84% We don't see what happens out of equilibrium. 00:53:23.380 --> 00:53:25.320 align:middle line:90% So we only see where it crosses. 00:53:25.320 --> 00:53:28.660 align:middle line:90% We see the fixed points. 00:53:28.660 --> 00:53:31.500 align:middle line:84% Well, let's give us a little more data then. 00:53:31.500 --> 00:53:33.570 align:middle line:84% Given a set of observations on prices 00:53:33.570 --> 00:53:36.820 align:middle line:84% and individual endowments, does there exist-- 00:53:36.820 --> 00:53:37.320 align:middle line:90% OK. 00:53:37.320 --> 00:53:40.380 align:middle line:90% So T now indicates the economy. 00:53:40.380 --> 00:53:42.930 align:middle line:84% It's as if we're going to go in the cross section, 00:53:42.930 --> 00:53:45.930 align:middle line:84% across several or many economies that 00:53:45.930 --> 00:53:49.900 align:middle line:84% vary in the individual ownership of endowments. 00:53:49.900 --> 00:53:55.110 align:middle line:84% So it's omega IT for the Tth economy observation. 00:53:55.110 --> 00:53:56.970 align:middle line:84% For any T that varies over I, we have 00:53:56.970 --> 00:53:59.880 align:middle line:84% a finite number of households, but T is varying in the sample. 00:53:59.880 --> 00:54:02.520 align:middle line:84% That's the number of economies in the sample. 00:54:02.520 --> 00:54:05.910 align:middle line:84% Those economies are all alike for preferences, 00:54:05.910 --> 00:54:07.440 align:middle line:90% that's assumed. 00:54:07.440 --> 00:54:12.660 align:middle line:84% But they differ in the ownership structure, the omega ITs. 00:54:12.660 --> 00:54:15.690 align:middle line:84% So then does there exist a sequence 00:54:15.690 --> 00:54:20.250 align:middle line:84% of these economies, each with its associated aggregate excess 00:54:20.250 --> 00:54:25.590 align:middle line:84% demand, but such that those prices that we see 00:54:25.590 --> 00:54:28.440 align:middle line:84% stuck into the aggregated excess demand equals 0. 00:54:28.440 --> 00:54:30.150 align:middle line:84% This last statement is just a statement 00:54:30.150 --> 00:54:32.535 align:middle line:84% that the price has come from a market. 00:54:32.535 --> 00:54:35.370 align:middle line:84% In other words, the data set we're going to see 00:54:35.370 --> 00:54:39.660 align:middle line:84% would be the prices that because it's a market 00:54:39.660 --> 00:54:44.072 align:middle line:84% or associated with zero excess demand, equilibrium prices. 00:54:44.072 --> 00:54:45.780 align:middle line:84% So we're going to see equilibrium prices, 00:54:45.780 --> 00:54:49.590 align:middle line:84% and we're also going to see individual endowments, 00:54:49.590 --> 00:54:52.540 align:middle line:84% assuming preferences are all the same. 00:54:52.540 --> 00:54:55.140 align:middle line:90% So I'll give you the punch line. 00:54:55.140 --> 00:54:57.060 align:middle line:90% That's Brown and Matzkin. 00:54:57.060 --> 00:55:00.000 align:middle line:84% The punch line is going to be what we've been saying 00:55:00.000 --> 00:55:02.190 align:middle line:90% throughout the lecture today. 00:55:02.190 --> 00:55:04.800 align:middle line:84% Namely, if we're given data like this, 00:55:04.800 --> 00:55:08.460 align:middle line:84% we can actually reject that the data could have 00:55:08.460 --> 00:55:11.070 align:middle line:90% come as a market equilibrium. 00:55:11.070 --> 00:55:13.110 align:middle line:84% That is to say, that the market the data 00:55:13.110 --> 00:55:17.790 align:middle line:84% couldn't potentially never have come from individual household 00:55:17.790 --> 00:55:20.670 align:middle line:84% maximization and market clearing. 00:55:20.670 --> 00:55:22.510 align:middle line:90% So it has content. 00:55:22.510 --> 00:55:24.780 align:middle line:84% Here's the differentiable version of it. 00:55:24.780 --> 00:55:28.080 align:middle line:84% As abstract as general equilibrium theory is, this 00:55:28.080 --> 00:55:31.470 align:middle line:84% Sonnenschein, Debreu, and Mantel is overturned. 00:55:31.470 --> 00:55:33.870 align:middle line:84% They're applying it to an exchange economy, 00:55:33.870 --> 00:55:35.800 align:middle line:84% where we see prices and endowments. 00:55:35.800 --> 00:55:38.520 align:middle line:84% They derive necessary and sufficient conditions 00:55:38.520 --> 00:55:42.330 align:middle line:84% for the equilibrium prices as a function of initial endowments, 00:55:42.330 --> 00:55:45.150 align:middle line:84% and in their language, they show that the economy 00:55:45.150 --> 00:55:47.970 align:middle line:90% can be generically identified. 00:55:47.970 --> 00:55:51.720 align:middle line:84% What they mean by that is there is an underlying economy that 00:55:51.720 --> 00:55:54.810 align:middle line:84% would have generated the data, but also 00:55:54.810 --> 00:55:57.570 align:middle line:84% that, in principle, one could have not 00:55:57.570 --> 00:55:59.820 align:middle line:84% been able to find such an economy. 00:55:59.820 --> 00:56:02.430 align:middle line:84% And then to summarize where we are, 00:56:02.430 --> 00:56:05.370 align:middle line:84% if you only had aggregate data, you're sunk. 00:56:05.370 --> 00:56:08.910 align:middle line:84% You're never going to get conditions 00:56:08.910 --> 00:56:10.710 align:middle line:90% that allow you to reject. 00:56:10.710 --> 00:56:12.900 align:middle line:84% But when you have individual data, 00:56:12.900 --> 00:56:15.630 align:middle line:84% you can get testable restrictions 00:56:15.630 --> 00:56:19.360 align:middle line:84% and bring back the content of general equilibrium theory. 00:56:19.360 --> 00:56:23.580 align:middle line:84% So again, this summarizes my deliberately provocative 00:56:23.580 --> 00:56:28.380 align:middle line:84% statement that you need micro data to do macro. 00:56:28.380 --> 00:56:31.650 align:middle line:84% And then they go on to talk about something 00:56:31.650 --> 00:56:35.070 align:middle line:84% you're familiar with, because we did it in the class. 00:56:35.070 --> 00:56:37.530 align:middle line:84% We're sharing in Indian villages. 00:56:37.530 --> 00:56:41.340 align:middle line:84% It's not uncommon to observe endowments, 00:56:41.340 --> 00:56:43.960 align:middle line:90% like crops and prices. 00:56:43.960 --> 00:56:45.840 align:middle line:84% We did a little bit of that, less so 00:56:45.840 --> 00:56:47.940 align:middle line:84% with the risk sharing, in which case 00:56:47.940 --> 00:56:51.150 align:middle line:90% their content works directly. 00:56:51.150 --> 00:56:53.430 align:middle line:84% If you really believed you were going from one village 00:56:53.430 --> 00:56:55.720 align:middle line:84% to another, each were self-contained. 00:56:55.720 --> 00:56:58.230 align:middle line:84% They had the same preferences and the premise 00:56:58.230 --> 00:57:00.480 align:middle line:84% would be we're seeing market outcomes, 00:57:00.480 --> 00:57:03.150 align:middle line:84% then their theory applies directly. 00:57:03.150 --> 00:57:06.930 align:middle line:84% You can even go to large economies, not villages, 00:57:06.930 --> 00:57:10.320 align:middle line:84% and think about types of individuals, 00:57:10.320 --> 00:57:14.250 align:middle line:84% finite number of types, but a large number of individuals 00:57:14.250 --> 00:57:19.870 align:middle line:84% of a given type and again you can use their artillery. 00:57:19.870 --> 00:57:21.820 align:middle line:84% Now, interestingly, lest you think 00:57:21.820 --> 00:57:24.910 align:middle line:84% economists have this all figured out, 00:57:24.910 --> 00:57:28.330 align:middle line:84% an interesting question is how to extend this to production. 00:57:28.330 --> 00:57:32.380 align:middle line:84% They refer to changes in fact or endowments, 00:57:32.380 --> 00:57:35.660 align:middle line:84% which have observable impact on factor prices. 00:57:35.660 --> 00:57:39.130 align:middle line:84% So that's an allusion to the trade lectures we did. 00:57:39.130 --> 00:57:41.080 align:middle line:84% And they haven't figured it out yet. 00:57:41.080 --> 00:57:44.440 align:middle line:84% They haven't figured out whether there 00:57:44.440 --> 00:57:46.100 align:middle line:90% are rejectable restrictions. 00:57:46.100 --> 00:57:50.210 align:middle line:84% Remember, when we did the model, the 2 by 2 case, 00:57:50.210 --> 00:57:55.460 align:middle line:84% we had constant returns to scale and a few other assumptions. 00:57:55.460 --> 00:57:58.780 align:middle line:84% So they're saying like drop all assumptions. 00:57:58.780 --> 00:58:01.960 align:middle line:84% Could we put restrictions on data 00:58:01.960 --> 00:58:07.330 align:middle line:84% and say we'd never see violations of some kind, 00:58:07.330 --> 00:58:09.250 align:middle line:90% and they haven't figured it out. 00:58:09.250 --> 00:58:13.500 align:middle line:84% So we're on the research frontier at this point. 00:58:13.500 --> 00:58:17.400 align:middle line:84% So that's what I have for today, again, very 00:58:17.400 --> 00:58:20.770 align:middle line:84% much part and parcel of the whole spirit of the class. 00:58:20.770 --> 00:58:23.730 align:middle line:84% Which is to think about economists 00:58:23.730 --> 00:58:29.070 align:middle line:84% as running experiments, and you saw three of them today. 00:58:29.070 --> 00:58:37.243 align:middle line:90%