WEBVTT 00:00:00.000 --> 00:00:03.493 align:middle line:90% [SOUND LOGO] 00:00:03.493 --> 00:00:11.980 align:middle line:90% 00:00:11.980 --> 00:00:14.290 align:middle line:84% ROBERT TOWNSEND: Thank you all for coming. 00:00:14.290 --> 00:00:20.670 align:middle line:84% So we're entering the last one-third of the class. 00:00:20.670 --> 00:00:26.250 align:middle line:84% I guess the best way to talk about that would be schedule. 00:00:26.250 --> 00:00:29.670 align:middle line:90% So we're down to these-- 00:00:29.670 --> 00:00:33.090 align:middle line:90% you just had your second exam. 00:00:33.090 --> 00:00:36.810 align:middle line:84% And the last and final problem set 00:00:36.810 --> 00:00:40.320 align:middle line:84% was released a bit late over the weekend. 00:00:40.320 --> 00:00:43.120 align:middle line:84% But anyway, you should have it now. 00:00:43.120 --> 00:00:47.610 align:middle line:84% So what we're going to do is the welfare theorems today. 00:00:47.610 --> 00:00:51.000 align:middle line:84% And then things like existence of competitive equilibria 00:00:51.000 --> 00:00:54.930 align:middle line:84% and Nash equilibria, Gorman aggregation, 00:00:54.930 --> 00:00:58.260 align:middle line:84% when we're allowed to look at that representative household 00:00:58.260 --> 00:00:58.875 align:middle line:90% or not. 00:00:58.875 --> 00:01:01.650 align:middle line:90% 00:01:01.650 --> 00:01:06.390 align:middle line:84% And thinking about whether models identify data. 00:01:06.390 --> 00:01:10.540 align:middle line:84% In fact, whether you could reject a model based on data. 00:01:10.540 --> 00:01:16.350 align:middle line:84% So these first four lectures are all in general equilibrium 00:01:16.350 --> 00:01:18.540 align:middle line:84% but nevertheless do link to other themes 00:01:18.540 --> 00:01:22.780 align:middle line:84% that we've been covering in the class. 00:01:22.780 --> 00:01:27.510 align:middle line:84% And then, the last few lectures go back to the welfare theorems 00:01:27.510 --> 00:01:30.120 align:middle line:84% and talk about how they can fail but, 00:01:30.120 --> 00:01:34.350 align:middle line:84% in particular, also how you can construct market remedies 00:01:34.350 --> 00:01:35.880 align:middle line:90% for failures. 00:01:35.880 --> 00:01:40.540 align:middle line:84% And then, we end with monetary theory. 00:01:40.540 --> 00:01:42.930 align:middle line:84% So about, I guess, seven or so lectures left. 00:01:42.930 --> 00:01:46.030 align:middle line:90% 00:01:46.030 --> 00:01:50.830 align:middle line:84% And then, if we look on the reading list, optimality 00:01:50.830 --> 00:01:52.090 align:middle line:90% and competitive equilibria. 00:01:52.090 --> 00:01:54.970 align:middle line:84% This was the welfare theorem lecture. 00:01:54.970 --> 00:01:57.050 align:middle line:84% And there's only one starred reading, 00:01:57.050 --> 00:02:00.520 align:middle line:90% which is Kreps, section 6.3. 00:02:00.520 --> 00:02:04.460 align:middle line:84% So again, I do recommend that you look at-- this 00:02:04.460 --> 00:02:06.020 align:middle line:90% is more textbook style. 00:02:06.020 --> 00:02:10.590 align:middle line:84% But it is written at a fairly straightforward level. 00:02:10.590 --> 00:02:12.125 align:middle line:84% And I think you will find it useful. 00:02:12.125 --> 00:02:15.090 align:middle line:90% 00:02:15.090 --> 00:02:20.640 align:middle line:84% So let me start in on lecture 16. 00:02:20.640 --> 00:02:24.240 align:middle line:84% So the fundamental welfare theorems, 00:02:24.240 --> 00:02:26.370 align:middle line:90% there are two of them. 00:02:26.370 --> 00:02:27.900 align:middle line:84% That's going to be the focus today. 00:02:27.900 --> 00:02:31.880 align:middle line:90% 00:02:31.880 --> 00:02:36.250 align:middle line:84% First one, competitive equilibria or while raising 00:02:36.250 --> 00:02:37.970 align:middle line:90% equilibria are Pareto optimal. 00:02:37.970 --> 00:02:41.540 align:middle line:90% 00:02:41.540 --> 00:02:45.600 align:middle line:84% We've defined both concepts earlier. 00:02:45.600 --> 00:02:48.060 align:middle line:84% We'll raise the equilibrium Pareto optimality. 00:02:48.060 --> 00:02:51.670 align:middle line:90% So that's review. 00:02:51.670 --> 00:02:53.950 align:middle line:84% I've mentioned that these welfare theorems are true, 00:02:53.950 --> 00:02:58.030 align:middle line:84% but we've never gone through exactly under what conditions 00:02:58.030 --> 00:03:01.725 align:middle line:90% and how the proofs go. 00:03:01.725 --> 00:03:03.100 align:middle line:84% That's the first welfare theorem. 00:03:03.100 --> 00:03:07.390 align:middle line:84% The second welfare theorem, any Pareto-optimal allocation 00:03:07.390 --> 00:03:10.480 align:middle line:84% can be supported as a competitive equilibrium 00:03:10.480 --> 00:03:11.340 align:middle line:90% with transfers. 00:03:11.340 --> 00:03:14.180 align:middle line:90% 00:03:14.180 --> 00:03:16.970 align:middle line:84% And again, we'll go through the proofs. 00:03:16.970 --> 00:03:21.440 align:middle line:84% Actually, two kinds of proofs of the second welfare theorem. 00:03:21.440 --> 00:03:24.215 align:middle line:84% One, I would say is constructive. 00:03:24.215 --> 00:03:26.510 align:middle line:84% It's matching up first-order conditions. 00:03:26.510 --> 00:03:31.060 align:middle line:84% And the second one is using separating hyperplanes 00:03:31.060 --> 00:03:33.820 align:middle line:90% in convex spaces. 00:03:33.820 --> 00:03:38.050 align:middle line:84% The second is more general than the first, in the sense 00:03:38.050 --> 00:03:43.510 align:middle line:84% that it applies to many environments. 00:03:43.510 --> 00:03:45.580 align:middle line:84% But the first constructive proof is 00:03:45.580 --> 00:03:49.550 align:middle line:90% providing a lot of intuition. 00:03:49.550 --> 00:03:52.510 align:middle line:84% So in some sense, when these welfare theorems hold, 00:03:52.510 --> 00:03:56.320 align:middle line:84% you can go back and forth between optimal allocations 00:03:56.320 --> 00:03:57.900 align:middle line:90% and competitive allocations. 00:03:57.900 --> 00:04:00.970 align:middle line:90% 00:04:00.970 --> 00:04:04.540 align:middle line:84% A particular competitive equilibrium is Pareto optimal, 00:04:04.540 --> 00:04:09.610 align:middle line:84% but that doesn't mean that you couldn't contemplate others. 00:04:09.610 --> 00:04:12.340 align:middle line:84% It's associated with a particular distribution 00:04:12.340 --> 00:04:17.260 align:middle line:84% of wealth from private ownership. 00:04:17.260 --> 00:04:20.140 align:middle line:84% And if you want to get to other redistribute wealth 00:04:20.140 --> 00:04:23.890 align:middle line:84% and benefit, some at the expense of others, 00:04:23.890 --> 00:04:26.680 align:middle line:84% find another Pareto-optimal allocation, 00:04:26.680 --> 00:04:30.920 align:middle line:84% there's also a way to support any additional one like that. 00:04:30.920 --> 00:04:34.650 align:middle line:84% But it is going to require these kinds of transfers. 00:04:34.650 --> 00:04:38.340 align:middle line:84% So that's the overall theme for today. 00:04:38.340 --> 00:04:42.500 align:middle line:84% So let me remind you, paste in some slides from earlier. 00:04:42.500 --> 00:04:47.150 align:middle line:84% This is village India, with income over time 00:04:47.150 --> 00:04:48.320 align:middle line:90% and over households. 00:04:48.320 --> 00:04:54.410 align:middle line:84% And you may remember this gradient, that in some sense, 00:04:54.410 --> 00:04:58.700 align:middle line:84% average income is going up as you go from the landless 00:04:58.700 --> 00:05:02.210 align:middle line:90% to the big landowners. 00:05:02.210 --> 00:05:06.560 align:middle line:84% That's an income, but on the same scale, so a bit harder 00:05:06.560 --> 00:05:10.610 align:middle line:84% to see, it is nevertheless present in consumption 00:05:10.610 --> 00:05:12.950 align:middle line:90% as well, with this upsloping. 00:05:12.950 --> 00:05:17.810 align:middle line:84% So in some sense, this cross-sectional gradient 00:05:17.810 --> 00:05:20.810 align:middle line:84% reflects the lambda weights, so to speak, 00:05:20.810 --> 00:05:23.540 align:middle line:84% when we solve for the optimal allocation 00:05:23.540 --> 00:05:28.190 align:middle line:84% of risk sharing in this village India environment. 00:05:28.190 --> 00:05:31.910 align:middle line:84% We maximize the lambda weighted sum of utilities 00:05:31.910 --> 00:05:36.380 align:middle line:84% and intuitively as well as, in particular, 00:05:36.380 --> 00:05:40.730 align:middle line:84% for particular functional forms, the higher are the lambdas, 00:05:40.730 --> 00:05:44.880 align:middle line:84% the higher would be average consumption. 00:05:44.880 --> 00:05:48.480 align:middle line:84% So that gradient reflects the lambda, so to speak. 00:05:48.480 --> 00:05:51.830 align:middle line:84% Now, we only looked at Pareto optimality 00:05:51.830 --> 00:05:55.070 align:middle line:84% when we did the optimal allocation of risk sharing. 00:05:55.070 --> 00:05:57.440 align:middle line:84% We didn't really link it up to competitive equilibria, 00:05:57.440 --> 00:05:59.720 align:middle line:84% but we're going to do that today. 00:05:59.720 --> 00:06:03.890 align:middle line:84% And in particular, we could ask whether the implicit lambdas 00:06:03.890 --> 00:06:06.830 align:middle line:84% in that gradient are related to wealth. 00:06:06.830 --> 00:06:09.500 align:middle line:84% Think about the village achieving its allocation 00:06:09.500 --> 00:06:13.070 align:middle line:84% of resources to competitive markets and, in particular, 00:06:13.070 --> 00:06:14.690 align:middle line:90% through private ownership. 00:06:14.690 --> 00:06:18.800 align:middle line:84% Then, we should see a correlation between the lambda 00:06:18.800 --> 00:06:22.020 align:middle line:84% weights and other variables that represent wealth. 00:06:22.020 --> 00:06:25.640 align:middle line:84% So I don't think I showed you this slide earlier. 00:06:25.640 --> 00:06:29.600 align:middle line:84% It is in the paper risk and insurance village India. 00:06:29.600 --> 00:06:32.090 align:middle line:84% And here, you can see a regression 00:06:32.090 --> 00:06:38.530 align:middle line:84% of that gradient, the lambdas in particular, that are backed out 00:06:38.530 --> 00:06:42.850 align:middle line:84% from the empirical work, from the optimal allocation of risk 00:06:42.850 --> 00:06:43.570 align:middle line:90% sharing. 00:06:43.570 --> 00:06:46.360 align:middle line:84% Those lambdas are correlated with things 00:06:46.360 --> 00:06:52.660 align:middle line:84% like landholding, where the value of the bull that they own 00:06:52.660 --> 00:06:57.350 align:middle line:84% and plow with and the value of inheritance. 00:06:57.350 --> 00:06:59.170 align:middle line:84% So that would seem to suggest there 00:06:59.170 --> 00:07:05.140 align:middle line:84% is some correlation between wealth and the lambda weights, 00:07:05.140 --> 00:07:14.740 align:middle line:84% although it's not uniform and not true at all in Shirapur. 00:07:14.740 --> 00:07:18.460 align:middle line:84% So it's mostly apparent in only one of the three villages, 00:07:18.460 --> 00:07:19.870 align:middle line:90% in Aurepalle. 00:07:19.870 --> 00:07:22.330 align:middle line:84% And what you don't know from this table 00:07:22.330 --> 00:07:28.600 align:middle line:84% is that the ownership of bulls and so on moves around 00:07:28.600 --> 00:07:29.630 align:middle line:90% over time. 00:07:29.630 --> 00:07:34.270 align:middle line:84% So it's not so representative of wealth anyway. 00:07:34.270 --> 00:07:38.770 align:middle line:84% So this might suggest that the allocation in those villages, 00:07:38.770 --> 00:07:42.910 align:middle line:84% even if achieved through competitive market forces, 00:07:42.910 --> 00:07:47.350 align:middle line:84% also includes some kind of redistribution 00:07:47.350 --> 00:07:50.230 align:middle line:84% from the upper-caste landowners to the poor. 00:07:50.230 --> 00:07:53.520 align:middle line:90% 00:07:53.520 --> 00:07:57.300 align:middle line:84% So let's turn to the welfare theorems. 00:07:57.300 --> 00:08:02.760 align:middle line:84% First one, suppose we had a competitive allocation, denoted 00:08:02.760 --> 00:08:08.190 align:middle line:84% x star, y star with price p and wealth w. 00:08:08.190 --> 00:08:13.270 align:middle line:84% And actually, this slide treats it more generally. 00:08:13.270 --> 00:08:16.390 align:middle line:84% A competitive equilibrium with transfers. 00:08:16.390 --> 00:08:20.380 align:middle line:84% A special case, of course, of a competitive equilibrium 00:08:20.380 --> 00:08:24.600 align:middle line:84% would be no transfers and just private ownership. 00:08:24.600 --> 00:08:26.560 align:middle line:84% But let's start with a competitive equilibrium 00:08:26.560 --> 00:08:28.660 align:middle line:90% with transfers. 00:08:28.660 --> 00:08:33.820 align:middle line:84% And if preferences are rational and locally nonsatiated, 00:08:33.820 --> 00:08:39.750 align:middle line:84% that allocation x star, y star is Pareto optimal or, for sure, 00:08:39.750 --> 00:08:43.320 align:middle line:84% competitive equilibrium or Pareto optimal. 00:08:43.320 --> 00:08:47.910 align:middle line:90% Strikingly, you don't need much. 00:08:47.910 --> 00:08:51.810 align:middle line:84% All you need is essentially rational preferences 00:08:51.810 --> 00:08:55.200 align:middle line:90% and local nonsatiation. 00:08:55.200 --> 00:08:58.110 align:middle line:84% And intuitively, the reason you need so little 00:08:58.110 --> 00:09:00.120 align:middle line:84% is that we're given a lot to begin with. 00:09:00.120 --> 00:09:03.970 align:middle line:84% We're given the existence of a competitive equilibrium 00:09:03.970 --> 00:09:05.640 align:middle line:90% with transfers. 00:09:05.640 --> 00:09:10.142 align:middle line:84% So we don't have to have all the assumptions that go with that 00:09:10.142 --> 00:09:12.070 align:middle line:90% that ensure that equilibrium. 00:09:12.070 --> 00:09:15.040 align:middle line:84% That's kind of taken as a given in the statement 00:09:15.040 --> 00:09:17.890 align:middle line:90% of the problem. 00:09:17.890 --> 00:09:20.590 align:middle line:84% In particular, we don't have to talk about the consumption 00:09:20.590 --> 00:09:24.130 align:middle line:84% sets of the agents, other than they 00:09:24.130 --> 00:09:27.010 align:middle line:84% have to be constructed in a way that 00:09:27.010 --> 00:09:29.530 align:middle line:84% is consistent with local nonsatiation. 00:09:29.530 --> 00:09:32.980 align:middle line:84% We don't have to say anything about production sets, 00:09:32.980 --> 00:09:35.800 align:middle line:84% other than evidently they're nonempty because there's 00:09:35.800 --> 00:09:42.220 align:middle line:84% a y star up here, although in principle, that could be 0. 00:09:42.220 --> 00:09:45.430 align:middle line:90% So very little is assumed. 00:09:45.430 --> 00:09:50.500 align:middle line:84% So here's an illustration in the Edgeworth box 00:09:50.500 --> 00:09:52.240 align:middle line:90% of the first welfare theorem. 00:09:52.240 --> 00:09:55.420 align:middle line:84% We start out with the endowment, private ownership. 00:09:55.420 --> 00:09:57.100 align:middle line:90% No production. 00:09:57.100 --> 00:10:03.390 align:middle line:84% And we find prices such that each of the two agents 00:10:03.390 --> 00:10:05.970 align:middle line:90% here are maximizing. 00:10:05.970 --> 00:10:11.550 align:middle line:84% And along their budget line, it's the same line for each, 00:10:11.550 --> 00:10:12.315 align:middle line:90% in some sense. 00:10:12.315 --> 00:10:15.540 align:middle line:90% 00:10:15.540 --> 00:10:18.090 align:middle line:84% Household 1 giving up Good 1, getting Good 2. 00:10:18.090 --> 00:10:21.150 align:middle line:84% Household 2 going in the opposite direction, 00:10:21.150 --> 00:10:24.060 align:middle line:84% giving up Good 2 and getting Good 1. 00:10:24.060 --> 00:10:26.050 align:middle line:90% And here's the equilibrium. 00:10:26.050 --> 00:10:28.650 align:middle line:84% And you can see how beautiful it is 00:10:28.650 --> 00:10:30.780 align:middle line:84% because the marginal rates of substitution 00:10:30.780 --> 00:10:33.720 align:middle line:90% match up with the price ratio. 00:10:33.720 --> 00:10:37.330 align:middle line:84% And we've achieved the competitive equilibrium. 00:10:37.330 --> 00:10:39.810 align:middle line:84% So competitive equilibrium is Pareto optimal. 00:10:39.810 --> 00:10:42.130 align:middle line:84% Of course, it's not the only one. 00:10:42.130 --> 00:10:46.500 align:middle line:84% There are many, many others, as you can see. 00:10:46.500 --> 00:10:47.970 align:middle line:84% But the first welfare theorem just 00:10:47.970 --> 00:10:52.980 align:middle line:84% starts with a particular competitive equilibrium, 00:10:52.980 --> 00:10:55.200 align:middle line:90% in this case, private ownership. 00:10:55.200 --> 00:10:57.315 align:middle line:84% Now, how do we prove the theorem? 00:10:57.315 --> 00:11:00.460 align:middle line:90% 00:11:00.460 --> 00:11:03.220 align:middle line:84% The picture makes it seem obvious. 00:11:03.220 --> 00:11:05.515 align:middle line:84% But we want the proof to be more general. 00:11:05.515 --> 00:11:08.530 align:middle line:90% 00:11:08.530 --> 00:11:13.560 align:middle line:84% So if preferences are locally nonsatiated, then, 00:11:13.560 --> 00:11:19.310 align:middle line:84% in particular, if the Xi star of the competitive equilibria 00:11:19.310 --> 00:11:25.950 align:middle line:84% is maximizing under Agent I's preferences, 00:11:25.950 --> 00:11:29.610 align:middle line:84% maximizing in the sense that it's the utility maximizing 00:11:29.610 --> 00:11:34.770 align:middle line:84% object in h and i's budget set, this is expenditures less than 00:11:34.770 --> 00:11:41.340 align:middle line:84% or equal to wealth, then for any other point Xi, which 00:11:41.340 --> 00:11:48.820 align:middle line:84% is at least as good in the preference as Xi star, 00:11:48.820 --> 00:11:51.790 align:middle line:84% it must be the case that the valuation 00:11:51.790 --> 00:11:56.680 align:middle line:84% of that other allocation, evaluated price's p, 00:11:56.680 --> 00:11:59.770 align:middle line:84% is not less than, is greater than 00:11:59.770 --> 00:12:03.940 align:middle line:84% or equal to the valuation of expenditures 00:12:03.940 --> 00:12:06.430 align:middle line:90% at the competitive equilibrium. 00:12:06.430 --> 00:12:08.170 align:middle line:90% That's kind a lot to take in. 00:12:08.170 --> 00:12:11.680 align:middle line:84% The key part of it is this implication, 00:12:11.680 --> 00:12:13.960 align:middle line:84% and I used to have the proof all written out. 00:12:13.960 --> 00:12:16.240 align:middle line:84% But it takes like six or seven lines. 00:12:16.240 --> 00:12:18.010 align:middle line:84% Let me just give you the intuition. 00:12:18.010 --> 00:12:20.260 align:middle line:90% Suppose it weren't true? 00:12:20.260 --> 00:12:27.830 align:middle line:84% Suppose that if pxi were strictly less than pxi star. 00:12:27.830 --> 00:12:31.320 align:middle line:84% Well, they're spending everything they have. 00:12:31.320 --> 00:12:35.570 align:middle line:84% So pxi star is their wealth, and they could 00:12:35.570 --> 00:12:38.480 align:middle line:90% spend less if they chose to. 00:12:38.480 --> 00:12:41.840 align:middle line:84% If pxi were less than pxi star, that 00:12:41.840 --> 00:12:45.810 align:middle line:84% would be a point where they're spending less. 00:12:45.810 --> 00:12:50.060 align:middle line:84% However, finally, we come back to the nonsatiation. 00:12:50.060 --> 00:12:54.350 align:middle line:84% Draw a little ball or circle around Xi. 00:12:54.350 --> 00:12:56.160 align:middle line:84% If you remember the statement, there 00:12:56.160 --> 00:12:58.410 align:middle line:84% has to be a bundle that's strictly preferred 00:12:58.410 --> 00:13:01.230 align:middle line:90% to Xi that's close by. 00:13:01.230 --> 00:13:07.450 align:middle line:84% Since it's close by, it is also spending less than Xi star. 00:13:07.450 --> 00:13:10.410 align:middle line:84% So now, there's a preferred point 00:13:10.410 --> 00:13:14.700 align:middle line:84% because Xi is at least as good as Xi star. 00:13:14.700 --> 00:13:19.170 align:middle line:84% This other point is going to strictly dominate Xi and hence 00:13:19.170 --> 00:13:22.240 align:middle line:84% dominate in the preference at Xi star. 00:13:22.240 --> 00:13:25.560 align:middle line:84% So there's a point in the carrier of the budget set 00:13:25.560 --> 00:13:29.040 align:middle line:84% which is strictly preferred to the chosen point Xi 00:13:29.040 --> 00:13:33.360 align:middle line:84% star, which contradicts utility maximization. 00:13:33.360 --> 00:13:35.130 align:middle line:84% Even took me quite a while to say 00:13:35.130 --> 00:13:38.400 align:middle line:84% it, but it's quite intuitive when you think about it. 00:13:38.400 --> 00:13:41.220 align:middle line:90% 00:13:41.220 --> 00:13:44.010 align:middle line:84% So this innocuous little Lemma is 00:13:44.010 --> 00:13:49.050 align:middle line:84% about to get used with the proof of the first welfare theorem, 00:13:49.050 --> 00:13:55.150 align:middle line:84% namely that a competitive equilibrium is Pareto optimal. 00:13:55.150 --> 00:13:56.980 align:middle line:90% Proof by contradiction. 00:13:56.980 --> 00:13:58.285 align:middle line:90% No, it's not. 00:13:58.285 --> 00:13:59.800 align:middle line:90% You say it is. 00:13:59.800 --> 00:14:03.440 align:middle line:84% But I'm going to assume it's not Pareto optimal. 00:14:03.440 --> 00:14:06.480 align:middle line:84% Well, any point that's not Pareto optimal means, 00:14:06.480 --> 00:14:09.440 align:middle line:84% of course, you guys all remember, 00:14:09.440 --> 00:14:12.770 align:middle line:84% that there's a feasible allocation, which 00:14:12.770 --> 00:14:15.050 align:middle line:84% has the property that it's at least as 00:14:15.050 --> 00:14:20.150 align:middle line:84% good as the star competitive allocation for all i 00:14:20.150 --> 00:14:25.700 align:middle line:84% and strictly dominates for at least one agent, in this case, 00:14:25.700 --> 00:14:26.885 align:middle line:90% denoted i prime. 00:14:26.885 --> 00:14:29.670 align:middle line:90% 00:14:29.670 --> 00:14:33.720 align:middle line:84% So that's what we mean by Xi star not being Pareto optimal. 00:14:33.720 --> 00:14:43.540 align:middle line:84% Now, clearly, for the guys, Xi prime, 00:14:43.540 --> 00:14:46.330 align:middle line:84% these bundles are strictly preferred to the one chosen. 00:14:46.330 --> 00:14:48.880 align:middle line:84% So they cannot be in the budget set. 00:14:48.880 --> 00:14:53.090 align:middle line:84% So we have a strict inequality in expenditures 00:14:53.090 --> 00:14:54.515 align:middle line:90% for those i prime guys. 00:14:54.515 --> 00:14:57.460 align:middle line:90% 00:14:57.460 --> 00:15:00.550 align:middle line:84% But what about these guys that were only weakly indifferent? 00:15:00.550 --> 00:15:01.940 align:middle line:90% That could be a problem. 00:15:01.940 --> 00:15:04.210 align:middle line:90% But no, we've got it covered. 00:15:04.210 --> 00:15:06.040 align:middle line:84% By the previous Lemma, we already 00:15:06.040 --> 00:15:11.460 align:middle line:84% know that expenditures are not less. 00:15:11.460 --> 00:15:16.050 align:middle line:84% So we have strictly greater, weakly greater 00:15:16.050 --> 00:15:22.020 align:middle line:84% expenditures of the alternative supposed dominating allocation 00:15:22.020 --> 00:15:26.010 align:middle line:84% with the given competitive allocation. 00:15:26.010 --> 00:15:29.670 align:middle line:84% Let's sum up the values over all the households 00:15:29.670 --> 00:15:33.270 align:middle line:84% I, and we're going to get a strict inequality. 00:15:33.270 --> 00:15:36.260 align:middle line:84% So if it were true that there exists an allocation, which 00:15:36.260 --> 00:15:39.200 align:middle line:84% Pareto dominates the competitive equilibrium, 00:15:39.200 --> 00:15:42.560 align:middle line:84% it would be associated at the equilibrium prices 00:15:42.560 --> 00:15:46.130 align:middle line:84% with the higher aggregate expenditure. 00:15:46.130 --> 00:15:51.620 align:middle line:84% Then, the next step is to review the definition of profit 00:15:51.620 --> 00:15:55.130 align:middle line:84% maximization in a competitive equilibrium, 00:15:55.130 --> 00:15:59.720 align:middle line:84% meaning at yj star, the valuation of profits 00:15:59.720 --> 00:16:03.830 align:middle line:84% is not less than any other feasible-- 00:16:03.830 --> 00:16:09.290 align:middle line:84% technologically feasible vector yj. 00:16:09.290 --> 00:16:14.310 align:middle line:84% And this is true for each firm j, hence true for all of them. 00:16:14.310 --> 00:16:17.120 align:middle line:90% So we can sum up over them. 00:16:17.120 --> 00:16:21.440 align:middle line:84% And now we take this statement for firms 00:16:21.440 --> 00:16:24.980 align:middle line:84% with the statement on the previous slide for households 00:16:24.980 --> 00:16:27.420 align:middle line:84% and start using them both simultaneously. 00:16:27.420 --> 00:16:32.860 align:middle line:84% So from the previous slide, we have the summation over I 00:16:32.860 --> 00:16:36.880 align:middle line:90% pxi strictly dominates pxi star. 00:16:36.880 --> 00:16:41.380 align:middle line:84% But we also had a competitive equilibrium given to us 00:16:41.380 --> 00:16:46.210 align:middle line:84% as an initial condition, therefore feasible. 00:16:46.210 --> 00:16:52.630 align:middle line:84% Therefore, xi star is equal to omega bar plus y star. 00:16:52.630 --> 00:16:54.440 align:middle line:90% And I should have pointed out. 00:16:54.440 --> 00:16:57.670 align:middle line:84% I think I just skipped over it too fast. 00:16:57.670 --> 00:17:00.190 align:middle line:84% It's easiest to think about prices 00:17:00.190 --> 00:17:02.600 align:middle line:90% as being strictly positive. 00:17:02.600 --> 00:17:08.450 align:middle line:84% So the only algebra going on here is to take dot products, 00:17:08.450 --> 00:17:12.500 align:middle line:84% take prices, put it outside the parentheses, 00:17:12.500 --> 00:17:15.349 align:middle line:84% then collect terms, put it back inside. 00:17:15.349 --> 00:17:21.619 align:middle line:84% So p dot xi, summation over I, but Xi 00:17:21.619 --> 00:17:25.560 align:middle line:84% is equal to a omega bar plus yj star. 00:17:25.560 --> 00:17:29.660 align:middle line:84% So if we put p outside and then multiplied it inside, 00:17:29.660 --> 00:17:35.130 align:middle line:84% we get price times the odd quantity object 00:17:35.130 --> 00:17:37.020 align:middle line:90% in each of the two terms. 00:17:37.020 --> 00:17:39.150 align:middle line:84% Now, this last inequality is the statement 00:17:39.150 --> 00:17:41.490 align:middle line:90% of profit maximization. 00:17:41.490 --> 00:17:45.600 align:middle line:84% Holding the valuation of the aggregate endowment of fixed py 00:17:45.600 --> 00:17:51.060 align:middle line:90% star j is no less than pyj. 00:17:51.060 --> 00:17:54.240 align:middle line:84% And so now, we again do this trick. 00:17:54.240 --> 00:17:57.480 align:middle line:84% We take prices, put in on the outside, 00:17:57.480 --> 00:18:01.200 align:middle line:84% and look at all terms inside the parentheses, 00:18:01.200 --> 00:18:07.650 align:middle line:84% and we've got the Xi over here and the omega bar and the yj 00:18:07.650 --> 00:18:08.230 align:middle line:90% here. 00:18:08.230 --> 00:18:10.200 align:middle line:84% So we collect those terms together, 00:18:10.200 --> 00:18:11.970 align:middle line:90% and we get this statement. 00:18:11.970 --> 00:18:14.280 align:middle line:84% But if this is greater than this, 00:18:14.280 --> 00:18:17.610 align:middle line:84% then when you subtract them, it's greater than 0. 00:18:17.610 --> 00:18:20.850 align:middle line:84% So we finally arrive at the contradiction. 00:18:20.850 --> 00:18:21.840 align:middle line:90% Why? 00:18:21.840 --> 00:18:26.690 align:middle line:84% Because this supposedly Pareto-dominating point 00:18:26.690 --> 00:18:29.330 align:middle line:84% has to be feasible in the sense of the resource 00:18:29.330 --> 00:18:34.560 align:middle line:84% constraint, which means this has to be 0 for each component. 00:18:34.560 --> 00:18:39.510 align:middle line:84% But by the argument achieved, it can't be 0 for every component 00:18:39.510 --> 00:18:42.310 align:middle line:84% or this could not be greater than 0. 00:18:42.310 --> 00:18:45.490 align:middle line:84% So it is not true that it's feasible, 00:18:45.490 --> 00:18:48.390 align:middle line:90% and that's the contradiction. 00:18:48.390 --> 00:18:51.970 align:middle line:84% So it's a big proof by contradiction. 00:18:51.970 --> 00:18:54.015 align:middle line:84% So something's happening along the way. 00:18:54.015 --> 00:18:59.390 align:middle line:90% 00:18:59.390 --> 00:19:02.370 align:middle line:84% We're going to go to the second welfare theorem now. 00:19:02.370 --> 00:19:07.280 align:middle line:84% Any Pareto-optimal allocation can be implemented 00:19:07.280 --> 00:19:11.390 align:middle line:90% as a competitive equilibrium. 00:19:11.390 --> 00:19:13.020 align:middle line:90% We can come back to this slide. 00:19:13.020 --> 00:19:14.710 align:middle line:90% But here's the picture. 00:19:14.710 --> 00:19:18.970 align:middle line:84% So we pick another Pareto-optimal point 00:19:18.970 --> 00:19:21.720 align:middle line:90% in that Edgeworth box. 00:19:21.720 --> 00:19:24.660 align:middle line:84% Not this one that ran through the endowment, 00:19:24.660 --> 00:19:27.920 align:middle line:90% but something southwest of it. 00:19:27.920 --> 00:19:28.920 align:middle line:90% So here's the endowment. 00:19:28.920 --> 00:19:30.700 align:middle line:90% It's still there. 00:19:30.700 --> 00:19:33.540 align:middle line:84% But now we're targeting this other Pareto-optimal 00:19:33.540 --> 00:19:35.920 align:middle line:90% allocation. 00:19:35.920 --> 00:19:39.070 align:middle line:84% And the theorem says there's a way 00:19:39.070 --> 00:19:42.790 align:middle line:84% to redistribute wealth in such a way to achieve it. 00:19:42.790 --> 00:19:44.920 align:middle line:84% Note the marginal rates of substitution 00:19:44.920 --> 00:19:48.520 align:middle line:84% all match up for Agent A and Agent B. 00:19:48.520 --> 00:19:50.620 align:middle line:90% They're equal to sum price. 00:19:50.620 --> 00:19:54.880 align:middle line:84% The price, p1 over p2, is captured by the slope 00:19:54.880 --> 00:19:58.570 align:middle line:90% of that, quote, "budget line." 00:19:58.570 --> 00:20:01.360 align:middle line:84% But I really want to say wealth line 00:20:01.360 --> 00:20:05.740 align:middle line:84% because it is not a line that runs through the privately 00:20:05.740 --> 00:20:07.800 align:middle line:90% owned endowments. 00:20:07.800 --> 00:20:10.280 align:middle line:84% So the previous slide, which I skipped, 00:20:10.280 --> 00:20:14.810 align:middle line:84% just says that those star allocations for A and B 00:20:14.810 --> 00:20:18.560 align:middle line:84% have the same slopes, which are their marginal rates 00:20:18.560 --> 00:20:20.030 align:middle line:90% of substitution. 00:20:20.030 --> 00:20:22.100 align:middle line:84% And the marginal rates of substitution 00:20:22.100 --> 00:20:24.500 align:middle line:84% match up since they're equal to each other. 00:20:24.500 --> 00:20:25.880 align:middle line:90% Just something called gamma. 00:20:25.880 --> 00:20:27.830 align:middle line:90% So that's the picture here. 00:20:27.830 --> 00:20:30.440 align:middle line:84% Margin rates of substitution are equal, 00:20:30.440 --> 00:20:33.290 align:middle line:84% equal to the slope of this wealth line, which 00:20:33.290 --> 00:20:36.670 align:middle line:90% has a slope of gamma. 00:20:36.670 --> 00:20:41.360 align:middle line:84% Now, the point is that if they had started 00:20:41.360 --> 00:20:47.760 align:middle line:84% from the endowment, this thing, agent B would not 00:20:47.760 --> 00:20:52.740 align:middle line:84% have enough wealth to get to this Pareto-optimal allocation. 00:20:52.740 --> 00:20:55.500 align:middle line:84% Conceptually, imagining somehow the slopes 00:20:55.500 --> 00:21:01.080 align:middle line:84% were the same if we had drawn at those prices 00:21:01.080 --> 00:21:05.790 align:middle line:84% gamma line, a budget line going through the endowment, 00:21:05.790 --> 00:21:07.680 align:middle line:90% Agent B has to be further out. 00:21:07.680 --> 00:21:09.570 align:middle line:90% He has to have more wealth. 00:21:09.570 --> 00:21:11.920 align:middle line:84% And where does that wealth come from? 00:21:11.920 --> 00:21:18.280 align:middle line:84% Well, Mr. A over here gets to make the sacrifice 00:21:18.280 --> 00:21:23.950 align:middle line:84% because he has gone southwest of where he would be in terms 00:21:23.950 --> 00:21:26.410 align:middle line:84% of the budget line through the endowment if we insisted 00:21:26.410 --> 00:21:29.290 align:middle line:84% that didn't go through the endowment. 00:21:29.290 --> 00:21:32.130 align:middle line:84% So all of that is summarized on this slide. 00:21:32.130 --> 00:21:36.390 align:middle line:84% If we start out with that price ratio-- 00:21:36.390 --> 00:21:38.310 align:middle line:90% by the way, gamma 1, why? 00:21:38.310 --> 00:21:43.300 align:middle line:84% Because the slope of this thing is p1 over p2. 00:21:43.300 --> 00:21:46.420 align:middle line:84% So we might as well set p2 equal to 1 as the numerator 00:21:46.420 --> 00:21:49.360 align:middle line:84% because we're not going to get any more 00:21:49.360 --> 00:21:51.790 align:middle line:90% determination than that. 00:21:51.790 --> 00:21:54.880 align:middle line:84% So it looks weird when you write it like this, 00:21:54.880 --> 00:22:00.506 align:middle line:84% but p equal p1 equal gamma p2 equal to 1 00:22:00.506 --> 00:22:03.710 align:middle line:90% are the prices we need. 00:22:03.710 --> 00:22:08.570 align:middle line:84% Agents are going to maximize at those prices, 00:22:08.570 --> 00:22:11.720 align:middle line:84% depending on what their wealth is. 00:22:11.720 --> 00:22:17.570 align:middle line:84% But the allocation at those prices for Agent B 00:22:17.570 --> 00:22:22.520 align:middle line:84% costs more than the valuation of Agent B's endowment 00:22:22.520 --> 00:22:27.260 align:middle line:90% at the price gamma 1. 00:22:27.260 --> 00:22:30.110 align:middle line:84% So what we propose to do to Agent 00:22:30.110 --> 00:22:34.460 align:middle line:84% B is to give them the difference, 00:22:34.460 --> 00:22:37.250 align:middle line:84% just give them enough extra wealth 00:22:37.250 --> 00:22:43.080 align:middle line:84% so that Agent B can afford to buy the star bundle, which 00:22:43.080 --> 00:22:46.980 align:middle line:84% requires more than the valuation of his wealth 00:22:46.980 --> 00:22:49.770 align:middle line:90% at his previous endowment. 00:22:49.770 --> 00:22:53.520 align:middle line:84% So we call this a transfer to B. And it's positive, 00:22:53.520 --> 00:22:56.310 align:middle line:84% as you could see from that diagram. 00:22:56.310 --> 00:22:58.050 align:middle line:90% And A makes the sacrifice. 00:22:58.050 --> 00:23:02.710 align:middle line:90% 00:23:02.710 --> 00:23:04.655 align:middle line:90% There seems to be a typo here. 00:23:04.655 --> 00:23:09.510 align:middle line:90% 00:23:09.510 --> 00:23:13.670 align:middle line:84% It's probably-- anyway, you can see it. 00:23:13.670 --> 00:23:16.030 align:middle line:90% The wealth transfer is the same. 00:23:16.030 --> 00:23:20.790 align:middle line:84% It goes from B to A, and this is written oddly 00:23:20.790 --> 00:23:24.360 align:middle line:84% as if it were B's endowment and B's expenditure. 00:23:24.360 --> 00:23:27.120 align:middle line:84% But it's really referring to the point on the box 00:23:27.120 --> 00:23:31.770 align:middle line:84% here, which is simultaneously the point for B and for A. 00:23:31.770 --> 00:23:38.460 align:middle line:84% And so we just move southwest, taxing A, and giving the tax 00:23:38.460 --> 00:23:41.760 align:middle line:90% to B as a physical transfer. 00:23:41.760 --> 00:23:45.000 align:middle line:84% So we found a price equilibrium with transfers 00:23:45.000 --> 00:23:49.080 align:middle line:84% that will support that target Pareto-optimal allocation. 00:23:49.080 --> 00:23:52.090 align:middle line:84% And it's possible to do any one of them. 00:23:52.090 --> 00:23:55.660 align:middle line:84% So if you go back here, this was the private ownership Pareto 00:23:55.660 --> 00:23:56.520 align:middle line:90% optimum. 00:23:56.520 --> 00:23:59.070 align:middle line:84% There are all these other Pareto-optimal allocations. 00:23:59.070 --> 00:24:04.740 align:middle line:84% The theorem says, give me any Pareto-optimal allocation, 00:24:04.740 --> 00:24:08.550 align:middle line:84% and I can redistribute wealth in such a way 00:24:08.550 --> 00:24:11.620 align:middle line:84% as to support it as a competitive equilibrium 00:24:11.620 --> 00:24:13.020 align:middle line:90% with transfers. 00:24:13.020 --> 00:24:16.200 align:middle line:84% Who gets taxed and who gets the transfers depends on which 00:24:16.200 --> 00:24:20.170 align:middle line:84% optimum we have under consideration. 00:24:20.170 --> 00:24:24.640 align:middle line:90% So this one takes more. 00:24:24.640 --> 00:24:29.250 align:middle line:84% We're going to have to start assuming stuff for the theorem 00:24:29.250 --> 00:24:30.090 align:middle line:90% to be-- 00:24:30.090 --> 00:24:34.470 align:middle line:84% for sufficient conditions for the theorem to be true. 00:24:34.470 --> 00:24:39.150 align:middle line:84% We start with the notation of our economy. 00:24:39.150 --> 00:24:46.710 align:middle line:84% Consumption sets for the agents, utility functions, endowments, 00:24:46.710 --> 00:24:52.650 align:middle line:84% production sets, shares in ownership of profits, 00:24:52.650 --> 00:24:56.190 align:middle line:84% which is a quick review of the notation you've seen before. 00:24:56.190 --> 00:24:59.420 align:middle line:84% We're going to have to assume that these consumption 00:24:59.420 --> 00:25:04.880 align:middle line:84% sets essentially can be formed by combinations 00:25:04.880 --> 00:25:06.690 align:middle line:90% of quasi-concave functions. 00:25:06.690 --> 00:25:10.370 align:middle line:84% This first line is the most awkward. 00:25:10.370 --> 00:25:13.220 align:middle line:84% And if you wish, you could just think 00:25:13.220 --> 00:25:20.090 align:middle line:84% about these being the non-negative orthant, 00:25:20.090 --> 00:25:24.940 align:middle line:84% that any positive consumption bundle is fine. 00:25:24.940 --> 00:25:27.100 align:middle line:84% I only put the most general version 00:25:27.100 --> 00:25:31.600 align:middle line:84% because I want to emphasize you need restrictions on things. 00:25:31.600 --> 00:25:35.860 align:middle line:84% Here's a statement about the production sets. 00:25:35.860 --> 00:25:40.750 align:middle line:84% Production sets Yj admit a concave transformation 00:25:40.750 --> 00:25:42.760 align:middle line:90% function called F. 00:25:42.760 --> 00:25:44.570 align:middle line:90% What the heck does that mean? 00:25:44.570 --> 00:25:49.600 align:middle line:84% Well, it means that for any vector Y in the production set, 00:25:49.600 --> 00:25:52.870 align:middle line:84% it can be represented through some function 00:25:52.870 --> 00:25:58.480 align:middle line:84% F, which is non-negative when evaluated at that Y. 00:25:58.480 --> 00:26:01.960 align:middle line:84% An example, because this is really obscure, 00:26:01.960 --> 00:26:05.950 align:middle line:84% suppose we have an input z mapped 00:26:05.950 --> 00:26:11.070 align:middle line:84% through production function little f to get output q. 00:26:11.070 --> 00:26:13.370 align:middle line:84% And we throw away some of the output. 00:26:13.370 --> 00:26:16.880 align:middle line:84% So we're not on the frontier, we're in the interior. 00:26:16.880 --> 00:26:23.814 align:middle line:84% Then if you rewrite this as f of z minus q, it's non-negative. 00:26:23.814 --> 00:26:26.470 align:middle line:90% 00:26:26.470 --> 00:26:34.520 align:middle line:84% So any input-output combination, zq, 00:26:34.520 --> 00:26:38.030 align:middle line:84% when evaluated under F, which takes this form, 00:26:38.030 --> 00:26:41.140 align:middle line:90% is non-negative. 00:26:41.140 --> 00:26:44.260 align:middle line:90% So it really is quite innocuous. 00:26:44.260 --> 00:26:46.060 align:middle line:84% But it's very convenient just to talk 00:26:46.060 --> 00:26:50.240 align:middle line:84% about this concave transformation. 00:26:50.240 --> 00:26:53.000 align:middle line:84% I should be emphasizing again, these 00:26:53.000 --> 00:26:55.010 align:middle line:90% are sufficient assumptions. 00:26:55.010 --> 00:26:59.390 align:middle line:84% Something about consumption sets being represented 00:26:59.390 --> 00:27:05.240 align:middle line:84% by quasi-concave, this thing being a concave transformation, 00:27:05.240 --> 00:27:07.340 align:middle line:84% preferences to get down to 3 are going 00:27:07.340 --> 00:27:12.320 align:middle line:84% to be concave and locally nonsatiated. 00:27:12.320 --> 00:27:14.210 align:middle line:84% Locally nonsatiated we had before. 00:27:14.210 --> 00:27:20.340 align:middle line:84% Concave is new, which you know from one 00:27:20.340 --> 00:27:22.890 align:middle line:84% of the early lectures is equivalent to preferences 00:27:22.890 --> 00:27:27.450 align:middle line:84% being rational, convex, continuous, and nonsatiated. 00:27:27.450 --> 00:27:30.570 align:middle line:84% And finally, we have an interior point 00:27:30.570 --> 00:27:35.950 align:middle line:84% so that none of these constraints or representations 00:27:35.950 --> 00:27:41.080 align:middle line:84% are binding at 0, namely non-negative, non-negative, 00:27:41.080 --> 00:27:43.420 align:middle line:84% and even in terms of the resource 00:27:43.420 --> 00:27:47.080 align:middle line:84% constraint through existing allocation, which 00:27:47.080 --> 00:27:52.610 align:middle line:84% doesn't use up all of the endowment plus production. 00:27:52.610 --> 00:27:57.560 align:middle line:84% That's a technical interior condition. 00:27:57.560 --> 00:28:01.910 align:middle line:84% If we have all of these things for this economy, 00:28:01.910 --> 00:28:06.160 align:middle line:84% then we get the second welfare theorem 00:28:06.160 --> 00:28:08.020 align:middle line:90% stated in terms of the lambdas. 00:28:08.020 --> 00:28:12.610 align:middle line:84% For any lambda, which is a Pareto weight vector, 00:28:12.610 --> 00:28:17.650 align:middle line:84% there exists a price vector p in this L dimensional Euclidean 00:28:17.650 --> 00:28:18.730 align:middle line:90% space. 00:28:18.730 --> 00:28:23.720 align:middle line:84% And our vector of wealth, one for each 00:28:23.720 --> 00:28:28.250 align:middle line:84% of the i agents such that x star, 00:28:28.250 --> 00:28:31.670 align:middle line:84% y star, p and w star is a Walrasian equilibrium 00:28:31.670 --> 00:28:35.170 align:middle line:90% with transfers. 00:28:35.170 --> 00:28:37.750 align:middle line:84% That's a formal statement of the theorem. 00:28:37.750 --> 00:28:40.780 align:middle line:90% Now, we're going to prove it. 00:28:40.780 --> 00:28:42.850 align:middle line:84% And again, in the proof, it's kind of easier 00:28:42.850 --> 00:28:47.010 align:middle line:84% to assume that those lambdas are strictly positive. 00:28:47.010 --> 00:28:51.020 align:middle line:84% And it will generalize to some of them being 0. 00:28:51.020 --> 00:28:53.010 align:middle line:90% But it just gets in the way. 00:28:53.010 --> 00:28:56.840 align:middle line:84% So let's make it a little bit easier on ourselves. 00:28:56.840 --> 00:29:01.610 align:middle line:84% The proof is constructive and a bit tedious, 00:29:01.610 --> 00:29:04.010 align:middle line:84% but it is straightforward enough. 00:29:04.010 --> 00:29:05.570 align:middle line:90% I'll try to guide you. 00:29:05.570 --> 00:29:11.090 align:middle line:84% Step 1, characterize the Pareto-optimal allocation. 00:29:11.090 --> 00:29:13.820 align:middle line:84% That's the given that we want to achieve 00:29:13.820 --> 00:29:16.970 align:middle line:84% with a competitive equilibrium of transfers. 00:29:16.970 --> 00:29:22.190 align:middle line:84% 2, characterize the conditions for a competitive equilibrium 00:29:22.190 --> 00:29:23.760 align:middle line:90% with transfer. 00:29:23.760 --> 00:29:28.130 align:middle line:84% And then, 3, kind of match up those conditions 00:29:28.130 --> 00:29:31.070 align:middle line:84% and show that if we have the conditions being satisfied 00:29:31.070 --> 00:29:32.960 align:middle line:84% for the optimal allocation, they're 00:29:32.960 --> 00:29:36.920 align:middle line:84% going to be satisfied for the competitive allocation, 00:29:36.920 --> 00:29:39.330 align:middle line:90% if we choose properly. 00:29:39.330 --> 00:29:45.280 align:middle line:84% This was the Pareto problem, written out now 00:29:45.280 --> 00:29:48.190 align:middle line:90% for the fully specified economy. 00:29:48.190 --> 00:29:51.550 align:middle line:84% To find a Pareto-optimal allocation, 00:29:51.550 --> 00:29:55.430 align:middle line:84% we're going to find the solution to maximizing 00:29:55.430 --> 00:29:58.700 align:middle line:84% a lambda-weighted sum of utilities, subject to resource 00:29:58.700 --> 00:30:01.400 align:middle line:84% feasibility and production constraints. 00:30:01.400 --> 00:30:04.520 align:middle line:90% Max, subject to 15 and 16. 00:30:04.520 --> 00:30:08.510 align:middle line:84% You've seen this many times before, in like six 00:30:08.510 --> 00:30:10.670 align:middle line:90% or seven applications already. 00:30:10.670 --> 00:30:12.270 align:middle line:90% This is a full statement. 00:30:12.270 --> 00:30:14.570 align:middle line:90% So the so-called Pareto problem. 00:30:14.570 --> 00:30:16.640 align:middle line:84% Sometimes I call it the math problem. 00:30:16.640 --> 00:30:19.460 align:middle line:84% And you may remember that solutions to this problem 00:30:19.460 --> 00:30:23.180 align:middle line:84% generate Pareto-optimal allocations 00:30:23.180 --> 00:30:26.570 align:middle line:84% and that any Pareto-optimal allocation can be achieved 00:30:26.570 --> 00:30:28.080 align:middle line:90% as a solution to this problem. 00:30:28.080 --> 00:30:32.030 align:middle line:84% So we're not losing anything by restricting our attention 00:30:32.030 --> 00:30:34.070 align:middle line:90% to these Pareto problems. 00:30:34.070 --> 00:30:37.190 align:middle line:84% Second step is to use the Lagrangian. 00:30:37.190 --> 00:30:40.400 align:middle line:84% And the reason I was emphasizing on the previous slide 00:30:40.400 --> 00:30:43.820 align:middle line:84% all the concavity and convexity is those, 00:30:43.820 --> 00:30:47.000 align:middle line:84% it's not just getting the necessary conditions 00:30:47.000 --> 00:30:50.210 align:middle line:84% for an optimum from the Lagrangian method, 00:30:50.210 --> 00:30:53.180 align:middle line:84% those conditions are also sufficient when 00:30:53.180 --> 00:30:55.550 align:middle line:90% we have this kind of concavity. 00:30:55.550 --> 00:30:57.600 align:middle line:90% I'll remind you momentarily. 00:30:57.600 --> 00:31:00.530 align:middle line:84% So we've got production constraints here. 00:31:00.530 --> 00:31:02.670 align:middle line:90% We've got resource constraints. 00:31:02.670 --> 00:31:05.360 align:middle line:84% So we're looking for an allocation consumption 00:31:05.360 --> 00:31:08.240 align:middle line:84% in the consumption set, positivity 00:31:08.240 --> 00:31:12.980 align:middle line:84% from those concave transformations, and resource 00:31:12.980 --> 00:31:14.210 align:middle line:90% clearing. 00:31:14.210 --> 00:31:16.340 align:middle line:90% Here's the review slide. 00:31:16.340 --> 00:31:18.440 align:middle line:84% We did this, the first time it came up 00:31:18.440 --> 00:31:21.920 align:middle line:84% was consumer optimization subject to a budget. 00:31:21.920 --> 00:31:25.100 align:middle line:84% And I said, oh, now we're going to learn a cool tool, the tool 00:31:25.100 --> 00:31:26.880 align:middle line:90% of constrained optimization. 00:31:26.880 --> 00:31:29.630 align:middle line:84% So we maximize the function subject to a series 00:31:29.630 --> 00:31:31.430 align:middle line:90% of constraints. 00:31:31.430 --> 00:31:35.720 align:middle line:84% We form the Lagrangian, took derivatives, 00:31:35.720 --> 00:31:39.800 align:middle line:84% and looked at the additional first-order conditions, 00:31:39.800 --> 00:31:42.410 align:middle line:84% which I'm about to show you again 00:31:42.410 --> 00:31:44.430 align:middle line:90% in the context of the problem. 00:31:44.430 --> 00:31:46.470 align:middle line:90% So that was a review slide. 00:31:46.470 --> 00:31:48.630 align:middle line:90% This is what we want to solve. 00:31:48.630 --> 00:31:52.350 align:middle line:84% And the Lagrangian for that is this problem. 00:31:52.350 --> 00:31:56.670 align:middle line:84% So we want to solve to maximize a lambda weighted sum 00:31:56.670 --> 00:32:02.880 align:middle line:84% of utilities by choice of an allocation Xi vector, 00:32:02.880 --> 00:32:09.390 align:middle line:84% with now we start writing out the L resource constraints. 00:32:09.390 --> 00:32:13.860 align:middle line:84% But we put a Lagrange multiplier gamma sub l in front of them, 00:32:13.860 --> 00:32:17.160 align:middle line:84% as we were instructed to do from the algorithm. 00:32:17.160 --> 00:32:22.820 align:middle line:84% We also put on to the Lagrangian these positivity-- 00:32:22.820 --> 00:32:28.520 align:middle line:84% these F's that represent the production function frontier. 00:32:28.520 --> 00:32:30.260 align:middle line:84% There are j of those, and we come up 00:32:30.260 --> 00:32:35.710 align:middle line:84% with some notation for the Lagrange multiplier. 00:32:35.710 --> 00:32:38.650 align:middle line:84% And then, when we differentiate this whole system, 00:32:38.650 --> 00:32:43.840 align:middle line:84% say with respect to the L-th good for the I-th household, 00:32:43.840 --> 00:32:46.210 align:middle line:84% we want to set the Lagrangian derivative 00:32:46.210 --> 00:32:52.150 align:middle line:84% to 0, which is the same thing as lambda i U i prime add Xi star. 00:32:52.150 --> 00:32:55.000 align:middle line:84% And where else does that Xil enter? 00:32:55.000 --> 00:32:57.250 align:middle line:84% It enters over here in the resource constraint. 00:32:57.250 --> 00:33:00.040 align:middle line:90% So it picks up a gamma. 00:33:00.040 --> 00:33:02.650 align:middle line:84% And now, I start putting stars on the solution 00:33:02.650 --> 00:33:06.430 align:middle line:84% because the solution will satisfy 00:33:06.430 --> 00:33:07.820 align:middle line:90% these first-order conditions. 00:33:07.820 --> 00:33:10.510 align:middle line:84% So there are special allocations and 00:33:10.510 --> 00:33:14.940 align:middle line:84% special Lagrange multipliers because they are solutions. 00:33:14.940 --> 00:33:20.610 align:middle line:84% Likewise, for the L-th good, for the J-th firm, 00:33:20.610 --> 00:33:23.280 align:middle line:84% set the derivative of the Lagrangian with 0. 00:33:23.280 --> 00:33:24.750 align:middle line:90% Where does that stuff happen? 00:33:24.750 --> 00:33:26.890 align:middle line:90% It happens here. 00:33:26.890 --> 00:33:30.150 align:middle line:84% So there's a picking up a gamma l. 00:33:30.150 --> 00:33:35.600 align:middle line:84% And over here, should we pick up a derivative. 00:33:35.600 --> 00:33:40.270 align:middle line:84% So this would be positive, positive equal to 0. 00:33:40.270 --> 00:33:42.460 align:middle line:84% So one of those terms has to be negative. 00:33:42.460 --> 00:33:46.600 align:middle line:84% We subtract it off, and we get this expression. 00:33:46.600 --> 00:33:51.280 align:middle line:84% The gamma l is the derivative of the function 00:33:51.280 --> 00:33:53.290 align:middle line:90% F with respect to yl. 00:33:53.290 --> 00:33:56.620 align:middle line:84% This is for the J-th firm, and we have this star 00:33:56.620 --> 00:34:00.800 align:middle line:90% on the Lagrange multiplier. 00:34:00.800 --> 00:34:04.030 align:middle line:84% So this equation 10 is a set of conditions. 00:34:04.030 --> 00:34:07.360 align:middle line:84% We need something, a technical condition 00:34:07.360 --> 00:34:11.739 align:middle line:84% that there can be something that these things are all-- 00:34:11.739 --> 00:34:12.429 align:middle line:90% sorry. 00:34:12.429 --> 00:34:17.280 align:middle line:84% Not technical, that the constraints are all satisfied, 00:34:17.280 --> 00:34:22.100 align:middle line:84% and the Lagrange multipliers are not non-negative. 00:34:22.100 --> 00:34:25.600 align:middle line:84% And then finally, this complementary slackness 00:34:25.600 --> 00:34:28.780 align:middle line:84% condition that the Lagrange multipliers 00:34:28.780 --> 00:34:34.780 align:middle line:84% times the respective constraints equals 0. 00:34:34.780 --> 00:34:37.420 align:middle line:84% Probably a lot to review, but it is 00:34:37.420 --> 00:34:40.510 align:middle line:84% in all this notation of the environment exactly what you've 00:34:40.510 --> 00:34:42.670 align:middle line:90% learned before. 00:34:42.670 --> 00:34:56.290 align:middle line:84% The point is that equations 10 and these others 00:34:56.290 --> 00:35:03.930 align:middle line:84% represent a solution to the maximum Pareto problem. 00:35:03.930 --> 00:35:06.150 align:middle line:84% And they're not just necessary, they're 00:35:06.150 --> 00:35:09.520 align:middle line:84% sufficient because of all the concavity. 00:35:09.520 --> 00:35:13.660 align:middle line:84% And in particular, these utility functions 00:35:13.660 --> 00:35:16.080 align:middle line:90% are strictly increasing. 00:35:16.080 --> 00:35:20.500 align:middle line:84% And I said earlier, everybody has a positive lambda. 00:35:20.500 --> 00:35:24.910 align:middle line:84% It's enough that at least one person has a positive lambda, 00:35:24.910 --> 00:35:29.770 align:middle line:84% then we're going to have to have the gammas being positive y. 00:35:29.770 --> 00:35:33.250 align:middle line:84% That's this thing because here's the marginal utility, which 00:35:33.250 --> 00:35:37.250 align:middle line:84% is positive, a lambda, which is positive. 00:35:37.250 --> 00:35:39.040 align:middle line:84% So if the left-hand side is positive, 00:35:39.040 --> 00:35:40.730 align:middle line:90% the right-hand side is positive. 00:35:40.730 --> 00:35:46.040 align:middle line:84% So that's where the positive gammas are coming from. 00:35:46.040 --> 00:35:48.550 align:middle line:84% So we know a little bit more about the solution. 00:35:48.550 --> 00:35:49.870 align:middle line:90% It's positive prices. 00:35:49.870 --> 00:35:54.760 align:middle line:84% And from this complementary slackness condition, 00:35:54.760 --> 00:35:58.090 align:middle line:84% if the prices are all positive, and this whole thing 00:35:58.090 --> 00:36:01.780 align:middle line:84% has to be 0, then the interior thing in brackets has to be 0. 00:36:01.780 --> 00:36:04.060 align:middle line:84% So the resource constraints are satisfied 00:36:04.060 --> 00:36:07.060 align:middle line:90% at equality, as is natural. 00:36:07.060 --> 00:36:09.250 align:middle line:90% So let's summarize. 00:36:09.250 --> 00:36:12.610 align:middle line:84% We've characterized the Pareto-optimal allocations, 00:36:12.610 --> 00:36:14.530 align:middle line:90% x star, y star. 00:36:14.530 --> 00:36:17.740 align:middle line:84% Also denoted x star, y star at lambda 00:36:17.740 --> 00:36:21.250 align:middle line:84% because we were given a particular Pareto-optimal 00:36:21.250 --> 00:36:24.370 align:middle line:84% allocation, hence a particular lambda 00:36:24.370 --> 00:36:27.910 align:middle line:84% weighted sum was used in the objective function. 00:36:27.910 --> 00:36:34.850 align:middle line:84% And it's the solution to 14, which is that Pareto problem. 00:36:34.850 --> 00:36:36.250 align:middle line:90% So it's Pareto optimal. 00:36:36.250 --> 00:36:40.860 align:middle line:84% It's feasible, and there are these gammas, 00:36:40.860 --> 00:36:45.180 align:middle line:84% which are strictly positive for every good l. 00:36:45.180 --> 00:36:49.260 align:middle line:84% And these Lagrange multiplies on the production 00:36:49.260 --> 00:36:55.490 align:middle line:84% sets such that x star, y star, gamma star, and phi star 00:36:55.490 --> 00:36:57.890 align:middle line:90% satisfy 19 and 21. 00:36:57.890 --> 00:37:04.790 align:middle line:84% 19 and 21 were, yeah, something happened in the process. 00:37:04.790 --> 00:37:05.425 align:middle line:90% This is 10. 00:37:05.425 --> 00:37:08.540 align:middle line:90% 00:37:08.540 --> 00:37:14.960 align:middle line:84% And 21 is the feasibility constraint. 00:37:14.960 --> 00:37:15.460 align:middle line:90% No. 00:37:15.460 --> 00:37:16.150 align:middle line:90% It's right here. 00:37:16.150 --> 00:37:16.650 align:middle line:90% 21. 00:37:16.650 --> 00:37:19.500 align:middle line:90% 00:37:19.500 --> 00:37:23.330 align:middle line:84% So much for characterizing the Pareto-optimal allocation. 00:37:23.330 --> 00:37:26.780 align:middle line:84% Let's look at the second part, characterizing the equilibrium 00:37:26.780 --> 00:37:28.910 align:middle line:90% with transfers, just separately. 00:37:28.910 --> 00:37:30.680 align:middle line:84% Forget the first part for a minute. 00:37:30.680 --> 00:37:34.790 align:middle line:84% We have a given while raising equilibrium with transfers, 00:37:34.790 --> 00:37:36.230 align:middle line:90% meaning what? 00:37:36.230 --> 00:37:39.830 align:middle line:84% Is x star, y star, p and w star which 00:37:39.830 --> 00:37:43.280 align:middle line:84% must be consistent with consumer optimization, profit 00:37:43.280 --> 00:37:47.120 align:middle line:84% maximization, resource constraints being satisfied. 00:37:47.120 --> 00:37:50.360 align:middle line:84% And since it's an equilibrium with transfers, not necessarily 00:37:50.360 --> 00:37:52.940 align:middle line:84% private ownership, we have to make sure 00:37:52.940 --> 00:37:55.340 align:middle line:84% that the distribution of wealths are feasible, 00:37:55.340 --> 00:37:57.680 align:middle line:84% that the wealths add up to the valuation 00:37:57.680 --> 00:38:00.050 align:middle line:90% of the endowment and profits. 00:38:00.050 --> 00:38:02.690 align:middle line:84% So let's just take this one at a time. 00:38:02.690 --> 00:38:07.670 align:middle line:84% Consumer op, firms op, and start look 00:38:07.670 --> 00:38:10.130 align:middle line:90% at the first-order conditions. 00:38:10.130 --> 00:38:13.950 align:middle line:84% Now, the third part of this is putting these pieces together. 00:38:13.950 --> 00:38:16.010 align:middle line:84% But at this point, we're lost, right? 00:38:16.010 --> 00:38:17.880 align:middle line:84% We don't remember where we started. 00:38:17.880 --> 00:38:20.630 align:middle line:84% We started from the second welfare theorem. 00:38:20.630 --> 00:38:25.130 align:middle line:84% So we were given a Pareto-optimal allocation. 00:38:25.130 --> 00:38:28.070 align:middle line:84% And we want to show that it can be achieved as a price 00:38:28.070 --> 00:38:29.600 align:middle line:90% equilibrium with transfers. 00:38:29.600 --> 00:38:33.650 align:middle line:84% But we want to find the components x star, y star, p 00:38:33.650 --> 00:38:37.070 align:middle line:84% and w star of the competitive equilibrium 00:38:37.070 --> 00:38:39.380 align:middle line:84% with transfers that's going to give us 00:38:39.380 --> 00:38:43.530 align:middle line:84% back exactly the Pareto-optimal allocations we started from. 00:38:43.530 --> 00:38:46.280 align:middle line:84% So let's start with the households. 00:38:46.280 --> 00:38:50.870 align:middle line:84% We have a set of conditions for the Pareto-optimal allocation, 00:38:50.870 --> 00:38:54.380 align:middle line:84% and which are given because we started with the Pareto 00:38:54.380 --> 00:38:57.150 align:middle line:84% optimal-- now, you're not used to seeing it like this. 00:38:57.150 --> 00:39:00.680 align:middle line:84% We're usually writing lambda i times the marginal utility is 00:39:00.680 --> 00:39:02.030 align:middle line:90% equal to gamma. 00:39:02.030 --> 00:39:03.980 align:middle line:90% Here it is. 00:39:03.980 --> 00:39:07.910 align:middle line:84% All I did was divide by lambda i and put it 00:39:07.910 --> 00:39:09.170 align:middle line:90% on the right-hand side. 00:39:09.170 --> 00:39:12.170 align:middle line:84% That was part of the necessary and sufficient conditions 00:39:12.170 --> 00:39:13.760 align:middle line:90% for the Pareto problem. 00:39:13.760 --> 00:39:14.720 align:middle line:90% That's this. 00:39:14.720 --> 00:39:16.250 align:middle line:90% So we're given this. 00:39:16.250 --> 00:39:20.570 align:middle line:84% And we want to find Lagrange multipliers mu and prices 00:39:20.570 --> 00:39:24.320 align:middle line:84% p so as to satisfy this optimizing first-order 00:39:24.320 --> 00:39:27.980 align:middle line:84% condition per household i in the competitive equilibrium 00:39:27.980 --> 00:39:28.940 align:middle line:90% with transfers. 00:39:28.940 --> 00:39:31.400 align:middle line:90% Now, what would you guess? 00:39:31.400 --> 00:39:34.780 align:middle line:84% Well, we want-- this is true, and we want this. 00:39:34.780 --> 00:39:38.560 align:middle line:84% So let's guess that the price that we want is 00:39:38.560 --> 00:39:43.110 align:middle line:84% the gamma, Lagrange multiplier on the resource constraint 00:39:43.110 --> 00:39:45.010 align:middle line:90% for good l. 00:39:45.010 --> 00:39:47.380 align:middle line:90% Pl equal gamma l. 00:39:47.380 --> 00:39:50.440 align:middle line:84% And let's guess that the mu that we want to find 00:39:50.440 --> 00:39:52.720 align:middle line:90% is simply 1 over lambda i. 00:39:52.720 --> 00:39:55.750 align:middle line:84% That's down here summarized at the bottom. 00:39:55.750 --> 00:39:59.290 align:middle line:84% And the final guess would be that the wealth allocation 00:39:59.290 --> 00:40:02.050 align:middle line:84% that we need in the Pareto-optimal allocation 00:40:02.050 --> 00:40:05.650 align:middle line:84% is just the valuation of expenditures 00:40:05.650 --> 00:40:08.650 align:middle line:84% at that Pareto-optimal allocation. 00:40:08.650 --> 00:40:11.380 align:middle line:90% I use my words carefully here. 00:40:11.380 --> 00:40:15.580 align:middle line:84% The valuation, though, is at the shadow prices gamma l. 00:40:15.580 --> 00:40:18.700 align:middle line:84% If you go back to the Pareto problem, 00:40:18.700 --> 00:40:22.630 align:middle line:84% these gammas were arbitrary Lagrange multipliers 00:40:22.630 --> 00:40:25.620 align:middle line:90% on the resource constraint. 00:40:25.620 --> 00:40:28.250 align:middle line:84% And we went to the math of all the first-order conditions. 00:40:28.250 --> 00:40:31.940 align:middle line:84% But really, what it is, is how much 00:40:31.940 --> 00:40:38.620 align:middle line:84% the solution that shadow price reflects the incremental gain 00:40:38.620 --> 00:40:44.410 align:middle line:84% in the objective function to weakening the resource 00:40:44.410 --> 00:40:48.380 align:middle line:90% constraint by a teeny amount. 00:40:48.380 --> 00:40:50.350 align:middle line:90% So it is like a shadow price. 00:40:50.350 --> 00:40:53.690 align:middle line:84% We often call the Lagrange multiplier shadow prices. 00:40:53.690 --> 00:40:56.800 align:middle line:84% It's really marginal utility pricing 00:40:56.800 --> 00:41:00.520 align:middle line:84% because we're incrementing the weighted sum of utilities 00:41:00.520 --> 00:41:07.400 align:middle line:84% a bit, hence like a derivative, as we weaken the constraint. 00:41:07.400 --> 00:41:12.830 align:middle line:84% So it's natural really that these Lagrange multipliers l, 00:41:12.830 --> 00:41:15.770 align:middle line:84% for good l, should be shadow prices. 00:41:15.770 --> 00:41:19.490 align:middle line:84% And the actual competitive equilibrium prices 00:41:19.490 --> 00:41:22.430 align:middle line:84% in the corresponding while raising in equilibrium 00:41:22.430 --> 00:41:24.560 align:middle line:90% with transfers. 00:41:24.560 --> 00:41:27.650 align:middle line:84% This thing down here, very simply written, 00:41:27.650 --> 00:41:31.790 align:middle line:90% is that mu i is 1 over lambda i. 00:41:31.790 --> 00:41:32.750 align:middle line:90% Oh, who cares? 00:41:32.750 --> 00:41:34.130 align:middle line:90% No, no, no, no, no, no. 00:41:34.130 --> 00:41:35.930 align:middle line:90% Don't think of it that way. 00:41:35.930 --> 00:41:38.210 align:middle line:84% Think of those beautiful graphs of Kansas 00:41:38.210 --> 00:41:40.310 align:middle line:90% and so on at the beginning. 00:41:40.310 --> 00:41:41.300 align:middle line:90% OK? 00:41:41.300 --> 00:41:44.870 align:middle line:84% So what this is saying is mu i, which 00:41:44.870 --> 00:41:46.790 align:middle line:84% is the marginal utility of income, 00:41:46.790 --> 00:41:53.408 align:middle line:84% the marginal utility of wealth in the target competitive-- 00:41:53.408 --> 00:41:55.200 align:middle line:84% we're raising an equilibrium with transfers 00:41:55.200 --> 00:41:56.580 align:middle line:90% that we're trying to find. 00:41:56.580 --> 00:42:03.070 align:middle line:84% The marginal utility of wealth at that location 00:42:03.070 --> 00:42:07.670 align:middle line:84% is inversely related to the lambda weights in the Pareto 00:42:07.670 --> 00:42:08.170 align:middle line:90% problem. 00:42:08.170 --> 00:42:11.080 align:middle line:90% 00:42:11.080 --> 00:42:13.990 align:middle line:84% So the higher the lambda weights, 00:42:13.990 --> 00:42:18.150 align:middle line:84% the lower would be the marginal utility of wealth. 00:42:18.150 --> 00:42:21.070 align:middle line:90% 00:42:21.070 --> 00:42:24.640 align:middle line:84% Remember the first two slides, I told you 00:42:24.640 --> 00:42:27.670 align:middle line:84% that the gradient in the consumption profile 00:42:27.670 --> 00:42:30.340 align:middle line:84% had something to do with the lambda weights. 00:42:30.340 --> 00:42:32.140 align:middle line:84% And we could look in the data to see 00:42:32.140 --> 00:42:33.535 align:middle line:90% if it was related to wealth. 00:42:33.535 --> 00:42:36.290 align:middle line:90% 00:42:36.290 --> 00:42:38.140 align:middle line:84% So this may look upside down, but that's 00:42:38.140 --> 00:42:41.620 align:middle line:84% only because mu i is the marginal utility of wealth. 00:42:41.620 --> 00:42:45.410 align:middle line:84% As wealth goes up, marginal utility goes down. 00:42:45.410 --> 00:42:47.380 align:middle line:84% So as lambda goes up, and they're 00:42:47.380 --> 00:42:50.740 align:middle line:84% more favored in the Pareto problem, 00:42:50.740 --> 00:42:53.350 align:middle line:84% the marginal utility of wealth is going down. 00:42:53.350 --> 00:42:55.370 align:middle line:90% And how does that happen? 00:42:55.370 --> 00:42:59.700 align:middle line:84% It happens because they get more wealth. 00:42:59.700 --> 00:43:06.060 align:middle line:84% So there's kind of a 1 to 1 mapping, a monotone mapping, 00:43:06.060 --> 00:43:08.350 align:middle line:84% between the lambdas in the Pareto problem 00:43:08.350 --> 00:43:12.890 align:middle line:84% and the wealth that corresponds to the competitive equilibrium 00:43:12.890 --> 00:43:14.960 align:middle line:90% with transfers. 00:43:14.960 --> 00:43:18.870 align:middle line:84% And here it is falling out in a very elegant way. 00:43:18.870 --> 00:43:22.300 align:middle line:90% 00:43:22.300 --> 00:43:25.770 align:middle line:84% And this other thing at the bottom, that Wi star, 00:43:25.770 --> 00:43:27.870 align:middle line:84% is the valuation of expenditures. 00:43:27.870 --> 00:43:33.450 align:middle line:84% Again, that's pretty much what we were doing in the box. 00:43:33.450 --> 00:43:36.090 align:middle line:84% We took a target allocation here, 00:43:36.090 --> 00:43:39.120 align:middle line:84% that it wasn't a competitive equilibrium 00:43:39.120 --> 00:43:40.740 align:middle line:90% with private ownership. 00:43:40.740 --> 00:43:45.120 align:middle line:84% But we could tax A and transfer to B 00:43:45.120 --> 00:43:51.000 align:middle line:84% so these stars here represented the target Pareto-optimal 00:43:51.000 --> 00:43:52.450 align:middle line:90% allocation. 00:43:52.450 --> 00:43:57.030 align:middle line:84% And we're valuating those expenditures 00:43:57.030 --> 00:44:02.080 align:middle line:84% at the prices corresponding to the budget line. 00:44:02.080 --> 00:44:04.530 align:middle line:84% So you can start them out with wealth 00:44:04.530 --> 00:44:09.280 align:middle line:84% being at carefully chosen prices, 00:44:09.280 --> 00:44:15.430 align:middle line:84% the valuation of the star Pareto-optimal allocation. 00:44:15.430 --> 00:44:18.430 align:middle line:84% And that's this part here at the bottom, 00:44:18.430 --> 00:44:20.230 align:middle line:84% that the wealth we give to Agent I 00:44:20.230 --> 00:44:24.790 align:middle line:84% is just the valuation of the Pareto-optimal allocation. 00:44:24.790 --> 00:44:28.450 align:middle line:84% Now also is utility maximizing allocation 00:44:28.450 --> 00:44:32.270 align:middle line:90% at the prices gamma, gamma star. 00:44:32.270 --> 00:44:35.390 align:middle line:90% We did all that for consumption. 00:44:35.390 --> 00:44:38.900 align:middle line:84% Something else happens automatically along the way, 00:44:38.900 --> 00:44:43.470 align:middle line:84% namely when you look at those first-order conditions, 00:44:43.470 --> 00:44:46.290 align:middle line:84% we can define the marginal rate of substitution 00:44:46.290 --> 00:44:52.550 align:middle line:84% of good l for good k for agent I. The lambdas cancel out, 00:44:52.550 --> 00:44:53.050 align:middle line:90% right? 00:44:53.050 --> 00:44:56.980 align:middle line:84% It's lambda I for a household i for any good l. 00:44:56.980 --> 00:44:58.990 align:middle line:84% So if we take the derivative of utility 00:44:58.990 --> 00:45:04.900 align:middle line:84% with respect to the two goods, in this case l and k, 00:45:04.900 --> 00:45:08.330 align:middle line:84% the lambda is in the numerator and denominator it cancels out. 00:45:08.330 --> 00:45:13.860 align:middle line:84% So this is the marginal rate of substitution corresponding 00:45:13.860 --> 00:45:16.350 align:middle line:84% to these optimizing first-order conditions for household 00:45:16.350 --> 00:45:19.065 align:middle line:90% I's, obviously equal to-- 00:45:19.065 --> 00:45:23.240 align:middle line:90% 00:45:23.240 --> 00:45:25.730 align:middle line:84% the I's, as I said, are canceling out, 00:45:25.730 --> 00:45:33.510 align:middle line:84% equal to the ratio of prices, back to some other household h. 00:45:33.510 --> 00:45:37.440 align:middle line:84% So this is the math of what you already saw in the picture 00:45:37.440 --> 00:45:40.890 align:middle line:84% that I've shown you three times, the marginal rates 00:45:40.890 --> 00:45:43.440 align:middle line:84% of substitution are equated across households 00:45:43.440 --> 00:45:46.350 align:middle line:84% and equal to the common price ratio. 00:45:46.350 --> 00:45:49.320 align:middle line:84% All of that coming from the first-order conditions 00:45:49.320 --> 00:45:52.980 align:middle line:90% that we've now formally derived. 00:45:52.980 --> 00:45:55.810 align:middle line:90% One more step, firms. 00:45:55.810 --> 00:45:59.200 align:middle line:90% So we have profit maximization. 00:45:59.200 --> 00:46:01.990 align:middle line:90% Oh, are you lost again? 00:46:01.990 --> 00:46:05.950 align:middle line:84% I say that a bit tongue in cheek because everything 00:46:05.950 --> 00:46:07.553 align:middle line:84% is equivalent to everything else. 00:46:07.553 --> 00:46:08.970 align:middle line:84% And it's easy to get turned around 00:46:08.970 --> 00:46:11.240 align:middle line:90% and forget our starting point. 00:46:11.240 --> 00:46:14.530 align:middle line:84% So we started with a Pareto-optimal allocation. 00:46:14.530 --> 00:46:16.600 align:middle line:84% We looked at the first-order conditions, which 00:46:16.600 --> 00:46:19.040 align:middle line:90% are necessary and sufficient. 00:46:19.040 --> 00:46:24.140 align:middle line:90% And we got this thing. 00:46:24.140 --> 00:46:27.190 align:middle line:90% So this is true. 00:46:27.190 --> 00:46:29.500 align:middle line:84% Now, we want to find the conditions for profit 00:46:29.500 --> 00:46:33.340 align:middle line:84% maximization in a competitive equilibrium with transfers. 00:46:33.340 --> 00:46:38.800 align:middle line:84% So we want this equation to be satisfied. 00:46:38.800 --> 00:46:43.510 align:middle line:84% As long as we can choose the gammas and the piece 00:46:43.510 --> 00:46:46.840 align:middle line:84% appropriately, we'll be done, and it's easy to choose them. 00:46:46.840 --> 00:46:49.570 align:middle line:84% P, we already pinned down already. 00:46:49.570 --> 00:46:50.530 align:middle line:90% p sub l. 00:46:50.530 --> 00:46:53.770 align:middle line:84% The price of good l is the shadow price 00:46:53.770 --> 00:46:56.590 align:middle line:90% derived previously gamma l. 00:46:56.590 --> 00:46:59.980 align:middle line:84% And it just remains to figure out what this delta l is. 00:46:59.980 --> 00:47:03.790 align:middle line:84% Can we find that Lagrange multiplier somehow? 00:47:03.790 --> 00:47:04.390 align:middle line:90% Searching? 00:47:04.390 --> 00:47:04.900 align:middle line:90% Sure. 00:47:04.900 --> 00:47:08.200 align:middle line:84% Let's let it equal to the associated Lagrange 00:47:08.200 --> 00:47:12.230 align:middle line:84% multiplier and the solution to the Pareto problem. 00:47:12.230 --> 00:47:14.660 align:middle line:90% And again, because all these-- 00:47:14.660 --> 00:47:18.740 align:middle line:84% if you can satisfy this because it's necessary and sufficient, 00:47:18.740 --> 00:47:21.350 align:middle line:90% then we have profit max. 00:47:21.350 --> 00:47:27.560 align:middle line:84% So we just pick the delta and the p to be what-- 00:47:27.560 --> 00:47:30.560 align:middle line:84% obviously, it would be true so that we 00:47:30.560 --> 00:47:33.730 align:middle line:90% have profit maximization. 00:47:33.730 --> 00:47:35.010 align:middle line:90% So we're almost there. 00:47:35.010 --> 00:47:37.680 align:middle line:90% We have a solution. 00:47:37.680 --> 00:47:40.090 align:middle line:84% We started with a target Pareto-optimal allocation. 00:47:40.090 --> 00:47:42.960 align:middle line:84% We have a solution to the household problem 00:47:42.960 --> 00:47:45.750 align:middle line:84% in the competitive equilibrium with transfers. 00:47:45.750 --> 00:47:48.200 align:middle line:84% We have a solution to the first problem. 00:47:48.200 --> 00:47:49.950 align:middle line:84% So what do we know about feasibility, 00:47:49.950 --> 00:47:55.080 align:middle line:84% and what do we know about the allocation of wealth? 00:47:55.080 --> 00:47:59.520 align:middle line:84% So we know what householdized wealth has to be, 00:47:59.520 --> 00:48:00.510 align:middle line:90% as I've been saying. 00:48:00.510 --> 00:48:06.990 align:middle line:84% It's just the valuation of the assignment, the Xi star 00:48:06.990 --> 00:48:12.380 align:middle line:84% assignment in the Pareto problem at the prices gamma equal to p. 00:48:12.380 --> 00:48:14.150 align:middle line:84% So that's the wealth of household I. 00:48:14.150 --> 00:48:16.235 align:middle line:84% Let's sum them up and make sure it's feasible. 00:48:16.235 --> 00:48:19.040 align:middle line:90% 00:48:19.040 --> 00:48:25.990 align:middle line:84% So we sum up the valuation of the target at prices p. 00:48:25.990 --> 00:48:27.910 align:middle line:90% But now we do this trick. 00:48:27.910 --> 00:48:29.320 align:middle line:90% Bring p outside. 00:48:29.320 --> 00:48:30.940 align:middle line:90% It's common. 00:48:30.940 --> 00:48:35.380 align:middle line:84% I suppose it's strictly positive in all the components too, 00:48:35.380 --> 00:48:37.270 align:middle line:90% just to make it easy. 00:48:37.270 --> 00:48:41.290 align:middle line:84% Then, this summation of-- because of we 00:48:41.290 --> 00:48:44.050 align:middle line:84% started with a Pareto optimum, so it 00:48:44.050 --> 00:48:46.520 align:middle line:90% had to be resource feasible. 00:48:46.520 --> 00:48:51.950 align:middle line:84% So now, we use that fact, that the summation over i Xi star 00:48:51.950 --> 00:48:55.400 align:middle line:84% must be equal to the summation over i 00:48:55.400 --> 00:49:00.330 align:middle line:90% of the endowment plus output. 00:49:00.330 --> 00:49:03.200 align:middle line:90% So demand is equal to supply. 00:49:03.200 --> 00:49:06.290 align:middle line:84% Demand is equal to endowments plus production. 00:49:06.290 --> 00:49:09.220 align:middle line:90% 00:49:09.220 --> 00:49:16.640 align:middle line:84% And then, you can rewrite the p dot yj's 00:49:16.640 --> 00:49:20.110 align:middle line:84% to be consistent with the claim on those profits 00:49:20.110 --> 00:49:24.770 align:middle line:84% under these ownership shares, theta ij. 00:49:24.770 --> 00:49:27.680 align:middle line:84% It probably would have been clearer if p were also 00:49:27.680 --> 00:49:29.700 align:middle line:90% still multiplying each part. 00:49:29.700 --> 00:49:30.380 align:middle line:90% But it's there. 00:49:30.380 --> 00:49:33.140 align:middle line:84% It's just outside the parentheses. 00:49:33.140 --> 00:49:35.930 align:middle line:90% And then, we group-- 00:49:35.930 --> 00:49:38.750 align:middle line:84% finally, we do put the p's on the inside. 00:49:38.750 --> 00:49:42.420 align:middle line:84% And we have what we want because this 00:49:42.420 --> 00:49:44.940 align:middle line:90% is total wealth in the economy. 00:49:44.940 --> 00:49:46.110 align:middle line:90% It's the total value. 00:49:46.110 --> 00:49:48.390 align:middle line:84% It's the valuation of the aggregate endowment 00:49:48.390 --> 00:49:51.140 align:middle line:90% plus profits. 00:49:51.140 --> 00:49:56.540 align:middle line:84% And it adds up to the sum of the wealths 00:49:56.540 --> 00:50:01.460 align:middle line:84% that we need, according to the solutions to the consumer 00:50:01.460 --> 00:50:03.180 align:middle line:90% maximization problem. 00:50:03.180 --> 00:50:04.970 align:middle line:90% So now we're done. 00:50:04.970 --> 00:50:07.273 align:middle line:84% This last slide is always a bit mysterious. 00:50:07.273 --> 00:50:09.440 align:middle line:84% It's like, well, why do we have to do anything else? 00:50:09.440 --> 00:50:12.170 align:middle line:84% Well, again, just remember the definition 00:50:12.170 --> 00:50:14.480 align:middle line:84% of a competitive equilibrium with transfers. 00:50:14.480 --> 00:50:16.940 align:middle line:84% Because it's transfers, it's specified 00:50:16.940 --> 00:50:18.590 align:middle line:84% not in terms of private ownership 00:50:18.590 --> 00:50:20.000 align:middle line:90% but in terms of wealth. 00:50:20.000 --> 00:50:21.680 align:middle line:84% So whatever those wealths are, they 00:50:21.680 --> 00:50:24.590 align:middle line:90% have to add up to the total. 00:50:24.590 --> 00:50:27.880 align:middle line:84% That's what this slide is all about. 00:50:27.880 --> 00:50:31.680 align:middle line:84% So that's what we call the constructive proof, 00:50:31.680 --> 00:50:37.560 align:middle line:84% is just using finite dimensional Euclidean spaces, 00:50:37.560 --> 00:50:40.680 align:middle line:84% putting in a lot of concavity, convexity, et cetera, 00:50:40.680 --> 00:50:43.740 align:middle line:84% then sort of matching up first-order conditions. 00:50:43.740 --> 00:50:46.260 align:middle line:84% But the theorem holds more generally. 00:50:46.260 --> 00:50:48.960 align:middle line:84% The second welfare theorem is true, 00:50:48.960 --> 00:50:52.650 align:middle line:84% even without making special assumptions about the commodity 00:50:52.650 --> 00:50:54.330 align:middle line:90% space and so on. 00:50:54.330 --> 00:50:56.010 align:middle line:84% A step along the way is something 00:50:56.010 --> 00:51:00.360 align:middle line:84% called the quasi-equilibrium with transfers, quasi 00:51:00.360 --> 00:51:05.430 align:middle line:84% meaning it's not quite fully consumer maximization. 00:51:05.430 --> 00:51:07.690 align:middle line:90% But it's going to be close. 00:51:07.690 --> 00:51:09.480 align:middle line:90% So here's the definition. 00:51:09.480 --> 00:51:14.520 align:middle line:84% An allocation, x star, y star and a price vector p, 00:51:14.520 --> 00:51:17.130 align:middle line:84% constitute a price quasi-equilibrium 00:51:17.130 --> 00:51:20.940 align:middle line:84% with transfers if we can find wealth levels that 00:51:20.940 --> 00:51:24.030 align:middle line:84% add up to the total wealth if each firm 00:51:24.030 --> 00:51:25.960 align:middle line:90% J is profit maximizing. 00:51:25.960 --> 00:51:26.760 align:middle line:90% So far so good. 00:51:26.760 --> 00:51:28.090 align:middle line:90% Nothing different. 00:51:28.090 --> 00:51:29.400 align:middle line:90% Here's the difference. 00:51:29.400 --> 00:51:32.670 align:middle line:84% Household i, this condition has to be true. 00:51:32.670 --> 00:51:37.230 align:middle line:84% If Xi star is the allocation we want 00:51:37.230 --> 00:51:40.950 align:middle line:84% in the definition of the equilibrium with transfer price 00:51:40.950 --> 00:51:43.470 align:middle line:84% quasi-equilibrium with transfers, 00:51:43.470 --> 00:51:46.620 align:middle line:84% if you start with that Xi star and there 00:51:46.620 --> 00:51:48.570 align:middle line:84% were an allocation that's strictly 00:51:48.570 --> 00:51:54.840 align:middle line:84% better for household i, then it costs at least as much 00:51:54.840 --> 00:51:58.350 align:middle line:84% as the wealth assignment in the price quasi-equilibrium 00:51:58.350 --> 00:51:59.050 align:middle line:90% with transfers. 00:51:59.050 --> 00:52:00.420 align:middle line:90% It's kind of a mouthful. 00:52:00.420 --> 00:52:04.530 align:middle line:84% One way to put this, though, is that Wi star 00:52:04.530 --> 00:52:08.640 align:middle line:84% is the valuation of expenditures at the equilibrium, 00:52:08.640 --> 00:52:10.860 align:middle line:90% at the equilibrium Xi star. 00:52:10.860 --> 00:52:14.820 align:middle line:84% So then this says pxi, the valuation of this arbitrary 00:52:14.820 --> 00:52:17.940 align:middle line:84% Xi, which is strictly preferred, the valuation 00:52:17.940 --> 00:52:22.200 align:middle line:84% is not less than the valuation of pxi star. 00:52:22.200 --> 00:52:26.310 align:middle line:84% That means that pxi star costs no more 00:52:26.310 --> 00:52:30.930 align:middle line:84% or is cost minimizing over the upper contour 00:52:30.930 --> 00:52:33.330 align:middle line:84% set of allocations that are associated 00:52:33.330 --> 00:52:35.280 align:middle line:90% with strictly more utility. 00:52:35.280 --> 00:52:38.175 align:middle line:84% So you've seen this before, way back when. 00:52:38.175 --> 00:52:41.970 align:middle line:84% When we did the consumer utility maximization problem, 00:52:41.970 --> 00:52:44.490 align:middle line:84% I pointed out that this is kind of a dual thing. 00:52:44.490 --> 00:52:47.280 align:middle line:84% Instead of maximizing utility subject to budgets, 00:52:47.280 --> 00:52:51.090 align:middle line:84% we minimized the budget necessary to achieve 00:52:51.090 --> 00:52:52.710 align:middle line:90% a given level of utility. 00:52:52.710 --> 00:52:54.540 align:middle line:90% In fact, we did a lot with that. 00:52:54.540 --> 00:52:59.860 align:middle line:84% We did Hicksian substitutions versus Walrasian substitutions 00:52:59.860 --> 00:53:00.360 align:middle line:90% and so on. 00:53:00.360 --> 00:53:02.560 align:middle line:90% We had a whole lecture on it. 00:53:02.560 --> 00:53:04.680 align:middle line:84% So here it is again, except so instead 00:53:04.680 --> 00:53:07.410 align:middle line:84% of assuming utility maximization, 00:53:07.410 --> 00:53:09.900 align:middle line:84% we're going to assume cost minimization. 00:53:09.900 --> 00:53:13.620 align:middle line:84% That's the only difference between a regular equilibrium 00:53:13.620 --> 00:53:15.960 align:middle line:84% with transfers and this quasi-equilibrium 00:53:15.960 --> 00:53:16.650 align:middle line:90% with transfers. 00:53:16.650 --> 00:53:21.010 align:middle line:84% The feasibility of the allocation is the same. 00:53:21.010 --> 00:53:23.130 align:middle line:84% So I'll show you the theorems now. 00:53:23.130 --> 00:53:27.660 align:middle line:84% But just to say, when we do the second welfare theorem, 00:53:27.660 --> 00:53:31.230 align:middle line:84% we're going to say that it's a price quasi-equilibrium 00:53:31.230 --> 00:53:32.130 align:middle line:90% with transfers. 00:53:32.130 --> 00:53:34.570 align:middle line:84% And then, given the structure of the economy, 00:53:34.570 --> 00:53:37.140 align:middle line:84% you may need a little bit more to establish 00:53:37.140 --> 00:53:39.780 align:middle line:84% that cost minimization corresponds 00:53:39.780 --> 00:53:42.090 align:middle line:90% with utility maximization. 00:53:42.090 --> 00:53:45.900 align:middle line:84% So suppose that the production sets are strict or convex, 00:53:45.900 --> 00:53:48.720 align:middle line:84% the preference relations are rational convex 00:53:48.720 --> 00:53:51.390 align:middle line:84% and satisfy local nonsatiation, then 00:53:51.390 --> 00:53:56.400 align:middle line:84% any Pareto-optimal allocation has a price p such that x star, 00:53:56.400 --> 00:53:59.130 align:middle line:84% y star p is a price quasi-equilibrium 00:53:59.130 --> 00:54:00.060 align:middle line:90% with transfers. 00:54:00.060 --> 00:54:03.900 align:middle line:84% So it's looking pretty similar, but we're 00:54:03.900 --> 00:54:05.740 align:middle line:84% going to prove this in a different way. 00:54:05.740 --> 00:54:07.530 align:middle line:84% And it's going to be a proof that works 00:54:07.530 --> 00:54:10.860 align:middle line:90% for quite arbitrary spaces. 00:54:10.860 --> 00:54:12.450 align:middle line:84% And the proof that's going to work 00:54:12.450 --> 00:54:14.970 align:middle line:84% has to do with separating hyperplanes. 00:54:14.970 --> 00:54:20.100 align:middle line:84% So let me remind you what that was about. 00:54:20.100 --> 00:54:22.920 align:middle line:84% Actually, this is not quite the right slide. 00:54:22.920 --> 00:54:27.300 align:middle line:84% This was a review slide about a supporting hyperplane. 00:54:27.300 --> 00:54:30.270 align:middle line:84% The better slide was, we had two convex sets 00:54:30.270 --> 00:54:32.610 align:middle line:90% that were tangent at a point. 00:54:32.610 --> 00:54:35.640 align:middle line:84% And then we were able to draw this hyperplane that 00:54:35.640 --> 00:54:40.200 align:middle line:84% would be tangent to the common slope of the two convex sets. 00:54:40.200 --> 00:54:42.270 align:middle line:84% So all you needed for the existence 00:54:42.270 --> 00:54:44.940 align:middle line:84% of supporting and separating hyperplanes 00:54:44.940 --> 00:54:49.000 align:middle line:84% is a presumption that the sets are convex. 00:54:49.000 --> 00:54:50.380 align:middle line:90% Here's a picture of it. 00:54:50.380 --> 00:54:52.990 align:middle line:84% We've seen versions of this before. 00:54:52.990 --> 00:54:56.080 align:middle line:90% Here's one production set. 00:54:56.080 --> 00:54:58.690 align:middle line:90% Here's another production set. 00:54:58.690 --> 00:55:03.860 align:middle line:84% Here's the endowment somewhere here, I think it was. 00:55:03.860 --> 00:55:07.540 align:middle line:84% And so we add up the production sets and the endowments, 00:55:07.540 --> 00:55:11.780 align:middle line:84% and we get this sort of outer frontier. 00:55:11.780 --> 00:55:16.450 align:middle line:84% The point being, it's defining a convex set. 00:55:16.450 --> 00:55:19.800 align:middle line:84% Things on the frontier but also interior. 00:55:19.800 --> 00:55:22.230 align:middle line:84% Any linear combinations of things in the interior 00:55:22.230 --> 00:55:24.740 align:middle line:90% is also in the interior. 00:55:24.740 --> 00:55:27.720 align:middle line:90% What's this other monster here? 00:55:27.720 --> 00:55:30.540 align:middle line:84% This thing looks like an indifference curve. 00:55:30.540 --> 00:55:32.660 align:middle line:84% But, in fact, we have many, many households. 00:55:32.660 --> 00:55:34.740 align:middle line:90% So what the heck is it? 00:55:34.740 --> 00:55:38.590 align:middle line:90% Well, it's this thing. 00:55:38.590 --> 00:55:40.630 align:middle line:90% It's the upper contour set. 00:55:40.630 --> 00:55:46.180 align:middle line:84% It's the set of allocations, which are weakly preferred 00:55:46.180 --> 00:55:50.050 align:middle line:84% to the allocation x star and summed up 00:55:50.050 --> 00:55:53.070 align:middle line:90% over all the households. 00:55:53.070 --> 00:55:57.680 align:middle line:84% So that's this thing that looks like an indifference curve. 00:55:57.680 --> 00:56:01.300 align:middle line:84% But because we have convexity, this upper contour set 00:56:01.300 --> 00:56:03.100 align:middle line:90% is also convex. 00:56:03.100 --> 00:56:06.160 align:middle line:90% So now, we have two convex sets. 00:56:06.160 --> 00:56:10.300 align:middle line:84% We have a supporting hyperplane for both of them, 00:56:10.300 --> 00:56:13.340 align:middle line:84% separating supporting hyperplane. 00:56:13.340 --> 00:56:16.290 align:middle line:84% And we're going to define the price system. 00:56:16.290 --> 00:56:18.110 align:middle line:84% So this is kind of a graphical way 00:56:18.110 --> 00:56:20.960 align:middle line:84% to think about that constructive proof 00:56:20.960 --> 00:56:23.360 align:middle line:84% without taking first-order conditions 00:56:23.360 --> 00:56:25.970 align:middle line:84% and matching up Lagrange multipliers. 00:56:25.970 --> 00:56:31.190 align:middle line:84% We're defining using the convexity and the separating 00:56:31.190 --> 00:56:34.020 align:middle line:84% supporting hyperplane theorems to get the prices. 00:56:34.020 --> 00:56:38.120 align:middle line:84% So in 1954, Debreu worked all this out. 00:56:38.120 --> 00:56:41.510 align:middle line:84% He called his equilibrium a valuation equilibrium, 00:56:41.510 --> 00:56:44.060 align:middle line:84% as distinct from a competitive equilibrium. 00:56:44.060 --> 00:56:47.330 align:middle line:84% Pareto optimality was defined the same way. 00:56:47.330 --> 00:56:49.410 align:middle line:90% The environment is similar. 00:56:49.410 --> 00:56:53.420 align:middle line:84% We have a finite number of households getting allocations 00:56:53.420 --> 00:56:55.130 align:middle line:90% in their consumption sets. 00:56:55.130 --> 00:56:59.000 align:middle line:84% But when it says the consumption set is a subset of this L, 00:56:59.000 --> 00:57:03.570 align:middle line:84% it does no longer mean a finite dimensional Euclidean space. 00:57:03.570 --> 00:57:06.440 align:middle line:84% It's an arbitrary linear space, conveniently 00:57:06.440 --> 00:57:08.450 align:middle line:90% denoted L for linear. 00:57:08.450 --> 00:57:10.850 align:middle line:84% Preferences have a complete ordering. 00:57:10.850 --> 00:57:12.290 align:middle line:90% That's rationality. 00:57:12.290 --> 00:57:15.110 align:middle line:84% Producers J have a production set, 00:57:15.110 --> 00:57:17.030 align:middle line:90% choosing objects in the set. 00:57:17.030 --> 00:57:20.360 align:middle line:84% Also a subset of the linear space. 00:57:20.360 --> 00:57:22.730 align:middle line:84% Market clearing looks pretty similar. 00:57:22.730 --> 00:57:27.140 align:middle line:84% x equal y plus 6e or x minus y equals 6e, 00:57:27.140 --> 00:57:29.180 align:middle line:84% where x is the summation over households, 00:57:29.180 --> 00:57:31.100 align:middle line:84% y is the summation over producers, 00:57:31.100 --> 00:57:33.860 align:middle line:90% and xi is the endowment. 00:57:33.860 --> 00:57:37.670 align:middle line:84% So we actually call a state of the economy something 00:57:37.670 --> 00:57:40.670 align:middle line:84% that's attainable being in the production sets, consumption 00:57:40.670 --> 00:57:45.580 align:middle line:84% sets, and satisfying the resource constraint. 00:57:45.580 --> 00:57:48.280 align:middle line:90% So then, we get to the theorems. 00:57:48.280 --> 00:57:50.200 align:middle line:84% First and second welfare theorems 00:57:50.200 --> 00:57:53.920 align:middle line:84% are similar, although the assumptions are a little bit 00:57:53.920 --> 00:57:54.820 align:middle line:90% different. 00:57:54.820 --> 00:57:57.460 align:middle line:90% Evaluation equilibrium, i.e. 00:57:57.460 --> 00:58:00.850 align:middle line:84% competitive equilibrium, is Pareto optimal. 00:58:00.850 --> 00:58:03.370 align:middle line:84% What does the price system look like? 00:58:03.370 --> 00:58:06.070 align:middle line:84% In these general linear spaces, it's 00:58:06.070 --> 00:58:09.040 align:middle line:84% just called a linear functional, meaning 00:58:09.040 --> 00:58:11.470 align:middle line:90% that it's retaining additivity. 00:58:11.470 --> 00:58:14.260 align:middle line:84% If you take the valuation of a weighted average of two 00:58:14.260 --> 00:58:16.000 align:middle line:84% bundles, it's the weighted average 00:58:16.000 --> 00:58:18.040 align:middle line:84% of the valuation of the two bundles. 00:58:18.040 --> 00:58:19.180 align:middle line:90% Seems quite natural. 00:58:19.180 --> 00:58:22.330 align:middle line:84% Obviously, for finite dimensional Euclidean spaces 00:58:22.330 --> 00:58:26.060 align:middle line:84% and the dot product, this holds automatically. 00:58:26.060 --> 00:58:28.790 align:middle line:84% This is a generalization of that idea. 00:58:28.790 --> 00:58:34.960 align:middle line:84% So a valuation equilibrium is attainable, maximizes utility, 00:58:34.960 --> 00:58:38.260 align:middle line:84% subject to the budget, and maximizes profits, 00:58:38.260 --> 00:58:43.660 align:middle line:84% where the price system is this valuation function v. This 00:58:43.660 --> 00:58:48.040 align:middle line:84% is literally utility maximization because it 00:58:48.040 --> 00:58:52.810 align:middle line:84% says that if it's in the budget, it is not preferred to Xi. 00:58:52.810 --> 00:58:55.630 align:middle line:84% If that weren't true, if you had a point in the budget that 00:58:55.630 --> 00:59:00.070 align:middle line:84% were strictly preferred to the allocation Xi0, 00:59:00.070 --> 00:59:02.020 align:middle line:90% then it would be chosen. 00:59:02.020 --> 00:59:06.600 align:middle line:84% So that would contradict utility maximization. 00:59:06.600 --> 00:59:10.410 align:middle line:84% The second welfare theorem of valuation equilibrium 00:59:10.410 --> 00:59:12.990 align:middle line:84% is a Pareto optimum welfare theorem. 00:59:12.990 --> 00:59:18.360 align:middle line:84% What we need, we need convexity of the consumption sets 00:59:18.360 --> 00:59:22.560 align:middle line:84% and a little bit more about convexity of preferences. 00:59:22.560 --> 00:59:24.840 align:middle line:90% We did not need this previously. 00:59:24.840 --> 00:59:27.840 align:middle line:84% When we had finite dimensional Euclidean spaces, 00:59:27.840 --> 00:59:30.450 align:middle line:84% I made a big deal of saying all we needed 00:59:30.450 --> 00:59:32.640 align:middle line:90% was local nonsatiation. 00:59:32.640 --> 00:59:35.070 align:middle line:90% We need no satiation here. 00:59:35.070 --> 00:59:38.940 align:middle line:84% We also need a little bit more about convexity of consumption 00:59:38.940 --> 00:59:40.210 align:middle line:90% sets and preferences. 00:59:40.210 --> 00:59:42.720 align:middle line:84% So you're trying for something bigger, something 00:59:42.720 --> 00:59:45.030 align:middle line:84% that works for a very, very large number 00:59:45.030 --> 00:59:46.740 align:middle line:84% of economic environments, you need 00:59:46.740 --> 00:59:48.220 align:middle line:90% to assume a little bit more. 00:59:48.220 --> 00:59:51.270 align:middle line:84% The second welfare theorem, any Pareto optimum 00:59:51.270 --> 00:59:53.950 align:middle line:84% can be achieved as a valuation equilibrium. 00:59:53.950 --> 00:59:57.960 align:middle line:84% So again, more things about consumption sets. 00:59:57.960 --> 01:00:02.550 align:middle line:84% When your combinations of points in the set are also in the set, 01:00:02.550 --> 01:00:03.660 align:middle line:90% sets closed. 01:00:03.660 --> 01:00:06.240 align:middle line:84% Production sets, when aggregated up, 01:00:06.240 --> 01:00:08.730 align:middle line:84% the aggregated production set is convex. 01:00:08.730 --> 01:00:12.930 align:middle line:84% And here, either L is finite dimensional or if infinite 01:00:12.930 --> 01:00:16.800 align:middle line:84% or other dimensional, Y has an interior point. 01:00:16.800 --> 01:00:19.050 align:middle line:84% I know this is really, really abstract. 01:00:19.050 --> 01:00:22.500 align:middle line:84% I'm going to show you a couple of applications momentarily. 01:00:22.500 --> 01:00:25.890 align:middle line:84% And the theorem is, assuming the 3, 4, 01:00:25.890 --> 01:00:31.300 align:middle line:84% and 5, every Pareto-optimal allocation, which is not 01:00:31.300 --> 01:00:36.040 align:middle line:84% a satiation point, is associated with a nontrivial continuous 01:00:36.040 --> 01:00:39.280 align:middle line:84% linear function, where we have a condition for households 01:00:39.280 --> 01:00:40.780 align:middle line:90% and a condition for firms. 01:00:40.780 --> 01:00:45.820 align:middle line:84% So here, the condition for households is that if Xi, 01:00:45.820 --> 01:00:49.360 align:middle line:84% we have the Pareto optimal allocation Pareto Xi0. 01:00:49.360 --> 01:00:52.030 align:middle line:84% We have an allocation that is strictly weakly preferred 01:00:52.030 --> 01:00:55.790 align:middle line:84% to Xi0 and is in the consumption set, 01:00:55.790 --> 01:01:00.160 align:middle line:84% then this implies the valuation of that other allocation Xi 01:01:00.160 --> 01:01:03.430 align:middle line:84% is not less than the valuation of Xi0. 01:01:03.430 --> 01:01:06.670 align:middle line:84% This corresponds to cost minimization. 01:01:06.670 --> 01:01:08.620 align:middle line:90% And again, just reverse it. 01:01:08.620 --> 01:01:12.790 align:middle line:84% If the valuation were less than Xi0, then 01:01:12.790 --> 01:01:16.750 align:middle line:84% that other point is, say, better, maybe indifferent. 01:01:16.750 --> 01:01:20.380 align:middle line:84% But because we have local nonsatiation, just 01:01:20.380 --> 01:01:23.200 align:middle line:84% like the Lemma, we could construct something close 01:01:23.200 --> 01:01:27.520 align:middle line:84% by which is strictly better and in the budget set for Xi0. 01:01:27.520 --> 01:01:29.050 align:middle line:84% So that would be a contradiction. 01:01:29.050 --> 01:01:31.720 align:middle line:84% So this is the cost minimization part. 01:01:31.720 --> 01:01:37.000 align:middle line:84% This is why it's going to be a quasi-equilibrium, not 01:01:37.000 --> 01:01:39.190 align:middle line:90% a full equilibrium. 01:01:39.190 --> 01:01:41.680 align:middle line:84% So I feel like it took a lot of work 01:01:41.680 --> 01:01:43.660 align:middle line:90% to get through all of this. 01:01:43.660 --> 01:01:45.970 align:middle line:84% The main takeaways are, there's a way 01:01:45.970 --> 01:01:49.090 align:middle line:84% to do things more generally with separating and supporting 01:01:49.090 --> 01:01:52.990 align:middle line:84% hyperplanes, as long as we have sufficient convexity. 01:01:52.990 --> 01:01:55.990 align:middle line:84% I took the time to take you through Debreu's article. 01:01:55.990 --> 01:01:59.500 align:middle line:84% It's really very similar, apart from changes in notation, 01:01:59.500 --> 01:02:03.250 align:middle line:84% to what we did with the finite dimensional Euclidean spaces. 01:02:03.250 --> 01:02:08.260 align:middle line:84% But it applies to other economies, 01:02:08.260 --> 01:02:10.480 align:middle line:84% in particular, competitive equilibria 01:02:10.480 --> 01:02:13.450 align:middle line:84% in the space of contracts, including 01:02:13.450 --> 01:02:16.030 align:middle line:84% the possibility of randomized allocations 01:02:16.030 --> 01:02:18.690 align:middle line:90% that we've used before. 01:02:18.690 --> 01:02:22.700 align:middle line:84% So I think I mentioned contracts with options, 01:02:22.700 --> 01:02:25.520 align:middle line:84% like CDs where you could withdraw early 01:02:25.520 --> 01:02:29.150 align:middle line:84% or pensions where you could decide to take a lump sum 01:02:29.150 --> 01:02:31.130 align:middle line:90% or take an annuity. 01:02:31.130 --> 01:02:35.150 align:middle line:84% I gave you those as examples of contracts with options. 01:02:35.150 --> 01:02:36.650 align:middle line:84% And then we had a couple of lectures 01:02:36.650 --> 01:02:38.330 align:middle line:90% talking about contracts. 01:02:38.330 --> 01:02:41.600 align:middle line:84% I'm sparing you working out through all 01:02:41.600 --> 01:02:46.670 align:middle line:84% of that notation how the contracts with lotteries 01:02:46.670 --> 01:02:51.020 align:middle line:84% and indeed without grids, so measures, 01:02:51.020 --> 01:02:55.160 align:middle line:84% can be mapped into Debreu's valuation equilibrium, 01:02:55.160 --> 01:02:59.340 align:middle line:84% and hence we get the welfare theorems. 01:02:59.340 --> 01:02:59.840 align:middle line:90% OK. 01:02:59.840 --> 01:03:03.010 align:middle line:90% So that's all for today. 01:03:03.010 --> 01:03:11.000 align:middle line:90%