WEBVTT 00:00:00.000 --> 00:00:01.992 align:middle line:90% [SQUEAKING] 00:00:01.992 --> 00:00:02.988 align:middle line:90% [RUSTLING] 00:00:02.988 --> 00:00:11.970 align:middle line:90% [CLICKING] 00:00:11.970 --> 00:00:16.110 align:middle line:84% ROBERT TOWNSEND: As usual, let's find out where we are. 00:00:16.110 --> 00:00:20.700 align:middle line:84% We did the first two sections of general equilibrium 00:00:20.700 --> 00:00:22.180 align:middle line:90% on welfare and existence. 00:00:22.180 --> 00:00:25.140 align:middle line:84% And now, we're going to do Gorman aggregation today, 00:00:25.140 --> 00:00:28.290 align:middle line:84% and then finish this segment with identification. 00:00:28.290 --> 00:00:31.390 align:middle line:84% We have two lectures to finish out the class. 00:00:31.390 --> 00:00:33.510 align:middle line:90% So that's where we are. 00:00:33.510 --> 00:00:37.590 align:middle line:84% In terms of the reading list for today, 00:00:37.590 --> 00:00:41.670 align:middle line:84% there's only one thing there, and it's 00:00:41.670 --> 00:00:44.610 align:middle line:84% the place from where I drew a lot of the material 00:00:44.610 --> 00:00:48.630 align:middle line:84% for the welfare theorem for the Gorman aggregation 00:00:48.630 --> 00:00:50.250 align:middle line:90% and my colleague Daron. 00:00:50.250 --> 00:00:53.400 align:middle line:84% It is starred, but it is also pretty 00:00:53.400 --> 00:00:57.100 align:middle line:84% extensively covered today as well. 00:00:57.100 --> 00:00:59.920 align:middle line:90% Then we have the study guide. 00:00:59.920 --> 00:01:04.950 align:middle line:84% So last time, we did concepts of existence 00:01:04.950 --> 00:01:07.020 align:middle line:90% in various distinct ways. 00:01:07.020 --> 00:01:09.750 align:middle line:84% There aren't too many review questions here. 00:01:09.750 --> 00:01:12.040 align:middle line:84% I didn't ask the first set of them. 00:01:12.040 --> 00:01:15.330 align:middle line:90% But we get down to Negishi. 00:01:15.330 --> 00:01:18.240 align:middle line:84% So how does Negishi use the first and second welfare 00:01:18.240 --> 00:01:23.220 align:middle line:84% theorems to generate a mapping in lambda weights, the fixed 00:01:23.220 --> 00:01:26.190 align:middle line:84% point of which is a competitive equilibrium? 00:01:26.190 --> 00:01:28.710 align:middle line:84% And since I was calling on everyone last time, 00:01:28.710 --> 00:01:32.550 align:middle line:90% let me just take volunteers. 00:01:32.550 --> 00:01:34.110 align:middle line:90% Hopefully, there are volunteers. 00:01:34.110 --> 00:01:39.470 align:middle line:90% 00:01:39.470 --> 00:01:40.520 align:middle line:90% AUDIENCE: I was-- 00:01:40.520 --> 00:01:41.478 align:middle line:90% ROBERT TOWNSEND: Go on. 00:01:41.478 --> 00:01:43.262 align:middle line:90% AUDIENCE: You go ahead. 00:01:43.262 --> 00:01:44.720 align:middle line:84% AUDIENCE: I was about to say that I 00:01:44.720 --> 00:01:47.450 align:middle line:84% was a little unclear on the proof overall, 00:01:47.450 --> 00:01:53.840 align:middle line:84% but I remember that we use the fact that for any lambda, 00:01:53.840 --> 00:01:57.470 align:middle line:84% the corresponding period of optimal allocation 00:01:57.470 --> 00:02:01.070 align:middle line:84% would be in equilibrium with transfers 00:02:01.070 --> 00:02:04.325 align:middle line:84% by the second welfare theorem, and we use that to prove 00:02:04.325 --> 00:02:07.120 align:middle line:90% Negishi's theorem. 00:02:07.120 --> 00:02:08.620 align:middle line:84% ROBERT TOWNSEND: Yeah, that's right. 00:02:08.620 --> 00:02:11.110 align:middle line:84% We used to have the second one heavily. 00:02:11.110 --> 00:02:14.200 align:middle line:84% The first one is that any competitive equilibrium 00:02:14.200 --> 00:02:19.110 align:middle line:84% is Pareto optimal, and hence, it corresponds to some lambda 00:02:19.110 --> 00:02:22.050 align:middle line:90% weights in the Pareto problem. 00:02:22.050 --> 00:02:29.840 align:middle line:84% So Negishi's mapping is to find a way 00:02:29.840 --> 00:02:33.620 align:middle line:84% to find the particular special lambdas 00:02:33.620 --> 00:02:38.330 align:middle line:84% of the competitive equilibrium with private ownership that is 00:02:38.330 --> 00:02:41.930 align:middle line:90% the fixed point of his mapping. 00:02:41.930 --> 00:02:44.240 align:middle line:84% I realized after class, in part talking 00:02:44.240 --> 00:02:53.130 align:middle line:84% with one of you who came to office hours, that I never-- 00:02:53.130 --> 00:02:57.060 align:middle line:84% I think I neglected to say the obvious. 00:02:57.060 --> 00:02:59.790 align:middle line:84% What I did say is to find the lambda such 00:02:59.790 --> 00:03:03.320 align:middle line:84% that the valuation of expenditures 00:03:03.320 --> 00:03:06.440 align:middle line:84% under the Pareto of optimum uses using the second welfare 00:03:06.440 --> 00:03:10.640 align:middle line:84% theorem is equal to the valuation of the endowment 00:03:10.640 --> 00:03:14.610 align:middle line:84% plus shares on profits that corresponds 00:03:14.610 --> 00:03:16.320 align:middle line:90% to the competitive equilibrium. 00:03:16.320 --> 00:03:19.460 align:middle line:90% 00:03:19.460 --> 00:03:23.530 align:middle line:84% But what I think I neglected to say was-- 00:03:23.530 --> 00:03:24.850 align:middle line:90% maybe I did say it. 00:03:24.850 --> 00:03:26.260 align:middle line:90% Bears repeating. 00:03:26.260 --> 00:03:29.200 align:middle line:84% That if you initially guess a lambda which 00:03:29.200 --> 00:03:33.730 align:middle line:84% is high than the value for a particular household, then 00:03:33.730 --> 00:03:36.460 align:middle line:84% the valuation of that household's expenditures 00:03:36.460 --> 00:03:44.750 align:middle line:84% will be large relative to the valuation of wealth 00:03:44.750 --> 00:03:48.440 align:middle line:84% in the private ownership economy. 00:03:48.440 --> 00:03:51.830 align:middle line:84% So sort of intuitively, then, you lower that lambda weight. 00:03:51.830 --> 00:03:54.530 align:middle line:84% That links up to something we learned when we did this 00:03:54.530 --> 00:03:56.300 align:middle line:84% the second welfare theorem, which 00:03:56.300 --> 00:04:00.440 align:middle line:84% is, those lambdas are inversely proportional to 00:04:00.440 --> 00:04:03.580 align:middle line:90% the marginal utility of wealth. 00:04:03.580 --> 00:04:09.450 align:middle line:84% So if the lambda is too high, the guy has too much-- 00:04:09.450 --> 00:04:13.120 align:middle line:84% the evaluation of expenditures is too high. 00:04:13.120 --> 00:04:17.130 align:middle line:84% So the marginal utility of wealth with concave utility 00:04:17.130 --> 00:04:18.730 align:middle line:90% is low. 00:04:18.730 --> 00:04:22.230 align:middle line:84% So you want to move in the direction of increasing 00:04:22.230 --> 00:04:26.790 align:middle line:84% the marginal utility of wealth, which effectively removes 00:04:26.790 --> 00:04:28.320 align:middle line:90% wealth by lowering lambda. 00:04:28.320 --> 00:04:31.520 align:middle line:90% 00:04:31.520 --> 00:04:34.030 align:middle line:90% So anyway, thank you. 00:04:34.030 --> 00:04:36.740 align:middle line:90% 00:04:36.740 --> 00:04:37.360 align:middle line:90% Next question. 00:04:37.360 --> 00:04:40.780 align:middle line:84% Define a mixed-strategy Nash equilibrium. 00:04:40.780 --> 00:04:43.780 align:middle line:84% AUDIENCE: I think I can do this one. 00:04:43.780 --> 00:04:46.300 align:middle line:84% Like, a mixed-strategy Nash equilibrium is-- 00:04:46.300 --> 00:04:49.480 align:middle line:84% it's a type of Nash equilibrium where at least some 00:04:49.480 --> 00:04:52.150 align:middle line:90% of the players are-- 00:04:52.150 --> 00:04:54.490 align:middle line:84% choose some of the options that are random? 00:04:54.490 --> 00:04:56.800 align:middle line:84% So there's some weighted probabilities 00:04:56.800 --> 00:05:00.190 align:middle line:84% where you can have some chance of doing option A, some chance 00:05:00.190 --> 00:05:02.170 align:middle line:90% of doing option B, and so on. 00:05:02.170 --> 00:05:08.530 align:middle line:84% And in order for it to work, the player who is doing mixed stuff 00:05:08.530 --> 00:05:10.677 align:middle line:84% has to be indifferent from all of the things 00:05:10.677 --> 00:05:12.760 align:middle line:84% with non-zero between all of the different options 00:05:12.760 --> 00:05:17.260 align:middle line:84% with non-zero probabilities, and it also has to be in such a way 00:05:17.260 --> 00:05:19.210 align:middle line:84% that, again, no one has any incentive 00:05:19.210 --> 00:05:21.682 align:middle line:90% to deviate from the strategy. 00:05:21.682 --> 00:05:22.640 align:middle line:90% ROBERT TOWNSEND: Right. 00:05:22.640 --> 00:05:24.530 align:middle line:84% The last thing, especially, is key. 00:05:24.530 --> 00:05:29.120 align:middle line:84% So the mixed-strategy Nash equilibrium is taking-- 00:05:29.120 --> 00:05:33.540 align:middle line:84% each player takes as given a configuration of mixed 00:05:33.540 --> 00:05:35.680 align:middle line:90% strategies of the others-- 00:05:35.680 --> 00:05:38.790 align:middle line:84% sort of star strategy, so to speak-- and then 00:05:38.790 --> 00:05:44.180 align:middle line:84% feels free to choose the best one for his or herself. 00:05:44.180 --> 00:05:47.370 align:middle line:84% But that will end up being exactly the star 00:05:47.370 --> 00:05:49.950 align:middle line:90% strategy to begin with. 00:05:49.950 --> 00:05:54.450 align:middle line:84% So it's, like, self-fulfilling in the space of strategies. 00:05:54.450 --> 00:05:54.960 align:middle line:90% Thank you. 00:05:54.960 --> 00:05:57.540 align:middle line:90% That was a good answer. 00:05:57.540 --> 00:06:00.750 align:middle line:90% And then the last one. 00:06:00.750 --> 00:06:02.700 align:middle line:84% How do we give an incentive for traders 00:06:02.700 --> 00:06:04.920 align:middle line:84% to live up to their promises and achieve 00:06:04.920 --> 00:06:07.640 align:middle line:90% over raising an equilibrium? 00:06:07.640 --> 00:06:12.890 align:middle line:84% So I did this in the context of the market for trades 00:06:12.890 --> 00:06:16.240 align:middle line:90% in US treasuries in New York. 00:06:16.240 --> 00:06:21.600 align:middle line:84% Which is a trade agreed to by treasury for liquidity. 00:06:21.600 --> 00:06:24.180 align:middle line:84% I mentioned that, in fact, there were trade 00:06:24.180 --> 00:06:27.540 align:middle line:84% fails where they don't live up to their agreement 00:06:27.540 --> 00:06:31.980 align:middle line:84% because the party selling the treasury doesn't own it, 00:06:31.980 --> 00:06:36.240 align:middle line:84% can't transfer it at the time, when a settlement is due 00:06:36.240 --> 00:06:38.820 align:middle line:84% or the person who promised to pay liquidity 00:06:38.820 --> 00:06:41.700 align:middle line:84% doesn't have sufficient liquidity. 00:06:41.700 --> 00:06:47.100 align:middle line:84% And then I mapped that into the Walrasian equilibrium 00:06:47.100 --> 00:06:47.780 align:middle line:90% framework. 00:06:47.780 --> 00:06:50.500 align:middle line:90% 00:06:50.500 --> 00:06:52.140 align:middle line:90% So let me ask it again. 00:06:52.140 --> 00:06:54.270 align:middle line:84% How do we give an incentive for trades-- 00:06:54.270 --> 00:06:58.740 align:middle line:84% trade-- for traders, I guess it should be, 00:06:58.740 --> 00:07:01.650 align:middle line:84% to live up to their promise and achieve a Walrasian 00:07:01.650 --> 00:07:02.490 align:middle line:90% equilibrium? 00:07:02.490 --> 00:07:05.050 align:middle line:90% 00:07:05.050 --> 00:07:05.905 align:middle line:90% Anybody remember? 00:07:05.905 --> 00:07:10.370 align:middle line:90% 00:07:10.370 --> 00:07:14.840 align:middle line:84% AUDIENCE: I remember there was a penalty for liquidity, right? 00:07:14.840 --> 00:07:21.900 align:middle line:84% If the creditor was negative, then it will have this utility. 00:07:21.900 --> 00:07:24.450 align:middle line:90% And so you're in-- 00:07:24.450 --> 00:07:27.680 align:middle line:84% so in the end, in order to achieve the [? variation ?] 00:07:27.680 --> 00:07:34.340 align:middle line:84% allocation, we need to let all of the downwards-- 00:07:34.340 --> 00:07:36.350 align:middle line:90% or there was a-- 00:07:36.350 --> 00:07:38.840 align:middle line:84% we need to let the [? operator ?] equal to 0 00:07:38.840 --> 00:07:43.700 align:middle line:84% so that there was no exercise demand, no-- 00:07:43.700 --> 00:07:45.380 align:middle line:90% yeah. 00:07:45.380 --> 00:07:46.160 align:middle line:90% I mean-- 00:07:46.160 --> 00:07:47.077 align:middle line:90% ROBERT TOWNSEND: Yeah. 00:07:47.077 --> 00:07:49.480 align:middle line:90% Yeah, that's good. 00:07:49.480 --> 00:07:51.120 align:middle line:84% AUDIENCE: We should we want to promise 00:07:51.120 --> 00:07:59.670 align:middle line:84% that if the endowment is equal to the allocations, the extent 00:07:59.670 --> 00:08:02.052 align:middle line:90% of the allocation. 00:08:02.052 --> 00:08:03.010 align:middle line:90% ROBERT TOWNSEND: Right. 00:08:03.010 --> 00:08:03.700 align:middle line:90% That's good. 00:08:03.700 --> 00:08:08.080 align:middle line:84% So the problem is, they're submitting bids, 00:08:08.080 --> 00:08:10.720 align:middle line:84% and we're looking for a Nash equilibrium. 00:08:10.720 --> 00:08:13.360 align:middle line:90% 00:08:13.360 --> 00:08:19.760 align:middle line:84% The bids consist of orders to buy and also orders to sell. 00:08:19.760 --> 00:08:24.060 align:middle line:84% We're shooting for the Walrasian equilibrium, 00:08:24.060 --> 00:08:25.840 align:middle line:84% in particular, the budget constraints. 00:08:25.840 --> 00:08:29.520 align:middle line:84% So we want, for each trader, that the valuation 00:08:29.520 --> 00:08:32.190 align:middle line:84% of what they're buying is at least 00:08:32.190 --> 00:08:34.679 align:middle line:84% matched by the valuation of what they're selling. 00:08:34.679 --> 00:08:39.679 align:middle line:84% In other words, they have sufficient liquidity. 00:08:39.679 --> 00:08:42.890 align:middle line:84% And if we didn't put in a penalty, everybody would 00:08:42.890 --> 00:08:47.130 align:middle line:90% overspend, because why not? 00:08:47.130 --> 00:08:51.020 align:middle line:90% 00:08:51.020 --> 00:08:53.860 align:middle line:84% So yeah, so then the penalty that you mentioned-- 00:08:53.860 --> 00:08:56.980 align:middle line:90% so beta is negative. 00:08:56.980 --> 00:08:59.230 align:middle line:84% When beta is negative, that means 00:08:59.230 --> 00:09:03.620 align:middle line:84% they're spending more than they have in revenue. 00:09:03.620 --> 00:09:04.830 align:middle line:90% That's a bad thing. 00:09:04.830 --> 00:09:08.540 align:middle line:84% So we effectively make the penalty beta-- 00:09:08.540 --> 00:09:11.480 align:middle line:84% sorry, I can't remember the notation. 00:09:11.480 --> 00:09:13.130 align:middle line:84% I think it was lambda times beta-- 00:09:13.130 --> 00:09:20.080 align:middle line:84% sufficiently high so then they do the right trade-off. 00:09:20.080 --> 00:09:22.840 align:middle line:84% They look at how much they would get in utility from buying 00:09:22.840 --> 00:09:27.760 align:middle line:84% this stuff directly and weigh that against the disutility 00:09:27.760 --> 00:09:29.670 align:middle line:90% of the penalty. 00:09:29.670 --> 00:09:34.120 align:middle line:84% The magnitude of the penalty had to do with the marginal utility 00:09:34.120 --> 00:09:37.360 align:middle line:84% of income, that Lagrange multiplier 00:09:37.360 --> 00:09:40.930 align:middle line:84% we keep focusing on in the competitive equilibrium 00:09:40.930 --> 00:09:44.030 align:middle line:90% and the budget constraint. 00:09:44.030 --> 00:09:48.640 align:middle line:84% So the margin utility of income represents the incremental gain 00:09:48.640 --> 00:09:51.520 align:middle line:84% from having epsilon more liquidity. 00:09:51.520 --> 00:09:53.290 align:middle line:84% And they're tempted to spend more 00:09:53.290 --> 00:09:57.430 align:middle line:84% than they have to go in that epsilon direction, 00:09:57.430 --> 00:10:03.610 align:middle line:84% but the penalty is large enough so as to dominate that gain, 00:10:03.610 --> 00:10:07.310 align:middle line:90% and they don't deviate. 00:10:07.310 --> 00:10:10.820 align:middle line:84% Well, in earmarking this question, 00:10:10.820 --> 00:10:13.820 align:middle line:84% I recognized that we rushed a bit at the end. 00:10:13.820 --> 00:10:17.700 align:middle line:84% In fact, I went over the class time by a couple of minutes. 00:10:17.700 --> 00:10:23.570 align:middle line:84% So anyway, hopefully, that was a useful review. 00:10:23.570 --> 00:10:25.505 align:middle line:84% Any other questions from last time? 00:10:25.505 --> 00:10:30.715 align:middle line:90% 00:10:30.715 --> 00:10:31.215 align:middle line:90% OK. 00:10:31.215 --> 00:10:33.980 align:middle line:90% 00:10:33.980 --> 00:10:36.530 align:middle line:84% So now, I want to get to aggregation. 00:10:36.530 --> 00:10:39.950 align:middle line:84% This is kind of an exciting lecture in the sense 00:10:39.950 --> 00:10:45.810 align:middle line:84% that it goes back and reviews some of the things 00:10:45.810 --> 00:10:47.130 align:middle line:90% we've been doing before-- 00:10:47.130 --> 00:10:51.240 align:middle line:84% certainly, the basics of consumer theory 00:10:51.240 --> 00:10:55.770 align:middle line:84% with utility maximization, some things about Lagrangians 00:10:55.770 --> 00:10:58.450 align:middle line:90% and envelope theorems. 00:10:58.450 --> 00:11:02.200 align:middle line:84% It also goes back to the first application 00:11:02.200 --> 00:11:06.040 align:middle line:84% we have had of consumers maximizing utility 00:11:06.040 --> 00:11:07.750 align:middle line:84% subject to budgets, and what happens 00:11:07.750 --> 00:11:09.770 align:middle line:90% when income changes, and so on. 00:11:09.770 --> 00:11:13.930 align:middle line:84% So we already talked about angles, curves, and Giffen 00:11:13.930 --> 00:11:15.200 align:middle line:90% goods, and so on. 00:11:15.200 --> 00:11:19.060 align:middle line:84% And so you have the micro background already. 00:11:19.060 --> 00:11:21.250 align:middle line:84% So you can judge the plausibility 00:11:21.250 --> 00:11:23.530 align:middle line:84% of the assumptions we're about to make in order 00:11:23.530 --> 00:11:25.690 align:middle line:90% to allow the aggregation. 00:11:25.690 --> 00:11:30.190 align:middle line:84% And the aggregation was used when we did, like, free trade. 00:11:30.190 --> 00:11:32.980 align:middle line:84% We would talk about a representative consumer, not 00:11:32.980 --> 00:11:35.500 align:middle line:84% just Robinson Crusoe, one man on the island, 00:11:35.500 --> 00:11:39.020 align:middle line:84% but someone representing the entire US population. 00:11:39.020 --> 00:11:41.470 align:middle line:90% So that is a big leap of faith. 00:11:41.470 --> 00:11:44.600 align:middle line:84% Today, we'll spell out what it takes to do that. 00:11:44.600 --> 00:11:47.020 align:middle line:84% So this is going to be Gorman aggregation, 00:11:47.020 --> 00:11:50.470 align:middle line:84% and we're going to do it in the positive sense of predicting 00:11:50.470 --> 00:11:53.650 align:middle line:84% what will happen and in the normative sense of making 00:11:53.650 --> 00:11:56.005 align:middle line:84% statements about consumer welfare. 00:11:56.005 --> 00:11:57.880 align:middle line:84% And then we're going to have indirect utility 00:11:57.880 --> 00:12:01.540 align:middle line:84% functions, Gorman Polar forms, and the rest I already 00:12:01.540 --> 00:12:02.810 align:middle line:90% mentioned. 00:12:02.810 --> 00:12:06.670 align:middle line:84% So some micro, but mostly macro economists, 00:12:06.670 --> 00:12:08.980 align:middle line:84% assume the representative consumer 00:12:08.980 --> 00:12:10.360 align:middle line:90% or representative agents. 00:12:10.360 --> 00:12:13.540 align:middle line:84% It kind of comes naturally, because they're 00:12:13.540 --> 00:12:19.520 align:middle line:84% worrying about aggregates, not distributional aspects. 00:12:19.520 --> 00:12:24.460 align:middle line:84% So that gives them and us a tractable way of understanding 00:12:24.460 --> 00:12:25.690 align:middle line:90% market demand-- 00:12:25.690 --> 00:12:27.490 align:middle line:90% and supply, for that matter-- 00:12:27.490 --> 00:12:30.880 align:middle line:84% and allows pretty easy welfare analysis 00:12:30.880 --> 00:12:32.450 align:middle line:90% because there's only one person. 00:12:32.450 --> 00:12:35.140 align:middle line:84% So those are all the advantages, but we'll 00:12:35.140 --> 00:12:37.150 align:middle line:90% see how realistic it is. 00:12:37.150 --> 00:12:41.470 align:middle line:84% So here's the utility maximization problem. 00:12:41.470 --> 00:12:46.750 align:middle line:84% Maximizing-- we'll suppress the labels here for households i 00:12:46.750 --> 00:12:47.930 align:middle line:90% and h and so on. 00:12:47.930 --> 00:12:53.000 align:middle line:84% So this household has a utility function represented by u of x, 00:12:53.000 --> 00:12:56.350 align:middle line:84% where x is in the consumption set and the valuation 00:12:56.350 --> 00:13:00.610 align:middle line:84% of expenditures at prices p cannot exceed wealth w. 00:13:00.610 --> 00:13:04.570 align:middle line:84% So we call the solution to this problem, after maxing it out-- 00:13:04.570 --> 00:13:09.170 align:middle line:84% the achieved maximized utility is called the value. 00:13:09.170 --> 00:13:13.390 align:middle line:84% And this function v is called the indirect utility function, 00:13:13.390 --> 00:13:14.780 align:middle line:90% v for value. 00:13:14.780 --> 00:13:17.830 align:middle line:84% So it takes as given the parameters 00:13:17.830 --> 00:13:20.110 align:middle line:84% that the household takes as given, 00:13:20.110 --> 00:13:23.920 align:middle line:84% namely, the price vector p and wealth w. 00:13:23.920 --> 00:13:26.680 align:middle line:84% So it is the maximal attainable utility 00:13:26.680 --> 00:13:30.640 align:middle line:84% in the budget set as a function of those parameters. 00:13:30.640 --> 00:13:35.140 align:middle line:84% And the bottom line here-- you can just see, if x of p and w 00:13:35.140 --> 00:13:37.780 align:middle line:84% is the maximizer, we just stick it back 00:13:37.780 --> 00:13:39.520 align:middle line:84% into the utility function, and that 00:13:39.520 --> 00:13:41.215 align:middle line:90% would be the optimized value. 00:13:41.215 --> 00:13:44.310 align:middle line:90% 00:13:44.310 --> 00:13:47.430 align:middle line:84% So there are very useful properties 00:13:47.430 --> 00:13:50.820 align:middle line:84% of this indirect utility function, 00:13:50.820 --> 00:13:53.740 align:middle line:84% and they are reviewed on this slide. 00:13:53.740 --> 00:13:56.460 align:middle line:84% Let's assume the consumption set is 00:13:56.460 --> 00:14:00.780 align:middle line:84% a subset of finite dimensional, L-dimensional Euclidean space. 00:14:00.780 --> 00:14:03.180 align:middle line:84% Utility functions are continuous, 00:14:03.180 --> 00:14:05.220 align:middle line:84% representing a preference relation that 00:14:05.220 --> 00:14:07.680 align:middle line:90% is rational and continuous. 00:14:07.680 --> 00:14:10.860 align:middle line:84% The indirect utility just defined. 00:14:10.860 --> 00:14:14.610 align:middle line:84% First of all, this v as a function of p and w 00:14:14.610 --> 00:14:18.780 align:middle line:84% is homogeneous of degree 0 in p and w, 00:14:18.780 --> 00:14:22.950 align:middle line:84% because as you already know, if you, for example, double prices 00:14:22.950 --> 00:14:26.340 align:middle line:84% and wealth, it doesn't change the picture of consumer 00:14:26.340 --> 00:14:30.030 align:middle line:84% maximization and you get the same solution. 00:14:30.030 --> 00:14:32.100 align:middle line:84% It's like, there's no money illusion here. 00:14:32.100 --> 00:14:35.580 align:middle line:84% If price is double, but wealth loss also doubles, 00:14:35.580 --> 00:14:39.540 align:middle line:84% you're just as well off as you were before, and not worse off. 00:14:39.540 --> 00:14:43.500 align:middle line:84% This indirect utility function has intuitive derivatives 00:14:43.500 --> 00:14:45.660 align:middle line:90% with respect to p and w. 00:14:45.660 --> 00:14:48.930 align:middle line:84% Increasing p makes you worse off, or at least not better 00:14:48.930 --> 00:14:50.280 align:middle line:90% off. 00:14:50.280 --> 00:14:53.610 align:middle line:84% Increasing w makes you better off. 00:14:53.610 --> 00:14:54.840 align:middle line:90% w is wealth. 00:14:54.840 --> 00:14:57.040 align:middle line:90% Prices are what you have to pay. 00:14:57.040 --> 00:15:00.180 align:middle line:84% In fact, that indirect utility will be strictly increasing 00:15:00.180 --> 00:15:03.690 align:middle line:84% in w when we have local non-satiation, which 00:15:03.690 --> 00:15:06.600 align:middle line:84% is what we're always going to be assuming here. 00:15:06.600 --> 00:15:13.380 align:middle line:84% The indirect utility function is quasi-convex, not concave-- 00:15:13.380 --> 00:15:15.900 align:middle line:90% convex in p and w. 00:15:15.900 --> 00:15:19.770 align:middle line:84% So again, when we did consumer theory, we had concave utility. 00:15:19.770 --> 00:15:22.350 align:middle line:84% Upper contour sets where quasi-convex, 00:15:22.350 --> 00:15:24.870 align:middle line:84% but the utility function was concave. 00:15:24.870 --> 00:15:27.720 align:middle line:84% This maximized utility over the budget 00:15:27.720 --> 00:15:31.440 align:middle line:84% is convex, or at least quasi-convex. 00:15:31.440 --> 00:15:36.750 align:middle line:84% Continuous, which is easy to assume, and differentiable. 00:15:36.750 --> 00:15:41.310 align:middle line:84% And when the indirect utility function is differentiable, 00:15:41.310 --> 00:15:44.610 align:middle line:90% we can use it to recover demand. 00:15:44.610 --> 00:15:46.620 align:middle line:84% And the proof is pretty easy, but here, 00:15:46.620 --> 00:15:49.180 align:middle line:84% it looks kind of amazing, I suppose, 00:15:49.180 --> 00:15:53.460 align:middle line:84% which is if we want the maximized quantity of good L 00:15:53.460 --> 00:15:59.100 align:middle line:84% for this consumer facing price vector p and wealth w, 00:15:59.100 --> 00:16:01.470 align:middle line:84% the answer will be take the derivative 00:16:01.470 --> 00:16:06.720 align:middle line:84% of the indirect utility function with respect to the L-th price, 00:16:06.720 --> 00:16:10.200 align:middle line:84% and divide by the derivative of the indirect utility 00:16:10.200 --> 00:16:13.620 align:middle line:84% function with respect to wealth, and stick a negative sign 00:16:13.620 --> 00:16:16.380 align:middle line:90% in front of all of that. 00:16:16.380 --> 00:16:18.570 align:middle line:90% So here's a little proof of it. 00:16:18.570 --> 00:16:21.810 align:middle line:84% We have the Lagrangian, which represents a utility 00:16:21.810 --> 00:16:27.140 align:middle line:84% maximization problem, with 0 being the shadow 00:16:27.140 --> 00:16:30.020 align:middle line:90% price of the budget and the L-- 00:16:30.020 --> 00:16:33.200 align:middle line:84% sorry, are lambda sub 0, and lambda sub L 00:16:33.200 --> 00:16:38.660 align:middle line:84% being potential shadow prices on non-negativity constraints. 00:16:38.660 --> 00:16:41.620 align:middle line:84% So we're going to use the envelope theorem, which 00:16:41.620 --> 00:16:44.800 align:middle line:84% to remind you, states that you can 00:16:44.800 --> 00:16:47.860 align:middle line:84% take a derivative with respect to the parameters 00:16:47.860 --> 00:16:51.820 align:middle line:84% of the optimization problem, in this case, p and w. 00:16:51.820 --> 00:16:56.290 align:middle line:84% And if you take the derivatives at the optimizing solution 00:16:56.290 --> 00:17:00.850 align:middle line:84% to get sort of the increment in utility associated 00:17:00.850 --> 00:17:03.170 align:middle line:90% with changing those parameters. 00:17:03.170 --> 00:17:07.510 align:middle line:84% So the increment in utility from solving the max problem 00:17:07.510 --> 00:17:12.119 align:middle line:84% at those parameters is also the derivative of the Lagrangian 00:17:12.119 --> 00:17:13.960 align:middle line:84% with respect to those parameters. 00:17:13.960 --> 00:17:15.670 align:middle line:84% So there's, like, two steps there-- 00:17:15.670 --> 00:17:18.609 align:middle line:84% how to represent the maximum value, namely, 00:17:18.609 --> 00:17:23.380 align:middle line:84% as a solution to the Lagrangian, and also the envelope theorem 00:17:23.380 --> 00:17:24.430 align:middle line:90% that tells you-- 00:17:24.430 --> 00:17:28.420 align:middle line:84% reminds you that the derivative of the optimized solution 00:17:28.420 --> 00:17:32.260 align:middle line:84% with respect to the parameter is, like, differentiating 00:17:32.260 --> 00:17:35.240 align:middle line:90% at the maximizing quantities. 00:17:35.240 --> 00:17:36.770 align:middle line:90% So too many words. 00:17:36.770 --> 00:17:40.630 align:middle line:84% So the first derivative here in equation 3, the derivative 00:17:40.630 --> 00:17:43.120 align:middle line:84% of the value function with respect to a particular price 00:17:43.120 --> 00:17:45.640 align:middle line:84% pl is the derivative of the Lagrangian with respect 00:17:45.640 --> 00:17:50.200 align:middle line:84% to-- at price pl evaluated at the maximum, namely, 00:17:50.200 --> 00:17:53.020 align:middle line:90% x sub p and w is the demand. 00:17:53.020 --> 00:17:55.300 align:middle line:84% And going back upstairs here, derivative 00:17:55.300 --> 00:17:58.162 align:middle line:84% of the Lagrangian with respect to pl, where does it appear? 00:17:58.162 --> 00:17:59.620 align:middle line:84% It only appears here in the budget. 00:17:59.620 --> 00:18:03.850 align:middle line:84% So you get a lambda 0, and p is linear in x. 00:18:03.850 --> 00:18:08.020 align:middle line:84% So you get the x sub l back, and that's this object. 00:18:08.020 --> 00:18:10.450 align:middle line:84% Likewise, the derivative with the indirect utility 00:18:10.450 --> 00:18:13.090 align:middle line:84% with respect to w, derivative of the Lagrangian with respect 00:18:13.090 --> 00:18:17.380 align:middle line:84% to w wealth, evaluated at the maximizing solution. 00:18:17.380 --> 00:18:19.210 align:middle line:84% There's only one place that w appears. 00:18:19.210 --> 00:18:20.150 align:middle line:90% It's up here. 00:18:20.150 --> 00:18:22.210 align:middle line:90% So you just get lambda 0. 00:18:22.210 --> 00:18:25.720 align:middle line:84% So lambda 0 is now in 4 and in 3. 00:18:25.720 --> 00:18:29.290 align:middle line:84% Substitute the expression in 4 into 3, 00:18:29.290 --> 00:18:33.340 align:middle line:90% and you get dv dp l equal to-- 00:18:33.340 --> 00:18:35.290 align:middle line:90% well, it's all written out here. 00:18:35.290 --> 00:18:37.840 align:middle line:84% Just basically taking the lambda 0, 00:18:37.840 --> 00:18:41.140 align:middle line:84% looking at the left-hand side, sticking it in here, 00:18:41.140 --> 00:18:43.270 align:middle line:90% and then repeating the x sub l. 00:18:43.270 --> 00:18:46.210 align:middle line:84% So now, solve this equation for x sub l 00:18:46.210 --> 00:18:49.000 align:middle line:84% and you get what we wanted, namely, 00:18:49.000 --> 00:18:51.520 align:middle line:84% the ratio of the derivative with respect to p 00:18:51.520 --> 00:18:54.230 align:middle line:84% divided by-- with respect to w with the minus 00:18:54.230 --> 00:18:55.330 align:middle line:90% sign in front of it. 00:18:55.330 --> 00:18:57.560 align:middle line:90% So we get what we wanted. 00:18:57.560 --> 00:18:58.420 align:middle line:90% Questions? 00:18:58.420 --> 00:19:01.200 align:middle line:90% 00:19:01.200 --> 00:19:04.970 align:middle line:84% So that's an interesting problem of the indirect utilities. 00:19:04.970 --> 00:19:09.620 align:middle line:84% Now, we're ready to define a particular version 00:19:09.620 --> 00:19:13.460 align:middle line:84% of the indirect utility function called Gorman Polar form, 00:19:13.460 --> 00:19:15.590 align:middle line:90% or Gorman form. 00:19:15.590 --> 00:19:19.220 align:middle line:84% This doesn't always happen, but it can happen. 00:19:19.220 --> 00:19:20.930 align:middle line:84% For particular utility functions, 00:19:20.930 --> 00:19:24.290 align:middle line:84% the indirect utility function in p and w 00:19:24.290 --> 00:19:26.810 align:middle line:84% has an intercept and a slope and. 00:19:26.810 --> 00:19:30.110 align:middle line:84% The intercept is a function of vector p. 00:19:30.110 --> 00:19:34.700 align:middle line:84% The slope b with respect to w is a function of vector p, 00:19:34.700 --> 00:19:36.350 align:middle line:90% and it's linear in w. 00:19:36.350 --> 00:19:38.270 align:middle line:84% And I'll show you an example right now, 00:19:38.270 --> 00:19:40.250 align:middle line:84% because this seems a bit mysterious. 00:19:40.250 --> 00:19:43.340 align:middle line:84% So let's go back to our favorite utility function, 00:19:43.340 --> 00:19:45.740 align:middle line:84% this sort of Cobb Douglass, with two goods, 00:19:45.740 --> 00:19:49.400 align:middle line:84% x1 and x2, raised to the power alpha and 1 minus alpha. 00:19:49.400 --> 00:19:53.000 align:middle line:84% You may remember that these alphas and 1 minus 00:19:53.000 --> 00:19:55.700 align:middle line:84% alpha represent the expenditure share 00:19:55.700 --> 00:19:58.670 align:middle line:90% in the optimized solution. 00:19:58.670 --> 00:20:02.240 align:middle line:90% So p1 times x1 is alpha times w. 00:20:02.240 --> 00:20:06.350 align:middle line:84% So this household spends alpha fraction of its wealth, w, 00:20:06.350 --> 00:20:12.290 align:middle line:84% on good 1, and spends 1 minus alpha times w on good 2. 00:20:12.290 --> 00:20:15.330 align:middle line:84% I'm just shifting these prices over here to the left-hand side 00:20:15.330 --> 00:20:16.460 align:middle line:90% so it's not just quantity. 00:20:16.460 --> 00:20:19.500 align:middle line:84% It's quantity times price or value. 00:20:19.500 --> 00:20:23.120 align:middle line:84% So with each Cobb Douglas, it's constant expenditure shares. 00:20:23.120 --> 00:20:25.410 align:middle line:90% More on that momentarily. 00:20:25.410 --> 00:20:28.160 align:middle line:84% And now, the definition of indirect utility 00:20:28.160 --> 00:20:32.190 align:middle line:84% says, just substitute in the maximized value. 00:20:32.190 --> 00:20:37.370 align:middle line:84% So we take Ux1, but use this expression for x1 00:20:37.370 --> 00:20:39.950 align:middle line:84% to the alpha x to this expression for x2 00:20:39.950 --> 00:20:41.450 align:middle line:90% to the 1 minus alpha. 00:20:41.450 --> 00:20:42.770 align:middle line:90% That's this business. 00:20:42.770 --> 00:20:46.970 align:middle line:84% And when you solve it out, you get of a of p, 00:20:46.970 --> 00:20:50.390 align:middle line:84% looking for it to be in this form. 00:20:50.390 --> 00:20:54.200 align:middle line:84% a of p is 0, and b of p is this coefficient, [INAUDIBLE] 00:20:54.200 --> 00:20:58.400 align:middle line:84% alpha 1 minus alpha p1 and p2 to these powers alpha 00:20:58.400 --> 00:20:59.900 align:middle line:90% and 1 minus alpha. 00:20:59.900 --> 00:21:01.130 align:middle line:90% That's b of p. 00:21:01.130 --> 00:21:03.810 align:middle line:84% It depends on p and on nothing else. 00:21:03.810 --> 00:21:06.290 align:middle line:90% And it is indeed linear in w. 00:21:06.290 --> 00:21:09.650 align:middle line:84% The linearity in w just comes from the property that 00:21:09.650 --> 00:21:12.530 align:middle line:90% alpha plus 1 minus alpha is 1. 00:21:12.530 --> 00:21:15.410 align:middle line:84% w the alpha times w to the 1 minus alpha. 00:21:15.410 --> 00:21:18.360 align:middle line:84% Just add up the power coefficients. 00:21:18.360 --> 00:21:22.010 align:middle line:84% So this is all, so far, for a single household, 00:21:22.010 --> 00:21:25.970 align:middle line:84% hence neglecting to put in notation h or i, 00:21:25.970 --> 00:21:28.290 align:middle line:84% or whatever label the household is. 00:21:28.290 --> 00:21:30.800 align:middle line:84% However, if this Cobb Douglas function 00:21:30.800 --> 00:21:33.560 align:middle line:84% were the same over all of the households, 00:21:33.560 --> 00:21:37.820 align:middle line:84% then we have a Gorman Polar form, 00:21:37.820 --> 00:21:41.930 align:middle line:84% where all of the households have the same intercept, a of p, 00:21:41.930 --> 00:21:44.360 align:middle line:90% and the same slope term, b of p. 00:21:44.360 --> 00:21:48.530 align:middle line:84% Likewise, if it were to be the case that the alpha coefficient 00:21:48.530 --> 00:21:50.390 align:middle line:84% varies across over the households, 00:21:50.390 --> 00:21:53.480 align:middle line:84% then this BSP is not going to be common over all 00:21:53.480 --> 00:21:54.230 align:middle line:90% of the households. 00:21:54.230 --> 00:21:57.620 align:middle line:84% It's going to depend on particular preferences. 00:21:57.620 --> 00:22:01.130 align:middle line:84% So maybe that's illustrative just how tenuous, already, 00:22:01.130 --> 00:22:02.930 align:middle line:90% the assumption is going to be. 00:22:02.930 --> 00:22:06.290 align:middle line:84% But it is going to be very powerful if it's true. 00:22:06.290 --> 00:22:10.950 align:middle line:84% So let's go at it and talk about aggregate demand. 00:22:10.950 --> 00:22:14.960 align:middle line:84% So we have an economy with I households, I agents. 00:22:14.960 --> 00:22:18.500 align:middle line:90% Their wealths are w1 through wI. 00:22:18.500 --> 00:22:20.210 align:middle line:90% A wealth profile. 00:22:20.210 --> 00:22:25.010 align:middle line:84% Let's define w bar to be the aggregate total wealth, just 00:22:25.010 --> 00:22:27.770 align:middle line:84% by summing up all of the individual wealths. 00:22:27.770 --> 00:22:31.550 align:middle line:84% And retain this notation that xI of p and wI 00:22:31.550 --> 00:22:34.400 align:middle line:84% is the demand function for agent I. 00:22:34.400 --> 00:22:39.020 align:middle line:84% The aggregate demand, x bar, by definition, 00:22:39.020 --> 00:22:42.410 align:middle line:84% will be the sum of the individual demands. 00:22:42.410 --> 00:22:47.630 align:middle line:84% And we write that notationally as x bar of the vector-- 00:22:47.630 --> 00:22:51.680 align:middle line:84% the parameters of the problem, namely, the common price vector 00:22:51.680 --> 00:22:53.930 align:middle line:90% and individual wealths. 00:22:53.930 --> 00:22:56.750 align:middle line:84% So that's the definition of aggregate demand. 00:22:56.750 --> 00:23:00.170 align:middle line:84% Nothing so far other than the definition. 00:23:00.170 --> 00:23:02.660 align:middle line:84% But the first question about aggregation 00:23:02.660 --> 00:23:08.060 align:middle line:84% is, can we find an aggregate demand function that is special 00:23:08.060 --> 00:23:12.260 align:middle line:84% in that it only depends on total wealth in the population, 00:23:12.260 --> 00:23:16.200 align:middle line:84% not how the wealth is distributed in the population? 00:23:16.200 --> 00:23:19.580 align:middle line:84% So the idea here is this x bar of L-- 00:23:19.580 --> 00:23:21.300 align:middle line:90% which, again, was just defined. 00:23:21.300 --> 00:23:26.990 align:middle line:84% But now, for the L-th good, x bar of L at p and w 00:23:26.990 --> 00:23:29.450 align:middle line:90% is the aggregate demand. 00:23:29.450 --> 00:23:33.800 align:middle line:84% We're looking for this function, capital X, now, of L-- 00:23:33.800 --> 00:23:36.270 align:middle line:84% capital, I guess, for big aggregate, 00:23:36.270 --> 00:23:39.860 align:middle line:84% which is a function of p, of course, but now w bar. 00:23:39.860 --> 00:23:41.810 align:middle line:90% So here's the content of it. 00:23:41.810 --> 00:23:44.810 align:middle line:84% Or looking at it this way, vector p, 00:23:44.810 --> 00:23:49.010 align:middle line:84% yes, but the wealth component in the aggregate demand 00:23:49.010 --> 00:23:51.950 align:middle line:84% depends on the sum of individual wealths only. 00:23:51.950 --> 00:23:54.590 align:middle line:84% It doesn't depend on the distribution. 00:23:54.590 --> 00:23:59.430 align:middle line:84% On this slide, wealth enter each agent's demand function. 00:23:59.430 --> 00:24:01.650 align:middle line:84% And there's no reason why it should 00:24:01.650 --> 00:24:06.180 align:middle line:84% be representable to depend only on the aggregate. 00:24:06.180 --> 00:24:09.210 align:middle line:84% But we're going to look for specifications of utility 00:24:09.210 --> 00:24:10.680 align:middle line:90% where that's true. 00:24:10.680 --> 00:24:14.520 align:middle line:84% And we haven't even asked whether that aggregate demand 00:24:14.520 --> 00:24:19.740 align:middle line:84% function, that capital X, comes from some underlying utility 00:24:19.740 --> 00:24:21.480 align:middle line:90% function as a solution. 00:24:21.480 --> 00:24:24.120 align:middle line:84% We're just looking for whether it depends only 00:24:24.120 --> 00:24:26.370 align:middle line:84% on aggregate wealth, or rather, might 00:24:26.370 --> 00:24:28.330 align:middle line:90% depend on the distribution. 00:24:28.330 --> 00:24:30.750 align:middle line:84% But this is a necessary condition for there 00:24:30.750 --> 00:24:33.630 align:middle line:84% to be a representative household that's 00:24:33.630 --> 00:24:36.660 align:middle line:84% maximizing some pseudo-utility function that's going 00:24:36.660 --> 00:24:38.620 align:middle line:90% to represent aggregate demand. 00:24:38.620 --> 00:24:40.870 align:middle line:84% So we're kind of doing one thing at a time. 00:24:40.870 --> 00:24:44.400 align:middle line:84% So what is a necessary condition for us 00:24:44.400 --> 00:24:47.010 align:middle line:84% to be able to aggregate up like that? 00:24:47.010 --> 00:24:49.920 align:middle line:84% And it is that the marginal impact 00:24:49.920 --> 00:24:54.060 align:middle line:84% of wealth changes on demands, say, for the L-th good, 00:24:54.060 --> 00:24:57.090 align:middle line:84% is the same over all of the households, 00:24:57.090 --> 00:25:00.150 align:middle line:84% and that is also true for all of the goods. 00:25:00.150 --> 00:25:04.650 align:middle line:84% So whatever good L you pick, whatever pairwise households I 00:25:04.650 --> 00:25:09.120 align:middle line:84% and J you pick, they need each to have the same income 00:25:09.120 --> 00:25:12.150 align:middle line:84% effect of how demand changes with wealth. 00:25:12.150 --> 00:25:16.140 align:middle line:84% And here's the picture, which you've kind of seen before. 00:25:16.140 --> 00:25:19.770 align:middle line:84% These are linear wealth expansion paths. 00:25:19.770 --> 00:25:23.460 align:middle line:84% We're holding prices fixed of the two goods, in this case. 00:25:23.460 --> 00:25:27.600 align:middle line:84% And for consumer-- one of these consumers, 00:25:27.600 --> 00:25:33.210 align:middle line:84% we're tracing out the optimized solution as we change wealth-- 00:25:33.210 --> 00:25:36.690 align:middle line:84% optimizing in the sense that the household has achieved 00:25:36.690 --> 00:25:39.090 align:middle line:84% maximal utility for any given wealth 00:25:39.090 --> 00:25:41.670 align:middle line:84% characterized by a tangency between the indifference 00:25:41.670 --> 00:25:43.470 align:middle line:90% curve and the budget line. 00:25:43.470 --> 00:25:45.960 align:middle line:84% The other households up here also 00:25:45.960 --> 00:25:48.000 align:middle line:90% has a linear expansion path. 00:25:48.000 --> 00:25:51.030 align:middle line:84% Now, it's true that this household-- 00:25:51.030 --> 00:25:55.260 align:middle line:84% I, it is-- has higher levels of demand for good 2 00:25:55.260 --> 00:26:01.350 align:middle line:84% relative to good 1 than is the case for household J. 00:26:01.350 --> 00:26:03.210 align:middle line:90% So they're not identical. 00:26:03.210 --> 00:26:06.480 align:middle line:84% But when you think about what happens when you redistribute 00:26:06.480 --> 00:26:10.230 align:middle line:84% wealth with these parallel linear expansion paths, 00:26:10.230 --> 00:26:11.970 align:middle line:90% the derivatives are the same. 00:26:11.970 --> 00:26:16.920 align:middle line:84% We could, for example, take some wealth away from household I. 00:26:16.920 --> 00:26:19.920 align:middle line:90% We would move southwest. 00:26:19.920 --> 00:26:23.250 align:middle line:84% We'd have the change in x1, x2, and the change 00:26:23.250 --> 00:26:26.340 align:middle line:84% in x1 little triangle over there, 00:26:26.340 --> 00:26:29.520 align:middle line:84% and give that wealth to household 00:26:29.520 --> 00:26:32.550 align:middle line:84% J, who would be moving northeast. 00:26:32.550 --> 00:26:35.520 align:middle line:84% But again, because these lines are parallel, 00:26:35.520 --> 00:26:40.030 align:middle line:84% we're going to have exactly the same changes [INAUDIBLE].. 00:26:40.030 --> 00:26:42.210 align:middle line:84% So the levels can be different, but the derivatives 00:26:42.210 --> 00:26:42.930 align:middle line:90% are the same. 00:26:42.930 --> 00:26:47.040 align:middle line:84% And that picture represents this condition 10, 00:26:47.040 --> 00:26:48.750 align:middle line:84% that the income effects are the same 00:26:48.750 --> 00:26:50.620 align:middle line:90% over all of the households. 00:26:50.620 --> 00:26:56.460 align:middle line:84% And that's a necessary condition for only aggregate wealth 00:26:56.460 --> 00:26:57.840 align:middle line:90% to matter. 00:26:57.840 --> 00:27:02.330 align:middle line:84% Individual demands are moving as you redistribute the wealth. 00:27:02.330 --> 00:27:06.300 align:middle line:84% But what one guy gives up is what the other guy gains. 00:27:06.300 --> 00:27:09.360 align:middle line:84% So the total demand isn't moving. 00:27:09.360 --> 00:27:12.450 align:middle line:84% Who gets what is moving, but not the total demand, 00:27:12.450 --> 00:27:16.480 align:middle line:90% as we redistribute the wealth. 00:27:16.480 --> 00:27:20.080 align:middle line:84% The next step is to find a utility function that's going 00:27:20.080 --> 00:27:23.080 align:middle line:90% to give that aggregate demand. 00:27:23.080 --> 00:27:25.960 align:middle line:84% We're going to call this guy the positive representative 00:27:25.960 --> 00:27:27.290 align:middle line:90% household. 00:27:27.290 --> 00:27:29.080 align:middle line:84% So let's have the consumption set 00:27:29.080 --> 00:27:33.310 align:middle line:84% be a subset of this L-dimension Euclidean space. 00:27:33.310 --> 00:27:38.710 align:middle line:84% The individual demands, xI of vector p and wI 00:27:38.710 --> 00:27:40.780 align:middle line:84% for households I over all of them 00:27:40.780 --> 00:27:42.670 align:middle line:90% to be the individual demands. 00:27:42.670 --> 00:27:47.080 align:middle line:84% We're looking for a preference ordering with a star on it 00:27:47.080 --> 00:27:50.890 align:middle line:84% on the aggregate consumption set, 00:27:50.890 --> 00:27:53.590 align:middle line:84% and a particular utility function, u star, 00:27:53.590 --> 00:27:56.620 align:middle line:84% that represents this preference ordering. 00:27:56.620 --> 00:28:00.580 align:middle line:84% And that object, consumption set and the preference ordering 00:28:00.580 --> 00:28:03.910 align:middle line:84% in particular, is said to represent or correspond 00:28:03.910 --> 00:28:06.760 align:middle line:84% with a positive representative household 00:28:06.760 --> 00:28:10.570 align:middle line:84% if and only if the aggregate demand, that capital 00:28:10.570 --> 00:28:15.400 align:middle line:84% X of p and w bar, comes from the utility maximization problem 00:28:15.400 --> 00:28:20.120 align:middle line:84% where the utility function is of the form u star. 00:28:20.120 --> 00:28:24.280 align:middle line:84% So again, we're used to utility maximization. 00:28:24.280 --> 00:28:26.860 align:middle line:90% u star is some utility function. 00:28:26.860 --> 00:28:28.480 align:middle line:90% We maximize it. 00:28:28.480 --> 00:28:31.810 align:middle line:84% Oh, subject to the budget, what is the budget? 00:28:31.810 --> 00:28:34.630 align:middle line:84% It's like giving this, quote, "person," 00:28:34.630 --> 00:28:37.210 align:middle line:84% this representative household, the entire wealth 00:28:37.210 --> 00:28:38.950 align:middle line:90% of the economy, w bar. 00:28:38.950 --> 00:28:41.770 align:middle line:84% And this X thing, in case you've forgotten, 00:28:41.770 --> 00:28:45.970 align:middle line:84% this capital X of p and w bar, was the object way back here 00:28:45.970 --> 00:28:49.780 align:middle line:84% that we were looking for, the aggregate demand 00:28:49.780 --> 00:28:52.540 align:middle line:84% as a function only of the sum of the wealths, 00:28:52.540 --> 00:28:54.970 align:middle line:90% or aggregate wealth. 00:28:54.970 --> 00:28:59.470 align:middle line:84% So under what conditions can we find this amazing u star? 00:28:59.470 --> 00:29:02.480 align:middle line:90% Yep, Gorman Polar form. 00:29:02.480 --> 00:29:05.620 align:middle line:84% So a necessary and sufficient condition 00:29:05.620 --> 00:29:08.350 align:middle line:84% to get that equation tend to be true 00:29:08.350 --> 00:29:13.180 align:middle line:84% is that everybody's preferences can be represented by 16, 00:29:13.180 --> 00:29:16.900 align:middle line:84% namely, now with the I's to be explicit, 00:29:16.900 --> 00:29:20.680 align:middle line:84% the indirect utility function for household or agent 00:29:20.680 --> 00:29:24.670 align:middle line:90% I has this intercept and slope. 00:29:24.670 --> 00:29:28.840 align:middle line:84% The intercept term can actually depend on i. 00:29:28.840 --> 00:29:32.110 align:middle line:84% We do allow that degree of heterogeneity. 00:29:32.110 --> 00:29:35.920 align:middle line:84% But the slope with respect to w and wI 00:29:35.920 --> 00:29:38.890 align:middle line:84% for household I does not depend on i. 00:29:38.890 --> 00:29:42.150 align:middle line:90% Common slope terms. 00:29:42.150 --> 00:29:44.120 align:middle line:84% So again, to write it out, the way 00:29:44.120 --> 00:29:48.530 align:middle line:84% the indirect utility functions for i and j respond to wealth 00:29:48.530 --> 00:29:49.380 align:middle line:90% is the same. 00:29:49.380 --> 00:29:52.550 align:middle line:84% Namely, it's bp, which is not quite the picture 00:29:52.550 --> 00:29:54.950 align:middle line:84% we were looking at, but it's getting close, 00:29:54.950 --> 00:29:57.860 align:middle line:90% as you can imagine. 00:29:57.860 --> 00:30:02.660 align:middle line:84% So if 16 is true over all households i, 00:30:02.660 --> 00:30:06.320 align:middle line:84% then we can find our amazing u star, 00:30:06.320 --> 00:30:08.330 align:middle line:84% or in particular, the indirect utility 00:30:08.330 --> 00:30:11.120 align:middle line:84% function corresponding to u star. 00:30:11.120 --> 00:30:14.780 align:middle line:84% And it's going to look a lot like the individual indirect 00:30:14.780 --> 00:30:17.120 align:middle line:84% utility functions in the sense that it's 00:30:17.120 --> 00:30:20.090 align:middle line:90% written down as a times b of w. 00:30:20.090 --> 00:30:24.830 align:middle line:84% The a star for the representative consumer 00:30:24.830 --> 00:30:28.670 align:middle line:84% is the sum of the aI of ps over i. 00:30:28.670 --> 00:30:30.080 align:middle line:90% So we just add them up. 00:30:30.080 --> 00:30:34.040 align:middle line:84% And the b star is the common b of p 00:30:34.040 --> 00:30:36.170 align:middle line:90% over all of the households i. 00:30:36.170 --> 00:30:39.920 align:middle line:84% So then this guy, represented by the indirect utility function, 00:30:39.920 --> 00:30:43.060 align:middle line:84% is the positive representative household. 00:30:43.060 --> 00:30:48.850 align:middle line:84% So it's an if and only thing, but let's just go one way. 00:30:48.850 --> 00:30:53.200 align:middle line:84% If and only if means if they have these polar forms, 00:30:53.200 --> 00:30:55.150 align:middle line:84% then it's Gorman-agreeable and we 00:30:55.150 --> 00:30:56.860 align:middle line:90% have a representative consumer. 00:30:56.860 --> 00:31:00.290 align:middle line:84% And also, if we have a representative consumer, 00:31:00.290 --> 00:31:01.900 align:middle line:84% then the underlying households have 00:31:01.900 --> 00:31:06.530 align:middle line:84% to have utility represented by 16. 00:31:06.530 --> 00:31:07.850 align:middle line:90% So let's just go one way. 00:31:07.850 --> 00:31:12.220 align:middle line:84% Suppose that preferences have indirect utility functions that 00:31:12.220 --> 00:31:13.480 align:middle line:90% satisfy 16. 00:31:13.480 --> 00:31:16.400 align:middle line:84% We'll go from the micro to the macro, so to speak. 00:31:16.400 --> 00:31:18.730 align:middle line:90% Now, we use Roy's identity. 00:31:18.730 --> 00:31:22.180 align:middle line:84% You were probably wondering why we ever had that. 00:31:22.180 --> 00:31:26.650 align:middle line:84% So demand of household i for good l at price vector 00:31:26.650 --> 00:31:31.510 align:middle line:84% p and wealth wi is just this ratio of derivatives 00:31:31.510 --> 00:31:33.440 align:middle line:90% with the minus sign out front. 00:31:33.440 --> 00:31:36.370 align:middle line:84% And now, we use the particular forms, namely, 00:31:36.370 --> 00:31:38.590 align:middle line:84% the derivative of the indirect utility function 00:31:38.590 --> 00:31:42.220 align:middle line:84% with respect to wi is namely just b of p, 00:31:42.220 --> 00:31:44.740 align:middle line:84% because that's that coefficient there. 00:31:44.740 --> 00:31:47.500 align:middle line:84% And the derivative of the indirect utility function 00:31:47.500 --> 00:31:50.470 align:middle line:90% with respect to p is what? 00:31:50.470 --> 00:31:51.550 align:middle line:90% Well, go back here. 00:31:51.550 --> 00:31:53.980 align:middle line:84% It's indirect utility function for household i. 00:31:53.980 --> 00:31:57.010 align:middle line:84% Take derivatives with respect to the intercept and the slope 00:31:57.010 --> 00:31:59.140 align:middle line:90% term with respect to p. 00:31:59.140 --> 00:32:00.580 align:middle line:84% So that's what's written out here, 00:32:00.580 --> 00:32:02.500 align:middle line:84% the derivative of the intercept and slope 00:32:02.500 --> 00:32:03.910 align:middle line:90% terms with respect to p. 00:32:03.910 --> 00:32:07.400 align:middle line:84% And of course, that w is hanging on there as a coefficient. 00:32:07.400 --> 00:32:09.250 align:middle line:90% So this must be true. 00:32:09.250 --> 00:32:13.840 align:middle line:84% Now, having derived the demands of household i, 00:32:13.840 --> 00:32:16.600 align:middle line:84% we can take the derivative with respect to wealth. 00:32:16.600 --> 00:32:19.070 align:middle line:84% And it's just going to appear here. 00:32:19.070 --> 00:32:23.980 align:middle line:84% So we get this times this as the way demand is changing. 00:32:23.980 --> 00:32:26.800 align:middle line:84% Well, one thing you can see, on the right-hand side 00:32:26.800 --> 00:32:28.180 align:middle line:90% here, there's no i. 00:32:28.180 --> 00:32:31.960 align:middle line:84% So the change in demands with respect to wealth 00:32:31.960 --> 00:32:34.900 align:middle line:84% are the same across all of the households, 00:32:34.900 --> 00:32:38.090 align:middle line:84% and equal to this common term on the right-hand side. 00:32:38.090 --> 00:32:41.440 align:middle line:84% So this gives us that with the indirect utility function, 00:32:41.440 --> 00:32:43.870 align:middle line:84% we have these constant income effects 00:32:43.870 --> 00:32:45.910 align:middle line:90% over all of the households. 00:32:45.910 --> 00:32:49.150 align:middle line:84% Now, the second thing we want to show is not just that, 00:32:49.150 --> 00:32:52.390 align:middle line:84% but that we can find this amazing representative 00:32:52.390 --> 00:32:53.630 align:middle line:90% consumer. 00:32:53.630 --> 00:32:59.470 align:middle line:84% So we want-- when we maximize whatever this v star is with 00:32:59.470 --> 00:33:01.150 align:middle line:90% respect to-- 00:33:01.150 --> 00:33:04.540 align:middle line:84% when we get this maximized solution for this star 00:33:04.540 --> 00:33:08.500 align:middle line:84% representative consumer and aggregate wealth w, 00:33:08.500 --> 00:33:11.185 align:middle line:84% it should represent the sum of the individual demands. 00:33:11.185 --> 00:33:14.070 align:middle line:90% 00:33:14.070 --> 00:33:18.900 align:middle line:84% So you have this expression already for-- 00:33:18.900 --> 00:33:24.300 align:middle line:84% let me remind you, the v star, the candidate, is 17. 00:33:24.300 --> 00:33:27.630 align:middle line:84% So we're going to prove that it works. 00:33:27.630 --> 00:33:32.340 align:middle line:84% Take v star indirect utility to be this a star, b star, 00:33:32.340 --> 00:33:35.490 align:middle line:90% w bar term. 00:33:35.490 --> 00:33:40.140 align:middle line:84% Then by Roy's identity, that's just another indirect utility 00:33:40.140 --> 00:33:41.320 align:middle line:90% function. 00:33:41.320 --> 00:33:46.020 align:middle line:84% So the demand that comes from it at the aggregate wealth w bar 00:33:46.020 --> 00:33:49.950 align:middle line:84% and price vector p would just be the ratio 00:33:49.950 --> 00:33:54.210 align:middle line:84% of these marginal effects with a negative sign 00:33:54.210 --> 00:33:55.830 align:middle line:90% out in front of it. 00:33:55.830 --> 00:33:58.900 align:middle line:84% And of course, we know the form of v star. 00:33:58.900 --> 00:34:02.370 align:middle line:84% So at an aggregated level, it's very much like before, 00:34:02.370 --> 00:34:08.320 align:middle line:84% 1 over bp, because that's just the wealth coefficient. 00:34:08.320 --> 00:34:11.400 align:middle line:84% And p enters into the intercept and slope 00:34:11.400 --> 00:34:14.100 align:middle line:84% term in this aggregated equation-- 00:34:14.100 --> 00:34:17.250 align:middle line:90% aggregate indirect utility 17. 00:34:17.250 --> 00:34:20.190 align:middle line:90% So we get this. 00:34:20.190 --> 00:34:23.820 align:middle line:84% Remember that the a star is the sum 00:34:23.820 --> 00:34:27.210 align:middle line:90% of the ai's by construction. 00:34:27.210 --> 00:34:30.739 align:middle line:84% So now, that is all written out very explicitly. 00:34:30.739 --> 00:34:33.020 align:middle line:90% Nothing going on with wealth. 00:34:33.020 --> 00:34:37.130 align:middle line:84% But now, we get to just take this summation 00:34:37.130 --> 00:34:39.000 align:middle line:90% and pull it outside. 00:34:39.000 --> 00:34:41.449 align:middle line:84% So we have the demand that we want 00:34:41.449 --> 00:34:46.429 align:middle line:84% is the summation over i of what's left over after pulling 00:34:46.429 --> 00:34:47.239 align:middle line:90% out the summation. 00:34:47.239 --> 00:34:49.489 align:middle line:90% We have the individual terms. 00:34:49.489 --> 00:34:52.980 align:middle line:84% But of course, that is nothing other than, again, 00:34:52.980 --> 00:34:57.500 align:middle line:84% from Roy's identity, the underlying demands, which, 00:34:57.500 --> 00:35:00.960 align:middle line:84% when we sum them up, is aggregate demand. 00:35:00.960 --> 00:35:05.900 align:middle line:84% So we've used this particular utility function, 17, 00:35:05.900 --> 00:35:10.610 align:middle line:84% to derive the aggregate demand, which is now 00:35:10.610 --> 00:35:13.700 align:middle line:84% proved to be the sum of the individual demands, 00:35:13.700 --> 00:35:15.980 align:middle line:84% and by this construction, depends only 00:35:15.980 --> 00:35:18.260 align:middle line:90% on aggregate wealth. 00:35:18.260 --> 00:35:20.510 align:middle line:90% The w bar is missing there. 00:35:20.510 --> 00:35:23.290 align:middle line:90% Questions? 00:35:23.290 --> 00:35:23.790 align:middle line:90% All right. 00:35:23.790 --> 00:35:28.850 align:middle line:84% So let me give you an example of the underlying utility 00:35:28.850 --> 00:35:31.610 align:middle line:90% functions and consumption sets. 00:35:31.610 --> 00:35:35.090 align:middle line:84% Suppose, being explicit about the heterogeneity, 00:35:35.090 --> 00:35:39.080 align:middle line:84% each household i's consumption set 00:35:39.080 --> 00:35:41.930 align:middle line:84% is a subset of this L-dimensional space. 00:35:41.930 --> 00:35:46.290 align:middle line:84% Namely, they're, like, subsistence or minimal bundles. 00:35:46.290 --> 00:35:53.130 align:middle line:84% So the amount of good l cannot be lower than some x lower bar 00:35:53.130 --> 00:35:54.990 align:middle line:90% l per household i. 00:35:54.990 --> 00:35:56.670 align:middle line:90% This is the heterogeneity. 00:35:56.670 --> 00:35:59.580 align:middle line:90% It can depend on household i. 00:35:59.580 --> 00:36:04.260 align:middle line:84% And let's have utility of household i over vector 00:36:04.260 --> 00:36:08.220 align:middle line:84% x just be the difference to a power 00:36:08.220 --> 00:36:13.760 align:middle line:84% between xl and those minimal subsistence bundles. 00:36:13.760 --> 00:36:16.730 align:middle line:84% So the heterogeneity is in the subsistence bundles, 00:36:16.730 --> 00:36:18.840 align:middle line:90% which can vary over households. 00:36:18.840 --> 00:36:21.170 align:middle line:90% But the sigmas are the same. 00:36:21.170 --> 00:36:22.160 align:middle line:90% It's not sigma i. 00:36:22.160 --> 00:36:26.330 align:middle line:84% It's sigma, common, across all of the households. 00:36:26.330 --> 00:36:29.050 align:middle line:84% So if we had an economy represented 00:36:29.050 --> 00:36:32.230 align:middle line:84% by these consumption sets and these preferences, 00:36:32.230 --> 00:36:36.290 align:middle line:84% we could look at their indirect utility. 00:36:36.290 --> 00:36:38.950 align:middle line:84% And if you do the math, you get this ugly thing. 00:36:38.950 --> 00:36:42.550 align:middle line:84% But it does have the right form, ai p plus bp 00:36:42.550 --> 00:36:45.640 align:middle line:90% times wi for household i. 00:36:45.640 --> 00:36:47.380 align:middle line:90% And then you add it up-- 00:36:47.380 --> 00:36:51.430 align:middle line:84% namely, add up this guy over households i. 00:36:51.430 --> 00:36:56.620 align:middle line:84% And you get the expression at the bottom of the slide. 00:36:56.620 --> 00:36:59.080 align:middle line:84% And in particular, the heterogeneity 00:36:59.080 --> 00:37:02.330 align:middle line:84% in the subsistence bundles is right here. 00:37:02.330 --> 00:37:04.210 align:middle line:90% So they're all different. 00:37:04.210 --> 00:37:06.610 align:middle line:90% They have different minimums. 00:37:06.610 --> 00:37:10.360 align:middle line:84% But miraculously, we're able to add up those minimums 00:37:10.360 --> 00:37:13.390 align:middle line:84% and call it kind of a pseudo-subsistence bundle 00:37:13.390 --> 00:37:16.810 align:middle line:84% for an aggregated representative household who 00:37:16.810 --> 00:37:20.950 align:middle line:84% would have the indirect utility that's written out here 00:37:20.950 --> 00:37:22.780 align:middle line:90% at the bottom of the slide. 00:37:22.780 --> 00:37:25.900 align:middle line:84% I don't have too many more examples in front of me today, 00:37:25.900 --> 00:37:27.490 align:middle line:90% but I mean, I do want to give-- 00:37:27.490 --> 00:37:29.800 align:middle line:84% I did want to give you one example, 00:37:29.800 --> 00:37:33.220 align:middle line:84% because this indirect utility space is a bit different 00:37:33.220 --> 00:37:34.540 align:middle line:90% and a bit abstract. 00:37:34.540 --> 00:37:38.740 align:middle line:84% So here, we actually start with underlying utility functions. 00:37:38.740 --> 00:37:41.980 align:middle line:90% 00:37:41.980 --> 00:37:44.450 align:middle line:90% I said all that in words. 00:37:44.450 --> 00:37:48.490 align:middle line:84% So now, let's go to the normative representative 00:37:48.490 --> 00:37:49.960 align:middle line:90% household. 00:37:49.960 --> 00:37:52.120 align:middle line:90% I should say, it's the same guy. 00:37:52.120 --> 00:37:54.010 align:middle line:90% It's the same person. 00:37:54.010 --> 00:37:56.740 align:middle line:90% It just has two characteristics. 00:37:56.740 --> 00:37:59.920 align:middle line:84% It can be used for positive economics in the sense 00:37:59.920 --> 00:38:04.420 align:middle line:84% that it's predictive of what the entire economy is doing, 00:38:04.420 --> 00:38:07.960 align:middle line:84% and it can also be used to make welfare statements 00:38:07.960 --> 00:38:11.920 align:middle line:84% by a rank ordering over the utility function. 00:38:11.920 --> 00:38:14.620 align:middle line:84% But there's not, like, one representative normative 00:38:14.620 --> 00:38:17.860 align:middle line:84% household and one representative positive household. 00:38:17.860 --> 00:38:19.840 align:middle line:84% It's a given representative household 00:38:19.840 --> 00:38:22.640 align:middle line:84% with both positive and normative characteristics, 00:38:22.640 --> 00:38:26.470 align:middle line:84% and we are about to go through the normative part. 00:38:26.470 --> 00:38:29.770 align:middle line:84% So let's have the individual demand functions-- 00:38:29.770 --> 00:38:31.760 align:middle line:84% this is-- the first little paragraph 00:38:31.760 --> 00:38:36.730 align:middle line:84% here is the positive part, that these demands can be derived 00:38:36.730 --> 00:38:40.270 align:middle line:84% as if through an aggregated demand function given 00:38:40.270 --> 00:38:42.380 align:middle line:90% the total wealth of the economy. 00:38:42.380 --> 00:38:45.160 align:middle line:84% Now, we want to ask the normative question. 00:38:45.160 --> 00:38:47.410 align:middle line:84% Given a set of individual demand functions, 00:38:47.410 --> 00:38:51.010 align:middle line:84% can we find this fictitious household i star such 00:38:51.010 --> 00:38:54.580 align:middle line:84% that Pareto optimal allocations in the economy 00:38:54.580 --> 00:38:58.480 align:middle line:84% can be thought of as coming from this representative household 00:38:58.480 --> 00:39:00.520 align:middle line:90% maximizing utility? 00:39:00.520 --> 00:39:03.405 align:middle line:84% I keep emphasizing fictitious, amazing, 00:39:03.405 --> 00:39:04.780 align:middle line:84% because I don't want you to think 00:39:04.780 --> 00:39:06.470 align:middle line:90% that this is a real person. 00:39:06.470 --> 00:39:08.720 align:middle line:90% It's a construct. 00:39:08.720 --> 00:39:11.920 align:middle line:84% So let's take our underlying economy, which 00:39:11.920 --> 00:39:16.390 align:middle line:84% we've been reading this line for the last two or three lectures. 00:39:16.390 --> 00:39:21.640 align:middle line:84% Underlying economy is script E characterized by consumption 00:39:21.640 --> 00:39:24.040 align:middle line:84% sets, utility functions over all of the households, 00:39:24.040 --> 00:39:27.100 align:middle line:84% production possibility sets over all of the firms, 00:39:27.100 --> 00:39:30.010 align:middle line:90% some aggregate endowment w bar. 00:39:30.010 --> 00:39:33.430 align:middle line:84% Now, finally, we define a normative representative 00:39:33.430 --> 00:39:36.820 align:middle line:84% household with two characteristics. 00:39:36.820 --> 00:39:41.320 align:middle line:84% If we started with an allocation x and y 00:39:41.320 --> 00:39:44.440 align:middle line:84% and we found an alternative allocation x prime, 00:39:44.440 --> 00:39:50.140 align:middle line:84% y prime, which Pareto dominates x, y, meaning 00:39:50.140 --> 00:39:53.860 align:middle line:84% at least as good for some and strictly better for others. 00:39:53.860 --> 00:39:58.120 align:middle line:84% Then under u star, evaluated that 00:39:58.120 --> 00:40:03.880 align:middle line:84% to some of the prime bundle is greater than u star evaluated 00:40:03.880 --> 00:40:05.380 align:middle line:90% at the original bundle. 00:40:05.380 --> 00:40:08.350 align:middle line:84% In other words, it's kind of intuitive, 00:40:08.350 --> 00:40:11.110 align:middle line:90% but it's not always true. 00:40:11.110 --> 00:40:13.840 align:middle line:84% Here, for the definition of Pareto dominance, 00:40:13.840 --> 00:40:16.450 align:middle line:84% we're looking at individual allocations over all 00:40:16.450 --> 00:40:19.360 align:middle line:84% the households and trying to find something 00:40:19.360 --> 00:40:21.280 align:middle line:84% that's at least as good for some and strictly 00:40:21.280 --> 00:40:22.790 align:middle line:90% better for the others. 00:40:22.790 --> 00:40:26.710 align:middle line:84% If we can find this amazing u star, then all we need to do 00:40:26.710 --> 00:40:29.350 align:middle line:84% is evaluate the aggregates, namely, 00:40:29.350 --> 00:40:33.040 align:middle line:84% the aggregate under x prime and aggregate under x. 00:40:33.040 --> 00:40:34.240 align:middle line:90% And what we need-- 00:40:34.240 --> 00:40:37.600 align:middle line:84% and apart from condition 2, the only thing we need-- 00:40:37.600 --> 00:40:40.760 align:middle line:84% is that u star assigns a higher value. 00:40:40.760 --> 00:40:44.260 align:middle line:84% In other words, it's easier to look for Pareto improvements 00:40:44.260 --> 00:40:46.960 align:middle line:84% just by looking at the value under 00:40:46.960 --> 00:40:51.430 align:middle line:84% the pseudo-utility function evaluated at the aggregates. 00:40:51.430 --> 00:40:55.360 align:middle line:84% So the second thing is similar, but not quite. 00:40:55.360 --> 00:40:59.410 align:middle line:84% Instead of starting with the underlying micro distributions, 00:40:59.410 --> 00:41:03.700 align:middle line:84% suppose that we had a feasible allocation x and y, as before, 00:41:03.700 --> 00:41:07.510 align:middle line:84% and an alternative feasible allocation, x prime, y prime. 00:41:07.510 --> 00:41:10.720 align:middle line:84% And we're given that under u star, 00:41:10.720 --> 00:41:14.050 align:middle line:84% the utility is higher at the aggregates. 00:41:14.050 --> 00:41:16.330 align:middle line:84% Then we want to claim that there is 00:41:16.330 --> 00:41:19.270 align:middle line:84% an alternative feasible allocation which 00:41:19.270 --> 00:41:20.740 align:middle line:90% Pareto dominates. 00:41:20.740 --> 00:41:23.290 align:middle line:84% In other words, is this xi allocation 00:41:23.290 --> 00:41:24.780 align:middle line:90% Pareto optimal or not? 00:41:24.780 --> 00:41:27.670 align:middle line:84% Not, if we can find another allocation, 00:41:27.670 --> 00:41:30.760 align:middle line:84% x prime, which dominates, but dominates 00:41:30.760 --> 00:41:31.960 align:middle line:90% in the sense of u star. 00:41:31.960 --> 00:41:34.360 align:middle line:84% And u star is just evaluating these things 00:41:34.360 --> 00:41:36.610 align:middle line:90% at the summed up aggregates. 00:41:36.610 --> 00:41:39.920 align:middle line:84% And actually, there's a subtlety here which is important. 00:41:39.920 --> 00:41:42.190 align:middle line:90% You can't just stop. 00:41:42.190 --> 00:41:43.900 align:middle line:90% You may have to redistribute. 00:41:43.900 --> 00:41:45.740 align:middle line:84% Well, here's what I mean by that. 00:41:45.740 --> 00:41:48.220 align:middle line:84% If this is true, there exists an alternative feasible 00:41:48.220 --> 00:41:50.170 align:middle line:90% allocation, tilde-- 00:41:50.170 --> 00:41:51.910 align:middle line:90% x tilde, y tilde-- 00:41:51.910 --> 00:41:57.250 align:middle line:84% such that the summation over x tilde of the demands and so on 00:41:57.250 --> 00:42:02.620 align:middle line:84% allocations is exactly the same aggregate as under the prime, 00:42:02.620 --> 00:42:04.640 align:middle line:90% and the y's aren't changing. 00:42:04.640 --> 00:42:06.610 align:middle line:84% So this kind of subtle step here, 00:42:06.610 --> 00:42:09.610 align:middle line:84% it doesn't mean, necessarily, that you 00:42:09.610 --> 00:42:13.660 align:middle line:84% can stop with x prime and y prime and claim 00:42:13.660 --> 00:42:17.230 align:middle line:84% victory, because if you stop there, 00:42:17.230 --> 00:42:20.380 align:middle line:84% it's not necessarily true that everyone's better off-- 00:42:20.380 --> 00:42:22.270 align:middle line:90% or no worse off. 00:42:22.270 --> 00:42:23.980 align:middle line:90% Some people may be damaged. 00:42:23.980 --> 00:42:26.920 align:middle line:84% In fact, maybe this is the time to talk about free trade. 00:42:26.920 --> 00:42:29.530 align:middle line:84% When we were looking at that lecture, 00:42:29.530 --> 00:42:32.590 align:middle line:84% we represented a country by some indifference curve 00:42:32.590 --> 00:42:35.410 align:middle line:84% as if from some u star representative 00:42:35.410 --> 00:42:38.110 align:middle line:84% household for the whole country to look 00:42:38.110 --> 00:42:39.340 align:middle line:90% at the direction of trade. 00:42:39.340 --> 00:42:41.540 align:middle line:84% So we were getting the country to agree 00:42:41.540 --> 00:42:44.750 align:middle line:84% we get demand in a positive sense 00:42:44.750 --> 00:42:47.630 align:middle line:84% by looking at that representative agent 00:42:47.630 --> 00:42:48.770 align:middle line:90% indifference curve. 00:42:48.770 --> 00:42:52.880 align:middle line:84% And it might have appeared that the whole country could be 00:42:52.880 --> 00:42:55.290 align:middle line:90% made better off by free trade. 00:42:55.290 --> 00:42:57.560 align:middle line:84% On the other hand, you saw in detail, 00:42:57.560 --> 00:42:59.420 align:middle line:90% in two different lectures-- 00:42:59.420 --> 00:43:02.300 align:middle line:84% three, really-- that not everyone 00:43:02.300 --> 00:43:05.300 align:middle line:90% is made off under free trade. 00:43:05.300 --> 00:43:08.030 align:middle line:84% Remember, the households that are relatively well 00:43:08.030 --> 00:43:12.710 align:middle line:84% endowed with a factor that's abundant are helped by exports. 00:43:12.710 --> 00:43:14.090 align:middle line:90% The other guys are hurt. 00:43:14.090 --> 00:43:15.440 align:middle line:90% Et cetera. 00:43:15.440 --> 00:43:18.050 align:middle line:84% So how can we claim that free trade makes everyone 00:43:18.050 --> 00:43:18.740 align:middle line:90% better off? 00:43:18.740 --> 00:43:22.380 align:middle line:84% We have to redistribute, which is not always done. 00:43:22.380 --> 00:43:24.950 align:middle line:84% Hence, people complain about free trade 00:43:24.950 --> 00:43:26.900 align:middle line:90% and they want tariffs. 00:43:26.900 --> 00:43:29.300 align:middle line:84% Anyway, I'm belaboring the fact that this lecture 00:43:29.300 --> 00:43:31.310 align:middle line:84% is consistent with the other ones, 00:43:31.310 --> 00:43:33.260 align:middle line:84% except now, we're finally in a position 00:43:33.260 --> 00:43:37.040 align:middle line:84% to be able to talk explicitly about redistribution 00:43:37.040 --> 00:43:41.360 align:middle line:84% as a necessary part of making everyone better off. 00:43:41.360 --> 00:43:43.460 align:middle line:84% In other words, to sum up, even when 00:43:43.460 --> 00:43:47.660 align:middle line:84% we have preferences aggregating, it is not necessarily the case 00:43:47.660 --> 00:43:50.600 align:middle line:84% that everyone is made better off under some alternative 00:43:50.600 --> 00:43:52.430 align:middle line:90% allocation xi prime. 00:43:52.430 --> 00:43:57.420 align:middle line:84% You have to redistribute the underlying bundles. 00:43:57.420 --> 00:43:59.610 align:middle line:84% So let's do the same definition, but do it 00:43:59.610 --> 00:44:04.690 align:middle line:84% in the space of indirect utility instead of direct utility. 00:44:04.690 --> 00:44:07.000 align:middle line:84% Suppose we're given a vector of wealth, 00:44:07.000 --> 00:44:09.040 align:middle line:84% as in the distribution of income, 00:44:09.040 --> 00:44:13.660 align:middle line:84% and some aggregate wealth that corresponds with that. 00:44:13.660 --> 00:44:15.700 align:middle line:90% p is a price vector. 00:44:15.700 --> 00:44:18.370 align:middle line:84% Define the indirect utility functions, again, 00:44:18.370 --> 00:44:20.290 align:middle line:90% as we have been doing. 00:44:20.290 --> 00:44:23.740 align:middle line:84% Now, we want to find this miraculous star 00:44:23.740 --> 00:44:28.780 align:middle line:84% person, this normative representative household, 00:44:28.780 --> 00:44:33.130 align:middle line:84% i star, that has an indirect utility function v 00:44:33.130 --> 00:44:35.740 align:middle line:84% star, evaluated at the price vector 00:44:35.740 --> 00:44:38.720 align:middle line:84% p and the aggregate wealth w bar. 00:44:38.720 --> 00:44:41.500 align:middle line:84% And there are two conditions, 1 and 2, 00:44:41.500 --> 00:44:45.130 align:middle line:84% that correspond with the one in the allocation space xi 00:44:45.130 --> 00:44:46.180 align:middle line:90% on the previous slide. 00:44:46.180 --> 00:44:49.630 align:middle line:84% Namely, if you have two configurations, 00:44:49.630 --> 00:44:52.750 align:middle line:90% you start with a p and w wealth. 00:44:52.750 --> 00:44:55.810 align:middle line:84% You're looking at an alternative, p prime, w prime, 00:44:55.810 --> 00:44:58.180 align:middle line:84% price vector in wealth, such that 00:44:58.180 --> 00:45:00.640 align:middle line:84% looking at the actual real indirect utilities 00:45:00.640 --> 00:45:03.490 align:middle line:84% of all the households, some people are no worse off 00:45:03.490 --> 00:45:05.590 align:middle line:84% and other people are strictly better off 00:45:05.590 --> 00:45:07.330 align:middle line:90% under the prime allocation. 00:45:07.330 --> 00:45:10.720 align:middle line:84% So that's the statement of Pareto dominance 00:45:10.720 --> 00:45:13.480 align:middle line:84% in the space of indirect utility functions. 00:45:13.480 --> 00:45:16.300 align:middle line:84% And what we want is a way to do it 00:45:16.300 --> 00:45:19.930 align:middle line:84% more easily through the star, the indirect utility 00:45:19.930 --> 00:45:22.150 align:middle line:84% of the normative representative household, 00:45:22.150 --> 00:45:26.020 align:middle line:84% which is just evaluating the alternative scenario, 00:45:26.020 --> 00:45:27.910 align:middle line:90% p prime and w prime. 00:45:27.910 --> 00:45:31.010 align:middle line:84% But it's w prime added up over all of the households. 00:45:31.010 --> 00:45:33.280 align:middle line:84% So we're just using aggregate wealth here 00:45:33.280 --> 00:45:35.800 align:middle line:84% on the left-hand side, and also aggregate wealth 00:45:35.800 --> 00:45:38.920 align:middle line:84% on the right-hand side, when we're thinking about trying to, 00:45:38.920 --> 00:45:41.110 align:middle line:84% quote, unquote, "Pareto dominate." 00:45:41.110 --> 00:45:45.550 align:middle line:84% So just like before, it's very intuitive, but amazing 00:45:45.550 --> 00:45:48.820 align:middle line:84% when it's true, that the way to look for Pareto dominating 00:45:48.820 --> 00:45:54.040 align:middle line:84% allocations is just to evaluate the aggregated allocations 00:45:54.040 --> 00:45:57.580 align:middle line:84% and see whether you get a higher value under v star. 00:45:57.580 --> 00:46:02.470 align:middle line:84% Likewise, going the other way, start with the aggregate wealth 00:46:02.470 --> 00:46:03.910 align:middle line:90% and price vector p. 00:46:03.910 --> 00:46:09.040 align:middle line:84% Look at an alternative scenario, p prime and w bar prime, such 00:46:09.040 --> 00:46:12.370 align:middle line:84% that the utility under v star is higher 00:46:12.370 --> 00:46:14.410 align:middle line:90% in the alternative allocation. 00:46:14.410 --> 00:46:18.310 align:middle line:84% If that's true, then there exists an alternative income 00:46:18.310 --> 00:46:21.880 align:middle line:84% distribution w prime which adds up 00:46:21.880 --> 00:46:24.910 align:middle line:84% to the aggregate wealth in the alternative-- 00:46:24.910 --> 00:46:28.150 align:middle line:84% the dominating allocation with the prime on it. 00:46:28.150 --> 00:46:30.910 align:middle line:90% Now, again, let's pause here. 00:46:30.910 --> 00:46:34.570 align:middle line:84% This alternative income means an income distribution 00:46:34.570 --> 00:46:37.660 align:middle line:84% that works under the prime scenario 00:46:37.660 --> 00:46:42.520 align:middle line:84% relative to the initial income distribution in the baseline 00:46:42.520 --> 00:46:43.940 align:middle line:90% without the prime. 00:46:43.940 --> 00:46:47.830 align:middle line:84% So we have to have the freedom to distribute wealth, 00:46:47.830 --> 00:46:51.460 align:middle line:84% not necessarily according to private ownership, when 00:46:51.460 --> 00:46:55.510 align:middle line:84% we change the prices and so on, but potentially, with transfers 00:46:55.510 --> 00:46:57.550 align:middle line:90% of wealth across the agents-- 00:46:57.550 --> 00:47:00.310 align:middle line:84% giving to some, taking away from others, 00:47:00.310 --> 00:47:02.680 align:middle line:84% as in the second welfare theorem, 00:47:02.680 --> 00:47:05.440 align:middle line:84% where we've seen this kind of redistribution before. 00:47:05.440 --> 00:47:09.670 align:middle line:84% So we find this alternative such that we're 00:47:09.670 --> 00:47:12.880 align:middle line:84% getting Pareto dominance in the underlying indirect utility 00:47:12.880 --> 00:47:13.930 align:middle line:90% functions. 00:47:13.930 --> 00:47:17.230 align:middle line:84% Not worse for anyone and strictly better for some. 00:47:17.230 --> 00:47:20.230 align:middle line:84% So how do we find this representative household, 00:47:20.230 --> 00:47:23.200 align:middle line:84% or the representative household with this normative property? 00:47:23.200 --> 00:47:27.580 align:middle line:84% Same guy, Gorman Polar form, although this is getting 00:47:27.580 --> 00:47:29.530 align:middle line:90% stated as a proposition. 00:47:29.530 --> 00:47:33.580 align:middle line:84% If we have this Gorman Polar form with this particular way 00:47:33.580 --> 00:47:37.720 align:middle line:84% that heterogeneity is entering into the indirect utility 00:47:37.720 --> 00:47:41.740 align:middle line:84% functions, then when we look at the indirect utility function 00:47:41.740 --> 00:47:45.940 align:middle line:84% under the aggregate, being in particular the sum of the ai 00:47:45.940 --> 00:47:50.770 align:middle line:84% plus b times w, that is the normative household. 00:47:50.770 --> 00:47:54.410 align:middle line:84% We already proved the positive part earlier. 00:47:54.410 --> 00:47:55.810 align:middle line:90% Now, it's normative. 00:47:55.810 --> 00:47:58.600 align:middle line:84% I put positive and normative both on the slide 00:47:58.600 --> 00:48:00.730 align:middle line:84% to make the point that, as I've said, 00:48:00.730 --> 00:48:03.400 align:middle line:84% we're not looking for a normative household that's 00:48:03.400 --> 00:48:05.590 align:middle line:84% different from the positive household. 00:48:05.590 --> 00:48:07.630 align:middle line:84% We're looking for this representative household 00:48:07.630 --> 00:48:09.220 align:middle line:90% with both attributes. 00:48:09.220 --> 00:48:11.200 align:middle line:84% We proved the positive thing earlier. 00:48:11.200 --> 00:48:13.780 align:middle line:84% Now, we need to prove the normative part. 00:48:13.780 --> 00:48:20.020 align:middle line:84% So suppose we have those Gorman Polar form, in particular, 00:48:20.020 --> 00:48:23.800 align:middle line:84% to figure out which direction we're going from the condition 00:48:23.800 --> 00:48:25.630 align:middle line:90% 1, this guy here. 00:48:25.630 --> 00:48:27.190 align:middle line:90% Condition 1. 00:48:27.190 --> 00:48:31.480 align:middle line:84% So we have a rank ordering under the actual indirect utility 00:48:31.480 --> 00:48:34.780 align:middle line:84% functions being explicit about household i. 00:48:34.780 --> 00:48:40.660 align:middle line:84% So under the vi's, not worse for some 00:48:40.660 --> 00:48:42.800 align:middle line:90% and strictly better for others. 00:48:42.800 --> 00:48:44.440 align:middle line:90% So we're given that. 00:48:44.440 --> 00:48:49.420 align:middle line:84% Now, we want to prove that when we evaluate under v star 00:48:49.420 --> 00:48:52.540 align:middle line:84% using the aggregates, we get strict dominance. 00:48:52.540 --> 00:48:56.650 align:middle line:84% So let's look at it. v star under the aggregate alternative 00:48:56.650 --> 00:49:00.760 align:middle line:84% prime allocation is-- and this is just now writing things out. 00:49:00.760 --> 00:49:06.930 align:middle line:84% This is the definition of the indirect utility function star, 00:49:06.930 --> 00:49:08.870 align:middle line:90% the one that's going to work. 00:49:08.870 --> 00:49:12.610 align:middle line:84% And then as we keep going across these equations, 00:49:12.610 --> 00:49:17.080 align:middle line:84% we recognize, again, the taking the summation over I 00:49:17.080 --> 00:49:19.990 align:middle line:84% out in front of all of it, that we're just adding up 00:49:19.990 --> 00:49:21.700 align:middle line:90% the indirect utility functions. 00:49:21.700 --> 00:49:23.920 align:middle line:84% But again, we're given not worse for some 00:49:23.920 --> 00:49:25.310 align:middle line:90% and strictly better for others. 00:49:25.310 --> 00:49:27.670 align:middle line:84% So when we add up the indirect utility functions 00:49:27.670 --> 00:49:30.010 align:middle line:84% under the prime allocation and compare it 00:49:30.010 --> 00:49:33.100 align:middle line:84% to the original allocation, it must be strict dominance. 00:49:33.100 --> 00:49:35.740 align:middle line:84% And then if you reverse the ordering of these 00:49:35.740 --> 00:49:39.460 align:middle line:84% vi's, collecting terms and going back toward the left-hand 00:49:39.460 --> 00:49:40.060 align:middle line:90% side-- 00:49:40.060 --> 00:49:42.580 align:middle line:84% not under prime, but under the baseline-- 00:49:42.580 --> 00:49:45.550 align:middle line:84% you would get this star indirect utility 00:49:45.550 --> 00:49:47.830 align:middle line:84% for the representative household. 00:49:47.830 --> 00:49:53.080 align:middle line:84% So the bottom line is the v star under prime strictly greater 00:49:53.080 --> 00:49:55.540 align:middle line:90% than v star under the baseline. 00:49:55.540 --> 00:49:58.280 align:middle line:90% And we proved one. 00:49:58.280 --> 00:50:00.610 align:middle line:84% The second thing we want to prove is this-- 00:50:00.610 --> 00:50:03.830 align:middle line:84% that we have an initial allocation 00:50:03.830 --> 00:50:07.720 align:middle line:84% and we're looking at a prime allocation that gives higher 00:50:07.720 --> 00:50:11.350 align:middle line:84% indirect utility for the aggregated household. 00:50:11.350 --> 00:50:13.810 align:middle line:84% And then we want to find a way to assign 00:50:13.810 --> 00:50:18.680 align:middle line:84% wealth that makes everyone better off, essentially. 00:50:18.680 --> 00:50:23.300 align:middle line:84% So again, from 2, we have this as a given, 00:50:23.300 --> 00:50:26.780 align:middle line:84% that v star returns a higher value under the prime relative 00:50:26.780 --> 00:50:28.550 align:middle line:90% to the baseline. 00:50:28.550 --> 00:50:30.590 align:middle line:84% We want to construct an alternative income 00:50:30.590 --> 00:50:34.340 align:middle line:84% distribution-- seems like that might be hard to do-- 00:50:34.340 --> 00:50:36.670 align:middle line:84% that makes everyone, in fact, I'm 00:50:36.670 --> 00:50:38.720 align:middle line:84% going to show you, strictly better off. 00:50:38.720 --> 00:50:41.180 align:middle line:84% Actually, strict dominance for everyone. 00:50:41.180 --> 00:50:45.680 align:middle line:84% And it's actually very constructive and intuitive. 00:50:45.680 --> 00:50:50.000 align:middle line:84% We're already given here that we have this equation 00:50:50.000 --> 00:50:54.480 align:middle line:84% 30 at the aggregates, the left-hand side 00:50:54.480 --> 00:50:56.110 align:middle line:84% being greater than the right-hand side. 00:50:56.110 --> 00:50:59.400 align:middle line:84% So take the difference and call it c. 00:50:59.400 --> 00:51:02.760 align:middle line:84% It's a positive constant now, the rest of the thing 00:51:02.760 --> 00:51:07.110 align:middle line:84% is just writing out what v star is under prime and v star 00:51:07.110 --> 00:51:11.520 align:middle line:84% is under the baseline, and then collecting terms. 00:51:11.520 --> 00:51:15.780 align:middle line:84% The only reason to do this part using the definition of v star 00:51:15.780 --> 00:51:18.690 align:middle line:84% is to go searching for the wealth levels 00:51:18.690 --> 00:51:24.980 align:middle line:84% that we want, namely, let's give each household their proportion 00:51:24.980 --> 00:51:25.480 align:middle line:90% of c. 00:51:25.480 --> 00:51:28.000 align:middle line:84% There are i households in the economy. 00:51:28.000 --> 00:51:30.400 align:middle line:90% Give each one c over i. 00:51:30.400 --> 00:51:34.160 align:middle line:84% So we start with the baseline in terms of the indirect utility 00:51:34.160 --> 00:51:34.660 align:middle line:90% function. 00:51:34.660 --> 00:51:39.100 align:middle line:84% We add on to the baseline this positive constant. 00:51:39.100 --> 00:51:43.420 align:middle line:84% And we want that to be the indirect utility 00:51:43.420 --> 00:51:46.160 align:middle line:84% per household i under the alternative. 00:51:46.160 --> 00:51:49.990 align:middle line:84% So now, we can just solve this equation for wi. 00:51:49.990 --> 00:51:52.690 align:middle line:90% It's almost mechanical. 00:51:52.690 --> 00:51:57.280 align:middle line:84% These are, though, the required new distributions of wealth 00:51:57.280 --> 00:51:58.430 align:middle line:90% that will work. 00:51:58.430 --> 00:51:59.560 align:middle line:90% Why does it work? 00:51:59.560 --> 00:52:03.200 align:middle line:84% Well, working backwards, if you give them this, 00:52:03.200 --> 00:52:05.390 align:middle line:90% then we have this equation. 00:52:05.390 --> 00:52:08.290 align:middle line:84% And you can also see, because we've just 00:52:08.290 --> 00:52:11.650 align:middle line:84% added a constant to the indirect utility function, 00:52:11.650 --> 00:52:14.320 align:middle line:84% that each household i under the alternative 00:52:14.320 --> 00:52:18.010 align:middle line:84% is having more utility than under the baseline. 00:52:18.010 --> 00:52:24.250 align:middle line:84% That's this vi of p prime w prime, vi of p and wi, 00:52:24.250 --> 00:52:28.180 align:middle line:90% plus this proportional constant. 00:52:28.180 --> 00:52:31.140 align:middle line:84% So it's almost trivial, because now, we've gone the other way. 00:52:31.140 --> 00:52:35.950 align:middle line:90% We proved 2, this thing. 00:52:35.950 --> 00:52:40.980 align:middle line:84% So we've succeeded, if they have these Gorman Polar forms, 00:52:40.980 --> 00:52:44.880 align:middle line:84% in constructing a pseudo-utility function 00:52:44.880 --> 00:52:48.360 align:middle line:84% over the indirect utility function over the aggregates. 00:52:48.360 --> 00:52:50.550 align:middle line:90% Questions? 00:52:50.550 --> 00:52:51.050 align:middle line:90% OK. 00:52:51.050 --> 00:52:54.680 align:middle line:84% So now let's go back to the Pareto problem. 00:52:54.680 --> 00:52:57.260 align:middle line:84% In particular, we did a version of this slide 00:52:57.260 --> 00:52:59.450 align:middle line:84% when we did the second welfare theorem. 00:52:59.450 --> 00:53:03.020 align:middle line:84% We have this underlying economy with consumption, preferences, 00:53:03.020 --> 00:53:06.380 align:middle line:84% production functions, aggregate, endowments, 00:53:06.380 --> 00:53:13.550 align:middle line:84% satisfying non-negative orthant, and differentiable convex 00:53:13.550 --> 00:53:16.910 align:middle line:84% production sets or concave transformations, 00:53:16.910 --> 00:53:19.760 align:middle line:84% non-negative aggregate resources, 00:53:19.760 --> 00:53:23.660 align:middle line:84% consumption sets having interior points, 00:53:23.660 --> 00:53:26.480 align:middle line:84% utility functions being strictly concave, 00:53:26.480 --> 00:53:31.100 align:middle line:84% and also an interior point with respect to aggregate resources. 00:53:31.100 --> 00:53:33.710 align:middle line:84% I'm rushing a bit, but only because you've 00:53:33.710 --> 00:53:36.190 align:middle line:90% seen this slide before. 00:53:36.190 --> 00:53:39.070 align:middle line:84% If all of that is true, then we can 00:53:39.070 --> 00:53:44.620 align:middle line:84% solve this Pareto problem to get Pareto optimal allocations. 00:53:44.620 --> 00:53:48.370 align:middle line:84% This is just maximizing lambda weighted sums of utilities 00:53:48.370 --> 00:53:50.740 align:middle line:84% subject to consumption being feasible, 00:53:50.740 --> 00:53:53.860 align:middle line:84% production being feasible, and the resource constraints 00:53:53.860 --> 00:53:55.060 align:middle line:90% being satisfied. 00:53:55.060 --> 00:53:56.920 align:middle line:90% Demand equals supply. 00:53:56.920 --> 00:53:59.440 align:middle line:84% I should say, allocations are feasible 00:53:59.440 --> 00:54:02.290 align:middle line:84% given endowments in the production set. 00:54:02.290 --> 00:54:03.760 align:middle line:90% Problem? 00:54:03.760 --> 00:54:05.770 align:middle line:90% What are the lambdas? 00:54:05.770 --> 00:54:08.860 align:middle line:84% We move the lambdas around, we get different Pareto optimum. 00:54:08.860 --> 00:54:11.740 align:middle line:84% That's what we did with the second welfare theorem. 00:54:11.740 --> 00:54:15.760 align:middle line:84% And we've used this with risk sharing in many applications 00:54:15.760 --> 00:54:17.260 align:middle line:90% previously. 00:54:17.260 --> 00:54:20.650 align:middle line:84% But we had to keep track of the lambdas. 00:54:20.650 --> 00:54:24.420 align:middle line:84% Here's what happens with the representative consumer. 00:54:24.420 --> 00:54:25.890 align:middle line:90% No more lambdas. 00:54:25.890 --> 00:54:28.290 align:middle line:90% Now, we just maximize utility. 00:54:28.290 --> 00:54:32.400 align:middle line:84% Well, this x is now aggregate consumption. 00:54:32.400 --> 00:54:34.920 align:middle line:84% Production sets are the same as before. 00:54:34.920 --> 00:54:38.580 align:middle line:84% Resource constraint's the same in the aggregate. 00:54:38.580 --> 00:54:40.740 align:middle line:90% I mean, x is aggregate. 00:54:40.740 --> 00:54:44.640 align:middle line:84% Aggregated x is entering into the pseudo utility function u 00:54:44.640 --> 00:54:47.340 align:middle line:84% star, and we want to find the highest u. 00:54:47.340 --> 00:54:54.180 align:middle line:84% Remember why-- because we already proved this, right? 00:54:54.180 --> 00:54:57.060 align:middle line:84% It's enough to look at you star in the aggregates 00:54:57.060 --> 00:55:00.900 align:middle line:84% in the sense of satisfying both these conditions. 00:55:00.900 --> 00:55:02.470 align:middle line:84% And that's what we're doing here. 00:55:02.470 --> 00:55:06.270 align:middle line:84% We're just looking in the space of aggregates 00:55:06.270 --> 00:55:09.630 align:middle line:84% under this pseudo-utility function u star. 00:55:09.630 --> 00:55:11.730 align:middle line:84% If we're not maximizing it, then when 00:55:11.730 --> 00:55:14.340 align:middle line:84% we move in a direction that improves it, 00:55:14.340 --> 00:55:16.290 align:middle line:84% we're actually improving the welfare of all 00:55:16.290 --> 00:55:17.880 align:middle line:90% of the individual households. 00:55:17.880 --> 00:55:21.330 align:middle line:84% We'll, in a sense, Pareto dominate. 00:55:21.330 --> 00:55:25.390 align:middle line:84% So simple to write, but extremely powerful. 00:55:25.390 --> 00:55:28.480 align:middle line:84% And this-- so this is contemporary macro. 00:55:28.480 --> 00:55:32.430 align:middle line:84% This is what is usually assumed in macroeconomics 00:55:32.430 --> 00:55:34.950 align:middle line:90% or aggregative economics. 00:55:34.950 --> 00:55:38.640 align:middle line:84% No reference to redistributions or wealth, underlying wealths, 00:55:38.640 --> 00:55:39.450 align:middle line:90% or whatever. 00:55:39.450 --> 00:55:42.420 align:middle line:84% So mesmerized by the individual household 00:55:42.420 --> 00:55:44.610 align:middle line:84% that we forget to spell out the underpinnings, 00:55:44.610 --> 00:55:48.120 align:middle line:84% and in particular, the assumptions that are needed. 00:55:48.120 --> 00:55:49.650 align:middle line:90% Are they satisfied? 00:55:49.650 --> 00:55:51.870 align:middle line:90% No, typically not. 00:55:51.870 --> 00:55:54.870 align:middle line:84% That previous lecture on micro data, 00:55:54.870 --> 00:55:59.100 align:middle line:84% we saw that income expansion paths are a special case. 00:55:59.100 --> 00:56:03.660 align:middle line:84% In particular, there are non-linear expansion paths. 00:56:03.660 --> 00:56:07.980 align:middle line:84% As we change income, expenditure shares go up for some goods. 00:56:07.980 --> 00:56:10.470 align:middle line:90% We call them luxury goods. 00:56:10.470 --> 00:56:13.380 align:middle line:84% Expenditure shares go down for other goods. 00:56:13.380 --> 00:56:15.090 align:middle line:90% We call them necessities. 00:56:15.090 --> 00:56:16.000 align:middle line:90% And so on. 00:56:16.000 --> 00:56:17.760 align:middle line:90% So that's what's in the data. 00:56:17.760 --> 00:56:20.590 align:middle line:90% Engel found this very early on. 00:56:20.590 --> 00:56:22.860 align:middle line:84% And we presented other applications, 00:56:22.860 --> 00:56:27.240 align:middle line:84% Giffen goods being an extreme version of the case when 00:56:27.240 --> 00:56:30.060 align:middle line:84% the income effects, through necessities, are 00:56:30.060 --> 00:56:32.760 align:middle line:84% so strong that they dominate the price effects. 00:56:32.760 --> 00:56:36.060 align:middle line:84% Well, maybe one can hope that as an approximation, 00:56:36.060 --> 00:56:38.220 align:middle line:84% even though we would not literally 00:56:38.220 --> 00:56:41.760 align:middle line:84% satisfy the assumptions, that those income effects 00:56:41.760 --> 00:56:43.440 align:middle line:90% come close to canceling out. 00:56:43.440 --> 00:56:45.480 align:middle line:90% That's kind of the best case. 00:56:45.480 --> 00:56:47.710 align:middle line:90% Macro with micro underpinnings. 00:56:47.710 --> 00:56:49.350 align:middle line:84% It's a reference to the trade lecture. 00:56:49.350 --> 00:56:51.720 align:middle line:84% I've already given you my spiel on that 00:56:51.720 --> 00:56:54.840 align:middle line:84% this afternoon about the gains to trade 00:56:54.840 --> 00:56:57.540 align:middle line:84% and the fact that we have losers, although the theorem is 00:56:57.540 --> 00:57:02.190 align:middle line:84% telling us that if we have representative consumer, then 00:57:02.190 --> 00:57:05.220 align:middle line:84% we're safe in looking at the max, both 00:57:05.220 --> 00:57:06.660 align:middle line:90% the positive and the normative. 00:57:06.660 --> 00:57:08.970 align:middle line:84% But it will require redistribution 00:57:08.970 --> 00:57:10.590 align:middle line:90% of a particular kind. 00:57:10.590 --> 00:57:15.840 align:middle line:84% And finally, the economic impact of COVID-19. 00:57:15.840 --> 00:57:18.600 align:middle line:84% I put that in there to remind you of things 00:57:18.600 --> 00:57:21.360 align:middle line:84% that we found in that lecture when I was reviewing 00:57:21.360 --> 00:57:23.020 align:middle line:90% what the impact has been. 00:57:23.020 --> 00:57:26.220 align:middle line:84% And we had relatively poor households 00:57:26.220 --> 00:57:29.580 align:middle line:84% who were unemployed cutting back on expenditure, 00:57:29.580 --> 00:57:32.040 align:middle line:84% or potentially increasing expenditure 00:57:32.040 --> 00:57:34.290 align:middle line:84% only if they got the government money, 00:57:34.290 --> 00:57:38.010 align:middle line:84% and also looked at higher-income households 00:57:38.010 --> 00:57:40.440 align:middle line:90% through JP Morgan's data set. 00:57:40.440 --> 00:57:44.650 align:middle line:84% And those guys had different expenditure patterns. 00:57:44.650 --> 00:57:49.170 align:middle line:84% So it does not seem as though these effects are canceling out 00:57:49.170 --> 00:57:52.395 align:middle line:84% when we actually look at the underlying micro data. 00:57:52.395 --> 00:57:54.540 align:middle line:84% I mean, to cancel out we wouldn't 00:57:54.540 --> 00:57:57.030 align:middle line:90% see anything happening-- 00:57:57.030 --> 00:57:58.440 align:middle line:90% well, I should be careful here. 00:57:58.440 --> 00:58:01.770 align:middle line:84% There is an aggregate adverse shock, obviously. 00:58:01.770 --> 00:58:06.010 align:middle line:84% We're all suffering from it, at least indirectly. 00:58:06.010 --> 00:58:07.990 align:middle line:90% So the aggregates are down. 00:58:07.990 --> 00:58:11.260 align:middle line:84% But then the question is, do other changes 00:58:11.260 --> 00:58:13.420 align:middle line:90% balance out at the aggregate? 00:58:13.420 --> 00:58:17.570 align:middle line:84% And the data I presented to you suggest that's not true. 00:58:17.570 --> 00:58:18.790 align:middle line:90% So questions? 00:58:18.790 --> 00:58:21.500 align:middle line:90% 00:58:21.500 --> 00:58:26.390 align:middle line:84% AUDIENCE: Can I ask a question about the slide 22? 00:58:26.390 --> 00:58:27.585 align:middle line:90% ROBERT TOWNSEND: 22? 00:58:27.585 --> 00:58:28.210 align:middle line:90% AUDIENCE: Yeah. 00:58:28.210 --> 00:58:37.315 align:middle line:90% 00:58:37.315 --> 00:58:38.190 align:middle line:90% ROBERT TOWNSEND: Yes? 00:58:38.190 --> 00:58:40.400 align:middle line:90% AUDIENCE: Yeah, here. 00:58:40.400 --> 00:58:42.920 align:middle line:90% I'm a little bit confusing that. 00:58:42.920 --> 00:58:48.050 align:middle line:84% Do we do assume that the sum of x prime 00:58:48.050 --> 00:58:50.930 align:middle line:90% will equal to the sum of x here? 00:58:50.930 --> 00:58:54.500 align:middle line:84% Because in the slide 21, we say that, 00:58:54.500 --> 00:58:57.380 align:middle line:84% at the [? center ?] of the allocation 00:58:57.380 --> 00:58:59.960 align:middle line:90% should be the same as original. 00:58:59.960 --> 00:59:02.240 align:middle line:90% And the sum of production-- 00:59:02.240 --> 00:59:05.820 align:middle line:84% the production is the equal to the original. 00:59:05.820 --> 00:59:13.850 align:middle line:84% So when you go from 21 to 22, do we still assume this? 00:59:13.850 --> 00:59:15.850 align:middle line:84% ROBERT TOWNSEND: It's only implicit in the sense 00:59:15.850 --> 00:59:22.610 align:middle line:84% that we're looking at individual wealths and aggregating it up. 00:59:22.610 --> 00:59:25.480 align:middle line:84% So I agree, it's probably not obvious, 00:59:25.480 --> 00:59:29.270 align:middle line:84% but there are just two representations, 00:59:29.270 --> 00:59:31.220 align:middle line:84% one in this space of indirect utility 00:59:31.220 --> 00:59:34.760 align:middle line:84% functions with the parameters p and w, 00:59:34.760 --> 00:59:40.010 align:middle line:84% and the other and the actual underlying allocations. 00:59:40.010 --> 00:59:44.360 align:middle line:84% Another way to put that is that if we-- where is it? 00:59:44.360 --> 00:59:45.680 align:middle line:90% If we went with-- 00:59:45.680 --> 00:59:46.700 align:middle line:90% can't find it. 00:59:46.700 --> 00:59:51.220 align:middle line:90% 00:59:51.220 --> 00:59:52.150 align:middle line:90% Yeah. 00:59:52.150 --> 00:59:56.930 align:middle line:84% Take, as an example, this utility function 00:59:56.930 --> 01:00:04.140 align:middle line:84% and the associated indirect utilities, these guys. 01:00:04.140 --> 01:00:05.512 align:middle line:90% You could go back and forth. 01:00:05.512 --> 01:00:07.470 align:middle line:84% You could do it one way or do it the other way. 01:00:07.470 --> 01:00:10.515 align:middle line:84% You're going to get the same answer. 01:00:10.515 --> 01:00:11.990 align:middle line:90% AUDIENCE: Oh, I get it. 01:00:11.990 --> 01:00:14.400 align:middle line:90% ROBERT TOWNSEND: OK. 01:00:14.400 --> 01:00:15.570 align:middle line:90% OK. 01:00:15.570 --> 01:00:20.300 align:middle line:84% So that's Gorman aggregation, with a lot 01:00:20.300 --> 01:00:21.590 align:middle line:90% of the pieces coming together. 01:00:21.590 --> 01:00:25.010 align:middle line:84% Next time, we're going to go back in the other direction. 01:00:25.010 --> 01:00:28.910 align:middle line:84% We're going to ask, if we don't do anything special 01:00:28.910 --> 01:00:32.840 align:middle line:84% like this parametrically, does economic theory 01:00:32.840 --> 01:00:35.670 align:middle line:84% put any restrictions on data at all? 01:00:35.670 --> 01:00:38.570 align:middle line:84% And the answer is going to be yes, 01:00:38.570 --> 01:00:43.200 align:middle line:84% so we'll talk about theory that can 01:00:43.200 --> 01:00:47.660 align:middle line:84% be rejected in the data, which is a good thing. 01:00:47.660 --> 01:00:50.490 align:middle line:90% Otherwise, there's no content. 01:00:50.490 --> 01:00:53.120 align:middle line:90% So that's all for today. 01:00:53.120 --> 01:00:55.330 align:middle line:90% Thank you for coming. 01:00:55.330 --> 01:01:01.369 align:middle line:90%