WEBVTT 00:00:00.000 --> 00:00:01.437 align:middle line:90% [SQUEAKING] 00:00:01.437 --> 00:00:03.353 align:middle line:90% [RUSTLING] 00:00:03.353 --> 00:00:06.227 align:middle line:90% [CLICKING] 00:00:06.227 --> 00:00:09.965 align:middle line:90% 00:00:09.965 --> 00:00:11.090 align:middle line:90% ROBERT TOWNSEND: All right. 00:00:11.090 --> 00:00:15.680 align:middle line:84% So let's take a look at where we are. 00:00:15.680 --> 00:00:19.270 align:middle line:84% On the calendar, we're at lecture 17 today. 00:00:19.270 --> 00:00:24.370 align:middle line:84% As I said last time, lectures 16, 17, 18 and 19 00:00:24.370 --> 00:00:27.460 align:middle line:84% are all about Walrasian equilibria 00:00:27.460 --> 00:00:29.210 align:middle line:90% and their properties. 00:00:29.210 --> 00:00:33.310 align:middle line:84% So we did the welfare theorems, which we can review again 00:00:33.310 --> 00:00:36.790 align:middle line:84% momentarily today about existence 00:00:36.790 --> 00:00:38.170 align:middle line:90% of all raising equilibria. 00:00:38.170 --> 00:00:42.250 align:middle line:84% And while we're at it, how we could interpret the Walrasian 00:00:42.250 --> 00:00:45.850 align:middle line:84% equilibrium as the outcome of the game using a Nash 00:00:45.850 --> 00:00:47.860 align:middle line:90% equilibrium notion. 00:00:47.860 --> 00:00:52.630 align:middle line:84% And then, we get to these two juxtaposed lectures, 18 and 19. 00:00:52.630 --> 00:00:55.510 align:middle line:84% Gorman aggregation has to do with when 00:00:55.510 --> 00:00:58.330 align:middle line:84% it looks as if there were far fewer households 00:00:58.330 --> 00:01:01.360 align:middle line:84% than there actually are, because we're able to add them up. 00:01:01.360 --> 00:01:03.880 align:middle line:84% That takes some pretty strong assumptions 00:01:03.880 --> 00:01:07.810 align:middle line:84% that we've already gone through in terms of linear income 00:01:07.810 --> 00:01:09.040 align:middle line:90% expansion paths. 00:01:09.040 --> 00:01:10.240 align:middle line:90% So we'll review that. 00:01:10.240 --> 00:01:13.060 align:middle line:84% And then, identification is, if you go the other way 00:01:13.060 --> 00:01:16.930 align:middle line:84% and assume almost nothing other than rationality, 00:01:16.930 --> 00:01:19.580 align:middle line:84% are you capable of rejecting the model? 00:01:19.580 --> 00:01:21.490 align:middle line:84% And then, the last two lectures are 00:01:21.490 --> 00:01:23.770 align:middle line:84% about failures of the welfare theorems 00:01:23.770 --> 00:01:27.070 align:middle line:84% and relatedly, monetary economics. 00:01:27.070 --> 00:01:29.520 align:middle line:90% We're moving right along. 00:01:29.520 --> 00:01:32.010 align:middle line:84% In terms of what you should be reading, 00:01:32.010 --> 00:01:33.600 align:middle line:84% ideally, I don't if you guys ever 00:01:33.600 --> 00:01:35.190 align:middle line:90% do read these start articles. 00:01:35.190 --> 00:01:39.160 align:middle line:84% But let me just point out that you should be. 00:01:39.160 --> 00:01:43.590 align:middle line:84% And in particular, today, lecture 17 today-- 00:01:43.590 --> 00:01:47.520 align:middle line:84% so on the course posting under Canvas, 00:01:47.520 --> 00:01:52.290 align:middle line:84% there'll be the Kreps section 6.4 on existence and number 00:01:52.290 --> 00:01:53.250 align:middle line:90% of equilibria. 00:01:53.250 --> 00:01:56.700 align:middle line:84% Some of the diagrams from today come from that chapter. 00:01:56.700 --> 00:01:59.850 align:middle line:84% Although obviously, there's more in there in the book 00:01:59.850 --> 00:02:01.230 align:middle line:90% than there is in the lecture. 00:02:01.230 --> 00:02:05.430 align:middle line:90% And also, this course material-- 00:02:05.430 --> 00:02:09.120 align:middle line:84% Asu's stuff from a different MIT course. 00:02:09.120 --> 00:02:12.720 align:middle line:84% She's got a really great lecture on existence 00:02:12.720 --> 00:02:13.740 align:middle line:90% of Nash equilibrium. 00:02:13.740 --> 00:02:15.720 align:middle line:90% And I borrowed that quite a bit. 00:02:15.720 --> 00:02:18.990 align:middle line:84% The entire lecture is posted on our website, 00:02:18.990 --> 00:02:22.270 align:middle line:84% but we will cover a big chunk of it today in class. 00:02:22.270 --> 00:02:25.080 align:middle line:84% So those are the two starred readings. 00:02:25.080 --> 00:02:29.350 align:middle line:84% And then, we get to a bit of a review. 00:02:29.350 --> 00:02:34.260 align:middle line:84% So this is a bit challenging, but I'll ask it anyway. 00:02:34.260 --> 00:02:37.830 align:middle line:84% It really has to do with stating the first welfare theorem, 00:02:37.830 --> 00:02:44.070 align:middle line:84% and then outlining for us the steps necessary to prove it. 00:02:44.070 --> 00:02:48.330 align:middle line:84% So Daniel, you want to take a crack at that by any chance? 00:02:48.330 --> 00:02:52.210 align:middle line:90% 00:02:52.210 --> 00:02:54.610 align:middle line:84% You remember the steps for proving the first welfare 00:02:54.610 --> 00:02:57.520 align:middle line:90% theorem? 00:02:57.520 --> 00:03:00.610 align:middle line:84% AUDIENCE: If I recall correctly, the first welfare theorem 00:03:00.610 --> 00:03:12.830 align:middle line:84% was that under the assumption of everybody being rational 00:03:12.830 --> 00:03:16.517 align:middle line:90% and [INAUDIBLE]. 00:03:16.517 --> 00:03:17.850 align:middle line:90% I think that might have been it. 00:03:17.850 --> 00:03:20.680 align:middle line:90% There is a lambda-weighted-- 00:03:20.680 --> 00:03:25.980 align:middle line:84% then, the economy will create a lambda-weighted-- 00:03:25.980 --> 00:03:27.660 align:middle line:84% it will maximize the lambda-weighted sum 00:03:27.660 --> 00:03:30.680 align:middle line:90% of the utilities of the people. 00:03:30.680 --> 00:03:34.890 align:middle line:84% And the proof-- yeah, I don't think I remember 00:03:34.890 --> 00:03:37.200 align:middle line:90% the steps of the proof. 00:03:37.200 --> 00:03:39.100 align:middle line:84% AUDIENCE: Can I take a shot at it? 00:03:39.100 --> 00:03:40.133 align:middle line:90% ROBERT TOWNSEND: Sure. 00:03:40.133 --> 00:03:41.550 align:middle line:84% AUDIENCE: If I remember correctly, 00:03:41.550 --> 00:03:44.730 align:middle line:84% the proof goes something like, for each person, 00:03:44.730 --> 00:03:46.620 align:middle line:84% because of the budget constraint, 00:03:46.620 --> 00:03:50.790 align:middle line:84% any outcome that would be strictly preferred 00:03:50.790 --> 00:03:53.700 align:middle line:84% by that person would require a larger budget. 00:03:53.700 --> 00:03:58.560 align:middle line:84% And so in order for something to be or not to-- 00:03:58.560 --> 00:04:01.830 align:middle line:84% for the allocation, whatever everyone ends up 00:04:01.830 --> 00:04:04.170 align:middle line:84% getting not to be free to optimal, 00:04:04.170 --> 00:04:06.930 align:middle line:84% it would require everyone getting-- 00:04:06.930 --> 00:04:09.990 align:middle line:90% it would require at least-- 00:04:09.990 --> 00:04:12.240 align:middle line:84% it would require all the budgets to either be the same 00:04:12.240 --> 00:04:12.990 align:middle line:90% or increased. 00:04:12.990 --> 00:04:17.220 align:middle line:84% But you can't-- and if you also factor in the price, 00:04:17.220 --> 00:04:21.480 align:middle line:84% the profit maximization part, you can do the same thing 00:04:21.480 --> 00:04:22.590 align:middle line:90% on the production side. 00:04:22.590 --> 00:04:25.890 align:middle line:84% And you basically end up getting an impossible situation, where 00:04:25.890 --> 00:04:28.020 align:middle line:84% you have to have more stuff being 00:04:28.020 --> 00:04:31.290 align:middle line:84% consumed than being produced because of market [INAUDIBLE].. 00:04:31.290 --> 00:04:34.420 align:middle line:84% ROBERT TOWNSEND: Yep, that's very good. 00:04:34.420 --> 00:04:37.970 align:middle line:84% That was a very, very good outline. 00:04:37.970 --> 00:04:40.750 align:middle line:84% So the first welfare theorem says that any Walrasian 00:04:40.750 --> 00:04:43.270 align:middle line:90% equilibrium is optimal. 00:04:43.270 --> 00:04:48.740 align:middle line:84% And the proof is by contradiction. 00:04:48.740 --> 00:04:50.680 align:middle line:90% Suppose it's not optimal. 00:04:50.680 --> 00:04:53.200 align:middle line:84% Then, there's something that's weakly 00:04:53.200 --> 00:04:57.880 align:middle line:84% preferred by some households and strictly preferred by others. 00:04:57.880 --> 00:05:03.580 align:middle line:84% And then, as was just said, the strictly preferred by others 00:05:03.580 --> 00:05:09.250 align:middle line:84% means it could not have been available, because it otherwise 00:05:09.250 --> 00:05:12.130 align:middle line:84% would have been purchased in Walrasian equilibria. 00:05:12.130 --> 00:05:15.790 align:middle line:84% And then, we had a little lemma that was quite subtle even 00:05:15.790 --> 00:05:20.360 align:middle line:84% for those guys who weekly prefer the alternative allocation, 00:05:20.360 --> 00:05:25.650 align:middle line:84% they too would be spending weekly more. 00:05:25.650 --> 00:05:29.540 align:middle line:84% And so we get the total expenditure 00:05:29.540 --> 00:05:32.370 align:middle line:84% summing up over all the households 00:05:32.370 --> 00:05:36.450 align:middle line:84% that would be necessary at the competitive prices 00:05:36.450 --> 00:05:39.540 align:middle line:84% to achieve the alternative allocation to be strictly 00:05:39.540 --> 00:05:43.590 align:middle line:84% greater than what is being spent in the competitive allocation. 00:05:43.590 --> 00:05:49.020 align:middle line:84% Then, we add that to profit maximization. 00:05:49.020 --> 00:05:56.840 align:middle line:84% And we get a relationship, a strict inequality relationship, 00:05:56.840 --> 00:06:00.350 align:middle line:84% which at the end of the day said that the allocation 00:06:00.350 --> 00:06:03.170 align:middle line:90% in the alternative-- 00:06:03.170 --> 00:06:07.010 align:middle line:84% allocation that's supposed to be Pareto dominating 00:06:07.010 --> 00:06:10.630 align:middle line:90% is not resource-feasible. 00:06:10.630 --> 00:06:13.100 align:middle line:84% But the definition of Pareto optimality 00:06:13.100 --> 00:06:20.310 align:middle line:84% is that there is a dominating-- if something's not optimal, 00:06:20.310 --> 00:06:23.410 align:middle line:84% it can be dominated by something that is feasible. 00:06:23.410 --> 00:06:26.700 align:middle line:84% So the feasibility part will be contradicted, 00:06:26.700 --> 00:06:29.490 align:middle line:90% and that was the proof. 00:06:29.490 --> 00:06:31.730 align:middle line:84% OK, so you'll see this theorem again 00:06:31.730 --> 00:06:36.890 align:middle line:84% when we get to that last lecture on monetary economics. 00:06:36.890 --> 00:06:43.060 align:middle line:84% It's going to turn out that under certain conditions 00:06:43.060 --> 00:06:48.380 align:middle line:84% like having overlapping generations of people 00:06:48.380 --> 00:06:55.710 align:middle line:84% so that no person lives forever, but time goes on forever, 00:06:55.710 --> 00:07:00.300 align:middle line:84% that that's enough to make the first welfare 00:07:00.300 --> 00:07:04.860 align:middle line:84% theorem fail, surprisingly, when we get into monetary economics. 00:07:04.860 --> 00:07:10.110 align:middle line:84% And the proof then begs the issue 00:07:10.110 --> 00:07:12.990 align:middle line:84% of where this proof is breaking down. 00:07:12.990 --> 00:07:16.770 align:middle line:84% And it's going to turn out that the valuation 00:07:16.770 --> 00:07:21.030 align:middle line:84% of the alternative bundle at the competitive equilibrium prices 00:07:21.030 --> 00:07:23.880 align:middle line:90% costs an infinite amount. 00:07:23.880 --> 00:07:26.100 align:middle line:84% So we implicitly assume everything 00:07:26.100 --> 00:07:29.970 align:middle line:84% is finite in this finite dimension of Euclidean space. 00:07:29.970 --> 00:07:33.390 align:middle line:90% Anyway, just to anticipate. 00:07:33.390 --> 00:07:39.390 align:middle line:84% Second thing I want to ask about is similar. 00:07:39.390 --> 00:07:44.880 align:middle line:84% Outline the steps of the proof of the second welfare theorem 00:07:44.880 --> 00:07:47.870 align:middle line:90% that we did in class last time. 00:07:47.870 --> 00:07:50.180 align:middle line:84% So I don't expect a lot of detail on this, 00:07:50.180 --> 00:07:56.090 align:middle line:90% but let's see who would like to. 00:07:56.090 --> 00:08:00.150 align:middle line:84% Yuen, can you volunteer for that? 00:08:00.150 --> 00:08:01.640 align:middle line:90% AUDIENCE: Sure, yeah. 00:08:01.640 --> 00:08:03.500 align:middle line:84% So the second welfare theorem states 00:08:03.500 --> 00:08:07.340 align:middle line:84% that any Pareto optimal allocation 00:08:07.340 --> 00:08:11.120 align:middle line:84% can be achieved by some price equilibrium, 00:08:11.120 --> 00:08:15.800 align:middle line:84% assuming that you can shift endowment point under exchange. 00:08:15.800 --> 00:08:20.780 align:middle line:84% And so in the Lagrangian maximization problem, 00:08:20.780 --> 00:08:23.030 align:middle line:84% basically, you're finding the actual endowment 00:08:23.030 --> 00:08:27.200 align:middle line:84% that will give you the correct price equilibrium 00:08:27.200 --> 00:08:31.500 align:middle line:84% corresponding to the desired Pareto optimal allocation. 00:08:31.500 --> 00:08:33.020 align:middle line:84% And you can do that just by first 00:08:33.020 --> 00:08:39.230 align:middle line:84% solving the Pareto problem to get the first order conditions 00:08:39.230 --> 00:08:40.070 align:middle line:90% reallocations. 00:08:40.070 --> 00:08:43.169 align:middle line:84% And then, you can solve the Walrasian equilibrium problem-- 00:08:43.169 --> 00:08:46.850 align:middle line:84% so the consumer and the firm side problem separately. 00:08:46.850 --> 00:08:49.400 align:middle line:84% And then, you just match the first order of conditions. 00:08:49.400 --> 00:08:51.650 align:middle line:84% So the appropriate parameters, like the shadow prices, 00:08:51.650 --> 00:08:55.640 align:middle line:84% need to equal the prices in the price equilibrium-- 00:08:55.640 --> 00:08:57.600 align:middle line:90% stuff like that. 00:08:57.600 --> 00:09:01.040 align:middle line:84% And you can show that all these things exist in our system. 00:09:01.040 --> 00:09:03.380 align:middle line:90% ROBERT TOWNSEND: Perfect, yep. 00:09:03.380 --> 00:09:06.870 align:middle line:84% The key is the Lagrangian method, 00:09:06.870 --> 00:09:12.830 align:middle line:84% which under convexity in sets and concavity in functions 00:09:12.830 --> 00:09:16.110 align:middle line:84% makes the first order conditions necessary and sufficient. 00:09:16.110 --> 00:09:19.220 align:middle line:84% So to find a solution, all you need to do 00:09:19.220 --> 00:09:24.060 align:middle line:84% is find variables and shadow prices that satisfy 00:09:24.060 --> 00:09:25.380 align:middle line:90% those first order conditions. 00:09:25.380 --> 00:09:29.970 align:middle line:84% We have the obvious candidate, because we start with a Pareto 00:09:29.970 --> 00:09:32.190 align:middle line:90% optimal allocation. 00:09:32.190 --> 00:09:36.000 align:middle line:84% So we just compare what we need for the Walrasian allocation 00:09:36.000 --> 00:09:39.830 align:middle line:84% to what we already have from the Pareto problem-- 00:09:39.830 --> 00:09:44.970 align:middle line:84% allocation, and choose prices, and Lagrange 00:09:44.970 --> 00:09:47.370 align:middle line:84% multipliers, and so on accordingly. 00:09:47.370 --> 00:09:50.050 align:middle line:90% So your answer was perfect. 00:09:50.050 --> 00:09:52.200 align:middle line:90% Thank you very much. 00:09:52.200 --> 00:09:52.700 align:middle line:90% All right. 00:09:52.700 --> 00:09:57.080 align:middle line:84% So that's so many review questions, 00:09:57.080 --> 00:10:02.540 align:middle line:84% because we really focused pretty narrowly on those two theorems 00:10:02.540 --> 00:10:03.170 align:middle line:90% in class. 00:10:03.170 --> 00:10:05.320 align:middle line:90% So all right. 00:10:05.320 --> 00:10:10.450 align:middle line:84% So then, the lecture today is existence. 00:10:10.450 --> 00:10:11.980 align:middle line:90% Sounds ominous. 00:10:11.980 --> 00:10:16.820 align:middle line:84% [CHUCKLES] What does it mean to not exist? 00:10:16.820 --> 00:10:19.690 align:middle line:84% Well, as per the welfare theorems, 00:10:19.690 --> 00:10:22.480 align:middle line:84% I said with the first welfare theorem, 00:10:22.480 --> 00:10:24.430 align:middle line:84% if a competitive equilibrium exists 00:10:24.430 --> 00:10:28.970 align:middle line:84% and it's optimal, how do we know it exists in the first place? 00:10:28.970 --> 00:10:30.190 align:middle line:90% That's one way to put it. 00:10:30.190 --> 00:10:33.310 align:middle line:84% Anyway, it's possible that many things we do 00:10:33.310 --> 00:10:35.920 align:middle line:84% are vacuous, because we assume a concept 00:10:35.920 --> 00:10:38.950 align:middle line:84% and assume an allocation that satisfies the concept. 00:10:38.950 --> 00:10:40.900 align:middle line:84% But we'd better go back and check. 00:10:40.900 --> 00:10:42.730 align:middle line:84% Along the way, there's more interesting 00:10:42.730 --> 00:10:44.950 align:middle line:84% stuff happening in this lecture, which 00:10:44.950 --> 00:10:48.430 align:middle line:84% has to do with reviewing the definition of a Walrasian 00:10:48.430 --> 00:10:51.760 align:middle line:84% equilibria and then seeing how sufficient assumptions 00:10:51.760 --> 00:10:54.760 align:middle line:84% for existence, but also in principle, 00:10:54.760 --> 00:10:57.040 align:middle line:84% how you might want to compute it, 00:10:57.040 --> 00:11:03.320 align:middle line:84% which brings us to modern day platforms and online markets. 00:11:03.320 --> 00:11:04.960 align:middle line:90% So it's not just abstract. 00:11:04.960 --> 00:11:07.780 align:middle line:84% It's actually something computer scientists 00:11:07.780 --> 00:11:09.880 align:middle line:84% and fintechs worry about quite a bit. 00:11:09.880 --> 00:11:13.660 align:middle line:84% And relatedly, we'll compare the Walrasian equilibria 00:11:13.660 --> 00:11:17.440 align:middle line:84% to the Nash equilibria, which is a different concept 00:11:17.440 --> 00:11:18.280 align:middle line:90% of an equilibrium. 00:11:18.280 --> 00:11:21.500 align:middle line:84% But it takes the same sufficient conditions. 00:11:21.500 --> 00:11:26.890 align:middle line:84% And again, that's a very, very useful tool to master. 00:11:26.890 --> 00:11:28.160 align:middle line:90% But we want to go beyond that. 00:11:28.160 --> 00:11:29.090 align:middle line:90% We want to use it. 00:11:29.090 --> 00:11:31.690 align:middle line:84% So I'm going to show you some pictures of what's 00:11:31.690 --> 00:11:35.320 align:middle line:84% going on in the New York financial markets and date, 00:11:35.320 --> 00:11:39.250 align:middle line:84% at least verbally, trade fails, and when those markets get 00:11:39.250 --> 00:11:42.550 align:middle line:84% messed up, as in the Coronavirus in March. 00:11:42.550 --> 00:11:44.470 align:middle line:84% And then, we'll think about, are there 00:11:44.470 --> 00:11:48.310 align:middle line:84% ways to clean up the markets and avoid the failures that 00:11:48.310 --> 00:11:50.330 align:middle line:90% have to do with market design. 00:11:50.330 --> 00:11:51.760 align:middle line:90% So that's the agenda for today. 00:11:51.760 --> 00:11:55.270 align:middle line:84% Mostly, it's conceptual and notational. 00:11:55.270 --> 00:11:57.760 align:middle line:84% And I have deliberately taken out 00:11:57.760 --> 00:12:00.520 align:middle line:84% a lot of the notation necessary to look 00:12:00.520 --> 00:12:03.380 align:middle line:84% at the actual mapping, because it gets quite tedious, 00:12:03.380 --> 00:12:03.880 align:middle line:90% actually. 00:12:03.880 --> 00:12:05.500 align:middle line:84% And it's not really the main point. 00:12:05.500 --> 00:12:08.920 align:middle line:90% OK, so here's the math tools. 00:12:08.920 --> 00:12:12.760 align:middle line:84% What do we mean by a fixed point of a function? 00:12:12.760 --> 00:12:15.700 align:middle line:84% We're going to want to solve nonlinear equations. 00:12:15.700 --> 00:12:19.780 align:middle line:84% We want to know whether or not there is a solution. 00:12:19.780 --> 00:12:23.590 align:middle line:84% Those could be excess demand equations in the Walrasian 00:12:23.590 --> 00:12:28.060 align:middle line:84% case, or reaction functions in the Nash case. 00:12:28.060 --> 00:12:29.660 align:middle line:90% We'll get there momentarily. 00:12:29.660 --> 00:12:35.980 align:middle line:84% So a function, f, maps some domain, A, back into itself. 00:12:35.980 --> 00:12:38.290 align:middle line:84% And we're looking for a particular value 00:12:38.290 --> 00:12:42.040 align:middle line:84% of x star in the domain such that when you stick it in, 00:12:42.040 --> 00:12:43.240 align:middle line:90% you get it back out. 00:12:43.240 --> 00:12:46.780 align:middle line:84% And what I mean by that is, at x star, f of x star 00:12:46.780 --> 00:12:48.490 align:middle line:90% is x star again. 00:12:48.490 --> 00:12:51.220 align:middle line:90% The more formal statement-- 00:12:51.220 --> 00:12:54.940 align:middle line:84% Brower's Fix Point Theorem-- is at the bottom of this slide. 00:12:54.940 --> 00:12:59.680 align:middle line:84% Let A be a subset of a finite dimensional L dimensional 00:12:59.680 --> 00:13:01.120 align:middle line:90% Euclidean space. 00:13:01.120 --> 00:13:04.810 align:middle line:84% And that quote "domain" being non empty, convex, 00:13:04.810 --> 00:13:10.120 align:middle line:84% and compact," and f maps that domain back into itself, 00:13:10.120 --> 00:13:15.490 align:middle line:84% A to A, f is continuous, that's enough conditions 00:13:15.490 --> 00:13:19.360 align:middle line:84% on f and on this set A such that there 00:13:19.360 --> 00:13:22.720 align:middle line:84% is a special point, x star, in A, 00:13:22.720 --> 00:13:24.610 align:middle line:84% and when you stick it into the function f, 00:13:24.610 --> 00:13:26.300 align:middle line:90% you get x star right back again. 00:13:26.300 --> 00:13:29.260 align:middle line:84% So what's going on is pretty well 00:13:29.260 --> 00:13:31.780 align:middle line:84% captured by this classic picture. 00:13:31.780 --> 00:13:35.830 align:middle line:84% The set A is just the integral 0 to 1. 00:13:35.830 --> 00:13:38.360 align:middle line:90% And that's the domain. 00:13:38.360 --> 00:13:42.310 align:middle line:84% And the range is the same, A to A, 0 to 1. 00:13:42.310 --> 00:13:45.250 align:middle line:84% And this function f is being drawn here 00:13:45.250 --> 00:13:47.530 align:middle line:90% as a continuous function. 00:13:47.530 --> 00:13:51.250 align:middle line:84% And the challenge would be, how could you possibly draw 00:13:51.250 --> 00:13:53.620 align:middle line:84% this function, which has to be continuous, 00:13:53.620 --> 00:13:57.850 align:middle line:84% in a way that it doesn't cross the green diagonal? 00:13:57.850 --> 00:13:59.500 align:middle line:90% This one clearly does. 00:13:59.500 --> 00:14:03.160 align:middle line:84% So where it crosses, by definition of the 45 degree 00:14:03.160 --> 00:14:08.050 align:middle line:84% line we found, x star, we get back what we put in. 00:14:08.050 --> 00:14:08.620 align:middle line:90% So, oh, no. 00:14:08.620 --> 00:14:10.240 align:middle line:90% I don't want to go down there. 00:14:10.240 --> 00:14:12.640 align:middle line:84% How about we hug the axis up here? 00:14:12.640 --> 00:14:17.320 align:middle line:84% But even then, where f of x is 1 for all x, 00:14:17.320 --> 00:14:21.130 align:middle line:84% it would still be a fixed point at the end point. 00:14:21.130 --> 00:14:23.770 align:middle line:90% So anyway, that's the intuition. 00:14:23.770 --> 00:14:28.397 align:middle line:84% It's pretty clearly sufficient to have the fixed point. 00:14:28.397 --> 00:14:29.230 align:middle line:90% And when you don't-- 00:14:29.230 --> 00:14:32.830 align:middle line:84% I'll show you a picture-- it's because it'll jump over. 00:14:32.830 --> 00:14:35.530 align:middle line:84% It will not be defined exactly where 00:14:35.530 --> 00:14:39.340 align:middle line:84% it would need to be defined to cross the 45 degree line. 00:14:39.340 --> 00:14:42.100 align:middle line:84% This is a generalization of Brower's theorem. 00:14:42.100 --> 00:14:45.610 align:middle line:84% It's Kakutani's Fixed Point Theorem. 00:14:45.610 --> 00:14:48.400 align:middle line:84% A-- still the same, a non empty subset 00:14:48.400 --> 00:14:51.940 align:middle line:84% of a finite dimensional Euclidean space. 00:14:51.940 --> 00:14:56.680 align:middle line:84% f, however, doesn't have unique single values in the range. 00:14:56.680 --> 00:14:59.200 align:middle line:84% It could be a whole set of values of f. 00:14:59.200 --> 00:15:01.720 align:middle line:90% It's called a correspondence. 00:15:01.720 --> 00:15:05.040 align:middle line:90% So we pick x in the domain. 00:15:05.040 --> 00:15:08.220 align:middle line:84% There's a plausible entire set of values-- 00:15:08.220 --> 00:15:11.790 align:middle line:84% f of x, which is a subset of A. So then, 00:15:11.790 --> 00:15:13.740 align:middle line:84% we just look at the conditions on A, 00:15:13.740 --> 00:15:17.100 align:middle line:84% which are already given to us to be non empty, compact, 00:15:17.100 --> 00:15:21.870 align:middle line:84% and convex, exactly as in Brower, f of x is non empty. 00:15:21.870 --> 00:15:24.900 align:middle line:84% Well, that's been implicitly assumed 00:15:24.900 --> 00:15:27.900 align:middle line:90% that it has a value for every x. 00:15:27.900 --> 00:15:29.220 align:middle line:90% And here's some more-- 00:15:29.220 --> 00:15:31.920 align:middle line:90% convex-valued and closed. 00:15:31.920 --> 00:15:34.180 align:middle line:84% So although it wasn't about this, the question 00:15:34.180 --> 00:15:38.020 align:middle line:84% he asked at the very beginning-- so it's like this. 00:15:38.020 --> 00:15:40.920 align:middle line:84% So f is convex-valued means take any two 00:15:40.920 --> 00:15:45.930 align:middle line:84% points in f of x, the convex combination is also in f of x. 00:15:45.930 --> 00:15:53.400 align:middle line:84% And closed simply means that if you have a sequence of points, 00:15:53.400 --> 00:15:58.920 align:middle line:84% xn and yn converging to x and y, with every point along the way 00:15:58.920 --> 00:16:01.680 align:middle line:84% in the sequence satisfying yn, an element 00:16:01.680 --> 00:16:06.840 align:middle line:84% of f of xn, then the limit y is an element of f of x. 00:16:06.840 --> 00:16:09.330 align:middle line:90% So again, it doesn't jump. 00:16:09.330 --> 00:16:12.060 align:middle line:84% And then, Kakutani tells us that these 00:16:12.060 --> 00:16:16.260 align:middle line:84% are sufficient assumptions for us to have a fixed point. 00:16:16.260 --> 00:16:18.870 align:middle line:84% I wish this had been written with a star here. 00:16:18.870 --> 00:16:22.170 align:middle line:84% If x star is an element of A, then x 00:16:22.170 --> 00:16:26.070 align:middle line:84% star is an element of f of x star, not equal. 00:16:26.070 --> 00:16:28.650 align:middle line:84% Because we have a correspondence, just 00:16:28.650 --> 00:16:30.210 align:middle line:90% an element of. 00:16:30.210 --> 00:16:33.870 align:middle line:84% So here is the jumpy picture that I promised you-- 00:16:33.870 --> 00:16:35.610 align:middle line:90% the two exceptions. 00:16:35.610 --> 00:16:40.320 align:middle line:84% F has to be convex-valued and continuous, in some sense, 00:16:40.320 --> 00:16:41.700 align:middle line:90% or have a closed graph. 00:16:41.700 --> 00:16:46.080 align:middle line:84% Here, it's not convex-valued at this particular point 00:16:46.080 --> 00:16:47.670 align:middle line:90% in the domain. 00:16:47.670 --> 00:16:50.280 align:middle line:84% It is not true that any linear combination 00:16:50.280 --> 00:16:53.250 align:middle line:84% of points in the range are also in the range. 00:16:53.250 --> 00:16:56.490 align:middle line:84% And in particular, there's nothing in the middle here. 00:16:56.490 --> 00:16:59.880 align:middle line:84% So it jumps over that 45 degree line. 00:16:59.880 --> 00:17:03.240 align:middle line:84% Likewise here, there's a sequence of points-- 00:17:03.240 --> 00:17:05.670 align:middle line:90% actually, unique points. 00:17:05.670 --> 00:17:09.190 align:middle line:84% Correspondence allows single value functions. 00:17:09.190 --> 00:17:12.180 align:middle line:84% But when you get here, the limit point 00:17:12.180 --> 00:17:20.609 align:middle line:84% is no longer consistent with f of x being a point of x. 00:17:20.609 --> 00:17:23.220 align:middle line:90% And the function drops down. 00:17:23.220 --> 00:17:26.099 align:middle line:90% x is here. f of x is here. 00:17:26.099 --> 00:17:27.150 align:middle line:90% That's fine. 00:17:27.150 --> 00:17:28.200 align:middle line:90% That jump is fine. 00:17:28.200 --> 00:17:30.900 align:middle line:84% But the problem is, this can be achieved by a limit. 00:17:30.900 --> 00:17:34.020 align:middle line:84% And the point in the limit is not in the pair. 00:17:34.020 --> 00:17:37.710 align:middle line:84% That's that little Cheerio with a hole. 00:17:37.710 --> 00:17:40.380 align:middle line:84% That's not an argument of f of x. 00:17:40.380 --> 00:17:44.910 align:middle line:84% So anyway, this is what can go wrong to motivate the quote 00:17:44.910 --> 00:17:46.930 align:middle line:90% "sufficient assumptions." 00:17:46.930 --> 00:17:50.230 align:middle line:84% Now, we get to the first application-- 00:17:50.230 --> 00:17:53.000 align:middle line:90% existence of Walrasian prices. 00:17:53.000 --> 00:17:55.390 align:middle line:90% We want sufficient conditions. 00:17:55.390 --> 00:17:57.280 align:middle line:84% Consider a pure exchange economy. 00:17:57.280 --> 00:17:59.200 align:middle line:90% Get rid of production. 00:17:59.200 --> 00:18:02.320 align:middle line:84% Have a strictly positive aggregate endowment. 00:18:02.320 --> 00:18:06.010 align:middle line:84% Assume that preferences are rational, locally nonsatiated, 00:18:06.010 --> 00:18:07.790 align:middle line:90% and continuous. 00:18:07.790 --> 00:18:10.240 align:middle line:84% In fact, preferences can be represented 00:18:10.240 --> 00:18:12.280 align:middle line:90% by utility functions. 00:18:12.280 --> 00:18:15.340 align:middle line:84% The consumption set is, say, all of the L 00:18:15.340 --> 00:18:18.650 align:middle line:90% dimensional non-negative space. 00:18:18.650 --> 00:18:21.110 align:middle line:90% So we define three things here. 00:18:21.110 --> 00:18:25.420 align:middle line:84% First of all, we're defining the maximizing demand. 00:18:25.420 --> 00:18:29.590 align:middle line:84% So we maximize utility function subject to consumption 00:18:29.590 --> 00:18:33.340 align:middle line:84% being in the consumption set, subject to not spending more 00:18:33.340 --> 00:18:37.180 align:middle line:84% than the valuation of the endowment at this price, p. 00:18:37.180 --> 00:18:41.080 align:middle line:84% That's the definition of demand for all prices 00:18:41.080 --> 00:18:43.120 align:middle line:90% p given parametrically. 00:18:43.120 --> 00:18:46.480 align:middle line:90% Z will call excess demand-- 00:18:46.480 --> 00:18:50.320 align:middle line:84% namely, not total demand, but the difference between demand 00:18:50.320 --> 00:18:51.550 align:middle line:90% and the endowment. 00:18:51.550 --> 00:18:54.220 align:middle line:84% This is typically positive if they're buying stuff. 00:18:54.220 --> 00:18:56.500 align:middle line:84% It could be negative if they're selling stuff. 00:18:56.500 --> 00:19:00.310 align:middle line:84% And yeah, we just call it excess demand. 00:19:00.310 --> 00:19:03.220 align:middle line:84% And then, we take the sum of the excess demands, 00:19:03.220 --> 00:19:05.050 align:middle line:90% summing over all the households. 00:19:05.050 --> 00:19:09.400 align:middle line:84% This is I, capital I. So this is the aggregate excess demand 00:19:09.400 --> 00:19:10.940 align:middle line:90% function. 00:19:10.940 --> 00:19:17.480 align:middle line:84% Now, if we can find a price, p, such that this z of p is 0, 00:19:17.480 --> 00:19:19.200 align:middle line:90% then we're done. 00:19:19.200 --> 00:19:20.030 align:middle line:90% Why? 00:19:20.030 --> 00:19:22.700 align:middle line:84% Because we found the p such that when 00:19:22.700 --> 00:19:25.820 align:middle line:84% we march across the different households, 00:19:25.820 --> 00:19:29.940 align:middle line:84% we have excess demand being positive for some people. 00:19:29.940 --> 00:19:33.230 align:middle line:84% But since it has to sum to 0, that excess demand 00:19:33.230 --> 00:19:35.460 align:middle line:90% has to be negative. 00:19:35.460 --> 00:19:39.440 align:middle line:84% So whatever gets supplied by households, some households, 00:19:39.440 --> 00:19:41.540 align:middle line:84% gets demanded by other households. 00:19:41.540 --> 00:19:44.780 align:middle line:84% By the way, all of this is written in vector notation, 00:19:44.780 --> 00:19:45.980 align:middle line:90% although it's not obvious. 00:19:45.980 --> 00:19:48.230 align:middle line:90% You have to be reminded of that. 00:19:48.230 --> 00:19:52.640 align:middle line:84% We want to be able to clear each and every one of the L markets 00:19:52.640 --> 00:19:54.670 align:middle line:90% with this price vector p. 00:19:54.670 --> 00:19:57.260 align:middle line:84% OK, so here's a picture where it works, 00:19:57.260 --> 00:19:59.210 align:middle line:90% and a picture where it doesn't. 00:19:59.210 --> 00:20:01.820 align:middle line:84% The aggregate endowment is over here. 00:20:01.820 --> 00:20:05.970 align:middle line:84% And we'll look, despite what I just said, at good 1. 00:20:05.970 --> 00:20:07.310 align:middle line:90% Take the price of good 1. 00:20:07.310 --> 00:20:09.920 align:middle line:84% Here's the demand for it, total demand. 00:20:09.920 --> 00:20:11.600 align:middle line:90% This is total supply. 00:20:11.600 --> 00:20:14.810 align:middle line:90% Excess demand hits 0 right here. 00:20:14.810 --> 00:20:15.560 align:middle line:90% We're done. 00:20:15.560 --> 00:20:16.850 align:middle line:90% We found a price-- 00:20:16.850 --> 00:20:19.520 align:middle line:84% this price, where excess demand is 0, 00:20:19.520 --> 00:20:21.720 align:middle line:84% because supply is equal to demand. 00:20:21.720 --> 00:20:25.070 align:middle line:84% But if it were to happen that somehow those demand curves 00:20:25.070 --> 00:20:30.140 align:middle line:84% jump, then for some prices, excess demand is positive 00:20:30.140 --> 00:20:31.710 align:middle line:90% and for others, it's negative. 00:20:31.710 --> 00:20:36.510 align:middle line:84% And we would not find 0 of that function. 00:20:36.510 --> 00:20:38.780 align:middle line:90% So when can this happen? 00:20:38.780 --> 00:20:42.260 align:middle line:84% I didn't draw the picture here, but if you take an Edgeworth 00:20:42.260 --> 00:20:45.890 align:middle line:84% box diagram, you can imagine that households 00:20:45.890 --> 00:20:50.630 align:middle line:84% might jump from wanting all of good x to wanting all of good y 00:20:50.630 --> 00:20:51.950 align:middle line:90% at a certain price. 00:20:51.950 --> 00:20:55.010 align:middle line:84% And maybe if you set it up even with more restrictions, 00:20:55.010 --> 00:20:58.520 align:middle line:84% they're not very happy with stuff in the middle, either-- 00:20:58.520 --> 00:21:02.750 align:middle line:84% for example, with not concave indifference curves, 00:21:02.750 --> 00:21:04.517 align:middle line:90% but convex indifference curves. 00:21:04.517 --> 00:21:06.350 align:middle line:84% The linear stuff won't work, because they'll 00:21:06.350 --> 00:21:07.850 align:middle line:90% be happy in the middle. 00:21:07.850 --> 00:21:10.370 align:middle line:84% But when it's a concave function, 00:21:10.370 --> 00:21:15.980 align:middle line:84% or convex indifference curves, or concave upper contour sets, 00:21:15.980 --> 00:21:19.160 align:middle line:84% then they will only want to go to extremes. 00:21:19.160 --> 00:21:21.200 align:middle line:84% And that's going to create a hole. 00:21:21.200 --> 00:21:24.650 align:middle line:90% So I think it's not written out. 00:21:24.650 --> 00:21:27.170 align:middle line:84% But you're going to see it indirectly in a minute. 00:21:27.170 --> 00:21:30.860 align:middle line:84% We need sufficient convexity of consumption sets, 00:21:30.860 --> 00:21:33.200 align:middle line:84% and concavity of utility functions, 00:21:33.200 --> 00:21:35.690 align:middle line:84% and if we had production, some convexity 00:21:35.690 --> 00:21:40.160 align:middle line:84% assumptions there in order to get the conditions to be 00:21:40.160 --> 00:21:42.050 align:middle line:90% able to use the fixed point. 00:21:42.050 --> 00:21:44.090 align:middle line:84% Remember, the fixed point assumes 00:21:44.090 --> 00:21:48.440 align:middle line:84% that functions are continuous and the domain 00:21:48.440 --> 00:21:51.890 align:middle line:84% of the underlying mapping is closed, 00:21:51.890 --> 00:21:53.960 align:middle line:90% and convex, and nonempty. 00:21:53.960 --> 00:21:57.840 align:middle line:84% Now, I could have put many more slides in here. 00:21:57.840 --> 00:22:00.230 align:middle line:84% In earlier years, I used to have them. 00:22:00.230 --> 00:22:02.000 align:middle line:90% It's surprisingly tedious. 00:22:02.000 --> 00:22:06.350 align:middle line:84% This was the work done by Arrow, and Debreu, and McKenzie 00:22:06.350 --> 00:22:08.660 align:middle line:90% in the '50s and the '60s. 00:22:08.660 --> 00:22:12.050 align:middle line:84% And the mapping is not as straightforward 00:22:12.050 --> 00:22:12.950 align:middle line:90% as you might hope. 00:22:12.950 --> 00:22:15.880 align:middle line:84% It's certainly nothing like these pictures. 00:22:15.880 --> 00:22:19.170 align:middle line:84% But I think the intuition is fine for today. 00:22:19.170 --> 00:22:21.670 align:middle line:84% Now, much more relevant since, we just did it, 00:22:21.670 --> 00:22:24.070 align:middle line:90% would be Negishi's mapping. 00:22:24.070 --> 00:22:27.490 align:middle line:84% Instead of looking for prices that clear markets, 00:22:27.490 --> 00:22:30.430 align:middle line:84% we're going to look for Pareto weights using 00:22:30.430 --> 00:22:33.340 align:middle line:84% the first and second welfare theorem. 00:22:33.340 --> 00:22:36.190 align:middle line:84% Again, exactly what we did last time. 00:22:36.190 --> 00:22:38.200 align:middle line:84% The first welfare theorem tells us 00:22:38.200 --> 00:22:41.080 align:middle line:84% that the competitive equilibrium will be Pareto optimal. 00:22:41.080 --> 00:22:43.630 align:middle line:84% And we're going to be having enough assumptions so 00:22:43.630 --> 00:22:45.700 align:middle line:84% that that theorem is true-- local non 00:22:45.700 --> 00:22:48.040 align:middle line:90% satiation and rationality. 00:22:48.040 --> 00:22:51.610 align:middle line:84% So we know, if it's optimal, there exists Pareto weights 00:22:51.610 --> 00:22:56.500 align:middle line:84% lambda such that when we solve the lambda weighted max 00:22:56.500 --> 00:23:00.640 align:middle line:84% problem, we will get that particular Pareto optimal 00:23:00.640 --> 00:23:01.600 align:middle line:90% allocation. 00:23:01.600 --> 00:23:04.570 align:middle line:84% Not any lambda-- particular lambdas 00:23:04.570 --> 00:23:08.650 align:middle line:84% that generate that particular Pareto optimal allocation 00:23:08.650 --> 00:23:11.600 align:middle line:84% corresponding to the Walrasian equilibrium. 00:23:11.600 --> 00:23:14.500 align:middle line:90% So that's Negishi's method. 00:23:14.500 --> 00:23:18.310 align:middle line:84% Instead of trying to find prices that clear markets, 00:23:18.310 --> 00:23:21.370 align:middle line:84% it's going to try to find Pareto weights-- 00:23:21.370 --> 00:23:23.770 align:middle line:84% in particular, Pareto weights such 00:23:23.770 --> 00:23:27.100 align:middle line:84% that we're searching as we vary the Pareto 00:23:27.100 --> 00:23:31.420 align:middle line:84% weights over the space of Pareto optimal allocations in such 00:23:31.420 --> 00:23:34.870 align:middle line:84% a way that the fixed point of a mapping 00:23:34.870 --> 00:23:38.950 align:middle line:84% will be the particular Pareto weights that generate 00:23:38.950 --> 00:23:40.870 align:middle line:90% the Walrasian allocation. 00:23:40.870 --> 00:23:46.960 align:middle line:84% And in particular, the valuation of the expenditures associated 00:23:46.960 --> 00:23:49.900 align:middle line:84% with the competitive equilibrium is exactly 00:23:49.900 --> 00:23:54.230 align:middle line:84% the valuation of wealth under the private ownership economy. 00:23:54.230 --> 00:23:56.890 align:middle line:84% So those are the words and the notations coming. 00:23:56.890 --> 00:23:58.420 align:middle line:90% It's three or four slides away. 00:23:58.420 --> 00:24:03.220 align:middle line:90% OK, so what does Negishi do in-- 00:24:03.220 --> 00:24:05.920 align:middle line:90% he actually adds up over firms. 00:24:05.920 --> 00:24:08.710 align:middle line:84% So we act like there's just one single firm. 00:24:08.710 --> 00:24:11.050 align:middle line:90% That's perfectly fine to do. 00:24:11.050 --> 00:24:13.960 align:middle line:84% It just makes the notation easier. 00:24:13.960 --> 00:24:16.630 align:middle line:84% And these assumptions are pretty much what 00:24:16.630 --> 00:24:19.360 align:middle line:84% we did last time when we did the Lagrangian 00:24:19.360 --> 00:24:20.950 align:middle line:90% and the second welfare theorem. 00:24:20.950 --> 00:24:23.740 align:middle line:84% The consumption sets are closed and convex. 00:24:23.740 --> 00:24:25.570 align:middle line:90% Let's say they contain 0. 00:24:25.570 --> 00:24:27.580 align:middle line:84% That's a little bit more special. 00:24:27.580 --> 00:24:31.090 align:middle line:84% Utility functions are continuous concave functions 00:24:31.090 --> 00:24:33.100 align:middle line:90% with local non satiation. 00:24:33.100 --> 00:24:36.310 align:middle line:84% We define an aggregate consumption set, 00:24:36.310 --> 00:24:40.570 align:middle line:84% the aggregate social endowment, the aggregated production 00:24:40.570 --> 00:24:43.870 align:middle line:84% of the representative firm, which is also convex 00:24:43.870 --> 00:24:46.540 align:middle line:84% and has a concave transformation, 00:24:46.540 --> 00:24:50.050 align:middle line:84% possibility of an action in the production set, 00:24:50.050 --> 00:24:51.820 align:middle line:90% and feasibility-- 00:24:51.820 --> 00:24:55.000 align:middle line:84% in particular, the existence of an interior point, 00:24:55.000 --> 00:24:59.770 align:middle line:84% and that the feasible set of allocations is compact. 00:24:59.770 --> 00:25:02.410 align:middle line:84% This compactness may seem strange. 00:25:02.410 --> 00:25:04.820 align:middle line:84% And I can come back to that if you want. 00:25:04.820 --> 00:25:07.990 align:middle line:84% Certainly, if you had an Edgeworth box economy, 00:25:07.990 --> 00:25:11.380 align:middle line:84% the set of aggregate resources is fixed and finite. 00:25:11.380 --> 00:25:14.260 align:middle line:84% Production creates the possibility of creating more, 00:25:14.260 --> 00:25:17.210 align:middle line:84% but you have a finite number of inputs, and so on. 00:25:17.210 --> 00:25:20.290 align:middle line:84% So in the end, you end up with a compact set. 00:25:20.290 --> 00:25:21.220 align:middle line:90% All right. 00:25:21.220 --> 00:25:24.640 align:middle line:90% So Negishi's theorem, 1960-- 00:25:24.640 --> 00:25:26.470 align:middle line:84% suppose we have a private ownership 00:25:26.470 --> 00:25:29.020 align:middle line:84% economy with this string of elements 00:25:29.020 --> 00:25:33.370 align:middle line:84% that you're now very familiar with that satisfy A.1 00:25:33.370 --> 00:25:37.760 align:middle line:84% through A.4, all the convexity, and concavity, and continuity, 00:25:37.760 --> 00:25:38.260 align:middle line:90% and so on. 00:25:38.260 --> 00:25:39.820 align:middle line:90% Then, voila. 00:25:39.820 --> 00:25:42.250 align:middle line:84% There exists a Walrasian equilibrium. 00:25:42.250 --> 00:25:44.740 align:middle line:90% x star, y star, p star. 00:25:44.740 --> 00:25:48.460 align:middle line:84% Let's assume utility functions are differentiable. 00:25:48.460 --> 00:25:49.600 align:middle line:90% F is differentiable. 00:25:49.600 --> 00:25:53.200 align:middle line:84% We did that anyway with the second welfare theorem. 00:25:53.200 --> 00:25:57.040 align:middle line:84% So the key here is that Pareto optimal allocations 00:25:57.040 --> 00:25:59.080 align:middle line:90% are indexed by lambda. 00:25:59.080 --> 00:26:04.420 align:middle line:84% This simple-looking problem, which is, pick a lambda vector, 00:26:04.420 --> 00:26:07.000 align:middle line:84% maximize lambda-weighted sums of utilities 00:26:07.000 --> 00:26:10.060 align:middle line:84% by choices of consumption and production 00:26:10.060 --> 00:26:13.990 align:middle line:84% that are in consumption sets, production sets, 00:26:13.990 --> 00:26:16.210 align:middle line:84% and satisfy the resource constraint 00:26:16.210 --> 00:26:18.130 align:middle line:90% for each of the L goods. 00:26:18.130 --> 00:26:20.140 align:middle line:84% You saw a slide like this last time 00:26:20.140 --> 00:26:22.450 align:middle line:90% with the second welfare theorem. 00:26:22.450 --> 00:26:25.180 align:middle line:84% Here, we're going to explicitly index 00:26:25.180 --> 00:26:29.540 align:middle line:84% the solution of a given lambda as x and y lambda. 00:26:29.540 --> 00:26:32.620 align:middle line:84% So for a given lambda, a particular target Pareto 00:26:32.620 --> 00:26:34.720 align:middle line:84% optimal allocation is the solution 00:26:34.720 --> 00:26:36.350 align:middle line:90% to this Pareto problem. 00:26:36.350 --> 00:26:38.590 align:middle line:90% Now again, said this earlier. 00:26:38.590 --> 00:26:42.040 align:middle line:84% We know there is a lambda, of course, that generates a Pareto 00:26:42.040 --> 00:26:46.000 align:middle line:84% optimal allocation that corresponds to the Walrasian 00:26:46.000 --> 00:26:47.110 align:middle line:90% equilibrium. 00:26:47.110 --> 00:26:49.870 align:middle line:84% And more generally, from the second welfare theorem, 00:26:49.870 --> 00:26:53.050 align:middle line:84% any Pareto optimal allocation hence 00:26:53.050 --> 00:26:56.500 align:middle line:84% has a particular lambda such that it 00:26:56.500 --> 00:27:00.760 align:middle line:84% can be achieved as a competitive equilibrium with transfers. 00:27:00.760 --> 00:27:03.790 align:middle line:84% So let's just search in the space of Pareto optimal 00:27:03.790 --> 00:27:06.940 align:middle line:84% allocations till we find the right one, 00:27:06.940 --> 00:27:09.160 align:middle line:90% find the Walrasian one. 00:27:09.160 --> 00:27:11.830 align:middle line:84% And in particular, the prices are 00:27:11.830 --> 00:27:14.810 align:middle line:90% going to happen along the way. 00:27:14.810 --> 00:27:17.390 align:middle line:84% They are the Lagrange multipliers associated 00:27:17.390 --> 00:27:18.710 align:middle line:90% with the resource constraints. 00:27:18.710 --> 00:27:23.190 align:middle line:84% This is exactly, in fact, the same notation from last time, 00:27:23.190 --> 00:27:27.050 align:middle line:84% except that this gamma, which is the price vector, 00:27:27.050 --> 00:27:31.040 align:middle line:84% is indexed by lambda when we matched up, remember, 00:27:31.040 --> 00:27:33.020 align:middle line:84% the consumer problem with the first order 00:27:33.020 --> 00:27:35.270 align:middle line:84% conditions in the Walrasian equilibrium 00:27:35.270 --> 00:27:38.250 align:middle line:84% with the first order conditions of the Pareto problem. 00:27:38.250 --> 00:27:41.780 align:middle line:84% And we also will get, from the Pareto problem, 00:27:41.780 --> 00:27:45.600 align:middle line:84% the consumer's allocation, the firm's allocation, 00:27:45.600 --> 00:27:47.700 align:middle line:84% these prices, again, as I just mentioned, 00:27:47.700 --> 00:27:49.430 align:middle line:90% and the wealth allocation. 00:27:49.430 --> 00:27:52.520 align:middle line:84% And the wealth for a given lambda for a given Pareto 00:27:52.520 --> 00:27:55.190 align:middle line:84% optimal allocation is just the valuation 00:27:55.190 --> 00:27:58.160 align:middle line:84% at these Lagrange multiplier prices 00:27:58.160 --> 00:28:02.330 align:middle line:84% of the target Pareto optimal x allocation. 00:28:02.330 --> 00:28:03.750 align:middle line:90% This is a given. 00:28:03.750 --> 00:28:07.730 align:middle line:84% I here is a household I. We're summing over all the L 00:28:07.730 --> 00:28:09.630 align:middle line:90% goods, finite number of them. 00:28:09.630 --> 00:28:14.150 align:middle line:90% And so again, add lambda x hat. 00:28:14.150 --> 00:28:16.670 align:middle line:90% Add lambda of L-- 00:28:16.670 --> 00:28:19.400 align:middle line:90% is the demand for the Lth good. 00:28:19.400 --> 00:28:21.140 align:middle line:84% This is the price of the Lth good. 00:28:21.140 --> 00:28:23.510 align:middle line:90% We sum over all of the L goods. 00:28:23.510 --> 00:28:25.070 align:middle line:90% We get total expenditures. 00:28:25.070 --> 00:28:27.290 align:middle line:90% And we assign that wealth level. 00:28:27.290 --> 00:28:29.090 align:middle line:90% You may remember that slide. 00:28:29.090 --> 00:28:30.810 align:middle line:90% I said, why do we need this? 00:28:30.810 --> 00:28:34.070 align:middle line:84% And I kept reminding you of the global algorithm, which 00:28:34.070 --> 00:28:37.430 align:middle line:84% is, we have to make sure the assignments of wealth that 00:28:37.430 --> 00:28:42.110 align:middle line:84% correspond with supporting prices of the Pareto problem 00:28:42.110 --> 00:28:45.230 align:middle line:84% sum up to the total social endowment. 00:28:45.230 --> 00:28:49.030 align:middle line:90% So you saw this before. 00:28:49.030 --> 00:28:50.960 align:middle line:90% What are we looking for? 00:28:50.960 --> 00:28:53.770 align:middle line:84% We're looking for an assignment of wealth such 00:28:53.770 --> 00:28:57.820 align:middle line:84% that the wealth is equal to the valuation of wealth 00:28:57.820 --> 00:28:59.630 align:middle line:84% in the private ownership economy. 00:28:59.630 --> 00:29:02.740 align:middle line:84% So this is just the endowment of household I 00:29:02.740 --> 00:29:07.300 align:middle line:84% plus household I's claim on the shares of the firm. 00:29:07.300 --> 00:29:11.540 align:middle line:84% You used to have theta Ij summation over j, 00:29:11.540 --> 00:29:13.700 align:middle line:84% but because for simplicity we assumed 00:29:13.700 --> 00:29:16.430 align:middle line:84% 1 representative aggregated firm, 00:29:16.430 --> 00:29:19.430 align:middle line:84% we only need theta I, not theta Ij. 00:29:19.430 --> 00:29:26.270 align:middle line:84% It's I's claim on the profits of the firm, the single firm. 00:29:26.270 --> 00:29:28.100 align:middle line:84% Anyway, that's private ownership. 00:29:28.100 --> 00:29:31.700 align:middle line:84% So we're looking for lambdas somehow. 00:29:31.700 --> 00:29:35.390 align:middle line:84% And again, I didn't go much further with the mapping. 00:29:35.390 --> 00:29:38.510 align:middle line:84% We're going to iterate on lambdas in such a way 00:29:38.510 --> 00:29:42.200 align:middle line:84% that eventually, we define the mapping in such a way 00:29:42.200 --> 00:29:45.650 align:middle line:84% that the lambda that we want will be a fixed 00:29:45.650 --> 00:29:48.890 align:middle line:84% point of the mapping and will satisfy 00:29:48.890 --> 00:29:52.400 align:middle line:84% the valuation of expenditures at that particular Pareto 00:29:52.400 --> 00:29:56.480 align:middle line:84% optimum is exactly the valuation of wealth 00:29:56.480 --> 00:29:59.130 align:middle line:84% in the private ownership economy. 00:29:59.130 --> 00:30:01.380 align:middle line:84% Again, the mapping is a bit more tedious 00:30:01.380 --> 00:30:05.170 align:middle line:90% than you might hope for. 00:30:05.170 --> 00:30:08.190 align:middle line:84% On the other hand, you can see clearly from the second welfare 00:30:08.190 --> 00:30:12.670 align:middle line:84% theorem how we were using convexity, and continuity, 00:30:12.670 --> 00:30:13.360 align:middle line:90% and so on. 00:30:13.360 --> 00:30:16.380 align:middle line:84% So perhaps it's not too surprising, 00:30:16.380 --> 00:30:18.750 align:middle line:84% at the end of the day, that we end up 00:30:18.750 --> 00:30:22.050 align:middle line:90% with sufficient assumptions to-- 00:30:22.050 --> 00:30:22.845 align:middle line:90% oh, here it is. 00:30:22.845 --> 00:30:25.470 align:middle line:90% Here I am talking away. 00:30:25.470 --> 00:30:31.470 align:middle line:84% Sufficient assumptions to get this particular lambda star 00:30:31.470 --> 00:30:35.060 align:middle line:90% as the fixed point we want. 00:30:35.060 --> 00:30:37.040 align:middle line:84% I said this in words, and I should have 00:30:37.040 --> 00:30:39.980 align:middle line:90% had this slide on the board. 00:30:39.980 --> 00:30:43.790 align:middle line:84% Valuation of expenditures equals the valuation 00:30:43.790 --> 00:30:47.920 align:middle line:84% of wealth in the private ownership economy. 00:30:47.920 --> 00:30:49.770 align:middle line:84% So there's really two ways to search 00:30:49.770 --> 00:30:52.140 align:middle line:90% for a competitive equilibrium. 00:30:52.140 --> 00:30:57.390 align:middle line:84% We can search in price space, or we can search in lambda space. 00:30:57.390 --> 00:31:01.350 align:middle line:84% It's curious that in both cases, we need a simplex. 00:31:01.350 --> 00:31:05.040 align:middle line:84% To remind you, a simplex is the space 00:31:05.040 --> 00:31:08.370 align:middle line:84% where each of the elements is non-negative greater than 00:31:08.370 --> 00:31:12.030 align:middle line:84% or equal to 0, and the sum of the elements adds up to 1. 00:31:12.030 --> 00:31:14.820 align:middle line:84% So prices add up to 1, because we 00:31:14.820 --> 00:31:18.840 align:middle line:84% have a degree of freedom for the normalization of prices, 00:31:18.840 --> 00:31:22.230 align:middle line:84% because of homogeneity of degrees 0, 00:31:22.230 --> 00:31:24.900 align:middle line:84% and homogeneity of degree 1, I guess. 00:31:24.900 --> 00:31:30.150 align:middle line:84% And also in the lambda weights, the lambdas are normalized. 00:31:30.150 --> 00:31:31.440 align:middle line:90% They're non-negative. 00:31:31.440 --> 00:31:33.270 align:middle line:90% And they sum up to 1. 00:31:33.270 --> 00:31:35.010 align:middle line:90% So that's another simplex. 00:31:35.010 --> 00:31:37.710 align:middle line:84% But on the other hand, the dimensions 00:31:37.710 --> 00:31:41.010 align:middle line:84% may be easier in the lambda space. 00:31:41.010 --> 00:31:45.330 align:middle line:84% Maybe there are more goods than there are types of households. 00:31:45.330 --> 00:31:50.130 align:middle line:84% We have inputs and outputs of all kinds. 00:31:50.130 --> 00:31:54.000 align:middle line:84% There may be, at least for economic modeling-- 00:31:54.000 --> 00:31:58.440 align:middle line:84% we can think about a finite number of types of households. 00:31:58.440 --> 00:32:01.560 align:middle line:84% Maybe when you think about wealth, we have rich and poor. 00:32:01.560 --> 00:32:03.390 align:middle line:90% We have medians. 00:32:03.390 --> 00:32:05.760 align:middle line:90% We have quartiles. 00:32:05.760 --> 00:32:08.460 align:middle line:84% When we thought about international trade, 00:32:08.460 --> 00:32:11.420 align:middle line:84% we had households differing by their endowments. 00:32:11.420 --> 00:32:12.420 align:middle line:90% Some are laborers. 00:32:12.420 --> 00:32:14.770 align:middle line:90% Some own the capital, and so on. 00:32:14.770 --> 00:32:19.200 align:middle line:84% So typically, the number of lambdas corresponding 00:32:19.200 --> 00:32:21.420 align:middle line:84% to the number of types of households 00:32:21.420 --> 00:32:24.840 align:middle line:84% is less than the number of prices we need. 00:32:24.840 --> 00:32:28.290 align:middle line:84% And it's just easier, therefore, to search and compute 00:32:28.290 --> 00:32:30.240 align:middle line:90% in the lambda space. 00:32:30.240 --> 00:32:32.820 align:middle line:84% Anyway, I thought you might appreciate this, 00:32:32.820 --> 00:32:35.940 align:middle line:84% because we just got done with the second welfare theorem. 00:32:35.940 --> 00:32:38.910 align:middle line:84% And this existence proof uses that. 00:32:38.910 --> 00:32:42.030 align:middle line:84% Now, lest we think it's just elegant, 00:32:42.030 --> 00:32:45.450 align:middle line:84% I turn to what's going on in computer science-- 00:32:45.450 --> 00:32:48.760 align:middle line:84% namely, how to find the Walrasian equilibrium. 00:32:48.760 --> 00:32:52.500 align:middle line:84% So Echenique and Wierman at Caltech, 00:32:52.500 --> 00:32:55.050 align:middle line:84% actually, have some really cool stuff. 00:32:55.050 --> 00:32:58.080 align:middle line:84% It can actually be pretty hard to find 00:32:58.080 --> 00:33:00.510 align:middle line:90% the competitive equilibrium. 00:33:00.510 --> 00:33:02.130 align:middle line:90% We draw simple pictures, right? 00:33:02.130 --> 00:33:04.980 align:middle line:90% But you have lots of households. 00:33:04.980 --> 00:33:08.850 align:middle line:84% And in particular, households could be posting information 00:33:08.850 --> 00:33:09.780 align:middle line:90% on websites. 00:33:09.780 --> 00:33:12.470 align:middle line:84% And we want to run some auction or something. 00:33:12.470 --> 00:33:14.220 align:middle line:84% And hopefully, there are enough households 00:33:14.220 --> 00:33:17.400 align:middle line:84% that we're trying to get to the Walrasian allocation. 00:33:17.400 --> 00:33:20.920 align:middle line:84% It's not easy to solve these things. 00:33:20.920 --> 00:33:23.850 align:middle line:84% In fact, in general, it looks like a very hard problem. 00:33:23.850 --> 00:33:27.940 align:middle line:84% And look, the examples of how to find it look relatively easy. 00:33:27.940 --> 00:33:30.930 align:middle line:84% So there's this big gap in the literature. 00:33:30.930 --> 00:33:33.690 align:middle line:84% Anyway, these guys discovered Negishi. 00:33:33.690 --> 00:33:36.540 align:middle line:90% Everybody goes back to 1960. 00:33:36.540 --> 00:33:40.380 align:middle line:84% And they have an algorithm that works 00:33:40.380 --> 00:33:43.350 align:middle line:84% to find the competitive equilibria using 00:33:43.350 --> 00:33:47.180 align:middle line:84% Negishi's algorithm that I just showed you. 00:33:47.180 --> 00:33:49.200 align:middle line:90% And this one is related. 00:33:49.200 --> 00:33:52.840 align:middle line:84% Computing Walrasian equilibria fast. 00:33:52.840 --> 00:33:56.080 align:middle line:84% [CHUCKLES] So again, there's stuff here. 00:33:56.080 --> 00:34:01.060 align:middle line:84% Although it's cool in the sense of writing a code 00:34:01.060 --> 00:34:02.290 align:middle line:90% that iterates. 00:34:02.290 --> 00:34:04.480 align:middle line:84% And you want to know the time that it 00:34:04.480 --> 00:34:06.320 align:middle line:90% takes to find the solution. 00:34:06.320 --> 00:34:09.190 align:middle line:84% And hopefully, that's not going up exponentially 00:34:09.190 --> 00:34:11.560 align:middle line:84% with the number of households, or the number of goods, 00:34:11.560 --> 00:34:12.909 align:middle line:90% or something. 00:34:12.909 --> 00:34:15.400 align:middle line:90% So they want it to work fast. 00:34:15.400 --> 00:34:19.300 align:middle line:84% So they don't want to evaluate derivatives all the time. 00:34:19.300 --> 00:34:21.429 align:middle line:84% So they figured out this algorithm that 00:34:21.429 --> 00:34:23.590 align:middle line:90% makes a call to the oracle. 00:34:23.590 --> 00:34:27.850 align:middle line:84% And the oracle, when consulted, will tell them the derivative 00:34:27.850 --> 00:34:30.020 align:middle line:90% of the excess demand curve. 00:34:30.020 --> 00:34:31.380 align:middle line:90% And so they've got this down. 00:34:31.380 --> 00:34:36.550 align:middle line:84% They only have to consult the oracle relatively few times. 00:34:36.550 --> 00:34:37.719 align:middle line:90% And it's polynomial. 00:34:37.719 --> 00:34:42.040 align:middle line:84% And so they can-- this algorithm works pretty quickly. 00:34:42.040 --> 00:34:44.139 align:middle line:84% This language of consulting oracles 00:34:44.139 --> 00:34:46.750 align:middle line:84% is used in computer science any time 00:34:46.750 --> 00:34:51.460 align:middle line:84% you want to go to an external data source. 00:34:51.460 --> 00:34:55.210 align:middle line:84% And could be rainfall, or stock prices, or whatever. 00:34:55.210 --> 00:34:57.670 align:middle line:84% And then, bring that into the code-- 00:34:57.670 --> 00:34:59.760 align:middle line:90% like The Matrix oracle. 00:34:59.760 --> 00:35:05.020 align:middle line:84% OK, let's go to another concept of equilibrium, 00:35:05.020 --> 00:35:09.340 align:middle line:84% Nash equilibria, and define a strategic form game. 00:35:09.340 --> 00:35:11.280 align:middle line:90% Let me back up a second. 00:35:11.280 --> 00:35:14.500 align:middle line:84% The Walrasian equilibrium is fine-- 00:35:14.500 --> 00:35:16.990 align:middle line:84% say, prices such that when everyone 00:35:16.990 --> 00:35:20.080 align:middle line:84% takes those prices as given and maximizes, 00:35:20.080 --> 00:35:22.570 align:middle line:84% then we look at the resource constraint, 00:35:22.570 --> 00:35:27.460 align:middle line:84% we find the prices such that excess demands are 0. 00:35:27.460 --> 00:35:30.550 align:middle line:84% But that does beg the issue of where the heck the prices 00:35:30.550 --> 00:35:32.050 align:middle line:90% are coming from. 00:35:32.050 --> 00:35:35.590 align:middle line:84% Even if we have algorithms for finding them, 00:35:35.590 --> 00:35:37.750 align:middle line:90% that's a step along the way. 00:35:37.750 --> 00:35:40.750 align:middle line:84% The strategic form games are ways 00:35:40.750 --> 00:35:45.100 align:middle line:84% to be very explicit about the strategies and outcomes 00:35:45.100 --> 00:35:47.050 align:middle line:90% that traders are using. 00:35:47.050 --> 00:35:49.690 align:middle line:84% And again, I'll link this up to the Walrasian price 00:35:49.690 --> 00:35:51.075 align:middle line:90% momentarily. 00:35:51.075 --> 00:35:54.110 align:middle line:84% OK, so here are some definitions. 00:35:54.110 --> 00:35:57.250 align:middle line:84% A strategic form game has a finite number 00:35:57.250 --> 00:36:00.430 align:middle line:84% of players, capital I, script I. There there's 00:36:00.430 --> 00:36:04.750 align:middle line:84% a set of possible actions for each player, S sub 00:36:04.750 --> 00:36:09.940 align:middle line:84% I. A particular action, little si, is in big SI. 00:36:09.940 --> 00:36:13.960 align:middle line:84% So that's a particular action in a set of all possible actions 00:36:13.960 --> 00:36:16.210 align:middle line:90% that define the game. 00:36:16.210 --> 00:36:20.080 align:middle line:84% And the payoff functions-- look like utility functions-- 00:36:20.080 --> 00:36:23.440 align:middle line:84% map the actions into real numbers. 00:36:23.440 --> 00:36:27.250 align:middle line:84% But this vector, capital S here, is actually 00:36:27.250 --> 00:36:30.850 align:middle line:84% the actions of all the players, unlike, say, in a Walrasian 00:36:30.850 --> 00:36:35.020 align:middle line:84% equilibrium, where prices are given and you choose demands. 00:36:35.020 --> 00:36:38.380 align:middle line:84% Here, we need to map explicitly the actions of each 00:36:38.380 --> 00:36:41.380 align:middle line:84% and every player in order to determine the outcome, 00:36:41.380 --> 00:36:45.190 align:middle line:84% and hence determine the utility of each player. 00:36:45.190 --> 00:36:47.290 align:middle line:90% Here's some convenient notation. 00:36:47.290 --> 00:36:51.190 align:middle line:84% S of minus I means everybody but I. 00:36:51.190 --> 00:36:54.670 align:middle line:84% So it's the particular actions of all players j 00:36:54.670 --> 00:36:57.430 align:middle line:90% excluding player I. 00:36:57.430 --> 00:37:01.690 align:middle line:84% Likewise, capital S sub I is the set of all possible actions 00:37:01.690 --> 00:37:06.160 align:middle line:84% over all players other than I. And finally, 00:37:06.160 --> 00:37:10.150 align:middle line:84% this notation S sub I S sub minus I 00:37:10.150 --> 00:37:12.670 align:middle line:90% is the strategy profile. 00:37:12.670 --> 00:37:15.400 align:middle line:84% It's particularly relevant for player I, 00:37:15.400 --> 00:37:20.080 align:middle line:84% because it has his or her own action in there, S sub I, 00:37:20.080 --> 00:37:23.260 align:middle line:84% but also as a summary, the actions of all 00:37:23.260 --> 00:37:26.310 align:middle line:90% the other players, S 0 I. 00:37:26.310 --> 00:37:30.910 align:middle line:84% OK, so these are the definitions of a strategic form game-- 00:37:30.910 --> 00:37:35.000 align:middle line:90% actions, utilities, and so on. 00:37:35.000 --> 00:37:39.520 align:middle line:84% So a strategy is how to play the game, how 00:37:39.520 --> 00:37:42.070 align:middle line:90% to choose particular actions. 00:37:42.070 --> 00:37:47.260 align:middle line:84% It's as if you were writing code to tell yourself 00:37:47.260 --> 00:37:52.000 align:middle line:84% what to do as a function of what everyone else is doing. 00:37:52.000 --> 00:37:54.180 align:middle line:90% So this can be easy or hard. 00:37:54.180 --> 00:37:57.150 align:middle line:84% Nowadays, you could program computers to play chess. 00:37:57.150 --> 00:38:00.220 align:middle line:84% But that's thought to be a pretty hard problem. 00:38:00.220 --> 00:38:03.690 align:middle line:84% But this is the way we think about agents choosing what 00:38:03.690 --> 00:38:06.430 align:middle line:90% to do in a strategic form game. 00:38:06.430 --> 00:38:10.530 align:middle line:84% So instead of jumping to a notion of a Nash equilibrium 00:38:10.530 --> 00:38:15.330 align:middle line:84% right away, let's think about probability measures 00:38:15.330 --> 00:38:17.905 align:middle line:90% over pure strategies. 00:38:17.905 --> 00:38:19.530 align:middle line:84% But let me just say from the beginning, 00:38:19.530 --> 00:38:21.290 align:middle line:90% why are we doing this? 00:38:21.290 --> 00:38:25.700 align:middle line:84% If you thought there's a finite discrete number of actions 00:38:25.700 --> 00:38:29.030 align:middle line:84% for two players, then player one could do one thing, 00:38:29.030 --> 00:38:32.750 align:middle line:84% and conditioned on that, player to would choose his thing. 00:38:32.750 --> 00:38:35.210 align:middle line:84% And again, conditioned on what player two is doing, 00:38:35.210 --> 00:38:39.120 align:middle line:84% player one wants to change his mind and do something else. 00:38:39.120 --> 00:38:40.970 align:middle line:84% So in principle, it could jump all around 00:38:40.970 --> 00:38:44.120 align:middle line:84% and never find a set of consistent actions. 00:38:44.120 --> 00:38:48.590 align:middle line:84% But if instead we think about each player choosing 00:38:48.590 --> 00:38:52.460 align:middle line:84% his or her action at random, then we're back to the simplex. 00:38:52.460 --> 00:38:54.530 align:middle line:84% Because the set of probability numbers 00:38:54.530 --> 00:38:57.603 align:middle line:84% are going to be non-negative and add up to 1. 00:38:57.603 --> 00:38:59.270 align:middle line:84% So you can see where we're going, right? 00:38:59.270 --> 00:39:02.270 align:middle line:84% We're going to map this into Kakutani. 00:39:02.270 --> 00:39:04.790 align:middle line:84% So that's why, right away, we start 00:39:04.790 --> 00:39:09.330 align:middle line:84% talking about probability measures over pure strategies. 00:39:09.330 --> 00:39:12.140 align:middle line:84% So for example, if you had two actions, 00:39:12.140 --> 00:39:14.600 align:middle line:84% you could think about this probability measure 00:39:14.600 --> 00:39:16.520 align:middle line:84% as assigning a probability of doing 00:39:16.520 --> 00:39:19.400 align:middle line:84% one thing or the other thing, or the probability 00:39:19.400 --> 00:39:21.360 align:middle line:84% over any finite number of things. 00:39:21.360 --> 00:39:26.540 align:middle line:84% And we call these probability measures mixed strategies, 00:39:26.540 --> 00:39:30.410 align:middle line:84% sigma being the mixed strategy of player 00:39:30.410 --> 00:39:33.800 align:middle line:84% I in the set of all probability measures 00:39:33.800 --> 00:39:36.650 align:middle line:90% that put mass over actions. 00:39:36.650 --> 00:39:42.020 align:middle line:84% So sigma sub I for player I, sigma without the I 00:39:42.020 --> 00:39:46.040 align:middle line:84% is a set of mixed strategies over everybody, 00:39:46.040 --> 00:39:48.230 align:middle line:90% over all the players. 00:39:48.230 --> 00:39:50.720 align:middle line:84% And it is being assumed here that there's 00:39:50.720 --> 00:39:54.980 align:middle line:84% no coordination, that they're randomizing independently. 00:39:54.980 --> 00:39:58.250 align:middle line:84% Sigma minus I is the mixed strategies 00:39:58.250 --> 00:40:00.800 align:middle line:90% of every player other than I. 00:40:00.800 --> 00:40:06.590 align:middle line:84% And you can think about the utility function of player I 00:40:06.590 --> 00:40:11.400 align:middle line:84% as a function of the vector of sigma that everyone is playing. 00:40:11.400 --> 00:40:11.900 align:middle line:90% Why? 00:40:11.900 --> 00:40:16.280 align:middle line:84% Because sigma includes not just I, but all the other players. 00:40:16.280 --> 00:40:19.040 align:middle line:84% It's a random way of choosing actions. 00:40:19.040 --> 00:40:22.640 align:middle line:84% And a function of a given set of configuration of actions 00:40:22.640 --> 00:40:25.130 align:middle line:84% will generate a payoff, and hence 00:40:25.130 --> 00:40:28.250 align:middle line:84% generate the utility of each agent I. 00:40:28.250 --> 00:40:30.230 align:middle line:84% So this is just expected utility. 00:40:30.230 --> 00:40:34.280 align:middle line:84% This is just a Von Neumann Morgenstern expected utility 00:40:34.280 --> 00:40:37.370 align:middle line:84% explicitly taking into account the random way 00:40:37.370 --> 00:40:39.680 align:middle line:90% that allocations are achieved. 00:40:39.680 --> 00:40:44.840 align:middle line:84% And I've been a bit vague about the strategy space, S. 00:40:44.840 --> 00:40:49.490 align:middle line:84% Sometimes, I said it was finite, like two actions. 00:40:49.490 --> 00:40:51.410 align:middle line:84% This bottom line specification here 00:40:51.410 --> 00:40:56.090 align:middle line:84% is like, there could be a continuum of possible actions. 00:40:56.090 --> 00:41:00.406 align:middle line:84% So we integrated rather than summing things up. 00:41:00.406 --> 00:41:04.970 align:middle line:84% OK, so let's finally define a Nash equilibrium. 00:41:04.970 --> 00:41:12.140 align:middle line:84% Mixed strategy Nash equilibrium is a strategy profile star 00:41:12.140 --> 00:41:16.220 align:middle line:84% for each and every player such that player 00:41:16.220 --> 00:41:20.390 align:middle line:84% I here takes as given the mixed strategy, 00:41:20.390 --> 00:41:23.600 align:middle line:84% the star mixed strategy of all the other players, 00:41:23.600 --> 00:41:27.170 align:middle line:84% and hopefully will find the sigma I star for him 00:41:27.170 --> 00:41:30.530 align:middle line:84% or herself is, in fact, not worse in utility terms 00:41:30.530 --> 00:41:33.870 align:middle line:84% than doing any other mixed strategy. 00:41:33.870 --> 00:41:37.500 align:middle line:84% So Nash is always, take as given what everyone else is doing. 00:41:37.500 --> 00:41:41.010 align:middle line:84% And then, you're fully free to optimize. 00:41:41.010 --> 00:41:43.450 align:middle line:84% And you can see where this is going. 00:41:43.450 --> 00:41:44.855 align:middle line:90% It's a bit more complicated. 00:41:44.855 --> 00:41:46.230 align:middle line:84% We're going to look for some kind 00:41:46.230 --> 00:41:48.730 align:middle line:84% of fixed point in these mixed strategies 00:41:48.730 --> 00:41:53.100 align:middle line:84% so that when we put sigma star in, we get sigma star back out. 00:41:53.100 --> 00:41:56.340 align:middle line:84% One curious thing here, not that it matters, 00:41:56.340 --> 00:41:59.820 align:middle line:84% but it's interesting to think about the out of equilibrium 00:41:59.820 --> 00:42:01.500 align:middle line:90% problem of a given agent. 00:42:01.500 --> 00:42:05.130 align:middle line:84% The agent takes as given the sigma I 00:42:05.130 --> 00:42:09.090 align:middle line:84% star of all the other players less himself, 00:42:09.090 --> 00:42:11.520 align:middle line:84% and then would in principle, be looking 00:42:11.520 --> 00:42:16.380 align:middle line:84% for a mixed strategy that might do better than sigma I star. 00:42:16.380 --> 00:42:20.160 align:middle line:84% But the mixed strategy would be putting mass 00:42:20.160 --> 00:42:24.270 align:middle line:84% on potential non-trivial number of actions. 00:42:24.270 --> 00:42:27.540 align:middle line:84% But if one action in the mixed strategy 00:42:27.540 --> 00:42:30.750 align:middle line:84% provides higher utility than the other, then player 00:42:30.750 --> 00:42:32.460 align:middle line:84% I would always choose that action. 00:42:32.460 --> 00:42:36.600 align:middle line:84% Why randomize over other things that give you less utility? 00:42:36.600 --> 00:42:40.050 align:middle line:84% So in fact, to try to guard against alternatives that 00:42:40.050 --> 00:42:44.700 align:middle line:84% do better, you only need to evaluate over pure strategies, 00:42:44.700 --> 00:42:46.320 align:middle line:90% not mixed strategies. 00:42:46.320 --> 00:42:49.470 align:middle line:84% But anyway, the definition is all on Nash 00:42:49.470 --> 00:42:54.030 align:middle line:84% that sigma I star here is going to weakly dominate 00:42:54.030 --> 00:42:59.160 align:middle line:84% any other pure strategy SI given that the other players are 00:42:59.160 --> 00:43:02.070 align:middle line:90% doing sigma star minus I. 00:43:02.070 --> 00:43:05.490 align:middle line:84% So that's the definition and a proposition. 00:43:05.490 --> 00:43:08.010 align:middle line:84% Now finally, we get the dividend. 00:43:08.010 --> 00:43:13.200 align:middle line:84% Every finite game has a mixed strategy Nash equilibrium. 00:43:13.200 --> 00:43:15.570 align:middle line:90% It's huge, hugely important. 00:43:15.570 --> 00:43:17.670 align:middle line:84% And again, you could write down a game. 00:43:17.670 --> 00:43:20.920 align:middle line:84% But how do we know that it even has Nash equilibrium? 00:43:20.920 --> 00:43:24.840 align:middle line:84% Now we know that if it's a game with a finite number of actions 00:43:24.840 --> 00:43:28.110 align:middle line:84% for every player, it does have a Nash equilibrium. 00:43:28.110 --> 00:43:30.540 align:middle line:84% It may not be a pure strategy equilibrium. 00:43:30.540 --> 00:43:32.820 align:middle line:90% It may involve randomization. 00:43:32.820 --> 00:43:34.410 align:middle line:90% But it exists. 00:43:34.410 --> 00:43:37.240 align:middle line:84% And to come to the fixed point part of it, 00:43:37.240 --> 00:43:40.740 align:middle line:84% we're looking for this profile sigma star 00:43:40.740 --> 00:43:44.670 align:middle line:84% such that for every player I, sigma I star weakly 00:43:44.670 --> 00:43:47.910 align:middle line:84% dominates any other strategy sigma I, 00:43:47.910 --> 00:43:50.670 align:middle line:84% including pure strategies, taking as 00:43:50.670 --> 00:43:52.600 align:middle line:84% given what the other players are doing. 00:43:52.600 --> 00:43:56.070 align:middle line:84% In other words, if you start with a sigma star, player 00:43:56.070 --> 00:44:00.510 align:middle line:84% I would be reacting to what the other players are doing 00:44:00.510 --> 00:44:03.510 align:middle line:84% and choose the best response given what 00:44:03.510 --> 00:44:04.920 align:middle line:90% the other players are doing. 00:44:04.920 --> 00:44:09.420 align:middle line:84% So indeed, this capital B is the best response set. 00:44:09.420 --> 00:44:13.380 align:middle line:84% There may be more than one utility maximizing response. 00:44:13.380 --> 00:44:14.670 align:middle line:90% If so, pick one. 00:44:14.670 --> 00:44:17.880 align:middle line:84% It should be utility maximizing per player I 00:44:17.880 --> 00:44:20.790 align:middle line:84% given the configuration of what everyone else is doing. 00:44:20.790 --> 00:44:25.080 align:middle line:84% So now, we can talk about the best response for player I, 00:44:25.080 --> 00:44:28.050 align:middle line:84% taking as given what other players are doing. 00:44:28.050 --> 00:44:31.110 align:middle line:84% It's a bit tricky notation here, because sigma I star 00:44:31.110 --> 00:44:32.610 align:middle line:90% is what we want to find. 00:44:32.610 --> 00:44:36.720 align:middle line:84% And this is the best response given the strategies 00:44:36.720 --> 00:44:38.910 align:middle line:90% of the others minus I. 00:44:38.910 --> 00:44:41.070 align:middle line:84% And to be consistent with the notation, 00:44:41.070 --> 00:44:43.330 align:middle line:90% this B has a minus I on it. 00:44:43.330 --> 00:44:46.350 align:middle line:84% But that does not mean that these are the responses 00:44:46.350 --> 00:44:48.360 align:middle line:90% of the other players j. 00:44:48.360 --> 00:44:51.720 align:middle line:84% It is just a way of denoting the best response for player 00:44:51.720 --> 00:44:54.510 align:middle line:84% I conditioned on what all the other guys are doing. 00:44:54.510 --> 00:44:56.850 align:middle line:84% So minus I and minus I both being 00:44:56.850 --> 00:44:58.130 align:middle line:90% consistent with each other. 00:44:58.130 --> 00:44:59.910 align:middle line:84% It's a bit weird in terms of notation. 00:44:59.910 --> 00:45:03.090 align:middle line:84% But so this B of minus I-- minus I 00:45:03.090 --> 00:45:05.760 align:middle line:84% means the best response for player I. 00:45:05.760 --> 00:45:09.090 align:middle line:84% We do that for all players I and define it here. 00:45:09.090 --> 00:45:10.230 align:middle line:90% This is BI. 00:45:10.230 --> 00:45:14.100 align:middle line:84% Should have been a minus I. I'll have to email Ashley. 00:45:14.100 --> 00:45:17.040 align:middle line:84% Anyway, so we have B of sigma to be 00:45:17.040 --> 00:45:20.310 align:middle line:84% the configuration of best responses for all 00:45:20.310 --> 00:45:21.450 align:middle line:90% of the players. 00:45:21.450 --> 00:45:24.720 align:middle line:84% And so we're looking for a Nash equilibrium 00:45:24.720 --> 00:45:27.240 align:middle line:84% to be a fixed point of this mapping. 00:45:27.240 --> 00:45:30.180 align:middle line:84% In particular, we can find a sigma such 00:45:30.180 --> 00:45:33.060 align:middle line:84% that given that sigma for each player, 00:45:33.060 --> 00:45:35.670 align:middle line:84% now knows what the other guys are doing in the mixed strategy 00:45:35.670 --> 00:45:40.260 align:middle line:84% sense, comes up with his or her own best mixed strategy as part 00:45:40.260 --> 00:45:42.210 align:middle line:90% of a best response. 00:45:42.210 --> 00:45:44.970 align:middle line:84% Each player is doing that and ends up 00:45:44.970 --> 00:45:48.780 align:middle line:84% doing what they all took as given to begin with-- namely, 00:45:48.780 --> 00:45:49.900 align:middle line:90% sigma star. 00:45:49.900 --> 00:45:53.740 align:middle line:84% So we just need this B thing to have good properties. 00:45:53.740 --> 00:45:57.390 align:middle line:84% And in particular, this B thing turns out 00:45:57.390 --> 00:46:01.140 align:middle line:84% to satisfy the conditions of Kakutani's theorem. 00:46:01.140 --> 00:46:05.730 align:middle line:84% It's operating over compact, convex, nonempty sets. 00:46:05.730 --> 00:46:10.620 align:middle line:84% Itself being nonempty, has a convex value correspondence 00:46:10.620 --> 00:46:12.550 align:middle line:90% and has a closed graph. 00:46:12.550 --> 00:46:16.290 align:middle line:84% So then, we invoke Kakutani and we're done. 00:46:16.290 --> 00:46:19.560 align:middle line:84% So the second or third illustration, now, 00:46:19.560 --> 00:46:21.780 align:middle line:84% of Kakutani's theorem, in this case, 00:46:21.780 --> 00:46:24.210 align:middle line:84% to establish the existence of a mixed strategy 00:46:24.210 --> 00:46:27.610 align:middle line:84% Nash equilibrium in a finite game-- 00:46:27.610 --> 00:46:30.650 align:middle line:90% are there questions? 00:46:30.650 --> 00:46:31.370 align:middle line:90% All right. 00:46:31.370 --> 00:46:37.800 align:middle line:84% So let's apply this artillery to US markets and talk about trade 00:46:37.800 --> 00:46:38.990 align:middle line:90% fails. 00:46:38.990 --> 00:46:44.930 align:middle line:84% So a trade fail is where a buyer of a treasury 00:46:44.930 --> 00:46:49.400 align:middle line:84% agrees to hand over liquidity-- money, 00:46:49.400 --> 00:46:53.630 align:middle line:84% excess reserves-- to the seller, who has a treasury. 00:46:53.630 --> 00:46:56.780 align:middle line:84% And likewise, the other guy, the seller, 00:46:56.780 --> 00:47:01.380 align:middle line:84% is happy to supply the treasury in return for the liquidity. 00:47:01.380 --> 00:47:03.440 align:middle line:90% So buyer and seller meet. 00:47:03.440 --> 00:47:05.450 align:middle line:84% They meet in over the counter market. 00:47:05.450 --> 00:47:08.270 align:middle line:84% It's not a centralized Walrasian market. 00:47:08.270 --> 00:47:12.100 align:middle line:84% And then, they're supposed to show up and settle the trade. 00:47:12.100 --> 00:47:14.990 align:middle line:84% Well, technically, the seller of the treasury 00:47:14.990 --> 00:47:17.150 align:middle line:84% may not yet own the treasury, but is 00:47:17.150 --> 00:47:19.980 align:middle line:84% planning to get a hold of it in one way or the other 00:47:19.980 --> 00:47:23.630 align:middle line:84% so as to be able to pass the treasury from him 00:47:23.630 --> 00:47:24.980 align:middle line:90% or herself to the buyer. 00:47:24.980 --> 00:47:28.040 align:middle line:84% And likewise, the buyer may or may not 00:47:28.040 --> 00:47:30.650 align:middle line:84% have liquidity at the time they strike the deal, 00:47:30.650 --> 00:47:35.660 align:middle line:84% but is planning, hopefully, to get a hold of the liquidity 00:47:35.660 --> 00:47:39.410 align:middle line:84% in order to be able to buy to pay for what they've already 00:47:39.410 --> 00:47:40.580 align:middle line:90% agreed to do. 00:47:40.580 --> 00:47:43.190 align:middle line:84% But it doesn't always work out that way. 00:47:43.190 --> 00:47:46.370 align:middle line:84% And there are settlement fails in the system. 00:47:46.370 --> 00:47:49.940 align:middle line:84% This happened, for example, with a computer outage. 00:47:49.940 --> 00:47:52.730 align:middle line:84% How can you settle trades when it's all electronic 00:47:52.730 --> 00:47:55.490 align:middle line:84% and there's nothing on the screen? 00:47:55.490 --> 00:47:59.690 align:middle line:84% 9/11 was another problem where there are a lot of disruptions. 00:47:59.690 --> 00:48:02.670 align:middle line:84% Now, they all have backup systems in New Jersey 00:48:02.670 --> 00:48:06.720 align:middle line:84% and at Sanford, but not at the time. 00:48:06.720 --> 00:48:09.740 align:middle line:84% So as I said, the seller may not have the securities. 00:48:09.740 --> 00:48:12.020 align:middle line:84% One possibility is that he was planning 00:48:12.020 --> 00:48:14.510 align:middle line:84% to get them from someone else, but that someone 00:48:14.510 --> 00:48:16.910 align:middle line:84% else who was selling the securities to him 00:48:16.910 --> 00:48:20.340 align:middle line:84% for him to sell it to the third party himself is failing. 00:48:20.340 --> 00:48:22.700 align:middle line:84% So we get what we call this daisy chain, 00:48:22.700 --> 00:48:25.470 align:middle line:90% a cascade of failures. 00:48:25.470 --> 00:48:28.520 align:middle line:84% So you really getting into the weeds in terms of-- it's 00:48:28.520 --> 00:48:29.600 align:middle line:90% not Walrasian. 00:48:29.600 --> 00:48:31.370 align:middle line:90% It's not simple Nash. 00:48:31.370 --> 00:48:35.150 align:middle line:84% It's a much more complicated institutional arrangement. 00:48:35.150 --> 00:48:38.340 align:middle line:84% But we're going to try to use the Nash equilibrium. 00:48:38.340 --> 00:48:40.820 align:middle line:84% Now, here's a picture of the trade fails. 00:48:40.820 --> 00:48:44.240 align:middle line:84% Although this was between 1990 and 2014, 00:48:44.240 --> 00:48:49.040 align:middle line:84% this paper from the New York Fed is intended to be illustrative 00:48:49.040 --> 00:48:53.510 align:middle line:84% of this surge of fails that happened in the great financial 00:48:53.510 --> 00:48:58.530 align:middle line:90% crisis of 2007, '07, '08. 00:48:58.530 --> 00:48:59.030 align:middle line:90% So OK. 00:48:59.030 --> 00:48:59.730 align:middle line:90% It's a spike. 00:48:59.730 --> 00:49:00.230 align:middle line:90% So what? 00:49:00.230 --> 00:49:05.270 align:middle line:84% We'll go over here, say, higher than 9,000. 00:49:05.270 --> 00:49:06.500 align:middle line:90% 9,000 what? 00:49:06.500 --> 00:49:08.240 align:middle line:90% 9,000 billion. 00:49:08.240 --> 00:49:09.900 align:middle line:90% We're talking trillions here. 00:49:09.900 --> 00:49:11.610 align:middle line:90% This is unbelievable. 00:49:11.610 --> 00:49:12.110 align:middle line:90% OK? 00:49:12.110 --> 00:49:13.443 align:middle line:90% It doesn't get bigger than this. 00:49:13.443 --> 00:49:16.640 align:middle line:84% In fact, the treasury market is the world's highest value 00:49:16.640 --> 00:49:18.980 align:middle line:90% market in the whole world. 00:49:18.980 --> 00:49:21.830 align:middle line:84% And now, we have a substantial fraction 00:49:21.830 --> 00:49:24.950 align:middle line:84% of trades that have been agreed to as failing. 00:49:24.950 --> 00:49:27.740 align:middle line:84% I wasn't able to update this slide, 00:49:27.740 --> 00:49:34.250 align:middle line:84% but I did see a paper just two days ago, which is basically, 00:49:34.250 --> 00:49:35.720 align:middle line:90% this year-- 00:49:35.720 --> 00:49:40.070 align:middle line:84% starting, say, January, February, March-- 00:49:40.070 --> 00:49:42.290 align:middle line:90% in March, the virus hit. 00:49:42.290 --> 00:49:45.420 align:middle line:84% And trade fails just skyrocketed. 00:49:45.420 --> 00:49:50.180 align:middle line:84% So the Fed had convinced itself that having enough reserves 00:49:50.180 --> 00:49:53.210 align:middle line:84% on its books would provide ample liquidity 00:49:53.210 --> 00:49:56.030 align:middle line:84% to the system sufficient so that we would never 00:49:56.030 --> 00:49:58.250 align:middle line:90% see trade sales again. 00:49:58.250 --> 00:50:00.190 align:middle line:90% They got it wrong. 00:50:00.190 --> 00:50:01.940 align:middle line:84% I'm not sure how much background you have. 00:50:01.940 --> 00:50:04.610 align:middle line:84% But the Fed is buying securities. 00:50:04.610 --> 00:50:07.070 align:middle line:84% They're buying mortgage backed securities 00:50:07.070 --> 00:50:08.720 align:middle line:90% that other people wouldn't hold. 00:50:08.720 --> 00:50:12.260 align:middle line:84% And for buying them, they credit buyers, broker dealers, 00:50:12.260 --> 00:50:15.230 align:middle line:84% and banks with accounts at the Federal Reserve. 00:50:15.230 --> 00:50:17.270 align:middle line:84% So those things are called reserves. 00:50:17.270 --> 00:50:20.150 align:middle line:84% They're well over excess reserves. 00:50:20.150 --> 00:50:23.360 align:middle line:84% That's where banks keep excess value. 00:50:23.360 --> 00:50:26.240 align:middle line:84% And they pay for stuff with those reserves. 00:50:26.240 --> 00:50:27.980 align:middle line:84% Banks don't borrow from one another 00:50:27.980 --> 00:50:30.740 align:middle line:84% very often anymore because of all the excess reserves, 00:50:30.740 --> 00:50:31.800 align:middle line:90% and so on. 00:50:31.800 --> 00:50:34.950 align:middle line:90% But in March, it all fell apart. 00:50:34.950 --> 00:50:37.590 align:middle line:84% And the Fed had to inject even more liquidity. 00:50:37.590 --> 00:50:40.640 align:middle line:84% So we're not quite ready for monetary economics yet, 00:50:40.640 --> 00:50:43.130 align:middle line:84% but we are ready to talk about how to model 00:50:43.130 --> 00:50:44.880 align:middle line:90% this market for securities. 00:50:44.880 --> 00:50:48.500 align:middle line:84% So let's think about it as a game, 00:50:48.500 --> 00:50:52.100 align:middle line:84% a strategic game played among traders. 00:50:52.100 --> 00:50:54.860 align:middle line:84% So there's going to be various goods, 00:50:54.860 --> 00:50:56.990 align:middle line:90% like liquidity, treasuries. 00:50:56.990 --> 00:50:59.780 align:middle line:84% We're back to our usual notation here. 00:50:59.780 --> 00:51:03.770 align:middle line:84% There are j goods, j equal 1 through not L, but k. 00:51:03.770 --> 00:51:05.750 align:middle line:84% And there's going to be a trading post-- 00:51:05.750 --> 00:51:07.610 align:middle line:90% a market, so to speak-- 00:51:07.610 --> 00:51:09.290 align:middle line:90% for each of the goods. 00:51:09.290 --> 00:51:13.910 align:middle line:84% And they're going to buy by paying unit of account 00:51:13.910 --> 00:51:16.310 align:middle line:84% to get the good, or sell the good, 00:51:16.310 --> 00:51:20.760 align:middle line:84% and return get unit of account into their accounts. 00:51:20.760 --> 00:51:26.520 align:middle line:84% So agent I, trader I, plays a strategy, 00:51:26.520 --> 00:51:30.380 align:middle line:90% which has four objects. 00:51:30.380 --> 00:51:33.980 align:middle line:84% Basically, can buy or sell, but even 00:51:33.980 --> 00:51:37.160 align:middle line:84% more than that, you could buy and specify 00:51:37.160 --> 00:51:40.610 align:middle line:90% a price pi for good j. 00:51:40.610 --> 00:51:43.850 align:middle line:84% You're willing to buy if the price is 00:51:43.850 --> 00:51:47.885 align:middle line:84% less than or equal to that price that you specified. 00:51:47.885 --> 00:51:49.010 align:middle line:90% These are consumers, right? 00:51:49.010 --> 00:51:51.290 align:middle line:84% So lower prices are a good thing. 00:51:51.290 --> 00:51:54.920 align:middle line:84% If you're willing to buy at pi j for good j, 00:51:54.920 --> 00:51:57.800 align:middle line:84% then you're willing to buy if the price were less. 00:51:57.800 --> 00:52:00.050 align:middle line:90% But you limit your exposure. 00:52:00.050 --> 00:52:03.050 align:middle line:84% Regardless of whether it's pi or less, 00:52:03.050 --> 00:52:05.630 align:middle line:84% you're only going to agree to buy 00:52:05.630 --> 00:52:07.880 align:middle line:90% at most qij units of the goods. 00:52:07.880 --> 00:52:10.730 align:middle line:84% So these are called limit prices. 00:52:10.730 --> 00:52:13.490 align:middle line:84% Then, that's the way the New York Stock Exchange works, 00:52:13.490 --> 00:52:14.960 align:middle line:90% et cetera. 00:52:14.960 --> 00:52:17.240 align:middle line:84% Not all of the exchanges work like this. 00:52:17.240 --> 00:52:20.810 align:middle line:84% The tildes here denote limit prices for sellers. 00:52:20.810 --> 00:52:27.200 align:middle line:84% So a seller of good j will say, well, I need a minimum price, 00:52:27.200 --> 00:52:29.570 align:middle line:90% p tilde ij. 00:52:29.570 --> 00:52:31.040 align:middle line:84% I'm willing to sell at that price. 00:52:31.040 --> 00:52:33.110 align:middle line:84% I'm willing to sell if the price is higher, 00:52:33.110 --> 00:52:35.900 align:middle line:84% but I'm not going to sell an infinite amount. 00:52:35.900 --> 00:52:40.370 align:middle line:84% I'm going to limit my sales to q tilde ij. 00:52:40.370 --> 00:52:43.520 align:middle line:84% So in this game, all the traders are 00:52:43.520 --> 00:52:45.740 align:middle line:90% submitting these limit orders. 00:52:45.740 --> 00:52:48.830 align:middle line:84% And they effectively could be on one side or the other, 00:52:48.830 --> 00:52:52.730 align:middle line:84% or be willing to do both depending on the prices 00:52:52.730 --> 00:52:53.960 align:middle line:90% that they need. 00:52:53.960 --> 00:52:58.610 align:middle line:84% So a strategy, S, consists of the set 00:52:58.610 --> 00:53:02.360 align:middle line:84% of possible actions, all these configurations of vectors 00:53:02.360 --> 00:53:05.630 align:middle line:84% for each of the capital N players. 00:53:05.630 --> 00:53:08.690 align:middle line:84% And then, we need to denote the outcome. 00:53:08.690 --> 00:53:11.460 align:middle line:84% The outcome function is this function g-- 00:53:11.460 --> 00:53:12.980 align:middle line:90% we'll put it over here-- 00:53:12.980 --> 00:53:18.110 align:middle line:84% that if everyone is S, being S1, S2, Sn for each of the players, 00:53:18.110 --> 00:53:21.830 align:middle line:84% this vector S of actions limit orders 00:53:21.830 --> 00:53:25.100 align:middle line:84% submitted by all of the end players maps 00:53:25.100 --> 00:53:28.100 align:middle line:84% into a particular outcome for player 00:53:28.100 --> 00:53:33.080 align:middle line:84% I, trader I, which is x, the excess demand 00:53:33.080 --> 00:53:36.330 align:middle line:90% or supply, and beta. 00:53:36.330 --> 00:53:38.360 align:middle line:90% And beta is a new object here. 00:53:38.360 --> 00:53:42.650 align:middle line:84% It's going to be the net credit that agent I receives 00:53:42.650 --> 00:53:47.270 align:middle line:84% when strategy profile S played by all the agents. 00:53:47.270 --> 00:53:50.780 align:middle line:84% Beta is going to be the total unit of account 00:53:50.780 --> 00:53:52.670 align:middle line:90% an agent spends. 00:53:52.670 --> 00:53:55.940 align:middle line:84% And if positive, if positive, beta means 00:53:55.940 --> 00:53:59.572 align:middle line:84% you spend less than what you get as a seller. 00:53:59.572 --> 00:54:02.030 align:middle line:84% You're a seller, you're getting revenues in unit of account 00:54:02.030 --> 00:54:02.810 align:middle line:90% from selling. 00:54:02.810 --> 00:54:05.720 align:middle line:84% How much depends on what everyone else put in. 00:54:05.720 --> 00:54:08.960 align:middle line:84% If you're buying, then you owe unit of account. 00:54:08.960 --> 00:54:11.870 align:middle line:84% How much depends on what everyone else is doing. 00:54:11.870 --> 00:54:16.370 align:middle line:84% Beta could reflect the fact that they sold more in value 00:54:16.370 --> 00:54:18.380 align:middle line:84% in unit of account than they demanded. 00:54:18.380 --> 00:54:20.460 align:middle line:84% That would be a good thing sort of. 00:54:20.460 --> 00:54:22.250 align:middle line:90% They're just underspending. 00:54:22.250 --> 00:54:24.980 align:middle line:84% The opposite is a disaster, where 00:54:24.980 --> 00:54:28.610 align:middle line:84% they've committed to spend unit of account they don't have. 00:54:28.610 --> 00:54:30.830 align:middle line:90% That's when beta goes negative. 00:54:30.830 --> 00:54:32.600 align:middle line:90% That's a fail. 00:54:32.600 --> 00:54:34.580 align:middle line:90% That's going to be a trade fail. 00:54:34.580 --> 00:54:37.260 align:middle line:84% They can't come up with the liquidity. 00:54:37.260 --> 00:54:41.730 align:middle line:84% Or well, we want to prevent that from happening. 00:54:41.730 --> 00:54:44.400 align:middle line:84% By the way, here's the simple-looking supply 00:54:44.400 --> 00:54:45.520 align:middle line:90% and demand schedules. 00:54:45.520 --> 00:54:47.880 align:middle line:84% These are consistent with the limit orders. 00:54:47.880 --> 00:54:51.330 align:middle line:84% For example, there's a maximum price 00:54:51.330 --> 00:54:52.980 align:middle line:84% that a household is willing to pay. 00:54:52.980 --> 00:54:54.750 align:middle line:90% We go over all the traders. 00:54:54.750 --> 00:54:57.630 align:middle line:84% The max over the max is the highest demand. 00:54:57.630 --> 00:54:59.580 align:middle line:84% And as price starts dropping, you'll 00:54:59.580 --> 00:55:02.770 align:middle line:84% pick up a few other traders who are willing to buy, 00:55:02.770 --> 00:55:05.290 align:middle line:84% but they limited their order in quantities. 00:55:05.290 --> 00:55:06.540 align:middle line:90% And the price drops some more. 00:55:06.540 --> 00:55:08.700 align:middle line:90% So you get these step functions. 00:55:08.700 --> 00:55:10.890 align:middle line:84% And likewise, for the suppliers, you 00:55:10.890 --> 00:55:12.240 align:middle line:90% get a step function going up. 00:55:12.240 --> 00:55:13.740 align:middle line:84% As the price gets higher and higher, 00:55:13.740 --> 00:55:15.030 align:middle line:90% you attract more sellers. 00:55:15.030 --> 00:55:17.170 align:middle line:84% But they have all submitted limit orders. 00:55:17.170 --> 00:55:19.500 align:middle line:84% So we're looking for this equilibrium 00:55:19.500 --> 00:55:21.240 align:middle line:84% in this single dimensional market. 00:55:21.240 --> 00:55:22.680 align:middle line:90% There could be ties. 00:55:22.680 --> 00:55:24.000 align:middle line:90% Supplies equal demand. 00:55:24.000 --> 00:55:28.140 align:middle line:84% We have to have a way of rationing or allocating the 00:55:28.140 --> 00:55:29.940 align:middle line:90% whatever is short or long here. 00:55:29.940 --> 00:55:32.890 align:middle line:84% In this case, there's more demand than supply. 00:55:32.890 --> 00:55:37.050 align:middle line:84% So we have to allocate that across the demanders. 00:55:37.050 --> 00:55:39.910 align:middle line:84% Anyway, these are meant to be intuitive pictures. 00:55:39.910 --> 00:55:43.140 align:middle line:84% So that's just describing that picture. 00:55:43.140 --> 00:55:47.430 align:middle line:84% Now, we want to make sure there are no fails. 00:55:47.430 --> 00:55:50.650 align:middle line:90% So we introduced this lambda. 00:55:50.650 --> 00:55:53.220 align:middle line:84% So lambda is going to be a penalty function. 00:55:53.220 --> 00:55:54.810 align:middle line:90% And how does it work? 00:55:54.810 --> 00:55:59.100 align:middle line:84% If beta is positive, then under their strategies 00:55:59.100 --> 00:56:01.540 align:middle line:84% and the strategies of everyone else, 00:56:01.540 --> 00:56:04.170 align:middle line:84% there's more revenue than costs, in which 00:56:04.170 --> 00:56:05.670 align:middle line:90% case, beta is positive. 00:56:05.670 --> 00:56:07.500 align:middle line:90% Positive's higher than 0. 00:56:07.500 --> 00:56:08.880 align:middle line:90% This whole term is 0. 00:56:08.880 --> 00:56:10.140 align:middle line:90% There's no penalty. 00:56:10.140 --> 00:56:12.840 align:middle line:84% But conversely, if beta is negative 00:56:12.840 --> 00:56:15.090 align:middle line:84% because their committed net expenditure is higher 00:56:15.090 --> 00:56:18.300 align:middle line:84% than their net revenue, then the minimum of this function 00:56:18.300 --> 00:56:20.080 align:middle line:90% is a negative number. 00:56:20.080 --> 00:56:22.270 align:middle line:90% And it's amplified by lambda. 00:56:22.270 --> 00:56:24.480 align:middle line:90% So lambda is the penalty cost. 00:56:24.480 --> 00:56:27.300 align:middle line:90% It incurs if you overspend. 00:56:27.300 --> 00:56:30.150 align:middle line:84% And of course, ex-ante, they're going 00:56:30.150 --> 00:56:32.610 align:middle line:84% to choose their strategies given the strategies 00:56:32.610 --> 00:56:35.430 align:middle line:84% that other people are adopting in such ways 00:56:35.430 --> 00:56:39.240 align:middle line:84% to maximize this total payoff, not just the utility of getting 00:56:39.240 --> 00:56:43.980 align:middle line:84% goods if they succeed in doing that, but also the disutility 00:56:43.980 --> 00:56:46.020 align:middle line:90% if they violate the budget. 00:56:46.020 --> 00:56:50.940 align:middle line:84% So there's a curious lemma here along the way 00:56:50.940 --> 00:56:54.510 align:middle line:90% that the betas end up being 0. 00:56:54.510 --> 00:56:58.200 align:middle line:84% In other words, in an active Nash equilibrium, 00:56:58.200 --> 00:56:59.490 align:middle line:90% everyone's on their budget. 00:56:59.490 --> 00:57:01.710 align:middle line:84% Oh, yeah, like the Walrasian thing, right? 00:57:01.710 --> 00:57:03.248 align:middle line:90% That's what we're looking for. 00:57:03.248 --> 00:57:05.040 align:middle line:84% We're going to try to support the Walrasian 00:57:05.040 --> 00:57:07.620 align:middle line:90% allocation momentarily. 00:57:07.620 --> 00:57:09.840 align:middle line:84% But one thing-- not the only thing which is true 00:57:09.840 --> 00:57:11.760 align:middle line:90% of the Walrasian allocation-- 00:57:11.760 --> 00:57:16.860 align:middle line:84% is that excess demands are 0 for each household. 00:57:16.860 --> 00:57:19.860 align:middle line:84% The valuation of what is sold is equal to the valuation 00:57:19.860 --> 00:57:21.940 align:middle line:90% of what is demanded. 00:57:21.940 --> 00:57:23.490 align:middle line:90% They're in their budget. 00:57:23.490 --> 00:57:24.540 align:middle line:90% Data I is 0. 00:57:24.540 --> 00:57:26.550 align:middle line:84% It turns out, the way this game is being 00:57:26.550 --> 00:57:29.010 align:middle line:90% defined, beta I must be 0. 00:57:29.010 --> 00:57:29.610 align:middle line:90% Why? 00:57:29.610 --> 00:57:32.100 align:middle line:90% Well, it's deceptively simple. 00:57:32.100 --> 00:57:35.220 align:middle line:84% Let's sum over the betas of all the households. 00:57:35.220 --> 00:57:36.930 align:middle line:90% That has to be 0. 00:57:36.930 --> 00:57:37.920 align:middle line:90% Why? 00:57:37.920 --> 00:57:40.080 align:middle line:84% Because it's all in unit of account 00:57:40.080 --> 00:57:43.680 align:middle line:84% if whatever is spent by me ends up 00:57:43.680 --> 00:57:46.770 align:middle line:84% in the lap of whoever sold me the goods. 00:57:46.770 --> 00:57:48.960 align:middle line:84% So the unit of account liquidity is just 00:57:48.960 --> 00:57:51.160 align:middle line:90% passing around the system. 00:57:51.160 --> 00:57:55.090 align:middle line:84% In other words, liquidity always flows so that it's balanced. 00:57:55.090 --> 00:57:57.690 align:middle line:84% So the sum of the liquidity deficits 00:57:57.690 --> 00:58:00.930 align:middle line:84% must be 0, because someone always 00:58:00.930 --> 00:58:02.790 align:middle line:90% ends up holding the liquidity. 00:58:02.790 --> 00:58:04.710 align:middle line:84% And then, you could try out what would happen 00:58:04.710 --> 00:58:07.650 align:middle line:90% if some beta I were not 0. 00:58:07.650 --> 00:58:10.020 align:middle line:84% Well, if this sum is 0 and sum beta 00:58:10.020 --> 00:58:14.550 align:middle line:84% I were positive, than if there's a beta I which is not 00:58:14.550 --> 00:58:16.380 align:middle line:84% 0-- say, less than 0-- then there's 00:58:16.380 --> 00:58:20.070 align:middle line:84% some other beta L which is greater than 0. 00:58:20.070 --> 00:58:21.750 align:middle line:84% Now, that trader didn't spend all 00:58:21.750 --> 00:58:24.480 align:middle line:84% of his unit of account valuation. 00:58:24.480 --> 00:58:26.520 align:middle line:84% So that couldn't be maximal, given 00:58:26.520 --> 00:58:29.460 align:middle line:84% the strategies all the other players are adopting. 00:58:29.460 --> 00:58:32.580 align:middle line:90% Player L would purchase more. 00:58:32.580 --> 00:58:34.380 align:middle line:90% So the betas have to be 0. 00:58:34.380 --> 00:58:39.840 align:middle line:84% And the final slide here is, how do we choose lambda? 00:58:39.840 --> 00:58:43.770 align:middle line:84% Or alternatively, given a competitive equilibrium, 00:58:43.770 --> 00:58:46.560 align:middle line:84% how do we find the corresponding Nash equilibrium 00:58:46.560 --> 00:58:48.510 align:middle line:90% in this game that achieves it? 00:58:48.510 --> 00:58:51.480 align:middle line:84% But we're given the prices and allocations 00:58:51.480 --> 00:58:53.500 align:middle line:90% of the competitive equilibrium. 00:58:53.500 --> 00:58:56.370 align:middle line:84% And in a competitive equilibrium, each trader's 00:58:56.370 --> 00:59:00.510 align:middle line:84% maximizing utility subject to budget defined by those given 00:59:00.510 --> 00:59:03.240 align:middle line:90% prices, vector p. 00:59:03.240 --> 00:59:07.030 align:middle line:84% That will define the marginal utility of income, mu, 00:59:07.030 --> 00:59:11.960 align:middle line:84% which coincidentally is exactly what we called it on Tuesday-- 00:59:11.960 --> 00:59:16.280 align:middle line:84% mu I, the marginal utility of income of trader I, OK? 00:59:16.280 --> 00:59:21.590 align:middle line:84% So what we want is to choose these penalties high enough 00:59:21.590 --> 00:59:24.980 align:middle line:84% so that the penalty is higher than the marginal utility 00:59:24.980 --> 00:59:26.630 align:middle line:90% of income. 00:59:26.630 --> 00:59:29.470 align:middle line:84% So mu I is the marginal utility of income. 00:59:29.470 --> 00:59:31.510 align:middle line:84% We're going to scale it by something 00:59:31.510 --> 00:59:36.230 align:middle line:84% called alpha in such a way that we get this inequality. 00:59:36.230 --> 00:59:39.880 align:middle line:84% So you contemplate not honoring the trade. 00:59:39.880 --> 00:59:43.090 align:middle line:84% That's going to be associated with this penalty cost lambda. 00:59:43.090 --> 00:59:45.790 align:middle line:84% And that's greater than the margin utility of income. 00:59:45.790 --> 00:59:47.680 align:middle line:84% You should have avoided that situation 00:59:47.680 --> 00:59:50.590 align:middle line:90% by not going over your budget. 00:59:50.590 --> 00:59:54.550 align:middle line:84% And so given the alpha, we can redefine the limit prices 00:59:54.550 --> 00:59:56.890 align:middle line:84% and the limit quantities in a way 00:59:56.890 --> 00:59:59.810 align:middle line:84% as to achieve a Walrasian equilibrium. 00:59:59.810 --> 01:00:03.550 align:middle line:84% So we've actually defined Walrasian equilibrium, Nash 01:00:03.550 --> 01:00:07.630 align:middle line:84% equilibrium, defined a game here in a market that potentially 01:00:07.630 --> 01:00:08.440 align:middle line:90% has a problem. 01:00:08.440 --> 01:00:11.110 align:middle line:84% Evidently, the penalties are not high enough. 01:00:11.110 --> 01:00:14.050 align:middle line:84% If we follow this model, there's a way 01:00:14.050 --> 01:00:17.890 align:middle line:84% to implement penalties to avoid trade fails so that the Nash 01:00:17.890 --> 01:00:20.050 align:middle line:84% equilibria of that institutional setup 01:00:20.050 --> 01:00:24.040 align:middle line:84% would correspond with the Walrasian equilibrium. 01:00:24.040 --> 01:00:28.320 align:middle line:84% OK, so thank you for coming today. 01:00:28.320 --> 01:00:35.000 align:middle line:90%