WEBVTT 00:00:00.000 --> 00:00:03.976 align:middle line:84% [SQUEAKING] [RUSTLING] [CLICKING] 00:00:03.976 --> 00:00:10.292 align:middle line:90% 00:00:10.292 --> 00:00:11.750 align:middle line:84% ROBERT TOWNSEND: Let me get started 00:00:11.750 --> 00:00:16.661 align:middle line:84% by asking if there are any questions from last time. 00:00:16.661 --> 00:00:20.900 align:middle line:84% You know I really like questions. 00:00:20.900 --> 00:00:24.590 align:middle line:84% AUDIENCE: So just to be clear, I do have a question. 00:00:24.590 --> 00:00:27.940 align:middle line:84% So just to be clear, for the pollution 00:00:27.940 --> 00:00:32.770 align:middle line:84% stuff for the last time, there are a bazillion prices, right? 00:00:32.770 --> 00:00:35.678 align:middle line:90% There's not just one price. 00:00:35.678 --> 00:00:37.720 align:middle line:84% ROBERT TOWNSEND: There was one price for permits. 00:00:37.720 --> 00:00:39.020 align:middle line:90% The model is simple. 00:00:39.020 --> 00:00:41.918 align:middle line:84% So there's only one dimension of pollution. 00:00:41.918 --> 00:00:42.460 align:middle line:90% AUDIENCE: OK. 00:00:42.460 --> 00:00:45.030 align:middle line:90% 00:00:45.030 --> 00:00:49.780 align:middle line:84% ROBERT TOWNSEND: We kind of did it as a partial equilibrium 00:00:49.780 --> 00:00:50.450 align:middle line:90% problem. 00:00:50.450 --> 00:00:53.170 align:middle line:84% So the consumers would take a price 00:00:53.170 --> 00:00:56.680 align:middle line:84% as given and deciding how many permits to sell. 00:00:56.680 --> 00:01:02.270 align:middle line:84% The single firm in that example would take prices as given 00:01:02.270 --> 00:01:05.690 align:middle line:90% and decide how much to produce. 00:01:05.690 --> 00:01:07.240 align:middle line:84% But in the equilibrium, there would 00:01:07.240 --> 00:01:10.020 align:middle line:90% be a common price for permits. 00:01:10.020 --> 00:01:10.780 align:middle line:90% AUDIENCE: OK. 00:01:10.780 --> 00:01:13.480 align:middle line:84% In that case, I'm a bit confused about something. 00:01:13.480 --> 00:01:18.760 align:middle line:84% If you have permits and stuff, if you have 1,000 people, 00:01:18.760 --> 00:01:20.770 align:middle line:84% if you have one person selling a permit, 00:01:20.770 --> 00:01:24.890 align:middle line:84% if that person sells a permit to have more pollution 00:01:24.890 --> 00:01:28.060 align:middle line:84% be polluted, then that pollution gets spread out over everyone. 00:01:28.060 --> 00:01:31.870 align:middle line:84% So the price that someone would be willing to sell 00:01:31.870 --> 00:01:33.680 align:middle line:84% would be much lower than the actual price 00:01:33.680 --> 00:01:35.980 align:middle line:84% that person carried out pollution or something. 00:01:35.980 --> 00:01:37.510 align:middle line:90% Or am I thinking about it wrong? 00:01:37.510 --> 00:01:38.200 align:middle line:90% ROBERT TOWNSEND: No, no, no. 00:01:38.200 --> 00:01:39.200 align:middle line:90% You're absolutely right. 00:01:39.200 --> 00:01:44.350 align:middle line:84% So that part is missing from the model 00:01:44.350 --> 00:01:48.100 align:middle line:84% because we only had one consumer or we talked about the consumer 00:01:48.100 --> 00:01:50.620 align:middle line:84% as a representative consumer and everybody wanting 00:01:50.620 --> 00:01:53.020 align:middle line:90% to do the same thing. 00:01:53.020 --> 00:01:56.410 align:middle line:84% But you're right, there could be a potential conflict 00:01:56.410 --> 00:02:06.980 align:middle line:84% of interest if you're all supposed to pay-- 00:02:06.980 --> 00:02:14.300 align:middle line:84% well, there's something called the Lindahl equilibrium that 00:02:14.300 --> 00:02:18.230 align:middle line:84% was just going to take us to far off the path of the lecture 00:02:18.230 --> 00:02:21.620 align:middle line:84% that actually elicits from each household 00:02:21.620 --> 00:02:24.380 align:middle line:84% their willingness to pay for pollution 00:02:24.380 --> 00:02:27.050 align:middle line:84% and then adds up those numbers to get 00:02:27.050 --> 00:02:28.550 align:middle line:84% the sum of the willingness to pay. 00:02:28.550 --> 00:02:32.420 align:middle line:84% And then, that number is, quote, "compared" 00:02:32.420 --> 00:02:40.590 align:middle line:84% to what the firms are willing to pay for the input costs. 00:02:40.590 --> 00:02:44.340 align:middle line:84% So that's the way the generalization would go. 00:02:44.340 --> 00:02:45.340 align:middle line:90% AUDIENCE: OK, thank you. 00:02:45.340 --> 00:02:45.990 align:middle line:90% ROBERT TOWNSEND: Yeah. 00:02:45.990 --> 00:02:47.120 align:middle line:90% It's a good question. 00:02:47.120 --> 00:02:48.130 align:middle line:90% Other questions. 00:02:48.130 --> 00:02:52.480 align:middle line:90% 00:02:52.480 --> 00:02:58.710 align:middle line:84% So just to remind you all, this is the last lecture 21, 00:02:58.710 --> 00:03:02.340 align:middle line:84% the last lecture I'm giving, although I will have class 00:03:02.340 --> 00:03:06.600 align:middle line:84% on Tuesday, which is another review session focusing mainly 00:03:06.600 --> 00:03:09.450 align:middle line:84% on the material that we've covered since the last time, 00:03:09.450 --> 00:03:12.240 align:middle line:84% trying as I usually do in those review sessions, 00:03:12.240 --> 00:03:14.500 align:middle line:90% to bring the pieces together. 00:03:14.500 --> 00:03:17.670 align:middle line:90% So that's the class schedule. 00:03:17.670 --> 00:03:20.850 align:middle line:84% Then, for the reading list today, for the lecture 00:03:20.850 --> 00:03:24.810 align:middle line:84% today, there are these two starred papers. 00:03:24.810 --> 00:03:29.310 align:middle line:84% And one is this Abel, Mankiw, Summers, Zeckhauser paper 00:03:29.310 --> 00:03:33.330 align:middle line:84% and how it's like one or two slides in the lecture today. 00:03:33.330 --> 00:03:35.880 align:middle line:84% We didn't spend a lot of time on it in the lecture, 00:03:35.880 --> 00:03:38.490 align:middle line:84% but it's listed there as a supplemental reading, 00:03:38.490 --> 00:03:40.590 align:middle line:90% along with my own paper. 00:03:40.590 --> 00:03:43.470 align:middle line:84% I'll spend a little more time on my own paper. 00:03:43.470 --> 00:03:47.670 align:middle line:84% Anyway, I'm kind of treating you guys as you are, 00:03:47.670 --> 00:03:50.460 align:middle line:90% as fully functioning economists. 00:03:50.460 --> 00:03:54.450 align:middle line:84% So you should be able to read the Abel, et al, paper 00:03:54.450 --> 00:03:57.960 align:middle line:84% and understand it without me going over it this lecture. 00:03:57.960 --> 00:04:00.740 align:middle line:84% And the previous one had that style to it. 00:04:00.740 --> 00:04:03.450 align:middle line:84% In fact, last time was the first time, I think, 00:04:03.450 --> 00:04:05.520 align:middle line:84% that I didn't finish the lecture material 00:04:05.520 --> 00:04:08.010 align:middle line:84% and asked you nevertheless to go ahead and understand 00:04:08.010 --> 00:04:10.950 align:middle line:84% the lecture and to read the starred reading. 00:04:10.950 --> 00:04:13.830 align:middle line:84% So I did not do this too, too much at the beginning 00:04:13.830 --> 00:04:14.820 align:middle line:90% of the course. 00:04:14.820 --> 00:04:16.589 align:middle line:84% But now that you're truly trained, 00:04:16.589 --> 00:04:18.570 align:middle line:90% I think you're more independent. 00:04:18.570 --> 00:04:22.019 align:middle line:84% Anyway, feel free to ask me questions about those 00:04:22.019 --> 00:04:24.180 align:middle line:84% at any time, including next time, 00:04:24.180 --> 00:04:26.430 align:middle line:84% if it comes up because it's material 00:04:26.430 --> 00:04:28.890 align:middle line:84% that we really haven't covered much in the lectures 00:04:28.890 --> 00:04:30.310 align:middle line:90% but I started. 00:04:30.310 --> 00:04:34.080 align:middle line:84% So meaning, I do expect you to understand it. 00:04:34.080 --> 00:04:38.860 align:middle line:84% And then, for the review sheets, we 00:04:38.860 --> 00:04:42.190 align:middle line:84% were doing the failures of the welfare theorems. 00:04:42.190 --> 00:04:44.110 align:middle line:84% The way these review questions are written, 00:04:44.110 --> 00:04:46.330 align:middle line:90% they're largely review. 00:04:46.330 --> 00:04:48.970 align:middle line:84% It's kind of like telling you what you need to know. 00:04:48.970 --> 00:04:52.810 align:middle line:84% It's hard to ask you when the answer is given, 00:04:52.810 --> 00:04:54.550 align:middle line:90% but I'll give it a try anyway. 00:04:54.550 --> 00:04:58.600 align:middle line:84% It says, you should know well the formal statement 00:04:58.600 --> 00:05:02.310 align:middle line:84% of the first welfare theorem and conditions 00:05:02.310 --> 00:05:03.390 align:middle line:90% under which it holds. 00:05:03.390 --> 00:05:07.470 align:middle line:84% So again, I think I'll just take volunteers. 00:05:07.470 --> 00:05:11.790 align:middle line:84% Can someone tell me the roughly or even more 00:05:11.790 --> 00:05:15.705 align:middle line:84% carefully the statement of the first welfare theorem? 00:05:15.705 --> 00:05:24.200 align:middle line:90% 00:05:24.200 --> 00:05:24.700 align:middle line:90% OK. 00:05:24.700 --> 00:05:26.080 align:middle line:90% There are two welfare theorems. 00:05:26.080 --> 00:05:28.197 align:middle line:90% AUDIENCE: OK. 00:05:28.197 --> 00:05:30.280 align:middle line:84% If I remember correctly, the first welfare theorem 00:05:30.280 --> 00:05:33.640 align:middle line:84% is something like assuming local nonsatiation 00:05:33.640 --> 00:05:36.940 align:middle line:84% and I don't think it's convexity, 00:05:36.940 --> 00:05:40.390 align:middle line:84% but there's second conditions that I forget. 00:05:40.390 --> 00:05:43.430 align:middle line:84% But to mean, local nonsatiation and something else. 00:05:43.430 --> 00:05:48.340 align:middle line:84% Then, the practical Walrasian, like, price equilibrium 00:05:48.340 --> 00:05:49.607 align:middle line:90% is pretty optimal. 00:05:49.607 --> 00:05:50.440 align:middle line:90% If I remember right. 00:05:50.440 --> 00:05:51.640 align:middle line:90% ROBERT TOWNSEND: Yeah. 00:05:51.640 --> 00:05:52.480 align:middle line:90% That's right. 00:05:52.480 --> 00:05:58.900 align:middle line:84% It was local nonsatiation and a statement about rationality 00:05:58.900 --> 00:06:03.860 align:middle line:90% of consumer preferences. 00:06:03.860 --> 00:06:06.350 align:middle line:84% So very, very weak, which is probably 00:06:06.350 --> 00:06:08.070 align:middle line:90% why they're hard to remember. 00:06:08.070 --> 00:06:08.570 align:middle line:90% Yeah. 00:06:08.570 --> 00:06:11.990 align:middle line:84% But the local nonsatiation is there. 00:06:11.990 --> 00:06:15.170 align:middle line:90% 00:06:15.170 --> 00:06:20.690 align:middle line:84% And then, the pollution we've already started talking about. 00:06:20.690 --> 00:06:24.230 align:middle line:90% Pollution permits. 00:06:24.230 --> 00:06:26.610 align:middle line:84% Why did we need those permits in the first place? 00:06:26.610 --> 00:06:28.190 align:middle line:90% What was it that we did wrong? 00:06:28.190 --> 00:06:32.480 align:middle line:84% Since the first welfare theorem is supposed to hold 00:06:32.480 --> 00:06:36.410 align:middle line:84% and even assuming local nonsatiation, 00:06:36.410 --> 00:06:38.525 align:middle line:90% what why did it fail in the end? 00:06:38.525 --> 00:06:41.660 align:middle line:90% 00:06:41.660 --> 00:06:47.100 align:middle line:84% AUDIENCE: Is it really like a market for pollution? 00:06:47.100 --> 00:06:50.180 align:middle line:84% So like, people's preferences don't get accounted for there. 00:06:50.180 --> 00:06:51.840 align:middle line:90% ROBERT TOWNSEND: Yep. 00:06:51.840 --> 00:06:52.500 align:middle line:90% Exactly. 00:06:52.500 --> 00:06:55.830 align:middle line:84% So it's a missing market problem. 00:06:55.830 --> 00:07:00.230 align:middle line:84% We assume their L goods and their markets in all of them 00:07:00.230 --> 00:07:02.450 align:middle line:84% when we do the proof of the first welfare theorem. 00:07:02.450 --> 00:07:05.570 align:middle line:84% In this case, we kind of left one out. 00:07:05.570 --> 00:07:11.220 align:middle line:84% So there aren't prices there to guide 00:07:11.220 --> 00:07:14.910 align:middle line:84% the solution of the market to the situation where 00:07:14.910 --> 00:07:17.370 align:middle line:84% the marginal rates of substitution over households 00:07:17.370 --> 00:07:22.580 align:middle line:84% are equated to the marginal rate of production across firms. 00:07:22.580 --> 00:07:24.740 align:middle line:84% So that that's the nature of the problem. 00:07:24.740 --> 00:07:27.510 align:middle line:90% 00:07:27.510 --> 00:07:30.845 align:middle line:84% And likewise, I thought I indicated one other. 00:07:30.845 --> 00:07:33.400 align:middle line:90% 00:07:33.400 --> 00:07:37.090 align:middle line:84% What about the second welfare theorem? 00:07:37.090 --> 00:07:39.950 align:middle line:84% Can anyone state it, including some of the assumptions that 00:07:39.950 --> 00:07:40.450 align:middle line:90% are needed? 00:07:40.450 --> 00:07:48.865 align:middle line:90% 00:07:48.865 --> 00:07:49.490 align:middle line:90% AUDIENCE: Sure. 00:07:49.490 --> 00:07:52.140 align:middle line:90% I can take a stab at this one. 00:07:52.140 --> 00:07:54.020 align:middle line:84% So the second welfare theorem says 00:07:54.020 --> 00:08:02.060 align:middle line:84% that basically any optimal point is supported by some while 00:08:02.060 --> 00:08:04.788 align:middle line:84% raising equilibrium up to transfers. 00:08:04.788 --> 00:08:06.830 align:middle line:84% And some assumptions that we need for that are we 00:08:06.830 --> 00:08:09.050 align:middle line:84% need convex open consumption sets. 00:08:09.050 --> 00:08:11.990 align:middle line:84% We need convex closed production sets 00:08:11.990 --> 00:08:15.260 align:middle line:84% with this concave transformation function. 00:08:15.260 --> 00:08:20.390 align:middle line:84% We need concave and locally nonsatiated preferences. 00:08:20.390 --> 00:08:24.770 align:middle line:84% And I think the last one is basically 00:08:24.770 --> 00:08:28.293 align:middle line:84% just like a budget constraint in some sort basically. 00:08:28.293 --> 00:08:29.210 align:middle line:90% ROBERT TOWNSEND: Yeah. 00:08:29.210 --> 00:08:30.890 align:middle line:90% It's some interiority condition. 00:08:30.890 --> 00:08:31.880 align:middle line:90% AUDIENCE: Yeah. 00:08:31.880 --> 00:08:32.797 align:middle line:90% ROBERT TOWNSEND: Yeah. 00:08:32.797 --> 00:08:34.429 align:middle line:90% That's right. 00:08:34.429 --> 00:08:38.450 align:middle line:84% So yeah, so this one takes a bit more in terms of assumptions. 00:08:38.450 --> 00:08:40.280 align:middle line:84% And then, I showed in the lecture 00:08:40.280 --> 00:08:44.230 align:middle line:84% two pictures of illustrative examples 00:08:44.230 --> 00:08:47.510 align:middle line:90% of where things could go wrong. 00:08:47.510 --> 00:08:51.820 align:middle line:84% One had nonconvexity in the preferences, 00:08:51.820 --> 00:08:54.400 align:middle line:84% and the other had nonconvexity in production. 00:08:54.400 --> 00:08:59.260 align:middle line:90% 00:08:59.260 --> 00:09:01.880 align:middle line:84% I don't know if it helps to give the math intuition. 00:09:01.880 --> 00:09:05.330 align:middle line:90% 00:09:05.330 --> 00:09:08.640 align:middle line:84% When we did the second welfare theorem, 00:09:08.640 --> 00:09:12.720 align:middle line:84% we were kind of getting those shadow prices by solving 00:09:12.720 --> 00:09:17.630 align:middle line:84% the Pareto problem and then essentially using them 00:09:17.630 --> 00:09:22.510 align:middle line:84% as candidates for the Walrasian or equilibrium 00:09:22.510 --> 00:09:24.880 align:middle line:84% or the equilibrium with transfers. 00:09:24.880 --> 00:09:29.810 align:middle line:84% And those shadow prices are marginal conditions. 00:09:29.810 --> 00:09:31.570 align:middle line:84% They are derivatives having to do 00:09:31.570 --> 00:09:37.070 align:middle line:84% with differentiating the Lagrangian with respect 00:09:37.070 --> 00:09:44.020 align:middle line:90% to constraints. 00:09:44.020 --> 00:09:47.860 align:middle line:84% Another way to put that is the candidate for prices 00:09:47.860 --> 00:09:51.320 align:middle line:84% are essentially margin utilities. 00:09:51.320 --> 00:09:54.530 align:middle line:84% Weighted margin utilities, at least in the Pareto problem, 00:09:54.530 --> 00:09:59.040 align:middle line:84% would be equated to a common Lagrange multiplier. 00:09:59.040 --> 00:10:02.880 align:middle line:84% And that's what I meant by marginal utility pricing, 00:10:02.880 --> 00:10:06.420 align:middle line:84% that the Lagrange multiplier becomes the price. 00:10:06.420 --> 00:10:10.390 align:middle line:84% It's kind of amazing because it's all teeny tiny local. 00:10:10.390 --> 00:10:13.560 align:middle line:84% Once we find the candidate we want, all we need to do 00:10:13.560 --> 00:10:17.080 align:middle line:84% is look at the derivatives for very small movements 00:10:17.080 --> 00:10:21.580 align:middle line:84% around that point and then kind of separate the problem 00:10:21.580 --> 00:10:23.850 align:middle line:90% globally. 00:10:23.850 --> 00:10:32.230 align:middle line:84% So it's not just that at those prices, 00:10:32.230 --> 00:10:33.493 align:middle line:90% households and firms are max-- 00:10:33.493 --> 00:10:35.410 align:middle line:84% I mean, that's true, but they're not only just 00:10:35.410 --> 00:10:39.130 align:middle line:84% maximizing locally, they could, in principle, 00:10:39.130 --> 00:10:41.950 align:middle line:84% choose things that are very far away from that local point that 00:10:41.950 --> 00:10:43.960 align:middle line:90% are global. 00:10:43.960 --> 00:10:47.050 align:middle line:84% And that was the point of those two figures 00:10:47.050 --> 00:10:50.170 align:middle line:84% with the nonconvexity in preferences 00:10:50.170 --> 00:10:52.600 align:middle line:84% or, alternatively, nonconvexity in production, 00:10:52.600 --> 00:10:55.610 align:middle line:84% you could see how locally things look fine. 00:10:55.610 --> 00:10:57.040 align:middle line:84% But if you drift pretty far away, 00:10:57.040 --> 00:10:59.830 align:middle line:84% you find another point that either 00:10:59.830 --> 00:11:02.890 align:middle line:84% maximizes profits or maximizing utility, 00:11:02.890 --> 00:11:05.550 align:middle line:90% depending on the example. 00:11:05.550 --> 00:11:08.000 align:middle line:84% So all of this stuff we're learning 00:11:08.000 --> 00:11:13.830 align:middle line:84% was kind of worked out by British economists 00:11:13.830 --> 00:11:17.950 align:middle line:84% and then formalized by Paul Samuelson. 00:11:17.950 --> 00:11:20.890 align:middle line:84% It's called the sort of marginal analysis 00:11:20.890 --> 00:11:22.970 align:middle line:90% way of doing economics. 00:11:22.970 --> 00:11:25.690 align:middle line:90% 00:11:25.690 --> 00:11:29.920 align:middle line:90% So that's good. 00:11:29.920 --> 00:11:32.410 align:middle line:84% So that that's essentially all I want 00:11:32.410 --> 00:11:38.300 align:middle line:84% to do in terms of reviewing from last class. 00:11:38.300 --> 00:11:40.239 align:middle line:84% Well, the other thing-- yeah, question. 00:11:40.239 --> 00:11:40.864 align:middle line:90% AUDIENCE: Yeah. 00:11:40.864 --> 00:11:43.420 align:middle line:90% 00:11:43.420 --> 00:11:45.790 align:middle line:84% When I mentioned the first welfare theorem, 00:11:45.790 --> 00:11:48.640 align:middle line:84% we just assumed that the preference are 00:11:48.640 --> 00:11:51.040 align:middle line:90% original and the nonsatiation. 00:11:51.040 --> 00:11:55.690 align:middle line:84% And we say that the production is a nonempty. 00:11:55.690 --> 00:11:59.500 align:middle line:84% And we don't consider whether the production function 00:11:59.500 --> 00:12:01.910 align:middle line:90% is convex or not. 00:12:01.910 --> 00:12:07.240 align:middle line:84% But I notice that if the production function is 00:12:07.240 --> 00:12:14.870 align:middle line:84% nonconvex, we see there might not have 00:12:14.870 --> 00:12:17.360 align:middle line:90% any competitive equilibrium. 00:12:17.360 --> 00:12:21.290 align:middle line:84% For example, when the production function is now 00:12:21.290 --> 00:12:24.800 align:middle line:90% convex, then the-- 00:12:24.800 --> 00:12:36.170 align:middle line:84% yeah, then the slide 3 showed that the point which consumers 00:12:36.170 --> 00:12:38.600 align:middle line:90% want to be maximizing utility. 00:12:38.600 --> 00:12:44.510 align:middle line:84% But at this point, it was not at the producers 00:12:44.510 --> 00:12:47.970 align:middle line:90% to maximize their profits. 00:12:47.970 --> 00:12:52.160 align:middle line:84% So in this situation, can we say that was at equilibrium? 00:12:52.160 --> 00:12:55.443 align:middle line:84% It looks like there was no equilibrium in this situation. 00:12:55.443 --> 00:12:56.360 align:middle line:90% ROBERT TOWNSEND: Yeah. 00:12:56.360 --> 00:12:57.440 align:middle line:90% It's a good question. 00:12:57.440 --> 00:13:00.420 align:middle line:84% And it gives me an opportunity to clarify. 00:13:00.420 --> 00:13:02.540 align:middle line:90% There are two theorems. 00:13:02.540 --> 00:13:05.780 align:middle line:84% One, if a competitive equilibrium exists, 00:13:05.780 --> 00:13:08.480 align:middle line:84% and it's optimal, that's the first welfare theorem. 00:13:08.480 --> 00:13:11.780 align:middle line:84% That's the one that only needs local nonsatiation 00:13:11.780 --> 00:13:14.630 align:middle line:90% and rationality. 00:13:14.630 --> 00:13:19.110 align:middle line:84% And the reason-- and it's true, when there's production, 00:13:19.110 --> 00:13:21.860 align:middle line:84% and there are no assumptions about production, 00:13:21.860 --> 00:13:24.530 align:middle line:84% but the intuition for why it works 00:13:24.530 --> 00:13:26.720 align:middle line:84% is you're given so much information already, 00:13:26.720 --> 00:13:31.190 align:middle line:84% namely this competitive equilibrium exists. 00:13:31.190 --> 00:13:36.730 align:middle line:84% So what's ever going on, firms were maximizing profits, 00:13:36.730 --> 00:13:40.850 align:middle line:84% and that added up to demand and so on. 00:13:40.850 --> 00:13:42.910 align:middle line:84% So all that stuff is kind of implicitly given 00:13:42.910 --> 00:13:46.780 align:middle line:84% in the statement of the problem, if the conditionality, 00:13:46.780 --> 00:13:51.010 align:middle line:84% if a competitive equilibrium exists, then something else 00:13:51.010 --> 00:13:52.030 align:middle line:90% is true. 00:13:52.030 --> 00:13:53.300 align:middle line:90% AUDIENCE: Oh, I get it. 00:13:53.300 --> 00:13:54.717 align:middle line:84% ROBERT TOWNSEND: Yeah, yeah, yeah. 00:13:54.717 --> 00:13:56.710 align:middle line:84% And then, I slipped up too when I 00:13:56.710 --> 00:14:00.400 align:middle line:84% was going through those examples because I'd 00:14:00.400 --> 00:14:04.360 align:middle line:84% forgotten that I'd switched to the second welfare theorem. 00:14:04.360 --> 00:14:06.760 align:middle line:84% So the second welfare theorem is the starting point 00:14:06.760 --> 00:14:07.810 align:middle line:90% is Pareto optimal. 00:14:07.810 --> 00:14:10.930 align:middle line:84% And I should have, when I went over those figures, 00:14:10.930 --> 00:14:15.190 align:middle line:84% emphasized how it was the single-best point 00:14:15.190 --> 00:14:19.780 align:middle line:84% in the whole space and could not be Pareto dominated, and then 00:14:19.780 --> 00:14:23.840 align:middle line:84% settled into the second part of the second welfare 00:14:23.840 --> 00:14:26.750 align:middle line:84% theorem, which is then you can maybe not 00:14:26.750 --> 00:14:30.287 align:middle line:84% decentralize if you lose the convexity. 00:14:30.287 --> 00:14:31.120 align:middle line:90% AUDIENCE: Thank you. 00:14:31.120 --> 00:14:31.600 align:middle line:90% ROBERT TOWNSEND: OK. 00:14:31.600 --> 00:14:32.140 align:middle line:90% I'm glad. 00:14:32.140 --> 00:14:33.880 align:middle line:90% I like questions. 00:14:33.880 --> 00:14:35.530 align:middle line:90% I know it's the last class. 00:14:35.530 --> 00:14:39.730 align:middle line:84% By the way, please come ready to ask questions next time. 00:14:39.730 --> 00:14:42.640 align:middle line:84% That's a much more fun way to review 00:14:42.640 --> 00:14:45.370 align:middle line:84% what we've done in the course than it is for me 00:14:45.370 --> 00:14:49.670 align:middle line:90% to lecture all over again. 00:14:49.670 --> 00:14:50.170 align:middle line:90% All right. 00:14:50.170 --> 00:14:53.380 align:middle line:84% That said, I should lecture all over again, I guess. 00:14:53.380 --> 00:14:56.350 align:middle line:90% Here's lecture 21. 00:14:56.350 --> 00:15:00.700 align:middle line:84% And again, without exaggerating, this one, like the last one, 00:15:00.700 --> 00:15:04.030 align:middle line:84% is kind of exciting, both for the topics that are covered 00:15:04.030 --> 00:15:07.780 align:middle line:84% and because it's building on material that you now 00:15:07.780 --> 00:15:11.470 align:middle line:84% know that was spread out throughout the course. 00:15:11.470 --> 00:15:13.270 align:middle line:84% I suppose a better title for this one 00:15:13.270 --> 00:15:17.200 align:middle line:84% would have been monetary economics or cryptocurrency. 00:15:17.200 --> 00:15:20.830 align:middle line:84% But Bubbles is a more appealing title. 00:15:20.830 --> 00:15:23.690 align:middle line:84% And as you will see, bubbles can exist, 00:15:23.690 --> 00:15:27.130 align:middle line:84% meaning something has value that has no value. 00:15:27.130 --> 00:15:29.120 align:middle line:90% And they can be optimal. 00:15:29.120 --> 00:15:32.840 align:middle line:84% And that also creates a scope for monetary policy. 00:15:32.840 --> 00:15:35.560 align:middle line:84% So these things look really different from the welfare 00:15:35.560 --> 00:15:37.450 align:middle line:90% theorems that we just did. 00:15:37.450 --> 00:15:40.600 align:middle line:84% And essentially, the reason that it's happening 00:15:40.600 --> 00:15:44.800 align:middle line:84% is either that we have a version of incomplete markets 00:15:44.800 --> 00:15:49.150 align:middle line:84% or we have an infinite number of commodities. 00:15:49.150 --> 00:15:51.360 align:middle line:84% But rather than reiterating those theorems, 00:15:51.360 --> 00:15:54.660 align:middle line:84% so this is the monetary economic slide. 00:15:54.660 --> 00:15:58.200 align:middle line:84% Let me tell you in words how we can get fiat money to have 00:15:58.200 --> 00:16:02.430 align:middle line:84% value, even when it's worth much more than the paper it's 00:16:02.430 --> 00:16:03.750 align:middle line:90% printed on. 00:16:03.750 --> 00:16:06.780 align:middle line:84% So you have these kinds of agents 00:16:06.780 --> 00:16:09.450 align:middle line:90% moving in opposite directions. 00:16:09.450 --> 00:16:11.790 align:middle line:84% Some are moving west, as in the top line. 00:16:11.790 --> 00:16:14.520 align:middle line:84% Others are moving east, as in the bottom line. 00:16:14.520 --> 00:16:18.520 align:middle line:84% And every period, they hit one of these trading posts. 00:16:18.520 --> 00:16:22.950 align:middle line:84% So the sequence experienced by the people traveling west, 00:16:22.950 --> 00:16:25.290 align:middle line:84% if they start with one, then next period they're 00:16:25.290 --> 00:16:27.720 align:middle line:84% going to have 0 goods, that's their endowment. 00:16:27.720 --> 00:16:29.790 align:middle line:90% Then one good and so on. 00:16:29.790 --> 00:16:32.010 align:middle line:84% And the guys going in the opposite direction 00:16:32.010 --> 00:16:33.180 align:middle line:90% have the opposite sequence. 00:16:33.180 --> 00:16:36.720 align:middle line:84% Whenever they're able to trade at these vertical lines, which 00:16:36.720 --> 00:16:40.560 align:middle line:84% are trading posts, one agent has one and the other has 0. 00:16:40.560 --> 00:16:43.620 align:middle line:84% And then, the basic idea is somehow 00:16:43.620 --> 00:16:48.210 align:middle line:84% the guy with 0 goods also has these pieces of paper 00:16:48.210 --> 00:16:49.590 align:middle line:90% called fiat money. 00:16:49.590 --> 00:16:51.060 align:middle line:90% And they give them up. 00:16:51.060 --> 00:16:54.450 align:middle line:84% They have value, and they get one sum amount of the good 00:16:54.450 --> 00:16:57.480 align:middle line:84% from the person who has the real commodity. 00:16:57.480 --> 00:17:00.390 align:middle line:84% So there's a trade going on between goods and money. 00:17:00.390 --> 00:17:03.420 align:middle line:84% So we have to extend the commodity space to include 00:17:03.420 --> 00:17:05.069 align:middle line:90% these pieces of paper. 00:17:05.069 --> 00:17:07.680 align:middle line:84% And then, the guy that basically got 00:17:07.680 --> 00:17:11.849 align:middle line:84% the money is willing to take it because he or she is 00:17:11.849 --> 00:17:17.190 align:middle line:84% about to experience in the next period 0 real endowment. 00:17:17.190 --> 00:17:20.900 align:middle line:84% But they can spend the money to a person with one who 00:17:20.900 --> 00:17:23.160 align:middle line:90% was in turn willing to take it. 00:17:23.160 --> 00:17:26.220 align:middle line:84% I'll show you this model at the end of class. 00:17:26.220 --> 00:17:27.609 align:middle line:90% And we'll work some things out. 00:17:27.609 --> 00:17:31.830 align:middle line:84% But that's an example of an economy with endowments 00:17:31.830 --> 00:17:36.690 align:middle line:84% and preferences that is pretty awful if there's no money. 00:17:36.690 --> 00:17:39.365 align:middle line:90% But it's much better with it. 00:17:39.365 --> 00:17:40.740 align:middle line:84% And then, yeah, at the end, we'll 00:17:40.740 --> 00:17:43.500 align:middle line:84% even talk about optimal monetary policy 00:17:43.500 --> 00:17:47.010 align:middle line:84% in that setup of why the Fed should be 00:17:47.010 --> 00:17:49.510 align:middle line:90% controlling the interest rate. 00:17:49.510 --> 00:17:52.890 align:middle line:84% This is a similar model down here, if you can see it. 00:17:52.890 --> 00:17:55.530 align:middle line:84% You have people going one direction or the other. 00:17:55.530 --> 00:17:58.900 align:middle line:84% When they meet, though, they stay together for two periods. 00:17:58.900 --> 00:18:01.380 align:middle line:84% So this person's endowment goes a 00:18:01.380 --> 00:18:04.920 align:middle line:84% in the first period, 0 endowment of the good 00:18:04.920 --> 00:18:05.850 align:middle line:90% in the second period. 00:18:05.850 --> 00:18:07.530 align:middle line:90% The opposite is true. 00:18:07.530 --> 00:18:10.080 align:middle line:90% This one has 0 and then B. 00:18:10.080 --> 00:18:13.320 align:middle line:84% So over these two periods, they can borrow and lend 00:18:13.320 --> 00:18:14.170 align:middle line:90% from each other. 00:18:14.170 --> 00:18:19.350 align:middle line:84% So this is just like a snapshot of a two-period intertemporal 00:18:19.350 --> 00:18:21.570 align:middle line:84% economy with borrowing and lending, 00:18:21.570 --> 00:18:24.240 align:middle line:84% which they will want to do because they 00:18:24.240 --> 00:18:27.090 align:middle line:84% have extreme endowment positions of the real good. 00:18:27.090 --> 00:18:28.560 align:middle line:90% But it doesn't end there. 00:18:28.560 --> 00:18:32.010 align:middle line:84% After they're paired like that, one goes to the east, 00:18:32.010 --> 00:18:33.510 align:middle line:90% the other goes to the west. 00:18:33.510 --> 00:18:38.680 align:middle line:84% And if you think like a is a very, very big number 00:18:38.680 --> 00:18:41.880 align:middle line:84% and b is a very small number, this economy 00:18:41.880 --> 00:18:45.150 align:middle line:84% starts to look like the one above it, 00:18:45.150 --> 00:18:47.070 align:middle line:84% where they're going from large number 00:18:47.070 --> 00:18:51.330 align:middle line:84% to something almost 0, large number to something almost 0. 00:18:51.330 --> 00:18:54.000 align:middle line:84% And the person below going in the other direction 00:18:54.000 --> 00:18:55.870 align:middle line:90% has the opposite pattern. 00:18:55.870 --> 00:19:01.160 align:middle line:84% So this economy displays both real credit, 00:19:01.160 --> 00:19:06.260 align:middle line:84% borrowing and lending and real goods, and fiat money. 00:19:06.260 --> 00:19:10.920 align:middle line:84% And of course, in the economy we live in, we arguably have both. 00:19:10.920 --> 00:19:12.810 align:middle line:84% We have fiat money, and we have borrowing 00:19:12.810 --> 00:19:14.950 align:middle line:90% and lending and credit markets. 00:19:14.950 --> 00:19:19.440 align:middle line:84% And then, finally, we go to this third economy, 00:19:19.440 --> 00:19:21.480 align:middle line:90% where there are two locations. 00:19:21.480 --> 00:19:25.230 align:middle line:84% Agent 1 is always residing in the first location. 00:19:25.230 --> 00:19:27.450 align:middle line:84% Agent 2 is always residing in the-- 00:19:27.450 --> 00:19:31.230 align:middle line:84% Agent 4, I'm sorry, is residing in the second location. 00:19:31.230 --> 00:19:35.310 align:middle line:84% And these other two agents labeled 2 and 3 switch 00:19:35.310 --> 00:19:36.060 align:middle line:90% locations. 00:19:36.060 --> 00:19:39.630 align:middle line:84% 2 goes from 1 to 2, then back to 1, et cetera. 00:19:39.630 --> 00:19:42.780 align:middle line:90% And 3 has the opposite pattern. 00:19:42.780 --> 00:19:46.500 align:middle line:84% You can kind of create this by taking this so-called turnpike 00:19:46.500 --> 00:19:50.580 align:middle line:84% and bending it around so that it meets like a merry-go-round. 00:19:50.580 --> 00:19:55.200 align:middle line:84% And that would create something very similar to this economy. 00:19:55.200 --> 00:19:59.430 align:middle line:84% So much more fun, you can see at date 1 and 3, 00:19:59.430 --> 00:20:02.650 align:middle line:84% Agents 1 and 2 are paired with each other. 00:20:02.650 --> 00:20:05.010 align:middle line:84% So you can imagine that borrowing and lending, 00:20:05.010 --> 00:20:08.490 align:middle line:84% simple two-period loans, are possible. 00:20:08.490 --> 00:20:13.270 align:middle line:84% And depending on endowments and preferences, desirable. 00:20:13.270 --> 00:20:16.770 align:middle line:84% So we have short-term credit, two-period debt, 00:20:16.770 --> 00:20:18.570 align:middle line:90% but also watch this. 00:20:18.570 --> 00:20:22.620 align:middle line:84% Now say 1 issues an IOU to 2, a promise 00:20:22.620 --> 00:20:24.930 align:middle line:90% to pay in the fourth period. 00:20:24.930 --> 00:20:31.380 align:middle line:84% 2 travels and passes it along to 4, 4 passes it along the 3, 00:20:31.380 --> 00:20:33.750 align:middle line:84% and 3 passes it along to 1, who was 00:20:33.750 --> 00:20:36.870 align:middle line:90% the initial issuer of the IOU. 00:20:36.870 --> 00:20:40.170 align:middle line:84% And there are several other network chains 00:20:40.170 --> 00:20:42.540 align:middle line:90% like that in this economy. 00:20:42.540 --> 00:20:44.790 align:middle line:90% And those would be debt. 00:20:44.790 --> 00:20:46.230 align:middle line:90% They're private IOUs. 00:20:46.230 --> 00:20:48.000 align:middle line:90% They're not fiat money. 00:20:48.000 --> 00:20:50.790 align:middle line:84% But they would facilitate exchange and act 00:20:50.790 --> 00:20:53.190 align:middle line:90% like privately issued monies. 00:20:53.190 --> 00:20:56.580 align:middle line:84% Now, you do have to redeem your force in the model 00:20:56.580 --> 00:20:59.670 align:middle line:84% to pay back the loan in the fourth period. 00:20:59.670 --> 00:21:03.370 align:middle line:84% But meanwhile, those IOUs circulate in the population. 00:21:03.370 --> 00:21:06.810 align:middle line:84% So this is a real economy, no fiat money. 00:21:06.810 --> 00:21:10.350 align:middle line:84% But there is a kind of money circulating private debt 00:21:10.350 --> 00:21:11.520 align:middle line:90% with high velocity. 00:21:11.520 --> 00:21:15.690 align:middle line:84% That IOU passes hands every single period. 00:21:15.690 --> 00:21:18.900 align:middle line:84% It has the highest single velocity in the whole economy. 00:21:18.900 --> 00:21:22.200 align:middle line:84% Velocity is the amount traded per unit time divided 00:21:22.200 --> 00:21:24.070 align:middle line:90% by the stock. 00:21:24.070 --> 00:21:26.950 align:middle line:84% Another implicit thing going on in this comparison 00:21:26.950 --> 00:21:30.370 align:middle line:84% is, depending how often agents meet with one another, 00:21:30.370 --> 00:21:34.420 align:middle line:84% you could evolve from a monetary economy 00:21:34.420 --> 00:21:37.180 align:middle line:84% with valued fiat money to the point 00:21:37.180 --> 00:21:41.560 align:middle line:84% that the credit markets get so good, especially with privately 00:21:41.560 --> 00:21:44.720 align:middle line:84% issued debt, that you don't need fiat money at all. 00:21:44.720 --> 00:21:46.210 align:middle line:84% So don't tell the Federal Reserve. 00:21:46.210 --> 00:21:48.190 align:middle line:84% They're not going to be happy about it. 00:21:48.190 --> 00:21:50.930 align:middle line:90% But that tension is there. 00:21:50.930 --> 00:21:55.180 align:middle line:84% And I think part of the issue with digital assets 00:21:55.180 --> 00:22:00.070 align:middle line:84% and cryptocurrency is that it's kind of filling in the market 00:22:00.070 --> 00:22:02.650 align:middle line:84% and can lead us to a situation where the fiat 00:22:02.650 --> 00:22:05.830 align:middle line:90% money is no longer valued. 00:22:05.830 --> 00:22:08.830 align:middle line:84% Well, I mean, you get the same thing in this model. 00:22:08.830 --> 00:22:14.900 align:middle line:84% a meets b for two periods, then three periods, then n periods. 00:22:14.900 --> 00:22:18.490 align:middle line:84% So they're together so much, the residual gap 00:22:18.490 --> 00:22:22.210 align:middle line:84% is not enough to create valued fiat money. 00:22:22.210 --> 00:22:25.720 align:middle line:84% So these slides then go back to the issue 00:22:25.720 --> 00:22:29.830 align:middle line:84% of either monetary policy or distributed ledgers. 00:22:29.830 --> 00:22:33.970 align:middle line:84% We talked about distributed ledgers throughout the class 00:22:33.970 --> 00:22:34.780 align:middle line:90% off and on. 00:22:34.780 --> 00:22:37.360 align:middle line:84% Maybe that wasn't effective because it pops up 00:22:37.360 --> 00:22:40.250 align:middle line:84% in different spots, depending on the lecture. 00:22:40.250 --> 00:22:43.150 align:middle line:84% But if you look at 1 passes it to 2, 00:22:43.150 --> 00:22:46.510 align:middle line:84% 2 passes it to 4, to 3, and back to 1, 00:22:46.510 --> 00:22:49.750 align:middle line:84% there's another path over here where 3 issues it to 4, 00:22:49.750 --> 00:22:56.110 align:middle line:84% 4 gives it to 2, 2 gives it to 1, and 1 passes it back to 3. 00:22:56.110 --> 00:22:59.210 align:middle line:84% It turns out that you don't need both of them. 00:22:59.210 --> 00:23:02.620 align:middle line:84% But if the amount of short-term circulating debt 00:23:02.620 --> 00:23:07.470 align:middle line:84% is not coordinated, then you can run into troubles. 00:23:07.470 --> 00:23:11.190 align:middle line:84% Another way to say that is there are multiple equilibria, 00:23:11.190 --> 00:23:13.050 align:middle line:84% a continuum of equilibria, but you 00:23:13.050 --> 00:23:15.720 align:middle line:84% need to coordinate on the amount of the privately 00:23:15.720 --> 00:23:20.230 align:middle line:84% issued currencies that are about to circulate in the economy. 00:23:20.230 --> 00:23:24.270 align:middle line:84% And if these are informationally decentralized locations, 00:23:24.270 --> 00:23:26.460 align:middle line:84% if they're not supposed to know what's 00:23:26.460 --> 00:23:28.530 align:middle line:84% going on in the other location, then there's 00:23:28.530 --> 00:23:30.820 align:middle line:90% no way they can coordinate. 00:23:30.820 --> 00:23:34.320 align:middle line:84% So one of the advantages of distributed ledgers 00:23:34.320 --> 00:23:39.180 align:middle line:84% would be, especially if you write smart contracts on them, 00:23:39.180 --> 00:23:42.610 align:middle line:84% that you can solve this coordination problem. 00:23:42.610 --> 00:23:45.180 align:middle line:84% And I think that's where we're going with digital assets, 00:23:45.180 --> 00:23:48.040 align:middle line:84% except people aren't thinking about it yet. 00:23:48.040 --> 00:23:50.100 align:middle line:84% We're going to end up with financial crises, 00:23:50.100 --> 00:23:52.770 align:middle line:84% not because digital assets are bad, 00:23:52.770 --> 00:23:55.540 align:middle line:84% but because there's a coordination problem. 00:23:55.540 --> 00:23:59.880 align:middle line:84% Fortunately, we have a solution in the ledgers. 00:23:59.880 --> 00:24:02.610 align:middle line:84% It's not crazy to link this to central banks 00:24:02.610 --> 00:24:06.000 align:middle line:84% because I mentioned once to you IOUs that 00:24:06.000 --> 00:24:08.160 align:middle line:90% circulated in Medieval England. 00:24:08.160 --> 00:24:09.780 align:middle line:90% Those were those willow sticks. 00:24:09.780 --> 00:24:11.730 align:middle line:84% Eventually, they were issuing pieces 00:24:11.730 --> 00:24:15.555 align:middle line:84% of paper, promissory notes as IOUs, that were circulating 00:24:15.555 --> 00:24:17.310 align:middle line:90% in the population. 00:24:17.310 --> 00:24:21.000 align:middle line:84% And in London, those bills of exchange were-- 00:24:21.000 --> 00:24:23.760 align:middle line:84% well, all of England and especially in London, 00:24:23.760 --> 00:24:25.690 align:middle line:84% there were markets for those bills. 00:24:25.690 --> 00:24:27.780 align:middle line:84% But occasionally, they would crash. 00:24:27.780 --> 00:24:29.880 align:middle line:84% And so that's actually what gave rise 00:24:29.880 --> 00:24:32.370 align:middle line:84% to one of the first central banks in the world, 00:24:32.370 --> 00:24:34.710 align:middle line:84% the Central Bank of England, where 00:24:34.710 --> 00:24:38.190 align:middle line:84% it would buy up those bills that were falling in price. 00:24:38.190 --> 00:24:40.260 align:middle line:84% That's the way monetary policy used 00:24:40.260 --> 00:24:42.480 align:middle line:84% to work when the Fed buys treasuries. 00:24:42.480 --> 00:24:44.730 align:middle line:84% But they may have created the wrong institution 00:24:44.730 --> 00:24:48.870 align:middle line:84% because the problem likely was the coordination problem, not 00:24:48.870 --> 00:24:52.740 align:middle line:84% the fact that the private markets were intrinsically 00:24:52.740 --> 00:24:53.970 align:middle line:90% flawed. 00:24:53.970 --> 00:24:56.700 align:middle line:84% And the other aspect that we're going to cover today 00:24:56.700 --> 00:24:59.670 align:middle line:84% is that fiat money can be introduced. 00:24:59.670 --> 00:25:01.860 align:middle line:84% Even though it's worth nothing, it's 00:25:01.860 --> 00:25:03.600 align:middle line:90% going to have a positive value. 00:25:03.600 --> 00:25:05.700 align:middle line:84% And that's going to create other problems. 00:25:05.700 --> 00:25:08.280 align:middle line:84% It's not like it's totally stable. 00:25:08.280 --> 00:25:11.310 align:middle line:84% It could be moving around, which brings us 00:25:11.310 --> 00:25:14.160 align:middle line:84% to the other kind of money, Bitcoin, 00:25:14.160 --> 00:25:16.170 align:middle line:84% which was introduced to be a money 00:25:16.170 --> 00:25:18.660 align:middle line:90% and to replace fiat money. 00:25:18.660 --> 00:25:21.600 align:middle line:84% That's what really made the central bank so annoyed 00:25:21.600 --> 00:25:24.270 align:middle line:84% with the whole Nakamoto experiment. 00:25:24.270 --> 00:25:26.100 align:middle line:90% They were threatened by Bitcoin. 00:25:26.100 --> 00:25:28.260 align:middle line:90% So a bit historical here. 00:25:28.260 --> 00:25:30.220 align:middle line:90% I kept it in for a reason. 00:25:30.220 --> 00:25:33.160 align:middle line:84% When I first started teaching, I grabbed this slide, 00:25:33.160 --> 00:25:35.370 align:middle line:84% which has to do with the price of Bitcoin 00:25:35.370 --> 00:25:42.180 align:middle line:84% moving from $100 to $147 in 2013. 00:25:42.180 --> 00:25:44.940 align:middle line:84% People were already talking about how ridiculous that was. 00:25:44.940 --> 00:25:48.040 align:middle line:90% And it wasn't tied to anything. 00:25:48.040 --> 00:25:51.330 align:middle line:84% So the value was completely indeterminate. 00:25:51.330 --> 00:25:54.780 align:middle line:84% But remember, fiat money is just like that. 00:25:54.780 --> 00:25:57.480 align:middle line:84% When you have the head of the Federal Reserve 00:25:57.480 --> 00:26:02.850 align:middle line:84% Bank of New York, Dudley, saying the problem with Bitcoin 00:26:02.850 --> 00:26:05.580 align:middle line:84% is that it has no fundamental value, 00:26:05.580 --> 00:26:08.700 align:middle line:84% don't throw stones if you live in glass houses 00:26:08.700 --> 00:26:11.050 align:middle line:84% because fiat money has the same problem. 00:26:11.050 --> 00:26:13.530 align:middle line:84% And it's just that in the US, fiat money 00:26:13.530 --> 00:26:15.570 align:middle line:90% is respected and not unstable. 00:26:15.570 --> 00:26:18.870 align:middle line:84% Anyway, back to the $100 to $147, 00:26:18.870 --> 00:26:20.950 align:middle line:84% I can't see your faces very well, 00:26:20.950 --> 00:26:24.900 align:middle line:84% but you may be a bit puzzled because if you looked at what 00:26:24.900 --> 00:26:30.640 align:middle line:90% the value is now, it's $19,372. 00:26:30.640 --> 00:26:34.690 align:middle line:84% So over here, at the low end, the price was so low, 00:26:34.690 --> 00:26:38.430 align:middle line:84% it looks like it's 0 now when we change the scale. 00:26:38.430 --> 00:26:41.070 align:middle line:84% And then, of course, it jumped up in '17, 00:26:41.070 --> 00:26:45.120 align:middle line:84% I guess it was, and hit about $19,000. 00:26:45.120 --> 00:26:47.220 align:middle line:90% And this is the most recent-- 00:26:47.220 --> 00:26:50.050 align:middle line:84% I kind of did this because I told you I was aware 00:26:50.050 --> 00:26:54.450 align:middle line:84% of the price three days ago was the highest on record. 00:26:54.450 --> 00:26:58.350 align:middle line:84% As of yesterday, it's back down about $200 00:26:58.350 --> 00:26:59.730 align:middle line:90% below the highest point. 00:26:59.730 --> 00:27:03.570 align:middle line:84% But essentially, these are almost identical spikes. 00:27:03.570 --> 00:27:05.510 align:middle line:84% Where do you think it's going to go? 00:27:05.510 --> 00:27:07.200 align:middle line:90% You want to buy? 00:27:07.200 --> 00:27:10.050 align:middle line:84% Well, what determines the value of this stuff? 00:27:10.050 --> 00:27:10.680 align:middle line:90% OK. 00:27:10.680 --> 00:27:14.520 align:middle line:84% So maybe only that it's used in exchange. 00:27:14.520 --> 00:27:16.480 align:middle line:90% Would that pin down the value? 00:27:16.480 --> 00:27:18.210 align:middle line:90% The answer is no. 00:27:18.210 --> 00:27:21.480 align:middle line:84% And now you're going to see the model. 00:27:21.480 --> 00:27:25.050 align:middle line:84% So this is the overlapping generations model. 00:27:25.050 --> 00:27:29.070 align:middle line:84% I mentioned last time we have an infinity of goods and infinity 00:27:29.070 --> 00:27:30.690 align:middle line:90% of households. 00:27:30.690 --> 00:27:31.770 align:middle line:90% Nobody lives forever. 00:27:31.770 --> 00:27:35.760 align:middle line:84% But there's one generation after another. 00:27:35.760 --> 00:27:38.520 align:middle line:84% We saw last time a version of that, 00:27:38.520 --> 00:27:40.850 align:middle line:84% that we could have a competitive equilibrium where 00:27:40.850 --> 00:27:43.550 align:middle line:84% the price of the second good was 1, everyone 00:27:43.550 --> 00:27:49.820 align:middle line:84% had 2 units of wealth, and they could all end up in autarky. 00:27:49.820 --> 00:27:51.710 align:middle line:84% And there was a way to Pareto dominate 00:27:51.710 --> 00:27:54.130 align:middle line:90% the competitive equilibrium. 00:27:54.130 --> 00:27:57.240 align:middle line:84% So that is true here in this is more general setup. 00:27:57.240 --> 00:28:00.090 align:middle line:84% A little bit of a better-known setup 00:28:00.090 --> 00:28:03.000 align:middle line:84% because this is Samuelson's model, which 00:28:03.000 --> 00:28:06.930 align:middle line:90% he created at MIT, by the way. 00:28:06.930 --> 00:28:10.170 align:middle line:84% You have two-- let's simplify the lifetime. 00:28:10.170 --> 00:28:12.060 align:middle line:90% You're either young or old. 00:28:12.060 --> 00:28:13.710 align:middle line:90% So no middle-aged people. 00:28:13.710 --> 00:28:16.200 align:middle line:90% No people living 100 years. 00:28:16.200 --> 00:28:17.800 align:middle line:90% They just live two periods. 00:28:17.800 --> 00:28:18.660 align:middle line:90% How many of them? 00:28:18.660 --> 00:28:22.180 align:middle line:84% We're going to put a little population growth in the model. 00:28:22.180 --> 00:28:26.940 align:middle line:84% So there's an initial population L0 at date 0. 00:28:26.940 --> 00:28:31.410 align:middle line:84% Then, L1 at date 1 is 1 plus n times L0. 00:28:31.410 --> 00:28:34.530 align:middle line:84% So there's a rate of population growth of n. 00:28:34.530 --> 00:28:39.060 align:middle line:84% L3 is 1 plus nL which is 1 plus n squared L1. 00:28:39.060 --> 00:28:44.130 align:middle line:84% So if the initial population is 1, 00:28:44.130 --> 00:28:49.050 align:middle line:84% then the total population satisfies this equation. 00:28:49.050 --> 00:28:50.832 align:middle line:84% It'll be interesting and annoying 00:28:50.832 --> 00:28:52.290 align:middle line:84% because we're going to have to keep 00:28:52.290 --> 00:28:54.690 align:middle line:84% track of the number of people all the time. 00:28:54.690 --> 00:28:56.920 align:middle line:84% And we are going to want to divide by the numbers. 00:28:56.920 --> 00:28:59.280 align:middle line:84% So the algebra gets to be a pain. 00:28:59.280 --> 00:29:02.970 align:middle line:84% But I'll show you, at least in one case, how to do it. 00:29:02.970 --> 00:29:05.760 align:middle line:90% And then it will be more clear. 00:29:05.760 --> 00:29:07.260 align:middle line:90% There's only one good period. 00:29:07.260 --> 00:29:10.830 align:middle line:84% But per period, there's an infinite number of periods. 00:29:10.830 --> 00:29:12.960 align:middle line:84% There's obviously an infinite number of people, 00:29:12.960 --> 00:29:15.600 align:middle line:84% even though no one lives more than two periods. 00:29:15.600 --> 00:29:19.590 align:middle line:84% There's a constant return to scale production technology 00:29:19.590 --> 00:29:24.420 align:middle line:84% that maps the level of the capital good, real physical 00:29:24.420 --> 00:29:28.440 align:middle line:84% savings, as it were, and the current labor supply 00:29:28.440 --> 00:29:31.410 align:middle line:90% into output of the single good. 00:29:31.410 --> 00:29:34.320 align:middle line:84% And because it's constant, returns to scale, 00:29:34.320 --> 00:29:38.310 align:middle line:84% we know that if we multiply the inputs by a scalar, 00:29:38.310 --> 00:29:41.830 align:middle line:84% the output gets multiplied by a scalar. 00:29:41.830 --> 00:29:44.640 align:middle line:84% We do the following trick and let the scalar 00:29:44.640 --> 00:29:47.250 align:middle line:90% be the population itself. 00:29:47.250 --> 00:29:48.720 align:middle line:90% We divide by labor. 00:29:48.720 --> 00:29:54.120 align:middle line:84% So L over L goes to 1, k over L is now termed little k. 00:29:54.120 --> 00:29:57.180 align:middle line:90% Its capital per capita. 00:29:57.180 --> 00:30:01.170 align:middle line:84% And output would scale by 1 over L. 00:30:01.170 --> 00:30:03.720 align:middle line:84% And that's like putting that L over here 00:30:03.720 --> 00:30:04.720 align:middle line:90% on the left-hand side. 00:30:04.720 --> 00:30:07.320 align:middle line:90% We have output per capita. 00:30:07.320 --> 00:30:11.460 align:middle line:84% So we can write basically a production function 00:30:11.460 --> 00:30:18.630 align:middle line:84% F, which maps output y, little y, from capital. 00:30:18.630 --> 00:30:20.010 align:middle line:90% So there are two goods here. 00:30:20.010 --> 00:30:21.270 align:middle line:90% There's labor and capital. 00:30:21.270 --> 00:30:23.780 align:middle line:90% They're both used in production. 00:30:23.780 --> 00:30:27.070 align:middle line:84% So the capital stock is brought in from the previous period 00:30:27.070 --> 00:30:29.590 align:middle line:90% and then used in production. 00:30:29.590 --> 00:30:32.080 align:middle line:84% The marginal product of capital is 00:30:32.080 --> 00:30:34.660 align:middle line:84% going to be the rental rate on capital 00:30:34.660 --> 00:30:36.730 align:middle line:84% or it looks like an interest rate. 00:30:36.730 --> 00:30:41.290 align:middle line:84% And the wage is going to be the residual of what 00:30:41.290 --> 00:30:43.670 align:middle line:90% isn't paid out in capital. 00:30:43.670 --> 00:30:45.760 align:middle line:84% I could have written down an explicit expression 00:30:45.760 --> 00:30:49.810 align:middle line:84% for the wage, which would be the marginal product of labor. 00:30:49.810 --> 00:30:52.000 align:middle line:84% But because we have constant returns to scale, 00:30:52.000 --> 00:30:54.130 align:middle line:90% we know there are no profits. 00:30:54.130 --> 00:30:57.010 align:middle line:84% And since there are no profits, the payments 00:30:57.010 --> 00:30:59.470 align:middle line:84% to the factors of production, labor and capital, 00:30:59.470 --> 00:31:01.630 align:middle line:90% adds up completely to output. 00:31:01.630 --> 00:31:03.990 align:middle line:90% And firms make no profits. 00:31:03.990 --> 00:31:06.720 align:middle line:84% So these are the expressions for the wage 00:31:06.720 --> 00:31:09.540 align:middle line:84% and for the rental rate on capital. 00:31:09.540 --> 00:31:13.320 align:middle line:84% But note, that in turn, because it's a function of k, 00:31:13.320 --> 00:31:16.770 align:middle line:84% It's output less whatever the marginal return on capital 00:31:16.770 --> 00:31:20.430 align:middle line:84% was times the total amount of capital involves k, 00:31:20.430 --> 00:31:22.200 align:middle line:90% and k involves r. 00:31:22.200 --> 00:31:25.710 align:middle line:84% So you can write this as a function of r. 00:31:25.710 --> 00:31:28.800 align:middle line:84% Here's a simple two-period max problem. 00:31:28.800 --> 00:31:32.370 align:middle line:84% Utility today plus beta times utility tomorrow. 00:31:32.370 --> 00:31:33.540 align:middle line:90% And how do they do that? 00:31:33.540 --> 00:31:35.610 align:middle line:84% They're endowed with 1 unit of labor today. 00:31:35.610 --> 00:31:36.930 align:middle line:90% They get the wage. 00:31:36.930 --> 00:31:41.550 align:middle line:84% And you can either eat it or save it, real physical savings, 00:31:41.550 --> 00:31:43.590 align:middle line:90% or call it capital stock. 00:31:43.590 --> 00:31:46.350 align:middle line:84% Likewise tomorrow, they don't have any income. 00:31:46.350 --> 00:31:46.890 align:middle line:90% They're old. 00:31:46.890 --> 00:31:48.120 align:middle line:90% They're not working. 00:31:48.120 --> 00:31:50.820 align:middle line:84% But they do have that saving, and the saving 00:31:50.820 --> 00:31:53.760 align:middle line:84% is going to earn interest at rate little r. 00:31:53.760 --> 00:31:57.390 align:middle line:84% So whatever capital is brought in to next period, when they're 00:31:57.390 --> 00:32:00.000 align:middle line:84% old, it's going to have a rate of return of rt 00:32:00.000 --> 00:32:03.480 align:middle line:84% plus 1 for the second period because they were born at t 00:32:03.480 --> 00:32:05.670 align:middle line:90% in they're old at t plus 1. 00:32:05.670 --> 00:32:08.580 align:middle line:84% So savings times 1 plus the interest rate 00:32:08.580 --> 00:32:10.350 align:middle line:90% would be their consumption. 00:32:10.350 --> 00:32:14.370 align:middle line:84% It kind of looks like Social Security, except here these 00:32:14.370 --> 00:32:17.010 align:middle line:84% are real physical savings, the only thing 00:32:17.010 --> 00:32:19.350 align:middle line:90% so far that they're able to do. 00:32:19.350 --> 00:32:22.350 align:middle line:84% No money, no Social Security, nothing. 00:32:22.350 --> 00:32:25.620 align:middle line:84% So we're going to get a savings function as a solution 00:32:25.620 --> 00:32:29.430 align:middle line:84% to this problem that's going to involve the prices, namely 00:32:29.430 --> 00:32:32.260 align:middle line:90% the wage, and the interest rate. 00:32:32.260 --> 00:32:34.120 align:middle line:84% Each household will take the prices 00:32:34.120 --> 00:32:37.660 align:middle line:84% as given, as we always do in a competitive equilibrium, 00:32:37.660 --> 00:32:38.920 align:middle line:90% and solve for s. 00:32:38.920 --> 00:32:42.520 align:middle line:84% But that s is, in turn, through the production function, 00:32:42.520 --> 00:32:44.530 align:middle line:84% going to generate the interest rate. 00:32:44.530 --> 00:32:49.040 align:middle line:84% And the level of capital will also determine the wage. 00:32:49.040 --> 00:32:50.860 align:middle line:90% So what is in equilibrium? 00:32:50.860 --> 00:32:53.260 align:middle line:84% If things are going to add up properly, 00:32:53.260 --> 00:32:55.600 align:middle line:84% then we take the savings function-- we just 00:32:55.600 --> 00:32:58.810 align:middle line:84% did this-- the savings function of an individual person at day 00:32:58.810 --> 00:33:02.350 align:middle line:84% t times the number of persons at day t, 00:33:02.350 --> 00:33:04.660 align:middle line:90% that's the population Lt. 00:33:04.660 --> 00:33:06.970 align:middle line:84% This is total savings, which is going 00:33:06.970 --> 00:33:11.215 align:middle line:84% to be the total capital stock used in production tomorrow. 00:33:11.215 --> 00:33:13.090 align:middle line:84% And that's where the interest rate came from. 00:33:13.090 --> 00:33:17.770 align:middle line:84% It's paid out, supplying a factor of production tomorrow, 00:33:17.770 --> 00:33:21.100 align:middle line:90% namely that capital. 00:33:21.100 --> 00:33:24.280 align:middle line:90% So this is market clearing. 00:33:24.280 --> 00:33:27.070 align:middle line:84% You might imagine from Vollrath's law, the only thing 00:33:27.070 --> 00:33:29.920 align:middle line:84% we need to worry about if we're going to clear savings 00:33:29.920 --> 00:33:31.480 align:middle line:84% equal to the capital stock, we're 00:33:31.480 --> 00:33:34.070 align:middle line:84% also going to clear the labor market and everything else. 00:33:34.070 --> 00:33:36.790 align:middle line:84% But let's just focus on the capital stock. 00:33:36.790 --> 00:33:42.940 align:middle line:84% Now, take this equation here and divide through by Lt. 00:33:42.940 --> 00:33:46.330 align:middle line:84% Then we have little s on the right and kt plus 1 00:33:46.330 --> 00:33:48.820 align:middle line:90% over Lt on the left. 00:33:48.820 --> 00:33:51.580 align:middle line:84% That's kind of weird because we'd 00:33:51.580 --> 00:33:54.970 align:middle line:84% want to normalize by the actual population in t plus 1, 00:33:54.970 --> 00:33:56.650 align:middle line:90% which is Lt plus 1. 00:33:56.650 --> 00:34:01.420 align:middle line:84% So this will be kt plus 1 over Lt multiplied by Lt plus 1 00:34:01.420 --> 00:34:02.912 align:middle line:90% over Lt plus 1. 00:34:02.912 --> 00:34:04.120 align:middle line:90% That doesn't change anything. 00:34:04.120 --> 00:34:04.945 align:middle line:90% That's just 1. 00:34:04.945 --> 00:34:08.620 align:middle line:84% But we take that thing in the denominator, the Lt plus 1, 00:34:08.620 --> 00:34:11.830 align:middle line:84% and stick it under the k, and that makes it a little k 00:34:11.830 --> 00:34:16.639 align:middle line:84% because that's the per capita capital stock at t plus 1. 00:34:16.639 --> 00:34:21.710 align:middle line:84% After you did that, with Lt plus 1 over Lt, 00:34:21.710 --> 00:34:25.040 align:middle line:84% and we know that's already going up at rate 1 plus n. 00:34:25.040 --> 00:34:27.679 align:middle line:84% So that's where this first term is coming from. 00:34:27.679 --> 00:34:30.800 align:middle line:84% And I know it's painful to hear me go over it 00:34:30.800 --> 00:34:33.110 align:middle line:84% verbally because it's all just algebra. 00:34:33.110 --> 00:34:37.040 align:middle line:84% But this little trick is going to get used four more times. 00:34:37.040 --> 00:34:39.889 align:middle line:84% So I wanted to draw your attention to it. 00:34:39.889 --> 00:34:44.389 align:middle line:84% The capital stock per capita needs 00:34:44.389 --> 00:34:47.120 align:middle line:84% to take into account the number of people. 00:34:47.120 --> 00:34:50.840 align:middle line:84% The number of capita, so to speak, is changing over time. 00:34:50.840 --> 00:34:53.000 align:middle line:84% That's kind of the math of this expression. 00:34:53.000 --> 00:34:56.120 align:middle line:84% Anyway, we have little s on the right. 00:34:56.120 --> 00:34:58.182 align:middle line:84% And we already know the wage is a function 00:34:58.182 --> 00:34:59.390 align:middle line:90% of the current capital stock. 00:34:59.390 --> 00:35:03.950 align:middle line:84% The interest rate is a function of capital stock next period. 00:35:03.950 --> 00:35:08.910 align:middle line:84% Hence, kt and kt plus 1 are in this expression. 00:35:08.910 --> 00:35:12.020 align:middle line:84% So we can effectively look for what 00:35:12.020 --> 00:35:15.110 align:middle line:84% happens if we start with initial capital 00:35:15.110 --> 00:35:21.410 align:middle line:84% stock in the whole economy at date 0, the genesis date, 00:35:21.410 --> 00:35:25.400 align:middle line:84% and stick that capital, what's going to be in, 00:35:25.400 --> 00:35:28.160 align:middle line:84% solve this equation, what's going to be the capital 00:35:28.160 --> 00:35:29.690 align:middle line:90% stock tomorrow? 00:35:29.690 --> 00:35:32.240 align:middle line:84% Well, it's meant to be-- the solution 00:35:32.240 --> 00:35:36.480 align:middle line:84% is meant to be along this line denoted as this curve. 00:35:36.480 --> 00:35:37.980 align:middle line:84% Looks like a difference equation, 00:35:37.980 --> 00:35:40.230 align:middle line:84% mapping current into future capital. 00:35:40.230 --> 00:35:42.630 align:middle line:84% If you start with current capital, 00:35:42.630 --> 00:35:47.550 align:middle line:84% tomorrow's capital at t plus 1, at 1, would be kt plus 1. 00:35:47.550 --> 00:35:49.090 align:middle line:90% Now you do this trick. 00:35:49.090 --> 00:35:53.310 align:middle line:84% Let's go back and hit the 45-degree line and start over. 00:35:53.310 --> 00:35:56.400 align:middle line:84% Because it's the 45-degree line, we're putting k1 down here 00:35:56.400 --> 00:35:58.410 align:middle line:90% on the x-axis. 00:35:58.410 --> 00:36:01.950 align:middle line:84% Start at k1, what capital we have in the third date, 00:36:01.950 --> 00:36:04.720 align:middle line:84% read it off the s-curve and so on. 00:36:04.720 --> 00:36:07.080 align:middle line:84% And you can see that it would evolve, 00:36:07.080 --> 00:36:10.020 align:middle line:84% if this line crosses the 45-degree line, 00:36:10.020 --> 00:36:12.760 align:middle line:90% to a steady state called k star. 00:36:12.760 --> 00:36:15.850 align:middle line:84% So after that, the economy isn't moving anymore. 00:36:15.850 --> 00:36:18.780 align:middle line:84% There are no more dates on anything. 00:36:18.780 --> 00:36:20.490 align:middle line:90% It's completely settled down. 00:36:20.490 --> 00:36:24.510 align:middle line:84% So this is a long-run steady state of the economy, 00:36:24.510 --> 00:36:26.790 align:middle line:90% with only real savings. 00:36:26.790 --> 00:36:30.960 align:middle line:84% Now, let's introduce pieces of paper that 00:36:30.960 --> 00:36:32.790 align:middle line:90% are intrinsically worthless. 00:36:32.790 --> 00:36:36.260 align:middle line:84% But imagine that they have value. 00:36:36.260 --> 00:36:40.730 align:middle line:84% And let's denote their value as the price Pt, 00:36:40.730 --> 00:36:44.750 align:middle line:84% and that is the price of money in terms of consumption. 00:36:44.750 --> 00:36:48.020 align:middle line:84% It's the opposite of what you would normally think. 00:36:48.020 --> 00:36:49.970 align:middle line:84% Normally, we think of consumption 00:36:49.970 --> 00:36:51.650 align:middle line:90% having a dollar price. 00:36:51.650 --> 00:36:57.100 align:middle line:84% This is dollars having a consumption price. 00:36:57.100 --> 00:36:58.850 align:middle line:90% And I'll keep pointing that out. 00:36:58.850 --> 00:37:04.510 align:middle line:84% So if that consumption price of money is changing over time, 00:37:04.510 --> 00:37:08.290 align:middle line:84% we put Pt as 1 over Pt, that would be appreciation. 00:37:08.290 --> 00:37:10.090 align:middle line:90% That's going to be a good thing. 00:37:10.090 --> 00:37:12.850 align:middle line:84% By the way, if it were a money price, a dollar 00:37:12.850 --> 00:37:16.450 align:middle line:84% price of consumption, then having the price level go up 00:37:16.450 --> 00:37:19.540 align:middle line:84% would be a bad thing because it would be depreciating-- 00:37:19.540 --> 00:37:21.507 align:middle line:90% it would be inflation. 00:37:21.507 --> 00:37:23.090 align:middle line:84% But here, it's a good thing because we 00:37:23.090 --> 00:37:28.530 align:middle line:84% flipped the numerator around to be the opposite. 00:37:28.530 --> 00:37:30.630 align:middle line:90% So this is a rate of return. 00:37:30.630 --> 00:37:35.180 align:middle line:84% This is a real positive rate of return if Pt plus 1 is over Pt. 00:37:35.180 --> 00:37:37.400 align:middle line:84% And maybe people want to hold this piece of paper. 00:37:37.400 --> 00:37:39.842 align:middle line:84% If they're going to hold these pieces of paper, 00:37:39.842 --> 00:37:42.050 align:middle line:84% they have to be indifferent to that and the other way 00:37:42.050 --> 00:37:45.020 align:middle line:84% to store value over time, which was namely 00:37:45.020 --> 00:37:47.480 align:middle line:90% physical savings carried over. 00:37:47.480 --> 00:37:50.330 align:middle line:84% They would get the interest rate, the rental rate on it 00:37:50.330 --> 00:37:52.830 align:middle line:84% next period and get the savings back. 00:37:52.830 --> 00:37:57.230 align:middle line:84% So if in an equilibrium, there's both real savings and money, 00:37:57.230 --> 00:37:59.420 align:middle line:90% then this equation has to hold. 00:37:59.420 --> 00:38:02.840 align:middle line:84% We can call the bubble, the title of the lecture, 00:38:02.840 --> 00:38:05.450 align:middle line:84% the value of money times its price. 00:38:05.450 --> 00:38:08.100 align:middle line:90% It's in real consumption terms. 00:38:08.100 --> 00:38:13.100 align:middle line:84% So if you just stick Pt plus 1 equals Bt plus 1 over n, 00:38:13.100 --> 00:38:17.630 align:middle line:84% then you can rewrite this equation too this way 00:38:17.630 --> 00:38:19.440 align:middle line:90% because n is constant. 00:38:19.440 --> 00:38:22.010 align:middle line:84% I'm not manipulating the amount of money in the economy. 00:38:22.010 --> 00:38:25.140 align:middle line:84% I'm keeping it constant, although the price is moving. 00:38:25.140 --> 00:38:27.740 align:middle line:84% So then, we have from 2 this equation 00:38:27.740 --> 00:38:30.500 align:middle line:84% about how the bubble, the real value of the bubble 00:38:30.500 --> 00:38:32.630 align:middle line:90% is changing over time. 00:38:32.630 --> 00:38:36.170 align:middle line:84% And it's related to the marginal product of capital. 00:38:36.170 --> 00:38:39.800 align:middle line:84% Furthermore, I could divide by the population. 00:38:39.800 --> 00:38:42.290 align:middle line:90% We'd have a bubble per capita. 00:38:42.290 --> 00:38:44.780 align:middle line:84% And I'll do the same trick I just did before, 00:38:44.780 --> 00:38:47.000 align:middle line:84% with apologies, but I'm not going 00:38:47.000 --> 00:38:49.190 align:middle line:90% to go through the algebra. 00:38:49.190 --> 00:38:52.670 align:middle line:84% Take the Lt, divide through, then 00:38:52.670 --> 00:38:55.760 align:middle line:84% you have a mismatch between t plus 1 and t. 00:38:55.760 --> 00:39:00.380 align:middle line:84% Then have Lt plus 1 divided by Lt plus 1, parse it out, 00:39:00.380 --> 00:39:03.260 align:middle line:84% and you get this thing, that the real per capita bubble is 00:39:03.260 --> 00:39:06.860 align:middle line:84% moving over time, according to the marginal product 00:39:06.860 --> 00:39:07.430 align:middle line:90% of capital. 00:39:07.430 --> 00:39:10.010 align:middle line:84% But that arithmetic, which over here 00:39:10.010 --> 00:39:17.570 align:middle line:84% created the 1 plus n hanging out there, that extra term, that 1 00:39:17.570 --> 00:39:20.360 align:middle line:90% plus n is in this math as well. 00:39:20.360 --> 00:39:22.440 align:middle line:90% And you're just dividing by it. 00:39:22.440 --> 00:39:24.890 align:middle line:84% So that's where the n is coming from, 00:39:24.890 --> 00:39:26.990 align:middle line:90% that rate of population growth. 00:39:26.990 --> 00:39:28.670 align:middle line:84% This is all a big premise right now. 00:39:28.670 --> 00:39:32.660 align:middle line:84% We're just saying, if there were money, if it had value, 00:39:32.660 --> 00:39:36.500 align:middle line:84% the value of it has to be increasing at the rate 00:39:36.500 --> 00:39:40.010 align:middle line:84% that they can earn money in another way, namely by storing. 00:39:40.010 --> 00:39:41.480 align:middle line:84% And so you're getting an equation 00:39:41.480 --> 00:39:45.110 align:middle line:84% for how the value of the bubble, hence money, 00:39:45.110 --> 00:39:47.510 align:middle line:90% must be evolving over time. 00:39:47.510 --> 00:39:51.590 align:middle line:84% Now, savings can be done in the two ways, as I've kept saying, 00:39:51.590 --> 00:39:53.270 align:middle line:90% but here's the equation. 00:39:53.270 --> 00:39:56.360 align:middle line:84% When they're young, we have the savings function 00:39:56.360 --> 00:39:58.670 align:middle line:84% of the young people, depending on the wage 00:39:58.670 --> 00:40:01.610 align:middle line:84% and tomorrow's interest rate, times a number of young people. 00:40:01.610 --> 00:40:03.450 align:middle line:90% But they can save in two ways. 00:40:03.450 --> 00:40:05.570 align:middle line:84% They can save in the real bubble or they 00:40:05.570 --> 00:40:07.850 align:middle line:90% can save in the capital stock. 00:40:07.850 --> 00:40:12.870 align:middle line:84% And now, again, doing that same math as always, 00:40:12.870 --> 00:40:16.490 align:middle line:90% start solving for Kt plus 1. 00:40:16.490 --> 00:40:19.040 align:middle line:84% Subtract B off of the right-hand side. 00:40:19.040 --> 00:40:22.940 align:middle line:84% Then divide through by the appropriate labor term, 00:40:22.940 --> 00:40:25.100 align:middle line:90% and you get this expression. 00:40:25.100 --> 00:40:27.230 align:middle line:84% That's the third time we're doing the algebra. 00:40:27.230 --> 00:40:28.910 align:middle line:84% It should kind of be obvious where 00:40:28.910 --> 00:40:35.600 align:middle line:84% it's coming from because Kt plus 1 is equal to Ls minus B. 00:40:35.600 --> 00:40:37.820 align:middle line:84% And the rest of it is finding that end term 00:40:37.820 --> 00:40:40.340 align:middle line:84% and dividing through by that term. 00:40:40.340 --> 00:40:43.250 align:middle line:84% So now, this is just another way of saying 00:40:43.250 --> 00:40:45.560 align:middle line:84% that the amount of capital next period 00:40:45.560 --> 00:40:47.780 align:middle line:84% is what it would have been before in the real economy 00:40:47.780 --> 00:40:50.690 align:middle line:84% without money, except now we have bubbles of money 00:40:50.690 --> 00:40:52.080 align:middle line:90% that have value. 00:40:52.080 --> 00:40:53.960 align:middle line:84% So we've got to subtract off the bubble. 00:40:53.960 --> 00:40:57.290 align:middle line:84% That's kind of preventing the capital stock from being 00:40:57.290 --> 00:41:00.050 align:middle line:90% as large as it would have been. 00:41:00.050 --> 00:41:04.820 align:middle line:84% So 3, this thing, gets rewritten at the top of the next slide 00:41:04.820 --> 00:41:10.010 align:middle line:84% to be the bubble b So now solve this thing for b, little b, 00:41:10.010 --> 00:41:11.480 align:middle line:90% bubble per capita. 00:41:11.480 --> 00:41:15.170 align:middle line:84% It's this thing, and that's one equation. 00:41:15.170 --> 00:41:18.530 align:middle line:84% And we also know something about bubbles over time. 00:41:18.530 --> 00:41:20.210 align:middle line:90% As I was saying, we did that. 00:41:20.210 --> 00:41:21.680 align:middle line:84% That was this thing at the bottom 00:41:21.680 --> 00:41:23.870 align:middle line:90% of the slide before last. 00:41:23.870 --> 00:41:26.090 align:middle line:90% So we have those two equations. 00:41:26.090 --> 00:41:28.310 align:middle line:84% This has to be true in a steady state. 00:41:28.310 --> 00:41:29.390 align:middle line:90% Should emphasize. 00:41:29.390 --> 00:41:32.520 align:middle line:84% There are t's all over the place here, but in a steady state, 00:41:32.520 --> 00:41:34.850 align:middle line:84% we're not going to have any evolution. 00:41:34.850 --> 00:41:36.750 align:middle line:84% So the t's are going to disappear. 00:41:36.750 --> 00:41:40.130 align:middle line:90% So Kt plus 1 will equal Kt. 00:41:40.130 --> 00:41:43.130 align:middle line:84% There'll be no t on the b, Kt, blah, blah, blah. 00:41:43.130 --> 00:41:46.100 align:middle line:84% So when I rewrote 3 hurriedly just then, 00:41:46.100 --> 00:41:47.900 align:middle line:90% I rewrote it without the t's. 00:41:47.900 --> 00:41:49.970 align:middle line:84% And that's what this thing is at the top. 00:41:49.970 --> 00:41:52.650 align:middle line:84% And with the t's, we have this thing, the second term, 00:41:52.650 --> 00:41:56.270 align:middle line:84% which I've just pasted in from previously. 00:41:56.270 --> 00:41:59.750 align:middle line:84% So if we're going to have a steady state, then these b's 00:41:59.750 --> 00:42:01.190 align:middle line:90% can't be moving around. 00:42:01.190 --> 00:42:05.310 align:middle line:84% So bt plus 1 over bt has got to be a constant. 00:42:05.310 --> 00:42:08.270 align:middle line:84% And if the left-hand side is the same, 00:42:08.270 --> 00:42:12.380 align:middle line:84% 1, then the right-hand side has to be the same, 1. 00:42:12.380 --> 00:42:14.540 align:middle line:84% Hence, the marginal product of capital 00:42:14.540 --> 00:42:17.820 align:middle line:84% has to be equal to the rate of population growth. 00:42:17.820 --> 00:42:21.340 align:middle line:84% So if there is a steady state with a real bubble, 00:42:21.340 --> 00:42:23.830 align:middle line:84% then the rate of population growth 00:42:23.830 --> 00:42:26.470 align:middle line:84% has to be equal to the marginal product of capital, 00:42:26.470 --> 00:42:29.210 align:middle line:90% this thing at the bottom here. 00:42:29.210 --> 00:42:31.240 align:middle line:84% There's a couple of other ways to see this. 00:42:31.240 --> 00:42:35.890 align:middle line:84% One, take this equation, but let me rewrite it verbally for you. 00:42:35.890 --> 00:42:39.310 align:middle line:84% We have output coming from the production function. 00:42:39.310 --> 00:42:41.930 align:middle line:84% Take this k and put it on the right-hand side. 00:42:41.930 --> 00:42:43.450 align:middle line:90% You can always eat the capital. 00:42:43.450 --> 00:42:45.460 align:middle line:84% You remember that medieval village economy 00:42:45.460 --> 00:42:47.860 align:middle line:84% with seed and storage and all of that, where 00:42:47.860 --> 00:42:49.480 align:middle line:90% we talked about the evolution? 00:42:49.480 --> 00:42:53.270 align:middle line:84% So you can always eat this, the harvest, or you can save it. 00:42:53.270 --> 00:42:54.290 align:middle line:90% So it's like that. 00:42:54.290 --> 00:42:56.260 align:middle line:90% So this k can be eaten. 00:42:56.260 --> 00:42:59.290 align:middle line:84% So you have total resources available today 00:42:59.290 --> 00:43:01.690 align:middle line:90% to be f of k plus k. 00:43:01.690 --> 00:43:03.460 align:middle line:90% And how do you use that stuff? 00:43:03.460 --> 00:43:05.500 align:middle line:90% You can eat it or save it. 00:43:05.500 --> 00:43:07.270 align:middle line:90% Eat it today, that's c. 00:43:07.270 --> 00:43:11.890 align:middle line:84% Tomorrow, tomorrow's k, would be k, except darn it. 00:43:11.890 --> 00:43:13.870 align:middle line:84% We got that population growth again. 00:43:13.870 --> 00:43:16.570 align:middle line:84% So if we're going to keep k constant in a steady state, 00:43:16.570 --> 00:43:20.470 align:middle line:84% the resource requirements to have a constant per capita 00:43:20.470 --> 00:43:23.320 align:middle line:84% capital stock tomorrow are that you 00:43:23.320 --> 00:43:27.310 align:middle line:84% have to augment the capital per capita today by n. 00:43:27.310 --> 00:43:29.150 align:middle line:90% Same trick, fourth time. 00:43:29.150 --> 00:43:31.660 align:middle line:84% So anyway, writing it that way, we 00:43:31.660 --> 00:43:33.950 align:middle line:90% want to maximize consumption. 00:43:33.950 --> 00:43:38.890 align:middle line:84% So you solve consumption equals to f of k plus k 00:43:38.890 --> 00:43:41.620 align:middle line:90% minus k times then 1 plus n. 00:43:41.620 --> 00:43:45.130 align:middle line:84% And where is that object, that c maximized? 00:43:45.130 --> 00:43:48.340 align:middle line:84% It's maximized by finding the first-order condition, 00:43:48.340 --> 00:43:51.490 align:middle line:84% and that's exactly where f prime of k is equal to n. 00:43:51.490 --> 00:43:55.720 align:middle line:90% So that's so-called golden rule. 00:43:55.720 --> 00:43:59.900 align:middle line:84% Distressed to see this morning that this k star is not, 00:43:59.900 --> 00:44:02.140 align:middle line:90% however, the previous k star. 00:44:02.140 --> 00:44:05.200 align:middle line:84% This is a new golden rule k star. 00:44:05.200 --> 00:44:08.350 align:middle line:84% Should have called the other k autarky for the economy 00:44:08.350 --> 00:44:09.220 align:middle line:90% without money. 00:44:09.220 --> 00:44:11.230 align:middle line:84% In this case, star would be the golden rule 00:44:11.230 --> 00:44:16.570 align:middle line:84% k for the steady state bubble equilibrium. 00:44:16.570 --> 00:44:19.270 align:middle line:84% So it doesn't have to happen that you 00:44:19.270 --> 00:44:21.490 align:middle line:90% will be at the steady state. 00:44:21.490 --> 00:44:26.770 align:middle line:84% If k were greater than k star, everyone could be better off. 00:44:26.770 --> 00:44:28.000 align:middle line:90% Why? 00:44:28.000 --> 00:44:31.780 align:middle line:84% It's as if, if you started at k greater than the golden rule, 00:44:31.780 --> 00:44:34.600 align:middle line:84% the economy's overaccumulated the capital. 00:44:34.600 --> 00:44:38.350 align:middle line:84% And then to maintain a constant per capita capital stock, 00:44:38.350 --> 00:44:40.600 align:middle line:84% you've got to take something out of production, 00:44:40.600 --> 00:44:43.720 align:middle line:84% not eat it, and squirrel it away for tomorrow. 00:44:43.720 --> 00:44:48.010 align:middle line:84% You could actually do better by increasing consumption 00:44:48.010 --> 00:44:50.860 align:middle line:84% of the current generation, having less capital 00:44:50.860 --> 00:44:53.890 align:middle line:84% stock to maintain, and then going out 00:44:53.890 --> 00:44:55.270 align:middle line:90% to the infinite future. 00:44:55.270 --> 00:45:00.490 align:middle line:84% None of the future generations are going to be worse off. 00:45:00.490 --> 00:45:03.460 align:middle line:84% This experiment is exactly like what 00:45:03.460 --> 00:45:07.030 align:middle line:84% we did last time with the infinite number of goods 00:45:07.030 --> 00:45:12.160 align:middle line:84% economy, where we can engineer a one-time improvement 00:45:12.160 --> 00:45:15.670 align:middle line:84% in at least one of the agents by creating something 00:45:15.670 --> 00:45:16.420 align:middle line:90% from nowhere. 00:45:16.420 --> 00:45:20.860 align:middle line:84% Basically, we move the overaccumulation of capital 00:45:20.860 --> 00:45:23.860 align:middle line:84% into the current generation and let them eat it. 00:45:23.860 --> 00:45:27.400 align:middle line:84% And then, subsequent generations don't suffer either 00:45:27.400 --> 00:45:30.250 align:middle line:84% because they don't have to try to maintain 00:45:30.250 --> 00:45:34.780 align:middle line:84% this per capita capital stock, which was too large. 00:45:34.780 --> 00:45:37.870 align:middle line:84% So is it possible that the economy 00:45:37.870 --> 00:45:40.180 align:middle line:90% can overaccumulate capital? 00:45:40.180 --> 00:45:41.970 align:middle line:90% Yes. 00:45:41.970 --> 00:45:45.090 align:middle line:84% So we're going to look at two steady states, the previous one 00:45:45.090 --> 00:45:46.590 align:middle line:84% without any money and the new one 00:45:46.590 --> 00:45:49.340 align:middle line:90% with the steady state bubble. 00:45:49.340 --> 00:45:51.620 align:middle line:84% An interesting thing about the steady state bubble, 00:45:51.620 --> 00:45:54.090 align:middle line:84% this one that we've already described, 00:45:54.090 --> 00:45:58.420 align:middle line:84% is that it has the property that it achieves the golden rule. 00:45:58.420 --> 00:46:01.840 align:middle line:84% And that's because-- you can kind of see it here-- 00:46:01.840 --> 00:46:04.990 align:middle line:84% when the bubble is constant and not depending on time, 00:46:04.990 --> 00:46:07.750 align:middle line:84% then f prime of k must be equal to n. 00:46:07.750 --> 00:46:11.620 align:middle line:84% So the golden rule is achieved in the steady state 00:46:11.620 --> 00:46:13.210 align:middle line:90% of the bubble equilibrium. 00:46:13.210 --> 00:46:14.920 align:middle line:90% And that was this top line. 00:46:14.920 --> 00:46:18.700 align:middle line:84% In a steady state with bubbles, the golden rule is achieved. 00:46:18.700 --> 00:46:20.410 align:middle line:84% And that's going to be this point. 00:46:20.410 --> 00:46:22.930 align:middle line:84% So that's k star, that's the golden rule 00:46:22.930 --> 00:46:26.020 align:middle line:84% k, corresponds to a particular bubble that 00:46:26.020 --> 00:46:27.750 align:middle line:90% solves the other equation. 00:46:27.750 --> 00:46:29.440 align:middle line:90% That's called a b star. 00:46:29.440 --> 00:46:31.420 align:middle line:84% This is what we would like to get to. 00:46:31.420 --> 00:46:34.090 align:middle line:84% And what happens in the economy with no money? 00:46:34.090 --> 00:46:37.820 align:middle line:84% You're down here at essentially an autarky value. 00:46:37.820 --> 00:46:41.260 align:middle line:84% So this is a perfectly fine competitive equilibrium. 00:46:41.260 --> 00:46:42.910 align:middle line:90% There is no money. 00:46:42.910 --> 00:46:47.650 align:middle line:84% But in it, the economy will have overaccumulated capital. 00:46:47.650 --> 00:46:51.040 align:middle line:84% And a Pareto improvement can be engineered 00:46:51.040 --> 00:46:53.320 align:middle line:84% by reducing the amount of capital. 00:46:53.320 --> 00:46:55.300 align:middle line:84% How does that happen in a steady state? 00:46:55.300 --> 00:46:59.420 align:middle line:84% You give other people a way to save in worthless stuff. 00:46:59.420 --> 00:47:01.570 align:middle line:84% So instead of real physical savings, 00:47:01.570 --> 00:47:04.015 align:middle line:84% they're scrambling to get these pieces of paper. 00:47:04.015 --> 00:47:05.890 align:middle line:84% The rate of return on the pieces of the paper 00:47:05.890 --> 00:47:08.110 align:middle line:84% has to be the same as physical savings. 00:47:08.110 --> 00:47:10.240 align:middle line:90% It has no intrinsic value. 00:47:10.240 --> 00:47:13.300 align:middle line:84% It has an extrinsic value because the price of money 00:47:13.300 --> 00:47:14.360 align:middle line:90% is positive. 00:47:14.360 --> 00:47:16.840 align:middle line:84% So in other words, these perfectly worthless pieces 00:47:16.840 --> 00:47:19.570 align:middle line:84% of paper, fiat money, have a value 00:47:19.570 --> 00:47:21.010 align:middle line:90% in a competitive equilibrium. 00:47:21.010 --> 00:47:24.400 align:middle line:84% And the money's actually helping to recover the Pareto optimum. 00:47:24.400 --> 00:47:27.580 align:middle line:84% So you see the connection to the welfare theorems here. 00:47:27.580 --> 00:47:30.010 align:middle line:84% We have an infinite number of people, 00:47:30.010 --> 00:47:32.200 align:middle line:90% an infinite number of periods. 00:47:32.200 --> 00:47:33.760 align:middle line:90% The welfare theorems fail. 00:47:33.760 --> 00:47:36.100 align:middle line:84% If you start adding up the value of wealth 00:47:36.100 --> 00:47:37.990 align:middle line:84% in the autarky equilibrium, it goes 00:47:37.990 --> 00:47:39.910 align:middle line:90% to infinity like last time. 00:47:39.910 --> 00:47:42.160 align:middle line:84% That's why the first welfare theorem fails. 00:47:42.160 --> 00:47:45.160 align:middle line:84% And the way to recover optimality 00:47:45.160 --> 00:47:47.830 align:middle line:84% is to create these worthless pieces of paper 00:47:47.830 --> 00:47:49.060 align:middle line:90% and let them trade. 00:47:49.060 --> 00:47:50.680 align:middle line:90% It's totally amazing. 00:47:50.680 --> 00:47:53.090 align:middle line:84% But it's an intrinsic part of money. 00:47:53.090 --> 00:47:58.010 align:middle line:84% However there's a downside here, which is it's unstable. 00:47:58.010 --> 00:48:00.310 align:middle line:84% So let me just describe what's going on 00:48:00.310 --> 00:48:04.120 align:middle line:84% without expecting you to be able to reproduce all the math. 00:48:04.120 --> 00:48:07.030 align:middle line:84% You're plotting, this was the autarky value. 00:48:07.030 --> 00:48:10.010 align:middle line:84% This is the golden rule value that we want. 00:48:10.010 --> 00:48:12.280 align:middle line:84% Now we want to look at starting from different points 00:48:12.280 --> 00:48:17.060 align:middle line:84% with the economy evolved to one or the other of those points. 00:48:17.060 --> 00:48:19.210 align:middle line:84% And this is an equation, depending 00:48:19.210 --> 00:48:22.060 align:middle line:84% on whether you're above it or below it for the capital stock 00:48:22.060 --> 00:48:26.140 align:middle line:84% to evolve, this equation, the BB, bubble bubble equation, 00:48:26.140 --> 00:48:28.930 align:middle line:84% has to do with whether you're to the left or the right of it, 00:48:28.930 --> 00:48:30.640 align:middle line:90% whether the bubble is evolving. 00:48:30.640 --> 00:48:33.820 align:middle line:84% And these arrows denote all the forces 00:48:33.820 --> 00:48:35.540 align:middle line:90% in the underlying economy. 00:48:35.540 --> 00:48:40.270 align:middle line:84% And if you're lucky and you start away from the golden rule 00:48:40.270 --> 00:48:43.300 align:middle line:84% steady state, you would evolve on a knife edge 00:48:43.300 --> 00:48:45.440 align:middle line:90% and eventually converge to it. 00:48:45.440 --> 00:48:46.790 align:middle line:90% So this is the good news. 00:48:46.790 --> 00:48:49.990 align:middle line:84% The economies that start with an appropriate amount of bubble 00:48:49.990 --> 00:48:55.000 align:middle line:84% or capital will evolve to this optimal golden rule, 00:48:55.000 --> 00:48:56.840 align:middle line:90% so steady state. 00:48:56.840 --> 00:48:58.750 align:middle line:84% But this is the really bad news-- 00:48:58.750 --> 00:49:02.960 align:middle line:84% this economy can also evolve to real savings and no money. 00:49:02.960 --> 00:49:05.340 align:middle line:90% And money inflates away. 00:49:05.340 --> 00:49:08.060 align:middle line:84% It could start with a positive value. 00:49:08.060 --> 00:49:09.080 align:middle line:90% This is an equilibrium. 00:49:09.080 --> 00:49:10.960 align:middle line:84% And it doesn't require stupid people. 00:49:10.960 --> 00:49:15.970 align:middle line:84% This is a fully rational expectation dynamic path, where 00:49:15.970 --> 00:49:18.340 align:middle line:90% the value of money goes to 0. 00:49:18.340 --> 00:49:21.130 align:middle line:84% OK, well, so why did I show you that? 00:49:21.130 --> 00:49:25.090 align:middle line:84% I can replace the word fiat money with cryptocurrency 00:49:25.090 --> 00:49:27.550 align:middle line:84% and ask what value it ought to have. 00:49:27.550 --> 00:49:30.550 align:middle line:84% It's perfectly plausible to me that you could 00:49:30.550 --> 00:49:32.380 align:middle line:90% see the price moving around. 00:49:32.380 --> 00:49:34.810 align:middle line:84% Maybe it will go to some constant value 00:49:34.810 --> 00:49:37.510 align:middle line:84% if this Bitcoin is also useful in exchange, 00:49:37.510 --> 00:49:40.510 align:middle line:84% the way money is in that overlapping generation's model. 00:49:40.510 --> 00:49:44.080 align:middle line:84% Maybe it's going to it's going to collapse again, 00:49:44.080 --> 00:49:45.670 align:middle line:90% and the price will go to 0. 00:49:45.670 --> 00:49:47.890 align:middle line:84% All those phenomenon are perfectly 00:49:47.890 --> 00:49:51.400 align:middle line:84% consistent with Bitcoin playing a useful role in exchange 00:49:51.400 --> 00:49:54.470 align:middle line:84% but not able to find a stable value. 00:49:54.470 --> 00:49:56.650 align:middle line:84% But again, the point is, and actually we've 00:49:56.650 --> 00:49:59.140 align:middle line:84% been emphasizing this, starting with fiat money 00:49:59.140 --> 00:50:01.780 align:middle line:84% the way Samuelson did, it's a property of any money. 00:50:01.780 --> 00:50:03.910 align:middle line:84% It's a property of fiat money and so on. 00:50:03.910 --> 00:50:06.340 align:middle line:84% The Fed worries a lot about forward guidance. 00:50:06.340 --> 00:50:08.230 align:middle line:84% They're constantly telling people 00:50:08.230 --> 00:50:10.880 align:middle line:84% what they predict to happen in the future 00:50:10.880 --> 00:50:13.030 align:middle line:90% so people don't panic. 00:50:13.030 --> 00:50:15.130 align:middle line:84% If you think fiat money is not going 00:50:15.130 --> 00:50:16.850 align:middle line:84% to be worth what it is in the future, 00:50:16.850 --> 00:50:18.400 align:middle line:84% you're not going to want to hold it. 00:50:18.400 --> 00:50:21.700 align:middle line:84% Oh, that's one explanation that traders 00:50:21.700 --> 00:50:25.000 align:middle line:84% have for what's going on with Bitcoin, namely, 00:50:25.000 --> 00:50:27.430 align:middle line:84% there's just an ever-increasing amount 00:50:27.430 --> 00:50:29.930 align:middle line:84% of government debt in the economy. 00:50:29.930 --> 00:50:34.090 align:middle line:84% The Fed has bought up all those securities effectively 00:50:34.090 --> 00:50:36.340 align:middle line:84% and issued accounts at the Federal Reserve 00:50:36.340 --> 00:50:38.410 align:middle line:90% in worthless pieces of paper. 00:50:38.410 --> 00:50:39.820 align:middle line:90% Are you scared now? 00:50:39.820 --> 00:50:42.580 align:middle line:84% So the market may be anticipating 00:50:42.580 --> 00:50:45.760 align:middle line:84% that, in fact, there aren't enough real resources 00:50:45.760 --> 00:50:46.990 align:middle line:90% to pay off the debt. 00:50:46.990 --> 00:50:49.660 align:middle line:84% The only way that the Fed and European 00:50:49.660 --> 00:50:52.570 align:middle line:84% governments are going to pay off the debt is to inflate. 00:50:52.570 --> 00:50:55.870 align:middle line:84% And if the market starts to expect inflation, 00:50:55.870 --> 00:51:00.070 align:middle line:84% fiat money is less valued, and they're shifting into Bitcoin. 00:51:00.070 --> 00:51:02.620 align:middle line:84% So that's what traders think is going on. 00:51:02.620 --> 00:51:04.510 align:middle line:84% I switched the story on you a bit 00:51:04.510 --> 00:51:06.880 align:middle line:84% because now we have fiat money competing 00:51:06.880 --> 00:51:08.998 align:middle line:90% with Bitcoin and vice versa. 00:51:08.998 --> 00:51:10.540 align:middle line:84% But I think you could imagine how you 00:51:10.540 --> 00:51:12.590 align:middle line:90% could get both in the model. 00:51:12.590 --> 00:51:15.550 align:middle line:84% So the way you get both in the model 00:51:15.550 --> 00:51:19.630 align:middle line:84% is to set up an underlying economic environment 00:51:19.630 --> 00:51:21.220 align:middle line:90% and have gaps. 00:51:21.220 --> 00:51:23.950 align:middle line:84% Say autarky households aren't smoothing, 00:51:23.950 --> 00:51:26.740 align:middle line:84% there's a lot of fluctuations, blah, blah, blah, blah, blah, 00:51:26.740 --> 00:51:28.210 align:middle line:90% fiat money can help smooth. 00:51:28.210 --> 00:51:31.220 align:middle line:84% But maybe fiat money doesn't smooth it out entirely. 00:51:31.220 --> 00:51:34.270 align:middle line:84% And then, you're left for tokens like Bitcoin 00:51:34.270 --> 00:51:36.640 align:middle line:84% to play a role on top of fiat money. 00:51:36.640 --> 00:51:39.160 align:middle line:84% So you could actually have both in the model. 00:51:39.160 --> 00:51:43.210 align:middle line:84% It's not true that one has to dominate the other. 00:51:43.210 --> 00:51:44.920 align:middle line:90% They could both coexist. 00:51:44.920 --> 00:51:47.590 align:middle line:84% And partly what we're seeing in these cryptosystems 00:51:47.590 --> 00:51:52.060 align:middle line:84% is crypto coins are a way of transferring fiat money around. 00:51:52.060 --> 00:51:54.880 align:middle line:84% You just create tokens that get transferred around. 00:51:54.880 --> 00:51:57.430 align:middle line:84% But they're claims on the underlying fiat money. 00:51:57.430 --> 00:52:00.010 align:middle line:84% But the tokens can have value because they 00:52:00.010 --> 00:52:02.680 align:middle line:84% facilitate exchange, even if they're not 00:52:02.680 --> 00:52:04.570 align:middle line:90% backed by the fiat money. 00:52:04.570 --> 00:52:06.490 align:middle line:90% So we could get both of them in. 00:52:06.490 --> 00:52:10.210 align:middle line:84% Now, this is pretty abstract, this overlapping generations 00:52:10.210 --> 00:52:10.900 align:middle line:90% model. 00:52:10.900 --> 00:52:13.390 align:middle line:84% You might say it's totally unrealistic. 00:52:13.390 --> 00:52:15.340 align:middle line:84% By the way, it's another interpretation 00:52:15.340 --> 00:52:18.160 align:middle line:84% of Social Security with pay as you go, 00:52:18.160 --> 00:52:22.240 align:middle line:84% namely the young pay in to Social Security, 00:52:22.240 --> 00:52:24.730 align:middle line:84% but it's not sitting there in some fund or something. 00:52:24.730 --> 00:52:26.350 align:middle line:90% It's not in a lockbox. 00:52:26.350 --> 00:52:29.920 align:middle line:84% It's just flowing right out to the old generation. 00:52:29.920 --> 00:52:32.590 align:middle line:84% So people have criticized Samuelson's model 00:52:32.590 --> 00:52:34.780 align:middle line:84% as a not realistic model of money 00:52:34.780 --> 00:52:37.150 align:middle line:84% because of the extreme assumptions 00:52:37.150 --> 00:52:41.300 align:middle line:84% about finite two-period lives and so on. 00:52:41.300 --> 00:52:43.810 align:middle line:90% But could it be applied? 00:52:43.810 --> 00:52:47.630 align:middle line:84% Well, these guys, Abel, Mankiw, Summers, and Zeckhauser said, 00:52:47.630 --> 00:52:50.380 align:middle line:84% look, we could actually look to see 00:52:50.380 --> 00:52:55.280 align:middle line:84% whether economies are on an efficient or inefficient path. 00:52:55.280 --> 00:52:58.450 align:middle line:84% And I thought I had started this on the reading list. 00:52:58.450 --> 00:53:01.500 align:middle line:90% So anyway, you can read this. 00:53:01.500 --> 00:53:02.538 align:middle line:90% Yeah, yeah, I did. 00:53:02.538 --> 00:53:03.580 align:middle line:90% It's on the reading list. 00:53:03.580 --> 00:53:05.680 align:middle line:84% You should be able to understand it. 00:53:05.680 --> 00:53:06.850 align:middle line:90% This, you won't understand. 00:53:06.850 --> 00:53:08.680 align:middle line:84% It's just explaining what they did. 00:53:08.680 --> 00:53:10.690 align:middle line:84% They didn't find any inefficiency, 00:53:10.690 --> 00:53:13.540 align:middle line:84% but a subsequent person, Geerolf, did. 00:53:13.540 --> 00:53:17.710 align:middle line:84% He adjusted what those guys did to take out land rent, 00:53:17.710 --> 00:53:20.320 align:middle line:84% having profits of entrepreneurs divvied up 00:53:20.320 --> 00:53:22.750 align:middle line:90% into opportunity cost wages. 00:53:22.750 --> 00:53:29.020 align:middle line:84% And he actually finds that Korea, Taiwan, Singapore, 00:53:29.020 --> 00:53:31.420 align:middle line:84% those East Asian growth economies, 00:53:31.420 --> 00:53:35.180 align:middle line:84% with all that capital, they were actually inefficient. 00:53:35.180 --> 00:53:37.450 align:middle line:84% They were accumulating too much capital. 00:53:37.450 --> 00:53:39.480 align:middle line:84% They were beyond the golden rule. 00:53:39.480 --> 00:53:41.290 align:middle line:90% It's really fascinating. 00:53:41.290 --> 00:53:42.070 align:middle line:90% All right. 00:53:42.070 --> 00:53:45.310 align:middle line:84% The other part is what I was alluding to at the beginning. 00:53:45.310 --> 00:53:49.900 align:middle line:84% If you create more and more real financial claims 00:53:49.900 --> 00:53:53.800 align:middle line:84% that can be traded, then fiat money can also be vulnerable. 00:53:53.800 --> 00:53:57.160 align:middle line:84% And you could convert those economies with money 00:53:57.160 --> 00:53:59.000 align:middle line:90% to economies without them. 00:53:59.000 --> 00:54:04.030 align:middle line:84% So when you have Bitcoin and tokens and cryptocurrencies 00:54:04.030 --> 00:54:06.070 align:middle line:84% and especially digital assets, which 00:54:06.070 --> 00:54:08.500 align:middle line:84% are claims on real capital circulating 00:54:08.500 --> 00:54:13.600 align:middle line:84% in the economy as money, you may end up with that model 00:54:13.600 --> 00:54:16.390 align:middle line:84% where there is no valued fiat money. 00:54:16.390 --> 00:54:19.240 align:middle line:84% And again, that would threaten central banks. 00:54:19.240 --> 00:54:21.640 align:middle line:84% But it doesn't mean it's a bad thing. 00:54:21.640 --> 00:54:24.130 align:middle line:84% In our terminology, it could be going 00:54:24.130 --> 00:54:26.860 align:middle line:84% toward a new Pareto-optimal allocation. 00:54:26.860 --> 00:54:30.110 align:middle line:84% Other ways to get money in the model, 00:54:30.110 --> 00:54:32.740 align:middle line:84% you could require it to pay taxes. 00:54:32.740 --> 00:54:34.900 align:middle line:84% So even if there's a last period, 00:54:34.900 --> 00:54:37.990 align:middle line:84% everyone has to pay it in, otherwise 00:54:37.990 --> 00:54:39.680 align:middle line:90% it could have 0 value. 00:54:39.680 --> 00:54:42.190 align:middle line:84% So the very first monetary models 00:54:42.190 --> 00:54:45.670 align:middle line:84% were finite-period models, but in the last period, 00:54:45.670 --> 00:54:48.280 align:middle line:84% people have to pay taxes in money. 00:54:48.280 --> 00:54:51.130 align:middle line:84% Some governor in Ohio got a smart idea which, 00:54:51.130 --> 00:54:53.440 align:middle line:90% is I guess he liked technology. 00:54:53.440 --> 00:54:57.100 align:middle line:84% He is allowing Ohio citizens to pay their state income 00:54:57.100 --> 00:54:59.050 align:middle line:90% taxes in Bitcoin. 00:54:59.050 --> 00:55:01.990 align:middle line:84% Maybe that'll stabilize the price. 00:55:01.990 --> 00:55:06.460 align:middle line:84% And you could require that money be used for goods. 00:55:06.460 --> 00:55:09.800 align:middle line:84% We have not been writing models like that. 00:55:09.800 --> 00:55:10.990 align:middle line:90% This is like fiat. 00:55:10.990 --> 00:55:14.170 align:middle line:84% Literally, by law, you're required to use money. 00:55:14.170 --> 00:55:17.110 align:middle line:84% And you have to hold it and have to have acquired it 00:55:17.110 --> 00:55:19.150 align:middle line:90% earlier in some way. 00:55:19.150 --> 00:55:21.760 align:middle line:84% These are called cash in advance models. 00:55:21.760 --> 00:55:26.500 align:middle line:84% So if you have certain types of goods in the economy that 00:55:26.500 --> 00:55:29.740 align:middle line:84% require cash, and those are valued goods 00:55:29.740 --> 00:55:32.710 align:middle line:84% with a margin utility of infinite at 0, 00:55:32.710 --> 00:55:34.150 align:middle line:84% then the value of cash isn't going 00:55:34.150 --> 00:55:36.940 align:middle line:84% to fall to 0 because if it were, you'd never 00:55:36.940 --> 00:55:38.330 align:middle line:90% be able to buy those goods. 00:55:38.330 --> 00:55:40.420 align:middle line:84% So that's kind of a way of, again, 00:55:40.420 --> 00:55:43.270 align:middle line:84% a mechanical way of making sure the value of money 00:55:43.270 --> 00:55:44.950 align:middle line:90% doesn't go to 0. 00:55:44.950 --> 00:55:48.700 align:middle line:84% Like it or not, I'm exposing you to different modeling 00:55:48.700 --> 00:55:50.650 align:middle line:90% strategies. 00:55:50.650 --> 00:55:54.640 align:middle line:84% So let me come back to the starting point, which 00:55:54.640 --> 00:55:58.610 align:middle line:84% are these agents that go 0, 1, 0 and 1, 0, 1. 00:55:58.610 --> 00:56:01.050 align:middle line:84% This is a spatial model of money. 00:56:01.050 --> 00:56:03.680 align:middle line:84% Why is the money going to be valued here? 00:56:03.680 --> 00:56:07.220 align:middle line:84% Because they're not all together to trade with one 00:56:07.220 --> 00:56:10.700 align:middle line:84% another at some initial date over all these future paths 00:56:10.700 --> 00:56:11.990 align:middle line:90% of consumption. 00:56:11.990 --> 00:56:15.440 align:middle line:84% They can only trade with each other when they meet. 00:56:15.440 --> 00:56:18.710 align:middle line:84% So this is an extreme version of incomplete markets. 00:56:18.710 --> 00:56:21.350 align:middle line:84% And you know what can happen to competitive equilibria 00:56:21.350 --> 00:56:23.180 align:middle line:90% with incomplete markets. 00:56:23.180 --> 00:56:24.860 align:middle line:90% They don't have to be optimal. 00:56:24.860 --> 00:56:29.600 align:middle line:84% Specifically, let's write down the problem of an agent of type 00:56:29.600 --> 00:56:31.460 align:middle line:90% of I. If you can see this. 00:56:31.460 --> 00:56:33.230 align:middle line:90% It's a little bit tiny. 00:56:33.230 --> 00:56:37.550 align:middle line:84% We have discounted utility of consumption. 00:56:37.550 --> 00:56:39.770 align:middle line:84% Consumption and money are non-negative. 00:56:39.770 --> 00:56:42.210 align:middle line:84% And we have this budget equation, 00:56:42.210 --> 00:56:43.610 align:middle line:90% which reads like this-- 00:56:43.610 --> 00:56:46.700 align:middle line:84% they have some endowment, maybe positive. 00:56:46.700 --> 00:56:49.670 align:middle line:84% This is now the dollar price, by the way, 00:56:49.670 --> 00:56:53.870 align:middle line:90% of goods, the usual price. 00:56:53.870 --> 00:56:56.540 align:middle line:84% They have the valuation of their endowment 00:56:56.540 --> 00:56:59.390 align:middle line:84% plus the money they brought in from the previous period. 00:56:59.390 --> 00:57:01.580 align:middle line:84% They may have to pay some lump-sum taxes. 00:57:01.580 --> 00:57:05.840 align:middle line:84% But then the residual can be used either to buy consumption 00:57:05.840 --> 00:57:08.970 align:middle line:84% or to take money over to the next state. 00:57:08.970 --> 00:57:11.480 align:middle line:84% So again, this is like sources and uses. 00:57:11.480 --> 00:57:13.730 align:middle line:90% And what's the solution to this? 00:57:13.730 --> 00:57:16.880 align:middle line:84% Solution, by the way, by manipulating the money. 00:57:16.880 --> 00:57:20.000 align:middle line:84% The way they determine how much they're eating today or eating 00:57:20.000 --> 00:57:23.180 align:middle line:84% tomorrow is to either cash out the money 00:57:23.180 --> 00:57:26.420 align:middle line:84% or acquire it and store it and carry it over. 00:57:26.420 --> 00:57:30.680 align:middle line:84% So not too surprisingly, you might expect, and it's true, 00:57:30.680 --> 00:57:34.400 align:middle line:84% that the marginal-- let me write it down here-- 00:57:34.400 --> 00:57:38.390 align:middle line:84% that with one exception, the marginal rate of substitution 00:57:38.390 --> 00:57:42.830 align:middle line:84% in consumption, of consumption at t minus 1 for consumption 00:57:42.830 --> 00:57:46.680 align:middle line:84% at t, that ought to equal the price ratio. 00:57:46.680 --> 00:57:51.410 align:middle line:84% So remember your basic lectures, consumer theory, two periods, 00:57:51.410 --> 00:57:53.900 align:middle line:84% money is a way to carry value over 00:57:53.900 --> 00:57:55.280 align:middle line:90% from one period to the next. 00:57:55.280 --> 00:57:58.700 align:middle line:84% It's like borrowing and lending, except that it 00:57:58.700 --> 00:58:02.760 align:middle line:84% appreciates or depreciates depending on the price level. 00:58:02.760 --> 00:58:05.540 align:middle line:84% So you would expect this equality to hold, 00:58:05.540 --> 00:58:09.002 align:middle line:84% that the tangency involves the marginal rate of substitution 00:58:09.002 --> 00:58:10.460 align:middle line:84% and consumption on the indifference 00:58:10.460 --> 00:58:14.540 align:middle line:84% curve equal to the price ratio implicit in the budget model. 00:58:14.540 --> 00:58:16.470 align:middle line:84% Now, I was covering up the inequality 00:58:16.470 --> 00:58:20.210 align:middle line:84% because it could be that you're holding money. 00:58:20.210 --> 00:58:22.040 align:middle line:84% The marginal utility of consumption 00:58:22.040 --> 00:58:24.680 align:middle line:84% today is greater than the marginal utility tomorrow 00:58:24.680 --> 00:58:25.760 align:middle line:90% discounted. 00:58:25.760 --> 00:58:27.770 align:middle line:90% So you'd like to eat more today. 00:58:27.770 --> 00:58:31.100 align:middle line:84% So you start driving that money down, and the money hits 0, 00:58:31.100 --> 00:58:32.600 align:middle line:90% and you can't go negative. 00:58:32.600 --> 00:58:35.510 align:middle line:84% And still, the margin utility today 00:58:35.510 --> 00:58:37.620 align:middle line:84% is greater than the margin utility tomorrow. 00:58:37.620 --> 00:58:40.310 align:middle line:84% So these guys actually-- where did it go? 00:58:40.310 --> 00:58:42.740 align:middle line:84% These guys here experienced that. 00:58:42.740 --> 00:58:44.450 align:middle line:90% They start at 1. 00:58:44.450 --> 00:58:46.040 align:middle line:90% They acquire money. 00:58:46.040 --> 00:58:47.900 align:middle line:84% They're going to have 0 next period. 00:58:47.900 --> 00:58:51.170 align:middle line:84% They start dumping the money, but they run out, 00:58:51.170 --> 00:58:53.160 align:middle line:90% and they can't go negative. 00:58:53.160 --> 00:58:55.490 align:middle line:84% So for them, they're at a corner. 00:58:55.490 --> 00:58:58.040 align:middle line:84% Guys that come in with money and spend it, 00:58:58.040 --> 00:59:00.950 align:middle line:84% because otherwise they have no endowment at all, 00:59:00.950 --> 00:59:03.110 align:middle line:84% are nevertheless not going to have enough money. 00:59:03.110 --> 00:59:05.210 align:middle line:84% So we can look for an equilibrium 00:59:05.210 --> 00:59:08.690 align:middle line:84% where for some periods and some types of agents, 00:59:08.690 --> 00:59:11.120 align:middle line:84% this is at a strict inequality, namely 00:59:11.120 --> 00:59:12.560 align:middle line:90% the guys that are hitting 0. 00:59:12.560 --> 00:59:15.140 align:middle line:90% And the other ones 0 endowment. 00:59:15.140 --> 00:59:17.150 align:middle line:84% And the other ones, it is that equality. 00:59:17.150 --> 00:59:19.710 align:middle line:90% Let me draw you the picture. 00:59:19.710 --> 00:59:22.730 align:middle line:84% So even though it's an infinite horizon economy, 00:59:22.730 --> 00:59:25.370 align:middle line:84% we can reduce it to a two-period economy 00:59:25.370 --> 00:59:29.570 align:middle line:84% and guess what the steady state equilibrium is 00:59:29.570 --> 00:59:31.500 align:middle line:90% going to look like. 00:59:31.500 --> 00:59:35.640 align:middle line:84% Guys that start with 1 unit would be here. 00:59:35.640 --> 00:59:37.290 align:middle line:84% If they just ate their endowment, 00:59:37.290 --> 00:59:40.950 align:middle line:84% they would have consumption of 1 today and none tomorrow. 00:59:40.950 --> 00:59:44.340 align:middle line:84% And then, moving along the budget line, 00:59:44.340 --> 00:59:48.480 align:middle line:84% at constant prices, they get money today, 00:59:48.480 --> 00:59:52.410 align:middle line:84% giving up consumption good today, and get money tomorrow. 00:59:52.410 --> 00:59:55.200 align:middle line:84% If the price level of money tomorrow, 00:59:55.200 --> 00:59:57.900 align:middle line:84% price of goods tomorrow is the price of goods today, 00:59:57.900 --> 01:00:01.510 align:middle line:84% then they're moving right along this budget line. 01:00:01.510 --> 01:00:03.572 align:middle line:90% And where would they end up? 01:00:03.572 --> 01:00:05.030 align:middle line:84% Well, here's an indifference curve. 01:00:05.030 --> 01:00:06.260 align:middle line:90% Here's a tangency. 01:00:06.260 --> 01:00:07.890 align:middle line:90% It's on the budget line. 01:00:07.890 --> 01:00:09.890 align:middle line:84% Now, how did I know it would be there? 01:00:09.890 --> 01:00:13.260 align:middle line:90% Namely, look at this expression. 01:00:13.260 --> 01:00:16.490 align:middle line:84% This is the marginal utility at t minus 1 to t 01:00:16.490 --> 01:00:18.830 align:middle line:84% or the marginal utility at t to t plus 1. 01:00:18.830 --> 01:00:20.360 align:middle line:90% There's a beta down here. 01:00:20.360 --> 01:00:22.130 align:middle line:90% Tomorrow doesn't count so much. 01:00:22.130 --> 01:00:24.110 align:middle line:90% Beta is between 0 and 1. 01:00:24.110 --> 01:00:26.630 align:middle line:84% So the utility, margin utility tomorrow, 01:00:26.630 --> 01:00:29.510 align:middle line:84% is always less than the margin utility today. 01:00:29.510 --> 01:00:31.820 align:middle line:84% The only thing that could compensate for it 01:00:31.820 --> 01:00:33.050 align:middle line:90% is the price movement. 01:00:33.050 --> 01:00:35.180 align:middle line:84% But if we start with constant prices, 01:00:35.180 --> 01:00:39.200 align:middle line:84% then this expression is only going 01:00:39.200 --> 01:00:43.040 align:middle line:84% to hold if we're able to adjust consumption. 01:00:43.040 --> 01:00:46.220 align:middle line:84% We want to get the margin utility up. 01:00:46.220 --> 01:00:50.360 align:middle line:84% Consumption at t is greater than at t plus 1 01:00:50.360 --> 01:00:54.170 align:middle line:84% in order to get the margin utility up to the point 01:00:54.170 --> 01:00:55.940 align:middle line:84% that the marginal rate of substitution 01:00:55.940 --> 01:00:59.980 align:middle line:84% is equal to the constant price ratio. 01:00:59.980 --> 01:01:04.600 align:middle line:84% Now, is this a good thing or a bad thing? 01:01:04.600 --> 01:01:07.600 align:middle line:84% Well, it is going to be a monetary equilibrium. 01:01:07.600 --> 01:01:09.700 align:middle line:90% The guess is good. 01:01:09.700 --> 01:01:14.100 align:middle line:84% So the path would be guys with 1 are 01:01:14.100 --> 01:01:19.440 align:middle line:84% going to go to c star and then c double star, then c star, then 01:01:19.440 --> 01:01:24.570 align:middle line:84% c double star, with money moving in the opposite direction 01:01:24.570 --> 01:01:27.780 align:middle line:84% from something positive, when they have 1 they acquire it, 01:01:27.780 --> 01:01:32.400 align:middle line:84% to 0, hitting the corner, back to positive amount again. 01:01:32.400 --> 01:01:35.190 align:middle line:84% So money is going to alternate over time 01:01:35.190 --> 01:01:36.390 align:middle line:90% at the individual level. 01:01:36.390 --> 01:01:38.370 align:middle line:84% Consumption is going to alternate over time. 01:01:38.370 --> 01:01:42.120 align:middle line:84% The money is going to have value at a constant price. 01:01:42.120 --> 01:01:45.280 align:middle line:84% But unlike the overlapping generations model, 01:01:45.280 --> 01:01:48.810 align:middle line:84% this one is not a Pareto-optimal allocation, 01:01:48.810 --> 01:01:50.280 align:middle line:90% even though we have money. 01:01:50.280 --> 01:01:52.200 align:middle line:84% You actually know the answer already. 01:01:52.200 --> 01:01:54.360 align:middle line:90% Let me remind you. 01:01:54.360 --> 01:01:56.790 align:middle line:84% How do we determine a Pareto optimum? 01:01:56.790 --> 01:02:00.240 align:middle line:84% Well, let's all the guys that start with, say, 0 01:02:00.240 --> 01:02:02.880 align:middle line:84% be called A types, and all the guys 01:02:02.880 --> 01:02:04.770 align:middle line:84% that start with 1 unit of the consumption 01:02:04.770 --> 01:02:06.360 align:middle line:90% are going to be called B types. 01:02:06.360 --> 01:02:09.142 align:middle line:84% And then, maximize the weighted sums of utility. 01:02:09.142 --> 01:02:10.350 align:middle line:90% We usually have lambdas here. 01:02:10.350 --> 01:02:11.160 align:middle line:90% I apologize. 01:02:11.160 --> 01:02:12.660 align:middle line:90% Now, they're w's. 01:02:12.660 --> 01:02:16.230 align:middle line:84% This is the weight on type A. wB is the weight on type B. 01:02:16.230 --> 01:02:19.920 align:middle line:84% We maximize the weighted sums of expected-- 01:02:19.920 --> 01:02:23.550 align:middle line:84% there's nothing random-- discounted utility 01:02:23.550 --> 01:02:24.900 align:middle line:90% for both of them. 01:02:24.900 --> 01:02:26.390 align:middle line:90% We get a first-order condition. 01:02:26.390 --> 01:02:27.390 align:middle line:90% You remember this stuff? 01:02:27.390 --> 01:02:28.480 align:middle line:90% We've done this before. 01:02:28.480 --> 01:02:30.420 align:middle line:90% This is the Pareto problem. 01:02:30.420 --> 01:02:34.920 align:middle line:84% Weighted margin utilities are going to be equated over agents 01:02:34.920 --> 01:02:37.740 align:middle line:84% and equal to a constant Lagrange multiplier. 01:02:37.740 --> 01:02:39.580 align:middle line:90% You've seen this before. 01:02:39.580 --> 01:02:41.490 align:middle line:84% This is a new application, that's all. 01:02:41.490 --> 01:02:44.400 align:middle line:84% So the marginal rate, solving this equation, 01:02:44.400 --> 01:02:46.980 align:middle line:84% the marginal rates of substitution over the agents 01:02:46.980 --> 01:02:48.630 align:middle line:90% have to be the same. 01:02:48.630 --> 01:02:50.410 align:middle line:84% But does it pin down the allocation? 01:02:50.410 --> 01:02:50.910 align:middle line:90% No. 01:02:50.910 --> 01:02:52.050 align:middle line:90% We're in an Edgeworth box. 01:02:52.050 --> 01:02:55.740 align:middle line:84% There's one good in the aggregate available today, 01:02:55.740 --> 01:02:58.530 align:middle line:84% one good in the aggregate available tomorrow. 01:02:58.530 --> 01:03:01.620 align:middle line:84% We know the Pareto optimum has to give 01:03:01.620 --> 01:03:04.790 align:middle line:90% all the guys equal consumption. 01:03:04.790 --> 01:03:07.970 align:middle line:84% You can satisfy this by having type A having 01:03:07.970 --> 01:03:11.810 align:middle line:84% constant consumption, type B having constant consumption, 01:03:11.810 --> 01:03:14.870 align:middle line:84% but those consumptions don't have to be equal to each other 01:03:14.870 --> 01:03:16.880 align:middle line:84% because things are going to cancel out 01:03:16.880 --> 01:03:19.640 align:middle line:84% of the numerator or the denominator. 01:03:19.640 --> 01:03:23.710 align:middle line:84% So this picture, marginal rate of substitution 01:03:23.710 --> 01:03:28.420 align:middle line:84% for day t and day tau is equal to each other for agents 01:03:28.420 --> 01:03:31.500 align:middle line:84% type A and type B. But to review, right, 01:03:31.500 --> 01:03:33.500 align:middle line:84% you remember this lecture when we did the Pareto 01:03:33.500 --> 01:03:36.120 align:middle line:90% optimum and your Edgeworth box? 01:03:36.120 --> 01:03:38.160 align:middle line:84% You can be anywhere on the contract curve, 01:03:38.160 --> 01:03:40.450 align:middle line:90% and it's all Pareto optimal. 01:03:40.450 --> 01:03:43.780 align:middle line:84% Anything that's off the diagonal is not Pareto optimal 01:03:43.780 --> 01:03:45.400 align:middle line:90% in this environment. 01:03:45.400 --> 01:03:47.770 align:middle line:84% And this one is off the diagonal. 01:03:47.770 --> 01:03:49.540 align:middle line:90% So it's not Pareto optimal. 01:03:49.540 --> 01:03:50.950 align:middle line:90% Where do you want to go? 01:03:50.950 --> 01:03:52.930 align:middle line:90% You kind of want to go here. 01:03:52.930 --> 01:03:55.030 align:middle line:90% And how would you engineer that? 01:03:55.030 --> 01:03:58.090 align:middle line:84% You have to generate a deflation so 01:03:58.090 --> 01:04:01.720 align:middle line:84% that they're getting a positive real rate of return on money. 01:04:01.720 --> 01:04:04.420 align:middle line:84% That's going to make them want to hold more money 01:04:04.420 --> 01:04:06.310 align:middle line:90% and eat less today. 01:04:06.310 --> 01:04:08.500 align:middle line:84% And that's going to put them at a tangency 01:04:08.500 --> 01:04:10.880 align:middle line:84% up here where they have equal consumption. 01:04:10.880 --> 01:04:13.370 align:middle line:84% So there is a Pareto-optimal allocation. 01:04:13.370 --> 01:04:16.120 align:middle line:84% It's a monetary equilibrium, but it's 01:04:16.120 --> 01:04:19.780 align:middle line:84% an equilibrium with an activist monetary policy that's 01:04:19.780 --> 01:04:24.400 align:middle line:84% withdrawing enough currency out of the system in lump-sum taxes 01:04:24.400 --> 01:04:27.050 align:middle line:90% to create a deflation. 01:04:27.050 --> 01:04:28.370 align:middle line:90% Deflation isn't bad here. 01:04:28.370 --> 01:04:31.640 align:middle line:84% It means that the purchasing power of money is going up. 01:04:31.640 --> 01:04:36.590 align:middle line:84% So that's all the rest of these slides I don't have time for. 01:04:36.590 --> 01:04:40.220 align:middle line:84% You are able to achieve these Pareto optimal allocations 01:04:40.220 --> 01:04:44.060 align:middle line:84% but only with an activist monetary policy, which 01:04:44.060 --> 01:04:47.120 align:middle line:84% is deliberately manipulating the amount of currency 01:04:47.120 --> 01:04:49.310 align:middle line:90% in the system. 01:04:49.310 --> 01:04:49.990 align:middle line:90% Questions. 01:04:49.990 --> 01:04:52.770 align:middle line:90% 01:04:52.770 --> 01:04:55.650 align:middle line:84% So welcome to the land of monetary economics. 01:04:55.650 --> 01:04:57.220 align:middle line:90% It's a lot of fun. 01:04:57.220 --> 01:04:59.370 align:middle line:84% So now you're able to see in this last lecture 01:04:59.370 --> 01:05:01.740 align:middle line:84% how real financial markets are linked 01:05:01.740 --> 01:05:06.960 align:middle line:84% to money markets, the role of cryptocurrency, controversies 01:05:06.960 --> 01:05:10.080 align:middle line:84% involving its price, and how we as economists can think 01:05:10.080 --> 01:05:12.990 align:middle line:84% about these values, and actually how we would 01:05:12.990 --> 01:05:14.710 align:middle line:90% recommend monetary policy. 01:05:14.710 --> 01:05:16.860 align:middle line:84% And I will add that if there was cryptocurrency, 01:05:16.860 --> 01:05:20.100 align:middle line:84% you would want an activist amount of manipulation 01:05:20.100 --> 01:05:22.110 align:middle line:84% of the amount of cryptocurrency in the system 01:05:22.110 --> 01:05:25.790 align:middle line:84% for the same reason that I just described to you. 01:05:25.790 --> 01:05:34.000 align:middle line:90%