WEBVTT 00:01.590 --> 00:05.540 Prof: So we have begun by talking about general 00:05.541 --> 00:08.901 equilibrium, and this was the background to 00:08.897 --> 00:11.877 the course so that you're all familiar. 00:11.880 --> 00:15.280 Those of you who haven't taken freshman economics or 00:15.281 --> 00:18.681 intermediate economics, what's the background that's 00:18.682 --> 00:20.552 required for the course? 00:20.550 --> 00:25.500 And the fact that general equilibrium is a part of finance 00:25.499 --> 00:27.349 is, as I say, a bit of an 00:27.349 --> 00:31.019 innovation because most finance courses are taught entirely 00:31.021 --> 00:33.301 independently of economic theory. 00:33.300 --> 00:37.330 But the two greatest Yale economists and two greatest Yale 00:37.333 --> 00:41.653 financial economists had finance and economics integrated, 00:41.650 --> 00:43.970 and I believe that's the only right way of looking at the 00:43.966 --> 00:44.376 problem. 00:44.380 --> 00:47.000 And as I said at the very beginning, I think the events of 00:47.003 --> 00:49.123 the last few years have confirmed that view. 00:49.120 --> 00:52.990 So let's take up the question today of why is the free market 00:52.994 --> 00:54.354 supposedly so good. 00:54.350 --> 00:58.930 So we worked out two examples last time, and the two examples 00:58.934 --> 01:03.294 are very similar to the first problem set that you did. 01:03.289 --> 01:06.929 In the first example you've got two agents A and B, 01:06.930 --> 01:10.360 and whenever we write agents A and B we really mean a million 01:10.361 --> 01:13.451 agents just like A and a million agents just like B. 01:13.450 --> 01:15.770 The heart of perfect competition in the economy is 01:15.771 --> 01:19.441 that there's lots of people, and so we can't--yeah, 01:19.441 --> 01:24.851 I think from now on hand them in at the end of class. 01:24.849 --> 01:30.029 So we mean a million people of type A and a million people type 01:30.025 --> 01:30.355 B. 01:30.360 --> 01:32.970 So the utility function of A, the welfare function of A is 01:32.971 --> 01:35.311 written there, and the welfare function of B 01:35.312 --> 01:37.902 is written and they each begin with endowments. 01:37.900 --> 01:40.600 So when these two million people come together they're 01:40.598 --> 01:43.858 going to be doing the same thing that you saw in the class on the 01:43.857 --> 01:44.567 first day. 01:44.569 --> 01:47.549 They're going to be haggling, and arguing, 01:47.550 --> 01:49.930 and the buyers are going to always say "the stuff's not 01:49.925 --> 01:52.215 worth very much and why should they pay you so much," 01:52.218 --> 01:54.228 and the sellers are going to say "it's worth a 01:54.232 --> 01:56.362 tremendous amount and they should pay even more," 01:56.364 --> 01:58.584 and eventually they're going to come to prices. 01:58.580 --> 02:02.190 And we discovered last time that the way of describing what 02:02.188 --> 02:05.358 prices and what final allocations they come to is by 02:05.361 --> 02:07.851 writing down a system of equations, 02:07.849 --> 02:11.379 and you did that all in your homework and we came up with 02:11.375 --> 02:12.315 this outcome. 02:12.318 --> 02:16.168 So similarly we did another problem in the last class and we 02:16.165 --> 02:19.945 wrote down different welfare functions or utility functions 02:19.947 --> 02:23.987 and different endowments and again we got the equilibrium. 02:23.990 --> 02:27.340 So we said that one of the amazing things is that these 02:27.342 --> 02:30.452 system of equations, the six equations for each 02:30.452 --> 02:34.142 economy that you solved in class always have a solution. 02:34.139 --> 02:38.559 So the people who discovered that discovered it at the Cowles 02:38.561 --> 02:39.521 Foundation. 02:39.520 --> 02:43.160 They were Ken Arrow, Gerard Debreu who was a Yale 02:43.157 --> 02:46.457 assistant professor, Ken Arrow who was visiting the 02:46.462 --> 02:49.592 Cowles Foundation in Chicago but was a Stanford professor, 02:49.590 --> 02:52.800 and Lionel McKenzie who actually taught at Rochester. 02:52.800 --> 02:55.730 The economic equations always have a solution. 02:55.729 --> 02:58.629 So what's so good about that economic solution? 02:58.628 --> 03:02.098 Adam Smith talked about the invisible hand, 03:02.100 --> 03:05.820 but there was no mathematics in Adam Smith. 03:05.818 --> 03:09.388 So why do we think equilibrium is such a good thing? 03:09.389 --> 03:12.509 After all, you saw in the example that there were three 03:12.508 --> 03:15.628 pairs of people who forlornly couldn't trade at all. 03:15.628 --> 03:18.978 So clearly not everybody gets to trade, discrimination 03:18.978 --> 03:19.608 happens. 03:19.610 --> 03:20.580 Some people get the stuff. 03:20.580 --> 03:22.320 Other people don't get the stuff. 03:22.318 --> 03:30.398 Well, the first approach of economists was that equilibrium 03:30.395 --> 03:34.985 maximizes the sum of utilities. 03:34.990 --> 03:43.280 So let's see how that works in this example. 03:43.280 --> 03:47.590 So this example's very special because everybody thinks that 03:47.591 --> 03:51.101 the good Y has constant marginal utility of 1. 03:51.098 --> 03:54.628 So that's going to play a big role in our optimization in 03:54.631 --> 03:56.651 maximizing the sum of utility. 03:56.650 --> 04:08.980 So if you want to maximize the sum of utilities you have to 04:32.795 --> 04:45.975 84 which is the sum of 4 and 80 and Y^(A) Y^(B) = 6,000. 04:45.980 --> 04:49.100 So the claim is that if all these two million people meet in 04:49.103 --> 04:51.913 a room like this and start shouting at each other, 04:51.910 --> 04:54.180 and trading and offering deals and stuff, 04:54.180 --> 04:57.420 it's going to come down to a final allocation which we've 04:57.416 --> 05:00.936 written over there were A gets 77 of the first good and B gets 05:00.944 --> 05:04.074 7 of the first good, and that final allocation is 05:04.069 --> 05:06.469 going to maximize the sum of utilities. 05:06.470 --> 05:08.120 So why should that be the case? 05:08.120 --> 05:09.830 In fact in this example it is the case. 05:09.829 --> 05:11.429 Well why is it the case? 05:11.430 --> 05:15.420 Because there's constant marginal utility of Y, 05:15.420 --> 05:19.560 so we don't have to worry about the Y allocation at all because 05:19.564 --> 05:23.244 if you increase Y^(A) and decrease Y^(B) they still have 05:23.242 --> 05:24.782 to add up to 6,000. 05:24.778 --> 05:28.398 You're going to be lowering A's utility and raising B's utility 05:28.403 --> 05:31.563 by the same amount so you won't have changed the sum of 05:31.560 --> 05:32.380 utilities. 05:32.379 --> 05:35.539 So the Y's are totally irrelevant to this maximizing 05:35.543 --> 05:36.973 the sum of utilities. 05:36.970 --> 05:40.540 So all we have to do is make sure that the X allocation, 05:40.538 --> 05:43.588 maximize the sum of utilities for the good X. 05:43.589 --> 05:48.949 So that you can see must be the case because we have to maximize 05:48.954 --> 05:52.964 the sum of two people's utilities or two million 05:52.956 --> 05:54.146 utilities. 05:54.149 --> 05:57.649 You can always plug in X^(B) as a function of X^(A), 05:57.646 --> 06:00.936 namely 84 [clarification: X^(B) = 84 - X^(A)]. 06:00.939 --> 06:07.219 So let's write it as a function of X^(A), 06:07.220 --> 06:12.130 and we can replace this maximization by recognizing that 06:12.127 --> 06:15.337 X^(B) is just a function of X^(A). 06:15.338 --> 06:19.578 In fact, it's a very special function where the derivative of 06:19.579 --> 06:22.949 X^(B) with respect to X^(A)-- so let's call the function, 06:22.949 --> 06:25.219 we'll call it that to denote that it's a function-- 06:25.220 --> 06:28.760 happens to equal -1. 06:28.759 --> 06:33.379 So we want to maximize this over X^(A) taking into account 06:33.377 --> 06:36.777 that X^(B) is just a function of X^(A), 06:36.779 --> 06:40.949 because to maximize you have to keep feasible. 06:40.949 --> 06:44.319 So we've now ignored all the constraints. 06:44.319 --> 06:47.969 Constraints have now disappeared provided we keep 06:47.971 --> 06:52.081 track of the fact that X^(B) is a function of X^(A). 06:52.079 --> 06:58.609 So why is it the case that the equilibrium 77 and 7 maximizes 06:58.608 --> 06:59.368 this? 06:59.370 --> 07:02.220 Well, the key is diminishing marginal utility. 07:02.220 --> 07:06.180 We know that the utility function for-- 07:06.180 --> 07:11.760 here's X--utility function for A is quadratic so it looks 07:11.764 --> 07:15.934 something like that, and the utility function for B 07:15.927 --> 07:17.917 also looks something like that. 07:17.920 --> 07:22.640 And so if you look at this function entirely as a function 07:22.641 --> 07:26.041 of A it's also going to look like that. 07:26.040 --> 07:40.470 So by diminishing marginal utility, the function, 07:40.468 --> 07:50.688 the sum of utilities is concave. 07:50.690 --> 07:52.980 Concave, remember, is a picture like that. 07:52.980 --> 07:55.330 So what is the essence of a picture like that? 07:55.329 --> 08:06.299 It means that setting derivative equal to zero implies 08:06.300 --> 08:09.820 you are at max. 08:09.819 --> 08:11.819 So that's not true of any function. 08:11.819 --> 08:14.059 If you have a function like that and then it goes up like 08:14.055 --> 08:16.205 that you could have the derivative of zero here and not 08:16.211 --> 08:18.801 really be at the max, but because there's diminishing 08:18.800 --> 08:21.850 marginal utility more and more of X does less and less good so 08:21.850 --> 08:22.700 it turns down. 08:22.699 --> 08:26.679 So all you have to do is figure out that the derivative is zero, 08:26.680 --> 08:30.280 which it is at this point, and then you know you're at the 08:30.281 --> 08:31.041 maximum. 08:31.040 --> 08:37.330 But that at X^(A) = 77 has to be the case because if you plug 08:37.330 --> 08:43.310 in X^(A) = 77 and you take the derivative of all this with 08:43.306 --> 08:47.706 respect to X^(A) of this whole thing, 08:47.710 --> 08:50.900 taking into account as before that X^(B) is a function of 08:50.897 --> 08:56.297 X^(A), you just get (100 - X^(A)) now 08:56.301 --> 09:05.601 it's just going to be (30 - X^(B)) times the derivative of 09:05.595 --> 09:10.645 X^(B) with respect to X^(A). 09:10.649 --> 09:11.249 Why is that? 09:11.250 --> 09:13.400 Because you know the chain rule, if I differentiate 09:13.404 --> 09:16.164 everything with respect to X^(A) and X^(B) is a function of X^(A) 09:16.163 --> 09:18.793 it's just 30 because I'm taking the derivative with respect to 09:18.792 --> 09:19.312 X^(B). 09:19.308 --> 09:23.338 So it's 30 times the derivative of X^(B) with respect to X^(A) 09:23.340 --> 09:27.040 minus the derivative with respect to X^(B) is X^(B) times 09:27.039 --> 09:30.409 the derivative of X^(B) with respect to X^(A), 09:30.409 --> 09:31.669 so it's that. 09:31.668 --> 09:39.348 But if you plug in X^(A) = 77 and X^(B) = 7, 09:39.350 --> 09:48.550 DX^(B), DX^(A) is -1 you just get 23 - 23 = 0, 09:48.548 --> 09:51.118 because the marginal utility of A is equal to the marginal 09:51.118 --> 09:52.468 utility of B at equilibrium. 09:52.470 --> 09:55.620 The sum of utilities is maximized, and that's the end of 09:55.621 --> 09:56.311 the story. 09:56.308 --> 09:59.268 So that you noticed in the problem set, something like 09:59.269 --> 09:59.659 that. 09:59.658 --> 10:01.168 You may not have given exactly this argument. 10:01.168 --> 10:03.988 That's what you were supposed to have sort of discovered in 10:03.990 --> 10:06.960 doing the problem set and now we've confirmed what you already 10:06.957 --> 10:07.977 pretty much knew. 10:07.980 --> 10:13.610 So are there any questions about that? 10:13.610 --> 10:16.480 It's just the simple generalization of our football 10:16.477 --> 10:19.577 ticket example where the football tickets end up in the 10:19.576 --> 10:22.326 hands of the people who want them the most, 10:22.330 --> 10:24.410 and A happens to like a lot of tickets, 10:24.408 --> 10:27.598 not just one, and so A's going to keep buying 10:27.601 --> 10:30.431 tickets until he's bought 77 of them. 10:30.428 --> 10:33.818 And by then the next football ticket is worth less than 23 to 10:33.820 --> 10:34.160 him. 10:34.158 --> 10:36.268 B, she likes football tickets a little less. 10:36.269 --> 10:38.049 She's not going to buy as many. 10:38.048 --> 10:42.458 By the time she gets down to 7 football tickets she's going to 10:42.462 --> 10:45.722 think--so she hasn't bought nearly as many. 10:45.720 --> 10:48.990 When she's gotten to 7 football tickets she's going to think the 10:48.989 --> 10:50.649 next one's worth less than 23. 10:50.649 --> 10:54.989 And since the price is 23 they're going to both stop 10:54.985 --> 10:57.125 there, but they're each going to stop 10:57.125 --> 11:00.055 at the same point where the last ticket is worth the same to each 11:00.058 --> 11:02.128 of them, and all the previous football 11:02.125 --> 11:04.785 tickets each bought is worth more to each of them. 11:04.788 --> 11:07.538 So there's no way of rearranging the tickets because 11:07.542 --> 11:10.842 if you did you'd have to take a football ticket that was worth 11:10.835 --> 11:13.855 more than 23 from one of them and give that ticket to the 11:13.856 --> 11:17.416 other one who by then would think it was worth less than 23. 11:17.419 --> 11:20.989 So that's the argument. 11:20.990 --> 11:21.690 Okay. 11:21.690 --> 11:25.680 So economists were beside themselves with their brilliance 11:25.683 --> 11:29.963 in having proved this theorem and given a mathematical form to 11:29.957 --> 11:31.567 the invisible hand. 11:31.570 --> 11:34.400 Whenever you take a verbal argument like Adam Smith and 11:34.403 --> 11:37.343 turn it into a mathematical argument you've understood it 11:37.341 --> 11:37.921 better. 11:37.918 --> 11:41.288 You understand why it's true and you also start to understand 11:41.293 --> 11:44.053 why in some circumstances it might not be true. 11:44.048 --> 11:46.898 So for 50 years, let's say between 30 and 50 11:46.899 --> 11:50.539 years this was the fundamental argument of economics and 11:50.544 --> 11:54.064 economic equilibrium that equilibrium was a good thing 11:54.057 --> 11:57.237 because it maximized the sum of utilities. 11:57.240 --> 12:00.220 But then little by little, starting with Irving Fisher and 12:00.220 --> 12:03.410 a bunch of people at the same time and a little bit later, 12:03.408 --> 12:06.178 so Hicks and Samuelson are famous for this, 12:06.178 --> 12:09.948 they began to wonder what kind of crazy utility functions are 12:09.947 --> 12:12.957 these where there's some mysterious good that has 12:12.961 --> 12:15.161 constant marginal utility of 1. 12:15.158 --> 12:18.808 So in modern terms how could you justify it? 12:18.808 --> 12:21.528 You could say, "Well, maybe there's a 12:21.532 --> 12:25.652 machine where we can calibrate exactly how happy people are and 12:25.649 --> 12:27.109 we can measure it. 12:27.110 --> 12:32.170 So for example, we can measure how aroused 12:32.166 --> 12:34.136 their skin is. 12:34.139 --> 12:39.549 The skin texture changes as you become less and more aroused. 12:39.549 --> 12:40.909 Maybe we could measure that. 12:40.908 --> 12:43.218 With MRIs we can measure brain waves. 12:43.220 --> 12:45.400 Maybe you can calibrate how many brain waves are spinning 12:45.404 --> 12:46.814 around and how happy somebody is, 12:46.808 --> 12:49.318 and maybe there's some mysterious good like food that 12:49.317 --> 12:51.147 has constant marginal utility." 12:51.149 --> 12:54.249 Well, that doesn't seem very persuasive to me and it wasn't 12:54.253 --> 12:56.933 to the economists of the beginning of the twentieth 12:56.928 --> 12:57.568 century. 12:57.570 --> 13:00.630 They all argued, starting with Irving Fisher, 13:00.629 --> 13:04.369 that it made no sense to think that we could actually measure 13:04.365 --> 13:06.675 utility, and more than that it made even 13:06.682 --> 13:09.692 less sense there'd be some good that had constant marginal 13:09.686 --> 13:10.526 utility of 1. 13:10.528 --> 13:13.898 So Fisher put everything on a symmetric footing and said, 13:13.898 --> 13:16.488 "Let's think of the two goods, X and Y, 13:16.485 --> 13:18.285 as more or less symmetric. 13:18.288 --> 13:20.838 I mean, there's no reason why one has to have some special 13:20.837 --> 13:21.417 role." 13:21.418 --> 13:23.318 So let's look at this equilibrium which we got the 13:23.317 --> 13:25.407 same way, and you did this on a problem set solving for 13:25.409 --> 13:26.029 equilibrium. 13:26.028 --> 13:28.408 Does this maximize the sum of utilities? 13:28.408 --> 13:32.638 Well, the answer is if it did maximize the sum of utilities at 13:32.639 --> 13:37.009 the final allocation we couldn't gain anything by switching, 13:37.009 --> 13:40.479 reducing 9 fifths a little bit and increasing 6 fifths a little 13:40.482 --> 13:40.822 bit. 13:40.820 --> 13:43.710 But what is the sum of utilities? 13:43.710 --> 13:53.410 So if you write W^(A) W^(B) it's going to be for one thing 13:53.410 --> 14:01.070 (3 fourths log X^(A)) (2 thirds log X^(B)). 14:01.070 --> 14:06.690 So let's look at the marginal utility to X^(A)--of good A. 14:06.690 --> 14:11.850 That's 3 fourths, so marginal utility of A,X 14:11.846 --> 14:18.556 marginal utility of B,X = 3 fourths times 1 over 9 fifths 14:18.561 --> 14:21.201 which is 5 ninths. 14:21.200 --> 14:28.290 That equals 15 over 36, which is 5 over 18 if I've done 14:28.292 --> 14:30.002 that right. 14:30.000 --> 14:34.760 It doesn't sound right. 14:34.759 --> 14:37.559 Student: It's 5 over 12. 14:37.559 --> 14:38.489 Prof: 5 over 12. 14:38.490 --> 14:39.280 Thank you. 14:39.279 --> 14:41.039 That sounds better. 14:41.038 --> 14:47.878 But let's look at--so this is the margin utility of X^(A) is 5 14:47.876 --> 14:49.106 twelfths. 14:49.110 --> 14:53.780 What's the marginal utility to B of X? 14:53.779 --> 14:59.279 Well, it equals 2 thirds times 1 over 6 fifths, 14:59.278 --> 15:03.818 which is 5 sixths, which is 10 over 18, 15:03.820 --> 15:06.450 which is 5 over 9. 15:06.450 --> 15:08.470 So these two numbers are different. 15:08.470 --> 15:10.930 The marginal utility to B is bigger than the margin 15:10.932 --> 15:13.942 utility--of X--is bigger than the margin utility to A of X. 15:13.940 --> 15:16.370 So clearly the final equilibrium allocation doesn't 15:16.365 --> 15:17.915 maximize the sum of utilities. 15:17.918 --> 15:19.628 By switching some goods from A to B you could make both people 15:19.629 --> 15:21.169 better off [clarification: you could increase the sum of 15:21.169 --> 15:22.679 both peoples' utilities, though one person would end up 15:22.682 --> 15:23.582 worse off in the process]. 15:23.580 --> 15:25.910 So that was a shock to economists. 15:25.908 --> 15:27.948 It meant that the argument they'd relied on, 15:27.950 --> 15:32.100 the thing which they were using to persuade policy people that 15:32.101 --> 15:34.691 the free markets were a good thing, 15:34.690 --> 15:36.810 is a false argument. 15:36.808 --> 15:38.708 It rests on a premise that's indefensible, 15:38.706 --> 15:41.386 namely that there's constant marginal utility and everybody 15:41.389 --> 15:42.269 can measure it. 15:42.269 --> 15:45.509 So they needed some other definition, some other way of 15:45.509 --> 15:49.049 capturing the mathematical idea that the invisible hand is a 15:49.048 --> 15:49.948 good thing. 15:49.950 --> 15:52.160 And the reason I'm spending so much time emphasizing this, 15:52.158 --> 15:53.738 even though you've seen this before, 15:53.740 --> 15:56.990 is because if economists made a mistake once what makes you 15:56.988 --> 15:59.338 think they haven't made another mistake? 15:59.340 --> 16:05.140 So now let's give the argument that in a weaker sense the 16:05.143 --> 16:09.773 invisible hand holds, that is that the free market 16:09.765 --> 16:13.425 comes to a very socially desirable equilibrium. 16:13.428 --> 16:16.248 And later in the course we're going to find that that argument 16:16.249 --> 16:17.219 is also too narrow. 16:17.220 --> 16:20.960 So my point that I'm going to gradually get to is that 16:20.961 --> 16:25.201 economists have constantly taken too narrow and too special a 16:25.196 --> 16:28.526 view of the world, and as you enlarge the view of 16:28.530 --> 16:30.830 the world you have, not by making things 16:30.831 --> 16:33.381 mathematically simpler, actually you have to make them 16:33.380 --> 16:35.060 mathematically slightly more complicated, 16:35.058 --> 16:38.348 as you enlarge the view of the world you have you get closer to 16:38.351 --> 16:41.701 the truth and you start to find that the free market isn't quite 16:41.697 --> 16:44.987 as wonderful as you thought at first and therefore there's room 16:44.990 --> 16:46.690 for government regulation. 16:46.690 --> 16:49.310 So it's not, as Paul Krugman argued, 16:49.308 --> 16:54.148 that economists are transfixed by mathematics and seduced into 16:54.152 --> 16:58.362 simple conclusions that the free market is perfect. 16:58.360 --> 17:01.520 On the contrary I would say they were afraid of too much 17:01.518 --> 17:03.698 mathematics, and by looking too narrowly 17:03.703 --> 17:06.303 they didn't realize what they could have just by being 17:06.304 --> 17:08.124 mathematically more sophisticated. 17:08.118 --> 17:09.828 So it's a failure of sophistication, 17:09.832 --> 17:11.302 not too much sophistication. 17:11.298 --> 17:14.148 So what did the economists argue? 17:14.150 --> 17:17.050 So the chief among them, Hicks, Samuelson, 17:17.048 --> 17:20.508 Pareto and Fisher, all these people basically came 17:20.510 --> 17:22.420 to the same conclusion. 17:22.420 --> 17:26.630 They said, "Well, it makes no sense to talk about 17:26.634 --> 17:28.944 the sum of utilities." 17:28.940 --> 17:35.470 Let's talk about Pareto Efficiency. 17:35.470 --> 17:36.850 So what's the general problem? 17:36.848 --> 17:41.878 The general problem is you start with an economy made up of 17:41.878 --> 17:43.958 all these individuals. 17:43.960 --> 17:47.250 Let's call them W^(A) and W^(B), 17:47.250 --> 17:51.720 but it could be as many as we want so let's just say W^(i)-- 17:51.720 --> 18:00.880 and E^(i), so E^(i) is the endowment of X and E^(i) of Y. 18:00.880 --> 18:04.490 So that's the economy. 18:04.490 --> 18:09.870 You start from the economy and you go to equilibrium. 18:09.868 --> 18:14.408 And equilibrium is a price vector P_X 18:14.410 --> 18:18.260 P_Y and allocations X^(i), 18:18.259 --> 18:25.209 Y^(i), i in I, such that summation over all 18:25.211 --> 18:34.321 the i's of X^(i) equals the summation for all the i's of 18:34.317 --> 18:37.957 E^(i)_X. 18:37.960 --> 18:42.210 So i's final consumption of X, plus j's final consumption of 18:42.209 --> 18:43.759 X, plus everybody else's final 18:43.760 --> 18:45.690 consumption of X, those are the sum of the 18:45.686 --> 18:47.186 X^(i)'s, is equal to the sum of 18:47.194 --> 18:48.304 everybody's endowment. 18:48.298 --> 18:53.268 And similarly for the Y's, [equals the sum] 18:53.272 --> 19:00.262 of E^(i)_Y and such that everybody is doing what's 19:00.259 --> 19:07.719 in their personal interest which is maximizing W^(i) of X, 19:07.720 --> 19:14.870 Y, so, the max of W^(i) of X, Y such that [(P_X 19:14.873 --> 19:18.863 times X) (P_Y times Y)] 19:18.859 --> 19:23.969 = [(P_X times E^(i)_X) 19:23.967 --> 19:28.827 (P_Y times E^(i)_Y)] 19:28.826 --> 19:34.946 is solved by X^(i), Y^(i) for all i. 19:34.950 --> 19:36.450 That's the definition of equilibrium. 19:36.450 --> 19:39.610 Everybody individually optimizes looking at his own 19:39.605 --> 19:41.745 budget set, the hell with everybody else, 19:41.753 --> 19:44.303 does the best in his interest, chooses X^(i), 19:44.298 --> 19:47.698 Y^(i) at the going prices, supply equals demand, 19:47.700 --> 19:49.850 that's how you get the final allocation. 19:49.848 --> 19:52.218 So that's the definition of equilibrium and now we're trying 19:52.215 --> 19:53.455 to argue that's a good thing. 19:53.460 --> 19:57.350 So what's our criterion? 19:57.348 --> 20:02.528 Well, what Pareto and Edgeworth and everybody else decided is, 20:02.528 --> 20:07.478 "Let's look at the welfare functions W^(A) and W^(B)." 20:07.480 --> 20:11.970 So if you take some allocation like the original allocation 2-1 20:11.965 --> 20:15.435 and 1-2 you'll have a welfare of each person, 20:15.440 --> 20:20.470 (3 fourths log 2) (one quarter log 1) that's the welfare of A, 20:20.470 --> 20:25.450 and the welfare of B will be (2 thirds log 1) (1 third log 2), 20:25.450 --> 20:30.110 so that'll be some point here. 20:30.108 --> 20:46.418 So this is welfare B, welfare A at initial endowment. 20:46.420 --> 20:50.350 And then if you look at the welfare at the final allocation, 20:50.348 --> 20:53.848 so, (3 fourths log 9 fifths) (1 quarter log 3 halves) you know 20:53.845 --> 20:56.075 it's going to be something like this. 20:56.079 --> 20:57.029 It's got to be up here. 20:57.029 --> 20:59.139 It's going to be better for both people. 20:59.140 --> 21:00.290 How do you know that? 21:00.288 --> 21:03.618 Because A had the choice of just buying his initial 21:03.623 --> 21:07.223 endowment--that's always affordable and he chose not to 21:07.222 --> 21:07.892 do it. 21:07.890 --> 21:10.330 He chose to do something else and he was better off and the 21:10.328 --> 21:12.308 same with B, so clearly the equilibrium 21:12.313 --> 21:15.493 allocation is going to be better than the initial endowment. 21:15.490 --> 21:34.050 And so the Pareto criterion is that an allocation X^(i)-hat, 21:34.048 --> 21:47.968 Y^(i)-hat, Pareto dominates an allocation-- 21:47.970 --> 21:57.180 so this is for all I--X^(i), let's put the i's on the top. 21:57.180 --> 22:03.290 If and only if everybody's better off: W^(i) (X^(i)-hat, 22:03.288 --> 22:09.988 Y^(i)-hat) is bigger than W^(i)(X^(i), Y^(i)). 22:09.990 --> 22:14.330 So we started with an allocation, the E allocation 22:14.330 --> 22:19.470 here, and then we moved to competitive equilibrium here and 22:19.470 --> 22:22.040 everybody was better off. 22:22.038 --> 22:25.968 So the equilibrium allocation Pareto dominates the initial 22:25.971 --> 22:26.871 allocation. 22:26.868 --> 22:31.458 But now the question is maybe there's some other allocation 22:31.459 --> 22:35.409 besides the competitive equilibrium allocation that 22:35.414 --> 22:38.664 dominates the equilibrium allocation. 22:38.660 --> 22:51.020 So the theorem is that if P, (X^(i) Y^(i)), 22:51.019 --> 23:06.909 i in I is an equilibrium for the economy E-- 23:06.910 --> 23:15.360 remember we defined the economy E, I better put E here-- 23:15.359 --> 23:28.459 then no allocation (X^(i)-hat, Y^(i)-hat), i in I, 23:28.462 --> 23:42.032 Pareto dominates (X^(i), Y^(i)) if summation X^(i)-hat 23:42.029 --> 23:50.129 equals summation E^(i)_X and 23:50.134 --> 23:54.504 summation-- got a little too tight here, 23:54.498 --> 23:57.888 sorry about this, Y^(i)-hat equals, 23:57.890 --> 24:02.060 this is over i, equals summation 24:02.058 --> 24:04.748 E^(i)_Y. 24:04.750 --> 24:06.730 So I can read this, it's really bad. 24:06.730 --> 24:08.730 It's a critical line right in the corner. 24:08.730 --> 24:13.140 So I'll just say it in words and then read it literally. 24:13.140 --> 24:16.030 So this allocation dominates this one. 24:16.028 --> 24:19.408 The competitive equilibrium dominated the initial endowment, 24:19.410 --> 24:22.670 but maybe there's some other way of rearranging the goods 24:22.673 --> 24:24.893 that gets you even further out here. 24:24.890 --> 24:27.760 The theorem says that's impossible. 24:27.759 --> 24:31.479 So it says if you start with a competitive equilibrium, 24:31.480 --> 24:35.660 with an allocation (X^(i), Y^(i)) for the economy E, 24:35.660 --> 24:40.800 the no allocation like that one, no allocation (X^(i)-hat, 24:40.798 --> 24:43.508 Y^(i)-hat) can Pareto dominate it. 24:43.509 --> 24:44.339 This couldn't happen. 24:44.338 --> 24:45.518 You'd have to be down here somewhere. 24:45.519 --> 24:49.199 That couldn't happen provided this other allocation used up 24:49.202 --> 24:51.492 the resources that were available. 24:51.490 --> 24:54.350 So the sum of the X's in this allocation is the sum of the 24:54.354 --> 24:57.294 initial endowments, and the sum of the Y's in this 24:57.290 --> 25:00.650 allocation is the sum of the initial endowments over the 25:00.648 --> 25:01.258 people. 25:01.259 --> 25:04.849 So of course if you add more goods to the economy you can 25:04.849 --> 25:08.449 make everybody better off, but using the existing goods in 25:08.451 --> 25:10.021 the economy, which is the sum of the 25:10.022 --> 25:12.102 E^(i)_X's and the sum of the E^(i)_Y's, 25:12.098 --> 25:14.578 there's no way to make everyone better off than at the 25:14.575 --> 25:15.785 competitive equilibrium. 25:15.788 --> 25:19.818 So it says literally if P (X^(i), Y^(i)) is an equilibrium 25:19.816 --> 25:21.896 for the economy E, which is that, 25:21.903 --> 25:23.633 which includes the initial endowments, 25:23.630 --> 25:27.700 the no allocation (X^(i)-hat, Y^(i)-hat) Pareto dominates 25:27.703 --> 25:31.053 (X^(I), Y^(I)) if the sum over I of 25:31.054 --> 25:36.404 X^(i)-hat equals the sum over i of E^(i)_X, 25:36.400 --> 25:43.420 and if the sum of the Y^(i)-hats over i equals the sum 25:43.424 --> 25:47.274 of the E^(i)_Y's. 25:47.269 --> 25:48.809 That's the theorem. 25:48.808 --> 25:52.598 So we've totally weakened the definition of equilibrium. 25:52.598 --> 25:59.138 So it might be that if this is the competitive allocation if 25:59.135 --> 26:05.225 you look at this line this is same sum of utilities, 26:05.230 --> 26:08.660 because if this slope is -1 and here's the utility to B and the 26:08.664 --> 26:11.994 utility to A everything on this line has the same utility. 26:11.990 --> 26:14.990 So that theorem would say anything else that's feasible 26:14.988 --> 26:16.208 had to be down here. 26:16.210 --> 26:19.010 The sum is less than that. 26:19.009 --> 26:20.269 But here we're not saying that. 26:20.269 --> 26:21.749 We're no longer saying that. 26:21.750 --> 26:23.330 Maybe you can increase the sum. 26:23.328 --> 26:26.428 From this equilibrium you could increase the sum by taking 26:26.430 --> 26:28.770 something away from A and giving it to B. 26:28.769 --> 26:30.029 You could increase the sum. 26:30.029 --> 26:32.719 So that would have been here. 26:32.720 --> 26:36.420 You increase the sum, so you no longer have maximized 26:36.416 --> 26:38.446 the sum, but the fact is to make B 26:38.454 --> 26:41.274 better off you had to make A worse off and that's the new 26:41.267 --> 26:44.977 criterion, Pareto Efficiency. 26:44.980 --> 26:48.980 So that's the theorem and now we're going to prove it so that 26:48.976 --> 26:53.036 you can see the whole logic of the free market just amounts to 26:53.039 --> 26:53.639 that. 26:53.640 --> 26:54.880 So are there any questions about this? 26:54.880 --> 26:58.490 You all supposedly saw this before, but that doesn't mean 26:58.492 --> 26:59.592 you really did. 26:59.588 --> 27:01.328 So does anyone have a question what's going on? 27:01.328 --> 27:07.868 This is a very basic incredibly important idea. 27:07.868 --> 27:11.308 All right, so I'll give you one more chance. 27:11.309 --> 27:13.559 Somebody must have a question. 27:13.559 --> 27:14.929 Yes? 27:14.930 --> 27:17.000 Student: Could you just re-say the theorem a little bit 27:17.002 --> 27:17.482 more slowly? 27:17.480 --> 27:18.020 Prof: Yes. 27:18.019 --> 27:21.609 Why don't I draw a picture, which is what I was going to do 27:21.612 --> 27:22.172 anyway? 27:22.170 --> 27:30.410 So let's draw a picture, and I'll bring the theorem back 27:30.406 --> 27:35.346 after having drawn the picture. 27:35.348 --> 27:39.148 So the picture is going to be due to Edgeworth. 27:39.150 --> 27:42.030 So he invented the indifference curve and he invented it in 27:42.029 --> 27:43.469 order to draw this picture. 27:43.470 --> 27:45.990 It was an incredibly clever picture. 27:45.990 --> 27:48.130 So he says, "Let's look at this economy," 27:48.131 --> 27:49.931 the one on the right that we start with. 27:49.930 --> 27:55.970 We put X^(A) here and X^(B) here and you notice that A 27:55.968 --> 28:00.068 begins with an endowment of (2,1). 28:00.068 --> 28:06.038 So let's put his endowment (2,1) here. 28:06.038 --> 28:15.348 So this is E^(A)_X = 2 and this is E^(A)_Y 28:15.346 --> 28:17.326 = 1, right? 28:17.329 --> 28:18.459 That makes sense. 28:18.460 --> 28:20.990 Now we can draw his indifference curves through 28:20.993 --> 28:21.383 that. 28:21.380 --> 28:23.990 So his indifference curves he likes X more than he likes Y, 28:23.990 --> 28:26.780 so the indifference curves are going to be sort of vertical. 28:26.778 --> 28:32.518 He doesn't like to give up X very much, so his indifference 28:32.522 --> 28:36.982 curves are going to look sort of like this. 28:36.980 --> 28:39.510 And the most important indifferent curves are the ones 28:39.511 --> 28:41.711 where he gets better off than he was before. 28:41.710 --> 28:47.150 So now in the end there's going to be a budget set which is X is 28:47.146 --> 28:54.056 way more expensive than Y, so there's going to be a budget 28:54.064 --> 28:57.914 set that looked like this. 28:57.910 --> 29:01.550 That's sort of the budget set that he faces. 29:01.548 --> 29:03.838 And he's not going to choose this. 29:03.838 --> 29:11.568 It turns out he chooses something like here which is he 29:11.571 --> 29:20.451 chooses a little bit less of X, 9 fifths, and a little bit more 29:20.451 --> 29:21.741 of Y. 29:21.740 --> 29:27.950 So here's X^(A) = 9 fifths which is a little less than 2, 29:27.952 --> 29:32.282 and X^(B) he chose was, I can't read it, 29:32.278 --> 29:36.048 3 halves which is more than 1. 29:36.049 --> 29:37.709 Are you with me? 29:37.710 --> 29:38.860 You asked the question. 29:38.859 --> 29:40.209 Does this picture make sense? 29:40.210 --> 29:41.120 Student: Yes. 29:41.118 --> 29:45.118 Prof: So he obviously did better than he did before, 29:45.122 --> 29:48.092 but we've got another guy to worry about. 29:48.088 --> 29:51.638 So how can you put the second guy in the same diagram? 29:51.640 --> 29:57.540 Well, let's just add the total endowment. 29:57.539 --> 29:58.599 Where's the total endowment? 29:58.599 --> 30:00.719 Well, it's 3 units of each. 30:00.720 --> 30:01.860 So here we've got 2. 30:01.859 --> 30:10.369 So this is 1,2, this must be 3. 30:10.368 --> 30:15.138 So the total endowment of good X is 3,2 1 is 3 and a good Y is 30:15.141 --> 30:16.081 3 as well. 30:16.079 --> 30:18.529 Here's 1, here's 2, here's 3. 30:18.529 --> 30:23.299 This is supposed to be a square. 30:23.299 --> 30:25.509 The aggregate endowment is here. 30:25.509 --> 30:38.259 This is the total endowment, 3 of each. 30:38.259 --> 30:41.729 What are the feasible allocations? 30:41.730 --> 30:44.060 So this is an extremely clever insight of Edgeworth's. 30:44.058 --> 30:47.358 He said, "Look, the total endowment is (3,3). 30:47.358 --> 30:51.548 If I tell you that A has got (2,1) (that's this point) what's 30:51.553 --> 30:54.283 left over, B must be consuming." 30:54.279 --> 30:58.139 So B's consuming the distance from here to here which is 30:58.144 --> 30:59.414 obviously (1,2). 30:59.410 --> 31:03.170 And if A ends up over here at 9 fifths, 31:03.170 --> 31:07.850 3 halves, that's from here to here is 9 fifths and 3 halves, 31:07.848 --> 31:11.808 since the total is (3,3) B's obviously consuming what's left 31:11.811 --> 31:13.861 over, so he's consuming this 31:13.864 --> 31:14.694 difference. 31:14.690 --> 31:18.160 So whatever A does you can see in the diagram looking at it the 31:18.156 --> 31:20.266 normal way, but if you just sort of twist 31:20.270 --> 31:22.750 your mind a little bit and look from this direction, 31:22.750 --> 31:25.220 once you know what A's doing you can figure out what B's 31:25.215 --> 31:28.605 doing, so each of these points defines 31:28.605 --> 31:30.815 a feasible allocation. 31:30.818 --> 31:33.868 And now what we want to say, the theorem says, 31:33.865 --> 31:37.175 look at the utilities that A and B got out of this 31:37.183 --> 31:38.203 allocation. 31:38.200 --> 31:40.660 A is on this indifference curve. 31:40.660 --> 31:41.470 You know what he's doing. 31:41.470 --> 31:43.610 B you know is coming from here. 31:43.609 --> 31:46.319 We know how he's doing. 31:46.318 --> 31:48.768 So look at the allocation they got. 31:48.769 --> 31:52.339 You can't make both people better off than that by choosing 31:52.337 --> 31:54.427 some other point in this square. 31:54.430 --> 31:58.190 So let's look at it from B's point of view. 31:58.190 --> 32:01.180 From B's point of view B likes more of everything. 32:01.180 --> 32:03.960 So if A gets less B's going to get more. 32:03.960 --> 32:08.060 So from B's point of view the further out you go this way the 32:08.056 --> 32:09.486 better it is for B. 32:09.490 --> 32:13.220 So the axes for B looks like that. 32:13.220 --> 32:24.260 So he's going to have some indifference curve that--B's 32:24.263 --> 32:33.063 indifference curve has to look like this. 32:33.058 --> 32:39.688 So here's where B started here, and then B's going to be better 32:39.688 --> 32:42.038 off doing like this. 32:42.038 --> 32:44.238 So B's indifference curve looks something like that. 32:44.240 --> 32:50.480 Those are B's indifference curves. 32:50.480 --> 32:53.800 So the further out you go this way the better off B is, 32:53.798 --> 32:56.148 so as you shrink what A gets, that obviously means B's 32:56.150 --> 32:59.650 getting more, and so B's getting happier and 32:59.652 --> 33:00.492 happier. 33:00.490 --> 33:04.040 So what is the definition of Pareto Efficiency? 33:04.038 --> 33:07.148 Whatever point we pick like this one to begin with: 33:07.150 --> 33:10.260 is there a way of making both people better off. 33:10.259 --> 33:12.139 Well the answer is the competitive equilibrium makes 33:12.137 --> 33:13.057 both people better off. 33:13.059 --> 33:13.779 Why is that? 33:13.778 --> 33:17.378 If you can see it from back there, the reason is that A went 33:17.384 --> 33:20.934 from this white indifference curve to a better indifference 33:20.928 --> 33:22.148 curve over here. 33:22.150 --> 33:26.370 B, whose budget line is the same budget line looked at from 33:26.369 --> 33:29.789 his point of view, he went from his indifference 33:29.790 --> 33:32.120 curve over here to one here. 33:32.118 --> 33:35.388 So he went from this yellow indifference curve to a better 33:35.390 --> 33:38.490 one, so both of them got better off going to this final 33:38.489 --> 33:40.669 allocation than they started with. 33:40.670 --> 33:43.890 And of course that had to be true because A could always have 33:43.894 --> 33:45.994 chosen to stay where he was at (2,1). 33:45.990 --> 33:48.240 He moved to (9 fifths, 3 halves) because it was 33:48.240 --> 33:51.030 better, and B could have stayed where she was at (1,2). 33:51.029 --> 33:54.449 She chose to go to (6 fifths, 3 halves) because it made her 33:54.452 --> 33:57.132 better off, so both of them must be better 33:57.134 --> 34:00.724 off at the final allocation than they were to begin with. 34:00.720 --> 34:02.310 What does the theorem say? 34:02.308 --> 34:04.378 The theorem says--remember, in pictures, 34:04.380 --> 34:08.200 it says whatever point we pick here that defines something for 34:08.195 --> 34:11.905 A and defines something for B, then we can look at the utility 34:11.907 --> 34:14.937 welfare functions of both and see what indifference curves 34:14.936 --> 34:15.676 they're on. 34:15.679 --> 34:18.939 The theorem says there's no other point in this whole 34:18.940 --> 34:22.770 diagram which puts both A and B on a higher indifference curve 34:22.766 --> 34:26.966 for each of them than they were at the competitive equilibrium. 34:26.969 --> 34:30.489 So if I read the definition again it says, 34:30.489 --> 34:33.839 if I start with an economy there with the endowments and 34:33.838 --> 34:36.318 the utility functions, so the endowments and the 34:36.322 --> 34:39.382 indifference curves, and I compute the competitive 34:39.380 --> 34:42.200 equilibrium like we did over there, 34:42.199 --> 34:45.909 then no allocation, that means no point in that box 34:45.914 --> 34:49.784 could make everybody better off than they were at the 34:49.777 --> 34:54.677 competitive equilibrium provided that new point is in the box. 34:54.679 --> 34:59.979 So it makes the sum of the X's equal to the sum of the 34:59.983 --> 35:05.693 endowments, and the sum of the Y's equal to the sum of the 35:05.688 --> 35:07.688 endowments of Y. 35:07.690 --> 35:08.790 This is the end of the proof. 35:08.789 --> 35:11.979 We already gave the proof according to Edgeworth. 35:11.980 --> 35:13.630 So what's the proof according to Edgeworth? 35:13.630 --> 35:15.930 He says, "It's all a matter of looking at it the 35:15.931 --> 35:16.731 right way." 35:16.730 --> 35:19.300 It's easy to look at the picture from A's point of view. 35:19.300 --> 35:23.480 You just look at the X-axis and the Y-axis. 35:23.480 --> 35:30.150 This is the Y-axis. 35:30.150 --> 35:30.860 What is this? 35:30.860 --> 35:37.960 This is X^(A)_Y. 35:37.960 --> 35:39.180 It was 3 halves. 35:39.179 --> 35:43.969 So this is the Y-axis going this way. 35:43.969 --> 35:48.019 This is the Y-axis and this is the X-axis going this way. 35:48.018 --> 35:51.528 For A it's very simple to look at the Y-axis and the X-axis for 35:51.532 --> 35:51.762 A. 35:51.760 --> 35:58.860 A's got its indifference curve getting better and better. 35:58.860 --> 36:00.990 So from A's point of view it's very simple, 36:00.989 --> 36:03.299 you look at the endowment, you draw the budget line 36:03.302 --> 36:05.452 through it, which is some linear line, 36:05.449 --> 36:08.969 and then A picks the best point on that budget set drawing the 36:08.967 --> 36:11.207 tangent, which is the white indifference 36:11.213 --> 36:12.823 curve to this point right here. 36:12.820 --> 36:14.720 That's the best he can do. 36:14.719 --> 36:19.569 Now the trick is to see that from B's point of view up here 36:19.574 --> 36:24.684 things look very similar because B's endowment is also at this 36:24.681 --> 36:25.521 point. 36:25.518 --> 36:31.048 He's got 1 unit up here of X and 2 units of Y. 36:31.050 --> 36:32.340 Why does he have that? 36:32.340 --> 36:35.450 Because the two together added up to 3, so if this is what A's 36:35.447 --> 36:38.047 endowment was then of course what's left over is B's 36:38.045 --> 36:38.755 endowment. 36:38.760 --> 36:40.370 So B has to go here. 36:40.369 --> 36:43.239 Now what does the budget set look like for B? 36:43.239 --> 36:46.019 Well it's also a linear thing with the same slope and it goes 36:46.016 --> 36:46.986 through this point. 36:46.989 --> 36:50.509 So from his point of view this pink line is also his budget 36:50.505 --> 36:50.865 set. 36:50.869 --> 36:55.459 It goes from here all the way down to where it hits his Y-axis 36:55.456 --> 36:56.356 down here. 36:56.360 --> 37:00.310 They've got the same budget set just looked at from opposite 37:00.313 --> 37:00.853 sides. 37:00.849 --> 37:03.469 Now what is competitive equilibrium and supply equals 37:03.465 --> 37:04.165 demand mean? 37:04.170 --> 37:08.140 It means that when A chooses this point as the best point, 37:08.139 --> 37:09.699 which happened to have been (9 fifths, 37:09.699 --> 37:14.509 3 halves), then B is happy to choose what's left over as his 37:14.510 --> 37:15.570 best point. 37:15.570 --> 37:18.500 So his optimal point, his indifference curve that's 37:18.496 --> 37:22.236 tangent to his budget set has to be the same point looked at from 37:22.242 --> 37:24.412 his origin instead of A's origin. 37:24.409 --> 37:25.469 That's the trick. 37:25.469 --> 37:30.049 So that's the key insight that Edgeworth had, 37:30.050 --> 37:33.840 his beautiful picture, and now the point is that if 37:33.844 --> 37:37.494 the white indifference curve looks like that, 37:37.489 --> 37:40.219 anything that makes A better off has to be on this side of 37:40.219 --> 37:43.379 the white indifference curve and the yellow indifference curve, 37:43.380 --> 37:45.950 which is B's indifference curve, anything that makes her 37:45.947 --> 37:48.607 better off has to be on this side of the yellow curve, 37:48.610 --> 37:51.330 but the white curve goes that way and the yellow curve goes 37:51.333 --> 37:51.853 that way. 37:51.849 --> 37:55.229 So there's no point that's simultaneously above A's 37:55.230 --> 37:59.220 indifference curve and also above B's indifference curve. 37:59.219 --> 38:01.679 So therefore nothing could Pareto dominate it. 38:01.679 --> 38:04.099 So that's the Edgeworth proof. 38:04.099 --> 38:04.859 Yes? 38:04.860 --> 38:06.780 Student: So no matter what our point's going to be on 38:06.782 --> 38:08.742 our pink budget line because that's > 38:08.737 --> 38:08.897 down. 38:08.900 --> 38:10.010 That's all the goods we have. 38:10.010 --> 38:11.070 Prof: Right. 38:11.068 --> 38:14.408 The pink budget line describes what you can afford to buy, 38:14.409 --> 38:17.699 describes the budget equation, this equation, 38:17.699 --> 38:27.499 so everyone's going to choose on the pink line. 38:27.500 --> 38:28.680 This point is feasible. 38:28.679 --> 38:31.929 B gets almost nothing and A gets almost everything. 38:31.929 --> 38:34.469 B's never going to let himself be forced down there because B 38:34.465 --> 38:36.195 can choose something on the pink line, 38:36.199 --> 38:42.209 so he's always going to choose something better than that. 38:42.210 --> 38:44.020 So that's proof number one. 38:44.018 --> 38:46.038 I'm just going to give proof number two. 38:46.039 --> 38:49.199 Most of you have seen this before, but I'm going to give 38:49.197 --> 38:52.467 another proof and then I'm going to see whether you really 38:52.469 --> 38:54.479 understand what all this means. 38:54.480 --> 39:06.050 So here's a second proof. 39:06.050 --> 39:08.230 This is a much better proof. 39:08.230 --> 39:11.090 I'm going to write this algebraically. 39:11.090 --> 39:15.960 The proof of Arrow 1951 and Debreu 1951 separately, 39:15.960 --> 39:20.150 so this is called the Fundamental Theorem of 39:20.148 --> 39:25.018 Economics, First Welfare Theorem of Economics. 39:25.018 --> 39:27.788 So all this pictorial stuff makes so many assumptions, 39:27.786 --> 39:30.606 there are only two kinds of traders, there are only two 39:30.606 --> 39:31.126 goods. 39:31.130 --> 39:35.250 They are consuming only on the boundaries. 39:35.250 --> 39:37.260 I mean, they're never consuming on the boundaries it's always in 39:37.255 --> 39:39.075 this, you know, there are just so 39:39.079 --> 39:42.279 many special assumptions and everything is two by two. 39:42.280 --> 39:43.870 It's just special case. 39:43.869 --> 39:46.919 So let's give a much more general proof of this theorem. 39:46.920 --> 39:49.810 So here we've got--two goods won't play any role. 39:49.809 --> 39:51.249 There could be any number of goods. 39:51.250 --> 39:54.230 There are two goods only here, but there are many traders, 39:54.228 --> 39:55.638 lots and lots of traders. 39:55.639 --> 40:01.349 So, what's the proof that (X^(i) hat, Y^(i) hat) couldn't 40:01.349 --> 40:03.489 Pareto dominate it? 40:03.489 --> 40:11.089 So supposed that we've got--this is equal to summation 40:11.092 --> 40:18.842 X^(i)-hat, and this is equal to summation Y^(i)-hat. 40:18.840 --> 40:24.710 So we've got another allocation which is feasible. 40:24.710 --> 40:27.650 So could it be that this other feasible allocations of 40:27.646 --> 40:30.836 X^(i)-hat's and Y^(i)-hats, could that make everybody 40:30.836 --> 40:31.566 better off? 40:31.570 --> 40:34.340 Could that make each i better off than (X^(i), 40:34.344 --> 40:34.904 Y^(i))? 40:34.900 --> 40:38.530 And this proof is so short and so beautiful, 40:38.530 --> 40:42.320 so elegant, and so convincing that it's mesmerized people now 40:42.315 --> 40:45.715 for over 50 years and it prevents them from seeing that 40:45.724 --> 40:49.514 there could be an even a more general situations where things 40:49.510 --> 40:50.900 aren't so great. 40:50.900 --> 40:52.250 So what is the proof? 40:52.250 --> 40:53.730 It's two lines. 40:53.730 --> 40:59.600 If (X^(i)-hat, Y^(i)-hat) is really better for 40:59.596 --> 41:05.586 i, then I claim--line one, W^(i) of (X^(i)-hat, 41:05.594 --> 41:07.554 Y^(i)-hat). 41:07.550 --> 41:10.680 Sorry, this is line zero. 41:10.679 --> 41:12.389 This is just repeating what's the case. 41:12.389 --> 41:15.949 W^(i) of X^(i)-hat, Y^(i)-hat greater than 41:15.949 --> 41:17.859 W^(i)(X^(i), Y^(i)). 41:17.860 --> 41:19.640 That's for all i. 41:19.639 --> 41:21.029 That's the claim. 41:21.030 --> 41:29.600 Could this happen that you've got a feasible (X^(i)-hat, 41:29.599 --> 41:33.029 Y^(i)-hat) and it makes everybody better off than 41:33.025 --> 41:34.235 (X^(i), Y^(i)). 41:34.239 --> 41:34.809 Could this happen? 41:34.809 --> 41:35.889 The answer's no. 41:35.889 --> 41:39.759 Line one is, if that were true then it would 41:39.760 --> 41:44.530 have to be the case that P_X times X^(i)-hat 41:44.530 --> 41:49.120 P_Y times Y^(i)-hat had to be greater than 41:49.119 --> 41:53.619 P_X times E^(i)_X P_Y 41:53.619 --> 41:56.499 times E^(i)_Y. 41:56.500 --> 41:57.540 Why is that? 41:57.539 --> 42:00.229 Because the budget set of Mr. 42:00.230 --> 42:05.980 i says he spends all his money on X and Y in his budget set and 42:05.983 --> 42:08.213 does the best he can. 42:08.210 --> 42:11.840 (X^(i), Y^(i)) solves this problem. 42:11.840 --> 42:15.350 So if this thing really makes him better off than what he 42:15.346 --> 42:19.096 chose, it can't be that it was affordable--otherwise he would 42:19.103 --> 42:20.923 have bought this instead. 42:20.920 --> 42:24.620 So it has to be that for i, this bundle cost more than he 42:24.619 --> 42:25.479 can afford. 42:25.480 --> 42:27.090 That's why he didn't choose it. 42:27.090 --> 42:33.410 So this relies on agent rationality. 42:33.409 --> 42:37.559 Everybody given his choices, he doesn't care about the rest 42:37.559 --> 42:39.919 of the world, but given his choice 42:39.920 --> 42:43.070 everybody's doing the best thing he can. 42:43.070 --> 42:45.500 Shiller would say, "People make mistakes. 42:45.500 --> 42:46.090 They're crazy. 42:46.090 --> 42:47.220 They have no idea what they're doing. 42:47.219 --> 42:47.999 They're stupid. 42:48.000 --> 42:51.860 Some guy tells them a story in the bar one night and they 42:51.858 --> 42:54.478 totally change their life around," 42:54.478 --> 42:56.958 so he'd say all this isn't true. 42:56.960 --> 43:00.090 All right, but anyway I actually believe that people 43:00.090 --> 43:03.780 have more sense about deciding for themselves what's good for 43:03.775 --> 43:07.885 them than third parties do about deciding what's good for them. 43:07.889 --> 43:10.209 So I don't want to challenge this. 43:10.210 --> 43:13.330 So if you don't challenge this you conclude that if something 43:13.329 --> 43:16.349 makes everybody better off each person must have found this 43:16.346 --> 43:19.096 allocation unaffordable otherwise he would have chosen 43:19.101 --> 43:19.571 it. 43:19.570 --> 43:22.380 [Audio dropped] over i of E^(i)_Y. 43:22.380 --> 43:25.660 So all I did was--that's true for every single person i. 43:25.659 --> 43:28.089 So I could add this inequality for i, 43:28.090 --> 43:31.690 plus this inequality for j, plus this inequality for all 43:31.690 --> 43:35.680 the people and then I use the distributive law to take the sum 43:35.681 --> 43:37.451 inside and I have this. 43:37.449 --> 43:42.349 But that is impossible and that's the end of the proof 43:42.349 --> 43:47.619 because the sum of this X^(i)-hat is equal to this sum. 43:47.619 --> 43:49.269 That was what we had supposed. 43:49.268 --> 43:53.738 You're just rearranging all the goods that are really there and 43:53.744 --> 43:56.274 this sum is the same as this sum. 43:56.268 --> 43:59.598 So therefore since the new allocation just rearranges the 43:59.601 --> 44:02.341 endowment differently from the equilibrium, 44:02.340 --> 44:04.520 and rearranges the X's maybe differently from the 44:04.518 --> 44:06.428 equilibrium, it has to be that actually this 44:06.429 --> 44:08.609 thing on the left is equal to this thing on the right. 44:08.610 --> 44:11.210 It's identical to it so it can't be greater and that's the 44:11.210 --> 44:11.940 contradiction. 44:11.940 --> 44:13.390 That's the end of the proof. 44:13.389 --> 44:18.829 So it's two lines, two lines to prove what Adam 44:18.826 --> 44:25.796 Smith spent 400 pages arguing that the free market is a good 44:25.797 --> 44:26.977 thing. 44:26.980 --> 44:29.830 So this is the basis for the idea that we shouldn't regulate, 44:29.833 --> 44:32.073 we shouldn't regulate, we shouldn't regulate. 44:32.070 --> 44:35.630 Sounds pretty convincing, so any questions about this? 44:35.630 --> 44:38.840 Anything you'd like to say? 44:38.840 --> 44:45.960 Are you convinced by this? 44:45.960 --> 44:50.470 What are some reasons you don't believe this, 44:50.465 --> 44:52.815 some obvious reasons? 44:52.820 --> 44:53.840 Yes? 44:53.840 --> 44:56.840 Student: Well, even if this proof makes sense 44:56.835 --> 45:00.125 a lot of regulation policies have to do with changing the 45:00.126 --> 45:02.886 prices of things anyway so we're not necessarily 45:02.887 --> 45:04.707 > 45:04.710 --> 45:08.280 with them at market > 45:08.280 --> 45:11.610 Prof: I'm glad you asked that question because it betrays 45:11.606 --> 45:13.976 a misunderstanding, so I'm glad you asked that 45:13.983 --> 45:14.673 question. 45:14.670 --> 45:22.400 So this theorem is correct given the assumptions. 45:22.400 --> 45:28.080 There's a little bit mistake in the reasoning he made. 45:28.079 --> 45:30.279 I want to repeat his question, change it a little bit so maybe 45:30.284 --> 45:31.914 he'll deny what this is what he was saying. 45:31.909 --> 45:35.269 But I believe what he just said is what really happens in a lot 45:35.271 --> 45:38.471 of regulation is they come in and they change the prices. 45:38.469 --> 45:40.719 They tax a good, they prevent people from 45:40.717 --> 45:42.907 trading so much so the price changes. 45:42.909 --> 45:44.339 They subsidize something. 45:44.340 --> 45:46.660 Something happens so that you get different prices, 45:46.661 --> 45:48.891 and of course at those different prices you get a 45:48.889 --> 45:50.049 different allocation. 45:50.050 --> 45:52.690 That's all true so far. 45:52.690 --> 45:56.110 And then he said because they're at different prices you 45:56.110 --> 45:59.530 can't compare the original equilibrium allocation to the 45:59.530 --> 46:01.460 new equilibrium allocation. 46:01.460 --> 46:05.330 Well, that's where he shortchanged this proof, 46:05.329 --> 46:10.159 this proof says however you get to a new allocation (X^(i)-hat, 46:10.159 --> 46:12.529 Y^(i)-hat), maybe it's because the government has intervened 46:12.534 --> 46:15.034 and changed all the prices and done a bunch of other stuff, 46:15.030 --> 46:17.910 but in the end after all that intervention the government is 46:17.909 --> 46:20.399 going to get you to a new allocation (X^(i)-hat, 46:20.400 --> 46:24.470 Y^(i)-hat), and that new allocation can't be better than 46:24.465 --> 46:25.865 the original one. 46:25.869 --> 46:29.399 So that's the force of the argument. 46:29.400 --> 46:32.720 No matter what the government does in the end the upshot is a 46:32.717 --> 46:34.927 new allocation (X^(i)-hat, Y^(i)-hat). 46:34.929 --> 46:36.159 We don't have to think about [how] 46:36.159 --> 46:36.719 it got there. 46:36.719 --> 46:39.609 That's where it got and according to this argument it 46:39.608 --> 46:40.608 can't be better. 46:40.610 --> 46:43.600 So the argument is correct and elegant. 46:43.599 --> 46:45.329 There must be something missing to the argument, 46:45.326 --> 46:47.376 some assumption you don't really believe if you doubt the 46:47.382 --> 46:47.862 argument. 46:47.860 --> 46:49.070 Yes? 46:49.070 --> 46:52.980 Student: What if you want an allocation that's better 46:52.981 --> 46:56.631 for one party even at the expense of the other party? 46:56.630 --> 46:58.560 Prof: What if you want an allocation? 46:58.559 --> 47:01.849 Student: It's socially desirable to have an allocation 47:01.853 --> 47:05.313 that's better for one party even if that comes at the expense of 47:05.311 --> 47:06.301 another party. 47:06.300 --> 47:10.760 Prof: So one argument you could say is that it may be 47:10.757 --> 47:14.837 in the equilibrium that A--so who's better off here? 47:14.840 --> 47:18.240 You notice that this equilibrium A ends up with more 47:18.242 --> 47:20.112 of everything than B does. 47:20.110 --> 47:22.970 So you could say this equilibrium is not very socially 47:22.965 --> 47:26.305 desirable because A ends up with more of everything than B does 47:26.307 --> 47:28.947 or at least as much of everything as B does, 47:28.949 --> 47:31.679 and that seems unjust and unfair and so we don't like the 47:31.677 --> 47:34.167 allocation, and so maybe we should move 47:34.168 --> 47:35.708 something from A to B. 47:35.710 --> 47:38.320 So that's an argument on the basis of justice. 47:38.320 --> 47:41.620 It says we can find juster allocations that are somehow 47:41.617 --> 47:44.727 socially more desirable, but this argument of Pareto 47:44.731 --> 47:46.871 Efficiency is about efficiency. 47:46.869 --> 47:51.969 Maybe you could hurt A to help B because that serves your 47:51.974 --> 47:56.444 desire for fairness, but that doesn't say anything 47:56.440 --> 48:00.270 about how efficient the allocation is. 48:00.268 --> 48:02.518 It's still true that the original allocation was 48:02.518 --> 48:05.438 efficient in the sense you couldn't make anyone better off. 48:05.440 --> 48:08.550 So the economists would say in order to hurt A and help B what 48:08.545 --> 48:11.595 you should do is take some of A's endowment away from him and 48:11.599 --> 48:14.549 give it to her and then let them trade to a new competitive 48:14.552 --> 48:17.252 equilibrium where she began with more than him. 48:17.250 --> 48:20.660 And so the just thing to do is to rearrange the resources at 48:20.655 --> 48:23.535 the beginning and still let people trade to a final 48:23.543 --> 48:24.413 allocation. 48:24.409 --> 48:27.719 So that's an argument from fairness, but it doesn't 48:27.719 --> 48:31.889 interfere, it doesn't contradict the economic argument that fair 48:31.891 --> 48:34.211 or unfair it's still efficient. 48:34.210 --> 48:36.830 We've done as well as we could making everyone as happy as we 48:36.827 --> 48:37.087 can. 48:37.090 --> 48:39.790 Maybe we can hurt someone and help someone else but we can't 48:39.786 --> 48:42.776 help everyone at the same time, whereas most regulation the 48:42.779 --> 48:46.029 argument in favor of regulation is you're helping everybody by 48:46.030 --> 48:48.040 regulating, and this says you can't ever do 48:48.043 --> 48:48.283 that. 48:48.280 --> 48:49.070 So what's missing? 48:49.070 --> 48:49.880 What's some of the assumptions? 48:49.880 --> 48:52.000 This should be an elementary question for you. 48:52.000 --> 48:53.060 Yes? 48:53.059 --> 48:55.979 Student: Is it violating one of the three assumptions 48:55.983 --> 48:57.473 that > 48:57.469 --> 48:58.609 , the market theory? 48:58.610 --> 49:00.650 Prof: No, it's got all those assumptions 49:00.648 --> 49:03.398 so there's something about the model that's too narrow and this 49:03.396 --> 49:04.236 should be easy. 49:04.239 --> 49:07.409 You're going to tell me obvious things where it's too narrow, 49:07.405 --> 49:09.775 so I'm going to just say where we're going. 49:09.780 --> 49:12.530 You're supposed to be telling me now obvious ways this is too 49:12.527 --> 49:12.937 narrow. 49:12.940 --> 49:14.120 I'm going to say, "You're right. 49:14.117 --> 49:14.537 You're right. 49:14.543 --> 49:15.923 You're right, but finance is different. 49:15.920 --> 49:18.520 We don't have to worry about those problems supposedly." 49:18.519 --> 49:21.279 So go ahead back there, you. 49:21.280 --> 49:22.430 Student: Externalities. 49:22.429 --> 49:25.879 Prof: So one critical shortcoming of this argument is 49:25.884 --> 49:27.704 that there are externalities. 49:27.699 --> 49:35.289 Suppose that X is cigarettes, and A buying and smoking more 49:35.288 --> 49:38.688 cigarettes makes B sick. 49:38.690 --> 49:40.200 So that's not part of this. 49:40.199 --> 49:41.339 Why is that not part of this? 49:41.340 --> 49:44.690 Because B's utility depends only on what B's consuming, 49:44.688 --> 49:45.988 not what A's doing. 49:45.989 --> 49:49.059 So there's no place in this model. 49:49.059 --> 49:51.589 It's too narrow to include externalities, 49:51.590 --> 49:55.660 so it doesn't capture the fact that he, 49:55.659 --> 49:58.739 by smoking, might make her sick by smoking so many cigarettes 49:58.744 --> 50:00.034 and that's not in here. 50:00.030 --> 50:03.770 And if you put the idea that B's utility, her utility might 50:03.771 --> 50:07.711 depend on what he's doing and not what she's doing the theorem 50:07.708 --> 50:08.868 won't be true. 50:08.869 --> 50:11.949 So there's a reason to tax pollution, and to tax cigarettes 50:11.949 --> 50:14.499 and all those things because of externalities. 50:14.500 --> 50:17.890 So that's a fundamental problem. 50:17.889 --> 50:18.669 Yes? 50:18.670 --> 50:20.680 Student: It's also assuming that all of our 50:20.677 --> 50:22.807 indifference curves are aligned in the same shape. 50:22.809 --> 50:24.759 Prof: Well, we're assuming diminishing 50:24.759 --> 50:26.889 marginal utility, but each person has a different 50:26.885 --> 50:28.255 indifference curve, so yes. 50:28.260 --> 50:31.210 We assume that they are all curved away from the origin, 50:31.210 --> 50:33.570 so we assume diminishing marginal utility. 50:33.570 --> 50:36.110 But I think that I'm almost prepared to believe, 50:36.112 --> 50:37.792 diminishing marginal utility. 50:37.789 --> 50:39.689 It's not literally true. 50:39.690 --> 50:46.250 Moving from one ticket to two tickets may be a huge gain in 50:46.251 --> 50:52.361 utility because now you can bring your friend along. 50:52.360 --> 50:54.900 Getting one ticket might not be worth much because you don't 50:54.900 --> 50:55.720 want to go alone. 50:55.719 --> 50:58.159 Getting the second one may add a huge amount of utility. 50:58.159 --> 51:00.629 That's an increasing marginal utility, but eventually after 51:00.625 --> 51:03.255 you've got enough tickets it's going to be diminishing marginal 51:03.260 --> 51:03.770 utility. 51:03.768 --> 51:06.668 So I believe on the whole that diminishing marginal utility is 51:06.670 --> 51:09.430 not such a bad assumption, but this clearly relies on 51:09.434 --> 51:12.524 that, so you're right but I don't think that's the critical 51:12.523 --> 51:13.113 problem. 51:13.110 --> 51:13.820 What else is there? 51:13.820 --> 51:14.550 Yes? 51:14.550 --> 51:15.670 Student: Credit markets. 51:15.670 --> 51:16.170 Prof: What? 51:16.170 --> 51:17.090 Student: Credit. 51:17.090 --> 51:19.180 Prof: Credit markets, now we're starting to talk a 51:19.181 --> 51:21.201 little bit about finance, so just hold that thought for 51:21.199 --> 51:21.759 one second. 51:21.760 --> 51:26.250 Anything else you can think of besides externalities that's 51:26.248 --> 51:29.808 terrible for--so let me transition this way. 51:29.809 --> 51:32.409 So of course people notice externalities. 51:32.409 --> 51:35.939 Another kind of regulation is perfect competition. 51:35.940 --> 51:38.080 We assumed everyone took the prices as given. 51:38.079 --> 51:41.509 There was no monopolist setting the price and refusing to 51:41.514 --> 51:44.094 bargain with people and stuff like that. 51:44.090 --> 51:47.400 So regulation could come about in order to enforce the 51:47.402 --> 51:50.902 competitive equilibrium to allow for perfect competition, 51:50.902 --> 51:52.842 so we assumed that already. 51:52.840 --> 51:55.240 That's one place for regulation, and a second place 51:55.235 --> 51:56.955 is because of these externalities. 51:56.960 --> 52:00.770 So for 50 to 100 years everybody has accepted those two 52:00.771 --> 52:01.621 arguments. 52:01.619 --> 52:04.789 Clearly there's a place for regulating free markets, 52:04.793 --> 52:08.283 ensuring free markets and stopping trusts and monopolies, 52:08.280 --> 52:10.460 and for stopping externalities. 52:10.460 --> 52:15.340 But then people said finance is different. 52:15.340 --> 52:17.820 There aren't any externalities. 52:17.820 --> 52:19.690 You're just trading stocks and bonds. 52:19.690 --> 52:20.700 There's no pollution. 52:20.699 --> 52:22.999 Nobody's going to get sick because someone else has a stock 52:23.003 --> 52:23.483 or a bond. 52:23.480 --> 52:25.770 Some people might be jealous but we're not really going to 52:25.773 --> 52:26.783 take that into account. 52:26.780 --> 52:30.190 We don't want to honor those kinds of feelings of jealousy. 52:30.190 --> 52:32.320 So the argument seemed to be when you get to finance you 52:32.317 --> 52:33.977 don't have to worry about externalities, 52:33.980 --> 52:36.770 and yes you have to worry about perfect competition but once 52:36.766 --> 52:39.646 you've got perfect competition that's the end of the story. 52:39.650 --> 52:42.160 So the rest of the course now is going to be, 52:42.159 --> 52:45.839 if you forget about externalities because you forget 52:45.838 --> 52:49.948 about pollution because we're trading pieces of paper, 52:49.949 --> 52:54.089 they're not polluting anything, why shouldn't the financial 52:54.088 --> 52:55.728 markets be efficient? 52:55.730 --> 52:58.450 Why doesn't this argument apply to the financial markets, 52:58.454 --> 53:01.084 and one of the critical financial markets is the credit 53:01.083 --> 53:01.623 market. 53:01.619 --> 53:04.589 So can you understand the credit market on just these 53:04.585 --> 53:07.775 terms and therefore argue that the market's efficient? 53:07.780 --> 53:10.590 So we're going to spend a large part of the course talking about 53:10.588 --> 53:10.898 that. 53:10.900 --> 53:14.060 So unless you had some particular point to raise now, 53:14.063 --> 53:17.473 I'm going to just use your comment as a good introduction 53:17.469 --> 53:19.599 to the next part of the course. 53:19.599 --> 53:24.079 So far there's nothing that seems to directly involve 53:24.077 --> 53:27.777 finance in what we've been talking about. 53:27.780 --> 53:29.950 So where does the finance come in? 53:29.949 --> 53:33.799 So Irving Fisher was the person who created the first really 53:33.797 --> 53:36.337 general equilibrium model of finance. 53:36.340 --> 53:39.530 So he was a Yale undergraduate. 53:39.530 --> 53:42.180 He was a superstar Yale undergraduate. 53:42.179 --> 53:47.529 He graduated first in his class from Yale. 53:47.530 --> 53:49.810 He was a math major. 53:49.809 --> 53:54.449 He decided that he wanted to build a financial equilibrium 53:54.447 --> 53:55.097 model. 53:55.099 --> 53:57.209 There were no economists there at his time. 53:57.210 --> 53:59.300 We're talking, in the late 1800s, 53:59.297 --> 54:02.177 and so he went to Gibbs, the famous physicist, 54:02.177 --> 54:05.527 one of America's most famous physicists at the time and said, 54:05.530 --> 54:11.320 "Can you advise an economics PhD?" 54:11.320 --> 54:12.680 And Gibbs said, "Well, if it's 54:12.675 --> 54:14.625 mathematical enough, it has a model and something, 54:14.632 --> 54:15.672 yes I can do it." 54:15.670 --> 54:18.610 And so Fisher's PhD dissertation was basically 54:18.605 --> 54:22.255 reinventing this general equilibrium model that I've just 54:22.257 --> 54:26.107 described and then making a machine to calculate equilibrium 54:26.105 --> 54:29.685 with water and pumps and water seeking its own level and 54:29.692 --> 54:31.782 solving for equilibrium. 54:31.780 --> 54:37.110 And we had this machine up until the 1970s in the Cowles 54:37.108 --> 54:40.208 Foundation when it got stolen. 54:40.210 --> 54:44.660 Some engineer named Sreenivasan about ten years ago was ready to 54:44.655 --> 54:47.065 rebuild the machine-- because we have the 54:47.067 --> 54:49.597 dissertation where it explains very carefully how Fisher built 54:49.596 --> 54:51.906 his machine, and he was going to rebuild it. 54:51.909 --> 54:55.809 And then he was offered ElBaradei's job. 54:55.809 --> 54:58.689 ElBaradei is the guy, the nuclear inspector who goes 54:58.693 --> 55:01.463 to Iraq and says they're building nuclear stuff or 55:01.463 --> 55:03.163 they're not building this. 55:03.159 --> 55:06.169 Anyway, so he was offered that job, and so he left Yale and 55:06.172 --> 55:08.512 then decided not to take the job after all. 55:08.510 --> 55:10.780 So we've still got ElBaradei in the job. 55:10.780 --> 55:13.930 But anyway, he left Yale and so we never got the machine built. 55:13.929 --> 55:14.819 He was a great guy. 55:14.820 --> 55:20.980 Anyway, so Fisher after writing his dissertation stayed at Yale 55:20.978 --> 55:25.548 as an assistant professor and he got tenure. 55:25.550 --> 55:28.200 He became the most famous economist probably in the 55:28.195 --> 55:29.355 country for a while. 55:29.360 --> 55:35.230 He invented finance, as I'm about to tell you, 55:35.230 --> 55:38.850 and in a way this model of finance I'm about to present 55:38.846 --> 55:42.856 which was in retrospect clearly the most interesting model of 55:42.864 --> 55:46.324 finance of its time, and he was a financial 55:46.320 --> 55:47.210 economist. 55:47.210 --> 55:50.960 He also was an entrepreneur, so you'll see later that he 55:50.963 --> 55:52.743 started his own company. 55:52.739 --> 55:57.299 It was a proto-computer company called Remington and he managed 55:57.304 --> 56:02.094 to make $10 million dollars for his company in the early 1900s. 56:02.090 --> 56:05.840 And he was a friend of Roosevelt's and he kept advising 56:05.840 --> 56:09.590 Roosevelt during the Depression to print more money. 56:09.590 --> 56:10.410 He said, "Print more money. 56:10.409 --> 56:10.869 Print more money. 56:10.869 --> 56:12.749 Print more money," and the Yale Library has all 56:12.746 --> 56:14.106 these letters he sent to Roosevelt. 56:14.110 --> 56:16.760 It also has Roosevelt's responses where you can tell 56:16.764 --> 56:19.054 that Roosevelt is paying no attention to him, 56:19.054 --> 56:21.194 but Fisher's ego seemed to be so big. 56:21.190 --> 56:23.470 Roosevelt says, "I'm very glad that the 56:23.469 --> 56:26.649 good professor has made such worthy recommendations and we'll 56:26.648 --> 56:28.768 certainly take them seriously." 56:28.768 --> 56:30.828 And then Fisher writes back and says, "I'm glad you're 56:30.826 --> 56:32.276 going to follow what I suggested." 56:32.280 --> 56:35.970 So anyway, Fisher, convinced that the economy was 56:35.972 --> 56:39.892 going to turn around, poured all his money into this 56:39.894 --> 56:42.284 computer company Remington. 56:42.280 --> 56:45.230 The company in the great stock market crash nearly went 56:45.233 --> 56:46.823 bankrupt and was leveraged. 56:46.820 --> 56:51.450 So he had to go to his wife who was very wealthy and borrow her 56:51.449 --> 56:53.989 money, and they lost that money. 56:53.989 --> 56:56.839 Then he went to her parents and borrowed all their money and 56:56.840 --> 56:57.710 lost that money. 56:57.710 --> 56:59.830 And finally he had lost everything of his money, 56:59.829 --> 57:01.989 his wife's money and his wife's parents' money and they were 57:01.985 --> 57:03.295 about to take his house from him, 57:03.300 --> 57:05.760 foreclose on his house, and so Yale was forced to buy 57:05.762 --> 57:08.132 his house for him so he could continue to teach. 57:08.130 --> 57:12.400 So he set a bad precedent for economists at Yale because 57:12.398 --> 57:17.288 whenever any economist at Yale has financial advice to give, 57:17.289 --> 57:20.899 someone always quotes Irving Fisher's 1929 line that the 57:20.902 --> 57:24.122 economy had reached a permanently high plateau. 57:24.119 --> 57:29.689 But anyway, despite all that his theories are well worth 57:29.686 --> 57:30.796 studying. 57:30.800 --> 57:32.900 He was famous for a few other things. 57:32.900 --> 57:36.380 Shiller is clearly trying to imitate Fisher. 57:36.380 --> 57:39.700 Fisher wrote a huge number of books, 50 books about 57:39.697 --> 57:40.557 everything. 57:40.559 --> 57:43.729 So he got tuberculosis and he survived and he wrote a series 57:43.733 --> 57:46.743 of books on good living and good health and how to combat 57:46.744 --> 57:47.664 tuberculosis. 57:47.659 --> 57:50.659 So he said you had to exercise, you had to get fresh air, 57:50.664 --> 57:52.494 and these were huge bestsellers. 57:52.489 --> 57:57.529 Then he said that drinking was a terrible thing. 57:57.530 --> 58:00.550 So he did an experiment in class where he would have his 58:00.550 --> 58:03.020 students do pushups, then he'd give them all a 58:03.021 --> 58:03.681 martini. 58:03.679 --> 58:05.529 Then he'd count how many pushups they could do. 58:05.530 --> 58:07.570 Then he'd give them a second martini and count how many 58:07.565 --> 58:08.465 pushups they could do. 58:08.469 --> 58:13.659 And he found that there was a 10% reduction for each martini. 58:13.659 --> 58:16.799 And so he said that prohibition--since the average 58:16.802 --> 58:20.782 business person had two drinks at lunch--that prohibition would 58:20.777 --> 58:22.507 increase output by 20%. 58:22.510 --> 58:27.910 And so he was one of the leading proponents of 58:27.909 --> 58:29.589 prohibition. 58:29.590 --> 58:34.690 So after the stock market crash he helped form the Cowles 58:34.686 --> 58:35.866 Foundation. 58:35.869 --> 58:40.979 So Cowles, who was a Yale undergraduate, 58:40.978 --> 58:47.918 had run a very famous macro forecasting company in the 58:47.920 --> 58:49.230 1920s. 58:49.230 --> 58:52.230 And in 1929 the stock market crashed. 58:52.230 --> 58:53.640 There were a whole series of these. 58:53.639 --> 58:54.659 They sent newsletters. 58:54.659 --> 58:56.699 The same kind of stuff they do now. 58:56.699 --> 58:59.829 He, unlike all the others, after the fact realized that he 58:59.826 --> 59:02.176 hadn't anticipated anything about the crash, 59:02.184 --> 59:04.384 much like nobody anticipated it now. 59:04.380 --> 59:06.960 And he was so embarrassed he went and collected, 59:06.960 --> 59:09.470 not only his old recommendations themselves, 59:09.469 --> 59:12.449 but all his competitors' and so he published a famous paper in 59:12.452 --> 59:15.242 which he argued that economists had no idea what they were 59:15.239 --> 59:17.469 talking about, in fact that they were frauds. 59:17.469 --> 59:20.499 And so he went to his old economist Irving Fisher at Yale 59:20.500 --> 59:22.910 and said, "Look, I just believe that 59:22.909 --> 59:25.519 economics without mathematics has no meaning. 59:25.518 --> 59:29.668 I want to start a mathematical wing of economics and I've got a 59:29.672 --> 59:31.082 lot of money." 59:31.079 --> 59:33.989 His family owned The Chicago Tribune and 59:33.990 --> 59:37.210 The Seattle Times and a whole bunch of other newspapers 59:37.208 --> 59:40.068 and there was a famous fashion model in the family, 59:40.070 --> 59:41.550 Fleur Cowles and stuff like that. 59:41.550 --> 59:45.140 So anyway, so with Cowles' money Fisher started the 59:45.137 --> 59:49.727 Econometric Society which is the most famous mathematical society 59:49.728 --> 59:52.338 in economics, Econometrica, 59:52.340 --> 59:55.220 which is the most famous journal in economics, 59:55.219 --> 59:58.669 and the Cowles Foundation which--they started off in 59:58.666 --> 1:00:01.366 Colorado Springs because Fisher said, 1:00:01.369 --> 1:00:03.539 "Oh, the weather's so good there everyone will want to go 1:00:03.536 --> 1:00:04.846 there and it's good for them." 1:00:04.849 --> 1:00:08.459 So nobody would go there so they had to move it to Chicago 1:00:08.460 --> 1:00:12.330 where the Tribune was and it was in Chicago from 1930 1:00:12.326 --> 1:00:15.636 to 1955, and then in 1955 it moved to 1:00:15.637 --> 1:00:16.167 Yale. 1:00:16.170 --> 1:00:18.700 And since then the Cowles Foundation has been at Yale, 1:00:18.699 --> 1:00:20.799 and you'll hear a lot more about it later. 1:00:20.800 --> 1:00:23.080 But anyway, so Cowles [correction: Fisher] 1:00:23.083 --> 1:00:26.203 with that background he set out to figure out how to turn 1:00:26.204 --> 1:00:28.644 economics-- in fact he invented economics, 1:00:28.643 --> 1:00:31.053 this model of equilibrium to study finance, 1:00:31.050 --> 1:00:33.470 but so far we don't have any finance in the model. 1:00:33.469 --> 1:00:35.269 So how can you put it in? 1:00:35.269 --> 1:00:36.619 Well, what is missing? 1:00:36.619 --> 1:00:41.269 What do I need to put in the model in order to turn it into 1:00:41.268 --> 1:00:42.068 finance? 1:00:42.070 --> 1:00:43.270 What would you say is missing? 1:00:43.268 --> 1:00:47.408 How would you turn this into finance? 1:00:47.409 --> 1:00:48.509 What other key thing--yes? 1:00:48.510 --> 1:00:51.290 Student: You need some element of time. 1:00:51.289 --> 1:00:54.999 Prof: So the first thing--of course you may have 1:00:54.996 --> 1:00:58.696 read ahead in the notes, but that's a brilliant insight 1:00:58.704 --> 1:00:59.944 if it's yours. 1:00:59.940 --> 1:01:06.950 You need time in the model. 1:01:06.949 --> 1:01:10.479 So far we just had apples and oranges or something being 1:01:10.481 --> 1:01:11.061 traded. 1:01:11.059 --> 1:01:14.829 So the first thing is Fisher said you need to put time in the 1:01:14.827 --> 1:01:15.327 model. 1:01:15.329 --> 1:01:18.639 What else is critically missing in this model, 1:01:18.635 --> 1:01:21.645 one other thing, the major other thing? 1:01:21.650 --> 1:01:22.280 Yeah? 1:01:22.280 --> 1:01:23.380 Student: Risk. 1:01:23.380 --> 1:01:23.950 Prof: What? 1:01:23.949 --> 1:01:24.569 Student: Risk. 1:01:24.570 --> 1:01:25.580 Prof: Well, that's good. 1:01:25.579 --> 1:01:26.949 There's risk. 1:01:26.949 --> 1:01:29.009 That's missing entirely. 1:01:29.010 --> 1:01:32.600 So Fisher actually couldn't figure out how to put that in 1:01:32.599 --> 1:01:35.419 the model, but we're going to get to that. 1:01:35.420 --> 1:01:36.350 That's very good. 1:01:36.349 --> 1:01:38.889 What else is missing? 1:01:38.889 --> 1:01:39.619 Yeah? 1:01:39.619 --> 1:01:44.579 Student: Impatience. 1:01:44.579 --> 1:01:46.739 Prof: Fisher, you'll see, had something to 1:01:46.740 --> 1:01:49.310 say about time and impatience that people always care more 1:01:49.307 --> 1:01:51.017 about the present than the future. 1:01:51.018 --> 1:01:53.958 But what other fundamental thing about--what is finance all 1:01:53.963 --> 1:01:54.373 about? 1:01:54.369 --> 1:01:56.379 When you think about finance what do you think about? 1:01:56.380 --> 1:02:00.190 Student: Savings or credit rates. 1:02:00.190 --> 1:02:05.530 Prof: So you think about credit, as she said, 1:02:05.525 --> 1:02:09.705 credit, interest rates, and what else? 1:02:09.710 --> 1:02:10.400 Student: Return. 1:02:10.400 --> 1:02:14.410 Prof: Return, that's quite related to that. 1:02:14.409 --> 1:02:16.579 So all this has to go in the model, but what fundamental 1:02:16.577 --> 1:02:18.977 object when you think--the first thing you think about finance 1:02:18.983 --> 1:02:19.973 what do you think of? 1:02:19.969 --> 1:02:20.879 Yes? 1:02:20.880 --> 1:02:21.680 Student: Money. 1:02:21.679 --> 1:02:24.569 Prof: Money, so that's not what I had in 1:02:24.574 --> 1:02:27.284 mind, but I'm glad you think about money. 1:02:27.280 --> 1:02:28.280 He's right. 1:02:28.280 --> 1:02:30.910 That is what most people think about, so money. 1:02:30.909 --> 1:02:33.649 So in this course we're not going to provide a theory of 1:02:33.650 --> 1:02:34.050 money. 1:02:34.050 --> 1:02:37.590 So Fisher did provide this theory of money and that's what 1:02:37.588 --> 1:02:40.628 he was talking to Roosevelt about all the time. 1:02:40.630 --> 1:02:43.190 We're going to talk about inflation and stuff like that, 1:02:43.190 --> 1:02:46.170 but we're not going to explain where the inflation comes from. 1:02:46.170 --> 1:02:49.160 So inflation is going to very important, but we're not going 1:02:49.157 --> 1:02:50.877 to talk about a theory of money. 1:02:50.880 --> 1:02:54.030 So what else is missing, something really basic though? 1:02:54.030 --> 1:02:54.560 Yes? 1:02:54.559 --> 1:02:57.639 Student: Institutions. 1:02:57.639 --> 1:03:03.969 Prof: So there are no banks and things like that, 1:03:03.972 --> 1:03:05.702 institutions. 1:03:05.699 --> 1:03:07.419 And still something. 1:03:07.420 --> 1:03:09.440 Come on what's the... 1:03:09.440 --> 1:03:10.100 Student: Loaning. 1:03:10.099 --> 1:03:11.719 Prof: What? 1:03:11.719 --> 1:03:12.749 Student: Loaning funds. 1:03:12.750 --> 1:03:14.860 Prof: Loaning funds, well somebody said credit. 1:03:14.860 --> 1:03:17.530 That's sort of loaning funds and interest rates. 1:03:17.530 --> 1:03:18.550 Student: Collateral. 1:03:18.550 --> 1:03:20.080 Prof: Collateral, that's something. 1:03:20.079 --> 1:03:21.799 I'm going to add that. 1:03:21.800 --> 1:03:22.370 Thank you. 1:03:22.369 --> 1:03:23.939 What a brilliant idea. 1:03:23.940 --> 1:03:28.190 What basic thing? 1:03:28.190 --> 1:03:28.690 Come on. 1:03:28.690 --> 1:03:29.230 Yes? 1:03:29.230 --> 1:03:30.160 Student: Wealth. 1:03:30.159 --> 1:03:31.499 Prof: Wealth, that's in the model. 1:03:31.500 --> 1:03:32.310 Where's wealth? 1:03:32.309 --> 1:03:33.559 It's already there. 1:03:33.559 --> 1:03:36.549 So when you take your endowment and you multiply it by the 1:03:36.547 --> 1:03:39.007 prices in your budget set that's your wealth. 1:03:39.010 --> 1:03:40.960 So we've got wealth. 1:03:40.960 --> 1:03:42.300 Student: Capital. 1:03:42.300 --> 1:03:46.980 Prof: Capital, so capital--and what is 1:03:46.976 --> 1:03:48.036 capital? 1:03:48.039 --> 1:03:49.049 What do you mean by capital? 1:03:49.050 --> 1:03:51.470 If it's money, I already told you that's not 1:03:51.474 --> 1:03:52.944 quite going to be there. 1:03:52.940 --> 1:03:53.630 Student: Contract. 1:03:53.630 --> 1:03:57.440 Prof: Contract, so we're getting there. 1:03:57.440 --> 1:03:59.000 Student: Assets. 1:03:59.000 --> 1:04:01.840 Prof: Assets, so assets--that's the main 1:04:01.840 --> 1:04:04.990 thing so I'm going to write it over here, assets. 1:04:04.989 --> 1:04:09.829 After all finance is a lot about the stock market, 1:04:09.829 --> 1:04:13.089 bond markets, mortgage markets. 1:04:13.090 --> 1:04:17.910 Those are all assets that you buy that pay dividends later. 1:04:17.909 --> 1:04:21.109 So assets are things that pay dividends at a later time. 1:04:21.110 --> 1:04:22.830 That's the critical thing that's missing. 1:04:22.829 --> 1:04:24.729 And when we talk about finance we want to know, 1:04:24.733 --> 1:04:26.683 what should the value of the stock market be? 1:04:26.679 --> 1:04:31.499 How much should a mortgage be worth? 1:04:31.500 --> 1:04:33.330 So those are the things we have to add. 1:04:33.329 --> 1:04:34.659 Now, how are we going to add them? 1:04:34.659 --> 1:04:37.029 It sounds like you might have to just totally redo the theory 1:04:37.034 --> 1:04:39.414 and start from scratch and do something totally different. 1:04:39.409 --> 1:04:41.189 How horrible that would be. 1:04:41.190 --> 1:04:43.510 It would mean I also wasted three lectures. 1:04:43.510 --> 1:04:46.820 So Fisher said, "You don't really have to 1:04:46.817 --> 1:04:49.167 change very much at all." 1:04:49.170 --> 1:04:51.700 So he said, "Number one, time." 1:04:51.699 --> 1:04:53.789 How do you put time in the model? 1:04:53.789 --> 1:05:14.389 Simply think of Y as the same good as X but one period later. 1:05:14.389 --> 1:05:16.919 So Fisher said, "Time isn't such a big 1:05:16.918 --> 1:05:17.338 deal. 1:05:17.340 --> 1:05:18.680 We've already got two goods. 1:05:18.679 --> 1:05:23.509 These crazy marginalists before me they had good X, 1:05:23.510 --> 1:05:27.120 which was interesting, and this very boring good Y and 1:05:27.123 --> 1:05:31.423 so it's not so clear how you can have time and everything in the 1:05:31.418 --> 1:05:33.088 model, but I've already got two 1:05:33.085 --> 1:05:34.905 interesting goods, X and Y. 1:05:34.909 --> 1:05:40.759 They're both entering with the same symmetric properties. 1:05:40.760 --> 1:05:43.570 Neither one of them seems more interesting or more important 1:05:43.567 --> 1:05:45.797 than the other one, so in fact maybe they're the 1:05:45.802 --> 1:05:46.472 same good. 1:05:46.469 --> 1:05:49.599 It's just--X and Y is the same good, it's one comes one period 1:05:49.601 --> 1:05:50.321 later." 1:05:50.320 --> 1:05:51.880 Now next class, we're going to talk about 1:05:51.878 --> 1:05:53.788 you're not going to value them the same amount. 1:05:53.789 --> 1:05:56.049 The utility for X and Y won't be the same. 1:05:56.050 --> 1:05:59.500 So here A obviously thinks more of X than he does of Y, 1:05:59.500 --> 1:06:02.250 and by the way, B, she also thinks more of X 1:06:02.248 --> 1:06:03.908 than she does about Y. 1:06:03.909 --> 1:06:05.309 So why is that? 1:06:05.309 --> 1:06:08.649 Well, if Y is the same as X but a year later--people are 1:06:08.650 --> 1:06:09.380 impatient. 1:06:09.380 --> 1:06:13.470 So the reason why A might like X better than Y and B also is 1:06:13.467 --> 1:06:17.277 not because they're different goods, but because they're 1:06:17.278 --> 1:06:18.248 impatient. 1:06:18.250 --> 1:06:19.490 So we're going to come to that later. 1:06:19.489 --> 1:06:24.639 So we can have the idea that they're two different time 1:06:24.644 --> 1:06:30.184 periods in the economy simply by change of notation and not 1:06:30.181 --> 1:06:33.621 introducing anything new at all. 1:06:33.619 --> 1:06:37.129 Well, what about assets? 1:06:37.130 --> 1:06:39.690 So I'm going to come back to that in the next class. 1:06:39.690 --> 1:06:40.780 What about assets? 1:06:40.780 --> 1:06:43.110 How can you introduce assets in to the economy? 1:06:43.110 --> 1:06:45.980 Well, Fisher says, "What is the essence of an 1:06:45.981 --> 1:06:46.451 asset? 1:06:46.449 --> 1:06:50.299 The essence of an asset is that it's something that you hold 1:06:50.295 --> 1:06:54.335 today because later it's going to deliver you money or goods or 1:06:54.335 --> 1:06:55.635 something." 1:06:55.639 --> 1:06:59.069 And we're not going to talk about money so we're imagining 1:06:59.072 --> 1:07:00.762 these are all real assets. 1:07:00.760 --> 1:07:03.820 So they're going to deliver goods in the future. 1:07:03.820 --> 1:07:07.670 So the definition of an asset is an asset, remember, 1:07:07.668 --> 1:07:10.008 we have to make it mathematical, 1:07:10.007 --> 1:07:14.987 so Fisher started this society to make economics mathematical. 1:07:14.989 --> 1:07:18.799 So an asset is literally something that delivers goods in 1:07:18.795 --> 1:07:19.675 the future. 1:07:19.679 --> 1:07:30.389 So an asset is defined by its payoffs, D_X and 1:07:30.385 --> 1:07:33.495 D_Y. 1:07:33.500 --> 1:07:37.240 So we can call that asset alpha, an asset alpha is defined 1:07:37.239 --> 1:07:39.339 by its payoffs, D_X and 1:07:39.338 --> 1:07:40.518 D_Y. 1:07:40.518 --> 1:07:43.398 So it's going to pay a certain amount of X and a certain amount 1:07:43.404 --> 1:07:43.734 of Y. 1:07:43.730 --> 1:07:45.240 That's all there is. 1:07:45.239 --> 1:07:48.249 Now, we're going to ask, do people actually know what 1:07:48.248 --> 1:07:50.098 the payoff of Y's going to be? 1:07:50.099 --> 1:07:52.319 If Y's in the future that means it hasn't happened yet. 1:07:52.320 --> 1:07:55.120 Do we really know what Y's going to be? 1:07:55.119 --> 1:07:59.589 And so maybe we know less about one asset and more about another 1:07:59.592 --> 1:08:03.712 asset, so maybe one asset is riskier than another asset. 1:08:03.710 --> 1:08:06.140 So someone said risk, obviously that's going to be a 1:08:06.137 --> 1:08:07.467 very fundamental question. 1:08:07.469 --> 1:08:12.249 But before we come to risk, if we don't have any risk yet, 1:08:12.250 --> 1:08:14.480 because Fisher didn't know how to put it in and we're going to 1:08:14.478 --> 1:08:16.158 start with his model where there's no risk, 1:08:16.158 --> 1:08:19.488 then if there's just today and tomorrow and everybody's 1:08:19.488 --> 1:08:21.438 rational, everybody, since there's no 1:08:21.440 --> 1:08:23.530 uncertainty, everybody wants to be able to 1:08:23.533 --> 1:08:26.163 anticipate exactly what the dividend's going to be. 1:08:26.158 --> 1:08:30.098 So the asset is no different from what its dividend is going 1:08:30.104 --> 1:08:30.644 to be. 1:08:30.640 --> 1:08:33.870 Maybe it's paying something today and also it's going to pay 1:08:33.872 --> 1:08:36.562 something tomorrow, so assets are defined by their 1:08:36.556 --> 1:08:38.636 dividends, nothing more than that. 1:08:38.640 --> 1:08:46.570 So the model now becomes a model where we have the W^(I) 1:08:46.570 --> 1:08:52.340 and then we have the goods as before, 1:08:52.340 --> 1:08:57.480 the endowments of the goods, but we've added a new thing 1:08:57.479 --> 1:09:02.899 which is we've added the assets D alpha X and D alpha Y. 1:09:02.899 --> 1:09:06.489 So the alpha's over all the assets, and we have to add 1:09:06.488 --> 1:09:10.008 everybody's ownership of the assets to begin with. 1:09:10.010 --> 1:09:11.630 So we have to have theta. 1:09:11.630 --> 1:09:22.630 I'm going to call it theta bar alpha, theta bar i alpha. 1:09:22.630 --> 1:09:26.180 So how much of asset alpha does i own? 1:09:26.180 --> 1:09:30.650 So that's the economy that we're going to start with. 1:09:30.649 --> 1:09:34.139 So we've made the model a bit more complicated. 1:09:34.140 --> 1:09:39.650 We started where we did before with utilities and endowments of 1:09:39.649 --> 1:09:40.359 goods. 1:09:40.359 --> 1:09:43.139 Now we're adding assets which payoff more goods, 1:09:43.136 --> 1:09:46.266 and of course people begin owning trees and stuff like 1:09:46.268 --> 1:09:46.798 that. 1:09:46.800 --> 1:09:48.010 So a tree is an asset. 1:09:48.010 --> 1:09:52.270 It's going to pay apples later on and various different people 1:09:52.265 --> 1:09:54.355 own various different trees. 1:09:54.359 --> 1:09:57.499 So to the old economy we add something new. 1:09:57.500 --> 1:10:01.900 And now Fisher's ingenious insight was that if you think 1:10:01.899 --> 1:10:04.519 about this, which we will for a little 1:10:04.524 --> 1:10:07.654 while, you'll be able to discover that you can simplify 1:10:07.648 --> 1:10:10.828 this economy back down to the economy we began with. 1:10:10.828 --> 1:10:13.888 So you can start talking about assets and interest and all 1:10:13.890 --> 1:10:17.060 kinds of others things using the same analysis as before. 1:10:17.060 --> 1:10:22.620 So with my last thought here, what does interest have to do 1:10:22.617 --> 1:10:26.447 with P_X over P_Y? 1:10:26.448 --> 1:10:27.698 What is P_X over P_Y? 1:10:27.699 --> 1:10:29.119 Suppose they're both apples. 1:10:29.119 --> 1:10:30.229 So X is an apple today. 1:10:30.229 --> 1:10:33.039 Y is an apple next year. 1:10:33.038 --> 1:10:34.758 What is P_X over P_Y? 1:10:34.760 --> 1:10:38.430 It says how much an apple is worth today relative to how much 1:10:38.434 --> 1:10:40.214 an apple is worth tomorrow. 1:10:40.210 --> 1:10:46.580 So this is going to turn out to be 1 the real interest rate, 1:10:46.577 --> 1:10:49.057 Fisher would call it. 1:10:49.060 --> 1:10:51.430 Because if P_X is more than P_Y, 1:10:51.430 --> 1:10:55.730 let's say P_X is 20% more than P_Y what it 1:10:55.731 --> 1:10:59.891 means is by giving up one apple today you can get 1.2 apples 1:10:59.893 --> 1:11:00.813 tomorrow. 1:11:00.810 --> 1:11:03.190 That's just what happens when you put money in the bank. 1:11:03.189 --> 1:11:06.489 You give up something today you get exactly the same thing back 1:11:06.493 --> 1:11:08.043 next period but more of it. 1:11:08.038 --> 1:11:11.238 So you see just by having time and the prices like we had 1:11:11.238 --> 1:11:14.438 before you're going to see very quickly how interest, 1:11:14.439 --> 1:11:17.609 assets, everything is going to start coming into play very 1:11:17.613 --> 1:11:18.173 quickly. 1:11:18.170 --> 1:11:21.430 So I didn't put the problem set on the web yet because I wasn't 1:11:21.431 --> 1:11:25.081 sure how far we'll get, but I'll put it on in a couple 1:11:25.081 --> 1:11:28.201 hours and we'll start again on Thursday. 1:11:28.199 --> 1:11:32.999