WEBVTT 00:00:00.000 --> 00:00:00.985 align:middle line:90% 00:00:00.985 --> 00:00:02.440 align:middle line:90% [SQUEAKING] 00:00:02.440 --> 00:00:04.380 align:middle line:90% [RUSTLING] 00:00:04.380 --> 00:00:07.775 align:middle line:90% [CLICKING] 00:00:07.775 --> 00:00:09.540 align:middle line:90% 00:00:09.540 --> 00:00:14.800 align:middle line:84% IAN BALL: So today, we're going to talk about signaling. 00:00:14.800 --> 00:00:19.480 align:middle line:84% And what we did last week was kind 00:00:19.480 --> 00:00:21.180 align:middle line:84% of a simple example of signaling, 00:00:21.180 --> 00:00:23.720 align:middle line:84% and today, we're going to do a more involved application 00:00:23.720 --> 00:00:25.040 align:middle line:90% of signaling. 00:00:25.040 --> 00:00:27.380 align:middle line:90% So what do we mean by signaling? 00:00:27.380 --> 00:00:28.840 align:middle line:90% What's the key idea here? 00:00:28.840 --> 00:00:33.600 align:middle line:84% Well, we're going to look at games where we have two players. 00:00:33.600 --> 00:00:40.040 align:middle line:90% We have a sender and a receiver. 00:00:40.040 --> 00:00:43.180 align:middle line:84% And the sender has some private information. 00:00:43.180 --> 00:00:48.280 align:middle line:90% 00:00:48.280 --> 00:00:50.600 align:middle line:84% And then the sender, knowing their private information, 00:00:50.600 --> 00:00:52.995 align:middle line:84% is going to make some choice about an actions. 00:00:52.995 --> 00:00:54.620 align:middle line:84% They're going to then choose an action. 00:00:54.620 --> 00:00:57.480 align:middle line:90% 00:00:57.480 --> 00:01:00.360 align:middle line:84% And we often call this private information, 00:01:00.360 --> 00:01:01.640 align:middle line:90% maybe the sender's type. 00:01:01.640 --> 00:01:07.100 align:middle line:84% This is what the sender knows that the receiver doesn't know. 00:01:07.100 --> 00:01:10.100 align:middle line:84% And then the receiver is going to observe the action 00:01:10.100 --> 00:01:12.500 align:middle line:84% that the sender chooses, but they're not 00:01:12.500 --> 00:01:15.700 align:middle line:84% going to directly observe the sender's private information. 00:01:15.700 --> 00:01:28.500 align:middle line:84% So the receiver observes the action, but not type. 00:01:28.500 --> 00:01:30.780 align:middle line:84% So they see what action the sender takes, 00:01:30.780 --> 00:01:33.500 align:middle line:84% but they can't observe the private information 00:01:33.500 --> 00:01:35.580 align:middle line:90% that the sender has. 00:01:35.580 --> 00:01:37.940 align:middle line:84% And then based upon that, they're 00:01:37.940 --> 00:01:39.320 align:middle line:90% also going to choose an action. 00:01:39.320 --> 00:01:49.300 align:middle line:90% 00:01:49.300 --> 00:01:51.780 align:middle line:84% So in the examples that we're going to look at, 00:01:51.780 --> 00:01:54.940 align:middle line:84% the receiver, when they're choosing their action, 00:01:54.940 --> 00:01:58.280 align:middle line:84% cares a lot about the sender's private information. 00:01:58.280 --> 00:02:02.400 align:middle line:84% And the receiver is going to try to learn or infer something 00:02:02.400 --> 00:02:04.520 align:middle line:84% about the sender's private information based 00:02:04.520 --> 00:02:07.080 align:middle line:84% upon the action that the sender chooses. 00:02:07.080 --> 00:02:09.800 align:middle line:84% And conversely, the sender cares a lot 00:02:09.800 --> 00:02:12.200 align:middle line:84% about what the receiver believes about them, 00:02:12.200 --> 00:02:15.200 align:middle line:84% and the sender is going to try to convey information 00:02:15.200 --> 00:02:18.660 align:middle line:84% about their private information through their choice of action. 00:02:18.660 --> 00:02:21.540 align:middle line:84% Now that's a bit abstract, so let's think of some examples. 00:02:21.540 --> 00:02:26.000 align:middle line:90% 00:02:26.000 --> 00:02:28.920 align:middle line:84% I don't know if any of you come across-- 00:02:28.920 --> 00:02:31.500 align:middle line:84% heard of the term signaling used colloquially 00:02:31.500 --> 00:02:34.260 align:middle line:84% and heard of examples where this term is applied. 00:02:34.260 --> 00:02:37.400 align:middle line:90% 00:02:37.400 --> 00:02:38.540 align:middle line:90% So a few examples. 00:02:38.540 --> 00:02:40.980 align:middle line:84% I guess a classic one would be luxury goods. 00:02:40.980 --> 00:02:43.680 align:middle line:90% 00:02:43.680 --> 00:02:46.840 align:middle line:84% So here, the story would be that the private information 00:02:46.840 --> 00:02:49.560 align:middle line:84% that the sender has, their type is maybe how rich they 00:02:49.560 --> 00:02:52.520 align:middle line:90% are, how much money they have. 00:02:52.520 --> 00:02:58.660 align:middle line:84% The action they're choosing is what brand of bag to buy. 00:02:58.660 --> 00:03:01.260 align:middle line:84% The receiver, then, doesn't directly 00:03:01.260 --> 00:03:03.220 align:middle line:84% know how rich the sender is, but they 00:03:03.220 --> 00:03:05.463 align:middle line:84% do observe what brand of item they choose. 00:03:05.463 --> 00:03:07.380 align:middle line:84% And the receiver is going to make an inference 00:03:07.380 --> 00:03:09.100 align:middle line:84% about how rich the person is based 00:03:09.100 --> 00:03:11.780 align:middle line:84% upon how luxurious the item is that they hold. 00:03:11.780 --> 00:03:15.080 align:middle line:84% And the sender, wanting to be perceived as rich, 00:03:15.080 --> 00:03:18.780 align:middle line:84% will then choose to buy something luxurious. 00:03:18.780 --> 00:03:20.223 align:middle line:90% Now, why does this work? 00:03:20.223 --> 00:03:21.640 align:middle line:84% Well, your concern might be, well, 00:03:21.640 --> 00:03:24.220 align:middle line:84% look, if everyone wants to be perceived as rich, 00:03:24.220 --> 00:03:27.422 align:middle line:84% why doesn't everyone just buy the luxury item? 00:03:27.422 --> 00:03:29.380 align:middle line:84% And the issue is that, well, they're expensive, 00:03:29.380 --> 00:03:31.000 align:middle line:84% and only some people can afford them. 00:03:31.000 --> 00:03:32.980 align:middle line:84% So a crucial aspect for these signals 00:03:32.980 --> 00:03:35.692 align:middle line:84% to be credible is that the action people are taking 00:03:35.692 --> 00:03:37.900 align:middle line:84% is generally going to have a different cost depending 00:03:37.900 --> 00:03:39.200 align:middle line:90% on their private information. 00:03:39.200 --> 00:03:41.140 align:middle line:84% If everyone was able to buy luxury items because they 00:03:41.140 --> 00:03:42.682 align:middle line:84% were so cheap, that wouldn't credibly 00:03:42.682 --> 00:03:43.767 align:middle line:90% convey that I was wealthy. 00:03:43.767 --> 00:03:46.100 align:middle line:84% But for a very expensive item, because only sufficiently 00:03:46.100 --> 00:03:47.475 align:middle line:84% wealthy people can buy it, that's 00:03:47.475 --> 00:03:49.480 align:middle line:90% going to convey information. 00:03:49.480 --> 00:03:52.960 align:middle line:84% Another classic story from the animal world, I guess, 00:03:52.960 --> 00:03:57.270 align:middle line:90% is a peacock's feathers. 00:03:57.270 --> 00:03:59.750 align:middle line:84% Any ideas about what's being conveyed here 00:03:59.750 --> 00:04:02.770 align:middle line:84% and what's the role of the peacock's feathers? 00:04:02.770 --> 00:04:06.110 align:middle line:90% 00:04:06.110 --> 00:04:11.010 align:middle line:84% Why do peacocks grow very, very ostentatious large feathers? 00:04:11.010 --> 00:04:14.090 align:middle line:84% Well, it's not because people really care about the feathers. 00:04:14.090 --> 00:04:16.310 align:middle line:84% The idea would be that this is a signal of health. 00:04:16.310 --> 00:04:19.110 align:middle line:84% So if we map this into our example, the sender, 00:04:19.110 --> 00:04:22.150 align:middle line:84% one peacock privately has private information 00:04:22.150 --> 00:04:23.670 align:middle line:90% about how healthy they are. 00:04:23.670 --> 00:04:25.870 align:middle line:84% They're trying to attract maybe a mate, 00:04:25.870 --> 00:04:28.870 align:middle line:84% and the mate cares about how healthy the peacock is. 00:04:28.870 --> 00:04:31.650 align:middle line:84% If the peacock grows these enormous feathers, 00:04:31.650 --> 00:04:32.650 align:middle line:90% that's extremely costly. 00:04:32.650 --> 00:04:33.730 align:middle line:90% It takes a lot of energy. 00:04:33.730 --> 00:04:36.250 align:middle line:90% You have to-- to grow this. 00:04:36.250 --> 00:04:38.855 align:middle line:84% So by growing very, very large, ostentatious feathers, 00:04:38.855 --> 00:04:40.230 align:middle line:84% they're conveying to the receiver 00:04:40.230 --> 00:04:41.810 align:middle line:84% that I must be very, very healthy. 00:04:41.810 --> 00:04:44.530 align:middle line:84% And the receiver doesn't care about the feathers, 00:04:44.530 --> 00:04:46.290 align:middle line:84% they just care about underlying health, 00:04:46.290 --> 00:04:49.310 align:middle line:84% and the feathers provide a signal of that. 00:04:49.310 --> 00:04:52.090 align:middle line:84% One classic example, which we'll go through today, 00:04:52.090 --> 00:04:55.930 align:middle line:90% would be college. 00:04:55.930 --> 00:04:59.290 align:middle line:84% So this is a classic signaling application. 00:04:59.290 --> 00:05:05.130 align:middle line:84% How does college fit into this sender/receiver model, 00:05:05.130 --> 00:05:06.770 align:middle line:90% or how could it? 00:05:06.770 --> 00:05:07.570 align:middle line:90% Yeah? 00:05:07.570 --> 00:05:11.328 align:middle line:84% AUDIENCE: --show a job recruiter that you are aspiring-- 00:05:11.328 --> 00:05:12.370 align:middle line:90% IAN BALL: Right, exactly. 00:05:12.370 --> 00:05:13.610 align:middle line:84% So here, your private information 00:05:13.610 --> 00:05:15.050 align:middle line:84% might be how smart you are, how diligent 00:05:15.050 --> 00:05:16.590 align:middle line:84% you are, how hardworking you are, 00:05:16.590 --> 00:05:18.850 align:middle line:84% we could have various kind of positive attributes. 00:05:18.850 --> 00:05:23.170 align:middle line:84% You're choosing whether to go to some difficult university 00:05:23.170 --> 00:05:25.210 align:middle line:90% and get some demanding degree. 00:05:25.210 --> 00:05:27.770 align:middle line:84% The receiver might directly care about what 00:05:27.770 --> 00:05:29.937 align:middle line:84% you learn in your degree, but they might not. 00:05:29.937 --> 00:05:31.770 align:middle line:84% It might be that your degree is actually not 00:05:31.770 --> 00:05:34.137 align:middle line:84% that relevant for what you're doing on the job, 00:05:34.137 --> 00:05:35.970 align:middle line:84% but they care about your degree because it's 00:05:35.970 --> 00:05:37.427 align:middle line:90% a signal of how smart you are. 00:05:37.427 --> 00:05:39.010 align:middle line:84% The idea would be that some people are 00:05:39.010 --> 00:05:42.053 align:middle line:84% able to get this degree and pass, and other people are not, 00:05:42.053 --> 00:05:43.970 align:middle line:84% and therefore, your ability to get this degree 00:05:43.970 --> 00:05:50.270 align:middle line:84% and take this action reveals information about your ability. 00:05:50.270 --> 00:05:55.978 align:middle line:84% So let's dig in a little bit deeper into this college example 00:05:55.978 --> 00:05:58.270 align:middle line:84% because that's going to be kind of the main application 00:05:58.270 --> 00:06:00.590 align:middle line:84% we're going to look at today, and this is often 00:06:00.590 --> 00:06:02.690 align:middle line:84% goes under the name of job market signaling. 00:06:02.690 --> 00:06:09.510 align:middle line:90% 00:06:09.510 --> 00:06:11.910 align:middle line:84% So this was introduced in this famous paper 00:06:11.910 --> 00:06:14.930 align:middle line:90% by Michael Spence in 1973-- 00:06:14.930 --> 00:06:17.770 align:middle line:84% I think it was actually his PhD dissertation, 00:06:17.770 --> 00:06:19.550 align:middle line:84% and he ultimately won the Nobel Prize 00:06:19.550 --> 00:06:21.388 align:middle line:84% for this idea, which we might go over today 00:06:21.388 --> 00:06:22.930 align:middle line:84% and you might say it's pretty simple. 00:06:22.930 --> 00:06:24.090 align:middle line:84% How did you get the Nobel Prize for that? 00:06:24.090 --> 00:06:26.090 align:middle line:84% But I think things after the fact seem simple, 00:06:26.090 --> 00:06:29.110 align:middle line:84% but coming up with them is often quite difficult. 00:06:29.110 --> 00:06:32.970 align:middle line:84% So let's just observe some facts about the world, 00:06:32.970 --> 00:06:34.330 align:middle line:90% about college education. 00:06:34.330 --> 00:06:36.730 align:middle line:84% There's something called the college wage premium. 00:06:36.730 --> 00:06:40.788 align:middle line:90% 00:06:40.788 --> 00:06:43.330 align:middle line:84% And this refers to the fact that at least all over the world, 00:06:43.330 --> 00:06:44.890 align:middle line:84% but let's focus in the United States, 00:06:44.890 --> 00:06:47.550 align:middle line:84% people who go to college or complete a college degree tend 00:06:47.550 --> 00:06:49.570 align:middle line:84% to earn more than people who don't. 00:06:49.570 --> 00:06:51.690 align:middle line:84% And this is something labor economists study a lot 00:06:51.690 --> 00:06:53.590 align:middle line:84% and try to give different estimates. 00:06:53.590 --> 00:06:56.173 align:middle line:84% You'll see a lot of estimates floating around depending on how 00:06:56.173 --> 00:06:58.730 align:middle line:84% they measure it or which cohort or which kind of person 00:06:58.730 --> 00:07:01.410 align:middle line:84% they're looking at, but maybe a rough estimate in the US might 00:07:01.410 --> 00:07:07.490 align:middle line:84% be something like maybe 500K to a million is maybe kind 00:07:07.490 --> 00:07:09.190 align:middle line:84% of the middle-of-the-road estimates, 00:07:09.190 --> 00:07:10.630 align:middle line:84% and this is in lifetime earnings. 00:07:10.630 --> 00:07:15.827 align:middle line:90% 00:07:15.827 --> 00:07:16.910 align:middle line:90% So what do I mean by this? 00:07:16.910 --> 00:07:18.785 align:middle line:84% I mean, if you just look at people in the US, 00:07:18.785 --> 00:07:20.730 align:middle line:84% you look at those who have college degrees, 00:07:20.730 --> 00:07:23.063 align:middle line:84% and you compute the average earnings of those people who 00:07:23.063 --> 00:07:25.271 align:middle line:84% have college degrees, and then you look at people who 00:07:25.271 --> 00:07:27.938 align:middle line:84% don't have college degrees, and you look at the average earnings 00:07:27.938 --> 00:07:30.350 align:middle line:84% of those people over their lifetime, and you compare it, 00:07:30.350 --> 00:07:32.570 align:middle line:84% you see that people who go to college, on average, 00:07:32.570 --> 00:07:35.877 align:middle line:84% earn maybe 500K to $1 million more over their lifetime. 00:07:35.877 --> 00:07:38.210 align:middle line:84% As I said, there's a few different estimates about this. 00:07:38.210 --> 00:07:41.210 align:middle line:90% Now the question is why. 00:07:41.210 --> 00:07:43.730 align:middle line:90% Any thoughts about why this is? 00:07:43.730 --> 00:07:48.350 align:middle line:84% Why do people go to college earn more? 00:07:48.350 --> 00:07:49.370 align:middle line:90% You're here at college. 00:07:49.370 --> 00:07:52.790 align:middle line:84% Maybe you partly want to earn more by coming to college. 00:07:52.790 --> 00:07:54.410 align:middle line:90% Any thoughts about why this is? 00:07:54.410 --> 00:07:57.190 align:middle line:90% 00:07:57.190 --> 00:08:00.790 align:middle line:84% So maybe-- we often separate this into two different stories. 00:08:00.790 --> 00:08:03.010 align:middle line:84% So one is what we might call a causal story. 00:08:03.010 --> 00:08:06.430 align:middle line:90% 00:08:06.430 --> 00:08:09.070 align:middle line:84% I'll just say causal, we don't have to put the word story. 00:08:09.070 --> 00:08:10.230 align:middle line:90% So is it causal? 00:08:10.230 --> 00:08:11.710 align:middle line:84% Is it causal, what I mean is, let's 00:08:11.710 --> 00:08:14.030 align:middle line:84% look at someone who doesn't go to college. 00:08:14.030 --> 00:08:16.010 align:middle line:84% If they chose to instead go to college, 00:08:16.010 --> 00:08:17.430 align:middle line:90% would their earnings go up? 00:08:17.430 --> 00:08:18.870 align:middle line:90% That would mean it's causal. 00:08:18.870 --> 00:08:20.450 align:middle line:90% Or maybe it's not causal. 00:08:20.450 --> 00:08:23.003 align:middle line:90% 00:08:23.003 --> 00:08:24.670 align:middle line:84% Maybe someone who doesn't go to college, 00:08:24.670 --> 00:08:26.503 align:middle line:84% if they did go to college, wouldn't actually 00:08:26.503 --> 00:08:27.510 align:middle line:90% increase their earnings. 00:08:27.510 --> 00:08:30.030 align:middle line:84% Now I think a lot of people would look at this fact 00:08:30.030 --> 00:08:32.330 align:middle line:84% and just immediately say, oh, it must be causal. 00:08:32.330 --> 00:08:34.850 align:middle line:84% If going to college-- if people who go to college earn more, 00:08:34.850 --> 00:08:36.490 align:middle line:84% then it must be that if I choose to go to college, 00:08:36.490 --> 00:08:38.032 align:middle line:84% that's going to increase my earnings, 00:08:38.032 --> 00:08:39.963 align:middle line:84% and this is kind of a classic fallacy-- 00:08:39.963 --> 00:08:41.630 align:middle line:84% or potential fallacy when people haven't 00:08:41.630 --> 00:08:44.110 align:middle line:84% thought so carefully about this data, and that's possible. 00:08:44.110 --> 00:08:45.005 align:middle line:90% It could be causal. 00:08:45.005 --> 00:08:46.630 align:middle line:84% But one explanation is it's not causal. 00:08:46.630 --> 00:08:47.380 align:middle line:90% How could that be? 00:08:47.380 --> 00:08:51.090 align:middle line:84% How could it be that people who go to college earn more money, 00:08:51.090 --> 00:08:53.990 align:middle line:84% yet going to college wouldn't actually increase my earning? 00:08:53.990 --> 00:08:57.610 align:middle line:90% How could that story make sense? 00:08:57.610 --> 00:08:59.490 align:middle line:84% Well, the story here would be that, well, 00:08:59.490 --> 00:09:01.610 align:middle line:84% the people who go to college are very different from the people 00:09:01.610 --> 00:09:02.710 align:middle line:90% who don't go to college. 00:09:02.710 --> 00:09:03.950 align:middle line:90% You have two pools of people. 00:09:03.950 --> 00:09:05.830 align:middle line:84% You have people who go to college and people who don't. 00:09:05.830 --> 00:09:07.788 align:middle line:84% People who go to college are just smarter, more 00:09:07.788 --> 00:09:10.090 align:middle line:84% ambitious on average or have some other trait 00:09:10.090 --> 00:09:12.350 align:middle line:90% that is hard to measure. 00:09:12.350 --> 00:09:15.083 align:middle line:84% They do earn more, but it's not because they go to college, 00:09:15.083 --> 00:09:16.750 align:middle line:84% they just happen to be going to college. 00:09:16.750 --> 00:09:19.070 align:middle line:84% And this is, I guess, the classic story of this. 00:09:19.070 --> 00:09:21.278 align:middle line:84% This is a problem with all these health studies where 00:09:21.278 --> 00:09:23.050 align:middle line:84% they say people who eat five almonds a day 00:09:23.050 --> 00:09:24.435 align:middle line:90% live five years longer. 00:09:24.435 --> 00:09:25.810 align:middle line:84% Eating five almonds a day doesn't 00:09:25.810 --> 00:09:27.450 align:middle line:84% make you live five years longer, but people who eat almonds 00:09:27.450 --> 00:09:28.670 align:middle line:84% are different in a lot of different ways. 00:09:28.670 --> 00:09:29.830 align:middle line:84% They eat healthily in other ways, 00:09:29.830 --> 00:09:32.163 align:middle line:84% they run, they exercise, they do all these other things. 00:09:32.163 --> 00:09:36.332 align:middle line:84% So it could be that though there's a big earnings gap, 00:09:36.332 --> 00:09:37.790 align:middle line:84% it's not because you go to college, 00:09:37.790 --> 00:09:39.190 align:middle line:84% it's just a spurious correlation. 00:09:39.190 --> 00:09:41.820 align:middle line:84% So that's one potential explanation. 00:09:41.820 --> 00:09:43.900 align:middle line:84% Another would be that it is causal. 00:09:43.900 --> 00:09:46.640 align:middle line:84% That if I, as an individual, choose to go to college, 00:09:46.640 --> 00:09:49.220 align:middle line:84% that's going to increase my expected earning. 00:09:49.220 --> 00:09:51.820 align:middle line:84% And I think the best evidence suggests that both-- there's 00:09:51.820 --> 00:09:53.400 align:middle line:84% an element of both of these at play. 00:09:53.400 --> 00:09:57.940 align:middle line:84% There's-- maybe this we might call selection, 00:09:57.940 --> 00:10:00.353 align:middle line:84% the people who go to college are a selected group. 00:10:00.353 --> 00:10:02.020 align:middle line:84% They're different on average than people 00:10:02.020 --> 00:10:03.140 align:middle line:90% who don't go to college. 00:10:03.140 --> 00:10:06.170 align:middle line:84% And I frame this as causal versus not causal. 00:10:06.170 --> 00:10:08.420 align:middle line:84% I think the best evidence suggests it's a bit of both. 00:10:08.420 --> 00:10:11.020 align:middle line:84% That when we look at this earning gap between people 00:10:11.020 --> 00:10:13.357 align:middle line:84% who go to college and people who don't, partly it's 00:10:13.357 --> 00:10:14.440 align:middle line:90% attributable to selection. 00:10:14.440 --> 00:10:16.240 align:middle line:84% Those people just have different skills, 00:10:16.240 --> 00:10:18.360 align:middle line:84% and partly it's a causal effect of college. 00:10:18.360 --> 00:10:20.120 align:middle line:84% But I want to drill down within causal, 00:10:20.120 --> 00:10:22.900 align:middle line:84% and I think there's actually two different mechanisms that 00:10:22.900 --> 00:10:23.920 align:middle line:90% could be at work here. 00:10:23.920 --> 00:10:26.320 align:middle line:84% So let's focus on the causal component. 00:10:26.320 --> 00:10:29.180 align:middle line:84% Let's just suppose that it is true that if I go to college, 00:10:29.180 --> 00:10:31.020 align:middle line:90% my earnings go up. 00:10:31.020 --> 00:10:32.700 align:middle line:90% Why might that be the case? 00:10:32.700 --> 00:10:36.160 align:middle line:84% Why my going to college cause me to earn more? 00:10:36.160 --> 00:10:36.660 align:middle line:90% Yeah? 00:10:36.660 --> 00:10:37.820 align:middle line:84% AUDIENCE: You're able to learn skills 00:10:37.820 --> 00:10:38.960 align:middle line:90% that employers might want. 00:10:38.960 --> 00:10:39.627 align:middle line:90% IAN BALL: Great. 00:10:39.627 --> 00:10:40.680 align:middle line:90% So. 00:10:40.680 --> 00:10:42.680 align:middle line:84% Certainly the story that people who run colleges 00:10:42.680 --> 00:10:44.240 align:middle line:84% would like to focus on is this story, 00:10:44.240 --> 00:10:47.080 align:middle line:90% that it's about learning skills. 00:10:47.080 --> 00:10:49.880 align:middle line:84% Sometimes this is called human capital formation, 00:10:49.880 --> 00:10:52.640 align:middle line:84% that you go to school, you learn how 00:10:52.640 --> 00:10:55.580 align:middle line:84% to code, you learn how to communicate with people, 00:10:55.580 --> 00:10:58.360 align:middle line:84% you learn knowledge, information about the world, you mature, 00:10:58.360 --> 00:11:00.878 align:middle line:84% various things, and these make you a more valuable employee. 00:11:00.878 --> 00:11:02.420 align:middle line:84% They allow you to be more productive. 00:11:02.420 --> 00:11:04.280 align:middle line:84% So this is sometimes-- we talk about this 00:11:04.280 --> 00:11:05.580 align:middle line:90% might be about productivity. 00:11:05.580 --> 00:11:08.520 align:middle line:90% 00:11:08.520 --> 00:11:10.480 align:middle line:84% But another possibility, which we're 00:11:10.480 --> 00:11:17.580 align:middle line:84% going to focus on today-- learning, is signaling. 00:11:17.580 --> 00:11:22.400 align:middle line:90% 00:11:22.400 --> 00:11:25.400 align:middle line:84% So the other possibility could be that it is causal. 00:11:25.400 --> 00:11:27.188 align:middle line:84% Going to college makes me earn more, 00:11:27.188 --> 00:11:29.480 align:middle line:84% but it's not because I actually learn anything valuable 00:11:29.480 --> 00:11:30.380 align:middle line:90% in college. 00:11:30.380 --> 00:11:33.360 align:middle line:84% It's simply because going to college proves to the employer 00:11:33.360 --> 00:11:36.220 align:middle line:84% that I'm smart, and therefore, the employer pays me more. 00:11:36.220 --> 00:11:37.140 align:middle line:90% It's still causal. 00:11:37.140 --> 00:11:39.640 align:middle line:84% If I didn't go to college, I wouldn't earn as much. 00:11:39.640 --> 00:11:41.100 align:middle line:84% So it is true that going to college 00:11:41.100 --> 00:11:43.520 align:middle line:84% changes my expected earnings, but it changes it 00:11:43.520 --> 00:11:45.760 align:middle line:84% not because I've actually acquired valuable skills, 00:11:45.760 --> 00:11:47.510 align:middle line:84% but because I've demonstrated that I could 00:11:47.510 --> 00:11:49.220 align:middle line:84% do a very difficult task, and that makes 00:11:49.220 --> 00:11:50.520 align:middle line:90% employers want to hire me. 00:11:50.520 --> 00:11:52.180 align:middle line:84% So one way of separating this, you 00:11:52.180 --> 00:11:58.142 align:middle line:84% could imagine that a school-- in the extreme case, you 00:11:58.142 --> 00:12:00.100 align:middle line:84% could imagine a school that teaches skills that 00:12:00.100 --> 00:12:02.350 align:middle line:84% really have no value, but are just really, really hard 00:12:02.350 --> 00:12:03.260 align:middle line:90% to learn. 00:12:03.260 --> 00:12:04.380 align:middle line:90% And you might have-- 00:12:04.380 --> 00:12:05.940 align:middle line:84% people who graduate from that school 00:12:05.940 --> 00:12:08.190 align:middle line:84% might actually still do well in the workforce. 00:12:08.190 --> 00:12:09.440 align:middle line:90% And maybe an example of that-- 00:12:09.440 --> 00:12:12.047 align:middle line:84% I mean, I think athletics are a valuable pursuit, 00:12:12.047 --> 00:12:14.380 align:middle line:84% but you might think that people who succeed in athletics 00:12:14.380 --> 00:12:17.422 align:middle line:84% are often hired by firms to work in finance, say. 00:12:17.422 --> 00:12:19.380 align:middle line:84% It's not that being good at running makes you-- 00:12:19.380 --> 00:12:21.520 align:middle line:84% or being good at weightlifting makes you better at finance, 00:12:21.520 --> 00:12:23.060 align:middle line:84% but it shows that you've done something very hard that 00:12:23.060 --> 00:12:25.643 align:middle line:84% reveals something maybe about your diligence, your work ethic, 00:12:25.643 --> 00:12:28.780 align:middle line:84% and that might be a reason why firms want to hire you. 00:12:28.780 --> 00:12:30.822 align:middle line:84% So in reality, I think both are probably present, 00:12:30.822 --> 00:12:33.030 align:middle line:84% and I think the best empirical evidence at the moment 00:12:33.030 --> 00:12:35.200 align:middle line:84% suggests both of these mechanisms are at play, 00:12:35.200 --> 00:12:36.912 align:middle line:84% but to highlight the signaling aspect, 00:12:36.912 --> 00:12:38.620 align:middle line:84% we're going to focus all in on signaling, 00:12:38.620 --> 00:12:40.620 align:middle line:84% and we're going to consider a model of education 00:12:40.620 --> 00:12:42.780 align:middle line:84% where education conveys no valuable skills. 00:12:42.780 --> 00:12:45.940 align:middle line:84% I want to say up front, this is not my view of education, 00:12:45.940 --> 00:12:48.080 align:middle line:90% but just to highlight this-- 00:12:48.080 --> 00:12:50.560 align:middle line:84% I think we all understand how learning skills increases 00:12:50.560 --> 00:12:51.460 align:middle line:90% your wage. 00:12:51.460 --> 00:12:54.100 align:middle line:84% I think this is the novel element of Spence's model, 00:12:54.100 --> 00:12:55.820 align:middle line:84% and this is what we want to look at. 00:12:55.820 --> 00:12:57.320 align:middle line:84% And I will say, there are people who 00:12:57.320 --> 00:12:59.613 align:middle line:84% argue that this is almost all of education. 00:12:59.613 --> 00:13:01.280 align:middle line:84% It's a debate, but there are some people 00:13:01.280 --> 00:13:03.363 align:middle line:84% who say this is really all that's what's going on. 00:13:03.363 --> 00:13:04.540 align:middle line:90% So let's go through this. 00:13:04.540 --> 00:13:06.983 align:middle line:90% 00:13:06.983 --> 00:13:08.900 align:middle line:84% So one-- I'll just maybe make one observation, 00:13:08.900 --> 00:13:11.817 align:middle line:84% one way that people try to argue it's all about signaling, 00:13:11.817 --> 00:13:13.400 align:middle line:84% one kind of interesting empirical test 00:13:13.400 --> 00:13:15.942 align:middle line:84% is they look at people who go to college for three and a half 00:13:15.942 --> 00:13:17.497 align:middle line:90% years and then drop out. 00:13:17.497 --> 00:13:19.080 align:middle line:84% Now if it's really about skills, you'd 00:13:19.080 --> 00:13:20.880 align:middle line:84% think someone who goes for three and a half years has acquired 00:13:20.880 --> 00:13:23.420 align:middle line:84% almost all of the skills as someone who goes for four years. 00:13:23.420 --> 00:13:24.780 align:middle line:84% But if it's about signaling, you might think, 00:13:24.780 --> 00:13:26.280 align:middle line:84% well, the fact that they dropped out 00:13:26.280 --> 00:13:28.180 align:middle line:84% reveals a lot about who they are as a person, 00:13:28.180 --> 00:13:29.920 align:middle line:84% and indeed, people who drop out just before graduating 00:13:29.920 --> 00:13:30.540 align:middle line:90% earn a lot less. 00:13:30.540 --> 00:13:32.620 align:middle line:84% And that's one argument for why signaling is at play, 00:13:32.620 --> 00:13:34.100 align:middle line:84% but we can debate about these things. 00:13:34.100 --> 00:13:34.600 align:middle line:90% OK. 00:13:34.600 --> 00:13:39.480 align:middle line:84% So now let's go through our formal model of signaling. 00:13:39.480 --> 00:13:43.560 align:middle line:84% So in our model we're going to have a sender and a receiver, 00:13:43.560 --> 00:13:46.690 align:middle line:90% just as I said before. 00:13:46.690 --> 00:13:48.440 align:middle line:84% And we're really going to simplify things. 00:13:48.440 --> 00:13:51.580 align:middle line:84% We're going to imagine the sender just has two types. 00:13:51.580 --> 00:13:57.400 align:middle line:84% They privately know they're type t, which is either l or h. 00:13:57.400 --> 00:13:59.660 align:middle line:84% They're either a low type or a high type. 00:13:59.660 --> 00:14:03.167 align:middle line:84% And we're going to think of this-- 00:14:03.167 --> 00:14:05.000 align:middle line:84% maybe we can think of this a bit abstractly, 00:14:05.000 --> 00:14:10.380 align:middle line:84% but a low type is someone who is going to find education more 00:14:10.380 --> 00:14:13.295 align:middle line:84% costly, and a high type is going to find education less costly. 00:14:13.295 --> 00:14:14.920 align:middle line:84% We can interpret this in a lot of ways. 00:14:14.920 --> 00:14:16.543 align:middle line:84% It could be about your intelligence, 00:14:16.543 --> 00:14:17.960 align:middle line:84% it could be about your work ethic, 00:14:17.960 --> 00:14:20.863 align:middle line:84% it could be about the quality of the school you went to before. 00:14:20.863 --> 00:14:22.780 align:middle line:84% There's a lot of things that can go into this, 00:14:22.780 --> 00:14:24.660 align:middle line:84% but we're just going to quite abstractly call 00:14:24.660 --> 00:14:26.720 align:middle line:90% it a low type and a high type. 00:14:26.720 --> 00:14:31.340 align:middle line:90% 00:14:31.340 --> 00:14:34.360 align:middle line:84% And what they're going to choose is 00:14:34.360 --> 00:14:38.080 align:middle line:84% they're going to choose a level of education. 00:14:38.080 --> 00:14:41.480 align:middle line:84% And in reality, education kind of usually comes in in units. 00:14:41.480 --> 00:14:44.680 align:middle line:84% Like, you can't-- we don't really think you get 862 days 00:14:44.680 --> 00:14:45.260 align:middle line:90% of education. 00:14:45.260 --> 00:14:47.160 align:middle line:84% We usually think you either get-- graduate high school 00:14:47.160 --> 00:14:49.140 align:middle line:84% or you graduate college, but to make things simple, 00:14:49.140 --> 00:14:50.848 align:middle line:84% we're going to make education continuous, 00:14:50.848 --> 00:14:53.880 align:middle line:84% e, and we're just going to say it's a number between 0 and 1. 00:14:53.880 --> 00:14:57.680 align:middle line:90% So you can think of this-- 00:14:57.680 --> 00:15:02.280 align:middle line:84% just a stylized model where you can get any continuous amount 00:15:02.280 --> 00:15:02.900 align:middle line:90% of education. 00:15:02.900 --> 00:15:07.760 align:middle line:84% This could be something like the fraction of your life you spend 00:15:07.760 --> 00:15:10.000 align:middle line:90% being educated or something. 00:15:10.000 --> 00:15:12.040 align:middle line:84% Now the receiver, we're going to think 00:15:12.040 --> 00:15:18.360 align:middle line:84% of this as maybe a firm or a market of firms. 00:15:18.360 --> 00:15:24.760 align:middle line:84% And they're going to observe e, but not t. 00:15:24.760 --> 00:15:27.980 align:middle line:84% When a firm chooses to hire you, they see what education you got. 00:15:27.980 --> 00:15:30.080 align:middle line:84% They don't directly observe this type t, 00:15:30.080 --> 00:15:32.220 align:middle line:84% whether you're a low or high type person. 00:15:32.220 --> 00:15:33.920 align:middle line:84% And then they're going to choose a wage. 00:15:33.920 --> 00:15:38.420 align:middle line:90% 00:15:38.420 --> 00:15:41.067 align:middle line:84% In reality, I think a lot of individual firms, 00:15:41.067 --> 00:15:42.900 align:middle line:84% the wage is roughly set and they're choosing 00:15:42.900 --> 00:15:43.760 align:middle line:90% whether to hire you or not. 00:15:43.760 --> 00:15:46.332 align:middle line:84% There's maybe some flexibility over wages, but that's one way 00:15:46.332 --> 00:15:47.040 align:middle line:90% we can think of-- 00:15:47.040 --> 00:15:49.457 align:middle line:84% I think it's helpful to think of this kind of as a market. 00:15:49.457 --> 00:15:52.340 align:middle line:84% So it may be that each individual firm is making 00:15:52.340 --> 00:15:54.143 align:middle line:84% a binary choice about whether to hire you, 00:15:54.143 --> 00:15:56.560 align:middle line:84% but then you go to the firm that gives you the best offer. 00:15:56.560 --> 00:16:00.900 align:middle line:84% So in the end, the outcome of the firm-- of the labor market 00:16:00.900 --> 00:16:03.280 align:middle line:84% is that you get hired at some place for some wage, 00:16:03.280 --> 00:16:04.660 align:middle line:84% and depending on how well you do, 00:16:04.660 --> 00:16:07.620 align:middle line:84% that wage is determined by which firm hires you. 00:16:07.620 --> 00:16:09.820 align:middle line:90% So now let's talk about payoffs. 00:16:09.820 --> 00:16:11.820 align:middle line:84% So we're going to assume the sender, 00:16:11.820 --> 00:16:19.100 align:middle line:90% their payoffs are like this. 00:16:19.100 --> 00:16:21.660 align:middle line:84% So the sender wants to get paid more. 00:16:21.660 --> 00:16:24.687 align:middle line:84% Their payoff is-- their utility is increasing their wage. 00:16:24.687 --> 00:16:26.020 align:middle line:90% They don't like going to school. 00:16:26.020 --> 00:16:27.800 align:middle line:84% Again, another maybe stylized assumption, 00:16:27.800 --> 00:16:30.230 align:middle line:84% some people enjoy going to school and that's why they go, 00:16:30.230 --> 00:16:31.970 align:middle line:84% but we're not going to have that in our model. 00:16:31.970 --> 00:16:34.345 align:middle line:84% We're going to assume that people just don't like school. 00:16:34.345 --> 00:16:36.970 align:middle line:84% And notice, the more they get educated, the more costly it is. 00:16:36.970 --> 00:16:38.950 align:middle line:84% So they'd always rather have less school 00:16:38.950 --> 00:16:41.430 align:middle line:84% than more school, other things being equal. 00:16:41.430 --> 00:16:44.610 align:middle line:84% But now their type t appears in the denominator here-- 00:16:44.610 --> 00:16:46.390 align:middle line:84% and maybe I should be more precise. 00:16:46.390 --> 00:16:50.010 align:middle line:84% Here, we have b, a natural if we did it any other way, 00:16:50.010 --> 00:16:52.430 align:middle line:84% the low type is lower than the high type. 00:16:52.430 --> 00:16:55.030 align:middle line:84% So by putting t in the denominator here, 00:16:55.030 --> 00:16:58.350 align:middle line:84% what I mean is if you're a high type, if t is higher, 00:16:58.350 --> 00:17:00.430 align:middle line:84% then the coefficient in front of this cost 00:17:00.430 --> 00:17:03.450 align:middle line:84% is lower, and therefore, the cost, 00:17:03.450 --> 00:17:05.790 align:middle line:84% or more importantly, the marginal cost of education 00:17:05.790 --> 00:17:08.869 align:middle line:90% is lower for the high type. 00:17:08.869 --> 00:17:10.190 align:middle line:90% Why might that be true? 00:17:10.190 --> 00:17:12.190 align:middle line:84% Well, maybe they don't have to study 00:17:12.190 --> 00:17:13.837 align:middle line:90% as hard to get into the school. 00:17:13.837 --> 00:17:15.670 align:middle line:84% Maybe once they get in, it's easier for them 00:17:15.670 --> 00:17:17.337 align:middle line:84% to get good grades and graduate, there's 00:17:17.337 --> 00:17:19.790 align:middle line:84% many stories we can tell, but, again, we're 00:17:19.790 --> 00:17:22.750 align:middle line:84% going to think of your type as representing how costly it 00:17:22.750 --> 00:17:25.473 align:middle line:90% is for you to get education. 00:17:25.473 --> 00:17:26.890 align:middle line:84% And then the receiver, we're going 00:17:26.890 --> 00:17:29.070 align:middle line:84% to use a bit of a weird payoff specification. 00:17:29.070 --> 00:17:33.650 align:middle line:90% 00:17:33.650 --> 00:17:36.890 align:middle line:84% The receiver's utility is going to be like this. 00:17:36.890 --> 00:17:38.750 align:middle line:84% Now, this may at first seem a bit strange. 00:17:38.750 --> 00:17:40.542 align:middle line:84% What we're saying is the receiver's utility 00:17:40.542 --> 00:17:43.130 align:middle line:84% is the negative of the difference between the wage they 00:17:43.130 --> 00:17:47.330 align:middle line:84% actually charge and the type of the applicant of the sender. 00:17:47.330 --> 00:17:49.410 align:middle line:84% And really, the reason we make that assumption 00:17:49.410 --> 00:17:57.010 align:middle line:84% is the implication of this is that the optimal wage w is 00:17:57.010 --> 00:18:02.770 align:middle line:90% equal to the expectation of t. 00:18:02.770 --> 00:18:06.010 align:middle line:84% And this is just an algebraic fact about the quadratic loss 00:18:06.010 --> 00:18:07.810 align:middle line:84% function, you can check this with algebra. 00:18:07.810 --> 00:18:10.970 align:middle line:84% So if I don't know what t is, with some probability, it's low; 00:18:10.970 --> 00:18:12.970 align:middle line:84% with some probability, it's high. 00:18:12.970 --> 00:18:17.170 align:middle line:84% The optimal wage for me to set is exactly my expectation of t. 00:18:17.170 --> 00:18:20.090 align:middle line:84% So if I know the person is low ability, it's optimal for me 00:18:20.090 --> 00:18:21.490 align:middle line:90% to set a wage of l. 00:18:21.490 --> 00:18:24.210 align:middle line:84% If I know they're a high type, h, it's optimal for me 00:18:24.210 --> 00:18:25.410 align:middle line:90% to set a wage of h. 00:18:25.410 --> 00:18:27.990 align:middle line:84% And if I think it's a 50/50, then I want to pay them exactly 00:18:27.990 --> 00:18:29.395 align:middle line:90% 1/2 l plus 1/2 h. 00:18:29.395 --> 00:18:30.770 align:middle line:84% Now you might say, wait a second, 00:18:30.770 --> 00:18:31.812 align:middle line:90% that's not what firms do. 00:18:31.812 --> 00:18:33.630 align:middle line:84% They pay the lowest wage they can subject 00:18:33.630 --> 00:18:35.250 align:middle line:84% to getting the worker, but again, we 00:18:35.250 --> 00:18:37.250 align:middle line:84% can think of this as the outcome of competition. 00:18:37.250 --> 00:18:39.270 align:middle line:84% So if there's many firms competing, 00:18:39.270 --> 00:18:42.590 align:middle line:84% if I charge-- if I pay a worker less than their expected 00:18:42.590 --> 00:18:46.062 align:middle line:84% ability, then maybe I'm going to lose out to another firm 00:18:46.062 --> 00:18:47.770 align:middle line:84% and the other firm is going to hire them. 00:18:47.770 --> 00:18:49.290 align:middle line:84% If I pay them more than their expected ability, 00:18:49.290 --> 00:18:50.332 align:middle line:90% then I'm overpaying them. 00:18:50.332 --> 00:18:53.630 align:middle line:84% So we can think of this as a reduced form representation 00:18:53.630 --> 00:18:56.830 align:middle line:90% of a competitive labor market. 00:18:56.830 --> 00:18:59.690 align:middle line:84% And I should say something about the prior here. 00:18:59.690 --> 00:19:03.510 align:middle line:84% Let's assume that h occurs with probability pi and l 00:19:03.510 --> 00:19:06.230 align:middle line:90% with 1 minus p. 00:19:06.230 --> 00:19:10.150 align:middle line:84% So in the population, fraction pi of people are of type h, 00:19:10.150 --> 00:19:12.870 align:middle line:84% are the high types, and fraction 1 minus pi of people 00:19:12.870 --> 00:19:14.010 align:middle line:90% are the low types. 00:19:14.010 --> 00:19:19.070 align:middle line:90% 00:19:19.070 --> 00:19:22.450 align:middle line:84% OK, so now we're ready to analyze this game. 00:19:22.450 --> 00:19:26.173 align:middle line:84% And the first step is always to write down what a strategy is. 00:19:26.173 --> 00:19:28.090 align:middle line:84% Notice, this game is a little more complicated 00:19:28.090 --> 00:19:30.490 align:middle line:84% than what we looked at before because now, education 00:19:30.490 --> 00:19:31.530 align:middle line:90% is something continuous. 00:19:31.530 --> 00:19:34.425 align:middle line:84% Before in the class on Thursday, we looked at this problem 00:19:34.425 --> 00:19:36.550 align:middle line:84% where you chose, whether to submit an essay or not, 00:19:36.550 --> 00:19:37.830 align:middle line:90% and that was a binary choice. 00:19:37.830 --> 00:19:39.210 align:middle line:90% I'll make one other observation. 00:19:39.210 --> 00:19:40.752 align:middle line:84% Last Thursday, we looked at something 00:19:40.752 --> 00:19:44.170 align:middle line:84% where your private type affected how much you wanted 00:19:44.170 --> 00:19:46.570 align:middle line:84% to get into a school, but not how costly it 00:19:46.570 --> 00:19:48.127 align:middle line:90% was to submit an essay. 00:19:48.127 --> 00:19:49.710 align:middle line:84% Now we're doing an opposite assumption 00:19:49.710 --> 00:19:52.750 align:middle line:84% where your type doesn't affect how much you want to get paid. 00:19:52.750 --> 00:19:55.090 align:middle line:84% Everyone wants to get paid more, but your type only 00:19:55.090 --> 00:19:57.350 align:middle line:84% affects how costly it is for you to get educated, 00:19:57.350 --> 00:19:59.670 align:middle line:84% so it's kind of the inverse of what we did last week. 00:19:59.670 --> 00:20:03.210 align:middle line:84% And I think this is probably a pretty good model of the labor 00:20:03.210 --> 00:20:07.130 align:middle line:84% market where most people want more higher wages, that's not 00:20:07.130 --> 00:20:08.970 align:middle line:90% really what distinguishes them. 00:20:08.970 --> 00:20:11.770 align:middle line:84% OK, so what is a strategy, or more precisely, what 00:20:11.770 --> 00:20:14.890 align:middle line:90% is an assessment? 00:20:14.890 --> 00:20:17.690 align:middle line:84% So remember, an assessment is a strategy profile together 00:20:17.690 --> 00:20:19.130 align:middle line:90% with a belief system. 00:20:19.130 --> 00:20:24.990 align:middle line:84% So we need to first say, let's look at the sender strategy. 00:20:24.990 --> 00:20:27.150 align:middle line:84% What is the strategy for the sender? 00:20:27.150 --> 00:20:30.350 align:middle line:84% Well, the sender chooses an action, a level of education 00:20:30.350 --> 00:20:33.310 align:middle line:84% after knowing or learning their private type t. 00:20:33.310 --> 00:20:36.430 align:middle line:84% So we need to specify what education the sender chooses 00:20:36.430 --> 00:20:38.750 align:middle line:84% when they're the low type and what education they 00:20:38.750 --> 00:20:40.030 align:middle line:90% choose when they're high type. 00:20:40.030 --> 00:20:43.930 align:middle line:84% So we'll represent a sender strategy as el, comma, eh. 00:20:43.930 --> 00:20:45.190 align:middle line:90% It's two numbers. 00:20:45.190 --> 00:20:47.470 align:middle line:84% el says if you're a low type sender, 00:20:47.470 --> 00:20:49.570 align:middle line:84% this is how much education I'm going to get. 00:20:49.570 --> 00:20:51.990 align:middle line:84% eh says if I'm a high type sender, 00:20:51.990 --> 00:20:55.790 align:middle line:84% this is how much education I'm going to get. 00:20:55.790 --> 00:20:59.390 align:middle line:84% And yeah, I say sender, you can think of this as the student. 00:20:59.390 --> 00:21:02.850 align:middle line:84% OK, what is the strategy for the receiver? 00:21:02.850 --> 00:21:08.430 align:middle line:90% 00:21:08.430 --> 00:21:09.670 align:middle line:90% This is a bit harder. 00:21:09.670 --> 00:21:12.710 align:middle line:84% Remember, a strategy is always a complete contingent plan. 00:21:12.710 --> 00:21:15.310 align:middle line:84% It says what action do you take as a function of what 00:21:15.310 --> 00:21:17.030 align:middle line:90% you observe. 00:21:17.030 --> 00:21:19.370 align:middle line:84% So in this case, the receiver observes your level 00:21:19.370 --> 00:21:21.570 align:middle line:84% of education, and they're going to choose 00:21:21.570 --> 00:21:24.350 align:middle line:84% what wage they set as a function of your education. 00:21:24.350 --> 00:21:29.850 align:middle line:84% So formerly, the receiver is choosing a function w from 0, 1, 00:21:29.850 --> 00:21:36.450 align:middle line:90% maybe I'll say to 0, infinity. 00:21:36.450 --> 00:21:44.170 align:middle line:84% So what this says is it assigns to each level of education 00:21:44.170 --> 00:21:49.590 align:middle line:84% e a wage w of e that the receiver or firm will pay you 00:21:49.590 --> 00:21:52.370 align:middle line:84% if they observe that your education level is e. 00:21:52.370 --> 00:21:55.690 align:middle line:84% And they have to do that for every single education level 00:21:55.690 --> 00:21:58.570 align:middle line:90% 0, 1. 00:21:58.570 --> 00:22:00.790 align:middle line:84% So that describes a strategy profile, 00:22:00.790 --> 00:22:02.930 align:middle line:84% but we're not done yet because an assessment also 00:22:02.930 --> 00:22:04.430 align:middle line:90% specifies beliefs. 00:22:04.430 --> 00:22:06.790 align:middle line:84% So now we also have to specify a belief system. 00:22:06.790 --> 00:22:13.050 align:middle line:90% 00:22:13.050 --> 00:22:16.270 align:middle line:84% Well, the sender already knows their type, 00:22:16.270 --> 00:22:18.860 align:middle line:84% so we don't need to say anything about the sender's type. 00:22:18.860 --> 00:22:20.980 align:middle line:84% We have to specify the receiver's beliefs 00:22:20.980 --> 00:22:22.940 align:middle line:90% about the sender's type. 00:22:22.940 --> 00:22:25.380 align:middle line:84% And once again, the receiver might 00:22:25.380 --> 00:22:27.420 align:middle line:84% learn about the sender's type through the level 00:22:27.420 --> 00:22:28.780 align:middle line:90% of education e. 00:22:28.780 --> 00:22:31.840 align:middle line:84% So a belief system is going to say, what do I as the firm 00:22:31.840 --> 00:22:35.320 align:middle line:84% believe about you as a function of what education I see? 00:22:35.320 --> 00:22:37.360 align:middle line:84% If I see that you've got a lot of education, 00:22:37.360 --> 00:22:39.040 align:middle line:84% what do I believe about your ability? 00:22:39.040 --> 00:22:41.120 align:middle line:84% If I see that you didn't get very much education, 00:22:41.120 --> 00:22:42.662 align:middle line:84% what do I believe about your ability? 00:22:42.662 --> 00:22:45.727 align:middle line:90% So a belief system formerly-- 00:22:45.727 --> 00:22:47.560 align:middle line:84% maybe if we write it-- let's start formally, 00:22:47.560 --> 00:22:49.185 align:middle line:84% and then we'll be a little less formal. 00:22:49.185 --> 00:22:57.220 align:middle line:90% 00:22:57.220 --> 00:22:59.060 align:middle line:84% So if we're formal about it, a belief system 00:22:59.060 --> 00:23:01.220 align:middle line:84% says for every level of education 00:23:01.220 --> 00:23:04.140 align:middle line:84% I see, I'm going to form a belief which 00:23:04.140 --> 00:23:06.620 align:middle line:84% is a probability distribution over your ability, 00:23:06.620 --> 00:23:07.520 align:middle line:90% either l or h. 00:23:07.520 --> 00:23:09.220 align:middle line:84% So the notation here just means this 00:23:09.220 --> 00:23:12.380 align:middle line:84% is the set of probability distributions over your ability 00:23:12.380 --> 00:23:14.660 align:middle line:84% being l and your ability being h. 00:23:14.660 --> 00:23:19.240 align:middle line:84% But we can specify a probability distribution here 00:23:19.240 --> 00:23:21.960 align:middle line:84% just by saying, what do I think is the probability that you're 00:23:21.960 --> 00:23:22.845 align:middle line:90% a high type? 00:23:22.845 --> 00:23:25.220 align:middle line:84% Because once we specify the probability you're high type, 00:23:25.220 --> 00:23:27.845 align:middle line:84% then the probability that you're low type is just 1 minus that. 00:23:27.845 --> 00:23:30.560 align:middle line:84% So we're going to think of this-- 00:23:30.560 --> 00:23:33.140 align:middle line:90% maybe we'll write b h e-- 00:23:33.140 --> 00:23:38.720 align:middle line:90% 00:23:38.720 --> 00:23:40.520 align:middle line:90% so we'll just use this notation. 00:23:40.520 --> 00:23:43.480 align:middle line:84% And this says, if I see that your education is e, 00:23:43.480 --> 00:23:46.240 align:middle line:84% this is the probability that I think you're a high type. 00:23:46.240 --> 00:23:47.918 align:middle line:84% And that fully specifies my beliefs 00:23:47.918 --> 00:23:49.960 align:middle line:84% because the probability I think you're a low type 00:23:49.960 --> 00:23:51.060 align:middle line:90% is always 1 minus that. 00:23:51.060 --> 00:23:54.120 align:middle line:84% So maybe I'll just write that because b 00:23:54.120 --> 00:23:57.920 align:middle line:84% l given e, the probability that I assign to you being a low type 00:23:57.920 --> 00:24:01.520 align:middle line:90% given e, is just 1 minus b h e. 00:24:01.520 --> 00:24:04.240 align:middle line:90% 00:24:04.240 --> 00:24:06.580 align:middle line:84% So technically, we have to specify both of these things, 00:24:06.580 --> 00:24:08.372 align:middle line:84% but we're only going to specify one of them 00:24:08.372 --> 00:24:12.600 align:middle line:84% because it's pinned down by the other. 00:24:12.600 --> 00:24:13.420 align:middle line:90% Great. 00:24:13.420 --> 00:24:17.540 align:middle line:84% So now, we have to fill all these things in. 00:24:17.540 --> 00:24:21.047 align:middle line:84% We have to specify the sender strategy, the receiver strategy, 00:24:21.047 --> 00:24:23.380 align:middle line:84% and then the belief system, which is a system of beliefs 00:24:23.380 --> 00:24:24.200 align:middle line:90% for the receiver. 00:24:24.200 --> 00:24:25.400 align:middle line:90% Any questions on this? 00:24:25.400 --> 00:24:31.660 align:middle line:90% 00:24:31.660 --> 00:24:33.160 align:middle line:90% OK. 00:24:33.160 --> 00:24:38.776 align:middle line:90% 00:24:38.776 --> 00:24:40.740 align:middle line:90% Now what do we need to check? 00:24:40.740 --> 00:24:43.320 align:middle line:84% So now we're going to look at our equilibrium conditions. 00:24:43.320 --> 00:24:51.500 align:middle line:90% 00:24:51.500 --> 00:24:53.920 align:middle line:84% And remember, we're looking for the conditions, 00:24:53.920 --> 00:24:55.700 align:middle line:90% and these are for pbE. 00:24:55.700 --> 00:24:59.100 align:middle line:84% Our equilibrium notion is perfect Bayesian equilibrium. 00:24:59.100 --> 00:25:03.240 align:middle line:84% And we have three conditions or, depending on how we write it, 00:25:03.240 --> 00:25:07.760 align:middle line:84% one is sequential optimality or sequential rationality. 00:25:07.760 --> 00:25:16.160 align:middle line:90% 00:25:16.160 --> 00:25:17.540 align:middle line:90% You might see both terms used. 00:25:17.540 --> 00:25:19.457 align:middle line:84% Sequential optimality, sequential rationality, 00:25:19.457 --> 00:25:20.740 align:middle line:90% they mean the same thing. 00:25:20.740 --> 00:25:22.100 align:middle line:90% And then belief consistency. 00:25:22.100 --> 00:25:32.960 align:middle line:90% 00:25:32.960 --> 00:25:34.600 align:middle line:84% And it's helpful to separate this 00:25:34.600 --> 00:25:43.000 align:middle line:84% into sequential optimality for the sender, 00:25:43.000 --> 00:25:45.000 align:middle line:90% and also for the receiver. 00:25:45.000 --> 00:25:47.440 align:middle line:84% So let's first just say what each of these things 00:25:47.440 --> 00:25:50.680 align:middle line:90% mean in words. 00:25:50.680 --> 00:25:52.240 align:middle line:84% Sequential optimality for the sender 00:25:52.240 --> 00:25:56.520 align:middle line:84% just means if I'm the sender, and I know my type t, 00:25:56.520 --> 00:25:58.760 align:middle line:84% my equilibrium level of education 00:25:58.760 --> 00:26:01.400 align:middle line:84% is optimal for me given what everything else that's 00:26:01.400 --> 00:26:02.760 align:middle line:90% happening in the game. 00:26:02.760 --> 00:26:09.820 align:middle line:84% So for the sender, this means for each t where, remember, 00:26:09.820 --> 00:26:11.820 align:middle line:84% t is my type, and it could be either low ability 00:26:11.820 --> 00:26:12.820 align:middle line:90% or high ability. 00:26:12.820 --> 00:26:21.740 align:middle line:84% So for each t, if my type is t, choosing et-- 00:26:21.740 --> 00:26:23.500 align:middle line:84% remember, this could either be el or eh 00:26:23.500 --> 00:26:34.140 align:middle line:84% depending on the value of t, is optimal given w. 00:26:34.140 --> 00:26:38.460 align:middle line:84% Remember, what is w here? w is the wage that the receiver sets. 00:26:38.460 --> 00:26:40.220 align:middle line:84% And which education level is optimal 00:26:40.220 --> 00:26:43.753 align:middle line:84% for me depends on w because what I'm thinking, what if I 00:26:43.753 --> 00:26:45.920 align:middle line:84% go to school a little bit less or a little bit more, 00:26:45.920 --> 00:26:47.018 align:middle line:90% what if I deviate? 00:26:47.018 --> 00:26:48.560 align:middle line:84% Well, to see whether that's worth it, 00:26:48.560 --> 00:26:50.760 align:middle line:84% I have to trade off the cost of, say, 00:26:50.760 --> 00:26:52.860 align:middle line:84% additional education against the effect that 00:26:52.860 --> 00:26:54.460 align:middle line:90% would have on my wage. 00:26:54.460 --> 00:26:56.700 align:middle line:84% And that's going to be governed by the strategy w 00:26:56.700 --> 00:26:57.720 align:middle line:90% that the receiver uses. 00:26:57.720 --> 00:27:00.020 align:middle line:84% So that's going to be relevant in determining 00:27:00.020 --> 00:27:03.420 align:middle line:90% whether et is optimal. 00:27:03.420 --> 00:27:06.540 align:middle line:90% And then for the receiver-- 00:27:06.540 --> 00:27:10.400 align:middle line:84% well, with the receiver, this is actually-- 00:27:10.400 --> 00:27:13.560 align:middle line:84% we can write this maybe a bit more easily. 00:27:13.560 --> 00:27:17.640 align:middle line:90% So for each e-- 00:27:17.640 --> 00:27:24.840 align:middle line:84% and e is in 0 or 1, the receiver's wage 00:27:24.840 --> 00:27:27.060 align:middle line:84% must be optimal given their beliefs. 00:27:27.060 --> 00:27:42.960 align:middle line:84% So we need choosing w of e is optimal given b h e. 00:27:42.960 --> 00:27:44.720 align:middle line:84% So what we're saying is I'm the receiver. 00:27:44.720 --> 00:27:47.680 align:middle line:90% I observe an education level e. 00:27:47.680 --> 00:27:52.200 align:middle line:84% That causes me to form beliefs b h given e about your type. 00:27:52.200 --> 00:27:54.160 align:middle line:84% And given those beliefs, the wage 00:27:54.160 --> 00:27:57.600 align:middle line:84% that I actually choose as the receiver is optimal. 00:27:57.600 --> 00:27:58.860 align:middle line:90% Can we write that out in math? 00:27:58.860 --> 00:28:01.600 align:middle line:84% I mean, given what we said above, 00:28:01.600 --> 00:28:04.520 align:middle line:84% we said that it's always optimal given my quadratic loss 00:28:04.520 --> 00:28:08.780 align:middle line:84% utility to always charge a wage equal to my updated 00:28:08.780 --> 00:28:10.660 align:middle line:90% expectation about your type. 00:28:10.660 --> 00:28:13.360 align:middle line:84% So can we write this condition in words? 00:28:13.360 --> 00:28:18.600 align:middle line:84% What does it mean for w of e to be optimal given b of h given e? 00:28:18.600 --> 00:28:21.180 align:middle line:90% 00:28:21.180 --> 00:28:22.600 align:middle line:90% What must w of e be? 00:28:22.600 --> 00:28:30.360 align:middle line:90% 00:28:30.360 --> 00:28:30.860 align:middle line:90% Yeah? 00:28:30.860 --> 00:28:32.038 align:middle line:90% AUDIENCE: Expected value 2. 00:28:32.038 --> 00:28:32.580 align:middle line:90% IAN BALL: OK. 00:28:32.580 --> 00:28:36.060 align:middle line:84% And how can we write that in terms of b of h given e? 00:28:36.060 --> 00:28:39.800 align:middle line:90% AUDIENCE: Um-- 00:28:39.800 --> 00:28:40.300 align:middle line:90% 00:28:40.300 --> 00:28:42.220 align:middle line:90% IAN BALL: So you're right. 00:28:42.220 --> 00:28:43.540 align:middle line:90% AUDIENCE: Wait, are all-- 00:28:43.540 --> 00:28:44.480 align:middle line:90% other numbers. 00:28:44.480 --> 00:28:48.860 align:middle line:90% So then the l top were-- 00:28:48.860 --> 00:28:53.280 align:middle line:84% Oh, is it h times b of h plus l times minus 1 minus-- 00:28:53.280 --> 00:28:54.280 align:middle line:90% IAN BALL: Exactly right. 00:28:54.280 --> 00:28:54.540 align:middle line:90% Yeah. 00:28:54.540 --> 00:28:56.240 align:middle line:84% So indeed, it is the expectation of t, 00:28:56.240 --> 00:28:59.140 align:middle line:84% but we have to be careful because our expectation of t 00:28:59.140 --> 00:29:01.760 align:middle line:84% is influenced by the education we see and our beliefs. 00:29:01.760 --> 00:29:03.840 align:middle line:84% So it's not just our prior expectation. 00:29:03.840 --> 00:29:06.170 align:middle line:84% And maybe I should be clear here, when I write this, 00:29:06.170 --> 00:29:08.730 align:middle line:90% this is the updated expectation. 00:29:08.730 --> 00:29:12.590 align:middle line:90% 00:29:12.590 --> 00:29:14.110 align:middle line:84% This is the expectation of t that 00:29:14.110 --> 00:29:16.350 align:middle line:84% reflects the new information that I have 00:29:16.350 --> 00:29:17.690 align:middle line:90% given your educational choice. 00:29:17.690 --> 00:29:19.425 align:middle line:84% And indeed, if seeing e caused me 00:29:19.425 --> 00:29:21.550 align:middle line:84% to believe that this is the probability that you're 00:29:21.550 --> 00:29:24.150 align:middle line:84% a high type and 1 minus that is the probability that you're 00:29:24.150 --> 00:29:26.670 align:middle line:84% a low type, well, now I can compute the expected value 00:29:26.670 --> 00:29:27.830 align:middle line:90% of your type. 00:29:27.830 --> 00:29:31.270 align:middle line:84% And, yeah, I should emphasize here that l and h are actually 00:29:31.270 --> 00:29:31.770 align:middle line:90% numbers. 00:29:31.770 --> 00:29:34.290 align:middle line:84% Sometimes we think of high type and low type as just labels, 00:29:34.290 --> 00:29:37.863 align:middle line:84% but here, a low type is like 0.7 and a high type is like 2.3, 00:29:37.863 --> 00:29:39.530 align:middle line:84% and these numbers actually have meaning. 00:29:39.530 --> 00:29:40.390 align:middle line:90% Yeah. 00:29:40.390 --> 00:29:41.030 align:middle line:90% OK. 00:29:41.030 --> 00:29:48.110 align:middle line:84% So indeed, we get h times p of h given e plus l times-- well, we 00:29:48.110 --> 00:29:50.870 align:middle line:84% could write b of l given e, but using the notation we're using, 00:29:50.870 --> 00:29:55.790 align:middle line:84% we'll just write 1 minus b of h given e. 00:29:55.790 --> 00:29:59.190 align:middle line:84% So that's pretty easy to write down. 00:29:59.190 --> 00:30:02.970 align:middle line:90% And now, belief consistency. 00:30:02.970 --> 00:30:05.730 align:middle line:84% Well, remember, belief consistency 00:30:05.730 --> 00:30:08.050 align:middle line:84% says the receiver has to be applying 00:30:08.050 --> 00:30:10.970 align:middle line:90% Bayes' rule wherever possible. 00:30:10.970 --> 00:30:13.650 align:middle line:84% And the receiver can only apply Bayes' rule 00:30:13.650 --> 00:30:15.290 align:middle line:84% when they see a level of education 00:30:15.290 --> 00:30:17.130 align:middle line:84% that someone is actually supposed to choose 00:30:17.130 --> 00:30:18.850 align:middle line:90% with positive probability. 00:30:18.850 --> 00:30:22.170 align:middle line:90% So we can separate-- 00:30:22.170 --> 00:30:25.450 align:middle line:84% sometimes this is called on-path and off-path. 00:30:25.450 --> 00:30:28.930 align:middle line:90% 00:30:28.930 --> 00:30:30.330 align:middle line:90% Let's say e prime. 00:30:30.330 --> 00:30:32.130 align:middle line:84% So if they see an education level 00:30:32.130 --> 00:30:35.450 align:middle line:84% e that's on-path, meaning that either the low type 00:30:35.450 --> 00:30:37.570 align:middle line:84% or the high type is supposed to choose this 00:30:37.570 --> 00:30:40.750 align:middle line:84% with positive probability under their equilibrium strategy, 00:30:40.750 --> 00:30:41.870 align:middle line:90% then we apply Bayes' rule. 00:30:41.870 --> 00:30:51.330 align:middle line:90% 00:30:51.330 --> 00:30:53.390 align:middle line:84% But if we see an off-path education, 00:30:53.390 --> 00:30:55.410 align:middle line:84% we see a level of education that is not 00:30:55.410 --> 00:30:59.410 align:middle line:84% supposed to be chosen by either type, 00:30:59.410 --> 00:31:03.810 align:middle line:84% and then there are no restrictions. 00:31:03.810 --> 00:31:07.830 align:middle line:90% 00:31:07.830 --> 00:31:11.030 align:middle line:84% And let's be a bit more precise here, which education levels are 00:31:11.030 --> 00:31:13.870 align:middle line:84% on-path given the strategy that we've 00:31:13.870 --> 00:31:16.270 align:middle line:90% written down for the sender? 00:31:16.270 --> 00:31:18.150 align:middle line:84% Well, there's only two that are on-path. 00:31:18.150 --> 00:31:20.550 align:middle line:84% It's exactly the numbers el and eh. 00:31:20.550 --> 00:31:23.470 align:middle line:84% Those are the only two education levels that the receiver expects 00:31:23.470 --> 00:31:25.310 align:middle line:90% to see in equilibrium. 00:31:25.310 --> 00:31:31.962 align:middle line:84% So these are precisely the cases where e is either el or eh. 00:31:31.962 --> 00:31:33.670 align:middle line:84% If the sender was using a mixed strategy, 00:31:33.670 --> 00:31:34.930 align:middle line:84% this would be a bit more complicated. 00:31:34.930 --> 00:31:37.270 align:middle line:84% There may be more education levels they might choose. 00:31:37.270 --> 00:31:41.070 align:middle line:84% But I, as the receiver, know that, well, if you're low type, 00:31:41.070 --> 00:31:43.028 align:middle line:84% you're supposed to choose this education level; 00:31:43.028 --> 00:31:44.528 align:middle line:84% if you're high type, you're supposed 00:31:44.528 --> 00:31:47.310 align:middle line:84% to choose this education level; and if I see any other education 00:31:47.310 --> 00:31:50.170 align:middle line:84% level, well, I wasn't supposed to see this in equilibrium, 00:31:50.170 --> 00:31:51.730 align:middle line:84% so I'm not able to apply Bayes' rule. 00:31:51.730 --> 00:31:53.480 align:middle line:84% Of course, this is quite a simplification. 00:31:53.480 --> 00:31:55.890 align:middle line:84% In reality, we see a lot of education levels, 00:31:55.890 --> 00:31:58.550 align:middle line:84% people's types are not binary, and we'll do extensions later, 00:31:58.550 --> 00:32:01.610 align:middle line:84% but for now, this is where we are. 00:32:01.610 --> 00:32:06.870 align:middle line:84% And then we have e is not an element of el and eh. 00:32:06.870 --> 00:32:14.490 align:middle line:90% 00:32:14.490 --> 00:32:16.990 align:middle line:90% OK. 00:32:16.990 --> 00:32:19.870 align:middle line:84% So to start dealing with Bayes' rule, 00:32:19.870 --> 00:32:22.213 align:middle line:84% it turns out that it's helpful to distinguish 00:32:22.213 --> 00:32:23.630 align:middle line:84% two different kinds of equilibria, 00:32:23.630 --> 00:32:25.422 align:middle line:84% and that's what we're going to look at now. 00:32:25.422 --> 00:32:30.130 align:middle line:84% So we're going to consider two kinds of equilibria what 00:32:30.130 --> 00:32:42.150 align:middle line:84% are called separating equilibria and a pooling equilibria. 00:32:42.150 --> 00:32:50.170 align:middle line:90% 00:32:50.170 --> 00:32:53.050 align:middle line:84% So what defines a separating equilibria 00:32:53.050 --> 00:32:58.443 align:middle line:90% is that el is not equal to eh. 00:32:58.443 --> 00:32:59.610 align:middle line:90% So let's think through this. 00:32:59.610 --> 00:33:00.443 align:middle line:90% What does this mean? 00:33:00.443 --> 00:33:01.710 align:middle line:90% What does the receiver learn? 00:33:01.710 --> 00:33:05.030 align:middle line:84% If we're in an equilibrium where the low type chooses 00:33:05.030 --> 00:33:07.110 align:middle line:84% one education level and the high type 00:33:07.110 --> 00:33:08.790 align:middle line:84% chooses a different education level, 00:33:08.790 --> 00:33:11.402 align:middle line:84% what can the receiver conclude when they see 00:33:11.402 --> 00:33:12.610 align:middle line:90% the sender's education level? 00:33:12.610 --> 00:33:13.410 align:middle line:90% What do they learn? 00:33:13.410 --> 00:33:17.750 align:middle line:90% 00:33:17.750 --> 00:33:20.430 align:middle line:84% Well, they perfectly learn the sender's type 00:33:20.430 --> 00:33:23.910 align:middle line:84% because in a separating equilibrium, 00:33:23.910 --> 00:33:25.210 align:middle line:90% the implication of that-- 00:33:25.210 --> 00:33:29.510 align:middle line:90% 00:33:29.510 --> 00:33:31.510 align:middle line:90% let's write it this way. 00:33:31.510 --> 00:33:34.830 align:middle line:84% If I'm the receiver, and I see that you chose a high level-- 00:33:34.830 --> 00:33:37.230 align:middle line:84% that you chose eh, I know for certain 00:33:37.230 --> 00:33:38.530 align:middle line:90% that you must be a high type. 00:33:38.530 --> 00:33:41.790 align:middle line:90% 00:33:41.790 --> 00:33:43.610 align:middle line:90% And conversely, 0 here. 00:33:43.610 --> 00:33:44.110 align:middle line:90% Why? 00:33:44.110 --> 00:33:47.670 align:middle line:84% Well, if I see eh, the low type never chooses eh. 00:33:47.670 --> 00:33:50.150 align:middle line:84% They always choose a different education level, el. 00:33:50.150 --> 00:33:52.930 align:middle line:84% So if I see eh, I know that you must be a high type, 00:33:52.930 --> 00:33:54.490 align:middle line:84% and therefore, I assign probability 1 00:33:54.490 --> 00:33:55.930 align:middle line:90% to you being a high type. 00:33:55.930 --> 00:33:58.450 align:middle line:84% Conversely, if I see a low education level, 00:33:58.450 --> 00:34:00.330 align:middle line:84% then I know that the only kind of person 00:34:00.330 --> 00:34:02.455 align:middle line:84% who would choose a low level of education, at least 00:34:02.455 --> 00:34:04.210 align:middle line:84% in this equilibrium, is the low type, 00:34:04.210 --> 00:34:07.250 align:middle line:84% so I assign 0 probability to being the high type. 00:34:07.250 --> 00:34:11.670 align:middle line:84% And then what about b of h given e for all other e? 00:34:11.670 --> 00:34:16.330 align:middle line:90% 00:34:16.330 --> 00:34:20.850 align:middle line:84% What about for e that aren't equal to el or eh? 00:34:20.850 --> 00:34:22.650 align:middle line:84% Well, there's no restriction on these. 00:34:22.650 --> 00:34:24.429 align:middle line:90% So these are unconstrained. 00:34:24.429 --> 00:34:26.449 align:middle line:84% And we'll fill these in as we want, 00:34:26.449 --> 00:34:28.070 align:middle line:84% but this is going to be unconstrained. 00:34:28.070 --> 00:34:33.409 align:middle line:90% 00:34:33.409 --> 00:34:36.210 align:middle line:84% OK, now let's look at the pooling equilibrium 00:34:36.210 --> 00:34:39.010 align:middle line:90% where el equals eh. 00:34:39.010 --> 00:34:41.690 align:middle line:84% And sometimes it's nice to give a label to this. 00:34:41.690 --> 00:34:44.570 align:middle line:90% So let's call this e star. 00:34:44.570 --> 00:34:46.730 align:middle line:84% So here, both the high types and the low types 00:34:46.730 --> 00:34:50.489 align:middle line:84% choose the same level of education, e star. 00:34:50.489 --> 00:34:55.000 align:middle line:84% So now, what is b of h given e star? 00:34:55.000 --> 00:35:10.340 align:middle line:90% 00:35:10.340 --> 00:35:13.380 align:middle line:84% Well, if everyone choose the same level of education, 00:35:13.380 --> 00:35:16.300 align:middle line:84% their choice of education hasn't revealed any new information 00:35:16.300 --> 00:35:17.250 align:middle line:90% to me. 00:35:17.250 --> 00:35:19.000 align:middle line:84% So my belief about them should be the same 00:35:19.000 --> 00:35:21.460 align:middle line:84% that it was, the same as my prior. 00:35:21.460 --> 00:35:25.160 align:middle line:84% And my prior belief was pi that they were high ability. 00:35:25.160 --> 00:35:27.440 align:middle line:90% So this must be pi. 00:35:27.440 --> 00:35:36.620 align:middle line:90% 00:35:36.620 --> 00:35:39.320 align:middle line:84% Because-- I mean, you could write out Bayes' rule, 00:35:39.320 --> 00:35:41.780 align:middle line:84% but I think the basic idea is if I thought 00:35:41.780 --> 00:35:44.300 align:middle line:84% that the probability that your high ability was pi, 00:35:44.300 --> 00:35:46.060 align:middle line:84% and then I know that whoever you are, 00:35:46.060 --> 00:35:48.300 align:middle line:84% you're going to choose the education level e star, 00:35:48.300 --> 00:35:49.758 align:middle line:84% then the fact that you chose e star 00:35:49.758 --> 00:35:52.300 align:middle line:84% doesn't give me any new information about your ability. 00:35:52.300 --> 00:35:54.880 align:middle line:90% So it's-- this must still be pi. 00:35:54.880 --> 00:35:59.160 align:middle line:84% And then again, we have flexibility 00:35:59.160 --> 00:36:07.340 align:middle line:84% is unconstrained for all other e. 00:36:07.340 --> 00:36:10.080 align:middle line:90% 00:36:10.080 --> 00:36:12.920 align:middle line:84% And part of constructing equilibria 00:36:12.920 --> 00:36:17.160 align:middle line:84% is going to be to think a bit about what those beliefs have 00:36:17.160 --> 00:36:18.400 align:middle line:90% to be. 00:36:18.400 --> 00:36:19.668 align:middle line:90% So let's start with pooling. 00:36:19.668 --> 00:36:21.460 align:middle line:84% I think pooling is actually an easier case, 00:36:21.460 --> 00:36:22.700 align:middle line:90% so let's start there. 00:36:22.700 --> 00:36:26.800 align:middle line:90% 00:36:26.800 --> 00:36:28.860 align:middle line:84% So let's try to construct our equilibrium. 00:36:28.860 --> 00:36:32.660 align:middle line:84% So case 1 is the pooling equilibrium. 00:36:32.660 --> 00:36:43.660 align:middle line:90% 00:36:43.660 --> 00:36:47.840 align:middle line:84% And for the sender, we know that they're choosing 00:36:47.840 --> 00:36:51.185 align:middle line:90% el equals eh equals e star. 00:36:51.185 --> 00:36:53.560 align:middle line:84% And we're going to try to solve for this value of e star, 00:36:53.560 --> 00:36:56.460 align:middle line:84% but we're just going to put it in as a placeholder for now. 00:36:56.460 --> 00:37:00.580 align:middle line:84% And now let's look at the receiver. 00:37:00.580 --> 00:37:02.920 align:middle line:84% The receiver needs to choose w of e-- 00:37:02.920 --> 00:37:08.820 align:middle line:90% 00:37:08.820 --> 00:37:11.480 align:middle line:84% let's think through as the analyst what we want to do here. 00:37:11.480 --> 00:37:15.460 align:middle line:84% We're trying to construct an equilibrium where 00:37:15.460 --> 00:37:17.340 align:middle line:84% both the low type and the high type 00:37:17.340 --> 00:37:20.380 align:middle line:84% choose this education level e star. 00:37:20.380 --> 00:37:24.180 align:middle line:84% And now we're going to try to choose what this wage is to make 00:37:24.180 --> 00:37:26.240 align:middle line:90% e star optimal for the sender. 00:37:26.240 --> 00:37:29.420 align:middle line:84% So any ideas about what might be-- 00:37:29.420 --> 00:37:33.260 align:middle line:84% Well, let's think a little bit about this wage. 00:37:33.260 --> 00:37:35.000 align:middle line:90% Let's separate into two cases. 00:37:35.000 --> 00:37:41.220 align:middle line:90% 00:37:41.220 --> 00:37:44.320 align:middle line:90% What is w of e star have to be? 00:37:44.320 --> 00:37:47.520 align:middle line:84% We're going to argue that if we're looking for an equilibrium 00:37:47.520 --> 00:37:51.520 align:middle line:84% where both the low type and the high type pool on this education 00:37:51.520 --> 00:37:53.320 align:middle line:84% level e star, and now we're going 00:37:53.320 --> 00:37:56.600 align:middle line:84% to try to fill in, in this equilibrium, what the receiver's 00:37:56.600 --> 00:37:57.813 align:middle line:90% wage strategy is. 00:37:57.813 --> 00:37:59.480 align:middle line:84% And I argue that here, we basically know 00:37:59.480 --> 00:38:03.920 align:middle line:84% what it has to be by the combination of belief 00:38:03.920 --> 00:38:09.040 align:middle line:84% consistency and sequential optimality. 00:38:09.040 --> 00:38:10.940 align:middle line:84% Well, we know what the beliefs are here. 00:38:10.940 --> 00:38:13.120 align:middle line:84% We just said the beliefs are that your ability is 00:38:13.120 --> 00:38:15.200 align:middle line:90% high probability pi. 00:38:15.200 --> 00:38:18.720 align:middle line:84% And if those are our beliefs, well, over-- 00:38:18.720 --> 00:38:19.820 align:middle line:90% where did we write it? 00:38:19.820 --> 00:38:22.340 align:middle line:84% Here, we said if those are your beliefs, 00:38:22.340 --> 00:38:24.560 align:middle line:84% we computed what the wage has to be. 00:38:24.560 --> 00:38:30.160 align:middle line:84% So let's plug in pi for this and 1 minus pi for this. 00:38:30.160 --> 00:38:34.720 align:middle line:84% And indeed, we get that this is h times pi 00:38:34.720 --> 00:38:37.860 align:middle line:90% plus l times 1 minus pi. 00:38:37.860 --> 00:38:43.200 align:middle line:90% 00:38:43.200 --> 00:38:46.020 align:middle line:90% So that's pinned down. 00:38:46.020 --> 00:38:48.380 align:middle line:84% And I think this is a key observation when analyzing 00:38:48.380 --> 00:38:49.620 align:middle line:90% these signaling games. 00:38:49.620 --> 00:38:53.260 align:middle line:84% The question is, I said that your beliefs 00:38:53.260 --> 00:38:54.800 align:middle line:90% off-path are unconstrained. 00:38:54.800 --> 00:38:57.340 align:middle line:84% So if you observe-- when the receiver observes 00:38:57.340 --> 00:38:59.443 align:middle line:84% a level of education that's different from e star, 00:38:59.443 --> 00:39:00.860 align:middle line:84% we can fill in whatever beliefs we 00:39:00.860 --> 00:39:03.660 align:middle line:84% want for that kind of receiver, but we can't 00:39:03.660 --> 00:39:05.580 align:middle line:90% fill in whatever wage we want. 00:39:05.580 --> 00:39:07.940 align:middle line:84% What is the range of wages we could possibly 00:39:07.940 --> 00:39:11.820 align:middle line:84% put in here that would be consistent with receiver 00:39:11.820 --> 00:39:13.900 align:middle line:90% optimality? 00:39:13.900 --> 00:39:16.620 align:middle line:84% How high or how low could these wages be? 00:39:16.620 --> 00:39:18.740 align:middle line:84% Well, what's the most pessimistic belief 00:39:18.740 --> 00:39:20.180 align:middle line:90% the receiver could form? 00:39:20.180 --> 00:39:23.040 align:middle line:84% What's the worst belief they could have about the-- 00:39:23.040 --> 00:39:25.100 align:middle line:84% they could think you're certainly a low type. 00:39:25.100 --> 00:39:26.660 align:middle line:84% And if they think you're certainly a low type, what 00:39:26.660 --> 00:39:28.280 align:middle line:90% is the optimal wage to charge? 00:39:28.280 --> 00:39:31.220 align:middle line:90% 00:39:31.220 --> 00:39:32.700 align:middle line:90% Just l. 00:39:32.700 --> 00:39:34.720 align:middle line:84% And if they think you're certainly a high type, 00:39:34.720 --> 00:39:36.620 align:middle line:90% then the optimal wage is h. 00:39:36.620 --> 00:39:38.587 align:middle line:84% And any belief they form is going 00:39:38.587 --> 00:39:40.420 align:middle line:84% to make it so that the optimal wage for them 00:39:40.420 --> 00:39:42.420 align:middle line:84% is going to be somewhere between l and h. 00:39:42.420 --> 00:39:44.120 align:middle line:84% It can't be lower than l because there's 00:39:44.120 --> 00:39:46.480 align:middle line:84% no belief that would make the expectation lower than l. 00:39:46.480 --> 00:39:48.200 align:middle line:84% It can't be higher than h because there's 00:39:48.200 --> 00:39:50.617 align:middle line:84% no belief that would make the wage the optimal wage higher 00:39:50.617 --> 00:39:57.420 align:middle line:84% than h, but we can actually choose anything between l and h. 00:39:57.420 --> 00:40:01.120 align:middle line:90% 00:40:01.120 --> 00:40:03.260 align:middle line:84% Because whatever choice we make between l and h, 00:40:03.260 --> 00:40:05.000 align:middle line:84% we can then construct some belief that 00:40:05.000 --> 00:40:07.140 align:middle line:90% justifies that wage choice. 00:40:07.140 --> 00:40:17.440 align:middle line:90% 00:40:17.440 --> 00:40:22.200 align:middle line:84% So let's try to think this through. 00:40:22.200 --> 00:40:26.800 align:middle line:84% If you're trying to construct a wage that makes-- 00:40:26.800 --> 00:40:28.720 align:middle line:84% we want to make this an equilibrium. 00:40:28.720 --> 00:40:33.480 align:middle line:84% So let's maybe think of it graphing it down here. 00:40:33.480 --> 00:40:36.200 align:middle line:90% Here's e star. 00:40:36.200 --> 00:40:37.500 align:middle line:90% Here's our wage w. 00:40:37.500 --> 00:40:40.080 align:middle line:90% 00:40:40.080 --> 00:40:43.560 align:middle line:84% The wage here is pinned down to be this. 00:40:43.560 --> 00:40:45.120 align:middle line:84% Maybe I'll give notation for this. 00:40:45.120 --> 00:40:51.380 align:middle line:90% 00:40:51.380 --> 00:40:52.480 align:middle line:90% I'll just write it down. 00:40:52.480 --> 00:40:54.200 align:middle line:90% I'll just-- that's here. 00:40:54.200 --> 00:40:58.860 align:middle line:90% 00:40:58.860 --> 00:41:01.900 align:middle line:84% And now the question is, what choices of wage 00:41:01.900 --> 00:41:05.900 align:middle line:84% should we make off-path to ensure that the sender actually 00:41:05.900 --> 00:41:08.460 align:middle line:84% wants to choose an education level of e star? 00:41:08.460 --> 00:41:10.640 align:middle line:84% So here, we're trying to construct an equilibrium. 00:41:10.640 --> 00:41:14.220 align:middle line:84% We don't want the sender to have any profitable deviations. 00:41:14.220 --> 00:41:17.700 align:middle line:84% So any ideas about what wage we would set off-path 00:41:17.700 --> 00:41:21.540 align:middle line:84% to encourage the sender to choose this education level e 00:41:21.540 --> 00:41:22.040 align:middle line:90% star? 00:41:22.040 --> 00:41:25.220 align:middle line:90% 00:41:25.220 --> 00:41:27.300 align:middle line:84% Well, what's the sender tempted to do? 00:41:27.300 --> 00:41:29.800 align:middle line:84% They're supposed to choose e star in equilibrium. 00:41:29.800 --> 00:41:32.157 align:middle line:84% They're attempting to choose a lower level of education 00:41:32.157 --> 00:41:34.240 align:middle line:84% because if they choose a lower level of education, 00:41:34.240 --> 00:41:35.980 align:middle line:90% that's less costly for them. 00:41:35.980 --> 00:41:37.940 align:middle line:84% So the only way to make sure that that's not 00:41:37.940 --> 00:41:39.570 align:middle line:84% a profitable deviation is to say, well, 00:41:39.570 --> 00:41:41.210 align:middle line:84% if you choose any less education, 00:41:41.210 --> 00:41:43.230 align:middle line:84% your wage is going to drop a lot. 00:41:43.230 --> 00:41:44.710 align:middle line:84% And let's look at the extreme case 00:41:44.710 --> 00:41:47.890 align:middle line:84% where the wage drops all the way to l. 00:41:47.890 --> 00:41:55.422 align:middle line:90% 00:41:55.422 --> 00:41:57.630 align:middle line:84% So we're going to think of a wage function that says, 00:41:57.630 --> 00:41:59.950 align:middle line:84% if you don't get enough education e star, 00:41:59.950 --> 00:42:03.270 align:middle line:84% you're going to get paid a very low wage of l. 00:42:03.270 --> 00:42:06.270 align:middle line:84% But if you go to college, if you achieve this certification, 00:42:06.270 --> 00:42:07.770 align:middle line:84% if you go to school for long enough, 00:42:07.770 --> 00:42:11.990 align:middle line:84% now you can get a job that pays a higher wage of this. 00:42:11.990 --> 00:42:14.430 align:middle line:84% And then what happens if e is even higher than that? 00:42:14.430 --> 00:42:16.590 align:middle line:84% Well, we want to make sure that no one wants 00:42:16.590 --> 00:42:19.510 align:middle line:84% to get even more education than necessary, so we might as well 00:42:19.510 --> 00:42:20.730 align:middle line:90% just keep the wage flat. 00:42:20.730 --> 00:42:24.590 align:middle line:90% 00:42:24.590 --> 00:42:26.590 align:middle line:84% And if we keep the wage flat, then we 00:42:26.590 --> 00:42:29.470 align:middle line:84% know that, well, if someone can get the same wage by choosing 00:42:29.470 --> 00:42:32.190 align:middle line:84% e star as choosing a higher level of education, 00:42:32.190 --> 00:42:34.630 align:middle line:84% they would never pay more to get an even higher level 00:42:34.630 --> 00:42:36.750 align:middle line:84% of education if the wage that they're going to get 00:42:36.750 --> 00:42:37.750 align:middle line:90% is exactly the same. 00:42:37.750 --> 00:42:40.730 align:middle line:90% 00:42:40.730 --> 00:42:44.850 align:middle line:84% So this is, I think, a good conjecture 00:42:44.850 --> 00:42:47.430 align:middle line:84% for what this equilibrium might be. 00:42:47.430 --> 00:42:51.850 align:middle line:90% 00:42:51.850 --> 00:42:54.770 align:middle line:84% So let's write this out more formally. 00:42:54.770 --> 00:43:00.870 align:middle line:84% What we have is w of e equals l if e is less than e star. 00:43:00.870 --> 00:43:04.970 align:middle line:90% 00:43:04.970 --> 00:43:07.570 align:middle line:90% Strictly less than e star. 00:43:07.570 --> 00:43:13.230 align:middle line:90% And it's h pi plus l1 minus pi. 00:43:13.230 --> 00:43:25.650 align:middle line:90% 00:43:25.650 --> 00:43:32.030 align:middle line:84% And then if you wanted to fill in beliefs, 00:43:32.030 --> 00:43:42.510 align:middle line:84% let's just be really precise, our beliefs are-- well, 00:43:42.510 --> 00:43:46.030 align:middle line:84% if e is less than e star, the firm charges-- 00:43:46.030 --> 00:43:47.817 align:middle line:90% pays a wage of l. 00:43:47.817 --> 00:43:50.150 align:middle line:84% The only way it's optimal for them to charge a wage of l 00:43:50.150 --> 00:43:52.430 align:middle line:84% is if they believe you're certain to be a low type. 00:43:52.430 --> 00:43:54.210 align:middle line:90% So we must have 0 here. 00:43:54.210 --> 00:44:00.670 align:middle line:90% 00:44:00.670 --> 00:44:02.390 align:middle line:84% And then if they see a level of education 00:44:02.390 --> 00:44:05.650 align:middle line:84% that's at least e star, they're paying exactly this wage. 00:44:05.650 --> 00:44:08.330 align:middle line:84% So what must their belief be if they see a level of education 00:44:08.330 --> 00:44:11.370 align:middle line:84% at least e star to make that wage optimal given their belief? 00:44:11.370 --> 00:44:14.150 align:middle line:90% 00:44:14.150 --> 00:44:16.430 align:middle line:90% Yeah? 00:44:16.430 --> 00:44:17.030 align:middle line:90% Let's think. 00:44:17.030 --> 00:44:19.850 align:middle line:84% If you're a high type, what's the optimal wage? 00:44:19.850 --> 00:44:22.710 align:middle line:90% 00:44:22.710 --> 00:44:26.350 align:middle line:84% If I know you're a high type, now my optimal wage is h. 00:44:26.350 --> 00:44:30.950 align:middle line:84% But the wage they're charging is this, 00:44:30.950 --> 00:44:34.370 align:middle line:84% which is h times pi plus l times 1 minus pi. 00:44:34.370 --> 00:44:38.710 align:middle line:84% So I need to find the belief that makes my expected-- 00:44:38.710 --> 00:44:43.490 align:middle line:84% so that makes the expected type equal to this. 00:44:43.490 --> 00:44:44.390 align:middle line:90% Pi, exactly. 00:44:44.390 --> 00:44:47.127 align:middle line:90% 00:44:47.127 --> 00:44:49.210 align:middle line:84% If I ever-- if there was some education level that 00:44:49.210 --> 00:44:50.752 align:middle line:84% made me believe you were a high type, 00:44:50.752 --> 00:44:52.790 align:middle line:84% then I'd want to charge you a high wage there, 00:44:52.790 --> 00:44:53.910 align:middle line:84% and now there might be a deviation 00:44:53.910 --> 00:44:55.930 align:middle line:84% because now someone would want to get that education to prove 00:44:55.930 --> 00:44:56.972 align:middle line:90% that they're a high type. 00:44:56.972 --> 00:44:59.450 align:middle line:84% So let's think through what's happening here. 00:44:59.450 --> 00:45:01.490 align:middle line:84% The firms are saying, look, there's 00:45:01.490 --> 00:45:05.050 align:middle line:84% some minimal education level e star that you need to get. 00:45:05.050 --> 00:45:06.965 align:middle line:84% If you get less than that education level, 00:45:06.965 --> 00:45:08.590 align:middle line:84% I don't think you're very high ability, 00:45:08.590 --> 00:45:10.382 align:middle line:84% I think you're certain to be a low ability, 00:45:10.382 --> 00:45:13.530 align:middle line:84% and I'm going to charge you a low wage, namely l. 00:45:13.530 --> 00:45:16.010 align:middle line:84% But if you get enough education-- let's 00:45:16.010 --> 00:45:18.090 align:middle line:84% say you graduate from college, then I'm 00:45:18.090 --> 00:45:20.230 align:middle line:84% going to form a more optimistic belief about you. 00:45:20.230 --> 00:45:23.170 align:middle line:84% I'm going to believe that you're just like anyone 00:45:23.170 --> 00:45:24.430 align:middle line:90% from the population. 00:45:24.430 --> 00:45:26.490 align:middle line:84% You're a high type with probability pi 00:45:26.490 --> 00:45:28.730 align:middle line:84% and you're low type with probability 1 minus pi. 00:45:28.730 --> 00:45:31.910 align:middle line:84% And because I believe that, I'm going to charge you 00:45:31.910 --> 00:45:34.830 align:middle line:90% a wage that's equal to this. 00:45:34.830 --> 00:45:37.190 align:middle line:84% So we have this two-tiered wage system 00:45:37.190 --> 00:45:39.715 align:middle line:84% where if you graduate from college, you get a wage premium, 00:45:39.715 --> 00:45:41.090 align:middle line:84% but at least in this equilibrium, 00:45:41.090 --> 00:45:43.530 align:middle line:84% further education doesn't further increase your wage. 00:45:43.530 --> 00:45:45.290 align:middle line:84% And that's consistent with some settings, 00:45:45.290 --> 00:45:47.090 align:middle line:84% we could also construct other equilibria. 00:45:47.090 --> 00:45:48.628 align:middle line:90% But we're not quite done yet. 00:45:48.628 --> 00:45:50.670 align:middle line:84% We're going to have to be careful about the value 00:45:50.670 --> 00:45:54.270 align:middle line:84% of e star to actually make this in equilibrium. 00:45:54.270 --> 00:45:56.730 align:middle line:84% So let's check-- let's see what we've chosen, 00:45:56.730 --> 00:45:58.790 align:middle line:90% what we've satisfied here. 00:45:58.790 --> 00:46:03.830 align:middle line:84% Belief consistency is good because the only time belief 00:46:03.830 --> 00:46:07.050 align:middle line:84% consistency has any bite here is when e equals e star. 00:46:07.050 --> 00:46:09.750 align:middle line:84% That's the only on-path level of education, 00:46:09.750 --> 00:46:11.350 align:middle line:84% and we're consistent with Bayes' rule, 00:46:11.350 --> 00:46:13.950 align:middle line:84% so we're good with belief consistency. 00:46:13.950 --> 00:46:16.550 align:middle line:84% And by construction, the receiver 00:46:16.550 --> 00:46:20.030 align:middle line:84% is always paying the wage that's equal to their updated 00:46:20.030 --> 00:46:21.950 align:middle line:84% expectation of the sender's type. 00:46:21.950 --> 00:46:24.110 align:middle line:90% So we're good for the receiver. 00:46:24.110 --> 00:46:28.270 align:middle line:84% The one thing we still have to check is sender optimality. 00:46:28.270 --> 00:46:33.305 align:middle line:84% So what could go wrong in this equilibrium? 00:46:33.305 --> 00:46:34.930 align:middle line:84% How could sender optimality potentially 00:46:34.930 --> 00:46:37.270 align:middle line:84% be violated if we choose e star wrong? 00:46:37.270 --> 00:46:42.370 align:middle line:90% 00:46:42.370 --> 00:46:47.970 align:middle line:84% Well, the problem is, if e star is really, really big, 00:46:47.970 --> 00:46:51.850 align:middle line:84% I might say, look, yes, I get a higher wage 00:46:51.850 --> 00:46:54.150 align:middle line:84% if I get at least e star of education, 00:46:54.150 --> 00:46:56.450 align:middle line:84% but that education is so costly to me 00:46:56.450 --> 00:46:59.450 align:middle line:84% that it's not worth me going to school to get that higher wage. 00:46:59.450 --> 00:47:02.898 align:middle line:84% So we need to check that e star is intuitively-- e star is going 00:47:02.898 --> 00:47:04.690 align:middle line:84% to have to be low enough that it's actually 00:47:04.690 --> 00:47:06.610 align:middle line:84% worthwhile for people to go to college. 00:47:06.610 --> 00:47:08.730 align:middle line:84% Just because there's a college wage premium, 00:47:08.730 --> 00:47:10.330 align:middle line:84% if the college wage premium is smaller 00:47:10.330 --> 00:47:12.750 align:middle line:84% than the cost of education, people aren't going to go. 00:47:12.750 --> 00:47:14.690 align:middle line:84% And in fact, if we look over here, 00:47:14.690 --> 00:47:16.770 align:middle line:84% the college wage premium is not too far 00:47:16.770 --> 00:47:17.970 align:middle line:90% from the cost of education. 00:47:17.970 --> 00:47:19.110 align:middle line:90% So that is an issue. 00:47:19.110 --> 00:47:21.890 align:middle line:84% We have to make sure that the wage premium exceeds 00:47:21.890 --> 00:47:23.150 align:middle line:90% the cost of education. 00:47:23.150 --> 00:47:25.110 align:middle line:84% So we have a final condition here. 00:47:25.110 --> 00:47:28.490 align:middle line:84% So we need to-- it remains to check sender optimality. 00:47:28.490 --> 00:47:37.390 align:middle line:90% 00:47:37.390 --> 00:47:38.650 align:middle line:90% So at first, it's tricky. 00:47:38.650 --> 00:47:42.890 align:middle line:84% We have to say, for both types, t equals l and h. 00:47:42.890 --> 00:47:45.710 align:middle line:90% 00:47:45.710 --> 00:47:56.690 align:middle line:84% Choosing e star is better than any other education e in 0, 1. 00:47:56.690 --> 00:48:01.062 align:middle line:90% 00:48:01.062 --> 00:48:02.950 align:middle line:84% Now this is a hard condition because it 00:48:02.950 --> 00:48:04.790 align:middle line:90% has to hold for both types. 00:48:04.790 --> 00:48:08.250 align:middle line:84% And we're comparing e star to any other level of education, 00:48:08.250 --> 00:48:10.590 align:middle line:84% and there's infinitely many other levels of education 00:48:10.590 --> 00:48:12.390 align:middle line:90% someone might choose. 00:48:12.390 --> 00:48:13.990 align:middle line:84% So the key to simplifying things is 00:48:13.990 --> 00:48:15.910 align:middle line:84% to say, well, instead of saying it's 00:48:15.910 --> 00:48:18.630 align:middle line:84% better than any other education, let's try 00:48:18.630 --> 00:48:21.410 align:middle line:90% to find the best deviation. 00:48:21.410 --> 00:48:25.030 align:middle line:90% 00:48:25.030 --> 00:48:26.698 align:middle line:84% I could deviate from e star to anything, 00:48:26.698 --> 00:48:28.240 align:middle line:84% and we have to make sure that I don't 00:48:28.240 --> 00:48:31.080 align:middle line:84% want to deviate from e star to any other level of education. 00:48:31.080 --> 00:48:33.840 align:middle line:84% But some of these other levels of education, it's 00:48:33.840 --> 00:48:35.600 align:middle line:84% clear that I'm never going to choose. 00:48:35.600 --> 00:48:38.515 align:middle line:84% So let's focus on the most attractive deviation, 00:48:38.515 --> 00:48:40.640 align:middle line:84% and as long as the most attractive deviation is not 00:48:40.640 --> 00:48:42.600 align:middle line:84% profitable, then no other deviation 00:48:42.600 --> 00:48:44.480 align:middle line:90% can be profitable either. 00:48:44.480 --> 00:48:47.280 align:middle line:84% So if I'm looking at this wage chart, 00:48:47.280 --> 00:48:49.920 align:middle line:84% this is the wage function I'm facing, 00:48:49.920 --> 00:48:52.600 align:middle line:84% and I'm choosing how much education to get. 00:48:52.600 --> 00:48:54.480 align:middle line:90% I'm already here at e star. 00:48:54.480 --> 00:48:56.313 align:middle line:84% What would be the most attractive deviation? 00:48:56.313 --> 00:48:58.063 align:middle line:84% I'm not saying it's profitable, but if I'm 00:48:58.063 --> 00:48:59.720 align:middle line:84% going to deviate, how should I deviate? 00:48:59.720 --> 00:49:02.420 align:middle line:84% What's the best possible deviation? 00:49:02.420 --> 00:49:04.240 align:middle line:90% Yeah? 00:49:04.240 --> 00:49:07.920 align:middle line:84% 0 is the only one that has a chance, because clearly, 00:49:07.920 --> 00:49:10.640 align:middle line:84% deviating above e star doesn't make any sense because I'm 00:49:10.640 --> 00:49:14.880 align:middle line:84% paying more in education and I'm not getting any wage benefit. 00:49:14.880 --> 00:49:17.500 align:middle line:84% So the only deviation that could be profitable-- 00:49:17.500 --> 00:49:19.560 align:middle line:84% and I should put "best" in parentheses. 00:49:19.560 --> 00:49:23.480 align:middle line:90% It could be that-- yeah, anyway. 00:49:23.480 --> 00:49:27.100 align:middle line:84% If I-- so I don't want to choose any education higher than e 00:49:27.100 --> 00:49:27.740 align:middle line:90% star. 00:49:27.740 --> 00:49:30.140 align:middle line:84% If I choose education strictly below e star, 00:49:30.140 --> 00:49:33.660 align:middle line:84% well, my wage is the same no matter what education I choose. 00:49:33.660 --> 00:49:35.800 align:middle line:84% So since I hate school, according to the model, 00:49:35.800 --> 00:49:37.800 align:middle line:84% I might as well choose the lowest possible level 00:49:37.800 --> 00:49:38.360 align:middle line:90% of education. 00:49:38.360 --> 00:49:39.900 align:middle line:84% So the only thing we have to check 00:49:39.900 --> 00:49:43.220 align:middle line:90% is a deviation of e equals 0. 00:49:43.220 --> 00:49:45.780 align:middle line:90% 00:49:45.780 --> 00:49:49.020 align:middle line:84% And now we have to check that both for the high type 00:49:49.020 --> 00:49:50.120 align:middle line:90% and for the low type. 00:49:50.120 --> 00:49:52.820 align:middle line:84% So let's write down the two conditions we need. 00:49:52.820 --> 00:49:56.700 align:middle line:84% So let's look at t equals l and t equals h. 00:49:56.700 --> 00:49:59.580 align:middle line:84% We want to compare what utility that person get 00:49:59.580 --> 00:50:01.940 align:middle line:90% if they choose e star versus 0. 00:50:01.940 --> 00:50:06.380 align:middle line:84% So if they choose e star, their utility is going to be-- 00:50:06.380 --> 00:50:10.260 align:middle line:90% I really need notation for this. 00:50:10.260 --> 00:50:13.060 align:middle line:90% Let me call this prior mean. 00:50:13.060 --> 00:50:17.720 align:middle line:84% Let me call this mu, mu for mean. 00:50:17.720 --> 00:50:20.220 align:middle line:84% So let me use-- just call that mu. 00:50:20.220 --> 00:50:25.840 align:middle line:84% So if I choose e star, I get a wage of mu minus 1 00:50:25.840 --> 00:50:31.080 align:middle line:90% over l e star squared. 00:50:31.080 --> 00:50:36.700 align:middle line:90% And that must be better than l. 00:50:36.700 --> 00:50:39.400 align:middle line:90% 00:50:39.400 --> 00:50:42.620 align:middle line:84% Maybe, just so we see what's going on here. 00:50:42.620 --> 00:50:45.280 align:middle line:90% 00:50:45.280 --> 00:50:47.680 align:middle line:84% OK, this is the key condition, let's go over it slowly. 00:50:47.680 --> 00:50:49.240 align:middle line:90% I'm a type l. 00:50:49.240 --> 00:50:53.080 align:middle line:84% If I do what I'm supposed to do, I get paid a wage of mu. 00:50:53.080 --> 00:50:55.520 align:middle line:84% I choose a level of education e star. 00:50:55.520 --> 00:50:59.220 align:middle line:84% My cost of that education is 1 over l times e star squared. 00:50:59.220 --> 00:51:02.880 align:middle line:84% That comes from the formula for my utility up here. 00:51:02.880 --> 00:51:05.520 align:middle line:84% And we've plugged in for my wage mu 00:51:05.520 --> 00:51:08.260 align:middle line:84% because that's the wage I get if I choose e star. 00:51:08.260 --> 00:51:09.960 align:middle line:90% So let's remind ourselves here. 00:51:09.960 --> 00:51:13.720 align:middle line:90% This is w of e star. 00:51:13.720 --> 00:51:16.000 align:middle line:84% If instead, I choose zero education, 00:51:16.000 --> 00:51:20.440 align:middle line:84% well, my wage is l because we see over here-- 00:51:20.440 --> 00:51:23.180 align:middle line:84% and maybe I'll write that down here. 00:51:23.180 --> 00:51:25.340 align:middle line:90% This is w of 0. 00:51:25.340 --> 00:51:28.860 align:middle line:84% This is the wage that's offered to me if I choose no education. 00:51:28.860 --> 00:51:31.460 align:middle line:84% And if I choose no education, this is the cost, 00:51:31.460 --> 00:51:34.280 align:middle line:84% but this cost is just 0, because if I choose no education, 00:51:34.280 --> 00:51:35.162 align:middle line:90% the cost is 0. 00:51:35.162 --> 00:51:37.120 align:middle line:84% So we don't even have to worry about this term. 00:51:37.120 --> 00:51:39.400 align:middle line:84% 0 squared is 0, I can just cross this out. 00:51:39.400 --> 00:51:41.980 align:middle line:90% 00:51:41.980 --> 00:51:43.820 align:middle line:84% And now let's go down to l equals h 00:51:43.820 --> 00:51:45.240 align:middle line:90% and write down the same formula. 00:51:45.240 --> 00:51:54.195 align:middle line:90% 00:51:54.195 --> 00:51:55.820 align:middle line:84% Now I think a lot of people are tempted 00:51:55.820 --> 00:51:59.040 align:middle line:84% to put an h on the right-hand side, but that's not correct. 00:51:59.040 --> 00:52:01.060 align:middle line:84% Because remember, what is l representing? 00:52:01.060 --> 00:52:02.960 align:middle line:90% l is the wage the firm charges. 00:52:02.960 --> 00:52:04.820 align:middle line:90% If you get no education. 00:52:04.820 --> 00:52:06.860 align:middle line:84% If I deviate and get no education, 00:52:06.860 --> 00:52:10.180 align:middle line:84% I'm offered a wage of l whether I'm a high type or low type. 00:52:10.180 --> 00:52:12.440 align:middle line:90% So this is coming from w of 0. 00:52:12.440 --> 00:52:14.140 align:middle line:84% It's not about my true type, it's 00:52:14.140 --> 00:52:15.960 align:middle line:84% about what the firm is actually doing. 00:52:15.960 --> 00:52:23.000 align:middle line:90% 00:52:23.000 --> 00:52:30.320 align:middle line:84% So here are the two conditions we have, 00:52:30.320 --> 00:52:33.320 align:middle line:84% and we need both of them to hold. 00:52:33.320 --> 00:52:34.880 align:middle line:84% It turns out, one of these conditions 00:52:34.880 --> 00:52:38.700 align:middle line:84% is going to be more demanding than the other. 00:52:38.700 --> 00:52:41.340 align:middle line:84% Which of these is harder to satisfy? 00:52:41.340 --> 00:52:46.098 align:middle line:90% 00:52:46.098 --> 00:52:47.140 align:middle line:90% We have two inequalities. 00:52:47.140 --> 00:52:47.838 align:middle line:90% Yeah? 00:52:47.838 --> 00:52:48.880 align:middle line:90% AUDIENCE: The one on top. 00:52:48.880 --> 00:52:49.060 align:middle line:90% IAN BALL: OK. 00:52:49.060 --> 00:52:49.880 align:middle line:90% And why? 00:52:49.880 --> 00:52:51.400 align:middle line:84% AUDIENCE: Because the 1 over l times 00:52:51.400 --> 00:52:54.480 align:middle line:84% e star squared term is greater than the 1 over h times 00:52:54.480 --> 00:52:55.500 align:middle line:90% the e star squared term. 00:52:55.500 --> 00:52:56.500 align:middle line:90% IAN BALL: Example right. 00:52:56.500 --> 00:52:59.338 align:middle line:84% So because l is smaller, 1 over l is bigger. 00:52:59.338 --> 00:53:00.880 align:middle line:84% So we're subtracting a bigger number, 00:53:00.880 --> 00:53:02.145 align:middle line:90% and that's harder to satisfy. 00:53:02.145 --> 00:53:03.020 align:middle line:90% And this makes sense. 00:53:03.020 --> 00:53:05.800 align:middle line:84% Let's think about what each of these inequalities say. 00:53:05.800 --> 00:53:07.960 align:middle line:84% If we move these things around, what we're saying 00:53:07.960 --> 00:53:11.160 align:middle line:84% is, let's think of moving the cost to this side 00:53:11.160 --> 00:53:13.040 align:middle line:90% and subtracting l on this side. 00:53:13.040 --> 00:53:14.700 align:middle line:90% Let me write that down here. 00:53:14.700 --> 00:53:16.140 align:middle line:90% This is the key condition. 00:53:16.140 --> 00:53:20.260 align:middle line:84% So the key condition is mu minus l 00:53:20.260 --> 00:53:24.960 align:middle line:84% is greater than or equal to 1 over l e star squared. 00:53:24.960 --> 00:53:27.780 align:middle line:84% And this, I think, is a very natural interpretation. 00:53:27.780 --> 00:53:31.540 align:middle line:84% Mu minus l is the, quote unquote, "college wage premium." 00:53:31.540 --> 00:53:34.420 align:middle line:84% This is the difference in wage I get if I choose an education 00:53:34.420 --> 00:53:37.320 align:middle line:84% level of e star versus education level of 0 00:53:37.320 --> 00:53:42.260 align:middle line:84% because if I choose e star, I get mu; if I choose 0, I get l. 00:53:42.260 --> 00:53:43.820 align:middle line:90% So this is the wage premium. 00:53:43.820 --> 00:53:46.740 align:middle line:84% And this condition says the wage premium is higher 00:53:46.740 --> 00:53:48.300 align:middle line:90% than the cost of education. 00:53:48.300 --> 00:53:50.980 align:middle line:84% And indeed, it's worth it for me to go to college 00:53:50.980 --> 00:53:53.580 align:middle line:84% if the wage premium I get from going to college-- that 00:53:53.580 --> 00:53:55.860 align:middle line:84% is, the additional wage I get from going to college 00:53:55.860 --> 00:53:57.620 align:middle line:84% more than outweighs the cost I have 00:53:57.620 --> 00:53:59.220 align:middle line:90% to bear of going to college. 00:53:59.220 --> 00:54:03.340 align:middle line:84% Now, we also need to check that the wage premium exceeds 00:54:03.340 --> 00:54:05.500 align:middle line:84% the cost of education for the high type. 00:54:05.500 --> 00:54:08.860 align:middle line:84% But for the high type, going to college is less costly. 00:54:08.860 --> 00:54:10.940 align:middle line:84% So as long as we check that the wage premium is 00:54:10.940 --> 00:54:12.510 align:middle line:84% worth it for the low type, well, it's 00:54:12.510 --> 00:54:14.510 align:middle line:84% certainly going to be worth it for the high type 00:54:14.510 --> 00:54:16.290 align:middle line:84% because they get the same wage premium, 00:54:16.290 --> 00:54:17.850 align:middle line:84% but they find education less costly, 00:54:17.850 --> 00:54:19.390 align:middle line:84% and that's exactly the observation 00:54:19.390 --> 00:54:20.550 align:middle line:90% that was made at the end. 00:54:20.550 --> 00:54:22.970 align:middle line:84% So this is our equilibrium condition. 00:54:22.970 --> 00:54:25.052 align:middle line:84% And as long as this is satisfied, 00:54:25.052 --> 00:54:27.010 align:middle line:84% then this is indeed going to be an equilibrium. 00:54:27.010 --> 00:54:31.310 align:middle line:90% 00:54:31.310 --> 00:54:34.230 align:middle line:84% So we actually have constructed many different equilibria 00:54:34.230 --> 00:54:36.172 align:middle line:84% parametrized by this level e star. 00:54:36.172 --> 00:54:37.630 align:middle line:84% There could be an equilibrium where 00:54:37.630 --> 00:54:40.528 align:middle line:84% we say if you go to school for 0.3-- for e equals 0.3, 00:54:40.528 --> 00:54:41.570 align:middle line:90% then you get a high wage. 00:54:41.570 --> 00:54:42.510 align:middle line:84% There could also be an equilibrium 00:54:42.510 --> 00:54:44.802 align:middle line:84% where if you go to school for 0.4, you get a high wage. 00:54:44.802 --> 00:54:46.552 align:middle line:84% And these are both conceivable equilibria, 00:54:46.552 --> 00:54:48.135 align:middle line:84% and there's a huge range of equilibria 00:54:48.135 --> 00:54:49.270 align:middle line:90% that we can construct here. 00:54:49.270 --> 00:54:50.890 align:middle line:90% Any questions on this? 00:54:50.890 --> 00:54:55.150 align:middle line:90% 00:54:55.150 --> 00:54:59.030 align:middle line:90% OK, so now, let's look for a-- 00:54:59.030 --> 00:55:00.730 align:middle line:84% let's look for a separating equilibria. 00:55:00.730 --> 00:55:02.250 align:middle line:84% So, so far, we've just done pooling. 00:55:02.250 --> 00:55:05.710 align:middle line:90% 00:55:05.710 --> 00:55:08.250 align:middle line:84% Let's look for a separating equilibria. 00:55:08.250 --> 00:55:24.250 align:middle line:90% 00:55:24.250 --> 00:55:26.790 align:middle line:84% Now let's look for a separating equilibria. 00:55:26.790 --> 00:55:34.930 align:middle line:90% 00:55:34.930 --> 00:55:40.970 align:middle line:84% So what that means is we have el is not equal to eh. 00:55:40.970 --> 00:55:43.050 align:middle line:84% So instead of just solving for e star 00:55:43.050 --> 00:55:46.030 align:middle line:90% we have to solve for el and eh. 00:55:46.030 --> 00:55:48.610 align:middle line:90% 00:55:48.610 --> 00:55:54.270 align:middle line:84% And now, let's try to think of what w is going to have to be. 00:55:54.270 --> 00:56:07.507 align:middle line:90% 00:56:07.507 --> 00:56:09.090 align:middle line:84% So I think on a problem set, if you're 00:56:09.090 --> 00:56:11.550 align:middle line:84% trying to construct equilibrium, it's always very helpful 00:56:11.550 --> 00:56:15.750 align:middle line:84% to separate what the firm-- or what the receiver does on-path 00:56:15.750 --> 00:56:17.870 align:middle line:84% from what the receiver does off-path. 00:56:17.870 --> 00:56:20.190 align:middle line:84% Because on-path, Bayes' rule applies, 00:56:20.190 --> 00:56:22.270 align:middle line:84% and we really have precise control 00:56:22.270 --> 00:56:24.230 align:middle line:90% over what the wage has to be. 00:56:24.230 --> 00:56:26.350 align:middle line:84% Off-path, Bayes rule can't be applied 00:56:26.350 --> 00:56:28.830 align:middle line:84% because we divide by 0 when we apply Bayes' rule, 00:56:28.830 --> 00:56:32.870 align:middle line:84% and we have a lot of flexibility in what this wage is. 00:56:32.870 --> 00:56:34.750 align:middle line:90% OK. 00:56:34.750 --> 00:56:37.050 align:middle line:84% So if we're looking for an equilibrium, 00:56:37.050 --> 00:56:39.750 align:middle line:84% a separating equilibrium, with these two levels of education, 00:56:39.750 --> 00:56:42.310 align:middle line:84% eh and el, what is the firm's wage 00:56:42.310 --> 00:56:46.090 align:middle line:84% going to have to be if they see an education level of el? 00:56:46.090 --> 00:56:47.590 align:middle line:84% Or maybe the first question is, what 00:56:47.590 --> 00:56:48.890 align:middle line:84% is their belief going to have to be, 00:56:48.890 --> 00:56:50.890 align:middle line:84% and then what is their wage going to have to be? 00:56:50.890 --> 00:56:57.270 align:middle line:90% 00:56:57.270 --> 00:56:59.150 align:middle line:84% Well, what do they learn if they see someone 00:56:59.150 --> 00:57:01.910 align:middle line:90% choose education level el? 00:57:01.910 --> 00:57:05.630 align:middle line:84% They learn that they must be a low type because the high types 00:57:05.630 --> 00:57:06.827 align:middle line:90% always choose eh. 00:57:06.827 --> 00:57:09.410 align:middle line:84% Notice, this is a key difference with the pooling equilibrium. 00:57:09.410 --> 00:57:11.950 align:middle line:84% If all the types choose the same level of education, 00:57:11.950 --> 00:57:14.810 align:middle line:84% I don't learn about your type by seeing your education level. 00:57:14.810 --> 00:57:17.610 align:middle line:84% But if different types choose different levels of education, 00:57:17.610 --> 00:57:19.890 align:middle line:84% now, your choice of education reveals 00:57:19.890 --> 00:57:22.050 align:middle line:84% a lot of information to me as the receiver. 00:57:22.050 --> 00:57:24.850 align:middle line:84% So when I el, I know that you're a low type 00:57:24.850 --> 00:57:27.050 align:middle line:84% and I must set the wage equal to l. 00:57:27.050 --> 00:57:33.070 align:middle line:84% And maybe in parentheses, this is because eh of el is 0. 00:57:33.070 --> 00:57:35.690 align:middle line:90% 00:57:35.690 --> 00:57:37.950 align:middle line:84% What about if I see an education level of eh? 00:57:37.950 --> 00:57:38.990 align:middle line:90% What do I conclude? 00:57:38.990 --> 00:57:42.050 align:middle line:90% 00:57:42.050 --> 00:57:42.610 align:middle line:90% Yeah? 00:57:42.610 --> 00:57:43.550 align:middle line:90% AUDIENCE: I want to say type h. 00:57:43.550 --> 00:57:44.758 align:middle line:90% IAN BALL: You must be type h. 00:57:44.758 --> 00:57:46.410 align:middle line:90% So now this is h. 00:57:46.410 --> 00:57:52.310 align:middle line:84% And now we have b of h given eh equals 1. 00:57:52.310 --> 00:57:55.690 align:middle line:90% 00:57:55.690 --> 00:58:00.450 align:middle line:84% And now the off-path wage, there's kind of a trick 00:58:00.450 --> 00:58:03.290 align:middle line:90% that we sometimes can use. 00:58:03.290 --> 00:58:05.510 align:middle line:84% If we really want to make this an equilibrium, 00:58:05.510 --> 00:58:06.990 align:middle line:84% we want to make it as unappealing 00:58:06.990 --> 00:58:10.590 align:middle line:84% as possible to deviate to any off-path level of education. 00:58:10.590 --> 00:58:12.570 align:middle line:84% So how could-- if we have flexibility here 00:58:12.570 --> 00:58:14.230 align:middle line:84% to choose this wage, how could we 00:58:14.230 --> 00:58:15.950 align:middle line:84% choose this wage to make deviations 00:58:15.950 --> 00:58:18.412 align:middle line:90% unappealing for the sender? 00:58:18.412 --> 00:58:20.370 align:middle line:84% And this may be the most realistic equilibrium, 00:58:20.370 --> 00:58:23.710 align:middle line:84% but it's kind of a good trick for computing equilibria. 00:58:23.710 --> 00:58:27.490 align:middle line:84% So 0 would really deter deviations, but we can't do 0. 00:58:27.490 --> 00:58:29.622 align:middle line:84% Why can't we necessarily-- why can't we do 0? 00:58:29.622 --> 00:58:31.330 align:middle line:84% That would violate one of our conditions. 00:58:31.330 --> 00:58:33.040 align:middle line:84% The problem is, it would violate r, 00:58:33.040 --> 00:58:35.290 align:middle line:84% and you may not have been here when we went over this, 00:58:35.290 --> 00:58:37.150 align:middle line:84% but this would violate r, and I think 00:58:37.150 --> 00:58:39.890 align:middle line:84% I just erased it, our receiver optimality condition. 00:58:39.890 --> 00:58:45.670 align:middle line:84% So off-path, Bayes' rule doesn't impose any constraints 00:58:45.670 --> 00:58:46.610 align:middle line:90% in our belief. 00:58:46.610 --> 00:58:48.230 align:middle line:84% But whatever our belief is, we still 00:58:48.230 --> 00:58:50.430 align:middle line:84% have to be behaving optimally given our belief. 00:58:50.430 --> 00:58:52.933 align:middle line:84% And the worst possible belief we could form off-path, worse 00:58:52.933 --> 00:58:54.350 align:middle line:84% from the standpoint of the sender, 00:58:54.350 --> 00:58:56.910 align:middle line:84% would to be certain that you're a low type. 00:58:56.910 --> 00:58:59.630 align:middle line:84% So if we're certain you're a low type, the wage we would charge 00:58:59.630 --> 00:59:00.590 align:middle line:90% is l. 00:59:00.590 --> 00:59:04.690 align:middle line:84% So you're right that a wage of 0 would be a stronger 00:59:04.690 --> 00:59:07.070 align:middle line:84% punishment for the sender for deviating, 00:59:07.070 --> 00:59:08.790 align:middle line:84% but it's not a credible punishment 00:59:08.790 --> 00:59:11.130 align:middle line:84% because even if the firm thinks you're always 00:59:11.130 --> 00:59:13.130 align:middle line:84% a low-ability person, they'd rather charge you 00:59:13.130 --> 00:59:15.710 align:middle line:84% a wage of l, which is strictly positive than a wage of 0, 00:59:15.710 --> 00:59:20.010 align:middle line:90% but that's a helpful comment. 00:59:20.010 --> 00:59:21.090 align:middle line:90% So here we are. 00:59:21.090 --> 00:59:27.090 align:middle line:84% Now I think we have some trouble because if I'm a low type, well, 00:59:27.090 --> 00:59:31.170 align:middle line:84% why should I choose el and get l if I could just-- well, 00:59:31.170 --> 00:59:33.990 align:middle line:84% any other education level I choose, I'll get l as well, 00:59:33.990 --> 00:59:37.290 align:middle line:84% so why don't I just deviate to education level equals 0? 00:59:37.290 --> 00:59:39.530 align:middle line:84% If I'm the low type, I think deviating 00:59:39.530 --> 00:59:42.810 align:middle line:90% to equals 0 looks pretty good. 00:59:42.810 --> 00:59:45.490 align:middle line:84% So in fact, we immediately see a strong constraint 00:59:45.490 --> 00:59:47.550 align:middle line:84% on what happens in a separating equilibrium. 00:59:47.550 --> 00:59:55.490 align:middle line:84% So first observation, note, must be 0. 00:59:55.490 --> 00:59:58.450 align:middle line:84% If el was not 0, then the low type 00:59:58.450 --> 01:00:01.270 align:middle line:84% would deviate to choose an education level of 0. 01:00:01.270 --> 01:00:03.470 align:middle line:84% And let's think through why this is. 01:00:03.470 --> 01:00:06.270 align:middle line:84% In a separating equilibrium, put yourself 01:00:06.270 --> 01:00:07.830 align:middle line:90% in the position of a low type. 01:00:07.830 --> 01:00:10.830 align:middle line:84% When you follow your equilibrium strategy and choose el, 01:00:10.830 --> 01:00:14.310 align:middle line:84% you're fully revealing to the firm that you're a low ability 01:00:14.310 --> 01:00:17.030 align:middle line:84% or that you're a low type sender, which 01:00:17.030 --> 01:00:19.630 align:middle line:84% is the worst possible thing they can believe about you. 01:00:19.630 --> 01:00:23.470 align:middle line:84% So it never makes sense to pay a positive cost to go to school, 01:00:23.470 --> 01:00:26.065 align:middle line:84% only to be believed that you're the worst possible type. 01:00:26.065 --> 01:00:28.190 align:middle line:84% So if I'm going to induce the worst possible belief 01:00:28.190 --> 01:00:30.310 align:middle line:84% about myself, I might as well pay no cost 01:00:30.310 --> 01:00:33.550 align:middle line:90% and not go to school at all. 01:00:33.550 --> 01:00:36.630 align:middle line:90% So now el equals 0. 01:00:36.630 --> 01:00:37.150 align:middle line:90% Good. 01:00:37.150 --> 01:00:39.830 align:middle line:90% 01:00:39.830 --> 01:00:42.930 align:middle line:90% And let's try to graph this. 01:00:42.930 --> 01:00:45.328 align:middle line:90% 01:00:45.328 --> 01:00:46.870 align:middle line:84% Let's do our graph again, and I think 01:00:46.870 --> 01:00:51.390 align:middle line:84% it's actually helpful to try to represent the sender optimality 01:00:51.390 --> 01:00:53.510 align:middle line:84% conditions on the graph, so let's try to do this. 01:00:53.510 --> 01:00:56.390 align:middle line:84% So we're going to look at a wage here. 01:00:56.390 --> 01:00:58.310 align:middle line:90% el is going to be 0. 01:00:58.310 --> 01:01:02.200 align:middle line:84% eh is not equal to el, so eh has to be strictly positive. 01:01:02.200 --> 01:01:06.920 align:middle line:84% So let's just put a guess of where eh is. 01:01:06.920 --> 01:01:09.640 align:middle line:84% And we said that the wage is going 01:01:09.640 --> 01:01:13.440 align:middle line:84% to be l If you deviate-- if you do something off-path. 01:01:13.440 --> 01:01:17.240 align:middle line:90% So here's l. 01:01:17.240 --> 01:01:18.840 align:middle line:90% Here's h. 01:01:18.840 --> 01:01:28.720 align:middle line:84% And to here, the wage is going to look like this. 01:01:28.720 --> 01:01:33.280 align:middle line:90% Now, at eh, the wage jumps to h. 01:01:33.280 --> 01:01:40.160 align:middle line:84% So this is an open circle, this is a closed circle here, 01:01:40.160 --> 01:01:42.000 align:middle line:84% and then the wage continues down. 01:01:42.000 --> 01:01:46.680 align:middle line:84% So this is, if we want to be really safe, what we would do. 01:01:46.680 --> 01:01:49.120 align:middle line:84% But to be a little more realistic, 01:01:49.120 --> 01:01:52.400 align:middle line:84% well, we want to make sure that no one wants 01:01:52.400 --> 01:01:54.540 align:middle line:84% to deviate to this high level of education. 01:01:54.540 --> 01:01:56.640 align:middle line:90% But this seems like-- 01:01:56.640 --> 01:01:58.100 align:middle line:84% I mean, it could be an equilibrium, 01:01:58.100 --> 01:01:59.660 align:middle line:84% but it seems kind of weird to say, 01:01:59.660 --> 01:02:01.160 align:middle line:84% if you get a really high education, 01:02:01.160 --> 01:02:02.600 align:middle line:90% we believe you're a low type. 01:02:02.600 --> 01:02:04.725 align:middle line:84% I mean, it's possible, but it seems a little weird. 01:02:04.725 --> 01:02:06.980 align:middle line:84% So I think often what we'll do is 01:02:06.980 --> 01:02:10.160 align:middle line:84% we'll choose the off-path beliefs like this 01:02:10.160 --> 01:02:13.780 align:middle line:84% so that actually, if you choose any education level above eh, 01:02:13.780 --> 01:02:15.780 align:middle line:90% we'll still pay you a wage of h. 01:02:15.780 --> 01:02:17.300 align:middle line:90% Now why is that OK? 01:02:17.300 --> 01:02:20.900 align:middle line:84% Well, if you can always get an education level of eh 01:02:20.900 --> 01:02:22.740 align:middle line:84% and get a wage of h, no one's going 01:02:22.740 --> 01:02:25.157 align:middle line:84% to ever want to deviate beyond eh to get the same wage 01:02:25.157 --> 01:02:26.240 align:middle line:90% that they would get at eh. 01:02:26.240 --> 01:02:29.280 align:middle line:84% So that going off your comment, to be really safe, 01:02:29.280 --> 01:02:32.300 align:middle line:84% we'd set we to bl, but really, the important thing 01:02:32.300 --> 01:02:33.517 align:middle line:90% is that it's l down here. 01:02:33.517 --> 01:02:36.100 align:middle line:84% Beyond here, we can make it h and we're not going to introduce 01:02:36.100 --> 01:02:38.500 align:middle line:90% any deviations, so let's-- 01:02:38.500 --> 01:02:44.420 align:middle line:84% bl for e less than eh, that's kind of the important thing. 01:02:44.420 --> 01:02:46.920 align:middle line:84% And let's try to do it like this. 01:02:46.920 --> 01:02:50.900 align:middle line:90% 01:02:50.900 --> 01:02:53.340 align:middle line:84% So notice that this graph looks a lot like what 01:02:53.340 --> 01:02:55.000 align:middle line:84% we had in the pooling equilibrium, 01:02:55.000 --> 01:02:56.208 align:middle line:90% but there's a key difference. 01:02:56.208 --> 01:03:00.640 align:middle line:84% So like the pooling equilibrium, if your education was too low, 01:03:00.640 --> 01:03:04.060 align:middle line:84% we paid you l, but in the pooling equilibrium, 01:03:04.060 --> 01:03:06.060 align:middle line:84% if you got to a high enough level of education, 01:03:06.060 --> 01:03:08.860 align:middle line:84% we paid you the average ability in the population. 01:03:08.860 --> 01:03:10.440 align:middle line:90% Now we pay you a lot more. 01:03:10.440 --> 01:03:11.880 align:middle line:90% We pay you h. 01:03:11.880 --> 01:03:14.500 align:middle line:84% Because now, if I see a high level of education, 01:03:14.500 --> 01:03:15.743 align:middle line:90% I draw a different inference. 01:03:15.743 --> 01:03:17.160 align:middle line:84% As the receiver, I say, oh, if you 01:03:17.160 --> 01:03:20.960 align:middle line:84% got a high level of education, you must be a high type person. 01:03:20.960 --> 01:03:22.820 align:middle line:84% So this wage is much, much higher, 01:03:22.820 --> 01:03:24.580 align:middle line:84% and we'll find that, not too surprisingly, 01:03:24.580 --> 01:03:26.122 align:middle line:84% eh is going to have to be pretty high 01:03:26.122 --> 01:03:29.280 align:middle line:84% as well to ensure that no one wants to deviate. 01:03:29.280 --> 01:03:29.780 align:middle line:90% OK. 01:03:29.780 --> 01:03:33.240 align:middle line:84% So we have our wage function written down. 01:03:33.240 --> 01:03:37.840 align:middle line:84% The wage function satisfies receiver optimality 01:03:37.840 --> 01:03:39.400 align:middle line:90% by our construction. 01:03:39.400 --> 01:03:43.080 align:middle line:90% Let's just check in words. 01:03:43.080 --> 01:03:45.920 align:middle line:84% Bayes rule applies only at these two points 01:03:45.920 --> 01:03:48.200 align:middle line:84% because at this point, at e equals 0, 01:03:48.200 --> 01:03:51.040 align:middle line:84% that's the level of education chosen by the low type. 01:03:51.040 --> 01:03:52.640 align:middle line:84% We believe that you're the low type, 01:03:52.640 --> 01:03:54.340 align:middle line:90% and we charge you a wage of l. 01:03:54.340 --> 01:03:55.540 align:middle line:90% That's great. 01:03:55.540 --> 01:03:59.135 align:middle line:84% Here, eh, If we see that, we believe you're a high type, 01:03:59.135 --> 01:04:01.260 align:middle line:84% and we charge you that-- we pay you the wage that's 01:04:01.260 --> 01:04:02.640 align:middle line:90% optimal given that belief. 01:04:02.640 --> 01:04:04.700 align:middle line:84% So Bayes' rule pins us down here. 01:04:04.700 --> 01:04:06.890 align:middle line:84% Everything outside of that is off-path, 01:04:06.890 --> 01:04:08.640 align:middle line:84% we had a lot of flexibility in our choice, 01:04:08.640 --> 01:04:10.820 align:middle line:84% and we made those choices to try to deter deviations 01:04:10.820 --> 01:04:12.700 align:middle line:90% by the sender. 01:04:12.700 --> 01:04:13.200 align:middle line:90% OK. 01:04:13.200 --> 01:04:16.340 align:middle line:84% So now, there's two things we have to check. 01:04:16.340 --> 01:04:22.340 align:middle line:84% We have to check that the low type-- so maybe I'll 01:04:22.340 --> 01:04:23.320 align:middle line:90% just put it over here. 01:04:23.320 --> 01:04:26.980 align:middle line:90% 01:04:26.980 --> 01:04:36.220 align:middle line:84% The low type prefers el to all other e, e prime, 01:04:36.220 --> 01:04:39.800 align:middle line:90% and the high type prefers eh. 01:04:39.800 --> 01:04:44.900 align:middle line:90% 01:04:44.900 --> 01:04:48.100 align:middle line:84% So this is what sender optimality means. 01:04:48.100 --> 01:04:52.695 align:middle line:84% The type l sender in equilibrium is choosing el, which is 0. 01:04:52.695 --> 01:04:54.320 align:middle line:84% And we have to check that that's better 01:04:54.320 --> 01:04:57.600 align:middle line:84% for them than deviating to any other level of education. 01:04:57.600 --> 01:04:59.880 align:middle line:84% Similarly, the high type, in equilibrium, 01:04:59.880 --> 01:05:02.420 align:middle line:84% is choosing eh, which is strictly positive. 01:05:02.420 --> 01:05:04.080 align:middle line:84% And we have to check that they prefer 01:05:04.080 --> 01:05:06.360 align:middle line:84% eh to any other level of education. 01:05:06.360 --> 01:05:08.340 align:middle line:84% Now I think this can be overwhelming at first 01:05:08.340 --> 01:05:10.560 align:middle line:84% because there's so many other levels of education 01:05:10.560 --> 01:05:12.120 align:middle line:84% that people might choose, but just 01:05:12.120 --> 01:05:15.600 align:middle line:84% like before, given this wage function, 01:05:15.600 --> 01:05:18.780 align:middle line:84% I think there's only one really candidate deviation. 01:05:18.780 --> 01:05:21.120 align:middle line:84% I think most of the deviations are clearly not going 01:05:21.120 --> 01:05:22.380 align:middle line:90% to be the optimal deviation. 01:05:22.380 --> 01:05:25.280 align:middle line:90% So let's say I'm a low type. 01:05:25.280 --> 01:05:26.240 align:middle line:90% Here I am. 01:05:26.240 --> 01:05:27.480 align:middle line:90% I'm down here. 01:05:27.480 --> 01:05:29.172 align:middle line:90% I'm getting a wage of l. 01:05:29.172 --> 01:05:31.380 align:middle line:84% I see that this is what the wage function looks like, 01:05:31.380 --> 01:05:33.480 align:middle line:84% and this is the choice that people actually face 01:05:33.480 --> 01:05:36.320 align:middle line:90% when they design their lives. 01:05:36.320 --> 01:05:41.600 align:middle line:84% What level of education might I be tempted to deviate to? 01:05:41.600 --> 01:05:42.100 align:middle line:90% Yeah? 01:05:42.100 --> 01:05:42.960 align:middle line:90% AUDIENCE: e of h? 01:05:42.960 --> 01:05:43.820 align:middle line:90% IAN BALL: eh, right? 01:05:43.820 --> 01:05:45.760 align:middle line:84% None of the other deviations make sense. 01:05:45.760 --> 01:05:48.440 align:middle line:84% If I deviate to here, I'm paying more 01:05:48.440 --> 01:05:50.640 align:middle line:84% and I'm not getting any wage return. 01:05:50.640 --> 01:05:53.427 align:middle line:84% If I deviate to here, well, I am getting a wage return, 01:05:53.427 --> 01:05:55.260 align:middle line:84% but I'm getting the same return I could have 01:05:55.260 --> 01:05:56.980 align:middle line:90% gotten by just going to eh. 01:05:56.980 --> 01:06:00.002 align:middle line:84% So if I want to deviate, that's really the key deviation. 01:06:00.002 --> 01:06:01.460 align:middle line:84% And similarly, now let's put myself 01:06:01.460 --> 01:06:07.460 align:middle line:84% in the position of someone of a high type who's choosing eh. 01:06:07.460 --> 01:06:10.380 align:middle line:84% They're here, and they're looking over the graph. 01:06:10.380 --> 01:06:14.460 align:middle line:84% What is the only reasonable or tempting deviation 01:06:14.460 --> 01:06:15.440 align:middle line:90% for the high type? 01:06:15.440 --> 01:06:20.980 align:middle line:90% 01:06:20.980 --> 01:06:21.480 align:middle line:90% Yeah? 01:06:21.480 --> 01:06:22.620 align:middle line:90% AUDIENCE: el? 01:06:22.620 --> 01:06:25.260 align:middle line:84% IAN BALL: Exactly. el, which, in this case, is 0. 01:06:25.260 --> 01:06:26.780 align:middle line:90% So el, for the same reason. 01:06:26.780 --> 01:06:28.123 align:middle line:90% I'm already getting a wage of h. 01:06:28.123 --> 01:06:30.040 align:middle line:84% There's no reason for me to go to school more. 01:06:30.040 --> 01:06:32.457 align:middle line:84% I don't like school, and it's not going to raise my wages, 01:06:32.457 --> 01:06:34.180 align:middle line:90% according to this strategy. 01:06:34.180 --> 01:06:35.620 align:middle line:84% If I go to less school, well, I'm 01:06:35.620 --> 01:06:38.042 align:middle line:84% going to get a big cut in wages, but I can go all the way 01:06:38.042 --> 01:06:40.500 align:middle line:84% down to here and I'm still going to get the same wage of l. 01:06:40.500 --> 01:06:42.940 align:middle line:84% So I might as well choose the lowest education level that 01:06:42.940 --> 01:06:45.740 align:middle line:90% ensures that I get that wage. 01:06:45.740 --> 01:06:47.720 align:middle line:84% So now let's write down these conditions. 01:06:47.720 --> 01:07:13.030 align:middle line:90% 01:07:13.030 --> 01:07:15.270 align:middle line:84% So really, what we need to check is-- 01:07:15.270 --> 01:07:17.610 align:middle line:84% and sometimes I write it like this, in shorthand. 01:07:17.610 --> 01:07:25.610 align:middle line:90% 01:07:25.610 --> 01:07:29.750 align:middle line:84% So what I mean by this is, this, I say, 01:07:29.750 --> 01:07:31.590 align:middle line:90% this means type h prefers. 01:07:31.590 --> 01:07:32.312 align:middle line:90% This is a weak. 01:07:32.312 --> 01:07:33.770 align:middle line:84% Maybe I'll say something like this. 01:07:33.770 --> 01:07:35.670 align:middle line:84% So I need to check is if I'm type h, 01:07:35.670 --> 01:07:38.950 align:middle line:84% I prefer choosing an education level of eh 01:07:38.950 --> 01:07:41.270 align:middle line:90% to an education level of el. 01:07:41.270 --> 01:07:43.770 align:middle line:84% Of course, I have to prefer it to any other education level, 01:07:43.770 --> 01:07:46.630 align:middle line:84% but we said this is really the deviation I'm tempted by. 01:07:46.630 --> 01:07:48.890 align:middle line:84% And then we have to check if I'm a low type l, 01:07:48.890 --> 01:07:52.170 align:middle line:84% I weakly prefer my own education level el 01:07:52.170 --> 01:07:55.570 align:middle line:84% to deviating to the education level eh. 01:07:55.570 --> 01:07:58.350 align:middle line:84% Now in general, I have to worry about every other deviation, 01:07:58.350 --> 01:08:00.130 align:middle line:84% but because of the special structure 01:08:00.130 --> 01:08:02.830 align:middle line:84% we've used when we define this wage function, 01:08:02.830 --> 01:08:04.670 align:middle line:84% we only have to check these two conditions. 01:08:04.670 --> 01:08:08.170 align:middle line:84% So let's just write this out in math. 01:08:08.170 --> 01:08:11.370 align:middle line:84% If I'm a high type and I choose eh, then I'm 01:08:11.370 --> 01:08:13.890 align:middle line:84% going to get a wage of h, but then I'm 01:08:13.890 --> 01:08:19.410 align:middle line:84% going to have to pay 1 over h eh squared. 01:08:19.410 --> 01:08:20.850 align:middle line:90% Maybe let's go down here. 01:08:20.850 --> 01:08:23.790 align:middle line:84% What if I'm a low type-- no, we'll just go through it. 01:08:23.790 --> 01:08:25.050 align:middle line:90% Let's see. 01:08:25.050 --> 01:08:31.510 align:middle line:84% Now if I deviate to el, well, then I get l. 01:08:31.510 --> 01:08:32.510 align:middle line:90% Maybe I'll write it out. 01:08:32.510 --> 01:08:38.510 align:middle line:84% It's 1 over h el squared, but that's just l because el is 0. 01:08:38.510 --> 01:08:47.029 align:middle line:90% 01:08:47.029 --> 01:08:53.710 align:middle line:84% And I think as usual, let's simplify this a little bit. 01:08:53.710 --> 01:08:58.810 align:middle line:84% That is, h minus l is greater than or equal to-- 01:08:58.810 --> 01:09:07.388 align:middle line:90% 01:09:07.388 --> 01:09:08.930 align:middle line:84% and this has the same interpretation. 01:09:08.930 --> 01:09:11.450 align:middle line:84% Now the wage premium is different 01:09:11.450 --> 01:09:12.930 align:middle line:90% in the separating equilibrium. 01:09:12.930 --> 01:09:15.229 align:middle line:84% The wage premium is now h minus l. 01:09:15.229 --> 01:09:18.149 align:middle line:84% And what this condition says is the wage premium 01:09:18.149 --> 01:09:22.630 align:middle line:84% is higher than the cost of education for the high type. 01:09:22.630 --> 01:09:25.550 align:middle line:84% So going to college and getting an education of eh 01:09:25.550 --> 01:09:28.990 align:middle line:84% is worth it for the high type because their cost of doing that 01:09:28.990 --> 01:09:31.069 align:middle line:84% is weakly smaller than the wage premium 01:09:31.069 --> 01:09:33.550 align:middle line:90% they'll get from doing that. 01:09:33.550 --> 01:09:36.630 align:middle line:84% Now let's look at the same condition for the low type. 01:09:36.630 --> 01:09:39.830 align:middle line:84% So for the low type, well, if they choose their equilibrium 01:09:39.830 --> 01:09:42.310 align:middle line:84% education level of 0, they don't pay anything 01:09:42.310 --> 01:09:44.370 align:middle line:90% and they get a wage of l. 01:09:44.370 --> 01:09:46.890 align:middle line:84% So we get-- and again, maybe I'll-- 01:09:46.890 --> 01:09:49.450 align:middle line:84% it's l, but if we really write it out, it's this. 01:09:49.450 --> 01:09:53.010 align:middle line:90% 01:09:53.010 --> 01:09:56.490 align:middle line:84% Where this just second term disappears because el is 0. 01:09:56.490 --> 01:09:59.990 align:middle line:84% And this has to be greater than or equal to what? 01:09:59.990 --> 01:10:03.630 align:middle line:84% Well, if they deviate and choose a high level of education, 01:10:03.630 --> 01:10:06.010 align:middle line:84% they're going to get the wage h, but then they 01:10:06.010 --> 01:10:11.650 align:middle line:90% have to pay 1 over l eh squared. 01:10:11.650 --> 01:10:14.470 align:middle line:90% And again, let's simplify. 01:10:14.470 --> 01:10:28.330 align:middle line:90% 01:10:28.330 --> 01:10:30.390 align:middle line:90% And what does this say in words? 01:10:30.390 --> 01:10:43.070 align:middle line:90% 01:10:43.070 --> 01:10:44.310 align:middle line:90% Can we interpret this? 01:10:44.310 --> 01:10:45.150 align:middle line:90% Yeah? 01:10:45.150 --> 01:10:48.510 align:middle line:84% AUDIENCE: The wage premium is smaller than the cost 01:10:48.510 --> 01:10:53.050 align:middle line:84% of, I guess, getting education eh for the low type. 01:10:53.050 --> 01:10:54.550 align:middle line:84% IAN BALL: For the low type, exactly. 01:10:54.550 --> 01:10:55.973 align:middle line:90% So-- and this is-- 01:10:55.973 --> 01:10:58.390 align:middle line:84% if you recall, it's similar when we talked about the essay 01:10:58.390 --> 01:11:00.710 align:middle line:84% problem last week, and we said the cost of the essay 01:11:00.710 --> 01:11:04.110 align:middle line:84% has to be cheap enough that the excited students take it, 01:11:04.110 --> 01:11:06.610 align:middle line:84% but expensive enough that the less excited students take it, 01:11:06.610 --> 01:11:08.110 align:middle line:84% that's exactly what's going on here. 01:11:08.110 --> 01:11:09.990 align:middle line:84% In order to sustain a separating equilibrium, 01:11:09.990 --> 01:11:11.890 align:middle line:84% because it's a separating equilibrium, 01:11:11.890 --> 01:11:13.690 align:middle line:84% the wage premium is going to be h minus l, 01:11:13.690 --> 01:11:16.453 align:middle line:84% because high education fully reveals you're a high type, 01:11:16.453 --> 01:11:18.370 align:middle line:84% low education fully reveals you're a low type, 01:11:18.370 --> 01:11:19.990 align:middle line:84% so that defines-- the fact that it's 01:11:19.990 --> 01:11:22.590 align:middle line:84% a separating equilibrium defines what the wage premium is 01:11:22.590 --> 01:11:23.430 align:middle line:90% going to be. 01:11:23.430 --> 01:11:25.263 align:middle line:84% But then we have to check that it's actually 01:11:25.263 --> 01:11:27.950 align:middle line:84% optimal for each type to follow what they're supposed to do. 01:11:27.950 --> 01:11:29.750 align:middle line:90% In this equilibrium. 01:11:29.750 --> 01:11:32.470 align:middle line:84% The high type people are supposed to get educated, 01:11:32.470 --> 01:11:36.230 align:middle line:84% so the level of education eh must be low enough 01:11:36.230 --> 01:11:38.790 align:middle line:84% that it's worth it for high ability or high type people 01:11:38.790 --> 01:11:41.170 align:middle line:84% to choose to get that education because the wage 01:11:41.170 --> 01:11:43.770 align:middle line:84% premium outweighs the cost for those types. 01:11:43.770 --> 01:11:47.110 align:middle line:84% At the same time, the level of education must be high enough. 01:11:47.110 --> 01:11:49.010 align:middle line:84% If it was too low, then even the low types 01:11:49.010 --> 01:11:50.130 align:middle line:84% would also choose that education, 01:11:50.130 --> 01:11:52.430 align:middle line:84% and we would no longer have a separating equilibrium. 01:11:52.430 --> 01:11:54.570 align:middle line:84% So the level of education must, at the same time, 01:11:54.570 --> 01:11:56.510 align:middle line:84% be high enough that for the low types, 01:11:56.510 --> 01:11:59.970 align:middle line:84% it's not worth going to education-- going to school 01:11:59.970 --> 01:12:04.090 align:middle line:84% because the cost exceeds the wage premium that they get. 01:12:04.090 --> 01:12:07.790 align:middle line:84% And if we do this, we can get a formula for what eh has to be. 01:12:07.790 --> 01:12:09.910 align:middle line:90% So maybe this is helpful. 01:12:09.910 --> 01:12:17.080 align:middle line:84% Let's pull the h over here, and what we get is eh 01:12:17.080 --> 01:12:23.290 align:middle line:84% needs to be between h, h minus l, 01:12:23.290 --> 01:12:28.410 align:middle line:84% and it has to be above l times h minus l. 01:12:28.410 --> 01:12:30.090 align:middle line:84% And let's just check that this works. 01:12:30.090 --> 01:12:32.890 align:middle line:84% Well, because l is smaller than h, 01:12:32.890 --> 01:12:35.490 align:middle line:84% this inequality does indeed-- is indeed 01:12:35.490 --> 01:12:39.750 align:middle line:84% satisfied for a certain range of types. 01:12:39.750 --> 01:12:45.870 align:middle line:90% I think it's helpful to compare. 01:12:45.870 --> 01:12:51.510 align:middle line:84% So this is-- let's compare what happened 01:12:51.510 --> 01:12:54.450 align:middle line:90% in the pooling equilibrium. 01:12:54.450 --> 01:12:55.690 align:middle line:90% So this is separating. 01:12:55.690 --> 01:13:00.070 align:middle line:90% 01:13:00.070 --> 01:13:01.910 align:middle line:84% So in separating, the low types got 01:13:01.910 --> 01:13:04.590 align:middle line:84% no education and the high types got eh. 01:13:04.590 --> 01:13:08.350 align:middle line:84% In the pooling equilibrium, both types got e star. 01:13:08.350 --> 01:13:12.210 align:middle line:84% And e star satisfied the following property. 01:13:12.210 --> 01:13:21.750 align:middle line:90% 01:13:21.750 --> 01:13:24.390 align:middle line:90% So how do these compare? 01:13:24.390 --> 01:13:30.550 align:middle line:84% If we look at these bounds, how does the pooling equilibrium 01:13:30.550 --> 01:13:31.050 align:middle line:90% compare? 01:13:31.050 --> 01:13:34.770 align:middle line:84% How does e star compare to eh if we look at these inequalities? 01:13:34.770 --> 01:13:36.800 align:middle line:90% Let's look. 01:13:36.800 --> 01:13:40.260 align:middle line:84% Mu is an average between h and l. 01:13:40.260 --> 01:13:42.840 align:middle line:84% So mu is certainly smaller than h. 01:13:42.840 --> 01:13:46.357 align:middle line:90% So this is smaller than this. 01:13:46.357 --> 01:13:48.440 align:middle line:84% So we see a fundamental difference between the two 01:13:48.440 --> 01:13:49.840 align:middle line:90% kinds of equilibria. 01:13:49.840 --> 01:13:52.440 align:middle line:84% In the separating equilibrium, the low types 01:13:52.440 --> 01:13:56.640 align:middle line:84% get no education and the high types get a lot of education. 01:13:56.640 --> 01:13:58.840 align:middle line:84% In the pooling equilibrium, both types 01:13:58.840 --> 01:14:01.420 align:middle line:84% get this lower intermediate level of education. 01:14:01.420 --> 01:14:05.440 align:middle line:84% That's e star is must be strictly lower than eh. 01:14:05.440 --> 01:14:07.020 align:middle line:90% So we have two worlds. 01:14:07.020 --> 01:14:09.540 align:middle line:84% We have one world-- and they're both equilibria. 01:14:09.540 --> 01:14:11.800 align:middle line:84% There's one world where getting educated 01:14:11.800 --> 01:14:14.960 align:middle line:84% conveys a lot of benefits, and people-- 01:14:14.960 --> 01:14:18.280 align:middle line:84% or I should say, getting very highly educated conveys a lot 01:14:18.280 --> 01:14:19.080 align:middle line:90% of benefits. 01:14:19.080 --> 01:14:20.380 align:middle line:90% People pay a huge cost. 01:14:20.380 --> 01:14:22.660 align:middle line:84% The high types pay a huge cost to get highly educated, 01:14:22.660 --> 01:14:25.120 align:middle line:90% and they get a huge return. 01:14:25.120 --> 01:14:26.960 align:middle line:84% There's also another equilibrium we 01:14:26.960 --> 01:14:30.180 align:middle line:84% could be in where people don't get as much as educated, 01:14:30.180 --> 01:14:33.000 align:middle line:84% there's not a huge benefit to education, but everyone gets it. 01:14:33.000 --> 01:14:36.660 align:middle line:84% And I think these actually are consistent with-- you might 01:14:36.660 --> 01:14:38.780 align:middle line:84% imagine in different countries, we might kind of 01:14:38.780 --> 01:14:40.697 align:middle line:84% reach different equilibria where education has 01:14:40.697 --> 01:14:44.420 align:middle line:90% a different role in society. 01:14:44.420 --> 01:14:46.202 align:middle line:90% Both of these are consistent. 01:14:46.202 --> 01:14:47.660 align:middle line:84% Everyone's behaving optimally given 01:14:47.660 --> 01:14:50.560 align:middle line:84% what everyone else is doing, but here, 01:14:50.560 --> 01:14:52.592 align:middle line:84% the beliefs that the market forms 01:14:52.592 --> 01:14:54.300 align:middle line:84% about your education and this equilibrium 01:14:54.300 --> 01:14:55.860 align:middle line:84% are different-- sorry, the beliefs, the market 01:14:55.860 --> 01:14:57.780 align:middle line:84% forms about your ability given your education 01:14:57.780 --> 01:14:59.620 align:middle line:84% are different than the beliefs the market forms 01:14:59.620 --> 01:15:01.120 align:middle line:84% in this equilibrium, and then people 01:15:01.120 --> 01:15:03.500 align:middle line:84% are responding accordingly, and it's mutually 01:15:03.500 --> 01:15:04.720 align:middle line:90% consistent in each case. 01:15:04.720 --> 01:15:07.580 align:middle line:90% 01:15:07.580 --> 01:15:10.080 align:middle line:84% Let me maybe just do one final thing. 01:15:10.080 --> 01:15:14.980 align:middle line:84% I think it's sometimes helpful to visualize these constraints 01:15:14.980 --> 01:15:16.140 align:middle line:90% geometrically. 01:15:16.140 --> 01:15:18.180 align:middle line:84% So I've written out the constraint 01:15:18.180 --> 01:15:20.040 align:middle line:84% for the separating equilibrium up here, 01:15:20.040 --> 01:15:22.740 align:middle line:84% but I want to try to visualize this. 01:15:22.740 --> 01:15:28.070 align:middle line:84% And I think a good way to visualize this is to say, 01:15:28.070 --> 01:15:29.820 align:middle line:84% well, we want to look at something that is 01:15:29.820 --> 01:15:31.343 align:middle line:90% called an indifference curve. 01:15:31.343 --> 01:15:33.260 align:middle line:84% And if you've studied other economics classes, 01:15:33.260 --> 01:15:34.320 align:middle line:90% you would have seen this. 01:15:34.320 --> 01:15:36.880 align:middle line:84% But what we want to say is, well, we're in the plain 01:15:36.880 --> 01:15:40.720 align:middle line:84% here where the horizontal axis is your level of education 01:15:40.720 --> 01:15:43.240 align:middle line:84% and the vertical axis is your wage. 01:15:43.240 --> 01:15:49.200 align:middle line:84% And we want to ask, what are the pairs of wage and education that 01:15:49.200 --> 01:15:52.280 align:middle line:90% give you the same utility? 01:15:52.280 --> 01:15:53.980 align:middle line:84% Well let's write the formula down. 01:15:53.980 --> 01:15:58.560 align:middle line:84% So for the low type, or for type t, 01:15:58.560 --> 01:16:02.480 align:middle line:84% an indifference curve would have the form, say, u bar equals 01:16:02.480 --> 01:16:07.520 align:middle line:90% wage minus 1 over t e squared. 01:16:07.520 --> 01:16:10.760 align:middle line:84% So what I'm saying is I want to look at all the pairs 01:16:10.760 --> 01:16:15.600 align:middle line:84% we that give type t the utility u bar. 01:16:15.600 --> 01:16:18.160 align:middle line:84% And if I solve-- if I write this out, what I get 01:16:18.160 --> 01:16:24.980 align:middle line:84% is w equals u bar plus 1 over t e squared. 01:16:24.980 --> 01:16:29.640 align:middle line:90% 01:16:29.640 --> 01:16:31.860 align:middle line:84% So what this tells me is w is going 01:16:31.860 --> 01:16:35.440 align:middle line:84% to be an increasing quadratic function of e squared. 01:16:35.440 --> 01:16:37.023 align:middle line:90% So let's try to write this down. 01:16:37.023 --> 01:16:38.440 align:middle line:84% It might look something like this. 01:16:38.440 --> 01:16:43.380 align:middle line:90% 01:16:43.380 --> 01:16:47.500 align:middle line:84% What this indicates is the set-- what I'm saying is if I moved 01:16:47.500 --> 01:16:52.060 align:middle line:84% along this curve, the utility that I would get-- 01:16:52.060 --> 01:16:53.580 align:middle line:90% say I'm the low type here-- 01:16:53.580 --> 01:16:55.458 align:middle line:90% is the same along this curve. 01:16:55.458 --> 01:16:58.000 align:middle line:84% Because if I move a little bit to the right along this curve, 01:16:58.000 --> 01:17:00.480 align:middle line:84% I'm getting more educated, that's more costly, 01:17:00.480 --> 01:17:03.140 align:middle line:84% but my wage is also increasing at just the right rate 01:17:03.140 --> 01:17:05.940 align:middle line:84% to ensure that my utility stays constant. 01:17:05.940 --> 01:17:07.940 align:middle line:84% And then we could draw another utility curve. 01:17:07.940 --> 01:17:09.980 align:middle line:84% By shifting the value of u bar, we 01:17:09.980 --> 01:17:13.620 align:middle line:84% could draw another utility curve like this and another one 01:17:13.620 --> 01:17:14.700 align:middle line:90% like this. 01:17:14.700 --> 01:17:17.740 align:middle line:84% So what this says is, these all give me the same utility, 01:17:17.740 --> 01:17:19.200 align:middle line:84% these all give me the same utility, 01:17:19.200 --> 01:17:21.657 align:middle line:84% but that utility is going to be higher than this utility. 01:17:21.657 --> 01:17:23.740 align:middle line:84% And similarly, these all give me the same utility, 01:17:23.740 --> 01:17:26.420 align:middle line:84% but a higher utility still than the previous case. 01:17:26.420 --> 01:17:29.520 align:middle line:90% 01:17:29.520 --> 01:17:32.640 align:middle line:84% Now, let's say these are the indifference 01:17:32.640 --> 01:17:36.340 align:middle line:90% curves for the low types. 01:17:36.340 --> 01:17:39.290 align:middle line:90% 01:17:39.290 --> 01:17:41.040 align:middle line:84% Now I want to draw the indifference curves 01:17:41.040 --> 01:17:43.200 align:middle line:90% for the high types. 01:17:43.200 --> 01:17:44.137 align:middle line:90% And this is tricky. 01:17:44.137 --> 01:17:45.720 align:middle line:84% Are the high types indifference curves 01:17:45.720 --> 01:17:47.640 align:middle line:84% going to be steeper than the low type 01:17:47.640 --> 01:17:49.727 align:middle line:90% or shallower than the low type? 01:17:49.727 --> 01:17:51.560 align:middle line:84% This is-- we can do the algebra, but I think 01:17:51.560 --> 01:17:55.477 align:middle line:90% it's good to think it through. 01:17:55.477 --> 01:17:57.060 align:middle line:84% So they're also going to be quadratic. 01:17:57.060 --> 01:17:59.380 align:middle line:84% They're going to have a similar shape. 01:17:59.380 --> 01:18:02.960 align:middle line:90% 01:18:02.960 --> 01:18:04.540 align:middle line:90% Is that shallower or steeper? 01:18:04.540 --> 01:18:06.200 align:middle line:90% What do you think? 01:18:06.200 --> 01:18:07.300 align:middle line:90% You have a half-chance. 01:18:07.300 --> 01:18:12.052 align:middle line:90% 01:18:12.052 --> 01:18:13.260 align:middle line:90% Well, let's think it through. 01:18:13.260 --> 01:18:14.380 align:middle line:90% What does this curve mean? 01:18:14.380 --> 01:18:18.120 align:middle line:84% It tells me, if I get this much more education, 01:18:18.120 --> 01:18:20.680 align:middle line:84% I have to get paid this much more to compensate me for it 01:18:20.680 --> 01:18:22.288 align:middle line:90% and keep me indifferent. 01:18:22.288 --> 01:18:24.080 align:middle line:84% So it is a high type need to be compensated 01:18:24.080 --> 01:18:28.940 align:middle line:84% more or less for the same level of education. 01:18:28.940 --> 01:18:30.620 align:middle line:90% AUDIENCE: More. 01:18:30.620 --> 01:18:32.700 align:middle line:84% IAN BALL: Actually less because a high type 01:18:32.700 --> 01:18:35.420 align:middle line:90% finds education less costly. 01:18:35.420 --> 01:18:38.940 align:middle line:84% So if I'm really good at school, and I go to school a little bit 01:18:38.940 --> 01:18:40.860 align:middle line:84% more, I don't need to get paid too much 01:18:40.860 --> 01:18:42.240 align:middle line:90% more to keep me indifferent. 01:18:42.240 --> 01:18:44.900 align:middle line:84% If I'm really good at school, I can go to school for 20 years 01:18:44.900 --> 01:18:47.060 align:middle line:84% and you just need to pay me a little bit. 01:18:47.060 --> 01:18:52.900 align:middle line:84% So here, I'm going to get curves like-- 01:18:52.900 --> 01:18:55.860 align:middle line:90% say something like this. 01:18:55.860 --> 01:18:57.440 align:middle line:90% No, these should not intersect. 01:18:57.440 --> 01:18:59.820 align:middle line:90% Sorry. 01:18:59.820 --> 01:19:04.700 align:middle line:90% Something like that. 01:19:04.700 --> 01:19:06.080 align:middle line:90% Let me-- I forget this last one. 01:19:06.080 --> 01:19:08.940 align:middle line:90% 01:19:08.940 --> 01:19:11.320 align:middle line:84% So these are the indifference curves for the high types. 01:19:11.320 --> 01:19:20.220 align:middle line:90% 01:19:20.220 --> 01:19:22.700 align:middle line:84% And in fact, using these curves, we 01:19:22.700 --> 01:19:25.190 align:middle line:84% can exactly represent graphically 01:19:25.190 --> 01:19:27.590 align:middle line:84% the levels of education that could support 01:19:27.590 --> 01:19:29.030 align:middle line:90% a separating equilibrium. 01:19:29.030 --> 01:19:33.490 align:middle line:84% So it's exactly going to be this range here. 01:19:33.490 --> 01:19:48.710 align:middle line:84% Do I have-- and let's just see it in words. 01:19:48.710 --> 01:19:53.590 align:middle line:84% This dotted line here is the indifference curve that passes 01:19:53.590 --> 01:19:56.310 align:middle line:90% through 0l for the low type. 01:19:56.310 --> 01:19:59.590 align:middle line:84% So these are the wage-education level pairs that make the low 01:19:59.590 --> 01:20:04.750 align:middle line:84% type indifferent with 0l, with no education and a wage of l. 01:20:04.750 --> 01:20:07.310 align:middle line:84% So if I want to have a separating equilibrium, 01:20:07.310 --> 01:20:10.350 align:middle line:84% I better make this eh high enough 01:20:10.350 --> 01:20:12.430 align:middle line:84% that the low type doesn't want to deviate to eh. 01:20:12.430 --> 01:20:15.030 align:middle line:84% And that precisely means eh has to be farther 01:20:15.030 --> 01:20:18.830 align:middle line:84% to the right of this point because if eh is over here, that 01:20:18.830 --> 01:20:21.430 align:middle line:84% tells me the low type strictly prefers 01:20:21.430 --> 01:20:23.690 align:middle line:84% to get no education and a wage of l 01:20:23.690 --> 01:20:27.490 align:middle line:84% than to get this level of education and a wage of h. 01:20:27.490 --> 01:20:30.890 align:middle line:84% On the other hand, the high type must weakly 01:20:30.890 --> 01:20:33.990 align:middle line:84% prefer to get high education than no education. 01:20:33.990 --> 01:20:37.130 align:middle line:90% 01:20:37.130 --> 01:20:39.760 align:middle line:90% Ah, I misdrew this. 01:20:39.760 --> 01:20:41.510 align:middle line:84% It's going to go all the way over to here. 01:20:41.510 --> 01:20:42.010 align:middle line:90% Sorry. 01:20:42.010 --> 01:20:49.810 align:middle line:90% 01:20:49.810 --> 01:20:54.210 align:middle line:84% So the high types indifference curve that passes through 0l 01:20:54.210 --> 01:20:58.650 align:middle line:90% goes all the way up to here. 01:20:58.650 --> 01:21:00.830 align:middle line:84% So just to maybe go through it one last time, 01:21:00.830 --> 01:21:02.410 align:middle line:90% let's make sure we understand. 01:21:02.410 --> 01:21:03.970 align:middle line:84% If I could get a level of education 01:21:03.970 --> 01:21:06.490 align:middle line:84% here and get paid a wage of h, that 01:21:06.490 --> 01:21:09.490 align:middle line:84% would be better than no education and a wage of l 01:21:09.490 --> 01:21:12.050 align:middle line:90% for both high and low types. 01:21:12.050 --> 01:21:14.890 align:middle line:84% So that couldn't be part of a separating equilibrium. 01:21:14.890 --> 01:21:17.330 align:middle line:84% If I chose eh to be way over here, 01:21:17.330 --> 01:21:20.650 align:middle line:84% then getting this level of education and a wage of h would 01:21:20.650 --> 01:21:23.590 align:middle line:90% be worse than 0l for both types. 01:21:23.590 --> 01:21:25.590 align:middle line:84% The only way I can have a separating equilibrium 01:21:25.590 --> 01:21:27.550 align:middle line:84% is if I lie exactly in this range. 01:21:27.550 --> 01:21:29.310 align:middle line:84% And if I lie in that range, I ensure 01:21:29.310 --> 01:21:33.650 align:middle line:84% that this additional education is worth it for the high type, 01:21:33.650 --> 01:21:35.830 align:middle line:84% but not worth it for the low type. 01:21:35.830 --> 01:21:38.830 align:middle line:84% And the crucial point here is I'm looking at the indifference 01:21:38.830 --> 01:21:42.470 align:middle line:84% curves that pass through the point 0l. 01:21:42.470 --> 01:21:44.910 align:middle line:84% So these are considering the pairs of education 01:21:44.910 --> 01:21:46.330 align:middle line:90% and wages that-- 01:21:46.330 --> 01:21:47.810 align:middle line:84% So let me just-- as I'm concluding, 01:21:47.810 --> 01:21:51.070 align:middle line:84% let me say it smoothly in words one last time. 01:21:51.070 --> 01:21:53.510 align:middle line:84% These are the pairs of wages in education 01:21:53.510 --> 01:21:56.430 align:middle line:84% that make the high type indifferent with this point, 01:21:56.430 --> 01:21:58.388 align:middle line:84% and these are the pairs of education and wages 01:21:58.388 --> 01:21:59.430 align:middle line:90% that make the high type-- 01:21:59.430 --> 01:22:01.970 align:middle line:84% that make the low type indifferent with this point. 01:22:01.970 --> 01:22:03.718 align:middle line:84% And everything in between is going 01:22:03.718 --> 01:22:05.510 align:middle line:84% to ensure that the high type finds it worth 01:22:05.510 --> 01:22:08.517 align:middle line:84% it to get educated and the low type finds it not worth it. 01:22:08.517 --> 01:22:10.850 align:middle line:84% We did the algebra, but it's just good to see the graph, 01:22:10.850 --> 01:22:12.433 align:middle line:84% and I think on the problem set, you'll 01:22:12.433 --> 01:22:15.980 align:middle line:84% be asked to illustrate this graphically. 01:22:15.980 --> 01:22:20.000 align:middle line:90%