WEBVTT 00:00:00.000 --> 00:00:01.984 align:middle line:90% [SQUEAKING] 00:00:01.984 --> 00:00:03.472 align:middle line:90% [RUSTLING] 00:00:03.472 --> 00:00:06.944 align:middle line:90% [CLICKING] 00:00:06.944 --> 00:00:11.260 align:middle line:90% 00:00:11.260 --> 00:00:12.010 align:middle line:90% GLENN ELLISON: OK. 00:00:12.010 --> 00:00:13.600 align:middle line:90% Let me go ahead and get started. 00:00:13.600 --> 00:00:18.400 align:middle line:84% So today, I'm moving on to the next unit on the syllabus 00:00:18.400 --> 00:00:19.670 align:middle line:90% on consumer search. 00:00:19.670 --> 00:00:21.880 align:middle line:84% And if you think back to what I was 00:00:21.880 --> 00:00:25.660 align:middle line:84% doing for the previous week and then what Tobias did last week, 00:00:25.660 --> 00:00:28.420 align:middle line:84% it's following what is very common in IO today, which 00:00:28.420 --> 00:00:31.600 align:middle line:84% is to put a tremendous emphasis on product differentiation 00:00:31.600 --> 00:00:33.163 align:middle line:84% as the source of markups in markets. 00:00:33.163 --> 00:00:36.250 align:middle line:90% 00:00:36.250 --> 00:00:41.260 align:middle line:84% And I think it's just helpful to think about, when we do that, 00:00:41.260 --> 00:00:46.100 align:middle line:84% we're both explaining markups using product differentiation 00:00:46.100 --> 00:00:48.850 align:middle line:84% and also, whenever we're making welfare calculations, 00:00:48.850 --> 00:00:52.150 align:middle line:84% we're attributing people's-- the product differentiation 00:00:52.150 --> 00:00:55.820 align:middle line:84% to people's underlying true preferences for goods. 00:00:55.820 --> 00:00:58.300 align:middle line:84% And we're assigning-- a significant part of welfare 00:00:58.300 --> 00:01:01.420 align:middle line:84% is then, what are people's epsilon ij's, and what 00:01:01.420 --> 00:01:02.745 align:middle line:90% is the average epsilon ij? 00:01:02.745 --> 00:01:04.870 align:middle line:84% And that's considered part of the utility function. 00:01:04.870 --> 00:01:08.100 align:middle line:84% That's something we consider when we're doing welfare. 00:01:08.100 --> 00:01:09.970 align:middle line:90% And I guess it's interesting. 00:01:09.970 --> 00:01:13.730 align:middle line:84% It reflects current IO, and it also reflects historical IO. 00:01:13.730 --> 00:01:15.650 align:middle line:84% If you look at Tirole's IO textbook, 00:01:15.650 --> 00:01:18.540 align:middle line:84% Tirole does not have a chapter on search. 00:01:18.540 --> 00:01:21.800 align:middle line:84% Tirole exclusively does product differentiation and repeated 00:01:21.800 --> 00:01:23.900 align:middle line:90% games as sources of markups. 00:01:23.900 --> 00:01:28.880 align:middle line:84% But intuitively, when you think about products-- 00:01:28.880 --> 00:01:31.820 align:middle line:84% and a lot of important products people buy today, 00:01:31.820 --> 00:01:34.910 align:middle line:84% it's unclear whether there really is much differentiation 00:01:34.910 --> 00:01:37.490 align:middle line:84% or whether the differentiation is at all sufficient to cause 00:01:37.490 --> 00:01:38.090 align:middle line:90% what-- 00:01:38.090 --> 00:01:39.960 align:middle line:84% the outcomes we are seeing in the market. 00:01:39.960 --> 00:01:44.480 align:middle line:90% So one example is credit cards. 00:01:44.480 --> 00:01:46.760 align:middle line:84% How many people have a really strong preference 00:01:46.760 --> 00:01:50.060 align:middle line:84% for whether they have a Capital One credit card or a Citibank 00:01:50.060 --> 00:01:53.060 align:middle line:84% credit card or a Bank of America credit card? 00:01:53.060 --> 00:01:55.700 align:middle line:84% Is it that I really like the color of the Citibank card 00:01:55.700 --> 00:01:58.200 align:middle line:84% much more than I like the color of the Bank of America card? 00:01:58.200 --> 00:02:01.370 align:middle line:84% Or is it the automated robot customer 00:02:01.370 --> 00:02:02.910 align:middle line:84% service agent I get with Citibank? 00:02:02.910 --> 00:02:04.670 align:middle line:84% Is that much better than the automated robot customer 00:02:04.670 --> 00:02:05.370 align:middle line:90% service agent-- 00:02:05.370 --> 00:02:08.491 align:middle line:84% I like the voice of this one, A versus B? 00:02:08.491 --> 00:02:09.949 align:middle line:84% It's very hard to think that that's 00:02:09.949 --> 00:02:11.600 align:middle line:90% what's driving the markups. 00:02:11.600 --> 00:02:12.860 align:middle line:90% Cell phone plans-- 00:02:12.860 --> 00:02:15.740 align:middle line:84% Verizon and T-Mobile will spend a lot of money 00:02:15.740 --> 00:02:17.990 align:middle line:84% having commercials with people talking about how great 00:02:17.990 --> 00:02:20.100 align:middle line:90% their service and reception is. 00:02:20.100 --> 00:02:23.820 align:middle line:84% But again, how many people have very strong preference for, 00:02:23.820 --> 00:02:26.990 align:middle line:84% I really like the Verizon service on my phone 00:02:26.990 --> 00:02:30.920 align:middle line:84% much more than I like the T-Mobile service on my phone? 00:02:30.920 --> 00:02:32.160 align:middle line:90% Mortgages, again. 00:02:32.160 --> 00:02:34.970 align:middle line:90% And maybe one other story. 00:02:34.970 --> 00:02:36.920 align:middle line:84% If you think about even goods where there's 00:02:36.920 --> 00:02:39.080 align:middle line:84% clear natural product differentiation, 00:02:39.080 --> 00:02:41.850 align:middle line:84% is that product differentiation really explaining the markup? 00:02:41.850 --> 00:02:45.470 align:middle line:84% So, for instance, I own a Hyundai Kona, 00:02:45.470 --> 00:02:49.880 align:middle line:90% which is a little, compact SUV. 00:02:49.880 --> 00:02:53.720 align:middle line:84% Let's suppose you-- you guys live in Cambridge or whatever, 00:02:53.720 --> 00:02:56.270 align:middle line:84% or in Somerville on top of Cambridge. 00:02:56.270 --> 00:02:58.280 align:middle line:84% Suppose you want to buy a Hyundai. 00:02:58.280 --> 00:02:59.070 align:middle line:90% What can you do? 00:02:59.070 --> 00:03:00.820 align:middle line:84% It turns out, there are no Hyundai dealers 00:03:00.820 --> 00:03:02.390 align:middle line:90% in the city of Cambridge. 00:03:02.390 --> 00:03:05.620 align:middle line:84% Instead you can go up to Arlington, you can get one. 00:03:05.620 --> 00:03:07.620 align:middle line:84% You could go out to Lynn, and you could get one. 00:03:07.620 --> 00:03:09.522 align:middle line:84% You could go down to Quincy and get one. 00:03:09.522 --> 00:03:11.480 align:middle line:84% And each of those places is, like, half an hour 00:03:11.480 --> 00:03:12.063 align:middle line:90% away from you. 00:03:12.063 --> 00:03:14.397 align:middle line:84% I guess Arlington is probably the easiest from Cambridge 00:03:14.397 --> 00:03:15.350 align:middle line:90% on public transport. 00:03:15.350 --> 00:03:20.920 align:middle line:84% And if you have a car way out here, there's dealers in Norwood 00:03:20.920 --> 00:03:24.190 align:middle line:84% and dealers in Framingham that you can get to on the highway. 00:03:24.190 --> 00:03:26.350 align:middle line:84% So for me, I could get to any of those car dealers 00:03:26.350 --> 00:03:27.910 align:middle line:90% in half an hour. 00:03:27.910 --> 00:03:31.030 align:middle line:84% And so if you think about it, the Hyundai Kona 00:03:31.030 --> 00:03:33.730 align:middle line:84% bought from the Arlington Hyundai dealer 00:03:33.730 --> 00:03:39.460 align:middle line:84% versus the Hyundai Kona bought from Boch Hyundai in Norwood, 00:03:39.460 --> 00:03:42.680 align:middle line:84% what's my preference for buying from dealer A versus dealer B? 00:03:42.680 --> 00:03:48.100 align:middle line:84% Again, it is the exact same car offered at both dealerships. 00:03:48.100 --> 00:03:49.600 align:middle line:84% If you think of it like, OK, you can 00:03:49.600 --> 00:03:51.490 align:middle line:84% think of this as geographic differentiation, 00:03:51.490 --> 00:03:56.260 align:middle line:84% where, for me, I'm equally good driving here or driving here. 00:03:56.260 --> 00:03:58.600 align:middle line:84% But there are some people who live right in Arlington 00:03:58.600 --> 00:04:01.270 align:middle line:84% for whom that is an annoying drive. 00:04:01.270 --> 00:04:06.160 align:middle line:84% But if you think about the p star equals c plus t model, 00:04:06.160 --> 00:04:08.710 align:middle line:84% what is t? t is the distance by which 00:04:08.710 --> 00:04:11.350 align:middle line:84% someone at one end of the line prefers their product 00:04:11.350 --> 00:04:14.260 align:middle line:84% to the people at the other end of the line. 00:04:14.260 --> 00:04:17.300 align:middle line:84% You can do that drive in 35, 40 minutes. 00:04:17.300 --> 00:04:19.180 align:middle line:84% What's your disutility of driving 00:04:19.180 --> 00:04:24.010 align:middle line:84% 35 or 40 minutes one time to go get your car 00:04:24.010 --> 00:04:25.960 align:middle line:90% or something like that? 00:04:25.960 --> 00:04:27.380 align:middle line:90% Is that $20? 00:04:27.380 --> 00:04:28.990 align:middle line:90% Is that $30? 00:04:28.990 --> 00:04:31.610 align:middle line:84% If you're a high-income person, is that $50 or whatever? 00:04:31.610 --> 00:04:35.500 align:middle line:84% But that would then say the prediction 00:04:35.500 --> 00:04:38.470 align:middle line:84% is that all these cars, the Hyundais at all the dealers, 00:04:38.470 --> 00:04:42.880 align:middle line:84% should be marked up by $40 or $50 over the dealer cost 00:04:42.880 --> 00:04:43.570 align:middle line:90% to the-- 00:04:43.570 --> 00:04:45.400 align:middle line:90% and we think that that's just-- 00:04:45.400 --> 00:04:47.350 align:middle line:84% counterfactually, that's just not correct. 00:04:47.350 --> 00:04:51.210 align:middle line:84% Buying a car, local dealer markups are $1,000 or something 00:04:51.210 --> 00:04:51.710 align:middle line:90% like that. 00:04:51.710 --> 00:04:54.960 align:middle line:84% And the c plus t model is not going to give us that. 00:04:54.960 --> 00:04:56.740 align:middle line:90% So how do we explain that? 00:04:56.740 --> 00:05:01.450 align:middle line:84% One idea that people have is that another important component 00:05:01.450 --> 00:05:03.820 align:middle line:84% of real-world markups could be search costs. 00:05:03.820 --> 00:05:08.620 align:middle line:84% And search costs, intuitively, the more complicated 00:05:08.620 --> 00:05:12.990 align:middle line:84% or difficult it is to find out what the prices are, 00:05:12.990 --> 00:05:14.610 align:middle line:84% the more people are going to give up 00:05:14.610 --> 00:05:16.480 align:middle line:84% without finding what the prices are, 00:05:16.480 --> 00:05:19.530 align:middle line:84% and the more that prevents the competition that Hotelling says 00:05:19.530 --> 00:05:22.560 align:middle line:84% should be there for those marginal consumers. 00:05:22.560 --> 00:05:25.950 align:middle line:84% In the Hotelling model, if you estimate that people-- 00:05:25.950 --> 00:05:27.960 align:middle line:90% that there's $1,000 markup-- 00:05:27.960 --> 00:05:32.940 align:middle line:84% let's suppose that you're doing a differentiated product 00:05:32.940 --> 00:05:35.107 align:middle line:84% estimation and you're imposing the supply side. 00:05:35.107 --> 00:05:37.440 align:middle line:84% I think Tobias pointed out that sometimes you just do it 00:05:37.440 --> 00:05:39.130 align:middle line:84% entirely with demand-side moments. 00:05:39.130 --> 00:05:41.520 align:middle line:84% Sometimes, you also impose the supply-side moments 00:05:41.520 --> 00:05:43.858 align:middle line:84% to explain what's going on and use the cost data. 00:05:43.858 --> 00:05:45.900 align:middle line:84% Well, if you're imposing the supply-side moments, 00:05:45.900 --> 00:05:48.360 align:middle line:84% you're saying there's a $1,000 differentiation between this 00:05:48.360 --> 00:05:50.820 align:middle line:84% Hyundai dealer and this Hyundai dealer, then you would get-- 00:05:50.820 --> 00:05:53.278 align:middle line:84% it would be terrible if one of those Hyundai dealers closed 00:05:53.278 --> 00:05:55.890 align:middle line:84% because all those people are getting $1,000 worth of utility 00:05:55.890 --> 00:05:58.348 align:middle line:84% from going to this one versus that one would have that huge 00:05:58.348 --> 00:06:00.780 align:middle line:84% loss in welfare, which we think is probably not a correct 00:06:00.780 --> 00:06:02.400 align:middle line:90% answer. 00:06:02.400 --> 00:06:05.070 align:middle line:84% So anyway, I'm going to start this with-- 00:06:05.070 --> 00:06:08.160 align:middle line:84% Diamond has this very nice paper from 1971, 00:06:08.160 --> 00:06:11.560 align:middle line:84% noting that this basic idea that the higher is 00:06:11.560 --> 00:06:13.480 align:middle line:84% search costs, the higher price is going to be 00:06:13.480 --> 00:06:15.910 align:middle line:84% seems harder to capture than you might have guessed. 00:06:15.910 --> 00:06:17.060 align:middle line:90% So here's his model. 00:06:17.060 --> 00:06:18.050 align:middle line:90% You have N firms. 00:06:18.050 --> 00:06:20.690 align:middle line:84% They're producing a homogeneous good, like the Hyundai Kona. 00:06:20.690 --> 00:06:21.190 align:middle line:90% Oh, no. 00:06:21.190 --> 00:06:22.607 align:middle line:84% Actually, it's not a Hyundai Kona. 00:06:22.607 --> 00:06:27.100 align:middle line:84% So this is going to be credit cards. 00:06:27.100 --> 00:06:31.030 align:middle line:84% So N firms produce a homogeneous good at cost c. 00:06:31.030 --> 00:06:35.650 align:middle line:84% So that cost c is it costs them 8.99 or 9% 00:06:35.650 --> 00:06:37.540 align:middle line:84% to loan you money on a credit card. 00:06:37.540 --> 00:06:40.060 align:middle line:90% And so c is 9%. 00:06:40.060 --> 00:06:42.130 align:middle line:84% They're a continuum of consumers, 00:06:42.130 --> 00:06:45.340 align:middle line:84% and they have identical multiunit demands D of p. 00:06:45.340 --> 00:06:48.100 align:middle line:84% So if they give me a credit card at 9%, 00:06:48.100 --> 00:06:50.150 align:middle line:90% I may borrow $10,000 on it. 00:06:50.150 --> 00:06:53.840 align:middle line:84% If they give me a credit card at 18%, I may borrow $3,000 on it. 00:06:53.840 --> 00:06:56.230 align:middle line:84% But there's some downward-sloping demand function 00:06:56.230 --> 00:06:57.813 align:middle line:84% for how many units you're going to buy 00:06:57.813 --> 00:07:00.880 align:middle line:84% from the firm as a function of the price. 00:07:00.880 --> 00:07:03.167 align:middle line:84% And I'm doing the credit cards rather than cars 00:07:03.167 --> 00:07:05.500 align:middle line:84% because it's important that you're buying more than one, 00:07:05.500 --> 00:07:09.040 align:middle line:84% whereas cars, typically, you buy exactly one. 00:07:09.040 --> 00:07:12.190 align:middle line:84% I'm going to assume that p minus c times D of p 00:07:12.190 --> 00:07:13.760 align:middle line:90% is a concave function. 00:07:13.760 --> 00:07:16.660 align:middle line:84% It's got some finite monopoly price pm. 00:07:16.660 --> 00:07:19.160 align:middle line:84% And I'm going to consider the following game. 00:07:19.160 --> 00:07:22.990 align:middle line:84% First, the firms simultaneously choose their prices. 00:07:22.990 --> 00:07:27.490 align:middle line:84% And then customers have to pay s every time 00:07:27.490 --> 00:07:28.785 align:middle line:90% they get a price quote. 00:07:28.785 --> 00:07:30.160 align:middle line:84% And we're going to want to assume 00:07:30.160 --> 00:07:34.270 align:middle line:84% that they search optimally, get some number of price quotes. 00:07:34.270 --> 00:07:36.460 align:middle line:84% And then after getting some number of price quotes, 00:07:36.460 --> 00:07:39.310 align:middle line:84% they buy D of p units from the lowest-priced firm 00:07:39.310 --> 00:07:41.230 align:middle line:90% that they found. 00:07:41.230 --> 00:07:45.730 align:middle line:84% And we're going to want to have some optimality in this-- 00:07:45.730 --> 00:07:47.740 align:middle line:84% search is somehow optimal, given how much 00:07:47.740 --> 00:07:49.575 align:middle line:84% the cost of every search is going to be. 00:07:49.575 --> 00:07:52.540 align:middle line:90% 00:07:52.540 --> 00:07:55.840 align:middle line:84% One immediate subtlety is we need some new solution concept 00:07:55.840 --> 00:07:57.100 align:middle line:90% other than Nash equilibrium. 00:07:57.100 --> 00:07:59.830 align:middle line:84% Because in a Nash equilibrium, everyone 00:07:59.830 --> 00:08:04.210 align:middle line:84% behaves as if they know the strategies of the other players. 00:08:04.210 --> 00:08:08.260 align:middle line:84% And so if you think about this as an N plus 1 player game, 00:08:08.260 --> 00:08:12.220 align:middle line:84% there's N firms and one consumer buying, 00:08:12.220 --> 00:08:14.350 align:middle line:84% then in Nash equilibrium, you're implicitly 00:08:14.350 --> 00:08:16.700 align:middle line:84% assuming that the consumer knows the firm's strategies. 00:08:16.700 --> 00:08:18.700 align:middle line:84% And if the consumer knows the firm's strategies, 00:08:18.700 --> 00:08:20.200 align:middle line:84% their strategies are their prices. 00:08:20.200 --> 00:08:23.140 align:middle line:84% And the consumer can just buy from the lowest-priced firm 00:08:23.140 --> 00:08:27.430 align:middle line:84% and not have to pay the s to find that firm. 00:08:27.430 --> 00:08:30.610 align:middle line:90% So what do we do? 00:08:30.610 --> 00:08:34.900 align:middle line:84% Diamond, essentially, in his paper, 00:08:34.900 --> 00:08:38.299 align:middle line:84% made this an N plus 2 player game. 00:08:38.299 --> 00:08:44.020 align:middle line:84% So you have the N firms, you have the one consumer, 00:08:44.020 --> 00:08:49.240 align:middle line:84% and then you have the one player I'll call "nature." 00:08:49.240 --> 00:08:52.510 align:middle line:90% And what the firms do is-- 00:08:52.510 --> 00:08:55.090 align:middle line:84% so first, stage one is the firm's-- 00:08:55.090 --> 00:08:58.120 align:middle line:90% they choose p1 up to pN. 00:08:58.120 --> 00:09:00.490 align:middle line:90% So that is the first thing-- 00:09:00.490 --> 00:09:03.910 align:middle line:90% actually, I need a timeline. 00:09:03.910 --> 00:09:10.780 align:middle line:84% So p equals 1, the firm's move, and they choose p1 up 00:09:10.780 --> 00:09:12.880 align:middle line:90% through pN. 00:09:12.880 --> 00:09:17.680 align:middle line:84% And then at stage two, nature moves. 00:09:17.680 --> 00:09:21.096 align:middle line:84% And what nature does is chooses a random permutation. 00:09:21.096 --> 00:09:25.380 align:middle line:90% 00:09:25.380 --> 00:09:29.790 align:middle line:84% So it's if nature builds a room, and this room 00:09:29.790 --> 00:09:34.920 align:middle line:90% has doors numbered 1 through N. 00:09:34.920 --> 00:09:37.440 align:middle line:84% And so the firms pick their prices, 00:09:37.440 --> 00:09:42.720 align:middle line:84% and then nature takes firm 1 and sticks it behind a closed door. 00:09:42.720 --> 00:09:44.610 align:middle line:90% Say it randomly picks door 7. 00:09:44.610 --> 00:09:47.880 align:middle line:84% And then it takes firm 2 and puts it behind another door, 00:09:47.880 --> 00:09:49.060 align:middle line:90% say, door 1. 00:09:49.060 --> 00:09:51.630 align:middle line:84% And so nature does this random permutation. 00:09:51.630 --> 00:09:58.640 align:middle line:84% And then t equals 3, the consumer starts opening doors. 00:09:58.640 --> 00:10:02.610 align:middle line:90% 00:10:02.610 --> 00:10:07.530 align:middle line:84% And each door you open, there's a cost of s. 00:10:07.530 --> 00:10:13.260 align:middle line:84% So in equilibrium, I'm going to know what p1, p2, and pN are. 00:10:13.260 --> 00:10:16.080 align:middle line:84% But I don't know which firm is behind which door. 00:10:16.080 --> 00:10:20.620 align:middle line:84% So then I open door number 1, see who's there. 00:10:20.620 --> 00:10:23.825 align:middle line:84% If I don't like that price, it seems high, I may open door 2. 00:10:23.825 --> 00:10:25.200 align:middle line:84% If it seems like it's still worth 00:10:25.200 --> 00:10:27.408 align:middle line:84% paying s to open another door, I'd open another door, 00:10:27.408 --> 00:10:29.370 align:middle line:90% and I continue like that. 00:10:29.370 --> 00:10:31.920 align:middle line:84% And so then Diamond asked, what's 00:10:31.920 --> 00:10:34.470 align:middle line:90% the equilibrium of that model? 00:10:34.470 --> 00:10:38.190 align:middle line:84% And the thought was that you might get an answer that says, 00:10:38.190 --> 00:10:39.960 align:middle line:84% when search costs are higher, prices 00:10:39.960 --> 00:10:41.520 align:middle line:90% are higher than they would-- 00:10:41.520 --> 00:10:44.550 align:middle line:84% without the random permutation, this is a Bertrand competition 00:10:44.550 --> 00:10:45.168 align:middle line:90% model. 00:10:45.168 --> 00:10:46.710 align:middle line:84% And so you might think that you would 00:10:46.710 --> 00:10:50.310 align:middle line:84% get a just Bertrand outcome, everyone's p star equal c. 00:10:50.310 --> 00:10:53.130 align:middle line:84% But then as search costs go up, maybe prices 00:10:53.130 --> 00:10:54.550 align:middle line:90% go up in equilibrium. 00:10:54.550 --> 00:10:58.407 align:middle line:84% And they do, but they do more than Diamond wanted. 00:10:58.407 --> 00:11:00.990 align:middle line:84% In some sense, he started this literature with this thing that 00:11:00.990 --> 00:11:03.830 align:middle line:84% sometimes people call the "Diamond paradox." 00:11:03.830 --> 00:11:07.780 align:middle line:84% If you take any search cost s that's strictly 00:11:07.780 --> 00:11:09.790 align:middle line:84% positive and less than the consumer surplus, 00:11:09.790 --> 00:11:15.910 align:middle line:84% the monopoly price, then any perfect Bayesian equilibrium 00:11:15.910 --> 00:11:18.910 align:middle line:84% in which any consumers search has all firms 00:11:18.910 --> 00:11:21.250 align:middle line:90% pricing at the monopoly price. 00:11:21.250 --> 00:11:24.250 align:middle line:84% So in some sense, the only equilibria of this model 00:11:24.250 --> 00:11:27.020 align:middle line:84% are it's a discontinuous jump in prices. 00:11:27.020 --> 00:11:29.020 align:middle line:84% As soon as there's any search cost, even 00:11:29.020 --> 00:11:31.600 align:middle line:84% a penny, the only equilibrium is everyone charges 00:11:31.600 --> 00:11:34.450 align:middle line:84% a monopoly price instead of everyone charges c. 00:11:34.450 --> 00:11:38.560 align:middle line:84% So it says, Bertrand competition is an extraordinarily fragile 00:11:38.560 --> 00:11:42.170 align:middle line:84% result. Bertrand competition relied on zero search costs. 00:11:42.170 --> 00:11:45.160 align:middle line:84% You put any search cost in, you don't just get a little markup, 00:11:45.160 --> 00:11:47.080 align:middle line:90% you get the monopoly markup. 00:11:47.080 --> 00:11:49.210 align:middle line:84% Now, we don't believe this, but I 00:11:49.210 --> 00:11:52.970 align:middle line:84% think it's good to see why this happens. 00:11:52.970 --> 00:11:56.800 align:middle line:84% So first, the argument that this is an equilibrium 00:11:56.800 --> 00:11:58.180 align:middle line:90% is straightforward. 00:11:58.180 --> 00:12:02.260 align:middle line:84% Let's suppose you have a world where all the firms charge pm, 00:12:02.260 --> 00:12:04.060 align:middle line:84% and you know that everyone charges pm. 00:12:04.060 --> 00:12:04.760 align:middle line:90% You've grown up. 00:12:04.760 --> 00:12:06.302 align:middle line:84% You've lived your life in this world. 00:12:06.302 --> 00:12:09.100 align:middle line:84% You know that, yes, everyone charges a monopoly price. 00:12:09.100 --> 00:12:11.255 align:middle line:84% You open the first-- you get assigned-- 00:12:11.255 --> 00:12:12.380 align:middle line:90% you have to play this game. 00:12:12.380 --> 00:12:14.320 align:middle line:84% You open a door, and you're like, oh, 00:12:14.320 --> 00:12:15.805 align:middle line:84% what does cell phone service cost? 00:12:15.805 --> 00:12:19.270 align:middle line:84% Oh, it's $59.99 a month, even though it has marginal cost 00:12:19.270 --> 00:12:20.840 align:middle line:90% of zero for them to provide it. 00:12:20.840 --> 00:12:21.740 align:middle line:90% What do you do? 00:12:21.740 --> 00:12:23.270 align:middle line:84% Do you keep opening the other doors? 00:12:23.270 --> 00:12:25.120 align:middle line:84% You're like, no, every cell phone company 00:12:25.120 --> 00:12:27.280 align:middle line:90% charges $59.99 a month. 00:12:27.280 --> 00:12:29.800 align:middle line:90% Why keep opening doors? 00:12:29.800 --> 00:12:32.380 align:middle line:84% I open the first door, I see the monopoly price, I pay it, 00:12:32.380 --> 00:12:34.660 align:middle line:90% and then I'm done. 00:12:34.660 --> 00:12:40.250 align:middle line:84% And could a firm cut its price and gain consumers? 00:12:40.250 --> 00:12:43.510 align:middle line:84% And the answer is no, because when consumers expect everyone 00:12:43.510 --> 00:12:45.460 align:middle line:84% to be charging the monopoly price, 00:12:45.460 --> 00:12:49.400 align:middle line:84% everyone stops at the first door and just buys from that firm. 00:12:49.400 --> 00:12:51.730 align:middle line:84% And so suppose you have a lower price-- 00:12:51.730 --> 00:12:55.840 align:middle line:84% well, then consumers-- and you get assigned to room 7. 00:12:55.840 --> 00:12:58.780 align:middle line:84% If a consumer comes to you first and opens your door, 00:12:58.780 --> 00:13:02.590 align:middle line:84% they'll be like, oh, bonus, I get to pay $35 a month 00:13:02.590 --> 00:13:04.580 align:middle line:90% instead of $59.99 a month. 00:13:04.580 --> 00:13:07.942 align:middle line:84% But it's only the consumers who open that door first 00:13:07.942 --> 00:13:08.650 align:middle line:90% who buy from you. 00:13:08.650 --> 00:13:09.660 align:middle line:84% And they would have bought from if you'd 00:13:09.660 --> 00:13:11.340 align:middle line:90% charged the monopoly price. 00:13:11.340 --> 00:13:14.190 align:middle line:84% So the firms don't get any extra consumers 00:13:14.190 --> 00:13:18.060 align:middle line:90% by charging a lower price. 00:13:18.060 --> 00:13:20.860 align:middle line:84% If everyone charged a monopoly price, consumers don't search. 00:13:20.860 --> 00:13:23.380 align:middle line:84% If no one searches, all firms charge monopoly price. 00:13:23.380 --> 00:13:24.460 align:middle line:90% That's an equilibrium. 00:13:24.460 --> 00:13:27.180 align:middle line:90% 00:13:27.180 --> 00:13:32.070 align:middle line:84% The less immediately obvious part of this argument 00:13:32.070 --> 00:13:35.820 align:middle line:84% is that there's no other pure strategy equilibrium. 00:13:35.820 --> 00:13:38.760 align:middle line:84% To think about that, just imagine 00:13:38.760 --> 00:13:44.070 align:middle line:84% we had some other equilibrium and the firms 00:13:44.070 --> 00:13:46.000 align:middle line:90% charged different prices. 00:13:46.000 --> 00:13:49.270 align:middle line:84% So maybe this is the monopoly price. 00:13:49.270 --> 00:13:51.360 align:middle line:84% And then some firms charge a monopoly price, 00:13:51.360 --> 00:13:53.630 align:middle line:84% but some firms charge less than the monopoly price. 00:13:53.630 --> 00:13:56.003 align:middle line:90% 00:13:56.003 --> 00:13:57.420 align:middle line:84% Well, the argument is going to be, 00:13:57.420 --> 00:13:59.370 align:middle line:84% you can't have an equilibrium like this 00:13:59.370 --> 00:14:02.730 align:middle line:84% where somebody charges a lower price than everybody else. 00:14:02.730 --> 00:14:07.680 align:middle line:84% Because if you're charging a lower price than everybody else, 00:14:07.680 --> 00:14:10.400 align:middle line:84% I could raise my price by epsilon. 00:14:10.400 --> 00:14:14.335 align:middle line:84% Again, anyone who opens the door and sees my-- 00:14:14.335 --> 00:14:16.710 align:middle line:84% so consumers were expecting that there was some firm that 00:14:16.710 --> 00:14:19.140 align:middle line:90% charged $35 a month. 00:14:19.140 --> 00:14:23.430 align:middle line:90% And I go to $35.50 a month. 00:14:23.430 --> 00:14:25.090 align:middle line:90% So consumers open that door. 00:14:25.090 --> 00:14:28.040 align:middle line:84% They may think, oh, no one was supposed to charge $35.50. 00:14:28.040 --> 00:14:30.660 align:middle line:84% There was supposed to be a firm at $35, a firm at $36, 00:14:30.660 --> 00:14:32.580 align:middle line:90% and a bunch of firms at $40. 00:14:32.580 --> 00:14:34.730 align:middle line:90% But I just found $35.50. 00:14:34.730 --> 00:14:35.950 align:middle line:90% What do I do? 00:14:35.950 --> 00:14:39.390 align:middle line:84% Well, the most you could expect to save by opening another door 00:14:39.390 --> 00:14:40.420 align:middle line:90% is $0.50. 00:14:40.420 --> 00:14:43.130 align:middle line:84% Because you thought that there was someone at $35. 00:14:43.130 --> 00:14:44.610 align:middle line:90% You didn't find the guy at $35. 00:14:44.610 --> 00:14:48.120 align:middle line:84% But that's the best you're going to think to do. 00:14:48.120 --> 00:14:51.710 align:middle line:84% As long as I keep the amount, I raise the price less than the s, 00:14:51.710 --> 00:14:55.780 align:middle line:84% anyone who opens my door first is still going to buy from me. 00:14:55.780 --> 00:14:57.580 align:middle line:90% So I don't lose any-- 00:14:57.580 --> 00:14:59.840 align:middle line:84% anyone who ever opens my door is going to buy from me. 00:14:59.840 --> 00:15:03.737 align:middle line:84% So therefore, I can't lose any consumers by this price range. 00:15:03.737 --> 00:15:06.070 align:middle line:84% Therefore, I'm always going to raise my price by epsilon 00:15:06.070 --> 00:15:08.160 align:middle line:90% over what people expect. 00:15:08.160 --> 00:15:10.500 align:middle line:84% And so that makes it impossible to have 00:15:10.500 --> 00:15:13.480 align:middle line:84% an equilibrium where somebody is below everybody else. 00:15:13.480 --> 00:15:15.300 align:middle line:84% It also even makes it impossible to have 00:15:15.300 --> 00:15:17.430 align:middle line:84% an argument where you have two people tied for the lowest 00:15:17.430 --> 00:15:17.930 align:middle line:90% price. 00:15:17.930 --> 00:15:21.040 align:middle line:84% Because again, I just make an epsilon increase. 00:15:21.040 --> 00:15:23.257 align:middle line:84% Anyone who opens my door, as long 00:15:23.257 --> 00:15:25.590 align:middle line:84% as I'm within epsilon of the lowest price in the market, 00:15:25.590 --> 00:15:26.965 align:middle line:84% they're all going to buy from me. 00:15:26.965 --> 00:15:30.233 align:middle line:84% So it's going to have to be that you can't have anyone-- 00:15:30.233 --> 00:15:31.650 align:middle line:84% the lowest-priced firms are always 00:15:31.650 --> 00:15:33.540 align:middle line:84% going to want to raise their price by epsilon if they're 00:15:33.540 --> 00:15:35.150 align:middle line:90% less than the monopoly price. 00:15:35.150 --> 00:15:38.220 align:middle line:84% Similarly, I can't have anyone charging greater 00:15:38.220 --> 00:15:39.330 align:middle line:90% than the monopoly price. 00:15:39.330 --> 00:15:42.600 align:middle line:84% Because if they're charging greater than the monopoly price 00:15:42.600 --> 00:15:46.343 align:middle line:84% and not selling anything, they make no money. 00:15:46.343 --> 00:15:47.760 align:middle line:84% If they are selling things, they'd 00:15:47.760 --> 00:15:51.390 align:middle line:84% be better off going down to the monopoly price itself. 00:15:51.390 --> 00:15:53.040 align:middle line:84% So the only equilibrium of this model 00:15:53.040 --> 00:15:55.235 align:middle line:84% is everyone charges monopoly prices. 00:15:55.235 --> 00:15:58.020 align:middle line:90% 00:15:58.020 --> 00:16:03.080 align:middle line:84% So I think it's a useful thought experiment, 00:16:03.080 --> 00:16:05.985 align:middle line:84% but it's obviously not what we want to do in practice. 00:16:05.985 --> 00:16:07.610 align:middle line:84% And so then where the search literature 00:16:07.610 --> 00:16:10.040 align:middle line:84% has gone is thinking about, what happens-- 00:16:10.040 --> 00:16:12.380 align:middle line:84% what can we do with search to get search 00:16:12.380 --> 00:16:15.710 align:middle line:84% to have a continuous effect like we think it probably 00:16:15.710 --> 00:16:16.975 align:middle line:90% does-- should have? 00:16:16.975 --> 00:16:18.475 align:middle line:84% And what's wrong with this argument? 00:16:18.475 --> 00:16:23.330 align:middle line:90% 00:16:23.330 --> 00:16:26.660 align:middle line:84% One way in which this literature has gone, 00:16:26.660 --> 00:16:28.490 align:middle line:84% and which is completely realistic, 00:16:28.490 --> 00:16:32.390 align:middle line:84% is models with heterogeneity in search costs, where there 00:16:32.390 --> 00:16:34.100 align:middle line:84% are some consumers who have higher search 00:16:34.100 --> 00:16:36.850 align:middle line:84% costs and some consumers who have lower search costs. 00:16:36.850 --> 00:16:40.250 align:middle line:90% 00:16:40.250 --> 00:16:41.810 align:middle line:90% And it turns out that-- 00:16:41.810 --> 00:16:44.090 align:middle line:84% it's a nice literature because it actually 00:16:44.090 --> 00:16:46.922 align:middle line:84% turns out it's going to explain two phenomena simultaneously. 00:16:46.922 --> 00:16:48.380 align:middle line:84% One is it's going to give us models 00:16:48.380 --> 00:16:51.050 align:middle line:84% that do have positive markups that are 00:16:51.050 --> 00:16:52.980 align:middle line:90% continuous in the search cost. 00:16:52.980 --> 00:16:54.680 align:middle line:84% And then the other thing it gets us 00:16:54.680 --> 00:16:58.400 align:middle line:84% is price dispersion, where different firms charge 00:16:58.400 --> 00:16:59.490 align:middle line:90% different prices. 00:16:59.490 --> 00:17:02.420 align:middle line:84% And I think in the car dealer example, 00:17:02.420 --> 00:17:04.670 align:middle line:84% you do expect that every different car dealer has 00:17:04.670 --> 00:17:07.609 align:middle line:84% a different price that they're willing to settle for on the car 00:17:07.609 --> 00:17:10.526 align:middle line:84% if you negotiate long enough and that every different firm 00:17:10.526 --> 00:17:12.234 align:middle line:84% is going to charge you a different price. 00:17:12.234 --> 00:17:15.079 align:middle line:90% 00:17:15.079 --> 00:17:16.859 align:middle line:90% So how would this model go? 00:17:16.859 --> 00:17:20.369 align:middle line:84% So again, let's stick with my credit card example. 00:17:20.369 --> 00:17:21.776 align:middle line:90% So I have N firms. 00:17:21.776 --> 00:17:23.359 align:middle line:84% They're all producing that homogeneous 00:17:23.359 --> 00:17:27.140 align:middle line:84% good at the same common, constant marginal cost. 00:17:27.140 --> 00:17:29.510 align:middle line:84% Again, I'm going to let them choose prices. 00:17:29.510 --> 00:17:32.130 align:middle line:84% Again, I'm going to have the same continuum of consumers. 00:17:32.130 --> 00:17:35.690 align:middle line:84% Each consumer has a multiunit demand D of p. 00:17:35.690 --> 00:17:38.270 align:middle line:84% And I'm going to have consumers searching optimally, 00:17:38.270 --> 00:17:40.760 align:middle line:90% as in the Diamond model. 00:17:40.760 --> 00:17:43.910 align:middle line:84% But instead of the Diamond model where everyone had search cost 00:17:43.910 --> 00:17:47.870 align:middle line:84% s, I'm going to have multiple types of consumers. 00:17:47.870 --> 00:17:50.490 align:middle line:84% I'm going to have a fraction mu of the consumers. 00:17:50.490 --> 00:17:53.326 align:middle line:84% Sometimes people refer to these guys as "shoppers." 00:17:53.326 --> 00:17:56.730 align:middle line:90% 00:17:56.730 --> 00:17:59.440 align:middle line:84% Shoppers have a negative search cost. 00:17:59.440 --> 00:18:02.790 align:middle line:84% So shoppers actually get utility out of comparing prices 00:18:02.790 --> 00:18:04.830 align:middle line:90% at different firms. 00:18:04.830 --> 00:18:07.080 align:middle line:84% And you may think that's unrealistic. 00:18:07.080 --> 00:18:10.170 align:middle line:84% But I mean, there are some people 00:18:10.170 --> 00:18:11.820 align:middle line:90% who like shopping as a hobby. 00:18:11.820 --> 00:18:13.320 align:middle line:84% There are some graduate students who 00:18:13.320 --> 00:18:16.263 align:middle line:84% love to tell me that, I booked this crazy plane ticket where 00:18:16.263 --> 00:18:18.180 align:middle line:84% I fly to here, and then I have a return ticket 00:18:18.180 --> 00:18:19.840 align:middle line:90% that's booked out of this city. 00:18:19.840 --> 00:18:22.690 align:middle line:84% But then I'm going to get off in this city and save that thing. 00:18:22.690 --> 00:18:24.940 align:middle line:90% And I'm saving $7 by doing this. 00:18:24.940 --> 00:18:27.630 align:middle line:84% And you have to think, did you spend, like, 00:18:27.630 --> 00:18:30.630 align:middle line:84% 14.5 hours figuring out some really clever way 00:18:30.630 --> 00:18:32.190 align:middle line:90% you could save $7? 00:18:32.190 --> 00:18:35.038 align:middle line:84% It must be that you got a great deal of utility 00:18:35.038 --> 00:18:37.080 align:middle line:84% out of figuring out all of the different ways one 00:18:37.080 --> 00:18:39.780 align:middle line:84% could fly from A to B and feeling that you were-- 00:18:39.780 --> 00:18:40.330 align:middle line:90% whatever. 00:18:40.330 --> 00:18:40.872 align:middle line:90% I don't know. 00:18:40.872 --> 00:18:44.280 align:middle line:84% But there are some people who just will enjoy shopping 00:18:44.280 --> 00:18:46.570 align:middle line:84% and will do more shopping than seems reasonable. 00:18:46.570 --> 00:18:50.490 align:middle line:84% And so I'll model them by having this negative search cost. 00:18:50.490 --> 00:18:52.740 align:middle line:84% And then I'm going to have a fraction, 1 minus mu 00:18:52.740 --> 00:18:55.500 align:middle line:84% of consumers, who have the more traditional positive search 00:18:55.500 --> 00:18:57.010 align:middle line:90% costs. 00:18:57.010 --> 00:18:59.040 align:middle line:84% And at least at first, I'm just going 00:18:59.040 --> 00:19:03.510 align:middle line:84% to assume that these people have search costs in some interval s 00:19:03.510 --> 00:19:04.890 align:middle line:90% lower bar, s upper bar. 00:19:04.890 --> 00:19:08.440 align:middle line:84% It could be distributed with some density. 00:19:08.440 --> 00:19:11.520 align:middle line:84% But they are all less than the consumer surplus, the monopoly 00:19:11.520 --> 00:19:12.318 align:middle line:90% price. 00:19:12.318 --> 00:19:14.610 align:middle line:84% The reason for putting this upper bound on search costs 00:19:14.610 --> 00:19:17.220 align:middle line:84% is, if your search costs are less than the consumer 00:19:17.220 --> 00:19:18.868 align:middle line:84% surplus of the monopoly price, then you 00:19:18.868 --> 00:19:20.910 align:middle line:84% run into this problem of just those people aren't 00:19:20.910 --> 00:19:22.117 align:middle line:90% going to search at all. 00:19:22.117 --> 00:19:23.950 align:middle line:84% And then they're not going to search at all, 00:19:23.950 --> 00:19:25.510 align:middle line:90% so then no market exists. 00:19:25.510 --> 00:19:27.210 align:middle line:84% So I want search costs low enough 00:19:27.210 --> 00:19:31.380 align:middle line:84% so that consumers are going to want to search and buy, 00:19:31.380 --> 00:19:32.770 align:middle line:90% even if the prices are high. 00:19:32.770 --> 00:19:35.353 align:middle line:84% And I don't have to worry about that edge case of what happens 00:19:35.353 --> 00:19:36.820 align:middle line:90% if prices are high-- too high. 00:19:36.820 --> 00:19:39.160 align:middle line:84% So this distribution of search costs. 00:19:39.160 --> 00:19:44.190 align:middle line:84% So I've just got some interval s lower bar, s upper bar, where 00:19:44.190 --> 00:19:45.510 align:middle line:90% I have a density of consumers. 00:19:45.510 --> 00:19:48.450 align:middle line:84% And then you can think of it as a point mass of consumers 00:19:48.450 --> 00:19:50.460 align:middle line:90% with negative search costs. 00:19:50.460 --> 00:19:56.220 align:middle line:84% And it's just a simplification, but it's 00:19:56.220 --> 00:19:58.720 align:middle line:84% a simplification that implies that these consumers, 00:19:58.720 --> 00:20:00.350 align:middle line:84% it's easy to solve for what they do. 00:20:00.350 --> 00:20:02.192 align:middle line:84% What they do is they visit every single firm 00:20:02.192 --> 00:20:04.150 align:middle line:84% and then buy from the lowest-priced firm that's 00:20:04.150 --> 00:20:06.680 align:middle line:90% out there. 00:20:06.680 --> 00:20:10.760 align:middle line:84% So what's Stahl's result on this model? 00:20:10.760 --> 00:20:13.430 align:middle line:84% Part a of the theorem, this model 00:20:13.430 --> 00:20:16.760 align:middle line:84% has no pure strategy equilibrium, 00:20:16.760 --> 00:20:20.150 align:middle line:84% but it does have at least one symmetric mixed strategy 00:20:20.150 --> 00:20:21.320 align:middle line:90% equilibrium. 00:20:21.320 --> 00:20:24.380 align:middle line:84% And in this symmetric mixed strategy equilibrium, 00:20:24.380 --> 00:20:28.340 align:middle line:84% the firms choose prices from an atomless distribution 00:20:28.340 --> 00:20:31.310 align:middle line:84% F that has some support p lower bar, p upper bar. 00:20:31.310 --> 00:20:34.670 align:middle line:84% And that support does not contain marginal cost. 00:20:34.670 --> 00:20:36.830 align:middle line:84% So what's going to typically happen here 00:20:36.830 --> 00:20:38.250 align:middle line:90% is, if you look at pricing-- 00:20:38.250 --> 00:20:42.416 align:middle line:90% 00:20:42.416 --> 00:20:43.700 align:middle line:90% did I give it a name? 00:20:43.700 --> 00:20:45.500 align:middle line:90% I did not give it a name. 00:20:45.500 --> 00:20:49.130 align:middle line:84% So we're going to have this distribution where there's-- 00:20:49.130 --> 00:20:52.610 align:middle line:84% you could price it marginal cost in Bertrand, but firms don't. 00:20:52.610 --> 00:20:55.730 align:middle line:84% Instead, there's some level p lower bar, 00:20:55.730 --> 00:20:58.670 align:middle line:84% and there's some level p upper bar. 00:20:58.670 --> 00:21:03.140 align:middle line:84% And then what firms do is there's some density. 00:21:03.140 --> 00:21:04.680 align:middle line:90% It might look like this. 00:21:04.680 --> 00:21:07.350 align:middle line:84% And so what firms do is they choose prices-- 00:21:07.350 --> 00:21:09.090 align:middle line:84% use a mixed strategy, choose prices 00:21:09.090 --> 00:21:11.260 align:middle line:84% from a density that looks like this. 00:21:11.260 --> 00:21:14.320 align:middle line:84% And so prices, from the consumer perspective, are random. 00:21:14.320 --> 00:21:17.760 align:middle line:84% And then, in some sense, this mixed strategy 00:21:17.760 --> 00:21:19.980 align:middle line:84% on the firm's part then rationalizes the search 00:21:19.980 --> 00:21:21.010 align:middle line:90% on the consumer part. 00:21:21.010 --> 00:21:24.450 align:middle line:84% Because if the firms are mixing over prices, 00:21:24.450 --> 00:21:25.960 align:middle line:90% you go and find a firm's price. 00:21:25.960 --> 00:21:28.252 align:middle line:84% What you're getting is one draw from the mixed strategy 00:21:28.252 --> 00:21:28.830 align:middle line:90% distribution. 00:21:28.830 --> 00:21:29.850 align:middle line:90% You may get a high draw. 00:21:29.850 --> 00:21:31.245 align:middle line:90% You may get a low draw. 00:21:31.245 --> 00:21:33.370 align:middle line:84% You may get a high draw and decide to search again. 00:21:33.370 --> 00:21:38.880 align:middle line:84% You may get a low draw and decide to stop your search. 00:21:38.880 --> 00:21:40.920 align:middle line:90% Clearly, if you draw-- 00:21:40.920 --> 00:21:43.895 align:middle line:84% if the first firm you draw actually has price p lower bar, 00:21:43.895 --> 00:21:45.270 align:middle line:84% you're always going to stop there 00:21:45.270 --> 00:21:48.390 align:middle line:84% because you know that's the best you can do. 00:21:48.390 --> 00:21:52.680 align:middle line:84% If the firm has price p upper bar, if your search cost is 00:21:52.680 --> 00:21:55.500 align:middle line:84% really high, like if your search cost is near s upper bar, 00:21:55.500 --> 00:21:57.210 align:middle line:90% you're going to buy-- 00:21:57.210 --> 00:21:58.980 align:middle line:84% the model is going to be such that you 00:21:58.980 --> 00:22:01.900 align:middle line:84% are going to buy from that price if you find it. 00:22:01.900 --> 00:22:04.020 align:middle line:84% But then someone with a lower s may 00:22:04.020 --> 00:22:06.510 align:middle line:84% decide not to buy there and only buy if they 00:22:06.510 --> 00:22:07.715 align:middle line:90% find this price or better. 00:22:07.715 --> 00:22:15.330 align:middle line:90% 00:22:15.330 --> 00:22:16.465 align:middle line:90% So, Remarks. 00:22:16.465 --> 00:22:18.090 align:middle line:84% Yeah, the mixed equilibrium can be seen 00:22:18.090 --> 00:22:19.215 align:middle line:90% as a potential explanation. 00:22:19.215 --> 00:22:20.220 align:middle line:90% Yes, question. 00:22:20.220 --> 00:22:21.060 align:middle line:90% AUDIENCE: Yes. 00:22:21.060 --> 00:22:25.210 align:middle line:84% So intuitively, why do you need the shoppers in this model? 00:22:25.210 --> 00:22:28.470 align:middle line:84% Because at first glance, it would seem like just 00:22:28.470 --> 00:22:31.770 align:middle line:84% by having the dispersion in search costs 00:22:31.770 --> 00:22:34.027 align:middle line:84% would generate a dispersion in prices. 00:22:34.027 --> 00:22:34.860 align:middle line:90% GLENN ELLISON: Yeah. 00:22:34.860 --> 00:22:37.620 align:middle line:84% Unfortunately, you get this same-- 00:22:37.620 --> 00:22:40.120 align:middle line:84% it's the same Diamond argument would work here. 00:22:40.120 --> 00:22:42.750 align:middle line:84% So let's suppose that we had these-- 00:22:42.750 --> 00:22:44.400 align:middle line:90% I took this away. 00:22:44.400 --> 00:22:48.000 align:middle line:84% And then imagine I tried to have an equilibrium like this. 00:22:48.000 --> 00:22:50.310 align:middle line:84% And then think about the firm that's 00:22:50.310 --> 00:22:51.915 align:middle line:90% charging price p lower bar. 00:22:51.915 --> 00:22:54.560 align:middle line:90% 00:22:54.560 --> 00:22:56.310 align:middle line:84% So to have a mixed strategy equilibrium, 00:22:56.310 --> 00:22:59.010 align:middle line:84% I have to be indifferent over all these prices. 00:22:59.010 --> 00:23:01.800 align:middle line:84% Let's suppose I deviate, and instead of charging p lower bar, 00:23:01.800 --> 00:23:05.580 align:middle line:84% I charge p lower bar plus epsilon. 00:23:05.580 --> 00:23:07.030 align:middle line:90% Again, the same argument. 00:23:07.030 --> 00:23:09.630 align:middle line:84% Any time anyone gets a quote from me 00:23:09.630 --> 00:23:14.580 align:middle line:84% and they find price p lower bar plus epsilon, as long 00:23:14.580 --> 00:23:21.480 align:middle line:84% as consumer surplus at p lower bar plus epsilon, 00:23:21.480 --> 00:23:25.440 align:middle line:84% consumer surplus at p lower bar minus consumer surplus 00:23:25.440 --> 00:23:29.710 align:middle line:84% at that is less than s, they're going to have to buy from me. 00:23:29.710 --> 00:23:34.750 align:middle line:84% Because the most they could gain by finding p lower bar is this. 00:23:34.750 --> 00:23:37.390 align:middle line:84% If this is less than s, they would stop. 00:23:37.390 --> 00:23:40.150 align:middle line:84% So you still get this epsilon over a cutting argument 00:23:40.150 --> 00:23:41.843 align:middle line:84% that says, this can't be an equilibrium, 00:23:41.843 --> 00:23:44.260 align:middle line:84% because if everyone knows that everyone charges at least p 00:23:44.260 --> 00:23:46.843 align:middle line:84% lower bar, then everyone would want to charge at least p lower 00:23:46.843 --> 00:23:47.930 align:middle line:90% bar plus epsilon. 00:23:47.930 --> 00:23:49.750 align:middle line:84% And if everyone knows everyone wants to charge at least p lower 00:23:49.750 --> 00:23:51.292 align:middle line:84% bar plus epsilon, then everyone wants 00:23:51.292 --> 00:23:53.860 align:middle line:84% to charge at least p lower bar plus 2 epsilon and the thing 00:23:53.860 --> 00:23:54.210 align:middle line:90% collapses. 00:23:54.210 --> 00:23:56.350 align:middle line:84% AUDIENCE: So you're going to want someone who's always 00:23:56.350 --> 00:23:57.150 align:middle line:90% willing to keep searching. 00:23:57.150 --> 00:23:57.820 align:middle line:84% GLENN ELLISON: So you want someone 00:23:57.820 --> 00:24:00.730 align:middle line:84% who is always willing to keep searching, that you may lose-- 00:24:00.730 --> 00:24:02.920 align:middle line:84% you need to explain, how could anyone ever charge 00:24:02.920 --> 00:24:04.280 align:middle line:90% the lowest possible price? 00:24:04.280 --> 00:24:07.570 align:middle line:84% And there, it could be, if you raise your price by epsilon, 00:24:07.570 --> 00:24:09.460 align:middle line:84% there are those shoppers who you're 00:24:09.460 --> 00:24:12.642 align:middle line:90% going to lose by doing that. 00:24:12.642 --> 00:24:14.350 align:middle line:84% AUDIENCE: So in this model, the consumers 00:24:14.350 --> 00:24:15.970 align:middle line:90% always know the distribution? 00:24:15.970 --> 00:24:17.553 align:middle line:84% GLENN ELLISON: So in this model, we're 00:24:17.553 --> 00:24:19.660 align:middle line:84% assuming the consumers know the distribution. 00:24:19.660 --> 00:24:21.760 align:middle line:90% That is, this is where-- 00:24:21.760 --> 00:24:23.748 align:middle line:84% obviously, you can have models where consumers 00:24:23.748 --> 00:24:25.040 align:middle line:90% have to learn the distribution. 00:24:25.040 --> 00:24:26.950 align:middle line:84% But here, we're sticking with the equilibrium 00:24:26.950 --> 00:24:29.358 align:middle line:84% notion of consumers are best responding to the firm's 00:24:29.358 --> 00:24:31.900 align:middle line:84% strategies, and the firms are best responding to the consumer 00:24:31.900 --> 00:24:33.200 align:middle line:90% strategies. 00:24:33.200 --> 00:24:33.760 align:middle line:90% Yeah. 00:24:33.760 --> 00:24:37.165 align:middle line:84% And I think that one doesn't strike me as so unreasonable 00:24:37.165 --> 00:24:38.890 align:middle line:84% that, like, on the credit card market, 00:24:38.890 --> 00:24:40.307 align:middle line:84% I know there are credit cards that 00:24:40.307 --> 00:24:42.890 align:middle line:84% offer a good deal and credit cards that offer a bad deal. 00:24:42.890 --> 00:24:45.760 align:middle line:84% And I go out there and I start clicking on credit cards, 00:24:45.760 --> 00:24:47.260 align:middle line:84% and I look at this credit card and I 00:24:47.260 --> 00:24:48.552 align:middle line:90% see what rate they're offering. 00:24:48.552 --> 00:24:51.230 align:middle line:84% And certainly, the credit card offers you get in the mail, 00:24:51.230 --> 00:24:53.740 align:middle line:84% they all have different rates on them. 00:24:53.740 --> 00:24:55.750 align:middle line:84% I kind of know what the distribution is, 00:24:55.750 --> 00:24:57.250 align:middle line:84% and I know what's going to take me-- 00:24:57.250 --> 00:25:00.290 align:middle line:84% I get some sense of how good I'm doing by following it. 00:25:00.290 --> 00:25:06.260 align:middle line:84% And like we said, they know the equilibrium strategies. 00:25:06.260 --> 00:25:08.740 align:middle line:84% They don't know the actual distribution. 00:25:08.740 --> 00:25:12.250 align:middle line:84% So if a firm deviates, it changes the distribution 00:25:12.250 --> 00:25:14.120 align:middle line:90% and consumers don't know that. 00:25:14.120 --> 00:25:17.810 align:middle line:84% So consumers know what's the normal distribution of prices. 00:25:17.810 --> 00:25:18.310 align:middle line:90% Yeah. 00:25:18.310 --> 00:25:21.250 align:middle line:90% 00:25:21.250 --> 00:25:23.032 align:middle line:90% So it's a mixed equilibrium. 00:25:23.032 --> 00:25:24.490 align:middle line:84% You can think of this as explaining 00:25:24.490 --> 00:25:28.090 align:middle line:90% why prices differ across firms. 00:25:28.090 --> 00:25:32.470 align:middle line:84% Prices in this model move continuously with search costs, 00:25:32.470 --> 00:25:34.420 align:middle line:84% and they move continuously with the fraction 00:25:34.420 --> 00:25:36.680 align:middle line:84% of consumers who are shoppers in the way you'd expect. 00:25:36.680 --> 00:25:41.800 align:middle line:84% So either as the fraction of shoppers goes to 1 or as 00:25:41.800 --> 00:25:45.890 align:middle line:84% s upper bar goes to 0, prices collapse to Bertrand pricing. 00:25:45.890 --> 00:25:53.110 align:middle line:84% So if this is what it looks like with what I've drawn, if I make 00:25:53.110 --> 00:25:56.887 align:middle line:84% s upper bar down here, so if I make all the consumer 00:25:56.887 --> 00:25:58.720 align:middle line:84% search costs tiny, what you're going to find 00:25:58.720 --> 00:26:02.140 align:middle line:84% is that, if this is c, the distribution looks like that. 00:26:02.140 --> 00:26:03.910 align:middle line:84% That is, the distribution collapses 00:26:03.910 --> 00:26:08.030 align:middle line:84% towards a point mass on p equals c as the s goes to 0. 00:26:08.030 --> 00:26:11.140 align:middle line:84% It also collapses towards a point mass on c 00:26:11.140 --> 00:26:13.800 align:middle line:84% as the fraction of shoppers goes to 1. 00:26:13.800 --> 00:26:18.660 align:middle line:90% 00:26:18.660 --> 00:26:20.520 align:middle line:84% These equilibria of this kind of model 00:26:20.520 --> 00:26:23.970 align:middle line:84% often have U-shaped distributions when N is large. 00:26:23.970 --> 00:26:26.790 align:middle line:84% In some sense, if you're a firm and you 00:26:26.790 --> 00:26:29.820 align:middle line:84% know you're competing for a large number of consumers, 00:26:29.820 --> 00:26:32.730 align:middle line:90% you have two good strategies. 00:26:32.730 --> 00:26:37.890 align:middle line:90% 00:26:37.890 --> 00:26:44.700 align:middle line:84% One strategy is typically priced near p lower bar 00:26:44.700 --> 00:26:46.563 align:middle line:84% and priced near p lower bar in hope 00:26:46.563 --> 00:26:48.480 align:middle line:84% to have the single lowest price and to get all 00:26:48.480 --> 00:26:50.550 align:middle line:90% the shoppers to buy from you. 00:26:50.550 --> 00:26:52.920 align:middle line:84% And then the other strategy is price up here 00:26:52.920 --> 00:26:56.160 align:middle line:84% near the monopoly price and figure that, I'm never 00:26:56.160 --> 00:26:57.430 align:middle line:90% going to win all the shoppers. 00:26:57.430 --> 00:26:59.888 align:middle line:84% I'm just going to get the people who come to my store first 00:26:59.888 --> 00:27:01.680 align:middle line:90% and have a high s. 00:27:01.680 --> 00:27:03.473 align:middle line:84% And so often, you get-- there's often 00:27:03.473 --> 00:27:04.890 align:middle line:84% less point of being in the middle, 00:27:04.890 --> 00:27:07.015 align:middle line:84% because in the middle, the shoppers-- 00:27:07.015 --> 00:27:08.640 align:middle line:84% you're never going to get the shoppers. 00:27:08.640 --> 00:27:10.320 align:middle line:84% And the nonshoppers, you would have gotten anyway 00:27:10.320 --> 00:27:11.230 align:middle line:90% with a higher price. 00:27:11.230 --> 00:27:14.590 align:middle line:84% So you often see these distributions that sort of look 00:27:14.590 --> 00:27:15.140 align:middle line:90% like that. 00:27:15.140 --> 00:27:15.807 align:middle line:90% They're bimodal. 00:27:15.807 --> 00:27:20.950 align:middle line:90% 00:27:20.950 --> 00:27:23.290 align:middle line:84% The limit as N goes to infinity is 00:27:23.290 --> 00:27:25.630 align:middle line:84% perhaps not what you might expect, 00:27:25.630 --> 00:27:26.920 align:middle line:90% but it is interesting itself. 00:27:26.920 --> 00:27:29.560 align:middle line:84% As N goes to infinity, with the way this distribution changes, 00:27:29.560 --> 00:27:31.660 align:middle line:90% is it goes like this. 00:27:31.660 --> 00:27:35.710 align:middle line:84% You have a small number who go for the nonshoppers. 00:27:35.710 --> 00:27:37.420 align:middle line:84% And then you have a huge mass of them 00:27:37.420 --> 00:27:39.410 align:middle line:90% up here at the monopoly price. 00:27:39.410 --> 00:27:44.500 align:middle line:84% So this says, in some sense, Tobias 00:27:44.500 --> 00:27:47.450 align:middle line:84% talked about cereal product proliferation 00:27:47.450 --> 00:27:49.742 align:middle line:84% and there being a million cereals or whatever. 00:27:49.742 --> 00:27:52.450 align:middle line:84% This says, if you have a million credit card companies out there, 00:27:52.450 --> 00:27:54.228 align:middle line:84% maybe what you would see is almost all 00:27:54.228 --> 00:27:56.770 align:middle line:84% of them are ripping people off and just charging the monopoly 00:27:56.770 --> 00:27:57.530 align:middle line:90% price. 00:27:57.530 --> 00:27:59.488 align:middle line:84% And there are a few of them out there competing 00:27:59.488 --> 00:28:01.220 align:middle line:90% for the sophisticated consumers. 00:28:01.220 --> 00:28:03.670 align:middle line:84% And we could see this bimodal thing, where most people are 00:28:03.670 --> 00:28:04.587 align:middle line:90% at the monopoly price. 00:28:04.587 --> 00:28:07.690 align:middle line:84% And in fact, this model does say, as N goes to infinity-- 00:28:07.690 --> 00:28:12.550 align:middle line:84% as N goes to infinity, they almost all end up up here. 00:28:12.550 --> 00:28:15.250 align:middle line:84% One argument for that is, if you think about it, 00:28:15.250 --> 00:28:17.800 align:middle line:84% let's suppose that you had even a constant fraction 00:28:17.800 --> 00:28:18.460 align:middle line:90% of the firms. 00:28:18.460 --> 00:28:23.020 align:middle line:84% Suppose that alpha N firms were charging 00:28:23.020 --> 00:28:25.270 align:middle line:90% close to the minimum price. 00:28:25.270 --> 00:28:28.870 align:middle line:84% Then, if you're a firm and you charge 00:28:28.870 --> 00:28:34.030 align:middle line:84% close to the minimum price, you have 1 over alpha N probability 00:28:34.030 --> 00:28:36.280 align:middle line:90% of being the lowest firm. 00:28:36.280 --> 00:28:38.290 align:middle line:90% But then because they're also-- 00:28:38.290 --> 00:28:40.480 align:middle line:84% because there are, like, alpha times N firms here, 00:28:40.480 --> 00:28:42.020 align:middle line:90% there are a lot of firms here. 00:28:42.020 --> 00:28:44.240 align:middle line:84% The distance between them is very, very small. 00:28:44.240 --> 00:28:46.640 align:middle line:84% So the price sensitivity is very, very large. 00:28:46.640 --> 00:28:49.600 align:middle line:84% So this point mass actually has to collapse towards-- 00:28:49.600 --> 00:28:52.480 align:middle line:84% this thing becomes very close to c. 00:28:52.480 --> 00:28:57.070 align:middle line:84% And so what you basically get, you get 1 over N consumers, 00:28:57.070 --> 00:29:01.780 align:middle line:84% and you get a 1 over N markup on them. 00:29:01.780 --> 00:29:04.453 align:middle line:84% And so you'd be getting 1 over N squared profits. 00:29:04.453 --> 00:29:10.240 align:middle line:90% 00:29:10.240 --> 00:29:12.190 align:middle line:90% And that's not going to work. 00:29:12.190 --> 00:29:14.560 align:middle line:84% Because if you had 1 over N squared profits, 00:29:14.560 --> 00:29:18.130 align:middle line:84% you could get 1 over N kind of profits by just pricing up here 00:29:18.130 --> 00:29:21.220 align:middle line:84% and getting the people with the high s's. 00:29:21.220 --> 00:29:24.130 align:middle line:84% And so it turns out that the equilibrium just 00:29:24.130 --> 00:29:27.700 align:middle line:84% is a very, very small point mass here and a very, very big one 00:29:27.700 --> 00:29:28.980 align:middle line:90% up there, when N gets big. 00:29:28.980 --> 00:29:35.640 align:middle line:90% 00:29:35.640 --> 00:29:37.072 align:middle line:90% So anyway, let's see. 00:29:37.072 --> 00:29:39.280 align:middle line:84% I guess I didn't actually say much about the theorem. 00:29:39.280 --> 00:29:42.280 align:middle line:84% All I said is that it doesn't have a pure strategy 00:29:42.280 --> 00:29:43.130 align:middle line:90% equilibrium. 00:29:43.130 --> 00:29:45.875 align:middle line:84% It does have a mixed strategy equilibrium. 00:29:45.875 --> 00:29:47.500 align:middle line:84% And in this mixed strategy equilibrium, 00:29:47.500 --> 00:29:49.630 align:middle line:84% you get something that looks like-- 00:29:49.630 --> 00:29:52.807 align:middle line:84% I just said, it has p lower bar greater than c. 00:29:52.807 --> 00:29:54.140 align:middle line:90% I didn't say much else about it. 00:29:54.140 --> 00:29:56.320 align:middle line:90% It's got a density. 00:29:56.320 --> 00:29:58.960 align:middle line:90% Why does that happen? 00:29:58.960 --> 00:30:01.760 align:middle line:84% The argument for part a, that it's no pure strategy 00:30:01.760 --> 00:30:04.630 align:middle line:84% equilibrium, is just like Diamond's. 00:30:04.630 --> 00:30:08.620 align:middle line:84% It's like Diamond's, most of the time, that you 00:30:08.620 --> 00:30:10.120 align:middle line:90% can't have an equilibrium. 00:30:10.120 --> 00:30:13.150 align:middle line:84% Again, suppose we tried to have an equilibrium 00:30:13.150 --> 00:30:15.610 align:middle line:84% where someone was at $35, someone was at $36, 00:30:15.610 --> 00:30:19.390 align:middle line:84% and then some other people were up here. 00:30:19.390 --> 00:30:22.160 align:middle line:84% You're the single lowest person in this model. 00:30:22.160 --> 00:30:25.300 align:middle line:84% You strictly want to increase your price in epsilon overcut. 00:30:25.300 --> 00:30:29.110 align:middle line:84% Because, again, if consumers expect this distribution, 00:30:29.110 --> 00:30:32.080 align:middle line:84% you're epsilon more than that, you still get all the shoppers, 00:30:32.080 --> 00:30:35.000 align:middle line:84% and all the nonshoppers still buy from you. 00:30:35.000 --> 00:30:36.815 align:middle line:84% And so then the only new case is, 00:30:36.815 --> 00:30:38.690 align:middle line:84% well, let's suppose we had-- like in Diamond, 00:30:38.690 --> 00:30:40.220 align:middle line:84% we have a whole bunch of firms all 00:30:40.220 --> 00:30:43.670 align:middle line:84% tied at exactly the same price, whether that's the monopoly 00:30:43.670 --> 00:30:45.110 align:middle line:90% price or something else. 00:30:45.110 --> 00:30:46.910 align:middle line:84% Well, then you get the Bertrand argument 00:30:46.910 --> 00:30:50.480 align:middle line:84% that you would epsilon undercut and get all the shoppers. 00:30:50.480 --> 00:30:52.730 align:middle line:84% So we have the same effect as Diamond, saying, 00:30:52.730 --> 00:30:55.250 align:middle line:84% you can't have people different from each other 00:30:55.250 --> 00:30:57.380 align:middle line:84% because the lowest firm always moves up. 00:30:57.380 --> 00:30:58.428 align:middle line:90% And then you have a new-- 00:30:58.428 --> 00:31:00.470 align:middle line:84% and then you go back to the Bertrand argument of, 00:31:00.470 --> 00:31:02.480 align:middle line:84% you can't have everyone the same at anything 00:31:02.480 --> 00:31:05.000 align:middle line:84% other than c because someone at the bottom goes down. 00:31:05.000 --> 00:31:07.160 align:middle line:84% Someone will go down and steal all the shoppers 00:31:07.160 --> 00:31:09.890 align:middle line:84% instead of splitting the shoppers with the other firms. 00:31:09.890 --> 00:31:13.690 align:middle line:84% So there's not going to be a pure strategy equilibrium here. 00:31:13.690 --> 00:31:18.840 align:middle line:90% 00:31:18.840 --> 00:31:20.740 align:middle line:84% What about mixed strategy equilibria? 00:31:20.740 --> 00:31:22.898 align:middle line:90% Well, in-- yes. 00:31:22.898 --> 00:31:23.940 align:middle line:90% AUDIENCE: Quick question. 00:31:23.940 --> 00:31:25.750 align:middle line:84% How does the shopper's utility work? 00:31:25.750 --> 00:31:29.130 align:middle line:84% Because if they get positive utility from opening doors, 00:31:29.130 --> 00:31:31.807 align:middle line:84% can't they just do that forever and get arbitrarily large-- 00:31:31.807 --> 00:31:32.640 align:middle line:90% GLENN ELLISON: Yeah. 00:31:32.640 --> 00:31:34.290 align:middle line:84% So I guess what I have to do is I have 00:31:34.290 --> 00:31:36.900 align:middle line:84% to have my shoppers-- you only get s the first time you open 00:31:36.900 --> 00:31:37.410 align:middle line:90% a door. 00:31:37.410 --> 00:31:40.620 align:middle line:90% You don't-- just-- 00:31:40.620 --> 00:31:41.670 align:middle line:90% yeah. 00:31:41.670 --> 00:31:43.420 align:middle line:84% Once a door's opened, it's already opened. 00:31:43.420 --> 00:31:47.280 align:middle line:84% You don't get new utility from opening it again, I guess. 00:31:47.280 --> 00:31:49.300 align:middle line:90% Again, it's a little-- 00:31:49.300 --> 00:31:51.540 align:middle line:84% it's mostly a construct to say that the shoppers just 00:31:51.540 --> 00:31:52.837 align:middle line:90% open every door. 00:31:52.837 --> 00:31:54.420 align:middle line:84% Sometimes, I guess the way people also 00:31:54.420 --> 00:31:57.600 align:middle line:84% think about these models is you may have people 00:31:57.600 --> 00:32:00.480 align:middle line:84% who know how to use some search engine that just gives them 00:32:00.480 --> 00:32:04.650 align:middle line:90% all the prices instant-- 00:32:04.650 --> 00:32:06.280 align:middle line:84% one search gives them all the prices, 00:32:06.280 --> 00:32:08.660 align:middle line:84% and that's where the shoppers are supposed to be. 00:32:08.660 --> 00:32:11.340 align:middle line:90% 00:32:11.340 --> 00:32:15.180 align:middle line:84% So anyway, in [14.]122, I talk about, 00:32:15.180 --> 00:32:19.150 align:middle line:84% this is a game with discontinuous-- 00:32:19.150 --> 00:32:22.270 align:middle line:84% profits do move discontinuously in price 00:32:22.270 --> 00:32:24.860 align:middle line:84% when you cross where another firm is in a search model. 00:32:24.860 --> 00:32:28.480 align:middle line:84% But Dasgupta Maskin's existence theorem 00:32:28.480 --> 00:32:32.032 align:middle line:84% for discontinuous games, the sum of the payoffs is continuous. 00:32:32.032 --> 00:32:33.490 align:middle line:84% So this is a classic example, where 00:32:33.490 --> 00:32:35.115 align:middle line:84% we know from general existence theorems 00:32:35.115 --> 00:32:36.890 align:middle line:84% that there must be some equilibrium. 00:32:36.890 --> 00:32:39.195 align:middle line:84% It's not pure, therefore it must be mixed. 00:32:39.195 --> 00:32:42.400 align:middle line:90% 00:32:42.400 --> 00:32:45.530 align:middle line:84% So we've got, some mixed equilibrium has to exist. 00:32:45.530 --> 00:32:50.530 align:middle line:84% If you think about firms mixing over some interval 00:32:50.530 --> 00:32:54.490 align:middle line:84% p lower bar, p upper bar, could p lower bar be c? 00:32:54.490 --> 00:32:57.340 align:middle line:84% p lower bar can't be c, because if p lower bar 00:32:57.340 --> 00:32:59.320 align:middle line:84% were c, firms charging p lower bar 00:32:59.320 --> 00:33:01.870 align:middle line:90% would be earning zero profits. 00:33:01.870 --> 00:33:03.820 align:middle line:84% If you're earning zero profits, again, you 00:33:03.820 --> 00:33:05.990 align:middle line:90% can go to c plus epsilon. 00:33:05.990 --> 00:33:07.720 align:middle line:84% Anyone who visits you first would 00:33:07.720 --> 00:33:09.670 align:middle line:84% have to buy at c plus epsilon, so you 00:33:09.670 --> 00:33:11.420 align:middle line:90% would get positive profits. 00:33:11.420 --> 00:33:13.810 align:middle line:84% So this equilibrium has to have a positive-- it has to be 00:33:13.810 --> 00:33:15.710 align:middle line:90% a positive profit equilibrium. 00:33:15.710 --> 00:33:21.340 align:middle line:84% So p lower bar is going to have to be strictly bigger than c. 00:33:21.340 --> 00:33:26.290 align:middle line:84% And then why is it a continuous distribution 00:33:26.290 --> 00:33:28.700 align:middle line:84% instead of a distribution that has maybe a point mass? 00:33:28.700 --> 00:33:31.690 align:middle line:84% There's a large mass of firms who charge here. 00:33:31.690 --> 00:33:34.813 align:middle line:84% Well, again, this is like an epsilon undercutting argument. 00:33:34.813 --> 00:33:37.480 align:middle line:84% If there was a large point mass of firms who all charged exactly 00:33:37.480 --> 00:33:40.790 align:middle line:90% $37, you would go to $36.99. 00:33:40.790 --> 00:33:45.310 align:middle line:84% And if in the realization of the randomizations, 00:33:45.310 --> 00:33:49.740 align:middle line:84% everyone randomized to $37, then by going to $36.99, 00:33:49.740 --> 00:33:52.240 align:middle line:84% you increase the probability that you're the low-priced firm 00:33:52.240 --> 00:33:53.690 align:middle line:90% and win the shoppers. 00:33:53.690 --> 00:33:56.050 align:middle line:84% So you can't have a point mass because that 00:33:56.050 --> 00:33:59.270 align:middle line:84% would create an incentive to epsilon undercut the point mass. 00:33:59.270 --> 00:34:01.683 align:middle line:84% So it's got to be some continuous distribution 00:34:01.683 --> 00:34:02.183 align:middle line:90% of prices. 00:34:02.183 --> 00:34:08.749 align:middle line:90% 00:34:08.749 --> 00:34:09.920 align:middle line:90% Any questions on that? 00:34:09.920 --> 00:34:14.270 align:middle line:90% 00:34:14.270 --> 00:34:18.080 align:middle line:84% So what does the density look like? 00:34:18.080 --> 00:34:21.199 align:middle line:90% The one case where it's-- 00:34:21.199 --> 00:34:24.290 align:middle line:90% the model can be hard to solve. 00:34:24.290 --> 00:34:28.040 align:middle line:84% One case where the model is easier to solve 00:34:28.040 --> 00:34:32.060 align:middle line:84% is I look at a special case where I just 00:34:32.060 --> 00:34:35.570 align:middle line:90% have a two-type version. 00:34:35.570 --> 00:34:38.550 align:middle line:84% So I have this two-type version of search costs. 00:34:38.550 --> 00:34:44.900 align:middle line:84% I have this fraction mu, have s prime less than 0, 00:34:44.900 --> 00:34:48.469 align:middle line:84% and I'm going have the fraction 1 minus mu have some number s 00:34:48.469 --> 00:34:50.120 align:middle line:90% that's greater than 0. 00:34:50.120 --> 00:34:53.300 align:middle line:84% So instead of having the heterogeneous s's 00:34:53.300 --> 00:34:55.250 align:middle line:84% on a positive interval, let me just 00:34:55.250 --> 00:34:57.800 align:middle line:90% make them all exactly the same. 00:34:57.800 --> 00:34:59.510 align:middle line:84% I think it's a worse model descriptively, 00:34:59.510 --> 00:35:02.750 align:middle line:84% but it's an easier model to solve. 00:35:02.750 --> 00:35:04.670 align:middle line:90% So what happens there? 00:35:04.670 --> 00:35:08.090 align:middle line:84% Well, one observation is, what are the profits if you 00:35:08.090 --> 00:35:10.792 align:middle line:90% charge price p lower bar? 00:35:10.792 --> 00:35:14.310 align:middle line:84% Oh, actually, sorry, first observation. 00:35:14.310 --> 00:35:15.600 align:middle line:90% Suppose I go to this. 00:35:15.600 --> 00:35:22.140 align:middle line:84% And why do I go to this two-type distribution? 00:35:22.140 --> 00:35:24.120 align:middle line:84% So I go to this two-type distribution 00:35:24.120 --> 00:35:28.080 align:middle line:84% because, when you have the two-type distribution, 00:35:28.080 --> 00:35:31.980 align:middle line:84% every nonshopper, every consumer with a positive s 00:35:31.980 --> 00:35:36.000 align:middle line:84% has to buy from the first firm that they visit. 00:35:36.000 --> 00:35:38.560 align:middle line:84% Why do they have to buy from the first firm that they visit? 00:35:38.560 --> 00:35:41.610 align:middle line:84% Well, suppose the first firm that they visit 00:35:41.610 --> 00:35:44.940 align:middle line:84% has price p upper bar and you ask, OK, 00:35:44.940 --> 00:35:48.450 align:middle line:84% I know that this is the worst price that any firm in the world 00:35:48.450 --> 00:35:50.260 align:middle line:90% offers, and I just found it. 00:35:50.260 --> 00:35:53.130 align:middle line:84% Do I buy from it or do I look for a better price? 00:35:53.130 --> 00:35:57.090 align:middle line:84% If everybody has the same s, the answer is yes, you buy anyway. 00:35:57.090 --> 00:35:58.710 align:middle line:84% Because if the answer was no, you 00:35:58.710 --> 00:36:01.680 align:middle line:84% don't buy if you find the worst possible price, 00:36:01.680 --> 00:36:05.190 align:middle line:84% then the firm who's pricing at p upper bar gets zero customers 00:36:05.190 --> 00:36:07.590 align:middle line:90% and is earning zero profits. 00:36:07.590 --> 00:36:13.770 align:middle line:84% So if we're in an equilibrium, it 00:36:13.770 --> 00:36:18.210 align:middle line:84% must be that the distribution has shrunk down 00:36:18.210 --> 00:36:23.970 align:middle line:84% to the width where if I find-- even if I find p upper bar, 00:36:23.970 --> 00:36:26.730 align:middle line:84% it must be that the consumer surplus from finding 00:36:26.730 --> 00:36:30.570 align:middle line:84% p lower bar minus the consumer surplus from finding p upper bar 00:36:30.570 --> 00:36:32.370 align:middle line:90% is at most s. 00:36:32.370 --> 00:36:37.438 align:middle line:84% Because if that wasn't true, no one would buy at p upper bar, 00:36:37.438 --> 00:36:39.730 align:middle line:84% and the p upper bar firm would be earning zero profits. 00:36:39.730 --> 00:36:41.800 align:middle line:84% So it must be the distribution just shrinks down. 00:36:41.800 --> 00:36:45.805 align:middle line:84% If s is small, maybe it shrinks down to something very tight. 00:36:45.805 --> 00:36:48.180 align:middle line:84% But it's got to be that this equilibrium distribution has 00:36:48.180 --> 00:36:50.940 align:middle line:84% to shrink in such a way that people always buy when they 00:36:50.940 --> 00:36:53.910 align:middle line:90% find the p upper bar price. 00:36:53.910 --> 00:36:56.490 align:middle line:84% And so if everyone buys when they find the p upper bar price, 00:36:56.490 --> 00:37:02.670 align:middle line:84% then we have these guys in equilibrium, these search all N. 00:37:02.670 --> 00:37:06.210 align:middle line:84% And these guys only search one firm, 00:37:06.210 --> 00:37:11.220 align:middle line:84% and they buy if p is less than or equal to p upper bar. 00:37:11.220 --> 00:37:13.710 align:middle line:84% So this makes it easier to write down the profit functions 00:37:13.710 --> 00:37:16.680 align:middle line:84% because we've gone from a complicated search problem 00:37:16.680 --> 00:37:18.790 align:middle line:84% to some consumers are buying from all N firms 00:37:18.790 --> 00:37:20.790 align:middle line:84% and some consumers are just looking at one place 00:37:20.790 --> 00:37:23.520 align:middle line:84% and buying there, as long as the price isn't above 00:37:23.520 --> 00:37:24.320 align:middle line:90% what was expected. 00:37:24.320 --> 00:37:26.910 align:middle line:90% 00:37:26.910 --> 00:37:28.590 align:middle line:84% So what do you get as profits if you 00:37:28.590 --> 00:37:30.180 align:middle line:84% buy-- if you find the p lower bar-- 00:37:30.180 --> 00:37:32.730 align:middle line:84% if you find the p lower bar-- sorry. 00:37:32.730 --> 00:37:34.780 align:middle line:84% If you're a firm, you set price p lower bar, 00:37:34.780 --> 00:37:35.920 align:middle line:90% what's your profits? 00:37:35.920 --> 00:37:39.060 align:middle line:84% It's p lower bar minus c D of p lower bar 00:37:39.060 --> 00:37:41.100 align:middle line:84% times the number of consumers you get, 00:37:41.100 --> 00:37:44.250 align:middle line:84% which is all the shoppers and a 1 over N share 00:37:44.250 --> 00:37:45.370 align:middle line:90% of the nonshoppers. 00:37:45.370 --> 00:37:47.370 align:middle line:84% Because if the nonshoppers go to any other firm, 00:37:47.370 --> 00:37:48.578 align:middle line:90% they will buy from that firm. 00:37:48.578 --> 00:37:51.518 align:middle line:90% 00:37:51.518 --> 00:37:54.600 align:middle line:84% But now in a mixed equilibrium, firms 00:37:54.600 --> 00:37:57.500 align:middle line:84% have to be indifferent over all the prices they're mixing over. 00:37:57.500 --> 00:38:02.280 align:middle line:84% So that means that at any other price, price minus cost times D 00:38:02.280 --> 00:38:05.580 align:middle line:84% of p times the number of consumers that you get, 00:38:05.580 --> 00:38:09.600 align:middle line:84% which is you get the people who visit you first 00:38:09.600 --> 00:38:14.700 align:middle line:84% and you get the shoppers, mu, if every other firm happens 00:38:14.700 --> 00:38:16.710 align:middle line:90% to choose a price above p. 00:38:16.710 --> 00:38:19.350 align:middle line:84% And that happens with probability 1 minus capital F 00:38:19.350 --> 00:38:22.500 align:middle line:90% of p to the N minus 1 power. 00:38:22.500 --> 00:38:25.710 align:middle line:84% So this is my profit if I set a price of p. 00:38:25.710 --> 00:38:28.710 align:middle line:84% This is my profit if I set a price of p lower bar. 00:38:28.710 --> 00:38:30.870 align:middle line:90% These have to be equal. 00:38:30.870 --> 00:38:35.490 align:middle line:84% And if you look at this equation, the capital F of p 00:38:35.490 --> 00:38:37.570 align:middle line:90% appears in exactly one place. 00:38:37.570 --> 00:38:38.880 align:middle line:90% So what can I say? 00:38:38.880 --> 00:38:43.440 align:middle line:84% First thing I can say is that I divide both sides by p minus c 00:38:43.440 --> 00:38:46.320 align:middle line:84% D of p, and I get mu times this equals 00:38:46.320 --> 00:38:50.070 align:middle line:90% this over p minus c D of p. 00:38:50.070 --> 00:38:54.000 align:middle line:84% And then I can subtract 1 minus mu over N from both sides. 00:38:54.000 --> 00:38:56.970 align:middle line:84% And then I can divide both sides by mu, 00:38:56.970 --> 00:38:59.460 align:middle line:84% and then I've got 1 minus f of p to the N 00:38:59.460 --> 00:39:02.170 align:middle line:84% minus 1 all alone on one side and stuff on the other. 00:39:02.170 --> 00:39:05.160 align:middle line:84% And then I take the N minus first root of both sides, 00:39:05.160 --> 00:39:07.630 align:middle line:84% and then I get 1 minus F of p equals something. 00:39:07.630 --> 00:39:09.780 align:middle line:84% And so I can solve this for F of p. 00:39:09.780 --> 00:39:11.430 align:middle line:84% So this is an equation we're just 00:39:11.430 --> 00:39:13.490 align:middle line:90% F of p appears exactly once. 00:39:13.490 --> 00:39:15.830 align:middle line:90% So I get F of p. 00:39:15.830 --> 00:39:19.430 align:middle line:84% I can always solve for what F of p must be, given any p lower bar 00:39:19.430 --> 00:39:23.340 align:middle line:84% and given any mu and all those other parameters. 00:39:23.340 --> 00:39:26.418 align:middle line:84% And then what is p upper bar going to be? 00:39:26.418 --> 00:39:27.960 align:middle line:84% p upper bar is going to have to solve 00:39:27.960 --> 00:39:30.990 align:middle line:84% this-- p minus c D of p upper bar equals 1 minus mu 00:39:30.990 --> 00:39:33.780 align:middle line:90% over N times this. 00:39:33.780 --> 00:39:37.470 align:middle line:84% And so then I can also use that to figure out p upper bar 00:39:37.470 --> 00:39:40.770 align:middle line:84% is a point where the revenue is equal to this 00:39:40.770 --> 00:39:43.590 align:middle line:90% divided by something or other. 00:39:43.590 --> 00:39:46.810 align:middle line:84% So if p lower bar is too big, that won't exist. 00:39:46.810 --> 00:39:49.350 align:middle line:84% But if p lower bar is small, that will exist. 00:39:49.350 --> 00:39:53.850 align:middle line:84% And you're going to find that I plug in a p lower bar-- 00:39:53.850 --> 00:39:57.150 align:middle line:84% say, I plug in a guess of what p lower bar is. 00:39:57.150 --> 00:39:59.040 align:middle line:84% And then given that p lower bar, I say, OK, 00:39:59.040 --> 00:40:01.560 align:middle line:84% if that's the p lower bar, this is what the distribution 00:40:01.560 --> 00:40:03.210 align:middle line:90% is going to look like. 00:40:03.210 --> 00:40:06.410 align:middle line:84% And this is what p upper bar would have to be. 00:40:06.410 --> 00:40:08.270 align:middle line:84% So for any guess of p lower bar, I 00:40:08.270 --> 00:40:11.785 align:middle line:84% can solve for what does capital F look like 00:40:11.785 --> 00:40:13.160 align:middle line:84% and what does little f look like? 00:40:13.160 --> 00:40:15.180 align:middle line:84% And where do we run out of probability? 00:40:15.180 --> 00:40:17.020 align:middle line:90% And it has to equal 1 up here. 00:40:17.020 --> 00:40:21.450 align:middle line:90% 00:40:21.450 --> 00:40:25.380 align:middle line:84% So is that thing that I solve for-- 00:40:25.380 --> 00:40:27.990 align:middle line:84% it's going to be a complicated formula, with something 00:40:27.990 --> 00:40:29.470 align:middle line:90% to the 1 over N minus 1 power. 00:40:29.470 --> 00:40:31.740 align:middle line:90% Is that an equilibrium? 00:40:31.740 --> 00:40:36.630 align:middle line:84% I can always make people indifferent over all prices 00:40:36.630 --> 00:40:38.880 align:middle line:84% in the interval p lower bar, p upper bar, 00:40:38.880 --> 00:40:42.480 align:middle line:84% but that's not sufficient to show that it's an equilibrium. 00:40:42.480 --> 00:40:44.280 align:middle line:84% Because what I have to show is that I've 00:40:44.280 --> 00:40:47.140 align:middle line:84% made the firms by construction in different overall prices. 00:40:47.140 --> 00:40:50.430 align:middle line:84% In this interval, I have to show that they don't prefer pricing 00:40:50.430 --> 00:40:52.590 align:middle line:90% lower or pricing higher. 00:40:52.590 --> 00:40:55.260 align:middle line:84% The constraint that turns out to be hard to satisfy 00:40:55.260 --> 00:40:57.765 align:middle line:84% is the price that they don't want to price higher. 00:40:57.765 --> 00:41:00.900 align:middle line:90% 00:41:00.900 --> 00:41:02.730 align:middle line:90% Why set price p upper bar? 00:41:02.730 --> 00:41:07.740 align:middle line:84% Why not go to p upper bar plus epsilon or all the way up to pm? 00:41:07.740 --> 00:41:11.403 align:middle line:84% Because, again, you know that there's zero probability 00:41:11.403 --> 00:41:14.070 align:middle line:84% you're going to get the shoppers if you're charging p upper bar. 00:41:14.070 --> 00:41:17.100 align:middle line:84% You're only going to get the people who open your door first. 00:41:17.100 --> 00:41:18.990 align:middle line:84% Why doesn't it then make sense to open your-- 00:41:18.990 --> 00:41:21.490 align:middle line:84% given that you only get the people who open your door first, 00:41:21.490 --> 00:41:23.530 align:middle line:90% why not go epsilon higher? 00:41:23.530 --> 00:41:26.510 align:middle line:84% And there could be two reasons why you don't go higher. 00:41:26.510 --> 00:41:29.860 align:middle line:84% One could be that p upper bar that you 00:41:29.860 --> 00:41:32.020 align:middle line:90% solved for was equal to pm. 00:41:32.020 --> 00:41:36.910 align:middle line:84% If the p upper bar was equal to pm, then obviously 00:41:36.910 --> 00:41:39.640 align:middle line:84% you don't want to go epsilon higher. 00:41:39.640 --> 00:41:41.230 align:middle line:84% The other thing that could happen 00:41:41.230 --> 00:41:48.130 align:middle line:84% is, when consumers see price p upper bar, the gain that they 00:41:48.130 --> 00:41:49.810 align:middle line:84% would get from searching again-- again, 00:41:49.810 --> 00:41:54.190 align:middle line:84% the gain that they would get from searching again is this. 00:41:54.190 --> 00:41:56.633 align:middle line:90% Sorry, it's not that. 00:41:56.633 --> 00:41:57.550 align:middle line:90% I will say what it is. 00:41:57.550 --> 00:42:03.820 align:middle line:84% But if consumers-- it's going to be the-- 00:42:03.820 --> 00:42:04.940 align:middle line:90% it's on the next slide. 00:42:04.940 --> 00:42:07.270 align:middle line:84% It's going to be the integral from p lower bar 00:42:07.270 --> 00:42:13.000 align:middle line:84% to p upper bar of consumer surplus of p minus consumer 00:42:13.000 --> 00:42:19.990 align:middle line:84% surplus at p upper bar F of p dp. 00:42:19.990 --> 00:42:24.250 align:middle line:90% If I go to search again-- 00:42:24.250 --> 00:42:26.860 align:middle line:84% I'll call this thing v of p upper bar. 00:42:26.860 --> 00:42:30.830 align:middle line:84% If I search again, I may find a lower price than p upper bar. 00:42:30.830 --> 00:42:34.150 align:middle line:84% And if I do, I gain more consumer surplus. 00:42:34.150 --> 00:42:37.610 align:middle line:84% And so this integral would tell me what is my gain from search. 00:42:37.610 --> 00:42:43.450 align:middle line:84% So it could be that v of p upper bar is exactly equal to s. 00:42:43.450 --> 00:42:45.790 align:middle line:84% And if v of p upper bar is equal to s, 00:42:45.790 --> 00:42:48.470 align:middle line:84% everybody who finds p upper bar would buy. 00:42:48.470 --> 00:42:50.350 align:middle line:84% But if you saw p upper bar plus epsilon, 00:42:50.350 --> 00:42:51.560 align:middle line:90% you'd be, that's ridiculous. 00:42:51.560 --> 00:42:54.520 align:middle line:84% I'm going to go take a new draw from the price distribution. 00:42:54.520 --> 00:42:56.230 align:middle line:84% So what we have to do, basically, 00:42:56.230 --> 00:43:00.590 align:middle line:84% is look for a p lower bar that makes one of two things true. 00:43:00.590 --> 00:43:03.800 align:middle line:84% Either it makes p upper bar equal to pm 00:43:03.800 --> 00:43:09.380 align:middle line:84% or it makes v of p upper bar equal to s. 00:43:09.380 --> 00:43:10.970 align:middle line:84% So anyway, to find the distribution, 00:43:10.970 --> 00:43:12.170 align:middle line:84% I just search for a p lower bar that 00:43:12.170 --> 00:43:13.510 align:middle line:90% does one of those two things. 00:43:13.510 --> 00:43:16.020 align:middle line:90% 00:43:16.020 --> 00:43:23.075 align:middle line:84% This one is actually not hard to find because, if you look at it, 00:43:23.075 --> 00:43:25.210 align:middle line:84% we're going to have this equation is true. 00:43:25.210 --> 00:43:26.960 align:middle line:84% So I could just plug in the monopoly price 00:43:26.960 --> 00:43:29.990 align:middle line:84% here and put monopoly price minus c D 00:43:29.990 --> 00:43:32.570 align:middle line:90% of pm times this equals this. 00:43:32.570 --> 00:43:34.070 align:middle line:84% And so the only thing that's unknown 00:43:34.070 --> 00:43:35.490 align:middle line:90% is the revenue at p lower bar. 00:43:35.490 --> 00:43:37.573 align:middle line:84% And so I can solve for the revenue at p lower bar. 00:43:37.573 --> 00:43:39.170 align:middle line:90% I can find p lower bar. 00:43:39.170 --> 00:43:41.253 align:middle line:84% So anyway, you find an equilibrium 00:43:41.253 --> 00:43:42.920 align:middle line:84% by looking for something that solves one 00:43:42.920 --> 00:43:44.446 align:middle line:90% of these two extra conditions. 00:43:44.446 --> 00:43:47.950 align:middle line:90% 00:43:47.950 --> 00:43:49.830 align:middle line:84% Questions on heterogeneous search models? 00:43:49.830 --> 00:43:53.800 align:middle line:90% 00:43:53.800 --> 00:43:55.480 align:middle line:90% Value of Price Search. 00:43:55.480 --> 00:43:59.890 align:middle line:84% I put this slide in because this is a "read it yourself at home." 00:43:59.890 --> 00:44:03.190 align:middle line:84% But just this is the way you normally 00:44:03.190 --> 00:44:04.960 align:middle line:84% think of the value of one price quote. 00:44:04.960 --> 00:44:06.585 align:middle line:84% The value of getting a new price quote, 00:44:06.585 --> 00:44:08.170 align:middle line:84% if you've already found a price p, 00:44:08.170 --> 00:44:10.780 align:middle line:84% is the integral from p lower bar to p, 00:44:10.780 --> 00:44:15.710 align:middle line:84% consumer surplus at x minus consumer surplus of p f of x dx. 00:44:15.710 --> 00:44:17.530 align:middle line:84% That's the natural representation 00:44:17.530 --> 00:44:20.020 align:middle line:84% of what getting one more price quote gets me. 00:44:20.020 --> 00:44:22.210 align:middle line:84% It gets me what I gain if I find the lower price 00:44:22.210 --> 00:44:26.140 align:middle line:84% quote times the density of the prices I'm drawing from. 00:44:26.140 --> 00:44:28.130 align:middle line:84% Something you see often when you're 00:44:28.130 --> 00:44:31.120 align:middle line:84% redoing this, when you're working at papers, is people 00:44:31.120 --> 00:44:33.350 align:middle line:84% often write v of p is the integral from p 00:44:33.350 --> 00:44:39.070 align:middle line:84% lower bar to p D of x F of x-- capital F of x dx. 00:44:39.070 --> 00:44:41.620 align:middle line:84% You can get from here to here by integrating by parts. 00:44:41.620 --> 00:44:43.120 align:middle line:84% You can also get from there to there 00:44:43.120 --> 00:44:45.260 align:middle line:84% by reversing the order of integration. 00:44:45.260 --> 00:44:47.917 align:middle line:84% People often write down this less-intuitive formula, 00:44:47.917 --> 00:44:49.750 align:middle line:84% just because it's well-known, without noting 00:44:49.750 --> 00:44:50.660 align:middle line:90% why they're doing it. 00:44:50.660 --> 00:44:54.610 align:middle line:84% So I put it on a slide and put demonstration 00:44:54.610 --> 00:44:56.110 align:middle line:84% of flipping the order of integration 00:44:56.110 --> 00:44:58.880 align:middle line:84% to prove those two things are equivalent. 00:44:58.880 --> 00:45:05.250 align:middle line:90% 00:45:05.250 --> 00:45:09.180 align:middle line:84% So second thing I want to do is search 00:45:09.180 --> 00:45:10.830 align:middle line:90% with differentiated products. 00:45:10.830 --> 00:45:15.060 align:middle line:84% I said that we want search costs because we think differentiation 00:45:15.060 --> 00:45:16.530 align:middle line:84% is not sufficient to explain what 00:45:16.530 --> 00:45:18.910 align:middle line:84% goes on in the world with real-world markups. 00:45:18.910 --> 00:45:21.510 align:middle line:84% But that doesn't mean differentiation is not real. 00:45:21.510 --> 00:45:26.010 align:middle line:84% And so while I think more people do search in that model 00:45:26.010 --> 00:45:27.930 align:middle line:84% like I just did, do the Bertrand search 00:45:27.930 --> 00:45:30.900 align:middle line:84% and then figure out what it does to markups, 00:45:30.900 --> 00:45:32.580 align:middle line:84% it's more realistic to have search 00:45:32.580 --> 00:45:35.070 align:middle line:84% added on top of a differentiated product model 00:45:35.070 --> 00:45:37.440 align:middle line:84% rather than searching a Bertrand model. 00:45:37.440 --> 00:45:40.050 align:middle line:84% So that's what I'm going to show for you here. 00:45:40.050 --> 00:45:44.340 align:middle line:84% So now, I'm going to have two firms 00:45:44.340 --> 00:45:48.030 align:middle line:90% with constant marginal cost c. 00:45:48.030 --> 00:45:50.700 align:middle line:84% And I'm going to do the Perloff-Salop 00:45:50.700 --> 00:45:55.860 align:middle line:84% version of Hotelling's model, where consumers 00:45:55.860 --> 00:45:57.515 align:middle line:90% have a two-dimensional type. 00:45:57.515 --> 00:46:02.470 align:middle line:90% 00:46:02.470 --> 00:46:06.350 align:middle line:90% So consumers have a-- 00:46:06.350 --> 00:46:09.380 align:middle line:90% yeah, so they have a theta-- 00:46:09.380 --> 00:46:14.540 align:middle line:84% consumer i has a theta i1, and they have a theta i2. 00:46:14.540 --> 00:46:19.220 align:middle line:84% The theta i1 is related to how much they like firm 1's product. 00:46:19.220 --> 00:46:23.450 align:middle line:84% The theta i2 is related to how much they like firm 2's product. 00:46:23.450 --> 00:46:27.380 align:middle line:84% I guess here, I did it with a v minus t, so this is your-- 00:46:27.380 --> 00:46:30.650 align:middle line:90% 00:46:30.650 --> 00:46:33.540 align:middle line:84% some people find both firms ideal. 00:46:33.540 --> 00:46:35.650 align:middle line:84% They're the ones who have theta is equal to 0. 00:46:35.650 --> 00:46:39.450 align:middle line:84% Some consumers really like firm 2, and they don't like firm 1. 00:46:39.450 --> 00:46:40.550 align:middle line:90% They're down here. 00:46:40.550 --> 00:46:43.260 align:middle line:84% Some consumers really like firm 1, don't like firm 2. 00:46:43.260 --> 00:46:44.250 align:middle line:90% They're up here. 00:46:44.250 --> 00:46:48.260 align:middle line:84% Some consumers really find both products far from their tastes. 00:46:48.260 --> 00:46:51.500 align:middle line:84% So it's the Perloff-Salop version of Hotelling, 00:46:51.500 --> 00:46:53.180 align:middle line:90% where I just have-- 00:46:53.180 --> 00:46:55.970 align:middle line:84% instead of having the theta i2 equals negative theta 00:46:55.970 --> 00:47:02.690 align:middle line:84% i1 or t minus theta i1, I make independent disutilities-- 00:47:02.690 --> 00:47:08.500 align:middle line:84% independent tastes for product 1 and product 2. 00:47:08.500 --> 00:47:12.880 align:middle line:84% So the firms simultaneously choose their prices, p1, p2. 00:47:12.880 --> 00:47:18.490 align:middle line:84% And what the consumers here have to do now is, when they pay s, 00:47:18.490 --> 00:47:21.490 align:middle line:84% they both learn the price and they 00:47:21.490 --> 00:47:25.120 align:middle line:84% learn whether they like the product or not. 00:47:25.120 --> 00:47:29.590 align:middle line:84% So this is the case where you know there are two firms-- 00:47:29.590 --> 00:47:32.650 align:middle line:84% you know there are two firms that are selling some product. 00:47:32.650 --> 00:47:35.320 align:middle line:84% But you have to actually go to the store 00:47:35.320 --> 00:47:38.740 align:middle line:84% and look at the product, at their version of the product, 00:47:38.740 --> 00:47:41.523 align:middle line:84% to decide whether you like it or not. 00:47:41.523 --> 00:47:42.940 align:middle line:84% Or you could think of it as, like, 00:47:42.940 --> 00:47:44.560 align:middle line:90% in the credit card example. 00:47:44.560 --> 00:47:45.940 align:middle line:90% I pay s. 00:47:45.940 --> 00:47:47.845 align:middle line:90% When I pay s, I learn-- 00:47:47.845 --> 00:47:51.010 align:middle line:90% 00:47:51.010 --> 00:47:52.810 align:middle line:84% I learn what the interest rate is. 00:47:52.810 --> 00:47:55.270 align:middle line:84% But then I also learn other features of the credit cards 00:47:55.270 --> 00:47:56.075 align:middle line:90% that are not-- 00:47:56.075 --> 00:47:58.450 align:middle line:84% which people have horizontally differentiated preference. 00:47:58.450 --> 00:48:00.370 align:middle line:84% Like, I do learn what color it is, 00:48:00.370 --> 00:48:04.300 align:middle line:84% and I do learn what their prepayment policy is 00:48:04.300 --> 00:48:06.610 align:middle line:84% and what day of the week they want their things due. 00:48:06.610 --> 00:48:10.600 align:middle line:84% Or I go to the car dealership, and I 00:48:10.600 --> 00:48:14.010 align:middle line:84% do learn how hard it is to drive that car. 00:48:14.010 --> 00:48:16.810 align:middle line:84% I do learn how clean is the service 00:48:16.810 --> 00:48:19.300 align:middle line:84% department at that store and other things 00:48:19.300 --> 00:48:20.270 align:middle line:90% or something like that. 00:48:20.270 --> 00:48:26.050 align:middle line:84% So I pay my s, I learn the price, 00:48:26.050 --> 00:48:27.860 align:middle line:84% and I learn whether I like it or not. 00:48:27.860 --> 00:48:30.640 align:middle line:84% And until I go there, my prior would be 00:48:30.640 --> 00:48:32.255 align:middle line:90% that my theta is uniform 0, 1. 00:48:32.255 --> 00:48:34.130 align:middle line:84% I know that this might be a good match to me. 00:48:34.130 --> 00:48:36.280 align:middle line:84% It might not be a good match to me, but I don't know until I-- 00:48:36.280 --> 00:48:38.030 align:middle line:84% I have to inspect the product to see that. 00:48:38.030 --> 00:48:41.190 align:middle line:90% 00:48:41.190 --> 00:48:44.500 align:middle line:84% Assume that the search cost is less than t over 2. 00:48:44.500 --> 00:48:46.170 align:middle line:84% And again, assume that consumers use 00:48:46.170 --> 00:48:49.470 align:middle line:84% some optimal sequential search procedure, which 00:48:49.470 --> 00:48:53.430 align:middle line:84% is they go visit one firm, they learn the theta, they then 00:48:53.430 --> 00:48:56.280 align:middle line:84% decide, given what I found about the price and the match quality 00:48:56.280 --> 00:48:58.728 align:middle line:84% to my tastes, do I go search for a second time, 00:48:58.728 --> 00:48:59.645 align:middle line:90% and then they're done. 00:48:59.645 --> 00:49:02.340 align:middle line:90% 00:49:02.340 --> 00:49:05.760 align:middle line:84% The theorem is, for large enough v, this model 00:49:05.760 --> 00:49:08.850 align:middle line:84% has a symmetric pure strategy Nash equilibrium. 00:49:08.850 --> 00:49:12.150 align:middle line:84% And that symmetric Nash equilibrium is price equals c 00:49:12.150 --> 00:49:17.430 align:middle line:84% plus 1 over 2 minus square root of 2s over t times t. 00:49:17.430 --> 00:49:21.270 align:middle line:84% So it's a c plus something times t markup. 00:49:21.270 --> 00:49:23.260 align:middle line:84% And it's a pure strategy equilibrium. 00:49:23.260 --> 00:49:27.850 align:middle line:84% So the Stahl-like dispersion has gone away. 00:49:27.850 --> 00:49:31.570 align:middle line:84% And here it's just search cost effect markups. 00:49:31.570 --> 00:49:37.050 align:middle line:84% So notice that the Perloff-Salop model with this game would have 00:49:37.050 --> 00:49:39.340 align:middle line:90% p star equals c plus 1/2 t. 00:49:39.340 --> 00:49:42.880 align:middle line:90% 00:49:42.880 --> 00:49:50.080 align:middle line:84% Here, if you set s equals 0, you get c plus 1/2 t. 00:49:50.080 --> 00:49:54.070 align:middle line:84% So when s goes to 0, this goes to Perloff-Salop. 00:49:54.070 --> 00:49:58.100 align:middle line:84% But when s is larger, like, imagine s is t over 2-- 00:49:58.100 --> 00:50:03.320 align:middle line:84% when s is t over 2, you get 2 times t over 2 is t over t. 00:50:03.320 --> 00:50:05.680 align:middle line:90% So it goes to c plus t. 00:50:05.680 --> 00:50:11.740 align:middle line:84% So price increases smoothly from c plus t over 2 to c 00:50:11.740 --> 00:50:15.400 align:middle line:90% plus t as search costs increase. 00:50:15.400 --> 00:50:20.290 align:middle line:84% This model has the "falling off a cliff" property that, 00:50:20.290 --> 00:50:22.840 align:middle line:84% if you make s-- so as s goes from-- 00:50:22.840 --> 00:50:25.860 align:middle line:90% 00:50:25.860 --> 00:50:33.850 align:middle line:84% if I graph a search cost, when s is 0, price is c plus t over 2. 00:50:33.850 --> 00:50:43.580 align:middle line:84% As s goes up from 0 to t over 2, it's just a 45-degree line. 00:50:43.580 --> 00:50:47.590 align:middle line:84% Prices increase and they reach c plus t. 00:50:47.590 --> 00:50:50.710 align:middle line:84% Once I pass this search cost, we fall off a cliff 00:50:50.710 --> 00:50:53.860 align:middle line:84% and the search disappears and no one ever buys. 00:50:53.860 --> 00:50:59.080 align:middle line:84% And the reason is because, when s is large enough, 00:50:59.080 --> 00:51:02.170 align:middle line:84% no one is going to get a second quote. 00:51:02.170 --> 00:51:04.180 align:middle line:84% So even if you find the first product 00:51:04.180 --> 00:51:07.030 align:middle line:84% and it's the worst possible product for you, and you know, 00:51:07.030 --> 00:51:10.630 align:middle line:84% OK, I hate firm 1's product, the most 00:51:10.630 --> 00:51:12.610 align:middle line:84% you're going to gain from getting firm 2's 00:51:12.610 --> 00:51:15.220 align:middle line:84% product in expectation is t over 2 00:51:15.220 --> 00:51:17.770 align:middle line:84% because firm 2 is disutility-- firm 2's product might 00:51:17.770 --> 00:51:18.940 align:middle line:90% be just as bad. 00:51:18.940 --> 00:51:20.560 align:middle line:90% Firm 2's product might be great. 00:51:20.560 --> 00:51:23.198 align:middle line:84% But you're only going to gain t over 2 on average. 00:51:23.198 --> 00:51:25.240 align:middle line:84% If you're only going to gain t over 2 on average, 00:51:25.240 --> 00:51:28.580 align:middle line:84% you won't do a second search if your s is bigger than t over 2. 00:51:28.580 --> 00:51:33.700 align:middle line:84% And so then once people have an s bigger than that, 00:51:33.700 --> 00:51:34.880 align:middle line:90% we're back to Diamond. 00:51:34.880 --> 00:51:36.250 align:middle line:90% Everyone only searches once. 00:51:36.250 --> 00:51:38.000 align:middle line:90% We go to the monopoly price. 00:51:38.000 --> 00:51:40.180 align:middle line:84% But when we get to the monopoly price 00:51:40.180 --> 00:51:43.900 align:middle line:84% without the multiunit demands, at the monopoly price, 00:51:43.900 --> 00:51:46.450 align:middle line:84% you don't want to be-- there's no point in buying 00:51:46.450 --> 00:51:49.150 align:middle line:84% at the monopoly price because the monopoly price is at least v 00:51:49.150 --> 00:51:49.780 align:middle line:90% minus t. 00:51:49.780 --> 00:51:52.400 align:middle line:84% Your expected utility's, at most, t over 2. 00:51:52.400 --> 00:51:54.100 align:middle line:84% In this model, at the monopoly price, 00:51:54.100 --> 00:51:57.667 align:middle line:84% you wouldn't bother searching to begin with. 00:51:57.667 --> 00:51:59.500 align:middle line:84% Because consumer surplus, the monopoly price 00:51:59.500 --> 00:52:01.605 align:middle line:84% is very low in this single-unit demand model. 00:52:01.605 --> 00:52:06.010 align:middle line:90% 00:52:06.010 --> 00:52:11.090 align:middle line:84% So intuition for why is it that equilibrium prices increase 00:52:11.090 --> 00:52:14.030 align:middle line:90% in search costs? 00:52:14.030 --> 00:52:16.120 align:middle line:84% If a firm is considering a price increase, 00:52:16.120 --> 00:52:17.870 align:middle line:84% they have to trade off the gains that they 00:52:17.870 --> 00:52:20.370 align:middle line:84% get from these inframarginal consumers who would have bought 00:52:20.370 --> 00:52:24.920 align:middle line:84% anyway against the losses from raising the price 00:52:24.920 --> 00:52:28.450 align:middle line:84% and having the marginal consumers go away. 00:52:28.450 --> 00:52:32.500 align:middle line:90% 00:52:32.500 --> 00:52:39.550 align:middle line:84% Here, as s gets bigger, there are fewer marginal consumers. 00:52:39.550 --> 00:52:42.970 align:middle line:84% When s is small, most consumers are going to visit both-- 00:52:42.970 --> 00:52:45.520 align:middle line:84% when s is small, I'm going to have 00:52:45.520 --> 00:52:48.910 align:middle line:84% some region like this, like, these consumers 00:52:48.910 --> 00:52:52.900 align:middle line:90% buy and these search again. 00:52:52.900 --> 00:52:55.090 align:middle line:84% Because if I'm firm 1, for someone 00:52:55.090 --> 00:52:57.520 align:middle line:84% to buy from me when s is tiny, why not 00:52:57.520 --> 00:52:58.562 align:middle line:90% get a second price quote? 00:52:58.562 --> 00:53:00.978 align:middle line:84% It's only if they really like me that they'll stop and buy 00:53:00.978 --> 00:53:01.600 align:middle line:90% from me. 00:53:01.600 --> 00:53:04.780 align:middle line:84% When s gets big, this line shifts out, 00:53:04.780 --> 00:53:07.100 align:middle line:84% and it becomes these guys all buy. 00:53:07.100 --> 00:53:09.010 align:middle line:84% And it's only if you have a really bad draw 00:53:09.010 --> 00:53:11.230 align:middle line:90% that you search again. 00:53:11.230 --> 00:53:17.890 align:middle line:84% And so what happens is just, when s gets big, 00:53:17.890 --> 00:53:21.790 align:middle line:84% there are fewer and fewer consumers who see both prices. 00:53:21.790 --> 00:53:24.620 align:middle line:84% Because there are fewer consumers who see both prices, 00:53:24.620 --> 00:53:28.490 align:middle line:84% there's a smaller gain to an epsilon price cut. 00:53:28.490 --> 00:53:30.820 align:middle line:84% And so your incentive to cut price 00:53:30.820 --> 00:53:32.870 align:middle line:84% is smaller because there are fewer consumers who 00:53:32.870 --> 00:53:34.840 align:middle line:90% actually see both prices. 00:53:34.840 --> 00:53:36.533 align:middle line:84% Whereas, you know in equilibrium, 00:53:36.533 --> 00:53:37.950 align:middle line:84% you're getting half the consumers. 00:53:37.950 --> 00:53:40.325 align:middle line:84% And on those half the consumers, if you raise your price, 00:53:40.325 --> 00:53:42.560 align:middle line:84% you're going to get more on all-- 00:53:42.560 --> 00:53:44.280 align:middle line:90% on roughly half of the people. 00:53:44.280 --> 00:53:47.162 align:middle line:84% So the number of marginal consumers goes down. 00:53:47.162 --> 00:53:49.370 align:middle line:84% The number of consumers you're getting in equilibrium 00:53:49.370 --> 00:53:50.580 align:middle line:90% stays the same. 00:53:50.580 --> 00:53:53.570 align:middle line:84% Therefore, your profit on the marginal consumers 00:53:53.570 --> 00:53:56.450 align:middle line:84% must be going up to make you not want to lose them. 00:53:56.450 --> 00:53:59.540 align:middle line:84% And so that's an intuition for why the markups have 00:53:59.540 --> 00:54:00.837 align:middle line:90% to go up because you-- 00:54:00.837 --> 00:54:02.420 align:middle line:84% the markups have to go up to the point 00:54:02.420 --> 00:54:03.980 align:middle line:84% where you don't want to lose that dwindling number 00:54:03.980 --> 00:54:05.146 align:middle line:90% of marginal consumers. 00:54:05.146 --> 00:54:07.836 align:middle line:90% 00:54:07.836 --> 00:54:09.205 align:middle line:90% Any questions on that? 00:54:09.205 --> 00:54:12.820 align:middle line:90% 00:54:12.820 --> 00:54:15.183 align:middle line:90% I'm trying to think. 00:54:15.183 --> 00:54:16.850 align:middle line:84% Why don't I go over this in a little bit 00:54:16.850 --> 00:54:19.460 align:middle line:84% of detail, just how the calculation in a model like this 00:54:19.460 --> 00:54:20.130 align:middle line:90% works. 00:54:20.130 --> 00:54:25.070 align:middle line:84% So imagine in this model you've got a consumer. 00:54:25.070 --> 00:54:27.170 align:middle line:84% I switched from the theta dimension 00:54:27.170 --> 00:54:29.100 align:middle line:90% to the match quality dimension. 00:54:29.100 --> 00:54:33.950 align:middle line:84% So imagine that a consumer visits one store. 00:54:33.950 --> 00:54:38.000 align:middle line:84% They find a match quality of x, by which I mean 1 minus theta i, 00:54:38.000 --> 00:54:42.620 align:middle line:84% and a price of p star, which is the price they expected to get. 00:54:42.620 --> 00:54:45.290 align:middle line:84% What's your expected gain from search? 00:54:45.290 --> 00:54:47.540 align:middle line:84% In equilibrium, you expect that the other firm is also 00:54:47.540 --> 00:54:48.980 align:middle line:90% charging p star. 00:54:48.980 --> 00:54:52.250 align:middle line:84% So the gain comes from getting a better match quality 00:54:52.250 --> 00:54:57.410 align:middle line:84% at the second firm, which is just an integral from x up to 1 00:54:57.410 --> 00:55:01.130 align:middle line:90% of u minus x du. 00:55:01.130 --> 00:55:07.250 align:middle line:84% So I might find a firm that's essentially no better. 00:55:07.250 --> 00:55:10.780 align:middle line:84% I might find a firm that's 1 minus x better. 00:55:10.780 --> 00:55:13.960 align:middle line:84% And I'm equally likely to find a firm that's any-- well, 00:55:13.960 --> 00:55:17.420 align:middle line:84% I may find a firm that's worse than what I found already. 00:55:17.420 --> 00:55:20.020 align:middle line:84% But if I found a firm that's worse, I can just ignore it. 00:55:20.020 --> 00:55:22.390 align:middle line:84% If I find something that's the same or better, 00:55:22.390 --> 00:55:23.600 align:middle line:90% I can buy from it. 00:55:23.600 --> 00:55:25.780 align:middle line:84% And so there's a 1 minus x chance 00:55:25.780 --> 00:55:27.760 align:middle line:90% that I'll find something better. 00:55:27.760 --> 00:55:30.040 align:middle line:84% The most by which I might increase my utility 00:55:30.040 --> 00:55:32.080 align:middle line:90% is 1 minus x times t. 00:55:32.080 --> 00:55:35.260 align:middle line:84% And so I get this integral, which is just-- 00:55:35.260 --> 00:55:38.880 align:middle line:84% it's the area of a triangle, 1/2 t 1 minus x squared. 00:55:38.880 --> 00:55:42.100 align:middle line:84% So that's my gain from search is the gain from improved match 00:55:42.100 --> 00:55:44.990 align:middle line:90% quality. 00:55:44.990 --> 00:55:47.090 align:middle line:84% When are you going to do a second search? 00:55:47.090 --> 00:55:49.400 align:middle line:84% Well, you're going to do the second search if-- 00:55:49.400 --> 00:55:51.860 align:middle line:84% suppose I find a match quality of x, 00:55:51.860 --> 00:55:55.530 align:middle line:84% and I actually find a price that's different from p1 star. 00:55:55.530 --> 00:55:59.810 align:middle line:84% So if the firm is charging even a higher price, 00:55:59.810 --> 00:56:01.647 align:middle line:84% it's like the match quality is worse. 00:56:01.647 --> 00:56:03.230 align:middle line:84% If the firm is charging a lower price, 00:56:03.230 --> 00:56:05.400 align:middle line:84% it's like the match quality that I found is better. 00:56:05.400 --> 00:56:06.780 align:middle line:90% And so I'm going to buy-- 00:56:06.780 --> 00:56:11.420 align:middle line:84% I got to search again if v of x plus p star minus p1 00:56:11.420 --> 00:56:15.755 align:middle line:84% is less than or equal to my search cost. 00:56:15.755 --> 00:56:19.440 align:middle line:84% In equilibrium, p1 equals p star. 00:56:19.440 --> 00:56:23.510 align:middle line:84% So this holds if 1/2 t 1 minus x star is less than s, 00:56:23.510 --> 00:56:27.590 align:middle line:84% which is if and only if x is at least 1 minus square root of 2s 00:56:27.590 --> 00:56:28.790 align:middle line:90% over t. 00:56:28.790 --> 00:56:33.387 align:middle line:84% I just solve this thing for x, moving the 2 divided 00:56:33.387 --> 00:56:35.720 align:middle line:84% by t over to the other side and then taking square roots 00:56:35.720 --> 00:56:39.312 align:middle line:90% and then doing 1 minus it. 00:56:39.312 --> 00:56:42.910 align:middle line:84% So the Nash equilibrium is always the solution 00:56:42.910 --> 00:56:45.790 align:middle line:84% to partial pi 1 with respect to p1, 00:56:45.790 --> 00:56:50.440 align:middle line:84% evaluated at both firms charging p star equals 0. 00:56:50.440 --> 00:56:54.190 align:middle line:84% And as I said, often, if you're looking 00:56:54.190 --> 00:56:57.058 align:middle line:84% for symmetric equilibria, rather than writing out best responses 00:56:57.058 --> 00:56:59.350 align:middle line:84% and intersecting them and solving two equations and two 00:56:59.350 --> 00:57:01.225 align:middle line:84% unknowns, you just take advantage of the fact 00:57:01.225 --> 00:57:03.130 align:middle line:84% that you know it's symmetric and just do, 00:57:03.130 --> 00:57:05.140 align:middle line:84% this first-order condition characterizes 00:57:05.140 --> 00:57:08.500 align:middle line:84% what the symmetric equilibrium p star must be. 00:57:08.500 --> 00:57:11.980 align:middle line:84% And so let's think about, well, if p star is in equilibrium, 00:57:11.980 --> 00:57:15.970 align:middle line:84% what happens if firm 1 cuts its price to p star minus dp? 00:57:15.970 --> 00:57:18.730 align:middle line:84% And let's think about, what are the gains and losses when 00:57:18.730 --> 00:57:21.280 align:middle line:90% I'm comparing profit at-- 00:57:21.280 --> 00:57:25.750 align:middle line:84% cutting my price to profit from keeping my price the same? 00:57:25.750 --> 00:57:29.650 align:middle line:84% And as I said, one loss that I get from a price cut 00:57:29.650 --> 00:57:31.720 align:middle line:84% is, I know in equilibrium, half the consumers 00:57:31.720 --> 00:57:32.870 align:middle line:90% are buying from me. 00:57:32.870 --> 00:57:36.190 align:middle line:84% So if I cut my price by dp, I'm giving dp away 00:57:36.190 --> 00:57:37.220 align:middle line:90% to all those people. 00:57:37.220 --> 00:57:41.060 align:middle line:84% That's a loss of 1/2 dp, because I'm getting half the consumers 00:57:41.060 --> 00:57:42.560 align:middle line:90% in equilibrium. 00:57:42.560 --> 00:57:46.280 align:middle line:90% Again, exploiting symmetry. 00:57:46.280 --> 00:57:48.170 align:middle line:90% What do I gain? 00:57:48.170 --> 00:57:48.710 align:middle line:90% OK. 00:57:48.710 --> 00:57:51.800 align:middle line:84% So I've got three different gains. 00:57:51.800 --> 00:57:57.240 align:middle line:84% First gain is, suppose that a consumer visits me first. 00:57:57.240 --> 00:58:01.760 align:middle line:84% That's half the consumers visit me first. 00:58:01.760 --> 00:58:04.540 align:middle line:84% When consumers visit me first in x space-- 00:58:04.540 --> 00:58:08.450 align:middle line:90% 00:58:08.450 --> 00:58:11.840 align:middle line:90% in x space, it looks like this. 00:58:11.840 --> 00:58:14.300 align:middle line:90% There's some x star. 00:58:14.300 --> 00:58:18.860 align:middle line:84% And here they buy from me, and here they search again. 00:58:18.860 --> 00:58:21.210 align:middle line:84% If my match quality is good, they buy. 00:58:21.210 --> 00:58:23.990 align:middle line:84% If my match quality is bad, they search again, 00:58:23.990 --> 00:58:27.530 align:middle line:90% where x is 1 minus theta. 00:58:27.530 --> 00:58:28.830 align:middle line:90% So match quality x. 00:58:28.830 --> 00:58:30.080 align:middle line:90% High match quality is you buy. 00:58:30.080 --> 00:58:32.000 align:middle line:84% Low match quality is you search again. 00:58:32.000 --> 00:58:34.460 align:middle line:84% So where I gain from the price cut 00:58:34.460 --> 00:58:39.270 align:middle line:84% is they visit me first with probability x star. 00:58:39.270 --> 00:58:42.140 align:middle line:84% They didn't like my product all that much. 00:58:42.140 --> 00:58:45.470 align:middle line:90% So they continue to search. 00:58:45.470 --> 00:58:48.020 align:middle line:84% But then it turns out that the amount 00:58:48.020 --> 00:58:50.210 align:middle line:84% that they like the other firm's product 00:58:50.210 --> 00:58:58.400 align:middle line:84% is just ever so slightly greater than x star and so close 00:58:58.400 --> 00:59:01.160 align:middle line:84% to x star that my dp price cut convinces them 00:59:01.160 --> 00:59:03.080 align:middle line:90% to buy from me instead of him. 00:59:03.080 --> 00:59:07.890 align:middle line:84% And that's going to happen with probability dp divided by t. 00:59:07.890 --> 00:59:09.890 align:middle line:84% So it's going to take this coincidence of things 00:59:09.890 --> 00:59:12.968 align:middle line:84% that they visit me, they then visit him. 00:59:12.968 --> 00:59:14.760 align:middle line:84% And then they like him just about the same, 00:59:14.760 --> 00:59:17.540 align:middle line:84% so my dp price cut makes a difference. 00:59:17.540 --> 00:59:22.332 align:middle line:84% There's also consumers who visit-- 00:59:22.332 --> 00:59:24.300 align:middle line:84% oh, sorry, I did that one backwards. 00:59:24.300 --> 00:59:27.230 align:middle line:84% So there are also consumers who visit the other firm first, 00:59:27.230 --> 00:59:29.030 align:middle line:84% don't like the other firm's product, 00:59:29.030 --> 00:59:31.530 align:middle line:84% visit you, like your product again, 00:59:31.530 --> 00:59:34.350 align:middle line:84% almost exactly as much as they like the other firm's product, 00:59:34.350 --> 00:59:36.840 align:middle line:84% and then the dp price cut makes a difference. 00:59:36.840 --> 00:59:42.980 align:middle line:84% So we get these two terms that are both 1/2 x star 1 over t dp. 00:59:42.980 --> 00:59:46.820 align:middle line:84% The third effect is what we'll call the search deterrence 00:59:46.820 --> 00:59:47.790 align:middle line:90% effect. 00:59:47.790 --> 00:59:49.550 align:middle line:84% The search deterrence effect only 00:59:49.550 --> 00:59:53.270 align:middle line:84% comes from consumers who visit me first. 00:59:53.270 --> 00:59:58.020 align:middle line:84% And they're consumers who visit me first and would have bought 00:59:58.020 --> 01:00:01.350 align:middle line:84% from firm 2 if they went and visited firm 2 01:00:01.350 --> 01:00:06.180 align:middle line:84% but who I convince not to visit firm 2 with my price cut. 01:00:06.180 --> 01:00:10.290 align:middle line:84% And so when I cut my price by dp, 01:00:10.290 --> 01:00:14.740 align:middle line:84% it's like I've made the match slightly better by some amount. 01:00:14.740 --> 01:00:19.110 align:middle line:84% It's like I increased the match quality to x star plus dp 01:00:19.110 --> 01:00:21.930 align:middle line:90% over t. 01:00:21.930 --> 01:00:24.780 align:middle line:84% So I've made my match slightly better. 01:00:24.780 --> 01:00:29.220 align:middle line:84% I've shifted the-- oops, I've made my price slightly better. 01:00:29.220 --> 01:00:37.560 align:middle line:84% So by making my price slightly better, 01:00:37.560 --> 01:00:40.240 align:middle line:90% I'm shifting this region back. 01:00:40.240 --> 01:00:46.280 align:middle line:84% So if the-- yeah, if my price is-- 01:00:46.280 --> 01:00:50.180 align:middle line:84% if my price is lower, the match quality in the-- 01:00:50.180 --> 01:00:55.520 align:middle line:84% quality space, they need to-- it's, like, x star shift back. 01:00:55.520 --> 01:00:57.295 align:middle line:84% Because if my price is better, they'll 01:00:57.295 --> 01:00:59.420 align:middle line:84% take it even if it's a slightly worse match quality 01:00:59.420 --> 01:01:01.250 align:middle line:90% in characteristic space. 01:01:01.250 --> 01:01:07.350 align:middle line:84% And so I shift that line to the left by this distance. 01:01:07.350 --> 01:01:09.825 align:middle line:84% And then if they would have bought from me anyway, 01:01:09.825 --> 01:01:11.700 align:middle line:84% it doesn't matter that I deterred the search. 01:01:11.700 --> 01:01:14.060 align:middle line:84% But if they would have bought from the other guy, 01:01:14.060 --> 01:01:15.390 align:middle line:90% then it is a gain. 01:01:15.390 --> 01:01:16.980 align:middle line:90% And so I get these two effects. 01:01:16.980 --> 01:01:20.420 align:middle line:84% One is the standard Bertrand effect-- 01:01:20.420 --> 01:01:23.333 align:middle line:84% or the standard Hotelling effect of the people 01:01:23.333 --> 01:01:25.250 align:middle line:84% who see both prices and they're close together 01:01:25.250 --> 01:01:26.370 align:middle line:90% by the epsilon price cut. 01:01:26.370 --> 01:01:27.680 align:middle line:90% I get those consumers. 01:01:27.680 --> 01:01:29.540 align:middle line:84% And then I have this extra effect 01:01:29.540 --> 01:01:32.400 align:middle line:90% of search deterrence effect. 01:01:32.400 --> 01:01:34.220 align:middle line:84% That is, by cutting my price, I keep people 01:01:34.220 --> 01:01:35.940 align:middle line:90% from searching the second firm. 01:01:35.940 --> 01:01:40.520 align:middle line:84% And so it must be that these three gains exactly 01:01:40.520 --> 01:01:43.537 align:middle line:90% offset this one loss. 01:01:43.537 --> 01:01:45.870 align:middle line:84% Anyway, and so what you do is, the first-order condition 01:01:45.870 --> 01:01:48.600 align:middle line:84% sets the gain equal to the loss, you solve, 01:01:48.600 --> 01:01:51.720 align:middle line:90% and it gives you that formula. 01:01:51.720 --> 01:01:54.570 align:middle line:90% So that's the mechanics-- 01:01:54.570 --> 01:01:56.590 align:middle line:84% that's the mechanics of finding an equilibrium. 01:01:56.590 --> 01:01:58.140 align:middle line:84% And I guess the key economic point 01:01:58.140 --> 01:02:01.800 align:middle line:84% is just there are those two tradeoffs-- the price comparison 01:02:01.800 --> 01:02:03.665 align:middle line:84% effect and the search deterrence effect. 01:02:03.665 --> 01:02:11.450 align:middle line:90% 01:02:11.450 --> 01:02:13.020 align:middle line:90% Fixed Sample Search. 01:02:13.020 --> 01:02:14.522 align:middle line:84% So anyway, any questions on search 01:02:14.522 --> 01:02:15.730 align:middle line:90% with differentiated products? 01:02:15.730 --> 01:02:18.500 align:middle line:90% 01:02:18.500 --> 01:02:23.110 align:middle line:84% So fixed sample search, in some applications, 01:02:23.110 --> 01:02:26.567 align:middle line:84% the model we've been doing so far has this idea of, you go out 01:02:26.567 --> 01:02:27.400 align:middle line:90% and you do a search. 01:02:27.400 --> 01:02:29.567 align:middle line:84% You see what you find, whether you like the product. 01:02:29.567 --> 01:02:32.960 align:middle line:84% You decide whether to stop here and buy or go do another search. 01:02:32.960 --> 01:02:36.640 align:middle line:84% There are some applications, like getting contractors 01:02:36.640 --> 01:02:39.580 align:middle line:84% to renovate your bathroom, that you can't do 01:02:39.580 --> 01:02:41.050 align:middle line:90% this sequential search process. 01:02:41.050 --> 01:02:45.355 align:middle line:84% It takes a long time to get a contractor to submit a bid. 01:02:45.355 --> 01:02:46.480 align:middle line:90% You call them on the phone. 01:02:46.480 --> 01:02:48.100 align:middle line:84% It takes the guy forever to come. 01:02:48.100 --> 01:02:49.520 align:middle line:90% He looks at your bathroom. 01:02:49.520 --> 01:02:50.900 align:middle line:90% He says, I'll get back to you. 01:02:50.900 --> 01:02:52.462 align:middle line:90% He doesn't get back to you. 01:02:52.462 --> 01:02:53.920 align:middle line:84% A month later, he finally works out 01:02:53.920 --> 01:02:55.490 align:middle line:90% what he's going to charge you. 01:02:55.490 --> 01:02:58.240 align:middle line:84% If you do that sequentially, you know your bathroom will never 01:02:58.240 --> 01:02:59.260 align:middle line:90% get renovated. 01:02:59.260 --> 01:03:02.590 align:middle line:84% So what you do in practice-- and as governments-- 01:03:02.590 --> 01:03:03.590 align:middle line:90% we do this all the time. 01:03:03.590 --> 01:03:05.882 align:middle line:84% We're renovating new space in the economics department. 01:03:05.882 --> 01:03:08.810 align:middle line:84% You try to get two or three contractors to bid. 01:03:08.810 --> 01:03:12.180 align:middle line:84% You say, OK, we're getting three bids. 01:03:12.180 --> 01:03:14.053 align:middle line:84% You may put a bid deadline on them. 01:03:14.053 --> 01:03:16.220 align:middle line:84% You may tell the guys, look, I'm getting three bids. 01:03:16.220 --> 01:03:17.975 align:middle line:84% I want all the bids in by December 1. 01:03:17.975 --> 01:03:19.850 align:middle line:84% If you don't get your thing in by December 1, 01:03:19.850 --> 01:03:21.570 align:middle line:90% I'm not going to consider you. 01:03:21.570 --> 01:03:24.380 align:middle line:84% By December 1, you get three bids and then you compare them 01:03:24.380 --> 01:03:25.470 align:middle line:90% and then you're done. 01:03:25.470 --> 01:03:28.610 align:middle line:84% But you're deciding on how many bids 01:03:28.610 --> 01:03:31.527 align:middle line:84% you're going to get before you see what the bids are. 01:03:31.527 --> 01:03:33.110 align:middle line:84% And that's the difference between what 01:03:33.110 --> 01:03:36.330 align:middle line:84% we call the fixed sample search model versus a sequential model. 01:03:36.330 --> 01:03:38.900 align:middle line:84% Sequential model, you treat them one at a time 01:03:38.900 --> 01:03:40.460 align:middle line:90% and decide whether to go on. 01:03:40.460 --> 01:03:46.030 align:middle line:84% The fixed sample one, you decide how many to get. 01:03:46.030 --> 01:03:50.950 align:middle line:84% And it turns out that these models work somewhat similarly. 01:03:50.950 --> 01:03:53.440 align:middle line:84% For a while, this fixed sample model 01:03:53.440 --> 01:03:55.930 align:middle line:84% was more popular in structural empirical work 01:03:55.930 --> 01:04:00.616 align:middle line:84% because the equations turn out to be a little bit-- 01:04:00.616 --> 01:04:02.800 align:middle line:84% if you think about it, in the sequential model, 01:04:02.800 --> 01:04:06.220 align:middle line:84% the consumer is doing this dynamic programming problem. 01:04:06.220 --> 01:04:07.960 align:middle line:84% In the fixed sample model, the consumer 01:04:07.960 --> 01:04:10.160 align:middle line:90% is choosing a finite integer. 01:04:10.160 --> 01:04:12.100 align:middle line:84% And so the choosing a finite integer 01:04:12.100 --> 01:04:14.860 align:middle line:84% is a much simpler strategy space for the consumer 01:04:14.860 --> 01:04:17.775 align:middle line:84% than this optimal dynamic programming thing. 01:04:17.775 --> 01:04:19.150 align:middle line:84% And so therefore, there are a lot 01:04:19.150 --> 01:04:21.793 align:middle line:84% of models that use the fixed sample search. 01:04:21.793 --> 01:04:22.960 align:middle line:90% Sometimes, it's appropriate. 01:04:22.960 --> 01:04:25.210 align:middle line:84% Sometimes, you think sequential seems better. 01:04:25.210 --> 01:04:27.940 align:middle line:90% But fixed does a lot of this. 01:04:27.940 --> 01:04:29.870 align:middle line:84% It behaves in a lot of ways similarly. 01:04:29.870 --> 01:04:31.828 align:middle line:84% So let me go over how you would solve that one. 01:04:31.828 --> 01:04:34.135 align:middle line:84% So again, the game is going to be-- 01:04:34.135 --> 01:04:36.785 align:middle line:90% 01:04:36.785 --> 01:04:38.270 align:middle line:84% oh, actually, timing of the game. 01:04:38.270 --> 01:04:39.460 align:middle line:90% I should have reversed this. 01:04:39.460 --> 01:04:41.920 align:middle line:84% First stage, consumers decide how many 01:04:41.920 --> 01:04:44.650 align:middle line:84% searches k they're going-- quotes they're going to get. 01:04:44.650 --> 01:04:46.520 align:middle line:84% Oh, no, actually, timing the game first. 01:04:46.520 --> 01:04:50.690 align:middle line:84% So first, the firms simultaneously choose prices. 01:04:50.690 --> 01:04:52.210 align:middle line:90% So I did the timing right. 01:04:52.210 --> 01:04:54.580 align:middle line:84% The firms are choosing prices, and they're 01:04:54.580 --> 01:04:56.120 align:middle line:90% selling a homogeneous good-- 01:04:56.120 --> 01:04:57.350 align:middle line:90% renovating your bathroom. 01:04:57.350 --> 01:05:00.160 align:middle line:90% Let's say it's homogeneous. 01:05:00.160 --> 01:05:01.970 align:middle line:84% You have heterogeneous consumers. 01:05:01.970 --> 01:05:04.390 align:middle line:84% They have search costs s that are distributed according 01:05:04.390 --> 01:05:08.950 align:middle line:84% to G on some interval s lower bar, s upper bar, like in Stahl. 01:05:08.950 --> 01:05:11.560 align:middle line:90% No shoppers here. 01:05:11.560 --> 01:05:13.510 align:middle line:84% And then the consumers are going to, 01:05:13.510 --> 01:05:16.000 align:middle line:90% again, have multiunit demands. 01:05:16.000 --> 01:05:19.000 align:middle line:84% And what the consumers do is they choose 01:05:19.000 --> 01:05:20.810 align:middle line:90% how many price quotes to get. 01:05:20.810 --> 01:05:24.490 align:middle line:84% So the timing of the game is, the firms choose their prices. 01:05:24.490 --> 01:05:27.740 align:middle line:84% Each consumer decides how many quotes they want. 01:05:27.740 --> 01:05:31.090 align:middle line:84% They then get the quotes by sampling randomly 01:05:31.090 --> 01:05:31.960 align:middle line:90% from the firms-- 01:05:31.960 --> 01:05:33.730 align:middle line:90% set of firms. 01:05:33.730 --> 01:05:36.130 align:middle line:84% And then the consumers buy from whichever 01:05:36.130 --> 01:05:37.425 align:middle line:90% firm gave them the best price. 01:05:37.425 --> 01:05:42.530 align:middle line:90% 01:05:42.530 --> 01:05:46.560 align:middle line:84% So again, as in that Stahl model I did, 01:05:46.560 --> 01:05:49.260 align:middle line:84% this is going to be a model that has a mixed strategy 01:05:49.260 --> 01:05:50.610 align:middle line:90% equilibrium. 01:05:50.610 --> 01:05:53.825 align:middle line:84% And it's going to have a mixed strategy equilibrium-- 01:05:53.825 --> 01:05:55.507 align:middle line:84% it doesn't look exactly like these, 01:05:55.507 --> 01:05:57.340 align:middle line:84% but it's going to be the same kind of thing. 01:05:57.340 --> 01:06:01.410 align:middle line:84% There's going to be some density f of p on some interval 01:06:01.410 --> 01:06:02.810 align:middle line:90% p lower bar, p upper bar. 01:06:02.810 --> 01:06:07.190 align:middle line:90% 01:06:07.190 --> 01:06:10.730 align:middle line:84% If that's what happens, firms mix over some interval p 01:06:10.730 --> 01:06:13.100 align:middle line:84% lower bar, p upper bar, what's your value 01:06:13.100 --> 01:06:17.030 align:middle line:90% from getting k price quotes? 01:06:17.030 --> 01:06:19.310 align:middle line:84% Your value from getting k price quotes 01:06:19.310 --> 01:06:21.950 align:middle line:84% is you're just going to integrate over from p lower bar 01:06:21.950 --> 01:06:27.830 align:middle line:84% to p upper bar, consumer surplus of x times f of 1 colon 01:06:27.830 --> 01:06:30.680 align:middle line:84% k of x dx, where f of 1 colon k I'm defining 01:06:30.680 --> 01:06:33.110 align:middle line:84% to be the first-order statistic of k 01:06:33.110 --> 01:06:34.860 align:middle line:90% draws from that distribution. 01:06:34.860 --> 01:06:38.060 align:middle line:84% So if I'm getting k draws from that distribution f, 01:06:38.060 --> 01:06:42.680 align:middle line:84% what is the density of the lowest of the k draws? 01:06:42.680 --> 01:06:45.860 align:middle line:84% And the density of the lowest of k draws from that distribution 01:06:45.860 --> 01:06:47.900 align:middle line:90% is-- 01:06:47.900 --> 01:06:49.370 align:middle line:90% what's this? 01:06:49.370 --> 01:06:51.650 align:middle line:84% Probability that the lowest draw is exactly 01:06:51.650 --> 01:06:57.090 align:middle line:84% x is k times f of x times 1 minus f to the k minus 1. 01:06:57.090 --> 01:07:00.620 align:middle line:84% So there's one firm offering price exactly x, 01:07:00.620 --> 01:07:04.520 align:middle line:84% which happens with probability k f of x, and then all other firms 01:07:04.520 --> 01:07:05.570 align:middle line:90% have higher prices. 01:07:05.570 --> 01:07:08.280 align:middle line:90% 01:07:08.280 --> 01:07:10.310 align:middle line:84% So if this is the distribution, and then I 01:07:10.310 --> 01:07:14.030 align:middle line:84% say, what's the distribution of the best of two 01:07:14.030 --> 01:07:16.940 align:middle line:84% draws in that distribution, it's got more mass here and less mass 01:07:16.940 --> 01:07:19.350 align:middle line:84% here because I'm more likely to find something big. 01:07:19.350 --> 01:07:21.200 align:middle line:84% If I do a hundred draws, I'm very 01:07:21.200 --> 01:07:24.410 align:middle line:84% likely to find the lowest price and very unlikely to find 01:07:24.410 --> 01:07:25.160 align:middle line:90% the highest price. 01:07:25.160 --> 01:07:27.800 align:middle line:84% This is the best in my sample of a hundred. 01:07:27.800 --> 01:07:29.300 align:middle line:90% So anyway, I'm choosing k. 01:07:29.300 --> 01:07:31.230 align:middle line:84% That affects the density of the lowest price. 01:07:31.230 --> 01:07:33.620 align:middle line:84% And then this becomes my utility that I'm 01:07:33.620 --> 01:07:35.480 align:middle line:84% going to get from buying from that firm, 01:07:35.480 --> 01:07:36.955 align:middle line:90% I guess, minus the search cost. 01:07:36.955 --> 01:07:39.740 align:middle line:90% 01:07:39.740 --> 01:07:43.010 align:middle line:84% Notice that this model has a natural decreasing 01:07:43.010 --> 01:07:45.740 align:middle line:84% marginal returns to search property. 01:07:45.740 --> 01:07:48.440 align:middle line:84% The value of the first search is bigger 01:07:48.440 --> 01:07:50.697 align:middle line:84% than the incremental value of the second search 01:07:50.697 --> 01:07:53.030 align:middle line:84% is bigger than the incremental value of the third search 01:07:53.030 --> 01:07:54.590 align:middle line:90% and so on. 01:07:54.590 --> 01:07:55.770 align:middle line:90% Why is that? 01:07:55.770 --> 01:07:58.100 align:middle line:84% Well, it's because the incremental value 01:07:58.100 --> 01:08:02.180 align:middle line:84% of the k-th search is just I'm integrating over all prices 01:08:02.180 --> 01:08:04.340 align:middle line:84% that I might find on the k-th search 01:08:04.340 --> 01:08:08.768 align:middle line:84% v of x times the density of the best price. 01:08:08.768 --> 01:08:10.310 align:middle line:84% So think about, I'm just conditioning 01:08:10.310 --> 01:08:13.650 align:middle line:84% on the best price I've found in the first k minus 1 searches. 01:08:13.650 --> 01:08:16.399 align:middle line:84% And so the best price I found in the first k minus 1 searches 01:08:16.399 --> 01:08:17.490 align:middle line:90% was x. 01:08:17.490 --> 01:08:20.540 align:middle line:84% I'm now going to get v of x from an extra search, where v of x 01:08:20.540 --> 01:08:24.550 align:middle line:84% is that quantity I gave you before, that integral. 01:08:24.550 --> 01:08:29.479 align:middle line:84% And so as v is an increasing function, 01:08:29.479 --> 01:08:31.520 align:middle line:84% the higher the best price I've found, 01:08:31.520 --> 01:08:33.740 align:middle line:90% the higher is my gain to search. 01:08:33.740 --> 01:08:37.939 align:middle line:84% But then as k increases, these distributions 01:08:37.939 --> 01:08:42.510 align:middle line:84% are decreasing in a first-order stochastic dominance sense. 01:08:42.510 --> 01:08:44.720 align:middle line:84% And so because I've already found a good price, 01:08:44.720 --> 01:08:47.569 align:middle line:84% the gain from a new search is reduced. 01:08:47.569 --> 01:08:51.590 align:middle line:90% So once I've already found-- 01:08:51.590 --> 01:08:53.460 align:middle line:84% if I've already done a thousand searches, 01:08:53.460 --> 01:08:55.460 align:middle line:84% I've probably already found something very close 01:08:55.460 --> 01:08:56.418 align:middle line:90% to the lowest possible. 01:08:56.418 --> 01:08:58.279 align:middle line:84% So the value of one extra search is small. 01:08:58.279 --> 01:09:01.790 align:middle line:84% And it just has that-- it's got that monotonicity property. 01:09:01.790 --> 01:09:03.606 align:middle line:84% Every extra search, it's more likely 01:09:03.606 --> 01:09:05.189 align:middle line:84% that the best price I've found is low. 01:09:05.189 --> 01:09:07.481 align:middle line:84% Therefore, it's more likely my gain from search is low. 01:09:07.481 --> 01:09:11.040 align:middle line:90% 01:09:11.040 --> 01:09:13.649 align:middle line:84% So what do we get in equilibrium? 01:09:13.649 --> 01:09:15.270 align:middle line:84% In equilibrium, we get a graph that 01:09:15.270 --> 01:09:21.180 align:middle line:84% looks like this, where, if your search cost is less than-- 01:09:21.180 --> 01:09:27.569 align:middle line:84% this line here would be at U of N minus U of N minus 1, 01:09:27.569 --> 01:09:31.260 align:middle line:84% or U sub N minus U sub N minus 1-- 01:09:31.260 --> 01:09:35.040 align:middle line:84% if your search cost is below that, you'll visit every firm. 01:09:35.040 --> 01:09:38.010 align:middle line:84% If your search cost is above that but less than U 01:09:38.010 --> 01:09:42.790 align:middle line:84% of N minus 1 minus U of N minus 2, you'll visit N minus 1 firms. 01:09:42.790 --> 01:09:46.090 align:middle line:84% If your search cost is higher, you'll visit N minus 2 firms, 01:09:46.090 --> 01:09:46.590 align:middle line:90% so on. 01:09:46.590 --> 01:09:47.189 align:middle line:90% You go all the way out. 01:09:47.189 --> 01:09:49.290 align:middle line:84% There's some consumers who visit one firm, 01:09:49.290 --> 01:09:51.915 align:middle line:84% and there may be some consumers with s is high enough that they 01:09:51.915 --> 01:09:53.533 align:middle line:90% would visit zero firms. 01:09:53.533 --> 01:09:55.200 align:middle line:84% So what we're going to get is the search 01:09:55.200 --> 01:09:58.647 align:middle line:84% costs are just going to divide up in these intervals. 01:09:58.647 --> 01:10:00.480 align:middle line:84% So there's going to be an interval of people 01:10:00.480 --> 01:10:04.080 align:middle line:84% who do N, N minus 1, N minus 2, and so on. 01:10:04.080 --> 01:10:09.270 align:middle line:84% And then remember, I had some density capital G on the search 01:10:09.270 --> 01:10:11.070 align:middle line:90% costs on some interval. 01:10:11.070 --> 01:10:15.480 align:middle line:84% So then integrating that density over these regions is going 01:10:15.480 --> 01:10:19.805 align:middle line:90% to give you the-- so if I-- 01:10:19.805 --> 01:10:21.180 align:middle line:84% don't want to write on my screen. 01:10:21.180 --> 01:10:26.650 align:middle line:90% 01:10:26.650 --> 01:10:29.920 align:middle line:84% I have these regions where you do N searches, 01:10:29.920 --> 01:10:35.170 align:middle line:84% N minus 1 searches, N minus 2 searches, 01:10:35.170 --> 01:10:38.050 align:middle line:84% all the way out to this region where you do one search. 01:10:38.050 --> 01:10:40.630 align:middle line:84% And then there's some density of search costs 01:10:40.630 --> 01:10:42.200 align:middle line:90% that looks like that. 01:10:42.200 --> 01:10:45.250 align:middle line:90% This is the g of s. 01:10:45.250 --> 01:10:47.590 align:middle line:84% The integral of g over this region 01:10:47.590 --> 01:10:52.130 align:middle line:84% is the mass qN of people who do exactly N searches. 01:10:52.130 --> 01:10:58.120 align:middle line:84% And the integral over this region, this is qN minus 1, 01:10:58.120 --> 01:11:00.617 align:middle line:84% the mass of people who do N minus 1 searches. 01:11:00.617 --> 01:11:02.200 align:middle line:84% And then I go all the way out to here. 01:11:02.200 --> 01:11:03.492 align:middle line:90% The integral over this region-- 01:11:03.492 --> 01:11:06.170 align:middle line:90% 01:11:06.170 --> 01:11:09.710 align:middle line:84% these are the people who do one search. 01:11:09.710 --> 01:11:11.810 align:middle line:84% So just integrating the density g 01:11:11.810 --> 01:11:15.110 align:middle line:84% over these boundary regions of how many searches you do 01:11:15.110 --> 01:11:17.990 align:middle line:84% gives me the number of people who do 01:11:17.990 --> 01:11:19.490 align:middle line:84% one search, the number of people who 01:11:19.490 --> 01:11:21.642 align:middle line:84% do two searches, the number of people 01:11:21.642 --> 01:11:23.600 align:middle line:84% who do N minus 1 searches, the number of people 01:11:23.600 --> 01:11:24.490 align:middle line:90% who do N searches. 01:11:24.490 --> 01:11:29.270 align:middle line:90% 01:11:29.270 --> 01:11:33.860 align:middle line:84% And then just like in Stahl, we're going to get-- 01:11:33.860 --> 01:11:37.145 align:middle line:84% the fact that-- if you think about Stahl, Stahl was-- 01:11:37.145 --> 01:11:42.200 align:middle line:84% the solvable equation of Stahl I did for you was q1 was mu, 01:11:42.200 --> 01:11:48.680 align:middle line:84% qN was 1 minus mu, all these other were 0. 01:11:48.680 --> 01:11:51.290 align:middle line:84% That's what I did for you, is there are a bunch of people 01:11:51.290 --> 01:11:54.590 align:middle line:84% doing one search, a bunch of people doing N searches, 01:11:54.590 --> 01:11:58.820 align:middle line:84% and then nobody else, in that one case I solved for you. 01:11:58.820 --> 01:12:02.900 align:middle line:84% In this model, all these q's are going to be non-zero, 01:12:02.900 --> 01:12:06.640 align:middle line:84% but you're going to solve this model in exactly the same way. 01:12:06.640 --> 01:12:10.443 align:middle line:84% You're going to say, I know what the distribution of-- 01:12:10.443 --> 01:12:11.610 align:middle line:90% I know what the profits are. 01:12:11.610 --> 01:12:13.490 align:middle line:84% In this case, the profits that I know 01:12:13.490 --> 01:12:19.640 align:middle line:84% is I know that the profits, if I charge the monopoly price, 01:12:19.640 --> 01:12:30.790 align:middle line:84% are going to be pm minus c D of pm times q1 times 1 over N. 01:12:30.790 --> 01:12:35.445 align:middle line:84% If I charge-- the p upper bar is going to be pm in this model, 01:12:35.445 --> 01:12:35.945 align:middle line:90% always. 01:12:35.945 --> 01:12:39.360 align:middle line:90% 01:12:39.360 --> 01:12:44.160 align:middle line:84% So if I charge the highest possible price, my profit is-- 01:12:44.160 --> 01:12:45.750 align:middle line:84% I'm going to sell to the consumer 01:12:45.750 --> 01:12:48.070 align:middle line:84% if and only if they visit me first 01:12:48.070 --> 01:12:50.370 align:middle line:84% and they're one of the consumers who decided 01:12:50.370 --> 01:12:52.180 align:middle line:90% to only get one price quote. 01:12:52.180 --> 01:12:55.420 align:middle line:84% So the high s people here decide to get exactly one price quote. 01:12:55.420 --> 01:12:59.010 align:middle line:84% So I'm going to sell to them if they get exactly one price quote 01:12:59.010 --> 01:13:01.140 align:middle line:90% and that one price quote is me. 01:13:01.140 --> 01:13:03.630 align:middle line:84% And then I'm going to rip them off with a monopoly price. 01:13:03.630 --> 01:13:09.180 align:middle line:84% And this is going to be my profit, pm minus c D of pm. 01:13:09.180 --> 01:13:13.860 align:middle line:84% What do I get if I charge any other price? 01:13:13.860 --> 01:13:21.210 align:middle line:84% Well, I'm going to get, again, p minus c D of pm per consumer. 01:13:21.210 --> 01:13:24.540 align:middle line:84% But then how many consumers am I going to get? 01:13:24.540 --> 01:13:28.740 align:middle line:84% Well, if a consumer visits me first-- 01:13:28.740 --> 01:13:31.590 align:middle line:84% if a consumer is doing one search and visits me first, 01:13:31.590 --> 01:13:33.540 align:middle line:90% they always buy. 01:13:33.540 --> 01:13:39.450 align:middle line:84% But then also, there are some consumers who visit two firms. 01:13:39.450 --> 01:13:43.290 align:middle line:84% They visit me with probability 2 over N, 01:13:43.290 --> 01:13:46.170 align:middle line:84% and then I sell to them with probability 1 01:13:46.170 --> 01:13:48.325 align:middle line:90% minus capital F of p. 01:13:48.325 --> 01:13:50.700 align:middle line:84% Because it has to be they visit two firms-- they visit me 01:13:50.700 --> 01:13:53.940 align:middle line:84% and somebody else charging an even higher price. 01:13:53.940 --> 01:13:58.050 align:middle line:84% And then there are also consumers who visit three firms. 01:13:58.050 --> 01:14:01.260 align:middle line:84% They visit me with probability 3 over N. 01:14:01.260 --> 01:14:03.180 align:middle line:84% And they buy from me with probability 1 01:14:03.180 --> 01:14:05.550 align:middle line:90% minus f of p squared. 01:14:05.550 --> 01:14:08.310 align:middle line:84% Because it has to be, they visit three firms, I'm included. 01:14:08.310 --> 01:14:11.940 align:middle line:84% And the other two firms turn out to have higher prices than me. 01:14:11.940 --> 01:14:17.520 align:middle line:84% Well, again, this becomes an n-th degree polynomial 01:14:17.520 --> 01:14:20.410 align:middle line:84% instead of something that just has 1 minus f in one place. 01:14:20.410 --> 01:14:21.930 align:middle line:90% But you can still-- 01:14:21.930 --> 01:14:24.360 align:middle line:84% this equals this lets you solve for f 01:14:24.360 --> 01:14:27.690 align:middle line:90% of p for every p less than pm. 01:14:27.690 --> 01:14:30.510 align:middle line:84% And so it's the same kind of argument as in Stahl. 01:14:30.510 --> 01:14:33.690 align:middle line:84% It's like, if you knew q1 through qN, 01:14:33.690 --> 01:14:38.010 align:middle line:84% you could solve for what capital F would have to be. 01:14:38.010 --> 01:14:40.660 align:middle line:84% But then in this model, there is a fixed-point calculation. 01:14:40.660 --> 01:14:44.700 align:middle line:84% You say, OK, here's my guess for what q1 through qN are. 01:14:44.700 --> 01:14:46.860 align:middle line:90% I guess q1 through qN. 01:14:46.860 --> 01:14:50.340 align:middle line:84% I figure out what capital F would have to be. 01:14:50.340 --> 01:14:53.140 align:middle line:84% Once I figured out what capital F would be, 01:14:53.140 --> 01:14:55.830 align:middle line:84% I do this computation of what the U's 01:14:55.830 --> 01:14:58.900 align:middle line:84% are to figure out how many consumers search N things. 01:14:58.900 --> 01:15:01.830 align:middle line:84% And I see if I reproduce the q1 through qN. 01:15:01.830 --> 01:15:04.170 align:middle line:84% And so numerically, I solve this by finding 01:15:04.170 --> 01:15:05.610 align:middle line:90% a fixed-point problem. 01:15:05.610 --> 01:15:07.680 align:middle line:90% I guess q1 through qN. 01:15:07.680 --> 01:15:09.460 align:middle line:84% I figure out what f would have to be. 01:15:09.460 --> 01:15:10.930 align:middle line:84% I figure out what U would then be. 01:15:10.930 --> 01:15:12.390 align:middle line:84% I figure out what the q's would then be, 01:15:12.390 --> 01:15:13.390 align:middle line:90% and I see if they match. 01:15:13.390 --> 01:15:15.900 align:middle line:84% If not, I have an optimizer moving 01:15:15.900 --> 01:15:19.380 align:middle line:84% the q's around to try to get the q's to match the q's I started 01:15:19.380 --> 01:15:20.825 align:middle line:90% with until I find a fixed point. 01:15:20.825 --> 01:15:23.850 align:middle line:90% 01:15:23.850 --> 01:15:25.650 align:middle line:84% Again, it's not something theorists 01:15:25.650 --> 01:15:28.330 align:middle line:84% do often because it doesn't have a nice closed form. 01:15:28.330 --> 01:15:31.410 align:middle line:84% But it is something that empirical people will often 01:15:31.410 --> 01:15:34.270 align:middle line:84% do if they're solving their model numerically anyway. 01:15:34.270 --> 01:15:36.900 align:middle line:84% It's easier than the dynamic programming thing 01:15:36.900 --> 01:15:38.800 align:middle line:90% that the other one has in it. 01:15:38.800 --> 01:15:42.250 align:middle line:90% 01:15:42.250 --> 01:15:43.486 align:middle line:90% Any questions? 01:15:43.486 --> 01:15:48.820 align:middle line:90% 01:15:48.820 --> 01:15:51.597 align:middle line:84% I have just two brief slides I'll do. and then the other one, 01:15:51.597 --> 01:15:52.180 align:middle line:90% I'm actually-- 01:15:52.180 --> 01:15:54.138 align:middle line:84% I think I'm going to let [? Rowey ?] talk about 01:15:54.138 --> 01:15:55.480 align:middle line:90% on Friday. 01:15:55.480 --> 01:15:57.920 align:middle line:90% So obfuscation. 01:15:57.920 --> 01:16:04.240 align:middle line:84% So an observation about all these search cost models is-- 01:16:04.240 --> 01:16:07.810 align:middle line:84% so far today, I've always taken the s as an exogenous parameter. 01:16:07.810 --> 01:16:11.230 align:middle line:84% And then profits of the firms have gone up in s. 01:16:11.230 --> 01:16:15.490 align:middle line:84% So that raised the question of, where do s's come from? 01:16:15.490 --> 01:16:19.030 align:middle line:84% Why are there search costs in models? 01:16:19.030 --> 01:16:23.900 align:middle line:84% And clearly, firms collectively are better off when s is bigger. 01:16:23.900 --> 01:16:27.333 align:middle line:84% The bigger is s, the higher are the firm profits. 01:16:27.333 --> 01:16:28.750 align:middle line:84% And so certainly, if all the firms 01:16:28.750 --> 01:16:30.400 align:middle line:84% could get together and agree, are 01:16:30.400 --> 01:16:32.390 align:middle line:84% we going to make searching for a mattress easier or are we 01:16:32.390 --> 01:16:34.550 align:middle line:84% going to make searching for a price on a mattress hard, 01:16:34.550 --> 01:16:35.530 align:middle line:84% all the firms will get together and say, 01:16:35.530 --> 01:16:37.613 align:middle line:84% let's make searching really hard for the consumers 01:16:37.613 --> 01:16:42.820 align:middle line:84% and we can all charge a monopoly price and be best off. 01:16:42.820 --> 01:16:44.590 align:middle line:84% But that raises the question of, does that 01:16:44.590 --> 01:16:48.120 align:middle line:84% require collusion among the firms to get s high? 01:16:48.120 --> 01:16:51.320 align:middle line:84% Or can firms also individually have an incentive 01:16:51.320 --> 01:16:56.400 align:middle line:84% to raise their s, even if it's just individually, 01:16:56.400 --> 01:16:57.767 align:middle line:90% do they want to raise s? 01:16:57.767 --> 01:16:59.600 align:middle line:84% And then you might think, well, that's going 01:16:59.600 --> 01:17:00.767 align:middle line:90% to be harder to get to work. 01:17:00.767 --> 01:17:02.360 align:middle line:84% Because if I raise my s and consumers 01:17:02.360 --> 01:17:04.730 align:middle line:84% see that I'm the annoying firm to visit, 01:17:04.730 --> 01:17:06.920 align:middle line:84% aren't they all going to go to the nonannoying firm 01:17:06.920 --> 01:17:09.540 align:middle line:84% first and pay less to get s there? 01:17:09.540 --> 01:17:11.870 align:middle line:84% And isn't that bad for me if I raise my search cost 01:17:11.870 --> 01:17:14.210 align:middle line:84% and I drive consumers to the other firms? 01:17:14.210 --> 01:17:18.890 align:middle line:84% And there are a few papers that discuss reasons why you actually 01:17:18.890 --> 01:17:21.650 align:middle line:84% can get it to be individually rational for firms 01:17:21.650 --> 01:17:25.550 align:middle line:84% to raise search costs and not just collectively rational. 01:17:25.550 --> 01:17:32.720 align:middle line:84% So one paper I put is this paper by Chris Wilson in the IJIO. 01:17:32.720 --> 01:17:36.410 align:middle line:84% And this considers this game, which has a surprising result. 01:17:36.410 --> 01:17:40.480 align:middle line:84% Firms simultaneously choose search costs, s1, s2, from s 01:17:40.480 --> 01:17:42.350 align:middle line:90% lower bar, s upper bar. 01:17:42.350 --> 01:17:45.680 align:middle line:84% The consumers and the firms observe what these search costs 01:17:45.680 --> 01:17:49.550 align:middle line:84% are, and then the firms choose an unobserved price, 01:17:49.550 --> 01:17:54.170 align:middle line:84% and then consumers follow a search and purchase policy 01:17:54.170 --> 01:17:56.810 align:middle line:90% with optimal sequential search. 01:17:56.810 --> 01:17:59.900 align:middle line:84% And it turns out that this model does not 01:17:59.900 --> 01:18:03.590 align:middle line:84% have an equilibrium where both firms set price s lower bar. 01:18:03.590 --> 01:18:06.020 align:middle line:84% So it is true that you try to have a firm set 01:18:06.020 --> 01:18:07.760 align:middle line:90% price s lower bar. 01:18:07.760 --> 01:18:10.250 align:middle line:84% You have this one firm, what it can 01:18:10.250 --> 01:18:12.530 align:middle line:90% do is it can raise its price. 01:18:12.530 --> 01:18:15.703 align:middle line:84% It can raise its price and be known as the annoying firm, 01:18:15.703 --> 01:18:18.120 align:middle line:84% so that all the consumers are going to go to firm 1 first. 01:18:18.120 --> 01:18:21.740 align:middle line:84% So if I'm firm 2, I raise my search cost, 01:18:21.740 --> 01:18:25.190 align:middle line:84% and then I get everyone to visit the other firm first. 01:18:25.190 --> 01:18:28.800 align:middle line:84% But then the other firm knows that it's getting visited first. 01:18:28.800 --> 01:18:31.250 align:middle line:84% And so because it knows that it's getting visited first, 01:18:31.250 --> 01:18:32.210 align:middle line:84% consumers are going to pay a search 01:18:32.210 --> 01:18:33.720 align:middle line:90% cost to go to the second firm. 01:18:33.720 --> 01:18:34.700 align:middle line:84% It's going to have a big incentive 01:18:34.700 --> 01:18:36.200 align:middle line:84% to raise its price because it knows, 01:18:36.200 --> 01:18:38.390 align:middle line:84% I can at least go s bigger than the other guy, 01:18:38.390 --> 01:18:40.640 align:middle line:84% and people are going to not go visit him. 01:18:40.640 --> 01:18:44.270 align:middle line:84% But then, of course, once it goes 01:18:44.270 --> 01:18:46.743 align:middle line:84% s bigger than the other guy, it's 01:18:46.743 --> 01:18:48.660 align:middle line:84% going to have an incentive to raise its price. 01:18:48.660 --> 01:18:50.330 align:middle line:84% Then firm 2 is also going to say, well, 01:18:50.330 --> 01:18:52.872 align:middle line:84% I'm only getting the people who visited firm 1 and then found 01:18:52.872 --> 01:18:55.072 align:middle line:84% out that firm 1's price was unexpectedly high. 01:18:55.072 --> 01:18:57.030 align:middle line:84% And so then they're going to raise their price. 01:18:57.030 --> 01:19:02.780 align:middle line:84% And you can create this asymmetric Bertrand-like 01:19:02.780 --> 01:19:05.240 align:middle line:84% competition, where the firms have an incentive 01:19:05.240 --> 01:19:06.170 align:middle line:90% to raise their price. 01:19:06.170 --> 01:19:07.587 align:middle line:84% And then because I know my rival's 01:19:07.587 --> 01:19:10.020 align:middle line:84% going to raise their price, I may raise my price too. 01:19:10.020 --> 01:19:12.800 align:middle line:84% And therefore, even though they're starting with him, 01:19:12.800 --> 01:19:14.840 align:middle line:84% he chooses a distribution-- a mixed strategy 01:19:14.840 --> 01:19:17.240 align:middle line:84% that's so high that people come to me anyway, 01:19:17.240 --> 01:19:18.890 align:middle line:90% and we're both better off. 01:19:18.890 --> 01:19:22.310 align:middle line:90% So that kind of story can work. 01:19:22.310 --> 01:19:26.390 align:middle line:84% The other story that I have in a paper with Alex Wolitzky 01:19:26.390 --> 01:19:31.100 align:middle line:84% is that, you can also do this when 01:19:31.100 --> 01:19:32.720 align:middle line:84% the cost of getting a price quote 01:19:32.720 --> 01:19:34.830 align:middle line:90% is not observable in advance. 01:19:34.830 --> 01:19:37.238 align:middle line:84% Suppose that it's only when you get to the car dealer 01:19:37.238 --> 01:19:39.530 align:middle line:84% that you learn whether it takes one hour to get a price 01:19:39.530 --> 01:19:41.510 align:middle line:84% quote, or five hours to get a price quote from that dealer, 01:19:41.510 --> 01:19:43.385 align:middle line:84% and how long they're going to make you haggle 01:19:43.385 --> 01:19:45.870 align:middle line:84% before they get the real price they're willing to offer. 01:19:45.870 --> 01:19:49.650 align:middle line:84% And we just discuss in our paper other ways you can do it. 01:19:49.650 --> 01:19:53.150 align:middle line:84% One way you can do this is with the convex disutility of time 01:19:53.150 --> 01:19:55.490 align:middle line:84% spent shopping, where after you've 01:19:55.490 --> 01:19:57.440 align:middle line:84% spent 5 hours in a car dealer, you now really 01:19:57.440 --> 01:20:00.065 align:middle line:84% don't want to spend 10 hours by going to the second car dealer. 01:20:00.065 --> 01:20:02.240 align:middle line:90% And there's a convex time cost. 01:20:02.240 --> 01:20:05.570 align:middle line:84% The other is if you add uncertainty over how efficient 01:20:05.570 --> 01:20:07.220 align:middle line:90% it could be to buy a car. 01:20:07.220 --> 01:20:10.760 align:middle line:84% And this one I think works well with getting a quote 01:20:10.760 --> 01:20:12.680 align:middle line:90% from a contractor. 01:20:12.680 --> 01:20:14.783 align:middle line:84% You get one contractor, and you spent 14 hours 01:20:14.783 --> 01:20:16.950 align:middle line:84% with this contractor trying to get a quote from him. 01:20:16.950 --> 01:20:18.440 align:middle line:84% You think getting a quote from another guy 01:20:18.440 --> 01:20:20.420 align:middle line:84% is also going to take you 14 hours because it 01:20:20.420 --> 01:20:21.690 align:middle line:90% must be that hard. 01:20:21.690 --> 01:20:24.470 align:middle line:84% And if consumers don't know how hard getting price quotes are, 01:20:24.470 --> 01:20:26.870 align:middle line:84% by making it more annoying, people 01:20:26.870 --> 01:20:30.080 align:middle line:84% are going to infer that the other dealers are also 01:20:30.080 --> 01:20:31.110 align:middle line:90% more annoying. 01:20:31.110 --> 01:20:33.770 align:middle line:84% And therefore, that's going to make you also individually raise 01:20:33.770 --> 01:20:34.970 align:middle line:90% search costs. 01:20:34.970 --> 01:20:36.830 align:middle line:84% Anyway, so that's the argument there 01:20:36.830 --> 01:20:40.880 align:middle line:84% for there are reasons why, even though you lose some customers, 01:20:40.880 --> 01:20:43.370 align:middle line:84% you may want to raise search costs. 01:20:43.370 --> 01:20:45.870 align:middle line:84% So let me say, plan for the rest of the week. 01:20:45.870 --> 01:20:48.210 align:middle line:84% I think I'm going to have [? Rowey ?] talk 01:20:48.210 --> 01:20:50.055 align:middle line:90% about multiproduct search. 01:20:50.055 --> 01:20:52.180 align:middle line:84% We don't have any problem set to go over on Friday. 01:20:52.180 --> 01:20:54.730 align:middle line:84% So I think he's going to talk about a couple recent papers. 01:20:54.730 --> 01:20:56.730 align:middle line:84% This paper on multiproduct search 01:20:56.730 --> 01:20:58.440 align:middle line:90% is likely to be one of them. 01:20:58.440 --> 01:21:03.000 align:middle line:84% And then next week, I'm going to jump ahead to empirical search. 01:21:03.000 --> 01:21:05.547 align:middle line:84% And these are the three papers I plan to talk about. 01:21:05.547 --> 01:21:07.380 align:middle line:84% There's no class Monday due to the holidays, 01:21:07.380 --> 01:21:09.597 align:middle line:90% so it's next Wednesday. 01:21:09.597 --> 01:21:11.430 align:middle line:84% And then if you want to read one in advance, 01:21:11.430 --> 01:21:17.076 align:middle line:84% I would read my paper with Sara on search and obfuscation. 01:21:17.076 --> 01:21:24.000 align:middle line:90%