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Lecture 11: Basic trade/spatial model — Transcript

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0:01Uh welcome back to um uh this continuing
0:05lectures lecture 11 on STEG's virtual
0:08course on key concepts in macro
0:10development. STEG is a research program
0:13run by CR on um structural
0:16transformation and economic growth
0:19uh funded by the foreign commonwealth
0:21and development office as part of the UK
0:23aid effort. Um and this course is
0:26offered uh free of charge obviously. Um
0:30um we're very excited. Uh I want to
0:32thank CPR uh for and the members of CPR
0:36uh Mandy Chan, uh Kirsty McNeel, and
0:38Lauren Wearing for helping organize and
0:42handle the logistics of this course.
0:44We're very excited today to have Melanie
0:45Morton. We're moving on to this new unit
0:48on uh spatial and trade frictions. And
0:51we're very excited to have Melanie
0:52Morton of Stanford uh giving an
0:55introduction to the spatial equilibrium
0:56model. Uh next week we will continue
0:59with more work with uh trade and spatial
1:02models. Um those of you in the front
1:06row, we'll go over the the sort of rules
1:08again. Uh if you're in the front row,
1:10please mute, but uh feel free to unmute
1:12and ask questions uh vocally. Uh
1:15Melanie's happy to take questions.
1:18Um there will be time at the end uh for
1:20questions as well. So clarifying
1:22questions are probably better during the
1:24talk. Um, and then those of you who are
1:27joining through the webinar, uh, please,
1:30um, feel free to ask questions through
1:34the Q&A. Uh, Melanie's, uh,
1:38TA is a graduate student at Stanford,
1:41um, named Sally Jean, and she will be,
1:45uh, answering questions. And Doug and I
1:46will also do our best to answer
1:48questions. And anything that we can't
1:50answer and anything at the end that gets
1:52asked we'll give to uh Melanie. Um I
1:57think that's everything. If you're in
1:58the front row I would ask you to please
2:01uh put your name, your what university,
2:03your affiliation and your uh country of
2:07origin uh to share that with the rest of
2:10the panelists. Angel will not be able to
2:12see raised hands. Uh so you know don't
2:14raise your hand. Uh if you're in the
2:15front row just interrupt. uh she also
2:17won't be able to see things in the chat.
2:19Um but um okay go ahead Melanie.
2:23Okay so good afternoon good morning hi
2:26everyone. Um it's wonderful to be here
2:30to give this lecture. I think this is a
2:31really great uh initiative. So I also
2:34want to give my thanks to uh Seer CPR
2:38and Stig and um of course Joe and Doug
2:42for all of this work uh organizing.
2:45So today uh we're going to talk through
2:47uh the spatial equilibrium model. And
2:49the spatial equilibrium model I think is
2:51um on one hand a very simple structure.
2:54So we're going to go through a very very
2:56simple like uh uh version of this model
2:59just to develop intuition. But what is
3:01really powerful about this model is that
3:03once you understand the simple intuition
3:06and how the different pieces uh go
3:07together, this very same workhorse model
3:10that we do today since can extend in
3:12very many different uh directions and
3:15then you're almost right at the frontier
3:16of uh research. So the goal today is to
3:19kind of go through a very simplified
3:20version on purpose so that we develop
3:22our intuition. The uh TA section on
3:25Monday is going to talk through each of
3:28the equations and it's going to have the
3:29goal of having you be able to code up um
3:34yourself the model on the computer so
3:36that you would be able to start using
3:38this model yourself. Um and then as we
3:40go through our sort of flag areas where
3:42you could imagine extending the model to
3:44make it a little more realistic. Uh and
3:46this would be kind of all you would need
3:47to be research ready to start uh using
3:50these tools for your uh own research.
3:53So why should we uh think about spatial
3:56equilibrium? So if you just sort of look
3:57around, everyone's probably from
3:59different parts of the world. Um but if
4:01you just sort of think about what you
4:02see where you live, there are often some
4:05puzzles. So I've put here two pictures
4:08from uh Indonesia. So on the left we
4:11have Jakarta. It's a large metropolis.
4:14You can see there are highrises. You can
4:16sort of see, for example, people driving
4:18cars. There's um a lot. It looks like a,
4:21you know, a city that could be almost
4:22anywhere in the world. And on the right,
4:25you can see the same uh a picture of
4:27someone who's working in the fields
4:29doing [snorts] labor in a way that's not
4:31very mechanized, sort of uh tilling
4:34their their crops. And these uh two
4:36pictures are going to be taken a few
4:38hundred miles apart from each other. So
4:40it sort of leads to a set of questions
4:42like what explains the coexistence even
4:45within the same country of such large
4:49differences in uh is it average income
4:52is it what's driving kind of these
4:54differences in how we see um something
4:57that looks like uh a difference in if we
4:59want to call it development or level of
5:00GDP across space even within the same
5:03country. And I think this is really
5:05fascinating because it sort of narrows
5:07in even more than thinking about the
5:09very large differences across country
5:11where we've had lots of lectures sort of
5:13understanding for example GDP
5:15differences across different countries
5:16in the world. If you start looking at
5:18spatial gaps within country, this is a
5:21setting where a lot of those variables
5:24such as the common political
5:26institution, the types um the average
5:28income of the country, all of that stuff
5:30is constant. But yet we still see these
5:32really phenomenal differences. So the
5:34spatial model is going to be a way to
5:36sort of ask is this an equilibrium that
5:40we have somewhere like Jakarta and we
5:41have somewhere like rule Java. Is it is
5:43that an equilibrium? Are we out of
5:45equilibrium that we have that? If it is
5:47an equilibrium, what is it that's
5:49driving that difference? If it's not an
5:52equilibrium, what potentially is going
5:54are there frictions? Are there other
5:56things that are going on that we could
5:57think about ways in which we could then
5:59think about policies to try and um you
6:02know have some way so that the picture
6:05on the right might get a little bit
6:06closer to the picture on the left. Okay.
6:08So this is a visual of uh Indonesia but
6:11you can sort of look at um data from
6:14just about anywhere in the world and
6:15you're going to see the same variation
6:18across space. So let me show you a few
6:20other examples and then we're going to
6:22sort of work through this uh simple
6:25workhouse model.
6:28So here is plotting uh income across
6:30income distribution across space for
6:32Tanzania. And what I'm going to do is
6:35sort of have sometimes you'll see two
6:37different versions of these sorts of
6:38figures. So let me just sort of show you
6:40what they show. On the left hand side
6:42what we've been plotting is nominal
6:44income um by district. And so what this
6:48is is the distribution if you think
6:50about average income within a district
6:53within a region. And what you can see
6:55here is that there are some people some
6:57regions where income is very high on
7:00average. And you can see that there's
7:01some regions again within the same
7:04country where average income is really
7:06low. Sometimes you'll see this uh
7:08summarized by the 9010 gap. So this is a
7:12um a measure of the dispersion. So what
7:14is the 90th percentile relative to the
7:1710th percentile? So for example, in this
7:19in this figure here, the region that's
7:22at the 90th percentile in Tanzania,
7:24people on average in that uh region have
7:27nominal income that's 4.3 times higher
7:30than people who are living in a region
7:33at the 10th percentile.
7:35So that's uh sort of something you'll
7:36see here. these distributions again this
7:39is already sort of averaging this is uh
7:41averaging out the variation within a
7:43region. This is just asking on average
7:46for example what is the average income
7:47in a region sort of averaging out the
7:49fact that even within a region you might
7:51have people who have higher incomes and
7:53people who have lower incomes.
7:56The figure on the map the the one on the
7:58right that shows the same data but just
8:00plotting it on a map so that you can
8:01sort of get a sense of you know where
8:03this what is the special what do the
8:04special patterns look like. So for
8:06example here you can see you know around
8:09uh uh Dar Salam parts of the country
8:12which are very rich you can see that
8:13average uh people who are living in
8:15those regions have higher income but
8:17what you also see is that there's a lot
8:18of variation it's not necessarily that
8:20all the low-income places are right
8:21beside each other there's sort of
8:23variation around the country and again
8:26this sort of leads back to those
8:27questions if we think people could move
8:30um what's stopping someone from moving
8:32from a place where they might have very
8:34low income to a place that they might
8:36have higher income. There's a set of
8:38questions about are these actually gaps
8:40that are exploitable? So should we
8:42expect someone who's living for example
8:44in the north if they move to Dar Salam
8:48would we expect them to have the average
8:50income or is there something else going
8:51on for example about the selection of
8:54the types of people who might be living
8:55in those cities that might mean that
8:57even if we encourage people to migrate
9:00they wouldn't gain the same average
9:01income. So these are all the sort of
9:04sets of questions that we can sort of
9:05start to think about and then uh try and
9:08understand. Let me just show you um one
9:10last figure which is the figure for the
9:12US. This is taken from um the Moretti
9:15handbook chapter um which is a really
9:18useful reference for going through the
9:20model that we're going to go through
9:21today. But again, what it's going to
9:23show you is this figure uh this
9:24histogram on the top and this is
9:27plotting the average wage of people who
9:30have a high school degree.
9:33Uh again it's across metropolitan areas.
9:36So this me is sort of looking for
9:38example if you're living in New York
9:39City versus you're living in San
9:40Francisco versus you might be living in
9:42Minneapolis or you're living in another
9:44city. What is the average wage of people
9:47across space? And then this figure on
9:49the bottom is showing that there's also
9:51a distribution if you look at people who
9:52have a college degree. So again, there's
9:54a distribution across space where even
9:57within the same country where we might
9:58think that a lot of those barriers to
10:01moving for example the legal barrier to
10:03being able to move those are much lower,
10:06we still see that there's a large
10:07dispersion of nominal wages across
10:10space. And this leads to lots of
10:12questions again. Is this an equilibrium?
10:15Is it not an equilibrium?
10:18Is this potentially a policy exploitable
10:20difference? Is there something else like
10:22selection that might be going on? Uh and
10:24so we sort of want to have a way to
10:26think about analyzing the contributions
10:29of all of these different channels uh to
10:31these gaps.
10:34Okay. So what we want to do today is
10:37these are the sorts of research
10:38questions that we might have. So you
10:40know what explains the difference in
10:41income across locations? What happens if
10:44we are going to open up a lot of new
10:46factories in one part of the country?
10:49What do we think should happen to this
10:50distribution of where how much people
10:53earn? How is that going to change? Who's
10:55going to benefit if there are
10:56productivity shocks? Some parts of the
10:58country and other parts of the country.
11:00We might have lots of questions about
11:02what happens to welfare, what happens to
11:04productivity. If we connect places, if
11:07we make it easier for either people or
11:10for goods to travel between two
11:12locations, do we expect that these wage
11:14gaps might go down? Um what can we sort
11:17of understand about some of the returns
11:20uh and the distribution of returns to
11:22investing in uh infrastructure?
11:25So we're going to use uh this workhorse
11:28model uh the Rosen rowback model. Um the
11:30Rosen rob model is um a key model in uh
11:35sort of labor and special economics. I'm
11:37going to talk through this one and it's
11:39very closely related to the trade model.
11:42So I'm going to talk at the end about
11:43how the intuition from the Rosen rowback
11:46model um kind of easily carries through
11:49uh to understanding trade. And indeed,
11:51if you really wanted to, you could sort
11:53of incorporate the labor mobility on one
11:56side and the goods mobility and you can
11:58easily put that exactly in the same
11:59model. Once you have the sort of
12:01intuition about how all the pieces fit
12:02together, uh it's quite easy to expand
12:05as you like. So what we're going to do
12:07is we're going to start off, we're going
12:08to think about what is an equ what does
12:11it mean to be in a spatial equilibrium.
12:13Um we'll start with a case where uh
12:16returns are exogenous and then we'll go
12:18and we'll understand how we could think
12:20about endogenous wages, endogenous
12:23housing prices uh things like that so
12:25that we can understand um you know this
12:27these general equilibrium spillovers for
12:30example when there's a productivity
12:31shock when there might be a change in
12:33amenities you could think about what
12:35might happen if it becomes easier to
12:36move uh etc etc.
12:40Um so what we're going to do today is
12:41we're going to work through that uh
12:43basic uh framework to get our develop
12:46our intuition and then the goal for the
12:48TA section. Um we have uh Sally is a
12:51fourth year PhD student at Stanford. Uh
12:54Sally's work um encompasses both spatial
12:57uh research is both spatial economics as
12:59well as the looking at uh work on the
13:02economics of the household. Um, and so
13:04what Sally is going to do is help you
13:07work through each of the equations in
13:09the model with the goal of what would
13:11you need to do to be able to put this on
13:13the computer to have this running on the
13:15computer. Um, she'll show some
13:17simulations and then once you have the
13:19simple model coded up, you can basically
13:21take it in whatever direction you want
13:22if you want to start using this uh for
13:24your own research.
13:27Okay, so let me maybe just stop and make
13:29sure there are no questions um before I
13:31keep going.
13:35Okay, please do feel free to interrupt.
13:40Okay, so I put up those two images. Um,
13:43I put up the uh Jakarta or you could
13:45look at the distribution in Tanzania or
13:47in the US. And you know what was sort of
13:49this puzzle is that we have the
13:51coexistence of places that on average
13:53have much higher incomes than places
13:55that are in the same country but have
13:57much lower incomes.
13:59So we can sort of think through lots of
14:01reasons why this might not be a puzzle.
14:03And then what's going to be um what's
14:05going to kind of be interesting is that
14:07we can then sort of start to piece um
14:09piece out those puzzles. So for example,
14:12you know, if we look back on on this
14:14difference here between
14:16um between Jakarta and rural Java, we
14:20might say, you know, one difference is
14:21that people are maximizing people are
14:24choosing where to live based off of
14:26their utility, not just off their wages.
14:28So perhaps it is the case that you earn
14:30much more money in Jakarta and you earn
14:34on average much less money if you're
14:36working in rural Java.
14:38But perhaps, you know, and you can sort
14:39of see in the background here, you know,
14:41maybe it's the case that there are some
14:42negative amenities about living in a big
14:44city. So maybe there are things like
14:46pollution. You know, maybe it's really
14:48crowded. Maybe um it's more difficult to
14:52access uh maybe you have to live in a
14:55housing that has uh lower average
14:57sanitation levels. So you could think
14:59about lots of other reasons which may go
15:02into either amenities or may go into
15:04preferences where people are not only
15:08considering the wage when they're making
15:10the decision about whether to live here
15:13or whether to live here. Okay. So one
15:16one one kind of concept we want to
15:17separate then is we want to separate out
15:19the difference between income and we
15:22want to separate out the difference
15:23between uh utility. And so we want to
15:26think about, you know, what else could
15:27be going on. Is it the case that you
15:30see, for example, in big cities that
15:32people are paying pay more in part to
15:35compensate for a quality of life that
15:37might be lower in some dimensions
15:39because um of congestion, pollution,
15:42etc., etc.
15:44Okay. So, one one possible explanation
15:46is um thinking about the difference
15:49between welfare and thinking about the
15:51difference between income. Another very
15:53obvious one you could look at here and
15:55you might say, you know, maybe the
15:57skills or the types of work that are
15:59available in somewhere like Jakarta
16:01versus somewhere like rural Java are
16:03just very different. So maybe it's the
16:05case that the types of people who have
16:07the skills to work inside highrises are
16:10going to be very different than the
16:12types of people who have the skills to
16:13be good uh agricultural workers or good
16:16farmers. And so part of what we are
16:18seeing the difference here is reflecting
16:20some selection when places have higher
16:23average wages. Maybe they're attracting
16:25people that have higher education or uh
16:28just different sets of skills. And
16:30again, part of what you're then
16:31reflecting is a different a difference
16:34in the return to a factor uh rather than
16:37an underlying spatial difference.
16:42So we've talked about utility, we've
16:43talked about amenities. Another big
16:45difference it could be that the cost of
16:47living is really different. So if you're
16:49living in Jakarta, maybe you're paying
16:5240% of your income in housing. If you're
16:54living in rural Java, you may have um a
16:58much lower proportional cost of your
17:00income into housing. So again, when
17:02we're thinking about the fact that
17:03people want to think about where to live
17:06based off at least their real wages, we
17:09would want to take into account
17:13the fact that there might be costs of
17:14living. So we might want to think about
17:16the difference between wages and the
17:19rents. Um we might also want to think
17:21about the difference between uh the cost
17:23of buying consumption goods. That might
17:25also differ. Although often we think
17:27that it's going to be probably a little
17:29bit cheaper to buy consumption goods
17:30inside cities. Uh if we think about
17:32stories where you know you have to pay
17:35transportation costs, maybe things
17:36already start in a city and then they
17:38sort of get trucked out. But so that
17:40might go some that might be something
17:41that goes the opposite direction. But we
17:43could think both about the cost of
17:44living in terms of you know paying for
17:46things like your your housing and we
17:48could think about the cost of living in
17:50terms of paying for other consumption
17:52goods that you want to consume.
17:54We've talked about um indiv uh we've
17:58talked about skill. There might be
18:00selection motivations here that help to
18:02explain this difference. Um and it might
18:05also be that there's just differences in
18:06what people like. So, you know, is it
18:08the case that people some people are
18:12happy to live in urban areas. Some
18:13people have very strong attachments back
18:15maybe to their families or where they're
18:18from. These strong attachments could be
18:20lots of other things. It might be things
18:22like links to informal risk sharing. It
18:25might be links to family members. It
18:27could be a whole lot of other things
18:28that are going on that might mean that
18:31um people have different uh ways in
18:34which they evaluate the same two
18:36decisions based on something that's
18:38specific to them.
18:40And then of course it might also be the
18:42case that we have um frictions. So one
18:44friction that might mean that um you
18:47know the reason that there's a a wage
18:49gap between somewhere like Jakarta and
18:51Java is maybe Ro Java is that it might
18:53be costly for people to move. People
18:55would like to move. We think if they did
18:57move that they would have higher
18:58incomes. They might even have higher
19:00utility but there might be something
19:02that's stopping them from moving. Uh in
19:05the case where people are allowed to
19:07move, the migration costs could be
19:09something like distance. It could be um
19:12thinking about you know you could think
19:14about linguistic or cultural gaps maybe
19:16falling you know falling into migration
19:18costs in some settings you might have
19:20restrictions on the internal mobility.
19:23So if you think about in China and the
19:24hook restrictions there might be some uh
19:28some barriers that mean that people
19:30might want to move but might have
19:31restrictions against moving. So again we
19:34might have migration costs the same
19:36intuition. Um when you think about trade
19:39models um if trade was perfectly if
19:42trade cost were zero you would be able
19:44to have uh equal uh returns to factors
19:46across space but there might be
19:48frictions coming through the inability
19:50uh to uh be able to trade perfectly. And
19:54then I've also put housing costs here um
19:57because a lot of work has sort of looked
19:59at the different role of for example how
20:02easy is it to expand housing in a
20:04location. How quickly do housing costs
20:07increase when more people want to live
20:09in that location? So how do we think
20:11about housing supply elasticities? So
20:15for example somewhere like San
20:17Francisco, it's surrounded by water on
20:18three sides. There's not a lot of area
20:20to move out. What does that mean? It
20:22might mean that um when more people want
20:24to move into San Francisco, the
20:26elasticity of housing prices to
20:29population is going to be higher than
20:31for example a place where you can just
20:32expand your housing supply. What does
20:35that mean? That might mean that um if we
20:37look at least at wages without
20:39accounting for those costs of living uh
20:41there's going to be uh big nominal gaps
20:44and that might mean that if we wanted to
20:46think about understanding why people
20:48might not be moving more we could think
20:50about the role of housing costs and
20:52housing supplies elasticities as being
20:55potentially uh a policy lever.
20:58Does anyone have any other sort of
21:00candidate hypotheses that they want to
21:02throw into the mix for reasons why we
21:05might still see differences in wages
21:07across space?
21:09Can I ask a question?
21:11Yes. Uh um would there be any benefit of
21:15thinking like these barriers as of some
21:17sort of capital that uh people uh
21:20acquire like I acquire capital of
21:23consuming goods that are available near
21:25the region where I was born or I acquire
21:28capital just to do the specific skills
21:30that are needed just in my region and
21:32that's why I don't we would we gain
21:35anything from looking at things this
21:37way.
21:38Yeah. So that I think probably you could
21:40think about that maybe you can think
21:42about um capital and consumption
21:44although that one
21:46um
21:48the one the area where this idea I think
21:50is a little bit um you see it a little
21:51bit more is on the production side maybe
21:53more than the consumption side. So for
21:55example maybe um and I think a really
21:58nice paper uh is Sam Bazy and
22:00co-author's work um in Indonesia
22:04where um for example there's different
22:06types of soil and if you grew up in an
22:09area that had a certain type of soil you
22:11get uh skills and expertise in farming
22:13for that type of soil and what they look
22:16at in this paper is they look at what
22:18happens in Indonesia um during
22:20colonization there was a period of force
22:23there was a period of sort of
22:24resettlement
22:25where people um could opt into being
22:27resettled outside of Java. And they look
22:30at how similar was the match in the
22:33origin soil in the destination soil of
22:35people who were getting resettled in
22:37this way. And they show that people who
22:39end up going to destinations that look
22:41more similar um to origins uh are going
22:45to uh more productive when they're in
22:47that destination. And so that's sort of
22:48this idea, right?
22:50part of your ability to be producing
22:53might depend on your knowledge or your
22:55skill. And it might be that there's a a
22:58potential spatial match between skills
23:00that you have when you start and when
23:01you go. You could think about it sort of
23:03in that direction. You could also think
23:05about it in things like human capital.
23:06You know, often in these models, if you
23:08think about indogenizing education, you
23:12might if you anticipate that you're
23:14going to have a large labor market and
23:16it's going to be there's going to be
23:17high returns to your skill. Therefore,
23:19you might want to invest in education
23:21more than if you anticipate that you
23:22won't have a large labor market. So
23:24therefore, it could also mean that
23:26you've had different levels of
23:27education, which might then affect um
23:30how easy it is for people from that
23:31location to go and move to other
23:32locations. Um so I think both uh both
23:35are there. On the consumption side, I
23:37think what we think about is a little
23:38bit more on things like um preferences.
23:41So for example, maybe just empirically,
23:44people seem to have very strong
23:45attachments to where they're born. Um,
23:48what does that mean? It probably is
23:50proxying for like your family
23:51connections and things like that. So,
23:53people who were born in one location
23:54seem to want to live there much more
23:56than people who were born in in another
23:57location. Um, and you see that both, for
24:01example, when people migrate once, they
24:03often come back to where they were. And
24:05so, it doesn't look like it's just a
24:06migration cost. It looks like it's
24:08something that's specific to where
24:10you're born. And so, that kind of maps a
24:12little bit to the consumption side. But,
24:14yeah, exactly. That could be these sorts
24:16of either production or consumption
24:17sites again might be something that keep
24:20people where they are and mean that it's
24:22either a skill or a preference. And so
24:25again, if we think about potential
24:26policies or potential equilibrium, we
24:30might want to account for that.
24:33Yes.
24:36Perfect. Okay. Okay. So, there's lots of
24:39explanations. So what we're going to do
24:40is we're going to put all of these are
24:43super easy to put into the spatial
24:44model. Um I'm going to give you a
24:46baseline model and I'll sort of flag you
24:48know where you can add each of these. I
24:50you can flag where you could add um it
24:53would be pretty easy to add a skill
24:55specific or you know capital again you
24:57could just think a little bit about
24:58that.
25:00Okay so let's start then going through
25:02the model. Um we're going to start with
25:04a case of exogenous prices because um
25:07what we want to understand is what does
25:09it really mean to be in a spatial
25:12equilibrium. So if we see that wages are
25:15not equalized, does that mean we're out
25:17of equilibrium? Does it mean that we're
25:18not out of equilibrium? You know, what
25:20should we actually expect to see in
25:22equilibrium? And so let's sort of work
25:24through that so that we can sort of
25:26understand these gaps and then we'll go
25:29and we'll endize for example the wage
25:32and the housing things like this so that
25:34we could then think about these other
25:36sets of questions is from an initial
25:38spatial equilibrium how do we adjust to
25:40a new spatial equilibrium if something
25:42fundamental in the economy is changing
25:47okay so and let's start again I think
25:49the once you have the intuition of these
25:51how these models
25:53um you're sort of able to do whatever
25:55you want. So the lecture today is on um
25:58purposefully quite simple because I
26:00think once you have the intuition you
26:02can really easily scale it up. Uh and so
26:04rather than um go through a lot of heavy
26:06math I want to kind of work on making
26:08sure everyone's understanding what's
26:09going on. Uh and then you can really
26:11easily just add in sort of the more
26:14complicated parts where you want to get
26:15there.
26:17Okay. So what is the baseline model? The
26:19baseline model is we're going to have
26:21two locations. We're going to have
26:22location A, location B.
26:27Each location is going to have a wage.
26:31It's going to have a cost of living and
26:34it's going to have a level of amenity.
26:37Okay, so wage, rents, amenities. Here,
26:40let's just assume that your nominal that
26:43your real wage is wage minus rents. Of
26:45course, um you could sort of and again
26:48we're writing down an indirect utility
26:49here. You can microfound something that
26:52looks like this in direct utility by
26:54giving people preferences for example
26:56over housing and non-housing consumption
26:58and giving them a budget constraint and
27:00then taking first order conditions. You
27:02can microfound something that looks like
27:04this. If you do that, what you would
27:06often end up with is one more
27:08coefficient here where you have a share
27:10of your income that you might spend on
27:12housing or the relative weight of
27:14housing versus non-housing in your
27:16utility function. But just to keep
27:18things simple, I have uh left that out.
27:21Okay. So, nominal wages are just wages
27:23minus rents. You get extra values of
27:25amenities. And then the piece that's
27:28actually the most important to
27:30understand is this little shock, the
27:33epsilon at the that affects your
27:35utility. And so this thing is important
27:39because what it is introducing is first
27:42of all heterogeneity across people. uh
27:45everyone agrees that the wage, the rent,
27:48the amenities, everyone agrees what they
27:50are and the simple model, everyone has
27:52the same valuation of those. The reason
27:54that people different is that they have
27:56different idiosyncratic preferences for
27:58locations. Okay. So we have um a piece
28:03that's common to A. So what I'm going to
28:05say is the indirect utility for person I
28:08who's living in location A is this
28:10common piece that everyone agrees on and
28:12their idiosyncratic piece. And then you
28:15can write the same thing for location B,
28:17the common piece and this idiosyncratic
28:21term. Um, and actually going now I'm
28:23just going to use these VA terms. And so
28:27anything that you think affects utility
28:29you can just stick in there. The rest of
28:31the model is just going to assume that
28:32there's something a bundle of u prices
28:36and wages at the destination. You can
28:38sort of put you can go back and put in
28:40whatever you whatever you want into that
28:41term.
28:43Okay. So these terms here
28:46are going to be quite important for both
28:48understanding conceptually what a
28:50spatial equilibrium is and then it's
28:52also important for separating out the
28:54idea of
28:56what is what is what's actual welfare of
28:59people who are choosing to live in
29:00different locations. How much of that is
29:02something that again is potentially
29:04policy relevant and then how much of
29:06it's just coming because people have
29:08these preferences where you know I might
29:11be someone who loves living in cities.
29:13No matter how much you pay me I don't
29:14want to move to rural Nebraska
29:19and so my idiosyncratic shock is just
29:21going to be super high. So I'm going to
29:22make decisions that look like I'm
29:24willing to trade off high income to live
29:26somewhere else. Um, and there's going to
29:28be very little you could actually do in
29:30terms of policy to change my mind if I
29:32have a really really strong preference
29:34uh for being in a city. Okay. So, these
29:36terms here
29:38um can be lots of different you can give
29:41a different interpretation to them. For
29:43now, let's just assume that they're
29:45preference terms. So, like I you know
29:47what I was just saying. I like living in
29:50San Francisco. I want to live there no
29:51matter. I'm willing to live there
29:53relative to living almost anywhere else
29:55because I I really like it. And so, so
29:57I'm willing I have a very high draw for
29:59this. If you don't want this to be a
30:01preference term, you could also think
30:03about this potentially being a
30:04productivity term in which it would map
30:07into something like I like San Francisco
30:09because I'm really productive at the
30:11type of work in San Francisco. And so
30:13that heterogeneity is instead of being
30:15just a preference uh dimension, it's
30:18going to come and be something like a
30:20productivity difference. Um, and in
30:22exactly the same term we'll talk about
30:24in a little bit, but just flagging it so
30:25you start to think about when you have a
30:27trade model, you have a term like this,
30:29which is thinking about sort of all of
30:31the goods that you could produce,
30:34um, how productive are you at in a
30:36location at producing each one, it's
30:38going to be kind of something random
30:39like that that again is going to sort of
30:41choose, you know, depending on what you
30:43can buy in and what you can buy out,
30:45you're going to make things that you're
30:46relatively productive at. And so the
30:49shocks and the trade model end up also
30:51sort of coming in and playing a really
30:52important role.
30:54Okay. So indirect utility. Um so we're
30:57going to have just something that's
30:58common to A, something that's common to
31:00B. Um you could put in whatever else you
31:03wanted there, the cost of living. You
31:04could put in um you know, you don't
31:06spend 100% of your income on on rent. So
31:09you would have uh a proportional pass
31:11through of your rent into your utility
31:13sort of whatever you want. But this is a
31:14good sort of place to start.
31:17[snorts]
31:20Okay. So if you can freely move, so
31:22let's say that there are there's no cost
31:25to moving and then this is our model.
31:31So our model is that you're maximizing,
31:35you're choosing where to live. These are
31:37your options.
31:39You can live in location A. You can get
31:41this common utility of A plus your
31:43drawer. You can live in location B. You
31:45can get the common piece of B plus your
31:47drawer and let's say people can freely
31:50move and so people are choosing to live
31:52in the location where we're there. These
31:55are four objects.
31:57What is going to be equalized across A
32:00and B in equilibrium?
32:02Let me just give you a minute to think
32:04about that.
32:06In equilibrium, which of these objects
32:08do we think are going to be equalized
32:10between A and B?
32:24Okay. Does anyone want to um
32:29take a guess? Does anyone want to give
32:31me some um arguments for either things
32:34being equalized or things not being
32:36equalized for any of these?
32:42So, everyone could do whatever they
32:44want. They can move wherever they want.
32:46Um, I guess at the moment, um, let's,
32:51uh,
32:52let's just I mean, let's just keep with
32:54this idea that we have exogenous
32:56returns. So, let's kind of rule these
32:57two out. Maybe let's just rule these two
32:59out for now because we haven't talked
33:00about the other pieces of the model. But
33:03I think what's kind of important is what
33:04do we think might be going on with what
33:07you could observe
33:09about what looks like the utility of
33:11living in each location and then what
33:13what welfare actually is based on a
33:15based on what people's choices are.
33:26I don't really get to see I can ask you
33:28to I can ask you to put your hands up
33:30but it might be uh might be gets to earn
33:34their privileges.
33:37Neither of these are necessarily going
33:38to be equalized. It's the marginal agent
33:40that matters. And so the marginal person
33:42is going to have welfare equalized
33:45between the two locations.
33:47Okay? So there's going to be one comment
33:48about the marginal, not the average.
33:51Okay. So if we think that the marginal
33:53guy is go or the marginal girl, the
33:54marginal person is going to be
33:56equalized,
33:58what does that mean then when we start
33:59looking we start looking at these
34:01pictures? um you know so you often hear
34:04stories about places that are
34:05underdeveloped, places that are more
34:07less developed. If it's the case um that
34:10we don't expect to see in equilibrium,
34:12anything like this equalized, you know,
34:14how do we understand or how do we think
34:15about whether or not we're in a special
34:17equilibrium when we just look at the raw
34:19data.
34:28So for example, sometimes people look at
34:30these wage gaps and they say the fact
34:31that we have these differences in wage
34:33gaps, that means we can't be in a
34:34spatial equilibrium because if we were
34:36in a spatial equilibrium, for example,
34:38real wages would be equalized.
34:41So maybe just thinking about that
34:43argument, if we were in a spatial
34:44equilibrium, real wages would be
34:46equalized. The fact that we see real
34:48wages not equalized,
34:50therefore we're not in a spatial
34:51equilibrium.
34:54Okay, I don't know how how well is
34:56working being interactive. Why don't you
34:57think about if do you agree with that
34:58statement? Do you not agree with that
34:59statement now? Then let's come back
35:01after we've gone through the model and
35:02then see if you agree with that
35:04statement or don't agree with that
35:05statement. Okay. So that's sort of what
35:08this slide is saying. It's saying you
35:10know if you see that real wages differ
35:15does that imply that we're not
35:17equilibrium?
35:20If you see that perhaps that could be
35:21wages and rents. If you see say you say
35:23like okay when real wages are not
35:24equalized because of compensating
35:27differentials with things like
35:28amenities. If you see then that um you
35:31know amenity adjusted real wages are
35:32equally uh uh different
35:39amenity adjusted real wages are
35:41different. Does that then imply we're
35:42not in equilibrium
35:46or can we still be in equilibrium even
35:48if those real wages or amenity adjusted
35:50real wages um are not equalized if we
35:54are in the case where we might not have
35:56equalized and so that's these these
35:58composite terms right these VAB if we
36:01are in the case where VA and VB are
36:03still different across space do we
36:05expect that average welfare for people
36:07living in each location
36:10do we expect that thing is going to be
36:12equalized
36:17or what even is this term here? Does
36:19anyone have any thoughts about this
36:21average welfare term
36:23that they want to think they want to
36:24share at this stage? Do we think average
36:27welfare is going to be equalized?
36:30You're it's correct that the the
36:31equilibrium is going to be when the
36:33marginal person is indifferent. But
36:35given um the marginal person being
36:38indifferent between two locations, can
36:40we say anything about the average
36:42welfare for people who have sorted into
36:44each of the locations?
36:46So the answer will depend on the joint
36:48distribution of the epsilons.
36:52But we can't say anything for sure
36:54without knowing that.
36:56Okay. Awesome. Okay. So we'll go through
36:58this intuition but the answer is exactly
37:00going to depend on the joint
37:01distribution of the epsilons. Sometimes
37:03you can say something about this.
37:04Sometimes the answer is yes average
37:06welfare is equalized. Um for example um
37:09a common way that we think about the
37:11distribution for these epsilon shocks
37:12we'll talk about is the fresh a
37:14distribution. Under the fresh a
37:16distribution you sort of have a knife
37:17edge property where the expectation of
37:20the shock condition on choosing it is
37:22going to be such that this term is
37:24actually equalized on average. people
37:26who are living in one location on
37:27average have the same welfare as people
37:29who are living on other locations. But
37:30that's a very knife edge property and
37:32you can come up with other other joint
37:34distributions between the shocks where
37:35you're not going to get that. Okay. So,
37:38so let's go through that. But that's
37:40exactly kind of what's going on. So it's
37:41going to be really difficult and which
37:43is why being careful and precise about
37:47what we think is going on with these uh
37:48error terms is a really big part of
37:51understanding um what you know what do
37:53we think is even equalized across space.
37:56What is this equilibrium concept? Um can
37:58we say anything about whether or not
38:00people would be better off moving on
38:02average and then how do we trace back uh
38:04that effect?
38:06Okay, any questions about these is the
38:08difference between these quantities
38:09before we keep going.
38:15Okay, we're going to and we're going to
38:17come back and we're going to endogenize
38:18wages and rents. Um so we will
38:21endogenize those. But sort of giving you
38:22a precursor, it's going to be very
38:24possible that you don't have um we're
38:27going to be able to have an equilibrium
38:29where we see different wages rents, we
38:31see different rail wages, we see
38:33different amenity adjusted rail wages.
38:35Therefore, we see different levels of
38:37these observed utilities and we also see
38:40different average uh welfares across
38:42space and then we're still in spatial
38:44equilibrium. Okay. And there can be
38:46other spatial equilibrium where we see
38:48that everything is equalized.
38:51Um but it's not necessarily the case
38:53that uh you're going to have any of
38:55these quantities equalized if you're in
38:57a spatial equilibrium.
39:00Okay. And it's and it's important just
39:02to have the differences in your mind.
39:03So, please do jump in if you want to
39:05because when we're really thinking about
39:07the impact of policy, usually we're
39:10probably talking about this concept.
39:12When you're plotting data, if you could
39:15figure out a way to observe income,
39:16you're probably thinking about this
39:18concept. If you can't observe things
39:20like amenities, you're probably thinking
39:21about these concepts. So, these are all
39:23different ways that we could think about
39:25what looking at what's going on in the
39:26data. And so it's helpful to sort of
39:28understand um these different components
39:31when we're thinking about um who what
39:34who is gaining what from living uh in
39:36what location.
39:39Okay. So um again let's just go through
39:42a very simple model. I think this
39:44picture here you can do a lot with even
39:46though it's very simple. So again, I
39:47want to build your intuition because if
39:49you have the intuition, you can then
39:51just sit down and you can go through
39:52very complicated math and you'll be able
39:54to understand what's going on in these
39:56models. So in our simple case, you're
40:00going to live in location A if your
40:05utility of being for person I if their
40:07utility of being in location A is higher
40:09than the utility of being in location B.
40:12Okay, so what does that mean? It means
40:14that you would live in location A if
40:16your idiosyncratic shock is bigger than
40:20your other idiosyncratic shock um with
40:25any gaps uh here. Okay. So for example,
40:29this is a line where let's assume that
40:32the two places have an equalized
40:35um observed utility term. So this is
40:38just showing this is just the line EA
40:41equals EB. Okay. So if that's the case,
40:44what you can see is that people who have
40:46a a realization of their idiosyncratic
40:49shop for location A. So anyone who is
40:51living up here, anyone who is in this
40:53part of the figure is going to choose to
40:56move to location A. Anyone who's in this
40:59part of the figure is going to choose to
41:00live live in location B. Okay. If it
41:04were the case that um location A on
41:07average had a higher utility than
41:09location B, that's really easy. you
41:11would just draw in a new line here. And
41:13so if it's the case that V epsilon B +
41:17VB minus VA, so VA is bigger than VB,
41:23that would just shift down this line.
41:25All that means is that
41:28these people here who would have lived
41:29in location B are now going to move to
41:32location A. Okay?
41:35Okay. And you could do the same thing if
41:36it were the case that on average utility
41:39was higher in location B. Then that
41:42would just shift up this this uh sort of
41:44marginal line here. You would be up
41:46here. And so now only people who have
41:49really high realizations of A are going
41:51to move there. Everyone else is going to
41:54move to location B because uh on average
41:57they earn more even if they have a um
42:01relative shock for location B that's
42:03lower than for location A.
42:06Okay. So what does that mean? What you
42:08can sort of see immediately from this is
42:10that these are the people who are going
42:12to sort into location A.
42:15These are the people who are going to
42:16sort into location B. So what we want to
42:20think about is you can think about the
42:22fact that you have to kind of take this
42:24this is what we were talking about the
42:25difference between welfare and util uh
42:27and average observed utility and actual
42:30welfare. So you can see all these people
42:32who live in A are going to be
42:33heterogeneous. They have different
42:34levels of their shock. And so when
42:37you're thinking about their average
42:39utility of being in A, it's going to be
42:41the average of the common piece plus the
42:43average of this idiosyncratic piece. If
42:45the utility of people in B, it's going
42:47to be like this. Um, so this figure
42:50again here I haven't put any
42:51distributional assumptions on on the
42:53epsilons. This is just sort of plotting
42:55it for the generalized model for any
42:57sort of value of different shocks. You
42:59could think about these being
43:00productivity shocks. So this
43:02automatically tells you something you
43:05know for example if this were were a
43:06productivity model
43:09and you imagine we start off in a case
43:12where A and B and equalized.
43:15So this is the case for VA equals VB.
43:18Let's imagine we're moving to a new
43:20equilibrium
43:21where VA is bigger than VB. These are
43:25going to be the marginal people here
43:28who would prefer to live in if wages
43:31were equalized they would have lived in
43:32location B. They now shift into location
43:35A. You can sort of automatically see
43:37something about the selection that's
43:38going on there. The people that are
43:39marginally switching are going to be
43:41those who have relatively lower product
43:43uh lower idiosyncratic draws for
43:46location A. That could mean that
43:48therefore they have lower idiosyncrat
43:50they like location A less. So they're
43:51going to be less attached to location A
43:53when they move in. If you wanted to
43:55interpret these shocks as productivity
43:57chores, what that means is that the
43:59people who would start living in
44:00location A are going to be less
44:02productive in location A because if they
44:05were really productive, they would have
44:06moved there already. These are only
44:08people who are moving because on average
44:10uh wages are higher.
44:14Um so any questions about sort of
44:16understanding understanding this figure?
44:25So again here I've drawn I've done the
44:27figure for um I've just put in a linear
44:30utility. The other way you sometimes see
44:32these sorts of models written
44:35is you might have something that's
44:36multiplicative. You can do the same
44:38thing right you would live in location
44:39A. If your utility of being in A is
44:42higher that means you live in A. if a is
44:46bigger than the ratio.
44:48Oops, this is a b.
44:51And so you just can draw exactly the
44:52same figure. Um, thinking about, you
44:54know, the slope of the line relative
44:56here, this this uh intercept is just
44:59shifting the line up and down and
45:00telling you who is either moving in or
45:02moving out.
45:05Okay. So, um, one way we can just, um,
45:09make progress, let's sort of, again,
45:10we're going to progressively build this
45:12up. I want to make sure that everyone's
45:14intuition is on the same page. So let's
45:16start by just assuming that the
45:18difference we'll we'll make a parametric
45:20assumption. Um let's start by assuming
45:23that the difference in your uh draws
45:25between location A b and a is a uniform
45:28distribution on support negative s to s.
45:31Okay. So let's start. This is negative
45:34s. This is s
45:37this is zero. Okay. Okay. So what we can
45:39do is we can if this were the distrib
45:41the joint distribution between A and B
45:43this also gives us a way to visualize
45:45this selection of who is choosing to
45:47move into location A or who is choosing
45:49to move into location B. So if it's the
45:52case that you had a picture like this
45:53again, you're going to live if you just
45:56do exactly the same simple um algebra,
45:59you're going to live in location A if uh
46:03the relative return is enough that you
46:05have a shock that's big enough given the
46:06g the gaps and differences. So for
46:09example, let's do the case where
46:11location A is bigger than location B.
46:15Okay. So let's imagine that this is here
46:20V A minus VB it's positive. Okay. So
46:24what does this say? It says that if this
46:27is the difference between A and B. Then
46:29these are the people here. Everyone who
46:32lives here is going to go work in
46:33location A. These are the people with
46:35the relatively small shock for location
46:37B. These are the people here who are
46:40going to go live in location B.
46:46So these people choose B, these people
46:48choose A. And then that's going to be
46:50one way that you could start then to
46:52think about the average of your um
46:54unobserved shot given that you choose
46:56it.
46:58Okay. So what happens if you have a
47:00support a uniform we're just doing a
47:02very simple case here where there's a
47:03uniform support on S negative S. What
47:06happens if instead you have uh this
47:09being a larger support? So you could
47:12imagine a world where people's
47:14preferences are pretty tightly
47:15distributed. You could imagine a world
47:17where people's preferences are less
47:19tightly distributed. So again, if we
47:21just draw in the same thing, this is
47:24zero. Let's draw in that line before
47:27that we had.
47:29So VA is bigger than VB. Okay.
47:34So who is going to live in location A?
47:37The people who are going to live in
47:39location A. If we're in the first world
47:42are all these people there, the people
47:45who are going to live in location A in
47:46the second world are these people here.
47:49Okay. So what you can see is that as
47:53you had uh preferences that were going
47:55to become more and more disperse, what
47:58you can sort of see hopefully
47:59graphically is that this relative
48:02importance of the actual difference in
48:04utilities across the two locations
48:06becomes less important for explaining
48:09uh people's locationational choices.
48:11Um, so maybe to see that super clearly,
48:13imagine that if you want to just sort of
48:16see that in a way that really convinces
48:18yourself if you don't quite see it,
48:21we have this is equal to zero. Let's
48:24imagine that the difference between
48:26location A and location B is exactly
48:28this line here. Okay, so this is VA is
48:32bigger than VB and let's say it's
48:34exactly equal to whatever that that
48:35difference there is on that uniform
48:37distribution. So if that were the case
48:39that we had um this is our realization
48:42this is our distribution of the shocks
48:44in that first world 100% of the people
48:47would go and live in location A but you
48:50can see in the second world
48:52you wouldn't have 100% oops you wouldn't
48:55have 100% of people living uh living
48:57there because you're going to see that
48:59some people so in the first case you
49:02would be like this in the second case
49:05you would have these people going but
49:07these people here would still choose to
49:09live in location B.
49:12Okay.
49:15So, what you can see hopefully here is
49:17that one of the really important
49:19parameters for thinking about how
49:20responsive people are to shocks. It's
49:22going to be something that measures the
49:25strength of how important their
49:26idiosyncratic piece of their realization
49:28is relative to the common uh el piece.
49:32So you can sort of see here that if you
49:34have preferences that are really
49:36dispersed, so the the support is really
49:38wide, what that means is that even if it
49:40looks like you could earn 10 times more
49:42in location A, you're going to have some
49:44people who have such a strong preference
49:46for living in location B that they're
49:49going to move there, even if it looks
49:51like there's a they're leaving a lot of
49:52money on the table. If it's a case, you
49:55know, that everyone, for example, almost
49:56exactly a agrees on where nice places
49:59are, where not nice places are, they
50:01don't have too much heterogeneity in
50:02their preferences, then you should
50:04expect to see many more people
50:05proportionally living in location A. And
50:08so what does this mean? This is um here
50:10it's I've sort of put it on a uniform
50:12distribution, but when you start using
50:15um for example extreme value
50:17distributions, this idea maps to what we
50:20call a shape parameter, which is
50:22measuring in some sense um the the
50:24variance of these shocks and it's going
50:26to be really important for thinking
50:28about things like what is your migration
50:29elasticity to a shock. So if it's the
50:32case that people uh agree that a play
50:36everyone agrees that a that they only
50:39want to get indirect utility you're
50:41going to see that people should be much
50:42more responsive to a change in that
50:44indirect utility. If it's the case that
50:46there's a lot of heterogeneity, it might
50:48look like people are not migrating in
50:50response to, for example, a productivity
50:51shock because what they're actually
50:53doing is they're accounting for both the
50:55piece that's common and their own
50:58idiosyncratic piece. And if there's a
50:59lot of distribution there, then what
51:01that means is that you might be less
51:03responsive. You're going to see that
51:04people are less responsive. Um, and so
51:07that's going to make in some sense the
51:08labor supply to that location uh look
51:12less responsive. For that reason, what
51:14you sometimes see is that this idea of
51:17the the variance of the underlying
51:19distributions, sometimes you'll see that
51:21referred to as a migration cost because
51:23it's something that makes people sort of
51:25sticky when they're wanting to move and
51:27respond to shocks.
51:30So, let me just pause there for a
51:32minute. Does that um you can sort of see
51:35two things from this figure. First, you
51:37can sort of see that the size of the the
51:39the underlying shape of the of the
51:41preferences is going to be really
51:42important for thinking about who wants
51:44to move, who doesn't want to move. The
51:46other thing I want you to see from this
51:48picture is when we remember we're
51:49talking about this welfare term.
51:53The welfare term is going to be
51:57what is
51:59the combined piece that's common and the
52:02piece that's not observed given you
52:04choose to live there. Okay? So you have
52:05to think about accounting for the
52:07selection of who is choosing to live
52:08there. For example, what is the average
52:11if you're looking at this one here? What
52:13is the average realization of the shock
52:15if you're moving in A? What is your
52:16average realization of the shock if
52:18you're moving in B? And we need a way to
52:19think about how to compute those to get
52:21average welfare for people who are
52:23living in the two locations.
52:30So let me sort of pause if there if
52:32there are any questions there before
52:34going on.
52:41Good. Okay. Okay. Cool. So, this kind of
52:45uniform difference in the two locations
52:47and if you want to read the Meridi
52:48handbook chapter, this is how he works
52:50through the model. Um, I put the
52:53references at the end. Um so if you want
52:56to use what's the advantage about making
52:58this kind of a little bit weird
53:00assumption that the differences in the
53:01preference shocks across the two
53:03locations is uniform. The main advantage
53:05is that you can go through and you can
53:07get a closed form for the model. Um and
53:09so if you really want to compute each of
53:11the endogenous quantities you can do
53:13that by making this quite um you can
53:17make this kind of simple distributional
53:18assumption. If that's something that
53:20helps you to like understand how all of
53:21these quantities um work, then have a
53:24look at the handbook chapter and you can
53:26work through that. You can sort of see
53:28though that the obvious challenge is
53:30that with only two locations, you know,
53:32we we're not going to be able to get too
53:33far. If we have our spatial equilibrium
53:36only being two locations in the real
53:38world, we probably think that there are
53:39more. So, let me put up this slide just
53:41to show you the intuition. Um and then
53:44the next slide I'm going to introduce
53:45the extreme value shocks which is a sort
53:48of more standard way to think about the
53:50distribution of these um idiosyncratic
53:53shocks that lets you sort of scale up
53:55from two to to n locations. Okay. So
53:59again what are we doing here? Person I
54:02will choose to live in location A if
54:04they're um if they're utility of being
54:08in A is bigger than the utility of being
54:10in B. What does that give us? If you
54:12just rearrange it, it just gives us
54:13something on the difference in the
54:15shocks. And if we've made a parametric
54:18assumption that this thing is
54:19distributed uniform, then it's really
54:22easy um to use the um formula for
54:27understanding what the proportion we
54:29want to sort of understand which of the
54:30people you know we have this figure
54:33here. We want to think about what
54:34proportion of people are going to have a
54:36shock such that they choose to live in a
54:38so we just calculate the probability
54:40that your your difference in your shocks
54:43is less than this value. Okay, if you've
54:45got this uniform distribution, we're
54:47just plugging in.
54:49This is our X.
54:52We have S to S, right? So, we're just
54:55saying X - A over B minus A. What we're
55:00doing there is we're doing VA minus VB
55:04minus min - S over S - - S. So what
55:11that's saying is that we can just
55:13rearranging that
55:17we have VA minus VB over 2S
55:21plus a half. Okay. And so this is
55:24exactly that intuition I was telling you
55:25before. What's the probability that you
55:28live in A? If it's the case that people
55:31just have infinite preferences and so S
55:33is really really large, what you're
55:35going to say is that just half the
55:36people live in A, half the people live
55:38in B, no matter what the differences in
55:41the utility of being in A and B are. If
55:44it's the case that people are this
55:46idiosyncratic piece is not too
55:48important, what you can see is that the
55:50smaller
55:52the in some sense the variance of that
55:54shock is, the more likely you are to
55:57live in location A. Okay.
56:00So this is sort of just exactly showing
56:02you for this particular distributional
56:04assumption how you can figure out what
56:07is the equilibrium allocation of people
56:10given the utilities of being in both
56:12locations. Um and then here it's again
56:15circling in and so showing you you know
56:18why this
56:20where the idiosyncratic distribution is
56:23going to come in and it's going to be
56:25really important for sort of
56:26understanding. You can sort of imagine
56:28holding things constant. You could take
56:30derivatives of this to ask what's the
56:32partial equilibrium response if for
56:35example VA goes up wages go up in
56:37location A. Of course we wouldn't want
56:39the partial equilibrium. we'd want to
56:41ch, you know, track through and
56:42understand, you know, if VA goes up,
56:44what happens in equilibrium to VB? And
56:46we'd want to work that out. But you can
56:48sort of see that super clearly from this
56:50um expression
56:52that the responsiveness of the
56:55population to changes and shocks is
56:57going to depend on the relative strength
56:59of your um underlying idiosyncratic
57:02shock.
57:06Okay. So, any questions about about
57:08that?
57:12It's okay.
57:13It's a little hard for me to see
57:14everyone. So, just um please don't be
57:16shy. Either talk to me or write
57:19something in the chat if you need to.
57:23Um okay. So,
57:26um
57:28okay. So, is it a spatial equilibrium?
57:30So what we just worked out is we said
57:32you know to order to figure out a
57:34spatial equilibrium um it's exactly this
57:37intuition that
57:41the equilibrium concept here it's going
57:43to be the marginal person who is going
57:45to be indifferent. You're going to
57:46migrate
57:48up until the point
57:52oops
57:54where
57:56on the margin everyone has these
57:58different um productivity drawers. So
58:00the marginal person who is indifferent
58:01between location A and location B is
58:04going to be the definition of the
58:05equilibrium. Um it's not going to be the
58:08case that the average is average person
58:10is going to be indifferent because
58:11people have these different uh
58:14idiosyncratic uh shocks. Um but it's
58:17going to be an equilibrium because
58:19everyone who has a shock higher than a
58:21relative shock higher than that marginal
58:23person is happy to live in location A.
58:25Everyone who has a relative shock lower
58:27than that marginal person wants to live
58:29in location B. No one wants to change
58:31location. And so if once we end up with
58:35an equilibrium where we would have
58:37equilibrium like constant prices, we'll
58:39sort of talk through the definition in a
58:40few slides. A set of um prices that
58:43support the number of people once you're
58:46there, no one wants to move. That's
58:47going to be our concept of of
58:48equilibrium. Okay.
58:51So, so this is sort of coming back to
58:53that slide here. We had different
58:56observable
58:57um I let me call it common utility. So
59:01these remember this VA was the bundle of
59:03wages minus rents plus amenities.
59:07You could put whatever other terms if
59:09you wanted to have the cost of living
59:10you could just put that cost of living
59:12term in. You know you can you can sort
59:13of see that you could easily put in
59:15different pieces here. You could you
59:17know you could imagine that people have
59:19type specific amenities that can easily
59:21go into that that kind of common VA
59:24piece. So you can have a difference in
59:27the observed average utility across
59:29locations. That would mean that more
59:32people on average should live in the
59:33place that has the higher real wages.
59:36The equilibrium is going to be based on
59:38the person who's the marginal mover. So
59:40it's going to be that the marginal mover
59:42is indifferent, but we're not going to
59:44have that the average migrant is
59:46indifferent because people have these
59:47different realizations. So when we look
59:49at the data, what we're going to see is
59:51that we'll see that there's a spatial
59:53difference in VA, VB.
59:55That's not going to let us answer
59:57exactly the question of are people
59:59better off living in location A versus
1:00:01location B. For that, what we want to
1:00:04think about is we want to look at
1:00:06what is the expectation of their shock
1:00:08conditional on choosing to live in
1:00:10location A. what is the expectation of
1:00:12the shock of location B for people uh um
1:00:17who choose to live in location B and
1:00:18that's going to be the thing that we'd
1:00:20want to calculate there's nothing that
1:00:22makes the model have that those two
1:00:23things are equal although often you get
1:00:26these knife edge case knife edge cases
1:00:28with for example the fresh air
1:00:30distribution where those two things end
1:00:32up being equal just because of the
1:00:34properties of the uh distribution
1:00:37okay so kind of coming back to that very
1:00:41first set of things that we're trying to
1:00:43think about wages. We can have a spatial
1:00:46equilibrium even if wages are different,
1:00:48even if rents are different, even if
1:00:50average um observed utility is different
1:00:53and even if average welfare of people
1:00:55living in the location is different. All
1:00:56of those things can be different and
1:00:58we're still in spatial equilibrium
1:01:00because no one wants to be living in a
1:01:02different place than the place that they
1:01:03have chosen.
1:01:10Okay.
1:01:12Okay. Any questions?
1:01:14Is it Sally? Is the chat okay?
1:01:18There's nothing.
1:01:19Yeah, most Doug and Joe are especially
1:01:22good at answering questions. [laughter]
1:01:24I see. Okay, good. Okay, so that was the
1:01:28case for two locations. What happens
1:01:30when you go to in locations? I think the
1:01:33really nice thing of being clear on the
1:01:35intuition of these models is that now
1:01:38it's pretty easy to go up to in
1:01:39locations. You don't really need to
1:01:41change how you're thinking about the
1:01:42framework. You know what we're going to
1:01:44now have is we're going to have like
1:01:46relative distributions. For example,
1:01:48what's your relative shock in living in
1:01:49location A? There's probably going to be
1:01:51it's not going to be exactly as simple
1:01:53as this, but there'll be some kind of
1:01:54you can think about it. There's some
1:01:55threshold. People who have really good
1:01:57location uh shocks are going to be the
1:01:58ones who move there. people who have
1:02:00lower shocks are not going to be the
1:02:01ones who move there. What we do to
1:02:03operationalize the idea that there are
1:02:05more than two locations in the world is
1:02:07that we make um some distributional
1:02:10assumptions. The ones that you're going
1:02:12to see the most often uh extreme value
1:02:15shocks. So you're either going to see
1:02:17type one shocks or type two shocks. So
1:02:21gumball shocks or fresh shocks. They are
1:02:24all they're both in the same family.
1:02:25They're related to each other. So the
1:02:27log of a type two shock is a type one
1:02:29shock. Um and so what you're going to
1:02:32see is that um this is the CDF for a
1:02:36type for a gumbo shock. This is a CDF
1:02:38for type two shock. What you're going to
1:02:40see is things that say you maximize for
1:02:43example VI plus
1:02:46um your shock for location I and then
1:02:48you'll say let's assume that those
1:02:50shocks are extreme value type one. Let's
1:02:52assume that those shocks are extreme
1:02:54value type two. Okay. So what's the
1:02:58difference between them? The difference
1:02:59is not a particularly big difference.
1:03:01It's just whether or not you're assuming
1:03:03that the shock is multiplicative or it's
1:03:06um additive. That's essentially what
1:03:08kind of comes down to when you see these
1:03:09models um written. But often when you
1:03:12write them in an additive term, this is
1:03:14a log additive thing anyway. And so it's
1:03:16going to be isomorphic to this way of
1:03:19writing it which is multiplicative. Um
1:03:22so what are these trucks? So this is you
1:03:25can assume that the shocks are iid. You
1:03:27can assume that there might be
1:03:28correlation across the people's shocks.
1:03:31Um you can put quite um you know people
1:03:35might have uh correlated you know
1:03:36someone might be absolute have absolute
1:03:39like someone might have just really high
1:03:41preferences in general. Um and so all of
1:03:44their shocks are really highly uh
1:03:45high-end levels and then correlated with
1:03:48one another. People might have low
1:03:49shocks. You can have different copulas
1:03:51that combine shocks from different
1:03:53locations. If you want to think about
1:03:54sort of decoupling um the absolute value
1:03:58of the shock from the relative value of
1:04:00the shock um this is a place where
1:04:02there's sort of quite a lot of work on
1:04:04making the right distributional
1:04:05assumption. Um so if you end up working
1:04:07these models you can do a little bit of
1:04:09reading around there but basically the
1:04:11intuition is going to go through kind of
1:04:13whatever shock you choose. So often
1:04:15people are choosing here it's common you
1:04:18just see iid shocks. What does that
1:04:21mean? You're getting each person
1:04:23independently draws one shock for living
1:04:26in San Francisco, one shock for living
1:04:28in New York, one shock for living in um
1:04:31Austin and then they choose how to live
1:04:35based on the realization of that shock
1:04:37and then these extreme value shocks are
1:04:38very nice because you can get closed
1:04:40form um predictions for how to solve
1:04:43those models. I'm not going to go
1:04:45through the math for how to do those um
1:04:47just you can I mean it's quite easy. you
1:04:50just sort of do exactly this idea.
1:04:53What's the probability that the shock is
1:04:55going to be less than um the shock you
1:04:58just want to calculate the probability
1:05:00that your shock that you choose is less
1:05:02than something else and then you have
1:05:04the CDF to do that. So you can sort of
1:05:06derive these migration probabilities
1:05:08fairly uh easily uh once you once you
1:05:10think about setting it up like that. But
1:05:13what does it mean? What you end up
1:05:14getting is that you end up getting um
1:05:17migration well I'll call them migration
1:05:19rates or uh location you know living
1:05:22location choices that have got these
1:05:24really nice closed forms. So you either
1:05:27get something that looks like a loit or
1:05:29you get this uh version that is the fra
1:05:31version where what we want to think
1:05:33about is we want to think about this
1:05:36observe this bundle of the utility of a
1:05:38location remember that's the function of
1:05:41for example the wage the rent the
1:05:43amenity
1:05:45um and you get the probability that the
1:05:48person wants to live in location A is a
1:05:51parametric uh function of whatever the
1:05:54bundle of how good A
1:05:56relative to what all your other choices
1:05:59were. Um how good it was to uh to live
1:06:02in any other location. Okay. So
1:06:04depending on whether or not you're going
1:06:06to be looking at applications that are
1:06:09to do with labor or to do with trade,
1:06:13you may be more likely to see formula
1:06:15that look like this or you may be more
1:06:17likely to see them that look like this.
1:06:19Um but they sort of come this is all
1:06:21derived from that distributional
1:06:23assumption on those shocks. Um, and so
1:06:26you can sort of have this intuition of,
1:06:28you know, what is in that shock that's
1:06:30maybe making it go be more dispersed?
1:06:32What is in that shock that's making it
1:06:34be less dispersed? The other nice thing
1:06:36about these shocks is that, for example,
1:06:38especially with a fresh, you can you can
1:06:40easily
1:06:43calculate this item, right? This is what
1:06:45I've been talking about. When you want
1:06:47to calculate the average welfare for
1:06:48people who've chosen to live in a
1:06:50location, you need to know the location,
1:06:52the piece that's common plus the mean of
1:06:55the shock given that you choose to live
1:06:56there. Um, and so what's kind of
1:06:59convenient is that it's very easy to
1:07:00calculate that thing. Once you've
1:07:02calculated that thing, you then um can
1:07:03like very easily talk about welfare
1:07:06without needing to do a lot of other
1:07:07simulations in your model. Okay. um it's
1:07:10a little bit knife edge because of
1:07:12course it's all coming from whatever
1:07:14distribution assumption you've made
1:07:16about those shocks. Um so whether or not
1:07:19you want to put a lot of uh faith in
1:07:21particular theoretical derivations
1:07:23depends on on your sort of like what you
1:07:25think about the underlying
1:07:27distributional assumption but you'll see
1:07:29a lot of work um that sort of uses these
1:07:31different properties um because it
1:07:33generates very clean empirical tests.
1:07:37But the basic idea is that you know you
1:07:38can work through that math if that's
1:07:39what you need for your setting. Um but
1:07:41the intuition of kind of what's going on
1:07:43in the background uh is exactly the same
1:07:45as a simple case with n equals 2. So I
1:07:48think it's often very helpful to work
1:07:49through the n equals2 case for your
1:07:51setting. Uh understand what's going on
1:07:54and then you can show that your n= n
1:07:57case is going to just work the same way
1:07:59but just with a few more bells and
1:08:00whistles.
1:08:02Melanie, can I ask a question?
1:08:04So the in the fchet case the theta sort
1:08:07of captures the diffusion of the
1:08:09distribution that you were discussing
1:08:11and the um the elasticity of
1:08:14this but in the gumball case there
1:08:17doesn't seem to be any kind of parameter
1:08:19to capture that.
1:08:21Yeah the gumball is the gumball is the
1:08:23case where the shape parameter is set to
1:08:24zero. Um and so the difference between
1:08:28you have three types of extreme value
1:08:30shocks. um you have the type one, type
1:08:32two, type three and the difference is
1:08:34based on whether or not there is a zero
1:08:36shape parameter or there are uh shape
1:08:39parameters that are positive or
1:08:41negative. And so in that sense um in the
1:08:44gumball case often you make an
1:08:46assumption about the variance of the
1:08:47shocks coming in. You you have the sort
1:08:49of the standard the standard thing where
1:08:51you can only identify the logic
1:08:52coefficients up to scale. um but it's
1:08:54not as intuitive in the expression there
1:08:56as what it is here where this this
1:08:59elasticity this theta is exactly mapping
1:09:02to that dispersion parameter uh here. So
1:09:05in that sense I think what you see in
1:09:07trade is that that dispersion parameter
1:09:09ends up being a really key parameter for
1:09:12the trade elasticity
1:09:14because it's exactly the trade if you
1:09:15think about what a trade elasticity is.
1:09:17You know how responsive uh flows to
1:09:20changes in for example trade costs
1:09:22that's telling you about how easily can
1:09:24people substitute out of producing
1:09:27something when they were producing it.
1:09:28That's telling you something about um
1:09:31and the model is telling you something
1:09:32about you know how different are
1:09:35people's rel idiosyncratic draws of
1:09:37producing in one location versus another
1:09:39location to be able to get you um so
1:09:42that you would have a productivity gain
1:09:43of like importing or exporting but
1:09:44you're right um in the gumball case you
1:09:46don't have that popping up partly
1:09:48because it's the type of it's the um
1:09:53it's the extreme value shock that has a
1:09:55shape parameter set to zero. I guess I
1:09:58was curious about what's the ad it's
1:10:00obviously that's a narrower that's a
1:10:02stronger assumption in some ways and
1:10:04what is there advantage is there
1:10:06something that advises you that the
1:10:07labor guys use that
1:10:10so I think um that's a good question I
1:10:13think um
1:10:18[snorts]
1:10:20yeah if I was thinking about this
1:10:22actually before when I was writing up
1:10:23these slides to try and think about if
1:10:24if I knew like a good answer for that.
1:10:27Um, it's
1:10:31one thing that's pretty nice about the
1:10:33logit is it's very easy to estimate it
1:10:34because you have um a well a well-
1:10:37definfined likelihood function. Um, so
1:10:39you can take the derivatives and things
1:10:40like that and you've got like well-
1:10:42definfined likelihood functions which
1:10:43make it very very easy to estimate. Um,
1:10:46it's a little less it's harder to define
1:10:49the likelihood function and all the
1:10:50cross derivatives for the fresh a case.
1:10:52So I think one reason that people use
1:10:53the lojit is that um the estimation of
1:10:57the parameters is pretty super simple
1:11:00and so people um kind of traditionally
1:11:04just had the log likelihood of the lojit
1:11:06function easily computable you have
1:11:08across partials and you can do that
1:11:10super easily where it's a little bit
1:11:12more involved to do that for the fresh a
1:11:15so I think that's potentially one reason
1:11:17but um
1:11:19if you go back and you sort of look at
1:11:21the extreme value theory there's not
1:11:23really a they're all sort of nested
1:11:25there and so
1:11:27um I don't know if there's like if it's
1:11:29just hysteresus that that was one that
1:11:31people started working on the this
1:11:34version of the shocks um
1:11:37the first you sort of see this in the
1:11:40trade so the early um eaten work on
1:11:44thinking about uh eaten and cordon work
1:11:46you sort of see these popping up that
1:11:47those some of the early derivision
1:11:49derivations of using the fresh
1:11:51uh and then I think the trade literature
1:11:53has gone that direction. The migration
1:11:55literature or the urban literature often
1:11:57just estimates kind of low and I guess
1:11:59also sometimes you see this in the IO
1:12:00literature as well. People are making um
1:12:03lowit type one assumptions. Um, but I
1:12:06think the two are not really that
1:12:07different in some sense. There's a shape
1:12:08parameter difference, but again, you can
1:12:11change around your parameters to make
1:12:13one equal to, you know, you can sort of
1:12:14redefine things and have the type two
1:12:16equal to the type one, but you would
1:12:18have a shape parameter in there.
1:12:22If anyone else has a better answer of uh
1:12:24the history of these type one, type two
1:12:26shocks, um, feel free to chime in. But I
1:12:28was actually um I think it's a good
1:12:30question, Jo. I was sort of thinking
1:12:31through this uh um like you know I spent
1:12:35when I first started working on these
1:12:36models I spent a lot of time doing all
1:12:38the derivations and thinking you know
1:12:40trying to understand what the
1:12:41differences were and that's my best
1:12:42understanding but I'm I'm very happy if
1:12:44I have missed a few things as well on on
1:12:47what else is going on.
1:12:50Okay.
1:12:52Cool. Okay. So basic idea yeah you'll
1:12:55either see lojit or you'll see fch
1:12:57shocks. I often end up working um with
1:12:59the fchet shocks because I quite like
1:13:02this idea. You can it's a little bit
1:13:05more involved in the loit shocks to
1:13:08um you can do it but for example
1:13:10calculating things like um you know
1:13:12often you want to think about if people
1:13:14are moving location you want to account
1:13:16for what is the change in their observed
1:13:17utility what's the change in their
1:13:19unobserved utility. Um you can calculate
1:13:21a lots of those quantities for both
1:13:23functions. um it's particularly easy to
1:13:25calculate things like the epsilon given
1:13:29that you choose i with a fresh a um just
1:13:32like the math ends up being a little bit
1:13:34easier. So I often end up writing models
1:13:36that has um that have this here. Um you
1:13:40can sort of work out what those un you
1:13:42know you just have to sort of do the
1:13:43conditional integrals and things like
1:13:45that um with the gumball version as
1:13:47well. But sort of my sense is that if
1:13:50you're writing something that's either
1:13:51trade or
1:13:54um sort of a spatial model, you're
1:13:56probably writing models that look
1:13:57multiplicative. If you're writing
1:13:59something that's an IO model or maybe
1:14:01like a labor model, you're probably
1:14:03writing um something that looks like a
1:14:05logit uh in this way.
1:14:08Okay. So, what does that give us? That
1:14:11gets us to understanding how we would un
1:14:14how we would think about the
1:14:16um equilibrium. You know, we figured out
1:14:20given wages, how do people make their
1:14:22decisions about where to live? The last
1:14:24piece of the puzzle is to think about
1:14:26how do those prices get determined in
1:14:28equilibrium.
1:14:30So I'm going to make a set of
1:14:31assumptions that are quite strict and
1:14:33again this very easy to relax any of
1:14:35these that you like. So you can just use
1:14:37this as a starting place because it's
1:14:39going to be very helpful, very easy to
1:14:41sort of visualize and draw this model on
1:14:44two figures. So I think that's helpful
1:14:47because it gives you a really clear
1:14:48sense of what's going on. If you want to
1:14:50extend it in any direction, it's very
1:14:51easy to go.
1:14:53So the baseline is options. Each person
1:14:56is homogeneous. They're going to supply
1:14:58one unit of labor wherever they're
1:15:00working. So for example, we rule out in
1:15:02the baseline model things like
1:15:04selection. We rule out unemployment. Um
1:15:07each city is producing a homogeneous
1:15:09good that's freely traded. That's just
1:15:11ruling out anything about cost of living
1:15:13differences through consumption. If you
1:15:15want to indogenize that, you just put a
1:15:18you layer trade model. Um, and it
1:15:21doesn't end up really affecting the
1:15:22migration decision because as you saw,
1:15:24all that really matters is that final
1:15:26bundle of goods and so you just need to
1:15:28know that. If there's a cost of living
1:15:30term as well inside of that, that's
1:15:31fine. Um, land is only a mobile factor.
1:15:35Um, let's assume that the quantity of
1:15:37land is fixed in each location. Let's
1:15:39start by saying labor is perfectly
1:15:41mobile. Um,
1:15:44in fact, today I've even written down a
1:15:45function, a production function that
1:15:47doesn't have capital in it. But if you
1:15:49look at the Meridian handbook, for
1:15:50example, there's capital, but there's an
1:15:52assumption that's capital is perfectly
1:15:54mobile and there's an international
1:15:55price of capital that ends up just
1:15:57giving you constant returns to scale in
1:16:00uh that ends up giving you a production
1:16:01function um where you just need to
1:16:03account for uh labor.
1:16:06Um,
1:16:07so if you want a couple of uh textbook
1:16:10treatments or handbook chapters, check
1:16:11out these two references. Um, and again,
1:16:14the idea here is to be simple, not
1:16:16because you can't put more in, but
1:16:18because um the intuition is going to be
1:16:20the same um sort of um for the simple
1:16:24case as for the more complicated case.
1:16:28Okay. So in the Rosen rowback model you
1:16:30have differences in wages, differences
1:16:32in the cost of living which we assume
1:16:34are just coming through rents, amenities
1:16:37and then um individual preferences.
1:16:42Okay. So what is the definition of a
1:16:44spatial equilibrium? So we have an
1:16:46economic environment. We're going to
1:16:47have what does that mean? We have
1:16:48exogenous productivities and amenities.
1:16:51And what we want to solve for is we want
1:16:53to solve for the number of workers who
1:16:54are in each location, the equilibrium
1:16:56wage, and the equilibrium rent.
1:17:00So what we do is we know labor supply.
1:17:03That's basically what we spent the first
1:17:05hour talking about.
1:17:07Um
1:17:09here, labor supply, labor demand. We're
1:17:12going to assume that firms pay marginal
1:17:13product. Just to sort of set things, um
1:17:16I've just put in here um a linear
1:17:19production function. If you want, for
1:17:21example, land or you want capital, you
1:17:24could just change that. It just changes
1:17:25your first order condition. Housing
1:17:27demand, each worker demands one unit of
1:17:29housing. Housing supply. Again, I've
1:17:32kind of put a reduce form inter uh value
1:17:34here where there's a positive housing
1:17:36supply elasticity. You can microfound
1:17:39whatever you want to be going on on the
1:17:41housing side, and that will just change
1:17:42your equilibrium uh there.
1:17:46Okay. So, what we're going to think
1:17:48through is how does the model work. I'm
1:17:50going to show you everything in diagrams
1:17:52because you get all of the intuition in
1:17:53diagrams. And then in the TA section,
1:17:55Sally is going to talk through exactly
1:17:57what you need to do to put those
1:17:59equations onto the computer. Once you
1:18:01have more than two locations, you're
1:18:02going to need to be simulating out your
1:18:04model. Um, it's not you're not going to
1:18:06be able to get closed forms for
1:18:07anything. So, you're going to be putting
1:18:08it on the computer. Um, and so she's
1:18:11going to talk through how what these
1:18:12equations are, how you put them on the
1:18:14computer, how you would then solve for
1:18:16your equilibrium, and then show some uh
1:18:18simulations from the model.
1:18:21So let's imagine that there's an
1:18:22increase, for example, in productivity
1:18:24in location A. What does that mean? It
1:18:27means that
1:18:30the value of being in location A goes
1:18:32up. So, we've just talked about how when
1:18:34value of living in one location goes up,
1:18:36the number of people, the probability
1:18:38that people live in A goes up. When that
1:18:40happens, we're going to see that rents
1:18:43are a function of the number of people
1:18:45who live on location. There's going to
1:18:47be an endogenous response on the housing
1:18:49market. When that happens, VA is a
1:18:53function both of wages and rents.
1:18:57When rents go up, that's going to reduce
1:19:02uh the value of living there. That means
1:19:05that not as many people go there as
1:19:08initially went there. So you're going to
1:19:10sort of have an iteration where you
1:19:12would reach a new equilibrium where the
1:19:14increase in wages partly has been
1:19:16capitalized into an increase in the
1:19:18housing market. And so in not in real
1:19:21wages and the terms of the real wages,
1:19:23you've had an increase in the nominal,
1:19:25but you've had an increase in the
1:19:26housing as well. how much that shock is
1:19:28fully capitalized into the housing
1:19:30market versus the n um is going to
1:19:32depend a little on the labor supply
1:19:34elasticity. How many people are willing
1:19:36to go there? So what's the division
1:19:38between the the shock getting fully
1:19:40capitalized into land versus uh some of
1:19:43it uh occurring in terms of an increase
1:19:44in real wages.
1:19:47Okay, so what does that look like? Um
1:19:49let's just sort of draw it on the
1:19:51figure.
1:19:52So um we can start off I've again just
1:19:57made the model very very simple so you
1:19:59understand the intuition but you start
1:20:01off there is a a wage we've assumed that
1:20:04the labor demand is
1:20:06perfectly elastic okay and there's a
1:20:08labor supply that depends the slope in
1:20:12some sense the labor supply elasticity
1:20:13that's where all of these um shape
1:20:16parameters so how much do people want to
1:20:18live in a location that has high wages
1:20:20how much heterogeneity that's going to
1:20:22enter into the li labor supply. So we're
1:20:25in this location. Um there's an increase
1:20:28in wages. So let's imagine that there's
1:20:31an increase in productivity. Labor
1:20:33demand goes up. So what does that mean?
1:20:36That there should be a movement along
1:20:38the labor supply curve. So what does
1:20:41that mean? It means that we should see
1:20:43in the housing market an increase
1:20:46in the demand for housing. Okay.
1:20:52So here we have rent on the quantity of
1:20:54labor housing. Here in the labor market
1:20:56we have wage on labor. What does that
1:20:58mean? As the rents go up that's going to
1:21:02shift back the labor supply curve
1:21:03somewhat because the labor supply
1:21:05remember is a function of rents, wages
1:21:08and amenities. So house prices have gone
1:21:10up. So what does that mean? And in some
1:21:13sense, the labor supply curve shifts a
1:21:16little bit in because we're no longer
1:21:18having holding the rents constant. Given
1:21:21that
1:21:23we would have slightly fewer people, so
1:21:25that would be a shift back in of the
1:21:27housing demand curve.
1:21:30So we go this way, we go this way. That
1:21:33means that rents are a little lower. So
1:21:34that would and in this picture here
1:21:36would put labor supply back out again.
1:21:40And you can see that if we continued
1:21:42iterating there, we would end up with
1:21:43some equilibrium where we have an
1:21:46equilibrium, higher rents, higher wages,
1:21:49and we have a larger number of people in
1:21:52that location, but there's going to be
1:21:54some general equilibrium adjustment such
1:21:56that the change in the labor is not the
1:21:59full partial equilibrium change in the
1:22:01labor. It's mitigated because there's
1:22:03some capitalization of that productivity
1:22:06into house prices.
1:22:11Um, so let me sort of leave that for a
1:22:14second. Is there are there any questions
1:22:15about about that figure?
1:22:23You can use exactly the same intuition
1:22:24if you want to to just um understand
1:22:26intuition. For example, if there's an
1:22:28amenity improvement. So an amenity
1:22:30improvement, what would that look like?
1:22:32Um
1:22:34that would look like,
1:22:36you know, you have your labor market,
1:22:38you have your housing market.
1:22:43An amenity improvement is something that
1:22:45makes the labor supply curve shift out.
1:22:48So that would just be shifting out here.
1:22:50The labor supply shifts out the housing
1:22:52demand.
1:22:54Housing supply as the housing demand
1:22:56goes as the rents go up, it shifts back
1:22:58in the labor supply a little bit. So for
1:23:01example, an amenity improvement, you
1:23:03should see a new equilibrium
1:23:06where there is higher rent but no change
1:23:08in wages. Okay? And you have an
1:23:10equilibrium a larger number of people
1:23:14living in that location.
1:23:18So you can basically work through in
1:23:20this you can work through on the figure
1:23:22um you know sort of any changes you
1:23:24could think through what might happen if
1:23:26there's a you could visually see if
1:23:28there's a change in the housing supply
1:23:29elasticity. So let's say we have new um
1:23:32policies that make it easier to expand
1:23:35housing supply. You could then easily
1:23:37show what would happen in terms of the
1:23:39relative pass through for example into
1:23:40rents. You could think about when are
1:23:42you going to have a proportionally
1:23:43larger population response for example
1:23:46to a productivity shock or when are you
1:23:48going to have a a smaller shock. We've
1:23:51been talking a lot about preferences
1:23:53that comes into the labor supply curve.
1:23:55You could think about other things that
1:23:57might affect labor supply. So you could
1:23:59think about for example if a place is
1:24:01really isolated it might be really hard
1:24:03for people to get there. So that's going
1:24:05to again you could think about looking
1:24:08at two different locations with
1:24:09different labor supply elasticities and
1:24:11I think almost all of the basic
1:24:12intuition you can get by just thinking
1:24:14about and in fact even from published
1:24:17papers that are published so recently
1:24:18you can get almost all of the
1:24:19comparative statics often by just
1:24:22drawing um a labor market a housing
1:24:24market and just thinking through what is
1:24:26which curves are shifting and which uh
1:24:29you know how do you end up in terms of
1:24:31what the new quantity is the new wages
1:24:33and the new house prices.
1:24:37Okay, so I think um let me talk then for
1:24:40I think we only actually have a few
1:24:41minutes left. So let me just mention I
1:24:44think I've talked through most of these
1:24:45things here. The spatial model that
1:24:47basic idea is just the building block
1:24:50that what you need uh for almost uh for
1:24:52many other uh applications. So I think
1:24:55one of the things that's great about the
1:24:56spatial model is that the same model is
1:24:58going to basically work for trade. It's
1:25:00going to work for commuting. It's going
1:25:01to work for migration. It's going to
1:25:02work for things within a country, things
1:25:04across a country. For trade, the
1:25:08interpretation is that instead of having
1:25:10preference shocks, locations have
1:25:12different um ability to produce
1:25:14different types of goods. And so what
1:25:17you are thinking about is if it gets
1:25:19easier to be importing. The reason you
1:25:21get productivity gains is that you allow
1:25:24people to stop producing stuff that
1:25:25they're relatively less important.
1:25:27They're less good at. You know, you your
1:25:30autoaki shifts up kind of who can
1:25:32produce what's the threshold for
1:25:34producing good stuff versus bad stuff.
1:25:36So instead of maximizing your utility,
1:25:38you're thinking about where to buy goods
1:25:40from um that have uh the cheapest cost.
1:25:43Next week's uh lecture is going to go
1:25:45and kind of follow up with trade.
1:25:47You can think about the shock being
1:25:49productivity and then it's really easy
1:25:51to think about models where you have
1:25:53sorting on productivity. So you can see
1:25:56the gaga some war you can see our uh JPE
1:25:58paper that has basically this model just
1:26:00reinterpreting
1:26:02those idiosyncratic shocks with
1:26:04preference shocks. And then if you want
1:26:06to think about things like frictions,
1:26:08what you can think about is um you know
1:26:11instead of just having this indirect
1:26:13utility, this common indirect utility
1:26:15term be just a function of the
1:26:16destination, it just differs because
1:26:19there's a term that's the cost between
1:26:21for example traveling between the origin
1:26:23and the destination.
1:26:26I'm going to skip over the empirical
1:26:27exercise because this is what Sally will
1:26:29go through in the TA section. If you
1:26:30want to look ahead at the notes, take a
1:26:32look at those. But she'll go through
1:26:33what the algorithm would be to solve it
1:26:35on the computer and then if you want to
1:26:37try that before Monday's TA section uh
1:26:39you can do that and then let me just
1:26:41leave a few minutes for questions now um
1:26:44to sort of wrap up you know here is a
1:26:46really versatile tool once you have the
1:26:48intuition you can expand the model in
1:26:50many different directions um you know
1:26:52you have the key ideas are to think
1:26:54about you know what is driving the
1:26:56different elasticities of labor housing
1:26:58and then if you want to work through and
1:27:00understand for example relative positive
1:27:02effects of a productivity shock into
1:27:04rents versus wages versus population.
1:27:07You can sort of understand really easily
1:27:09even just by kind of drawing graphs
1:27:12about how those depend on different uh
1:27:14elasticities. So, let me um leave it. I
1:27:17know we only have a minute or two now
1:27:18for questions, but um leave anything
1:27:20that might want to come up for for now.
1:27:23Thanks, Melanie. Yeah, let's uh just
1:27:25take a couple of questions because we're
1:27:28roughly out of time. We've gone over and
1:27:30others. So we'll take a few questions.
1:27:32Let's start with the front row.
1:27:34So a couple of slides back you talk
1:27:37about, you know, just solving
1:27:39[clears throat] for the share of the
1:27:43population in each location and the
1:27:44rents in different locations on the
1:27:46computer. Is that because
1:27:50there isn't a kind of closed form
1:27:52solution for those once you've set uh
1:27:55you know amenities, productivity and um
1:27:59the elasticity of rent.
1:28:01Yeah. So, um what you need to know is um
1:28:04I mean if you have just two locations
1:28:06and you have for example the uniform
1:28:09distribution on the difference of the
1:28:10shocks between the two locations that's
1:28:13a case where if you look at the Meridian
1:28:14handbook chapter you can solve out
1:28:17everything in closed form for all of the
1:28:19equilibrium prices and quantities. Once
1:28:21you have more than two locations, it
1:28:24becomes quite difficult because the
1:28:26population who wants to live in a
1:28:27location is now um a parame a very you
1:28:32know for example if you're working with
1:28:33the fresh a it's the vi divided by this
1:28:37thing here. So you need to kind of go
1:28:40through all of those
1:28:42um if you want to think about what the
1:28:45change of population is, what the
1:28:46equilibrium, things like that, I it's
1:28:48almost impossible to get that in close
1:28:50form. And so what you end up showing is
1:28:51sort of um how do those how do those um
1:28:55elasticities look like for example with
1:28:56changes in rents once you put them on
1:28:58the computer. You can probably get sign
1:29:00you can get sign effects but it's going
1:29:01to be really hard to get them quantities
1:29:04in closed form. And so for that reason,
1:29:06you'll see those presented by
1:29:07simulation. If you then also of course
1:29:09want to have a more complicated model,
1:29:11maybe you want to have correlation
1:29:12between people's drawers, things like
1:29:14that, you're going to get into a case
1:29:15where it's going to get really difficult
1:29:16to um
1:29:19to be able to solve any of that stuff in
1:29:21close form. Because if you think about
1:29:23it, basically the rents in one location,
1:29:25they're a function of rents in every
1:29:26other location. They're a function of
1:29:28wages in every other location, they're a
1:29:30function of amenities in every other
1:29:31location. So um you just have a lot of
1:29:34interactions and equilibrium between all
1:29:36of uh those quantities.
1:29:38Great. Thanks.
1:29:39We have a very interesting question in
1:29:40the Q&A for a development
1:29:43related question. Emanuel Bazid Mongu
1:29:47asks, I am wondering how land tenure
1:29:49insecurity a friction in say location B
1:29:52can be incorporated in the model if
1:29:54people are making decisions of moving
1:29:56from location A.
1:29:57Ah so that's super interesting. Where
1:29:59would you think about putting it in?
1:30:01This is somebody from the webinar, so
1:30:03he's not going to be able to answer.
1:30:03Ah, from the webinar. Okay. So, so um,
1:30:08so there's a couple ways you could think
1:30:09about that, but you, so our very basic
1:30:13model that we've talked about. So, we've
1:30:15talked about the utility of being in
1:30:18location A. For example, there's an
1:30:21amenity of location A, there's a wage of
1:30:24location A, and there's some cost of
1:30:26living of location A. One way that you
1:30:29could think about this is that you might
1:30:31say that
1:30:33um
1:30:35you could probably think about it
1:30:36different ways. One kind of way that one
1:30:38immediate way to think about it is the
1:30:40amenity of living in a location depends
1:30:42if you have land in that location or
1:30:44not. So for someone who um it sounds
1:30:48like what you're saying is that if you
1:30:52one way you could think about it is if
1:30:54if you have land tenure in security uh
1:30:57you only get access to use the land
1:30:59maybe if you have like some um some
1:31:02right to that land and maybe that right
1:31:03to the land is not something that's
1:31:05transferable across people. One way you
1:31:07could think about that is you could
1:31:08think about one way you could put it in
1:31:10is saying that you could put it in the
1:31:13amenity term that you could say that the
1:31:15amenity of being in location A depends
1:31:16on for example your land status in
1:31:19location A. And so someone who has the
1:31:22right to land on location A versus
1:31:24someone who doesn't have the right to
1:31:25land would have a different valuation.
1:31:27You might want to put it into the wage
1:31:28term or something in the productivity
1:31:30term where you might say that the wages
1:31:33that you earn are a function. Maybe you
1:31:36put it in sort of you earn a wage per
1:31:39per amount of land and an amount per
1:31:41amount of land. That might differ
1:31:43between people, for example, who are
1:31:44historically from that location versus
1:31:46people that are not who might have land
1:31:48rights. It might be that you think about
1:31:50having um
1:31:53this thing be an indicator based on, you
1:31:55know, does your family have land? Does
1:31:56it not? That either might load onto
1:31:59something that looks like preferences or
1:32:00it might load onto something that look
1:32:02like wages. It may or may not matter to
1:32:04separate out too much for the
1:32:05productivity for the migration for the
1:32:07location choice decision.
1:32:09um what it's going to probably do is
1:32:10it's going to make people look uh less
1:32:13likely to move away because they if they
1:32:16move away they might give up their land.
1:32:18Um and if you're interested in that it's
1:32:20a really nice if you want to sort of
1:32:21look at a paper that has that idea in it
1:32:23but it hasn't got all the spatial model
1:32:25in it but the idea is there um you can
1:32:28look at um Gonzalez Navaro um and
1:32:31co-authors they look
1:32:34at what happened after land reform where
1:32:37people could then start migrating after
1:32:39land reform because they had more secure
1:32:41property rights what you see is it looks
1:32:42like um it's some it's either something
1:32:45you know it's reduced the amenity of
1:32:46staying in a location or it's reduced
1:32:48the migration cost. You see that after
1:32:50land reform in Mexico, you see that
1:32:51people are
1:32:53um so this is an A paper um with other
1:32:57co-authors. Um you see that people have
1:33:01land right reform, therefore they have
1:33:02privatized land rights, therefore they
1:33:04start to migrate. And I think that
1:33:06concept is there in the data and then
1:33:08exactly how you wanted to model it might
1:33:10depend about how complicated you wanted
1:33:12to make your indogenous land market. Do
1:33:15you think it's coming through
1:33:16productivity? Do you think it's coming
1:33:17through just preferences for location?
1:33:20Um, you know, how do you think about who
1:33:22has access to that and who doesn't? Um,
1:33:25and you could sort of model it in a
1:33:26couple of different ways. Um, but once
1:33:29you had it modeled, it would then just
1:33:31be part of that uh common utility term
1:33:35um when you're figuring out, you know,
1:33:36where do people want to live? And it
1:33:38might be something that explains a
1:33:40heterogeneous response to a productivity
1:33:42shock based on this other term which is
1:33:44important to account for. So, I think
1:33:46it's a really interesting question to
1:33:48look at. I think it's probably really
1:33:49really important um given that there are
1:33:51really big differences in formal land
1:33:54rights and people's right to be claiming
1:33:56access to the land if they're not
1:33:58physically on it. And so, when we start
1:34:00thinking about how responsive are you to
1:34:02shocks and spacing spatially, that might
1:34:05indeed be a kind of friction that means
1:34:08that people don't want to move away
1:34:09because they're giving up something that
1:34:11would otherwise be an option value for
1:34:13them.
1:34:14I think it's a really interesting
1:34:15question. Yeah.
1:34:16Yeah. Let's get one last uh question
1:34:18here from uh in the Q&A and we'll
1:34:20probably have to have a brief answer
1:34:22because
1:34:24we're running short on time even if we
1:34:27can stay I don't want to keep Mandy and
1:34:30CPR there late in the evening on a
1:34:31Friday. Uh it's Tomas Biors says um I
1:34:34would like to hear her thoughts your
1:34:36thoughts Melanie on the ID the
1:34:38assumption of shocks which is probably
1:34:40less stringent in the trade context than
1:34:42in the migration context.
1:34:44So sometimes there are testable
1:34:46predictions about whether or not that
1:34:48assumption so the IID assumption the
1:34:50pure vanilla iid assumption is probably
1:34:52a little bit too strong. It's very easy
1:34:55to have um correlated shocks within
1:34:58person um where you have um I mean one
1:35:02way to think about it is you um kind of
1:35:06have something that's making people
1:35:07absolutely you know there's an absolute
1:35:09advantage margin and then a comparative
1:35:10advantage margin. What do I mean by
1:35:12that? That you can really easily show
1:35:13the fresh kind of as isomeorphic to
1:35:15having one more parameter which is just
1:35:17scaling up the level of shocks but then
1:35:19it's still sort of the partials are
1:35:21still um um iid. they're just different
1:35:24people have more likely to have high
1:35:25shocks for uh good shocks. I think the
1:35:28other way you can do it is you can think
1:35:29about nesting the shocks. So for example
1:35:32um we have a paper looking at migration
1:35:33from Mexico to the US where we um nest
1:35:38we sort of write it down where you first
1:35:40of all get a shock about whether or not
1:35:41you want to be living in Mexico or
1:35:43living in the US. Um and then the second
1:35:46shock is what destination in the US you
1:35:48want to go to. And that's important
1:35:50because for example a lot of the
1:35:53elasticity is going to depend on how
1:35:55much are you willing to substitute on
1:35:57that extensive margin. So like if you
1:35:59are really if you just want to migrate
1:36:02and you're someone who's going to
1:36:03migrate no matter what you're going to
1:36:04be quite willing to substitute within
1:36:06destination in the US. So it's going to
1:36:08be important to get the magnitudes uh
1:36:10correct. So I think I think the IID
1:36:12assumption by itself is too strong but
1:36:13there's a couple of really easy ways to
1:36:16um expand it. um you can do things like
1:36:19nested shocks, you can do things like
1:36:21this correlation within person types of
1:36:22shocks. You then have um other ways as
1:36:25using copulars. So if you look at the
1:36:27Lagos war paper, they have a really nice
1:36:29discussion of thinking about um two
1:36:33shocks that are um attach you know the
1:36:36the joint distribution depends on the
1:36:38specific copular that is a function of
1:36:40some parameters that you can estimate
1:36:41from data. And so that's going to give
1:36:43you again a little bit more flexibility
1:36:45and what it breaks is the necessarily
1:36:48the linkage between sort of absolute and
1:36:50comparative advantage which ends up
1:36:52being quite important um in lots of
1:36:54these models. So if you're interested in
1:36:56that um I would suggest that with
1:36:57reading uh the legos and wallpaper um
1:37:00and then a lot of the tools that are
1:37:03developed for example on IO you can you
1:37:06don't often see people using those in
1:37:07the migration but you can port a lot of
1:37:09those over to spatial models pretty
1:37:10easily as well. Okay. Um, thank you very
1:37:13much Melanie. Wonderful. I remind
1:37:16everybody, let's thank Mandy again and
1:37:19Kirsty and Lauren from CR. I want to
1:37:22remind you all that everybody who's
1:37:23lecturing u Melanie as one of them, you
1:37:27know, is doing this out of the goodness
1:37:28of their heart. So, let's um unmute and
1:37:30give Melanie a round.
1:37:37Thanks. And I I should also say um
1:37:39especially for folks who um you know I
1:37:42know people are dialing in from all
1:37:43around the world like if you're working
1:37:44on these sorts of models um you know
1:37:47it's always great to sort of see what
1:37:48kind of work people are doing there's
1:37:49lots of really exciting ways to extend
1:37:50them. So if you have an idea or you have
1:37:52a question and you're working on it and
1:37:54and you would like to get some comments
1:37:56about that I'd be super happy to uh to
1:37:59hear them and talk to you about them. So
1:38:02wonderful. Thank you. We'll see you all
1:38:04next week hopefully.
1:38:05Bye bye. Thank you.
1:38:08Okay. Thanks everyone.
1:38:09Thanks Melanie.

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