The Decline of the Rust Belt: A Dynamic Spatial
Equilibrium Analysis∗
Chamna Yoon†
February 15, 2012
Abstract
One of the most striking patterns of the U.S. economy over the past 50 years has been the
decline of industrial cities in the Midwest and Northeast, also known as the Rust Belt. The
goal of this paper is to provide an understanding of what triggered this economic change,
and why the adjustment process took the form it did. I address two empirical issues. First,
I measure the extent to which the decline of the Rust Belt is attributable to the transition
of the U.S. economy to a service sector, and to the reduced area-specific advantage of the
Rust Belt. Second, I assess the extent to which the increased share of less educated in the
Rust Belt results from higher mobility costs for the less educated or higher preference for
less expensive housing. To perform the quantitative assessment, I build a dynamic spatial
equilibrium model that allows me to address these empirical issues in a unified, coherent
framework.
Keywords: labor mobility, the Rust Belt, local labor market, housing market
JEL Classification: J1, J6, R1, R2
∗Work in Progress. I would like to acknowledge the valuable advice and suggestions provided by KennethWolpin and Holger Sieg. I thank Albert Saiz for generously providing me Land Use Data (GIS). This research wassupported in part by the National Science Foundation through XSEDE resources provided by the XSEDE ScienceGateways program (TG-SES120008). I also thank the Pittsburgh Supercomputing Center for granting and helpingme access to their machines. I thank the participants of Penn’s Empirical Micro Lunch.
†University of Pennsylvania. [email protected]
1
1 Introduction
One of the most striking patterns of the U.S. economy over the past 50 years has been the
decline of industrial cities in the Midwest and Northeast, also known as the Rust Belt.1 The
goal of this paper is to provide an understanding of what triggered this economic change, and
why the adjustment process took the form it did. There have been a number of explanations
offered in the literature for the decline. However, they have not been placed within a compre-
hensive framework that enables a quantitative assessment of their relative contributions to the
decline. This paper addresses this issue by building and estimating an economic model that
allows for each of the factors discussed in the literature to play a role.
The Rust Belt cities have experienced a relative decline in population, a change in the compo-
sition of the population in terms of educational attainment, falling wages and declining land
values and housing rents. In 1960, 28 percent of the U.S. population lived in the Rust Belt, but
by 2010 that figure had fallen to 19 percent. In addition, although in 1960 average wages and
housing rents were higher in the Rust Belt than in other areas by 10 and 2 percent respectively,
today the wage gap between the Rust Belt and other areas has disappeared and housing rents
in the Rust Belt are 20 percent lower. Similarly, in 1960 the share of the non-college educated
in the Rust Belt was 4 percent higher than that of elsewhere in the U.S., but today that figure
has more than doubled.
A prominent explanation for the decline of the Rust Belt was a fall in the Rust Belt’s com-
parative production advantages due to two major exogenous structural changes in the U.S.
economy. First, due to technological changes and to the globalization of the economic envi-
ronment, the U.S. economy shifted from manufacturing to services; between 1950 and 2000,
service sector employment grew from 57 to 75 percent of total employment. This structural
change had a greater impact on manufacturing-oriented regions, especially the Rust Belt. Sec-
ond, improvements in transportation and communication technology reduced manufacturers’
gains from locating in the Rust Belt cities that had previously attracted people and firms be-
cause of their easier access to waterways and well developed railroads (Glaeser and Ponzetto,
2007).2
Responses to these exogenous changes exacerbated their impact. As people moved out of the
Rust Belt, productivity decreased further, inducing additional population loss. For example,
1The Rust Belt conventionally includes Illinois, Indiana, Michigan, Ohio, Pennsylvania, West Virginia, andWisconsin.
2Transportation costs have been reduced to less than one tenth of that in 1900 (Glaeser and Kohlhase, 2003).
2
population outflow decreased the tax base to finance investments in locally provided factors of
production, such as physical infrastructure. In addition, as the concentration of manufacturing
jobs fell, the productivity gain from knowledge spillovers (agglomeration) among workers and
from sharing resources among firms diminished.
As noted above, the population outflow from Rust Belt cities was accompanied by a compo-
sitional change reflecting selective migration. On one hand, the benefit of out-migration may
have been lower for the less educated because they face poorer opportunities elsewhere. On
the other hand, the benefit of not migrating may have been higher because declining housing
rents disproportionately affect the non-college educated who spend a larger fraction of their
income on housing consumption (Glaeser and Gyourko, 2005).3
In this paper, I address two empirical issues. First, to what extent is the decline of the Rust
Belt attributable to the transition of the U.S. economy to a service sector and to what extent
to the reduced area-specific advantage of the Rust Belt? I compare the evolution of the U.S.
economy to economies where one of those factors is set to their 1960 levels. Second, I assess
the extent to which the increased share of less educated in the Rust Belt results from higher
mobility costs for the less educated or higher preference for less expensive housing. I compare
the evolution of the U.S. economy to economies where mobility costs are uniform across
demographic groups and where housing preferences are the same across demographic groups.
This research contributes to a growing empirical literature on local labor markets. First, it
analyzes the incidence of local labor demand shocks. The previous literature relied on a
relatively simple framework (Topel, 1986, Notowidigdo, 2011). In contrast, I adopt a fully
dynamic framework in which people’s dynamic considerations, such as their expectations
about future wages and rents, and mobility cost are explicitly modeled. Second, this paper
is related to the migration literature in which the determinant of moving decisions and the
returns to migration in terms of lifetime wages are studied. Kennan and Walker (2011) focus
on the micro behavior of migrants, hence ignoring important macroeconomic aspects, such
as general equilibrium effects through the labor and housing markets and aggregate uncer-
tainties in the economy. I extend their framework to study the migration decision in response
to macroeconomic changes. Third, this paper is also related to labor adjustment to labor de-
mand shocks. Lee and Wolpin (2006) and Artuç et al. (2010) focused on the labor adjustment
to sector shocks. I incorporate geographic dimension to their approach to address the issues
related to the decline of the Rust Belt.3Among renters, non-college educated spent 4 percent point more on housing consumption from 1980 to 2010.
3
To perform the quantitative assessment, I build a dynamic spatial equilibrium model that al-
lows me to address these empirical issues in a unified, coherent framework. A typical static
framework, in which people are assumed to be perfectly mobile, cannot be used to address
these empirical issues because any shock to the local economy would be adjusted instan-
taneously. However, in reality, worker mobility is limited and heterogeneous; hence local
shocks to demand for labor will generate an adjustment process that is spread through time
and unequally distributed across demographic groups. My framework incorporates dynamic
adjustments in population, wages and housing rent to the long-run structural changes that
impacted the Rust Belt.
The following are the general features of the model I estimate. There are two regions in this
economy: the Rust Belt and the remaining U.S. In each region, there are three production
sectors: the manufacturing sector, service sector, and housing sector. Manufacturing goods
and service are produced using non-college educated labor, college educated labor, capital,
and locally provided productivity factors (e.g. infrastructure). The latter is provided by local
governments using revenue from property and income taxes. The overall productivity of
these sectors in each region can be affected by location-specific technological change, and
sector-biased aggregate shocks. Housing services are produced by using capital and land,
and consumed locally. The housing rental price is determined by the aggregate demand for
and supply of the housing service in each city.
Individuals have a forecast of how wages and housing rents will evolve in the future, and
choose optimally among six discrete alternatives at each age: a pair of two location alterna-
tives and three work alternatives. They also decide on their consumption of housing services
at the competitively determined, city-specific housing rents. The housing expenditure share
may differ across education levels. An individual receives a wage offer from each city and
sector in each period which depends on the competitively determined city- and sector-specific
wage rate and the individual’s accumulated sector specific human capital. The level of an in-
dividual’s human capital depends on accumulated work experience in each sector. Transitions
among alternatives involve a mobility cost which can differ across demographic groups.
I estimate the parameters of the model using a simulated minimum distance method in which
the distance between sample aggregate statistics and their simulated analog is minimized.
Specifically, I use data on employment and wages from the Current Population Survey, on
city- and sector-specific output and capital from the Bureau of Economic Analysis, on sector
and regional transition from the National Longitudinal Survey of Youth, and on housing rent
from the Census.
4
I will use the estimated model to simulate counterfactual experiments to quantitatively assess
the relative importance of the explanations previously mentioned. First, I assess the causes
of the decline of the Rust Belt. There were two factors that can account for the decline of the
Rust Belt, the decline of the manufacturing sector and the decrease in the value of area-specific
attributes. I will first simulate an economy where both factors are set to their 1960 levels, that
is, an economy without growth in the long run. The experiments relax the two factors in
turn. Second, to understand the reason why the Rust Belt population became less educated,
I compare the evolution of the U.S. economy to economies where mobility costs are uniform
across demographic groups and where housing preferences are the same across demographic
groups. Lastly, I quantitatively evaluate the impact of previously suggested policies. For
example, I assess how subsidies for people to move to reduce mobility costs would affect the
speed of adjustment.
The rest of the paper is organized as follows. In the next section, I provide a brief descriptive
history of the decline of the Rust Belt. The model is presented in Section 3, along with the
solution algorithm. Section 4 introduces the estimation procedure. Section 5 presents the
results of estimation and counterfactual experiments. I then summarize and conclude.
2 A Brief Descriptive History
In this section, I provide a brief descriptive history of the Rust Belt using March Current
Population Surveys from 1962 to 2010 and Censuses from 1940 and 2010. The Rust Belt cities
have experienced a relative decline in population as seen in Figure 1. In 1960, 28 percent of
the U.S. population lived in the Rust Belt, but by 2010 that figure had fallen to 19 percent.
The relative decline in population was severe between 1970 and 1990; the population share
dropped from 28 percent to 21 percent.
Changes over the last 50 years in the log real hourly wages for both areas are shown in Figure
2. Although in 1960 average wages were higher in the Rust Belt than in other areas by 10
percent, today the wage gap between the Rust Belt and other areas has disappeared.
Figure 3 shows the ratio of goods sector employment and service sector employment. The
ratio is higher in the Rust Belt by 20 percent in 1960. As the U.S. economy shifts from the
goods-sector to service-sector, the share of goods-sector decreased in both regions. However,
the gap between two regions also decreased rapidly between 1970 and 1990.
5
Figure 1: Population Share of the Rust Belt
.2.2
2.2
4.2
6.2
8.3
popu
latio
n sh
are
1920 1940 1960 1980 2000 2020year
rustbelt Median spline
The decline of the Rust Belt has involved a relative increase in the proportion of low skilled
labor in the Rust Belt. Figure 4 shows the proportion of non-college educated people in each
region. In 1960 the share of the non-college educated in the Rust Belt was 4 percent higher
than that of elsewhere in the U.S., but today that figure has more than doubled.
Figure 2: Mean Log Real Hourly Wages
22.
12.
22.
32.
4lo
g re
al h
ourly
wag
e
1960 1970 1980 1990 2000 2010year
outside rustbelt
Table 1 shows the median housing rents from 1960 to 2010. Although in 1960 housing rents
were higher in the Rust Belt than in other areas by 2 percent, today housing rents in the Rust
Belt are 20 percent lower.
6
Figure 3: Employment Shares of Goods Producing Sector
.2.4
.6.8
11.
2go
ods/
serv
ice
1940 1960 1980 2000 2020year
outside rustbelt
Figure 4: Share of Non-College Educated People
1.02
1.04
1.06
1.08
1.1
1.12
ratio
.4.5
.6.7
.8.9
shar
e of
uns
kille
d
1940 1960 1980 2000 2020year
outside rustbeltrustbelt/outside
Table 1: Median Real Housing RentsOutside Rust Belt Rust Belt/Outside
1960 2120 2150 1.021970 2474 2377 0.961980 2828 2656 0.941990 3299 2775 0.842000 3774 3437 0.912010 4558 3580 0.79
7
3 Model
Consider an economy with two cities indexed by j ∈ {1 : Rust Belt,2 : Remaining U.S.}. Thereare two tradable sectors and a housing sector in each city4. The economy is endowed with
fixed amount of domestic capital K and time-varying amount of developed land area Lt = L1t +
L2t . Traded goods and capital flow freely around the world, hence their prices are exogenously
determined. At any calendar time t the population consists of overlapping generations of
individuals of age 25-64. There are two types of workers, unskilled and skilled, and the total
number of each type of worker is exogenously given5.
3.1 Technology
3.1.1 Tradable Sectors
There are two tradable sectors, the goods sector (G) and the service sector (R), each producing
output Y using unskilled labor U and skilled labor S, physical capital K, and a locally provided
productivity factor M. Specifically, production of sector i located in city j at time t, valued at
the sector’s period t real price p, is given by
pitYijt = A
ijt
(Uijt)αi1t (
Sijt)αi2t (
Kijt)αi3t (
Mjt)1−αi1t−αi2t−αi3t
where Aijt is the real total factor productivity of the sector i in city j. Each sector is subject to an
aggregate productivity shock ζ. The productivity differences across the cities are determined
by the value of city-specific attributes B and agglomeration externality E.6 Specifically, the
real total factor productivity Aijt is given by
Aijt = pitζ
itB
ijt E
ijt
4Manufacturing sector consists of the mining, construction and manufacturing industry categories, the servicesector of the transportation and utilities, trade, finance, insurance and other service industry categories excludingreal estate industry. Housing sector consists of real estate industry.
5Conceptually the number of workers should be continuum. However, to make the notation be consistent withthat of numerical algorithm section, I assume there are finite number of workers hereafter.
6See Davis and Weinstein (2002)
8
The sector-specific real productivity shock, zit = pitζ
it, evaluated at constant dollars, is assumed
to follow a joint first-order Vector Autoregressive process in growth rates.7
logzit+1 − logzit = φi0 + ∑k=G,R
φi1
(logzkt − logzkt−1
)+ ηit+1 (i = G, R) (1)
By normalizing Bi2t = 1, Bi1t measures the relative value of area-specific attributes in city 1.
Bijt =
exp(
βit)
j = 1
1 j = 2
Reflecting the reduced manufacturers’ gains from locating in the Rust Belt due to improve-
ments in transportation and communication technology, βit is assumed to be constant up to
1970 and then to follow linear trends until 1990 and then different linear trends thereafter.8
Specifically,
βit =
βi0 if t < 1970
βi0 + βi1 (t− 1970) if 1970≤ t < 1990
βi0 + 20βi1 + β
i2 (t− 1990) if 1990≤ t ≤ 2010
Following Lucas and Rossi-Hansberg (2002), the agglomeration externality depends on the
aggregate skill density in the city.
Eijt =[(
uGjt)γi1 (
uRjt)γi2 (
sGjt)γi3 (
sRjt)γi4]γi5
,4
∑k=1
γik = 1
where uijt and sijt are the density of human capital in sector i in city j coming from unskilled
workers and skilled workers respectively
uijt =UijtLjt
sijt =SijtLjt
where Ljt is the land size of the city j at time t.
3.1.2 Housing Services
In each city j, housing service can be produced using the following production function7I do not distinguish between relative product price changes and Hicks-neutral technological change.8The breaks in time trends are necessary to capture the rapid decline of the goods sector in the Rust Belt
between 1970 and 1990.
9
H jt =(
KHjt)λ(
Ljt)1−λ
where KHjt is the aggregate physical capital employed in housing service sector in city j at
time t and Ljt the exogenous supply of developed land in city j at time t.9
3.2 Choice Set
At each age, from a = 25− 64, individuals choose among six discrete alternatives: a pair (I, J)of two location alternatives J ∈ {1,2} and three work alternatives I ∈ {O : out of labor force, G, R}.They also decide on their consumption level of local housing service (h) and a numeraire(b).
I define the following dichotomy variables to denote individual decision.
dia = 1{Ia = i}
dja = 1{Ja = j}
dija = 1{Ia = i, Ja = j}
3.3 State Space
At any time t, agents in the economy form a common forecast of the evolution of future skill
rental prices and housing rents. Let Ωat be the individual state spaces at age a and time t.
The individual state space consists of current and past skill rental prices, current and past
housing rental prices, current idiosyncratic shocks, years of work experience, past decisions,
skill type. The population consists of nθ discrete types of individuals who permanently differ
in preferences and skill endowments.10 The probability that an individual is of type θ depends
on the individual’s skill type κ ∈ {unskilled, skilled}. In what follows, I drop the θ subscriptswhen the meaning is clear.
3.4 Preferences
Each individual receives a utility flow that depends on her consumption of local housing ser-
vice (h), consumption of a numeraire (b), choice of current and past city-sector pair (da,da−1),9I ignore the labor input for the housing service production to simplify the analysis, since the share of labor
input in housing sector is less than 5%.10I estimate the model with three unobservable types, having found substantial improvement in fit over two
types.
10
skill type (κ). Specifically, the flow utility of κ type individual at each age a is given by
Ua = ∑i,j
ωijdija + u (h,b;κ)− Cost (da,da−1;κ)
where Cost (·) is the psychic cost of switching residential location or sector and ωij the non-pecuniary benefits associated with choosing (i, j) pair. Specifically,
ωij = ωiκ + ωjθ (i = G, R j = 1,2)
ωOj = ωOjκ + e
Oja (j = 1,2)
The consumption branch of utility function has a Cobb-Douglas form11
u (h,b;τ) = hµb1−µ
where µ is the skill type-specific housing expenditure share.
3.5 Budget Constraints
The budget constraint for a κ-type single individual is
b +
[2
∑j=1
(1 + τ jPt
)pHjt d
ja
]h =
2
∑j=1
∑i=G,R
(1− τ jI
)(wijat + yκt
)dija
where wijat is the real wage an individual of age a receives from working in city j and sector i
at time t, τ jPt the local property tax, τjI the local income tax and yκt the type-specific non-labor
income in period t.
3.6 Wage Offers
I follow the Ben-Porath-Griliches specification of the wage function. Labor income is given by
the product of rental price of skill and individual skill level
wijκat = rijκt f
ija
11I follow Davis and Ortalo-Magné (2011).
11
where rijκt is the type-specific skill rental price in sector i and city j at time t. fija is the choice
specific skill level, and depends on characteristics such as unobserved type (θ) and sector-
specific experience accumulated up to age a − 1. It also depends on age-varying shocks toskill eija which are serially independent.
f ija = exp
bi1θ +(
∑k=G,R
bik2 xka
)bi3+ e
ija
xka is the experience level in sector k, and evolves as follows
xka+1 =
xka + 1 if dka = 1xka otherwise3.7 Local Governments
The local governments levy property tax and income tax based on the exogenously given rate
τjPt and τ
jI , and spend the revenues REV
jPt and REV
jIt to invest in infrastructure. The accrued
rents from providing infrastructure REV jMt are also spent on the investment. Specifically,
REV jMt = ∑i=G,R
pitYijt
(1− αi1 − αi2 − αi3
)REV jt = REV
jPt + REV
jIt + REV
jMt
The quality of local infrastructure is determined by the per capita expenditure of the local
government and evolves as follows,
Mjt+1 = (1− δ)Mjt +
REV jtN jt
(2)
where N jt is the population size of city j in period t.
3.8 Capital and Land Ownership
There are remaining rentals paid to capital and land in this economy. π fraction of the total
rental income is distributed to skilled workers, and the remaining to unskilled workers. Work-
ers, within the two skill groups, own identical diversified portfolios of the domestic capital
12
and land, and hence have equal share of domestic capital and land. Let ΓKt and ΓLt denote the
total rents at time t for domestic capital and land respectively. Specifically,
ΓKt = rKt K
ΓLt =2
∑j=1
(pHjt H
jt − rKt K
Hjt
)= (1− λ)
2
∑j=1
pHjt Hjt
Then, the type-specific non-labor income in each period is given by
yst =π(ΓKt + Γ
Lt)
Nst
yut =(1− π)
(ΓKt + Γ
Lt)
Nut(3)
where Nκt is the exogenously given total number of κ-type workers in this economy.
3.9 Market Clearing
At any time t all the local labor markets and housing markets are cleared with the equilibrium
prices and allocations
Pt =[rG1ut ,r
R1ut ,r
G2ut ,r
R2ut ,r
G1st ,r
R1st ,r
G2st ,r
R2st , p
H1t , p
H2t ,yst,yut
]Qt =
[UM1t ,U
R1t ,U
G2t ,U
R2t ,S
G1t ,S
R1t ,S
G2t ,S
R2t , H
1t , H
2t , N
1st, N
1ut
]Furthermore, following objects are set to satisfy the budget balancing requirement.[
M1t+1, M2t+1
]
13
Let Nuat and Nsat be the total number of unskilled and skilled individuals respectively who
are aged a at time t, aggregate skill supplies are given by
Uijt =64
∑a=25
Nuat
∑n=1
f ijnatdijnat
Sijt =64
∑a=25
Nsat
∑n=1
f ijnatdijnat
N jut =64
∑a=25
Nuat
∑n=1
dijnat (4)
N jst =64
∑a=25
Nsat
∑n=1
dijnat
The demand side of the model is essentially static, and hence the aggregate skill demand is
determined by equating the marginal revenue product of aggregate skill for each city and
sector to its current skill rental price. The amount of capital used in each sector at time t is
given by equating the marginal revenue product of capital to the exogenous rental price of
capital, rKt . Specifically,
∂pitYijt
(zit,U
ijt ,S
ijt ,K
ijt , M
jt
)∂Uijt
= rijut i = G, R j = 1,2
∂pitYijt
(zit,U
ijt ,S
ijt ,K
ijt , M
jt
)∂Sijt
= rijst i = G, R j = 1,2 (5)
∂pitYijt
(zit,U
ijt ,S
ijt ,K
ijt , M
jt
)∂Kijt
= rKt i = G, R j = 1,2
At each time t, the eight excess demand function (exd) satisfy[Uijt]
Demand−[Uijt]
Supply= exdijut
(Pt; Z̃t, r̃Kt , τ̃t,Ω̃t,Ψ
)= 0 i = G, R j = 1,2[
Sijt]
Demand−[Sijt]
Supply= exdijst
(Pt; Z̃t, r̃Kt , τ̃t,Ω̃t,Ψ
)= 0 i = G, R j = 1,2
where Z̃t is the vector of current and past real productivity shocks, r̃Kt the vector of the current
and past capital rental prices, τ̃t the vector of the current and past tax rates, Ω̃t the state space
vector at time t over all individuals in the economy and Ψ the set of model parameters.
14
The housing demand in the city j is given by:
H jdt =64
∑a=25
Nat
∑n=1
hnatdjnat j = 1,2
Supply side of housing market is static. By equating the marginal revenue product of capital
to the exogenous rental price of capital, rKt , housing supply is given by
H jst =
(λpHjt
rKt
) λ1−λ
Ljt j = 1,2
At each time t, the housing demand and supply in each city should be equal.
H jdt = Hjst j = 1,2
Let H jt be the equilibrium housing quantity in city j.
3.10 Forecasting Rule
The relevant aggregate state variables are skill rental prices, gross housing rental prices and
non-labor incomes:
Pt =[rG1ut ,r
R1ut ,r
G2ut ,r
R2ut ,r
G1st ,r
R1st ,r
G2st ,r
R2st , p
1Ht, p
2Ht,yst,yut
]I assume the following VAR12 process13 in growth rates:
log Pt+1 − log Pt = Φ0 + Φ1 (log Pt − log Pt−1) + Φ2 (log Zt+1 − log Zt) (6)
where Zt =[zGt ,z
Rt]′.
3.11 Solution Algorithm
The solution algorithm is an extension of the method developed in Lee and Wolpin (2006).
Given parameters of the model, observed sequences of output in each sector, the rental price
12Parsimonious specifications can be considered by imposing some restrictions on the VAR process.13I am agnostic about the workers knowledge over the exogenous evolution of capital rental price, property tax
rates and total population.
15
of capital, supply of land in each city and local property and income tax rates, the algorithm
consists of the following steps:
1. Choose a set of parameters for the equilibrium process (6) and for the aggregate shock
process (1).
2. Solve the optimization problem for each cohort that exists from t = 1 through t = T.
The maximization problem can be cast as a finite horizon dynamic programming problem.
The value function can be written as the maximum over alternative-specific value functions,
Vija (Ωat), i.e., the expected discounted value of alternative ij, that satisfy the Bellman equation,
namely
Va (Ωat) = maxi,j
[Vija (Ωat)
]Vija (Ωat) = max
b,hUija (b, h;Ωat) + ρEV
(Ωa+1,t+1|d
ijat = 1,Ωat
)
The solution of the optimization problem is in general not analytic. In solving the model
numerically, the solution consists of the values of EV(
Ωa+1,t+1|dijat = 1,Ωat
)for all i and j and
elements of Ωat.14 The solution method proceeds by backward recursion.
3. Guess an initial set of values for aggregate prices and local infrastructures at t = 1, say
(P1)0 =
[(rijκ1)0
ij,(
pHj1)0
, (yκ1)0]
and(
Mj1)0
. Given this initial guess, I proceed as Gauss-
Seidel algorithm: (a) Update skill rental prices to be the marginal product of aggregate skill
to have(
rijκ1)1
. (b) Using(
rijκ1)1
, calculate the non-labor income (yκt)1. (b) Using
(rijκ1)1
and (yκ1)1 calculate the housing expenditure, and equate the supply and demand of housing
service to have(
pHj1)1
. Specifically,
(a) Given (P1)0 and the distribution of state variables for each cohort alive at that time and
between age 25 and 64, simulate a sample of agents’ chosen alternatives at t = 1 by drawing
from the distribution of the idiosyncratic shocks to preferences and skills. Using (4), calculate
aggregate skill levels, housing demand and populations of two types of skills in each city-
sector. Given aggregate skill supplies, equate the marginal product of capital in each of four
city-sectors to the rental price of capital, which are data. Equate the two production functions
14To circumvent the “curse of dimensionality”, I adopt the approximation method developed by Keane andWolpin (1994).
16
to the actual output in the two sectors.
∂pitYijt
(zit,U
ijt ,S
ijt ,K
ijt , M
jt
)∂K jit
= rKt i = G, R j = 1,2
2
∑j=1
pitYijt
(zit,U
ijt ,S
ijt ,K
ijt , M
jt
)= outputi i = G, R
where Mjt is predetermined for t = 2, however, use (M1)0 for period one. Solve the equations
for the optimal capital input in each city-sector and for the two aggregate shocks, say (Zi1)1.
Calculate the marginal product of the skill, at the calculated value of skill, capital and shocks.
Let(
rijκ1)1
denote the updated skill rental prices at period one.
(b) Non-labor incomes are also functions of skill rental prices. Using this relation and 3, I can
update the period one value of non-labor income, say (yκ1)1.
(c) Housing expenditure in city j in t = 1(
HEj1)
is a function of skill rental prices and aggre-
gate skill quantities. Using(
rijκ1)1
, (yκ1)1, calculate the housing expenditure, say
(HEj1
)1. The
housing demand and supply are given by
H jd1 =
(HEj1
)1(1 + τ jP1
)pHj1
, j = 1,2
H js1 =
(λpHj1
rK1
) λ1−λ
Lj1, j = 1,2
Thus, the equilibrium housing price is given by
(pjH1)1
=
(
HEj1)1(
1 + τ jP1)
Lj1
1−λ(
rK1λ
)λ
Using(
rij1t)1
ij, and
(pHj1)1
, calculate the total revenues of the local governments in period one,
REV j1 . The quality of second period local infrastructure is given by
Mj2 = (1− δ)Mj1 +
REV j1N j1
17
(only for t = 1) Since Mj1 is predetermined, there is no equilibrium restriction to pin-down this
value. Thus, I impose the following additional restriction that the value of infrastructure in
the first two periods are the same:
Mj1 = Mj2, j = 1,2
Thus, use M2 as the updated guess for the first period value, say (M1)1. From (a)-(d), we have
(P1)1 and (M1) which will, in general, differ from the initial guesses.
4. Update the initial guesses for the prices and the quality of local infrastructure to be equal
to (P1)1 and (M1)
1. Repeat step 3 until the sequences of prices, quality and aggregate shocks
converge, say to (P1)∗, (M1)
∗ and (Z1)∗.
5. Guess and initial set of values for the period two prices, say (P2)0 = (P1)
∗. Repeat step 3-4
for t = 2 to obtain (P2)∗ and (Z2)
∗.
6. Repeat step 5 for t = 3, ..., T.
7. Using the calculated series of equilibrium skill rental prices and aggregate shocks, estimate
(1), the VAR governing aggregate shocks, and (6), the process governing the equilibrium
prices.
8. Using theses estimates, repeat until the series of prices and aggregates shocks converge.
4 Estimation Method
The model parameters are estimated by simulated minimum distance (SMD) method. Specifi-
cally, the SMD estimator minimizes a weighted distance measure between sample aggregated
statistics and their simulated analogs. The weights are given by the inverse of estimated
variances of the sample statistics.
The data moments come from the several sources. The March Current Population Surveys
over the period 1968-2011 and the National Longitudinal Surveys 1979 youth cohort over the
period 1979-1993 provide information on life cycle employment and schooling choices, and
on wages; various U.S. Censuses from 1960 to 2010 on housing consumption; and the Bureau
of Economic Analysis (BEA) provides data on sectoral capital stocks and outputs.
The following is a list of aggregate statistics available from various sources.
18
1. Career decisions
CPS data
(a) The proportion of individuals choosing each of the six alternatives by year (1968-
2010) and age (25-64).
(b) The proportion of individuals choosing each of the six alternatives by year and skill
type (unskilled, skilled).
(c) The proportion of individuals choosing each of the six alternatives by year and past
choice.
NLSY79 data
(a) The proportion of individuals choosing each of the six alternatives by experience
and skill type.
2. Wages
CPS data
(a) The mean, median, and 10th and 90th percentiles of the log hourly real wage by
region- and sector-categories and year.
(b) The mean, median, and 10th and 90th percentiles of the log hourly real wage by
the two skill types and year.
(c) The variance of the log hourly real wage by region- and sector-categories and year.15
(d) The variance of the log hourly real wage by the two skill types and year.
NLSY79 data
(a) The mean log hourly real wage by work experience and skill types.
3. Mean non-labor income by year and skill types.
4. Housing expenditure
(a) The mean, median of real housing rent by region and year.
(b) The mean, median of real housing rent by skill type and year.
5. Skill type distribution
(a) Distribution of skill over cities by year and age.
15I also allow for log-normally distributed measurement error in the reported hourly wage rate.
19
6. Career transitions
CPS data
(a) One-period joint transitions between two location alternatives by year (1982-2010)
and skill type.
(b) One-period joint location transitions by age and skill type
(c) One-period joint transitions between two sectors by year.16
(d) One-period joint sectoral and home transitions by age and skill type (matched CPS)
Census data
(a) Five-period joint transitions between two location alternatives by decade (1970-
2010) and skill type.
NLSY79 data
(a) Distribution of years of work experience in each sector.
7. Location- and sector specific capital and output: by year17
5 Results
5.1 Parameter Estimates
The parameter estimates are shown in this section.18 I normalize some parameters because
skill is not observable, but must be inferred from wages. As a result, the constant terms in the
skill production functions cannot be separately identified from the level of skill rental prices.
I normalize the constant term in each sector skill production function for type one person
to zero. The non-pecuniary benefit associated with employment in the goods sector is also
normalized to zero.
Table 2 shows the estimates for production parameters. The share of unskilled labor had
diminished in both sectors during this period; α1 had decreased in goods and service sector.
16A number of years are missing because identifies are that match household between 2 years are not available.The missing transitions are between 1971 and 1972, 1972 and 1973, 197y6 and 1977, 1985 and 1986, 1995 and 1996.
17Imputed following Desmet and Rossi-Hansberg (2010)18The variance-covariance matrix of the parameter estimates is given by
(G′W−1G
)−1, where G is the matrix ofderivative of the moments with respect to the parameters and W is the variance-covariance matrix of the moments.Off-diagonal elements are ignored.
20
Table 2: Production Parameters
Production Function Production ShockGoods Service Goods Service Goods Service
α10 0.369 0.371 β0 0.030 -0.006 φ0 −0.081 −0.027α11 0.3 0.167 β1 -0.001 -0.009 φG −1.319 −1.011α20 0.105 0.205 β2 -0.042 -0.034 φR 1.924 1.441α21 0.3 0.475 σGG 0.009α30 0.3 0.418 σGR 0.024α31 0.33 0.345 σRR 0.020
Table 3: Utility ParametersType1 Type2 Type3 No College College
ω2 13.18 -84.35 30.00 ωκ 19107 35709ωR 884 ση 12872 13842
µ 0.33 0.27
The estimate for β shows that the Rust Belt had comparative advantage in production of
goods before 1970, i.e., βG0 is positive. However, the advantage had decreased; in 2010 it is
less productive in producing goods. The estimate for φ0 is lower in the goods sector than in
service sector; the growth rate in the aggregate productivity was lower in the goods sector.
Table 3 shows the estimate for utility parameters. The non-pecuniary benefits associated with
working in the service sector are larger than in the goods sector. Type 2 people have larger
non-pecuniary benefits from living in the Rust Belt than from living elsewhere. Unskilled
worker’s expenditure share on housing service is higher than that of skilled worker.
The mobility costs are presented in Table 4. Sector switching costs are higher for unskilled
people. The location switching costs are 23420 and 27364 for unskilled and skilled labor
respectively. Since the difference in the wage between the skilled people and unskilled people
is larger than the difference in the mobility costs, net benefit of migration is higher for the
skilled people.
Table 5 shows the estimates for skill production function parameters. Experience obtained in
a given sector is more transferable to other sector for skilled people than for unskilled people.
21
Table 4: Sector Switching Costs
Unskilled SkilledT\T+1 Goods Services Home Goods Services HomeGoods 0 8306
136950 9534
11833Service 8910 0 4176 0Home 13695 0 11833 0
Table 5: Skill Production Functions
i Type ProbabilityGoods Services Unskilled Skilled
bi11 0.000 0.000 0.07 0.18bi12 0.018 -0.01 0.23 0.15bi12 -0.012 0.010 0.70 0.67
Unskilled Skilledbik2 Goods Services Goods Services
k =Goods 0.070 0.014 0.078 0.036k =Service 0.007 0.045 0.024 0.091
bi3 0.57 0.84 0.64 0.59σ1e 0.45 0.41 0.51 0.44σ2e 0.47 0.42 0.67 0.44
5.2 Model Fit
Figure 5 shows that the model is able to fit the data well. It compares the relative share of
unskilled people in the Rust Belt over the time in the actual data to that from the estimated
model. As seen, the model can capture the rapid increase in the less educated people in the
Rust Belt between 1970 and 1990. In 1990 the share of unskilled people (non-college educated)
is 10 percent higher in the Rust Belt than in other U.S. areas.
22
Figure 5: Relative Share of Unskilled People: Rust BeltOutside
65 70 75 80 85 90 95 100 105 1101
1.02
1.04
1.06
1.08
1.1
1.12
1.14
1.16
1.18
1.2Relative skill dist
datamodel
6 Conclusion
In this paper, I have studied what triggered the economic change in the Rust Belt, and why the
adjustment process took the form it did. I specifically addressed two empirical issues. First,
to what extent is the decline of the Rust Belt attributable to the transition of the U.S. economy
to a service sector and to what extent to the reduced area-specific advantage of the Rust Belt?
Second, I assessed the extent to which the increased share of less educated in the Rust Belt
results from higher mobility costs for the less educated or higher preference for less expensive
housing. To this end, I have developed and estimated a multi-sector multi-city competitive
equilibrium model of the local labor markets with both idiosyncratic shocks to preferences,
and skill and aggregate production shocks. The model was able to fit the data well in many
aspects.
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Appendix
A1. Identification
Identification is achieved by a combination of functional form and distributional assumptions,
along with exclusion restrictions. In terms of the latter, production function parameters are
identified because current and past cohort sizes and rental prices of capital, assumed exoge-
nous, are valid instruments for input level. Identification of the wage offer parameters follows
from standard selection correction arguments, namely from distributional assumptions and
from the existence of variables that affect choices that are not in the wage offer functions.
Identification of utility function parameters follows from the existence of variables in the wage
function that does not enter the utility function, for example, sector-specific work experience.
I do not estimate the (subjective) discount factor, which, in prior partial equilibrium structural
estimation problems, has proven difficult to pin down, It is instead fixed at 0.95, a 5 percent
discount rate, which is close to the implied interest rate given that the rental price of capital
in the data is 0.15 and given a 10 percent annual depreciation rate.
Let me heuristically explain how the identification is achieved. First order condition of profit
maximization identifies the factor shares as follows
αi1 =pitY
ijt
rijutUijt
, αi2 =pitY
ijt
rijstSijt
, αi3 =pitY
ijt
rKt Kijt
Then, I can Identify real TFP Ajit as a Solow residual.
Aijt =pitY
ijt(
Uijt)αi1 (Sijt )αi2 (Kijt )αi3 (Mjt)1−αi1−αi2−αi3
25
By comparing TFP of the same sector i in two different cities, I can identify the agglomeration
effects and city-and sector-specific productivity trends.
log(
Ai1tAi2t
)= βi0
+ βi1 (t− 1960)
+ γi5
[γi1 log
(uG1tuG2t
)+ ... + γi4 log
(sR1tsR2t
)]Then, the identification of parameters in utility function, cost function and skill production
function would be similar to that of usual dynamic discrete choice model.
A2. Implicit User Cost of Housing
Following Poterba (1992), I calculate the user cost of housing for a house of market value V
from the expression,
R =[(
1− τy)(
i + τj)+ ξ]
V
where τy is marginal income tax rate, i interest rate, τj is property tax rate, and ξ is a parameter
that captures risk premium and depreciation. I set ξ = −0.02 following (Poterba, 1992). I setthe marginal tax rate from based on tax brackets.
A3. Data Inputs
Developed Land
The amount of land built up for residential purposes in 1976, 1992 and 2001 is calculated based
on the data developed in Overman et al. (2008) and by Albert Saiz19. 1976 and 1992 data are
constructed from two publicly-available remote-sensing data sets20. I impute the amount of
residential land for other periods from the information on the number of housings units. U.S.
Census Bureau provides the unit of housing for individual states from 1940.
19University of Pennsylvania, Wharton20The most recent of these two remote-sensing data sets, the 1992 National Land Cover Data is derived mainly
from 1992 Landsat5 Thematic Mapper satellite imagery.
26
Cohort Size
Cohort size is obtained from Vital Statistics of the United States and from U.S. Census Bureau
reports.
Skill Type Distribution
I define skilled worker as one with at least one year of college education. The distribution of
skill type for each cohort is estimated from CPS and U.S. Census.
Tax Rates
I set 3% for state income tax for both areas. Property tax (for renter) is assumed to be 35% for
both areas.
Capital Stock
Following Garofalo and Yamarik (2002) and Desmet and Rossi-Hansberg (2010), I approximate
non-residential capital at the state level by using sectoral non-residential capital stock data at
the U.S. level and allocating it to the different states proportional to their sectoral weights.
Sectoral non-residential capital stock data is available from the National Economic Accounts
from the BEA. I allocate the sectoral capital stock to the states in function of their shares of
sectoral earnings. Data on sectoral earnings both at the state and the U.S. levels come from
the Regional Economic Accounts of the Bureau of Economic Analysis.
27
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