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. chamna@sas.upenn.edu

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