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Public Debt, Private Pain:
Regional Borrowing, Default, and Migration
WP 21-13
Grey Gordon
Federal Reserve Bank of Richmond
Pablo Guerron-Quintana
Boston College
Public Debt, Private Pain:
Regional Borrowing, Default, and Migration∗
Grey Gordon
Federal Reserve Bank of Richmond
Research Department
greygordon@gmail.com
Pablo Guerron-Quintana
Boston College
Department of Economics
pguerron@gmail.com
July 30, 2021
Abstract
How do local government borrowing, default, and migration interact? We
find in-migration results in excessive debt accumulation due to a key externality:
Immigrants help repay previously-issued debt. In addition to providing direct
IV evidence on this mechanism, we show cities are heavily indebted and remain
so even after large population growth, resulting in boom defaults. While default
rates are currently low, default risk has increased secularly despite the secular
decline in interest rates, which we show lowered default risk else equal. Our
quantitative model implies large interest rate declines in the Great Recession
and COVID-19 crisis prevented default.
JEL Codes: E21, F22, F34, R23, R51
Keywords: migration, population, debt, default, cities, bankruptcy, Detroit,
regional
∗
We thank our discussant Pablo D’Erasmo as well as Marco Bassetto, Susanto Basu, Jeff
Brinkman, Lorenzo Caliendo, Jerry Carlino, Ryan Chahrour, Yoosoon Chang, Satyajit Chatterjee,
David Childers, Daniele Coen-Pirani, Hal Cole, Jonas Fisher, Aaron Hedlund, Ivan Ivanov, Dirk
Krueger, Amanda Michaud, Leonard Nakamura, Jim Nason, Jaromir Nosal, Santiago Pinto, Esteban Rossi-Hansberg, Pierre-Daniel Sarte, Sam Schulhofer-Wohl, Tony Smith, Nora Traum, Marcelo
Veracierto, and Mark Wright for valuable discussions. We also thank participants at various seminars and conferences. Alexey Khazanov and Michelle Liu provided excellent research assistance.
A previous version of this paper circulated under the title “On Regional Borrowing, Default, and
Migration.” The views expressed are those of the authors and do not necessarily reflect those of the
Federal Reserve Bank of Richmond or the Federal Reserve System.
1
1
Introduction
Municipal debt is a $1.9 trillion source of financing for local governments (Mayo et
al., 2020). Normally considered a safe investment, yields exploded both in the Great
Recession and COVID-19.1 Since municipal bond yields reflect actual default risk
(Schwert, 2017), their sharp increase suggests local government finances are more
vulnerable than they appear in noncrisis periods. Both crises precipitated large-scale
interventions, which at least for the COVID-19 crisis, seem to have worked (Haughwout et al., 2021). At the same time, local government finances depend not only on
national economic conditions, but also on migration decisions, with each entrant to
the city reducing debt per person and increasing the tax base (and each departure
doing the opposite). This link can be clearly seen in major urban centers where
population growth or decline can be tremendous. The leading example of the latter,
Detroit, had a 35% reduction in population from 1986 to 2013, which undoubtedly
contributed to its 2013 bankruptcy. The remote work era that COVID-19 may usher
in has potential to induce massive population shifts with commensurate implications
for city finances. Despite this, there is currently limited empirical evidence and no
model linking city finances, migration, and default. This paper fills these gaps in
the literature.
We first use a two-period Lucas (1972)-type islands model to highlight a key
mechanism, specifically, an overborrowing externality. Each island represents a local
economy and has households who make migration decisions, a per person endowment, and a planner who issues debt in the first period (transferring the proceeds to
households) and repays it in the second (using lump-sum taxes). The key assumption is that the local planner maximizes the welfare of current residents. The model
reveals that, relative to an economy-wide planner, local planners have an incentive
to overborrow. The reason is simple: new arrivals in the second period will help
repay debt issued in the first period, and the planner does not directly value their
utility.
With this externality in mind, we turn to the data where we expect to find, and
do find, excessive debt. Using comprehensive datasets on city finances, population,
migration, and labor productivity, we establish six stylized facts. (While technically
1
For instance, the BBB-AAA spread increased from 0.6% in 2007Q1 to a peak of 4.3% in 2009Q1;
while the AAA spread over 10-year Treasuries rose from 0.71% on February 21, 2020, to 2.74% by
March 23, 2020. (Authors’ calculations using data from Haver.)
2
incorrect, we will refer to cities and municipalities interchangeably.)2 First, city
debt and default risk have been increasing over time. Second, cities of all types are
heavily indebted. Third, cities of all types are near state-imposed borrowing limits.
Fourth, cities respond to arguably exogenous variation (constructed using a Bartik
style shift-share instrument) in in-migration rates by increasing debt. Fifth, cities
remain leveraged even after booms in population and productivity, leading to boom
defaults. Lastly, default risk is highly senstive to interest rate movements.
We extend the two-period model to a full-blown quantitative model with an
infinite horizon, production, government services, housing, borrowing limits, and
default. After showing the economy can be centralized (at a local level), we demonstrate the calibrated model reproduces the data’s stylized facts. We then use the
model to understand the observed path of the economy in the Great Recession and
COVID-19. In both cases, the observed decline in real interest rates proves crucial
in preventing an onslaught of municipal defaults.
Related literature
Our paper builds on the large sovereign default literature begun by Eaton and
Gersovitz (1981), which has focused almost exclusively on nation states. Some of the
key references are Arellano (2008), Chatterjee and Eyigungor (2012), Hatchondo and
Martinez (2009), and Mendoza and Yue (2012). (The handbook chapter Aguiar et
al. (2016) provides a thorough description of the literature.) Epple and Spatt (1986)
is an exception that argues states should restrict local debt because default by one
local government makes other local governments appear less creditworthy. Such a
force is not at work in our model because we assume full information. We contribute
to this literature by showing migration strongly influences debt accumulation and
can result in boom defaults.
Our work also connects to a vast literature on intranational migration. The empirical work and to a lesser extent theoretical is surveyed in Greenwood (1997).
Two seminal papers in this literature, Rosen (1979) and Roback (1982), employ a
static model with perfectly mobile labor. This implies every region provides individuals with the same utility. While this indifference condition allows for elegant
characterizations of equilibrium prices and rents, it also means government policies
are completely indeterminate: every debt, service, or tax choice results in the same
utility. Our model breaks this result by assuming labor is imperfectly mobile, which
2
A municipality is a city, town, or village that is incorporated into a local government.
3
lets it match both the sluggish population adjustments and the small correlations
between productivity and migration rates observed in the data.
More recently, Armenter and Ortega (2010), Coen-Pirani (2010), Van Nieuwerburgh and Weill (2010), Kennan and Walker (2011), Davis et al. (2013), and Caliendo
et al. (2017) have analyzed determinants of migration and its consequences in the
U.S. Kennan and Walker (2011) use a structurally estimated model of migration
decisions and find expected income differences play a key role, providing external
evidence of the model’s productivity-driven migration decisions. Outside the U.S.,
recent research has been focused on migration in the EU (Farhi & Werning, 2014;
Kennan, 2013, 2017). All these papers abstract from debt. To our knowledge, ours
is the only quantitative model of regional borrowing and migration, let alone having
default.
A few papers in this literature have discussed the potential for local governments
to overborrow because of migration. Bruce (1995) and Schultz and Sjöström (2001)
prove that overborrowing generally does occur. However, both of their models are
two-period models with costless moving, and our theoretical results imply this is
not an innocuous assumption. Additionally, we show empirically and quantitatively
the role of overborrowing in reproducing many of the data’s features.
Finally, building on earlier versions of this paper, Alessandria et al. (2020) features a sovereign government (Spain in their calibration) facing in-migration and issuing debt. Similar to our finding that in-migration induces overborrowing, they find
in-migration causes increased indebtedness relative to a no-in-migration economy.
A key difference from our paper is that in-migration, in their model, is exogenous
and out-migration is not allowed: in contrast, in-migration and out-migration in this
paper are jointly determined given the entire distribution of available locations.
The rest of the paper is organized as follows. Section 2 presents a simple model
that highlights the overborrowing mechanism inherent in models with in-migration
and borrowing. Section 3 documents key stylized facts. Section 4 lays out the quantitative model, and Section 5 describes the calibration. Section 6 shows the quantitative model reproduces the stylized facts and analyzes the Great Recession and
COVID-19 shocks. Section 7 concludes. The appendices report data details, the
computational algorithms, and proofs.
4
2
The overborrowing mechanism
First, we highlight how migration influences borrowing decisions and efficiency using
a two-period model. To focus purely on the role of borrowing, we assume there is
full commitment to repay debt and, hence, no default.
The economy is comprised of a unit measure of islands and a unit measure of
households. Assume islands are homogeneous, and consider an arbitrary one. In
the first (second) period, the island has a per person nonstochastic endowment of y1
(y2 ). The local government issues −b2 debt per person (b2 > 0 means assets) at price
q̄. Total debt issuance is −b2 n1 , where n1 is the initial measure of households on
the island. At the beginning of the second period, households draw an idiosyncratic
utility cost of moving φ ∼ F (φ) with a density f and then decide whether to migrate.
If they migrate, they pay φ and obtain expected utility J, which is an equilibrium
object.
Households value consumption according to u(c1 ) + βu(c2 ), where c1 (c2 ) is
consumption in the first (second) period. Household utility in the second period is
u(c2 ) if they stay and J − φ if they move, so migration decisions follow a cutoff
rule in φ with indifference at J − u(c2 ). Consequently, the outflow rate is o2 =
F (J − u(c2 )). The inflow rate is given by i2 = īI(u(c2 )), where I is a differentiable,
increasing function, and ī is an equilibrium object that ensures aggregate inflows
equal aggregate outflows. (Consequently, inflows can depend on the distribution of
utility across islands, but that information must be summarized in ī.) The population
law of motion is n2 = (1 + i2 − o2 )n1 . We assume the migration decision is noisy in
the sense that F (0) > 0, so that some people will move even if u(c2 ) = J.
After all migration has taken place, the government pays back its total obligation, −b2 n1 , by taxing the n2 households lump sum. Consequently, per person
consumption in the second period is c2 = y2 + b2 n1 /n2 . The government’s problem
may be written
Z
max u(c1 ) + β
b2
s.t. c1 + q̄b2 = y1 ,
max {u(c2 ), J − φ} dF (φ)
c2 = y2 + b2
i2 = īI(u(c2 )),
n1
,
n2
n2 = n1 (1 − o2 + i2 ),
c1 , c2 ≥ 0,
(1)
o2 = F (J − u(c2 )).
An equilibrium is a pair {i, J} with optimal migration, consumption, and borrowing decisions such that
5
1. total inflows equal total outflows, i
R
I(u(c2,j ))n1,j dj =
R
F (J − u(c2,j ))n1,j dj,
and
2. the expected utility of moving is consistent, J =
R
iI(u(c
))n
u(c2,j ) R iI(u(c2,j ))n1,jdi dj
2,i
1,i
(where j indexes islands).
Proposition 1 gives the Euler equation for government bonds (all proofs are in
Appendix C).
Proposition 1. The local government’s Euler equation is
1 − o2
b2 ∂n2
0
0
u (c1 )q̄ = β
u (c2 ) 1 −
.
1 − o2 + i2
n2 ∂b2
(2)
The Euler equation reflects two competing forces. One is an externality seen in
the term
1−o2
1−o2 +i2
≤ 1. Because the planner does not value the utility of new entrants
and because new entrants bear
i2
1−o2 +i2
of the debt burden (which is their share of the
second period population), the marginal cost associated with an additional unit of
1−o2
1−o2 +i2 . All else equal, higher
1−o2
(β 1−o
) and increases borrowing.
2 +i2
debt—holding fixed migration rates—is 1 − 1−oi22+i2 or
in-migration lowers the effective discount factor
Clearly, then, the assumption that the planner only values current residents plays a
key role. But note that it is also the most natural assumption: if households could
vote on the planner’s policy in the first period, they would unanimously approve it
because it maximizes their welfare.
Before discussing the second force, we emphasize that this externality is really
about in-migration, not out-migration. Consider an extreme case where half the
population leaves o2 = 1/2 but no one arrives i2 = 0. In that case, the overborrowing
term
1−o2
1−o2 +i2
equals 1, i.e., there is no extra discounting in the Euler equation. The
reason is that while half of current residents leave and pay nothing, the half who
remain must pay double, and this offsets in the Euler equation. On the other hand,
if no one leaves o2 = 0 but the population doubles through in-migration i2 = 1,
then the overborrowing term
1−o2
1−o2 +i2
equals 1/2, implying an effective 50% discount
on debt issuance. In intermediate cases where o2 and i2 are positive, in-migration
has a first-order effect while out-migration has only a second-order effect. While the
externality is primarily driven by in-migration, out-migration still has a first-order
effect on debt per person and hence will play a key role in default decisions.
The Euler equation’s other, potentially offsetting force, is seen in the term
1−
b2 ∂n2
n2 ∂b2 ,
which is one minus the elasticity of the next period’s population with
6
respect to savings. It reflects that for each person attracted to the island through
less borrowing, the overall debt burden per person falls. (Conversely, if b2 > 0, each
additional entrant reduces assets per person, which discourages savings.) Hence, a
rational government, internalizing the effects of city finances on migration decisions,
should exercise more financial discipline (else equal) to attract individuals to the
islands to reduce debt per person. However, migration decisions in the quantitative
model must be very noisy to match migration patterns in the data; consequently,
1−o
migration will strongly effect local government debt ( 1−o+i
1), but local govern-
ment debt will only weakly effect migration (1− nb ∂n
∂b ≈ 1). Intuitively, we also expect
that government debt has only second-order effects on migration: many of us know
the typical population, temperature, income, and housing prices to expect in a city,
but how many of us know the city’s outstanding debt amount? Consequently, we
expect the desire to overborrow will generally be stronger than this debt-disciplining
effect.
To this point, we have claimed that cities overborrow, implicitly having in mind
the solution to a social planner problem, which we now state. Let ĉ1,i , ĉ2,i denote
the optimal consumption (in periods 1 and 2, respectively) of household i ∈ [0, 1],
and let φi denote the moving cost shock realization the household receives. With
homogeneous islands, the endowments are the same irrespective of moving decisions,
with y1 (y2 ) the first (second) period endowment. Taking migration decisions mi as
given, the planner’s objective function is
Z
max
αi (u(ĉ1,i ) + β(u(ĉ2,i ) − mi φi ))di,
ĉ1,i ≥0,ĉ2,i ≥0
(3)
where αi is the Pareto weight on household i. The resource constraint is given by
Z
Z
ĉ1,i di + q̄ ĉ2,i di = y1 + q̄y2 .
(4)
We say an allocation is constrained efficient if it solves the planner problem with
migration decisions given for some Pareto weights.
Optimality requires that marginal rates of substitution must be equated across
individuals, i.e., βu0 (ĉ2,i )/u0 (ĉ1,i ) = βu0 (ĉ2,j )/u0 (ĉ1,j ) for almost all i, j. Using the
resource constraint, it is easy to show these must also equal q̄, i.e.,
u0 (ĉ1,i )q̄ = βu0 (ĉ2,i ).
(5)
In comparing (5) with the local government’s Euler equation (2), it is clear that
overborrowing will occur if the optimal bond choice b2 is close to zero: in that case,
7
the incentive to attract people—reflected in the term 1 −
b2 ∂n2
n2 ∂b2 —is
close to zero,
while the externality of new entrants shouldering the burden—reflected in
1−o2
1−o2 +i2 —
is not. With q̄ = βu0 (y2 )/u0 (y1 ), implementing the constrained efficient allocation
requires b2 = 0, which results in overborrowing as formalized in Proposition 2:3
Proposition 2. If q̄ = βu0 (y2 )/u0 (y1 ), equilibrium is not constrained efficient.
Moreover, at the constrained efficient allocation, governments would strictly prefer
to borrow.
To summarize, we showed cities have an incentive to overborrow. The mechanism
is an externality created through in-migration: new entrants will help repay debt
issued today. While in theory, migration can act as a disciplining force, we argued
that in practice this is unlikely. We now turn to the data.
3
Empirical patterns of debt, migration, and default
In this section, we will establish six stylized facts using a variety of data sources
described in Appendix A. First, debt and default risk have been increasing over
time. Second, virtually all cities, large and small, productive and unproductive, are
indebted. Third, almost all cities are close to state-imposed borrowing limits. Fourth,
in response to in-migration, expenditures per person and debt per person increase.
Fifth, cities default after both booms and busts in population and productivity. Last,
municipal bond default risk is sensitive to interest rate changes. Before establishing
the facts, however, we begin with a quick overview of municipal debt and default.
3.1
An overview of municipal debt and default
Two broad types of municipal debt exist: general obligation debt (GO) and nongeneral obligation debt (Non-GO). Non-GO debt is typically attached to a specific
revenue stream, e.g., bonds for construction of a toll road that will be paid for
using revenue from the toll road. In contrast, GO debt is fully backed by taxes.
Additionally, Novy-Marx and Rauh (2011), Novy-Marx and Rauh (2009), and Rauh
(2017) have shown state and local governments have a third large type of “debt,”
which is unfunded pension obligations.
How much debt local governments can accumulate is typically constrained by
state-imposed borrowing limits that vary in type and degree. For example, California
3
This inefficiency result may be surprising in light of the seminal paper by Tiebout (1956). It
arises because he assumes costless and fully-directed mobility, which we do not require. We prove
in Proposition 4 in the appendix that efficiency can hold in this case, which requires an “infinite”
elasticity of in-migration to debt.
8
(CA) limits are tied to spending or revenue that year. In contrast, most of the states
restrict debt based on a percentage of property valuations, but the percentages can
differ substantially from as little as 0.5% (IL) to 10% (NY). Table 3 in the appendix
reports some of these, and reveals almost all the states have known exceptions, which
usually include debt related to education, water supplies, and referendum-approved
debt.
When GO and non-GO debt becomes excessive, local governments do have access, potentially, to Chapter 9 bankruptcy. This substantially differs from consumer
and corporate bankruptcy, with one key distinction being that the municipality
must be insolvent, either “unable to pay its debts” or “generally not paying its
debts.”4 The latter statement, which in isolation would appear to be nonrestrictive,
is strengthened by the additional requirement that filers must negotiate in good
faith with creditors. A second key distinction is that local governments are allowed
to keep essentially all their assets in default to prevent creditors from infringing on
the local government’s sovereignty.5
While bankruptcy allows discharge of GO and non-GO debt, pension obligations
are commonly viewed as nondefaultable, either explicitly protected by state constitutions or otherwise protected by contract law (Brown & Wilcox, 2009). In practice,
CA cities that went through bankruptcy did not have their pension obligations reduced (Myers, 2019). In Detroit, a worst-case example, there were in fact some
pension cuts: 4.5% directly with cessation of Cost of Living Adjustments (COLAs)
(Stempel, 2016). However, viewing pension obligations as nondefaultable seems to
be a reasonable approximation.
3.2
Stylized fact #1: Debt and default are increasing over time
Historically, municipal bond default has been rare. For bonds rated by Moody’s,
Moody’s (2013) report there have been 73 municipal bond defaults between 1970
and 2011, a default rate of 0.012%.
However, these low default rates belie the severity of the situation for several
reasons. First, one reason is that default rates have been trending upward over time.
Using a large municipal-bond dataset from Mergent, Figure 1 reports the time-series
4
11 U.S.C. § 101(32)(C).
Chapter 9 “is significantly different in that there is no provision in the law for liquidation of the
assets of the municipality and distribution of the proceeds to creditors. Such a liquidation or dissolution would undoubtedly violate the Tenth Amendment to the Constitution and the reservation
to the states of sovereignty over their internal affairs” (United States Courts, 2018).
5
9
path of default rates, both as a frequency of outstanding bond counts (unweighted)
and as a percent of outstanding debt (weighted). The top left panel uses all bonds,
while the top right panel uses only GO bonds. Until the late 2000s, default rates
for all bonds were generally below 0.05%. However, in the Great Recession, these
doubled or even tripled and have remained elevated. For GO bonds, the upward trend
is less obvious, but the period of relative stability from 2011 to 2019 saw default
rates of around 0.01%, whereas during mid 2000s that rate was zero. It’s also worth
noting both series show substantial sensitivity of default rates to aggregate risk.
This increase in default risk is perhaps more clearly seen by looking at interest
rate spreads over time. For instance, when we compare the weighted P90 yield to
maturity (YTM) over a “riskfree” P10 YTM, the spread has risen from 2% during
the Great Moderation to close to 4% in the 2010s.6 A similar story appears looking
at GO bonds, or the P75-P10 difference, or even to a lesser extent the P25-P10.
(Later, we will look specifically at the Great Recession and COVID-19.) Whether
we look at default rates or interest rate spreads, the story is the same: Default risk
is increasing over time.
Default rate
All bonds
0.15
Weighted
Unweighted
Percent
Percent
0.15
Default rate
GO bonds
0.1
0.05
YTM spread
6
4
0.05
2000
2010
0
1990
2020
YTM spreads over P10
All bonds, weighted
6
P90-P10
P75-P10
P25-P10
2
0
1990
0.1
YTM spread
0
1990
Weighted
Unweighted
2000
2010
2020
4
2000
2010
2020
YTM spreads over P10
GO bonds, weighted
P90-P10
P75-P10
P25-P10
2
0
1990
2000
2010
2020
Note: Default series excludes Puerto Rico.
Figure 1: YTM spreads and default rates over time
6
We use the P10 YTM of municipal bonds as our measure of the riskfree rate because municipal
bonds have a number of key differences from Treasuries that make them not very comparable. Chief
among these differences are that municipal bonds tend to be callable (i.e., they can be refinanced
at lower rates), and they tend to be tax-exempt.
10
There are also other factors that have masked the amount of fiscal stress. As
in Flint, cities can avoid default but come under state management and thereby
lose fiscal autonomy. In fact, Kleine and Schulz (2017) report that Michigan (MI)
had 11 cities (4%), one township, and one county under state oversight due to a
financial emergency (p. 9). Additionally, real interest rates—which we will show
strongly effect municipal bond risk premia—have fallen secularly over time.
Lastly, debt has grown tremendously over the past few decades. For instance,
the average debt per person across counties was $3085 in 2016 but $2107 in 1996 in
constant 2012 dollars—nearly a 50% increase. Similarly, debt for cities grew by 51%
from $496 to $750 per person. With the increased debt burden comes increased risk
that local governments will not be able to repay.
3.3
Stylized fact #2: Cities of all types are indebted
Our second stylized fact is that all types of cities are indebted. This can be seen
in Figure 2, which presents the empirical relationship of log debt per person (p.p.)
and log expenditures (p.p.) One can see cities of all sizes have debt: there are not
cities clustering at low debt levels. Additionally, the linear relationship is stark, with
more expenditures strongly positively correlated with more debt. The quantitative
model will generate both of these features of the data through (1) the overborrowing
incentive already seen in the theoretical model paired with (2) borrowing constraints
that relax as income and/or expenditures increase.
-5
0
5
10
Log debt and expenditures per person in 2011
2
4
6
Log expenditures p.p.
Fitted values
8
10
Log debt p.p.
The area of each observation is proportional to its population. The linear fit is weighted by population.
Figure 2: Debt and expenditures per person
11
3.4
Stylized fact #3: Cities of all types are close to borrowing limits
While we saw above that cities of all types are indebted, we have not established
precisely how indebted they are relative to their state-imposed maximum borrowing
capacity. Because most of the borrowing limits are expressed in terms of assessed
property value (which is not in our dataset), in general it is hard for us to directly
look at this question.
However, we can directly assess how close cities are to their borrowing limits for
two states. CA uses expenditures to constrain debt, and so we can tell how close
cities are to their limit as displayed in the top panel of Figure 3. The graph reveals
that many cities—including very large ones—are borrowing beyond the revenue per
person limit (recall, the limit allows exceptions for spending on special projects and
borrowing authorized by referendum). The other state we could check was MI using taxable-valuation data from Kleine and Schulz (2017), and the bottom panel of
Figure 3 displays how close MI cities are to their limits. Again, many cities are at,
near, or above the limit including Detroit and Flint. Even the wealthiest cities (as
measured by property values) are borrowing. In summary, it seems cities regularly
borrow, and many of them borrow as much as they legally can. This evidence suggests the borrowing limits are binding, consistent with the model prediction that
cities have a strong incentive to borrow.
Borrowing and statutory limits in Michigan (2011,2012)
Log debt (FY end) per person
4
6
8
10
10
Borrowing and statutory limits in California (2011)
Detroit
Log debt per person in 2011
4
6
8
Los Angeles
San Diego
Stockton Vallejo
Flint
2
2
San Bernardino
6
7
8
Log revenue per person
Log debt (FY end) per person
9
10
9
Limit (with exceptions)
10
11
12
Log taxable property value in 2012
Log debt per person in 2011
13
Log debt limit in 2012
Figure 3: Statutory borrowing limits and closeness to the limits
3.5
Stylized fact #4: In-migration increases expenditures and debt
So far, we have shown cities are highly indebted, but we have not looked at the
link between migration and debt accumulation. We now establish this link directly
using an instrumental variable approach. In particular, we construct an instrument
12
for in-migration using the shift-share approach, a common technique in empirical
studies of migration (see, e.g., Altonji and Card, 1991 and Card, 2001). For each
county c, we construct the share of individuals at a reference time τ > 0 periods ago
who live in c but arrived from county o at some time in the past.7 Then, with the
share θc,o,t−τ , we compute the total outflows Oo,t from each county o at time t. Given
this aggregate shift from o, we then apply the share θc,o,t−τ to get expected inflows
θc,o Oo,t at time t from county o to county c. Summing across all other counties gives
the total expected inflows and a total expected migration rate:
X
îc,t =
θc,o,t−τ Oo,t /Nc,t ,
(6)
o6=c
where Nc,t is the population in county c at the start of time t. If shocks to county o
are orthogonal to shocks in c (after controlling for time effects), the expected inflow
rate îc,t should only affect local variables through its effect on ic,t , as required for a
valid instrument.
Given this, we run the specification
yc,f +t = α + βic,t + γyc,t−l + ζ$c +
X
µt̃ 1[t = t̃] + uc,t ,
(7)
t̃
where $c is a productivity fixed effect and µt are time effects. We instrument the
actual in-migration rate ic,t with the predicted one îc,t . We use τ = 5 in (6); in (7),
we use l = 5 with f ranging from 0 to 20.8
Figure 4 plots the semi-elasticity coefficient β at different horizons (f in equation
7) for each of debt, expenditures, and revenue as the dependent variable (always
in log per person). Full estimates are reported in tables in Appendix C. The top
(bottom) panels give the IV (OLS) estimates, and the gray bands provide 95%
confidence intervals from robust standard errors. Evidently, the IV estimates are far
from the OLS and, for expenditures and revenue, differently signed (indicating the
presence of endogeneity and the necessity of IV).
Since debt—whose maturity is around ten years—is mostly predetermined, on
impact debt should change little with a valid instrument, and one can see that it
does.9 In contrast, ten years ahead there is a statistically significant effect, which
7
See the appendix (Section C.6) for details.
We focus on 5-year intervals because the Census of Governments data is most comprehensive
in years ending in 1 or 6.
9
In the Census of Governments data, the median (mean) of the ratio of retired to total long-term
debt is 0.09 (0.14), implying a maturity of 7 to 11 years. In our Mergent bond data, the median
(mean) maturity is 9 (10.1) years.
8
13
is also economically significant: the standard deviation of ic,t is 0.022, so the point
estimate implies a one standard deviation increase in ic,t increases debt per person by
approximately 31% (= e12.39×0.022 ). There are also significant effects for revenue and
expenditures per person, with a semi-elasticity of around 10 and 12.5, respectively.
The larger increase in expenditures, paired with expenditures typically being larger
than revenue, eventually results in debt growth, consistent with the theory.10
Note: Gray bands denote 95% confidence intervals.
Figure 4: Estimated effects due to the overborrowing externality
3.6
Stylized fact #5: Cities default after busts and booms
We now seek to establish a key, surprising, and novel finding, which is that cities
default after both busts and booms. To this end, our sample is Detroit (MI), Flint
(MI), Harrisburg (PA), San Bernardino (CA), Stockton (CA), Vallejo (CA), Chicago
(IL), and Hartford (CT), cities that have defaulted or been reported as having
financial difficulties in the last few years.11
The data point to heterogeneous paths to default or, more generally, fiscal stress.
Figure 5 reveals some cities experience unusually large population growth before default, while others experience large losses. Given that population growth directly reduces debt per person, it may be surprising that default could occur after such large
booms. However, the model will generate this through the overborrowing external10
We do not report estimates for deficits because they are very noisy. The noise is due in large
part to lagged deficits containing little information, unlike lagged debt stocks or expenditures.
11
News coverage on these and other cities is listed in Appendix A.
14
ity that keeps debt per person high especially when cities are growing. Productivity
growth has a similar though less drastic pattern. Unsurprisingly, some cities experience fiscal stress after adverse productivity shocks. However, other cities experience fiscal stress after either stable or positive productivity gains (Chicago, Vallejo,
Hartford). From the lens of consumption-smoothing models, the latter observation
is surprising: In response to positive productivity shocks, agents should deleverage.
However, we will show the model can generate defaults after productivity booms (as
well as busts) due to overborrowing.
Note: Changes are log differences relative to 1986 except for Chicago, which is relative to 1987; the
TFP measure is net of time effects; circles denote periods of acute fiscal stress such as defaults,
bankruptcies, or emergency manager takeovers (the last only for Flint); “other cities” is not the
universe of cities but covers 64% to 74% of the U.S. population over the time range; fiscal variables
are in 2012 dollars per person; the interquartile range is given by the shaded area.
Figure 5: Case Study – Cities under Financial Stress
Figure 5 also reveals that cities experiencing stress have large debts, debtexpenditure ratios, and budgets. For example, while the average city owed less than
$1,000 in 2011, Chicago and Detroit owed about $8,000 and $12,000, respectively.
In some cases, financial maneuvering has been used to underplay the amount of
debt: Harrisburg’s massive debt in the early 1990s plummeted due to a sale of its
incinerator project to a “special district” under its control, while still guaranteeing
the debt (Murphy, 2013).12 While the average debt position of all cities looks flat
12
Faulk and Killian (2017) show that having more special districts is positively correlated with
increased local government debt, suggesting this type of behavior is not unique.
15
because it is so much smaller than the case study cities’ average, it has increased by
50% since the 1980s. Defaulters also have large expenditures and taxes per person.
The model will also generate large expenditures for defaulters, in part because the
borrowing constraint limits debt relative to expenditures (as in the data): To loosen
this constraint, cities spend more. The volatility of Hartford’s expenditures come
from it relying on large, volatile state support amounts (Rojas & Walsh, 2017),
which is suggestive of being borrowing constrained.13
3.7
Stylized fact #6: Default risk is elastic to interest rate changes
Interest rates, including municipal bond rates, have declined secularly for decades.
We will now show that default risk is sensitive to interest rate changes, implying
that this secular decline in interest rates has masked the worsening state of local
government finances. We will do this by first considering one piece of narrative
evidence and second by considering evidence from a vector autoregression model.
The left panel of figure 6 reports the spread of BBB, A, and AA municipal bond
yield to maturity (YTM) over AAA municipal bonds. To the extent that default
risk is tail risk, one should expect increases in default risk to primarily show up in
larger BBB spreads, and one can clearly see the spikes in 2009 and 2020. There is
one other noticeable spike, which occurs in the latter part of 2013 into 2014. What
was its cause?
The most likely culprit for this spike is the sharp increase in long-term interest rates known as the taper tantrum. This period, beginning in May 2013 (and
seemingly coinciding with the May FOMC statement), saw the ten-year Treasury
note rise from 1.6% to 2.9% in four months at a time when the fundamentals of the
economy appeared relatively stable (and short-rates were unchanged). This rapid increase in interest rates, both in the ten-year Treasury and in AAA municipal bonds,
is evident in the linearly-detrended series in the right panel of figure 6. In our view,
these nearly-doubled debt servicing costs of all municipalities resulted in increased
default risk that was concentrated specifically in the riskier municipal bond tranches.
While this anecdotal evidence may be unsatisfactory, it is also born out in impulse responses from a structural vector autoregression (SVAR). To this end, we
estimate a VAR using monthly data on industrial production, the yield-to-maturity
for AAA (ytmAAA), the consumer price index, and the spread between the return
on municipal bond with category A, AA, or BBB and the ytmAAA. We use the
13
Typical state support is only $100-$300 per person.
16
Figure 6: Muni bond yields and spreads over time
ytmAAA as our interest rate measure rather than a Treasury because municipal
bonds are typically callable (i.e., they can be prematurely repayed) and are taxexempt. The sample runs from January 2005 and October 2020, and the VAR uses
6 lags. Figure 7 shows the response to an orthogonalized shock to the ytmAAA.14
It is not difficult to see that the exogenous increases in the AAA yield lead to a persistent response of the spreads. Consistent with interest rate increases inducing an
increase in tail risk, BBB spreads are the most sensitive to the rise in interest rates,
reaching a peak of 8 basis points around five months after the exogenous innovation.
That is, the BBB rate almost doubles the initial jump in the risk-free rate (10 basis
points). In contrast, the less risky spread AA barely increases, and its response is
not statistically significant.
3.8
Summary of empirical findings
We have documented a number of empirical findings that indicate local governments
have a strong incentive to borrow. We saw this in cities of all types being heavily
indebted and close to their borrowing limits. We also saw it in default that occurs after population booms, where large population growth—whose direct effect is
to decrease debt per person—was actually associated with stable or growing debt.
The overborrowing mechanism we highlighted in Section 2 provides an explanation
for why we observe this large indebtedness, and we provided direct evidence on
this mechanism, showing larger in-migration rates induce more spending and more
14
More concretely, we use a Cholesky decomposition of the forecast errors from the VAR with
the variables ordered as in the main text. Then we plot the IRFs to a shock to the second variable,
i.e., the yield to maturity AAA.
17
Spreads AA
Spreads A
1.5
4
1
Basis points
Basis points
3
0.5
0
2
1
-0.5
0
-1
5
10
15
20
5
Months
Spreads BBB
10
10
15
20
Months
Yield to Maturity AAA
8
10
Basis points
Basis points
6
4
2
0
8
6
4
-2
2
-4
0
-6
5
10
15
20
Months
5
10
15
20
Months
Note: Dashed lines corresponds to bootstrapped 68% confidence intervals.
Figure 7: Impulse response functions to a risk-free rate (AAA) shock
borrowing. Moreover, while default rates and municipal bond risk premia have historically been very low, we showed that they (1) have been trending upward, (2)
are sensitive to interest rate movements, and (3) have been depressed by a secular
decline in interest rates. Together, these findings suggest it is important, and will be
increasingly important, to understand the relationship of debt, default, and migration. To this end, we turn now to the quantitative model, showing that the patterns
we have observed in the data can be captured in an internally-consistent way. We
will then look at the model’s policy implications and glean some predictions for the
future of the U.S. economy.
4
The quantitative model
We first provide an overview of the model and its timing. Then, we describe the
household, firm, and government problems. Finally, we define equilibrium.
4.1
Overview and timing
We model municipalities as a unit measure of islands. Each island consists of a continuum of households (whose measure in the aggregate is one), a local government,
and a neoclassical firm. Each local government is a sovereign entity that issues debt,
taxes its residents, and provides government services. Households consume, work,
and decide whether to stay on the island or migrate to another one.
18
The timing of the model is as follows. At the beginning of the period, all shocks
are realized. Upon observing them, households make migration decisions. After migration occurs, the local government chooses its policies, including debt issuance.15
Finally, households make consumption and labor decisions simultaneously with firms
while taking prices and government policies as given.
4.2
Households
Define the state vector of a generic island as x := (b, n, z, ω), where b is assets per
person measured before migration, n is the population before migration, z is the
island’s productivity, and ω is a permanent island type that we will call “weather.”
The weather variable is a location-specific fixed effect that captures (in reduced
form) weather, location, history, or any other immutable, location-specific factor.
Including it allows the model to match the variance of population across cities
without producing a counterfactually large correlation between productivity and
in-migration. We assume z follows a finite-state Markov chain.
Households, knowing x, decide whether to stay m = 0 or move m = 1. If they
stay, they expect to receive lifetime utility S(x) (specified below). If they move,
they are assigned to another island, receive J in expected lifetime utility, and pay
an idiosyncratic utility cost φ ∼ F (φ). Their problem is
V (φ, x) = max (1 − m)S(x) + m(J − φ).
(8)
m∈{0,1}
The moving decision follows a reservation strategy R(x) with m = 1 when φ < R(x).
The utility conditional on staying is
S(x) = max
c≥0,h≥0
u(c, g(x), h, ω) + βEφ0 ,x0 |x V (φ0 , x0 )
(9)
s.t. c + r(x)h = w(x) + π(x) − T (x),
where w(x) is the island’s wage; π(x) is the per person profit from the island’s
firm; g(x) is government services; T (x) are lump-sum taxes (which we will show is
virtually equivalent to using property taxes); and h is a housing good, owned by the
firm and rented to households at price r(x). The expectation term Eφ0 ,x0 |x embeds
household beliefs about the local government’s policies.
If a household decides to move, they migrate to island x at rate i(x) and must
15
This timing means that unanticipated changes in government policies do not immediately alter
the population. We view this “sticky population” assumption as reasonable in that migration is a
time-consuming process that often involves searching for a new job, finishing a school year, selling
an existing home, and finishing rental agreements.
19
stay there for at least one period. The inflow rate at an island of type x is
Z
exp(λS(x))
i(x) =
,
nF (R(x))dµ(x) R
exp(λS(x))dµ(x)
(10)
where µ is the invariant distribution of islands.16 This inflow rate is a continuous
analogue of a logit-style, discrete choice framework.17
By construction, the measure of households leaving equals the measure entering
R
R
in aggregate, i(x)dµ(x) = nF (R(x))dµ(x). If λ = 0, households are uniformly
assigned to each island (“random search”). As λ → ∞, the city with the largest
utility S(x) receives all the inflows (“directed search”). Given these inflows, the
expected value of moving in equilibrium is
Z
i(x)
J = S(x) R
dµ(x),
i(x)dµ(x)
(11)
and the law of motion for population is
ṅ(x) = n(1 − F (R(x))) + i(x),
(12)
where ṅ denotes the population after migration has taken place.
4.3
Firms
Each island has a firm that operates a linear production technology zL and owns
the island’s housing stock H̄. Alternatively, H̄ may be thought of as the island’s
land. We assume H̄ is in fixed supply and homogeneous across islands to prevent
adding an extra state variable, but our inclusion of weather ω will capture some of
this fixed heterogeneity across islands. Firms solve
ṅ(x)π(x) = max zL − w(x)L + r(x)H,
(13)
L,H≤H̄
taking w and r competitively, and the solution of this problem gives labor demand,
Ld (and the housing supply, H = H̄). Since ṅ(x) denotes the number of households
remaining after migration and each household inelastically supplies one unit of labor,
16
This rule has the same form as in the two-period model. In particular, one can take I(S(x)) =
exp(λS(x)).
17
With a finite number of choices indexed by x, the usual specification would be written
maxx S(x) + εx /λ where each εx is i.i.d. with a Type 1 extreme value distribution. Then the
probability of choosing x is proportional to exp(λS(x)). The problem that arises with a continuum
of choices is that E[maxx S(x) + εx /λ] becomes infinite since εx has unbounded support. What we
need in the continuous case is the notion of a Gumbel process. As the technical details for this are
quite involved, we discuss how to micro-found (10) in Appendix C.
20
labor market clearing requires
Ld (x) = ṅ(x).
(14)
It is worth making a few observations about the firm problem. First, in equilibrium, per person profits π equal rH/ṅ. Consequently, by making local residents
the firm shareholders, we are effectively assuming each gets the rent associated with
owning an equal share of the housing/land stock. Second, if there were property
taxes, say via τ r(x)H for τ ∈ [0, 1], the taxes would reduce these rents by τ rH/ṅ in
the same way that the lump-sum tax T in (9) does. For this reason, we can interpret
the lump-sum tax T as a property tax. Last, we have assumed there are no agglomeration or congestion effects in the production function (or that they are both present
and cancel).18 Their absence could result in the model under- or over-predicting the
relationship between population and productivity. However, the model generates a
signficant positive correlation between city density and productivity like that found
in the data (Glaeser, 2010). Also, the model has congestion externalities in the form
of reduced housing per person and agglomeration effects in that local governments
provide a partly nonrival service, as will be discussed shortly.
4.4
Local governments
Each local government decides the level of services g ≥ 0 it wishes to provide.
These services are potentially nonrival in that, to provide g services to each of the ṅ
households, the government must only invest ṅ1−η g units of the consumption good
where η ∈ [0, 1] is a parameter. The government pays for these services using tax
revenue T ṅ or, potentially, debt issuance. The government chooses a new level of debt
per person −b0 , implying a total obligation next period of −b0 ṅ. The discount price
it receives on this pledge is q(b0 , ṅ, z, ω), which depends on the debt level, population
after migration ṅ (which equals the next period’s population before migration n0 ),
productivity, and weather, as all of these potentially influence repayment rates.19
We assume a portion γ is nondefaultable, and we will calibrate it to match the share
of unfunded pension obligations.
In keeping with the statutory borrowing limits discussed in Section 3, we impose
18
A simple way to introduce agglomeration is with the modified production function zLṅ$ , where
N is population and $ > 1. Duranton and Puga (2004) provide microfoundations for this type of
agglomeration.
19
Capeci (1994) and Schwert (2017) provide empirical evidence on the link between municipal
default risk and interest rates. Our use of short-term debt significantly simplifies the computation
as long-term debt models suffer from convergence problems (Chatterjee & Eyigungor, 2012).
21
a borrowing limit
− b0 ≤ δg ṅ−η ,
(15)
where δ ∈ R+ controls how tight the limit is. Hence, we require total debt issued in
a period −b0 ṅ to be less than a fraction δ of total expenditures g ṅ1−η . While this
limit is qualitatively closer to the CA limit than the other states’ limits, government
expenditures are very positively correlated with housing rents in the model, so it
also effectively captures a limit based on housing value. Additionally, exemptions
in many states allow for spending on projects, which this form permits. Given the
large variation in laws across states, we will choose δ to match observed debt levels
rather than trying to choose it based on statutory law.
To define the government’s problem, we need to specify how the economy will
respond to deviations in government policies. To this end, we assume that wages and
the rental rate adjust dynamically in response to the government policies (d, g, b0 ,
and T ) clearing the labor and housing markets and that households and firms optimize given those prices and implied profits. Formally, we assume that c, h, r, w, π, Ld
always solve the following equations:
uc r = uh
,
w=z
, and
c + rh = w + π − T ṅπ = zLd − wLd + rH
|
{z
} |
{z
}
Household optimization
Firm optimization
ṅ = Ld ,
ṅh = H
| {z }
.
(16)
Market clearing
Note that bankruptcy d and debt issuance b0 only affects households indirectly
through T and g. Letting U denote the indirect flow utility associated with g and
T , it is easy to show
U (g, T, ṅ, z, ω) = u(z − T, g, H/ṅ, ω).
(17)
To receive bankruptcy protection in the U.S., local governments must be insolvent and negotiate in good faith with creditors, as discussed in Section 3.1. We
interpret these statutory requirements as follows. First, we assume a municipality
is insolvent if its debt service exceeds a fixed fraction κ of the expected lifetime
income of residents. Specifically, the municipality is insolvent if debt service per
person exceeds κz̄(z) for z̄(·) defined recursively by
z̄(z) = z + q̄ Ez 0 |z z̄(z 0 ).
(18)
When insolvent and filing for bankruptcy, the city pays the greater of κz̄(z)ṅ and
−γbn, making γ fraction of its debt nondefaultable. Additionally, we assume filing
22
for bankruptcy entails a cost ι proportional to the municipality’s total income. Last,
if the city files for bankruptcy when solvent, the city must repay the full debt
amount (and the filing cost—consequently, no solvent city files for bankruptcy).
Consequently, the total payment required in bankruptcy is
p(b, n, z, ω) := max{γ(−bn), min{−bn, κz̄(z)ṅ(b, n, z, ω)}}.
(19)
Since all the costs and benefits of default occur within a period in our model, the
default decision—conditional on endogenous variables—is static and given by
d(x) = 1[p(x) + ιz ṅ(x) < −bn].
(20)
This modeling of bankruptcy has a number of important features. First, the
financial gain possible in bankruptcy hinges on the relative bargaining power of
sovereigns and creditors as captured by the parameter κ. Second, the use of lifetime
income, rather than current income, means that temporarily low income does not
make the city insolvent, which is essential for preventing the model from predicting
counterfactually-massive waves of default after large, but short-lived, shocks. Third,
solvency hinges not just on the size of the debt stock, but also on its rollover cost.
E.g., if real interest rates are zero, i.e., q̄ = 1, then the municipality is never insolvent,
and it should not be because the municipality can roll over its debt at zero cost.20
Now we can state the government’s problem as
S̃(b, n, z, ω) =
max
d∈{0,1},g≥0,T ≤z,b0
U (g, T, n0 , z, ω) + βEφ0 ,z 0 |z Ṽ (φ0 , b0 , n0 , z 0 , ω)
s.t. gn0(1−η) + q(b0 , n0 , z, ω)b0 n0 = T n0 + (1 − d)bn + d(−p(b, n, z, ω) − ιzn0 )
0
(21)
0−η
−b ≤ δgn
0
n = ṅ(b, n, z, ω)
with Ṽ (φ, x) = max{S̃(x), J − φ}. We use the tilde on the value functions to distinguish them from the household value functions in (8) and (9), but equilibrium will
require that the values coincide.
4.5
Debt pricing
With risk-neutral debt pricing, bond prices must be given by
q(b0 , n0 , z, ω) = q̄ Ez 0 |z [1 − d(x0 ) + d(x0 )p(x0 )/(−b0 n0 )].
20
(22)
That is, if they can roll over at the risk-free rate. A perhaps more theoretically appealing
modification would be to use the discount rates implied by the sovereign’s current and expected
borrowing. However, such a modification also induces convergences problems as is typically encountered in long-term debt models (Chatterjee & Eyigungor, 2012).
23
This links default rates and spreads very tightly, resulting in spreads being smaller
than default rates, whereas in the data the reverse is true. Given this discrepancy, we
will focus on matching small default rates rather than large spreads. To some extent,
we could match both by using an extremely risk-averse pricing kernel. However,
to properly match both would require incorporating aggregate risk and, perhaps
especially, disaster risk. So we follow the bulk of the sovereign debt literature and
use risk-neutrality.
4.6
Equilibrium
A steady-state recursive competitive equilibrium is value functions S, V, S̃, Ṽ ; an
expected value of moving J; household policies c, h, m; government policies g, T, b0 , d;
prices and profit q, w, r, π; labor demand Ld ; a law of motion for population ṅ; and a
distribution of islands µ, such that (1) household policies c, h and migration decisions
are optimal taking V , S, J, prices and government policies as given; (2) government
policies g, T, b0 , d are optimal taking Ṽ , S̃, J, the population law of motion ṅ(x),
and prices q as given; (3) firms optimally choose Ld (x) taking w(x), r(x) as given
and optimal per person profits are π(x); (4) bond prices are given by equation 22;
(5) beliefs are consistent: S = S̃ and V = Ṽ ; (6) the distribution of islands µ is
invariant; (7) and J and ṅ are consistent with µ and household and government
policies.
4.7
Centralization
Proposition 3 shows the government, household, and firm problem may be centralized into a single problem, which we use as the basis for computation:
Proposition 3. Suppose Ŝ satisfies
Ŝ(b, n, z, ω) =
max
c>0,g≥0,d,b0
u(c, g, H/n0 , ω) + βEφ0 ,z 0 |z max{Ŝ(b0 , n0 , z 0 , ω), J − φ0 }
s.t. c + n0−η g + q(b0 , n0 , z, ω)b0 = z + (1 − d)
bn
p(b, n, z, w)
+ d(−
− ιz)
n0
n0
n0 = ṅ(b, n, z, ω)
−b0 ≤ δg ṅ−η
(23)
with associated optimal policies
c(x), g(x), d(x), b0 (x).
Then (1) Ŝ is a solution to
the household problem, and c is its optimal consumption policy; (2) Ŝ is a solution
to the government problem, and g, d, b0 are its optimal policies; and (3) there exists
prices r, w such that labor and housing markets clear and firms optimize.
24
In what follows, we will use S in place of Ŝ.
We lack a proof of equilibrium uniqueness. However, we investigate uniqueness
quantitatively by using 100 randomly drawn initial guesses for the equilibrium objects. Each guess converged to the same equilibrium values, and so at least computationally there is no evidence of indeterminacy at the calibrated values. See Appendix
C.4 for more details.
5
Calibration
We now discuss the model’s calibration and its fit of targeted and untargeted moments. A model period corresponds to a year in the data.
5.1
Productivity
As productivity (TFP) plays a vital role in the model, it is necessary to have a
process that accurately captures location-specific productivity dynamics. For our
TFP measure, we use real annual payrolls per employee. Let TFP for a county-year
pair be denoted zit . We specify
log zit = ςi + $t + z̃it
(24)
and obtain the residual z̃it using a fixed-effects regression. To discretize the fixed
effects ςi , we nonparametrically break the estimates into bins corresponding to 010%, 10-50%, 50-90%, 90-99%, and 99-100%. The estimated fixed effects averaged
within these bins are −0.34, −0.13, 0.09, 0.37, and 0.65, respectively. We discard the
time effects $t as we will only consider steady states or specific paths for aggregate
TFP.
For the residual TFP z̃it , we use an AR(2) specification, which allows more
persistent movements in TFP that better capture decade-long persistent movements
in productivity such as what occurred in Detroit and Flint (see Figure 5). Restricting
the sample to cities of at least one million residents, the estimated first and second
AR coefficients are 0.73 (0.02) and 0.23 (0.03), respectively, with an innovation
variance 0.001 (= 2×10−5 ).21 We describe our discretization process in the appendix.
5.2
Preferences and moving costs
We set β = 0.96 and assume the flow utility exhibits constant relative risk aversion
over a Cobb-Douglas aggregate of consumption, government services, and housing
21
We use large cities to reduce the role of measurement error.
25
plus a taste shifter for weather:
(c1−ζg −ζh g ζg hζh )1−σ
+ ω.
(25)
1−σ
As ζg and ζh are relatively small, the constant relative risk aversion over consumpu(c, g, h, ω) =
tion is approximately σ, which we take to be 2. The free parameters ζg and ζh are
estimated jointly, strongly controlling the mean level of government expenditures
and housing expenditures, respectively. We take the weather term ω—which is fixed
over time for any given region but heterogeneous across regions—to be normally distributed with mean zero (a normalization) and standard deviation σω . We discipline
σω by matching the standard deviation of log population across cities.
We assume the moving cost φ is distributed as φ, −φ, and Logistic(µφ , ςφ ) with
probability pφ /2, pφ /2, and 1 − pφ , respectively. Having the ±φ shock means that,
for a sufficiently large φ, every island’s departure rate is in [pφ /2, 1 − pφ /2], which
ensures some minimal stability in calibrating the model. We set pφ = 10−4 and take
R
φ arbitrarily large giving V (φ, x)dF (φ), the expected utility of being in an island
with state x, equal to
pφ
2
(J + S(x)) + (1 − pφ )(S(x) + ςφ log(1 + e(S(x)−J−µφ )/ςφ ))
(26)
plus a constant that we offset via a normalization.22
We jointly estimate the parameters controlling moving costs (µφ , ςφ ) and how directed moving is (λ). We identify them using three moments: the mean and standard
deviation of out-migration rates, and the productivity-fixed-effect regression coefficient in a regression of log population on productivity residuals z̃i,t , productivity
fixed effects ςi , and a constant.
We discipline how rival public goods are, as governed by η, using the coefficient
of a regression of log population on log expenditures (1.118).
5.3
Borrowing and default
We set q̄ to give a risk-free interest rate of 4%, the recent average. We choose the
borrowing limit δ to match the data’s total debt to GDP ratio of 0.125, equal to
an explicit debt to GDP ratio of 0.089 plus unfunded pension debt to GDP of
0.036. The debt measure is gross in that we do not deduct the value of any assets
because assets cannot be seized in a Chapter 9 bankruptcy (as we noted in Section
22
The constant is φpφ /2. We subtract β times it from flow utility in (25) each period.
26
3.1). The default cost κ is chosen to match a 0.03% default rate.23 To calibrate the
cost of bankruptcy ι, we use information from Detroit’s bankruptcy and arrive at
ι = 0.125%.24
5.4
Fit of targeted and untargeted moments
Table 1 reports the targeted and untargeted statistics alongside the jointly calibrated
parameter values. The model closely matches all of the targeted statistics. The
estimated debt limit, δ, allows cities to borrow up to 153% of their expenditures,
which is not very far from CA’s statutory limit of 100% (plus exceptions).
Targeted Statistics
R
Default rate (×100)
ddµ
R
R
Debt / GDP −bndµ/ z ṅdµ
R
R
Gov. expenditures / GDP g ṅR1−η dµ/ R z ṅdµ
Housing expenditures / GDP rH̄dµ/ z ṅdµ
Std. deviation ofRlog n
*Out rate mean F (R)/ndµ
*Out rate st. dev.
*Population reg. coef., log z FE
Regression coef., log expenditures on log n
Data
Model
Parameter
Value
0.030
0.125
0.084
0.125
1.843
0.065
0.023
4.224
1.118
0.028
0.125
0.082
0.125
1.848
0.064
0.022
4.222
1.034
κ
δ
ζg
ζh
σω
µφ
ςφ
λ
η
0.006
1.528
0.069
0.112
0.331
22.445
6.452
0.603
0.316
Untargeted Statistics
Data
Model
Autocorrelation of log n
Std. deviation of net migration rates
Correlation of log expenditures and log n
Std. deviation of log expenditures
*In rate mean
*In rate st. dev.
*In rate reg. coef., log z
*Out rate reg. coef., log z
0.999
0.021
0.858
2.388
0.065
0.022
0.000
-0.006
1.000
0.023
0.995
1.921
0.066
0.030
0.023
-0.005
Note: * means the data is county-level; the regressions are specified in Section C.5.
Table 1: Calibration targets and parameter values
Utility from weather plays a large role in location decisions, with a rough calculation giving the lifetime consumption equivalent variation of permanently moving
from the median weather ω = 0 to 2σω at 66%. The importance of weather for
utility helps the model match the very low (in fact, negative) correlation between
in-migration and productivity and, simultaneously, the large dispersion in population. The large value for µφ with a correspondingly large ςφ makes out-migration
23
While high for Moody’s rated bonds, it is not high for the Mergent dataset (see Figure 1).
Detroit’s cost the city $178M on its $18.5B bankruptcy (1% per unit of debt) (Reuters, 2014).
Using Detroit’s 1% legal cost per unit of debt and a 12.5% debt-gdp ratio, we set ι = 0.125% =
1% ∗ 12.5%.
24
27
largely dependent on moving cost shocks rather than fundamentals like productivity,
letting the model match the weak relationships between productivity and migration
rates.
The model gets most of the untargeted predictions qualitatively correct while
missing on a few statistics. The model recreates the very slow population adjustments seen in the data with log population autocorrelation exceeding the data’s
0.999. It also matches the small correlations between migration rates and productivity and migration rates and population. Last, it reproduces the dispersion in net
migration rates and the large dispersion in government expenditures. Table 4 in the
appendix provides additional untargeted moments.
6
Quantitative results
With the calibrated model, we first show the model can replicate all the stylized
facts we established in Section 3 (except for the increasing trend, which the model
is not designed to address).25 Having established the empirical stylized facts hold
in the model, we then turn to the models predictions to answer two key questions.
First, why were there so few defaults in the Great Recession? Second, what can we
expect from the reallocation and productivity shocks caused by COVID-19?
6.1
Stylized facts in the model
In establishing the stylized facts hold in the model, we begin with the key model
mechanism, showing that in-migration induces more expenditures, revenue, and debt
per person. We then show cities of all types are heavily indebted; that they are close
to their borrowing limits; and that the model generates default after booms and
busts. Finally, we establish that default risk is sensitive to interest rate changes.
6.1.1
The model mechanism in action
Figure 8 plots the response of a few key variables in response to two types of shocks:
a shock to in-migration rates and a shock to interest rates. To model an exogenous
in-migration rate shock, we proportionately scale up or down the inflows i(x) in
(10) by 1+ī, and we assume that ī decays annually at rate 0.956, based on a fiveyear autocorrelation of in-migration rates equal to 0.8. The results for a positive
(negative) shock are displayed in the blue solid (dashed) lines. Consistent with the
empirical evidence, more in-migration leads to increases in debt, expenditures, and
25
The increase in indebtedness and default risk over time, which is paralleled in the consumer
context, likely does not have a simple explanation.
28
revenue per person. Additionally, the model predicts that higher in-migration rates
reduce default rates on impact—reflecting the direct effect of less debt per person—
but increase default rates in the future as the government’s debt grows. With the
increased default risk, spreads increase. Less obvious is that out-migration rates fall
on impact but rise a few years out.
Figure 8: Model mechanisms
More in-migration increases borrowing and expenditures in the quantitative
model for two first-order reasons and two second-order reasons. First, as highlighted
in Section 2, the overborrowing externality increases, which has a first-order effect on debt accumulation. However, it also has a second-order effect in borrowingconstrained cities increasing their expenditures to slacken the borrowing constraint.
The second first-order effect is that a larger population optimally creates substitution into expenditures because the public goods are partially nonrival. This also has a
second-order effect in that increased expenditures slacken the borrowing constraint,
enabling more debt accumulation. Evidently, these forces work in conjunction and
help the model generate a large correlation between debt and expenditures like in
the data.
6.1.2
All city types are heavily indebted and close to borrowing limits
The model replicates the stark relationships among debt, expenditures, and borrowing constraints that we showed in Sections 3.3 and 3.4. This can be seen most clearly
in Figure 9, which provides a scatter plot of cities debt and expenditures along with
the borrowing constraint. It shows cities of all types—large and small, productive
29
and unproductive, with great and poor weather—are indebted and close to their
borrowing limits, which were our number two and three stylized facts, respectively.
Note: circle areas are proportional to population.
Figure 9: Model distribution of cities relative to their borrowing limit
Why are all cities indebted and close to their borrowing limits? The answer
lies in the high level of in-migration rather than in its dynamics. In the data and
model, in-migration rates at a local level are around 6.5%. Loosely speaking, this
means for every dollar of debt issued per person, 6.5 cents are paid for by new
entrants. This is such a large “discount” on debt issuance that it dominates any
consumption-smoothing motives, including a desire to save for a rainy day.
The model does have a restraining force in it, which is that debt accumulation
deters entrants to the city (the term ∂n2 /∂b2 in equation 2). However, to match
the data’s large out-migration rates, most of the migration decision is attributed to
idiosyncratic, person-specific factors—as seen in large µφ and ςφ —rather than local
fundamentals. Consequently, the elasticity is quite small, and the model is almost bifurcated: migration strongly affects local government decisions, but local government
decisions weakly affect migration. Moreover, this is reinforced in equilibrium because
all governments over-accumulate debt, which depresses any incentive to move to a
more fiscally responsible city—such cities simply do not exist in the model. Hence,
the main restraining force in the model is essentially inoperable, resulting in cities
of all types being nearly indebted as they can possibly be.
30
6.1.3
Cities default after busts and booms
We now show the model generates boom and bust defaults, like in the data. We
consider default episodes, looking at windows around the time of default. These are
displayed in Figure 10 where we have broken default episodes into three cases: an
average default event (blue line), a default during a technology boom (red dashed
line), and a default during a technology bust (green dotted line). Formally, a boom
(bust) is defined as having log productivity growth in the ten years preceding default
above the 75th (below the 25th) percentile, and we compute the respective averages
in the top and bottom quartile to construct the time series. Unsurprisingly, the model
does generate default after long periods of productivity decline. But importantly, the
“boom” defaults do in effect follow periods of substantial productivity growth—a
feature that is not necessitated by our definition of booms.
Considering first average default episodes, one sees they coincide with slightly
growing productivity followed by a sharp decline in productivity (a drop close to
10%) at the time of default. Additionally, the shock is such that the drop in TFP
is expected to last a long time. Although on average population increases slightly
predefault, cities see their population decline postdefault, losing about 5% of their
inhabitants within five years.
Because the mean default episodes average over boom and bust defaults, they
hide a large amount of heterogeneity. Looking at bust defaults, one finds a prolonged
decline in population leading up to default, qualitatively similar to the experience
of Flint and Detroit. Facing this shrinking population and persistently adverse productivity, the sovereign initially holds taxes and debt per person stable and cuts
expenditures, resulting in a modest primary surplus. In the few years before default, interest rates increase, reflecting the increased default risk. Following default,
the municipality deleverages by sharply reducing expenditures.
Looking at boom defaults, the population and productivity growth is strong until
just a year before default, like in Vallejo, CA. With the boom, the cities do not pay
down debt, and even run a substantial deficit shortly before default. Consequently,
debt per person grows as interest payments pile up, and this is despite substantial
population growth that (else equal) reduces debt per person. The debt growth is
paired with a noticeable increase in expenditures and taxes. Interest rates trend
upward, showing that the city is taking on increasing (albeit small) amounts of risk.
When a substantial negative productivity shock hits, in-migration plummets and
debt per person increases, triggering default.
31
Log pop. change
TFP change
In-migration rate
Out-migration rate
Income
Debt
Expenditures
Debt-expend. ratio
Taxes
Primary deficit
Interest rates
Overborrowing term
Note: all financial variables are 2010 dollars per person.
Figure 10: Default episodes
While boom defaults are triggered by a decline in productivity, a necessary ingredient is that the city must be leveraged enough to make default worthwhile, which
is where overborrowing plays a crucial role in generating boom defaults. To see this,
consider the “overborrowing term” (1 − o)/(1 − o + i). For boom defaults, this falls
to as low as 0.85, implying a massive 15% discount on debt issuance. This overborrowing incentive dwarfs the usual consumption smoothing motive, keeping cities in
debt even after long periods of growing productivity and leaving them vulnerable
to adverse shocks.
6.1.4
Default risk is sensitive to interest rate movements
We close this section by establishing the last stylized fact, which is that default risk
is sensitive to interest rate. We turn again to Figure 8, which displays the response
of a 1 percentage point risk-free rate increase (decrease) in the red circled (dashed)
lines. The effect decays at rate 0.63 (which comes from a regression of detrended
AAA yield to maturity on its twelve-month lag).
On impact, the increase in rates increases the default rate of bonds appreciably,
as well as spreads. Default rates increase because larger interest rates increase debt
service costs, which makes more cities insolvent and able to benefit from bankruptcy.
32
Formally, a lower q̄ lowers z̄(z), reducing—for insolvent or nearly insolvent cities—
required debt payments and inducing bankruptcies. Spreads increase because of
higher future default risk that sovereigns only partially undo through deleveraging.
Of note, the plotted spreads series is ex-ante in that it is a function of q(b0 , x).
Consequently, the surprise losses incurred from the unanticipated shock are not
displayed.
Cities rely on a combination of expenditure cuts and tax increases to deleverage.
Using this combination is optimal from a utility perspective because c and g are
complements, so reducing both g and c modestly is superior to letting one or the
other fully absorb the impact. The impact on migration is not noticeable because
(1) the interest rate only change taxes by a small magnitude, and (2) the interest
rate change has no substantial redistributive effect since all cities are indebted.
6.2
The Great Recession
Having established the empirical stylized facts hold in the model, we now try to
answer why there were so few municipal defaults in the Great Recession. We will
model the Great Recession as a perfect foresight transition following a one-time
unanticipated shock.
The Great Recession resulted in a large and essentially permanent drop in real
GDP per capita. Relative to the pre-2009 linear trend, the drop was 12% on impact
and grew to almost 20% by 2020Q1, as seen in Figure 15 in Appendix D. The full
impact of this decline, however, was not felt by local governments because of the
American Reinvestment and Recovery Act of 2009 (ARRA). The ARRA provided
large federal transfers that bolstered state and local government (SLG) revenue. In
fact, as we document in Appendix D, transfers caused SLG total revenue to GDP
to rise until 2011, despite large declines in tax revenue.
In light of this, we will assume a pass-through from GDP declines to model
TFP declines that is less than 100%. Specifically, since local government budgets
are 8.2% of GDP and state to local transfers were 3.2% from 2006 to 2009, we’ll
assume a pass-through of 61% = 1 − 3.2%/8.2%. Combining the declines in real
GDP deviations from trend with a 61% pass-through results in the aggregate TFP
$t declines reported in Table 8 in the appendix.
Of course, the Great Recession also exhibited steep declines in risk-free real
interest rates. These real rates, measured using five-year TIPS yields, fell by more
than 3pp from 2006 to 2012 as can be seen in Appendix D. Given the empirical
33
sensitivity of default to interest rates that we already established, we incorporate
these declines as well. In particular, we compute two transitions, one incorporating
these declines as reported in Table 8, and one holding the real rate constant.
Figure 11 plots the response of key variables over the transition path. The most
glaring observation is that the default rate initially rises from a few basis points to
2pp in the absence of real rate declines (blue lines in Figure 11). Is this hypothetical
prediction reasonable? As seen in Figure 1, in the early days of the 2007-2008 financial crisis, spreads of higher-yield municipal bonds relative to lower-yield were as
little as 1%. As the spillover into unemployment and a housing crisis became clear,
the spreads rose to 5% in anticipation of a wave of default. While default rates
rose—from a low of 0.01% to as much as 0.16%—they never reached anything close
to 2%. From the model lens, this is because the massive drop in real rates reduced
default rates by a huge amount.
In contrast to default rates, spreads remain low and stable for many years. Part
of this is a time aggregation issue: spreads are purely forward-looking, so the high
default rates on impact are decoupled from spreads on impact. More substantively,
municipalities deleverage substantially, reducing debt by about 10% absent any interest rate decline. They do this by increasing taxes noticeably when the shock hits.
This painful deleveraging runs counter to common policy advice that says governments should borrow more in downturns. And, indeed, governments have incentive
to consumption smooth by borrowing more. Nevertheless, this effect is dominated
by (1) spreads moving against them if they do not deleverage; (2) the exogenous
borrowing limit, which effectively tightens in response to a decrease in optimal expenditures (see equation 15); and (3) effective impatience due to the overborrowing
externality.
A final aspect from the transition is that arrival rates (and departure rates, not
pictured) are little changed from the shock. This is due to the nature of the aggregate
shock, which effects all cities proportionally. We will revisit this when analyzing the
COVID shock.
6.3
COVID-19
We now turn to address an important contemporaneous question: what are the midterm and long-term effects on municipalities from COVID-19 and the unprecedented,
coincident policy interventions? Unfortunately, the ongoing uncertainty and lack of
multiyear data on outcomes means we will not be able to provide a precise answer.
34
Figure 11: Great Recession transitions with and without interest rate reductions
Our approach is rather to feed in a few shocks that capture the defining features of
2020-2021: a drop in GDP, a decline in real interest rates, and a motive to relocate
out of large cities. Given the rapid onset of COVID-19, we will also consider the
consequences of cities being able to only sluggishly adjust their tax policies. We’ll
consider each of these in turn.
6.3.1
Aggregate productivity decline
We begin by assuming that aggregate productivity exp($) falls by 10%. Subsequently, we assume productivity reverts to its steady state at a rate of 0.95, that
is, $t = 0.95$t−1 . Given the uncertainty behind COVID-19 and its impact on the
economy, we view this persistence as a reasonable starting point.
Following the shock, default rates rise by an order of magnitude, to nearly 0.4%
(blue line in Figure 12). Like in the Great Recession, despite the consumption
smoothing motive, taxes increase on impact—we will consider an alternative case
where they cannot—to bring debt to a more sustainable level. Two years after the
shock, debt has shrunk by 7 percent and remains protracted for many years following
the drop in productivity. The reasons are as before: an effectively tightened borrowing limit from the optimal g decrease and spreads moving against the sovereign if
they do not deleverage. Also as before, spreads remain low, mainly due to their
forward-looking nature. (When we look at exogenous taxes, however, substantial
recovery rates will play a key role in preventing large spread increases.)
35
Figure 12: Transitions with and without exogenous taxes and interest rate declines
6.3.2
Exacerbating factor: Tax inflexibility
Given the rapid onset and unanticipated nature of the COVID-19 crisis, local government budgets could not adjust immediately to the shock. We now consider the
possibility that taxes cannot adjust on impact. Specifically, we will think about the
case where the first period tax rates must be proportional to housing expenditures,
i.e., T = τ rh. (We will continue to assume that residents take T —rather than τ —as
given.) In this case, it is not hard to show that c = z − T and uc /uh = 1/r must
still hold. Combining these equations, one has c as a function of τ , z, and parameters, with g given as a residual (conditional on a choice of b0 and d) from the local
government’s budget constraint. We choose τ = 0.696 to deliver similar allocations
to the benchmark.26
As the optimal response to the shock is to raise taxes, the inability to do so leads
to worse outcomes (red dashed lines in Figure 12). Here, default rates increase on
impact but are much higher the next year as cities did not raise taxes and deleverage
as much as they should have. Reflecting this, spreads also increase noticeably. The
default rate increase of 180 bp is far larger than the 10 bp increase in spreads due
to the large recovery rates, which are roughly 94% (= 1 − 10/180).
26
Note T ṅ is used to finance government spending g ṅ1−η and debt service, which on average is
(1/q̄ − 1)(−ḃṅ). To not be too distortionary, we want T ṅ = τ rhṅ = τ r roughly equal to g ṅ1−η +
1−η
ṅ
(1/q̄ − 1)(−ḃṅ). Expressed relative to GDP Y , we need τ rh
= gṅY
+ (1/q̄ − 1) −Yḃṅ . These
Y
relative-to-GDP quantities are targeted, so we want τ × 0.125 = 0.082 + 0.04 × 0.125, which implies
τ = 0.696. Note that this is 70% of housing expenditures, not property values.
36
6.3.3
Mitigating factor #1: Real interest rate declines
As in the Great Recession, real interest rates declined substantially in 2020 and
2021. To capture this, we assume that the real interest rate falls by 1.5 pp, onetime and permanently. (While the permanent aspect is an exaggeration, given the
massive decline in thirty-year rates, this may well approximate expectations.)
The decline in real rates drastically reduces default even when taxes cannot
increase after the shock (green circled lines in Figure 12). The reasons are the same
as before and are discussed in Section 6.1.4. Consequently, despite the very large and
sudden shock, it is likely default rates will remain low due to the accommodation of
real interest rates.
6.3.4
Mitigating factor #2: Redistributive effects
In addition to these aggregate shocks, COVID-19 has also reduced the attractiveness
of living in densely populated regions. Unlike the other shocks, this has first-order
redistributive effects. In the model, we can capture this by reducing the value of
being in “high-weather” states—i.e., states with ω large. Because we discretized our
ω state into three point {ω, 0, ω}, we implement this idea by reducing ω by a given
percentage. This decline lasts for a year and then recovers at the rate 0.95.
Figure 13: Negative TFP transitions with and without redistributive weather shock
Figure 13 reports the transitions. A key takeaway is that larger declines in ω
lead to lower default rates. As people move from higher population to lower population cities, debt per person at the lower population cities declines. Because, as we
already established, smaller to medium-sized cities are more likely to default than
37
the largest cities, this shift from high population to lower population cities induces
a composition effect that lowers default rates.
While default rates, measured as the rate at which cities default, decline, the migration shift from larger to smaller cities increases the typical size of bankrupt cities
and the size of filings. The first claim follows immediately from the log population
conditional on default in the bottom right panel. The second follows from the first
in conjunction with an almost constant debt per person conditional on default (as
seen in the bottom, middle panel). Hence, while default rates of cities in general are
smaller, we may see defaults by larger cities involving correspondingly more debt.
7
Conclusion
Borrowing, migration, and default are intimately connected. Theoretically, we demonstrated that migration tends to result in overborrowing. Empirically, we documented
that defaults can occur after booms or busts in labor productivity and population,
in-migration leads to indebtedness, that many cities appear to be at or near stateimposed borrowing limits, and that default risk is highly sensitive to interest rate
movements. Our quantitative model was able to capture these stylized facts, in large
part due to the overborrowing externality. Given the increase in debt and default
over time, and the fiscal stress created by the Great Recession and continuing with
COVID-19, understanding regional borrowing, default, and migration will remain a
high priority.
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United States Courts. (2018). Chapter 9 – bankruptcy basics [Accessed: 2018-02-06].
Van Nieuwerburgh, S., & Weill, P.-O. (2010). Why has house price dispersion gone up?
Review of Economic Studies, 77, 1567–1606.
41
A
Additional data details [Not for Publication]
This appendix describes our data sources, definitions of key variables, and cleaning
procedures in Sections A.1, A.2, and A.3. Section A.4 gives a collection of statutory
borrowing limits, and Section A.5 records newspaper headlines on local government
finances.
A.1
Census County Business Patterns data
To construct TFP measures, we use data from the Census’ County Business Patterns
(CBP) database from 1986 to 2014. The main measures we use are the payroll
variable ap (converted to constant dollars using the standard CPI series obtained
from FRED) and the mid-March employment variable emp, along with the FIPS
codes. In the CBP database, missing or bad values are assigned a value of zero,
so we treat ap and emp as missing whenever they are 0. Our overall productivity
measure zit is ap/emp. The data includes disaggregated employment levels by sectors
(NAICS and SIC), so we keep only the observations corresponding to aggregates.
The panel includes 91,800 year-county nonmissing observations for zit .
A.2
Annual Survey of State & Local Government Finances data
For our data on government finances and population, we use the Annual Survey of
State & Local Government Finances (IndFin) compiled by the Census Bureau. Every
five years (in years ending in two or seven), the aim is to construct a comprehensive
record of state and local finances. (In practice, surveys are sent out for most cities
and not all are returned, but the coverage is good enough to cover 64-74% of the
U.S. population depending on the year.) In intervening years, a nonrepresentative
sample is selected from the population. Some of the larger cities are “jacket units,”
and instead of surveys, the Census sends its own workers to record the data. The data
are aggregated at different levels, with “cities”—i.e., municipalities and townships—
counties, and states. We consider two samples: one corresponding to cities (typecode
equal to 1 or 2) and the other to data aggregated at a county level (the aggregation
of typecode values 1 through 5). Some of the data go back to 1967. However, the
first population records begin in 1986 (survey year 1987), so we restrict ourselves to
the 1987-2012 survey years.
The population is not recorded in each year (the data for it does not necessarily correspond to the survey year but are given by yearpop), and so we construct
estimates. We restrict the sample so that each city/county has at least two popula-
42
tion measures. We fill in missing observations using linear interpolation of the log
population. We also allow for some extrapolation, but do not allow extrapolation
beyond five years.
The raw sample consists of 390,557 year-county or year-city observations. We
then use the sample restrictions as described in Table 2. We compute implied interest
rates via the interest paid during the year over the total debt, short and long term:
100* totalinterestondebt / (stdebtendofyear + totallongtermdebtout). All
financial variables are converted to real 2012 dollars using the CPI.
Sample selection condition
Obs.
Raw
Require nonmissing name, require no id changes (idchanged=0), drop
if “no data” (zerodata>0)
Require yearpop not missing, drop if datayearcode=“N”, require at
least two population observations
Drop observations with nonmissing investment annual returns on trust
funds exceeding 30%
Dropping missing population estimates
Dropping observations where population growth rates could not be estimated
Dropping Louisville, KY observations before 2003
Require annual population growth rates of less than 25%
Require revenue per person of less than $25,000
Require debt per person of less than $30,000
Require accounting identity for the evolution of long-term debt to nearly
hold
Require estimated interest rates be less than 40% annually
390557
387963
387219
386723
386515
386511
386494
386494
386090
376778
363365
362122
Table 2: Sample selection in IndFin
A.3
Migration data
Our migration data comes from the IRS. Up to 2010, we use the county-to-county
migration flows as harmonized by Hauer and Byars (2019); from 2011 on, we construct our own.
A.4
State-imposed borrowing limits
Table 3 reports state-imposed borrowing limits for a collection of states.
43
State
Limit
Known exceptions
CA
Indebtedness less than revenue that
year
Limits range from 0.5%-3% of assessed
value (1/3 of market value)
2% of assessed value
5% of property valuation last year
10% (5% for townships and school districts) of assessed value
“Net debt” less than 3% of market
value of taxable property
Roughly 10% of the property valuation
over the previous five years
Net indebtedness less than 5.5% (or
10.5% with vote) of tax valuation
5% of taxable property value
Authorization by referendum, special
projects, and public school spending.
Schools have debt limits of 13.8% value
of taxable property.
Some revenue bonds (see note).
IL
IN
MA
MI
MN
NY
OH
WI
Approval of voters for more school district debt; most revenue bonds.
Charter can increase to 3.67%, “first
class” cities have a 2% limit.
Debt related to water supplies and
sewers.
Self-supporting projects for water facilities, airports, etc.
Schools have a 10% debt limit, may issue $1 million without approval.
Sources are as follows: CA’s is Harris (2002); MA’s is MCTA (2009); MN’s is Bubul (2017);
IL, IN, MI, and WI’s is Faulk and Killian (2017); NY’s is ONYSC (2018); OH’s is OMAC,
2013, p. 50. Revenue bonds are municipal bonds that are paid by revenue from a particular
project (they are non-GO bonds).
Table 3: Sample of statutory borrowing limits by state
44
A.5
Cities making headlines
Here, we document some cities/municipalities experiencing financial difficulties as
reported by different media outlets. In quotations, we include excerpts of these news.
To retrieve the source, the interested reader should click on the city’s name.
U.S. Virgin Islands: “With just over 100,000 inhabitants, the protectorate now
owes north of $2 billion to bondholders and creditors. That is the biggest per capita
debt load of any U.S. territory or state - more than $19,000 for every man, woman
and child scattered across the island chain of St. Croix, St. Thomas and St. John.
The territory is on the hook for billions more in unfunded pension and healthcare
obligations.”
Chicago: “Chicago’s finances are already sagging under an unfunded pension liability Moody’s has pegged at $32 billion and that is equal to eight times the city’s
operating revenue. The city has a $300 million structural deficit in its $3.53 billion
operating budget and is required by an Illinois law to boost the 2016 contribution to
its police and fire pension funds by $550 million.
Cost-saving reforms for the city’s other two pension funds, which face insolvency in
a matter of years, are being challenged in court by labor unions and retirees.
State funding due Chicago would drop by $210 million between July 1 and the end
of 2016 under a plan proposed by Illinois Governor Bruce Rauner.”
Detroit: “‘It is indeed a momentous day,’ U.S. Bankruptcy Judge Steven Rhodes
said at the end of a 90-minute summary of his ruling. ‘We have here a judicial
finding that this once-proud city cannot pay its debts. At the same time, it has an
opportunity for a fresh start. I hope that everybody associated with the city will recognize that opportunity.’
In a surprise decision Tuesday morning, Rhodes also said he will allow pension cuts
in Detroit’s bankruptcy. He emphasized that he won’t necessarily agree to pension
cuts in the city’s final reorganization plan unless the entire plan is fair and equitable.
‘Resolving this issue now will likely expedite the resolution of this bankruptcy case,’
he said.”
Flint: “Flint once thrived as the home of the nation’s largest General Motors plant.
The city’s economic decline began during the 1980s, when GM downsized. In 2011,
45
the state of Michigan took over Flint’s finances after an audit projected a $25 million
deficit. In order to reduce the water fund shortfall, the city announced that a new
pipeline would be built to deliver water from Lake Huron to Flint. In 2014, while it
was under construction, the city turned to the Flint River as a water source. Soon
after the switch, residents said the water started to look, smell and taste funny. Tests
in 2015 by the Environmental Protection Agency (EPA) and Virginia Tech indicated
dangerous levels of lead in the water at residents’ homes.”
Hartford: “Hartford’s biggest bond insurer said it had offered to help the city postpone payments on as much as $300 million in outstanding debt, in a move designed
to help prevent a bankruptcy filing for Connecticut’s capital. Under Assured Guaranty’s proposal, debt payments due in the next 15 years would instead be spread
out over the next 30 years without bankruptcy or default. The city would issue new
longer-dated bonds and use the proceeds to make the near-term debt payments.”
Puerto Rico:
“The Puerto Rican government failed to pay almost half of $2
billion in bond payments due Friday, marking the commonwealth’s first-ever default
on its constitutionally guaranteed debt.”
New Jersey and other states: “The particular factors are as diverse as the
states. But one thing is clear: More states are facing financial trouble than at any
time since the economy began to emerge from the Great Recession, according to experts who say the situation will grow more dire as the Trump administration and
GOP leaders on Capitol Hill try to cut spending and rely on states to pick up a
greater share of expensive services like education and health care.”
On the State Crisis: “States and cities around the country will soon book similar
losses because of new, widely followed accounting guidelines that apply to most governments starting in fiscal 2018 – a shift that could potentially lead to cuts to retiree
heath benefits.”
Illinois: “After decades of historic mismanagement, Illinois is now grappling with
$15 billion of unpaid bills and an unthinkable quarter-trillion dollars owed to public
employees when they retire.”
46
B
Computation [Not for Publication]
This appendix describes the computational algorithms used.
B.1
Discretization of the AR(2) process
To discretize the AR(2), we cast it in the form of a VAR(1) and then follow Gordon
(2021). This method increases efficiency by dropping low-probability states and then
suitably adjusting the transition matrix.27 We use Tauchen (1986) as the underlying
tensor-grid method with a “coverage” (i.e., a support) of two unconditional standard
deviations. The algorithm delivers 58 discrete (z̃it , z̃it−1 ) states. These, combined
with the five permanent productivity states and three weather states, make 870
exogenous states.
B.2
Equilibrium computation
To compute the equilibrium, we guess on two objects: the expected utility conditional
on moving J and the average inflows over a “normalization” term for the logit
probabilites,
R
i :=
nF (R(x))dµ(x)
.
exp(λ(S(x) − maxx S(x))dµ(x)
(27)
Subtracting off maxx S(x) prevents overflows in the computation. Note that knowing
i, i(x) can be obtained via
i(x) = i exp(λ(S(x) − max S(x))).
x
B.2.1
(28)
Solving for the law of motion and value and price functions
With the (J, i), we solve for the value function S(x), the law of motion ṅ(x), and
the price schedule q(b0 , ṅ, z) as follows:
1. Construct discrete grids of of debt per person B, population N , productivity
Z, and weather W.
For B, we use 20 linearly spaced points from -0.2 to 0. Since average income
across cities is normalized to 1 and the debt-output ratio is around .02, this
allows for a given city to hold roughly four times as much debt as the average, and it is not binding in the benchmark. (This grid is coarse relative
to those used in Bewley-Huggett-Aiyagari type models, but the dispersion in
debt holdings is much more concentrated for cities.) For N , we use 64 log27
Specifically, we use the “TT0” refinement and the theshold value π = 10−6 .
47
linearly spaced grid points over ±5 ∗ 1.8 since the standard deviation of the
log population is roughly 1.8. For Z, we discretize the process as described
in Section B.1 and tensor product it with the nonparametrically discretized
permanent shocks. For W, we use a three-point discretization {−2σω , 0, 2σω }
with Tauchen’s method.
2. Fix tolerances (tolq , toln , tolS ).
We use (tolq , toln , tolS ) = (10−6 , 10−5 , 10−6 ).
3. Guess on S(x), ṅ(x), q(b0 , ṅ, z, ω).
The initial guess we use is S(x) = 0, ṅ(x) = n, and q(b0 , ṅ, z, ω) = q̄.
4. Solve for the implied S ∗ (x) and associated policies.
To determine the optimal value, we first do a grid search over the discrete
bond states. When a discrete bond state, say Bi , is less than 0.01% away from
the maximum, we then search for a local maximum in (Bi−1 , Bi ) and (Bi , Bi+1 )
(whenever applicable). In doing this search, we use Brent’s method. Whenever
we interpolate, we use linear interpolation.
Conditional on debt and default outcomes, we solve for c, g, h using the analytic
solution of the intratemporal problem.
5. Compute an update q ∗ (b0 , ṅ, z, ω).
6. Compute an update ṅ∗ (x) using S ∗ (x) and J.
7. Determine whether the convergence criteria ||q ∗ − q||∞ < tolq , ||ṅ∗ − ṅ||∞ <
toln , and ||S ∗ − S||∞ < tolS · ||S||∞ are satisfied. If so, stop. Otherwise, update
the guesses and go to Step 4.
B.2.2
Solving for the invariant distribution and key equilibrium object
updates
Given the converged values for ṅ(x) and the bond policy b0 (x), we compute the
∗
invariant distribution µ(x) and updates J ∗ , i , q̄ ∗ as follows:
1. Fix a tolerance tolµ .
We use tolµ = 10−10 .
48
2. Guess on µ.
Our initial guess is µ(0, 1, z, 1) = P(z) with µ = 0 elsewhere. (Consequently,
the mass of households is 1 initially.) On subsequent invariant distribution
computations, we use the previously computed µ.
3. Using ṅ(x), µ(x), and the bond and default policies, compute an update on
the invariant distribution µ∗ (x).
We use linear interpolation to distribute the mass from µ to µ∗ . (An important
advantage of linear interpolation is that it keeps the number of households the
R
R
same on each iteration, i.e., µ(x)ndx = µ∗ (x)ndx.)
4. Determine whether the convergence criteria ||µ∗ − µ||∞ < tolµ is satisfied. If
so, continue to the next step. Otherwise, update the guess µ := µ∗ and go to
Step 3.
∗
5. For the updates J ∗ and i , use the values associated with the computed invariant distribution µ.
B.2.3
Solving for the key equilibrium objects
∗
With the initial guesses J, i and the updates J ∗ , i , produce new initial guesses as
follows:
1. Fix tolerances (tolJ , toli ).
We use (tolJ , toli ) = (10−6 , 10−6 ).
∗
2. Check whether |J ∗ − J| < tolJ · |J| and |i − i| < toli . If so, STOP: an
equilibrium has been computed. Otherwise, go on.
3. Update the equilibrium values.
Using the new guesses on J, i, resolve for the value functions, price functions,
law of motion, invariant distribution, and key equilibrium objects as described
in Sections B.2.1 and B.2.2. Then go to step 2.
49
C
Omitted proofs and results [Not for Publication]
This section contains additional theoretic, empirical, and quantitative results, as
well as omitted proofs from the two-period model.
C.1
Microfounding inflow rates
To microfound inflow rates, we must use the notion of a Gumbel process. The
beginning of the theory seems to be quite recent and due to Malmberg (2013).
Here we follow the definition in Maddison et al. (2015):
Definition 1 (Maddison et al., 2015). Let L(I) be a sigma-finite measure on sample
space Ω, I ⊆ Ω measurable, and GL (I) a random variable. GL = {GL (I)|I ⊂ Ω} is
a Gumbel process if
1. GL (I) ∼ Gumbel(log L(I))
2. GL (I) ⊥ GL (I c )
3. for measurable A, B ∈ Ω, then GL (A ∪ B) = max{GL (A), GL (B)}.
Essentially, what the Gumbel process does is assign an infinitesimally small taste
shock to any of the continuum of choices. The taste shock is small enough that the
maximum over a continuum of choices is well-defined but large enough to influence
the choices themselves.
For our purposes, x = (b, n, z) ∈ R × R+ × R+ =: X. Let X denote the Borel
σ-algebra of X with a Borel measure of islands µ. Then every X ∈ X is measurable
with respect to µ. The sample space is X.
To formalize the inflow rates, we do the following. Define L : X → R via
R
L(X) = X exp(λS(x) + c)dµ(x) where c is a constant. (Then L is absolutely continuous with respect to µ.) In the Gumbel process, L(X) will be the σ-finite measure
on the sample space X . Then a Gumbel process GL with base measure L has random
utility GL (X) ∼ Gumbel(log L(X)) for each X ∈ X with the additional restrictions
that GL (X) is independent of GL (X c ) and GL (A ∪ B) = max{GL (A), GL (B)}.
The last restriction says, essentially, that if options in A and B are both available,
whichever is best (taking into account random utility) will be chosen. In the finite
case, this amounts to the optimal value being the maximum over the finite set. The
probability that the optimum over X is contained in some set X ∈ X is equivalent
to the event that GL (X) ≥ GL (X c ). Malmberg (2013) showed P(GL (X) ≥ GL (X c ))
50
is L(X ∩ X)/L(X) (and here L(X ∩ X)/L(X) = L(X)/L(X)). And L(X) by defiR
nition is X exp(λS(x) + c)dµ(x). Therefore, the probability that the max is in X
R
R
is X exp(λS(x))dµ(x)/ X exp(λS(x)dµ(x). Consequently, the argmax has a probability density (formally, a Radon-Nikodym derivative) of
R
exp(λS(x))
.
exp(λS(x))dµ(x)
There-
fore, by a law of large numbers, the measure going to each island with type x
exp(λS(x))
is ī R exp(λS(x))dµ(x)
where ī is the total measure of in-migrants (equivalently, out-
migrants).
C.2
The centralized problem
We now give the proof that the model can be centralized at a local level:
Proof of Proposition 3. Consider an arbitrary choice (c, g, d, b0 ) in the centralized
problem. At this choice, household and firm optimization and market clearing will be
satisfied if we take r = uh /uc , w = z, Ld = ṅ, h = H̄/ṅ, π solving ṅπ = z ṅ−wṅ+rH,
and T solving ṅc + rhṅ = wṅ + π ṅ − T ṅ (see equation 16). Eliminating profit from
the consumption equation, one has c = z −T (and clearly h = H̄/ṅ). Hence, the flow
utility associated with this allocation is u(z − T, g, H̄/ṅ, ω), which according to (17)
is the same as U (g, T, ṅ, z, ω). Hence, an arbitrary choice delivers u(c, g, H̄/ṅ, ω)
flow utility in the centralized problem, which—when supported using the above
prices and allocations—is the same as U (g, T, ṅ, z, ω). Moreover, at these prices and
allocations, b0 , d, T is feasible for the government as guaranteed by Walras’s law.28
Then, since the centralized planner is maximizing the same flow utility, discounting,
and expectations as the government, optimal choices for the centralized planner
must simultaneously solve the government’s problem. Hence, the optimal choices
from the centralized problem can be supported as a decentralized equilibrium using
the prices r, w, firm allocation Ld , household housing consumption allocation h, firm
profits π, and taxes T .
C.3
Two-period model proofs and omitted results
This subsection provides omitted proofs establishing the Euler equation and constrained inefficiency. It also includes a constrained efficiency result in the case of
costless and fully-directed migration.
28
One can verify this easily. For instance, if d = 0, then the centralized budget constraint reads
ṅc + ṅ1−η g + qb0 ṅ = z ṅ + ḃṅ. Using c = z − T to eliminate c, one finds ṅ1−η g + qb0 ṅ = T ṅ + ḃṅ,
which is the government’s budget constraint.
51
C.3.1
The Euler equation
Proof of Proposition 1. The objective function may be written
!
Z J−u(c2 )
(J − φ)f (φ)dφ
u(c1 ) + β (1 − o2 )u(c2 ) +
(29)
−∞
Using Leibniz’s rule,
∂c1
0 = u0 (c1 )
+β
∂b2
∂u(c2 )
−∂o2 ∂(J − u(c2 ))
(1 − o2 )
+ u(c2 )
+
(J − φ)f (φ)
∂b2
∂b2
∂b2
!
φ=J−u(c2 )
∂u(c2 )
−∂o2 ∂(J − u(c2 ))
+ β (1 − o2 )
+ u(c2 )
+
u(c2 )f (J − u(c2 ))
∂b2
∂b2
∂b2
∂u(c2 )
∂F (J − u(c2 )) ∂(J − u(c2 ))
+ β (1 − o2 )
− u(c2 )
+
u(c2 )f (J − u(c2 ))
∂b2
∂b2
∂b2
∂u(c2 )
∂(J − u(c2 )) ∂(J − u(c2 ))
+ β (1 − o2 )
− u(c2 )f (J − u(c2 ))
+
u(c2 )f (J − u(c2 ))
∂b2
∂b2
∂b2
∂c2
+ β(1 − o2 )u0 (c2 )
∂b2
∂b2 nn12
= −q̄u0 (c1 ) + β(1 − o2 )u0 (c2 )
∂b2
n1
∂n−1
0
0
2
= −q̄u (c1 ) + β(1 − o2 )u (c2 )
+ b2 n1
n2
∂b2
n
1
−2 ∂n2
0
0
= −q̄u (c1 ) + β(1 − o2 )u (c2 )
+ b2 n1 (−1)n2
n2
∂b2
n1 1 ∂n2
n1
− b2
= −q̄u0 (c1 ) + β(1 − o2 )u0 (c2 )
n2
n2 n2 ∂b2
n
b
1
2 ∂n2
0
0
= −q̄u (c1 ) + β (1 − o2 )u (c2 ) 1 −
n2
n2 ∂b2
n1
b2 ∂n2
0
0
= −q̄u (c1 ) + β
(1 − o2 )u (c2 ) 1 −
n1 (1 − o2 + i2 )
n2 ∂b2
1 − o2
b2 ∂n2
u0 (c2 ) 1 −
= −q̄u0 (c1 ) + β
1 − o2 + i2
n2 ∂b2
∂c1
= u0 (c1 )
∂b2
∂c1
= u0 (c1 )
∂b2
∂c
1
= u0 (c1 )
∂b2
∂c1
= u0 (c1 )
∂b2
Consequently, the Euler equation reads
1 − o2
b2 ∂n2
0
q̄u (c1 ) = β
u (c2 ) 1 −
.
1 − o2 + i2
n2 ∂b2
0
52
(30)
C.3.2
Constrained inefficiency
Proof of Proposition 2. With no cross-sectional heterogeneity, constrained efficiency
requires (5). If q̄ = βu0 (y2 )/u0 (y1 ), then this requires
u0 (y2 + b2 nn21 )
u0 (y2 )
u0 (c2 )
= 0
= 0
.
u0 (y1 )
u (c1 )
u (y1 − q̄b2 )
(*)
Evidently, this requires b2 = 0. However, b2 = 0 is not compatible with the government’s Euler equation. In particular, at b2 = 0 and at q̄, the government Euler
equation can be written
u0 (y2 )
1 − o2 u0 (c2 )
=
.
u0 (y1 )
1 − o2 + i2 u0 (c1 )
(**)
Hence, if i2 > 0, then (*) and (**) cannot simultaneously hold. And in fact, some
people will enter (i.e., i2 is greater than 0) because in the constrained efficient
allocation c2 = y2 for every island and so J = u(c2 ) and—given this—some people
will move since F (0) > 0 (i.e., migration is noisy). Hence, the constrained efficient
allocation cannot be supported as an equilibrium.
For the claim that at the constrained efficient allocation governments would
strictly prefer to borrow, note the Euler equation at the constrained efficient allocation is not satisfied with
1 − o2
1 − o2
⇔1>
.
1 − o2 + i 2
1 − o2 + i2
The way to equate marginal utilities would then be to increase c1 by borrowing.
u0 (y1 )q̄ > βu0 (y2 )
C.3.3
Constrained efficiency under costless and fully-directed migration
Our constrained inefficiency result established in Proposition 2 may be surprising in
light of the seminal paper by Tiebout (1956). Tiebout showed that, under certain
assumptions, equilibria are efficient when local governments compete for workers.
One of his key assumptions, which is not met here, is that of costless and fullydirected mobility. In fact, the equilibrium can be efficient if migration is perfectly
directed. To see why, consider trying to implement an efficient allocation that implies
b2 = 0. For the reasons described above, the Euler equation (2) would typically imply
this is impossible. However, if inflow rates “punish” any debt accumulation by falling
to zero in a nondifferentiable way, the Euler equation no longer characterizes the
optimal choice, and the equilibrium can be efficient. We prove this in Proposition 4.
Proposition 4. If migration is completely directed with (1) I(u(c2 )) = 0 for c2 < y2 ,
(2) the right-hand derivative of I(·) at u(y2 ) infinite, and (3) I(·) differentiable
53
elsewhere, then an equilibrium with q̄ = βu0 (y2 )/u0 (y1 ) exists and it is constrained
efficient.
Proof of Proposition 4. Under these assumptions, the Pareto optimal allocation is
c1 = y1 , c2 = y2 , with households moving whenever φ < 0 and staying whenever
φ > 0 (with indifference elsewhere). Note that in contrast to the hypothesis of
Proposition 2, inflow rates are assumed to be not differentiable at b2 = 0, which
means the Euler equation is not valid at that point.
We will prove the existence of a symmetric equilibrium with q̄ = βu0 (y2 )/u0 (y1 )
both of which have b2 = 0 as optimal. We will do so by establishing that at this
price b2 < 0 is not optimal, that b2 > 0 is not optimal, and that an optimal choice
exists (in which case it must be b2 = 0). This will then support the allocation
(c1 , c2 ) = (y1 , y2 ) (and the migration decisions).
For use below, we note that whenever the derivative ∂n2 /∂b2 exists, one has
b2 ∂n2
b2
∂c2
=
n1 īI0 (u(c2 )) + f (J − u(c2 )) u0 (c2 )
n2 ∂b2
n2
∂b2
2
(31)
n1
0
0
= b2 u (c2 )
īI (u(c2 )) + f (J − u(c2 ))
n2
Because I is increasing and f is positive, this has the same sign as b2 .
First we will show that b2 < 0 is not optimal by showing the Euler equation does
not hold there. Given no inflows for b2 < 0, borrowing is not optimal because the
Euler equation (which is valid everywhere except at b2 = 0) requires
u0 (c2 ) 1 − o2
b2 ∂n2
u0 (c2 )
u0 (y2 )
1−
β 0
= q̄ = β 0
≥β 0
.
u (y1 )
u (c1 ) 1 − o2 + i2
n2 ∂b2
u (c1 )
{z
}|
|
{z
}
=1
(32)
≥1 since b2 <0
However, with b2 < 0, c1 > y1 and c2 < y2 , which gives a contradiction.
Now we will show that b2 > 0 is not optimal. The Euler equation in this case
reads
β
b2 ∂n2
u0 (y2 )
u0 (c2 ) 1 − o2
u0 (c2 )
=
q̄
=
β
1
−
≤
β
.
u0 (y1 )
u0 (c1 ) 1 − o2 + i2
n2 ∂b2
u0 (c1 )
{z
}|
|
{z
}
≤1
(33)
≤1 since b2 >0
However, with b2 > 0, c1 < y1 and c2 > y2 , which gives a contradiction.
Since b2 < 0 and b2 > 0 are not optimal, all that remains to show is that an
optimal choice exists. Without loss of generality, we can restrict the choice set to
b2 ∈ [−δ, δ] for δ arbitrarily small such that every choice is feasible. Then, with
a continuous objective function being maximized over a compact set, a maximum
54
exists, which must be b2 = 0.
C.4
Quantitative testing of indeterminacy
To test for indeterminacy, we proceed by drawing 100 random starting guesses for J
and ī uniformly distributed about ±50% of the benchmark’s computed equilibrium
values. (For the definition of ī, see Appendix B.2.) We then compute the implied
equilibrium solution. Figure 14 shows a scatter plot of the guesses, and also reveals
that they all converge to the same solution (up to small numerical differences).
This suggests that the equilibrium is unique in a wide range about the computed
benchmark equilibrium.
Figure 14: Quantitative testing of indeterminacy
C.5
Additional calibration results
The cross-sectional regression specifications in Table 1 are as follows. For the row
”Regression coef. log expendituers on log n”, the specification is
log xi,t = α + β log ni,t + i,t ,
where xi,t is total expenditures, and ni,t is population. For the rows with “rate reg.
coef., log z,” the specification is
yi,t = α + β log zi,t + γ log ni,t + i,t .
55
The dependent variable is either in-migration rates or out-migration rates, as specified in the row. In all regressions, the sample is restricted to t = 2011.
Table 4 provides additional untargeted statistics coming from richer regressions.
The regression coefficients correspond to a regression of the form
yi,t = α + βςi + γ z̃i,t + δ log ni,t + i,t
with ςi and z̃i,t being the fixed effect and residual productivity from (24) (and
ni,t population). Again, the sample is restricted to 2011. Overall, the underlying
elasticities are not very different from those in the data.
Untargeted Statistics
Data
Model
*In rate reg. coef., log n
*In rate reg. coef., log z FE
*In rate reg. coef., log z res
*Out rate reg. coef., log n
*Out rate reg. coef., log z FE
*Out rate reg. coef., log z res
-0.000
-0.011
0.034
-0.000
-0.013
0.015
-0.013
0.006
0.072
-0.011
0.000
-0.019
Note: * means the underlying data is county-level.
Table 4: Additional untargeted moments
C.6
Additional estimation results
We construct the share of individuals at a reference time τ > 0 periods ago who live
in c but arrived from county o at some time in the past as follows. Given inflows
Ic,o,t , a population measured at the start of the period Nc,t , and an out-migration
rate δc,t , we construct θc,o,t under the assumption that every individual in c has
the same probability δt of leaving. Let Nc,o,t denote the stock of individuals in c
from o and time t. First, we obtain Nc,o,1 by assuming the county’s population is
in proportion to its inflows in t = 1, Nc,o,1 =
PIc,o,1 Nc,1 .
o Ic,o,1
Second, we obtain Nc,o,t
for t ≥ 2 recursively using Nc,o,t = Nc,o,t−1 (1 − δc,t−1 ) + Ic,o,t−1 . The share is then
P
θc,o,t = Nc,o,t / o Nc,o,t .
Tables 5, 6, and 7 report the estimates underlying Figure 4.
56
Debt
F5.Debt
F10.Debt
F15.Debt
F20.Debt
In-migration rate
3.505
(0.9)
10.36
(1.9)
12.39
(2.8)
6.613
(1.9)
1.615
(0.3)
Lagged debt
0.676
(77.8)
0.496
(51.4)
0.342
(30.2)
0.271
(20.4)
0.239
(10.0)
Lagged out-migration rate
-2.619
(-0.8)
-8.023
(-1.7)
-8.720
(-2.3)
-3.286
(-1.1)
0.468
(0.1)
Prod. FE
0.484
(18.9)
0.761
(18.1)
0.939
(20.9)
0.969
(16.3)
0.846
(5.7)
Constant
2.398
(41.6)
3.628
(49.8)
4.887
(63.3)
5.460
(57.1)
5.786
(33.0)
Observations
R2
First-stage F
31830
0.609
14.78
20993
0.414
32.32
14303
0.269
30.47
8142
0.222
36.30
2196
0.161
13.75
Note: robust standard errors are used; year effects are included; “debt” is log of per
person, real debt measured at the end of the fiscal year.
Table 5: IV regressions capturing the externality effects on debt
Exp.
F5.Exp.
F10.Exp.
F15.Exp.
F20.Exp.
In-migration rate
12.57
(3.6)
11.10
(4.4)
10.09
(3.4)
7.754
(3.3)
5.383
(1.9)
Lagged exp.
0.844
(42.6)
0.814
(54.0)
0.768
(40.5)
0.656
(43.0)
0.656
(24.8)
Lagged out-migration rate
-10.88
(-3.6)
-9.509
(-4.4)
-8.415
(-3.3)
-6.529
(-3.4)
-4.350
(-1.8)
Prod. FE
0.0215
(1.4)
0.0353
(2.7)
0.00232
(0.1)
-0.00792
(-0.3)
0.0101
(0.2)
Constant
1.245
(6.9)
1.368
(9.1)
1.936
(10.5)
2.910
(21.0)
3.092
(13.6)
Observations
R2
First-stage F
32145
0.423
21.42
21172
0.371
41.88
14421
0.268
39.45
8221
0.253
39.98
2209
0.336
14.58
Note: robust standard errors are used; year effects are included; “expenditures” is
log of per person, real expenditures.
Table 6: IV regressions capturing the externality effects on expenditures
57
Rev.
F5.Rev.
F10.Rev.
F15.Rev.
F20.Rev.
In-migration rate
10.30
(5.0)
12.48
(5.2)
8.460
(4.3)
7.766
(4.3)
4.917
(1.9)
Lagged rev.
0.850
(62.3)
0.838
(47.6)
0.779
(52.1)
0.667
(44.2)
0.679
(25.1)
Lagged out-migration rate
-8.966
(-5.0)
-10.75
(-5.1)
-7.169
(-4.2)
-6.510
(-4.2)
-3.746
(-1.7)
Prod. FE
0.0192
(1.7)
0.0312
(2.2)
0.00170
(0.1)
-0.0142
(-0.6)
0.00546
(0.1)
Constant
1.220
(10.1)
1.181
(7.1)
1.917
(13.8)
2.840
(21.6)
2.941
(12.8)
Observations
R2
First-stage F
32145
0.531
21.69
21172
0.334
43.88
14421
0.345
41.08
8221
0.260
41.01
2209
0.359
14.48
Note: robust standard errors are used; year effects are included; “revenue” is log of
per person, real revenue.
Table 7: IV regressions capturing the externality effects on revenue
Hi
58
D
The Great Recession [Not for Publication]
In this appendix, we provide more background and rationale for the shocks we feed
into the model for the Great Recession period.
To begin with, Figure 15 shows log real GDP per capita in relation to a pre-2009
linear trend. The drop in GDP p.c. was on the order of 12% on impact, and—relative
to trend—continued through 2020Q1 reaching almost 20%. Against this headwind,
the American Reinvestment and Recovery Act of 2009 (ARRA) was a sweeping
legislation that included provisions to bolster state and local finances. This funding flowed directly to states and more indirectly to local governments. To see this,
consider the time series of a few key state and local government (SLG) variables
provided in NIPA in Figure 16. Expressed relative to GDP and in differences from
2006, federal transfers to state and local governments (SLG) rose from zero (i.e.,
they were at their 2006 levels) to almost 1% in late 2009 into 2011. This more than
offset the decline in SLG tax revenue, which fell -0.5% relative to GDP, resulting
in an overall boost of SLG revenue amounting to 0.25% that lasted into 2011 . Unfortunately, this data does not separate the revenues of state and local governments
with one exception, in that it reports the transfers from state to local governments.
These rose slightly in 2009 and 2010 before falling substantially starting in 2011.
Log real GDP p.c. deviations from pre-2009 linear trend
1950q1
1960q1
1970q1
1980q1 1990q1
Time
2000q1
Log real GDP p.c.
2010q1
2020q1
-.3
9.5
-.2
10
Deviation
-.1
10.5
0
11
.1
11.5
Log real GDP p.c. and pre-2009 linear trend
1950q1
Trend
Source: authors' calculations using FRED series A939RX0Q048SBEA.
1960q1
1970q1
1980q1 1990q1
Time
2000q1
2010q1
2020q1
Source: authors' calculations using FRED series A939RX0Q048SBEA.
Figure 15: Log real GDP relative to trend
From a local government perspective, state to local transfers, while exhibiting
some dip despite the ARRA, have been a steady percentage of GDP. Hence, in
mapping this to the model, we assumed a pass-through from GDP declines to model
TFP declines that is less than 100%. Specifically, we used that local government
budgets are 8.2% of GDP, and state to local transfers were 3.2% from 2006 to 2009,
59
0
-.005
0
.05
.1
.005
.15
SLG variables as a share of GDP
.01
SLG variables as a share of GDP, difference from 2006
2005q1
2010q1
2015q1
2020q1
1950q1
1960q1
1970q1
Time
Federal to SLG
SLG Rev.
SLG Tax Rev.
State to Local
1980q1 1990q1
Time
Federal to SLG
SLG Rev.
For scale, 2020 has not been plotted
2000q1
2010q1
2020q1
SLG Tax Rev.
State to Local
For scale, 2020 has not been plotted
Figure 16: Federal, state, and local government transfers and revenue
suggesting a pass-through of 61% = 1 − 3.2%/8.2%.
Of course, the Great Recession also exhibits steep declines in risk-free real interest rates. These real rates, measured using five-year TIPS yields, are plotted in
differences from 2006 in Figure 17. It reveals real rates declined by more than 3pp
from 2006 to 2012. Table 8 summarizes the declines in productivity and interest
rates that we feed into the model.
-.04
-.03
meanrealrD06
-.02
-.01
0
Average annual real rate (5-year TIPS), difference from 2006
2005
2010
2015
2020
Year
Figure 17: Real interest rates, difference from 2006
Table 8 reports the transition variables we use in the experiments.
60
Parameter
Year
2007 (steady state)
2008 (shock, 1st period)
2009
2010
2011
2012
2013
2014+ (new steady state)
$
q̄ −1 − 1
0
-0.03
-0.07
-0.07
-0.08
-0.08
-0.09
-0.09
0.040
0.030
0.028
0.020
0.013
0.005
0.010
0.020
Table 8: Transition variables for the Great Recession
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0030
Malmberg, H. (2013). Random choice over a continuous set of options (Licentiate Thesis).
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MCTA. (2009). Chapter 9 borrowing [Accessed: 04-17-2018].
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