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On Regional Borrowing, Default,
and Migration
WP 19-04
Grey Gordon
Federal Reserve Bank of Richmond
Pablo Guerron-Quintana
Boston College
On 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
February 7, 2019
Working Paper No. 19-04
Abstract
Migration plays a key role in city finances with every new entrant reducing debt per person
and every exit increasing it. We study the interactions between regional borrowing, migration,
and default from empirical, theoretical, and quantitative perspectives. Empirically, we document that in-migration rates are positively correlated with deficits, that many cities appear to
be at or near state-imposed borrowing limits, and that defaults can occur after booms or busts
in productivity and population. Theoretically, we show that migration creates an externality
that results in over-borrowing, and our quantitative model is able to rationalize many features
of the data because of it. Counterfactuals reveal (1) Detroit should have deleveraged in the financial crisis to avoid default; (2) a return to the high-interest rate environment prevailing in
the 1990s has only small long-run effects on city finances; and (3) anticipated bailouts double
default rates.
JEL Codes: E21, F22, F34, R23, R51
∗
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, Daniele Coen-Pirani, Hal Cole, Jonas Fisher, Aaron
Hedlund, Dirk Krueger, Amanda Michaud, Leonard Nakamura, Jim Nason, Jaromir Nosal, Sam Schulhofer-Wohl, Tony
Smith, Nora Traum, Marcelo Veracierto, and Mark Wright for valuable discussions. We also thank seminar participants
at Boston College; CIDE (Mexico); Duke; ESPOL (Ecuador); the Federal Reserve Banks of Kansas City, Philadelphia, and
Richmond; Indiana U.; NCSU; U. of Wisconsin; the ADEMU Prague Macro Workshop; SED Toronto 2014; ITAM-PIER;
and LACEA/LAMES 2018 for comments. Alexey Khazanov and Michelle Liu provided excellent research assistance. 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
The local government finances of Detroit and Flint, Michigan, have received significant attention in
the media, and for good reason. The two cities have both experienced shrinking populations (with
declines exceeding 30% since 1986) and wages (decreasing 10% and 30%, respectively) that have
placed significant strain on their finances. Detroit responded to these challenges with burgeoning
debt—which grew from $2,000 per person to $12,500 in 2012 dollars—and a default in 2013, while
Flint kept debt low but was forced to raise taxes from $3,000 to $4,500 per person. Which of these
paths was best? Could anything else have been done? Were these debt levels optimal or too high?
Currently, there is scant empirical evidence and no model linking city finances, migration, and
default. This paper fills these gaps in the literature in two ways: first, by merging panel datasets
on local government finances, labor productivity, and migration to document patterns of cities in
general and defaulting cities in particular; and, second, by proposing and analyzing a rich and
novel general equilibrium model that successfully captures these patterns.
To guide our investigation of the data, we first consider a comparatively simple two-period
Lucas (1972)-type islands model. Each island represents a local economy and has a continuum of
households who make migration decisions, an exogenously given per person endowment that is
location-specific, 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. There is a potentially offsetting effect, namely, that as borrowing increases, the island’s attractiveness to newcomers declines and this increases debt per person. However, we show that in general equilibrium with two heterogeneous island types, the equilibrium is
not constrained efficient—i.e., not efficient taking migration decisions as given—and is therefore
not Pareto efficient, either.
The theory suggests that cities should accumulate large amounts of debt, that cities with larger
in-migration rates all else equal should accumulate more debt, and that states or federal governments should have policies in place that restrict municipal borrowing. (For our purposes, we will
refer to cities and municipalities interchangeably.)1 Using comprehensive datasets on city finances,
population, migration, and labor productivity as well as institutional details, we find support for
all of these predictions. Results from fixed effects regressions reveal a positive correlation between
deficits and in-migration rates. Moreover, we show many states have restrictions on municipal borrowing. While the form of most of the states’ borrowing limits precludes us from measuring how
close cities are to the limits, for California and Michigan we can tell. And there, it seems most cities
are close to, at, or even above the limit.
We further look at the relationship between borrowing, migration, and default by conduct1
A municipality is a city, town, or village that is incorporated into a local government.
2
ing a case study of municipal defaults—which are rare overall but have doubled in number since
the early 2000s. The study reveals that defaulters have larger debt, expenditures, and deficits than
typical cities. However, heterogeneous paths lead to default: Defaults have happened during challenging times characterized by population declines and low productivity (as in Detroit and Flint),
but also during productivity or population booms (as in San Bernardino, Stockton, and Vallejo, Callifornia).2 Theoretically, the bust defaults are unsurprising: Negative, mean-reverting shocks should
result in borrowing for consumption smoothing purposes with persistently negative ones leading
to default. However, based on that intuition, the boom defaults are very surprising. Yet, our model
offers an explanation: In response to rapid population growth, cities over-borrow expecting future
entrants to help repay the debt. Then, with a high amount of leverage, default looks attractive when
a negative shock eventually occurs.
With these empirical regularities in mind, we extend the two-period model to capture them,
allowing for a decentralized economy with an infinite horizon, production, government services,
a taste shifter we think of as weather, housing, borrowing limits, and default. After showing the
economy can be centralized (at a local level), we demonstrate the calibrated model is capable of
matching a host of statistics including the mean and standard deviation of in- and out-migration
rates, mean default rates, interest rates on municipal debt, local debt-GDP ratios, the standard deviation of log population, correlations between productivity and migration rates, and population
autocorrelations. The model also reproduces the patterns observed in the fixed effects regressions
and the proximity of cities to borrowing limits in the cross-section. Importantly, the model generates both boom and bust defaults.
Having established the model’s success in matching relevant features of municipal borrowing,
migration, and default, we turn to its counterfactual predictions. Feeding in the estimated productivity process for Detroit—which shows a rapid decline beginning in 2006 and continuing through
2012—gives an alternative, and optimal, path for its economy. The results show, perhaps surprisingly, that Detroit’s debt levels in the early 2000s (around $7,000 per person in 2010 dollars) were
nearly optimal. However, when the financial crisis hit, it should have deleveraged by around $750
per person via large cuts to expenditures and smaller cuts to taxes. In contrast, Detroit raised taxes
in 2009 and 2010 and expenditures and debt per person rose almost continually from 2006 to 2011,
precipitating default.
We also investigate the consequences of three policy changes. First, we look at increasing the
risk-free municipal bond interest rate from 4% (its real value in 2010) to 6.5% (its actual value in
the early 1990s). We show raising rates causes migration away from high-debt, low-productivity
cities to low-debt, high-productivity ones, providing a slight boost to aggregate output but with
few other consequences in the long run. Second, we look at elimination of state-imposed borrowing limits. We show this has small effects precisely because private credit markets constraint city
borrowing virtually as much as the state-imposed constraints. Finally, we investigate the conse2
Technically, Flint did not default. However, in 2002 and 2011, the state appointed emergency financial managers
who took over the city finances. Moreover, its 2014 water crisis can be seen as a type of default.
3
quences of bailouts in our calibrated model. We find ε-bailouts—defined as the smallest transfer
sufficient to avoid default—double default rates but otherwise have small effects. The reason is
ε-bailouts are not attractive because (1) they give the same utility as default in the bailout period
and, (2) to match both large debt positions and low default rates in the data, the calibration makes
default costly (equal to 8.4% of annual income). Hence, while bailouts do create a moral hazard
problem and lead to some increase in default rates, the moral hazard is mitigated by low ex-post
returns from being bailed out.
Since our model features local governments competing (through migration) via taxes and spending, it connects to large literatures on tax competition and fiscal federalism as surveyed by Wilson
(1999) and Weingast (2009). In influential work, Tiebout (1956) showed this competition can lead to
efficiency, and we show that—in a case where migration reacts to policies in an extreme way—there
can be efficiency. Outside of this extreme case, however, we prove the equilibrium is generally inefficient. Interestingly, the inefficiency does not come from either of the two common sources Wilson
(1999) highlights.3
A few papers in this literature have discussed the potential for local governments to overborrow because of migration. In particular, Bruce (1995) and Schultz and Sjöström (2001) prove that
overborrowing generally does occur. However, both of their models are two-period, partial equilibrium models with costless moving. And, in fact, we show that the Bruce (1995) and Schultz
and Sjöström (2001) results need not go through in general equilibrium: With symmetry, general
equilibrium bond prices undo the incentive to overborrow. (But overborrowing does occur with a
modicum of heterogeneity.) More importantly, we contribute to this literature by showing empirically and quantitatively the role of overborrowing in reproducing many of the data’s features.
Our paper also builds on the large sovereign default literature begun by Eaton and Gersovitz
(1981), which has focused almost exclusively on nation states.4 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
3
The first source is fiscal externalities induced by tax bases being linked across regions. Gelbach (2004), Akcigit,
Baslandze, and Stantcheva (2016), Moretti and Wilson (2017), and Coen-Pirani (2018) provide recent examples of this
where migration is affected by tax progressivity. In our model, inefficiency is not driven by changes in tax bases per
se since out-migration, by itself, does not lead to inefficiency. The other source of inefficiency comes from pecuniary
externalities induced by general equilibrium effects (a recent example of this type is Fajgelbaum, Morales, Serrato, and
Zidar, 2015). We show inefficiency results in both closed and open economy versions (the latter having exogenous bond
prices) of our model, so this also is not a driving force behind our results.
4
Some of the key references here are Arellano (2008); Hatchondo and Martinez (2009); Chatterjee and Eyigungor
(2012); and Mendoza and Yue (2012). The handbook chapter Aguiar, Chatterjee, Cole, and Stangebye (2016) provides a
thorough description of the literature.
4
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 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, Fisher, and Veracierto (2013), Caliendo, Parro, RossiHansberg, and Sarte (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 productivitydriven migration decisions. Outside the U.S., much recent research has been focused on migration
in the European Union (Kennan, 2013, 2017; Farhi and Werning, 2014). All these papers abstract
from debt. To our knowledge, ours is the only quantitative model of regional borrowing and migration, let alone having default.
The rest of the paper is organized as follows. Section 2 presents a simple model that highlights
the relation between borrowing and migration and guides our empirical investigation. Section 3
contains some relevant features about cities and municipalities in the U.S., which will guide us in
building the quantitative model described in Section 4. Section 5 gives the calibration and model
validation, and Section 6 assesses what leads to default, what (if anything) Detroit should have
done differently, and the counterfactuals. Section 7 concludes.
2
On borrowing and migration
Before turning to the data, 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.
Consider an arbitrary island. In the first (second) period, the island has a per person nonstochastic
endowment of y1 (y2 ). The local planner / 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
5
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.5
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
n1
,
n2
i2 = īI(u(c2 )),
n2 = n1 (1 − o2 + i2 ),
(1)
c1 , c2 ≥ 0,
o2 = F (J − u(c2 )).
Proposition 1 gives the Euler equation for government bonds (all proofs are in Appendix C).
Proposition 1. The local government’s Euler equation is
b2 ∂n2
1 − o2
0
u (c2 ) 1 −
u (c1 )q̄ = β
.
1 − o2 + i2
n2 ∂b2
0
(2)
1−o2
The Euler equation reflects two competing forces. One is an externality seen in the term 1−o
≤
2 +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 debt—holding fixed migration rates—is 1 −
1−o2
1−o2 +i2 .
All else equal, higher in-migration lowers the effective discount factor
i2
1−o2 +i2
1−o2
(β 1−o
)
2 +i2
or
and in-
creases borrowing. Clearly, then, the assumption that the planner only values current residents
plays a key role. But note that 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.
2
The other force is seen in the term 1− nb22 ∂n
∂b2 , which is one minus the elasticity of the next period’s
population with 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 all else equal to attract individuals to the islands to reduce debt per person.
Consider two equilibrium concepts. A closed economy equilibrium is a 3-tuple {i, J, q̄} with optimal migration, consumption, and borrowing decisions such that
R
R
1. total inflows equal total outflows, i I(u(c2,j ))n1,j dj = F (J − u(c2,j ))n1,j dj;
2. the expected utility of moving is consistent, J =
5
R
iI(u(c
))n
u(c2,j ) R iI(u(c2,j ))n1,jdi dj; and
2,i
1,i
This is relevant because a few times we will assume cross-sectional homogeneity in terms of fundamentals to facilitate the proofs, and we want to ensure migration happens in this case.
6
3. the bond market is in zero net supply,
R
b2,j n1,j dj = 0.
An open economy equilibrium differs in that q̄ is taken parametrically and the bond market clearing
is not required.
Is the equilibrium optimal from a societal perspective? To answer this, we need a social planner
problem. To this end, 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.
Let y1,i denote the endowment agent i will receive in the first period (which depends on their initial
s denote the output the household will receive if she decides to
island placement). Similarly, let y2,i
m the output in expectation associated with its moving. Taking migration
stay on her island and ȳ2,i
decisions as given, the planner’s objective function is
Z
max
ĉ1,i ≥0,ĉ2,i ≥0
αi (u(ĉ1,i ) + β(u(ĉ2,i ) − mi φi ))di
(3)
where αi is the Pareto weight on household i. We will consider two formulations of the resource
constraint, an open economy resource constraint given by
Z
Z
ĉ1,i di + q̄
Z
ĉ2,i di =
Z
y1,i di + q̄
s
m
((1 − mi )y2,i
+ mi ȳ2,i
)di
(4)
s
m
((1 − mi )y2,i
+ mi ȳ2,i
)di.
(5)
and a closed economy resource constraint given by
Z
Z
ĉ1,i di =
Z
y1,i di and
Z
ĉ2,i di =
If the planner can choose migration decisions, then mi ∈ [0, 1] should be added as a choice variable
m will be identically equal to the maximum second-period endowment value.6
and ȳ2,i
Definition 1. An allocation is constrained efficient if it solves the planner problem with migration decisions given for some Pareto weights.
For either resource constraint, 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. With the
open economy constraint, it is easy to show these must also equal q̄, i.e.,
u0 (ĉ1,i )q̄ = βu0 (ĉ2,i )
(6)
for almost all i. In comparing (6) 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, the incentive
b2 ∂n2
n2 ∂b2 —is close to zero, while the externality of new
1−o2
entrants shouldering the burden—reflected in 1−o
—is not. (On the other hand, if debt issuance
2 +i2
to attract people—reflected in the term 1 −
6
There are alternative ways to think of allowing migration here, such as a constrained efficient notion where migration
decisions depend on J, but these are beside the point for our purposes.
7
is large, b2 0, then attracting new entrants is of primary importance and there could be underborrowing.) Absent cross-sectional heterogeneity and with q̄ = βu0 (y2 )/u0 (y1 ), implementing
the constrained efficient allocation requires b2 = 0. In this case, the externality dominates and the
efficient allocation cannot be implemented, which is formalized in Proposition 2:
Proposition 2. Suppose there is no cross-sectional heterogeneity in y1 and y2 . Then, if q̄ = βu0 (y2 )/u0 (y1 ),
the open economy equilibrium is not constrained efficient. Moreover, at the constrained efficient allocation,
governments would strictly prefer to borrow.
Tiebout (1956) showed that, under certain assumptions, equilibria are efficient when local governments compete for workers. One of his key assumptions that is not met here is that of costless
and fully directed mobility. In fact, the equilibrium can be Pareto efficient if migration is fully directed. To see why, consider trying to implement a Pareto optimal allocation with 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 3.
Proposition 3. Suppose there is no cross-sectional heterogeneity in y1 and y2 . 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 elsewhere, then a symmetric open economy equilibrium with q̄ = βu0 (y2 )/u0 (y1 ) and a closed
economy equilibrium exist and they are Pareto optimal.
Note that we do not need any special restrictions on the moving cost distribution as the overbor1−o2
rowing externality reflected in 1−o
goes away if i2 = 0. In general, the more elastic net migration
2 +i2
is to bond holdings, i.e., the larger ∂n2 /∂b2 is, the less incentive the government has to overborrow.
With a closed economy, there is a different way that the economy can be efficient. In particular,
if islands are homogeneous, then the desire for all the islands to overborrow can result in lower
equilibrium bond prices / higher interest rates that exactly offset that desire. This result is stated
in Proposition 4.
Proposition 4. If there is no cross-sectional heterogeneity in endowments and initial populations, then any
symmetric closed economy equilibrium is Pareto optimal and has q̄ = β(1 − F (0))u0 (y2 )/u0 (y1 ).
Note the equilibrium q̄ in Proposition 4 includes a 1 − F (0) term. Because i2 = o2 absent heterogeneity, this term decreases the equilibrium price (relative to an equilibrium without migration) in
a way that exactly offsets the externality reflected in the Euler equation’s (1 − o2 )/(1 − o2 + i2 ) =
1 − F (0) in term. However, with an arbitrarily small amount of heterogeneity, a single price cannot
perfectly offset this externality as Proposition 5 shows.7
Proposition 5. Suppose there are two island types. If both types have the same first period endowments and
population but different second period endowments, then the closed economy equilibrium is not constrained
efficient.
7
The reason two types are required is simply for ease in characterizing asset positions (as in that case b2 > 0 for one
island implies b2 < 0 for the other). However, we suspect it holds far more generally.
8
3
Data and institutions
The theoretic model predicts a dynamic relationship between income, migration, and debt that results in overborrowing. We now investigate these relationships using data collected from a variety
of sources described in Appendix A. The goal of this section is not to identify causal effects of, e.g.,
migration on borrowing or vice versa. Rather, our goal is to provide suggestive evidence of the
relationships between migration, debt, and default while also establishing some stylized facts that
will be useful for constructing and validating the full model.
3.1
Borrowing and migration
Table 1 reports the results of fixed effects regressions using U.S. county-level data on deficits per
person and migration rates (we only have migration data at the county level, so we use county-level
deficits to be consistent). The results coincide well with the implications of the two-period Euler
equation (2) established in the previous section. Specifically, a regression of deficits on the overborrowing term
1−o
1−o+i
show a statistically significant negative correlation between discount rates
and deficits, consistent with the theory. For in-migration rates—which in large part determine the
strength of the overborrowing externality—there is also a statistically significant positive correlation, consistent with the theory. The magnitude is such that going from a 0% to 10% in-migration
rate would cause deficits to increase by almost $500 per person (all measures are in 2012 dollars).
Moreover, this is overcoming any consumption smoothing effect due to productivity shocks: Positive productivity shocks should increase in-migration while decreasing borrowing, resulting in a
negative correlation.
(1)
Deficit
Overborrowing term (1 − o)/(1 − o + i)
(2)
Deficit
(3)
Deficit
(4)
Deficit
-4527.0∗∗∗
(1314.0)
-1964.1
(1706.0)
4564.4∗∗∗
(1188.6)
In-migration rate
-2220.1∗
(1155.1)
Out-migration rate
Net-migration rate
Observations
R2
7682
0.018
7682
0.019
7682
0.016
3284.6∗∗∗
(818.7)
2503.2∗∗
(1063.4)
7682
0.019
7682
0.019
Standard errors in parentheses
Constant and year dummies included in estimation
∗
(5)
Deficit
p < 0.10, ∗∗ p < 0.05, ∗∗∗ p < 0.01
Table 1: Fixed-effects regressions of deficits per person on migration rates
9
The out-migration rate is negatively correlated with deficits. If causal, this would work against
the theory. However, the quantitative model will also give a negative coefficient, and the reason
is twofold. First, a negative productivity shock increases out-migration while simultaneously increasing borrowing (for consumption smoothing purposes), which generates positive comovement
between out-migration rates and deficits. Second, the overborrowing externality, as reflected in
1−o
1−o+i ,
is far less influenced by out-migration than in-migration, which should make it harder to
detect. For example, since i is around 6% in the data, a 10 percentage point increase in the outmigration rate (say from 0 to 0.1) makes the overborrowing term go from around 0.943 to 0.938. In
contrast, if o is 6% and i increases from 0 to 10%, the overborrowing term should go from 1 to 0.904.
Higher net-migration rates also should cause increased borrowing since the effective discount factor can be written (1 − o)/(1 + net) where net is the net-migration rate, and this is borne out in the
fixed-effects regressions.
In the final specification, we include net-migration rates alongside the overborrowing term.
This specification is designed to control for effects that may increase deficits not due to overborrowing but rather due to financing of large capital projects, for example. In this specification, the
overborrowing term is still negatively correlated with deficits but not at standard confidence levels.
While these regressions should not be interpreted causally, they are consistent with the model’s
theoretical predictions and indicate a connection between borrowing and migration. Later, we will
use these regression results in validating our calibrated model.
3.2
Borrowing limits
One of the model’s predictions is that a supra-local planner would generally like to restrict local
government borrowing, and that this constraint would generally be binding.8 In fact, many states
do have statutory limits on how much cities can borrow. To show the variety both in sizes and
types of limits, we report in Table 2 borrowing limits for nine states. The table reveals states have
implemented a variety of rules. E.g., California (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 one-third of 0.5% (IL) to 10% (NY). Almost all
the states have known exceptions, and these usually include debt related to education and or water
supplies or voter-approved debt. Qualitatively, the rules in CA and the spending-based exceptions
in other states could produce an incentive to have big budgets in order to borrow more, and the
quantitative model will allow for this feature of the law.
Are these constraints binding? Here we are limited because our main dataset does not have
data on taxable property valuations, and many states use this information in their borrowing limit
formulation. However, California 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 1. The graph reveals that many cities—
8
While we did not state this as a separate proposition, this is the case in the environment of Proposition 2. There, the
constrained efficient allocation cannot be supported because, at b2 = 0, the local government would strictly prefer to
borrow. If borrowing were forbidden, then the allocation would be supported as an equilibrium.
10
State
Limit
Known exceptions
CA
Indebtedness less than revenue that year
IL
Limits range from 0.5%-3% of assessed value
(1/3 of market value)
2% of assessed value (usually 100% of market value)
5% of taxable property valuation last year
10% (5% for townships and school districts)
of assessed value (usually 50% of market)
Authorization by referendum, special
projects, and public school spending.
Schools have debt limits of 13.8% value
of taxable property.
Some revenue bonds.
IN
MA
MI
MN
“Net debt” less than 3% of market value of
taxable property
NY
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
OH
WI
Approval of voters for more school district debt; revenue bonds (see definition
in notes) generally excepted.
Charter can authorize higher rates up
to 3.67%, “first class” cities have a 2%
limit.
Debt related to water supplies and sewers.
Self-supporting projects for water facilities, airports, public attractions, et al.
School’s have a 10% debt limit, may issue $1 million without voter 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 2: Sample of statutory borrowing limits by state
11
Log debt (FY end) per person
4
6
8
10
Borrowing and statutory limits in California (2011)
Los Angeles
San Diego
Stockton Vallejo
2
San Bernardino
6
7
8
Log revenue per person
Log debt (FY end) per person
9
10
Limit (with exceptions)
10
Borrowing and statutory limits in Michigan (2011,2012)
Log debt per person in 2011
4
6
8
Detroit
2
Flint
9
10
11
12
Log taxable property value in 2012
Log debt per person in 2011
Log debt limit in 2012
Figure 1: Statutory borrowing limits and closeness to the limits
12
13
including very large ones—are borrowing beyond the revenue per person limit (which could reflect spending on special projects or borrowing authorized by referendum). Additionally, we were
able to find taxable valuation data for Michigan (from Kleine and Schulz, 2017), and the bottom
panel of Figure 1 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 are overborrowing and that states are optimally restricting
them. Additionally, the quantitative model—where we know overborrowing does occur and can
tell explicitly whether the limits are binding—will produce similar patterns and generate a positive
(though small) welfare gain from having borrowing limits.
3.3
A case study of default and financial distress
To uncover some stylized facts on municipal defaults and financial distress, we turn to a case study
of local governments. 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. (News coverage on these and other
cities is listed in Appendix A.)
As emphasized in the introduction, the population growth in the top left panel of Figure 2 data
reveal heterogeneous paths to default or, more generally, fiscal stress. San Bernardino, Stockton,
and Vallejo all experience unusually large population growth leading up to default. Detroit and
Flint, and to a lesser extent Chicago, Hartford, and Harrisburg, experience the opposite. Similarly,
the top right panel reveals that cities encounter fiscal stress during periods of adverse productivity shocks (Detroit, Flint, Stockton, San Bernardino) or after unusually large productivity gains
(Chicago, Vallejo, Hartford). From the lens of standard default models, the latter observation is
surprising; but we will show our model can generate defaults after productivity and population
booms (as well as busts).
The middle left panel in Figure 2 displays the dynamics of debt in 2012 dollars per person since
the mid-1980s. One can see that financially struggling cities tend to have debt far above average.
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 separate government entity (Murphy, 2013).9 While the average debt position of all cities looks flat because it is so much smaller than the case study cities’ average, it has
9
In particular, it was sold to the Harrisburg Authority, a municipal authority with the power to issue debt (Murphy,
2013, p.4). The sale occurred in 1993, but Harrisburg “continued to operate the facility” and has guaranteed debt issuance
of the authority totaling at least $299 million (Murphy, 2013, p.5). These guarantees do not show up as debt in our data at
the city level. Faulk and Killian (2017) show empirically that having more special districts (and the Harrisburg Authority
is classified as one of these) is positively correlated with increased local government debt in four of the five states they
consider, which suggests this type of behavior is not unique.
13
Log population change*
Residual TFP change*
0.4
0.1
0
0.2
-0.1
0
-0.2
-0.2
-0.3
-0.4
-0.4
1990
1995
2000
2005
2010
1990
1995
Debt
2000
2005
2010
2005
2010
2005
2010
Taxes
4000
15000
3500
3000
10000
2500
2000
1500
5000
1000
500
0
1990
1995
2000
2005
2010
1990
Expenditures
1995
2000
Interest rates
8000
12
7000
10
8
6000
6
5000
Other cities
Chicago
Detroit
Flint
Harrisburg
San Bernardino
Stockton
Vallejo
Hartford
4
4000
2
3000
0
2000
1000
1990
1995
2000
2005
2010
1990
1995
2000
Note: Changes are log differences relative to 1986 except for Chicago, which is relative to 1987;
circles denote periods of acute fiscal stress such as defaults, bankruptcies, or emergency manager
takeovers (the last only for Flint); triangles denote acute fiscal stress periods that occur after 2011;
“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 2010 dollars per person; the interquartile range is given by the shaded
area.
Figure 2: Case Study – Cities under Financial Stress
14
increased by 56% since the 1980s, going from $469 per person to $732.
Regarding inlays and outlays (the middle right panel and lower left panel, respectively), we
observe that defaulters’ expenditures tend to outstrip their tax revenues, but not always. Furthermore, defaulters have large expenditures per person. In contrast, typical cities seem to run close to
balanced budgets, which is consistent with the comparatively low average debt per person. Hartford’s large tax revenue shortfall (with expenditures often around $6,000 and with tax revenue
closer to $2,500) has been offset with large cash infusions by the state (Rojas and Walsh, 2017).
While Connecticut’s support for Hartford has been exceptional, we will analyze the consequences
of this bailout-like behavior in Section 6.10
The data reveal a secular decline in interest rates over the past decades and show the financial
crisis pushed up the borrowing costs of defaulters in our sample.11 (The data for “other cities”
is plotted only in years ending in two or seven, which is when the coverage is almost universal.
Consequently, the high frequency variation is missed; see Appendix A.2 for details.) This run up
in interest rates likely contributed to the wave of defaults at the end of our sample. Furthermore, the
recent increase in the federal fund rates raises the question of whether increased borrowing costs
will have significant deleterious effects on city finances. We will also investigate this in Section 6.
In summary, our case studies uncover the existence of multiple paths to default. These paths
are quite heterogeneous with defaults happening during booms and busts. As we will show, the
model we will propose is rich enough to capture these heterogeneous default episodes.
3.4
Default rates, trends, and institutional details
We close this section by discussing overall default rates and trends, as well as few relevant institutional details. According to Moody’s (2012), there have been 73 municipal bond defaults (of bonds
rated by Moody’s) since 1970. This figure translates into a default rate of roughly one municipal
default every 4.3 years. These very low default rates imply that municipal bonds generally carry
low interest rates, as was seen in Figure 2.
However, this low default rate belies the precarious nature of city finances. For instance, as happened in Flint, cities can come under state management and thereby lose control of their finances.
In fact, Kleine and Schulz (2017) report that Michigan in and just before 2017 had 11 cities (4%),
one township, and one county under state oversight due to a financial emergency (p. 9).
Additionally, the last decade has seen a substantial increase in default rates. Figure 3 shows the
default rates for speculative-grade municipal bonds over the past 40 years, and these rates have
doubled since the onset of the Great Recession (going from 1% during 1991-2007 to 2% during
2008-2012). These results show that while overall default rates are low, fiscal stress is substantial
and potentially growing.
10
State support of local governments is generally on the order of $100-$300 per person, an order of magnitude smaller
than Connecticut’s support for Hartford. E.g., median and mean non-tax revenue was $129 and $256, respectively, in
2011.
11
Instrumental-variable-based evidence for the link between municipal default risk and interest rates can be found in
Capeci (1994).
15
5
4
Per cent
3
2
1
0
1972
1975
1978
1981
1984
1987
1990
1993
1996
1999
2002
2005
2008
2011
Figure 3: Default Rates on Speculative Grade Bonds (Source: Moody’s)
Before moving to the quantitative model, we would comment on one institutional aspect that
will impact modeling decisions and the calibration. Specifically, municipal bankruptcy (i.e., Chapter 9 bankruptcy) allows municipalities to discharge their debt while keeping essentially all their
assets (in contrast to consumer and firm bankruptcy under Chapter 7).12 Because of this, we will
focus on gross debt (not debt net of assets) in the calibration and treat bankruptcy as a complete
discharge of debt.
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 in the U.S. 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. The government is a sovereign entity that issues debt, taxes its residents, and provides government services. Households consume, work, and crucially decide whether to stay on the island or
migrate to another one. Finally, there is a financial intermediary who buys portfolios of municipal
debt as well as a risk-free bond.
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 govern12
The legal reason for this difference is state sovereignty over local affairs: 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). This was seen in the
courts’ rejection of creditor demands that Detroit sell part of its art museum collection (US Bankruptcy Court, City of
Detroit Case No. 13-53846).
16
ment chooses its policies, including debt issuance.13 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, f, ω), where b is assets per person measured before migration, n is the population before migration, z is the island’s productivity, f ∈
{0, 1} indicates whether the government is in a state of default, and ω is a fixed island type we
will think of as weather. Including “weather” 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|z). The
dependence of φ on z allows us to capture, in a reduced form way, the notion that high-income
workers are more mobile than low-income ones; however, the estimated dependence of F on z
turns out to be very small. Their problem is
V (φ, x) = max (1 − m)S(x) + m(J − φ).
m∈{0,1}
(7)
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 u(c, g(x), h, ω) + βEφ0 ,x0 |x V (φ0 , x0 )
c≥0,h≥0
(8)
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. We
assume u is continuously differentiable, strictly concave, strictly increasing, and satisfies the Inada
conditions in its first three arguments.
If a household decides to move, they migrate to island x at rate i(x) and must stay there for at
least one period. We assume that this rate is given by
Z
i(x) =
nF (R(x)|z)dµ(x)
13
R
exp(λS(x))
exp(λS(x))dµ(x)
(9)
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.
17
where µ is the invariant distribution of islands.14 By construction, the measure of households leavR
R
ing equals the measure entering 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
S(x) receives all the inflows (“directed search”). Note that these inflows are what would arise from
using Type 1 extreme value shocks (as in Kennan and Walker, 2011, and many others) were there
a finite number of islands.15 Given these inflows, the expected value of moving in equilibrium is
Z
J=
S(x) R
i(x)
dµ(x)
i(x)dµ(x)
(10)
and the law of motion for population is
ṅ(x) = n(1 − F (R(x)|z)) + i(x),
(11)
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 (1 − κ(x))zL − w(x)L + r(x)H.
L,H≤H̄
(12)
taking w and r competitively, and the solution of this problem gives labor demand, Ld (and the
housing supply, H = H̄). The term κ(x) = κ max{d(x), f } is a pecuniary cost of default. Since ṅ(x)
denotes the number of households remaining after migration and each household inelastically
supplies one unit of labor, labor market clearing requires
Ld (x) = ṅ(x).
(13)
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 (8) does. For this reason, we
loosely 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
14
This rule has the same form as in the two-period model. In particular, one can take I(S(x)) = exp(λS(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)), as it is in our formulation. However, a
continuum of x choices makes E[maxx S(x) + εx /λ] infinite, and so it is difficult to micro-found our inflow assumption.
15
18
cancel).16 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 non-rival 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 non-rival 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 default flag f (which
is a component of x) indicates the city’s standing with creditors. If f = 0, then the city is in good
standing with its creditors and can borrow and default. If f = 1, it is in bad standing and cannot
borrow (and has no debt). In the case of default or bad standing, firm output drops by κ and the
government returns to good standing f 0 = 0 and no debt b0 = 0 with probability χ. A government
with f = 0 that repays its debt −bn 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.17
In keeping with the statutory borrowing limits discussed in Section 3, we impose a borrowing
limit b0 ≥ b(z, ṅ, g). For the quantitative work, we further assume that
− b0 ≤ g ṅ−η δ
(14)
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 . Note that this limit is qualitatively closer to
the standard limits in CA than the other states. However, the 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.
However, the estimated value will be not far from California’s statutory limit of δ = 1.
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 r and w 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
16
A simple way to introduce agglomeration is with the modified production function zLṅ$ , where N is population
and $ > 1. Duranton and Puga (2004) provide micro-foundations for this type of agglomeration.
17
Capeci (1994) provides empirical evidence on the link between municipal default risk and interest rates. This bond
pricing framework was first introduced in Eaton and Gersovitz (1981). Our use of short-term debt significantly simplifies
the computation as long-term debt models suffer from convergence problems (Chatterjee and Eyigungor, 2012).
19
always solve the following equations
ˆ
w = (1 − κ̄d)z
ṅ = Ld ,
,
,
and
ˆ d − wLd + rH
ṅh = H
c + rh = w + π − T ṅπ = (1 − κ̄d)zL
{z
} |
{z
}
| {z }
|
uc r = uh
,
Firm optimization
Household optimization
(15)
Market clearing
where dˆ := max{d, f }. Letting U denote the indirect flow utility associated with g, T, and d, it is
easy to show
ˆ ṅ, z, ω) = u((1 − κd)z
ˆ − T, g, H/ṅ, ω).
U (g, T, d,
(16)
Now we can state the government’s problem as
(
S̃(x) =
maxd∈{0,1} (1 − d)S N (ḃ(x), ṅ(x), z, ω) + dS D (ḃ(x), ṅ(x), z, ω) if f = 0
S D (ḃ(x), ṅ(x), z, ω)
(17)
if f = 1
where ḃ(x) := bn/ṅ(x) gives assets per person after migration and
S̃ N (ḃ, ṅ, z, ω) =
max
g≥0,T ≤z,b0 ∈B
U (g, T, 0, ṅ, z, ω) + βEφ0 ,z 0 |z Ṽ (φ0 , b0 , ṅ, z 0 , 0, ω)
(18)
s.t. g ṅ1−η + q(b0 , ṅ, z, ω)b0 ṅ = T ṅ + ḃṅ
b0 ≥ b(z, ṅ, g)
and
S̃ D (ḃ, ṅ, z, ω) =
max
g≥0,T ≤(1−κ̄)z
U (g, T, 1, ṅ, z, ω) + βEφ0 ,z 0 |z
s.t. g ṅ
1−η
χṼ (φ0 , 0, ṅ, z 0 , 0, ω)
+(1 − χ)Ṽ (φ0 , ḃ, ṅ, z 0 , 1, ω)
!
(19)
= T ṅ
with Ṽ (φ, x) = max{S̃(x), J − φ}. (Note that in (18) we used the identity bn = ḃṅ.) For the quantitative work and most of the theoretical results, we restrict the bond choice b0 to be in a finite set B
that includes 0. For the derivation of the Euler equation, however, we treat B as an interval [b, ∞)
with b < 0.
4.5
Financial intermediaries
Following Chatterjee, Corbae, Nakajima, and Ríos-Rull (2007), we posit a competitive financial intermediary who purchases measures of contracts. To write the intermediary’s problem, we think
of governments and the intermediary as choosing contracts indexed by (b0 , n0 , z, ω)—note n0 = ṅ.
From the government’s perspective, a contract costs Q(b0 , n0 , z, ω) := q(b0 , n0 , z, ω)b0 n0 in the current
period, yields b0 n0 if the city does not default, and yields zero otherwise. Because the intermediP
ary purchases a measure of contracts, a law of large numbers gives a certain yield z 0 P(z 0 |z)(1 −
d(b0 , n0 , z 0 , ω))b0 n0 . The intermediary behaves competitively, taking Q and d as given.
20
The intermediary chooses a measure M 0 over contracts to maximize the net present value of
dividends D discounted at rate q̄. His problem is
W (B, M ) = max
D + q̄W (B 0 , M 0 )
D,B 0 ,M 0
Z
Z X
0
0 0
0 0 0
P(z|z−1 )(1 − d(b, n, z, ω))(bn)dM (b, n, z−1 , ω),
s.t. D + q̄B + Q(b , n , z, ω)dM (b , n , z, ω) = B +
z
(20)
with a no-Ponzi condition.
In equilibrium, we require that (1) the intermediary issues zero dividends each period D =
0; (2) the intermediary has zero wealth W = 0; (3) contract markets clear; and (4) the risk-free
bond market clears. We assume the risk-free bond is in net supply B, and so bond market clearing
requires B 0 = B = B. We further assume each contract is in zero net supply, which means the
intermediary must be the counterparty to every contract purchased by cities. Formally, contract
markets clear if
0
M (B, N , z, ω) = −
Z
1[b0 (b,n,z,0,ω)∈B,ṅ(b,n,z,0,ω)∈N ] (1 − d(b, n, z, ω))µ(db, dn, z, 0, ω)
(21)
for all z, ω and all B ×N in a product Borel σ-algebra. Our assumption that the economy is closed is
particularly appropriate for municipal bonds, as approximately 97% of them are held domestically
(Cohen and Eappen, 2015). An attractive implication of the zero-dividend condition is that we do
not need to specify who owns the intermediary.
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 q̄, q, w, r, π;
labor demand Ld ; the intermediary policies and value function D, M 0 , B 0 , W ; an intermediary state
M ; 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) the intermediary policies D, M 0 , B 0 are optimal given W, q̄, q, and d; (5) beliefs
are consistent: S(x) = S̃(x) and V (φ, x) = Ṽ (φ, x); (6) the distribution of islands µ is invariant;
(7) the intermediary’s portfolio is time-invariant, M = M 0 (B, M ); (8) J and ṅ are consistent with
µ and household and government policies; (9) the intermediary makes zero profits, D(B, M ) = 0
and W (B, M ) = 0; and (10) markets clear.
4.7
Centralization and the Euler equation
To characterize equilibrium, we first simplify the equilibrium conditions by providing sufficient
conditions for the intermediary’s problem, bond-market clearing, and contract market-clearing to
21
be satisfied. We then show how the government and household problems may be centralized into
a single problem. Last, we derive the Euler equation.
Proposition 6. If prices satisfy
q(b0 , n0 , z, ω) = q̄Ez 0 |z (1 − d(b0 , n0 , z 0 , ω)).
(22)
and if
Z
B=
(1 − d(b, n, z, ω))bnµ(db, dn, dz, 0, dω)
(23)
then there exist prices and an optimal policy M 0 , with M 0 invariant, such that contract markets and the
risk-free bond market clear and zero profits obtain (provided the other equilibrium conditions are met).
R
Note that if there is no default, this simply says B = bndµ, which means in equilibrium the
R
intermediary holds a portfolio of city assets / debt ( bndµ) that corresponds to aggregate debt
demand (−B).
Proposition 7 shows the government, household, and firm problem may be centralized into a
single problem, which we use as the basis for computation.
Proposition 7. Suppose Ŝ satisfies
(
Ŝ(x) =
maxd∈{0,1} (1 − d)Ŝ N (ḃ(x), ṅ(x), z, ω) + dŜ D (ḃ(x), ṅ(x), z, ω) if f = 0
Ŝ D (ḃ(x), ṅ(x), z, ω)
(24)
if f = 1
where
Ŝ N (ḃ, ṅ, z, ω) =
max
c>0,g≥0,b0 ∈B
u(c, g, H/ṅ, ω) + βEφ0 ,z 0 |z max{Ŝ(b0 , ṅ, z 0 , 0, ω), J − φ0 }
(25)
s.t. ṅc + ṅ1−η g + q(b0 , ṅ, z, ω)b0 ṅ = z ṅ + ḃṅ
b0 ≥ b(z, ṅ, g)
and
Ŝ D (ḃ, ṅ, z, ω) = max u(c, g, H/ṅ, ω) + βEφ0 ,z 0 |z
c>0,g≥0
s.t. ṅc + ṅ
1−η
χ max{Ŝ(0, ṅ, z 0 , 0, ω), J − φ0 }
+(1 − χ) max{Ŝ(ḃ, ṅ, z 0 , 1, ω), J − φ0 }
!
(26)
g = (1 − κ)z ṅ
with associated optimal policies c(x), g(x), d(x), b0 (x) (with d and b0 arbitrary for f = 1). Then (1) Ŝ is a
solution to the household problem and c, l are optimal policies; (2) Ŝ is a solution to the government problem
and g, d, b0 are optimal policies; and (3) there exists prices r, w such that labor and housing markets clear and
firms optimize.
In what follows, we will use S, S N , S D in place of Ŝ, Ŝ N , Ŝ D , respectively.
Proposition 8 characterizes the government Euler equation:
22
Proposition 8. Consider the S N problem in (25) at some state (ḃ, n0 , z). Suppose that at the optimal choices,
the borrowing constraint is not binding and that locally about b0 repaying is strictly preferred. If in addi0
0
tion SḃN (b0 nn00 , n00 , z 0 , ω), SṅN (b0 nn00 , n00 , z 0 , ω), and ṅb (b0 , n0 , z 0 , 0, ω) exist locally for n00 := ṅ(b0 , n0 , z 0 , 0, ω)
about b0 , then the Euler equation satisfies
uc q̄ = βEz 0 |z uc
1 − o0
1 + i0 − o0
0
b0
0
N
0n
00 0
1 − 00 ṅb + (1 − o )Sṅ b 00 , n , z , ω ṅb
n
n
(27)
where o0 is the outflow rate F (R(b0 , n0 , z 0 , 0)|z 0 ) and i0 = i(b0 , n0 , z 0 , 0)/n0 is the inflow rate next period.
Relative to the two-period model Euler equation in (2), there is an additional term connected to SṅN .
In the two-period model, the level of population only effects utility through its effect on debt per
person. Here, there are additional effects through reduced housing per person (H/ṅ) and partially
non-rival government services (if η > 0).
One result we would like to have is a proof of equilibrium uniqueness. Unfortunately, we have
not been able to show uniqueness even in the general version of the two-period model. This is
potentially worrisome in that, given the link between migration and borrowing we demonstrated,
expectations of how many people are moving might feed into debt accumulation decisions and justify those expectations. 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.6 for more details.
5
Calibration and estimation
In this section, we show that the model can reproduce a large number of key empirical moments,
targeted and untargeted. The data, including definitions, construction, and cleaning of key variables, are described in Appendix A. We take a model period to be a year.
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. To this end, we begin by constructing a TFP series
in the data using the County Business Patterns (CBP), which is an annual panel dataset published
by the Census covering the universe of counties (not cities, unfortunately) dating back to 1986.18
For our TFP measure, we use real annual payrolls per employee.
Let the log of TFP for a county-year pair be denoted zit . We specify
zit = ςi + $t + z̃it
(28)
18
In fact, it goes back as far as 1946 but the data are not easily accessible. The sample is available at https://www.
census.gov/programs-surveys/cbp/about.html.
23
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 to 0-10%, 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.
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 2). 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 ).19 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 CobbDouglas aggregate of consumption, government services, and housing plus a taste shifter for weather:
u(c, g, h, ω) =
(c1−ζg −ζh g ζg hζh )1−σ
+ ω.
1−σ
(29)
As ζg and ζh are relatively small, the constant relative risk aversion over consumption 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 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 is distributed
φ
φ|z ∼
Logistic(µφ − βφ log z, ςφ )
−φ
w.p. pφ /2
w.p. 1 − pφ .
(30)
w.p. pφ /2
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. Having the mean
of the logistic distribution be contingent on z is meant to capture the idea that high-productivity
individuals have lower moving costs, but the estimated value of βφ is approximately zero. We set
R
pφ = 10−4 and take φ arbitrarily large giving V (φ, x)dF (φ|z), 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 +βφ log(z))/ςφ ))
(31)
19
We use large cities to reduce the role of measurement error. Weighting by population gives essentially the same
estimates (.68,.18, and .001, respectively), while unweighted estimates have smaller persistence parameters (the estimates
are .60,.19, and .004, respectively), consistent with attenuation bias.
24
plus a constant that we offset via a normalization.20
The parameters controlling moving costs µφ , βφ , ςφ and the parameter λ controlling how directed moving is (see (9)) are jointly estimated. We identify the cost parameters using mean departure and arrival rates as well as coefficients from the regressions in Table 3 of productivity and
population on outflow rates in the data. βφ controls how the outflow rate varies with productivity,
so we use it to target the regression coefficient for log productivity on outflow rates. ςφ controls
how much the migration decisions vary with fundamentals, so we use the standard deviation of
out-migration rates to discipline it. µφ controls the overall level of outflow rates, so we use the mean
departure rate of 6.4% to discipline it. We identify λ by matching the regression coefficient for log
TFP fixed effect on log population.
Log TFP
Log TFP residual
Log TFP fixed effect
Log population
Constant
Observations
R2
∗
(1)
In mig. rate
(2)
Out mig. rate
-0.00456
0.00288
(3)
In mig. rate
(4)
Out mig. rate
0.00481
0.00220
-0.00376∗∗∗
0.102∗∗∗
2736
0.058
-0.0000821
0.0821∗∗∗
-0.00385∗∗∗
0.0933∗∗∗
0.0261∗∗∗
-0.0155∗∗∗
0.00134∗∗∗
0.0515∗∗∗
2736
0.001
2736
0.058
2736
0.017
p < 0.10, ∗∗ p < 0.05, ∗∗∗ p < 0.01
Table 3: In- and out-migration rate determinants
The borrowing limit δ and risk-free bond net supply B jointly determine the overall level of
debt in the economy and the equilibrium interest rate associated with q̄. Consequently, we use
them to target a mean risk-free interest rate of 4% (the recent average, see Figure 2) and the data’s
total debt to GDP ratio of 0.089.21 Since η controls the relative price of government services as
population grows, we use the coefficient of a regression of log population on log expenditures
(1.113) to discipline it. Finally, we calibrate the default cost κ to match default rates in the data.
Since it is difficult to separately identify κ and χ (the probability of returning from autarky), we set
χ = 1, which concentrates all default costs in the period of default.
5.3
Fit of targeted and untargeted moments
Table 4 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 128% of their expenditures, which is fairly close to California’s statutory limit
20
The constant is φpφ /2. We subtract β times it from flow utility in (29) each period.
As discussed in Section 3.4, we use a gross debt measure since assets are not seized in a Chapter 9 bankruptcy.
Additionally, we do not include debt from the underfunding of pensions because they are only underfunded at a local
level if one assumes risk-free discount rates when in fact local governments have generally earned much higher returns.
We discuss this more with appropriate references in Appendix A.2.
21
25
of 100% (plus exceptions). The consumption-equivalent flow cost of default, κ, is estimated to be
around 3%.22 The utility shares ζg and ζh are close to the shares observed in the data. Migration is
partially directed with λ > 0. Weather plays a large role, with a rough calculation giving the lifetime consumption equivalent variation of permanently moving from the median weather ω = 0 to
2σω at 67%.23 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
largely dependent on moving cost shocks rather than fundamentals like productivity. The value
for βφ is nearly zero. Finally, the provision of public goods displays some rivalry with η = 0.33.
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.
As an additional validation step, we rerun the fixed effects regressions from Table 1 on simulated data from the model. To convert the numeraire to dollars for this example and in the remainder of the paper, we multiply by 50,000, giving average output per person in our benchmark
as $68,200. This is between the median and mean household income in 2010 but closer to the median.24 The results are presented in Table 5, which reveals the model has similar patterns with the
overborrowing term (1 − o)/(1 − o + i) and in-migration significantly moving borrowing. One difference is that out-migration, while having the same negative sign as in the data, is much more
negative. This reflects an optimal response present in the model that is less dramatic in the data. In
particular, when bad productivity causes populations to shrink, the model suggests cities should
cut deficits sharply to prevent exploding debt per person, but in the data cities seem not to do this
(at least to such an extent). Overall, the magnitudes (except for out-migration) are fairly similar
to the data. This feature will reappear in comparing Detroit’s optimal and actual path. The R2 is
naturally larger than in the data as there are fewer determinants of spending in the model, but still
the regressors explain at most 15% of the variation.
A final validation step is to compare the pattern of cities in relation to the borrowing constraint,
which is done in Figure 4. The pattern is remarkably similar to that for California seen in Figure
1. (Note we use a CA-type borrowing limit that ties debt to spending rather than debt to property
22
We targeted a default rate of 0.0003. Since there are around 35,000 cities/townships/villages in our data, this implies
10.5 defaults a year, which is exaggerated. In large part, we do this for computational reasons since the simulations, which
already use 16 million cities, must scale even larger to generate
reasonable sample sizes conditional on default.
P
23
Approximating lifetime utility as ω/(1 − β) + E0 β t c1−σ
/(1 − σ) gives the consumption equivalent variation
t
P t 0 1−σ
E0
β (ct )
/(1−σ)
0
1
γ as γ = 2σω
− 1 where ct and ct are the optimal plans in the original and new environment,
P t 1 1−σ
+E0
β (ct )
/(1−σ)
1−β
respectively. Assuming these consumption plans equal 1 (the median unconditional value of output per person in our
normalization), plugging in the parameters gives 2/3 or 67%.
24
In 2010, median household income was $49,276 while GDP per household was $117,538. There is not a single appropriate measure to look at because the model has far less inequality than in the data. But any desired conversion can
be done via dividing by 50,000 and multiplying by the desired number.
26
Targeted Statistics
Data
Model
Parameter
Value
Default rate (×100) ddµ
Interest rate 1/q̄
R −1 R
Debt / GDP −bndµ/ z ṅdµ
R
R
Gov. expenditures / GDP g ṅR1−η dµ/ Rz ṅdµ
Housing expenditures
/ GDP rH̄dµ/ z ṅdµ
R
*Out rate mean F (R)/ndµ
*Out rate st. dev.
Std. deviation of log n
*Population reg. coef., log z FE
*Out rate reg. coef., log z
Regression coef., log expenditures on log n
0.030
0.040
0.089
0.082
0.125
0.064
0.022
1.846
4.224
0.003
1.113
0.049
0.040
0.089
0.074
0.125
0.063
0.022
1.797
4.152
-0.005
1.049
κ
δ
B
ζg
ζh
µφ
ςφ
σω
λ
βφ
η
0.084
1.286
-0.120
0.066
0.112
22.513
6.504
0.326
0.582
0.000
0.325
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
*Out rate reg. coef., log z res
*In rate reg. coef., log z res
*In rate mean
*In rate st. dev.
*In rate reg. coef., log n
0.999
0.021
0.858
2.388
0.005
0.026
0.066
0.028
0.001
1.000
0.022
0.994
1.896
-0.019
0.070
0.066
0.029
-0.013
R
Note: Parameters are listed beside statistics they strongly influence; * means the underlying data is county-level; in all cases, income measures are at the county level;
debt measures are gross (i.e., excluding any assets); and the debt and government expenditures are the sum of measures at county, city, and special district levels (see the
appendix for more details).
Table 4: Calibration targets and parameter values
(1)
Deficit
Overborrowing term
In-migration rate
Out-migration rate
Net-migration rate
(2)
Deficit
(3)
Deficit
(4)
Deficit
-10802
(5)
Deficit
-16276
9665
-38402
8406
R2
-4340
0.153
0.152
0.079
0.150
0.154
Note: A constant is included in each regression; the largest standard error is 58.8, and all the values are significant at a 1% level;
each regression has 15,200,000 observations.
Table 5: Fixed-effects regressions using model data
27
Borrowing and statutory limits
9.4
9.2
Log debt choice per person
9
8.8
8.6
8.4
8.2
8
7.8
Borrowing
Borrowing limit
7.6
7.6
7.8
8
8.2
8.4
8.6
8.8
9
9.2
Log expenditures per person
Note: A 1% subsample is used (to limit file size). ________________________
Figure 4: Model distribution of cities relative to their borrowing limit
values like in MI.) Specifically, almost all the cities are close to the statutory borrowing limit, though
only a few are exactly at it. As this pattern is nontargeted, it lends additional credence to the model’s
other predictions and the counterfactuals that we now analyze.
6
Model predictions and counterfactuals
With our calibrated model in hand, we first ask what leads to and triggers default. We then turn to
counterfactuals, looking at the elimination of state-imposed borrowing limits and the effects of a
return to higher interest rates. Finally, we use the model to assess what, if anything, Detroit should
have done differently, and the consequences of bailing out it and other cities.
6.1
What leads to and triggers default?
Default is a rare event in both the data and model, but the model-generated data, via an unrestricted
sample size, let us determine what leads to and triggers default. We begin answering these questions by comparing the unconditional marginal distributions with the distributions conditional on
a default in three years, both plotted in Figure 5.
At first glance, cities more or less likely to default look the same, as the overall means and
standard deviations are fairly close to each other. However, on further inspection, a few important
28
differences emerge, especially with respect to migration flows. In particular, default is predicted
by higher out-migration rates and higher in-migration. The combined effect is a smaller overborrowing term (1 − o)/(1 − o + i), giving greater borrowing incentive for cities prone to default, and
because of that one might expect the debt-expenditure ratio to be larger for cities likely to default.
However, this is not the case because all cities are close to their borrowing limits.
Importantly, default in the model can happen in all types of cities except the largest and least
productive. This means the model is capable of producing default in big, average productivity cities
(like Detroit); medium-sized, high-productivity cities (like Stockton); and small, low-productivity
cities (like Harrisburg). Default also can happen when a city is shedding population at 5% a year
(like Flint or Detroit); growing at 5% a year (like San Bernardino or Stockton); or with a stable
population (like Harrisburg).
An alternative way to assess what leads to default is to consider default events, i.e., windows
before and after default. These are displayed in Figure 6. We break 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 circled line).25 On average, default episodes coincide with
slightly growing productivity followed by a sharp decline in productivity (a drop close to 12%).
Although on average population increases slightly pre-default, cities see their population decline
post-default, losing about 10% of their inhabitants within five years. This is driven by an uptick in
the out-migration rate and a sharper contraction in the in-migration rate.
Because the mean default episodes average over boom and bust defaults, they hide a large
amount of heterogeneity. Looking specifically at bust defaults, one finds a prolonged decline in
population with a 10% population loss over the 10 years leading up to default, qualitatively similar
to the experience of Flint and Detroit. Facing this shrinking population, the sovereign holds debt
per person constant, which requires running a significant primary surplus of almost $200 per year.26
This surplus is financed with cuts to expenditures and taxes. In the few years before default, interest
rates do increase, reflecting the increased default risk, but the largest interest rates are still low, not
unlike in the data (as can be seen in Figure 2).
Looking at boom defaults (red dashed lines), the first thing to notice is the model can generate
them (which was not clear a priori). The population and productivity growth is strong until just a
year before default, like in Vallejo, CA. With the boom, the cities have virtually no primary deficit,
which results in growing debt per person as interest payments pile up, and this is despite substantial population growth that all else equal reduces debt per person. The debt growth is paired
with a noticeable increase in expenditures and taxes. Interest rates trend upwards, showing that
the city is taking on increasing (albeit small) amounts of risk. When an unusually large decline
in productivity occurs, in-migration plummets and debt per person increases, triggering default.
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. This is where overborrowing plays
25
A boom (bust) is formally defined as having log productivity growth in the 10 years preceding default above the
75th (below the 25th) percentile.
26
The primary deficit is g ṅ1−η − T ṅ.
29
Probability
Probability
0.15
0.1
0.05
0.15
0.1
0.05
0
0
-0.4
-0.2
0
0.2
0.4
0.6
-6
-4
-2
Log TFP
0.15
2
4
0.1
Probability
Probability
0
Log population
0.1
0.05
0.05
0
0
-0.05
0
0.05
0.1
0.05
Net migration rate
0.1
0.15
In-migration rate
Probability
Probability
0.15
0.1
0.05
0.1
0.05
0
0
0.02
0.04
0.06
0.08
0.1
0.12
0.08
Out-migration rate
Probability
Probability
0.1
0.11
0.4
0.2
0.1
0
0.2
0
0.065 0.07 0.075 0.08 0.085 0.09
0.075
Tax-output ratio
0.08
0.085
Expenditure-output ratio
Probability
0.1
Probability
0.09
Debt-output ratio
0.05
0
Cond.
Uncond.
0.15
0.1
0.05
0
0.85
0.9
0.95
1.1
Overborrowing term
1.15
1.2
1.25
Debt-expend. ratio
Figure 5: Distributions, unconditional and conditional on a default in three years
a crucial role in generating boom defaults: By keeping cities in debt even after long periods of
growing productivity, even high-productivity cities are at risk of default.
A striking difference between the boom and bust defaults appears in the incentive to borrow
as reflected in the “overborrowing term” (1 − o)/(1 − o + i). In boom defaults, increases in inmigration drive down the overborrowing term, which increases borrowing incentives. The converse is true for bust defaults. Additionally, the magnitudes of the changes are large: Recalling that
1−(1−o)/(1−o+i) is the share of debt subsidized (i.e., paid for) by new entrants, in boom (bust) defaults this subsidy increases (decreases) by 2 percentage points. While the debt paths show trends
qualitatively consistent with these changes, the dynamics are muted in part because cities are close
30
Log population change
In-migration rate
TFP change
0.1
0.09
0.08
0
0
0.07
-0.1
-0.1
0.06
-0.2
0.05
-0.2
-10
-5
0
5
-10
-5
10 4
Out-migration rate
0
5
-10
-5
0
5
0
5
0
5
Debt
Income
5.2
0.076
4000
5
0.074
3000
0.072
4.8
0.07
4.6
2000
0.068
4.4
1000
0.066
-10
-5
0
5
4.2
-10
Expenditures
-5
0
0
-10
5
Debt-expend. ratio
-5
Taxes
4000
3800
1.2
3600
3400
1.1
3000
1
2000
3200
0.9
3000
1000
-10
-5
0
5
0.8
-10
Primary deficit
2500
0
5
-10
Interest rates
Mean
Mean | boom
Mean | bust
2000
-5
-5
Overborrowing term
0.95
4.3
0.94
1500
4.2
0.93
1000
4.1
500
0.92
0
-10
-5
0
Years since default
5
4
-10
-5
0
5
Years since default
-10
-5
0
5
Years since default
Note: all financial variables are per person. _______________________________
Figure 6: Default episodes
to their debt limits: The debt-expenditure ratio is close to its limit of δ (1.286) in every year leading
up to default.
In answer to the questions of what leads to and triggers default, the model reveals much the
same answer as the data: The causes are myriad. Bust defaults lead to shrinking populations and
primary surpluses financed by cuts to taxes and larger cuts to expenditures. In contrast, boom defaults are characterized by growing populations, growing debt per person, and growing taxes and
expenditures. Boom defaults are possible in the model because of large overborrowing incentives
31
that keep cities leveraged even after a decade of productivity and income growth.
6.2
Disentangling the roles of migration, borrowing limits, credit markets, and general equilibrium effects
The analysis of what drives defaults showed that while defaulters tend to have large debt-expenditure
ratios and greater overborrowing incentives than typical cities, the dynamics of debt per person
are surprisingly muted. The reason given above was that debt is constrained by the exogenous
borrowing limit, which keeps it from from moving much. However, the model has three external
forces that restrict borrowing: Exogenous borrowing limits, a Laffer curve for borrowing in private
credit markets, and the endogenously-determined risk-free interest rate. Additionally, the level of
in-migration and its elasticity with respect to debt act as internal restraints on borrowing. We now
try to assess the role of each of these forces before turning to the counterfactuals.
6.2.1
External restraints on borrowing
To start, we first eliminate the borrowing constraint in partial equilibrium, i.e., holding q̄ fixed at
the benchmark. The percent change relative to the benchmark is displayed in the third column
(labeled “(∞, ·)”) of Table 6. One can see there is very little change from the benchmark except that
the default rate increases somewhat. This is because private credit markets are restraining cities. To
show this, we make default infeasible by setting the default cost κ̄ = 1 in addition to having δ = ∞.
This eliminates the Laffer curve for debt by turning the borrowing schedule −q(b0 , ṅ, z, ω)b0 , which
has a peak, into just −q̄b0 , which does not. Those results are displayed in the fifth column, labeled
“(∞, 1)”. Now debt per person expands by 65.9%, reaching the maximum debt amount allowed
by the computational method ($10,000 per person).27 One interesting feature of these results is
that private credit markets restrain borrowing significantly despite small equilibrium default rates.
Hence, in our calibration the threat of default, rather than outright losses from default, constrains
borrowing.
In fact, our calibration suggests that private credit markets restrain municipal borrowing more
than the exogenous borrowing limit. This can be seen in the fourth column, labeled (·, 1), where
we have made default costs infinite but kept the borrowing limit at the benchmark value. In this
case, debt per person increases by 12.5%. Note that in this case aggregate debt over aggregate
expenditures rises from 1.201 to 1.285. Since the benchmark value of δ = 1.286 constrains the
debt-expenditure ratio in individual cities to 1.286, this shows that without the Laffer curve from
borrowing in private credit markets, virtually every city goes to the borrowing limit in the model.
A final force in the model that restrains borrowing is the general equilibrium response of the
risk-free interest rate. In particular, it is sufficient to discuss the most extreme case of (δ, κ̄) = (∞, 1)
in the rightmost column of Table 6. In it, we see that while the overborrowing incentives are very
large, they can be undone by a strong general equilibrium response that increases the risk-free rate
27
The equilibrium is solved for using discrete state space techniques, and as part of this the grid for debt per person
specifies both a minimum ($0) and maximum ($10,000).
32
(δ, κ̄)
General equilibrium effects
Bench.
(∞, ·)
(·, 1)
(∞, 1)
(∞, 1)
−
No
No
No
Yes
68.2
62.9
16.0
5.06
5.30
6.08
0.562
4.00
-0.2
-0.2
1.0
-0.3
-0.3
-0.2
-0.2
-0.0
0.1
-0.4
0.7
5.4
5.6
12.5
-0.0
-0.0
0.2
0.1
0.8
-0.0
1.7
65.9
-0.0
-0.0
0.9
0.6
2.5
0.4
4.5
-0.6
0.2
121.4
0.049
12.2
-100.0
-100.0
-100.0
Aggregate measures
Output per person
Consumption per person
Consumption per person s.d.
Expenditures per person
Taxes per person
Debt per person
Weather
Interest rate (%)
Average across cities
Default rate ×100
Note: the benchmark financial variables are in thousands of dollars; nonbenchmark are values are percent deviation from the benchmark; a ·
means the value is the same as in the benchmark.
Table 6: Changes due to exogenous and endogenous borrowing constraints
33
from 4% to 8.8%, thereby keeping debt levels approximately at the benchmark level.28 From these
experiments, we conclude that the strong overborrowing incentives seen in the two-period model
are present in the quantitative model, but that they are significantly restrained by both exogenous
and endogenous (i.e., Laffer curve) credit limits.
6.2.2
Internal restraints on borrowing
In the two-period model, there were two forces that determined how much the city wanted to
overborrow. First was the magnitude of the overborrowing term (1 − o)/(1 − o + i). Second was
the elasticity of population with respect to debt accumulation. We change the magnitudes of these
forces by increasing and decreasing the mean moving costs µφ (which primarily affects the first
force) and the search directedness parameter λ (which substantially affects both) holding the riskfree price q̄ fixed. The results are presented in Table 7.
The first thing to notice is that—in each case—changes in debt per person have the same sign
and roughly the same magnitude as changes in expenditures per person and output per person.
To a large extent this is caused by all cities having a large incentive to overborrow that pushes
them close to their borrowing limits. Because the exogenous borrowing limit restricts the debtexpenditure ratio, debt and expenditures positively comove. Additionally, credit markets keep the
debt-output ratio from getting too large, and so debt and output also tend to comove. Last, output
and expenditures comove because government services are a normal good. Consequently, debt,
expenditures, and output move together irrespective of whether the city is constrained by the exogenous borrowing limit or endogenous ones. The lesson here is that modest changes in either the
overborrowing term or the population elasticity do not restrain borrowing because there is a large
incentive to borrow that modest changes do not overcome.
Keeping this in mind, consider specifically the impacts of a decrease in moving costs. This increases in- and out-migration, which also increases the overborrowing term and, in the two-period
model, leads to more debt accumulation. However, debt per person decreases because output decreases, and as just discussed, debt and output comove. Smaller moving costs reduce output because, in our calibration, migration is very noisy: Many people leave high-productivity islands for
no reason other than that they wanted to, i.e., their moving cost draw φ was substantially negative.
(Recall that the data dictate this feature because of the near-zero correlation between out-migration
and log productivity.) If the moving costs are even lower, that means even more people are leaving
cities for purely idiosyncratic reasons rather than fundamentals.29
28
If the market perfectly cleared, the debt per person average change would literally be zero, but here aggregate debt
declines by 0.6%. This particular experiment has more numerical error than the others because, without exogenous or
endogenous borrowing limits, borrowing becomes very elastic to guesses on the risk-free rate.
29
These comparative statics hinge on the calibrated values. For instance, consider the setup in the two-period model
without borrowing and with random search (λ = 0)—so that J is fixed—and no dispersion in φ with φ = µφ . Then
efficiency—in terms of total output—dictates that people should move whenever S < J and stay whenever S > J.
Decreasing µφ from a positive value to zero improves sorting and increases output: People who previously stayed at
islands with S < J because moving was too costly now leave; but, people at islands with S > J stay because in
expectation they would be worse off from a move. On the other hand, decreasing µφ further to a negative number
would worsen output: Now even people with S > J move simply to obtain utility J − µφ > J.
34
Pct. change from bench.
µφ
General equilibrium effects
λ
Bench.
↓
↑
↓
↑
−
No
No
No
No
68.2
62.9
16.0
5.06
5.30
6.08
0.562
-2.1
-2.1
0.0
-1.9
-1.9
-0.8
-4.2
1.7
1.7
1.3
1.6
1.6
0.3
3.3
-1.2
-1.2
-1.4
-1.2
-1.2
-1.3
-1.9
1.2
1.2
0.5
1.2
1.2
1.2
2.0
0.049
6.56
6.31
1.80
29.6
22.8
23.3
-7.7
-20.5
-20.9
-21.2
8.3
1.9
-0.9
-0.8
-4.3
16.2
0.6
0.5
4.7
Aggregate measures
Output per person
Consumption per person
Consumption per person s.d.
Expenditures per person
Taxes per person
Debt per person
Weather
Average across cities
Default rate ×100
In-migration rate (%)
Out-migration rate (%)
Log population s.d.
Note: the benchmark financial variables are in thousands of dollars;
non-benchmark are values are percent deviation from the benchmark.
Table 7: Changes due to moving costs and search directedness
35
Now consider an increase in search directedness (i.e., λ ↑), which has three effects. First, it makes
moving more attractive, which tends to increase in-migration and, consequently, increase debt. Second, it increases output, which, based on arguments above, also tends to increase debt. Third, it
makes in-migration more elastic to city debt positions, which reduces borrowing by “punishing”—
in the sense of lower in-migration—cities that accumulate debt. Here, quantitatively, the first two
forces dominate locally around the calibrated levels because λ is small, as required to match the
data’s small correlations between in-migration and productivity. For very large values of λ (which
would imply counterfactually large correlations), the third force would likely dominate (as suggested by the efficiency result obtained assuming perfect search directedness in Proposition 3).
6.3
Optimal borrowing limits?
With a better understanding of the model mechanisms, we now turn to the model’s counterfactuals.
One prediction of the theoretical model is that a supralocal government may wish to constrain
local government borrowing. And, as we showed, CA, MI, and several other states have imposed
municipal borrowing limits. Using the model, we now explore what would be the impact of lifting
these limits by setting δ to a large value. In contrast to the earlier exercise (in Section 6.2), we allow
for general equilibrium effects. The results are displayed in column δ = ∞ of Table 8.
To understand why output goes down, one must remember there are essentially two types of
cities in the model: Cities constrained by the exogenous borrowing limit and cities constrained by
the private credit markets. The first tend to be high-productivity cities (because their bond prices
are better) while the latter tend to have low productivity. By eliminating the exogenous limit, highproductivity cities accumulate more debt. This reduces household incentives to relocate to highproductivity cities, which results in lower output. Everything that comoves with output, such as
consumption, expenditures, and taxes, also falls. Moreover, welfare, as measured by J (see equation 10), also falls (because J is negative in the benchmark, a positive percent change means a lower
J). At face value, this measure—which is potentially problematic as it ignores transitions costs—
indicates that borrowing limits are in fact optimal. However, the magnitudes of the changes are
small because, as previously discussed, private credit markets significantly constrain city borrowing.
6.4
A return to high interest rates
In light of the recent increases in the federal funds rate, we assess the impact of a higher risk-free
rate on municipalities. In particular, we increase the risk-free interest rate in the model from 4%
to 6.5%—the latter corresponding to the interest rates in the early 1990s as can be seen in Figure
2—by reverse-engineering a risk-free bond supply B that delivers a 6.5% interest rate. The results
are displayed in Table 8 in the column labeled q −1 ↑.
More expensive borrowing reduces the incentives to issue debt, and debt per person declines
by 16%. The higher debt service costs encourage migration away from high-debt, low-productivity
36
Pct. change from bench.
General equilibrium effects
Bench.
q̄ −1 ↑
δ=∞
−
Yes
Yes
Yes
68.2
62.9
16.0
5.06
5.30
6.08
-12.9
0.562
4.00
0.000
0.2
0.1
-0.0
-0.0
1.4
-16.0
0.2
0.1
62.6
0.000∗
-0.2
-0.2
0.9
-0.3
-0.3
-0.2
0.0
-0.2
0.1
0.000∗
-0.1
-0.2
0.2
-0.1
0.1
-0.1
0.0
-0.1
8.4
0.057∗
0.049
0.000
-13.0
0.000∗
12.2
0.000∗
157.9
0.064∗
πb =
1
2
Aggregate measures
Output per person
Consumption per person
Consumption per person s.d.
Expenditures per person
Taxes per person
Debt per person
Utility conditional on moving (J)
Weather
Interest rate (%)
Tax for financing bailouts (%)
Average across cities
Default rate ×100
Bailout rate ×100
Note: The benchmark financial variables are in thousands of dollars; non-benchmark are values are percent deviation from the
benchmark unless marked by ∗ . The experiments are as follows:
q̄ −1 ↑ increases the risk-free interest rate by 2.5 percentage points;
δ = ∞ removes the borrowing limit; πb = 12 is anticipated bailouts
where bailouts occur 50% of the time.
Table 8: Counterfactual experiment results
37
cities, which increases aggregate output and “weather” slightly (both of which only reflect sorting).
Naturally, then, consumption also increases.30 Default rates fall slightly in the long run, while still
remaining low overall. From these results, we conclude that, while there will be some effects from
a return to high interest rates, the overall impact is likely to be small, at least in the long run.
6.5
What should Detroit have done differently?
Since our model has city planners who maximize the welfare of current residents, the planner’s response to shocks gives an optimal strategy for dealing with adverse (or beneficial) shocks. Hence,
by feeding in Detroit’s observed shock path in our simulation, our model provides a counterfactually optimal path for the city including policy prescriptions.31 We conduct this simulation by
feeding in the estimated fixed and residual productivity series when starting the economy with
the initial debt and population observed in the data. (For robustness, Appendix C.8 gives the paths
where the simulation starts after a long burn-in.)
The simulation results for Detroit are displayed in Figure 7, and they show that Detroit’s modelimplied long-run level of population is much lower than its 1986 level. From 1986 to 2011, the model
predicts a 0.75 log decrease or a 53% decline in population. While the decline in the data over this
period was 0.45 log point or 36% (see Figure 2), we note that Detroit’s population has continued
in its downward march in the data. In the model, the levels of debt, expenditures, and taxes stabilize after a few years of debt accumulation, and the city then optimally runs a primary surplus of
around $500 per person. The optimal model-implied debt level in the early 2000s of around $7,000
per person is not far from the value Detroit actually held, which was around $7,000 in 2001. However, Detroit’s debt continued to grow, and at the start of the financial crisis was around $10,000
per person. More importantly, when productivity sharply fell in the financial crisis, precipitating
the General Motors bankruptcy, the optimal response was to cut expenditures and—to a lesser
extent—taxes, reducing debt per person by $750. In contrast, Detroit raised taxes and expenditures
simultaneously, ultimately increasing debt per person by $2,500.
While Detroit is a special case in terms of the magnitude of changes it underwent, its suboptimal response to adverse productivity shocks is not. In particular, one signficant disagreement
between the model and data is the correlation between deficits and out-migration. In the fixedeffects regressions of Table 1, the negative correlation between between deficits and out-migration
is only weakly negative. Emblematic of this weak correlation is that Detroit’s out-migration in the
financial crisis was paired with larger deficits (as seen in Figure 2). In contrast, the same fixedeffects regressions on model data (in Table 5) reveal a strong negative correlation between deficits
and out-migration: When adverse shocks hit and increase out-migration, the optimal response is
30
Unlike in the counterfactuals seen thus far, output and debt do not comove. The reason is the interest rate increase
endogenously pushes the largest and most productive cities away from their borrowing constraint, which breaks the
tight link between expenditures, output, and debt. We show this in Figure 10 in Appendix C.7.
31
Since we have productivity data only at the county level, we use Wayne county’s measure in place of Detroit’s. We
also assume that Detroit has the best discretized “weather” state—in this case, clearly one should understand we are
not talking about weather per se, but rather some fixed characteristic that resulted in Detroit’s very large population.
38
Log population change
TFP change
0
In-migration
Out-migration
Net-migration
2
0
-0.5
Migration rates
-0.05
0
-0.1
-2
-1
1990
8.5
2000
2010
10 4Output
1990
Gov. budget items
6000
7.5
5000
Debt
Expenditures
Taxes
4000
7
2000
2010
7000
8
1990
2000
2010
1000
2000
2010
Primary Deficit
0
-1000
1990
Debt-expend. ratio
1.2
1990
2000
2010
Interest rates
1990
2000
2010
Overborrowing term
1.004
4.05
1.002
1
1
4
0.998
0.996
0.8
0.994
3.95
1990
2000
2010
1990
2000
2010
1990
2000
2010
Note: all financial variables are per person. ___________________
Figure 7: The optimal response to Detroit’s productivity shocks
39
to significantly deleverage. While Detroit’s residents in the model unanimously favor the record
primary surpluses of $800 per person, one can see that in practice these policies would not be
particularly popular, especially at a time when output has fallen by more than $10,000 per person.
6.6
Bailouts (or, what should the U.S. have done differently?)
The dire effects of the financial crisis not only precipitated Detroit’s bankruptcy, but also led to
bailouts for General Motors and Chrysler. This led to discussion on whether Detroit should be
bailed out, to which the governor of Michigan replied “bankruptcy is there to deal with the debt
question” (Condon, 2013). Moreover, bailouts are not just a Detroit issue. For instance, Hartford’s
finances exhibit massive swings in expenditures while taxes and debt are steady (as can be seen in
Figure 2) because of repeated, but uncertain, cash infusions from the Connecticut state government.
We now examine a general bailout policy that applies to any city at risk of default. Specifically, we
will focus on ε-bailouts as made precise by the following definition:
Definition 2. An ε-bailout is an unconditional transfer of resources b (measured per person and after migration) that makes the government indifferent between repaying and default (and we assume they repay in
such a case). Formally, b(x) is given by S N (ḃ(x) + b(x), ṅ(x), z, ω) = S D (ṅ(x), z, ω).
We assume bailouts occur with probability πb (i.i.d.) known to the local government. We also
assume that households make migration decisions before seeing whether the city is bailed out or
not. Hence, ṅ and ḃ are functions of x (as before) and do not depend on whether a bailout is realized
or not. To pay for the bailouts, we suppose a federal government taxes purchases of the risk-free
bond at rate τ . These assumptions change debt pricing and the intermediary’s problem and require
a federal government budget constraint, and we describe these modifications in Appendix C.5.
As can be seen in column πb =
1
2
of Table 8, anticipated bailouts that occur with 50% probability
more than double default rates. However, apart from this change, the effects on the other statistics
are quite weak with output, consumption, and expenditures all falling. Again, the exogenous borrowing limits play a significant role: Cities would like to borrow significantly, but loosening the
borrowing constraint by increasing expenditures (at the expense of consumption) is distortionary.
Additionally, with ε-bailouts, utility in the bailout period is the same as utility in default, which is
very low. This also limits any utility gain from being bailed out to excessive spending in the periods
leading up to the bailout.
To examine the moral hazard effects, we again turn to default episodes. These are displayed in
Figure 8, which plots the benchmark’s default (blue lines) and anticipated bailout (green circled
lines) episodes, as well as surprise bailouts (red dashed lines) that are measure zero events. Anticipated bailouts do not lead to excessive borrowing in anticipation of a bailout as cities at risk of
default are already at or very near to their borrowing limit. The surprise bailouts, which show just
the cash-injection effects without any moral hazard or general equilibrium effects, reveal effects
on debt, taxes, expenditures, and deficits, but all those changes do little to change the population
dynamics. Overall, we find the effects of ε-bailouts to be quite small.
40
Log population change
In-migration rate
TFP change
0
0.075
0
0.07
-0.05
0.065
-0.05
0.06
-0.1
-0.1
0.055
-0.15
0.05
-5
0
5
10
-5
0
10 4
Out-migration rate
5
10
5
10
5
10
5
10
Debt
4000
4.7
0.072
0
Income
4.8
0.074
-5
3000
4.6
0.07
0.068
4.5
2000
4.4
1000
4.3
0
-5
0
5
10
-5
Expenditures
0
5
10
-5
Debt-expend. ratio
3600
0
Taxes
3500
1.2
3000
3400
1.1
3200
1
3000
0.9
2500
2000
1500
1000
-5
0
5
10
-5
0
5
10
-5
Overborrowing term
Interest rates
Primary deficit
0
4.6
2500
Default
Surprise bailout
Anticip. bailout
0.945
2000
4.4
1500
0.94
1000
0.935
4.2
500
0.93
0
4
-5
0
5
Years since default
10
-5
0
5
Years since default
10
-5
0
5
10
Years since default
Figure 8: Comparison of Default and Bailout Episodes
7
Conclusion
Borrowing, migration, and default are intimately connected. Theoretically, we demonstrated that
migration tends to result in over borrowing. Empirically, we documented that defaults can occur
after booms or busts in labor productivity and population, that in-migration rates are negatively
correlated with deficits, and that many cities appear to be at or near state-imposed borrowing
limits. We interpreted these empirical findings loosely in light of the theoretical results, but our
quantitative model was able to formalize these explanations by replicating these and many other
41
features of the data.
Our counterfactuals revealed a number of interesting results. One novel result was that Detroit
should have cut spending, taxes, and debt in response to its large negative TFP shocks in 2007, 2008,
and 2009, thereby avoiding default. We also found that bailing out Detroit and cities like it does not
generate large moral hazard costs as long as the bailouts are not generous. With regards to monetary policy, we found a return to a high interest rate environment, while slightly improving GDP
via migration incentives, has fairly small effects. Last, we found that eliminating state-imposed
borrowing limits should have small effects because private credit markets restrain cities nearly as
much as the state-imposed limits.
While our paper has focused on municipalities, it should also prove useful for analyzing states
and nations. The drastic population flight from Puerto Rico, a less well-known population decline
in Greece, and large in-migration to Spain are three such applications where our framework could
prove useful.
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Chapter 9 – bankruptcy basics.
http://www.uscourts.gov/
services-forms/bankruptcy/bankruptcy-basics/chapter-9-bankruptcy-basics, 2018. Accessed: 2018-02-06.
S. Van Nieuwerburgh and P.-O. Weill. Why has house price dispersion gone up? Review of Economic
Studies, 77:1567–1606, 2010.
B. R. Weingast. Second generation fiscal federalism: The implications of fiscal incentives. Journal
of Urban Economics, 65(3):279 – 293, 2009. doi: https://doi.org/10.1016/j.jue.2008.12.005.
J. D. Wilson. Theories of tax competition. National Tax Journal, 52(2):269–304, 1999.
45
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 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
non-missing 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 population 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 9. We compute annual trust returns as totinstrustinvrev
divided by insurtrustcashsec less the CPI-measured inflation rate. 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.
46
Sample selection condition
Obs.
Raw
Require non-missing 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 non-missing 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 9: Sample selection in IndFin
By following the IndFin definitions, our debt measure excludes any implicit debt in the form of
underfunded pensions. In attempting to construct an alternate measure that included this “debt,”
we found that whether there is any underfunded pension liabilities or not hinges on the discount
rate assumed in accounting. If one assumes Treasury bill rates, then there is underfunding. However, many pension assets (75% according to Biggs, 2016) are invested in stocks, private equity, and
hedge funds, and the average return on pension assets has historically been much larger. If one
uses the expected returns, the liabilities turn into an asset for local pensions; see Table 1 of Rauh
(2017). Since the model has no aggregate risk, the impact of assuming a risk-free or risky rate only
affects the target for the debt-output ratio, and we suspect changes the model little. Because of this,
we ignore any underfunding of pensions, effectively treating them as fully funded. Additionally, a
municipality’s assets generally cannot be seized in default. Because of this, we look at gross debt,
not liabilities minus assets.
A.3
Migration data
We retrieve information on migration flows from the Census database on county-level migration.
The database contains data on migration in five-year intersecting intervals starting from 2006-2010
originating from the American Community Survey (ACS) and the Puerto Rico Community Survey
(PRCS). The respondents are asked where they lived one year prior to the survey.
The datasets include population moving within the same county, from a different county but
same state, from a different state, from abroad, and non-movers. As a measure of migration inflows,
we are using the sum of population moving from a different county within the same state, from
a different state, and from abroad. Thus, we do not count as migrants those who move within the
47
same county. Outflows are calculated analogously.
The data have limitations, especially for small counties. Geographies with populations of 65,000
or more have annual estimates produced, three years of data are needed for geographies of 20,00065,000, and five years of data for smaller geographies. We address this issue by, e.g., taking the
city-county-migration database and creating, using 2010-2014 migration data that corresponds to
the year 2012 estimate, five variable types, assigning the 2010-2014 data as *_F2 for 2010, *_F1 for
2011, * for 2012, *_L1 for 2013, and *_L2 for 2014 where * is one of inflow, outflow, or population.
We then compute a weighted average for each year via 14 (*_F2+*_L2) + 12 (*_F1+*_L1) + * divided by
2.5. (When one or more of the variables were missing, we computed the weighted average using
only the nonmissing data.)
A.4
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 90minute 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.”
48
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, 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."
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 follow Gordon (2019), which exploits information about the autocorrelation function to discretize efficiently. We use a “coverage” (i.e., the support in Tauchen’s 1986
method) of two unconditional standard deviations, treating any discretized state (z̃it , z̃it−1 ) occur49
ring with probability less than 10−6 in the invariant distribution as irrelevant and dropping it from
the set of discretized states (with appropriate reweighting of transition probabilities). 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 three objects: the expected utility conditional on moving
J, the risk-free price q̄,32 and the average inflows over a “normalization” term for the logit probabilites,
R
i :=
nF (R(x)|z)dµ(x)
,
exp(λ(S(x) − maxx S(x))dµ(x)
(32)
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
(33)
Solving for the law of motion and value and price functions
With the tuple (J, i, q̄), 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 4 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 log-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 ) = (10−3 , 10−5 , 10−7 ).
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.
32
For the bailout case, we guess on q̄(1−τ ) rather than q̄ as this is the relevant object for households and cities. Having
computed the equilibrium value of q̄(1 − τ ), we then compute τ using the invariant distribution and bailout amounts
and use it to recover q̄.
50
4. Solve for S N (ḃ, ṅ, z, ω) via grid search and update S D (ḃ, ṅ, z, ω).
For this, we use the analytic solution—conditional on b0 —of the intratemporal problem. Whenever we interpolate, we use linear interpolation.
5. Solve for d(x) and an update S ∗ (x) by comparing S N and S D evaluated at (ḃ(x), ṅ(x), z, ω).
6. Compute an update q ∗ (b0 , ṅ, z, ω) using d(x).
7. Compute an update ṅ∗ (x) using S ∗ (x) and J.
8. 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 as S := S ∗ ,
ṅ := 21 ṅ∗ + 12 ṅ, and q := q ∗ 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), the bond policy b0 (ḃ, ṅ, z, ω) (from the S N problem), and the
∗
default decision d(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 .
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. For all x, compute the out-migration rate o(x) := F (R(x)|z) where R(x) = S(x) − J.
∗
4. Using o(x), ī, µ(x), compute updates ṅ∗ (x), J ∗ , i .
5. Using ṅ∗ (x), and µ(x), the bond and default policies, compute an update on the invariant
distribution µ∗ (x).
Again, 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 same on each
R
R
iteration, i.e., µ(x)ndx = µ∗ (x)ndx.) We do not linearly interpolate the bond policies,
which are discontinuous because they are computed via grid search. Rather, we compute the
four sets of linear interpolation weights and knots for ḃ and ṅ then evaluate b0 (ḃ, ṅ, z) four
distinct times at the knots.
6. 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.
51
∗
7. For the updates J ∗ and i , use the values associated with the computed invariant distribution
µ. For an “update” on q̄, there is no natural fixed point update. However, construct a “pseudoR
update” by defining q̄ ∗ := q̄ − 10(B − (1 − d(x))(bn)dµ(x)).
R
Note that if cities are borrowing too little in that (1−d(x))(bn)dµ(x) > B, this makes q̄ ∗ > q̄,
lowering interest rates and making borrowing more attractive. The reason we multiply by 10
R
is because B is on the order of .001-.004, so if (1 − d(x))(bn)dµ(x) = 0 this implies q̄ ∗ would
be increased by .01-.04 (i.e., a few percentage point increase in the interest rate).
B.2.3
Solving for the key equilibrium objects
∗
With the initial guesses J, i, q and the updates J ∗ , i , q̄ ∗ , produce new initial guesses as follows:
1. Fix tolerances (tolJ , toli , tolq ). Fix an initial step size δq̄ > 0. Fix ξJ > 0 and ξi > 0.
We use (tolJ , toli , tolq ) = (10−7 , 10−5 , 10−3 ). Our δq̄ varies based on the experiment (with
larger values when the equilibrium value is suspected to be far away), but in the benchmark
we use δq̄ = .01. Our initial values for ξJ and ξi are 1.5 and 1, respectively.
∗
2. Check whether |J ∗ − J| < tolJ · |J|,|i − i| < toli · max{|i|, .01}, and |q ∗ − q| < tolq . If so, STOP:
an equilibrium has been computed. Otherwise, go on.
∗
3. Save the d∗J := J ∗ − J, di∗ := i − i, d∗q̄ := q ∗ − q and—before doing so—store the previous
changes (except on the first iteration) as dJ , di , and dq̄ .
4. Update the equilibrium values:
(a) J according to J := ξJ J ∗ + (1 − ξJ )J
∗
(b) i according to i := ξi i + (1 − ξi )i
(c) q according to q := q + sign(q ∗ − q) min{δq̄ , |q ∗ − q|}.
5. If not the first iteration and if dq̄ d∗q̄ < 0, replace δq with δq := 0.5 · δq .
The update on δq is necessary. To speed the computation, we also increase (decrease) ξJ when
dq̄ d∗q̄ ≥ 0 (dq̄ d∗q̄ < 0) and similarly for ξi , but this is not necessary.
If convergence has not been obtained, the new guesses on J, i, q are used to solve 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.
C
Omitted proofs and results [Not for Publication]
This section contains omitted proofs from the two-period model and additional theoretic and quantitative results.
52
C.1
Two-period model proofs
We first give the two-period model proofs.
C.1.1
The Euler equation
Proof of Proposition 1. The objective function may be written
Z
u(c1 ) + β
!
J−u(c2 )
(J − φ)f (φ)dφ
(1 − o2 )u(c2 ) +
(34)
−∞
Using Leibniz’s rule,
∂c1
0 = u (c1 )
+β
∂b2
0
∂u(c2 )
−∂o2 ∂(J − u(c2 ))
(1 − o2 )
+ u(c2 )
+
(J − φ)f (φ)
∂b2
∂b2
∂b2
!
φ=J−u(c2 )
∂c1
∂u(c2 )
−∂o2 ∂(J − u(c2 ))
0
= u (c1 )
+ β (1 − o2 )
+ u(c2 )
+
u(c2 )f (J − u(c2 ))
∂b2
∂b2
∂b2
∂b2
∂c1
∂u(c2 )
∂F (J − u(c2 )) ∂(J − u(c2 ))
0
= u (c1 )
+ β (1 − o2 )
− u(c2 )
+
u(c2 )f (J − u(c2 ))
∂b2
∂b2
∂b2
∂b2
∂u(c2 )
∂(J − u(c2 )) ∂(J − u(c2 ))
∂c1
0
+ β (1 − o2 )
− u(c2 )f (J − u(c2 ))
+
u(c2 )f (J − u(c2 ))
= u (c1 )
∂b2
∂b2
∂b2
∂b2
∂c1
∂c2
= u0 (c1 )
+ β(1 − o2 )u0 (c2 )
∂b2
∂b2
n1
∂b
2 n2
0
0
= −q̄u (c1 ) + β(1 − o2 )u (c2 )
∂b2
n1
∂n−1
0
0
2
= −q̄u (c1 ) + β(1 − o2 )u (c2 )
+ b2 n1
n2
∂b2
n1
−2 ∂n2
0
0
+ b2 n1 (−1)n2
= −q̄u (c1 ) + β(1 − o2 )u (c2 )
n2
∂b2
n
n
1
∂n
1
1
2
= −q̄u0 (c1 ) + β(1 − o2 )u0 (c2 )
− b2
n2
n2 n2 ∂b2
n1
b2 ∂n2
0
0
= −q̄u (c1 ) + β (1 − o2 )u (c2 ) 1 −
n2
n2 ∂b2
n1
b2 ∂n2
(1 − o2 )u0 (c2 ) 1 −
= −q̄u0 (c1 ) + β
n1 (1 − o2 + i2 )
n2 ∂b2
1 − o2
b2 ∂n2
= −q̄u0 (c1 ) + β
u0 (c2 ) 1 −
1 − o2 + i2
n2 ∂b2
Consequently, the Euler equation reads
q̄u0 (c1 ) = β
1 − o2
b2 ∂n2
u0 (c2 ) 1 −
.
1 − o2 + i2
n2 ∂b2
53
(35)
C.1.2
Constrained efficiency
Proof of Proposition 2. With no cross-sectional heterogeneity, constrained efficiency with the open
economy constraint requires (6). If q̄ = βu0 (y2 )/u0 (y1 ), then this requires
u0 (y2 + b2 nn21 )
u0 (y2 )
u0 (c2 )
=
=
u0 (y1 )
u0 (c1 )
u0 (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
1 − o2 u0 (c2 )
u0 (y2 )
=
.
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
u0 (y1 )q̄ > βu0 (y2 )
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.
Proof of Proposition 3. 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)—these conditions are the same for both the open economy (with the specified q̄) and
closed economy constraints. 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 open economy equilibrium with q̄ = βu0 (y2 )/u0 (y1 )
and a closed economy equilibrium with the same price, 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
∂c2
b2
=
n1 īI0 (u(c2 )) + f (J − u(c2 )) u0 (c2 )
n2 ∂b2
n2
∂b2
2
n1
= b2 u0 (c2 )
īI0 (u(c2 )) + f (J − u(c2 ))
n2
Because I is increasing and f is positive, this has the same sign as b2 .
54
(36)
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
b2 ∂n2
u0 (y2 )
u0 (c2 ) 1 − o2
u0 (c2 )
1−
β 0
= q̄ = β 0
≥β 0
.
u (y1 )
u (c1 ) 1 − o2 + i2
n2 ∂b2
u (c1 )
|
{z
}|
{z
}
=1
(37)
≥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
u0 (y2 )
u0 (c2 )
u0 (c2 ) 1 − o2
b2 ∂n2
β 0
≤β 0
= q̄ = β 0
1−
.
u (y1 )
u (c1 ) 1 − o2 + i2
n2 ∂b2
u (c1 )
|
{z
}|
{z
}
≤1
(38)
≤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 exists, which must be b2 = 0.
Proof of Proposition 4. Absent cross-sectional heterogeneity, Pareto efficiency in the closed economy
dictates that moving should occur if φ < 0 and staying if φ > 0 (with indifference for φ = 0) and
that c1 = y1 and c2 = y2 .
In a symmetric equilibrium, bond market clearing dictates b2 = 0 for all islands. Consequently,
c2 = y2 and c1 = y1 for all islands. Since there is no cross-sectional heterogeneity in endowments,
this means in equilibrium J = u(c2 ). Therefore, the reservation cutoff for moving is φ = 0, and each
island experiences outflows o2 = F (0). Because there is no heterogeneity in initial populations,
outflows and inflows are the same, so i2 = F (0) as well. Consequently, if q̄ is given by
1 − o2 u0 (c2 )
q̄ = β
1 − o2 + i2 u0 (c1 )
b2 ∂n2
u0 (y2 )
1−
= β(1 − F (0)) 0
,
n2 ∂b2
u (y1 )
then b2 = 0, c2 = y2 , c1 = y1 is optimal and constitutes an equilibrium. Moreover, the equilibrium
is Pareto optimal.
To prove Proposition 5, we first establish the following lemma:
Lemma 1. If there are two island types with homogeneous first-period endowments and heterogeneous
second-period endowments, then in equilibrium the island with a larger second period endowment has strictly
greater second period consumption and strictly borrows.
55
Proof. For use below, we first note that
n1 ∂[(1 − F (J − u(c2 )) + īI(u(c2 ))]
∂n2
=
∂b2
∂b2
= n1 [−f (J − u(c2 ))(−u0 (c2 )) + īI0 (u(c2 ))u0 (c2 )]
= n1 u0 (c2 )(f (J − u(c2 )) + īI0 (u(c2 )))
n1
.
n2
∂c2
∂b2
Defining
n2 ∂n2
n1 ∂b2
n1
n2
0
0
n1 u (c2 )(f (J − u(c2 )) + īI (u(c2 )))
=
n1
n2
g(c2 ) :=
= n1 u0 (c2 )(f (J − u(c2 )) + īI0 (u(c2 ))),
we have the Euler equation as
1 − o2
b2 n1
0
q̄u (c1 ) = β
u (c2 ) 1 −
g(c2 )
1 − o2 + i2
n2 n2
0
and note that g(c2 ) ≥ 0.
Now, use tildes to denote variables associated with the island having higher endowments in
the second period.
For contradiction, suppose that b̃2 ≥ 0. Then b2 ≤ 0 and c̃2 > c2 and c̃1 < c1 . Because c̃2 > c2 , ĩ2
is larger and õ2 is smaller than i2 and o2 , respectively. Consequently,
β 1 − õ2
1
1
β 1 − o2
β
β
=
=
<
i2
q̄ 1 − õ2 + ĩ2
q̄ 1 + ĩ2
q̄ 1 + 1−o
q̄ 1 − o2 + i2
2
1−õ2
From the Euler equations, this implies
u0 (c̃2 )
u0 (c̃1 )
n1 b̃2
1−
g(c̃2 )
ñ2 ñ2
!
u0 (c2 )
> 0
u (c1 )
n1 b2
1−
g(c2 ) .
n2 n2
Because b̃2 ≥ 0 and b2 ≤ 0 and g ≥ 0, one necessarily has
u0 (c̃2 )
u0 (c2 )
>
.
u0 (c̃1 )
u0 (c1 )
or
u0 (c2 )
u0 (c1 )
>
.
u0 (c̃1 )
u0 (c̃2 )
Since the islands have the same initial endowment and b̃2 ≥ 0, c̃1 < c1 and so the lefthand side
is less than 1. That means the righthand side must also be less than 1, which implies c2 > c̃2 . But
ỹ2 > y2 and b̃2 ≥ b2 , so clearly c̃2 ≥ c2 . This contradicts b̃2 ≥ 0. Therefore, b̃2 < 0.
56
Now, with b̃2 < 0, one has b2 > 0. From the Euler equation,
β u0 (c2 )
1 − o2 + i2
=
0
q̄ u (c1 )
1 − o2
−1
n1 b2
1−
g(c2 )
n2 n2
This implies
u0 (c̃2 ) 1 − õ2
u0 (c̃1 ) 1 − õ2 + ĩ2
!
n1 b̃2
u0 (c2 ) 1 − o2
n1 b2
1−
g(c̃2 ) = 0
1−
g(c2 )
ñ2 ñ2
u (c1 ) 1 − o2 + i2
n2 n2
u0 (c̃2 )
u0 (c̃1 )
u0 (c2 )
u0 (c1 )
u0 (c̃2 )
u0 (c2 )
u0 (c̃1 )
u0 (c1 )
=
1−
n1 b2
n2 n2 g(c2 )
1−õ2
1−õ2 +ĩ2
1−
n1 b̃2
ñ2 ñ2 g(c̃2 )
1
1−
n1 b2
n2 n2 g(c2 )
1−
n1 b̃2
ñ2 ñ2 g(c̃2 )
i
=
1−o2
1−o2 +i2
2
1+ 1−o
2
1
ĩ
2
1+ 1−õ
2
Suppose for contradiction that c2 ≥ c̃2 . Then the numerator of the lefthand side is weakly
greater than 1. Since b̃2 < 0, c̃1 > c1 so the denominator of the lefthand side is strictly less than
1. Therefore, the lefthand side is strictly greater than 1. Additionally, since c2 ≥ c̃2 , i2 /(1 − o2 ) ≥
ĩ2 /(1 − õ2 ). Consequently, 1/(1 + i2 /(1 − o2 )) ≤ 1/(1 + ĩ2 /(1 − õ2 )). So, the only way the equality
could hold is if
1−
n1 b2
n2 n2 g(c2 )
1−
n1 b̃2
ñ2 ñ2 g(c̃2 )
>1
However, since g ≥ 0 and b̃2 < 0, this is impossible. Therefore, it must be the case that c2 < c̃2 .
Proof of Proposition 5. Consider two islands of different types. Let the allocations etc. associated
with the one having higher endowments in the second period be denoted with tildes.
By Lemma 1, c̃2 > c2 , b̃2 < 0, and consequently b2 > 0. So, ĩ2 is larger and õ2 is smaller than i2
and o2 , respectively. Consequently,
1 − õ2
1
1
1 − o2
=
<
=
i
2
ĩ2
1 − o2 + i2
1 + 1−o2
1 − õ2 + ĩ2
1 + 1−õ
2
Now, with b̃2 < 0, one has b2 > 0. From the Euler equation,
q̄ u0 (c1 )
1 − o2
=
0
β u (c2 )
1 − o2 + i2
57
n1 b2
1−
g(c2 )
n2 n2
So, the lefthand side, and hence the marginal rate of substitution, will differ across islands if
1 − õ2
1 − õ2 + ĩ2
!
n1 b̃2
1 − o2
n1 b2
1−
g(c̃2 ) 6=
1−
g(c2 )
ñ2 ñ2
1 − o2 + i2
n2 n2
With b̃2 < 0 and b2 > 0, this must be the case, and, in particular,
u0 (c̃1 ) q̄
1 − õ2
=
0
u (c̃2 ) β
1 − õ2 + ĩ2
!
n1 b̃2
1 − o2
n1 b2
u0 (c1 ) q̄
1−
g(c̃2 ) >
1−
g(c2 ) = 0
ñ2 ñ2
1 − o2 + i2
n2 n2
u (c2 ) β
From the Euler equations,
u0 (c̃1 )
u0 (c1 )
>
.
u0 (c̃2 )
u0 (c2 )
Hence, the equilibrium is not constrained efficient because consumption growth is not equated
across individuals (even excluding individuals who migrate).
C.2
The intermediary’s problem
We now give the results used to simplify the intermediary’s problem.
For the intermediary problem, we consider a slightly more general formulation from that in
the main text, allowing for a proportional tax on risk-free bond holdings τ . Consequently, the intermediary’s problem is
W (B, M ) = max
D + q̄(1 − τ )W (B 0 , M 0 )
D,B 0 ,M 0
Z
0
s.t. D + q̄(1 + τ )B + Q(b0 , n0 , z, ω)dM 0 (b0 , n0 , z, ω) =
Z X
B+
P(z|z−1 )(1 − d(b, n, z))(bn)dM (b, n, z−1 , ω)
(39)
z
We will first prove the following, with Proposition 6 following as an immediate corollary:
Proposition 9. If prices satisfy
q(b0 , n0 , z, ω) = q̄(1 + τ )Ez 0 |z (1 − d(b0 , n0 , z 0 , ω))
(40)
B = (1 − d(b, n, z, ω))(bn)µ(db, dn, dz, 0, dω)
(41)
and if
then there exist prices and an optimal policy M 0 , with M 0 invariant, such that contract markets and the
risk-free bond market clear and zero profits obtain (provided the other equilibrium conditions are met).
Proof of Proposition 9. To characterize the intermediary’s problem, first consider its first-order conditions (FOCs). The FOC for B 0 is trivially satisfied for any B 0 .
58
Suppress dependence on ω—it enters the same way as z everywhere, but with ω 0 = ω always.
Equivalently, expand the definition of z to a pair giving TFP and ω. From the FOC for M (b0 , n0 , z)
one must have
Q(b0 , n0 , z) = q̄(1 + τ )
X
P(z 0 |z)(1 − d(b0 , n0 , z 0 ))(b0 n0 ).
(42)
z0
Replacing Q(b0 , n0 , z) with q(b0 , n0 , z)b0 n0 and simplifying for b0 6= 0, this becomes
q(b0 , n0 , z) = q̄(1 + τ )Ez 0 |z (1 − d(b0 , n0 , z 0 )).
(43)
q(b0 , n0 , z) = q̄(1 + τ )Ez 0 |z (1 − d(b0 , n0 , z 0 )).
(44)
Hence, in equilibrium, if prices satisfy the above equations, then the indifferent is intermediary
over all feasible contract, bond holding, and dividend distribution schemes.
Contract market clearing as stated in (21) dictates what M 0 must be as a function of the distribution µ, household policies b0 , d, and law of motion ṅ. Additionally, B 0 (B, M ) must equal B for
risk-free bond market clearing. The other equilibrium conditions pertaining to intermediary are
that (1) M must be invariant (M 0 = M ) and (2) that the intermediary makes zero profits. To satisfy
the first, we take M = M 0 .
To satisfy zero profits, first, note that for any B ∩{0} = ∅ and any integrable function g(b0 , n0 , z 0 ),
XZ
P(z 0 |z)g(b0 , n0 , z 0 )1[(b0 ,n0 )∈B×N ] M 0 (db0 , dn0 , z)
z
Z
=−
g(b0 , n0 , z 0 )P(z 0 |z)1[(b0 (b,n,z,0),ṅ(b,n,z,0))∈B×N ] (1 − d(b, n, z))dµ(b, n, z, 0)
Z
=−
(45)
g(b0 , n0 , z 0 )µ(db0 , dn0 , z 0 , 0)
B×N
which follows from µ being invariant and P(z 0 |z)1[(b0 (b,n,z,0),ṅ(b,n,z,0))∈B×N ] (1 − d(b, n, z)) being the
transition probability from b, n, z to b0 , n0 , z 0 as long as b0 6= 0 (for b0 = 0, there is an additional
arrival coming from cities that transition from f = 1 to f = 0).
Now we need to show the intermediary makes zero profits supposing that the other conditions
hold. Consider the intermediary’s budget constraint assuming zero dividends and bond market
clearing:
(1 − q̄(1 + τ ))B
Z
Z X
= Q(b0 , n0 , z)dM 0 (b0 , n0 , z) −
P(z|z−1 )(1 − d(b, n, z))(bn)dM (b, n, z−1 ).
z
59
(46)
If this holds, then zero dividends is feasible and hence optimal. Then one has
Z
Q(b0 , n0 , z)dM 0 (b0 , n0 , z)
XZ
= q̄(1 + τ )
(1 − d(b0 , n0 , z 0 ))(b0 n0 )P(z 0 |z)dM 0 (b0 , n0 , z)
z0
XZ
= −q̄(1 + τ )
(47)
0
0
0
0 0
0
0
0
(1 − d(b , n , z ))(b n )µ(db , dn , z , 0)
z0
Z
(1 − d(b0 , n0 , z 0 ))(b0 n0 )µ(db0 , dn0 , dz 0 , 0)
= −q̄(1 + τ )
where the first equality follows from equilibrium pricing of Q(b0 , n0 , z), the second from (45), and
the third is just different notation.
R
R
From the hypothesis that B = (1−d(b, n, z))(bn)µ(db, dn, dz, 0) and (47), we have Q(b0 , n0 , z)dM 0 (b0 , n0 , z) =
−q̄(1 + τ )B. So, (46) will hold as long as
B=−
Z X
P(z|z−1 ) ((1 − d(b, n, z))bn) dM (b, n, z−1 )
(48)
z
From M being invariant, we can replace M with M 0 so this condition is equivalent to
B=−
Z X
P(z 0 |z) (1 − d(b0 , n0 , z 0 ))b0 n0 dM 0 (b0 , n0 , z)
(49)
z0
Rewriting and using (45), this is
B=−
XZ
P(z 0 |z) (1 − d(b0 , n0 , z 0 ))b0 n0 dM 0 (b0 , n0 , z)
z0
X Z
=−
−
(1 − d(b , n , z ))b n µ(db , dn , z , 0)
0
0
0
0 0
0
0
0
(50)
z0
Z
=
(1 − d(b0 , n0 , z 0 ))b0 n0 µ(db0 , dn0 , dz 0 , 0)
which holds from the hypothesis.
Because the q(b0 , n0 , z) prices are pinned down by q̄(1 + τ ), exactly what q̄ and τ are is irrelevant
for them. However, given the equilibrium value of q̄(1 + τ ) and a τ , one immediately has q̄. Note
that τ is only used to finance bailouts, and in this case it is determined by how many resources are
needed in equilibrium to bail out cities (as implied by the government budget constraint).
Proof of Proposition 6. The result follows from Proposition 9 taking τ = 0.
60
C.3
The centralized problem
We now give the proof that the model can be centralized at a local level.
Proof of Proposition 7. 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 ,
ˆ Ld = ṅ, h = H̄/ṅ, π solving ṅπ = (1 − κ̄d)z
ˆ ṅ − wṅ + rH, and T solving
w = (1 − κ̄d)z,
ṅc + rhṅ = wṅ + π ṅ − T ṅ (see equation 15). Eliminating profit from the consumption equation,
ˆ − T (and clearly h = H̄/ṅ). Hence, the flow utility associated with this allocaone has c = (1 − κ̄d)z
ˆ − T, g, H̄/ṅ, ω), which according to (16) is the same as U (g, T, d,
ˆ ṅ, z, ω). Hence,
tion is u((1 − κ̄d)z
an arbitrary choice delivers u(c, g, H̄/ṅ, ω) flow utility in the centralized problem, which—when
ˆ ṅ, z, ω). Moreover, at
supported using the above prices and allocations—is the same as U (g, T, d,
these prices and allocations, b0 , d, T is feasible for the government as guaranteed by Walras’s law.33
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.4
The full-model Euler equation
We now give the derivation of the full-model’s Euler equation.
Proof of Proposition 8. Fix an ω (weather) and suppress dependence on it.
With repayment strictly preferred, default rates are zero the next period, q(b0 , ṅ, z) = q̄. Also,
note that for the same reason, S N inherits the differentiability of S. Additionally, the continuation
utility of the decentralized problem S N (ḃ, n0 , z)—to avoid confusion, we are using n0 as the state
rather than ṅ—may be written
βEφ0 ,z 0 |z max S
N
b0 n0
0 0 0
0
0
, ṅ(b , n , z , 0), z , J − φ dF (φ0 ),
ṅ(b0 , n0 , z 0 , 0)
(51)
which is valid for any b0 locally. Equivalently,
Z
βEz 0 |z
∞
R(b0 ,n0 ,z 0 ,0)
SN
!
Z R(b0 ,n0 ,z 0 ,0)
b0 n0
, ṅ(b0 , n0 , z 0 , 0), z 0 dF (φ0 |z 0 ) +
(J − φ0 )dF (φ0 |z 0 ) .
ṅ(b0 , n0 , z 0 , 0)
−∞
(52)
One can verify this easily. For instance, if f = d = 0, then the centralized budget constraint reads ṅc+ ṅ1−η g+qb0 ṅ =
ˆ − T as was found above with dˆ = 0 to eliminate c, one finds ṅ1−η g + qb0 ṅ = T ṅ + ḃṅ,
z ṅ + ḃṅ. Using c = (1 − κ̄d)z
which is the government’s budget constraint.
33
61
Using Leibniz’s rule and noting S N = J − R, the derivative with respect to b0 is
0
0
0
0
βEz 0 |z (1 − F (R(b , n , z , 0)|z ))
SḃN
n0
∂ ṅ−1
+ b0 n0
ṅ
∂b0
+
SṅN
∂ ṅ
∂b0
.
(53)
0 0
bn
0 0 0
0
where the arguments for SḃN and SṅN are ( ṅ(b0 ,n
0 ,z 0 ,0) , ṅ(b , n , z , 0), z ) and the argument for ṅ is
(b0 , n0 , z 0 , 0). The envelope condition gives SḃN (b0 , n0 , z) = uc . Also, ∂ ṅ−1 /∂b0 = −ṅ−2 ṅb0 . Plugging
these in,
0
n
N ∂ ṅ
0 0 1 ∂ ṅ
βEz 0 |z (1 − F (R(b , n , z , 0)|z )) uc
− b n 2 0 + Sṅ 0
ṅ
ṅ ∂b
∂b
0
0
n
b ∂ ṅ
∂ ṅ
βEz 0 |z (1 − F (R(b0 , n0 , z 0 , 0)|z 0 )) uc
1−
+ SṅN 0
0
ṅ
ṅ ∂b
∂b
0
0
0
0
(54)
So, the FOC for b0 is
n0
b0 ∂ ṅ
N ∂ ṅ
uc q̄ = βEz 0 |z (1 − F (R(b , n , z , 0)|z )) uc
+ Sṅ 0
1−
ṅ
ṅ ∂b0
∂b
0
0
0
0
(55)
Using the i0 and o0 notation from the hypothesis, note that ṅ(b0 , n0 , z 0 , 0)/n0 is 1+i0 −o0 and F (R(b0 , n0 , z 0 , 0)|z 0 ) =
o0 . Hence,
uc q̄ = βEz 0 |z
C.5
1 − o0
1 + i0 − o0
b0 ∂ ṅ
0
N ∂ ṅ
1−
uc + (1 − o )Sṅ 0 .
ṅ ∂b0
∂b
(56)
Modifications necessary for bailouts
In having bailouts financed by a bailout tax, the intermediary (who is the only purchaser of the
risk-free bond) has a new budget constraint:
Z
Z X
0 0
0 0 0
D+q̄(1+τ )B + Q(b , n , z, ω)dM (b , n , z, ω) = B+
P(z|z−1 )(1−d(b, n, z))(bn)dM (b, n, z−1 , ω).
0
z
(57)
This changes the equilibrium bond prices in Proposition 6 to
q(b0 , n0 , z, ω) = q̄(1 + τ )Ez 0 |z (1 − d(b0 , n0 , z 0 , ω)),
but the condition B =
R
(58)
(1 − d(b, n, z, ω))bnµ(db, dn, dz, 0, ω) is unchanged. The federal govern-
ment’s budget constraint holds provided
Z
τ q̄(−B) =
C.6
b(x)ṅ(x)πb 1[S N (ḃ, ṅ, z, ω) < S D (ḃ, ṅ, z, ω)]dµ(x).
(59)
Quantitative testing of indeterminacy
To test for indeterminacy, we proceed by drawing 100 random starting guesses for J, ī, and q̄ uniformly distributed about ±50% of the benchmark’s computed equilibrium values. (For the defini62
tion of ī, see Appendix B.2.) We then compute the implied equilibrium solution. Figure 9 shows a
scatter plot of the guesses in three-dimensional space, 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.
1.6
1.4
Guess
Solution
1.2
1
0.8
0.6
0.4
15
-5
-10
10
-15
5
-20
Figure 9: Quantitative testing of indeterminacy
C.7
Proximity to borrowing limits for a higher risk-free rate
Analogously to Figure 4, Figure 10 shows the proximity of cities relative to their borrowing limits
after a 2.5% increase in the risk-free interest rate. The largest cities are the ones that deleverage first
in response to the interest rate increase.
C.8
Detroit’s optimal path after its population stabilizes
Figure 11 presents Detroit’s optimal response to shocks after allowing the population to stabilize
(which takes almost 200 years and entails the population shrinking by 85% relative to 1987). Overall, the response is similar to that seen in Figure 7 with a large deleveraging occurring in 2008 with
cuts to expenditures that outweigh the tax cuts.
63
Borrowing and statutory limits
9.5
9
Log debt choice per person
8.5
8
7.5
7
6.5
Borrowing
Borrowing limit
6
7.6
7.8
8
8.2
8.4
8.6
8.8
9
9.2
Log expenditures per person
Note: A 1% subsample is used (to limit file size). ________________________
Figure 10: Model distribution of cities relative to their borrowing limit after an interest rate increase
C.9
Partial equilibrium results
Table 10 gives the partial equilibrium results (i.e., holding q̄ fixed) for the counterfactuals.
64
Log population change
TFP change
0.02
Migration rates
2
0
0
-0.02
-0.05
-0.04
-0.1
0
-0.06
-2
1990
2000
2010
10 4Output
1990
2000
2010
1990
Gov. budget items
0
8
7000
-200
7.5
6000
7
5000
1990
2000
2010
1990
Debt-expend. ratio
1.2
Debt
Expenditures
Taxes
2000
2010
-400
-600
2010
Interest rates
2000
Primary Deficit
8000
8.5
In-migration
Out-migration
Net-migration
1990
2000
2010
Overborrowing term
0.99
4.05
0.985
1.18
4
0.98
1.16
3.95
1990
2000
2010
0.975
1990
2000
2010
1990
2000
2010
Note: All financial variables are per person. ___________________
Figure 11: The optimal response to Detroit’s productivity shocks
65
Pct. change from bench.
General equilibrium effects
Bench.
q̄ −1 ↑
δ=∞
−
No
No
No
68.2
62.9
16.0
5.06
5.30
6.08
-12.9
0.562
4.00
0.000
0.2
0.1
-0.0
-0.0
1.4
-16.0
0.2
0.1
62.6
0.000∗
-0.2
-0.2
1.0
-0.3
-0.3
-0.2
-0.0
-0.2
-0.0
0.000∗
0.2
0.2
0.7
0.2
0.2
0.7
0.0
0.1
1.4
0.057∗
0.049
0.000
-13.0
0.000∗
12.2
0.000∗
197.7
0.074∗
πb =
1
2
Aggregate measures
Output per person
Consumption per person
Consumption per person s.d.
Expenditures per person
Taxes per person
Debt per person
Utility conditional on moving (J)
Weather
Interest rate (%)
Tax for financing bailouts (%)
Average across cities
Default rate ×100
Bailout rate ×100
Note: The benchmark financial variables are in thousands of dollars; non-benchmark are values are percent deviation from the
benchmark unless marked by ∗ . The experiments are as follows:
q̄ −1 ↑ increases the risk-free interest rate by 2.5 percentage points;
δ = ∞ removes the borrowing limit; πb = 12 is anticipated bailouts
where bailouts occur 50% of the time.
Table 10: Counterfactuals in partial equilibrium
66
@FedFRASER