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Public Debt, Private Pain: Regional Borrowing, Default, and Migration WP 21-13 Grey Gordon Federal Reserve Bank of Richmond Pablo Guerron-Quintana Boston College Public Debt, Private Pain: Regional Borrowing, Default, and Migration∗ Grey Gordon Federal Reserve Bank of Richmond Research Department greygordon@gmail.com Pablo Guerron-Quintana Boston College Department of Economics pguerron@gmail.com July 30, 2021 Abstract How do local government borrowing, default, and migration interact? We find in-migration results in excessive debt accumulation due to a key externality: Immigrants help repay previously-issued debt. In addition to providing direct IV evidence on this mechanism, we show cities are heavily indebted and remain so even after large population growth, resulting in boom defaults. While default rates are currently low, default risk has increased secularly despite the secular decline in interest rates, which we show lowered default risk else equal. Our quantitative model implies large interest rate declines in the Great Recession and COVID-19 crisis prevented default. JEL Codes: E21, F22, F34, R23, R51 Keywords: migration, population, debt, default, cities, bankruptcy, Detroit, regional ∗ We thank our discussant Pablo D’Erasmo as well as Marco Bassetto, Susanto Basu, Jeff Brinkman, Lorenzo Caliendo, Jerry Carlino, Ryan Chahrour, Yoosoon Chang, Satyajit Chatterjee, David Childers, Daniele Coen-Pirani, Hal Cole, Jonas Fisher, Aaron Hedlund, Ivan Ivanov, Dirk Krueger, Amanda Michaud, Leonard Nakamura, Jim Nason, Jaromir Nosal, Santiago Pinto, Esteban Rossi-Hansberg, Pierre-Daniel Sarte, Sam Schulhofer-Wohl, Tony Smith, Nora Traum, Marcelo Veracierto, and Mark Wright for valuable discussions. We also thank participants at various seminars and conferences. Alexey Khazanov and Michelle Liu provided excellent research assistance. A previous version of this paper circulated under the title “On Regional Borrowing, Default, and Migration.” The views expressed are those of the authors and do not necessarily reflect those of the Federal Reserve Bank of Richmond or the Federal Reserve System. 1 1 Introduction Municipal debt is a $1.9 trillion source of financing for local governments (Mayo et al., 2020). Normally considered a safe investment, yields exploded both in the Great Recession and COVID-19.1 Since municipal bond yields reflect actual default risk (Schwert, 2017), their sharp increase suggests local government finances are more vulnerable than they appear in noncrisis periods. Both crises precipitated large-scale interventions, which at least for the COVID-19 crisis, seem to have worked (Haughwout et al., 2021). At the same time, local government finances depend not only on national economic conditions, but also on migration decisions, with each entrant to the city reducing debt per person and increasing the tax base (and each departure doing the opposite). This link can be clearly seen in major urban centers where population growth or decline can be tremendous. The leading example of the latter, Detroit, had a 35% reduction in population from 1986 to 2013, which undoubtedly contributed to its 2013 bankruptcy. The remote work era that COVID-19 may usher in has potential to induce massive population shifts with commensurate implications for city finances. Despite this, there is currently limited empirical evidence and no model linking city finances, migration, and default. This paper fills these gaps in the literature. We first use a two-period Lucas (1972)-type islands model to highlight a key mechanism, specifically, an overborrowing externality. Each island represents a local economy and has households who make migration decisions, a per person endowment, and a planner who issues debt in the first period (transferring the proceeds to households) and repays it in the second (using lump-sum taxes). The key assumption is that the local planner maximizes the welfare of current residents. The model reveals that, relative to an economy-wide planner, local planners have an incentive to overborrow. The reason is simple: new arrivals in the second period will help repay debt issued in the first period, and the planner does not directly value their utility. With this externality in mind, we turn to the data where we expect to find, and do find, excessive debt. Using comprehensive datasets on city finances, population, migration, and labor productivity, we establish six stylized facts. (While technically 1 For instance, the BBB-AAA spread increased from 0.6% in 2007Q1 to a peak of 4.3% in 2009Q1; while the AAA spread over 10-year Treasuries rose from 0.71% on February 21, 2020, to 2.74% by March 23, 2020. (Authors’ calculations using data from Haver.) 2 incorrect, we will refer to cities and municipalities interchangeably.)2 First, city debt and default risk have been increasing over time. Second, cities of all types are heavily indebted. Third, cities of all types are near state-imposed borrowing limits. Fourth, cities respond to arguably exogenous variation (constructed using a Bartik style shift-share instrument) in in-migration rates by increasing debt. Fifth, cities remain leveraged even after booms in population and productivity, leading to boom defaults. Lastly, default risk is highly senstive to interest rate movements. We extend the two-period model to a full-blown quantitative model with an infinite horizon, production, government services, housing, borrowing limits, and default. After showing the economy can be centralized (at a local level), we demonstrate the calibrated model reproduces the data’s stylized facts. We then use the model to understand the observed path of the economy in the Great Recession and COVID-19. In both cases, the observed decline in real interest rates proves crucial in preventing an onslaught of municipal defaults. Related literature Our paper builds on the large sovereign default literature begun by Eaton and Gersovitz (1981), which has focused almost exclusively on nation states. Some of the key references are Arellano (2008), Chatterjee and Eyigungor (2012), Hatchondo and Martinez (2009), and Mendoza and Yue (2012). (The handbook chapter Aguiar et al. (2016) provides a thorough description of the literature.) Epple and Spatt (1986) is an exception that argues states should restrict local debt because default by one local government makes other local governments appear less creditworthy. Such a force is not at work in our model because we assume full information. We contribute to this literature by showing migration strongly influences debt accumulation and can result in boom defaults. Our work also connects to a vast literature on intranational migration. The empirical work and to a lesser extent theoretical is surveyed in Greenwood (1997). Two seminal papers in this literature, Rosen (1979) and Roback (1982), employ a static model with perfectly mobile labor. This implies every region provides individuals with the same utility. While this indifference condition allows for elegant characterizations of equilibrium prices and rents, it also means government policies are completely indeterminate: every debt, service, or tax choice results in the same utility. Our model breaks this result by assuming labor is imperfectly mobile, which 2 A municipality is a city, town, or village that is incorporated into a local government. 3 lets it match both the sluggish population adjustments and the small correlations between productivity and migration rates observed in the data. More recently, Armenter and Ortega (2010), Coen-Pirani (2010), Van Nieuwerburgh and Weill (2010), Kennan and Walker (2011), Davis et al. (2013), and Caliendo et al. (2017) have analyzed determinants of migration and its consequences in the U.S. Kennan and Walker (2011) use a structurally estimated model of migration decisions and find expected income differences play a key role, providing external evidence of the model’s productivity-driven migration decisions. Outside the U.S., recent research has been focused on migration in the EU (Farhi & Werning, 2014; Kennan, 2013, 2017). All these papers abstract from debt. To our knowledge, ours is the only quantitative model of regional borrowing and migration, let alone having default. A few papers in this literature have discussed the potential for local governments to overborrow because of migration. Bruce (1995) and Schultz and Sjöström (2001) prove that overborrowing generally does occur. However, both of their models are two-period models with costless moving, and our theoretical results imply this is not an innocuous assumption. Additionally, we show empirically and quantitatively the role of overborrowing in reproducing many of the data’s features. Finally, building on earlier versions of this paper, Alessandria et al. (2020) features a sovereign government (Spain in their calibration) facing in-migration and issuing debt. Similar to our finding that in-migration induces overborrowing, they find in-migration causes increased indebtedness relative to a no-in-migration economy. A key difference from our paper is that in-migration, in their model, is exogenous and out-migration is not allowed: in contrast, in-migration and out-migration in this paper are jointly determined given the entire distribution of available locations. The rest of the paper is organized as follows. Section 2 presents a simple model that highlights the overborrowing mechanism inherent in models with in-migration and borrowing. Section 3 documents key stylized facts. Section 4 lays out the quantitative model, and Section 5 describes the calibration. Section 6 shows the quantitative model reproduces the stylized facts and analyzes the Great Recession and COVID-19 shocks. Section 7 concludes. The appendices report data details, the computational algorithms, and proofs. 4 2 The overborrowing mechanism First, we highlight how migration influences borrowing decisions and efficiency using a two-period model. To focus purely on the role of borrowing, we assume there is full commitment to repay debt and, hence, no default. The economy is comprised of a unit measure of islands and a unit measure of households. Assume islands are homogeneous, and consider an arbitrary one. In the first (second) period, the island has a per person nonstochastic endowment of y1 (y2 ). The local government issues −b2 debt per person (b2 > 0 means assets) at price q̄. Total debt issuance is −b2 n1 , where n1 is the initial measure of households on the island. At the beginning of the second period, households draw an idiosyncratic utility cost of moving φ ∼ F (φ) with a density f and then decide whether to migrate. If they migrate, they pay φ and obtain expected utility J, which is an equilibrium object. Households value consumption according to u(c1 ) + βu(c2 ), where c1 (c2 ) is consumption in the first (second) period. Household utility in the second period is u(c2 ) if they stay and J − φ if they move, so migration decisions follow a cutoff rule in φ with indifference at J − u(c2 ). Consequently, the outflow rate is o2 = F (J − u(c2 )). The inflow rate is given by i2 = īI(u(c2 )), where I is a differentiable, increasing function, and ī is an equilibrium object that ensures aggregate inflows equal aggregate outflows. (Consequently, inflows can depend on the distribution of utility across islands, but that information must be summarized in ī.) The population law of motion is n2 = (1 + i2 − o2 )n1 . We assume the migration decision is noisy in the sense that F (0) > 0, so that some people will move even if u(c2 ) = J. After all migration has taken place, the government pays back its total obligation, −b2 n1 , by taxing the n2 households lump sum. Consequently, per person consumption in the second period is c2 = y2 + b2 n1 /n2 . The government’s problem may be written Z max u(c1 ) + β b2 s.t. c1 + q̄b2 = y1 , max {u(c2 ), J − φ} dF (φ) c2 = y2 + b2 i2 = īI(u(c2 )), n1 , n2 n2 = n1 (1 − o2 + i2 ), c1 , c2 ≥ 0, (1) o2 = F (J − u(c2 )). An equilibrium is a pair {i, J} with optimal migration, consumption, and borrowing decisions such that 5 1. total inflows equal total outflows, i R I(u(c2,j ))n1,j dj = R F (J − u(c2,j ))n1,j dj, and 2. the expected utility of moving is consistent, J = R iI(u(c ))n u(c2,j ) R iI(u(c2,j ))n1,jdi dj 2,i 1,i (where j indexes islands). Proposition 1 gives the Euler equation for government bonds (all proofs are in Appendix C). Proposition 1. The local government’s Euler equation is 1 − o2 b2 ∂n2 0 0 u (c1 )q̄ = β u (c2 ) 1 − . 1 − o2 + i2 n2 ∂b2 (2) The Euler equation reflects two competing forces. One is an externality seen in the term 1−o2 1−o2 +i2 ≤ 1. Because the planner does not value the utility of new entrants and because new entrants bear i2 1−o2 +i2 of the debt burden (which is their share of the second period population), the marginal cost associated with an additional unit of 1−o2 1−o2 +i2 . All else equal, higher 1−o2 (β 1−o ) and increases borrowing. 2 +i2 debt—holding fixed migration rates—is 1 − 1−oi22+i2 or in-migration lowers the effective discount factor Clearly, then, the assumption that the planner only values current residents plays a key role. But note that it is also the most natural assumption: if households could vote on the planner’s policy in the first period, they would unanimously approve it because it maximizes their welfare. Before discussing the second force, we emphasize that this externality is really about in-migration, not out-migration. Consider an extreme case where half the population leaves o2 = 1/2 but no one arrives i2 = 0. In that case, the overborrowing term 1−o2 1−o2 +i2 equals 1, i.e., there is no extra discounting in the Euler equation. The reason is that while half of current residents leave and pay nothing, the half who remain must pay double, and this offsets in the Euler equation. On the other hand, if no one leaves o2 = 0 but the population doubles through in-migration i2 = 1, then the overborrowing term 1−o2 1−o2 +i2 equals 1/2, implying an effective 50% discount on debt issuance. In intermediate cases where o2 and i2 are positive, in-migration has a first-order effect while out-migration has only a second-order effect. While the externality is primarily driven by in-migration, out-migration still has a first-order effect on debt per person and hence will play a key role in default decisions. The Euler equation’s other, potentially offsetting force, is seen in the term 1− b2 ∂n2 n2 ∂b2 , which is one minus the elasticity of the next period’s population with 6 respect to savings. It reflects that for each person attracted to the island through less borrowing, the overall debt burden per person falls. (Conversely, if b2 > 0, each additional entrant reduces assets per person, which discourages savings.) Hence, a rational government, internalizing the effects of city finances on migration decisions, should exercise more financial discipline (else equal) to attract individuals to the islands to reduce debt per person. However, migration decisions in the quantitative model must be very noisy to match migration patterns in the data; consequently, 1−o migration will strongly effect local government debt ( 1−o+i 1), but local govern- ment debt will only weakly effect migration (1− nb ∂n ∂b ≈ 1). Intuitively, we also expect that government debt has only second-order effects on migration: many of us know the typical population, temperature, income, and housing prices to expect in a city, but how many of us know the city’s outstanding debt amount? Consequently, we expect the desire to overborrow will generally be stronger than this debt-disciplining effect. To this point, we have claimed that cities overborrow, implicitly having in mind the solution to a social planner problem, which we now state. Let ĉ1,i , ĉ2,i denote the optimal consumption (in periods 1 and 2, respectively) of household i ∈ [0, 1], and let φi denote the moving cost shock realization the household receives. With homogeneous islands, the endowments are the same irrespective of moving decisions, with y1 (y2 ) the first (second) period endowment. Taking migration decisions mi as given, the planner’s objective function is Z max αi (u(ĉ1,i ) + β(u(ĉ2,i ) − mi φi ))di, ĉ1,i ≥0,ĉ2,i ≥0 (3) where αi is the Pareto weight on household i. The resource constraint is given by Z Z ĉ1,i di + q̄ ĉ2,i di = y1 + q̄y2 . (4) We say an allocation is constrained efficient if it solves the planner problem with migration decisions given for some Pareto weights. Optimality requires that marginal rates of substitution must be equated across individuals, i.e., βu0 (ĉ2,i )/u0 (ĉ1,i ) = βu0 (ĉ2,j )/u0 (ĉ1,j ) for almost all i, j. Using the resource constraint, it is easy to show these must also equal q̄, i.e., u0 (ĉ1,i )q̄ = βu0 (ĉ2,i ). (5) In comparing (5) with the local government’s Euler equation (2), it is clear that overborrowing will occur if the optimal bond choice b2 is close to zero: in that case, 7 the incentive to attract people—reflected in the term 1 − b2 ∂n2 n2 ∂b2 —is close to zero, while the externality of new entrants shouldering the burden—reflected in 1−o2 1−o2 +i2 — is not. With q̄ = βu0 (y2 )/u0 (y1 ), implementing the constrained efficient allocation requires b2 = 0, which results in overborrowing as formalized in Proposition 2:3 Proposition 2. If q̄ = βu0 (y2 )/u0 (y1 ), equilibrium is not constrained efficient. Moreover, at the constrained efficient allocation, governments would strictly prefer to borrow. To summarize, we showed cities have an incentive to overborrow. The mechanism is an externality created through in-migration: new entrants will help repay debt issued today. While in theory, migration can act as a disciplining force, we argued that in practice this is unlikely. We now turn to the data. 3 Empirical patterns of debt, migration, and default In this section, we will establish six stylized facts using a variety of data sources described in Appendix A. First, debt and default risk have been increasing over time. Second, virtually all cities, large and small, productive and unproductive, are indebted. Third, almost all cities are close to state-imposed borrowing limits. Fourth, in response to in-migration, expenditures per person and debt per person increase. Fifth, cities default after both booms and busts in population and productivity. Last, municipal bond default risk is sensitive to interest rate changes. Before establishing the facts, however, we begin with a quick overview of municipal debt and default. 3.1 An overview of municipal debt and default Two broad types of municipal debt exist: general obligation debt (GO) and nongeneral obligation debt (Non-GO). Non-GO debt is typically attached to a specific revenue stream, e.g., bonds for construction of a toll road that will be paid for using revenue from the toll road. In contrast, GO debt is fully backed by taxes. Additionally, Novy-Marx and Rauh (2011), Novy-Marx and Rauh (2009), and Rauh (2017) have shown state and local governments have a third large type of “debt,” which is unfunded pension obligations. How much debt local governments can accumulate is typically constrained by state-imposed borrowing limits that vary in type and degree. For example, California 3 This inefficiency result may be surprising in light of the seminal paper by Tiebout (1956). It arises because he assumes costless and fully-directed mobility, which we do not require. We prove in Proposition 4 in the appendix that efficiency can hold in this case, which requires an “infinite” elasticity of in-migration to debt. 8 (CA) limits are tied to spending or revenue that year. In contrast, most of the states restrict debt based on a percentage of property valuations, but the percentages can differ substantially from as little as 0.5% (IL) to 10% (NY). Table 3 in the appendix reports some of these, and reveals almost all the states have known exceptions, which usually include debt related to education, water supplies, and referendum-approved debt. When GO and non-GO debt becomes excessive, local governments do have access, potentially, to Chapter 9 bankruptcy. This substantially differs from consumer and corporate bankruptcy, with one key distinction being that the municipality must be insolvent, either “unable to pay its debts” or “generally not paying its debts.”4 The latter statement, which in isolation would appear to be nonrestrictive, is strengthened by the additional requirement that filers must negotiate in good faith with creditors. A second key distinction is that local governments are allowed to keep essentially all their assets in default to prevent creditors from infringing on the local government’s sovereignty.5 While bankruptcy allows discharge of GO and non-GO debt, pension obligations are commonly viewed as nondefaultable, either explicitly protected by state constitutions or otherwise protected by contract law (Brown & Wilcox, 2009). In practice, CA cities that went through bankruptcy did not have their pension obligations reduced (Myers, 2019). In Detroit, a worst-case example, there were in fact some pension cuts: 4.5% directly with cessation of Cost of Living Adjustments (COLAs) (Stempel, 2016). However, viewing pension obligations as nondefaultable seems to be a reasonable approximation. 3.2 Stylized fact #1: Debt and default are increasing over time Historically, municipal bond default has been rare. For bonds rated by Moody’s, Moody’s (2013) report there have been 73 municipal bond defaults between 1970 and 2011, a default rate of 0.012%. However, these low default rates belie the severity of the situation for several reasons. First, one reason is that default rates have been trending upward over time. Using a large municipal-bond dataset from Mergent, Figure 1 reports the time-series 4 11 U.S.C. § 101(32)(C). Chapter 9 “is significantly different in that there is no provision in the law for liquidation of the assets of the municipality and distribution of the proceeds to creditors. Such a liquidation or dissolution would undoubtedly violate the Tenth Amendment to the Constitution and the reservation to the states of sovereignty over their internal affairs” (United States Courts, 2018). 5 9 path of default rates, both as a frequency of outstanding bond counts (unweighted) and as a percent of outstanding debt (weighted). The top left panel uses all bonds, while the top right panel uses only GO bonds. Until the late 2000s, default rates for all bonds were generally below 0.05%. However, in the Great Recession, these doubled or even tripled and have remained elevated. For GO bonds, the upward trend is less obvious, but the period of relative stability from 2011 to 2019 saw default rates of around 0.01%, whereas during mid 2000s that rate was zero. It’s also worth noting both series show substantial sensitivity of default rates to aggregate risk. This increase in default risk is perhaps more clearly seen by looking at interest rate spreads over time. For instance, when we compare the weighted P90 yield to maturity (YTM) over a “riskfree” P10 YTM, the spread has risen from 2% during the Great Moderation to close to 4% in the 2010s.6 A similar story appears looking at GO bonds, or the P75-P10 difference, or even to a lesser extent the P25-P10. (Later, we will look specifically at the Great Recession and COVID-19.) Whether we look at default rates or interest rate spreads, the story is the same: Default risk is increasing over time. Default rate All bonds 0.15 Weighted Unweighted Percent Percent 0.15 Default rate GO bonds 0.1 0.05 YTM spread 6 4 0.05 2000 2010 0 1990 2020 YTM spreads over P10 All bonds, weighted 6 P90-P10 P75-P10 P25-P10 2 0 1990 0.1 YTM spread 0 1990 Weighted Unweighted 2000 2010 2020 4 2000 2010 2020 YTM spreads over P10 GO bonds, weighted P90-P10 P75-P10 P25-P10 2 0 1990 2000 2010 2020 Note: Default series excludes Puerto Rico. Figure 1: YTM spreads and default rates over time 6 We use the P10 YTM of municipal bonds as our measure of the riskfree rate because municipal bonds have a number of key differences from Treasuries that make them not very comparable. Chief among these differences are that municipal bonds tend to be callable (i.e., they can be refinanced at lower rates), and they tend to be tax-exempt. 10 There are also other factors that have masked the amount of fiscal stress. As in Flint, cities can avoid default but come under state management and thereby lose fiscal autonomy. In fact, Kleine and Schulz (2017) report that Michigan (MI) had 11 cities (4%), one township, and one county under state oversight due to a financial emergency (p. 9). Additionally, real interest rates—which we will show strongly effect municipal bond risk premia—have fallen secularly over time. Lastly, debt has grown tremendously over the past few decades. For instance, the average debt per person across counties was $3085 in 2016 but $2107 in 1996 in constant 2012 dollars—nearly a 50% increase. Similarly, debt for cities grew by 51% from $496 to $750 per person. With the increased debt burden comes increased risk that local governments will not be able to repay. 3.3 Stylized fact #2: Cities of all types are indebted Our second stylized fact is that all types of cities are indebted. This can be seen in Figure 2, which presents the empirical relationship of log debt per person (p.p.) and log expenditures (p.p.) One can see cities of all sizes have debt: there are not cities clustering at low debt levels. Additionally, the linear relationship is stark, with more expenditures strongly positively correlated with more debt. The quantitative model will generate both of these features of the data through (1) the overborrowing incentive already seen in the theoretical model paired with (2) borrowing constraints that relax as income and/or expenditures increase. -5 0 5 10 Log debt and expenditures per person in 2011 2 4 6 Log expenditures p.p. Fitted values 8 10 Log debt p.p. The area of each observation is proportional to its population. The linear fit is weighted by population. Figure 2: Debt and expenditures per person 11 3.4 Stylized fact #3: Cities of all types are close to borrowing limits While we saw above that cities of all types are indebted, we have not established precisely how indebted they are relative to their state-imposed maximum borrowing capacity. Because most of the borrowing limits are expressed in terms of assessed property value (which is not in our dataset), in general it is hard for us to directly look at this question. However, we can directly assess how close cities are to their borrowing limits for two states. CA uses expenditures to constrain debt, and so we can tell how close cities are to their limit as displayed in the top panel of Figure 3. The graph reveals that many cities—including very large ones—are borrowing beyond the revenue per person limit (recall, the limit allows exceptions for spending on special projects and borrowing authorized by referendum). The other state we could check was MI using taxable-valuation data from Kleine and Schulz (2017), and the bottom panel of Figure 3 displays how close MI cities are to their limits. Again, many cities are at, near, or above the limit including Detroit and Flint. Even the wealthiest cities (as measured by property values) are borrowing. In summary, it seems cities regularly borrow, and many of them borrow as much as they legally can. This evidence suggests the borrowing limits are binding, consistent with the model prediction that cities have a strong incentive to borrow. Borrowing and statutory limits in Michigan (2011,2012) Log debt (FY end) per person 4 6 8 10 10 Borrowing and statutory limits in California (2011) Detroit Log debt per person in 2011 4 6 8 Los Angeles San Diego Stockton Vallejo Flint 2 2 San Bernardino 6 7 8 Log revenue per person Log debt (FY end) per person 9 10 9 Limit (with exceptions) 10 11 12 Log taxable property value in 2012 Log debt per person in 2011 13 Log debt limit in 2012 Figure 3: Statutory borrowing limits and closeness to the limits 3.5 Stylized fact #4: In-migration increases expenditures and debt So far, we have shown cities are highly indebted, but we have not looked at the link between migration and debt accumulation. We now establish this link directly using an instrumental variable approach. In particular, we construct an instrument 12 for in-migration using the shift-share approach, a common technique in empirical studies of migration (see, e.g., Altonji and Card, 1991 and Card, 2001). For each county c, we construct the share of individuals at a reference time τ > 0 periods ago who live in c but arrived from county o at some time in the past.7 Then, with the share θc,o,t−τ , we compute the total outflows Oo,t from each county o at time t. Given this aggregate shift from o, we then apply the share θc,o,t−τ to get expected inflows θc,o Oo,t at time t from county o to county c. Summing across all other counties gives the total expected inflows and a total expected migration rate: X îc,t = θc,o,t−τ Oo,t /Nc,t , (6) o6=c where Nc,t is the population in county c at the start of time t. If shocks to county o are orthogonal to shocks in c (after controlling for time effects), the expected inflow rate îc,t should only affect local variables through its effect on ic,t , as required for a valid instrument. Given this, we run the specification yc,f +t = α + βic,t + γyc,t−l + ζ$c + X µt̃ 1[t = t̃] + uc,t , (7) t̃ where $c is a productivity fixed effect and µt are time effects. We instrument the actual in-migration rate ic,t with the predicted one îc,t . We use τ = 5 in (6); in (7), we use l = 5 with f ranging from 0 to 20.8 Figure 4 plots the semi-elasticity coefficient β at different horizons (f in equation 7) for each of debt, expenditures, and revenue as the dependent variable (always in log per person). Full estimates are reported in tables in Appendix C. The top (bottom) panels give the IV (OLS) estimates, and the gray bands provide 95% confidence intervals from robust standard errors. Evidently, the IV estimates are far from the OLS and, for expenditures and revenue, differently signed (indicating the presence of endogeneity and the necessity of IV). Since debt—whose maturity is around ten years—is mostly predetermined, on impact debt should change little with a valid instrument, and one can see that it does.9 In contrast, ten years ahead there is a statistically significant effect, which 7 See the appendix (Section C.6) for details. We focus on 5-year intervals because the Census of Governments data is most comprehensive in years ending in 1 or 6. 9 In the Census of Governments data, the median (mean) of the ratio of retired to total long-term debt is 0.09 (0.14), implying a maturity of 7 to 11 years. In our Mergent bond data, the median (mean) maturity is 9 (10.1) years. 8 13 is also economically significant: the standard deviation of ic,t is 0.022, so the point estimate implies a one standard deviation increase in ic,t increases debt per person by approximately 31% (= e12.39×0.022 ). There are also significant effects for revenue and expenditures per person, with a semi-elasticity of around 10 and 12.5, respectively. The larger increase in expenditures, paired with expenditures typically being larger than revenue, eventually results in debt growth, consistent with the theory.10 Note: Gray bands denote 95% confidence intervals. Figure 4: Estimated effects due to the overborrowing externality 3.6 Stylized fact #5: Cities default after busts and booms We now seek to establish a key, surprising, and novel finding, which is that cities default after both busts and booms. To this end, our sample is Detroit (MI), Flint (MI), Harrisburg (PA), San Bernardino (CA), Stockton (CA), Vallejo (CA), Chicago (IL), and Hartford (CT), cities that have defaulted or been reported as having financial difficulties in the last few years.11 The data point to heterogeneous paths to default or, more generally, fiscal stress. Figure 5 reveals some cities experience unusually large population growth before default, while others experience large losses. Given that population growth directly reduces debt per person, it may be surprising that default could occur after such large booms. However, the model will generate this through the overborrowing external10 We do not report estimates for deficits because they are very noisy. The noise is due in large part to lagged deficits containing little information, unlike lagged debt stocks or expenditures. 11 News coverage on these and other cities is listed in Appendix A. 14 ity that keeps debt per person high especially when cities are growing. Productivity growth has a similar though less drastic pattern. Unsurprisingly, some cities experience fiscal stress after adverse productivity shocks. However, other cities experience fiscal stress after either stable or positive productivity gains (Chicago, Vallejo, Hartford). From the lens of consumption-smoothing models, the latter observation is surprising: In response to positive productivity shocks, agents should deleverage. However, we will show the model can generate defaults after productivity booms (as well as busts) due to overborrowing. Note: Changes are log differences relative to 1986 except for Chicago, which is relative to 1987; the TFP measure is net of time effects; circles denote periods of acute fiscal stress such as defaults, bankruptcies, or emergency manager takeovers (the last only for Flint); “other cities” is not the universe of cities but covers 64% to 74% of the U.S. population over the time range; fiscal variables are in 2012 dollars per person; the interquartile range is given by the shaded area. Figure 5: Case Study – Cities under Financial Stress Figure 5 also reveals that cities experiencing stress have large debts, debtexpenditure ratios, and budgets. For example, while the average city owed less than $1,000 in 2011, Chicago and Detroit owed about $8,000 and $12,000, respectively. In some cases, financial maneuvering has been used to underplay the amount of debt: Harrisburg’s massive debt in the early 1990s plummeted due to a sale of its incinerator project to a “special district” under its control, while still guaranteeing the debt (Murphy, 2013).12 While the average debt position of all cities looks flat 12 Faulk and Killian (2017) show that having more special districts is positively correlated with increased local government debt, suggesting this type of behavior is not unique. 15 because it is so much smaller than the case study cities’ average, it has increased by 50% since the 1980s. Defaulters also have large expenditures and taxes per person. The model will also generate large expenditures for defaulters, in part because the borrowing constraint limits debt relative to expenditures (as in the data): To loosen this constraint, cities spend more. The volatility of Hartford’s expenditures come from it relying on large, volatile state support amounts (Rojas & Walsh, 2017), which is suggestive of being borrowing constrained.13 3.7 Stylized fact #6: Default risk is elastic to interest rate changes Interest rates, including municipal bond rates, have declined secularly for decades. We will now show that default risk is sensitive to interest rate changes, implying that this secular decline in interest rates has masked the worsening state of local government finances. We will do this by first considering one piece of narrative evidence and second by considering evidence from a vector autoregression model. The left panel of figure 6 reports the spread of BBB, A, and AA municipal bond yield to maturity (YTM) over AAA municipal bonds. To the extent that default risk is tail risk, one should expect increases in default risk to primarily show up in larger BBB spreads, and one can clearly see the spikes in 2009 and 2020. There is one other noticeable spike, which occurs in the latter part of 2013 into 2014. What was its cause? The most likely culprit for this spike is the sharp increase in long-term interest rates known as the taper tantrum. This period, beginning in May 2013 (and seemingly coinciding with the May FOMC statement), saw the ten-year Treasury note rise from 1.6% to 2.9% in four months at a time when the fundamentals of the economy appeared relatively stable (and short-rates were unchanged). This rapid increase in interest rates, both in the ten-year Treasury and in AAA municipal bonds, is evident in the linearly-detrended series in the right panel of figure 6. In our view, these nearly-doubled debt servicing costs of all municipalities resulted in increased default risk that was concentrated specifically in the riskier municipal bond tranches. While this anecdotal evidence may be unsatisfactory, it is also born out in impulse responses from a structural vector autoregression (SVAR). To this end, we estimate a VAR using monthly data on industrial production, the yield-to-maturity for AAA (ytmAAA), the consumer price index, and the spread between the return on municipal bond with category A, AA, or BBB and the ytmAAA. We use the 13 Typical state support is only $100-$300 per person. 16 Figure 6: Muni bond yields and spreads over time ytmAAA as our interest rate measure rather than a Treasury because municipal bonds are typically callable (i.e., they can be prematurely repayed) and are taxexempt. The sample runs from January 2005 and October 2020, and the VAR uses 6 lags. Figure 7 shows the response to an orthogonalized shock to the ytmAAA.14 It is not difficult to see that the exogenous increases in the AAA yield lead to a persistent response of the spreads. Consistent with interest rate increases inducing an increase in tail risk, BBB spreads are the most sensitive to the rise in interest rates, reaching a peak of 8 basis points around five months after the exogenous innovation. That is, the BBB rate almost doubles the initial jump in the risk-free rate (10 basis points). In contrast, the less risky spread AA barely increases, and its response is not statistically significant. 3.8 Summary of empirical findings We have documented a number of empirical findings that indicate local governments have a strong incentive to borrow. We saw this in cities of all types being heavily indebted and close to their borrowing limits. We also saw it in default that occurs after population booms, where large population growth—whose direct effect is to decrease debt per person—was actually associated with stable or growing debt. The overborrowing mechanism we highlighted in Section 2 provides an explanation for why we observe this large indebtedness, and we provided direct evidence on this mechanism, showing larger in-migration rates induce more spending and more 14 More concretely, we use a Cholesky decomposition of the forecast errors from the VAR with the variables ordered as in the main text. Then we plot the IRFs to a shock to the second variable, i.e., the yield to maturity AAA. 17 Spreads AA Spreads A 1.5 4 1 Basis points Basis points 3 0.5 0 2 1 -0.5 0 -1 5 10 15 20 5 Months Spreads BBB 10 10 15 20 Months Yield to Maturity AAA 8 10 Basis points Basis points 6 4 2 0 8 6 4 -2 2 -4 0 -6 5 10 15 20 Months 5 10 15 20 Months Note: Dashed lines corresponds to bootstrapped 68% confidence intervals. Figure 7: Impulse response functions to a risk-free rate (AAA) shock borrowing. Moreover, while default rates and municipal bond risk premia have historically been very low, we showed that they (1) have been trending upward, (2) are sensitive to interest rate movements, and (3) have been depressed by a secular decline in interest rates. Together, these findings suggest it is important, and will be increasingly important, to understand the relationship of debt, default, and migration. To this end, we turn now to the quantitative model, showing that the patterns we have observed in the data can be captured in an internally-consistent way. We will then look at the model’s policy implications and glean some predictions for the future of the U.S. economy. 4 The quantitative model We first provide an overview of the model and its timing. Then, we describe the household, firm, and government problems. Finally, we define equilibrium. 4.1 Overview and timing We model municipalities as a unit measure of islands. Each island consists of a continuum of households (whose measure in the aggregate is one), a local government, and a neoclassical firm. Each local government is a sovereign entity that issues debt, taxes its residents, and provides government services. Households consume, work, and decide whether to stay on the island or migrate to another one. 18 The timing of the model is as follows. At the beginning of the period, all shocks are realized. Upon observing them, households make migration decisions. After migration occurs, the local government chooses its policies, including debt issuance.15 Finally, households make consumption and labor decisions simultaneously with firms while taking prices and government policies as given. 4.2 Households Define the state vector of a generic island as x := (b, n, z, ω), where b is assets per person measured before migration, n is the population before migration, z is the island’s productivity, and ω is a permanent island type that we will call “weather.” The weather variable is a location-specific fixed effect that captures (in reduced form) weather, location, history, or any other immutable, location-specific factor. Including it allows the model to match the variance of population across cities without producing a counterfactually large correlation between productivity and in-migration. We assume z follows a finite-state Markov chain. Households, knowing x, decide whether to stay m = 0 or move m = 1. If they stay, they expect to receive lifetime utility S(x) (specified below). If they move, they are assigned to another island, receive J in expected lifetime utility, and pay an idiosyncratic utility cost φ ∼ F (φ). Their problem is V (φ, x) = max (1 − m)S(x) + m(J − φ). (8) m∈{0,1} The moving decision follows a reservation strategy R(x) with m = 1 when φ < R(x). The utility conditional on staying is S(x) = max c≥0,h≥0 u(c, g(x), h, ω) + βEφ0 ,x0 |x V (φ0 , x0 ) (9) s.t. c + r(x)h = w(x) + π(x) − T (x), where w(x) is the island’s wage; π(x) is the per person profit from the island’s firm; g(x) is government services; T (x) are lump-sum taxes (which we will show is virtually equivalent to using property taxes); and h is a housing good, owned by the firm and rented to households at price r(x). The expectation term Eφ0 ,x0 |x embeds household beliefs about the local government’s policies. If a household decides to move, they migrate to island x at rate i(x) and must 15 This timing means that unanticipated changes in government policies do not immediately alter the population. We view this “sticky population” assumption as reasonable in that migration is a time-consuming process that often involves searching for a new job, finishing a school year, selling an existing home, and finishing rental agreements. 19 stay there for at least one period. The inflow rate at an island of type x is Z exp(λS(x)) i(x) = , nF (R(x))dµ(x) R exp(λS(x))dµ(x) (10) where µ is the invariant distribution of islands.16 This inflow rate is a continuous analogue of a logit-style, discrete choice framework.17 By construction, the measure of households leaving equals the measure entering R R in aggregate, i(x)dµ(x) = nF (R(x))dµ(x). If λ = 0, households are uniformly assigned to each island (“random search”). As λ → ∞, the city with the largest utility S(x) receives all the inflows (“directed search”). Given these inflows, the expected value of moving in equilibrium is Z i(x) J = S(x) R dµ(x), i(x)dµ(x) (11) and the law of motion for population is ṅ(x) = n(1 − F (R(x))) + i(x), (12) where ṅ denotes the population after migration has taken place. 4.3 Firms Each island has a firm that operates a linear production technology zL and owns the island’s housing stock H̄. Alternatively, H̄ may be thought of as the island’s land. We assume H̄ is in fixed supply and homogeneous across islands to prevent adding an extra state variable, but our inclusion of weather ω will capture some of this fixed heterogeneity across islands. Firms solve ṅ(x)π(x) = max zL − w(x)L + r(x)H, (13) L,H≤H̄ taking w and r competitively, and the solution of this problem gives labor demand, Ld (and the housing supply, H = H̄). Since ṅ(x) denotes the number of households remaining after migration and each household inelastically supplies one unit of labor, 16 This rule has the same form as in the two-period model. In particular, one can take I(S(x)) = exp(λS(x)). 17 With a finite number of choices indexed by x, the usual specification would be written maxx S(x) + εx /λ where each εx is i.i.d. with a Type 1 extreme value distribution. Then the probability of choosing x is proportional to exp(λS(x)). The problem that arises with a continuum of choices is that E[maxx S(x) + εx /λ] becomes infinite since εx has unbounded support. What we need in the continuous case is the notion of a Gumbel process. As the technical details for this are quite involved, we discuss how to micro-found (10) in Appendix C. 20 labor market clearing requires Ld (x) = ṅ(x). (14) It is worth making a few observations about the firm problem. First, in equilibrium, per person profits π equal rH/ṅ. Consequently, by making local residents the firm shareholders, we are effectively assuming each gets the rent associated with owning an equal share of the housing/land stock. Second, if there were property taxes, say via τ r(x)H for τ ∈ [0, 1], the taxes would reduce these rents by τ rH/ṅ in the same way that the lump-sum tax T in (9) does. For this reason, we can interpret the lump-sum tax T as a property tax. Last, we have assumed there are no agglomeration or congestion effects in the production function (or that they are both present and cancel).18 Their absence could result in the model under- or over-predicting the relationship between population and productivity. However, the model generates a signficant positive correlation between city density and productivity like that found in the data (Glaeser, 2010). Also, the model has congestion externalities in the form of reduced housing per person and agglomeration effects in that local governments provide a partly nonrival service, as will be discussed shortly. 4.4 Local governments Each local government decides the level of services g ≥ 0 it wishes to provide. These services are potentially nonrival in that, to provide g services to each of the ṅ households, the government must only invest ṅ1−η g units of the consumption good where η ∈ [0, 1] is a parameter. The government pays for these services using tax revenue T ṅ or, potentially, debt issuance. The government chooses a new level of debt per person −b0 , implying a total obligation next period of −b0 ṅ. The discount price it receives on this pledge is q(b0 , ṅ, z, ω), which depends on the debt level, population after migration ṅ (which equals the next period’s population before migration n0 ), productivity, and weather, as all of these potentially influence repayment rates.19 We assume a portion γ is nondefaultable, and we will calibrate it to match the share of unfunded pension obligations. In keeping with the statutory borrowing limits discussed in Section 3, we impose 18 A simple way to introduce agglomeration is with the modified production function zLṅ$ , where N is population and $ > 1. Duranton and Puga (2004) provide microfoundations for this type of agglomeration. 19 Capeci (1994) and Schwert (2017) provide empirical evidence on the link between municipal default risk and interest rates. Our use of short-term debt significantly simplifies the computation as long-term debt models suffer from convergence problems (Chatterjee & Eyigungor, 2012). 21 a borrowing limit − b0 ≤ δg ṅ−η , (15) where δ ∈ R+ controls how tight the limit is. Hence, we require total debt issued in a period −b0 ṅ to be less than a fraction δ of total expenditures g ṅ1−η . While this limit is qualitatively closer to the CA limit than the other states’ limits, government expenditures are very positively correlated with housing rents in the model, so it also effectively captures a limit based on housing value. Additionally, exemptions in many states allow for spending on projects, which this form permits. Given the large variation in laws across states, we will choose δ to match observed debt levels rather than trying to choose it based on statutory law. To define the government’s problem, we need to specify how the economy will respond to deviations in government policies. To this end, we assume that wages and the rental rate adjust dynamically in response to the government policies (d, g, b0 , and T ) clearing the labor and housing markets and that households and firms optimize given those prices and implied profits. Formally, we assume that c, h, r, w, π, Ld always solve the following equations: uc r = uh , w=z , and c + rh = w + π − T ṅπ = zLd − wLd + rH | {z } | {z } Household optimization Firm optimization ṅ = Ld , ṅh = H | {z } . (16) Market clearing Note that bankruptcy d and debt issuance b0 only affects households indirectly through T and g. Letting U denote the indirect flow utility associated with g and T , it is easy to show U (g, T, ṅ, z, ω) = u(z − T, g, H/ṅ, ω). (17) To receive bankruptcy protection in the U.S., local governments must be insolvent and negotiate in good faith with creditors, as discussed in Section 3.1. We interpret these statutory requirements as follows. First, we assume a municipality is insolvent if its debt service exceeds a fixed fraction κ of the expected lifetime income of residents. Specifically, the municipality is insolvent if debt service per person exceeds κz̄(z) for z̄(·) defined recursively by z̄(z) = z + q̄ Ez 0 |z z̄(z 0 ). (18) When insolvent and filing for bankruptcy, the city pays the greater of κz̄(z)ṅ and −γbn, making γ fraction of its debt nondefaultable. Additionally, we assume filing 22 for bankruptcy entails a cost ι proportional to the municipality’s total income. Last, if the city files for bankruptcy when solvent, the city must repay the full debt amount (and the filing cost—consequently, no solvent city files for bankruptcy). Consequently, the total payment required in bankruptcy is p(b, n, z, ω) := max{γ(−bn), min{−bn, κz̄(z)ṅ(b, n, z, ω)}}. (19) Since all the costs and benefits of default occur within a period in our model, the default decision—conditional on endogenous variables—is static and given by d(x) = 1[p(x) + ιz ṅ(x) < −bn]. (20) This modeling of bankruptcy has a number of important features. First, the financial gain possible in bankruptcy hinges on the relative bargaining power of sovereigns and creditors as captured by the parameter κ. Second, the use of lifetime income, rather than current income, means that temporarily low income does not make the city insolvent, which is essential for preventing the model from predicting counterfactually-massive waves of default after large, but short-lived, shocks. Third, solvency hinges not just on the size of the debt stock, but also on its rollover cost. E.g., if real interest rates are zero, i.e., q̄ = 1, then the municipality is never insolvent, and it should not be because the municipality can roll over its debt at zero cost.20 Now we can state the government’s problem as S̃(b, n, z, ω) = max d∈{0,1},g≥0,T ≤z,b0 U (g, T, n0 , z, ω) + βEφ0 ,z 0 |z Ṽ (φ0 , b0 , n0 , z 0 , ω) s.t. gn0(1−η) + q(b0 , n0 , z, ω)b0 n0 = T n0 + (1 − d)bn + d(−p(b, n, z, ω) − ιzn0 ) 0 (21) 0−η −b ≤ δgn 0 n = ṅ(b, n, z, ω) with Ṽ (φ, x) = max{S̃(x), J − φ}. We use the tilde on the value functions to distinguish them from the household value functions in (8) and (9), but equilibrium will require that the values coincide. 4.5 Debt pricing With risk-neutral debt pricing, bond prices must be given by q(b0 , n0 , z, ω) = q̄ Ez 0 |z [1 − d(x0 ) + d(x0 )p(x0 )/(−b0 n0 )]. 20 (22) That is, if they can roll over at the risk-free rate. A perhaps more theoretically appealing modification would be to use the discount rates implied by the sovereign’s current and expected borrowing. However, such a modification also induces convergences problems as is typically encountered in long-term debt models (Chatterjee & Eyigungor, 2012). 23 This links default rates and spreads very tightly, resulting in spreads being smaller than default rates, whereas in the data the reverse is true. Given this discrepancy, we will focus on matching small default rates rather than large spreads. To some extent, we could match both by using an extremely risk-averse pricing kernel. However, to properly match both would require incorporating aggregate risk and, perhaps especially, disaster risk. So we follow the bulk of the sovereign debt literature and use risk-neutrality. 4.6 Equilibrium A steady-state recursive competitive equilibrium is value functions S, V, S̃, Ṽ ; an expected value of moving J; household policies c, h, m; government policies g, T, b0 , d; prices and profit q, w, r, π; labor demand Ld ; a law of motion for population ṅ; and a distribution of islands µ, such that (1) household policies c, h and migration decisions are optimal taking V , S, J, prices and government policies as given; (2) government policies g, T, b0 , d are optimal taking Ṽ , S̃, J, the population law of motion ṅ(x), and prices q as given; (3) firms optimally choose Ld (x) taking w(x), r(x) as given and optimal per person profits are π(x); (4) bond prices are given by equation 22; (5) beliefs are consistent: S = S̃ and V = Ṽ ; (6) the distribution of islands µ is invariant; (7) and J and ṅ are consistent with µ and household and government policies. 4.7 Centralization Proposition 3 shows the government, household, and firm problem may be centralized into a single problem, which we use as the basis for computation: Proposition 3. Suppose Ŝ satisfies Ŝ(b, n, z, ω) = max c>0,g≥0,d,b0 u(c, g, H/n0 , ω) + βEφ0 ,z 0 |z max{Ŝ(b0 , n0 , z 0 , ω), J − φ0 } s.t. c + n0−η g + q(b0 , n0 , z, ω)b0 = z + (1 − d) bn p(b, n, z, w) + d(− − ιz) n0 n0 n0 = ṅ(b, n, z, ω) −b0 ≤ δg ṅ−η (23) with associated optimal policies c(x), g(x), d(x), b0 (x). Then (1) Ŝ is a solution to the household problem, and c is its optimal consumption policy; (2) Ŝ is a solution to the government problem, and g, d, b0 are its optimal policies; and (3) there exists prices r, w such that labor and housing markets clear and firms optimize. 24 In what follows, we will use S in place of Ŝ. We lack a proof of equilibrium uniqueness. However, we investigate uniqueness quantitatively by using 100 randomly drawn initial guesses for the equilibrium objects. Each guess converged to the same equilibrium values, and so at least computationally there is no evidence of indeterminacy at the calibrated values. See Appendix C.4 for more details. 5 Calibration We now discuss the model’s calibration and its fit of targeted and untargeted moments. A model period corresponds to a year in the data. 5.1 Productivity As productivity (TFP) plays a vital role in the model, it is necessary to have a process that accurately captures location-specific productivity dynamics. For our TFP measure, we use real annual payrolls per employee. Let TFP for a county-year pair be denoted zit . We specify log zit = ςi + $t + z̃it (24) and obtain the residual z̃it using a fixed-effects regression. To discretize the fixed effects ςi , we nonparametrically break the estimates into bins corresponding to 010%, 10-50%, 50-90%, 90-99%, and 99-100%. The estimated fixed effects averaged within these bins are −0.34, −0.13, 0.09, 0.37, and 0.65, respectively. We discard the time effects $t as we will only consider steady states or specific paths for aggregate TFP. For the residual TFP z̃it , we use an AR(2) specification, which allows more persistent movements in TFP that better capture decade-long persistent movements in productivity such as what occurred in Detroit and Flint (see Figure 5). Restricting the sample to cities of at least one million residents, the estimated first and second AR coefficients are 0.73 (0.02) and 0.23 (0.03), respectively, with an innovation variance 0.001 (= 2×10−5 ).21 We describe our discretization process in the appendix. 5.2 Preferences and moving costs We set β = 0.96 and assume the flow utility exhibits constant relative risk aversion over a Cobb-Douglas aggregate of consumption, government services, and housing 21 We use large cities to reduce the role of measurement error. 25 plus a taste shifter for weather: (c1−ζg −ζh g ζg hζh )1−σ + ω. (25) 1−σ As ζg and ζh are relatively small, the constant relative risk aversion over consumpu(c, g, h, ω) = tion is approximately σ, which we take to be 2. The free parameters ζg and ζh are estimated jointly, strongly controlling the mean level of government expenditures and housing expenditures, respectively. We take the weather term ω—which is fixed over time for any given region but heterogeneous across regions—to be normally distributed with mean zero (a normalization) and standard deviation σω . We discipline σω by matching the standard deviation of log population across cities. We assume the moving cost φ is distributed as φ, −φ, and Logistic(µφ , ςφ ) with probability pφ /2, pφ /2, and 1 − pφ , respectively. Having the ±φ shock means that, for a sufficiently large φ, every island’s departure rate is in [pφ /2, 1 − pφ /2], which ensures some minimal stability in calibrating the model. We set pφ = 10−4 and take R φ arbitrarily large giving V (φ, x)dF (φ), the expected utility of being in an island with state x, equal to pφ 2 (J + S(x)) + (1 − pφ )(S(x) + ςφ log(1 + e(S(x)−J−µφ )/ςφ )) (26) plus a constant that we offset via a normalization.22 We jointly estimate the parameters controlling moving costs (µφ , ςφ ) and how directed moving is (λ). We identify them using three moments: the mean and standard deviation of out-migration rates, and the productivity-fixed-effect regression coefficient in a regression of log population on productivity residuals z̃i,t , productivity fixed effects ςi , and a constant. We discipline how rival public goods are, as governed by η, using the coefficient of a regression of log population on log expenditures (1.118). 5.3 Borrowing and default We set q̄ to give a risk-free interest rate of 4%, the recent average. We choose the borrowing limit δ to match the data’s total debt to GDP ratio of 0.125, equal to an explicit debt to GDP ratio of 0.089 plus unfunded pension debt to GDP of 0.036. The debt measure is gross in that we do not deduct the value of any assets because assets cannot be seized in a Chapter 9 bankruptcy (as we noted in Section 22 The constant is φpφ /2. We subtract β times it from flow utility in (25) each period. 26 3.1). The default cost κ is chosen to match a 0.03% default rate.23 To calibrate the cost of bankruptcy ι, we use information from Detroit’s bankruptcy and arrive at ι = 0.125%.24 5.4 Fit of targeted and untargeted moments Table 1 reports the targeted and untargeted statistics alongside the jointly calibrated parameter values. The model closely matches all of the targeted statistics. The estimated debt limit, δ, allows cities to borrow up to 153% of their expenditures, which is not very far from CA’s statutory limit of 100% (plus exceptions). Targeted Statistics R Default rate (×100) ddµ R R Debt / GDP −bndµ/ z ṅdµ R R Gov. expenditures / GDP g ṅR1−η dµ/ R z ṅdµ Housing expenditures / GDP rH̄dµ/ z ṅdµ Std. deviation ofRlog n *Out rate mean F (R)/ndµ *Out rate st. dev. *Population reg. coef., log z FE Regression coef., log expenditures on log n Data Model Parameter Value 0.030 0.125 0.084 0.125 1.843 0.065 0.023 4.224 1.118 0.028 0.125 0.082 0.125 1.848 0.064 0.022 4.222 1.034 κ δ ζg ζh σω µφ ςφ λ η 0.006 1.528 0.069 0.112 0.331 22.445 6.452 0.603 0.316 Untargeted Statistics Data Model Autocorrelation of log n Std. deviation of net migration rates Correlation of log expenditures and log n Std. deviation of log expenditures *In rate mean *In rate st. dev. *In rate reg. coef., log z *Out rate reg. coef., log z 0.999 0.021 0.858 2.388 0.065 0.022 0.000 -0.006 1.000 0.023 0.995 1.921 0.066 0.030 0.023 -0.005 Note: * means the data is county-level; the regressions are specified in Section C.5. Table 1: Calibration targets and parameter values Utility from weather plays a large role in location decisions, with a rough calculation giving the lifetime consumption equivalent variation of permanently moving from the median weather ω = 0 to 2σω at 66%. The importance of weather for utility helps the model match the very low (in fact, negative) correlation between in-migration and productivity and, simultaneously, the large dispersion in population. The large value for µφ with a correspondingly large ςφ makes out-migration 23 While high for Moody’s rated bonds, it is not high for the Mergent dataset (see Figure 1). Detroit’s cost the city $178M on its $18.5B bankruptcy (1% per unit of debt) (Reuters, 2014). Using Detroit’s 1% legal cost per unit of debt and a 12.5% debt-gdp ratio, we set ι = 0.125% = 1% ∗ 12.5%. 24 27 largely dependent on moving cost shocks rather than fundamentals like productivity, letting the model match the weak relationships between productivity and migration rates. The model gets most of the untargeted predictions qualitatively correct while missing on a few statistics. The model recreates the very slow population adjustments seen in the data with log population autocorrelation exceeding the data’s 0.999. It also matches the small correlations between migration rates and productivity and migration rates and population. Last, it reproduces the dispersion in net migration rates and the large dispersion in government expenditures. Table 4 in the appendix provides additional untargeted moments. 6 Quantitative results With the calibrated model, we first show the model can replicate all the stylized facts we established in Section 3 (except for the increasing trend, which the model is not designed to address).25 Having established the empirical stylized facts hold in the model, we then turn to the models predictions to answer two key questions. First, why were there so few defaults in the Great Recession? Second, what can we expect from the reallocation and productivity shocks caused by COVID-19? 6.1 Stylized facts in the model In establishing the stylized facts hold in the model, we begin with the key model mechanism, showing that in-migration induces more expenditures, revenue, and debt per person. We then show cities of all types are heavily indebted; that they are close to their borrowing limits; and that the model generates default after booms and busts. Finally, we establish that default risk is sensitive to interest rate changes. 6.1.1 The model mechanism in action Figure 8 plots the response of a few key variables in response to two types of shocks: a shock to in-migration rates and a shock to interest rates. To model an exogenous in-migration rate shock, we proportionately scale up or down the inflows i(x) in (10) by 1+ī, and we assume that ī decays annually at rate 0.956, based on a fiveyear autocorrelation of in-migration rates equal to 0.8. The results for a positive (negative) shock are displayed in the blue solid (dashed) lines. Consistent with the empirical evidence, more in-migration leads to increases in debt, expenditures, and 25 The increase in indebtedness and default risk over time, which is paralleled in the consumer context, likely does not have a simple explanation. 28 revenue per person. Additionally, the model predicts that higher in-migration rates reduce default rates on impact—reflecting the direct effect of less debt per person— but increase default rates in the future as the government’s debt grows. With the increased default risk, spreads increase. Less obvious is that out-migration rates fall on impact but rise a few years out. Figure 8: Model mechanisms More in-migration increases borrowing and expenditures in the quantitative model for two first-order reasons and two second-order reasons. First, as highlighted in Section 2, the overborrowing externality increases, which has a first-order effect on debt accumulation. However, it also has a second-order effect in borrowingconstrained cities increasing their expenditures to slacken the borrowing constraint. The second first-order effect is that a larger population optimally creates substitution into expenditures because the public goods are partially nonrival. This also has a second-order effect in that increased expenditures slacken the borrowing constraint, enabling more debt accumulation. Evidently, these forces work in conjunction and help the model generate a large correlation between debt and expenditures like in the data. 6.1.2 All city types are heavily indebted and close to borrowing limits The model replicates the stark relationships among debt, expenditures, and borrowing constraints that we showed in Sections 3.3 and 3.4. This can be seen most clearly in Figure 9, which provides a scatter plot of cities debt and expenditures along with the borrowing constraint. It shows cities of all types—large and small, productive 29 and unproductive, with great and poor weather—are indebted and close to their borrowing limits, which were our number two and three stylized facts, respectively. Note: circle areas are proportional to population. Figure 9: Model distribution of cities relative to their borrowing limit Why are all cities indebted and close to their borrowing limits? The answer lies in the high level of in-migration rather than in its dynamics. In the data and model, in-migration rates at a local level are around 6.5%. Loosely speaking, this means for every dollar of debt issued per person, 6.5 cents are paid for by new entrants. This is such a large “discount” on debt issuance that it dominates any consumption-smoothing motives, including a desire to save for a rainy day. The model does have a restraining force in it, which is that debt accumulation deters entrants to the city (the term ∂n2 /∂b2 in equation 2). However, to match the data’s large out-migration rates, most of the migration decision is attributed to idiosyncratic, person-specific factors—as seen in large µφ and ςφ —rather than local fundamentals. Consequently, the elasticity is quite small, and the model is almost bifurcated: migration strongly affects local government decisions, but local government decisions weakly affect migration. Moreover, this is reinforced in equilibrium because all governments over-accumulate debt, which depresses any incentive to move to a more fiscally responsible city—such cities simply do not exist in the model. Hence, the main restraining force in the model is essentially inoperable, resulting in cities of all types being nearly indebted as they can possibly be. 30 6.1.3 Cities default after busts and booms We now show the model generates boom and bust defaults, like in the data. We consider default episodes, looking at windows around the time of default. These are displayed in Figure 10 where we have broken default episodes into three cases: an average default event (blue line), a default during a technology boom (red dashed line), and a default during a technology bust (green dotted line). Formally, a boom (bust) is defined as having log productivity growth in the ten years preceding default above the 75th (below the 25th) percentile, and we compute the respective averages in the top and bottom quartile to construct the time series. Unsurprisingly, the model does generate default after long periods of productivity decline. But importantly, the “boom” defaults do in effect follow periods of substantial productivity growth—a feature that is not necessitated by our definition of booms. Considering first average default episodes, one sees they coincide with slightly growing productivity followed by a sharp decline in productivity (a drop close to 10%) at the time of default. Additionally, the shock is such that the drop in TFP is expected to last a long time. Although on average population increases slightly predefault, cities see their population decline postdefault, losing about 5% of their inhabitants within five years. Because the mean default episodes average over boom and bust defaults, they hide a large amount of heterogeneity. Looking at bust defaults, one finds a prolonged decline in population leading up to default, qualitatively similar to the experience of Flint and Detroit. Facing this shrinking population and persistently adverse productivity, the sovereign initially holds taxes and debt per person stable and cuts expenditures, resulting in a modest primary surplus. In the few years before default, interest rates increase, reflecting the increased default risk. Following default, the municipality deleverages by sharply reducing expenditures. Looking at boom defaults, the population and productivity growth is strong until just a year before default, like in Vallejo, CA. With the boom, the cities do not pay down debt, and even run a substantial deficit shortly before default. Consequently, debt per person grows as interest payments pile up, and this is despite substantial population growth that (else equal) reduces debt per person. The debt growth is paired with a noticeable increase in expenditures and taxes. Interest rates trend upward, showing that the city is taking on increasing (albeit small) amounts of risk. When a substantial negative productivity shock hits, in-migration plummets and debt per person increases, triggering default. 31 Log pop. change TFP change In-migration rate Out-migration rate Income Debt Expenditures Debt-expend. ratio Taxes Primary deficit Interest rates Overborrowing term Note: all financial variables are 2010 dollars per person. Figure 10: Default episodes While boom defaults are triggered by a decline in productivity, a necessary ingredient is that the city must be leveraged enough to make default worthwhile, which is where overborrowing plays a crucial role in generating boom defaults. To see this, consider the “overborrowing term” (1 − o)/(1 − o + i). For boom defaults, this falls to as low as 0.85, implying a massive 15% discount on debt issuance. This overborrowing incentive dwarfs the usual consumption smoothing motive, keeping cities in debt even after long periods of growing productivity and leaving them vulnerable to adverse shocks. 6.1.4 Default risk is sensitive to interest rate movements We close this section by establishing the last stylized fact, which is that default risk is sensitive to interest rate. We turn again to Figure 8, which displays the response of a 1 percentage point risk-free rate increase (decrease) in the red circled (dashed) lines. The effect decays at rate 0.63 (which comes from a regression of detrended AAA yield to maturity on its twelve-month lag). On impact, the increase in rates increases the default rate of bonds appreciably, as well as spreads. Default rates increase because larger interest rates increase debt service costs, which makes more cities insolvent and able to benefit from bankruptcy. 32 Formally, a lower q̄ lowers z̄(z), reducing—for insolvent or nearly insolvent cities— required debt payments and inducing bankruptcies. Spreads increase because of higher future default risk that sovereigns only partially undo through deleveraging. Of note, the plotted spreads series is ex-ante in that it is a function of q(b0 , x). Consequently, the surprise losses incurred from the unanticipated shock are not displayed. Cities rely on a combination of expenditure cuts and tax increases to deleverage. Using this combination is optimal from a utility perspective because c and g are complements, so reducing both g and c modestly is superior to letting one or the other fully absorb the impact. The impact on migration is not noticeable because (1) the interest rate only change taxes by a small magnitude, and (2) the interest rate change has no substantial redistributive effect since all cities are indebted. 6.2 The Great Recession Having established the empirical stylized facts hold in the model, we now try to answer why there were so few municipal defaults in the Great Recession. We will model the Great Recession as a perfect foresight transition following a one-time unanticipated shock. The Great Recession resulted in a large and essentially permanent drop in real GDP per capita. Relative to the pre-2009 linear trend, the drop was 12% on impact and grew to almost 20% by 2020Q1, as seen in Figure 15 in Appendix D. The full impact of this decline, however, was not felt by local governments because of the American Reinvestment and Recovery Act of 2009 (ARRA). The ARRA provided large federal transfers that bolstered state and local government (SLG) revenue. In fact, as we document in Appendix D, transfers caused SLG total revenue to GDP to rise until 2011, despite large declines in tax revenue. In light of this, we will assume a pass-through from GDP declines to model TFP declines that is less than 100%. Specifically, since local government budgets are 8.2% of GDP and state to local transfers were 3.2% from 2006 to 2009, we’ll assume a pass-through of 61% = 1 − 3.2%/8.2%. Combining the declines in real GDP deviations from trend with a 61% pass-through results in the aggregate TFP $t declines reported in Table 8 in the appendix. Of course, the Great Recession also exhibited steep declines in risk-free real interest rates. These real rates, measured using five-year TIPS yields, fell by more than 3pp from 2006 to 2012 as can be seen in Appendix D. Given the empirical 33 sensitivity of default to interest rates that we already established, we incorporate these declines as well. In particular, we compute two transitions, one incorporating these declines as reported in Table 8, and one holding the real rate constant. Figure 11 plots the response of key variables over the transition path. The most glaring observation is that the default rate initially rises from a few basis points to 2pp in the absence of real rate declines (blue lines in Figure 11). Is this hypothetical prediction reasonable? As seen in Figure 1, in the early days of the 2007-2008 financial crisis, spreads of higher-yield municipal bonds relative to lower-yield were as little as 1%. As the spillover into unemployment and a housing crisis became clear, the spreads rose to 5% in anticipation of a wave of default. While default rates rose—from a low of 0.01% to as much as 0.16%—they never reached anything close to 2%. From the model lens, this is because the massive drop in real rates reduced default rates by a huge amount. In contrast to default rates, spreads remain low and stable for many years. Part of this is a time aggregation issue: spreads are purely forward-looking, so the high default rates on impact are decoupled from spreads on impact. More substantively, municipalities deleverage substantially, reducing debt by about 10% absent any interest rate decline. They do this by increasing taxes noticeably when the shock hits. This painful deleveraging runs counter to common policy advice that says governments should borrow more in downturns. And, indeed, governments have incentive to consumption smooth by borrowing more. Nevertheless, this effect is dominated by (1) spreads moving against them if they do not deleverage; (2) the exogenous borrowing limit, which effectively tightens in response to a decrease in optimal expenditures (see equation 15); and (3) effective impatience due to the overborrowing externality. A final aspect from the transition is that arrival rates (and departure rates, not pictured) are little changed from the shock. This is due to the nature of the aggregate shock, which effects all cities proportionally. We will revisit this when analyzing the COVID shock. 6.3 COVID-19 We now turn to address an important contemporaneous question: what are the midterm and long-term effects on municipalities from COVID-19 and the unprecedented, coincident policy interventions? Unfortunately, the ongoing uncertainty and lack of multiyear data on outcomes means we will not be able to provide a precise answer. 34 Figure 11: Great Recession transitions with and without interest rate reductions Our approach is rather to feed in a few shocks that capture the defining features of 2020-2021: a drop in GDP, a decline in real interest rates, and a motive to relocate out of large cities. Given the rapid onset of COVID-19, we will also consider the consequences of cities being able to only sluggishly adjust their tax policies. We’ll consider each of these in turn. 6.3.1 Aggregate productivity decline We begin by assuming that aggregate productivity exp($) falls by 10%. Subsequently, we assume productivity reverts to its steady state at a rate of 0.95, that is, $t = 0.95$t−1 . Given the uncertainty behind COVID-19 and its impact on the economy, we view this persistence as a reasonable starting point. Following the shock, default rates rise by an order of magnitude, to nearly 0.4% (blue line in Figure 12). Like in the Great Recession, despite the consumption smoothing motive, taxes increase on impact—we will consider an alternative case where they cannot—to bring debt to a more sustainable level. Two years after the shock, debt has shrunk by 7 percent and remains protracted for many years following the drop in productivity. The reasons are as before: an effectively tightened borrowing limit from the optimal g decrease and spreads moving against the sovereign if they do not deleverage. Also as before, spreads remain low, mainly due to their forward-looking nature. (When we look at exogenous taxes, however, substantial recovery rates will play a key role in preventing large spread increases.) 35 Figure 12: Transitions with and without exogenous taxes and interest rate declines 6.3.2 Exacerbating factor: Tax inflexibility Given the rapid onset and unanticipated nature of the COVID-19 crisis, local government budgets could not adjust immediately to the shock. We now consider the possibility that taxes cannot adjust on impact. Specifically, we will think about the case where the first period tax rates must be proportional to housing expenditures, i.e., T = τ rh. (We will continue to assume that residents take T —rather than τ —as given.) In this case, it is not hard to show that c = z − T and uc /uh = 1/r must still hold. Combining these equations, one has c as a function of τ , z, and parameters, with g given as a residual (conditional on a choice of b0 and d) from the local government’s budget constraint. We choose τ = 0.696 to deliver similar allocations to the benchmark.26 As the optimal response to the shock is to raise taxes, the inability to do so leads to worse outcomes (red dashed lines in Figure 12). Here, default rates increase on impact but are much higher the next year as cities did not raise taxes and deleverage as much as they should have. Reflecting this, spreads also increase noticeably. The default rate increase of 180 bp is far larger than the 10 bp increase in spreads due to the large recovery rates, which are roughly 94% (= 1 − 10/180). 26 Note T ṅ is used to finance government spending g ṅ1−η and debt service, which on average is (1/q̄ − 1)(−ḃṅ). To not be too distortionary, we want T ṅ = τ rhṅ = τ r roughly equal to g ṅ1−η + 1−η ṅ (1/q̄ − 1)(−ḃṅ). Expressed relative to GDP Y , we need τ rh = gṅY + (1/q̄ − 1) −Yḃṅ . These Y relative-to-GDP quantities are targeted, so we want τ × 0.125 = 0.082 + 0.04 × 0.125, which implies τ = 0.696. Note that this is 70% of housing expenditures, not property values. 36 6.3.3 Mitigating factor #1: Real interest rate declines As in the Great Recession, real interest rates declined substantially in 2020 and 2021. To capture this, we assume that the real interest rate falls by 1.5 pp, onetime and permanently. (While the permanent aspect is an exaggeration, given the massive decline in thirty-year rates, this may well approximate expectations.) The decline in real rates drastically reduces default even when taxes cannot increase after the shock (green circled lines in Figure 12). The reasons are the same as before and are discussed in Section 6.1.4. Consequently, despite the very large and sudden shock, it is likely default rates will remain low due to the accommodation of real interest rates. 6.3.4 Mitigating factor #2: Redistributive effects In addition to these aggregate shocks, COVID-19 has also reduced the attractiveness of living in densely populated regions. Unlike the other shocks, this has first-order redistributive effects. In the model, we can capture this by reducing the value of being in “high-weather” states—i.e., states with ω large. Because we discretized our ω state into three point {ω, 0, ω}, we implement this idea by reducing ω by a given percentage. This decline lasts for a year and then recovers at the rate 0.95. Figure 13: Negative TFP transitions with and without redistributive weather shock Figure 13 reports the transitions. A key takeaway is that larger declines in ω lead to lower default rates. As people move from higher population to lower population cities, debt per person at the lower population cities declines. Because, as we already established, smaller to medium-sized cities are more likely to default than 37 the largest cities, this shift from high population to lower population cities induces a composition effect that lowers default rates. While default rates, measured as the rate at which cities default, decline, the migration shift from larger to smaller cities increases the typical size of bankrupt cities and the size of filings. The first claim follows immediately from the log population conditional on default in the bottom right panel. The second follows from the first in conjunction with an almost constant debt per person conditional on default (as seen in the bottom, middle panel). Hence, while default rates of cities in general are smaller, we may see defaults by larger cities involving correspondingly more debt. 7 Conclusion Borrowing, migration, and default are intimately connected. Theoretically, we demonstrated that migration tends to result in overborrowing. Empirically, we documented that defaults can occur after booms or busts in labor productivity and population, in-migration leads to indebtedness, that many cities appear to be at or near stateimposed borrowing limits, and that default risk is highly sensitive to interest rate movements. Our quantitative model was able to capture these stylized facts, in large part due to the overborrowing externality. Given the increase in debt and default over time, and the fiscal stress created by the Great Recession and continuing with COVID-19, understanding regional borrowing, default, and migration will remain a high priority. References Aguiar, M., Chatterjee, S., Cole, H., & Stangebye, Z. (2016). Quantitative models of sovereign default crises (J. Taylor & H. Uhlig, Eds.). In J. Taylor & H. Uhlig (Eds.), Handbook of macroeconomics. North-Holland. Alessandria, G., Bai, Y., & Deng, M. (2020). Migration and sovereign default risk. Journal of Monetary Economics, 113, 1–22. Altonji, J., & Card, D. (1991). The effects of immigration on the labor market outcomes of less-skilled natives, In Immigration, trade, and the labor market. National Bureau of Economic Research. Arellano, C. (2008). Default risk and income fluctuations in emerging economies. American Economic Review, 98 (3), 690–712. Armenter, R., & Ortega, F. (2010). Credible redistributive policies and migration across US states. Review of Economic Dynamics, 13 (2), 403–423. Brown, J. R., & Wilcox, D. W. (2009). Discounting state and local pension liabilities. American Economic Review: Papers & Proceedings, 99 (2), 538–542. 38 Bruce, N. (1995). A fiscal federalism analysis of debt policies by sovereign regional governments. The Canadian Journal of Economics, 28, S195–S206. Caliendo, L., Parro, F., Rossi-Hansberg, E., & Sarte, P.-D. (2017). The impact of regional and sectoral productivity changes on the U.S. economy. The Review of Economic Studies. https://doi.org/10.1093/restud/rdx082 Capeci, J. (1994). Local fiscal policies, default risk, and municipal borrowing costs. Journal of Public Economics, 53, 73–89. Card, D. (2001). Immigrant inflows, native outflows, and the local labor market impacts of higher immigration. Journal of Labor Economics, 19 (1), 22–64. Chatterjee, S., & Eyigungor, B. (2012). Maturity, indebtedness and default risk. American Economic Review, 102 (6), 2674–2699. Coen-Pirani, D. (2010). Understanding gross worker flows across U.S. states. Journal of Monetary Economics, 57 (7), 769–784. Davis, M. A., Fisher, J. D., & Veracierto, M. (2013). Gross migration, housing, and urban population dynamics (Working Paper WP 2013-19). Federal Reserve Bank of Chicago. Duranton, G., & Puga, D. (2004). Micro-foundations of urban agglomeration economies (J. V. Henderson & J.-F. Thisse, Eds.). In J. V. Henderson & J.-F. Thisse (Eds.), Handbook of regional and urban economics. Amsterdam, North-Holland. Eaton, J., & Gersovitz, M. (1981). Debt with potential repudiation: Theoretical and empirical analysis. Review of Economic Studies, 48 (2), 289–309. Epple, D., & Spatt, C. (1986). State restrictions on local debt. Journal of Public Economics, 29, 199–221. Farhi, E., & Werning, I. (2014). Labor mobility within currency unions (Working Paper). Harvard University. Faulk, D., & Killian, L. (2017). Special districts and local government debt: An analysis of “old northwest territory” states. Public Budgeting & Finance, 112–134. Glaeser, E. L. (Ed.). (2010). Agglomeration economics. The University of Chicago Press. Greenwood, M. J. (1997). Chapter 12 internal migration in developed countries, In Handbook of population and family economics. Elsevier. https://doi.org/https://doi.org/10. 1016/S1574-003X(97)80004-9 Hatchondo, J. C., & Martinez, L. (2009). Long-duration bonds and sovereign defaults. Journal of International Economics, 79, 117–125. Haughwout, A. F., Hyman, B., & Shachar, O. (2021). The option value of municipal liquidity: Evidence from federal lending cutoffs during covid-19 [Available at SSRN: https://ssrn.com/abstract=3785577 or http://dx.doi.org/10.2139/ssrn.3785577]. Kennan, J. (2013). Open borders. Review of Economic Dynamics, 16, L1–L13. Kennan, J. (2017). Open borders in the European Union and beyond: Migration flows and labor market implications [Mimeo]. Mimeo. 39 Kennan, J., & Walker, J. R. (2011). The effect of expected income on individual migration decisions. Econometrica, 79 (1), 211–251. Kleine, R., & Schulz, M. (2017). Service solvency: An analysis of the ability of michigan cities to provide an adequate level of public services (MSU Extension White Paper). Michigan State University Extension. Lucas, R. J. (1972). Expectations and the neutrality of money. Journal of Economic Theory, 4 (2), 103–124. Mayo, R., Moore, R., Ricks, K., & St. Onge, H. (2020). State and local government finances summary: 2018. U.S. Census Bureau. Mendoza, E. G., & Yue, V. Z. (2012). A general equilibrium model of sovereign default and business cycles. The Quarterly Journal of Economics, 127 (2), 889–946. Moody’s. (2013). US municipal bond defaults and recoveries, 1970-2012. Moody’s Investors Service. Murphy, E. M. (2013). Securities exchange act of 1934, release no. 69515 [Accessed: 02-212018]. Myers, S. (2019). Public employee pensions and municipal insolvency [Mimeo]. Mimeo. Novy-Marx, R., & Rauh, J. (2011). Public pension promises: How big are they and what are they worth? The Journal of Finance, 66 (4), 1211–1249. Novy-Marx, R., & Rauh, J. D. (2009). The Liabilities and Risks of State-Sponsored Pension Plans. Journal of Economic Perspectives, 23 (4), 191–210. Rauh, J. D. (2017). Hidden debt, hidden deficits: 2017 edition (Technical Report). Hoover Instiution. Reuters. (2014). Fees, expenses for detroit bankruptcy hit nearly $178 million [Accessed: July 13, 2021]. Roback, J. (1982). Wages, rents, and the quality of life. Journal of Political Economy, 90 (6), 1257–78. Rojas, R., & Walsh, M. W. (2017). Hartford, with its finances in disarray, veers toward bankruptcy [Accessed: 02-20-2018]. Rosen, S. (1979). Wage-based indexes of urban quality of life (P. N. Miezkowski & M. R. Straszheim, Eds.). In P. N. Miezkowski & M. R. Straszheim (Eds.), Current issues in urban economics. Baltimore, MD, Johns Hopkins University Press. Schultz, C., & Sjöström. (2001). Local public goods, debt and migration. Journal of Public Economics, 80, 313–337. Schwert, M. (2017). Municipal bond liquidity and default risk. The Journal of Finance, 72 (4), 1683–1722. https://doi.org/https://doi.org/10.1111/jofi.12511 Stempel, J. (2016). Detroit defeats pensioners’ appeal over bankruptcy cuts. [Accessed: February 26, 2020]. Tiebout, C. M. (1956). A pure theory of local expenditures. Journal of Political Economy, 64 (5), 416–424. https://doi.org/10.1086/257839 40 United States Courts. (2018). Chapter 9 – bankruptcy basics [Accessed: 2018-02-06]. Van Nieuwerburgh, S., & Weill, P.-O. (2010). Why has house price dispersion gone up? Review of Economic Studies, 77, 1567–1606. 41 A Additional data details [Not for Publication] This appendix describes our data sources, definitions of key variables, and cleaning procedures in Sections A.1, A.2, and A.3. Section A.4 gives a collection of statutory borrowing limits, and Section A.5 records newspaper headlines on local government finances. A.1 Census County Business Patterns data To construct TFP measures, we use data from the Census’ County Business Patterns (CBP) database from 1986 to 2014. The main measures we use are the payroll variable ap (converted to constant dollars using the standard CPI series obtained from FRED) and the mid-March employment variable emp, along with the FIPS codes. In the CBP database, missing or bad values are assigned a value of zero, so we treat ap and emp as missing whenever they are 0. Our overall productivity measure zit is ap/emp. The data includes disaggregated employment levels by sectors (NAICS and SIC), so we keep only the observations corresponding to aggregates. The panel includes 91,800 year-county nonmissing observations for zit . A.2 Annual Survey of State & Local Government Finances data For our data on government finances and population, we use the Annual Survey of State & Local Government Finances (IndFin) compiled by the Census Bureau. Every five years (in years ending in two or seven), the aim is to construct a comprehensive record of state and local finances. (In practice, surveys are sent out for most cities and not all are returned, but the coverage is good enough to cover 64-74% of the U.S. population depending on the year.) In intervening years, a nonrepresentative sample is selected from the population. Some of the larger cities are “jacket units,” and instead of surveys, the Census sends its own workers to record the data. The data are aggregated at different levels, with “cities”—i.e., municipalities and townships— counties, and states. We consider two samples: one corresponding to cities (typecode equal to 1 or 2) and the other to data aggregated at a county level (the aggregation of typecode values 1 through 5). Some of the data go back to 1967. However, the first population records begin in 1986 (survey year 1987), so we restrict ourselves to the 1987-2012 survey years. The population is not recorded in each year (the data for it does not necessarily correspond to the survey year but are given by yearpop), and so we construct estimates. We restrict the sample so that each city/county has at least two popula- 42 tion measures. We fill in missing observations using linear interpolation of the log population. We also allow for some extrapolation, but do not allow extrapolation beyond five years. The raw sample consists of 390,557 year-county or year-city observations. We then use the sample restrictions as described in Table 2. We compute implied interest rates via the interest paid during the year over the total debt, short and long term: 100* totalinterestondebt / (stdebtendofyear + totallongtermdebtout). All financial variables are converted to real 2012 dollars using the CPI. Sample selection condition Obs. Raw Require nonmissing name, require no id changes (idchanged=0), drop if “no data” (zerodata>0) Require yearpop not missing, drop if datayearcode=“N”, require at least two population observations Drop observations with nonmissing investment annual returns on trust funds exceeding 30% Dropping missing population estimates Dropping observations where population growth rates could not be estimated Dropping Louisville, KY observations before 2003 Require annual population growth rates of less than 25% Require revenue per person of less than $25,000 Require debt per person of less than $30,000 Require accounting identity for the evolution of long-term debt to nearly hold Require estimated interest rates be less than 40% annually 390557 387963 387219 386723 386515 386511 386494 386494 386090 376778 363365 362122 Table 2: Sample selection in IndFin A.3 Migration data Our migration data comes from the IRS. Up to 2010, we use the county-to-county migration flows as harmonized by Hauer and Byars (2019); from 2011 on, we construct our own. A.4 State-imposed borrowing limits Table 3 reports state-imposed borrowing limits for a collection of states. 43 State Limit Known exceptions CA Indebtedness less than revenue that year Limits range from 0.5%-3% of assessed value (1/3 of market value) 2% of assessed value 5% of property valuation last year 10% (5% for townships and school districts) of assessed value “Net debt” less than 3% of market value of taxable property Roughly 10% of the property valuation over the previous five years Net indebtedness less than 5.5% (or 10.5% with vote) of tax valuation 5% of taxable property value Authorization by referendum, special projects, and public school spending. Schools have debt limits of 13.8% value of taxable property. Some revenue bonds (see note). IL IN MA MI MN NY OH WI Approval of voters for more school district debt; most revenue bonds. Charter can increase to 3.67%, “first class” cities have a 2% limit. Debt related to water supplies and sewers. Self-supporting projects for water facilities, airports, etc. Schools have a 10% debt limit, may issue $1 million without approval. Sources are as follows: CA’s is Harris (2002); MA’s is MCTA (2009); MN’s is Bubul (2017); IL, IN, MI, and WI’s is Faulk and Killian (2017); NY’s is ONYSC (2018); OH’s is OMAC, 2013, p. 50. Revenue bonds are municipal bonds that are paid by revenue from a particular project (they are non-GO bonds). Table 3: Sample of statutory borrowing limits by state 44 A.5 Cities making headlines Here, we document some cities/municipalities experiencing financial difficulties as reported by different media outlets. In quotations, we include excerpts of these news. To retrieve the source, the interested reader should click on the city’s name. U.S. Virgin Islands: “With just over 100,000 inhabitants, the protectorate now owes north of $2 billion to bondholders and creditors. That is the biggest per capita debt load of any U.S. territory or state - more than $19,000 for every man, woman and child scattered across the island chain of St. Croix, St. Thomas and St. John. The territory is on the hook for billions more in unfunded pension and healthcare obligations.” Chicago: “Chicago’s finances are already sagging under an unfunded pension liability Moody’s has pegged at $32 billion and that is equal to eight times the city’s operating revenue. The city has a $300 million structural deficit in its $3.53 billion operating budget and is required by an Illinois law to boost the 2016 contribution to its police and fire pension funds by $550 million. Cost-saving reforms for the city’s other two pension funds, which face insolvency in a matter of years, are being challenged in court by labor unions and retirees. State funding due Chicago would drop by $210 million between July 1 and the end of 2016 under a plan proposed by Illinois Governor Bruce Rauner.” Detroit: “‘It is indeed a momentous day,’ U.S. Bankruptcy Judge Steven Rhodes said at the end of a 90-minute summary of his ruling. ‘We have here a judicial finding that this once-proud city cannot pay its debts. At the same time, it has an opportunity for a fresh start. I hope that everybody associated with the city will recognize that opportunity.’ In a surprise decision Tuesday morning, Rhodes also said he will allow pension cuts in Detroit’s bankruptcy. He emphasized that he won’t necessarily agree to pension cuts in the city’s final reorganization plan unless the entire plan is fair and equitable. ‘Resolving this issue now will likely expedite the resolution of this bankruptcy case,’ he said.” Flint: “Flint once thrived as the home of the nation’s largest General Motors plant. The city’s economic decline began during the 1980s, when GM downsized. In 2011, 45 the state of Michigan took over Flint’s finances after an audit projected a $25 million deficit. In order to reduce the water fund shortfall, the city announced that a new pipeline would be built to deliver water from Lake Huron to Flint. In 2014, while it was under construction, the city turned to the Flint River as a water source. Soon after the switch, residents said the water started to look, smell and taste funny. Tests in 2015 by the Environmental Protection Agency (EPA) and Virginia Tech indicated dangerous levels of lead in the water at residents’ homes.” Hartford: “Hartford’s biggest bond insurer said it had offered to help the city postpone payments on as much as $300 million in outstanding debt, in a move designed to help prevent a bankruptcy filing for Connecticut’s capital. Under Assured Guaranty’s proposal, debt payments due in the next 15 years would instead be spread out over the next 30 years without bankruptcy or default. The city would issue new longer-dated bonds and use the proceeds to make the near-term debt payments.” Puerto Rico: “The Puerto Rican government failed to pay almost half of $2 billion in bond payments due Friday, marking the commonwealth’s first-ever default on its constitutionally guaranteed debt.” New Jersey and other states: “The particular factors are as diverse as the states. But one thing is clear: More states are facing financial trouble than at any time since the economy began to emerge from the Great Recession, according to experts who say the situation will grow more dire as the Trump administration and GOP leaders on Capitol Hill try to cut spending and rely on states to pick up a greater share of expensive services like education and health care.” On the State Crisis: “States and cities around the country will soon book similar losses because of new, widely followed accounting guidelines that apply to most governments starting in fiscal 2018 – a shift that could potentially lead to cuts to retiree heath benefits.” Illinois: “After decades of historic mismanagement, Illinois is now grappling with $15 billion of unpaid bills and an unthinkable quarter-trillion dollars owed to public employees when they retire.” 46 B Computation [Not for Publication] This appendix describes the computational algorithms used. B.1 Discretization of the AR(2) process To discretize the AR(2), we cast it in the form of a VAR(1) and then follow Gordon (2021). This method increases efficiency by dropping low-probability states and then suitably adjusting the transition matrix.27 We use Tauchen (1986) as the underlying tensor-grid method with a “coverage” (i.e., a support) of two unconditional standard deviations. The algorithm delivers 58 discrete (z̃it , z̃it−1 ) states. These, combined with the five permanent productivity states and three weather states, make 870 exogenous states. B.2 Equilibrium computation To compute the equilibrium, we guess on two objects: the expected utility conditional on moving J and the average inflows over a “normalization” term for the logit probabilites, R i := nF (R(x))dµ(x) . exp(λ(S(x) − maxx S(x))dµ(x) (27) Subtracting off maxx S(x) prevents overflows in the computation. Note that knowing i, i(x) can be obtained via i(x) = i exp(λ(S(x) − max S(x))). x B.2.1 (28) Solving for the law of motion and value and price functions With the (J, i), we solve for the value function S(x), the law of motion ṅ(x), and the price schedule q(b0 , ṅ, z) as follows: 1. Construct discrete grids of of debt per person B, population N , productivity Z, and weather W. For B, we use 20 linearly spaced points from -0.2 to 0. Since average income across cities is normalized to 1 and the debt-output ratio is around .02, this allows for a given city to hold roughly four times as much debt as the average, and it is not binding in the benchmark. (This grid is coarse relative to those used in Bewley-Huggett-Aiyagari type models, but the dispersion in debt holdings is much more concentrated for cities.) For N , we use 64 log27 Specifically, we use the “TT0” refinement and the theshold value π = 10−6 . 47 linearly spaced grid points over ±5 ∗ 1.8 since the standard deviation of the log population is roughly 1.8. For Z, we discretize the process as described in Section B.1 and tensor product it with the nonparametrically discretized permanent shocks. For W, we use a three-point discretization {−2σω , 0, 2σω } with Tauchen’s method. 2. Fix tolerances (tolq , toln , tolS ). We use (tolq , toln , tolS ) = (10−6 , 10−5 , 10−6 ). 3. Guess on S(x), ṅ(x), q(b0 , ṅ, z, ω). The initial guess we use is S(x) = 0, ṅ(x) = n, and q(b0 , ṅ, z, ω) = q̄. 4. Solve for the implied S ∗ (x) and associated policies. To determine the optimal value, we first do a grid search over the discrete bond states. When a discrete bond state, say Bi , is less than 0.01% away from the maximum, we then search for a local maximum in (Bi−1 , Bi ) and (Bi , Bi+1 ) (whenever applicable). In doing this search, we use Brent’s method. Whenever we interpolate, we use linear interpolation. Conditional on debt and default outcomes, we solve for c, g, h using the analytic solution of the intratemporal problem. 5. Compute an update q ∗ (b0 , ṅ, z, ω). 6. Compute an update ṅ∗ (x) using S ∗ (x) and J. 7. Determine whether the convergence criteria ||q ∗ − q||∞ < tolq , ||ṅ∗ − ṅ||∞ < toln , and ||S ∗ − S||∞ < tolS · ||S||∞ are satisfied. If so, stop. Otherwise, update the guesses and go to Step 4. B.2.2 Solving for the invariant distribution and key equilibrium object updates Given the converged values for ṅ(x) and the bond policy b0 (x), we compute the ∗ invariant distribution µ(x) and updates J ∗ , i , q̄ ∗ as follows: 1. Fix a tolerance tolµ . We use tolµ = 10−10 . 48 2. Guess on µ. Our initial guess is µ(0, 1, z, 1) = P(z) with µ = 0 elsewhere. (Consequently, the mass of households is 1 initially.) On subsequent invariant distribution computations, we use the previously computed µ. 3. Using ṅ(x), µ(x), and the bond and default policies, compute an update on the invariant distribution µ∗ (x). We use linear interpolation to distribute the mass from µ to µ∗ . (An important advantage of linear interpolation is that it keeps the number of households the R R same on each iteration, i.e., µ(x)ndx = µ∗ (x)ndx.) 4. Determine whether the convergence criteria ||µ∗ − µ||∞ < tolµ is satisfied. If so, continue to the next step. Otherwise, update the guess µ := µ∗ and go to Step 3. ∗ 5. For the updates J ∗ and i , use the values associated with the computed invariant distribution µ. B.2.3 Solving for the key equilibrium objects ∗ With the initial guesses J, i and the updates J ∗ , i , produce new initial guesses as follows: 1. Fix tolerances (tolJ , toli ). We use (tolJ , toli ) = (10−6 , 10−6 ). ∗ 2. Check whether |J ∗ − J| < tolJ · |J| and |i − i| < toli . If so, STOP: an equilibrium has been computed. Otherwise, go on. 3. Update the equilibrium values. Using the new guesses on J, i, resolve for the value functions, price functions, law of motion, invariant distribution, and key equilibrium objects as described in Sections B.2.1 and B.2.2. Then go to step 2. 49 C Omitted proofs and results [Not for Publication] This section contains additional theoretic, empirical, and quantitative results, as well as omitted proofs from the two-period model. C.1 Microfounding inflow rates To microfound inflow rates, we must use the notion of a Gumbel process. The beginning of the theory seems to be quite recent and due to Malmberg (2013). Here we follow the definition in Maddison et al. (2015): Definition 1 (Maddison et al., 2015). Let L(I) be a sigma-finite measure on sample space Ω, I ⊆ Ω measurable, and GL (I) a random variable. GL = {GL (I)|I ⊂ Ω} is a Gumbel process if 1. GL (I) ∼ Gumbel(log L(I)) 2. GL (I) ⊥ GL (I c ) 3. for measurable A, B ∈ Ω, then GL (A ∪ B) = max{GL (A), GL (B)}. Essentially, what the Gumbel process does is assign an infinitesimally small taste shock to any of the continuum of choices. The taste shock is small enough that the maximum over a continuum of choices is well-defined but large enough to influence the choices themselves. For our purposes, x = (b, n, z) ∈ R × R+ × R+ =: X. Let X denote the Borel σ-algebra of X with a Borel measure of islands µ. Then every X ∈ X is measurable with respect to µ. The sample space is X. To formalize the inflow rates, we do the following. Define L : X → R via R L(X) = X exp(λS(x) + c)dµ(x) where c is a constant. (Then L is absolutely continuous with respect to µ.) In the Gumbel process, L(X) will be the σ-finite measure on the sample space X . Then a Gumbel process GL with base measure L has random utility GL (X) ∼ Gumbel(log L(X)) for each X ∈ X with the additional restrictions that GL (X) is independent of GL (X c ) and GL (A ∪ B) = max{GL (A), GL (B)}. The last restriction says, essentially, that if options in A and B are both available, whichever is best (taking into account random utility) will be chosen. In the finite case, this amounts to the optimal value being the maximum over the finite set. The probability that the optimum over X is contained in some set X ∈ X is equivalent to the event that GL (X) ≥ GL (X c ). Malmberg (2013) showed P(GL (X) ≥ GL (X c )) 50 is L(X ∩ X)/L(X) (and here L(X ∩ X)/L(X) = L(X)/L(X)). And L(X) by defiR nition is X exp(λS(x) + c)dµ(x). Therefore, the probability that the max is in X R R is X exp(λS(x))dµ(x)/ X exp(λS(x)dµ(x). Consequently, the argmax has a probability density (formally, a Radon-Nikodym derivative) of R exp(λS(x)) . exp(λS(x))dµ(x) There- fore, by a law of large numbers, the measure going to each island with type x exp(λS(x)) is ī R exp(λS(x))dµ(x) where ī is the total measure of in-migrants (equivalently, out- migrants). C.2 The centralized problem We now give the proof that the model can be centralized at a local level: Proof of Proposition 3. Consider an arbitrary choice (c, g, d, b0 ) in the centralized problem. At this choice, household and firm optimization and market clearing will be satisfied if we take r = uh /uc , w = z, Ld = ṅ, h = H̄/ṅ, π solving ṅπ = z ṅ−wṅ+rH, and T solving ṅc + rhṅ = wṅ + π ṅ − T ṅ (see equation 16). Eliminating profit from the consumption equation, one has c = z −T (and clearly h = H̄/ṅ). Hence, the flow utility associated with this allocation is u(z − T, g, H̄/ṅ, ω), which according to (17) is the same as U (g, T, ṅ, z, ω). Hence, an arbitrary choice delivers u(c, g, H̄/ṅ, ω) flow utility in the centralized problem, which—when supported using the above prices and allocations—is the same as U (g, T, ṅ, z, ω). Moreover, at these prices and allocations, b0 , d, T is feasible for the government as guaranteed by Walras’s law.28 Then, since the centralized planner is maximizing the same flow utility, discounting, and expectations as the government, optimal choices for the centralized planner must simultaneously solve the government’s problem. Hence, the optimal choices from the centralized problem can be supported as a decentralized equilibrium using the prices r, w, firm allocation Ld , household housing consumption allocation h, firm profits π, and taxes T . C.3 Two-period model proofs and omitted results This subsection provides omitted proofs establishing the Euler equation and constrained inefficiency. It also includes a constrained efficiency result in the case of costless and fully-directed migration. 28 One can verify this easily. For instance, if d = 0, then the centralized budget constraint reads ṅc + ṅ1−η g + qb0 ṅ = z ṅ + ḃṅ. Using c = z − T to eliminate c, one finds ṅ1−η g + qb0 ṅ = T ṅ + ḃṅ, which is the government’s budget constraint. 51 C.3.1 The Euler equation Proof of Proposition 1. The objective function may be written ! Z J−u(c2 ) (J − φ)f (φ)dφ u(c1 ) + β (1 − o2 )u(c2 ) + (29) −∞ Using Leibniz’s rule, ∂c1 0 = u0 (c1 ) +β ∂b2 ∂u(c2 ) −∂o2 ∂(J − u(c2 )) (1 − o2 ) + u(c2 ) + (J − φ)f (φ) ∂b2 ∂b2 ∂b2 ! φ=J−u(c2 ) ∂u(c2 ) −∂o2 ∂(J − u(c2 )) + β (1 − o2 ) + u(c2 ) + u(c2 )f (J − u(c2 )) ∂b2 ∂b2 ∂b2 ∂u(c2 ) ∂F (J − u(c2 )) ∂(J − u(c2 )) + β (1 − o2 ) − u(c2 ) + u(c2 )f (J − u(c2 )) ∂b2 ∂b2 ∂b2 ∂u(c2 ) ∂(J − u(c2 )) ∂(J − u(c2 )) + β (1 − o2 ) − u(c2 )f (J − u(c2 )) + u(c2 )f (J − u(c2 )) ∂b2 ∂b2 ∂b2 ∂c2 + β(1 − o2 )u0 (c2 ) ∂b2 ∂b2 nn12 = −q̄u0 (c1 ) + β(1 − o2 )u0 (c2 ) ∂b2 n1 ∂n−1 0 0 2 = −q̄u (c1 ) + β(1 − o2 )u (c2 ) + b2 n1 n2 ∂b2 n 1 −2 ∂n2 0 0 = −q̄u (c1 ) + β(1 − o2 )u (c2 ) + b2 n1 (−1)n2 n2 ∂b2 n1 1 ∂n2 n1 − b2 = −q̄u0 (c1 ) + β(1 − o2 )u0 (c2 ) n2 n2 n2 ∂b2 n b 1 2 ∂n2 0 0 = −q̄u (c1 ) + β (1 − o2 )u (c2 ) 1 − n2 n2 ∂b2 n1 b2 ∂n2 0 0 = −q̄u (c1 ) + β (1 − o2 )u (c2 ) 1 − n1 (1 − o2 + i2 ) n2 ∂b2 1 − o2 b2 ∂n2 u0 (c2 ) 1 − = −q̄u0 (c1 ) + β 1 − o2 + i2 n2 ∂b2 ∂c1 = u0 (c1 ) ∂b2 ∂c1 = u0 (c1 ) ∂b2 ∂c 1 = u0 (c1 ) ∂b2 ∂c1 = u0 (c1 ) ∂b2 Consequently, the Euler equation reads 1 − o2 b2 ∂n2 0 q̄u (c1 ) = β u (c2 ) 1 − . 1 − o2 + i2 n2 ∂b2 0 52 (30) C.3.2 Constrained inefficiency Proof of Proposition 2. With no cross-sectional heterogeneity, constrained efficiency requires (5). If q̄ = βu0 (y2 )/u0 (y1 ), then this requires u0 (y2 + b2 nn21 ) u0 (y2 ) u0 (c2 ) = 0 = 0 . u0 (y1 ) u (c1 ) u (y1 − q̄b2 ) (*) Evidently, this requires b2 = 0. However, b2 = 0 is not compatible with the government’s Euler equation. In particular, at b2 = 0 and at q̄, the government Euler equation can be written u0 (y2 ) 1 − o2 u0 (c2 ) = . u0 (y1 ) 1 − o2 + i2 u0 (c1 ) (**) Hence, if i2 > 0, then (*) and (**) cannot simultaneously hold. And in fact, some people will enter (i.e., i2 is greater than 0) because in the constrained efficient allocation c2 = y2 for every island and so J = u(c2 ) and—given this—some people will move since F (0) > 0 (i.e., migration is noisy). Hence, the constrained efficient allocation cannot be supported as an equilibrium. For the claim that at the constrained efficient allocation governments would strictly prefer to borrow, note the Euler equation at the constrained efficient allocation is not satisfied with 1 − o2 1 − o2 ⇔1> . 1 − o2 + i 2 1 − o2 + i2 The way to equate marginal utilities would then be to increase c1 by borrowing. u0 (y1 )q̄ > βu0 (y2 ) C.3.3 Constrained efficiency under costless and fully-directed migration Our constrained inefficiency result established in Proposition 2 may be surprising in light of the seminal paper by Tiebout (1956). Tiebout showed that, under certain assumptions, equilibria are efficient when local governments compete for workers. One of his key assumptions, which is not met here, is that of costless and fullydirected mobility. In fact, the equilibrium can be efficient if migration is perfectly directed. To see why, consider trying to implement an efficient allocation that implies b2 = 0. For the reasons described above, the Euler equation (2) would typically imply this is impossible. However, if inflow rates “punish” any debt accumulation by falling to zero in a nondifferentiable way, the Euler equation no longer characterizes the optimal choice, and the equilibrium can be efficient. We prove this in Proposition 4. Proposition 4. If migration is completely directed with (1) I(u(c2 )) = 0 for c2 < y2 , (2) the right-hand derivative of I(·) at u(y2 ) infinite, and (3) I(·) differentiable 53 elsewhere, then an equilibrium with q̄ = βu0 (y2 )/u0 (y1 ) exists and it is constrained efficient. Proof of Proposition 4. Under these assumptions, the Pareto optimal allocation is c1 = y1 , c2 = y2 , with households moving whenever φ < 0 and staying whenever φ > 0 (with indifference elsewhere). Note that in contrast to the hypothesis of Proposition 2, inflow rates are assumed to be not differentiable at b2 = 0, which means the Euler equation is not valid at that point. We will prove the existence of a symmetric equilibrium with q̄ = βu0 (y2 )/u0 (y1 ) both of which have b2 = 0 as optimal. We will do so by establishing that at this price b2 < 0 is not optimal, that b2 > 0 is not optimal, and that an optimal choice exists (in which case it must be b2 = 0). This will then support the allocation (c1 , c2 ) = (y1 , y2 ) (and the migration decisions). For use below, we note that whenever the derivative ∂n2 /∂b2 exists, one has b2 ∂n2 b2 ∂c2 = n1 īI0 (u(c2 )) + f (J − u(c2 )) u0 (c2 ) n2 ∂b2 n2 ∂b2 2 (31) n1 0 0 = b2 u (c2 ) īI (u(c2 )) + f (J − u(c2 )) n2 Because I is increasing and f is positive, this has the same sign as b2 . First we will show that b2 < 0 is not optimal by showing the Euler equation does not hold there. Given no inflows for b2 < 0, borrowing is not optimal because the Euler equation (which is valid everywhere except at b2 = 0) requires u0 (c2 ) 1 − o2 b2 ∂n2 u0 (c2 ) u0 (y2 ) 1− β 0 = q̄ = β 0 ≥β 0 . u (y1 ) u (c1 ) 1 − o2 + i2 n2 ∂b2 u (c1 ) {z }| | {z } =1 (32) ≥1 since b2 <0 However, with b2 < 0, c1 > y1 and c2 < y2 , which gives a contradiction. Now we will show that b2 > 0 is not optimal. The Euler equation in this case reads β b2 ∂n2 u0 (y2 ) u0 (c2 ) 1 − o2 u0 (c2 ) = q̄ = β 1 − ≤ β . u0 (y1 ) u0 (c1 ) 1 − o2 + i2 n2 ∂b2 u0 (c1 ) {z }| | {z } ≤1 (33) ≤1 since b2 >0 However, with b2 > 0, c1 < y1 and c2 > y2 , which gives a contradiction. Since b2 < 0 and b2 > 0 are not optimal, all that remains to show is that an optimal choice exists. Without loss of generality, we can restrict the choice set to b2 ∈ [−δ, δ] for δ arbitrarily small such that every choice is feasible. Then, with a continuous objective function being maximized over a compact set, a maximum 54 exists, which must be b2 = 0. C.4 Quantitative testing of indeterminacy To test for indeterminacy, we proceed by drawing 100 random starting guesses for J and ī uniformly distributed about ±50% of the benchmark’s computed equilibrium values. (For the definition of ī, see Appendix B.2.) We then compute the implied equilibrium solution. Figure 14 shows a scatter plot of the guesses, and also reveals that they all converge to the same solution (up to small numerical differences). This suggests that the equilibrium is unique in a wide range about the computed benchmark equilibrium. Figure 14: Quantitative testing of indeterminacy C.5 Additional calibration results The cross-sectional regression specifications in Table 1 are as follows. For the row ”Regression coef. log expendituers on log n”, the specification is log xi,t = α + β log ni,t + i,t , where xi,t is total expenditures, and ni,t is population. For the rows with “rate reg. coef., log z,” the specification is yi,t = α + β log zi,t + γ log ni,t + i,t . 55 The dependent variable is either in-migration rates or out-migration rates, as specified in the row. In all regressions, the sample is restricted to t = 2011. Table 4 provides additional untargeted statistics coming from richer regressions. The regression coefficients correspond to a regression of the form yi,t = α + βςi + γ z̃i,t + δ log ni,t + i,t with ςi and z̃i,t being the fixed effect and residual productivity from (24) (and ni,t population). Again, the sample is restricted to 2011. Overall, the underlying elasticities are not very different from those in the data. Untargeted Statistics Data Model *In rate reg. coef., log n *In rate reg. coef., log z FE *In rate reg. coef., log z res *Out rate reg. coef., log n *Out rate reg. coef., log z FE *Out rate reg. coef., log z res -0.000 -0.011 0.034 -0.000 -0.013 0.015 -0.013 0.006 0.072 -0.011 0.000 -0.019 Note: * means the underlying data is county-level. Table 4: Additional untargeted moments C.6 Additional estimation results We construct the share of individuals at a reference time τ > 0 periods ago who live in c but arrived from county o at some time in the past as follows. Given inflows Ic,o,t , a population measured at the start of the period Nc,t , and an out-migration rate δc,t , we construct θc,o,t under the assumption that every individual in c has the same probability δt of leaving. Let Nc,o,t denote the stock of individuals in c from o and time t. First, we obtain Nc,o,1 by assuming the county’s population is in proportion to its inflows in t = 1, Nc,o,1 = PIc,o,1 Nc,1 . o Ic,o,1 Second, we obtain Nc,o,t for t ≥ 2 recursively using Nc,o,t = Nc,o,t−1 (1 − δc,t−1 ) + Ic,o,t−1 . The share is then P θc,o,t = Nc,o,t / o Nc,o,t . Tables 5, 6, and 7 report the estimates underlying Figure 4. 56 Debt F5.Debt F10.Debt F15.Debt F20.Debt In-migration rate 3.505 (0.9) 10.36 (1.9) 12.39 (2.8) 6.613 (1.9) 1.615 (0.3) Lagged debt 0.676 (77.8) 0.496 (51.4) 0.342 (30.2) 0.271 (20.4) 0.239 (10.0) Lagged out-migration rate -2.619 (-0.8) -8.023 (-1.7) -8.720 (-2.3) -3.286 (-1.1) 0.468 (0.1) Prod. FE 0.484 (18.9) 0.761 (18.1) 0.939 (20.9) 0.969 (16.3) 0.846 (5.7) Constant 2.398 (41.6) 3.628 (49.8) 4.887 (63.3) 5.460 (57.1) 5.786 (33.0) Observations R2 First-stage F 31830 0.609 14.78 20993 0.414 32.32 14303 0.269 30.47 8142 0.222 36.30 2196 0.161 13.75 Note: robust standard errors are used; year effects are included; “debt” is log of per person, real debt measured at the end of the fiscal year. Table 5: IV regressions capturing the externality effects on debt Exp. F5.Exp. F10.Exp. F15.Exp. F20.Exp. In-migration rate 12.57 (3.6) 11.10 (4.4) 10.09 (3.4) 7.754 (3.3) 5.383 (1.9) Lagged exp. 0.844 (42.6) 0.814 (54.0) 0.768 (40.5) 0.656 (43.0) 0.656 (24.8) Lagged out-migration rate -10.88 (-3.6) -9.509 (-4.4) -8.415 (-3.3) -6.529 (-3.4) -4.350 (-1.8) Prod. FE 0.0215 (1.4) 0.0353 (2.7) 0.00232 (0.1) -0.00792 (-0.3) 0.0101 (0.2) Constant 1.245 (6.9) 1.368 (9.1) 1.936 (10.5) 2.910 (21.0) 3.092 (13.6) Observations R2 First-stage F 32145 0.423 21.42 21172 0.371 41.88 14421 0.268 39.45 8221 0.253 39.98 2209 0.336 14.58 Note: robust standard errors are used; year effects are included; “expenditures” is log of per person, real expenditures. Table 6: IV regressions capturing the externality effects on expenditures 57 Rev. F5.Rev. F10.Rev. F15.Rev. F20.Rev. In-migration rate 10.30 (5.0) 12.48 (5.2) 8.460 (4.3) 7.766 (4.3) 4.917 (1.9) Lagged rev. 0.850 (62.3) 0.838 (47.6) 0.779 (52.1) 0.667 (44.2) 0.679 (25.1) Lagged out-migration rate -8.966 (-5.0) -10.75 (-5.1) -7.169 (-4.2) -6.510 (-4.2) -3.746 (-1.7) Prod. FE 0.0192 (1.7) 0.0312 (2.2) 0.00170 (0.1) -0.0142 (-0.6) 0.00546 (0.1) Constant 1.220 (10.1) 1.181 (7.1) 1.917 (13.8) 2.840 (21.6) 2.941 (12.8) Observations R2 First-stage F 32145 0.531 21.69 21172 0.334 43.88 14421 0.345 41.08 8221 0.260 41.01 2209 0.359 14.48 Note: robust standard errors are used; year effects are included; “revenue” is log of per person, real revenue. Table 7: IV regressions capturing the externality effects on revenue Hi 58 D The Great Recession [Not for Publication] In this appendix, we provide more background and rationale for the shocks we feed into the model for the Great Recession period. To begin with, Figure 15 shows log real GDP per capita in relation to a pre-2009 linear trend. The drop in GDP p.c. was on the order of 12% on impact, and—relative to trend—continued through 2020Q1 reaching almost 20%. Against this headwind, the American Reinvestment and Recovery Act of 2009 (ARRA) was a sweeping legislation that included provisions to bolster state and local finances. This funding flowed directly to states and more indirectly to local governments. To see this, consider the time series of a few key state and local government (SLG) variables provided in NIPA in Figure 16. Expressed relative to GDP and in differences from 2006, federal transfers to state and local governments (SLG) rose from zero (i.e., they were at their 2006 levels) to almost 1% in late 2009 into 2011. This more than offset the decline in SLG tax revenue, which fell -0.5% relative to GDP, resulting in an overall boost of SLG revenue amounting to 0.25% that lasted into 2011 . Unfortunately, this data does not separate the revenues of state and local governments with one exception, in that it reports the transfers from state to local governments. These rose slightly in 2009 and 2010 before falling substantially starting in 2011. Log real GDP p.c. deviations from pre-2009 linear trend 1950q1 1960q1 1970q1 1980q1 1990q1 Time 2000q1 Log real GDP p.c. 2010q1 2020q1 -.3 9.5 -.2 10 Deviation -.1 10.5 0 11 .1 11.5 Log real GDP p.c. and pre-2009 linear trend 1950q1 Trend Source: authors' calculations using FRED series A939RX0Q048SBEA. 1960q1 1970q1 1980q1 1990q1 Time 2000q1 2010q1 2020q1 Source: authors' calculations using FRED series A939RX0Q048SBEA. Figure 15: Log real GDP relative to trend From a local government perspective, state to local transfers, while exhibiting some dip despite the ARRA, have been a steady percentage of GDP. Hence, in mapping this to the model, we assumed a pass-through from GDP declines to model TFP declines that is less than 100%. Specifically, we used that local government budgets are 8.2% of GDP, and state to local transfers were 3.2% from 2006 to 2009, 59 0 -.005 0 .05 .1 .005 .15 SLG variables as a share of GDP .01 SLG variables as a share of GDP, difference from 2006 2005q1 2010q1 2015q1 2020q1 1950q1 1960q1 1970q1 Time Federal to SLG SLG Rev. SLG Tax Rev. State to Local 1980q1 1990q1 Time Federal to SLG SLG Rev. For scale, 2020 has not been plotted 2000q1 2010q1 2020q1 SLG Tax Rev. State to Local For scale, 2020 has not been plotted Figure 16: Federal, state, and local government transfers and revenue suggesting a pass-through of 61% = 1 − 3.2%/8.2%. Of course, the Great Recession also exhibits steep declines in risk-free real interest rates. These real rates, measured using five-year TIPS yields, are plotted in differences from 2006 in Figure 17. It reveals real rates declined by more than 3pp from 2006 to 2012. Table 8 summarizes the declines in productivity and interest rates that we feed into the model. -.04 -.03 meanrealrD06 -.02 -.01 0 Average annual real rate (5-year TIPS), difference from 2006 2005 2010 2015 2020 Year Figure 17: Real interest rates, difference from 2006 Table 8 reports the transition variables we use in the experiments. 60 Parameter Year 2007 (steady state) 2008 (shock, 1st period) 2009 2010 2011 2012 2013 2014+ (new steady state) $ q̄ −1 − 1 0 -0.03 -0.07 -0.07 -0.08 -0.08 -0.09 -0.09 0.040 0.030 0.028 0.020 0.013 0.005 0.010 0.020 Table 8: Transition variables for the Great Recession Appendix references Bubul, S. J. (2017). Debt and borrowing, In Handbook for minnesota cities. Saint Paul, MN, League for Minnesota Cities. Faulk, D., & Killian, L. (2017). Special districts and local government debt: An analysis of “old northwest territory” states. Public Budgeting & Finance, 112–134. Gordon, G. (2021). Efficient VAR discretization. Economics Letters, 204, 109872. Harris, R. (2002). California constitutional debt limits and municipal lease financing [Accessed: 2018-02-20]. Hauer, M., & Byars, J. (2019). IRS county-to-county migration data, 1990-2010. Demographic Research, 40 (40), 1153–1166. Maddison, C. J., Tarlow, D., & Minka, T. (2015). A* sampling. https://arxiv.org/abs/1411. 0030 Malmberg, H. (2013). Random choice over a continuous set of options (Licentiate Thesis). Stockholm University. MCTA. (2009). Chapter 9 borrowing [Accessed: 04-17-2018]. OMAC. (2013). Municipal debt in Ohio, the guide (tech. rep.) [Accessed: 2018-02-20]. Ohio Municipal Advisory Council. Accessed: 2018-02-20. ONYSC. (2018). Constitutional debt limit [Accessed: 04-17-2018]. Tauchen, G. (1986). Finite state Markov-chain approximations to univariate and vector autoregressions. Economics Letters, 20 (2), 177–181. Tiebout, C. M. (1956). A pure theory of local expenditures. Journal of Political Economy, 64 (5), 416–424. https://doi.org/10.1086/257839 61