metadata: authors: 0: author: Guo,Si
1: author: Pei,Yun
2: author: Xie,Leiyu
count: World
docna: 0: docna: A Dynamic Model of Fiscal Decentralization
            and Public Debt Accumulation
docty: Policy Research Working Paper
subtopic: Democratic Government,Public Sector Administrative and Civil
            Service Reform,Public Sector Administrative & Civil
            Service Reform,De Facto Governments,Economic Adjustment and Lending,Public Finance Decentralization and
            Poverty Reduction,Public Sector Economics,Macro-Fiscal Policy,Financial Sector Policy,Public Financial Management
teratopic: Macroeconomics and Economic Growth,Governance,Public Sector Development
abstracts: cdata!: This paper develops a dynamic
            infinite-horizon model with two layers of governments to
            study theoretically and quantitatively how fiscal
            decentralization affects local and central government debt
            accumulation and spending. In the model, the central
            government makes transfers to local governments to offset
            vertical and horizontal fiscal imbalances. But the
            anticipation of transfers lowers local governments’ expected
            cost of borrowing and leads to overborrowing ex ante. Absent
            commitment, the central government over-transfers to reduce
            local governments’ future need to borrow, and in the
            equilibrium both local and central debts are inefficiently
            high. Consistent with empirical evidence, when fiscal
            decentralization widens vertical fiscal imbalances, local
            governments become more reliant on transfers, and both local
            and central debts rise. Applied to Spain, the model explains
            39 percent of the rise in total government debt when the
            vertical fiscal imbalances widened during 1988–1996, and 18
            percent of the fall in debt when the imbalances narrowed
            during 1996–2006.
repnme: repnme: A Dynamic Model of Fiscal Decentralization
            and Public Debt Accumulation
display_title: A Dynamic Model of Fiscal
            Decentralization and Public Debt Accumulation
listing_relative_url: /research/2022/02/33723413/dynamic-model-fiscal-decentralization-public-debt-accumulation
topicv3: Inter-governmental Fiscal Relations
            m1325996 981,Public Finance Management m1326316 716,Fiscal Decentralization m1327382 829,Financial Sector Development m1327681 1920,Public Administration m1327723 723
publishtoextweb_dt: 2022-02-07T00:00:00Z
dois: 10.1596/1813-9450-9927
historic_topic: Macroeconomics and Economic Growth,Governance,Public Sector Development
origu: Off of Sr VP Dev Econ/Chief Econ (DECVP)
no_of_pages: 91
author_keys: 0: author_key: 2028432
model_extractions: Policy Research Working Paper 9927 A Dynamic Model of Fiscal Decentralization and Public Debt Accumulation _Si Guo Yun Pei Zoe Xie_ South Asia Region Office of the Chief Economist February 2022
Policy Research Working Paper 9927

# **Abstract**

This paper develops a dynamic infinite-horizon model with two layers of governments to study theoretically and quantitatively how fiscal decentralization affects local and central government debt accumulation and spending. In the model, the central government makes transfers to local governments to offset vertical and horizontal fiscal imbalances. But the anticipation of transfers lowers local governments’ expected cost of borrowing and leads to _overborrowing_ ex ante. Absent commitment, the central government over-transfers to reduce local governments’ future need to borrow, and in the equilibrium both local and central debts are inefficiently high. Consistent with empirical evidence, when fiscal decentralization widens vertical fiscal imbalances, local governments become more reliant on transfers, and both local and central debts rise. Applied to Spain, the model explains 39 percent of the rise in total government debt when the vertical fiscal imbalances widened during 1988–1996, and 18 percent of the fall in debt when the imbalances narrowed during 1996–2006.

This paper is a product of the Office of the Chief Economist, South Asia Region. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://www.worldbank.org/prwp. The authors may be contacted at sguo@imf.org, yunpei@buffalo.edu, and lxie@worldbank.org.

_The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent._ Produced by the Research Support Team
# A Dynamic Model of Fiscal Decentralization and Public Debt Accumulation<sup>∗</sup>

Si Guo Yun Pei International Monetary Fund University at Buffalo World Bank; Federal Reserve Bank of Atlanta Keywords: Fiscal decentralization, Public debt, Time-consistent policy, Soft budget constraint JEL classifications: E61, E62, H74

> ∗This paper was previously titled “Decentralization and Overborrowing in a Fiscal Federation.” We would like to thank Luc Eyraud and Lusine Lusinyan for sharing their data, and thank Gustavo Leyva for his help understanding some literature in Spanish. We thank audience and discussants at Atlanta Fed, Tsinghua Macro Workshop, SED, Econometric Society (Davis), EEA, Fed Macro & International Meetings, Philadelphia Macro Workshop, and the IADB for helpful comments. This paper has benefited from discussions with Manuel Amador, Jinhui Bai, Huixin Bi, Toni Braun, Lei Fang, Dirk Foremny, Grey Gordon, Jonathan Heathcote, Roozbeh Hosseini, Anastasios Karantounias, Karen Kopecky, Federico Mandelman, Fernando Martin, Víctor Ríos-Rull, Juan Rubio-Ramirez, Philippe Wingender, and Kei-Mu Yi. The views expressed in this paper are those of the authors and do not necessarily represent the views of the IMF, its Executive Board, IMF management, the Federal Reserve System, the Federal Reserve Banks, the IBRD/World Bank or its executive directors.

> †Emails: Guo (sguo@imf.org). Pei (yunpei@buffalo.edu). Xie (xiexx196@gmail.com; lxie@worldbank.org).
_“The creation of Debt should always be accompanied with the means of extinguishment.”_ — Alexander Hamilton, Report on Public Credit, 1790

# 1 Introduction

In many countries, revenue collection power is largely held by the central government, while spending responsibilities are disproportionately assigned to local governments.<sup>1</sup> Because of this asymmetry— known as “vertical fiscal imbalances,” local governments’ own revenue often falls short of their spending, and the gap is filled by intergovernmental transfers from the central government and local governments’ borrowing. Existing empirical findings suggest that widening vertical fiscal imbalances in a country are associated with worsening aggregate fiscal performance, such as faster public debt accumulation (see e.g., Rodden 2002, Eyraud and Lusinyan 2013). As an illustration, Panel A of Figure 1 shows the gaps in local government finances for three OECD countries, and Panel B shows that total (central and local) government debt tends to rise faster when vertical fiscal imbalances widen.<sup>2</sup> The theoretical literature on the soft budget constraint problem offers one channel for this relationship: rising vertical fiscal imbalances worsen local governments’ soft budget constraint problem and lead to rising local government debt (Boadway and Shah 2007, Chapter 5).<sup>3</sup> But because this literature uses mostly static or two-period models, it does not have anything to say about central government debt,<sup>4</sup> which accounts for the majority of government debt changes in the data.<sup>5</sup> Even less is known about the size of the effect of vertical fiscal imbalances on government debt accumulation.

In this paper, we quantify the effect of vertical fiscal imbalances on total government debt accumulation by developing an infinite-horizon model. The infinite-horizon is a valid choice because it allows us to consider central government debt alongside local government debt. The model delivers a novel result in the equilibrium: the central government over-transfers to local governments, to the extent that the marginal utility from local government spending is smaller than the marginal utility from central government spending. Applying the model to fiscal decentralization, wider vertical fiscal imbalances make local governments more reliant on transfers and exacerbate the soft budget

> 1We use “central government” to refer to the national government in a country, and use “local government” to collectively refer to the subnational government such as a regional, provincial, or municipal government.

> 2Appendix A includes plots for 25 other OECD countries and shows that changes in total government debt comove with vertical fiscal imbalances in a large set of countries over time.

> 3In their survey of the soft budget constraint concept, Kornai et al. (2003) define soft budget constraint as the phenomenon that arises when “one or more supporting organizations (S-organizations) are ready to cover all or part of the deficit” of an organization that has a budget constraint (a BC-organization). In our setting, local governments are the BC-organizations, and the central government is the S-organization.

> 4In a two-period model, central government debt is zero at the end of the second period by default. This means that even though it is possible to model central government borrowing between the first and second periods, a two-period model does not allow the accumulation of debt over time.

> 5For example, for G-7 countries, the change in central government debt-to-GDP ratio during 1995–2015 ranges from 35% (Canada) to 133% (Japan); the change in subnational government debt-to-GDP ratio is much smaller, ranging from 2% (United Kingdom) to 13% (Japan).

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Given the inherent time-inconsistency problem in the model, we focus on the time-consistent equilibrium. We characterize the Markov-perfect equilibrium, where the central government has no commitment over its future transfer, spending, and debt policies. We show two theoretical results in this equilibrium. First, the central government prefers debt-contingent transfer to uniform lump-sum transfer, because without commitment to future policies it does not consider the ex ante distortionary effect of debt-contingent transfer. As a result, the local governments overborrow relative to the efficient allocation benchmark—a common result in the soft budget constraint literature.

The second and more novel result is that in the time-consistent equilibrium, the central government over-transfers such that at the steady state, the marginal utility from local government spending is smaller than the marginal utility from central government spending. This is in sharp contrast with the common result in the literature, where the size of federal transfers is chosen to equalize the marginal utility of spending across different regions or different levels of governments.<sup>6</sup> The reason lies in the infinite-horizon feature of our model. In a two-period model with soft budget constraints, the central government moves last and does not need to take into account the impact of its decisions on the future. It simply chooses the amount of transfers that equates the marginal utilities from different types of public spending. In an infinite-horizon model, the central government considers the impact of its decisions on local governments’ behaviors in the future. In particular, in our model, the central government today understands that local governments overborrow because of their anticipation of future debt-contingent transfers. Without the ability to commit to lower future debt-contingent transfers, the central government finds it optimal to over-transfer today to reduce local governments’ need to borrow. In the equilibrium, the over-transfer exacerbates ex ante overborrowing by local governments, and worsens the vertical fiscal imbalances.<sup>7</sup> Given the equilibrium properties, the model predicts that widening vertical fiscal imbalances are associated with faster government debt accumulation, which is consistent with past empirical findings. In the model, when vertical imbalances widen, local governments become more reliant on intergovernmental transfers to finance local spending. The anticipation of more transfers, and in particular more debt-contingent transfers, exacerbates local governments’ overborrowing. At the same time, central government debt rises to finance the higher transfers. In many countries, fiscal decentralization reforms tend to widen the vertical imbalances, because decentralization on the spending side often outpaces the revenue side. A number of international organizations have advocated for “balanced” fiscal decentralization (e.g., Lam et al. 2017; Sow and Razafimahefa 2017; Cibils and Ter-Minassian 2015). We formalize the idea and calculate the revenue decentralization needed to keep the observed expenditure decentralization “debt-neutral” or balanced.

> 6For example, in Sanguinetti and Tommasi (2004), the size of central-to-local transfers is chosen to equalize each region’s marginal utility from private consumption (which is comparable to local government spending in our model) and the marginal utility from central government consumption. In models without central government consumption, such as Goodspeed (2002a) and Kothenburger (2007), central-to-local transfer is chosen to equalize the marginal utility from public spending in each region, after accounting for the possible externality of public goods or voting preference.

> 7We also consider the case of (time-inconsistent) Ramsey allocation. We find that because the central government has commitment to future policies, it optimally **under-transfers** relative to the efficient allocation, which stands in contrast with the over-transfer result in the time-consistent Markov equilibrium.

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To quantify the effects of vertical fiscal imbalances on government debt accumulation, we calibrate the model to Spain. Like many other countries, the local governments in Spain rely heavily on central government transfers, and the regions with higher debt tend to receive more transfers.<sup>8</sup> The country has gone through several episodes of fiscal decentralization which changed the vertical fiscal imbalances. Empirically, as vertical imbalances widened, total government debt rose, which is consistent with the empirical relationship documented in the cross-country empirical studies in the literature.

Using the calibrated model, we find sizeable effects of changes in vertical fiscal imbalances on government debt accumulation in Spain. To take advantage of different fiscal decentralization reforms in our analysis, we look at two distinct episodes prior to the European debt crisis. In the first episode (1988–1996), faster decentralization of spending than revenue led to widening vertical imbalance and rising total government debt. In the second episode (1996–2006), decentralization of revenue caught up, so the vertical imbalances shrank, and total government debt fell. Quantitatively, we find that decentralization explains 39% of the debt increase in the first episode and 18% of the fall in the second episode.

Our analysis is closely related to the public economics literature on the soft budget constraint and common pool problems.<sup>9</sup> Our paper complements this literature by using an infinite-horizon model. With the exception of Velasco (2000), most of the past literature uses static or two-period models. The infinite-horizon setup allows us to introduce central government debt and to study the dynamic interaction between current and future central government policies. An important implication of infinite-horizon is that absent commitment to future policies, the central government over-transfers to offset distortions from future central government policies, to the extent that the marginal utility from local government spending is smaller than (rather than equal to) the marginal utility from central government spending. Introducing central government debt also allows us to quantitatively study the impact of fiscal decentralization reforms on public debt accumulation.

A few papers in this literature also study the effects of fiscal decentralization on fiscal outcomes. The most related studies are Bellofatto and Besfamille (2018) and Garcia-Milà et al. (2002). Bellofatto and Besfamille (2018) use a three-period model to compare two different revenue arrangements for refinancing local projects: centralized versus decentralized tax revenue. Their focus is the impact of different fiscal arrangements on the provision of local public goods rather than on aggregate fiscal performance, and hence government debt is not modeled. Garcia-Milà et al. (2002) use a two-period model to study the impact of soft budget constraint and revenue decentralization on local government borrowing, when expenditure is already decentralized. In their model, whether or not the local governments can raise tax revenue affect their debt choices between the first and second period, 8For other countries, Pettersson-Lidbom (2010) finds that Swedish local governments that received the highest numbers of discretionary transfers also had the largest accumulation of debt. Nicolini et al. (2002) show that the expectation of bailouts increases Argentina’s provincial governments’ incentives to run budget deficits. In Appendix B.2, we discuss anecdotal examples from Spain, Hungary, Italy, Germany, and several other countries where transfers are redistributive and approximately dependent on debt or deficit.

> 9See, for example, Kornai (1986); Qian and Roland (1998); Velasco (2000); Sanguinetti and Tommasi (2004); Besfamille and Lockwood (2008); Bellofatto and Besfamille (2018); and Goodspeed (2016) for a review.

4
while central government is assumed to maintain a balanced budget. Compared to these papers, we use an infinite-horizon model to study the effect of fiscal decentralization on both local and central government debt accumulation. The inclusion of central government debt is important given that it is empirically more relevant in explaining the movements in total government debt. Additionally, our model allows us to study the full spectrum of spending and revenue decentralization in a quantitative exercise, instead of the two polar cases of full decentralization and full centralization.

Conceptually, the model shares common elements of the literature on time-inconsistency and time-consistent policies.<sup>10</sup> We apply the idea of lack of commitment and policy distortion to the context of fiscal federalism. Importantly, we model the borrowing decisions of both central and local governments, and illustrate how central and local government debts interact with each other. In the model, to reduce local governments’ overborrowing, the central government without commitment over-transfers. The ability to borrow gives the central government the capacity to make even higher transfers than a government with a balanced budget. The higher transfer further increases local governments’ overborrowing, which leads to even higher total transfer and central government debt. As such, the interaction between central and local debt amplifies the aggregate effects of vertical imbalances on total debt.

The rest of the paper proceeds as follows: Section 2 presents the dynamic infinite-horizon model and characterizes the equilibrium and optimal policies. Section 3 calibrates the model to Spain. Section 4 contains quantitative applications to fiscal decentralization reforms and comparisons with other fiscal systems. Extensions and robustness checks to the baseline model are contained in this section. Section 5 concludes.

# 2 Model

In this section, we present an infinite-horizon model with two layers of governments, vertical fiscal imbalances and inter-governmental transfers. As a benchmark, we first characterize the allocations under a single consolidated government. We then consider the decentralized environment with two layers of governments making their own decisions separately. The allocations will depend on whether the central government has the ability to commit to its future transfers.

## **2.1 Model environment**

We consider a small open economy. Time is discrete and infinite. There is a central government and a continuum of local regions indexed by _i ∈_ [0,1]. Each region has one local government and a mass one of identical households. In each period, the representative household of a region receives endowment income _y_ , pays tax _e_ and consumes the residual. The tax revenue is shared: _f_ for the local governments and _e − f_ for the central government. The sharing of spending responsibilities between the central and local governments is captured by a preference parameter _θ_ to be elaborated 10See, for example, Klein, Krusell, and Ríos-Rull (2008); Martin (2011); Song, Storesletten, and Zilibotti (2012); Bianchi (2016); Karantounias (2017); Bianchi and Mendoza (2018).

5
later. Given these, each (local or central) government makes its spending and borrowing decisions each period, and the central government also determines transfers to local governments.

**Preferences.** The representative household living in region _i_ has the preference

where _ct_ is the per capita central government spending and _gi_ , _t_ is the per capita public spending of local government _i_ , and

**Assumption 1.** The utility functions _u_ ( _·_ ) and _v_ ( _·_ ) are smooth, increasing and concave.

Both levels of governments are benevolent and utilitarian. Local government _i_ chooses _gi_ , _t_ and local debt issuance _bi_ , _t_ +1 to maximize (1), subject to its local budget constraint. The central government maximizes the average utilities of households living in all regions, subject to a central budget constraint. We take household income _y_ and total government tax revenue _e_ as exogenous, so households’ utility from private consumption, _w_ ( _y − e_ ), drops out from the central and local governments’ welfare maximization problems.

While the additive preference form in (1) over public goods provided by different levels of governments and the utility weight _θ_ have been widely adopted in the literature,<sup>11</sup> they can be microfounded as follows when _u_ ( _·_ ) and _v_ ( _·_ ) are logarithmic and the elasticity of substitution is one. Sup ̄ pose there is a unit mass of varieties of public goods, indexed by _ω ∈_ [0,1]. Goods _ω ∈_ [0, _w_ ) are provided by the central government (e.g., national defense), and goods _ω ∈_ [ _w_ ̄ ,1] are provided by the region’s local government (e.g., local roads and fire stations). Households in each region derive utility from the basket of public goods: ̄ where _q_ ( _ω_ ) is the quantity of good _ω_ . A decrease in _w_ means more goods (or spending “lines”) are provided locally. This can be understood for example as the decentralization of education and health care spending in many countries. Under unitary elasticity of substitution between goods ( _σ_ = 1) and log utility, the preference becomes separable as in (1), with the parameter _θ_ = _w_ ̄<sup>1/</sup><sup>_σ_</sup> +( _<u>w</u>_ ̄<sup>1</sup> 1<sup>/</sup> _−_<sup>_σ_</sup> _w_ ̄ )<sup>1/</sup><sup>_σ_=</sup><sup>_w_ ̄</sup> being the utility weight for goods provided by the central government. Appendix B.1 provides the details of the transformation.

**Vertical fiscal imbalances.** To make the model suitable for studying inter-governmental transfers, we assume that local governments’ share of total government revenue is not high enough to cover their spending obligations. This describes the vertical fiscal imbalances in many countries (Figure 1) and provides the motivation for central-to-local transfers. The precise mathematical assumption 4. for vertical fiscal imbalances is given later by Assumption

> 11See, for example, Alesina and Tabellini (1990); Goodspeed (2002b); Besley and Coate (2003); Kothenburger (2007); Breuillé and Vigneault (2010).

6
**Two types of local governments.** We further introduce a cross-region (or “horizontal”) fiscal gap by modeling two types of local governments: the unconstrained (or type “n”) and the constrained (or type “o”). Unconstrained local governments _i ∈_ [0, _μ_ ], where _μ ∈_ (0,1), have the full autonomy to borrow subject to an exogenous interest rate schedule. Constrained local governments _i ∈_ ( _μ_ ,1] face a constraint on borrowing, _b_<sup>_o_</sup> _i_ , _t_<sup>_≤Bo_.To make notations simpler, we assume</sup><sup>_Bo_= 0 in this section, but</sup> results apply for any _B_<sup>_o_</sup> _≥_ 0. When _β_ is small enough, which is guaranteed by Assumption 3 later, _b_<sup>_o_</sup> _i_ , _t_<sup>= 0 always holds.12Regions are otherwise identical.This difference in borrowing introduces a</sup> **cross-region fiscal gap** , as regions with more public debt also have higher debt service burdens.<sup>13</sup>

We denote the average debt of unconstrained local governments by _b_<sup>_n_</sup> _t_<sup>,and</sup><sup>_bo_</sup> _t_<sup>forconstrained</sup> local governments. Because all local governments of the same type are identical, we have _bi_ , _t_ = _b_<sup>_n_</sup> _t_ for _i ∈_ [0, _μ_ ] and _bi_ , _t_ = _b_<sup>_o_</sup> _t_<sup>= 0for</sup><sup>_i∈_(</sup><sup>_μ_,1].Thenationwide(acrossallregions)localdebtisthen</sup> _bt_ = _μb_<sup>_n_</sup> _t_<sup>.</sup> We introduce two types of local governments to create a link between local governments’ debt and their transfer income. If all regions are identical, the central government can simply use a lumpsum uniform transfer to replicate the allocation under a single consolidated government. With two types of regions, the central government has incentives to provide more transfers to the more indebted regions to offset the cross-region fiscal gap. This creates a positive correlation between the debt burden and transfer, which is the source of distortion in our model and is also supported empirically as we document in Section 3.1.

**Government Debt.** The central government and unconstrained local governments borrow from international financial markets, subject to exogenous and increasing interest rate schedules. The central government faces the gross interest rate schedule _S_ ( _dt_ ), where _dt_ is the central government debt. The unconstrained local government faces the gross interest rate schedule _R_ ( _b_<sup>_n_</sup> _i_ , _t_<sup>) for debt</sup><sup>_bn_</sup> _i_ , _t_<sup>.</sup> The separate interest rate schedules for the central and local governments make the analysis more transparent, and the distinction is not essential to the theoretical results. We make the following assumption on the interest rate schedules to ensure interior solutions.<sup>14</sup>

**Assumption 2.** Both interest rate schedules _R_ ( _·_ ) and _S_ ( _·_ ) are increasing and convex functions, and there exist _B_<sup> ̄</sup> , _D>_<sup> ̄</sup> 0 such that _R_ ( _B_<sup> ̄</sup> ) = _S_ ( _D_<sup> ̄</sup> ) = 1/ _β_ .

**Transfers.** In each period, the central government optimally chooses the amount of transfers to each local government. At time _t_ , for each region _i_ , the only relevant _individual_ state variable is its own

> 12The lack of autonomy to borrow for some local governments can be understood either as the result of financial frictions or self-imposed fiscal discipline. In Appendix C and for all the proofs of propositions (Appendix E), we present a more general case where the constrained local governments face a constraint on borrowing _b_<sup>_o_</sup> _i_ , _t_<sup>_≤Bo_with the exogenous debt</sup> limit _B_<sup>_o_</sup> _≥_ 0.

> 13In our setup, the difference in debt service burden across regions directly comes from the assumption that regions’ abilities to borrow differ. In Appendix F.1 we show an alternative setup, where both types of regions can borrow but with different interest rate schedules. We show that the different interest schedules can also lead to the cross-region difference in debt burden, and that similar theoretical results hold.

> 14The theoretical results in the paper only require increasing interest rate schedules with ranges large enough to ensure an interior solution. The convexity assumption is one way to ensure the range of interest rate levels is large enough. In Appendix B.3, we provide two ways to motivate the increasing interest rate schedules.

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debt stock _bi_ , _t_ . The relevant _aggregate_ state variables are the average debt of the unconstrained regions and the central government debt ( _b_<sup>_n_</sup> _t_<sup>,</sup><sup>_dt_).</sup> In general, we can write the transfers received by region _i_ at time _t_ , _Ti_ , _t_ , as a function of these state variables. For tractability we adopt a first-order approximation: where _αt_<sup>0,</sup><sup>_α_</sup> _t_<sup>1,</sup><sup>_α_</sup> _t_<sup>2and</sup><sup>_α_</sup> _t_<sup>3are the (time-varying) coefficients chosen by the central government.Denote</sup> _αt_<sup>0=</sup><sup>_τt_and</sup><sup>_T_</sup> _t_<sup>_u_=</sup><sup>_α_</sup> _t_<sup>1</sup><sup>_bn_</sup> _t_<sup>+</sup><sup>_α_</sup> _t_<sup>2</sup><sup>_dt_+</sup><sup>_α_</sup> _t_<sup>3, this becomes</sup> which is a more general case for the affine tax/transfer functions used in the literature (e.g., Lucas and Stokey 1983, Aiyagari et al. 2002, Werning 2007, and Bianchi 2016).

Effectively, the central government has two transfer instruments: _Tt_<sup>_u_and</sup><sup>_τt_,whichitdecides</sup> upon knowing the aggregate states ( _b_<sup>_n_</sup> _t_<sup>,</sup><sup>_dt_).Each unconstrained region receives</sup><sup>_τtbn_</sup> _i_ , _t_<sup>+</sup><sup>_T_</sup> _t_<sup>_u_and each</sup> constrained region receives _Tt_<sup>_u_.Here</sup><sup>_T_</sup> _t_<sup>_u_can be interpreted as the uniform transfer identical across all</sup> regions, and _τt_ is the rate at which the central government compensates for the debt service burden of unconstrained regions. Both instruments reduce the vertical fiscal imbalances between the central and local governments. Additionally, any positive _τt_ also reduces the cross-region fiscal gap for given _b_<sup>_n_</sup> _t_<sup>_>_0.As such, the transfer function can be interpreted as a form of redistributive transfer, whereby</sup> the central government has a tendency to transfer funds from fiscally rich to fiscally poor regions.<sup>15</sup> We further impose some lower bounds on both transfer instruments:

**Assumption 3.** _Tt_<sup>_u_and</sup><sup>_τt_are subject to lower bounds:</sup>

- (a) _Tt_<sup>_u≥T_ ̄, where</sup><sup>_T_ ̄</sup><sup>_≥_0 is a constant;</sup>

- (b) _τt ≥ τ_  ̄, where _τ_  ̄ < 0 is a constant and satisfies _βR_ (0)(1 _− τ_  ̄) _<_ 1.

The lower bound of uniform transfer, _T_<sup> ̄</sup> , captures that some basic locally-provided public goods are usually partially financed by federal transfers to ensure minimum quality and quantity (e.g., education and medical care). It is easy to see that _T_<sup> ̄</sup> and local governments’ own revenue _f_ are substitutable: changing ( _f_ , _T_<sup> ̄</sup> ) to ( _f_ + _T_<sup> ̄</sup> ,0) would only decrease the equilibrium intergovernmental transfers by _T_<sup> ̄</sup> , without altering other elements of the equilibrium allocation. For our theoretical properties to hold, we only need _T_<sup> ̄</sup> to be non-negative (that is, _Tt_<sup>_u≥T_ ̄can be relaxed to</sup><sup>_T_</sup> _t_<sup>_u≥_0), so</sup> that the central government cannot tax away local governments’ own revenue. In the quantitative exercise, we calibrate _T_<sup> ̄</sup> such that the simulated moment for total transfers matches the data.<sup>16</sup> The assumption _τt ≥ τ_ ̄ (where _τ<_ ̄ 0) can be replaced with a stronger and more conventional assumption _τt ≥_ 0 without affecting the rest of the paper. Later we will show that the Ramsey optimal

> 15In Appendix B.2, we discuss anecdotal examples from Spain, Germany, and several other countries that transfers are redistributive and approximately dependent on debt or deficit. We also present empirical evidence showing that the Spanish central government implicitly takes into account local governments’ debt when it decides the amount of transfers.

> 16In Appendix F.2, we relax Assumption 3(a). We instead assume that the central government faces an adjustment cost of changing _T_<sup>_u_</sup> , and show that the theoretical properties of the model still hold.

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policy entails a zero _τt_ at the steady state. With the lower bound at _τ_ ̄ instead of zero, it is clear that the zero optimal _τ_ is not a corner solution.

The following assumption states that the vertical fiscal imbalances are too large to be offset by the minimum uniform transfer alone when local government debt _b_<sup>_n_</sup> = _B_<sup> ̄</sup> and central government debt _d_ = _D_<sup> ̄</sup> .

**Assumption 4.** ( **Vertical fiscal imbalances** ) _f_ + _T_<sup> ̄</sup> is small relative to _e_ such that

where _D_<sup> ̄</sup> satisfies _S_ ( _D_<sup> ̄</sup> ) = 1/ _β_ .

**Timing.** Figure 2 illustrates the timeline during period _t_ . Knowing the _aggregate_ states ( _b_<sup>_n_</sup> _t_<sup>,</sup><sup>_dt_)de-</sup> termined in period _t −_ 1 and central revenue _e − f_ , the central government moves first ( _Stackelberg leader_ ) to choose its spending _ct_ , transfer rate _τt_ , and uniform transfer _Tt_<sup>_u_.The deficit is financed by</sup> new debt _dt_ +1 issued at the interest rate _S_ ( _dt_ +1). An unconstrained local government _i ∈_ [0, _μ_ ] with outstanding debt _b_<sup>_n_</sup> _i_ , _t_<sup>receives its own revenue</sup><sup>_f_and transfer</sup><sup>_Ti_,</sup><sup>_t_=</sup><sup>_τtbn_</sup> _i_ , _t_<sup>+</sup><sup>_T_</sup> _t_<sup>_u_, chooses local public</sup> spending _gi_<sup>_n_</sup> , _t_<sup>, and finances its deficit by issuing new debt</sup><sup>_bn_</sup> _i_ , _t_ +1<sup>at the interest rate</sup><sup>_R_(</sup><sup>_bn_</sup> _i_ , _t_ +1<sup>).A con-</sup> strained local government carries no debt and spends all its revenue on public spending _gi_<sup>_o_</sup> , _t_<sup>=</sup><sup>_f_+</sup><sup>_T_</sup> _t_<sup>_u_.</sup>

**Figure 2:** Timeline within period

( _b_<sup>_n_</sup> _t_<sup>,</sup><sup>_dt_)</sup> Central government Local governments ( _b_<sup>_n_</sup> _t_ +1<sup>,</sup><sup>_dt_+1)</sup> _t_ decides transfer policy ( _τt_ , _Tt_<sup>_u_)</sup> repays _bi_ , _t_ , gets transfer _τtbi_ , _t_ + _Tt_<sup>_u_</sup> _t_ + 1 repays _dt_ , borrows _dt_ +1/ _S_ ( _dt_ +1), spends _ct_ borrows _bi_ , _t_ +1/ _R_ ( _bi_ , _t_ +1), spends _gi_ , _t_

**Note on notation.** In our model, all regions of the same type are identical. As such, the many small local governments collapse into one _representative_ constrained local government with population size _μ_ and one _representative_ unconstrained local government with size 1 _− μ_ . In the rest of the paper, wherever appropriate, we drop the subscript _i_ .

# **2.2 Efficiency benchmark: System with a consolidated government**

Before we move to the equilibrium with two layers of government, we first characterize the allocation under a consolidated government that makes decisions for both central and local governments. The benevolent consolidated government maximizes the present value of the weighted average utility of all households. We use this “consolidated government allocation” as the efficiency benchmark.<sup>17</sup>

**Definition 1.** Given ( _b_<sup>_n_</sup> 0<sup>,</sup><sup>_d_0) and the interest rate schedules, the consolidated government’s problem</sup>

> 17We assume that the consolidated government issues two types of debt: local debt and central debt. This assumption makes the consolidated government’s problem a closer benchmark to the equilibrium model with two layers of governments where both types of debt are issued (Sections 2.3-2.4).

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consists of choosing a sequence of debt and spending _{b_<sup>_n_</sup> _t_ +1<sup>,</sup><sup>_dt_+1,</sup><sup>_g_</sup> _t_<sup>_n_,</sup><sup>_g_</sup> _t_<sup>_o_,</sup><sup>_ct}∞_</sup> _t_ =0<sup>that solves</sup> subject to the consolidated budget constraint and the constrained local governments’ no-borrowing condition, where _gt_<sup>_n_and</sup><sup>_g_</sup> _t_<sup>_o_are the local spend-</sup> ing of the unconstrained and constrained local governments, respectively.

Here we assume that governments do not internalize the _slope_ of their interest rate schedules, which means that governments take their interest rates as given. As such, there are no derivatives of the interest rate functions _R_ ( _·_ ) and _S_ ( _·_ ) in the optimality conditions below.<sup>18</sup> This modeling choice greatly simplifies notations, without changing the theoretical properties of the Markov equilibrium that we will discuss later. More importantly, when governments internalize interest rate schedules, there is a _pecuniary externality_ , which potentially also generates overborrowing (e.g., Kim and Zhang 2012). We shut down this channel to highlight that overborrowing in our model is not a result of the pecuniary externality. In Section 4.3.1, we present an extension where governments internalize the slope of their interest rate schedules as a robustness exercise.

**Proposition 1.** The consolidated government’s optimal allocation satisfies

At the steady state, _b_<sup>_n_</sup> _t_<sup>=</sup><sup>_B_ ̄,</sup><sup>_dt_=</sup><sup>_D_ ̄, and</sup><sup>_S_(</sup><sup>_dt_) =</sup><sup>_R_(</sup><sup>_bn_</sup> _t_<sup>).Condition (3) indicates that the consoli-</sup> dated government achieves **perfect resource sharing** between the central and local levels and there is no cross-region difference in the quantity of locally-provided public goods. Under log utility, (3) implies that the central-local government spending ratio _ct_ / _gt_<sup>_n_=</sup><sup>_ct_/</sup><sup>_g_</sup> _t_<sup>_o_=</sup><sup>_θ_/(1</sup><sup>_−θ_).It is easy to see</sup> that the consolidated government allocation is **efficient** .

# **2.3 Distortion without a consolidated government**

Without a consolidated government, each local or central government makes its own decisions. Central-to-local transfers distort the unconstrained local governments’ borrowing decisions, as their higher borrowings today lead to higher transfers tomorrow.

Given the aggregate states ( _b_<sup>_n_</sup> _t_<sup>,</sup><sup>_dt_) and individual state</sup><sup>_bn_</sup> _i_ , _t_<sup>, an unconstrained local government</sup><sup>_i_</sup> chooses its spending _gi_<sup>_n_</sup> , _t_<sup>and debt</sup><sup>_bn_</sup> _i_ , _t_ +1<sup>to maximize the welfare of its residents.Its problem written</sup>

> 18 More specifically, when a local government does not internalize its interest rate schedule, its marginal interest cost of higher _bi_ is simply _Ri_ = _R_ ( _bi_ ). In comparison, for a local government that internalizes the interest rate schedule, the marginal cost of higher _bi_ is _Ri_ +<sup>_<u>∂R</u>_</sup> _∂bi_<sup>_<u>i</u>bi_, where the extra term represents the increased interest cost of</sup><sup>_bi_.</sup>

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recursively is subject to the budget constraint Taking interest rates and the central government’s transfer policies ( _τt_ , _Tt_<sup>_u_)asgiven,itsoptimal</sup> choice of _b_<sup>_n_</sup> _i_ , _t_ +1<sup>is characterized by the Euler equation,</sup> Comparing (8) and (5), as long as _τt_ +1 _>_ 0, the unconstrained governments **overborrow** relative to the local debt level chosen by a consolidated government. Given this distortionary effect of _τ_ , the central government faces the following trade-off: On the one hand, the transfer through _τ_ provides more ex post debt relief to the unconstrained regions, who have higher debt service burdens than the constrained regions. On the other hand, all things equal, a positive _τ_ lowers local governments’ borrowing cost and induces overborrowing. Because of this trade-off, there is a classic time-inconsistency problem. At time _t_ , the central government always wants to promise a low future transfer rate _τt_ +1 (as well as _τt_ +2, _τt_ +3...) to reduce overborrowing ceteris paribus. When time _t_ + 1 comes, however, the central government would like to renege the promise of a low _τt_ +1 and make more transfers to the unconstrained regions.

With this time-inconsistency problem, it matters whether the central government can commit to future policies. We first look at the equilibrium in which the central government does not have the ability to pre-commit to future policies _{τt_ , _Tt_<sup>_u_,</sup><sup>_ct_,</sup><sup>_dt_+1</sup><sup>_}t>t_</sup> 0<sup>atanytime</sup><sup>_t_0.Wethencompareit</sup> with the equilibrium in which the central government can make commitment about future policies (the Ramsey solution). We also discuss ways to implement the consolidated government allocation regardless of commitment.

# **2.4 Equilibrium without commitment**

## **2.4.1 Equilibrium definition**

We define the equilibrium recursively (without time subscripts), using prime to denote next period values and dropping the subscript _i_ for the local government. Given the beginning-of-period states of the economy ( _b_<sup>_n_</sup> , _d_ ), the central government chooses spending _c_ , debt _d_<sup>_′_</sup> , transfer rate _τ_ and uniform transfer _T_<sup>_u_</sup> to maximize the welfare of its residents, subject to the unconstrained local governments’ Euler equation (8). We follow Klein et al. (2008) to define the Markov-perfect equilibrium where policy functions depend differentiably on current states ( _b_<sup>_n_</sup> , _d_ ) and central government’s optimal policy is time-consistent.

For convenience, define the following functions of (representative) local and central government 11
spending, derived from their budget constraints where as defined before _μ_ is the measure of unconstrained regions, and _μb_<sup>_n_</sup> is the total amount of local government debt given constrained regions have zero debt.

**Definition 2.** A Markov-perfect equilibrium consists of a value function _V_ , central government’s policy rules _{_ Φ<sup>_τ_</sup> , Φ<sup>_T_</sup> , Φ<sup>_d_</sup> _}_ , and a policy function Φ<sup>_b_</sup> for the unconstrained local government’s debt, such that for all aggregate states ( _b_<sup>_n_</sup> , _d_ ), _τ_ = Φ<sup>_τ_</sup> ( _b_<sup>_n_</sup> , _d_ ), _T_<sup>_u_</sup> = Φ<sup>_T_</sup> ( _b_<sup>_n_</sup> , _d_ ), _d_<sup>_′_</sup> = Φ<sup>_d_</sup> ( _b_<sup>_n_</sup> , _d_ ) and _b_<sup>_n′_</sup> = Φ<sup>_b_</sup> ( _b_<sup>_n_</sup> , _d_ ) solve

and the central government’s value function satisfies the functional equation In this equilibrium, the expected transfers tomorrow follow _τ_<sup>_′_</sup> = Φ<sup>_τ_</sup> ( _b_<sup>_n′_</sup> , _d_<sup>_′_</sup> ) and _T_<sup>_u′_</sup> = Φ<sup>_T_</sup> ( _b_<sup>_n′_</sup> , _d_<sup>_′_</sup> ). Because each local government is infinitesimally small, it takes _{τ_ , _τ_<sup>_′_</sup> , _T_<sup>_u_</sup> , _T_<sup>_u′_</sup> _}_ as given. However, when the central government decides _τ_ and _T_<sup>_u_</sup> today, it understands that its choices will affect not only _b_<sup>_n′_</sup> , _d_<sup>_′_</sup> , but also _τ_<sup>_′_</sup> and _T_<sup>_u′_</sup> through Φ<sup>_τ_</sup> ( _b_<sup>_n′_</sup> , _d_<sup>_′_</sup> ) and Φ<sup>_T_</sup> ( _b_<sup>_n′_</sup> , _d_<sup>_′_</sup> ).

In the equilibrium, policy rules Φ = _{_ Φ<sup>_τ_</sup> , Φ<sup>_T_</sup> , Φ<sup>_d_</sup> , Φ<sup>_b_</sup> _}_ solve the central government’s problem each period, given the expectation that the central government’s choices in the next period and after will also follow Φ. In other words, if the central government today “announces” that Φ will be implemented tomorrow and thereafter (even though it does not have commitment or credibility), then when tomorrow comes, it will indeed find Φ optimal. This makes this equilibrium time-consistent.

# **2.4.2 Equilibrium characterization**

The equilibrium can be characterized by conditions (12)–(13) and<sup>19</sup>

> 19 _n_ Simplified notations, e.g., _ungg_ for _ugg_ ( _G_ ), are used to make the optimality conditions more compact. We use prime to denote future variables, and subscript for partial derivatives, e.g., Φ _d_<sup>_T′_denotes</sup><sup>_∂_Φ</sup><sup>_T_(</sup><sup>_bn′_,</sup><sup>_d′_)/</sup><sup>_∂d′_.Appendix D.1 gives</sup> the derivation of these optimality conditions.

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where and _λ_ and _ζ_ are the Lagrange multipliers on constraints (12) and (13), respectively. The policy derivatives such as Φ<sup>_T′_and</sup><sup>_d′_on future policies.</sup> _b_<sup>_n_and Φ</sup><sup>_τ_</sup> _d_<sup>_′_capture the effects of today’s choices of</sup><sup>_bn′_</sup> We can obtain some sharper characterizations of the Markov equilibrium at the steady state, given 1-4. Assumptions

**Lemma 1.** At the Markov equilibrium steady state, the unconstrained regions have lower public spending than the constrained regions: _g_<sup>_n_</sup> _< g_<sup>_o_</sup> .

_Proof._ See Appendix E.1.

Because the unconstrained regions have larger debt service burdens than the constrained regions, the central government always wants to transfer more to the unconstrained regions than the constrained ones. This can only be achieved by transferring through _τ_ . But because _τ_ encourages the unconstrained regions to borrow more, these regions end up with even larger debt service burdens. As a result, at the steady state, the spending gap between the two types of regions is not eliminated.

**Proposition 2.** At the Markov equilibrium steady state, the central government gives the minimum uniform transfer _T_<sup>_u_</sup> = _T_<sup> ̄</sup> and sets region-specific transfer _τ>_ 0.<sup>20</sup>

_Proof._ See Appendix E.3.

Intuitively, because the central government wants to transfer more to the unconstrained regions, it will always try to minimize the size of _T_<sup>_u_</sup> and instead transfer through _τ_ . The central government understands that _τ_ is distortionary as higher _τt_ leads to higher _b_<sup>_n_</sup> _t_<sup>.Ifthecentralgovernmentcan</sup> make policy commitment, it would like to pre-commit a low _τt_ in period _t −_ 1. But because the central government here does not have commitment, it takes _b_<sup>_n_</sup> _t_<sup>as given when it chooses</sup><sup>_τt_at time</sup><sup>_t_,</sup> and hence it does not take into account the ex ante overborrowing effect of _τt_ on _b_<sup>_n_</sup> _t_<sup>.In other words,</sup>

> 20It can be proved that Propositions 2 and 3 still hold when governments can internalize the slopes of interest rate schedules, i.e., when the optimality conditions (14)–(17) contain terms such as _Rb_ and _Sd_ . We report the quantitative results under this alternate assumption in Section 4.3.1.

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even though the central government knows that its choice of _τ_ is distortionary ex ante, it does not internalize this distortionary effect, exactly because it cannot make an ex ante commitment about _τ_ .

The policy constraint _T_<sup>_u_</sup> _≥ T_<sup> ̄</sup> _≥_ 0 is important for the equilibrium. Without it, the central government would want to do more cross-region redistribution by using a negative _T_<sup>_u_</sup> (i.e., a lump sum tax on all regions) to finance a higher _τ_ , which encourages local governments to borrow more ex ante, and leads to even higher _τ_ and lower _T_<sup>_u_</sup> . Equilibrium is thus either non-existing or exhibits an unrealistically high level of local debt.

**Proposition 3.** ( **Over-transfer** ) At the Markov equilibrium steady state, the average marginal utility of regional government spending is smaller than the marginal utility of central government spending, after adjusting for the utility weight _θ_ :

_Proof._ Appendix E.2 provides the full proof. Here we offer a heuristic explanation for the proof. From condition (15), because _u_<sup>_n_</sup> _gg_<sup>_<_0andtheLagrangemultiplier</sup><sup>_ζ≥_0,itiseasytoseethatthe</sup> essence of proving Proposition 3 is to show that the Lagrange multiplier for the representative unconstrained region’s Euler equation (12) is positive: _λ>_ 0. Condition (12) can be replaced with two inequalities: If we can show that only (19) is potentially binding around the steady state while (20) is redundant, we can come to the conclusion that _λ_ is positive around the steady state. To see why (20) is redundant, note that in the absence of (12), the central government would always want to choose allocations such that _u_<sup>_n_</sup> _g_<sup>=</sup><sup>_βR′un_</sup> _g_<sup>_′_, which is sufficient to ensure (20) holds for any transfer rate</sup><sup>_τ′ >_0.</sup> The central government’s over-transfer result in Proposition 3 is _not_ a direct implication of local governments’ overborrowing. In the consolidated government benchmark, **perfect resource sharing** in condition (3) implies that the ( _θ_ -adjusted) average marginal utility of local governments (“MULG”) exactly equals the ( _θ_ -adjusted) marginal utility of the central government (“MUCG”). That is, the left-hand side of (18) would exactly equal the right-hand side. In the Markov equilibrium, because transferring through _τ_ is distortionary, one may (mistakenly) think that the central government would _under-transfer_ , to the extent that MULG _>_ MUCG. To the contrary, Proposition 3 states that the central government _over-transfers_ in the Markov equilibrium, to the extent that MULG _<_ MUCG.

The reason behind the over-transfer result includes two layers. First, despite the distortionary effect of _τt_ , the central government does not have incentives to reduce _τt_ from the level required for MULG = MUCG. This is because in the Markov equilibrium, the time- _t_ central government takes the local debt _b_<sup>_n_</sup> _t_<sup>asgiven,andhenceneglectsthe</sup><sup>_exante_overborrowingeffectof</sup><sup>_τt_on</sup><sup>_bn_</sup> _t_<sup>.Second</sup> and more importantly, the central government has an incentive to increase _τt_ to be above the level 14
required for MULG = MUCG. This is because the marginal cost of local borrowing is higher for the entire fiscal federation ( _R_ ( _b_<sup>_n_</sup> _t_ +1<sup>)) than for the unconstrained regions ((1</sup><sup>_−τt_+1)</sup><sup>_R_(</sup><sup>_bn_</sup> _t_ +1<sup>)).This dif-</sup> ference gives the central government an incentive to lower local debt _b_<sup>_n_</sup> _t_ +1<sup>.But without commitment,</sup> the central government cannot promise lower future transfer rate _τt_ +1 to reduce the unconstrained regions’ _incentive to borrow_ . Instead, it increases the current transfer rate _τt_ to relax the unconstrained regions’ budget constraint at time _t_ and hence their _need to borrow_ . The higher transfers are financed by lower central government spending (and higher central government debt), which explains why MULG < MUCG.<sup>21</sup> The over-transfer result of Proposition 3 is somewhat unique to our infinite-horizon environment. In a typical static or two-period model setting, the central government’s lack of commitment is modeled as the central government moving after local governments or private agents. In the last period of such models, as the last mover, the central government has no future (local or central) governments’ actions to take into account. That is, its optimization problem does not have local governments’ Euler equation (12) as a constraint. It thus sets the transfer in the last period to achieve the within-period perfect resource sharing, MULG = MUCG.<sup>22</sup> With infinite horizons, however, the central government takes into account how its choice of ( _τ_ , _T_<sup>_u_</sup> ) affects future (central and local) governments’ choices. The impact of raising _τ_ on local government’s future debt choice _b_<sup>_′_</sup> , which is captured by the right hand side of (14), drives the over-transfer result.<sup>23</sup>

# **2.4.3 Effects of fiscal decentralization**

How do fiscal decentralization reforms affect aggregate fiscal performance? In the model, a revenue decentralization (larger _f_ ) results in lower total government debt, whereas a spending decentralization (higher 1 _− θ_ ) leads to higher total government debt. We use numerical examples to help illustrate the effects of fiscal decentralization on the steady state debt levels. The examples are based on the calibration in Section 3.

A **revenue decentralization** increases local governments’ revenue _f_ and decreases the central government’s revenue _e − f_ one-to-one. This narrows the vertical fiscal imbalances. All else equal, the central government wants to reduce its transfers to local governments. Because the equilibrium

> 21Appendix D.1.1 provides an alternative way to understand the intuition of the over-transfer result, based on the Generalized Euler Equations.

> 22This can be seen from Equation (14) when _λ_ = 0. In the literature, similar “equal marginal utilities” results can be found in many static or two-period settings of a fiscal or monetary union. For example, in Chari and Kehoe (2007), the central monetary authority can use monetary policy to partially inflate away regions’ nominal debt payment. When the central monetary authority moves last (i.e., no commitment), it chooses the inflation level such that the marginal output loss from higher inflation equals the marginal benefits from the regions’ lower debt payments. In Akai and Sato (2011) and Kothenburger (2007), the central government does not provide central public goods directly but derives utility from the public goods provided by local governments. In the equilibrium without commitment, the central government chooses transfers such that the marginal utilities from local public goods are equalized across regions.

23In Appendix G, we show that in a three-period model, under some conditions, there can be over-transfer in the initial period. However, in the final period of the three-period model, MULG = MUCG still has to hold. Similarly, for models with more than three (but finite) periods, there can be over-transfer in the initial periods, but over-transfer will weaken and eventually disappear in the final period. That is, the over-transfer property, if it holds, is only _temporary_ in a finite-period model. In contrast, in our infinite-horizon model, over-transfer occurs in every period around the steady state.

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government becomes too financially stretched to further increase the transfers.

**Remark.** Before we end this section, we want to highlight that the only assumption we have made on the interest rate schedules _S_ ( _·_ ) and _R_ ( _·_ ) is that they are increasing and convex functions (Assumption 2). In particular, we do **not** require that the central government’s interest rate schedule _S_ ( _·_ ) is more favorable than local governments’ _R_ ( _·_ ). As such, the over-transfer result (Proposition 3) and the impact of fiscal decentralization are **not** because the central government wants to replace local debt with cheaper central debt to reduce net interest cost. Even if we do not allow the central government to borrow (i.e., _dt ≡_ 0 for any _t_ ), the propositions in Section 2.4.2 still hold, and a revenue (spending) decentralization still reduces (increases) local government debt.<sup>24</sup>

# **2.5 Equilibrium with commitment (Ramsey)**

In the Ramsey problem, the central government chooses the entire sequence of policies and allocations at time 0, and sticks to the choices at any time _t ≥_ 0. Because of this commitment to future policies, the Ramsey central government takes into account how time- _t_ policy affects local governments’ decisions at time _t −_ 1 (as well as _t −_ 2, _t −_ 3, ...). This is the key difference from the Markov equilibrium. Formally,

**Definition 3.** Given initial debt positions ( _b_<sup>_n_</sup> 0<sup>,</sup><sup>_d_0), the optimal (Ramsey) policy of a central govern-</sup> ment with commitment consists of a sequence of debt and transfer policies _{τt_ , _Tt_<sup>_u_,</sup><sup>_bn_</sup> _t_ +1<sup>,</sup><sup>_dt_+1</sup><sup>_}∞_</sup> _t_ =0<sup>that</sup> solves

subject to the unconstrained regions’ Euler equation and transfer policy constraint for _all_ time _t ≥_ 0 where the budget constraints of the central and local governments are implicit in the spending functions _C_ , _G_<sup>_n_</sup> , and _G_<sup>_o_</sup> as defined in (9)–(11).

Let _β_<sup>_t_</sup> _γt_ and _β_<sup>_t_</sup> _ζt_<sup>_r_be the Lagrange multipliers on the two constraints above respectively.Written</sup>

> 24The only change to the derivations in Section 2.4.2 is that Equation (17), the first-order condition with respect to _d′_ , is replaced with _d_<sup>_′_</sup> = 0.

17
recursively, the first-order conditions of the Ramsey problem are<sup>25</sup> The distinctions between the Ramsey equilibrium and the equilibrium without commitment (Markov) can be seen, for example, by comparing (14) with (23). Both conditions characterize the direct effect of increasing _τ_ . Because the Ramsey central government can pre-commit to future policies, it takes into account that an increase in _τ_ raises the unconstrained regions’ borrowing in the previous period. This “commitment effect” is captured by the _γ_<sup>_−_</sup> term in (23), where _γ_<sup>_−_</sup> is the shadow value of loosening the unconstrained local governments’ Euler equation in the previous period. By contrast, in the Markov equilibrium, the central government does not consider how its choice today affects local governments’ borrowings in the previous period, and so there is no _γ_<sup>_−_</sup> term in (14).<sup>26</sup> This distinction gives rise to the potential time-inconsistency problem of the Ramsey allocation. At time 0, when the central government chooses the sequence of optimal transfer plan _{τt_ , _Tt_<sup>_u}∞_</sup> _t_ =0<sup>, it</sup> understands that the time- _t_ policy _τt_ affects unconstrained local governments’ debt _b_<sup>_n_</sup> _t_<sup>.However, this</sup> _τt_ may be suboptimal at time _t_ after _b_<sup>_n_</sup> _t_<sup>is realized, because the effect of</sup><sup>_τt_on</sup><sup>_bn_</sup> _t_<sup>is already foregone.</sup> In other words, the policy that is optimal before observing _b_<sup>_n_</sup> _t_<sup>may differ from the optimal plan after</sup> _b_<sup>_n_</sup> _t_<sup>is realized.</sup> We show that the Ramsey equilibrium steady state exhibits _under-transfer_ , in contrast to the _overtransfer_ property in the Markov equilibrium steady state.

**Proposition 4.** ( **Under-transfer** ) At the Ramsey equilibrium steady state, the central government sets the distortionary transfer rate _τ_ = 0 and the uniform transfer _T_<sup>_u_</sup> _> T_<sup> ̄</sup> . Additionally,

_Proof._ See Appendix E.4.<sup>27</sup>

> 25 _−_ Prime indicates future values, and “ ” indicates variables of the previous period. Writing these first-order conditions recursively is more compact and makes for easier comparison with the Markov equilibrium conditions. Appendix D.2 gives the derivation of these conditions.

> 26Another difference between the Ramsey and Markov equilibria is that, compared to the Ramsey optimality conditions, the optimality conditions of the Markov equilibrium have additional terms with policy derivatives, e.g., the term Ω _b′_ that collects policy derivatives. These policy derivatives reflect that the central government without commitment considers how its policy choices today affect future central governments’ choices. These policy derivative terms are absent in the Ramsey equilibrium because the Ramsey central government can directly choose and commit to all future policies.

> 27When governments internalize the slope of their interest rate schedules, _τ_ = 0 may not hold, but _T u > T_  ̄ and (27) still hold. We provide more details on this alternative setting in Section 4.3.1.

18
Here we give some intuition for Proposition 4. The Ramsey central government takes into account the overborrowing effect of _τ_ (the term with _γ_<sup>_−_</sup> in condition 23). Proposition 4 shows that this additional cost is large enough to offset the incentive to use _τ_ to close the cross-region fiscal gap. Note that _τ_ = 0 is not a corner solution as _τ_ is bounded below by a negative value (Assumption 3) and _not_ by 0. With _τ_ = 0, it is then obvious that the Ramsey central government cannot redistribute between the two types of regions. As long as there is a cross-region fiscal gap, local public spending in the two regions cannot be equalized ( _g_<sup>_n_</sup> = _g_<sup>_o_</sup> ), and the Ramsey solution is not efficient. It is worth noting that because the unconstrained regions overborrow when _τ>_ 0, using _τ_ to transfer will only _widen_ the cross-region spending gap, and the Ramsey central government internalizes that.

Note that (27) establishes that MULG > MUCG. That is, at the steady state, the Ramsey central government “under-transfers”: its optimal choice of uniform transfer _T_<sup>_u_</sup> is smaller than the level needed to achieve perfect resource sharing. To see why, note that a marginal increase in _T_<sup>_u_</sup> has three effects, captured by terms in the condition (24). First and foremost, it shifts resources from central to local governments (the left-hand side of 24). If this is the only effect from a higher _T_<sup>_u_</sup> , then the optimal _T_<sup>_u_</sup> would achieve perfect resource sharing (MULG=MUCG). Second, a higher _T_<sup>_u_</sup> relaxes the minimum uniform transfer constraint _T_<sup>_u_</sup> _≥ T_<sup> ̄</sup> , captured by the _ζ_<sup>_r_</sup> term in (24). This effect is zero at the steady state ( _ζ_<sup>_r_</sup> = 0), because with _τ_ = 0 the vertical fiscal imbalances require a large _T_<sup>_u_</sup> which makes _T_<sup>_u_</sup> _≥ T_<sup> ̄</sup> non-binding. Third, for any marginal increase in _Tt_<sup>_u_at time</sup><sup>_t_, the</sup> unconstrained regions want to smooth spending inter-temporally by increasing _b_<sup>_n_</sup> _t_<sup>and decreasing</sup> _b_<sup>_n_</sup> _t_ +1<sup>.These effects are captured on the right-hand side of (24) by the terms containing the shadow</sup> values _γ_ and _γ_<sup>_−_</sup> of relaxing the unconstrained regions’ Euler equation. At the steady state, the net effect of higher _T_<sup>_u_</sup> through the changes in local governments’ borrowings is _−γu_<sup>_n_</sup> _gg_<sup>+</sup><sup>_γR_(1</sup><sup>_−τ_)</sup><sup>_un_</sup> _gg_ and is negative (i.e., a net social cost) when _τ_ = 0.<sup>28</sup> Because of this additional social cost of higher _T_<sup>_u_</sup> , the central government sets _T_<sup>_u_</sup> below the efficient level and hence MULG > MUCG.

**Corollary 1.** The Ramsey steady state local and central government debt levels are the same as in the consolidated government allocation, _b_<sup>_n_</sup> = _B_<sup> ̄</sup> and _d_ = _D_<sup> ̄</sup> .

Because (26) is the same as (4), central debt at the Ramsey steady state is the same as under the consolidated government. It follows from _τ_ = 0 that (25) is identical to (5) at the steady state, and so the Ramsey steady state _b_<sup>_n_</sup> is also the same as the level under the consolidated government.

**Summary of theoretical results.** Table 1 summarizes the theoretical results.

# **2.6 Implementation of consolidated government allocation**

As illustrated in Table 1, neither the Markov nor the Ramsey equilibrium can achieve the consolidated government allocation. In this section, we show that a _prudential_ tax, combined with the

> 28Appendix E.4 shows that _γ>_ 0 around the steady state. Note that _τ_ = 0 is an equilibrium outcome and does not necessarily imply that constraint (21) is redundant. This is because without this constraint, the central government does not take into account the effect of _τ_ on local borrowing, and is always willing to choose _τ>_ 0 to mitigate the horizontal fiscal imbalance. As such, in the Ramsey equilibrium, the binding part of the Euler equation is _u_<sup>_n_</sup> _g_ , _t_<sup>_≤βR_</sup> _t_ +1<sup>(1</sup><sup>_−τ_</sup> _t_ +1<sup>)</sup><sup>_un_</sup> _g_ , _t_ +1<sup>,</sup> and so _γ>_ 0.

19
**Table 1:** Summary of Theoretical Results at Steady State

||Consolidated<br>|Equilibrium without|consolidated government|
|---|---|---|---|
||government<br>allocation|Non-commitment<br>(Markov)|Commitment<br>(Ramsey)|
|Transfer policy|-|_T_<sup>_u_</sup>= <sup> ̄</sup>_T_,_τ>_0|_T_<sup>_u_</sup>_>_ <sup> ̄</sup>_T_,_τ_=0|
|Debt|No overborrowing<br>_b_<sup>_n_</sup>= <sup> ̄</sup>_B_,_d_= <sup> ̄</sup>_D_|Overborrowing<br>_b_<sup>_n_</sup>_>_ <sup> ̄</sup>_B_|No overborrowing<br>_b_<sup>_n_</sup>= <sup> ̄</sup>_B_,_d_= <sup> ̄</sup>_D_|
|Horizontal spending gap|None,_g_<sup>_n_</sup>=_g_<sup>_o_</sup>|_g_<sup>_n_</sup>_< g_<sup>_o_</sup>|_g_<sup>_n_</sup>_< g_<sup>_o_</sup>|
|Vertical sendin a|None|Over-transfer|Under-transfer|
|pg gp|MULG = MUCG|MULG < MUCG|MULG > MUCG|

Note: This table summarizes the theoretical results of Propositions 1–4, Lemma 1, and Corollary 1. MULG (MUCG) stands for the _θ_ -adjusted average marginal utility of local (central) governments: MULG= (1 _− θ_ )[ _μu_<sup>_n_</sup> _g_ + (1 _− μ_ ) _u_<sup>_o_</sup> _g_ ]; MUCG= _θvc_ . We assume governments do not internalize the slopes of interest rate schedules.

existing transfer instruments ( _τt_ , _Tt_<sup>_u_), can achieve this efficient allocation.The optimal policy with</sup> this prudential tax is time-consistent, which means that the central government does not have incentives to deviate from it even without commitment to future policies. Intuitively, similar to Bianchi (2016), the prudential tax offsets the ex ante overborrowing distortion created by _τt_ , and allows the central government to use the transfer instruments _τt_ and _Tt_<sup>_u_to exactly offset both the vertical and</sup> horizontal fiscal imbalances.

We endow the central government with an additional instrument: a prudential tax _taxt_ on local government borrowing, to be implemented ex ante. The tax proceeds are rebated to unconstrained local governments in a lump sum fashion. Figure 4 illustrates the timeline with the additional instrument.

**Figure 4:** Timeline with prudential tax

|Unconstrained local|(_b_<sup>_n_</sup><br>_t_<sup>,</sup><sup>_dt_)</sup>|Central government|Unconstrained local<br>(_b_<sup>_n_</sup><br>_t_+1<sup>,</sup><sup>_dt_+1)</sup>|
|---|---|---|---|
|_t−_1<br>receives(1_−taxt−_1)_b_<sup>_n_</sup><br>_i_,_t_<sup>/</sup><sup>_Ri_,</sup><sup>_t_</sup>|_t_|decides(_taxt_,_τt_,_T_<sup>_u_</sup><br>_t_<sup>)</sup>|repays_b_<sup>_n_</sup><br>_it_<br>_t_+1|
|from debt issuance||<br><br>repays_dt_|gets transfer_τtb_<sup>_n_</sup><br>_i_,_t_<sup>+</sup><sup>_Tu_</sup><br>_t_<br>,|
|||borrows_dt_+1/_St_+1|receives(1_−taxt_)_b_<sup>_n_</sup><br>_i_,_t_+1<sup>/</sup><sup>_Ri_,</sup><sup>_t_+1</sup><br>from debt issuance|

Formally, the unconstrained local governments’ problem is similar as before, except that their budget constraint (7) becomes The lump-sum rebate _Treb_<sup>_n_</sup> , _t_<sup>=</sup><sup>_taxt· b_</sup> _t_<sup>_n_</sup> +1<sup>/</sup><sup>_R_(</sup><sup>_bn_</sup> _t_ +1<sup>) is equal to the total tax receipt (hence no</sup><sup>_i_on</sup><sup>_bn_</sup> _t_ +1<sup>)</sup> equally distributed among unconstrained regions, and is taken as given by local governments. Constrained local governments’ revenue and spending are the same as before, as they are not subject to 20
the prudential tax or receiving any rebate.

The central government’s Ramsey problem with prudential tax (R2) becomes _∞_ max ∑︁ _β_<sup>_t_</sup> _{_ (1 _− θ_ )[ _μu_ ( _gt_<sup>_n_) + (1</sup><sup>_−μ_)</sup><sup>_u_(</sup><sup>_g_</sup> _t_<sup>_o_)] +</sup><sup>_θv_(</sup><sup>_ct_)</sup><sup>_}_</sup> (R2) _{ct_ , _gt_<sup>_n_,</sup><sup>_g_</sup> _t_<sup>_o_,</sup><sup>_bn_</sup> _t_ +1<sup>,</sup><sup>_dt_+1,</sup><sup>_τt_,</sup><sup>_T_</sup> _t_<sup>_u_,</sup><sup>_taxt}∞_</sup> _t_ =0 _t_ =0 subject to and _Tt_<sup>_u≥T_ ̄.Equations (29)–(31) are the Euler equation and budget constraints of the</sup><sup>_representative_</sup> local governments. Equation (32) is the resource constraint for the entire government sector. Notice that (30) comes from aggregating the budget constraints of all unconstrained regions (28), and is identical to the budget constraint without prudential tax because total tax receipt equals total rebate.

**Proposition 5.** Suppose the central government can implement a prudential tax _taxt_ on local debt _b_<sup>_n_</sup> _t_ +1<sup>at time</sup><sup>_t_, in addition to the transfer instruments</sup><sup>_τt_and</sup><sup>_T_</sup> _t_<sup>_u_.Then the Ramsey optimal policy that</sup> solves (R2) can achieve the consolidated government allocation and is time-consistent.

_Proof._ We show that the consolidated government allocation satisfies (29)–(32) with appropriately constructed sequence _{tax_<sup>ˆ</sup> _t_ , ˆ _τt_ , _T_<sup>ˆ</sup> _t_<sup>_u}_.Given (</sup><sup>_bn_</sup> 0<sup>,</sup><sup>_d_0), the consolidated government allocation, denoted</sup> by _{b_<sup>_n_</sup> _t_<sup>_*_,</sup><sup>_d*_</sup> _t_<sup>,</sup><sup>_g_</sup> _t_<sup>_n*_,</sup><sup>_g_</sup> _t_<sup>_o*_,</sup><sup>_c*_</sup> _t_<sup>_}_, automatically satisfies the resource constraint (32).Choose</sup><sup>_{τ_ˆ</sup><sup>_t_,</sup><sup>_T_ˆ</sup> _t_<sup>_u}_such that</sup> the local governments’ budget constraints (30)–(31) are satisfied Construct _{tax_<sup>ˆ</sup> _t}_ such that It is then straightforward to verify that Euler equation (29) becomes the same as the consolidated government’s optimality condition with respect to _b_<sup>_n_</sup> _t_ +1<sup>(condition 5).Therefore,the consolidated</sup> government allocation _{b_<sup>_n_</sup> _t_<sup>_*_,</sup><sup>_d*_</sup> _t_<sup>,</sup><sup>_g_</sup> _t_<sup>_n*_,</sup><sup>_g_</sup> _t_<sup>_o*_,</sup><sup>_c*_</sup> _t_<sup>_}_andtheconstructedpolicysequence</sup><sup>_{_</sup> _tax_<sup>ˆ</sup> _t_ , ˆ _τt_ , _T_<sup>ˆ</sup> _t_<sup>_u}_to-</sup> gether solve this new Ramsey problem (R2). The time-consistency property follows immediately from the consolidated government allocation being efficient and hence time-consistent.

# 3 Calibration to Spain

In this section, we describe the calibration of our model to Spain, which we use for quantitative applications in the next section. We calibrate the model to data moments in Spain for two reasons.

21
First, Spain went through several well-documented fiscal decentralization reforms, and the decentralization on the revenue and spending sides was asymmetric. This allows us to test and quantify the model predicted relationship between vertical fiscal imbalances and total government debt accumulation (Section 2.4.3). Second, despite the fiscal decentralization reforms, Spain’s regional governments had little autonomy on revenue policies (e.g., deciding tax rates) but more autonomy to decide spending. This is consistent with our modeling choices of endogenous government spending and exogenous pre-transfer revenue. We focus on the period before the European debt crisis and Global Financial Crisis by excluding post-2006 data.<sup>29</sup>

# **3.1 Background: fiscal decentralization in Spain**

**Fiscal decentralization.** Since the creation of 17 financially autonomous regions (the Autonomous Communities or ACs) in 1978, the funding of these regions has been periodically reformed.<sup>30</sup> During the 1980s and 1990s, the country went through several fiscal decentralization reforms to gradually increase regional governments’ shares of tax revenue and spending responsibilities. On the spending side, changes in the national versus regional spending responsibilities were mainly driven by shifts in the provision of health care and education from central to the regions at different points in time. On the revenue side, some tax revenue items were ceded to regional governments to increase regional governments’ own revenue. From 1986 onward, the fiscal arrangements between the central and regional governments were renegotiated about every 5 years.

We treat the regional and sub-regional governments in the data as the local governments in our model. Panels a and b of Figure 5 show that the local governments’ pre-transfer revenue and spending as shares of total government revenue and spending both went up after the mid-1980s. Two observations are worth noting. First, the local governments’ share of total revenue was always smaller than their share of total spending, indicating vertical fiscal imbalances. Second, the paces of revenue and spending decentralization are different. Overall, local governments’ spending share rose twice as much as revenue share during 1988–2006. The increase in local governments’ spending share was broadly steady during the entire period, while the rise in their revenue share accelerated after the renegotiation round in 1996.

The different paces of decentralization in revenue and spending have led to changing vertical fiscal imbalances. For an empirical measure, we follow the literature (see e.g., Eyraud and Lusinyan 2013) to define _vertical fiscal imbalance_ (VFI) as the share of the local governments’ own spending (including debt interest payment) that is not financed by its own revenue:

> 29We choose to focus on patterns up until 2006 because the European debt crisis and Global Financial Crisis from 2007 onward likely disrupted the intergovernmental fiscal relations in many countries, while the sharp decline in economic activity and the bailout of the financial sector led to drastic increases in public debt in many countries that were largely unrelated to vertical fiscal arrangements. Among related works, Eyraud and Lusinyan (2013), Sorribas-Navarro (2011) also stop data in 2006 or 2007, as do many other papers that look at economic patterns and trends of European countries before the crisis, e.g., García-Santana et al. (2020).

> 30Garcia-Milà and McGuire (2007) provide a thorough review of the fiscal decentralization process in Spain.

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to help them with debt service costs. In addition to the empirical evidence, there is also anecdotal evidence to support the positive relationship between debt and transfers in Spain and several other countries. The details are included in Appendix B.2.

# **3.2 Calibration**

We calibrate the parameters by matching the Markov equilibrium steady state moments to the empirical counterparts in Spain.<sup>34</sup> All debt, spending, revenue, and transfer moments are normalized by trend national GDP. To quantify the impact of decentralization on debt changes, we look at two separate episodes: the rising vertical fiscal imbalances and debt during 1988–1996, and the falling imbalances and debt during 1996–2006. Accordingly, we calibrate the model separately to the economies in 1988, 1996, and 2006.

We introduce two additional parameters to make the model more flexible to fit the data. First, we allow the constrained regions to borrow, subject to a debt limit _b_<sup>_o_</sup> _≤ B_<sup>_o_</sup> . When _B_<sup>_o_</sup> is small enough,<sup>35</sup> which is the case in our calibration, the debt limit always binds and each constrained region’s debt is _B_<sup>_o_</sup> in equilibrium. These regions then borrow at the interest rate _R_ ( _B_<sup>_o_</sup> ) in equilibrium. This modification allows us to better match the constrained regions’ debt level observed in the data. Second, to better match the unconstrained regions’ debt level, we introduce an exogenous parameter _ε_ that adjusts the distortionary effect of debt-dependent transfer rate _τ_ . More details about _ε_ will be given later. Appendix C provides the general model with these two additional parameters. The model presented in Section 2 is a special case of the general model with _B_<sup>_o_</sup> = 0 and _ε_ = 1, which we use to illustrate the main mechanisms of the model. We emphasize that these two additional parameters **do not** change the theoretical results of Section 2. Appendix E shows the proofs of the theoretical results with _ε ∈_ (0,1] and _B_<sup>_o_</sup> _≥_ 0.

## **3.2.1 Exogenously set parameters**

We set some parameters exogenously using data moments. Table 2 summarizes these externally calibrated parameters.

**Constrained regions’ share (** 1 _− μ_ **) and debt limit (** _B_<sup>_o_</sup> **).** The constrained regions in our model can be best represented by the regions with low and stable debt levels in the data. We rank all 15 Spanish regions (excluding Navarra and the Basque country) by their debt level and debt volatility. We use the 7 regions with the lowest average debt levels and debt volatility to proxy for the constrained regions.<sup>36</sup> During 1994–2006, the shares of national GDP and population of these 7 regions were about 20%, which we use as the value for 1 _− μ_ . The average local public debt of these 7 regions was _B_<sup>_o_</sup> . stable over time at around 4% of their GDP, which we use for the constrained regions’ debt limit

> 34The model is solved by approximating the equilibrium policy functions as functions of current states. Appendix H provides more details.

> 35Appendix E gives the criterion for how small _Bo_ needs to be.

> 36During 1994–2006, the average regional debt-regional GDP ratio for the constrained regions was 4%, compared to 11.5% for the unconstrained regions. The standard deviation of regional debt-regional GDP ratio during the same period had an average of 1.1 among the constrained regions, compared to 2.1 for the unconstrained regions.

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**Table 2:** Externally Calibrated Parameters

|Parameters|Description||Value||
|---|---|---|---|---|
|_β_|Time discount factor||0.95||
|_ψd_|Central interest elasticity||0.03||
|_ψb_|Local interest elasticity||0.03||
|_μ_|Unconstrained region’s Pareto weight||0.80||
|_B_<sup>_o_</sup>|Constrained region’s debt limit||0.04||
|||1988|1996|2006|
|_e_|Total government revenue|0.3117|0.3835|0.4041|
|_f_|Local government revenue|0.0430|0.0638|0.1327|
| ̄_T_|Lower bound on uniform transfer|0.032|0.07|0.0485|

Note: Revenue and transfer statistics are normalized by national GDP.

In general, changing the values of _μ_ and _B_<sup>_o_</sup> does not significantly affect the quantitative results once the other parameters of the model are also re-calibrated. As a robustness check, in Section 4.3.3, we present an alternate classification of constrained regions and the corresponding set of _μ_ and _B_<sup>_o_</sup> , which yield similar quantitative results.

**Time discount (** _β_ **) and interest rate schedules (** _R_ ( _·_ ), _S_ ( _·_ ) **).** The annual time discount rate _β_ is set at 0.95. We use log utility functions. The exogenous interest rate schedules for local (unconstrained and constrained) and central debt are defined à la Schmitt-Grohe and Uribe (2003):

and satisfy Assumption 2. Grande et al. (2013) estimate that a 1-percentage-point increase in public debt-to-GDP ratio corresponds to a 3-basis-point increase in interest rates among OECD countries during 1995–2011. Based on this estimate, we set the elasticity parameter for central debt _ψd_ = 0.03. There is no consensus on the interest rate elasticity of local debt, and so in the baseline, we set _ψb_ equal to _ψd_ . In Section 4.3.2, we use alternative values of _ψb_ and _ψd_ for sensitivity analysis.

**Revenue (** _e_ , _f_ **) and transfer (** _T_<sup> ̄</sup> **).** Data on government revenue, inter-governmental transfer, and central and local (regional) government debt come from Eyraud and Lusinyan (2013) and IMF’s Government Finance Statistics (GFS), and are normalized by trend national GDP. The lower bound _T_  ̄ on uniform transfer is not directly observable in the data. Sorribas-Navarro (2011) documents that over 1987–1996, about 85% of central-to-regional transfers (grants) in Spain were non-discretionary (formula-based), and this ratio fell to 81% during 2002–2006. This non-discretionary transfer corresponds to the uniform transfer _T_<sup>_u_</sup> in our model. Because _T_<sup>_u_</sup> = _T_<sup> ̄</sup> (Proposition 1), _T_<sup> ̄</sup> is then calculated as the product of the total central-to-regional transfer from the data and the share of nondiscretionary transfers reported by Sorribas-Navarro (2011).<sup>37</sup>

> 37In the data, a government’s fiscal balances (flow) do not necessarily match the change in debt (stock). For example, the central government’s budget equation (in nominal terms) _transfer + other non-interest expenditure + interest expense = revenue + change in debt_ does not hold in the data. One reason for this statistical discrepancy is that we use gross debt instead of net debt to calculate _change in debt_ , as is customary in the literature, due to the lack of accurate public asset data.

26
# **3.2.2 Endogenously determined parameters**

Given the exogenously set parameters, we then jointly calibrate the remaining parameters.

**Transfer negotiation.** In our model, the central government can achieve larger redistribution between constrained and unconstrained regions by raising _τ_ . In reality, redistribution may face political challenges. A region may choose to stay out of a fiscal federation (autarky) if it feels the net intergovernmental transfers it receives is much lower than other regions. To ensure that all regions stay in the fiscal federation, the cross-region difference in transfer (or equivalently, the extent of cross-region redistribution) cannot be too big relative to the region’s cost of being in autarky.<sup>38</sup> We introduce a negotiation term in the transfer function to capture this “participation constraint” in a reduced-form way. This change will allow us to better match the data moments of local and central 2. debt, but will not affect the theoretical results presented in Section In particular, we replace the transfer function (2) with

where _b_<sup> ̄</sup> _t_ = _μb_<sup>_n_</sup> _t_<sup>+ (1</sup><sup>_−μ_)</sup><sup>_Bo_is the cross-region average debt, and</sup><sup>_τt_(</sup><sup>_b_ ̄</sup><sup>_t−bi_,</sup><sup>_t_) is the difference between</sup> the average and region _i_ ’s transfer incomes. Parameter _ε ∈_ [0,1] captures the negotiation power of central versus regions: a region gets 1 _− ε_ portion of the cross-region transfer difference. When _ε_ is equal to 1, the central government has all the negotiating power, and the transfer rate _τ_ is _fully_ redistributive. Whenever 0 _< ε<_ 1, _τ_ is only partially redistributive. When _ε_ = 0, transfer has no redistributive effect at all. Accordingly, the unconstrained local government’s Euler equation becomes We let data inform us on the value of _ε_ by jointly calibrating it with other parameters. In Section 4.3.4, we explore the quantitative effects of _ε_ .

**Joint calibration.** We have four parameters to be calibrated internally: _B_<sup> ̄</sup> and _D_<sup> ̄</sup> in the interest rate schedules, the central government’s share of spending responsibility _θ_ , and the negotiation parameter _ε_ in the transfer function. We jointly calibrate them to match four steady-state moments from the model: (1) total local government debt, (2) central government debt, (3) central-local spending ratio, and (4) total central-to-local transfer. For the year 1988, we use the four moments to pin down the four parameters. For later years, we fix _ε_ to the 1988 value and use the first three moments to calibrate the other three parameters.<sup>39</sup>

Intuitively, all else equal, a higher _B_<sup> ̄</sup> or _D_<sup> ̄</sup> raises debt levels; a larger _θ_ raises the central-local

> Another source of discrepancy is the valuation effect of debt and assets. We add a residual to balance each government’s budget in the data, and carry the residual term as an exogenous item to the budget constraint in our calibration and simulation exercises.

> 38This “participation constraints” problem is discussed, for example, in Persson and Tabellini (1996), who show that the threat of secession gives rise to a limit on the extent of regional redistribution. See also Balcells et al. (2015) and Porta (2015) on Catalonia’s complaints about the uneven inter-governmental transfers in Spain.

> 39 In a previous version we allowed _ε_ to change after 1988 and calibrated four parameters using all four moments in each of the three years, and found that the variation of calibrated _ε_ is marginal in the three years.

27
**Table 3:** Internally Calibrated Parameters

|Parameters|Target|C|alibrated Val|ues|
|---|---|---|---|---|
|||1988|1996|2006|
| ̄_B_|Local government debt|0.0128|0.114|0.116|
| ̄_D_|Central government debt|0.3|0.585|0.315|
|_θ_|Central-local spending ratio|0.789|0.653|0.515|
|_ε_|Central-to-local transfer|0.014|0.014*|0.014*|

Note: Debt and transfer levels are normalized by national GDP. *We assume _ε_ for 1996 and 2006 is the same as 1988.

**Table 4:** Targeted and Untargeted Moments

||19|88|19|96|2|006|
|---|---|---|---|---|---|---|
|Moments|Data|Model|Data|Model|Data|Model|
||Ta|rgeted Mo|ments||||
|Local government debt|0.062|0.062|0.130|0.130|0.122|0.122|
|Central government debt|0.338|0.337|0.615|0.614|0.336|0.336|
|Central-local spending ratio|3.63|3.63|1.85|1.85|1.05|1.05|
|Central-to-local transfer|0.038|0.039|0.077|0.080*|0.056|0.055*|
||Unt|argeted M|oments||||
|Local spending|0.060|0.062|0.125|0.129|0.166|0.162|
|Central spending|0.217|0.224|0.231|0.239|0.174|0.170|
|Vertical fscal imbalance|32.2|34.0|52.2|53.1|21.3|21.1|
|Local interest payment|0.0037|0.0032|0.0085|0.0066|0.0030|0.0062|
|Central interest payment|0.026|0.017|0.043|0.031|0.013|0.017|

Note: All variables except central-local spending ratio and vertical fiscal imbalance are normalized by national GDP. Out of the untargeted moments, there is one degree of freedom, given the targeted moments and the central and local governments’ budget constraints. Additionally for 1996 and 2006, because _ε_ is assumed the same as 1988, there is one more degree of freedom(*).

spending ratio; a higher _ε_ increases the distortion of transfers, raises total transfer, and leads to higher central and local government debt levels.<sup>40</sup> Table 3 reports these internally calibrated parameters, and Table 4 presents the targeted and untargeted moments. We match the targeted moments well. In particular, the calibrated model captures three properties in the data: debt levels increased from 1988 to 1996 and fell from 1996 to 2006; total transfer also increased during the first period and fell during the second period; and central-local spending ratio fell during both periods. For the untargeted moments, the model also does reasonably well in matching untargeted moments in the calibrated years.

**Degree of freedom.** Out of the untargeted moments in Table 4, there is one degree of freedom,

> 40We also tried an alternative calibration strategy where we set _ε_ = 1 and jointly calibrated the share of unconstrained regions _μ_ (instead of taking it directly from data) with the other parameters. _ε_ = 1 means _τ_ is fully redistributive and distortionary, while a smaller _μ_ means that a smaller share of regions are subject to the distortionary effect of _τ_ , and so the aggregate distortion is limited. As such, we can loosely match data moments of total transfer and central debt using _ε_ = 1 and a very small _μ_ . However, we find that because of the large distortionary effect of _τ_ when _ε_ = 1, unconstrained regions’ debt in the model is implausibly high—about 10 times higher than the level observed in the data.

28
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# 4 Quantitative Applications

This section uses the calibrated model for two applications. First, we quantify the effects of fiscal decentralization on public debt accumulation, under the calibrated Markov equilibrium where the central government cannot pre-commit to future transfer policies. Second, we quantitatively compare the Markov equilibrium to the Ramsey equilibrium and the allocation under a consolidated government. We provide extensions and sensitivity analyses at the end of the section.

## **4.1 Fiscal decentralization**

As discussed in Section 2.4.3, the model predicts that a revenue decentralization alone narrows vertical fiscal imbalances, lowers transfers, and reduces government debt, whereas a spending decentralization widens vertical fiscal imbalances, raises transfers, and increases government debt. In this section, we perform quantitative experiments to see how much of the changes in government debt in Spain during 1988–2006 can be attributed to the _unbalanced_ fiscal decentralization reforms. Figure 5 shows that the rise and fall in total government debt coincide with major changes in the decentralization reforms in Spain. For this reason, we divide 1988–2006 into two episodes: 1988–1996 characterized by faster spending than revenue decentralization and rising total government debt, and 1996–2006 with revenue catching up with spending decentralization and falling total government debt.

Table 5 summarizes the results. To construct the counterfactuals, we change only the parameters related to fiscal decentralization and keep all other parameters unchanged. More specifically, for the 1988–1996 period, we start with the model economy calibrated to data moments in 1988, and then change the government revenue and spending responsibility parameters ( _e_ , _f_ , _θ_ ) to their 1996 values to get a “counterfactual 1996” economy. Comparing the actual change in debt during 1988– 1996 (Column 1 to Column 2) with the counterfactual change (Column 1 to Column 3) tells us the effects of decentralization on debt. For example, the total government debt-to-GDP ratio increased from 0.399 to 0.744 in the data, and to 0.534 in the counterfactual experiment. This indicates that the unbalanced decentralization, in this case, the faster spending than revenue decentralization, generates 39% of the rise in total government debt during 1988–1996. Similarly, for the 1996–2006 period, we start with the model economy calibrated to data moments in 1996, and then change ( _e_ , _f_ , _θ_ ) to their 2006 values to obtain the “counterfactual 2006” economy. The comparison between the actual change in debt during 1996–2006 (Column 4 to Column 5) and the counterfactual change (Column 4 to Column 6) tells us the contribution of fiscal decentralization to debt changes during the second period. Figure 8 shows the actual and counterfactual total government debt in each period and illustrates the change in debt due to fiscal decentralization.

Overall, the unbalanced revenue and spending decentralization explains 39% of the actual changes in total government debt from 1988 to 1996 and 18% from 1996 to 2006. The rest are explained by changes in uniform transfer and interest rate schedule parameters _B_<sup> ̄</sup> and _D_<sup> ̄</sup> . Decentralization explains a smaller proportion of the change in debt during 1996–2006 than during 1988–1996. One 30
**Figure 8:** Illustration of Counterfactual Experiment

<!-- Start of picture text --> (a) 1988–1996 period (b) 1996–2006 period<br>0.8 0.8<br>1996<br>0.75 1996 debt due to<br>0.7 decentralization<br>0.7 (18%)<br>2006 counterfactua<br>0.65<br>0.6<br>0.6<br>1996 counterfactual<br>0.5<br>debt due to 0.55<br>decentralization<br>(39%) 0.5<br>0.4 1988<br>2006<br>0.45<br>0.3 0.4<br>1986 1988 1990 1992 1994 1996 1998 1994 1996 1998 2000 2002 2004 2006 2008 2010<br>Time Time<br>Change only (e,f,<br>) to 2006<br>) to 1996<br>Change only (e,f,<br>Change all parameters to<br>calibrated 1996 values<br>Change all parameters to<br>calibrated 2006 values<br>Total government debt Total government debt<br><!-- End of picture text --> Note: Counterfactual experiments change decentralization parameters ( _e_ , _f_ and _θ_ ) to the end year values, and keep all other parameters at their starting year values. The experiments illustrate the change in debt that is explained by decentralization.

**Table 5:** Counterfactual Experiment Effects of fiscal decentralization on debt levels

|||1988–19|96||1996–20|06|
|---|---|---|---|---|---|---|
||(1)|(2)|(3)|(4)|(5)|(6)|
|Moments|1988|1996|Counterfactual<br>1996 (%)|1996|2006|Counterfactual<br>2006 (%)|
|Local debt|0.062|0.130|0.149 (129%)|0.130|0.122|0.103 (344%)|
|Central debt|0.337|0.614|0.385 (17%)|0.614|0.336|0.589 (9%)|
|Total government debt|0.399|0.744|0.534 (**39%**)|0.744|0.458|0.692 (**18%**)|

Note: Total government debt (also known as “general government debt”) is the sum of local and central government debt. Counterfactual experiments use revenue ( _e_ and _f_ ) and spending responsibility share ( _θ_ ) values from end year, and keep all other parameters at their starting year values. Numbers in **parentheses** show the percent of debt changes that is explained by decentralization. All debt levels are normalized by national GDP.

possible reason is that there was increased scrutiny over government borrowing during the later period in order to meet EU regulations, such as the Stability and Growth Pact. That made it harder for all levels of governments to borrow and could have weakened the effect of soft budget constraint.

For both episodes, fiscal decentralization explains a larger proportion of changes in local debt and a smaller portion for central debt. In an alternative calibration presented in section 4.3.2, we use a more elastic local interest rate schedule ( _ψb > ψd_ ), as motivated by anecdotal evidence. The more elastic local interest rate increases the cost of local relative to central borrowing. As such, fiscal decentralization explains larger changes in central government debt than in the baseline.

**Debt-neutral (balanced) fiscal decentralization.** Spending decentralization alone raises debt, so how much revenue decentralization is needed to make a decentralization reform debt-neutral? Figure 9 plots the relationship between revenue and spending decentralization such that the total gov-

31
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Table 6 provides a numerical illustration of the theoretical results summarized in Table 1, using the parameters calibrated to data moments in 1996.

Consistent with the theoretical results, under a consolidated government, the steady state unconstrained local government debt _b_<sup>_n_</sup> equals _B_<sup> ̄</sup> , and central debt _d_ equals _D_<sup> ̄</sup> . The two types of local governments have the same spending, _g_<sup>_n_</sup> = _g_<sup>_o_</sup> , and the adjusted ratio of local-central marginal utilities, (1 _− θ_ )[ _μu_<sup>_n_</sup> _g_<sup>+ (1</sup><sup>_−μ_)</sup><sup>_uo_</sup> _g_<sup>]/(</sup><sup>_θvc_), is equal to 1, i.e., MULG = MUCG.</sup> In the non-commitment economy (Markov equilibrium), the steady state total government debt is about 9% of GDP higher than the level under a consolidated government. Consistent with Lemma 1, unconstrained local governments have lower spending, _g_<sup>_n_</sup> _< g_<sup>_o_</sup> . The adjusted local-central marginal utility ratio is less than 1, i.e., MULG < MUCG, because the central government without commitment over-transfers relative to the level of transfers required for MULG = MUCG (Proposition 3).

In the commitment (Ramsey) economy, because the central government has commitment to future policies, it internalizes the overborrowing incentives from positive _τ_ and so it sets _τ_ = 0 at the steady state. Transfers in this economy are solely made through uniform transfer _T_<sup>_u_</sup> at the steady state, and MULG > MUCG (Proposition 4). Without any overborrowing incentives ( _τ_ = 0), both local and central debt levels are the same as in the consolidated government allocation (Corollary 1), and the cross-region spending gap is smaller than in the Markov equilibrium.

In terms of consumption-equivalent welfare, because of the larger debt and cross-region spending gap, the Markov economy has 0.98% lower welfare than the economy with a consolidated government. By contrast, because there is no overborrowing in the Ramsey economy, the welfare difference from the consolidated government allocation is much smaller (0.002%). Furthermore, a 5.1% prudential tax on local government debt, as described in Section 2.6, can help implement the consolidated government allocation.

# **4.3 Extension and robustness**

## **4.3.1 Extension: internalizing interest rate schedules**

In the baseline model, we assume that governments do not internalize the slopes of their interest rate schedules. The assumption gives us clean theoretical results, and allows us to differentiate from the literature on the pecuniary externality of decentralized borrowing.

In this section, we relax this assumption by allowing governments to internalize the slopes of their interest rate schedules and re-calibrate the model economy. Overall, relative to the baseline model, the only difference in the theoretical results is that the Ramsey steady state _τ_ is no longer zero. All other theoretical results in the Markov equilibrium hold as before.<sup>41</sup> Below we briefly describe the changes to the model and discuss the quantitative results.

**Model with extension.** When the consolidated government internalizes the slopes of the interest rate schedules, an extra term with the derivative of the interest rate schedules shows up in the opti-

> 41The proofs for this extension are similar to the proofs in Appendix E.

33
**Table 6:** Comparison of Steady States

|Moments|Consolidated<br>government<br>allocation|Non-<br>commitment<br>(Markov)|Commitment<br>(Ramsey)|
|---|---|---|---|
|Uniform transfer,_T_<sup>_u_</sup>|–|0.07|0.078|
|Distortionary transfer|–|0.01|0|
|Total central-local transfer|–|0.08|0.078|
|Unconstrained local debt,_b_<sup>_n_</sup>|0.114|0.152|0.114|
|Central debt,_d_|0.585|0.614|0.585|
|Cross-region spending gap,_g_<sup>_o_</sup>-_g_<sup>_n_</sup>|0|0.0057|0.0038|
|Adjusted local-central marginal<br>utility ratio, MULG/MUCG|1|0.98|1.00013|
|Consumption equivalent welfare change<br>relative to consolidated govt allocation (%)|–|-0.98|-0.002|

Note: Parameters calibrated to the 1996 economy are used. Transfer, debt, and spending are normalized by national GDP. Consumption equivalent welfare change calculates the percent of local and central spending that the household is willing to forgo.

mality conditions (4)–(5): The additional terms show up because the consolidated government internalizes that higher debt increases the marginal cost of borrowing. As such, optimal debt levels will be lower in this alternative setup.

In the economy without a consolidated government, each local government internalizes the interest rate cost of higher borrowing. The representative unconstrained region’s Euler equation also has an additional term with the derivative of its interest rate schedule The central government internalizes the interest rate schedules of both local and central debt, and as a result, there are additional terms with _Sd_ or _Rb_ in the central government’s optimality conditions. **Quantitative results.** We re-calibrate the extended model and show that, similar to the baseline model, (1) there are sizable overborrowing and over-transfer in the Markov equilibrium, and (2) fiscal decentralization changes debt levels in quantitatively meaningful ways.

The results in Table 7 show that similar to the baseline model, the Markov equilibrium has higher local and central debt than the consolidated government allocation. Also similar to the baseline, the Markov government over-transfers (MULG < MUCG) and the Ramsey government under-transfers (MULG > MUCG). Two things differ qualitatively from the baseline. First, the debt levels un34
der the consolidated government are lower than (instead of equal to) the re-calibrated parameters _B_ ̄ (=0.236) and _D_ ̄ (=1.063), because the central and local governments internalize the change in interest rates from additional borrowing. Second, the Ramsey central government chooses a positive (instead of 0) distortionary transfer, and as a result, local debt is slightly higher than (instead of equal to) in the consolidated government allocation.

The quantitative effects of fiscal decentralization in the extended model are summarized in Appendix I.2. Overall, the results are similar with the baseline model: decentralization explains 32% of the increase in total government debt during 1988–1996 and 9% of the debt decrease during 1996– 2006.

**Table 7:** Comparison of Steady States when governments internalize interest rate schedules

|Moments|Consolidated<br>government<br>allocation|Non-<br>commitment<br>(Markov)|Commitment<br>(Ramsey)|
|---|---|---|---|
|Uniform transfer,_T_<sup>_u_</sup>|–|0.07|0.0797|
|Distortionary transfer|–|0.0121|0.0001|
|Total central-local transfer|–|0.0821|0.0798|
|Unconstrained local debt,_b_<sup>_n_</sup>|0.1213|0.152|0.1216|
|Central debt,_d_|0.593|0.615|0.593|
|Cross-region spending gap,_g_<sup>_o_</sup>-_g_<sup>_n_</sup>|0|0.0053|0.0039|
|Adjusted local-central marginal<br>utility ratio, MULG/MUCG|1|0.978|1.00015|
|Consumption equivalent welfare change<br>relative to consolidated govt allocation (%)|–|-0.64|-0.0054|

Note: Assuming governments internalize interest rate schedules. Parameters are re-calibrated such that the simulated Markov equilibrium moments match the targeted data moments in 1996. Transfer, debt, and spending are normalized by national GDP. Consumption equivalent welfare change calculates the percent of local and central spending that the household is willing to forgo.

# **4.3.2 Alternative calibration of interest rate elasticities** _ψb_ **and** _ψd_

In the baseline calibration, we assume that the elasticities of local and central debt interest rate schedules are the same: _ψb_ = _ψd_ = 0.03. However, it is also plausible that local governments face a _more_ elastic interest rate schedule than the central government, as the market for local government debt is usually less liquid. In an alternative calibration, we let _ψb_ = 0.05 and _ψd_ = 0.01 and re-calibrate the model.<sup>42</sup> As _ψb_ becomes larger (and _ψd_ becomes smaller), local (central) debt interest rates become more (less) sensitive to changes in local (central) debt. So local (central) debt will be less (more) responsive to fiscal decentralization than in the baseline calibration.

Table 8 shows that in terms of the changes in central and local debt, this alternative calibration performs better than the baseline calibration. For example, fiscal decentralization explains 23% 42Appendix I.3 presents re-calibrated parameter values for each year.

35
(97%) of the actual change in central (local) debt during 1988–1996, compared with 17% (129%) in the baseline. Overall, this alternative calibration delivers similar changes in total government debt as the baseline: decentralization accounts for 37% of total government debt changes during 1988–1996, and 24% during 1996–2006.

**Table 8:** Counterfactual Experiment with alternative local and central debt interest elasticities

|||1988–19|96||1996–20|06|
|---|---|---|---|---|---|---|
||(1)|(2)|(3)|(4)|(5)|(6)|
|Moments|1988|1996|Counterfactual<br>|1996|2006|Counterfactual<br>|
||||1996 (%)|||2006 (%)|
|Local debt|0.062|0.130|0.128 (97%)|0.130|0.122|0.116 (179%)|
|Central debt|0.338|0.615|0.401 (23%)|0.615|0.336|0.559 (20%)|
|Total government debt|0.400|0.745|0.529 (**37%**)|0.745|0.458|0.675 (**24%**)|

Note: Using _ψb_ = 0.05 and _ψd_ = 0.01. All other model parameters are re-calibrated. Calibration strategy is identical to the baseline. Total government debt (also known as “general government debt”) is the sum of local and central government debt. Counterfactual experiments use revenue ( _e_ and _f_ ) and spending responsibility share ( _θ_ ) values from end year, and keep all other parameters at their starting year values. Numbers in **parentheses** show the percent of debt changes that is explained by decentralization. All debt levels are normalized by national GDP.

# **4.3.3 Alternative calibration of constrained regions’ share and debt limit:** 1 _− μ_ **and** _B_<sup>_o_</sup>

In the baseline calibration, we use half of the regions (7 out of 15) with the lowest debt level and volatility in the data to proxy for the _constrained regions_ in the model. This gives the share of constrained regions 1 _− μ_ = 0.2 and their debt limit _B_<sup>_o_</sup> = 0.04. As an alternative calibration, we use 9 out of 15 regions with the lowest and most stable debt levels to proxy for the group of constrained regions. This yields 1 _− μ_ = 0.4 and _B_<sup>_o_</sup> = 0.046. With this set of ( _μ_ , _B_<sup>_o_</sup> ), we re-calibrate the model. We find that changing _μ_ and _B_<sup>_o_</sup> has little effect on the results from the counterfactual experiment once parameters are re-calibrated to match the moments in each year. Appendix I.4 reports the full calibration and results. To summarize, decentralization explains 39% of the increase in total government debt from 1988 to 1996, and 15% of the decrease in total government debt from 1996 to 2006.

# **4.3.4 Effect of a larger negotiation parameter** _ε_

The jointly calibrated negotiation parameter _ε_ has a sizeable impact on the Markov equilibrium debt levels. Figure 10 shows that holding other parameters at their calibrated values, both the central and total local debt levels increase with _ε_ . This is because as _ε_ increases, _τ_ becomes more redistributive, and so the central government is more willing to increase _τ_ to subsidize unconstrained regions. Higher _ετ_ encourages local governments to borrow more, and the central government also has to borrow more to help finance the higher total transfers.

A natural question is how _ε_ affects the results from the counterfactual experiment. We perform the counterfactual exercises using a larger _ε_ value (0.1) than the baseline (0.014), keeping all other 36
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such that the simulated moments in 1996 match the data moments in 1996. To give more details, the new government revenues _{e_ , _f }_ can be directly taken from their 1996 values. The new spending responsibility parameter _θ_ and other parameters _{B_<sup> ̄</sup> , _D_<sup> ̄</sup> , _ε}_ are not directly observed from the data, so we jointly calibrate them to target the total local government debt, central government debt, central-local spending ratio, and central-to-local transfer in the simulated 1996 economy along the transition path. Table I.6 reports the calibrated parameter values and moments.

We then conduct the counterfactual exercise. Calibration in this alternative approach is complicated because it entails calibrating to a point on the simulated transition path. For this reason, we only simulate until 1996 and do the 1988–1996 counterfactual as an illustration. For this counterfactual exercise, we keep all the parameters at their 1988 levels except for the fiscal decentralization changes to _e_ , _f_ , and _θ_ . Table I.6 compares the calibrated and counterfactual moments, and Figure I.1 plots the simulated path of total debt and compares it to the counterfactual path. Overall, the unbalanced fiscal decentralization explains 18% of the increase in total debt during 1988-1996, somewhat smaller than the results using our baseline calibration.

# 5 Conclusion

This paper develops a dynamic infinite-horizon model of fiscal federation with vertical fiscal imbalances. Our model captures the main ingredients of the typical soft budget constraint problem: local governments have the autonomy to decide their spending and borrowing; and central-to-local transfers serve as a “common pool” for local governments. Compared to literature, our main contributions are the expansion of the analysis to infinite horizons and the incorporation of central government debt. From the theoretical perspective, these changes allow us to discuss the interaction between central government’s policy today and tomorrow, in addition to the interaction between central and local governments. From the quantitative perspective, these changes allow us to quantify the impact of fiscal decentralization reforms on both local and central government debt. Central government debt is often omitted in this literature due to the balanced-budget assumption, even though its change is empirically more relevant than local government debt.

In the model, when the central government cannot pre-commit the amount of future transfers, local governments have overborrowing incentives because they expect the central government to transfer more to regions that are more indebted—a common result in the literature. The more novel result from our infinite-horizon model is that the central government _over-transfers_ to local governments, to the extent that residents’ marginal utilities from local public consumption is lower than from central public consumption. We apply the framework to show that, consistent with the empirical patterns documented in the literature, when a fiscal decentralization reform increases vertical fiscal imbalances, local governments become more reliant on transfers and total government debt rises. Quantitatively, we find that fiscal decentralization in Spain explains 39% of the increase in total government debt during 1988–1996 and 18% of the decrease during 1996–2006.

The experience of Spain is not unique. Fiscal decentralization is often unbalanced. In many coun38
tries, the decentralization in spending is faster than revenue, for economic or political reasons.<sup>43</sup> This gives rise to large local fiscal gaps to be filled by intergovernmental transfers, which are often associated with common pool and soft budget constraint problems. Notable examples include the Bremen and Saarland’s long-term reliance on transfers following the initial bailout in Germany, and more recently, the rising local government debt and the debt swap program in China, just to name a few. The policy lesson we draw here is that, to avoid a deterioration in aggregate fiscal performance, a country should implement _balanced_ fiscal decentralization, such that any additional spending responsibilities assigned to the local level come with a sufficient increase in revenue allocated to the local governments.

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> 43Economically, public finance literature following Musgrave (1959) argues that centralized tax collection can minimize inefficient tax competition among regional governments, while Tiebout (1956) and Oates (1972) advocate for decentralized spending to cater to different tastes of local residents. Politically, a central government may have strong incentives to hold a large share of revenue to weaken the political power of local governments. For example, Robinson and Torvik (2009) argue that soft budget constraint is politically rational because it increases the probability of politicians’ political survival.

39
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41
**Appendix for “Decentralization and Overborrowing in a Fiscal Federation”**

Si Guo, Yun Pei, and Zoe Xie<sup>44</sup>

|**Table of Contents**||
|---|---|
|**A Empirical evidence on VFI and debt**|**3**|
|**B Model assumptions: Remarks**|**5**|
|B.1<br>Micro-foundation of_θ_in a special case . . . . . . . . . . . . . . . . . . . . .|. . . . . .<br>5|
|B.2<br>Motivation of transfer function (2) . . . . . . . . . . . . . . . . . . . . . . .|. . . . . .<br>6|
|B.3<br>Exogenous interest rate schedules . . . . . . . . . . . . . . . . . . . . . . . .|. . . . . .<br>11|
|**C General model**|**13**|
|**D Derivations**|**15**|
|D.1 Markov optimality conditions and GEEs . . . . . . . . . . . . . . . . . . . .|. . . . . .<br>15|
|D.2 Ramsey optimality conditions . . . . . . . . . . . . . . . . . . . . . . . . . .|. . . . . .<br>20|
|**E Proofs**|**22**|
|E.1<br>Proof of Lemma 1 (Cross-region spending gap in Markov) . . . . . . . . .|. . . . . .<br>22|
|E.2<br>Proof of Proposition 3 (Over-transfer in Markov) . . . . . . . . . . . . . . .|. . . . . .<br>23|
|E.3<br>Proof of Proposition 2 (Minimum uniform transfer in Markov). . . . . . .|. . . . . .<br>24|
|E.4<br>Proof of Proposition 4 (Under-transfer in Ramsey) . . . . . . . . . . . . . .|. . . . . .<br>25|
|**F Model extensions**|**27**|
|F.1<br>Allowing both types of local governments to borrow . . . . . . . . . . . . .|. . . . . .<br>27|
|F.2<br>Adding adjustment cost on_T_<sup>_u_</sup><br>_t_<sup>. . . . . . . . . . . . . . . . . . . . . . . . . . </sup>|<sup>. . . . . .</sup><br>33|
|**G A three-period model**|**36**|
|**H Computational appendix**|**39**|
|**I**<br>**Additional quantitative results**|**40**|
|I.1<br>Additional moments for Section 4.1 . . . . . . . . . . . . . . . . . . . . . . .|. . . . . .<br>40|

> 44GUO: International Monetary Fund (sguo@imf.org). PEI: University at Buffalo, SUNY (yunpei@buffalo.edu). XIE: World Bank; Federal Reserve Bank of Atlanta (xiexx196@gmail.com). The views expressed in this paper are those of the authors and do not necessarily represent the views of the IMF, its Executive Board, IMF management, the Federal Reserve System, the Federal Reserve Banks, the IBRD/World Bank or its executive directors.

1
|I.2|Quantitative results for Section 4.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . .<br>41|
|---|---|
|I.3|Quantitative results for Section 4.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . .<br>42|
|I.4|Quantitative results for Section 4.3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . .<br>43|
|I.5|Quantitative results for Section 4.3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . .<br>44|
|I.6|Quantitative results for Section 4.3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . .<br>45|

2
# A Empirical evidence on VFI and debt

Figure 1 Panel B illustrates the relationship between a country’s vertical fiscal imbalances (VFI) and changes in total government debt over time, for Spain, Germany, and Portugal. Figure A.1 in this appendix supplements those figures and shows the same general pattern among 25 other OECD countries.

In addition, we run a fixed effect panel regression on these 28 countries to establish positive correlation between VFI and change in total government debt. We follow the regression specification in Eyraud and Lusinyan (2013), but instead of primary deficit, we use change in total government debt-to-GDP ratio as the dependent variable: where ∆ _dit_ is the change in country _i_ ’s debt-to-GDP ratio between years _t −_ 1 and _t_ ; VFI _t_ is the country’s VFI in year _t_ ; _Xit_ is a set of control variables, which include expenditure decentralization, lag of debt-to-GDP ratio, lag of output gap, a measure of trade openness, population, and rule of law; _αi_ are country fixed effects, _θt_ are time dummies, and _εit_ is the error term. We follow Eyraud and Lusinyan (2013) and Aldasoro and Seiferling (2014) in deciding the control variables. Both papers empirically study the relationship between a country’s VFI and primary deficit or change of government debt. Table A.1 shows that a 10 percentage point increase in VFI is associated with a 1.2 to 1.3 percentage point acceleration in the growth of total government debt as percentage of GDP.

**Table A.1:** Dependent Variable: Change of total government debt (% GDP) ∆ _dit_ = _<u>β</u> ×_ VFI _<u>it</u>_ <u>+</u> _<u>γ</u> × Xit_ <u>+</u> _αi_ <u>+</u> _θt_ <u>+</u> _εit_

||(1)|(2)|(3)|(4)|
|---|---|---|---|---|
|VFI|0.12**|0.13***|0.13***|0.12**|
|Expenditure decentralization||-0.32***|-0.38***|-0.31***|
|Lag of debt-to-GDP ratio|||-0.11***|-0.13***|
|Lag of output gap|||-0.20|0.1|
|Openness (trade as % GDP)||||-0.02|
|Population (million)||||0.04|
|Rule of law||||-4.5***|
|Observations|356|356|327|273|
|Countries|28|28|25|25|
|Overall R<sup>2</sup>|0.40|0.45|0.52|0.51|
|Within R<sup>2</sup>|0.23|0.29|0.38|0.32|

Note: Fixed effects estimation with robust standard errors clustered at the country level. *, **, *** for significance at the 10, 5, and 1 percent level, respectively. Sample restricted to 1985–2007 if available.

3
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# B Model assumptions: Remarks

## **B.1 Micro-foundation of** _θ_ **in a special case**

While all the results in Section 2 work for any utility functions that are smooth, increasing and concave (Assumption 1), the preference parameter _θ_ has a micro-foundation with log utility functions and unitary elasticity of substitution among varieties of goods (Section 2.1). This section provides a derivation of this microfoundation.

̄ Assume there are infinitely many varieties of public goods, each indexed by _ω ∈_ [0,1]. Goods _ω ∈_ [0, _w_ ) are provided by the central government. Goods _ω ∈_ [ _w_ ̄ ,1] are provided by the local governments.

The household’s preference over the basket of public goods is 1 _<u>σ−</u>_ <u>1</u> _σ−σ_ 1 where _γ>_ 0 is the household’s risk aversion. The basket of public goods is given by _C_ = [︁∫︀0<sup>_q_(</sup><sup>_ω_)</sup> _σ dω_ ]︁ , where _σ>_ 0 is the elasticity of substitution and _q_ ( _ω_ ) denotes the public spending on good _ω_ . Let _c_ and _g_ denote the central and local governments’ total spending on public goods respectively. Because all types of goods _ω ∈_ [0,1] have the same weight in the consumption basket, each central and local governments optimally chooses to evenly distribute its total spending on each type of goods.

Therefore, where _θ_ = _w_ ̄<sup>1/</sup><sup>_σ_</sup> +( _<u>w</u>_ ̄<sup>1</sup> 1<sup>/</sup> _−_<sup>_σ_</sup> _w_ ̄ )<sup>1/</sup><sup>_σ_.In the special case when</sup><sup>_σ_=</sup><sup>_γ_= 1, we have</sup> where _A_ ( ̄ _w_ ) = _w_ ̄ log( ̄ _w_ ) + (1 _− w_ ̄ ) log(1 _− w_ ̄ ). This is essentially equivalent to utility function (1) by assuming both _u_ ( _·_ ) and _v_ ( _·_ ) are log functions. The parameter _θ_ can be interpreted as the utility weight of central public goods.

5
# **B.2 Motivation of transfer function (2)**

The transfer function (2) in our model assumes that at least part of the central government transfers is dependent on regions’ debt burden. In this section, we provide some justifications for this assumption. We first provide anecdotal country examples where transfers are directly dependent on debt burden. We then summarize the empirical evidence on the relationship between regional government debt burden and intergovernmental transfers in Spain. Finally, we show that with some simplifying assumptions, the “debt-dependent transfer” feature in function (2) can be mathematically derived from other (more complicated) transfer schemes with greater real world relevance.

## **B.2.1 Anecdotal country cases**

We provide a brief review of country examples where federal transfers reward the less fiscally disciplined regions that have higher debt burden or higher deficits. The purpose of this review is to show that in many countries, regional debt burden is indeed a factor when the central government decides the amount of transfers. We include past “deficits” in addition to “debt” in our discussion because debt is the accumulation of deficits over time.

The anecdotal examples discussed here broadly fall into two categories. Some of these examples present evidence that the central government favors the local governments that are more indebted (or have been running larger deficits) through higher transfers in “normal” times. Other examples are interpreted as “bailout”, a term that is often linked to emergency transfers during “crisis” times. Although the debt-dependent transfers in our model occur in every period, as we will see, the distinction between bailouts in crisis times and the debt-dependent transfers in normal times is often blurred in real world.

One of the most relevant examples is the “deficit grant” program in **Hungary** . The initial intention of the program was to help local governments to overcome the deficits accrued through “no fault of their own” and avoid local government bankruptcy. The grants were only made to assist local governments for mandatory spending. However, judging whether a local government’s deficits were accumulated through exogenous shocks or its lack of fiscal discipline can be difficult in practice. As documented in Wetzel and Papp (2003), the deficit grants softened the budget constraints of local governments, because local governments “can increase their grant revenues through behavior changes,” such as raising less revenue and/or increasing expenditures on mandatory services, with the expectation that “deficit grants may potentially fill any gap.” According to Wetzel and Papp (2003), about one third of the local governments were receiving deficit grants in 1999. A similar program in _India_ is documented by McCarten (2003).

Another example is the intergovernmental transfers in the public health sector in **Italy** . Bordignon and Turati (2009) document that prior to the reform in the mid-1990s, the central government systematically used additional transfers to cover local governments’ debt accumulated from excessive health spending in previous years. On average, about 50% of the health deficits accumulated by local governments during 1995-1999 were later offset by additional central government transfers by 2002. Although the term “bailout” is used to describe these transfers, they do not necessarily happen under emergency, as commented by Bordignon and Turati (2009): “Bailing out of regional health deficits is not an exceptional occurrence in Italy. With an average delay of 2-3 years, the Central government usually ‘finds out’ that it has made a ‘mistake’ in computing the health needs of regions, and covers (part of) the past health deficits of regions by issuing government bonds and by taking care of Local Health Units debts.”<sup>45</sup>

> 45Similarly, Pedreja-Chaparro et al. (2006) document that the central government in Sweden bailed out local governments 1697 times during 1974-1992. This high frequency makes it hard to justify that all bailouts happened under emer-

6
The additional federal transfers to the two indebted **German** regions (Lander) – Bremen and Saarland – provide another example where federal transfers are de facto dependent on local governments’ debt burdens. As noted in Rodden (2005b), Bremen and Saarland avoided expenditure adjustments and accumulated large amount of public debt in the 1980s, betting higher federal transfers to pay for their regional public debt. The Federal Constitutional Court’s decision in 1992 confirmed that the federal government has the obligation to make additional transfers to these two regions to help relieve the regional fiscal burden and maintain the norm of public service. Following the court decision, additional federal supplementary transfers were provided to Bremen and Saarland annually.

There are two points to be emphasized in Germany’s case. First, the additional transfers to Bremen and Saarland are “debt-dependent”: the court decision in 1992 was explicit that the debt burden is an important factor in deciding whether additional federal transfers are justified. As noted in Von Hagen et al. (2000), the court decision was based on the argument that a state with “extreme budgetary hardship” is entitled to receive additional transfers, and “the decision clarified an emergency exists in a Lander if the deficit/expenditure ratio and interest payment/tax ratios are more than twice the Land average” (Rodden 2005b). Second, in the case of Bremen and Saarland, the distinction between crisis time bailout and debt-dependent transfers in normal times is also blurred. Even prior to the court decision in 1992, (smaller) annual grants had been provided to these two regions since 1987 to mitigate the debt burden (Von Hagen et al. 2000). After the court decision in 1992, even though the additional supplementary transfers from the federal government were initially planned for 5 years, they were extended several times and the payment continued till 2005 (Stehn and Fedelino 2009). Arguably, the long time span of the supplementary transfers makes it less convincing to argue the transfers should be solely interpreted as an emergency bailout.

In the **United States** , the type of debt-dependent transfers analogous to our model is the interest rate subsidy provided by the federal government. In the United States, the federal interest rate subsidy is provided in two ways: the indirect subsidy through the income tax exemption of municipal bond interest payment; and the direct interest rate subsidy on municipal bond issuance, such as the Build America Bond (BAB) program (Luby et al. 2021). Although the indirect subsidy through tax exemption is less discretionary (because the effective subsidy rates are determined by the bond purchasers’ federal income tax rates rather than the federal government’s willingness to subsidize), the BAB program is clearly a discretionary program aiming to help local governments secure infrastructure financing. Similar interest subsidy programs were also established in **other countries** by extended bodies of the federal government, such as municipal banks, development banks and municipal funds (see Peterson 2000 for a review). These development financing entities are usually owned by the central government, hence the subsidies they provide are eventually borne by the central government budget.

# **B.2.2 Debt-dependent transfers in Spain**

# **A. Anecdotal evidence**

In **Spain** , although there is no comprehensive documentation of a formal program similar to the “deficit grants” in Hungary or Italy, Garcia-Milà et al. (2002) document that “for some regions and types of expenditures additional grants were given to cover the historical debt that had been accumulated... this is a dollar-fordollar increase in grants in response to borrowing, at least for part of the debt.” This should not be surprising, given the central-local fiscal relationship in Spain shares many similarities with Italy, especially in the area of health spending.

gency.

7
There were also at least two waves of explicit bailouts of local governments in Spain. The first wave was in 1983 when the central government assumed part of the local government debt, as documented in PedrejaChaparro et al. (2006). The second wave took place in 2012. In response to the crisis and mounting regional government debt, two regional liquidity funds were set up by the central government to provide cheaper loans to regional governments: the Fund for Financing Payments to Suppliers (“FFPP” in Spanish) that lent to local governments to pay for arrears owed to local suppliers; and the Autonomous Liquidity Fund (“FLA” in Spanish) designed to help with regional governments’ difficulties in debt refinancing. In essence, local governments that were facing surging interest rates or even losing access to the financial market can borrow cheap loans from the FFPP and FLA funded by the central government.<sup>46</sup>

# **B. Empirical evidence**

Empirical studies on the causal link between central government transfer and local governments’ debt burden mainly follow two approaches. The first approach uses econometric methods to estimate or test whether higher debt burden (or higher deficits in the past) indeed results in higher transfers. Solé-Ollé and SorribasNavarro (2012) use the municipal level panel data in Catalonia region during 1988-2006 to study how municipal governments respond to shocks to revenue, expenditure and grants. They find that a positive expenditure shock to municipal government is followed by an increase in grants from upper level governments. As stated by the authors, these grants are usually discretionary grants, and they “might have generated a moral-hazard problem, with municipalities having the incentive to undertake excessive expansion since additional capital spending will always be financed (at least in the long run) by higher tiers of government.” Sorribas-Navarro (2011) uses regional fiscal data in Spain during 1986-2006 to document that regions with larger debt burden tend to receive more federal transfers. The potential endogeneity problem is addressed by (1) adding cyclical economic indicators as control variables, such as unemployment, GDP, and population net flows; and (2) using the share of debt at a variable interest rate interacted with changes in interbank interest rates as an instrumental variable.<sup>47</sup> The second approach follows Rodden (2005a) to see if the financial market distinguishes regional governments based on their debt service abilities. If regions’ debt service abilities do not affect their debt ratings, it implies that the financial market believes that the central government will eventually help indebted regional governments with their debt burden, either through emergency bailout or periodic transfers in normal times. Rodden (2005a) compares the dispersion of regional debt ratings in countries where regional governments face “harder” budget constraints (Canada and the U.S.) and “softer” constraints (Germany and Spain) in the late 1990s. He finds that in Canada and the U.S., regional debt ratings are more dispersed, and the cross-region rating difference can be largely explained by the heterogeneity in regional debt burden (proxied by regional debt-to-own revenue ratio). To the contrary, the debt ratings of German Landers and Spanish regions are much less dispersed and closely follow their sovereign ratings, and debt-to-own revenue ratio is a weak factor

> 46There are two reasons why the FFPP and FLA can provide cheap loans to regions. The first is that the FFPP and FLA can borrow from banks under the central government’s guarantee, which lowered the funding cost of the FFPP and FLA. The second is that the central government also made direct budgetary contributions to the two funds. For example, the initial budgetary contribution to the FFPP announced in 2012 was €6 billion (Minguez et al. 2012).

> 47The main concern about endogeneity is that a negative shock to a region (e.g., flood in a region) can increase the region’s debt (due to higher expenditure) while trigger additional federal transfers through insurance channels. This may create an upward bias to the estimated impact of regional debt on federal transfers. The ideal instrumental variable should be correlated with debt burden but uncorrelated with other shocks that affect transfers. As discussed in SorribasNavarro (2011), the accumulation of debt comes from two parts: primary deficits and interest payment. The instrument adopted (share of debt at a variable interest rate interacted with interbank interest rates) would be correlated with interest payment (and hence debt burden), but plausibly uncorrelated with other regional shocks such as regional flood.

8
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summarized by the equation below: where _Tt_<sup>_u_</sup> _−_ 1<sup>and</sup><sup>_T_</sup> _i_<sup>_d_</sup> , _t−_ 1<sup>are the uniform and discretionary transfers at</sup><sup>_t−_1,</sup><sup>_α∈_[0,1] is a constant,</sup><sup>_fi_,</sup><sup>_t−_1is the</sup> regional revenue, and _rt−_ 1 _bi_ , _t−_ 1 is the net interest payment on debt, where net interest rate _r_ is related to the gross interest rate _R_ in the paper by _r_ = 1 _−_ 1/ _R_ .

The term _C_ ( _·_ ) is the central government’s estimated cost of local public services in period _t −_ 1, which will be elaborated later. Thus, (B.2) means if the central government estimates that the local government’s total revenue is insufficient to cover the cost of providing public services at _t −_ 1, it seeks to increase the transfers for period _t_ by a factor of _α_ , either as a result of the goodwill of the central government or the bargaining between the two levels of governments. Note that _C_ ( _·_ ) takes into account not only the expenses on salaries and goods and services (captured by _gi_ , _t−_ 1), but also the interest payment of the debt associated with spending. The reason is that for some local expenditure items (especially capital spending items such as building a hospital or highway), upfront spending financed by debt is usually incurred. The interest cost associated with this type of capital spending should be counted as a cost of local public service.<sup>49</sup> As discussed, estimating the cost of public service is difficult. In practice, past expenditure is an important factor in assessing the cost of service. For simplicity, assume the central government’s estimate of the cost of local public service follows where _δi_ , _t−_ 1 is an i.i.d shock with zero mean.

Substituting the local government’s budget constraint, the discretionary transfer that local government _i_ receives can be transformed to Equation (B.3) implies that for local government _i_ , increasing borrowing _bi_ , _t_ (decided at _t −_ 1) by one dollar to finance higher spending _gi_ , _t−_ 1 will result in an increase in discretionary transfer _Ti_<sup>_d_</sup> , _t_<sup>by</sup><sup>_α_dollar.</sup> More formally, aggregating (B.3) over _t_ = 1,2,... and substituting into (B.1), we have The first term on the right-hand side of (B.4) shows that transfers are partially dependent on debt outstanding.

> 49For example, in the United States, the criteria for state and local governments to apply for federal cost reimbursement grants is guided by the Office of Management and Budget (OMB) Circular A-87 and the associated guidance, which states that certain types of interest cost incurred by state and local governments should be allowable for the purpose of federal cost reimbursement.

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# **B.3 Exogenous interest rate schedules**

The theoretical results in the paper rely on the exogenous interest rate schedules _R_ ( _·_ ) and _S_ ( _·_ ) being _increasing_ in debt outstanding.<sup>50</sup> This assumption is widely used in international finance literature following SchmittGrohe and Uribe (2003) to ensure that the steady state is not dependent on initial conditions. The exponential functional form of _R_ ( _·_ ) and _S_ ( _·_ ) is only used for numerical exercises and is not a necessary assumption for the model properties to hold. Below we offer two ways to motivate the assumption that interest rates are increasing in debt. In the first approach, the increasing interest rate schedule comes from investors’ preference for portfolio diversification. In the second approach, the increasing interest rate schedule comes from financial intermediation frictions. Despite the different mechanisms, both approaches yield increasing interest rate schedules that are consistent with our model assumption.

**The first way** to motivate the positive slopes of interest rate schedules follows Chang et al. (2015), where international investors favor international portfolio diversification. Assume there is a continuum of international investors who can invest in Spanish government bonds _Dt_ and global bonds _Dt_<sup>_*_.51Theinternational</sup> investors’ problem is:

subject to the international investors’ budget constraint, In this problem, _Ct_<sup>_*_and</sup><sup>_Y_</sup> _t_<sup>_*_aretheinternationalinvestors’consumptionandincome,</sup><sup>_Dt_and</sup><sup>_D_</sup> _t_<sup>_*_aretheir</sup> holdings of Spanish and global bonds, and _St_ and _St_<sup>_*_are the respective gross interest rates.The parameter</sup> _ψ_ ̄ denotes the steady-state share of Spanish bonds in international investors’ portfolio. We assume _ψ_ ̄ _∈_ (0,1) so that investors prefer some degree of portfolio diversification. The parameter Ω _>_ 0 controls the size of the adjustment cost deviating from this steady-state portfolio allocation.

Denote _ηt_ as the multiplier to the budget constraint and _ψt_ = _DtD_ + _<u>tDt</u>_<sup>_~~*~~_, the optimality conditions are:</sup> Along the balanced growth path,<sup>_<u>ηt</u>_</sup> _η_<sup><u>+</u></sup> _t_<sup><u>1</u></sup> _→ β_ and the world interest rate _St_<sup>_*_=</sup> _β_<sup><u>1</u>.Hence,theSpanishbonds’</sup> interest rate is which is an increasing function of _Dt_ +1.

**The second way** to motivate the positive slopes of interest rate schedules is through financial intermediation frictions. Below we follow the setup of Gabaix and Maggiori (2015), though other ways to model financial frictions (e.g., Fanelli and Straub (forthcoming)) can also work.

> 50Assumption 2 states that both _R_ ( _·_ ) and _S_ ( _·_ ) are increasing and convex. However, convexity is not a necessary condition for our theoretical results to hold. What is essential is that the interest rate schedules are increasing functions with ranges large enough to ensure an interior solution. Convexity is a sufficient (but not necessary) condition to ensure the ranges of the interest rate schedules are large enough.

> 51The global bonds here can be understood as German government bonds or U.S. treasury bonds.

11
We assume that the government bond market is only accessible to a unit mass of “financiers” (e.g., banks and investment funds), reflecting the real world environment that the retail bond market either does not exist or exists in a small scale in many countries. Each financier has zero capital to start with and lives for only one period. In each period, a financier can draw deposits _qt_ from households at interest rate _St_<sup>_*_and use the</sup> proceeds to invest in government bonds at interest rate _St_ . Here _St_ will be endogenously determined. _St_<sup>_*_can</sup> be exogenously tied to global interest rates or central bank policy rates. At time _t_ , a financier’s value can be denoted as<sup>52</sup> It is clear that if the financial market is frictionless and there is no restrictions on the range of _qt_ that a financier can choose from, the non-arbitrage condition ensures that _St_ = _St_<sup>_*_.</sup> The key assumption here is that there are some forms of financial frictions that can limit financiers’ capacity of intermediating deposits into investments in government bonds. When this happens, the non-arbitrage condition _St_ = _St_<sup>_*_maynotholdanymore.FollowingGabaixandMaggiori(2015),theparticulartypeof</sup> financial friction we consider here is that the financiers can divert some of the invested assets away. More specifically, for a financier that draws deposits _q_ from households and uses the proceeds to invest in government bonds, the financier has the option to divert Γ _q_<sup>2</sup> and walk away. Because households anticipate that a financier has the option to divert assets and walk away, any deposit contract between households and financiers has to ensure that the value of any financier is larger than the assets that can be diverted away, Therefore, the financier will choose _qt_ to maximize his value, This problem yields the solution that each financier will choose That is, because of the friction (B.8), each financier’s investment in government bonds will be limited and hence the spread between _St_ and _St_<sup>_*_may not necessarily be eliminated.53Integrating over the unit mass of</sup> financiers, we have Therefore, the interest rate of government bonds is an increasing function of bond supply. The premium (over the risk-free rate _St_<sup>_*_) comes from the financial intermediation cost.</sup>

> 52Here we assume that the government bonds are one-period bonds sold at face value, so _qt_ deposits can be used to purchase _qt_ units of bonds. The principal and coupon payments ( _Stqt_ in total) are made in the next period. Changing the bond type to zero-coupon bonds with selling price _S_<sup><u>1</u></sup> _t_<sup>will not change the essence of the result.</sup>

> 53When the friction (B.8) does not apply, financiers can hold unlimited long or short positions of government bonds, arbitraging away any spread between _St_ and _St_<sup>_*_.</sup>

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# C General model

This section lays out the general model that we use for the calibration and quantitative exercises in Sections 3 and 4. To make exposition easier and to highlight economic mechanisms, a special case of this general model is used in Section 2. In the following sections of the Appendix, we show derivations and proofs for the general model, which also apply to the special case.

This model is more general than the model presented in Section 2 in two ways:

1. the constrained regions face a debt limit _b_<sup>_o_</sup> _i_ , _t_ +1<sup>_≤Bo_; and for</sup><sup>_β_and</sup><sup>_Bo≥_0 small enough (Assumption</sup> E.1 stated in Appendix E), the debt limit constraint is always binding: _b_<sup>_o_</sup> _i_ , _t_ +1<sup>=</sup><sup>_Bo_.</sup>

2. the transfer function contains a negotiation parameter _ε_ , such that region _i_ receives total transfer where _bi_ , _t_ is region _i_ ’s debt stock and _b_<sup> ̄</sup> _t_ = _μb_<sup>_n_</sup> _t_<sup>+ (1</sup><sup>_−μ_)</sup><sup>_Bo_is the cross-region average debt.</sup> The model in Section 2 is a special case when _B_<sup>_o_</sup> = 0 and _ε_ = 1. Below we use highlights to indicate the differences from this model.

**Consolidated government’s problem.** Given ( _b_<sup>_n_</sup> 0<sup>,</sup><sup>_d_0),</sup><sup>_bo_</sup> 0<sup>=</sup><sup>_Bo_andtheinterestrateschedules,theconsoli-</sup> dated government’s problem consists of choosing a sequence of debt and spending _{b_<sup>_n_</sup> _t_ +1<sup>,</sup><sup>_bo_</sup> _t_ +1<sup>,</sup><sup>_dt_+1,</sup><sup>_g_</sup> _t_<sup>_n_,</sup><sup>_g_</sup> _t_<sup>_o_,</sup><sup>_ct}∞_</sup> _t_ =0 that solves

subject to the consolidated budget constraint and constrained regions’ debt limit

**Without a consolidated government:**

**Constrained local government** is subject to the debt limit _b_<sup>_o_</sup> _i_ , _t_ +1<sup>_≤Bo_,whichbindsgivenAssumptionE.1.</sup> So the constrained region’s debt _b_<sup>_o_</sup> _i_ , _t_ +1<sup>=</sup><sup>_Bo_,and we do not need to keep track of it as a state variable.The</sup> constrained region’s spending is

**Unconstrained local government’s problem** is similar to the model in Section 2 except for the modified transfer function. Given the aggregate states ( _b_<sup>_n_</sup> _t_<sup>,</sup><sup>_dt_) and individual state</sup><sup>_bn_</sup> _i_ , _t_<sup>, an unconstrained local government</sup> _i_ chooses its spending _gi_<sup>_n_</sup> , _t_<sup>anddebt</sup><sup>_bn_</sup> _i_ , _t_ +1<sup>tomaximizethewelfareofitsresidents,takingthesequencesof</sup> central government policies and interest rates as given. Its problem written recursively is

subject to the budget constraint where _b_<sup> ̄</sup> _t_ = _μb_<sup>_n_</sup> _t_<sup>+ (1</sup><sup>_−μ_)</sup><sup>_Bo_.Its optimal choice of</sup><sup>_bn_</sup> _i_ , _t_ +1<sup>is characterized by the Euler equation,</sup> 13
**Central government’s problem** (in a Markov or Ramsey equilibrium) is similar to the model in Section 2, but with the modified local governments’ budget constraints (C.2) and (C.1) and the central government’s budget constraint:

Using the modified budget constraints to define the spending of the representative local and central governments as functions of transfer and debt, as in Section 2.4: A **Markov-perfect equilibrium** in the general model consists of a value function _V_ , central government’s policy rules _{_ Φ<sup>_τ_</sup> , Φ<sup>_T_</sup> , Φ<sup>_d_</sup> _}_ , and a policy function Φ<sup>_b_</sup> for the unconstrained local government’s debt, such that for all aggregate states ( _b_<sup>_n_</sup> , _d_ ), _τ_ = Φ<sup>_τ_</sup> ( _b_<sup>_n_</sup> , _d_ ), _T_<sup>_u_</sup> = Φ<sup>_T_</sup> ( _b_<sup>_n_</sup> , _d_ ), _d_<sup>_′_</sup> = Φ<sup>_d_</sup> ( _b_<sup>_n_</sup> , _d_ ) and _b_<sup>_n′_</sup> = Φ<sup>_b_</sup> ( _b_<sup>_n_</sup> , _d_ ) solve _τ_ , _T_ max<sup>_u_</sup> , _b_<sup>_n′_</sup> , _d_<sup>_′_</sup> (1 _− θ_ ) [︁ _μu_ (︁ _G_<sup>_n_</sup> ( _b_<sup>_n_</sup> , _b_<sup>_n′_</sup> , _τ_ , _T_<sup>_u_</sup> ))︁ + (1 _− μ_ ) _u_ ( _G_<sup>_o_</sup> ( _τ_ , _T_<sup>_u_</sup> ))]︁ + _θv_ (︀ _C_ ( _b_<sup>_n_</sup> , _d_ , _d_<sup>_′_</sup> , _τ_ , _T_<sup>_u_</sup> ))︀ + _βV_ ( _bn′_ , _d′_ ) subject to the representative unconstrained region’s Euler equation and a policy constraint, _ug G_<sup>_n_</sup> ( _b_<sup>_n_</sup> , _b_<sup>_n′_</sup> , _τ_ , _T_<sup>_u_</sup> ) = _βR_ ( _b_<sup>_n′_</sup> )(1 _− ε_ Φ<sup>_τ_</sup> ( _b_<sup>_n′_</sup> , _d_<sup>_′_</sup> )) _ug G_<sup>_n_</sup> ( _b_<sup>_n′_</sup> , Φ<sup>_b_</sup> ( _b_<sup>_n′_</sup> , _d_<sup>_′_</sup> ), Φ<sup>_τ_</sup> ( _b_<sup>_n′_</sup> , _d_<sup>_′_</sup> ), Φ<sup>_T_</sup> ( _b_<sup>_n′_</sup> , _d_<sup>_′_</sup> ))(C.8) (︁ )︁ (︁ )︁ _T_<sup>_u_</sup> _≥ T_ ̄ (C.9) and the central government’s value function satisfies the functional equation _V_ ( _b_<sup>_n_</sup> , _d_ ) = (1 _− θ_ ) _μu G_<sup>_n_</sup> ( _b_<sup>_n_</sup> , Φ<sup>_b_</sup> ( _b_<sup>_n_</sup> , _d_ ), Φ<sup>_τ_</sup> ( _b_<sup>_n_</sup> , _d_ ), Φ<sup>_T_</sup> ( _b_<sup>_n_</sup> , _d_ )) + (1 _− μ_ ) _u G_<sup>_o_</sup> (Φ<sup>_τ_</sup> ( _b_<sup>_n_</sup> , _d_ ), Φ<sup>_T_</sup> ( _b_<sup>_n_</sup> , _d_ )) [︁ (︁ )︁ (︁ )︁]︁ + _θv C_ ( _b_<sup>_n_</sup> , _d_ , Φ<sup>_d_</sup> ( _b_<sup>_n_</sup> , _d_ ), Φ<sup>_τ_</sup> ( _b_<sup>_n_</sup> , _d_ ), Φ<sup>_T_</sup> ( _b_<sup>_n_</sup> , _d_ )) + _βV_ (Φ<sup>_b_</sup> ( _b_<sup>_n_</sup> , _d_ ), Φ<sup>_d_</sup> ( _b_<sup>_n_</sup> , _d_ )) (︁ )︁ The **Ramsey problem:** Given initial debt positions ( _b_<sup>_n_</sup> 0<sup>,</sup><sup>_d_0), the optimal (Ramsey) policy of a central gov-</sup> ernment with commitment consists of a sequence of debt and transfer policies _{τt_ , _Tt_<sup>_u_,</sup><sup>_bn_</sup> _t_ +1<sup>,</sup><sup>_dt_+1</sup><sup>_}∞_</sup> _t_ =0<sup>that solves</sup> _∞_ max ∑︁ _β_<sup>_t_{︀</sup> (1 _− θ_ )[ _μu_ ( _G_<sup>_n_</sup> ( _b_<sup>_n_</sup> _t_<sup>,</sup><sup>_b_</sup> _t_<sup>_n_</sup> +1<sup>,</sup><sup>_τt_,</sup><sup>_T_</sup> _t_<sup>_u_)) + (1</sup><sup>_−μ_)</sup><sup>_u_(</sup><sup>_Go_(</sup><sup>_τt_,</sup><sup>_T_</sup> _t_<sup>_u_))] +</sup><sup>_θv_(</sup><sup>_C_(</sup><sup>_b_</sup> _t_<sup>_n_,</sup><sup>_dt_,</sup><sup>_dt_+1,</sup><sup>_τt_,</sup><sup>_T_</sup> _t_<sup>_u_))</sup> }︀ _t_ =0 subject to the following constraints for _all_ time _t ≥_ 0 where the budget constraints of central and local governments are implicit in the spending functions _G_<sup>_n_</sup> , _G_<sup>_o_</sup> and _C_ .

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# D Derivations

We present all derivations for the general model with debt limit _b_<sup>_o_</sup> _i_ , _t_<sup>_≤Bo_,</sup><sup>_Bo≥_0facedbytheconstrained</sup> regions, and _ε ∈_ (0,1] parameter in the transfer function. **The model presented in Section 2 is a special case when** _B_<sup>_o_</sup> = 0 **and** _ε_ = 1 **.**

## **D.1 Markov optimality conditions and GEEs**

This section derives the Markov optimality conditions (14)–(17) and Generalized Euler Equations (GEEs) for the general case _B_<sup>_o_</sup> _≥_ 0 and _ε ∈_ (0,1]. For expositional convenience, denote the unconstrained regions’ Euler equation (12) by _η_ ( _b_<sup>_n_</sup> , _b_<sup>_n′_</sup> , _d_<sup>_′_</sup> , _τ_ , _T_<sup>_u_</sup> ) = 0, and use _ηbn_ , _ηbn′_ , _ηd′_ , _ητ_ and _ηT_ to denote partial derivatives.

where _ζ ≥_ 0 is the Lagrange multiplier on the constraint (13). Envelope conditions are substitute the FOCs into the Envelope conditions, Differentiate _η_ with respect to _b_<sup>_n_</sup> and _d_ , substitute into _Vbn_ and _Vd_ , substitute into the last two FOCs, Expanding the auxiliary functions in (D.1), (D.2), (D.5) and (D.6), we get the optimality conditions of 15
the Markov equilibrium: where Next, we combine the optimality conditions to get the GEEs. From Proposition 2, the constraint on _T_<sup>_u_</sup> is binding, so Φ<sup>_T_</sup> _b_<sup>_n_= Φ</sup><sup>_T_</sup> _d_<sup>= 0 and the</sup><sup>_ζ_terms drop out in (D.5) and (D.6).From (D.1),</sup> Substitute the expression for _λ_ into (D.5) and (D.6), we get the two GEEs, 16
The auxiliary functions are _G_<sup>_n_</sup> _b_<sup>_n_</sup> = _−_ 1 + _ετ G_<sup>_n_</sup> _τ_ = _εb_<sup>_n_</sup> + (1 _− ε_ ) _b_<sup> ̄</sup> _G_<sup>_n_</sup> _T_ = 1 1 _G_<sup>_n_</sup> = _b_<sup>_n′_</sup> _R_ ( _b_<sup>_n′_</sup> ) _G_<sup>_o_</sup> _b_<sup>_n_</sup> = 0 _G_<sup>_o_</sup> _τ_ = _εB_<sup>_o_</sup> + (1 _− ε_ ) _b_<sup> ̄</sup> _G_<sup>_o_</sup> _T_ = 1 _Cbn_ = _−μτ Cτ_ = _−_ [ _μb_<sup>_n_</sup> + (1 _− μ_ ) _B_<sup>_o_</sup> ] _CT_ = _−_ 1 _Cd_ = _−_ 1 1 _Cd′_ = _S_ ( _d_<sup>_′_</sup> ) _ηbn_ = _u_<sup>_n_</sup> _gg_<sup>_G_</sup> _b_<sup>_nn_</sup> _ητ_ = _u_<sup>_n_</sup> _gg_<sup>_G_</sup> _τ_<sup>_n_</sup> _ηT_ = _u_<sup>_n_</sup> _gg_<sup>_G_</sup> _T_<sup>_n_</sup> _ηbn′_ = _u_<sup>_n_</sup> _gg_<sup>_Gn_</sup> _b_<sup>_n′_+</sup><sup>_βεR_(</sup><sup>_bn′_)Φ</sup><sup>_τ_</sup> _b_<sup>_n′un_</sup> _g_<sup>_′−βR_(</sup><sup>_bn′_)(︀</sup> 1 _− ετ_<sup>_′_)︀</sup> _u_<sup>_n_</sup> _gg_<sup>_′_</sup> _G_<sup>_n_</sup> _b_<sup>_n′_+</sup><sup>_G_</sup> _τ_<sup>_n′_Φ</sup><sup>_τ_</sup> _b_<sup>_n′_+</sup><sup>_G_</sup> _T_<sup>_n′_Φ</sup> _b_<sup>_Tn′_+</sup><sup>_Gn_</sup> _b_<sup>_n′′_Φ</sup> _b_<sup>_bn′_</sup> [︁ ]︁ _ηd′_ = _βεR_ ( _b_<sup>_n′_</sup> )Φ<sup>_τ_</sup> _d_<sup>_′un_</sup> _g_<sup>_′−βR_(</sup><sup>_bn′_)(︀</sup> 1 _− ετ_<sup>_′_)︀</sup> _u_<sup>_n_</sup> _gg_<sup>_′_</sup> [︁ _G_<sup>_n_</sup> _τ_<sup>_′_Φ</sup><sup>_τ_</sup> _d_<sup>_′_+</sup><sup>_G_</sup> _T_<sup>_n′_Φ</sup> _d_<sup>_T′_+</sup><sup>_Gn_</sup> _b_<sup>_n′′_Φ</sup> _d_<sup>_b′_</sup> ]︁ Note that because individual local government treats the average local debt _b_<sup> ̄</sup> as given, there is no additional term in _G_<sup>_n_theeffectof</sup><sup>_bn_on</sup><sup>_b_ ̄.SomeexpressionsareslightlydifferentintheextensionofSection</sup> _b_<sup>_n_from</sup> 4.3.1, because the governments internalize the slopes of their interest rate schedules (captured by highlighted parts): 17
# **D.1.1 Economic intuition from Markov GEEs**

The GEEs derived here provide another way to understand the intuition behind Proposition 3. Rewriting (D.11): This condition characterizes the marginal welfare effects of increasing the transfer rate _τ_ . The signs in parentheses in the GEEs are based on calibrated values of Section 3. Using Implicit Function Theorem, ratios of _η_ -derivatives have the interpretation of marginal effects holding other variables of the local government’s Euler equation constant. For example, _−ητ_ / _ηbn′_ gives the effect of changing _τ_ on _b_<sup>_n′_</sup> holding constant _b_<sup>_n_</sup> and _d_<sup>_′_</sup> .<sup>54</sup> The right-hand side of (D.13) comprises of three marginal welfare effects of increasing the transfer rate _τ_ . First, intra-temporally, a higher transfer _τ_ shifts spending from central to local governments in the current period. Second, _τ_ changes _b_<sup>_n′_</sup> (by _−ητ_ / _ηbn′ <_ 0 amount), which shifts spending inter-temporally. Third, _τ_ changes _τ_<sup>_′_</sup> through changing _b_<sup>_n′_</sup> , which affects within-period spending sharing between central and local governments in the next period.

The consolidated government achieves perfect resource sharing: _u_<sup>_n_</sup> _g_<sup>=</sup><sup>_uo_</sup> _g_<sup>=</sup><sup>_θ_/(1</sup><sup>_−θ_)</sup><sup>_vc_(MULG = MUCG).</sup> This essentially corresponds to the first line of (D.13). In addition, at the steady state, the third line of (D.13) also collapses into the perfect resource sharing condition. This leaves the second line of (D.13) as the major difference between the consolidated government allocation and the non-commitment (Markov) equilibrium. Specifically, the second line demonstrates that the central government’s marginal value of increasing _b_<sup>_n′_</sup> is different from the marginal value perceived by local governments. To see this, we decompose the inter-temporal effect of raising _b_<sup>_n′_</sup> (second line of D.13) into three parts: the marginal benefit from higher local spending today, the marginal cost of lower local spending tomorrow, and the additional marginal cost to the central which captures the marginal welfare effects of increasing _τ_ , through changing _d_<sup>_′_</sup> while holding _b_<sup>_n′_</sup> constant. For the purpose here we only need to focus on (D.13).

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government from larger transfers tomorrow: The first two terms on the right-hand side of (D.15) constitute the unconstrained local government’s Euler equation and capture the local government’s trade-offs of increasing _b_<sup>_n′_</sup> . The central government, by contrast, internalizes an additional cost of increasing _b_<sup>_n′_</sup> : it raises the total transfers paid out tomorrow. When _τ_<sup>_′_</sup> = 0, as in the consolidated government case, (D.15) coincides with the unconstrained local government’s Euler equation and is zero, and so at the steady state (D.13) implies perfect resource sharing. When _τ_<sup>_′_</sup> _>_ 0, as in the Markov equilibrium, the distortionary transfer drives a wedge between the central and local governments’ perceived marginal values of _b_<sup>_n′_</sup> : while the local governments think the borrowing cost is cut by _τ_<sup>_′_</sup> _>_ 0, the central government recognizes that larger borrowing implies larger transfers tomorrow, and the total borrowing cost paid by the entire fiscal system is still _R_ ( _b_<sup>_n′_</sup> ). Thus the central government wants a lower _b_<sup>_n′_</sup> than the level determined by the unconstrained local government’s inter-temporal condition.

To achieve this, the central government has to use higher distortionary transfers to relax the unconstrained local government’s budget constraint in the current period and reduce its need to borrow. In the equilibrium, the transfer level is higher than that implied by the within-period term of (D.13) alone. As a result, the weighted marginal utility of local spending is too low relative to that of central spending, _μu_<sup>_n_</sup> _g_<sup>+ (1</sup><sup>_−μ_)</sup><sup>_uo_</sup> _g_<sup>_<_</sup> _θ_ /(1 _− θ_ ) _vc_ . In other words, compared with the consolidated government allocation, which is the efficient allocation, the central government **over-transfers** (Proposition 3).

The over-transfer, however, does not reduce local government debt in equilibrium, because of the ex ante overborrowing effect. When the unconstrained local governments anticipate a larger transfer today, they borrow more in the previous period. Because the central government in this time-consistent equilibrium takes ex ante incentives as forgone, it does not take into account this ex ante effect of larger transfers. The larger transfer, aimed to lower local debt, ends up raising local debt. In other words, **over-transfer** exacerbates **overborrowing** .

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# **D.2 Ramsey optimality conditions**

This section derives the Ramsey optimality conditions for the general case _B_<sup>_o_</sup> _≥_ 0 and _ε ∈_ (0,1]. Denote the unconstrained local government’s Euler equation at time _t_ by _η_ ̃( _b_<sup>_n_</sup> _t_<sup>,</sup><sup>_bn_</sup> _t_ +1<sup>,</sup><sup>_τt_,</sup><sup>_T_</sup> _t_<sup>_u_,</sup><sup>_bn_</sup> _t_ +2<sup>,</sup><sup>_τt_+1,</sup><sup>_T_</sup> _t_<sup>_u_</sup> +1<sup>) = 0withthe</sup> Lagrange multiplier _β_<sup>_t_</sup> _γt_ . Let _β_<sup>_t_</sup> _ζt_<sup>_r_be the Lagrange multiplier on the constraint</sup><sup>_T_</sup> _t_<sup>_u≥T_ ̄.The time-</sup><sup>_t_optimality</sup> conditions for _t>_ 0 are: Expanding the auxiliary functions and writing the above conditions recursively: Setting _B_<sup>_o_</sup> = 0 and _ε_ = 1, we get conditions (23)–(26) in Section 2.5.

The auxiliary functions are listed below. The highlighted parts are the additional terms applicable only when local and central governments internalize the slopes of their interest rate schedules (Section 4.3.1).

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The model presented in Section 2 is a special case with _B_<sup>_o_</sup> = 0 and _ε_ = 1 in the above auxiliary functions.

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# E Proofs

We present all proofs for the general model with debt limit _b_<sup>_o_</sup> _i_ , _t_<sup>_≤Bo_,</sup><sup>_Bo≥_0 faced by the constrained regions</sup> and the parameter _ε ∈_ (0,1] in the transfer function. **The model presented in Section 2 is a special case when** _B_<sup>_o_</sup> = 0 **and** _ε_ = 1 **.**

## **Preamble**

The general model requires a modified assumption that replaces Assumption 3(b).

**Assumption E.1.** _β_ and _B_<sup>_o_</sup> are small enough such that given _ε ∈_ (0,1], _βR_ ( _B_<sup>_o_</sup> )(1 _− ετ_  ̄) _<_ 1, where _τ<_  ̄ 0 is the lower bound of _τ_ .

This assumption ensures that _b_<sup>_o_</sup> _i_ , _t_<sup>_≤Bo_binds and constrained local governments’ debt is</sup><sup>_bo_</sup> _i_ , _t_<sup>=</sup><sup>_Bo_in equilib-</sup> rium. In addition, Assumption 4 is restated for the general case as

**Assumption E.2.** ( **Vertical fiscal imbalance with** _B_<sup>_o_</sup> ) _f_ + _T_<sup> ̄</sup> is small relative to _e_ such that

where _D_<sup> ̄</sup> satisfy _S_ ( _D_<sup> ̄</sup> ) = 1/ _β_ .

When _B_<sup>_o_</sup> = 0, this modified assumption becomes Assumption 4.

## **E.1 Proof of Lemma 1 (Cross-region spending gap in Markov)**

At the steady state, the unconstrained local government’s Euler equation (C.3) implies _βR_ ( _b_<sup>_n_</sup> _ss_<sup>)(1</sup><sup>_−ετss_) = 1.</sup> Given the assumptions that _βR_ ( _B_<sup>_o_</sup> )(1 _− ετ_ ̄) _<_ 1 (Assumption E.1), _τ ≥ τ_ ̄ and _R_ ( _·_ ) is an increasing function, it is easy to see that _b_<sup>_n_</sup> _ss_<sup>_> Bo_.</sup> and for the special case _B_<sup>_o_</sup> = 0, this implies _b_<sup>_n_</sup> _ss_<sup>_>_0.</sup> It follows then at the steady state, where the first inequality follows from _b_<sup>_n_</sup> _ss_<sup>_> Bo≥_0, and the second inequality follows from</sup> which makes use of the unconstrained local government’s Euler equation (C.3) at the steady state.

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# **E.2 Proof of Proposition 3 (Over-transfer in Markov)**

The optimality condition (D.8): From the Kuhn Tucker Theorem, the multiplier _ζ ≥_ 0. Thus the following lemma is sufficient to prove Proposition 3.

**Lemma E.1.** The Lagrange multiplier _λ>_ 0 at the steady state.

_Proof._ The proof will be organized as follows: we first construct an alternate central government’s problem where the implementability constraint (C.8) is relaxed to an inequality (so that the associated Lagrange multipliers are non-negative). Next, we show that the solution to the alternate problem, with non-negative Lagrange multipliers, also solves the original problem around the steady state. Finally, we show that the Lagrange multipliers must be strictly positive.

- (1) First, we set up the alternate problem:

**The alternate central government’s problem.** The alternate central government’s problem is similar to the original central government’s problem defined in Section 2.4, except that the implementability constraint is an inequality.

and the central government’s value function satisfies the functional equation Denote the Lagrange multiplier of constraint (E.1) by _λ_<sup> ̃</sup> . The optimality conditions of the alternative problem include conditions (D.7)-(D.10) (after replacing _λ_ with _λ_<sup> ̃</sup> ) and the complementary slackness condition from the Kuhn-Tucker Theorem,

- (2) Next, we show that around the steady state, the solution to the alternate problem also satisfies the constraint (E.1) with equality (or equivalently, constraint (C.8) of the general model), so that the solution to the alternate problem also solves the original problem.

We prove it by contradiction. Suppose for **<u>contradiction</u>** that there is a solution to the alternate problem such that the constraint (E.1) is not binding: Then from (E.2), we have _λ_<sup> ̃</sup> = 0. Plug it into the first order conditions of the alternate problem (still 23
conditions D.7-D.10), we have Define _μ_ ˆ = _μb_<sup>_<u>μ</u>_</sup><sup><u>[</u></sup><sup>_εbn_</sup> +(<sup>_n_</sup><sup><u>+(</u></sup> 1 _−_<sup>1</sup><sup>_−_</sup> _μ_ )<sup>_ε_</sup> _B_<sup><u>)</u></sup><sup>_b_ ̄</sup><sup>_<u>o</u>_</sup><sup><u>]</u>.We can rewrite (E.4) as</sup> From Lemma 1, since _b_<sup>_n_</sup> _> B_<sup>_o_</sup> around the steady state, _b_<sup>_n_</sup> _> b_<sup> ̄</sup> _≡ μb_<sup>_n_</sup> + (1 _− μ_ ) _B_<sup>_o_</sup> for any _μ ∈_ (0,1). So we have _μ_ ˆ _∈_ ( _μ_ ,1) for any _ε ∈_ (0,1]. Further, from Lemma 1, _u_<sup>_n_</sup> _g_<sup>_> uo_</sup> _g_<sup>aroundthesteadystate.Hence</sup> the left-hand side of (E.8) is greater than the left-hand side of (E.5), and so _ζ>_ 0, which further implies _T_<sup>_u_</sup> = _T_<sup> ̄</sup> .

Because _T_<sup> ̄</sup> is not big enough to resolve the vertical fiscal imbalance (Assumption E.2), the distortionary transfer rate _τ_ must be strictly positive around the steady state, otherwise the total transfers will not be large enough to make (E.5) and (E.8) hold.

_T_<sup>_u_</sup> = _T_<sup> ̄</sup> further implies that Φ<sup>_T_</sup> _b_<sup>_n_= 0 and Φ</sup><sup>_T_</sup> _d_<sup>= 0.Then we can rewrite (E.6) as</sup> With _τ_<sup>_′_</sup> _>_ 0 around the steady state,<sup>55</sup> the right-hand side of (E.9) is strictly positive, therefore the lefthand side of (E.9) must also be strictly positive: This contradicts (E.3) when _τ_<sup>_′_</sup> _>_ 0 around the steady state. Hence, the solution of the alternate problem must satisfy (E.1) with equality around the steady state. As such, it also solves the original problem around the steady state if we let _λ_ = _λ_<sup> ̃</sup> .

- (3) Last, we show that the non-negative _λ_<sup> ̃</sup> must be strictly positive, hence _λ_ is also strictly positive. Follow the same proof in Step (2), it is easy to see that when _λ_<sup> ̃</sup> = 0, (E.10) has to hold. This contradicts with (E.1). Hence _λ_<sup> ̃</sup> must be strictly positive.

# **E.3 Proof of Proposition 2 (Minimum uniform transfer in Markov)**

Define _μ_ ˆ = _μb_<sup>_<u>μ</u>_</sup><sup><u>[</u></sup><sup>_εbn_</sup> +(<sup>_n_</sup><sup><u>+(</u></sup> 1 _−_<sup>1</sup><sup>_−_</sup> _μ_ )<sup>_ε_</sup> _B_<sup><u>)</u></sup><sup>_b_ ̄</sup><sup>_<u>o</u>_</sup><sup><u>]</u>.We can rewrite (D.7) as</sup>

> 55Note that both _T u_ = _T_  ̄ and _τ ′ >_ 0 here follow from the contradiction and are not part of Proposition 3. The properties that _T_<sup>_u_</sup> = _T_<sup> ̄</sup> and _τ>_ 0 around the steady state are formally proved in Proposition 2.

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At the steady state, The first equality is from (D.8). The first inequality follows because: at the steady state for any _μ ∈_ (0,1) and given _ε ∈_ (0,1], _μ> μ_ ˆ as _b_<sup>_n_</sup> _ss_<sup>_> Bo≥_0, and</sup><sup>_un_</sup> _g_<sup>_> uo_</sup> _g_<sup>as</sup><sup>_g_</sup> _ss_<sup>_n< g_</sup> _ss_<sup>_o_(Lemma 1).The second equality follows from</sup> (E.11). The second inequality comes from _λ>_ 0 (Proposition 3), _μ> μ_ ˆ and _u_<sup>_n_</sup> _gg_<sup>_<_0.Hence,</sup><sup>_ζ>_0,andthe</sup> constraint _T_<sup>_u_</sup> _≥ T_<sup> ̄</sup> must be binding.

To show _τ>_ 0 at the steady state, note that _u_<sup>_o_</sup> _g_<sup>_< un_</sup> _g_<sup>(Lemma 1) and (E.11) together imply</sup> as _λ>_ 0 and _u_<sup>_n_Since we have established</sup><sup>_Tu_=</sup><sup>_T_ ̄, if</sup><sup>_τ≤_0, Assumption E.2 would imply (1</sup><sup>_−θ_)</sup><sup>_uo_</sup> _gg_<sup>_<_0.</sup> _g_<sup>_> θvc_,</sup> which is a contradiction. Therefore, it must be the case that _τ>_ 0.

# **E.4 Proof of Proposition 4 (Under-transfer in Ramsey)**

- In the first part we prove _τ_ = 0 at the steady state.

At the steady state, the left-hand side of (D.18) can be simplified to where we use the unconstrained region’s Euler equation at the steady state _βR_ ( _b_<sup>_n_</sup> )(1 _− ετ_ ) = 1.

Similarly, the right-hand side of (D.18) at the steady state becomes where _βR_ ( _b_<sup>_n_</sup> )(1 _− ετ_ ) = 1 is repeatedly used. Equating the left- and right-hand sides, we can get _τ_ = 0 at the steady state.

- Next, we show _T_<sup>_u_</sup> _> T_<sup> ̄</sup> in the Ramsey steady state equilibrium. We first prove the following lemma:

**Lemma E.2.** The Lagrange multiplier _γ>_ 0 at the steady state.<sup>56</sup>

_Proof._ By a similar argument as in Section E.2, we can construct an alternate problem (to the Ramsey ̃ problem) with inequalities in the implementability constraints and Lagrange multiplier _γ ≥_ 0. Then we can show the solution to this alternate problem also solves the original problem, and so the Lagrange multiplier of the original problem _γ ≥_ 0.

> 56Note that _γ>_ 0 holds in this case (despite the absence of distortionary transfer in equilibrium, i.e., _τ_ = 0) because of the horizontal fiscal imbalance between regions. If there is only one type of region ( _μ_ = 0 or 1 or _B_<sup> ̄</sup> = _B_<sup>_o_</sup> ), then _τ_ = 0 implies _γ_ = 0 and the Ramsey equilibrium would achieve the (efficient) consolidated government allocation.

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It remains to show that _γ_ = 0. Suppose for **contradiction** _γ_ = 0, then (D.16)-(D.17) become Given _τ_ = 0 at the steady state, the unconstrained region’s Euler equation implies _βR_ ( _b_<sup>_n_</sup> _ss_<sup>) = 1, so</sup><sup>_b_</sup> _ss_<sup>_n_=</sup> _B> B_ ̄<sup>_o_</sup> (Assumptions 2 and E.1). It follows then _μ> μ_ ˆ around the steady state for any _ε ∈_ (0,1] and _μ ∈_ (0,1). Because _b_<sup>_n_</sup> _ss_<sup>_> Bo_and both types of regions get the same transfer</sup><sup>_Tu_, we have</sup><sup>_g_</sup> _ss_<sup>_n< g_</sup> _ss_<sup>_o_and</sup> _u_<sup>_n_So the left-hand side of (E.14) is greater than the left-hand side of (E.13).This implies</sup><sup>_ζr>_0,</sup> _g_<sup>_> uo_</sup> _g_<sup>.</sup> and _ζ_<sup>_r_</sup> _>_ 0 also means _T_<sup>_u_</sup> = _T_<sup> ̄</sup> , and since we have shown _τ_ = 0, the transfer that local governments get is only _T_ ̄ . Hence, under the assumption that _f_ + _T_ ̄ is small relative to _e_ (Assumption E.2), local government spendings _g_<sup>_n_</sup> and _g_<sup>_o_</sup> are low relative to central government consumption _c_ . But this contradicts with (E.15). Thus _γ>_ 0.

Combining (D.16)–(D.17), we have Because _γ>_ 0 and _ζ_<sup>_r_</sup> _≥_ 0, the right hand side of (E.16) is strictly positive.

On the left-hand side of (E.16), because _τ_ = 0 in the Ramsey steady state, from the Euler equation of the unconstrained local government, it must be that _b_<sup>_n_</sup> = _B_<sup> ̄</sup> in the Ramsey steady state. Because _B> B_<sup> ̄</sup><sup>_o_</sup> from Assumption E.1, we have _b_<sup>_n_</sup> _> B_<sup>_o_</sup> in the Ramsey steady state. Therefore, in order to make the left hand side of (E.16) strictly positive, we need Suppose for **contradiction** _T_<sup>_u_</sup> = _T_<sup> ̄</sup> . Given _τ_ = 0, the only transfer to any local government is _T_<sup> ̄</sup> . Then Assumption E.2 ( _f_ + _T_<sup> ̄</sup> is small relative to _e_ ) implies (1 _− θ_ ) _u_<sup>_o_</sup> _g_<sup>_> θvc_, which contradicts (E.17).Therefore,</sup> it must be the case _T_<sup>_u_</sup> _> T_<sup> ̄</sup> . This also implies that _ζ_<sup>_r_</sup> = 0.

- Lastly, given _τ_ = 0, (D.17) at the steady state becomes

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# F Model extensions

## **F.1 Allowing both types of local governments to borrow**

In this subsection, we relax the assumption in the benchmark model that only the unconstrained regions’ local governments can borrow. We assume that both types of regions can borrow, but they face different interest schedules. We show that the constrained regions, which face a higher interest schedule, would have lower local government debt levels than the unconstrained regions.<sup>57</sup> The central government will always want to use debt-dependent transfers to subsidize the unconstrained regions (which are more indebted) and the main results in our paper carry through. For exposition purposes, we still use the terms “constrained” and “unconstrained”, although now both types of local governments can borrow freely. In particular, we make the following assumption to replace Assumption 2:

**Assumption F.1.** Local governments in both types of regions can borrow.

(a) The unconstrained and constrained local governments face the interest schedules _R_<sup>_n_</sup> ( _·_ ) and _R_<sup>_o_</sup> ( _·_ ), respectively. Both _R_<sup>_n_</sup> ( _·_ ) and _R_<sup>_o_</sup> ( _·_ ) are increasing and convex functions, and there exist _B_<sup> ̄</sup><sup>_n_</sup> and _B_<sup> ̄</sup><sup>_o_</sup> such that _R_<sup>_n_</sup> ( _B_<sup> ̄</sup><sup>_n_</sup> ) = _R_<sup>_o_</sup> ( _B_<sup> ̄</sup><sup>_o_</sup> ) = 1/ _β_ ; (b) _B_<sup> ̄</sup><sup>_o_</sup> is sufficiently small relative to _B_<sup> ̄</sup><sup>_n_</sup> . A sufficient condition when _τt ∈_ [0, _τmax_ ] is that _B_<sup> ̄</sup><sup>_o_</sup> _<_ (1 _− τmax_ ) _B_<sup> ̄</sup><sup>_n_</sup> .

Assumption F.1(a) allows the constrained and unconstrained regions to borrow freely under different interest schedules. Assumption F.1(b) ensures that the “constrained” regions’ interest schedule is high enough, so that (as we will show later) they have lower debt and higher spending in equilibrium than the “unconstrained” regions. The underlying reasons for the difference in interest schedules may be due to differences in credit risks, targeted investor base or tax treatment. We do not distinguish the specific reasons behind it.

The additional technical difficulty is that when both types of regions can borrow, there is an additional state variable, which is the debt stock of the constrained regions ( _b_<sup>_o_</sup> ). The additional state variable not only increases the dimensionality of the problem, but also brings additional derivative terms of the policy functions, such as the derivatives of the the unconstrained regions’ spending and borrowing choices with respect to the constrained regions’ debt. To keep the model tractable, we make the following simplifying assumption that only the unconstrained regions receive the debt-dependent transfers _τb_<sup>_n_</sup> . As we will show later, this assumption means that we do not need to keep track of the constrained regions’ debt stock _b_<sup>_o_</sup> as a state variable in solving the Markov equilibrium.<sup>58</sup>

**Assumption F.2.** Constrained local governments do not face the distortionary transfer scheme _τ_ .

Although allowing only one type of regions to receive debt-dependent transfers is a technical simplification, Assumption F.2 also has its real-world relevance. For example, Rodden (2005b) documents a similar system in Germany, where under their equalization scheme, only some states are eligible for discretionary transfers. The equalization scheme, which is an intergovernmental redistribution scheme, includes two parts: formula-based transfers and discretionary transfers. The formula-based part divides all German states into two categories based on their tax revenues: those with lower-than-average tax revenues (the _recipient_ states), and those with higher tax revenues (the _paying_ states). Only the _recipient_ states are eligible to receive the

> 57In the benchmark model presented in Section 2, the heterogeneity in the level of local borrowing is achieved by assuming that the constrained regions cannot borrow, either due to self-prudence or financial market frictions.

> 58With Assumption F.2, _bo_ is no longer a state variable for the central government’s and the unconstrained regions’ problems, although it is still a state variable for the constrained regions’ problem.

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discretionary transfers, which also include the bailouts to debt-distressed states. A _paying_ state would not qualify for the discretionary transfers, even if the state is heavily indebted.

With Assumptions F.1 and F.2, the local governments’ Euler equations are The budget constraints are

**Definition F.1.** A Markov-perfect equilibrium consists of a value function _V_ , central government’s policy rules _{_ Φ<sup>_τ_</sup> , Φ<sup>_T_</sup> , Φ<sup>_d_</sup> _}_ , and policy functions _{_ Φ<sup>_b_</sup> , Φ<sup>_o_</sup> _}_ for two types of local government’s debt, such that for all aggregate states ( _b_<sup>_n_</sup> , _b_<sup>_o_</sup> , _d_ ), _τ_ = Φ<sup>_τ_</sup> ( _b_<sup>_n_</sup> , _b_<sup>_o_</sup> , _d_ ), _T_<sup>_u_</sup> = Φ<sup>_T_</sup> ( _b_<sup>_n_</sup> , _b_<sup>_o_</sup> , _d_ ), _d_<sup>_′_</sup> = Φ<sup>_d_</sup> ( _b_<sup>_n_</sup> , _b_<sup>_o_</sup> , _d_ ), _b_<sup>_n′_</sup> = Φ<sup>_b_</sup> ( _b_<sup>_n_</sup> , _b_<sup>_o_</sup> , _d_ ) and _b_<sup>_o′_</sup> = Φ<sup>_o_</sup> ( _b_<sup>_n_</sup> , _b_<sup>_o_</sup> , _d_ ) solve

subject to the local regions’ Euler equations and a policy constraint, and the central government’s value function satisfies the functional equation

# **Derivation of Markov optimality conditions**

Denote the local governments’ Euler equations (F.1) and (F.2) by _η_<sup>_n_</sup> ( _b_<sup>_n_</sup> , _b_<sup>_n′_</sup> , _b_<sup>_o′_</sup> , _d_<sup>_′_</sup> , _τ_ , _T_<sup>_u_</sup> ) = 0 and 28
_η_<sup>_o_</sup> ( _b_<sup>_o_</sup> , _b_<sup>_n′_</sup> , _b_<sup>_o′_</sup> , _d_<sup>_′_</sup> , _τ_ , _T_<sup>_u_</sup> ) = 0, and the corresponding multipliers by _λ_<sup>_n_</sup> and _λ_<sup>_o_</sup> .

where _ζ ≥_ 0 is the Lagrange multiplier on the _T_<sup> ̄</sup> constraint (F.3). Envelope conditions are substitute the FOCs into the Envelope conditions, Differentiate _η_<sup>_n_</sup> and _η_<sup>_o_</sup> with respect to _b_<sup>_n_</sup> , _b_<sup>_o_</sup> and _d_ , substitute into _Vbn_ , _Vbo_ and _Vd_ , 29
substitute into the last three FOCs, Expanding the auxiliary functions in (F.4)-(F.5) and (F.9)-(F.11), we get the following Markov optimality conditions.

where 30
The auxiliary functions are _G_<sup>_n_</sup> _b_<sup>_n_</sup> = _−_ (1 _− τ_ ) _G_<sup>_n_</sup> _τ_ = _b_<sup>_n_</sup> _G_<sup>_n_</sup> _T_ = 1 1 _G_<sup>_n_</sup> = _b_<sup>_n′_</sup> _R_<sup>_n_</sup> ( _b_<sup>_n′_</sup> ) _G_<sup>_o_</sup> _b_<sup>_o_</sup> = _−_ 1 _G_<sup>_o_</sup> _τ_ = 0 _G_<sup>_o_</sup> _T_ = 1 1 _G_<sup>_o_</sup> = _b_<sup>_o′_</sup> _R_<sup>_o_</sup> ( _b_<sup>_o′_</sup> ) _Cbn_ = _−μτ Cbo_ = 0 _Cτ_ = _−μb_<sup>_n_</sup> _CT_ = _−_ 1 _Cd_ = _−_ 1 1 _Cd′_ = _S_ ( _d_<sup>_′_</sup> ) _ηb_<sup>_nn_</sup> = _u_<sup>_n_</sup> _gg_<sup>_G_</sup> _b_<sup>_nn_</sup> _ητ_<sup>_n_</sup> = _u_<sup>_n_</sup> _gg_<sup>_G_</sup> _τ_<sup>_n_</sup> _ηT_<sup>_n_</sup> = _u_<sup>_n_</sup> _gg_<sup>_G_</sup> _T_<sup>_n_</sup> _ηb_<sup>_nn′_</sup> = _u_<sup>_n_</sup> _gg_<sup>_Gn_</sup> _b_<sup>_n′_+</sup><sup>_βRn_(</sup><sup>_bn′_)Φ</sup><sup>_τ_</sup> _b_<sup>_n′u_</sup> _g_<sup>_n′−βRn_(</sup><sup>_bn′_)(︀</sup> 1 _− τ_<sup>_′_)︀</sup> _u_<sup>_n_</sup> _gg_<sup>_′_</sup> [︁ _G_<sup>_n_</sup> _b_<sup>_n′_+</sup><sup>_G_</sup> _τ_<sup>_n′_Φ</sup><sup>_τ_</sup> _b_<sup>_n′_+</sup><sup>_G_</sup> _T_<sup>_n′_Φ</sup><sup>_T_</sup> _b_<sup>_n′_+</sup><sup>_Gn_</sup> _b_<sup>_n′′_Φ</sup> _b_<sup>_bn′_</sup> ]︁ _ηb_<sup>_no′_</sup> = _βR_<sup>_n_</sup> ( _b_<sup>_n′_</sup> )Φ<sup>_τ_</sup> _b_<sup>_o′un_</sup> _g_<sup>_′−βRn_(</sup><sup>_bn′_)(︀</sup> 1 _− τ_<sup>_′_)︀</sup> _u_<sup>_n_</sup> _gg_<sup>_′_</sup> [︁ _G_<sup>_n_</sup> _τ_<sup>_′_Φ</sup><sup>_τ_</sup> _b_<sup>_o′_+</sup><sup>_G_</sup> _T_<sup>_n′_Φ</sup><sup>_T_</sup> _b_<sup>_o′_+</sup><sup>_Gn_</sup> _b_<sup>_n′′_Φ</sup> _b_<sup>_bo′_</sup> ]︁ _ηd_<sup>_n′_</sup> = _βR_<sup>_n_</sup> ( _b_<sup>_n′_</sup> )Φ<sup>_τ_</sup> _d_<sup>_′un_</sup> _g_<sup>_′−βRn_(</sup><sup>_bn′_)(︀</sup> 1 _− τ_<sup>_′_)︀</sup> _u_<sup>_n_</sup> _gg_<sup>_′_</sup> [︁ _G_<sup>_n_</sup> _τ_<sup>_′_Φ</sup><sup>_τ_</sup> _d_<sup>_′_+</sup><sup>_G_</sup> _T_<sup>_n′_Φ</sup><sup>_T_</sup> _d_<sup>_′_+</sup><sup>_Gn_</sup> _b_<sup>_n′′_Φ</sup> _d_<sup>_b′_</sup> ]︁ _ηb_<sup>_oo_</sup> = _u_<sup>_o_</sup> _gg_<sup>_G_</sup> _b_<sup>_oo_=</sup><sup>_−uo_</sup> _gg ητ_<sup>_o_</sup> = _u_<sup>_o_</sup> _gg_<sup>_G_</sup> _τ_<sup>_o_= 0</sup> _ηT_<sup>_o_</sup> = _u_<sup>_o_</sup> _gg_<sup>_G_</sup> _T_<sup>_o_</sup> _ηb_<sup>_on′_</sup> = _−βR_<sup>_o_</sup> ( _b_<sup>_o′_</sup> ) _u_<sup>_o_</sup> _gg_<sup>_′_</sup> _G_<sup>_o_</sup> _T_<sup>_′_Φ</sup><sup>_T_</sup> _b_<sup>_n′_+</sup><sup>_Go_</sup> _b_<sup>_o′′_Φ</sup> _b_<sup>_on′_</sup> [︁ ]︁ _ηb_<sup>_oo′_</sup> = _u_<sup>_o_</sup> _gg_<sup>_Go_</sup> _b_<sup>_o′−βRo_(</sup><sup>_bo′_)</sup><sup>_uo_</sup> _gg_<sup>_′_</sup> _−_ 1 + _G_<sup>_o_</sup> _T_<sup>_′_Φ</sup> _b_<sup>_To′_+</sup><sup>_Go_</sup> _b_<sup>_o′′_Φ</sup> _b_<sup>_oo′_</sup> [︁ ]︁ _ηd_<sup>_o′_</sup> = _−βR_<sup>_o_</sup> ( _b_<sup>_o′_</sup> ) _u_<sup>_o_</sup> _gg_<sup>_′_</sup> [︁ _G_<sup>_o_</sup> _T_<sup>_′_Φ</sup><sup>_T_</sup> _d_<sup>_′_+</sup><sup>_Go_</sup> _b_<sup>_o′′_Φ</sup> _d_<sup>_o′_</sup> ]︁

# **Outline of the Proofs of Propositions 2 and 3 when both types of regions can borrow**

We first show **Lemma 1** ( _gss_<sup>_o> g_</sup> _ss_<sup>_b_).</sup> At the steady state, the local governments’ Euler equations (F.1) and (F.2) are 31
which implies _R_<sup>_n_</sup> ( _b_<sup>_n_</sup> _ss_<sup>)(1</sup><sup>_−τss_) =</sup><sup>_Ro_(</sup><sup>_bo_</sup> _ss_<sup>) =</sup> _β_<sup><u>1</u>.It follows then at the steady state,</sup> Therefore, _gss_<sup>_o> g_</sup> _ss_<sup>_n_is equivalent to</sup><sup>_B_ ̄</sup><sup>_o<_(1</sup><sup>_−τss_)</sup><sup>_bn_</sup> _ss_<sup>.The latter is guaranteed by Assumption F.1(b).</sup>

# Next we show **Propositions 2 and 3** .

It is easy to see that if the central government always finds it optimal to choose _Tt_<sup>_u_=</sup><sup>_T_ ̄inequilibrium</sup> (which is to be verified later), the constrained regions’ maximization problem is independent of the central government and the unconstrained regions’ choices. That is, if _Tt_<sup>_u_=</sup><sup>_T_ ̄holds, (</sup><sup>_bo_</sup> _t_<sup>,</sup><sup>_g_</sup> _t_<sup>_o_) will be determined by</sup> the constrained regions’ budget constraint and Euler equation rather than _τt_ and _b_<sup>_n_</sup> _t_<sup>.Meanwhile, the central</sup> government’s choice of _τt_ and the unconstrained regions’ choice of _b_<sup>_n_</sup> _t_<sup>and</sup><sup>_g_</sup> _t_<sup>_n_will be independent of</sup><sup>_bo_</sup> _t_<sup>.</sup> Therefore, as long as _Tt_<sup>_u_=</sup><sup>_T_ ̄in equilibrium, the constrained regions’ Euler equation (F.2) can be dropped</sup> from the central government’s problem, Φ<sup>_n_</sup> _b_<sup>_o_= Φ</sup><sup>_τ_</sup> _b_<sup>_o_= Φ</sup><sup>_T_</sup> _b_<sup>_o_= 0 and Φ</sup><sup>_o_</sup> _b_<sup>_n_= Φ</sup><sup>_o_</sup> _d_<sup>= 0.Thus, the optimality condi-</sup> tions (F.12)-(F.16) can be simplified to The rest of the proofs of Propositions 2 and 3 follow the proofs in the benchmark model. Intuitively, the vertical fiscal imbalance entails central government’s transfers to regions, either through _τ_ or _T_<sup>_u_</sup> . Because the constrained regions’ debt burden is smaller than the unconstrained regions (in absence of any transfers through _τ_ ), the constrained regions’ public spending will be higher than the unconstrained regions. Hence the central government always prefers using _τ_ , which can channel more transfers to the unconstrained regions. That is why the central government finds it optimal to set _T_<sup>_u_</sup> = _T_<sup> ̄</sup> .

32
# **F.2 Adding adjustment cost on** _Tt_<sup>_u_</sup>

In this subsection, we relax the assumption that the uniform transfer is bounded below ( _Tt_<sup>_u≥T_ ̄).Instead, we</sup> assume that the central government incurs a cost when its uniform transfer deviates from _T_<sup> ̄</sup> , and this cost is quadratic,<sup><u>1</u></sup> 2<sup>_φ_(</sup><sup>_T_</sup> _t_<sup>_u−T_ ̄)2.We show that the main results in the paper continue to hold here.</sup> The budget constraints are Unconstrained local:

**Definition F.2.** A Markov-perfect equilibrium consists of a value function _V_ , central government’s policy rules _{_ Φ<sup>_τ_</sup> , Φ<sup>_T_</sup> , Φ<sup>_d_</sup> _}_ , and a policy function Φ<sup>_b_</sup> for the unconstrained local government’s debt, such that for all aggregate states ( _b_<sup>_n_</sup> , _d_ ), _τ_ = Φ<sup>_τ_</sup> ( _b_<sup>_n_</sup> , _d_ ), _T_<sup>_u_</sup> = Φ<sup>_T_</sup> ( _b_<sup>_n_</sup> , _d_ ), _d_<sup>_′_</sup> = Φ<sup>_d_</sup> ( _b_<sup>_n_</sup> , _d_ ) and _b_<sup>_n′_</sup> = Φ<sup>_b_</sup> ( _b_<sup>_n_</sup> , _d_ ) solve

_τ_ , _T_ max<sup>_u_</sup> , _b_<sup>_n′_</sup> , _d_<sup>_′_</sup> (1 _− θ_ ) [︁ _μu_ (︁ _G_<sup>_n_</sup> ( _b_<sup>_n_</sup> , _b_<sup>_n′_</sup> , _τ_ , _T_<sup>_u_</sup> ))︁ + (1 _− μ_ ) _u_ ( _G_<sup>_o_</sup> ( _τ_ , _T_<sup>_u_</sup> ))]︁ + _θv_ (︀ _C_ ( _b_<sup>_n_</sup> , _d_ , _d_<sup>_′_</sup> , _τ_ , _T_<sup>_u_</sup> ))︀ + _βV_ ( _bn′_ , _d′_ ) subject to the representative unconstrained region’s Euler equation, _ug G_<sup>_n_</sup> ( _b_<sup>_n_</sup> , _b_<sup>_n′_</sup> , _τ_ , _T_<sup>_u_</sup> ) = _βR_ ( _b_<sup>_n′_</sup> )(1 _−_ Φ<sup>_τ_</sup> ( _b_<sup>_n′_</sup> , _d_<sup>_′_</sup> )) _ug G_<sup>_n_</sup> ( _b_<sup>_n′_</sup> , Φ<sup>_b_</sup> ( _b_<sup>_n′_</sup> , _d_<sup>_′_</sup> ), Φ<sup>_τ_</sup> ( _b_<sup>_n′_</sup> , _d_<sup>_′_</sup> ), Φ<sup>_T_</sup> ( _b_<sup>_n′_</sup> , _d_<sup>_′_</sup> ))(F.22) (︁ )︁ (︁ )︁ and the central government’s value function satisfies the functional equation _V_ ( _b_<sup>_n_</sup> , _d_ ) = (1 _− θ_ ) _μu G_<sup>_n_</sup> ( _b_<sup>_n_</sup> , Φ<sup>_b_</sup> ( _b_<sup>_n_</sup> , _d_ ), Φ<sup>_τ_</sup> ( _b_<sup>_n_</sup> , _d_ ), Φ<sup>_T_</sup> ( _b_<sup>_n_</sup> , _d_ )) + (1 _− μ_ ) _u G_<sup>_o_</sup> (Φ<sup>_τ_</sup> ( _b_<sup>_n_</sup> , _d_ ), Φ<sup>_T_</sup> ( _b_<sup>_n_</sup> , _d_ )) [︁ (︁ )︁ (︁ )︁]︁ + _θv C_ ( _b_<sup>_n_</sup> , _d_ , Φ<sup>_d_</sup> ( _b_<sup>_n_</sup> , _d_ ), Φ<sup>_τ_</sup> ( _b_<sup>_n_</sup> , _d_ ), Φ<sup>_T_</sup> ( _b_<sup>_n_</sup> , _d_ )) + _βV_ (Φ<sup>_b_</sup> ( _b_<sup>_n_</sup> , _d_ ), Φ<sup>_d_</sup> ( _b_<sup>_n_</sup> , _d_ )) (︁ )︁

## **Derivation of Markov optimality conditions**

Denote the unconstrained local governments’ Euler equation (F.22) by _η_ ( _b_<sup>_n_</sup> , _b_<sup>_n′_</sup> , _d_<sup>_′_</sup> , _τ_ , _T_<sup>_u_</sup> ) = 0 and the corresponding multiplier by _λ_ .

Envelope conditions are substitute the FOCs into the Envelope conditions, 33
Differentiate _η_ with respect to _b_<sup>_n_</sup> and _d_ , substitute into _Vbn_ and _Vd_ , substitute into the last two FOCs, Expanding the auxiliary functions in (F.23)-(F.24) and (F.27)-(F.28), we get the following Markov optimality conditions.

where The auxiliary functions are 34
# **Proofs of propositions in the Markov equilibrium**

It is easy to see that **Lemma 1** in the paper continues to hold. In addition, following the same steps as in the proof of Lemma E.1, it is straightforward to show _λ ≥_ 0 (we do not even need _λ_ to be strictly positive here). Proposition 2 in the original model becomes:

**Proposition 2a** At the Markov equilibrium steady state, the central government gives the uniform transfer _T_<sup>_u_</sup> _< T_<sup> ̄</sup> and sets region-specific transfer _τ>_ 0.

At the steady state, The first equality is from (F.30). The first inequality follows because: at the steady state for _μ ∈_ (0,1), _μ_ ˆ ( _b_<sup>_n_</sup> _ss_<sup>,</sup><sup>_Bo_)</sup><sup>_>_</sup> _μ_ as _b_<sup>_n_</sup> _ss_<sup>_> Bo≥_0, and</sup><sup>_un_</sup> _g_<sup>_> uo_</sup> _g_<sup>as</sup><sup>_g_</sup> _ss_<sup>_n< g_</sup> _ss_<sup>_o_(Lemma 1).The second equality follows from (F.33).The second</sup> inequality comes from _λ ≥_ 0, _μ> μ_ ˆ and _u_<sup>_n_</sup> _gg_<sup>_<_0.Hence, it implies</sup><sup>_θvcφ_(</sup><sup>_Tu−T_ ̄)</sup><sup>_<_0, and</sup><sup>_Tu<T_ ̄.</sup> To show _τ>_ 0 at the steady state, note that _u_<sup>_o_</sup> _g_<sup>_< un_</sup> _g_<sup>(Lemma 1) and (F.30) together imply</sup> as _T_<sup>_u_</sup> _< T_<sup> ̄</sup> , _λ ≥_ 0 and _u_<sup>_n_</sup> _gg_<sup>_<_0.Sincewehaveestablished</sup><sup>_Tu<T_ ̄,soif</sup><sup>_τ≤_0,AssumptionE.2wouldimply</sup> (1 _− θ_ ) _u_<sup>_o_</sup> _g_<sup>_> θvc_, which is a contradiction.Therefore, it must be the case that</sup><sup>_τ>_0.</sup>

**Remark** . This is consistent with the intuition in the paper. Without the minimum bound on _T_<sup>_u_</sup> , the Markov central government wants to underprovide the uniform transfer in order to finance a higher _τ_ , in the presence of heterogeneity in regional debt.

# **Proof of Proposition 3 (Over-transfer in Markov)**

_Proof._ We can rewrite (F.30) as 35
# G A three-period model

This section presents a three-period version of the infinite-horizon model in Section 2. We show that the **overtransfer** result does not hold in period 2 (the final period), though it can hold in the earlier period under some conditions. This is different from the infinite-horizon model, where over-transfer occurs in every period around the steady state.

The timing of the model is as follows. There are three periods: period 1a, period 1b, and period 2. The central and local governments’ actions in each period are listed below:

- Period 1a. Local governments start with initial debt _b_ 1. The central government starts with revenue _e − f_ , chooses the transfer rate _τ_ 1, makes the transfers _τ_ 1 _b_ 1 and spends _c_ 1 = _e − f − τ_ 1 _b_ 1.

- Period 1b. Local governments receive own revenues _f_ and transfers _τ_ 1 _b_ 1, choose spending _g_ 1 and borrowing<sup>_<u>b</u>_</sup> _R_<sup><u>2</u>, and repay debt</sup><sup>_b_1.</sup>

- Period 2. The central government receives revenue _e − f_ , chooses its spending _c_ 2 and transfers _τ_ 2 _b_ 2. Local governments receive own revenues _f_ , transfers _τ_ 2 _b_ 2, repay _b_ 2 and spend _g_ 2.

For simplicity, we assume the central government does not borrow, and all local governments are identical. We also assume that transfers only consist of distortionary transfers _τb_ . The interest rate is constant, _R_ = _β_<sup><u>1</u>.</sup> The central government’s inability to commit to transfer policies is reflected in the timing of the model. In this timing, the central government chooses _τ_ 1 after _b_ 1 has been given in period 1a. Similarly, _τ_ 2 is chosen in period 2 after _b_ 2 is determined in period 1b.

We solve the model backward from period 2.

## **<u>Period 2</u>**

### **Local Government**

The local government’s problem is trivial in period 2. Because this is the terminal period, no investor is willing to lend and all debt has to be repaid. Therefore, given _b_ 2 and _τ_ 2, the local government simply consumes whatever resources it has, _g_ 2( _b_ 2, _τ_ 2) = _f − b_ 2 + _τ_ 2 _b_ 2.

### **Central Government**

For any given _b_ 2, the central government in period 2 chooses the transfer rate _τ_ 2 to solve, max _τ_ 2<sup>_θv_(</sup><sup>_e−f−τ_2</sup><sup>_b_2) + (1</sup><sup>_−θ_)</sup><sup>_u_(</sup><sup>_f−b_2 +</sup><sup>_τ_2</sup><sup>_b_2)</sup> The first-order condition with respect to _τ_ 2 is As this is the terminal period, the only effect of the transfer _τ_ 2 _b_ 2 is on closing the central-local spending gap. The central government subsidizes local governments to equalize ( _θ_ -weighted) marginal utilities, so there is **no over-transfer in period 2** . Assuming log utility and simplifying, we get 36
# **<u>Period 1b</u>**

## **Local Government**

For any given _b_ 1 and central government policies _τ_ 1 and _τ_ 2, the local government in period 1b solves the following problem, The first-order condition with respect to _b_ 2 is Simplifying, Note that<sup>_<u>∂b</u>_</sup><sup><u>2</u></sup> _∂τ_ 2<sup>_>_0, which implies</sup><sup>**overborrowing**by the local government (relative to the case when there is</sup> no distortionary transfer).

What we have described so far in periods 1b and 2 is similar to other two-period or static models in the literature on fiscal decentralization and soft budget constraint.<sup>59</sup> The central government chooses _τ_ 2 after local governments decide _b_ 2, so it does not consider the impact of its policy _τ_ 2 on local government’s incentive to borrow. In this case, the local government overborrows and the central government transfers to equalize marginal utilities, i.e., no over-transfer.

Next we consider the impact of _τ_ 1. Substituting _τ_ 2 (equation G.2) into _b_ 2( _τ_ 1, _τ_ 2, _b_ 1), we get As _b_ 2 shows up on both sides of the equation, equation (G.5) defines _b_ 2 as an implicit function of _τ_ 1 for any given _b_ 1.

**Lemma G.1.** If (1 _− θ_ ) _e − f>_ 0 and _b_ 1 _>_ 0, then _∂τ_<sup>_<u>∂b</u>_</sup><sup><u>2</u></sup> 1<sup>_<_0.</sup>

_Proof._ Implicitly differentiate (G.5) with respect to _τ_ 1, therefore, Lemma G.1 says that, ceteris paribus, transfers in the current period relax local governments’ budget constraint and reduce their (need for) borrowings into the next period.

# **<u>Period 1a</u>**

> 59See Goodspeed et al. (2017) for a review.

37
# **Central Government**

For any initial local debt position _b_ 1 and local government policy function _b_ 2( _τ_ 1), the central government chooses _τ_ 1 to solve Note that we have already substituted equations (G.3) and (G.4) into the objective function. The first-order condition with respect to _τ_ 1 is

**Proposition G.1.** If (1 _− θ_ ) _e − f>_ 0, _b_ 1 _>_ 0 and _b_ 2 _>_ 0, then the central government **over-transfers in period 1a** , i.e., (1 _− θ_ ) _ug_ 1 _< θvc_ 1 .

_Proof._ We can rewrite equation (G.6) as The inequality follows because (1 _− θ_ ) _β ∂τ_<sup>_<u>∂b</u>_</sup><sup><u>2</u></sup> 1<sup>_<_0 due to Lemma G.1; and [</sup><sup>_−ug_1+</sup><sup>_ug_2]</sup><sup>_>_0, because local gov-</sup> ernment’s optimality condition is _ug_ 1 = _βR_ (1 _− τ_ 2) _ug_ 2 = (1 _− τ_ 2) _ug_ 2 _< ug_ 2, as _τ_ 2 _>_ 0 if (1 _− θ_ ) _e − f>_ 0 and _b_ 2 _>_ 0 (equation G.2). From equation (G.7), when the local government starts off the period with positive debt ( _b_ 1 _>_ 0), the central government wants to over-transfer, i.e., (1 _− θ_ ) _ug_ 1 _< θvc_ 1 .

# **Discussions.**

Proposition G.1 states that the central government may “over-transfer” in period 1a, consistent with Proposition 3 and the intuition in the paper. In period 1a, the central government is not able to commit to its (future) choice of _τ_ 2, but it understands that a positive _τ_ 2 may induce local governments’ overborrowing (a high _b_ 2). Thus in order to reduce the the amount of local borrowing _b_ 2, the central government increases its transfers in period 1b ( _τ_ 1 _b_ 1), so that local governments’ budget constraint in period 1 is relaxed and the need for a high _b_ 2 is lessened.

However, the central government does not over-transfer in period 2, as shown in equation (G.1). This is in contrast to Proposition 3, where the over-transfer property applies in every period. Intuitively, in our threeperiod model, period 2 is the final period, and the central government’s choice of _τ_ 2 has no impact on the amount of local debt (which has to be zero) by the end of the period. If we extend the model to be more than three (but finite) periods, the over-transfer property will be present in the first period but gradually weaken, until it disappears in the final period.

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# H Computational appendix

This section outlines the procedures we use to compute the Markov-perfect equilibrium steady state used in the quantitative section. We focus on policy functions that depend **differentiably** on current states ( _b_<sup>_n_</sup> , _d_ ). Using Proposition 2 to simplify the problem, we do not need to solve for the uniform transfer _Tt_<sup>_u_.This leaves</sup> three policy functions to solve for: _b_<sup>_n′_</sup> = Φ<sup>_b_</sup> ( _b_<sup>_n_</sup> , _d_ ), _d_<sup>_′_</sup> = Φ<sup>_d_</sup> ( _b_<sup>_n_</sup> , _d_ ), and _τ_ = Φ<sup>_τ_</sup> ( _b_<sup>_n_</sup> , _d_ ).

We solve a system of equations for the policy function and their derivatives with respect to each state. We adopt a local approximation method similar to Klein et al. (2008). We use a linear approximation and assume policy functions are linear functions of states around the steady state.<sup>60</sup> We solve for the three policy functions and their first-order derivatives using a system of nine equations: Local government’s Euler equation (8), central government’s GEEs (D.11) and (D.12), and their derivatives with respect to _b_<sup>_n_</sup> and _d_ where Φ<sup>_b_Φ</sup><sup>_b_with respect to</sup><sup>_d_, and functional derivatives are computed</sup> _d_<sup>is the derivative of policy function</sup> as follows, for example, where _∂_ Euler/ _∂b_<sup>_n_</sup> is approximated by for some tiny value ∆, and second derivatives are set to zero.

> 60We checked higher-order (2nd, 3rd) approximations in Mathematica and found that the estimated steady states and first-order derivatives are similar to what we get using a linear approximation.

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# I Additional quantitative results

This section includes additional quantitative results not included in the main text.

## **I.1 Additional moments for Section 4.1**

Table I.1 complements Table 5 and provides additional moments for the baseline counterfactual exercise of Section 4.1.

**Table I.1:** Baseline Counterfactual Experiments: Additional Moments

|||1988–19|96||1996–20|06|
|---|---|---|---|---|---|---|
||(1)|(2)|(3)|(4)|(5)|(6)|
|Moments|1988|1996|Counterfactual<br>1996 (%)|1996|2006|Counterfactual<br>2006 (%)|
|Central-local spending ratio|3.63|1.85|1.83|1.85|1.05|1.06|
|Local spending|0.062|0.129|0.124|0.129|0.162|0.191|
|Central spending|0.224|0.239|0.226|0.239|0.170|0.202|
|Central-to-local transfer|0.039|0.080|0.085|0.080|0.055|0.071|

Note: Counterfactual experiments use revenue ( _e_ and _f_ ) and spending responsibility share ( _θ_ ) values from end year, and keep all other parameters at their starting year values. All spending and transfer levels are normalized by national GDP.

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# **I.2 Quantitative results for Section 4.3.1**

Table I.2 shows the calibrated parameter values and the effects of fiscal decentralization on debt levels when governments internalize interest rate schedules.

**Table I.2:** Counterfactual Experiment <u>governments internalize interest rate schedules</u>

|||Calibrated V|alues||||
|---|---|---|---|---|---|---|
|Parameters|1988|1996|2006||||
| ̄_B_|0.0646|0.236|0.237||||
| ̄_D_|0.587|1.063|0.601||||
|_θ_|0.792|0.654|0.515||||
|_ε_|0.018|0.018*|0.018*||||
|||1988–19|96||1996–20|06|
||(1)|(2)|(3)|(4)|(5)|(6)|
|Moments|1988|1996|Counterfactual<br>1996 (%)|1996|2006|Counterfactual<br>2006 (%)|
|Local debt|0.062|0.130|0.131 (102%)|0.130|0.122|0.115 (193%)|
|Central debt|0.338|0.615|0.380 (15%)|0.615|0.336|0.603 (4%)|
|Total government debt|0.400|0.745|0.511 (32%)|0.745|0.458|0.718 (9%)|

Note: Assuming governments internalize interest rate schedules and re-calibrating each year. Calibration strategy is identical to the baseline. Total government debt (also known as “general government debt”) is the sum of the local and central government debt. Counterfactual experiments use revenue ( _e_ and _f_ ) and spending responsibility share ( _θ_ ) values from end year, and keep all other parameters at their starting year values. Numbers in **parentheses** show the percent of debt changes that is explained by decentralization. All debt levels are normalized by national GDP. *We assume _ε_ for 1996 and 2006 is the same as 1988.

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# **I.3 Quantitative results for Section 4.3.2**

Table I.3 shows the values of internally calibrated parameters with alternative local and central debt interest elasticities.

**Table I.3:** Internally Calibrated Parameters with alternative local and central debt interest elasticities

|Parameter|Target|C|alibrated Va|lues|
|---|---|---|---|---|
|||1988|1996|2006|
| ̄_B_|Local government debt|0.0345|0.129|0.126|
| ̄_D_|Central government debt|0.220|0.538|0.280|
|_θ_|Central-local spending ratio|0.791|0.653|0.515|
|_ε_|Central-to-local transfer|0.014|0.014*|0.014*|

Note: Using _ψb_ = 0.05 and _ψd_ = 0.01 and re-calibrating each year. Calibration strategy is identical to the baseline. Debt and transfer levels are normalized by national GDP. *We assume _ε_ for 1996 and 2006 is the same as 1988.

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# **I.4 Quantitative results for Section 4.3.3**

Table I.4 shows the calibrated parameter values and the effects of fiscal decentralization on debt levels with alternative calibration of _μ_ and _B_<sup>_o_</sup> .

**Table I.4:** Counterfactual Experiment with alternative calibration of _<u>μ</u>_ and _B_<sup>_o_</sup>

||C|alibrated V|alues||||
|---|---|---|---|---|---|---|
|Parameters|1988|1996|2006||||
| ̄_B_|0.00358|0.137|0.139||||
| ̄_D_|0.316|0.594|0.319||||
|_θ_|0.788|0.652|0.514||||
|_ε_|0.018|0.018*|0.018*||||
|||1988–19|96||1996–20|06|
||(1)|(2)|(3)|(4)|(5)|(6)|
|Moments|1988|1996|Counterfactual<br>1996 (%)|1996|2006|Counterfactual<br>2006 (%)|
|Local debt|0.062|0.130|0.147 (126%)|0.130|0.122|0.104 (332%)|
|Central debt|0.338|0.615|0.389 (18%)|0.615|0.336|0.597 (6%)|
|Total government debt|0.400|0.745|0.536 (**39%**)|0.745|0.458|0.701 (**15%**)|

Note: Using _μ_ = 0.6 and _B_<sup>_o_</sup> = 0.046 and re-calibrating each year. Calibration strategy is identical to the baseline. Total government debt (also known as “general government debt”) is the sum of the local and central government debt. Counterfactual experiments use revenue ( _e_ and _f_ ) and spending responsibility share ( _θ_ ) values from end year, and keep all other parameters at their starting year values. Numbers in **parentheses** show the percent of debt changes that is explained by decentralization. All debt levels are normalized by national GDP. *We assume _ε_ for 1996 and 2006 is the same as 1988.

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# **I.5 Quantitative results for Section 4.3.4**

Table I.5 shows the effects of fiscal decentralization on debt levels with an alternative value of _ε_ .

**Table I.5:** Counterfactual Experiment

|||w|ith_ε_=0.1||||
|---|---|---|---|---|---|---|
|||1988–19|96||1996–2|006|
||(1)|(2)|(3)|(4)|(5)|(6)|
|Moments|1988|1996|Counterfactual|1996|2006|Counterfactual|
||||1996 (%)|||2006 (%)|
|Local debt|0.149|0.284|0.398 (184%)|0.284|0.180|0.092 (185%)|
|Central debt|0.482|1.029|0.863 (70%)|1.029|0.593|0.635 (90%)|
|Total government debt|0.631|1.313|1.261 (**92%**)|1.313|0.773|0.727 (**109%**)|

Note: Using _ε_ = 0.1, and all other parameters identical to the baseline. Moments are _not_ matched to data. Total government debt (also known as “general government debt”) is the sum of the local and central government debt. Counterfactual experiments use revenue ( _e_ and _f_ ) and spending responsibility share ( _θ_ ) values from end year, and keep all other parameters at their starting year values. Numbers in **parentheses** show the percent of debt changes that is explained by decentralization. All debt levels are normalized by national GDP.

44
<!-- Start of picture text --> 0.8<br>——— total debt<br>+++*+++* counterfactual total debt<br>O7<br>0.6<br>0.5<br>1988 461989 1999 1991 1992 1993 1994 1995 1996<br><!-- End of picture text -->
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