A Multi-Sectorial Model of the linkage between Fiscal Policy and Public Debt Sustainability

Universita degli studi di Pisa

dipartimento di economia e management

Scuola Superiore Sant'Anna

di studi universitari e di perferionamento

Corso di Laurea Magistrale in Economics

A Multi-Sectorial Model of the

linkage between Fiscal Policy and

Public Debt Sustainability

tesi di laurea magistrale

Relatore:

Prof. Davide Fiaschi

Candidato:

Gabriele Macci

Contents

3 Sustainable fiscal policies 13

4 Nash bargaining and government size 17

5 Conclusions 20

A The optimal saving problem 23

B Public debt sustainability 25

C Derivation of the conditions for the maximum net revenues fiscal

pol-icy 27

D Public Debt-to-Income partial derivatives 29

E Code for the evaluation of the Nash bargaining solution 31

Abstract

The aim of this work is to present and investigate a model that departs from three considerations. First, fiscal policy and debt sustainability are two sides of the same coin: the former is the policy instrument that government manages in the present time to achieve its social objectives, whereas, the latter is the repercussion of the former in the long run. Second, fiscal policy has a direct impact on industries capital accumulation because it affects the available income and, therefore, their savings; such an impact can be evaluated through the intertemporal optimization of industries saving decisions. Finally, fiscal policy has to be characterized on both the revenue and the spending side: revenues are determined through a unique tax rate, while spending is represented as an industry-based subsidy with a fixed fraction, φ, that must be re-invested in capital. The production function in each industry is assumed AK and investment is subjected to quadratic adjustment costs.

The consequences of assuming a convex adjustment cost function are twofold: the model does not display growth in the long run and the invested capital due to the public incentivization program hyper-crowds out private saving, consequently the optimal φ is 0. Another advantage, due to the simplicity of the framework, is that the notion of debt sustainability, contrary to the majority of literature on fiscal policy sustainability, is defined unambiguously therefore it is always possible to determine whether a certain public debt can be sustained. In section 4, the model is solved as a Nash bargaining problem by assuming an explicit welfare function and it is found that the optimal government size results undetermined whenever the optimal φ is 0. However, in the conclusions, it is proposed a way to modify slightly the model in order to overcome the problem of the hyper-crowding-out and letting the designed public program of capital incentivization work effectively.

1

Introduction

Designing a wise fiscal policy that is able to accomplish government objectives and, at the same time, to be compatible with public accounting is a crucial task of any treasury ministry.

Historically, long before the birth of the welfare state, warfare has been the leading cause of sovereign debt defaults (Reinhart and Rogoff, 2009): the radical and immedi-ate objective of defeating the enemy clashes with the huge financial distress to afford the expenses of a fighting army. In accordance with this evidence, Dincecco (2009) argues that in European polities, since the xvii century, the major tuning points in both collecting revenues and in limiting monarch expenses are associated with politi-cal transformation due to revolutions, coops and wars; thus letting European history appear as a spiral animated by continuous changes in social objectives and concerns of public finance sustainability .

On the other hand, during the xx century, along with war reparations, welfare program emerged as another fundamental cause in explaining confidence crisis and defaults on public debt. This new element has provided additional fuel to the old question of assessing whether certain fiscal and transfer policies are compatible with public finance sustainability. The assessment is not an easy task to perform because three sets of problems must be addressed when one wants to fulfil it.

First of all, sustainability is not always a well-defined notion when applied to public finances for several reasons. Differently from common debtors, a State is connoted with sovereignty, therefore, it might be tricky to design a legally binding contract whenever the State is entitled to change the rules on the fly and refuse to recognize creditors rights, as in the case of theAncien R´egime. A further difference with households’ or corporates’ accounting is that State time-horizon is naturally unlimited, as a conse-quence, even though nearly every public bond has a finite maturity, in principle, its intertemporal solvency must be determined ad infinitum. Accordingly, even if short-, medium- and long-term indicators of fiscal policy sustainability can be defined, their

effectiveness is somehow ambiguous (Talvi and Vegh, 1998). Dealing with unlimited time intervals poses another more technical problem: the final level of debt, ad in-finitum, does not map injectively to the functional space of transversality conditions (Blanchard et al., 1990). Therefore it is generally ambiguous which is the constant level of debt-to-GDP ratio that a sustainable fiscal policy has to target. For instance, the same transversality condition could fit with a final ratio either equal, lower or higher than the one of the initial time, or it could be compatible even with a zero ratio.

The last point inherent to the issue of establishing the meaning of sustainability is related with the No-Ponzi-Game condition. Even though this intertemporal budget constraint is a necessary condition in a steady-state non-stochastic olg environment, Blanchard and Weil (2001) showed that rolling over the debt could be feasible in a stochastic framework, even under the restrictive assumption of a dynamically efficient economy. Furthermore, when these games are feasible they might, or they might not, be Pareto-improving.

However, one should observe that the powerful result of Blanchard and Weil (2001) is achieved in a plain exogenous growth context. Indeed, the model complexity is the second set of problems one must address. Models become incredibly complex to treat whenever one attempts to consider any endogenous effect either of fiscal policy on production, with its consequential impact on the steady-state level and/or on the rate of growth of the economy, or of variations of debt levels on the interest rate1_.

Shifting the perspective from partial to general equilibrium models might pave the way to subcases of dynamic inefficiency in which Ponzi-Games become feasible and therefore even some particularly expansive fiscal policies turn out to be sustainable.

Another issue related to the complexity of the modelling framework is that on mere national accounting (NA) grounds increasing taxes is equivalent to decreasing spend-ing. In order to break this symmetry, which leads to substantial policy undeterminacy, one needs to amend the pure accounting-based model either with a theory of public

1_{See for instance (Barro and Sala-i-Martin, 2004, Chapter 3), whose general exposition is,}

spending/transfers or with some considerations related with distortionary taxation. Finally, in a model with heterogeneous agents, one has to evaluate a policy vector which accounts for the diversity in the economy and, possibly, for the interaction of the agents with each other and with the government.

The last set of problems an economist has to deal with to assess the compatibility of a certain fiscal policy with public finances emerges directly from the interactions among agents and with the government. Fiscal policy must be acknowledged being the area of macroeconomic policy most directly interrelated with politics. Accordingly, only as a first approximation, its macroeconomic effects can be disentangled from its re-distributive consequences, but in order to develop a comprehensive investigation these two aspects must be combined together. A classical reference is the theory of optimal tax smoothing due to Barro (1979) which states that deficits are varied in order to maintain expected constancy in tax rates, implying a countercyclical response of debt to temporary income movements. Departing from this contribution, several theories tried to reconcile theoretical predictions with empirical facts that record a long story of huge public debts leading, sometimes, to defaults such as in the experience of some Latin American countries. Among these theories2_{, Drazen and Eslava (2006) propose to}

bridge public deficits and elections exploiting the notion of pork-barrel cycles, Alesina and Drazen (1991) considerations on delayed debt stabilization introduce the “war of attrition” explanation and Velasco (2000) advances a similar justification to argue why fragmented government tend to default more frequently than centralized ones. The common feature of all these explanations is questioning the assumption stating that to the infinite-horizon accounting perspective discussed above has to correspond a similar benevolent view of the decision-maker. As a consequence, even if a fiscal policy might be the result of a majority voting system, still, in a political economy perspective, it could be affected by some shortsightedness problems and thus not being sustainable

2_{See Alesina and Passalacqua (2016) for a recent comprehensive survey of the political economy}

of government debt and the survey contained in Eslava (2011) to see what aggregation mechanisms play a role in converting conflicting interests into different economic policies enforced by the State authority.

ad infinitum.

The scope of the current dissertation is to elaborate a multi-sectorial model dealing extensively with the first two sets of questions and to establish eventually which fiscal policies are sustainable given the fundamentals of the economy.

In particular, the theoretical structure is presented in section 2: the model is enough simple to define unambiguously what “debt sustainability” means and, on the other hand, it is sufficiently sophisticated to endogenize capital accumulation with fiscal policies and debt sustainability. In section 3 the region of sustainable fiscal policies is derived. In the current framework, the political economy of fiscal policies — the third set of issues to address — is treated marginally in section 4 and only when the region of sustainable fiscal policies is already determined. Accordingly, in section 4 the Pareto-optimal policy is identified in the case that industries perform a Nash bargaining over the given sustainable policy region. Finally, section 5 concludes and points out three possible lines for future research.

2

The model

2.1

Industries

In the economy there are I sectors, {1, ..., I}, whose production function is an AK, Yi = AiKi. This class of technologies was introduced by Romer (1986) as a response

to the neoclassical growth model inability to generate sustained economic growth3_.

Capital is not subject to diminishing returns and the linearity of output in Ki allows

to build a tractable model. Although the functional form is assumed to be the same in every sector, observe that productivity might vary among industries so as to relax the constraint that an additional capital units has the same impact throughout the economy. Technological progress is assumed to be absent and thus output only evolves according to changes in capital, every industry is supposed to be endowed with an initial given capital expressed in the vector ¯K0 = [ ¯K1(0), ..., ¯KI(0)].

The instantaneous profits of each industry depend on four components: net revenues determined by the exploited technology combined with taxes, a subsidy component bestowed by the government to industries, the cost of purchasing new capital and finally, the adjustment costs due to investment. More precisely, profits in industry i are:

Πi(t) = (1 − τ )PiAiKi(t) + αiτ PiAiKi(t) − φαiτ PiAiKi(t) − PikSi(t) − C(φαiτ PiAiKi(t) + PikSi) =

[1 − τ (1 − αi(1 − φ))]PiAiKi(t) − PikSi− C(φαiτ PiAiKi(t) + PikSi)

(1)

with τ, αi, φ, Pi, Pik time-invariant. The general tax rate on gross profits is τ , Pi is the

price of the production in sector i and Pk

i is the cost of purchasing an additional unit

of capital in sector i. Equation (1) is in nominal terms, since working on real terms in a multi-sectorial model necessitates to deflate for a proper price index. The second addend on the rhs of (1) represents the amount of subsidies that government accords to sector i, the subsidy can be interpreted as a tax-credit since it is proportional to taxes payed by sector i, it depends on the industry-specific coefficient αi ≥ 0. The

3_{In the present model, there is no reference to households neither as a supplier of labor nor as}

sum φαiτ PiAiKi(t) + PikSi is the nominal investment at time t and it will be thereafter

denoted as Ii(t). The term φαiτ PiAiKi reads as the amount of the subsidy that the

government requires the industry to invest in capital accumulation. Therefore one can interpret the subsidization mechanism as an outright grant partially supplied in the form of capital grant, for a fraction φ. This pattern grabs the essential features of procurement contracts: paying back a taxpayer by hiring her for some public work whose contract terms require the private party to invest in some new capital. Having this feature in the model allows breaking the NA-symmetry between increasing taxes and reducing public spending to obtain a well-defined fiscal policy. On the contrary, Si(t) is the real investment component determined by private saving decisions. Finally

C(·) denotes the adjustment cost of nominal investment:

C(I) = a1· I + a2· I2 (2)

the adjustment costs are basically due to installation and purchase costs. As pointed out in Gould (1968), the origin of the purchase costs may be both internal and external to the firm, therefore the formulation in terms of nominal investment appears to be reasonable4_{. In other contributions, such as Barro and Sala-i-Martin (2004),}

adjust-ment costs depends on the relative size of expansion C _KI
, such specification, however, lead to non-linear differential equations and to difficulties in establishing existence and uniqueness of the optimal growth path (Gould, 1968). Considering adjustment costs in function of gross investment —including capital replacement —implies that any indus-try faces a positive cost even if it is not expanding its capital stock. Consequently, if the cost of adjustment function is such that average cost increases the higher the level of investment I, this will introduce decreasing returns to scale thereby giving sectors a determined capital level of equilibrium even if the production function is homoge-neous of degree one; this result is uncommon for the literature departing from Romer

4_{Actually, by writing C(φα}

iτ PiAiKi/Pik+ Si), it is possible to interpret the adjustment cost of

investment in real terms. However, the results of the model do not change significantly while the exposition becomes more convoluted.

To keep notation simple, the adjustment cost function is assumed to be homogeneous across sectors and a1, a2 are time-invariant.

(1986) contribution. Specifying the higher-order term of equation (2) in a quadratic form permits us, in addition to draw a limit to endogenous economic growth, to avoid nonlinearities in the Bellman differential equation (A.1) in appendix A.

Real capital accumulation in industry i is expressed in net terms since it is decreased by the depreciation term δiKi:

˙ Ki(t) = −δiKi+ Ii/Pik = −δiKi+  φαiτ PiAiKi Pk i + Si  (3)

symmetrically with the assumption of heterogeneous productivities, the model allows depreciation rates δi to be different across sectors. Observe that the gross investment

is considered in real terms, thereby it is divided by the price of capital in sector i.

Let us define as Ri = φαiτ PiAi the marginal accumulation of capital due to public

subsidies, with r the discount rate that has the same value of the interest rate and with Di —following the derivation in appendix A and equation (A.6) —the equlibrium level

of capital5 in sector i: Di = [1 − τ (1 − αi)]PiAi− (1 + a1)[φαiτ PiAi+ r + δi] 2a2δi[φαiτ PiAi+ r + δi] = [1 − τ (1 − αi)]PiAi− (1 + a1)[Ri+ r + δi] 2a2δi[Ri+ r + δi] (4)

According to the computations in appendix A, the optimal private saving trajec-tory, S_i∗, which maximizes profits in (1) over an infinite time-horizon subject to equa-tion (3), is obtained by posing total nominal gross investment, I_i∗, constant over time.

5_{Along the model let us assume that the equilibrium capital in every industry, for a given tax rate}

τ , is positive at least for some values of αi and φ: ∃(αi, φ) ∈ [0, 1]2 : ∀i, Di > 0, i.e. [1 − τ (1 −

αi)]PiAi− (1 + a1)[φαiτ PiAi+ r + δi] > 0, ∀i.

Consequently the optimal saving, investment and capital trajectories are:            I_i∗(t) = δiDi S_i∗(t) = (δi− Ri)Di− Ri(Ki(0) − Di)e −δit Pk i K_i∗(t) = Di+ ( ¯Ki(0) − Di)e−δit (5) (6) (7) First of all, observe that the the intertemporal optimization resulting in equations from (5) to (7) is explicitly defined over infinite time-horizon, consistently with public debt management as exposed in section 2.2. Secondly, notice that the fiscal policy vector referred to industry i, i.e. [τ, αi, φ], directly affects the equilibrium level of

cap-ital Di and, along the transition to the steady-state, the growth rate of the sectorial

economy. On the other side, capital determines production and indirectly tax-revenues, thus the fiscal policy vector is endogenous to the economy and affects circularly capital accumulation and debt sustainability. Accordingly with the considerations of Gould (1968) about quadratic adjustment cost functions, no growth is displayed in the long-run because limt→∞Ki∗(t) = Di, on top of that, observe that the capital of equilibrium,

Di, does not depend on the initial capital level ¯Ki(0).

A considerable advantage of the current model formulation is that the value function for sector i derived under optimal intertemporal optimization has close form:

Vi Ki(0)
= [1 − τ (1 − αi)]PiAiDi− (1 + a1)δiDi− a2δ 2 iDi2 r + [1 − τ (1 − αi)]PiAi K¯i(0) − Di
δi+ r (8)

This has a fundamental importance because, as Kenneth Arrow pointed out, it allows performing welfare analysis and deriving explicit optimal policies. Finally, observe both the capital of equilibrium, Di, and the value function, Vi(·), do not depend on

Pk

i . Actually the cost of capital only affects the repartition between voluntary

invest-ment due to savings, Si, and investment due to the tax credit scheme devised by the

2.2

Public debt

Assuming that government is legally bounded to its own obligations and that, whenever a bond is issued, the terms of contract can be effectively enforced are two necessary precondition to proceed in the exposition of the model without ambiguities. As antici-pated at the end of the last section, fiscal policy can be seen as the merging of the fiscal policies at industry level, i.e. one can define the fiscal policy vector as [τ, φ, α1, ..., αI].

For simplicity, government performs only three distinct operations: collecting tax-revenues from industries, transferring back some amount of resources to sectors through the mechanism described in section 2.1 and issuing public bonds on the world bond market. Under the assumptions that the national economy is small and open, that there is no uncertainty on future and that sovereign debt default is ruled out in any form, the bond interest rate r is fixed (see for instance Velasco, 2000). Therefore, the government fiscal policy vector [τ, φ, α1, ..., αI] has a direct impact on the stock of

public bonds, B, and this influence is expressed directly in the law of motion of public debt: ˙ B(t) = r · B(t) − τX i PiAiKi(t) + τ X i αiPiAiKi(t) = = r · B(t) − τX i (1 − αi)PiAiKi(t) (9)

A necessary condition for debt sustainability in a model with no growth in the long-run is that ˙B(t) is definitively lower or equal to 0 from some t0 > 0 onward. Actually, this requirement is equal to impose the No-Ponzi-Game condition and, furthermore, to extinguish public debt in a finite time (see Appendix B for a proof). Given a certain structure of the economy, as shown in Appendix B, a public debt at initial time, B0, is

sustainable only when exists at least one fiscal policy vector,[τ, φ, α1, ..., αI] ∈ [0, 1]I+2

such that B(0) ≤ τ r X i (1 − αi) · PiAi· [δiDi+ rKi(0)] r + δi (10)

Similarly, defining the total initial nominal income as YN

0 =

P

iPiAiKi(0) and

ratio is: B(0) YN 0 ≤ τ r X i (1 − αi) r + δi γi  δiDi ¯ Ki(0) + r  (11)

for some feasible fiscal policy vector [τ, φ, α1, ..., αI] ∈ [0, 1]I+2. Most notably, the ratio δiDi

¯

Ki(0) and γi are the two crucial parameters in determining the sustainability of large

debt-to-income ratios. The fraction δiDi

¯

Ki(0) stands for the residual growth in sector i

adjusted for the depreciation rate δi. Fon instance, when the initial capital is equal to

the equilibrium capital, the rhs of equation (11) simplifies to τr

P

i(1 − αi)γi. Another

example is considering, ceteris paribus, an industry whose productivity Ai stands out

against other sectors: the lower ¯Ki(0) is for a given Di and the greater the sustainable

ratios will be. In analogy, for a δiDi

¯

Ki(0) relatively low w.r.t. other sectors, ceteris paribus,

the higher Ai is —therefore the fraction γi —the greater the ratios the economy is be

able to afford are. Finally in both equation (10) and (11), the greater the absolute difference lhs−rhs is, other things equal, the larger expansionary fiscal policies are sustainable.

3

Sustainable fiscal policies

A fiscal policy is generally defined as a vector [τ, φ, α1, ..., αI] in the [0, 1]I+2 space.

However, it is possible to adopt a useful simplification observing:          ∂Di ∂φ = − [1 − τ (1 − αi)]PiAiRi 2a2δi(Ri+ r + δi)2 < 0 ∂Vi ∂φ = − [αiτ [1 − τ (1 − αi)]Pi2A2i] 2 φ 2a2r(r + δi)(r + δi+ Ri) < 0 (12) (13)

The responsibility of the negative sign in both equation (12) and (13) is due to the quadratic adjustment cost function in (2). Because of the introduction of dis-economies of scale in investment, the public scheme of capital accumulation does not work effec-tively and the model dispays hyper-crowding-out of private investment. To understand why this happens, it is sufficient to notice from equation (1) that the public capital grant appears in instantaneous profits just as a cost, that is ∂Πi

∂φ < 0. Although one

may say capital accumulates faster in equation (3) when φ increases, this acceleration is only linear, whereas the costs of a faster investment are convex. The hyper-crowding-out effect results, therefore, from combining together equation (6) with (12) so that on the optimal trajectory ∂I_∂φ∗ = ∂(Ri· ¯Ki∗)

∂φ −

∂S_i∗ ∂φ < 0.

As a consequence of (12) and (13), for any plausible welfare objective —maximizing individual or aggregate value functions, obtaining a production as large as possible, etcetera... —one has that the optimal φ is equal to 0. Thus Ri becomes 0, private

saving equals total investment, S_i∗ = I_i∗ = δiDi and equation (4) simplifies to:

Di =

[1 − τ (1 − αi)]PiAi− (1 + a1)(r + δi)

2a2δi(r + δi)

(4-bis)

Notice that now Di is always increasing in αi. Also equations (5), (7)-(8) and (10)-(11)

simplify accordingly. A further simplification, that will result essential in the sustain-ability analysis, is that the fiscal policy vector turns into a one component less vector: [τ, α1, ..., αI] ∈ [0, 1]I+1. From now on, let us assume φ is 0 and that all the related

a non-zero φ see the conclusion section.

Given the above simplification, the vector ¯K(0) of initial capitals, and τ ∈ T —with ¯

K(0) and T fulfilling some regularity conditions stated in appendix C— it is possible to characterize the maximum sustainable debt level directly from the maximization of equation (10) w.r.t [α1, ..., αI] ∈ [0, 1]I:            αnr i (τ ) = (2τ − 1)PiAi+ 1 + a1− 2a2r ¯Ki(0)
(r + δi) 2τ PiAi , ∀i ∈ {1, ..., I} Bmax 0 = 1 8a2r X i " PiAi− 1 + a1− 2a2r ¯Ki(0)
(δi+ r) r + δi #2 (14) (15)

First of all, as one might expect, the maximum level of sustainable debt6, Bmax

0 ,

is independent from τ and only depends on the structural variables of the economy. The [αnr

1 (τ ), ..., αInr(τ )] can be interpreted as the subsidy policy that maximizes net

revenues from every sector. Actually, notice that the equilibrium capital, Di, is not

maximum for αi = αnri (τ ) since equation (4-bis) is monotonic in α, therefore maximum

production —and maximum gross revenues, ceteris paribus —is achieved for αi = 1, ∀i,

that is when the effective tax rate, τ (1 − αi), is 0. On the other hand, the effective tax

rate for αi = αnri (τ ) is independent from the particular fiscal policy since it holds:

τ 1 − αnr

i (τ )
=

PiAi+ 1 + a1− 2a2r ¯Ki(0)
(r + δi)

2PiAi

(16)

which is comprised in the (0, 1) interval whenever the vector [αnr

1 (τ ), ..., αnrI (τ )] exists

in the domain of valid fiscal policies (see appendix C for a proof).

Once that the maximum sustainable debt is determined, it is possible to characterize the region of feasible fiscal policies under the assumption that the initial public debt level, B(0), is lower than Bmax

0 .

6_{See appendix D for some similar considerations about the sustainability of a certain public}

B0MAX B1 B2 B3 B4 Α₁ Α₂

Figure 1: Admissible tax credits for industry 1 and 2 given a sample of sustainable initial debts values Bmax

0 > B1 > B2 > B3 > B4. The yellow area denotes all the possible Pareto-improvements

w.r.t. αnr

1 and αnr2 .

Given the production parameters and the prices of the economy, the admissible policy region in the space of all the αi is an I-dimensional ellipse for any τ ∈ T satisfying

the conditions of appendix C. For a two industry exemplification, one can plot the admissible tax credit region as in figure 1 and observe that for increasing initial public debt the policy region narrows, until when B0 = B0max the region collapses to a point.

It is important to point out that the eccentricity of the ellipse depends solely on the industries relative prices and production technologies, the area (or volume, when I > 2) depends on B(0) and τ and, finally, the center is [αnr

1 (τ ), ..., αInr(τ )]. More precisely,

the smaller the initial debt is, the larger the feasible policy region is. Similarly, the higher is τ the more the centre of the ellipse goes to “northeast” and its area narrows. Finally, the point αnr

i (0) = 1, ∀i is feasible only when B(0) ≤ 0, that is when the public

debt turns into public assets and τ can be set to 0.

For any reasonable welfare objective, furthermore, whatever is the optimal fiscal policy [τ∗, α∗₁(τ ), ..., α∗_I(τ )] that government will choose among those available in the I-dimensional ellipse, it will be such that α∗_i(τ ) ≥ αnr

i because ∂Vi

∂αi is always positive

when Di > 0 and the same is true for the capital of equilibrium.

Consequently, for any welfare objective, it is possible to establish in each sector i the minimum level of the capital of equilibrium and of the value function whenever B(0)

is sustainable: Dmin i = PiAi− 1 + a1+ 2a2r ¯Ki(0)
(r + δi) 4a2δi(r + δi) Vmin i = P_i2A2_i − 2 1 + a1− 2a2r ¯Ki(0)
(r + δi)PiAi 16a2r(r + δi)2 + + 1 + a1(a1 + 2) + 12a2r ¯Ki(0) · 1 + a1− a2r ¯Ki(0)
(r + δi) 2 16a2r(r + δi)2 (17) (18)

In the {α1× α2}-plane of figure 1, thus, the Pareto-improving region is represented by

the cone with vertex in Bmax

4

Nash bargaining and government size

As shown in the previous section, whenever B0 < B0max, there is a public finance surplus

that government can distribute according to its social objectives.

As anticipated in section 3, a reasonable welfare objective would be to design a fiscal policy so as to optimize an aggregate expression of industries value functions. In particular, if government’s objective is to maximize:

max [τ,α1,...,αI] ( Y i h Vi ¯Ki(0), αi iβi ) (19)

a feasible resolution strategy is to look for the Nash bargaining solution (see, for in-stance, Myerson, 1991, page 375 ff.)7_{. First of all, a bargaining power coefficient β}

i,

time-invariant, should be assigned to every industry i such thatP

iβi = 1. There could

be at least three ways to attribute it. First, βi might be set equal to γi, a possible

interpretation could be that government cares about each sector proportionally to the amount of taxes that are collected from it. Secondly, the bargaining power could be commensurate to the relative initial capital level in each industry. The interpretation would be similar to the previous one, but now government regards only about the ini-tial capital and not its productivity. Therefore, in the latter case an inefficient sector highly capital-intensive would have more bargaining power than in the former scenario. Finally, the most simple way to fix all the βi is giving an identical bargaining power to

every sector, thus the unique policy parameter in solving (19) is the value function as defined in equation (8).

Once the bargaining power coefficients are given, two more elements have to be defined: the feasibility region — compact and convex—where to look for the Nash bargaining solution and the outside option in case government cannot find an agreement

7_{The Nash bargaining solution is independent from whether it is government to make the choice}

on how to re-allocate the public finance surplus or the redistribution is operated as a real bargaining among groups.

Interestingly, even though the political economy of debt management is not in the scope of the current model, given this procedural equivalence the model could be easily extended toward a political economy direction.

on how to share the public finance surplus. Defining the outside option of every industry is not straightforward, however, a natural candidate is [αnr

1 (τ ), ..., αnrI (τ )], so that if

government does not come up with a decision on how to distribute the public finance surplus, simply, the surplus will not be allocated.

On the contrary, for a certain τ , the fiscal policy region, Fτ, is defined as the

sub-region of the poi0ts [α1, ..., αI] such that the vector [α01, ..., α 0

I] ∈ Fτ if and only if

∀i, α0

i ≥ αnri (τ ) and the inequality (10) is fulfilled. In the case of a two-sector economy,

Fτ is represented in figure 1 as the intersection of the yellow area with the appropriate

ellipse and the point Bmax

0 is the outside option fo the corresponding τ .

Following Myerson (1991), it is possible to rewrite equation (19) as:

max τ ∈T ,[α1,...,αI]∈Fτ ( Y i h Vi ¯Ki(0), αi  − Vi ¯Ki(0), αnri iβi ) = max τ ∈T ( max [α1,...,αI]∈Fτ ( Y i h Vi ¯Ki(0), αi  − Vi ¯Ki(0), αnri iβi )) = max τ ∈T ( Y i h Vi ¯Ki(0), α∗i(τ )  − Vi ¯Ki(0), αinr iβi ) (20)

Unfortunately, equation (20), does not have close-form solution and, therefore, it is only possible to evaluate numerically8 (see figure 2 for a representation in a two-sector economy). Actually, the sole analytical conclusion is that the optimal fiscal policy vector will always lie on the frontier of the I-dimensional ellipse because ∂Vi

∂αi is

always positive. On the other hand, from the numerical evaluation emerges that for any admissible tax rate, τ ∈ T , the corresponding optimal subsidies, [α∗₁(τ ), ..., α∗_I(τ )], are such that the value function Vi K¯i(0), α∗i(τ )
remains constant in every sector i:

∀{τ1, τ2} ∈ T and ∀i ∈ {1, ..., I} : Vi K¯i(0), α∗i(τ1)
= Vi K¯i(0), α∗i(τ2)

As a consequence, the optimal fiscal policy is not unique. Indeed, for any τ ∈ T

NB OO

Α₁ Α₂

Figure 2: In a two-sector economy, on the {α1× α2}-plane, given τ , the Nash bargaining solution

(nb) and the outside option (oo) are represented. The tangente curve to the ellipse in nb is the iso-welfare curve which is, obviously, increasing in αi. The point (nb) is Pareto-optimal.

exists an optimal fiscal policy that satisfies equation (20). Therefore, as in most of the literature, there is perfect symmetry between rising taxes or reducing subsidies, and vice versa. Since a measure of public sector magnitude can be the sum of its financial inflows and outflows, P

iτ PiAiKi +

P

iαiτ PiAiKi, a direct consequence of

this symmetry is that an optimal government size does not exist. As a matter of fact, calling τmin the minimum admissible tax rate in T , knowing that the maximum

allowable tax rate is 1, then the public sector size could result in any value in the interval: " X i τminPiAiKi+ X i α∗_i(τmin)τminPiAiKi , X i PiAiKi+ X i α∗_i(1)PiAiKi #

This result, combined with the observation that αi and τ appear in equation (8)

only paired up together when φ = 0, implies that the the Nash bargaining effective tax rate, τ (1 − α∗_i(τ )), in every sector i is independent from the effectively adopted fiscal policy. The reason behind this result, again, is due to the simplification operated in section 3 to set φ = 0 so as to break down the asymmetric relation between τ and αi.

5

Conclusions

To conclude, even if the optimal government size resulting from the Nash bargaining in section 4 is undetermined, the model displays several interesting properties. Despite the general simplicity of the proposed framework, first, fiscal policy and debt dynamic are successfully endogenized with respect to the process of capital accumulation. This is an important aspect because it allows to investigate both the direct and indirect role of government fiscal policy on debt sustainability. The direct effect is due to the uncompensated dynamics of tax revenues and public spending when [τ, α1, ..., αi] are

altered, whereas, the indirect response is related with the annexed compensation of public cash flows owning to the change in industries paths of capital accumulation, that, in general, are affected by the fiscal policy vector [τ, φ, α1, ..., αI], including φ.

In spite of the widespreaded ambiguities when defining the notion of public debt sustainability in several models, the current framework attributes a precise meaning to this notion. As a consequence, given the structural variables of the economy, it is always possible to establish whether a certain public debt and debt-to-income ratio is sustainable or not. In particular, as a consequence of relaxing the assumption of homogenous sectors, when B(0) = Bmax

0 , the feasible fiscal policies, one only for a given

τ ∈ T , —recall that the ellipse determined in equation (10) collapses to a singleton —cannot be such that all αi are equal to zero. In other words, the maximum net

revenues fiscal policies always distort the effective net tax rate, so that some sectors always have a net fiscal pressure higher than others, but for veryad hoc parameters values.

The last point in favor of the current modelling design is the proposal, in section 2, to include in this class of models some public policies aimed to incentivize capital accumulation. Actually, this theory is merely an intuition and some further research would be required to obtain some sensible results, since the value of φ in the optimal fiscal policy is always set to 0 for the time being. There are two possible ways to extend this result for other values of φ: changing the convexity of the cost of adjustment function and allowing φ to have a potentially positive return in the instantaneous

profit function (1). The former direction does not look very promising because the adjustment cost function in (2), as already pointed out in Gould (1968), seems to be reasonable enough and because it guarantees the analytical tractability of the model. The latter strategy goes in the direction of investigating a profit function like:

Πi(t) = F(1 − τ )PiAiKi(t), φ + αiτ PiAiKi(t)+

−φαiτ PiAiKi(t) − PiKSi− C(φαiτ PiAiKi(t) + PiKSi)

with F0_φ > 0 and F00_φ< 0. In this way, the hyper-crowding-out effect due to the public investment incentivization program would be mitigated with a crowding-in compo-nent. The assumptions on F might be corroborated observing that the re-invested component of the subsidy is associated with some public procurement contract. As a consequence, the industry i would have new profit-opportunities that do not crowd-out with its private businesses and therefore, justify, both the new investment based on the incentivization scheme and the private saving. Additionally, including either of these modifications in the model design would permit to break the symmetry between the revenue and spending side of government so as to find an optimal government size. Even more interestingly, the latter approach would most probably redefine the shape of the I-dimensional region of feasible subsidies [α1, ..., αI] into a generic I-dimensional

conic —an ellipse, a parabola (either convex or concave) or even an hyperbola.

A second line for future research origins from observing that both the optimal fiscal policy in the Nash bargaining and the maximum net revenue policy [αnr

1 , ..., αnrI ] are

time-inconsistent since they depend on Ki(0). Therefore, if the government can ever

revise its policy after time 0, the optimal initial fiscal policy would be sub-optimal and therefore another policy would be adopted. Since this is true ∀t > 0, the re-determined fiscal policy would fail to remain optimal for future times as well. A possible way, admittedly weak, to justify the validity of such an inconsistent fiscal policy is to think that government either is committed or must face insuperable institutional and political difficulties to change its policy (Persson and Tabellini, 2000, chapter 11). However, the alternative approach of finding a close-loop solution seems a bit puzzling at this moment.

introduction, properly considering the conflicting interest among groups is fundamen-tal when one deals with public debt issues. Certainly, the Nash bargaining framework proposed in section 4 goes in this direction, especially because the bargaining coeffi-cients can be regarded as determined by a political process —either with elections or with side-payments. However, some further interactions among industries might be considered also on the side of the profit function so as to transform the model in a differential game.

A

The optimal saving problem

The Bellman equation to find the optimal private saving path associated to the model depicted in section 2.1, for a generic group i (let us drop, for simplicity, subscript i and time t in the notation), taking as given all other parameters, is:

max

S { [1 − τ (1 − α)]P AK − (φατ P AK + P k_S)

−C(φατ P AK + Pk_{S) − rV + (φατ P AK + P}k_{S − δK)V}0_{(K)} = 0}

(A.1)

Since the production function and equation (1) are, respectively, linear and concave in K and equation (1) and (3) are, respectively, concave and linear in S, the F.O.C. are necessary and sufficient and the optimal path, if exists, is unique. The F.O.C. are:

         V0(K) − Pk_{− C}0_{(S) = 0} [1 − τ (1 − α)]P A − [1 + a1+ 2a2(φατ P A + PkS)](φατ P A) + −(r + δ)V0 + (φατ P AK + Pk_{S − δK)V}00_{(K) = 0} (A.2) (A.3)

following Chang (2004) in observing that ∂V_∂t0(K) = V00K = (I − δK)V˙ 00 and plugging equation (A.2) into equation (A.3) one gets a differential equation in φατ P AK + PK_S,

that by substituting the total nominal gross investment term I reads:

C00(I) ˙I − (r + δ)Pk_{(1 + C}0_{(I)) + [1 − τ (1 − α)]P A =}

2a2PkI − 2a˙ 2I(R + r + δ) − (1 + a1)(R + r + δ) = 0

(A.4)

the first equality is due to write explicitly C(I) as in equation (2). Solving equa-tion (A.4) in I and applying equaequa-tion (4) to define D, one finds that the general solution as a function of I(0) is:

I(t) = δD + eR+r+δP K t(I(0) − δD) (A.5)

first-order differential equation in K one obtains (see: Sydsæter et al., 2008, page 204): K(t) = D(1 + δ) − I(0) + [ ¯K(0) − D] · e−δt+ [I(0) − δD] · P k R + r + δ(1 + Pk₎ · e R+r+δ P K t (A.6)

Finally, applying the transversality condition (see: Gould, 1968; Chang, 2004, page 123):

lim t→+∞e −rt VI(0), K(0), K(t)= Z +∞ 0 e−rsΠ(s) ds = 0 (A.7)

and solving for I(0) under the standard assumption that δ < r, one obtains I∗(t) = I(0) = δD as in equation (5). Thus equation (6) is easily recoverend and by substituting I(0) back into equation (A.6) one obtains equation (7). Finally, equation (8) is obtained by solving the integral in (A.7) when I(0) = δD.

B

Public debt sustainability

B(t) = ert " B(0) −τ r X i (1 − αi)PiAi[δiDi+ r ¯Ki(0)] r + δi # (B.1)

Observe that (B.1) is a monotonic function of time, therefore imposing ˙B(t) ≤ 0 since t0 > 0 onward is true iff hB(0) − τ_rP

i

(1−αi)PiAi[δiDi+r ¯Ki(0)]

r+δi

i

≤ 0 which is exactly equa-tion (10).

The No-Ponzi-Game condition rules out the possibility of rolling over public debt, in the limit ad infinitum it reads:

lim

t→+∞B(t)e

−rt_{≤ 0}

(B.2)

substituting equation (B.1) into (B.2) results again in equation (10).

Finally, public debt is considered extinguished in a given time interval when the stock of public bonds is non-positive during that time, B ≤ 0 (when B < 0 is verified, government is a net creditor in the world bond market). Since B(t) is monotonic, when-ever public debt extinguishes it will not reappear at a later time. Suppose B(0) ≥ 0, there are two cases: i) equation (10) is satisfied with equality, then B(t) goes to zero immediately and stays there later on; ii) equation (10) is verified as a strict inequality, thus B(t) becomes immediately negative and keeps decreasing (accumulating foreign assets) along time.

However, both the scenarios to be feasible require perfect financial markets; in partic-ular, sectors must be assumed not to be financially constraint. As a matter of fact, to extinguish instantly a public debt B(0) > 0 the tax rate, τ should be such that:

τ = _P B(0) · (1 + r)

i(1 − αi)PiAiK¯i(0)

unfortunately, the rhs of (B.3) could be larger than 1 and therefore not representing a proper fiscal policy since instantaneous tax revenues cannot exceed the whole instan-taneous production of the economy. As a consequence, to keep the plausibility of the model in the case that (B.3) is greater than 1, which always appear to be true when the public debt-to-income ratio is larger than 1, sectors need to be assumed financially unconstrained. Assuming industries are financially unconstrained would allow to de-sign a financial instrument that dispenses them with enough money to refund B(0) and that diverts future net revenues (taxes minus subsidies) from government balance sheet to financial institutions.

C

Derivation of the conditions for the maximum

net revenues fiscal policy

Equation (14) can be rewritten as:

αnr

i (τ ) = 1 +

−PiAi+ 1 + a1− 2a2r ¯Ki(0)
(r + δi)

2τ PiAi

(C.1)

and represents a valid fiscal policy only when αnr

i (τ ) ∈ [0, 1]. Let us state under what

conditions equation (14) is a valid policy:

αnr

i (τ ) ≥ 0, ∀i is fulfilled only when in every industry the following two conditions

are fulfilled in every industry:

τ ≥ PiAi− 1 + a1− 2a2r ¯Ki(0)
(r + δi) 2PiAi (C.2a) ¯ Ki(0) ≤ PiAi+ (1 + a1)(r + δi) 2a2r(r + δi) (C.2b) αnr

i (τ ) ≤ 1, ∀i is always verified as a strict inequality because of the combination of

the statement in note 5 with the additional simplification of φ = 0. Thus, PiAi− (1 + a1)(r + δi) > 0 together with ¯Ki(0) > 0 imply that

the last term of (C.1) is always negative;

The condition (C.2b) is needed in order to assure that exists a τ smaller than 1 fulfilling equation (C.2a). Notice that condition (C.2b) is not effectively very stringent since the capital of equilibrium Di is equal to [1−τ (1−αi)]PiAi

−(1+a1)(r+δi)

2a2δi(r+δi) which is about 1/δi times

larger than the rhs of the inequality (C.2b); for instance, suppose δi = 0.05 than the

condition (C.2b) requires, approximately, ¯Ki(0) to be no less than ₂₀1 of its equilibrium

capital.

While condition (C.2b) is generally not very stringent, the condition (C.2a) might be non-fulfilled for very reasonable parameter values. However, fortunately, since the effective tax rate in equation (16) is independent from the fiscal policy vector

A

B Α1

Α₂

Figure 3: Two-sector economy, representation of fiscal policies on the {α1× α2}-plane. Fiscal policy

in A is invalid since at least one component is negative whereas the policy vector in B is valid. The effective tax rate in each sector, τ (1 − αnr

i ), is identical.

[τ, αnr

1 (τ ), ..., αnrI (τ )] as long as conditions (C.2b) is satisfied it is always possible to

increase the rate from τ to τ0 such that: max i  PiAi− (1 + a1− 2a2r ¯Ki(0)(r + δi) 2PiAi  < τ0 < 1 (C.3)

Let denote as T the interval specified in (C.3).

For a two-sector economy, supposing τ0 = τ · maxi

n

PiAi−(1+a1−2a2r ¯Ki(0)(r+δi)

2τ PiAi

o for i = 2, the corresponding shift of the maximum net revenue policy is represented in the {α1× α2}-plane with the corresponding movement from point A to B in figure 3.

Observe that A is an invalid fiscal policy since both αnr

1 and αnr2 are negative. On the

contrary for τ0 the subsidy to sector 1 becomes positive while the subsidy to sector 2, for whom the condition (C.2a) is violated more seriously than for industry1, is null.

D

Public Debt-to-Income partial derivatives

Dividing both sides of equation (15) one obtains the maximum public debt-income ratio that can be tolerated in the economy:

Bmax 0 Yn 0 = 1 8a2r X i PiAi− 1 + a1− 2a2r ¯Ki(0)
· (r + δi)
2 Yn 0 (r + δi)2 (D.1)

More interestingly, the partial derivatives of the ratio in (D.1) w.r.t. the structural variables of the economy are9:

• ∂ ∂a1 _Bmax 0 YN 0 

< 0 but can change sign when the magnitude of the linear coefficient of the adjustment cost function is comparable to the price of capitals, PK

i , in the

economy (see figure 4a, obtained for PK i = 1); • ∂ ∂a2  Bmax 0 YN 0 

< 0 but can change sign for ¯Ki(0) much greater than Di, indeed a2can

be regarded as the scale parameter for the capital of equilibrium (see equation 4-bis). Consequently, given a certain ¯Ki(0), increasing a2 will scale down Di so to

obtain the result in figure 4b; • ∂ ∂r _Bmax 0 YN 0 

is ambiguous right around the most relevant interval [0.05, 0.2]. In-deed this partial derivative results in a polynomial of degree 3, suggesting the complexity of dealing accurately with the variation due to the interest rate (see figure 4c); • ∂ ∂Ai _Bmax 0 Y0N 

> 0 but can be negative when Ai is low w.r.t. other industries and

γi is high, since this implies an increase in the share of low-productivity sectors

that generates an effect similar to the one described at the end of section 2.2 (see figure 4d); • ∂ ∂Ki(0) _Bmax 0 YN 0 

< 0 under the assumption Di > 0 since the ratio _Kδi_iD₍₀₎i decreases in

an effect very similar to the one related to the derivative w.r.t. Ai (see figure 4e);

9_{The results have been obtained through numerical simulation because the analytical approach}

would prove inconclusive, apart for the derivative w.r.t. Ki(0). In this case, Ki(0) is without the

0.2 0.4 0.6 0.8 a1 (a) ∂ ∂a1 _Bmax 0 YN 0  0.2 0.4 0.6 0.8 a2 (b) ∂ ∂a2 _Bmax 0 YN 0  0.2 0.4 0.6 0.8 r (c) _∂r∂ Bmax0 YN 0  0.1 0.2 0.3 Ai (d) _∂A∂ i _Bmax 0 YN 0  KiH0L (e) ∂ ∂Ki(0) _Bmax 0 YN 0  Figure 4: Representation of the partial derivatives of the debt-to-income ratio w.r.t. the structural variables of the economy. The plots are obtained through numerical simulation, but for Ki(0) in

figure 4e.

Two-sector economy, representation of fiscal policies on the {α1× α2}-plane. Fiscal policy in A is invalid since at least one component is negative whereas the policy vector in B is valid. The effective tax rate in each sector, τ (1 − αnr

E

Code for the evaluation of the Nash bargaining

solution

The code in the next three pages was written for Mathematica R _{language and its aim}

is to inform the reader on what is the numerical evidence supporting the considerations that follow equation (20) in section 4

number 2;to set the number of sectors  20 100;initializing the tax rate

a1 20 100;the linear part of the adjustment cost function

a2 1 100;the quadratic part of the adjustment cost function

r 5 100;the interest rate

P 1;the price level, set equal in both sectors

in 80 1000; to initialize Ki0 as a fraction of Di initialization

NFindInstance1 P Aa1 1 r  0 &&A  0 &&r 2 && r&&K0 0 &&PA1 a1 2a2 r K0 r  2 AP && K01 1  inP A1 a1 r  2a2 r,

A,,K0,number;find A1,2,1,2 and K1,20

myD_{1 }_{1 }_P A_{1 }a1 r_{  2}a2 r_{ .}initialization; DomyDi myDi .i,i,number;Di in function of i K0 ConstantArray,number; DoK0i myDi .iin,i,number; A  ConstantArray,number;

DoAi A.initializationi1, i,number;  ConstantArray,number;

Doi .initializationi2, i,number; maxDebt 1 8a2 r 

SumPAi  1 a1 2a2 r K0i ri  ri^2,

i,number;maximum sustainable debt maxDebt ConstantArray,number;

DomaxDebti 

P Ai 2  1  1 a1 2a2 r K0i  ri 

2 PAi, i,number; find iNR_{ for the initial tax rate}

DomyDi myDi . imaxDebti, i,number;find DiMIN

IfmaxDebt1 maxDebt2,j 1,j 2;

determine which sector subsidy, eventually, might go to 0 for min_ min P Aj  1 a1 2a2 r K0j rj

2   P Aj ;find min

frac 99 100;

B0maxDebtfrac; initialize B0 as a fraction of B0MAX_  45 100; set the bargaining power 

frame 199;

set the fineness of sampling in Tmin,1 as 1minframe counter 0; to record results

p1 ConstantArray,frame; record the value function in sector 1 p2 ConstantArray,frame; record the value function in sector 2

DoFORcycle over  start

myV1_ : Simplify1 1 P A1 1 1 P A1  1 a1 r1  2a2 1 r_{1  1 }a1 _{1 1 }_{1 }P A1  1 a1 r1  2a2 1 r1  a2 1^2  1 1 P A1  1 a1 r1  2a2 1 r1^2 r 1 1 P A1 K01  1 1 P A1  1 a1 r_{1  2}a2 1 r_{1 }

r1;value function in sector 1 myV2_ : Simplify1 1 P A2 1 1 P A2  1 a1 r2  2a2 2 r2  1 a1 2 1 1 P A2  1 a1 r2  2a2 2 r2  a2 2^2  1 1 P A2  1 a1 r2  2a2 2 r2^2 r 1 _{1 }P A2 K02  1 _{1 }P A2  1 a1 r2  2a2 2 r2 

r2; value function in sector 2 maxDebt ConstantArray,number;find iNR_{}

DomaxDebti 

P Ai 2  1  1 a1 2a2 r K0i  ri 

2 PAi, i,number;

myD_{1 }_{1 }_P A_{1 }a1 r_{  2}a2 r_{;find Di}MIN_{}

DomyDi myDi .i,i,number;

DomyDi myDi . imaxDebti, i,number; func Solve r 1 r_{1} 1 1P A1 1 1 1 1P A1  1 a1 r_{1  2}a2 1 r_{1 }r K01  1 r2 1 2P A2 2 1 1 2P A2  1 a1 r2  2a2 2 r2  r K02 B0 0,₂;define the ellipse

max ConstantArray,number;

find the most north and east point on the ellipse

DoIfi 1,max1 

1. Solve2.func2 maxDebt2,12,

max2  _2.func2 .₁maxDebt1, i,number; nash ConstantArray,number;

IfMinChopmaxDebt  0,find numerically 1_ nash1 

Re₁. FindRootDmyV1₁ myV1maxDebt1^myV2₂.func

2 myV2maxDebt2^1 ,1  0, 1, 0.05,

nash1  Re₁. FindRootDmyV1₁ myV1maxDebt1^ myV22.func2 myV2maxDebt2^1 ,1  0,

₁,maxDebt1 _max1 maxDebt1; nash2 2.func2 .1nash1;find 2

nashCurve2. SolvemyV11 myV1maxDebt1^

myV2₂ myV2maxDebt2^1  V0 0,₂2 . V0myV11 myV1maxDebt1^myV22.func2 

myV2maxDebt2^1  .₁nash1; construct the isowelfare curve passing through the Nash

bargaining solution point countercounter 1;

p1counter myV1nash1;record value function for sector 1 p2counter myV2nash2;record value function for sector 2

ClearmyV1,myV2;

,,min, 1,1 min frame;FORcycle over  end Printp1,p2verify that they do not change over time

policy in A is invalid since at least one component is negative whereas the policy vector in B is valid. The effective tax rate in each sector, τ (1 − αnr

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