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Is the public sector of your country a diffusion

borrower? Empirical evidence from Brazil

Leno S. Rochaa,∗, Frederico S. A. Rochab, Tharsis T. P. Souzac

aBrazilian Treasury Secretariat, 70.048-900 Brasilia-DF, BrazilbDepartment of Computer Science, University of Utah, Salt Lake City, USA

cDepartment of Computer Science, UCL, Gower Street, London, WC1E 6BT, UK

Abstract

We propose a diffusion process to describe the global dynamic evolution of creditoperations at a national level given observed operations at a subnational levelin a sovereign country. Empirical analysis with a unique dataset from Brazilianfederate constituents supports the conclusions. Despite the heterogeneity ob-served in credit operations at a subnational level, the aggregated dynamics at anational level were accurately described with the proposed model. Results mayguide management of public finances, particularly debt manager authorities incharge of reaching surplus targets.

Keywords: Public Debt, Diffusion Process, Gompertz Function

1. Introduction

Recent empirical literature supports a negative impact of public debt oneconomic growth in both developed and emerging markets [1, 2, 3, 4]. Indeed,poorly managed public debt has been an important factor in inducing and prop-agating recent economic crises [5, 6].

While the linkage between public debt and economic growth for nationaleconomies has long been studied, the investigation considering the public debtat a subnational level has been largely neglected in the literature [7]. Matz andMitze (2015) [7] showed a link between regional public debt and economic growthfor German federal states. Further, Jenkner and Lu (2014) [8] and Buiatti,Carmeci, and Mauro (2014) [9] provided empirical evidence that regional fiscalimbalance influences public debt at a national level.

In this work, we consider a division of a sovereign state in administrativeunits where each one can potentially contract credit operations with the approvalof the central government. The study of aggregated regional debt dynamics isimportant as it impacts the public debt at the national level. We discuss the

∗Corresponding authorEmail address: leno.rocha@tesouro.gov.br (Leno S. Rocha)

Preprint submitted to Physica A November 20, 2021

aggregated subnational debt related to credit operations pleaded by regionalgovernments and we propose a diffusion process [10] to model the number andmonetary volume of such operations. We describe mechanisms in financing thatmotivate the modeling of the response variables to be a sigmoid curve annually.

A case study with federated entities in Brazil confirms this sigmoid patternand showcases the adequacy of the proposed diffusion processes. The applica-bility of the proposed methods in other sovereign countries is also considered.Results may be applied in distinct countries where credit pleas at the subna-tional level are legitimate via consent of a central public finance management,as is the case for the United Kingdom and Japan, for instance.

2. Methodology

Federated entities or administrative divisions of a sovereign state may per-form credit operations for local financing purposes. Let N(t) be the aggregatednumber of credit operations pleaded by federated entities in a given year, wheret ≥ 0 represents the continuous time since the first fiscal day of the year. Wepropose to model the evolution of N(t) as a learning curve following a diffusionprocess with a sigmoid trajectory.

The velocity with which entities of the public sector have credit operationsplaintiffs during the year is assumed to be intrinsically related to a learningprocess where the number of credit pleas may be low at the beginning of theyear further accelerating with time until it reaches a climax point. By the endof the year, the evolution of the number of credit operations may decelerate torespect limits and fulfill conditions established by law. These constraints, inaddition to the credit volume limiters, lead us to propose the following diffusionprocess:

N(t) = g(t){mn − [N(t)]n}/n, (1)

wherein the point superscript to N(t) represents the time derivative of thisvariable; g(t) is the coefficient of diffusion; m represents the potential maximumnumber of pleas, i.e., the saturation level or the carrying capacity [10]; and n isa parameter which characterizes the distance between m and N(t).

With respect to the diffusion coefficient, we assume that the number ofoperations pleaded is linearly related to the understanding of the process bythe parties involved, so that g(t) = wN(t), in which w is a learning speedparameter. Hence, we solve the differential Eq. 1, as shown in Appendix A, tofind the logistic function:

N(t) = m{1 + (m− 1) exp[−wm(t− 1)]}−1. (2)

Considering that the course of the events concerning credit operations canlast months, and the object, value, financial conditions and other characteristicsof the pleas may change significantly, it may become more interesting to theborrower to file a new plea. This situation characterizes a variable mortalityrate phenomenon, which was studied initially by Gompertz [11]. Thus, we

2

extend the logistic model from Eq. 1 to incorporate the Gompertz function, aspresented in Appendix B, which leads to:

N(t) = m exp[−b exp(−ct)], (3)

where b = −(lnm)exp(w) and c = w. It is convenient to let the functionhave more degrees of freedom, using the parameters m, b and c independently,so that more adherence to the data can be achieved. The resulting modelis better suited to fit sigmoid curves with different levels of acceleration andrelaxation. Moreover, it presents a good compromise between descriptive powerand parsimony while having only three parameters for calibration.

3. Case Study: Credit Operations in Brazil

Federated units in Brazil are forbidden to issue bonds [12], but they cantake loans to fund projects of their interest. In order to accomplish creditoperations with financial institutions, federated entities must respect limits andfulfill a series of conditions determined by law. The Brazilian National TreasurySecretariat (STN) is in charge of checking these limits and conditions for thecredit pleas. Contracting credit operations by federal entities in Brazil is acontinuous flow process and the pleas can be sent at any time for STN’s approval.

In this context, two main general mechanisms make the time path of theTreasury’s analysis response to be a sigmoid curve annually. The first is theneed to learn the new procedures to plead credit operations and to be informedabout the updated budgetary information that may change every year. Thesecond mechanism is related to several credit volume limiters, such as the fiscalsurplus target, and the amount of money in the market available for loans. Thefirst mechanism shapes the process as a learning curve, in which the number ofcredit pleas may be low at the beginning of the year, while the second mechanismimposes limits and therefore defines the level of saturation of the process.

Given Eq. 2, we defineN(t) to be the aggregated number of credit operationsof Brazilian states and municipalities. The parameter m is related to creditconstraints scattered in legal rules such as: (i) constitutional limitations forthe credit operations of the public sector1; (ii) limits and conditions for thecredit operations of the governments 2; (iii) rules of contingency for credit tothe public sector 3; (iv) fiscal space remaining from the surplus target; and (v)discretionary limit of external operations 4.

The information about the credit operations of the federate entities was col-lected from the Brazilian Treasury Secretariat database [16]. These data providespecific information about the date and the value of each loan thus allowing the

1Defined in the Article 52 of the Brazilian Federal Constitution of 1988 [13].2Resolution of the Senate n. 43 of 2001 [14].3Defined in the Resolution of the National Monetary Council n. 2.827 of 2001 [14].4Defined by the External Financing Commission according to the Article 7 of the Decree

n. 3.502 of the year 2000 [15].

3

empirical verification of the trajectory of both the number of requirements andthe claimed financial volume. The credit data is given at state and municipalitygranularities. Here, we consider the public debt of a state as the total debt ofits corresponding federated unit plus the debt of its municipalities.

Fig. 1 demonstrates that most of the operations are performed by a fewstates. A Pareto relationship is observed where 80% of the number of creditoperations came from approximately 20% of the Brazilian states between 2002and 2015. An asymmetric relationship also follows in monetary terms with thesame share of states being responsible for 40% of the total pecuniary volumein the period analyzed. The values of the credit operations also present hetero-geneity. Fig. 2 shows the complementary cumulative distribution of the valueof a credit operation performed by a state. The distribution of values is hugelyasymmetric which demonstrates the disparity among values of credit operationsin the states of Brazil.

Figure 1: Approximately 20% of the Brazilian states were responsible for more than 80%of the total number of credit operations between 2002 and 2015. The same share of statesaccounted for more than 40% of the monetary volume in the same period.

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Figure 2: Demonstration of heterogeneity in the value of credit operations. This figure showsthe complementary of the cumulative distribution function of the value of credit operationsin log-log scale.

4. Results and Discussion

The heterogeneity observed at the state level imposes challenges in the mod-eling of the trajectory of the number and volume of credit operations. Despitethis complexity, we show that the observation of a sigmoid pattern at the ag-gregated national level results in trajectories fit with high accuracy with theproposed models.

We fit both the logistic (Eq. 2) and Gompertz (Eq. 3) models to the yearlynumber of credit operations. The former model performed better for all yearsand its results are shown in Fig. 3. The figure shows the curves for the yearlynumber of credit operations that were assented by the government and the totalnumber of credit pleas (operations rejected included). The adherences of thelogistic and Gompertz models to the data, measured by the average of the R2

of the fitted curves, were above 96% and 98%, respectively. Model calibrationwas performed using a nonlinear least squares fitting procedure implemented inthe cftool of the software Matlab r.

Notice that, in general, the yearly number of credit pleas increases slowlyat the beginning of each year then it accelerates nearly reaching the yearlycapacity. It then starts to slow down thus presenting a behavior well describedby a sigmoid function. In some years, the curves do not reach a flat region ofsaturation (e.g. 2015 and 2005), indicating that the number of pleaded credit

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0 100 200 300 400Time (elapsed days)

0

200

400

6002002

Total operation pleasAnalitical Solution - operation pleasTotal assented operationAnalitical Solution - assented operation

0 100 200 300 4000

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20002014

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8002015

R2 = 98%

R2 = 99% R2 = 99.1%

R2 = 99.7%

R2 = 99.1%

R2 = 99.1%

R2 = 99.7%

R2 = 99.7%

R2 = 99.6%

R2 = 99.4%

R2 = 99%

R2 = 99%

R2 = 99.2%

R2 = 99.3%

R2 = 99.4%

R2 = 99.8%

R2 = 95.2%

R2 = 93.6%

R2 = 99.6%

R2 = 99.3%

R2 = 98.7%

R2 = 99.3%

R2 = 99.8%

R2 = 99.9%

R2 = 99.7%

R2 = 99.5%

R2 = 98%

R2 = 96.4%

Figure 3: Demonstration that the amount of public credit pleas per year follows a sigmoiddiffusion process. This figure shows the total number of Brazilian credit pleas by year alongwith the analytic solution provided by the derived Gompertz model.

operations might not have reached the government’s budgetary expectations.Fig. 4 shows the derivatives of the number of credit operations (N(t)) in

time, which were fit via central finite differences. Again, we clearly observethat each year is characterized by a period of acceleration followed by a peakthat precedes an ending period of relaxation. The peak of acceleration in thenumber of credit operations consistently differs by year. Bearing in mind thesmall sample of these responses, however, we observe that years with a peakat the beginning of the period tend to follow years with a peak at the end ofthe period and vice-versa. On the other hand, there is no clear tendency whenpeaks occur in the middle of the year. These empirical observations indicate

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0 200 400Time (elapsed days)

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Figure 4: Derivative of the number of credit operations (N(t)). Clear peaks at the center andperiods of acceleration and relaxation in the tails. Empirical derivatives estimated via centralfinite differences.

that the aggregated number of credit operations at a subnational level presentseasonal patterns which can be explained by political-economic cycles [17, 18],for instance, related to election periods.

From 2002 to 2015, the variability of credit operations pleas in monetaryterms by year is considerable, especially due to outlier loans of very high value,contrasting with the wide majority of operations of small value pleaded bysmall states. This dispersion prevents the usage of the model of the number ofoperations to describe the dynamics in monetary terms by simply multiplyingthe number of credit operations by the yearly mean value of the pleas. However,

7

0 100 200 300 400Time (elapsed days)

0

2

4

6 ×109 2002

Total value of operation pleasAnalitical Solution - values pleadedTotal value of operations assentedAnalitical Solution - assented operations

0 100 200 300 4000

0.5

1

1.5

2 ×109 2003

0 100 200 300 4000

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1

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3 ×1010 2015

R2 = 98.2%

R2 = 98.7%

R2 = 95.3%

R2 = 98%

R2 = 97.8%

R2 = 97.2%

R2 = 99.2%

R2 = 98.6%

R2 = 90.6%

R2 = 93.4%

R2 = 99.2%

R2 = 97.6%

R2 = 98.7%

R2 = 95.9%

R2 = 97.7%

R2 = 98.9%

R2 = 98%

R2 = 99.2%

R2 = 97.8%

R2 = 98%

R2 = 97.5%

R2 = 97.9%

R2 = 97.9%

R2 = 98.9%

R2 = 96.9%

R2 = 96.8%

R2 = 96.6%

R2 = 97.9%

Figure 5: Demonstration that the Gompertz model explains to some extent the public creditpleas also in monetary terms. Figure shows the total credit pleaded per year along with theanalytical solution provided by the derived Gompertz model.

we observe that the aggregated volume of credit operations in monetary termsalso follows a sigmoid curve to some extent. This observation enables the usageof the proposed models by simply replacing the number of credit pleas N(t)with its corresponding value in monetary terms.

Fig. 5 shows the results of the fitting further demonstrating that the sigmoidmodeling can also be applied to the pecuniary aspect of the credit operations,though with smaller quality, since the model does not capture the “jumps” dueto the presence of outliers in the pleaded values.

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5. Conclusions

We proposed a diffusion process to describe the dynamic evolution of thenumber and volume of credit operations aggregated from a subnational level.We uncovered a sigmoid pattern in the dynamics of credit operations furthervalidated with empirical data from Brazilian federate constituents. Despite theheterogeneity observed at the subnational level, the aggregated dynamics at thenational level were accurately described with the proposed diffusion processes.Moreover, even with the particular behavior presented in each year with respectto acceleration, peak and capacity, all the curves were meaningfully adherentto sigmoid trajectories in terms of the number and monetary volume of creditoperations. These results enable macroeconomic policy-making without a cum-bersome analysis at a subnational level.

With the calibration of the proposed diffusion model, it is possible to obtaininsights into the evolution of public credit operations in a given year, which al-lows improvements of economic nature, like the commitment to surplus targets.Possible extensions of this analysis may take into account market expectationsand goals set by the government. Key variables to take into consideration in aGompertz parameter prediction exercise may include: GDP, credit risk ratingsand dummy variables for elections and anti-cyclical government interventions.

The proposed model might be useful to explain credit operations of othercountries. The sigmoid pattern is likely to be observed in other sovereign statesin which local government rely upon credit operations, like the United Kingdom[19] and Japan [20] [21], where at least the limitation on loan amounts andbudgetary principle of annuity are immediate similarities that apply. Furtherresearch along with the availability of consolidated data on other countries wouldallow for the applicability and extension of proposed models.

Acknowledgments

F.S.A.R. is grateful to CAPES (Coordination for Enhancement of HigherEducation Personnel) and to the University of Utah for the financial support.T.T.P.S. acknowledges financial support from CNPq - The Brazilian NationalCouncil for Scientific and Technological Development.

References

[1] D. Matesanz, G. J. Ortega, Sovereign public debt cri-sis in Europe. A network analysis, Physica A: Statisti-cal Mechanics and its Applications 436 (2015) 756 – 766.doi:http://dx.doi.org/10.1016/j.physa.2015.05.052.

[2] U. Panizza, A. F. Presbitero, Public debt and economic growth: Isthere a causal effect?, Journal of Macroeconomics 41 (2014) 21 – 41.doi:http://dx.doi.org/10.1016/j.jmacro.2014.03.009.

9

[3] G. Bua, J. Pradelli, A. F. Presbitero, Domestic public debt in low-incomecountries: Trends and structure, Review of Development Finance 4 (1)(2014) 1 – 19. doi:http://dx.doi.org/10.1016/j.rdf.2014.02.002.

[4] L. Carranza, C. Daude, A. Melguizo, Public infrastructure in-vestment and fiscal sustainability in latin america: incompati-ble goals?, Journal of Economic Studies 41 (1) (2014) 29–50.doi:http://dx.doi.org/10.1108/JES-03-2012-0036.

[5] IMF guidelines for public debt management, http://bit.ly/231Id9s,Last accessed on Mar 31, 2016 (2014).

[6] S. Spilioti, G. Vamvoukas, The impact of government debt on eco-nomic growth: An empirical investigation of the greek market,The Journal of Economic Asymmetries 12 (1) (2015) 34 – 40.doi:http://dx.doi.org/10.1016/j.jeca.2014.10.001.

[7] T. Mitze, F. Matz, Public debt and growth in german federal states: Whatcan europe learn?, Journal of Policy Modeling 37 (2) (2015) 208 – 228.doi:http://dx.doi.org/10.1016/j.jpolmod.2015.02.003.

[8] E. Jenkner, Z. Lu, Subnational credit risk and sovereign bailouts – Whopays the premium?, IMF working paper WP14/20.

[9] C. Buiatti, G. Carmeci, L. Mauro, The origins of the public debt of Italy:Geographically dispersed interests?, Journal of Policy Modeling 36 (1)(2014) 43–62.

[10] R. Shone, Economic Dynamics: Phase diagrams and their economic appli-cation, Cambridge University Press, 2002.

[11] B. Gompertz, On the nature of the function expressive of the law of humanmortality, and on a new mode of determining the value of life contingencies,Philosophical transactions of the Royal Society of London (1825) 513–583.

[12] Brazilian Federal Senate, Resolution 43, http://bit.ly/1Ud7xJg, Lastaccessed on Mar 31, 2016 (2001).

[13] Constitution of the Federative Republic of Brazil,http://bit.ly/1GLK9tA, Last accessed on Mar 31, 2016 (1988).

[14] Brazilian National Monetary Council, Resolution no 2.827,http://bit.ly/23WyNl7, Last accessed on Apr 25, 2016 (2001).

[15] Brazilian Presidency, Decree no 3.502, http://bit.ly/1QxvJPu, Last ac-cessed on Mar 31, 2016 (2000).

[16] SADIPEM, Brazilian National Treasury Secretariat: Historical data,https://sadipem.tesouro.gov.br/, Last accessed on Mar 31, 2016(2015).

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[17] S. N. Sakurai, N. Menezes-Filho, Opportunistic and partisan election cyclesin Brazil: new evidence at the municipal level, Public Choice 148 (1) (2010)233–247. doi:10.1007/s11127-010-9654-1.

[18] A. Alesina, G. D. Cohen, N. Roubini,Macroeconomic Policy and Elections in OECD Democracies, WorkingPaper 3830, National Bureau of Economic Research (September 1991).doi:10.3386/w3830.URL http://www.nber.org/papers/w3830

[19] United Kingdom Goverment, Department of Communities and Local Gov-ernment, http://bit.ly/21deqLE, Last accessed on Mar 31, 2016 (2015).

[20] Japan Finance Organization for Municipalities,http://www.jfm.go.jp/en/about/financing.html, Last accessedon Apr 6, 2016 (2016).

[21] Ministry of Internal Affairs and Communications, Lo-cal Government Bond System and Market in Japan,http://www.jlgc.org.uk/en/pdfs/MIC%20LGB.pdf, Last accessedon Apr 6, 2016 (2016).

Appendix A. Solution of the logistic diffusion process

Recall Eq. 1, the differential equation of generalized growth, and consider itwith t ∈ R

+ and n ∈ N∗:

N(t) =dN(t)

dt= g(t){mn − [N(t)]n}/n (A.1)

It is possible, using a notation abuse, to rewrite it as:

ndN(t)

N(t){mn − [N(t)]n}= wdt (A.2)

Making use of partial fractions, we have:

nm−n

{

1

N(t)+

[N(t)]n−1

mn − [N(t)]n

}

dN(t) = wdt (A.3)

By the chain rule, d[N(t)]n = n[N(t)]n−1dN(t) and performing the integra-tion, it follows:

n ln |N(t)| − ln |[N(t)]n −mn| = wmn(t− t0) (A.4)

Once m ≥ N(t) ≥ 0, we can eliminate the modulus notation, and doing theexponentiation we have:

[N(t)]n

mn − [N(t)]n=

1

mn − 1exp[wmn(t− t0)] (A.5)

11

in which t0 is the initial moment, that for methodological definition is the firstof the year, i.e, t0 = 1. Finally, isolating the function of interest we have:

N(t) = m{1 + (mn − 1) exp[−wmn(t− 1)]}−1

n (A.6)

Assuming n = 1, which makes the growth expressed in the differential Equa-tion 1 to be proportional to an unidimensional distance, one arrives at the for-mula of Equation 2.

Appendix B. Derivation of Gompertz equation

Writing powers via logarithms and making n → 0, which corresponds to agrowth proportional to a logarithmic distance [11] in the differential Equation1 , we have:

limn→0

N(t) = wN(t) limn→0

exp(n lnm)− exp[n lnN(t)]

n(B.1)

Resorting to power series of the Napier’s exponential function, it follows:

N(t) = wN(t) limn→0

1

n

{

∞∑

i=0

(n lnm)i

i!−

∞∑

i=0

[n lnN(t)]i

i!

}

(B.2)

Sorting out the two first elements of each summation, we have:

N(t) = wN(t)

{

ln

[

m

N(t)

]

+ limn→0

∞∑

i=2

ni−1{

(lnm)i/i!− [lnN(t)]i/i!}

}

(B.3)

The limit of the remaining summation tends to zero, so that we have, asalready stated, a logarithmic distance:

N(t) = wN(t) ln

[

m

N(t)

]

. (B.4)

Rewriting the Equation B.4, again with notation abuse, follows:

d{ln[N(t)/m]}

ln[N(t)/m]= −wdt. (B.5)

Integrating the Equation B.5 we have:

ln | ln[N(t)/m]| − ln | lnm| = −w(t− 1). (B.6)

Performing two exponentiations, we arrive at the Gompertz Equation:

∣

∣

∣

∣

ln[N(t)/m]

lnm

∣

∣

∣

∣

= exp[−w(t− 1)] (B.7)

N(t) = m exp{−(lnm)exp[−w(t− 1)]} (B.8)

12

In the original Gompertz function, b = − (ln m) exp(w) and c = w, as can beseen in Equation B.8 . But as previously mentioned, we allowed the parametersto be independent in order to improve the adherence to the data. Adopting thisrelaxation one arrives at the Equation 3.

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