Functional trinity of public finance in an emerging economy
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Abstract
Two challenges in the literature have inspired this paper. First, the highly contradictory puzzle of debates on the “public budget–economic growth” correlation, explainable in great measure by the inadequacy of involved modeling tools (assumptions and specifications) with studied samples. Regarding this issue, the paper proposes three temporal scales of analysis: (1) the global historical trajectory of the modern society, concerning which the Wagner–Kaldor theorem has a leading explicative position; (2) the given structural state–society relationship, for which Barro–Armey–Rahn–Scully (BARS), Laffer in narrower sense (LINS), and the public budget balance restriction (BBR) curves are more suited; and (3) the short–medium horizon, where the Keynesian and postKeynesian framework remains relevant. Another starting point of this paper refers to the second scale. While the BARS and LINS curves were extendedly examined, the BBR one benefited of attention only during the last period in the context of the sovereign debt crisis. Besides, all these functions were studied separately. Since in the real economy they interact continuously, we integrate BARS, LINS, and BBR curves into a unitary model with a “compatibility restriction” as technical solution. Data for an emergent mediumsized EU country (Romanian case study) illustrate this approach.
Keywords
Public budget BARS LINS BBR Collinearity Compatibility restriction Model simulationsAbbreviations
 atax
average legal taxation
 BARS
Barro–Armey–Rahn–Scully
 BB
public budget balance
 BBR
public budget balance restriction
 BE
volume of the public budget expenditures
 BR
volume of the public budget revenues
 cbb
ratio of BB to GDP
 cbbo
deviation of the effective cbb from cbbr
 cbbr
ceiling for cbb
 cbe
ratio of BE to GDP
 IOT
contribution to IGB of nonbudget factors
 ck
ratio of PK to PGDP
 cnfbr
the coefficient of the nonfiscal budget revenues
 GCB
general consolidated budget
 GDP
gross domestic product at current prices
 GDPc
gross domestic product at constant prices
 GFCF
gross fixed capital formation at current prices
 GFCFc
gross fixed capital formation at constant prices
 IGB
economic growth index
 IGDPc
index of the GDP at constant prices
 IGFCFc
index of the gross fixed capital formation at constant prices
 letg
legaleffective taxation gap
 LINS
Laffer in narrower sense
 LOT
other factors (than atax) which influence the legt
 NFBR
nonfiscal public budget revenues
 oatax
optimal level of atax
 OBB
other factors (than pdg) which influence the budget deficit within cbbr
 ocbe
optimal level of cbe
 opdg
optimal level of pdg
 PD
public debt
 pdg
ratio of PD to GDP
 PGDP
GDP deflator
 PK
fixed capital formation deflator
 rgfcfc
ratio of the gross fixed capital formation to the gross domestic product at constant prices
 t
time factor
 TAXT
total of the collected taxes
 TAXTL
total of the legal taxation
 Wb
part of the social welfare dependent on the GDP
1 Background
The literature on government finances has gravitated toward the following issues: (1) the impact of public budget expenditures on the degree of society’s general development, especially the productive outcome of the economy; (2) the collected taxes as the most important component of the public budget’s income; and (3) the capacity of government to obtain credit from financial markets as a complementary resource for its own expenses.
The corresponding literature is characterized by strikingly contradictory viewpoints.
Some authors have commented that government expenditure is almost neutrally related to aggregate output: besides the famous Harberger (1964, 2003) conjecture (Engen and Skinner 1996; Mark et al. 1997; Mendoza et al. 1997; Wasylenko 1997; Hines 2002; Xing 2011). Some studies have considered this impact negative: Mark et al. (1997), Beach (1998), Poulson and Kaplan (2008), McBride (2012), Pettinger (2017). The opinions positing the positive influence of government spending on economic growth predominate, and such a role is attributed to the socalled productive items of the public budget (Kneller et al. 1999); notably, Asimakopoulos et al. (2016) included them into their macroeconomic production function.
Divergent opinions have also been formulated concerning public budget taxes. Mertens and Ravn (2013) identified immediate effects on output induced by changes in average tax rates. Engen and Skinner (1996) posited, “the tax level is likely to exert a modest, but cumulatively important, influence on longterm growth rates. (p. 36).” Structurally, many studies have outlined the unequal consequences of various taxes on output. Therefore, although taxation of income and profit, payroll and labor, property, and social security contributions would cause distortions, no such effect was identified for other taxes (Kneller et al. 1999). Concerning capital taxation, the theoretical modeling indicated a complicated sinusoidal evolution, from level zero (Judd 1985; Chamley 1986) to relatively high rates (Piketty and Saez 2013). Lee and Gordon (2005) found that the corporate tax rate is negatively correlated with economic growth, but other tax variables are not significantly associated.
The variegated picture of the research conclusions on the “public budget–economic growth” tandem was described in overviews, such as Huang (2006), Kenny and Winer (2006), Afonso et al. (2008), Myles (2009), Magazzino and Forte (2010), MartinezVazquez et al. (2011), Magazzino (2012), McBride (2012), Nygård and Revesz (2015), Afonso and Alves (2016), Dobrescu (2016a), Winer (2016), Churchill et al. (2017), Fisman et al. (2017), and GomezPuig and SosvillaRivero (2017).
How can such interpretative diversity be explained? The epistemological premises of various studies and the real specificities of studied experiences cannot be ignored. Working on this paper, we concluded that only these circumstances are not sufficient.
A significant cognitive impact arises from underestimating the temporal scale to which the examined problems refer. Or concerning government finance, the delimitation of three perspectives seems essential.
The first one considers the global historical trajectory of modern society, for example, the Wagnerian hypothesis (Wagner 1883). With roots in classical economics, the anticipation that economic development will be accompanied by an increasing share of government expenditure in national income has been confirmed by authorized statistical series. Inherent parallel growth of taxation—admitted still by Wagner—was comprehensively discussed by Kaldor (1962, 1963a, b). The syntagm “Wagner–Kaldor law” is justified.
Another perspective concerns the limited temporal segment, provisionally named a “given structural state–society relationship,” as a distinct component of the larger notions, such as human social environment (Barnett and Casper 2001), social capital (Field 2008), social structure in a networkbased perspective (Mukherjee 2007), or a given historical context (Dobrescu 2016a). The binomial state–society is dynamic, its forms and ampleness modifying along global history and within the same social system. Such changes occur periodically, and existing intervals with sufficiently stable features include: the dominant behaviors of people as taxpayers and beneficiaries of public spending. In this paper, we shall refer exactly to these intervals.
Finally, note the short–medium horizon, and the problems of such a time span being prioritary for macroeconomic policies. Keynesian and postKeynesian models remain relevant for such intervals.
In our opinion, if empirical research does not account for these temporal scales, its consistency is inevitably affected.
The trap of applying or checking the theorems on improper samples of observations was unavoidable. Among such temptations, especially risky would be to relate the Wagner–Kaldor law to limited historical information or, by contrast, to expostulate the Keynesian schemes to long and very long series.
Similar problems occur when PANEL analyses involve data belonging to economies in different development stages. The assertion—sometimes deduced from such studies—that data corroborate the Wagner–Kaldor law for lowincome countries while rejecting it in the case of rich ones, overlooks that this law envisages the global historical trajectory of society and not its various segments.
A possible disregard of the temporal scale of analysis can also be observed in model specification. The econometrics of the Wagner–Kaldor law is again edifying and intended for a relationship on very long horizon between output and ratio of government spending to gross domestic product (GDP). In its formalization, the main difficulty stems from the nonunitary evolution of the involved variables. Although the potential output registered an obvious increasing trend, the share of government expenditure in GDP cannot extend unlimitedly. At superior development stages, despite the increasing percapita income, the social acceptability of taxation decelerates, tending toward an asymptotical level. (Normally, it is situated under unity, which is its maximum potential limit.) This trend seems to be linked to the extending capacity of the private sector to provide some public goods and services, and enforcing democratic control on government decisions. Consequently, in modeling the size of the government in relation to economic growth long term, the threshold regression methods are more suitable (Christie 2011) as well as nonlinear functions (Kuckuck 2012; Hajamini and Falahi 2014; Aghion et al. 2016), including the logistic one (Florio and Colautti 2002; Magableh 2006). In this context, notably, is the conclusion drawn by Fölster and Henrekson (2001): The relationship between government expenditure and growth is negative in rich countries.
 (a)
State intervention propensity, especially concerning the government spending,
 (b)
Tax acceptability, and
 (c)
Risk aversion to public indebtedness.
All these conditions are sociological reactive processes [as characterized in Mahoney (2000)]. Any change in the institutional framework of public budget expenditures, taxation, or public debt evolution is followed—as a rule—by counter reactions (i.e., upback or pushback) on the side of domestic economic agents, foreign markets, civic organizations, mass media, political formations, and public opinion in general.
The tendency toward equilibrium of society generates some steadystate levels of leading indicators, defining quantitatively the aforementioned behaviors: the share of public budget expenditures in GDP, degree of tax acceptability, and public debt size. Obviously, these levels are specific for each given structural state–society relationship.
Admittedly, the dynamics of actual processes around the steadystate levels is submitted concomitantly—similarly to other economic dependences—to the scale efficiency and diminishing marginal returns’ theorems. The advantages of scale efficiency appear more intensive when the involved indicators are situated at the left side of the steadystate levels (low government spending, taxation, and public debt), whereas the diminishing marginal returns become prevalent at the right side of these points (higher government spending, taxation, and public debt). Sociologically, the first slope could be considered a selfreinforcing sequence, and the second a suigeneris selfundermining one. The Ushaped form of the main relationships of government finance therefore approximates the real socioeconomic life. A deeper discussion of these problems exceeds the intended framework of this paper and was evoked only as assumptions of the subsequent analyses.
In the empirical research, a database was used for an emergent mediumsized EU economy (Romania, as a case study), with series for two and half decades, namely, 1990–2016. Despite the transformations induced by the transitional processes, after the first package of reforms in 1990s, the structural state–society relationship became, to a great measure, congruent with the functional market economy’s requirements. In principle, therefore, such a database can be admitted as relevant to the paper’s goal. The specificities of an emergent economy could be a profitable challenge to a scientific plan because they extend the cognitive area of an issue developed, until now, preponderantly on advanced occidental countries.
More concretely, the involved inpaper indicators refer to the general consolidated budget (GCB), which aggregates data for the central and local public budgets, the social insurance budget, and other similar funds. The classification of GCB revenues and expenditures adopted by the 2012 version of the Romanian macromodel (Dobrescu 2013a, b, 2015b) has been used. This classification is presented in Additional file 1: Appendix A, and the corresponding statistical series are detailed in Additional file 2: Appendix B.
This paper continues with a description of the main functional relationships characterizing government finances, including the collinearity problem associated with the Ushaped form of these relationships. More specifically, the Barro–Armey–Rahn–Scully (BARS) curve, the Laffer in narrower sense (LINS) curve, the specifications for nonfiscal public budget revenues (NFBR), and the public budget balance restriction (BBR) are examined separately (like in the literature) and as an integrated model. With such an aim, we introduced a compatibility restriction derived from the unitary framework, within which these functions intimately interact under a given structural state–society relationship.
Several conclusions and further possible research opportunities complete the paper.
2 Methods
2.1 Macromodel specification
Obviously, the specification of an applicative model must consider the peculiarities of the simulated economy.
2.2 BARS curve
The interpretation of the impact of public budget expenditures on economic growth as an inverted Ushaped dependence was consecrated as the BARS curve—an acronym of its founders’ names: Scully (1989, 1995, 1998), Barro (1990, 1991), Armey (1995), and Rahn and Fox (1996). Subsequent contributions to this topic belong to Tanzi and Schuknecht (1996), Chao and Grubel (1998), Gwartney et al. (1998), Tanzi and Schuknecht (2000), Tanzi (2005), Bania et al. (2007), Afonso et al. (2008), Altunc and Aydın (2013), Facchini and Melki (2013), and Di Liddo et al. (2015). By applying a nonlinear panel GMM approach on the database of 129 countries (World Bank statistics), Asimakopoulos and Karavias (2016) revealed a consistent nonlinear relationship between government spending and economic growth. Short bibliographical overviews are presented in Magazzino and Forte (2010), Dobrescu (2015a).
2.3 LINS curve
The interpretation of taxation as costs of W_{b} outlines the significance of the “Wicksellian connection” (Winer 2016). When the taxpayers perceive that legal taxes surpass the quality and quantity of the supplied public goods and services, their enforcing opposition to such a system is expectable.
The modern standard model was initiated by Allingham and Sandmo (1972), who attempted an early conceptualization of this mechanism by defining an analytical tool based on contributions from Becker, Tulkens, Jacquemin, Arrow, Mossin, Von NeumannMorgenstern (Allingham and Sandmo 1972). Shortly after this notable contribution, Yitzhaki (1974) added to its specification the penalty for discovered evasion. As a result, the complete model is sometimes called “AllinghamSandmoYitzhaki” (Slemrod and Yitzhaki 2002). Among the alternate approaches, the prospect theory (Kahneman and Tversky 1979) and experimental applications represented also important contributions. The topic has been debated in numerous studies, for example: Becsi (2000), Kirchler and Braithwaite (2007), Bruhin et al. (2010), Iyer et al. (2010), Kleven et al. (2010), Alm (2012), Laffer (2012), Piolatto and Rablen (2013), Holter et al. (2014), Myles (2014a, b), Isakov and Pekarski (2015), and Pappadà and Zylberberg (2016). Also notable is the increasing interest in the study of this topic on transitional economies: Johnson et al. (1997) (in connection with unofficial economy) (Papava 2002; Walewski 2009; Dobrescu 2016b).
The inverted Ushaped curve of collected taxes as a function of compulsory rates was associated in the beginning of the 1970s with Arthur Laffer, although such dependencies—at least intuitively—had been previously identified (see Laffer 2004; Bartlett 2012a, b, c). Following Feldstein’s papers (1995a, b, 1999) on the reaction of companies and households to taxation, the concepts of “tax gap” (Andreoni et al. 1998; James and Alley 2002), “tax compliance gaps” (Gemmell and Hasseldine 2013), and “the legaleffective tax gap (letg)” (Dobrescu 2016b) were formulated. This formalization has been referred to as the LINS curve.
 (a)
“tax acceptability” which cumulates the “tax morale” (Torgler and Schaltegger 2006; Myles 2014a) as a voluntary compliance and the government capacity to impose the fiscal rules or “fiscal coercion” (Akhand and Hubbard 2016), “economic deterrence” (Devos 2007), and
 (b)
“tax aversion” (Soldatos 2015) or “tax repulsiveness” (Dobrescu 2016b). Schematically, this interaction can be described as follows. Voluntary compliance accommodates with increasing taxes as long as the growth of cbe based on them is perceived by public opinion as providing adequate benefits for society. At higher levels of taxation, however, such a perception weakens, or disappears, transforming into an amplified repulsiveness (which stems from the natural propensity of economic agents to dispose autonomously of their own income as much as possible). Tax repulsiveness induces an accentuated opposition against related regulations, boosting legal, and illegal tax evasion and other civil disobediences. The capacity of the government to impose the rules has, of course, unsurpassable socioeconomic and institutional limitations.
The LINS curve involves λ_{1} > 0 and λ_{2} < 0. The ratio λ_{1}/(− 2*λ_{2}) can therefore be observed as an optimal level of the atax, under a given structural state–society relationship. Similar to the BARS curve, in the case of LOT, the econometric specifications are accommodated to the available statistics and features of the economy in cause.
2.4 BBR curve
As an outturn account, budget balance (the difference between public budget expenditures and public budget revenues) can be positive (a surplus), null, or negative (a deficit). For both nonzero cbb, constraints exist. The positive one is capped by the society’s availability to finance other countries instead of using their own budget resources for internal needs. The negative one encounters the sensitivity of the potential lenders concerning the dimension of public debt (PD) compared with national output (pdg = PD/GDP). The existence of a ceiling for cbb (noted cbbr) seems therefore admissible.
Opinions diverge regarding the macroeconomic implications of the pdg level. High public debt was frequently accepted as having, in general, a negative impact on economic growth (Bose et al. 2003; Greiner 2014; Pescatori et al. 2014; Stone 2014; Trecroci and Salotti 2014). There are also morenuanced opinions, in the sense that an increasing relative public debt could be positive in the short run but has a negative effect in the long run (Elmendorf and Mankiw 2000; GomezPuig and SosvillaRivero 2017). Some studies have revealed an ambiguous relationship between public debt and growth and, in any case, a high heterogeneity of this correlation across countries and time periods (Kumar and Woo 2010; DeLong and Summers 2012; Lof and Malinen 2013; Panizza and Presbitero 2013).
Since budget deficits are recorded with a negative algebraic sign, their dependence on public debt displays a Ushaped curve. In the case of potential surpluses, when the balance becomes positive, the resulting curve is an inverted U shape.
Obviously, such a statistical approximation must be made with caution because the maximum bearable level of public debt is conditioned by numerous internal and external circumstances—from natural and accumulated endowment, active population, performance of the institutional framework, and general development degree of the given economy up to the current and predictable regional or global geostrategic context. Additionally, independent from the econometric estimations, such ceiling parameters can be officially adopted, either at the national (United States Public Debt Act) or international (the Maastricht Treaty criteria in European Union) levels.
If extrastatistical reasons or unpredictable factors induce lesser or greater levels of cbb, in comparison with the estimated cbbr, the difference (cbbo) must be established exogenously.
2.5 Additional relationships
To coagulate an integrated public budget model, the aforementioned functional trinity, namely, the BARS, LINS, BBR curves, must be supplemented with some auxiliary relationships.
BARS function (5) contains the factor IGFCFc(− 1). For a static simulation, this indicator could be considered as a given, but for a dynamic one such an assumption fails; therefore, it must be transformed into an endogenous variable.
2.6 Collinearity problem
The usual specifications of the BARS, LINS, and BBR curves involve a high correlation between series cbe, atax, pdg, and their quadratic forms. In our application, the Pearson–Galtung coefficients for the corresponding pairs are as follows: 0.99913 cbe–cbe^{2}, 0.99843 atax–atax^{2}, and 0.98579 pdg–pdg^{2}. As expected, in regressions, the estimators register a great variance inflation factor.
An extended literature has gravitated toward this problem. (Some synthetic overviews can be found in Wold et al. 1984; Belsley 1991; Dalal and Zickar 2012.) Among the sources of collinearity, the polynomial specification was especially examined (Bradley and Srivastava 1977; Shacham and Brauner 1997; Yu and Tempe 2000; Seber and Lee 2003; Piegorsch 2015; Helwig 2017).
Several methods have been advanced to identify and quantify the size (hence, the gravity) of such a phenomenon (Farrar and Glauber 1967; Kumar 1975; O’Hagan and McCabe 1975; Wichers 1975; Belsley 1991; Goldberger 1991; Gujarati 2003; Seber and Lee 2003; Paul 2006; O’Brien 2007; Robinson and Schumacker 2009; Wooldridge 2009; Baguley 2012; Dallal 2012b; Greene 2012; Dormann et al. 2013; Uriel 2013; Piegorsch 2015; Hansen 2018). The most used is probably the variance inflation factor (VIF).
Many studies have discussed the possible remedies for collinearity, preponderantly referring to the individual relationships.
“Do nothing” is one of them (Gujarati 2003; Allison 2012). A possible motivation could be that, as a rule, the collinearity does not alter the cumulated effect of the respective explanatory variables, or in a lot of applications, the principal goal is to obtain a computational instrument for extrasample estimations. Subtextually a constancy of the collinearity structure is presumed, which is uncertain or admissible only in the proximity of the used sample; otherwise, evidently, “do nothing” becomes—at least—questionable.
A respecification of the model (Farrar and Glauber 1967; Yu and Tempe 2000; Gujarati 2003; Paul 2006; O’Brien 2007; Baguley 2012; Greene 2012; Uriel 2013) is another advocated approach. If the objective and relevance of the research in cause allow such an operation, it would be categorically preferable. The simplest method is the amputation of redundant variables. However, to cut an essential variable only for the sake of collinearity could be too high a cognitive cost; this is the reason why modeling is compromised.

increase the sample when possible (Farrar and Glauber 1967; Yu and Tempe 2000; Gujarati 2003; Paul 2006; Baguley 2012; Greene 2012; Uriel 2013);

datatransforming procedures (Bradley and Srivastava 1977; Mandel 1985; Gujarati 2003; Seber and Lee 2003; Baguley 2012; Dallal 2012a; Uriel 2013; Piegorsch 2015; Iacobucci et al. 2016) as mean centering, standardized data, and replacement by ratios or logarithms;

subset selection (Hastie et al. 2008) to eliminate the data that are “impeachable” for collinearity.
The available a priori information usable for introducing constraints in the estimation procedure should not be neglected (Farrar and Glauber 1967; Gujarati 2003; Uriel 2013). The sources of such a possibility are multiple: the theoretical foundations of the model, the crosssectional sample, or the data provided by other empirical studies.
In this manner, the collinearity problem disappears.
A consistent package of technical solutions was developed that aimed to eliminate or attenuate the implications of collinearity. A brief and illustrative enumeration is available: the principal components analysis (Gujarati 2003; Paul 2006; Hastie et al. 2008; Chen 2009, 2012; Dallal 2012b; Greene 2012; Piegorsch 2015), partial least squares (Yu and Tempe 2000; Chen 2009, 2012), staged regression (Chen 2009, 2012), and the robust conjecture approach (Chen 2012).
In the literature, the utilization of the ridge regression has been extended (Hoerl and Kennard 1970; Mandel 1985; Gujarati 2003; Seber and Lee 2003; Paul 2006; O’Brien 2007; Hastie et al. 2008; Chen 2009, 2012; Younker 2012; Piegorsch 2015; van Wieringen 2018) and that of other similar constraints, such as the least absolute shrinkage and selection operator—Lasso (Seber and Lee 2003; Hastie et al. 2008; Younker 2012; Piegorsch 2015), and the least angle regression—LARS (Younker 2012). To interpret these procedures as a purely numerical operation with doubtful semantics would be tempting, but superficial. More reasonable, in our opinion, would be to admit that changes made in the covariance matrix correspond to some implicit sample modifications. A reexamination from such a viewpoint of the ridgetype techniques would be a promising scientific challenge.
In the case of the present application, we tried to avoid the socalled do nothing approach by trying to use the recommended methods for compressing VIF. Since the principal objective of analysis was a numerical approximation of the parameters defining Ushaped slopes (BARS, LINS, and BBR curves), the elimination of one from collinear explicative variables failed ab initio. Significance for our research a priori information was not found. The ridge regression proved irrelevant.
Estimations by a meancentering procedure compared with the original data
Estimator  Coeff.  SE  Prob.  Uncentr.VIF.  Coeff.  SE  Prob.  Uncentr.VIF 

Original data  Meancent. data  
BARS  
c(500)  0.59269  0.13466  2.00E−04  NA  
c(1)  3.89484  1.08741  0.0017  1831.267  − 0.982  0.51415  0.07  1.88359 
c(2)  − 6.1117  2.05932  0.0071  822.2038  19.3968  13.5041  0.166  1.66784 
c(3)  0.13639  0.06035  0.0341  52.11735  0.11111  0.05853  0.072  1.20984 
c(4)  0.29429  0.14137  0.0492  208.0542  0.33406  0.13519  0.022  1.32065 
LINS  
c(501)  0.60997  0.01506  0  NA  
c(5)  3.11878  0.25275  0  144.9425  − 0.5907  0.23838  0.022  2.26558 
c(6)  − 3.8849  0.62866  0  180.2087  9.84449  3.27783  0.007  1.54104 
c(7)  − 0.3209  0.19454  0.1132  6.385676  − 0.4657  0.15021  0.005  2.43814 
BBR  
c(502)  − 0.0323  0.00432  0  NA  
c(8)  − 0.2283  0.05239  0.0002  18.80119  − 0.0004  0.03858  0.993  1.13524 
c(9)  0.41726  0.15136  0.0112  18.82267  0.59931  0.25075  0.026  1.16495 
c(10)  0.00437  0.00508  0.3983  1.005027  0.00512  0.00516  0.332  1.02795 
Thus, in the case of the BARS and LINS curves, the algebraic signs became contrary to those theoretically expected (and attested by original series). An essential coefficient of a BBR curve is insignificant. Therefore, the meancentering procedure did not work.
In conclusion, the trials to ameliorate VIF by usual collinearity remedies failed, and this phenomenon raises an important question: Maybe the polynomial form of the explicative variables is congenitally improper for our application. Such a specification can—of course—be replaced, but it would be undesirable because the parabola has indisputable interpretative pluses.

the modification (from sample to sample) of separate estimators c(1) and c(2) and the ratios ρ;

the stability degree of these indicators within each set of samples and overall; the coefficient of variation (CV in module) is used for such a purpose.
Regarding c(1) therefore, the coefficients of variation oscillate around 8–10% for BARS and LINS curves, and between 22 and 25% for BBR one. The volatility is sensibly higher in the case of c(2) in all these functions.
To notice that the volatility of ρ ratios does not exceed the acceptable (from a practical viewpoint) borders and the sample sets differ very little among themselves.

Estimators c(1) and c(2) register as really high coefficients of variation for all the curves, which can occur either from the available data or the adopted functional form of regression;

by contrast, the ratios ρ are characterized by sensibly lower such coefficients overall and within each sample’s set.
The last conclusion is essential. If the ratios ρ are relatively stable, then the optimal points of the BARS, LINS, and BBR curves are again relatively stable for the samples (having of course, a similar informational structure as the primary series), or just these optimal points are of the greatest interest for macroeconomic analysis and simulations. In other words, the more stable the ratios ρ are, so much the less would be justified to abandon the polynomial specification. Even if we did not intend, the “do nothing” option seems reasonable in our application. Some authors have already outlined that if the high VIFs are caused by the inclusion of powers or products of other variables, the multicollinearity could be ignored (Allison 2012).
In the case of the functional trinity of government finance, another computational problem is crucial. The real socioeconomic processes reflected by the BARS, LINS, and BBR curves are not isolated. By contrast, they are profoundly interconnected. The following chapter estimates them as a system of equations. The collinearity question will again be evoked.
3 Results and discussion
Econometric Eqs. (5), (7), (10), (13), (14), and (15) (supplemented by the corresponding accounting identities, some dummy variables, and the Ar terms) were integrated into a unitary model. This operation was accomplished in two steps.
First, these equations were solved as such, without any additional constraint. As a starting point, the iterative least squares were applied (Additional file 3: Appendix CSys2018OLS1). Since this method does not account for the possible interequation correlations, a seemingly unrelated regression (Additional file 3: Appendix CSys2018SUR1) was subsequently used.
Estimations OLS1SUR1
Estimator  Iterative least squares  Seemingly unrelated regression  

Coefficient  t statistic  Prob.  Coefficient  t statistic  Prob.  
c(1)  3.894841  3.581765  0.0005  3.344425  3.419789  0.0008 
c(2)  − 6.11171  − 2.96783  0.0036  − 5.15028  − 2.78416  0.0062 
c(3)  0.136385  2.260074  0.0255  0.14171  2.630572  0.0096 
c(4)  0.294288  2.081649  0.0394  0.369918  2.904524  0.0044 
c(5)  2.865476  11.7154  0  2.798618  12.87796  0 
c(6)  − 3.30885  − 5.51001  0  − 3.1713  − 5.96151  0 
c(7)  − 0.39527  − 2.25594  0.0258  − 0.372  − 2.43269  0.0164 
c(8)  − 0.16799  − 4.3002  0  − 0.17296  − 6.09464  0 
c(9)  0.266685  2.39636  0.018  0.283238  3.710497  0.0003 
c(10)  0.073309  2.603768  0.0103  0.056011  2.779252  0.0063 
c(11)  0.317634  3.663902  0.0004  0.360568  5.249144  0 
c(12)  0.604139  5.234602  0  0.556234  6.127794  0 
c(13)  − 0.51399  − 3.34185  0.0011  − 0.58888  − 4.78107  0 
c(14)  1.557579  23.31209  0  1.571832  32.80355  0 
c(15)  − 0.57783  − 8.59337  0  − 0.5922  − 12.3275  0 
c(16)  − 0.46212  − 8.21488  0  − 0.46603  − 11.6207  0 
c(17)  0.036793  4.726899  0  0.037518  6.377313  0 
c(18)  − 0.86622  − 5.60487  0  − 0.88646  − 8.23769  0 
c(19)  0.132421  3.88641  0.0002  0.118528  5.414735  0 
Although the BARS, LINS, and BBR relationships are gathered into a unique system, they are not explicitly interconnected. The second step of the present computational approach addresses this problem. The general equilibrium theory is useful in such a matter. If an integrated system has several optimizing entities, a structural compatibility is ultimately achieved among them. Why not consider the public finance blocks similarly? Since these blocks constitute a unitary system, rejecting the supposition that their dynamic trajectories are also interdependent would be difficult. More specifically, it would be implausible to assert that—for a given structural state–society relationship—the dominant social preference concerning taxation would have no link to the dominant social choice for public expenditures.
This reasoning can be extended to budget balance, significantly influenced by the optimizing behavior of the potential lenders.
This expression could be considered a “compatibility restriction” of the public budget model. Notably, (16) is not a simply computational artifice; by contrast, it comes from consistent socioeconomic reasons. Such a condition restrains the range of “estimating ambiguity” inducible by collinearity. Certainly, this solution requires further research, which would be justified not only for this discussed application but also for other economic problems approachable in a similar manner.
Estimations SUR2
Estimator  Coefficient  t statistic  Prob.  Estimator  Coefficient  t statistic  Prob. 

c(1)  2.381749  3.837335  0.0002  c(11)  0.362474  5.268048  0 
c(2)  − 3.12049  − 3.29485  0.0012  c(12)  0.553518  6.1059  0 
c(3)  0.161709  3.167595  0.0019  c(13)  − 0.59205  − 4.78954  0 
c(4)  0.442898  3.877219  0.0002  c(14)  1.572393  33.14472  0 
c(5)  2.87114  13.78983  0  c(15)  − 0.59283  − 12.469  0 
c(6)  − 3.35769  − 6.63002  0  c(16)  − 0.4648  − 11.7486  0 
c(7)  − 0.33017  − 2.24893  0.026  c(17)  0.037762  6.314574  0 
c(8)  − 0.16583  − 5.62014  0  c(18)  − 0.88956  − 8.04251  0 
c(9)  0.26444  3.313027  0.0012  c(19)  0.122241  5.380544  0 
c(10)  0.05932  2.829249  0.0053 
In the static framework, the BARS, LINS, and BBR curves are calculated for a given temporal interval, changing their determinants within a large economically plausible range. The other explicative variables were fixed (maintained at the basic year level). The used estimators were provided by OLS1, SUR1, and SUR2.
The modification of the IGDPc depends on the cbe configures; therefore, there is a clear inverted Ushaped BARS curve. Related to public spending and constancy of the investment influence, the economic growth reaches the maximum (1.053378) for cbe = 0.38 in the proximity of ocbe (= 0.381631). Notably, such a level of cbe is higher than the actual one (0.338161 in 2016).
Again, an inverted Ushaped curve is obtained, with the maximum (= 0.601526) for atax = 0.43 (oatax = 0.427548), compared with 0.3869 in 2016.
As expected, this time the simulations reveal a Ushaped curve. The maximal public budget deficit as ratio to the GDP (− 0.02819) is slightly lower than the Maastricht recommendation (− 0.03).
In conclusion, the static simulations confirm the presence of the BARS, LINS, and BBR curves in the statistical database.
Undoubtedly, revealing how the model reacts as an integrated dynamic system to modifications in taxation and relative public debt would be of interest. Are the BARS, LINS, and BBR functions still valid under the intertemporal effect of the involved model factors? With this aim, the system was solved for ten postsample successive intervals, and a horizon long enough to identify the dynamic properties of the model was observed. These intervals are labeled posts_{i}, where i = 0 for the basic year, and 1…10 for the subsequent years. Simulations are operated using the estimators obtained from the SUR2 variant.
As a type of inertial scenario, the first application computes the system, starting with the database for 2016 as a basic year (i = 0). To avoid the impression of an authentic forecast, the subsequent intervals are noted as 1, 2,…, 10.
Inertial scenario
Interval  IGDPc  letg  cbb  cbe  IGFCFc  cnfbr  pdg 

Basic  1.047964  0.63233  − 0.02742  0.338161  1.0626  0.038667  0.413896 
posts1  1.048035  0.596436  − 0.02333  0.33113  1.146855  0.042033  0.437231 
posts2  1.060107  0.596842  − 0.01991  0.324878  1.124344  0.039052  0.457137 
posts3  1.054851  0.597222  − 0.02054  0.324569  1.109352  0.037958  0.477681 
posts4  1.052763  0.597577  − 0.02079  0.326235  1.070691  0.039244  0.498468 
posts5  1.047515  0.59791  − 0.01695  0.32532  1.057708  0.042033  0.515422 
posts6  1.045492  0.598222  − 0.01342  0.324528  1.069497  0.044652  0.528844 
posts7  1.04749  0.598517  − 0.01374  0.325935  1.074827  0.045628  0.542584 
posts8  1.049199  0.598794  − 0.01382  0.324997  1.057347  0.044502  0.556403 
posts9  1.046376  0.599056  − 0.0104  0.31921  1.050474  0.042033  0.566804 
posts10  1.043429  0.599304  − 0.00743  0.314002  1.063473  0.039697  0.574236 
The inertial scenario is useful as a referential or experimental subject for other applications; however, as such, it does not allow the identification of Ushaped trajectories appropriate for the main functional relationships of government finance. More suited for such a purpose are the simulations involving a significant modification of the main determinants of the model.
The dynamic simulation reconfirms the LINS curve.
Consequently, the BARS curve is also present in the dynamic simulations.
The dynamic simulations also attest the BBR curve.
Obviously, the present exercises cannot be considered as practical recommendations, although such suggestions emerge. Our only goal was to illustrate the behavioral characteristics of the discussed model. Its operationalization would require transformations; among them, the explicit representation of the interconnections among the BARS, LINS, and BBR curves with other macroeconomic dependencies is essential.
4 Conclusions
 1.
the impact of public expenditures on general human development and, in particular, economic growth;
 2.
the volume of taxes collected; and
 3.
the degree of government accessibility to borrowing additional resources when its spending exceeds the current public budget revenues.
 1.
the global historical trajectory of modern society, concerning which the Wagner–Kaldor theorem continues to have a leading explicative position;
 2.
the given structural state–society relationship as a component of the larger notion “given social context” or “given historical context”; and
 3.
the shortmedium horizon, where the Keynesian and postKeynesian framework remains relevant.
Each of these perspectives is characterized by specific requirements concerning the modeling work. Contradictory results provided in the empirical literature are explainable, at least partially, by the utilization of inadequate methodological instruments (theoretical hypotheses and functional specifications) on the available samples.
Our research focused on the second temporal scale, for which analysis—in mirror with the enounced functional triangle—the BARS, LINS, and BBR, curves were considered adequate. In addition to the literature that examined these relationships separately, this paper attempts to integrate them into a unitary system.
This paper assumes that under a given structural state–society relationship, the social propensity for government spending, tax acceptability, and risk aversion to public indebtedness move—more or less transparently—toward steadystate levels approximated by the optimal points of the BARS, LINS, and BBR as Ushaped (as such or inverted) curves.
In accordance with the general equilibrium paradigm, such points cannot be viewed separately because they are inherently interdependent. It seems natural to presume, for example, that the dominant social preferences concerning taxation are relatively concordant with the dominant social choices for public expenditures. Obviously, this reasoning can be extended to a balanced budget, which is significantly influenced by risk aversion to public indebtedness and optimizing behavior of potential lenders.
Consequently, the model specification was completed with an equation interlinking the optimal points of the BARS, LINS, and BBR functions, and this was admitted as a “compatibility restriction” of the system.
By using the database for an emergent mediumsized EU economy, the econometric analysis was finalized by model simulations in static and dynamic conditions. In static conditions, the resulting BARS, LINS, and BBR curves were determined for a given temporal interval, changing only their determinants of interest (with the other explicative variables remaining at their basic year level). In the second case, the entire system was solved for ten postsample successive intervals, a horizon long enough to identify the dynamic properties (including the intertemporal effects) of all the involved indicators. Both types of simulations revealed the presence of the BARS, LINS, and BBR curves.
Certain crucial issues require, however, additional clarification. A deepened characterization of the given structural state–society relationship requires further multidisciplinary research from both the qualitative socioeconomic and quantitative statistical perspectives. A rigorous mathematical foundation of the “compatibility restriction” is also of great interest. Notably, the inclusion in the model of the possible variability of the optimal points of BARS, LINS, and BBR curves would be especially opportune under the present intensity of technological, demographic, and institutional changes.
Notes
Authors’ contributions
The author read and approved the final manuscript.
Acknowledgements
The author thanks D. Jula, M. Matei, B. Păuna, C. Stănică for their help.
Competing interests
The author declares that he/she has no competing interests.
Availability of data and materials
Please see Additional files (Appendices) for data.
Funding
The author declares that there is not any funding for this research.
Publisher’s Note
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Supplementary material
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