The effect of the behavior of an average consumer on the public debt dynamics
Roberto De Luca, Marco Di Mauro, Angelo Falzarano, Adele Naddeo
aa r X i v : . [ q -f i n . E C ] A p r The effect of the behavior of an average consumer on the public debt dynamics
Roberto De Luca , Marco Di Mauro , Angelo Falzarano , and Adele Naddeo Dipartimento di Fisica E.R.Caianiello, Universit´a di Salerno, Fisciano (SA) - 84084, Italy Dipartimento di Scienze Economiche e Statistiche,Universit´a di Napoli Federico II, Napoli - 80126, Italy and INFN Sezione di Napoli, Napoli - 80126, Italy
An important issue within the present economic crisis is understanding the dynamics of thepublic debt of a given country, and how the behavior of average consumers and tax payers in thatcountry affects it. Starting from a model of the average consumer behavior introduced earlier by theauthors, we propose a simple model to quantitatively address this issue. The model is then studiedand analytically solved under some reasonable simplifying assumptions. In this way we obtain acondition under which the public debt steadily decreases.
PACS numbers: 89.65Gh
I. INTRODUCTION
Understanding consumption is a key issue in macroe-conomics. The links between individual consumption de-cisions and outcomes for the whole economy have beendeeply investigated by A. Deaton [1], who was awardedthe Nobel prize in 2015 [2]. In fact, since the begin-ning, research focused on the relation between income(and the interest rate) and consumption. Within thisframework, Friedman’s permanent income hypothesis [3]and Modigliani’s life-cycle theory [4–6] contributed tosuggest how income variables might enter the consump-tion function. In particular life-cycle theory provides auseful framework for thinking about saving and relatedmechanism, and paves the way for undestanding the linkbetween saving and growth of a country. Furthermore,the theory predicts that saving rates are higher in morerapidly growing economies, even in the short run, there-fore establishing a correlation between growth and sav-ings rates. However, more recent studies pointed outthat the range of applicability of the above mechanism isquite limited [7–10], so that it alone does not succeed inexplaining the international correlation between savingand economic growth.Macroeconomists and policy-makers have traditionallybeen concerned with the issue of the sustainability ofpublic debt in developing and emerging market countries.However, starting from the appearance of the global fi-nancial crisis, the attention has been shifted to developedeconomies, which suffer from rising debt-Gross DomesticProduct (GDP) ratios in the face of stagnant or con-tracting output. As a matter of fact, in most Europeancountries debt is at an unprecedented level in the lastfifty years. In some cases, the increases since 2007 haveexceeded 20 percentage points of GDP while the level ofthe debt was already high [11].The correct definition and measurement of public debtwere given by [12, 13]. The relation between public debtand economic growth is the object of much work today[14–19]. If debt growth rates are lower than those ofGDP, the debt is not bad. The economic growth restoresthe income part of the budget, which is used to pay inter- est on debt. Vice versa, with low economic growth rates,public debt becomes a serious macroeconomic problemfor the country. In such a case governments have to paya large amount of public revenue to the creditors, whichresults in fewer resources for education and public in-vestments. An unsustainable level of debt is obtained incorrespondence to a high debt to GDP ratio [18]. In par-ticular, interest rates rise on the basis of a greater prob-ability of a default on debt obligations, with a negativeinfluence on private investment and consumption. Thedebt-relief Laffer curve [20] provides information aboutthe negative impact of high debt on output growth; inparticular a point can be found, where outstanding debtis so large that output growth gets reduced and the prob-ability of debt repayment lowers. In this scenario, infla-tion could be a viable solution to the government debtproblem [21, 22], while the effectiveness of fiscal consoli-dation and of austerity requirements, such as the reduc-tion in governments expenditure, is widely debated (seee.g. [11] and references therein).Actually, new provisions in the Stability and GrowthPact (SGP) require European countries with a debt toGDP ratio higher than 60 per cent to act to decrease itin the next few years. It is thus important to assess underwhich conditions this is possible. This is the purpose ofthe present paper, in which a simple model is proposedto try to find such conditions.Since the model uses the representative agent frame-work, it is subject to all its limitations, as it does not takeinto account the differences between individuals. The useof the representative agent in economy has been criticizedmainly in [23]. Progress in providing alternatives to thead hoc assumptions of the representative agent approachhas recently been made by using the tools of StatisticalMechanics (see e.g. [24] and references therein).Another related limiting aspect of the present analysisconcerns the absence of private enterprises in this sce-nario. In fact, the presence of this type of agent cangenerate earnings that can substantially help in payingback public debt. However, if we assume that the eco-nomic behavior of private enterprises can be included inthe representative agent picture, the present model couldbe still adopted in this more general setting.In a previous paper [25] the authors proposed a sim-ple mathematical model, based on a hydrodynamic ana-log, to quantitatively describe the time evolution of theamount of money an average consumer decides to spend,depending on his/her available budget. In the presentpaper we couple this model, or better, its difference equa-tion version, to the standard equation for the time evo-lution of the public debt given in ref. [12]. The resultingdynamical model is then analytically solved under somereasonable simplifying assumptions. The result is a rangeof the parameters of the model for which the public debtsteadily decreases. The model is presented in Section 2,and its dynamics is studied in Section 3. Finally, in Sec-tion 4, some conclusions are drawn, and some commentsand perspectives of our work are outlined.
II. THE MODEL
The typical strategy adopted by a Government to de-crease its public debt is to burden the average consumerwith additional taxation, which may also be applied tothe amount of money present, at a certain date, on theconsumer’s bank account. Let us write the tax payed bya single average consumer in the k-th year by τ k = αp a + βδ km b k + γc k , (1)where p a is the yearly income, which we assume fixed, b k is the average yearly bank deposit, and c k is the yearlyexpenditure of the consumer. α , β and γ are the aver-age taxation rates for these quantities, respectively. TheKronecker delta δ km picks a fraction of the amount b k only in the m-th year.The dynamics of the public debt D k . calculated at theyear k and normalized to the number N of consumers, isdescribed by the difference equation [12] g k + rD k − = τ k + D k − D k − , (2)where D k − D k − is the yearly pro capite deficit at thetime k , r is the fixed rate of return on public and privatedebts, g k is the pro capite amount of money the Stateutilizes for funding education, health, welfare, and allpublic activities. Plugging (1) into (2) we get D k − (1 + r ) D k − = g k − αp a − βb k δ km − γc k (3)To describe the behavior of the consumer we may adoptthe leaking bucket model [25]: b k − b k − = y k − c k (4)where y k is the expendable part of the salary, i.e. y k = p a − τ k . Substituting this definition into (4), we get p a − ( αp a + βδ km b k + γc k ) − c k = b k − b k − (5) Substituting c k = ab k as in [25] we get(1 + γ ) ab k + (1 + βδ km ) b k − b k − = (1 − α ) p a . (6)This equation should of course be coupled to the publicdebt evolution equation (3). Let us now study the latterwith the simplifying assumption g k = g = const. andthe functional relation c k = ab k : D k − (1 + r ) D k − = ( g − αp a ) − βb k δ km − γab k (7)or D k = (1 + r ) D k − + ∆ k (8)where we defined ∆ k = ( g − αp a ) − βb k δ km − γab k . Thesolution of this difference equation is D k = (1 + r ) k " D + k X i =1 ∆ i (1 + r ) i (9)The sum in the above equation is S k = k X i =1 ∆ i (1 + r ) i (10)= ( g − αp a ) (1 + r ) k − r (1 + r ) k − γa k X i =1 b i (1 + r ) i − βb m (1 + r ) m so that D k = (1 + r ) k (cid:20) D + ( g − αp a ) (1 + r ) k − r (1 + r ) k − γa k X i =1 b i (1 + r ) i − βb m (1 + r ) m (11)In the above two equations it is understood that the lastterm is present only if k ≥ m . We notice that the firstequality in eq. (10) is similar to the formula for the priceof a coupon bond [26].Therefore, following this model, we can argue that theGovernment choice of the values of α , β , γ and m affectsboth the average consumer’s budget according to Eq. (6),and the public debt evolution, as specified in (11). III. MAXIMUM STATE EXPENSE IN PAYINGBACK PUBLIC DEBT
In this section we study the dynamics governed by Eqs.(6) and (11) under some specific simplifying assumptions.First of all, we exclude the possibility of direct taxationof the amount b k , so that β = 0. Also, we take α = γ , we assume that all consumers have no debts, i.e. a positivebudget, cfr. [25] since in many countries these rates are in fact quite close.In this way, Eqs. (6) and (11) get rewritten respectivelyas (1 + α ) ab k + b k − b k − = (1 − α ) p a (12)and D k = (1 + r ) k D + ( g − αp a ) (1 + r ) k − r (13) − αa (1 + r ) k k X n =1 b n (1 + r ) . In Fig.1 we show the dynamics of the consumer budgetdescribed by Eq.(12), where we set a = 0 .
15 and p a =100, and we assume that α = 0 .
25, as is reasonable foran average consumer in a european country. We noticethat the fixed point b λ = 20 is reached after few steps. kb H k L FIG. 1: Consumer’s budget dynamics described by Eq.(12)for α = 0 . a = 0 .
15 and p a = 100, for the initial conditions b = 18 (continuous line), b = 20 (dotted line) and b = 22(dashed line). Let us find the fixed point of Eq.(12). The condition b λ +1 = b λ gives b λ = r − α α p a a (14)Assuming that the average consumer does indeed startwith an initial budget close to b λ , we may rewrite Eq.(13) as follows D k = (1 + r ) k D + ( g − αp a ) (1 + r ) k − r − αa (1 + r ) k b λ k X n =1 r ) n = (1 + r ) k D + (cid:18) g − αp a α (cid:19) (1 + r ) k − r (15)The public debt starts decreasing if there is some value of k such that D k +1 < D k . This implies the k − independentrelation p a > α α ( rD + g ) . (16) If this condition is met, then, a k − independent decreaseof the public debt is possible, therefore reducing the debtto GDP ratio, as required by SGP. For example, in acountry where α = 1 / r = 1 /
20, Eq. (16) reads p a > (cid:18) D
20 + g (cid:19) . (17)If D / ≪ g , we have p a > . g , which indicates thatthe value of g cannot be higher of (2 / p a , in order tohave decreasing values of D k for k > IV. THE R ˆOLE OF THE PUBLICEXPENDITURE
In this section we consider our model under some moregeneral assumptions. In particular we relax the assump-tion that the public expenditure g k is constant. Moreoverwe consider a more general income-expenditure relationfor the consumer, i.e. c k = a n b nk , n ≥ , (18)where we excluded a linear behavior in order to modelthe existence of a threshold. In this case the stable fixedpoint is given by b λ = n r − α α p a a . (19)Assuming as above that the consumer is in proximity ofthis fixed point we get, in place of Eq. (15): D k = (1 + r ) k D − α (1 + α ) r p a (cid:2) (1 + r ) k − (cid:3) +(1 + r ) k k X i =1 g i (1 + r ) i . (20)Notice that the assumption of being at the fixed pointhas again erased the dependence on the coefficient a n .The condition D k +1 < D k in this case gives2 αp a α > g + rD + k − X j =1 g j +1 − g j (1 + r ) j , (21)where g is the expenditure at year 1. Let us now assumethat the expenditure in the subsequent years has the form g j = ( j − G + g (22)where g >
0. In this way g j +1 − g j = ∆ G , so thatthe sign of ∆ G determines wether the public expendituregrows or decreases over the years. The condition (23)then specializes to2 αp a α > g + rD + ∆ G (1 + r ) k − − r (1 + r ) k − , (23)where k ≥
1. For | ∆ G | ≪ g the last term in the rhsis negligible, therefore our conclusions of the previoussection are still valid. V. DISCUSSION AND CONCLUSIONS
In this paper, building on a simple quantitative de-scription of the behavior of an average consumer over abrief period [25], and on Barro’s theory of the public debt[12], we propose a model for the description of the influ-ence of the former on the dynamics of the public debt,with the aim of establishing the condition for its decrease.Our result are clearly limited in scope by the use of theconcept of average consumer and by the fact that it isvalid only over a short period. We obtained analyticalresults under the simplifying assumptions of equal aver-age taxation rates for the yearly income and expenditureof the consumer, respectively. Furthermore the possibil-ity of direct taxation of consumer’s average yearly bankdeposit has been excluded. Besides extending the model to overcome its shortcomings, it would be interesting tofurther relax the above simplifying assumptions and seewhen and how the condition we found gets enhanced.An interesting issue to investigate is how the aboveresults would get modified by generalizing the leakingbucket model for the average consumer behavior [25], insuch a way to include the effect of different goods on thelevel of consumer demand.Another interesting generalization of the present modelcould be the inclusion of private enterprises as a differentagent in the economic landscape.The proposed model could finally be rephrased interms of the debt to GDP ratio in order to study theimpact of fiscal consolidations on public debt dynamicsand how it is related to fiscal multipliers [11].
Author contribution statement
All the authors contributed equally to the paper. [1] A. Deaton,
Understanding Consumption , ClarendonPress, Oxford, 1992.[2] [3] M. Friedman,
A theory of the consumption function ,Princeton University Press, Princeton, 1957.[4] F. Modigliani,
Social Research , (1966) 160.[5] F. Modigliani, in: Induction, growth and trade: Essaysin Honour of Sir Roy Harrod , W. A. Eltis et al. (Eds.),Clarendon Press, London, 1970.[6] F. Modigliani,
American Economic Review (1986),297.[7] C. D. Carroll, L. H. Summers, in: National saving andeconomic performance , B. D. Bernheim and J. B. Shoven(Eds.), Chicago University Press, Chicago, 1991.[8] B. Bosworth, G. Burtless, G. Sabelhaus,
Brookings Pa-pers on Economic Activity (1991), 183.[9] C. Paxson, European Economic Review (1996), 255.[10] A. Deaton, C. Paxson, Review of Economics and Statis-tics (2000), 212.[11] J. Boussard, F. de Castro, M. Salto, Fiscal Multipliersand Public Debt Dynamics in Consolidations , EuropeanEconomy - Economic Papers 2008-2015, European Com-mission, n. 460 (2012).[12] R. J. Barro,
Journal of Political Economy , (1979) 940.[13] J. De Haan, De Economist (1987), 367.[14] A. Greiner, B. Fincke,
Public Debt, Sustainability andEconomic Growth: Theory and Empirics , Springer(2015). [15] T. Herndon, M. Ash, R. Pollin,
Camb. Jour. Economics (2014), 257.[16] U. Panizza, A. F. Presbitero, Swiss Jour. Econ. Stat. , (2013), 175.[17] D. J. Smyth, Y. Hsing, Contemp. Econ. Policy (1995),51.[18] A. Nastansky, H. G. Strohe, Econometrics (2015),9.[19] A. Caruso, L. Reichlin, G. Ricco,
The Legacy Debtand the Joint Path of Public Deficit and Debt in theEuro Area , European Economy Discussion Papers, n. 010(2015).[20] D. Miles, A. Scott, F. Breedon,
Macroeconomics: Under-standing the Global Economy , Wiley (2012).[21] C. M. Reinhart, K. S. Rogoff,
This Time is Different:Eight Centuries of Financial Folly , Princeton UniversityPress, Princeton, 2009.[22] C. M. Reinhart, M. Sbrancia,
The liquidation of govern-ment debt , NBER Working Paper Series, n. 16893 (2011).[23] A. P. Kirman,
J.Econ.Persp. (1992), 117.[24] A. De Martino, M. Marsili, J. Phys. A: Math. Gen. (2006), R465.[25] R. De Luca, M. Di Mauro, A. Falzarano and A. Naddeo, Eur. Phys. J.
B 89 (2016), 184.[26] B. E. Baaquie,