There is a long period since the problem of public debt sustainability captures the attention of economists. However, there is no unanimity concerning an adequate unique sustainability indicator or function generally accepted. Just in this line of elaborating new models and improving methodologies in order to quantify the impact of various factors on public debt sustainability is our paper. Moreover, last years, during its pre- and post-accession into EU period, Romanian economy is facing to numerous problems. Among these, the public debt sustainability plays a central role, its implications practically expanding on all fields connected to the economic dynamics.public debt, sustainability function, contour plot, primary deficit, interest rate

Albu, Lucian-Liviu

English

Research Papers in Economics

A Simulation Model of Public Debt Sustainability 
 
Lucian-Liviu Albu 
(Institute for Economic Forecasting, Bucharest) 
 
 
 
Abstract 
 
There is a long period since the problem of public debt sustainability captures the attention of 
economists. However, there is no unanimity concerning an adequate unique sustainability 
indicator or function generally accepted. Just in this line of elaborating new models and 
improving methodologies in order to quantify the impact of various factors on public debt 
sustainability is our paper. Moreover, last years, during its pre- and post-accession into EU 
period, Romanian economy is facing to numerous problems. Among these, the public debt 
sustainability plays a central role, its implications practically expanding on all fields 
connected to the economic dynamics. 
 
Key words: public debt, sustainability function, contour plot, primary deficit, interest rate 
JEL Codes: C15, C51, H63, H68 
 
 
 
1. Public debt equations 
 
To quantify the dynamics of public sector debt, often it starts from the well-known 
definition relation of the government’s budget constraint. So, the change in public 
sector debt, D, between two periods (years) t and t-1, is given by the following 
relation: 
 
D t  –  D t – 1  =  i t D t – 1  +  Π t  +  a t D t – 1  –  ∆B t      ( 1 )  
 
where i is the average nominal interest rate on public sector debt, Π is the primary 
deficit (net of interest payments), a is the revaluation effect on existing debt (in 
Romania this was supposed to be integrally due to the depreciation of ROL) and ∆B is 
the direct financing of budget from the Central Bank. In order to estimate parameters i 
and a, we also used the following relations: it = Dbt / Dt-1, where Db is the effective 
interest paid on public debt, and respectively at = (Dt / Dt-1) [1 - (CSt-1 / CSt)], CS 
being the exchange rate (ROL/USD or ROL/EUR) at the end of year (for details see 
Albu, 2002). 
 
Dividing both sides of equation (1) by nominal GDP, Yt, and manipulating we obtain: 
 
d t  –  d t – 1  =  ( i t  +  a t  –  g t )  [ d t – 1 / ( 1  +  g t ) ]  +  π t  –  b t    (2) 
 
  1where dt and dt–1 are the public sector debt to GDP ratio in two consecutive years, t 
and t –1, π is the primary public sector deficit as a percent of GDP, g is the nominal 
GDP growth rate between years t and t-1 and b is ∆B/Y. Alternatively we can 
approximate the nominal growth rate g as the sum of the change in GDP deflator p 
and the real GDP growth rate q and rewrite equation (2) as follows: 
 
d t  –  d t – 1  =  ( is t  –  q t)  [ d t – 1 / ( 1  +  g t ) ]  +  π t  –  b t     (3) 
 
where is could be defined as the real effective average interest rate on public sector 
debt (it is equal to the average real interest rate, i-p, plus the revaluation effect, a). 
To see what the dynamics of debt accumulation involves, we can solve equation (3) 
recursively to obtain 
 
d T
  =  d 0  v 
T  +  Σ (π 
m  –  b m ) v 
T  – 
 m     (m = 1, 2, …, T)      (4) 
 
where v = (1 + is + p) / (1 + q + p), while it has been assumed, in order to simplify 
calculations, that the real effective interest rate, is, the real growth rate, q, and the 
change in the GDP deflator, p, are constant: ist = is, qt = q , pt = p. Using equation (4) 
we can predict the debt ratio to GDP ratio for some future moments T, making 
assumptions about the relevant parameters. A high real growth rate relative to the 
effective real interest rate tends to reduce the debt to GDP ratio, d, while persistent 
primary deficits net of (real) Central Bank financing tend to increase it. 
 
Actions in Romania to diminish inflation in order to stabilise economy and to achieve 
the conditions to be accepted in future period within European Monetary Union, 
restricts its ability to increase the direct financing of budget deficits by Central Bank, 
while it also implies that (real) interest rates will have to tend to European levels. A 
further safe, and quite helpful – regarding calculations – assumption to make is that 
the growth rate q will be equal to the average effective real interest rate, is, on public 
debt. It has also a theoretical reason: it corresponds to the “golden rule of 
accumulation” of optimum growth theory. In fact, it is speaking about the approach of 
accumulating debt problem by using equations with finite differences in conditions of 
indeterminacy in the case g = i, while the method that we shall use in next section of 
paper, starting from equation (3) and solving it recursively to obtain equation (4), 
avoids it (OECD, 1989). Under the assumption q = is, equation (4) could be written as 
follows: 
 
d T
  =  d 0  +  Σ (π 
m  –  b m )
              ( 5 )  
 
In fact, dT will always tend to infinity for a very large T, unless the “average” future primary 
deficit is zero. An interesting, and empirically appealing, case arises, when the primary deficit 
is positive but declining. It can be shown (using the so-called d’Alambert’s theorem on the 
convergence of infinite series) that dT will converge to a finite limit for a very large T, if the 
primary deficit, π-b, is declining at a constant rate. If q>is, it can be shown from equation (4) 
that dT will always be bounded, provided that primary deficits remain bounded. In a special 
case, in which the primary deficit, π-b, is constant, dT will converge to  (π-b) / (1-v) for a very 
  2large T. It should be noted, however, that this limit would be a very large one (and may not be 
practically sustained). Finally, if q<is, the debt to GDP ratio increases without limit (this is the 
so-called Domar’s law). 
 
 
2. Dynamics of debt in long term and sustainability function 
 
One of the most important results of our investigation is the so-called sustainability 
function, f(π, b, is, q, p, d), which must tend to zero in dynamics (or at least to a very 
small constant value), as a fundamental condition for sustainability: 
 
f1 (π, b, is, q, p, d) = [ ( π  -  b ) / d ] +  ( is  -  q ) / ( 1 + p + q )        (6) 
 
or 
 
f2 (π, b, is, q, p, d) = [ (π  -  b ) / d ] +  ( is  -  q ) / ( 1 + p + q + pq)       (7) 
 
The computed values of function f during last fifteen years demonstrate that the 
sustainability function entered an oscillating regime, its values being situated around 
equilibrium. However, some risks of new jumps in the future continue to exist. That is 
why we shall concentrate on the behaviour of terms in sustainability function and 
implicitly on virtuous or vicious circle that could occur in the economic dynamics. 
 
First term of sustainability function express the impact of the direct governmental 
policies (budgetary policies) and respectively those of central monetary authorities 
(monetary policies). Second term, expressed by the ratio (is-q)/(1+p+q), or more 
precisely by (is-q)/(1+p+q+pq), describes the behaviour of the real economy. 
 
To study the behaviour of real economy, we used two partial models in order to 
simulate the following correlations: investment rate – growth rate and respectively 
investment rate – investment efficiency. The main hypotheses on which are based the 
models are referring to the existence of a direct positive correlation either between 
investment rate (α) and GDP growth rate (q), on the one hand, and between 
investment rate and its efficiency (η), on the other hand. 
 
Other hypothesis is that, at limit, in case of an investment efficiency equal to the 
interest rate (noted by i or is), the investment process is stopped, i.e. α = 0 (in this 
limit-case, the economic agents will be stimulated to place their savings in bank, 
economic investment as an alternative ensuring no supplementary money return). 
 
In order to illustrate the mentioned hypotheses, in Figure 1 there is presented the 
output of simulation in case of the two partial theoretic models, their parameters being 
estimated on data covering last fifteen years of (1993-2007) and conforming to the 
following equations of regression: 
 
qest t   = a α t–1    +     b             ( 8 )  
 
  3ηest t   = c α t–1  +  is t–1                         (9) 
 
where a, b, and c are estimated coefficients. In case of estimated efficiency (ηest) we 
considered the definition relation of efficiency in real terms, ηt = ∆Yd / I t-1 = (Yd t - 
Ydt-1) / It-1 (where Yd is the disposable income in private sector and households after 
the extraction of all taxes, Tx, i.e. Yd=Y-Tx, and I is investment) and the dynamics of 
prices as well as. On the theoretical graphs of the GDP growth rate and investment 
efficiency (qT and respectively ηT), corresponding to a hypothetic variation of the 
investment rate (αM) within 0-0.35, we noted also some significant values such as: 
the minimum level of investment rate under which the GDP growth rate becomes 
negative (αcr); the average investment rate for the considered period (αM) and 
respectively the average saving rate (αΕM); the theoretic efficiency corresponding to 
the average saving rate (ηTEM); and the average rate of interest on public debt 
computed implicitly on the “is” base (isM).  
 
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
0.05
0
0.05
0.1
b
0
qT( ) α
αcr αM
αT
 
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
0
0.5
1
1.5
isM
ηTEM
ηT( ) α
αEM αM
αT
 
 
Figure 1. 
 
 
  4In order to study the sustainability behaviour on the real side of economy, we 
combined the two partial models. After some algebraic operations and using the so-
called technique backward perfect foresight, we can write explicitly the interest rate 
function, R, as follows: 
 
R (q, tx, ∆tx) = [qa
2 (1-tx+∆tx) + ∆txa
2] / [-Kq
2 + K(a+2b)q - ab - Kb
2]      (10) 
 
where  
 
K = [(kE – 1 ) a] / (qE – b)                 (11) 
 
qE is GDP growth rate corresponding to the saving rate (used to replace the 
investment rate), according to the first partial model; kE - the ratio between the 
efficiency corresponding to the level of brute savings (used to replace the volume of 
brute investments) and the interest rate, according to the second partial model; 
tx=Tx/Y. Now, considering, by simplification reasons, ∆tx=0, qE=q and the following 
relation of kE: 
 
kE = 1 + [ (e α  c )   /   i s   ]         ( 1 2 )  
 
where e is the report between savings and investments (or in equilibrium case e=1) 
 
ke 1 + [ (α c) / is ]                            (13) 
 
we obtained the following expressions for the function of interest rate: 
 
R( ) , , qk Et x
. . qa
2 () 1t x
.
. () kE 1 a
qb
q
2 . .
. () kE 1 a
qb
() a . 2b q . ab .
. () kE 1 a
qb
b
2
              (14) 
 
Re( ) , , qk et x
. . qa
2 () 1t x
.
. () ke 1 a
qb
q
2 . .
. () ke 1 a
qb
() a . 2b q . ab .
. () ke 1 a
qb
b
2
        (15)
 
 
However, in line with the sustainability function, we are interested in the difference is-
q, noted this time as G and having the following two forms (the second one is in case 
of fulfilling the equilibrium condition between saving and investment): 
 
G( ) , , qk Et x
. . qa
2 () 1t x
.
. () kE 1 a
qb
q
2 . .
. () kE 1 a
qb
() a . 2b q . ab .
. () kE 1 a
qb
b
2
q
   (16) 
 
Ge( ) , , qk et x
. . qa
2 () 1t x
.
. () ke 1 a
qb
q
2 . .
. () ke 1 a
qb
() a . 2b q . ab .
. () ke 1 a
qb
b
2
q
      (17)
  
  5 
Graphical representation of simulations is shown in Figure 2. Some conclusions could 
be extracted from the simulation model, as follows: 
 
•  The optimum level for the sustainability function, G, is obtained for a growth 
rate, q, of 3.6%. 
•  In case of growth rates larger than 7% or less than 1.5-2% the sustainability is 
dramatically compromised. 
•  In case of interest function, the optimum level (minimum) is obtained for a 
growth rate, q, of 2.4%. 
•  In case of a growth rate of 7% the corresponding interest rate continues to be 
below 15%. 
 
Focussing more on the origin neighbourhood zone permitted a better specification of 
the structure of local map, local behaviour, and some of characteristic mechanisms 
that govern the dynamics of system, being this time plausible from a normal economic 
viewpoint. The conclusion is that the dynamics of the sustainability function, despite 
of the imposed simplifying hypotheses, demonstrates a vast complexity. The change 
of values for certain fundamental parameters till in the neighbourhood of some zones 
of turbulence, as those near the asymptotes, could attract the system to enter regimes 
of chaotic behaviour, i.e. non-predictable ones. From the viewpoint of the 
policymakers, these could be roads toward errors and uncontrolled measures, returns 
and abrupt changes, either in legislation or in practice, and could create a negative 
impact on long-run economic evolution. 
 
0.04 0.02 0 0.02 0.04 0.06 0.08
0.05
0
0.05
0.1
0.15
0.2
R( ) , , qcrR1 kE tx
R( ) , , qcrR2 kE tx
R( ) , , qk Et x
G( ) , , qk Et x
qcrR1 qcrR2
q
 
 
Figure 2. 
  6 
 
Relating to the question “how is the imagine of sustainability function?”, two 3D 
graphical representations and respectively two contour plot maps in case of f1 are 
shown in Figures 3 and 4. 
 
 
0
5
10
15
20
0
5
10
15
20
0
5
, , qpf 1  
0 5 10 15
20
0 5 10 15 20
0
2
4
6
8
, , qi sf 1
 
 
Figure 3. 
 
 
0.13 0.088 0.046 0.004 0.038 0.08
1
2
9
8
8
7
7
7
6
6
6
5
5
5
4
4
4
3
3
3
2
2
2
1
1
1
1
0
0
0
0
0
0
  1
  1
, , qpf 1  
0.13 0.088 0.046 0.004 0.038 0.08
0
10
20 8.5 8
8 8 7.5
7.5 7.5 7
7 7 6.5
6.5 6.5
6
6 6
5.5
5.5 5.5
5
5 5
4.5
4.5
4.5
4
4
4
3.5
3.5
3.5
3
3
3
2.5
2.5
2
2
2
1.5 1.5
, , qi sf 1  
 
Figure 4. 
 
 
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  8

A simulation model of public debt sustainability

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