Fiscal and Debt Sustainability of the State
5.1 Theoretical Framework for Examining Fiscal and Debt Sustainability
The issue of fiscal sustainability and solvency is usually addressed by analyzing the variables such as growth rate of GSDP, average interest rate on public debt and growth rate of public debt etc. The concept of solvency and sustainability are closely related in the sense that an unsustainable time path will ultimately threaten the solvency of a state (Rajaraman et al., 2005). The earlier statement of debt dynamics by Domar (1944) remains the simplest guide for the policy maker for fiscal and debt sustainability which states that “If the government finances a part of the expenditure (amounting to a given fraction of full employment output) through borrowings, in a growing economy, public debt and the government interest outgo as a proportion of GDP will be stable in the long run provided that growth rate exceeds the interest rate.” Subsequent restatements in terms of infinite horizon constraint on the present discounted value (PDV) of debt have not changed the fundamental Domar condition for stabilization of debt as a ratio to GDP (Rajaraman, 2005; Rakshit, 2005; Rath, 2005). Fiscal and debt sustainability of the state have been analysed with the help of the following equations as provided below.
According the Domar’s model for solvency of public debt, 0 PDtt
D - ...(i) (1+r)
=
∑
Here, D0 = Present stock of outstanding debt PDt = Primary deficit for the time period t r = interest rate on public debt
The above equation implies that for solvency, present outstanding stock of public debt must be equal to the summation of discounted primary surplus of future years expressed in terms of present value. Primary deficit incurred in a particular year can be expressed as,
t t t-1
PD = D -(1+r)D ...(ii)
Equation (ii) simply states that primary deficit plus interest on past debt (rDt-1) has to be financed by a built-up debt itself (Dt –Dt-1) (Lahiri and Kannon, 2004).
Now, equation (i) can be rewritten by replacing the value of PDt as obtained from equation (ii)
t t-1
0 t
D -(1+r)D
D - ...(iii) (1+r)
=
∑
Let us, consider that the variable Dt is growing at the rate of k, so that D = (1+k)Dt t-1
Replacing the value of Dt in equation (iii)
t-1 t-1
0 t
t-1
0 t
t-1
0 t
t-1 0
0 t
t-1
0 0
(1+k) D - (1+r)D D = -
(1+r) (k-r)D D = -
(1+r)
D = (r-k) D ...(iv) (1+r)
(1+k) D D =(r-k)
(1+r) (r-k) (1+k)
D = D ...(v)
(1+r) (1+r)
⇒
⇒
⇒
⇒
⇒
∑
∑
∑
∑
∑
D = 0 if r = k ...(vi)0
⇒
The above equation implies that for solvency of public debt, the interest rate on public debt must be equal to growth rate of public debt. Apart from solvency, it is also necessary to examine the conditions for sustainability of public debt.
To examine sustainability, the equation (ii) can be expressed as
t t-1 t
D = (1+r)D +PD ...(1) Dividing both sides by Yt
t t-1 t
t t t
t t-1 t
D (1+r)D PD
= ...(2)
Y Y Y
D 1+r D PD
= ...(3)
Y 1+g Y Y
+
⇒ +
Writing dt = t
D Y
as the debt-GSDP ratio and pd = t
t
PD Y
t t-1 t
d = 1+r d + pd ...(4) 1+g
⇒
Now, pdt can be assumed as pd as the ratio of primary deficit to GSDP is targeted to a constant value (Rath, 2005). Now, equation (4) can be rewritten as
t t-1
d = 1+r d + pd...(5) 1+g
⇒
Equation 5 is a first order difference equation. On solving the equation, it is found,
t
t 0
1+g 1+r 1+g
d = [d - pd] + pd...(6)
g-r 1+g g-r
dt tends to 1+g g-r PD
if and only if 1+r t
1+g
tends to zero as t tends to infinity.
This is possible if 0 < 1+r < 1
1
(1+r) < (1+g) r < g
g
+
⇒
⇒
i.e., interest rate on public debt must be less than the annual growth rate of GSDP. Domar model concludes that for solvency and sustainability of public debt, the following condition must be satisfied, i.e., growth rate of public debt (k) ≤ interest rate on public debt (r) <
growth rate of GSDP (g) when an economy is running by the accumulation of primary deficit.
It is also necessary to determine the conditions for sustainability of public debt when the rate of interest on public debt is greater than the growth rate of GSDP. For doing this, equation (4) can also be expressed as
t t-1 t
d = (r-g)d + pd ...(7)
From the above equation, it is evident that when r > g, for the sustainability of public debt, i.e., to keep dt = dt-1 or for achieving a stable constant debt-GSDP ratio for the future, there must be targeted primary surplus to GSDP ratio. This can be derived in the following manner:
t t-1 t
d = 1+r d + pd 1+g
t t-1 t
d = 1+r d - pd 1+g
if there is primary surplus ps = 1+r d - d
1+g
in static sense
ps = r-g d ...(8) 1+g
Therefore, when r > g, for an economy to achieve debt sustainability, the following conditions must be satisfied
r-g debt
ps = ...(9) 1+g GSDP
From equation (9), it is possible to determine amount of primary surplus required when r > g.
It is also necessary to determine the amount of fiscal deficit for debt sustainability. The sustainability condition can also be derived from the concept of fiscal deficit (Rajaraman et al. 2005). Fiscal deficit is nothing but total net borrowings of the government as given in equation 10 as produced below:
t t t-1
t t-1 t
(Fiscal Deficit) = D - D ...(10) D = D + (FD) ...(11)
⇒
Diving both side by Yt
t t-1 t
t t t
t t-1 t
t t-1 t
D D (FD)
= + ...(12)
Y Y Y
D D (FD)
= + ...(13)
Y (1+g)Y Y
D D FD
- = ...(14) in static sense
Y (1+g)Y Y
D 1 D
1- =
Y 1+g Y
⇒
⇒
⇒
⇒
g FD
= fd...(15) where fd =
1+g GSDP
It implies for debt sustainability;
Debt 1+g Fiscal Deficit
= ...(16)
GSDP g GSDP
The above equation gives the relationship between the fiscal deficit and debt-GSDP ratio. It tells about the amount of fiscal deficit an economy can incur with a given growth rate. To illustrate, if there is a 3 percent limit on the fiscal deficit as a percent of GSDP, following the Maastricht Treaty imposed by the consensus on EU member countries, and if g is 10 percent, d will stabilize at 33 percent of GSDP.
The above theoretical framework provides an idea about the crucial variables and their relationship which are used for studying the fiscal and debt sustainability of the state. It is found from the above discussion that deficit indicators have a significant impact on the sustainability of fiscal position of a state. The next section of the chapter is carried out to examine fiscal sustainability of the state with the help of the deficit indicators.