To guage the dynamic relationship between growth and debt, an intertemporal theoretical model is developed. The model is derived from (Zaidi, 1985; Greene and Khan, 1990; Raheem 1994) with adjustment to reflect a developing economy like Nigeria. The take-off point is to assume a neoclassical production function in the fashion of “augmented” Solow model as given below.
Q = f ( K L A ) ……… (3.3)
Where Q = output, and L = labour, K = capital, A = Exogenous productivity term
Equation 3.3 states that output is produced using capital and labour including a productivity
capital with positive productivity. Capital goods are mainly imported and some inputs are domestically produced through the investment process. The labour input is either skilled or unskilled labour. The supply of unskilled labour is not a constraint in developing countries given the huge unemployment pool, while the demand for most skilled labour is satisfied from
domestic sources.
The aggregate neoclassical production function could therefore be simplified to this form: Qt = a Kt ……… (3.4)
Where ‘a’ is the incremental capital-output ratio and Qt is output tied to the capital stock(Kt). The production function is well behaved and satisfying all the neoclassical properties in a scenerio of fixed real wages. A substantial pool of technologically unemployed workers permits this relationship to be maintained over time. By this assumption, the change in aggregate output is a constant fraction of the change in capital stock
Qt = a Kt ……… (3.5)
Where Qt denotes dQ/dt and the dot on a variable indicates the time rate of change of that variable. The law of motion of capital is represented by
Kt = It – ó Kt……… (3.6)
Where ó is the depreciation rate and It is investment or the flow of newly installed capital. According to the neoclassical accelarator investment theory, investment occurs to enlarge the stock of capital to produce more output. The decision to invest is to correct any discrepancy between desired capital stock and actual capital stock, thus:
It = Kd t – Kt-1 = A (Qt – Qt-1) ……. (3.7)
Where Kd t is desired capital stock and Kt-1 is actual capital stock in the previous year, “A” is the accelarator coefficient. Total investment at any given point in time is made up of public and private investments, Ig and Ip , respectively.
It = Ip + Ig …….. ……….(3.8)
The neoclassical growth framework put emphasis on savings accumulation as the prime mover of investment. Saving (S) can come from domestic or foreign sources( Sd and Sf , respectively); domestically sourced savings are of two types: private and public savings, Sp and Sg. Thus:
St = Sp + Sg + Sf …………. (3.9)
are conditioned by some determinants, such as the level of national output, growth of income and income distribution patterns, etc. Given the trend variation in private savings, the relationship between private savings and domestic income can be represented thus:
Sp = s0 + s1 u …….. (3.10)
Where u is the capacity utilisation in the economy and is defined as the ratio between actual output (Q) and potential output. The potential output will be the highest level of Q that can reasonably be produced with existing capacity.
Public savings are clearly determined by government current revenue in relation to government expenditure. The public sector earns income from taxes(and levies), T, and its investments, H; and spends it on various items of current goods and services, G, transfers and net subsidies, W as well as interest payment on internal debt, J. The differences constitute public savings, thus:
Sg = T + H + G – W – J (3.11)
When transformed, becomes
Sg = s0 + z1 u – v (3.12)
Where v is the expenditure components From the basic national income identity,
Ct + Ip + Ig + Gt+ (Xt – Mt ) = Yt = Ct + St + Tt + r Dt-1 (3.13)
Where Y is national income, C is private consumption, G is government routine spending, and (X-M) is net exports, T is taxes, r is interest rate on foreign debt, and D is the stock of debt. By substracting C from both sides and rearranging gives:
Mt – Xt = (Ip – St ) + (Ig + Gt – Tt) = dt - r Dt-1 (3.14)
dt is net foreign borrowing. Hence, the current account balance(i.e., net foreign borrowing) or foreign savings is:
Sf = dt = Mt – Xt + r Dt-1 (3.15)
Equation 3.15 sets foreign savings equal to current account balance which also equals net foreign borrowings. The current account deficit or net foreign borrowing, ceteris paribus, will be higher the greater is the volume of capital accumulation (import), the lower is private savings, the higher is foreign interest rate and the larger the fiscal deficits.
Furthermore, it is assumed that debt-service payment due is instantaneously met out of output or GDP. Therefore, real national income (Y) is the difference between total domestic output and interest payments on external debt, i.e.,
Yt = Qt – rDt-1 (3.16)
In normalised form, equations (3.16) and (3.5) can be collapsed and differenced with respect to time as is done in Greene and Khan (1990); this gives:
y = fk k – rD – rD (3.17)
where fk = dq/dk > 0, is the marginal product of capital and a dot on the variable denotes a time derivative. As a way of relating total investment and savings in an economy or when national income is in equilibrium, aggregate investment must be equal to aggregate savings, thus combining equations 3.8 and 3.9, we have
It = St or Ip + Ig = Sp + Sg + Sf (3.18)
Equation 3.18 above must hold ex-post in order to satisfy macroeconomic equilibrium such as in equation 3.13.
Therefore, substituting equations (3.10), (3.12), and ( 3.15) into (3.18) gives y = { s0 + ( z1 + s1) u – v} + (M – X) + r D (3.19)
reconciling equation (3.17) and (3.19), and obtaining the change in the variables and dividing both sides by ‘y’ gives growth in income as:
y/y = fk/ y { s0 + ( z1 + s1) u – v} + [ fk – r]D/y - r D/y (3.20)
Equations 3.19 and 3.20 yield some conditions that are vital to the theoretical basis for determining the sustainability of a given debt profile. Some of these conditions correspond to those in Zaidi (1985), Greene and Khan(1990), Degefe(1992), Raheem(1994) and Cohen(1996). Equations 3.19 and 3.20 imply that ceteris paribus, external debt build up should not pose a problem to growth of the economy if it reflects increased investment, rising public and private savings, low interest rates on foreign debt, capital is used efficiently and export is growing over import. It is expected that the marginal product of capital (fk) should exceed or at least equal the interest rate ( r ) on foreign borrowing. The debt profile will be sustainable if the marginal benefit from a unit borrowed is greater than or equal to the opportunity cost of borrowing. If the rate of return on debt finance domestic investment exceeds the cost of borrowing it is expected that the resulting output growth would contribute not only to closing the savings gap but also to
repaying the external loans incurred. Thus, economic growth will be enhanced or affected by the extent to which a country reduces its external debt, increased efficiency of investment, savings propensity is promoted, and fiscal deficit is reduced. Finally, equation 3.20 also shows that there will be a decline in the growth rate of the economy when debt to GDP ratio increases over time.
4. METHODOLOGY AND EMPIRICAL ANALYSIS