# External Debt, Growth and Debt-Service Capacity in Sub-Saharan Africa: A Theoretical and Empirical Analysis

Scott, Gerald, American Economist

Introduction

The purpose of this paper is firstly, to use a simple neoclassical production function to show how capital imports can increase output in a low income nation, such as those in Sub-Saharan Africa (SSA). Secondly we will show how this increase in output, which has resulted from capital imports, can potentially be distributed between debt service payments and consumption. Finally, we will attempt to provide empirical estimates on both the impact of capital imports on economic growth, and the effect of growth on consumption and debt service payments.

In neoclassical analysis foreign borrowing or capital import is profitable if the productivity of such capital is greater than the cost of borrowing. This means that the rate of growth of domestic output will exceed the contribution made by domestic investments alone as long as the productivity of capital imports is greater than the interest paid on borrowed capital.

Capital Imports, Growth and Consumption

In the tradition of neo-classical one sector models, output depends on labor and capital i.e.

Where

Q = q(K, L) Q = Output K = Capital L = Labor Force (1)

We assume that labor force and domestic capital are both constant, so that we can focus exclusively on capital imports. We will use K to represent capital imports. The production function in equation 1 exhibits the following property

[Delta]Q/[Delta]K [is greater than] 0 [[Delta].sup.2]Q/[[Delta].sup.2] [is less than] 0

i.e. the marginal product of capital imports is positive and declining.

Once capital imports increase output there is a surplus available for debt service payments or domestic savings. This surplus is the difference between output and consumption, i.e.

Where

S = Q - C S = Surplus C = Consumption (2)

Consumption in turn depends on output, i.e., with

C = c[q(K,L)] (3)

[Delta]C/[Delta]Q [is greater than] 0

i.e. as output rises up to some point so does consumption. From e.g. (1) (2) and (3)

S = q(K, L) - c[q(K, L)] (4)

The question is how much of the output is available for debt-service payments and domestic savings, once capital imports have increased output.

From equation (4)

[Delta]S/[Delta]K = [Delta]Q/[Delta]K - [Delta]C/[Delta]Q x [Delta]Q/[Delta]K (5)

Multiplying and dividing equation 5 by K and Q respectively, we get

[Delta]S/[Delta]K x K/Q = [Delta]Q/[Delta]K K/Q - [Delta]C/[Delta]Q [Delta]Q/[Delta]K x K/Q (6)

Dividing Equation (6) by S, gives

[Delta]S/[Delta]K K/S x 1/Q = [Delta]Q/[Delta]K K1/QS - [Delta]C/[Delta]Q [Delta]Q/[Delta]K K/Q 1/S (7)

This can be reduced to

[[Epsilon].sub.SK] = [[Epsilon].sub.QK] Q/S - [Delta]C/[Delta]Q [E.sub.QK] Q/S (8)

When [[Epsilon].sub.SK] = Elasticity of the surplus with respect to capital imports [[Epsilon].sub.QK] = Elasticity of output with respect to capital imports.

Equation (8) can be rewritten as

[[Epsilon].sub.SK] = [[Epsilon].sub.QK] (Q/S) (1 - [Delta]C/[Delta]Q) (9)

[[Epsilon].sub.QK] = [is greater than] 0 and 0 [is less than or equal to] [Delta]C/Q [is less than or equal to] 1

At low levels of income and high levels of poverty [Delta]C/[Delta]Q can be equal to unity, so that all the change in output is totally consumed. If this is the case then, even though capital imports increase output, there is no surplus left for debt service.

The surplus depends not only on [[Epsilon].sub.SK] but also on [[Epsilon].sub.QK]. In very poor LDCs, because of institutional and structural rigidities, and market distortions of various forms, the response of output to capital imports is severely constrained. With very limited infrastructures and unavailability of essential complementary inputs such as skilled labor for example, the effectiveness or productivity of capital is severely limited. At the same time, because consumption hovers on the subsistence, the consumption elasticity of output is expected to be high and may likely be unity. …

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