APPENDIX: 1. Dynamic properties with balanced budget
Under the assumption of a balanced budget the dynamic structure of the system (20) is a fourth order system, which may be expressed in linearized form about the steady-state equilibrium
where the elements appearing in the matrix are evaluated at steady state. The dynamic properties of the economy depend upon the eigenvalues of the characteristic equation of (A1), namely
where the element a14 ≡—λ(2i′–i"b).
Assuming 2i′–i"b > 0, the following properties can be established:
(1) The product of the four roots is positive, implying that there are either 0, 2 or 4 positive roots.
(2) The sum of the roots is positive, ruling out the case of 0 positive roots.
(3) The coefficient of μ in (A2) is negative, ruling out the possibility of all roots being positive.
We are therefore left with the case of 2 positive and 2 negative roots, which may be ordered as follows
The dynamics is therefore a saddle point, with the stock variables k and z evolving gradually over time, and the shadow prices λ, q being allowed to undergo instantaneous jumps in response to new information.
The quantities ϕ(μ1), ϕ(μ2), as defined in (A2) are both critical parts of the solution. By direct evaluation of the characteristic equation, one can establish that
where μ1, μ2, are ordered as above.
We shall focus our analysis on stable adjustment paths beginning from given initial capital stock ko and stock of national debt zo. The solutions for k, z, λ, and q along such paths are
One can eliminate eμ1t, eμ2t, to define the two locuses
These are both three dimensional planes relating the respective instantaneous shadow prices to the slowly evolving dynamic variables k and z. They are the analogues to the usual two dimensional stable adjustment paths associated with saddle points.
The solutions reported in equations (A5) - (A8) form the basis for the analysis of the short-run dynamics in response to the various disturbances presented in Section IV. The different shocks identified differ simply in terms of how they impact on the long-run equilibrium stock of capital and national debt.
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The authors are grateful to Mohsin Khan for useful comments on an earlier draft. Stephen Turnovsky is Professor of Economics at the University of Washington.
Reference should be made to a recent paper by Otani and Villanueva (1988) which analyzes the accumulation of capital and external debt in a neoclassical growth model. However, that paper adopts a very different approach and emphasizes different issues (i.e., the role of human capital formation) from those addressed in the present paper.
There are precedents for the type of debt supply function used in this paper. Eaton and Turnovsky (1983) incorporate such a function in their analysis of exchange rate dynamics under covered interest parity. Obstfeld (1982) also considers an upward sloping debt function in his intertemporal optimizing analysis of terms of trade shocks. However, most previous studies have not embedded such a debt supply function in an elaborate macroeconomic growth model such as the present one.
The model can also be used, as illustrated below, to study questions relating to the issue of the debt overhang, or for understanding the dynamics of a policy of debt relief.
Note that since the model is real, there are no prices or nominal variables that need be considered.
Even though in some countries the private sector has borrowed abroad, implicit or explicit government guarantees have essentially underwritten this debt, making private debt indistinguishable from government debt, insofar as the foreign creditor is concerned.
Stiglitz and Weiss (1981) have shown that even in cases of individual borrowing risk premia or credit constraints or both may exist because of informational asymmetries.
This enables us to incorporate the Eaton and Gersovitz (1981a,b) argument.
This formulation which postulates the cost of debt to depend upon the absolute level of foreign debt is similar to that adopted by Eaton and Turnovsky (1983) and Obstfeld (1982). Alternative specifications which scale the absolute level of national debt by variables such as output or the capital stock will be considered later. Such specifications are intended to endogenize creditworthiness of the debt or country; see for example, Sachs (1984). However, as will be seen below, our essential qualitative results are not altered by these modifications.
It should also be noted at this point that the risk premium attached to the home country could alternatively be modelled as a function of total government debt. However, as Dooley (1987) points out, this reformulation does not lead to any substantive change in the analysis.
For simplicity, labor is assumed to be fixed. Since in a developing country context the endogeneity of labor is not likely to be a critical issue, this assumption is not viewed as being particularly restrictive.
If b > 0 the consumers are creditors while if b < 0 then they are debtors. Examination of the budget constraint shows that if consumers are creditors, then acquisition of increasingly costly debt by the government adds to disposable income and vice versa.
Subscripts and primes (') denote derivatives.
Note that the cost of debt depends on whether the consumer is a net creditor or a net debtor. In the former case the marginal cost exceeds the interest rate; in the latter case, the opposite is true.
Note that this specification implies that in the case where disinvestment may occur that C(I) < 0 for low rates of disinvestment. This may be interpreted as reflecting the revenue obtained as capital is sold off. The possibility that all changes in capital are costly can be incorporated by introducing sufficiently large fixed costs, so that C(0) > 0. This does not alter our analysis in any substantive way.
This is ensured by an appropriate adjustment in q at each point in time.
Hereafter the labor variable will be suppressed for convenience.
In Section IV below, we shall also discuss a form of debt financing. In order to be sustainable in the long run, this needs to be accompanied by a once-and-for-all change in lump sum taxes.
If the debt supply function is horizontal, i.e., i1 = 0 and i(z) = io, then any increase in the external interest rate would translate into an equiproportionate rise in domestic interest rates.
Assuming a convex debt function of the form i = io + i1zα, α > 1, this will be so if the ratio
These results may be usefully compared with the long-run effects of an increase in the foreign interest rate under the limiting assumption of uncovered interest parity. In such a case, the domestic interest rate rises by the same amount as does the foreign interest rate, leading to a larger fall in the domestic capital stock than in the present case. The stock of external debt can be shown to decline by an amount which is proportional to the reduction in the capital stock, with the resulting effect on the long-run trade balance being ambiguous, depending upon the stock of external debt.
For example, for the constant elasticity convex debt function i = io + i1zα, α ≥ 1, the quantity ω—ω′
For this case, the production function is changed to f(k, θ).
For obvious reasons we restrict our discussion to the plausible case where the increase in the foreign interest rate leads to an increase in the long-run domestic interest rate and a corresponding decline in the long-run capital stock. The perverse case, where the long-run interest rate declines can be analyzed similarly but is of little practical interest
This contrasts with the dynamics under the limiting case of uncovered interest parity when the paths followed by z and k can both be shown to be monotonic.
As discussed in the Appendix, the dynamic structure of the balanced-budget variant involves four differential equations. With debt finance, the resulting dynamics is of the fifth order; the additional source of dynamics is the evolution of the stock of government debt
The technical details are discussed in the Appendix.
For expositional convenience we restrict our discussion to the case where the debt schedule is linear.
The impact of changes in the zo, i.e., changes in the stock of debt held by the country, can be used to study the dynamics of the effects of debt forgiveness schemes. In this model, interest relief schemes are equivalent to negative foreign interest rate shocks.
This would transform (1) to