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# A2 A Simple Growth Model

## A2.1 Analytical Framework

The model developed here is a very simple neoclassical growth model. The production function in this economy is given by

where y is per capita real GDP, k (= K/L) is the capital/labor ratio, and f () is the per capita production function. Labor supply is exogenous and increases at the rate of n (>0). We implicitly introduce public investment. For simplicity, public capital and private capital are perfect substitutes. k may be regarded as total capital. Capital will not depreciate. The production function satisfies the usual neoclassical properties. Under perfect competition, the real yield on capital (r) is equal to the marginal product of capital.

Let g denote per capita real government expenditures excluding interest payments to public debt, t denote per capita tax revenues, and b denote per capita outstanding public debt. Then, the government budget constraint is given, in per capita terms, by

where D means the derivative with respect to time. In addition, let gc = (1 — a)g denote per capita government consumption and ag denote per capita government investment. The propensity to invest from government expenditures (0 < a < 1) is assumed to be given.

The value of household assets is the sum of the value of government liabilities and the capital stock. For simplicity, we assume that public bonds and capital are perfect substitutes for saving.

All per capita private savings s must be absorbed in either private capital accumulation (Dk — ag + nk) or additional public bonds in per capita terms. Thus,

For normative analysis, the method of formulating private saving is important. A natural assumption would be to allow the savings rate to vary as individuals maximize an intertemporal utility function. The framework of full optimization with an infinite horizon would lead to the neutrality proposition of alternative financing of government expenditures; that is, to a first approximation where the choice between current taxation and debt issuance in order to finance a given government expenditure stream is irrelevant to the determination of the level of aggregate demand. Thus, if the horizon is infinite, the concept of optimal deficit would lose its policy meaning (unless distortionary taxes are explicitly introduced).

This is because in such a circumstance private saving and government deficits are perfect substitutes. Whatever the level of government deficits, the private sector can always realize the optimal (first best) state by adjusting private savings. We examined this debt neutrality theorem in Chap. 4. Thus, in order to discuss the meaningful government deficit problem in terms of macro IS balance, from now on we assume that private savings are determined rather myopically. The private saving rate out of real disposable income a is fixed:

Thus, the dynamic system of this economy may be summarized by Eqs. (6.A2), (6.A3), (6.A4), and (6.A5), and the asset accumulation equation

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