Debt, Deficits, and Finite Horizons

The paper’s starting point is that “there is in general no simple aggregate consumption function in an economy composed of finitely lived agents” because agents differ in both their wealth (by age) and their propensity to consume (by horizon), making exact or approximate aggregation generally impossible (Introduction, p. 2, citing Modigliani [1966]). Diamond [1965] avoided this by using a simple two-period population structure. This paper instead makes an assumption that permits aggregation while keeping horizons genuinely finite: “the central assumption is that agents face, throughout their life, a constant instantaneous probability of death p,” so that agents of different ages have different wealth levels but an identical horizon and propensity to consume (Sec. I, pp. 2-3). Letting p → 0 recovers the infinite-horizon case as a limit, so the horizon index p can be varied continuously between the two polar cases.

Individual consumption is c(s,t) = (p+θ)(w(s,t) + h(s,t)): a constant propensity (p+θ) applied to the sum of non-human wealth and human wealth, where θ is the subjective discount rate; aggregating across cohorts gives “aggregate consumption is a linear function of total aggregate wealth” (Sec. I, p. 7, Eq. 5). Critically, “the discount rate used to discount non-interest income is higher than the interest rate” whenever horizons are finite (p > 0): human wealth discounts future labor income at (r+p) rather than r, because each agent’s own claim on future income is itself subject to the probability of death (Sec. I, p. 7). In the infinite-horizon case (p = 0), this collapses to the standard Hall [1978] consumption-smoothing equation, in which non-human wealth drops out of the equation describing the rate of change of consumption; with p > 0, non-human wealth re-enters that equation (Eq. 8-9).

The paper shows that the steady-state capital stock K is such that the net marginal product of capital r(K) lies strictly between 0 and (θ+p); when p = 0, “the modified golden rule obtains” (r(K) = θ), but “when p is positive, r(K) is larger than θ,” meaning steady-state capital is lower than in the infinite-horizon case for the same θ (Sec. II, pp. 9-10). The reason given is that with flat labor income over life, “in order to generate positive aggregate capital, agents must be saving initially… consumption must be increasing. Thus r must be larger than θ,” and this required excess of r over θ is shown to be an increasing function of p – “the shorter the horizon, the lower the level of capital” (Sec. II, p. 10).

The paper notes that the answer depends on which discount rate is treated as “correct”: “If we use the subjective discount rate θ as the relevant rate, r(K) > θ and the economy has too small a capital stock. If, however, we use the effective subjective discount rate θ+p, r(K) < θ+p and the economy has too large a capital stock” (Sec. II, p. 10). The paper does not resolve this ambiguity further in the no-government case; it is revisited once declining labor income is introduced, where the equilibrium can become genuinely (Diamond-)inefficient.

With a fixed world interest rate r and no capital, the paper finds that “if r is greater than θ, individual consumption is increasing, agents are accumulating over their life and the level of foreign assets is positive[;] if r is smaller than θ, agents are decumulating and, as a result, the level of foreign assets is negative” – the country is a net debtor in steady state – while “if r = θ, the value of F is zero.” An increase in the horizon parameter p “pushes the level of foreign assets towards zero, reducing it if positive, increasing it if negative: shorter horizons imply smaller accumulation or decumulation” (Sec. II, pp. 11-13).

With isoelastic (rather than logarithmic) utility, “the lower [the elasticity of substitution] … the lower steady state capital” – the propensity to consume becomes a function of the whole expected path of future interest rates rather than a constant, but the qualitative finite-horizon result survives (Sec. III, pp. 14-15). Allowing labor income to decline exponentially through life (to capture a “saving for retirement” motive) raises the effective discount rate applied to human wealth still further, and the paper finds that “an increase in [the rate of income decline] increases steady state capital” – the opposite sign from the pure finite-horizon effect (Sec. III, p. 17). “To summarize, the effect of finite horizons per se is to decrease capital accumulation. The effect of declining labor income, of saving for retirement, is however to increase it. The net effect is ambiguous and the steady state can be inefficient” (Sec. III, p. 18).

Considering a tax cut at time t offset by a tax increase of equal present value at a later time t+τ, the paper derives that the resulting effect on human wealth and current consumption is -dT_t(1 - e^{-pτ}), so that “unless p = 0, a decrease in taxes today increases human wealth and consumption[;] the longer taxes are deferred, the larger the effect.” This result “comes from the different discount rates in the government budget constraint and in the definition of human wealth[, which] in turn reflects the fact that taxes are partly shifted to future generations” – a currently alive taxpayer may die, in the model’s actuarial sense, before the future tax increase is levied (Sec. IV, pp. 19-20).

The paper constructs a single index g that collects every channel through which current and anticipated taxes, spending, and debt affect aggregate demand, and shows for a stylized deficit-then-surplus tax path (taxes proportional to accumulated debt) that “the index of fiscal policy therefore increases over time” – rising monotonically from an initial value g₀, which depends on the anticipated sequence of deficits, toward a strictly larger steady-state value g∞ (Sec. IV, pp. 20-22, Eq. 30). The intuition offered is that in steady state, the taxes needed to service the debt are valued in human wealth at the lower rate r/(r+p), rather than fully offsetting the debt, leaving a net positive effect on demand that persists rather than fading out as deficits give way to surpluses.

In the closed economy, “the levels of capital and of consumption are decreasing functions of the level of debt and the government can choose any level of capital by an appropriate choice of debt,” since issuing debt is partly perceived as net wealth and crowds out capital accumulation until a lower steady state is reached (Sec. V, pp. 22-23). In the open economy, “government debt displaces foreign assets in the agents’ wealth[; t]he displacement is one for one if r = 0, but may be much larger if r is greater than 0,” and “by its debt policy, the government can choose any level of private consumption” (Sec. V, pp. 24-25).

The paper shows the entire dynamic path of foreign-asset decumulation induced by the deficit is governed by the single index g rather than by the current deficit alone, so that “although fiscal deficits will be associated with current account deficits, there is no simple relation between the two.” At the moment the deficit policy begins, the paper derives that the pass-through from the rate of change of government debt to the rate of change of foreign assets is (dF/dD) = p/(a+p), so “the effect is less than one for one[;] it tends to one as p tends to infinity, as the horizon of agents shortens” (Sec. VI, pp. 26-27).

With a zero steady-state world interest rate and labor income fluctuating cyclically, the paper shows that absent fiscal policy, “aggregate consumption is cyclical” even though each surviving individual’s own consumption path is flat, because successive cohorts are born into different income conditions; “if the social welfare function is concave in individual utilities, it is desirable to smooth consumption across cohorts,” and the paper shows this can be achieved exactly by taxing away each period’s deviation of income from its mean. Under this policy, “government debt and foreign assets follow symmetric but opposite paths,” with debt displacing foreign assets one for one, and the paper notes that “this change in portfolio composition has no effect, in an open economy, on consumption or the interest rate[;] this would not be the case if the same debt policy was pursued in a closed economy,” where the resulting variation in the capital stock would also move the interest rate, making optimal policy “substantially more difficult” to characterize there (Sec. VI, pp. 28-30).

The paper’s own conclusion states that its purpose “was to characterize rigorously the effects of intertemporal reallocations of taxes when agents have finite horizons,” and that “the aggregate consumption function developed here seems well adapted to the task[;] the index of fiscal policy should also prove useful both conceptually and empirically.” The paper is explicit that this was obtained only by making simplifying assumptions – lump-sum taxation and a constant employment level foremost among them – “which need to be removed to obtain a more realistic characterization of the effects of debt and deficits” (Conclusion, p. 30).