National Debt in a Neoclassical Growth Model

Diamond states his model is “designed to serve two purposes, to examine long-run competitive equilibrium in a growth model and then to explore the effects on this equilibrium of government debt” (p. 1126). He credits Samuelson [1958] with having worked out interest-rate determination in a “single-commodity world without durable goods,” where rates are set purely by consumption loans between different-aged individuals; Diamond’s innovation is to introduce a durable capital good into production, so that individuals can instead (or additionally) “provide for their retirement years by lending to entrepreneurs” who use that saving as productive capital (p. 1126). He also builds on Modigliani’s [1961] work on government debt in an aggregate growth model, aiming to re-examine Modigliani’s conclusions in a setting with individually optimizing consumers, explicit taxation to finance debt service, and capital-stock effects on output made explicit.

The economy has an infinite horizon, a constant-returns-to-scale aggregate production function F(K, L) with no depreciation (capital and output are the same, storable commodity), and individuals who live for exactly two periods, working in the first and retired in the second, each with an ordinal utility function U(e¹, e²) over their two periods’ consumption (p. 1127). The labor force (equal to each period’s births) grows at a constant rate n, so L_t = L(1+n)^t, and there are no bequests – an assumption Diamond flags as important, since a link between debt and bequests could alter his conclusions (p. 1127, n. 2).

Diamond shows that on any “Golden Age Path” (a path holding the capital-labor ratio, and hence the capital-output ratio and marginal product of capital, constant), the capital-labor ratio that maximizes steady-state per-capita consumption satisfies F_K = n – the standard Golden Rule condition – and this choice is independent of how that consumption is then divided between the young and old generations alive together (pp. 1128-1129). Optimally dividing that consumption between generations requires e²/e¹ = (1+n) × (marginal utility ratio), which is “the allocation that would occur if consumption decisions were individually made employing a rate of interest for consumer decisions equal to the rate of growth” – exactly Samuelson’s biological optimum, reappearing here because holding the capital-labor ratio fixed makes the division-of-consumption problem formally identical to Samuelson’s single-factor (labor-only) model (p. 1129).

Each individual born in period t works for a wage w_t equal to the marginal product of labor, allocates it between current consumption and saving to maximize lifetime utility given the interest rate r_{t+1} on one-period loans, and consumes his savings plus accrued interest in retirement; entrepreneurs demand capital up to the point where its marginal product equals the interest rate, so the equilibrium interest rate equals the marginal product of capital, r_{t+1} = f’(K_{t+1}/L_{t+1}) (pp. 1130-1131). Aggregate individual savings functions s(w_t, r_{t+1}) constitute the capital market’s supply curve; combined with the demand curve given by capital’s marginal-product schedule, this yields a difference equation, r_{t+1} = ψ(w_t), that traces out the economy’s entire time path once a stability condition on relative demand/supply slopes is assumed (Sec. 8-9, pp. 1131-1134).

Diamond shows that “the competitive solution need not occur at an interest rate exceeding the Golden Rule level,” so the competitive solution “may be dynamically inefficient since there exists a time after which the capital-labor ratio will exceed the Golden Rule level by a nonvanishing amount” (p. 1133), a possibility he credits Koopmans [1957] with first raising for economies with infinitely many decision-makers and Phelps [1961, unpublished] with formally showing implies inefficiency. This is a striking result because it means the “invisible hand” of a fully competitive market – with no monopoly, no externality, no taxation – can nonetheless durably over-accumulate capital relative to what is Pareto-improvable, purely because the economy’s infinite horizon of new, not-yet-born generations defeats the usual finite-agent efficiency theorems.

Using Cobb-Douglas utility U = β log e¹ + (1−β) log e², under which saving is independent of the interest rate, and Cobb-Douglas production y = Ak^α, Diamond derives a closed-form long-run equilibrium interest rate and shows it equals the Golden Rule (population growth) rate n only for one special knife-edge relationship among the parameters α, β, and n (pp. 1134-1135). For any other combination of the capital share α and the saving parameter β, “different economies… can clearly have interest rates either larger or smaller than n,” concretely demonstrating that dynamic inefficiency is not a pathological special case but the generic outcome for arbitrary (if simplified) preferences and technology.

Diamond restricts his analysis to the “differential incidence” question – comparing debt finance versus tax finance for a given level of government expenditure – rather than “balanced-budget incidence” (comparing different combined levels of spending and financing) (pp. 1135-1136). He shows that simultaneously issuing debt and purchasing capital makes the government a mere financial intermediary with no long-run economic effect, so the substantive comparison is between an original equilibrium and one where the government finances a fixed, non-productive expenditure (e.g., a “windfall” transfer such as veterans’ bonuses) via a growing, one-period-maturity debt-to-labor ratio rather than solely by lump-sum taxes on the young (pp. 1136-1137).

Diamond shows algebraically that a permanent external debt-to-labor ratio g₁ requires per-worker taxes of (r−n)g₁ to service, and this tax shifts the capital-market supply curve, moving the long-run equilibrium interest rate further from the Golden Rule (growth) rate; in the “normal” (dynamically efficient) case, where r initially exceeds n, external debt therefore unambiguously lowers the long-run utility of an individual living in equilibrium (Sec. 13, pp. 1138-1141, esp. Eq. 20). Diamond decomposes this welfare change into a direct tax effect (positive or negative depending on the sign of r−n) and a “purely domestic” factor-price effect from the interest rate moving away from the Golden Rule level – “the existence of external debt increases the difference between” the equilibrium interest rate and wage relative to no-debt (p. 1138). In the dynamically inefficient case (r below n initially), the two effects can work in opposite directions, so there is no unambiguous conclusion.

Diamond shows that internal debt has the same tax effects on individual savings decisions as external debt, but adds a further channel: government bonds directly substitute for physical capital in individual wealth portfolios (since the equilibrium condition becomes savings = capital stock + debt outstanding, S_t = K_{t+1} + G_{t+1}), so a given increase in internal debt causes a larger fall in the capital stock than the same increase in external debt (Sec. 14, pp. 1141-1143, esp. Eqs. 21-27). “Comparing internal with external debt, we see that they both require taxes to be paid by each worker, while internal debt has a further effect in that it substitutes pieces of paper for physical capital in the portfolios of wealth owners, thus reducing output” (p. 1138/1141) – and in the efficient case this additional capital displacement causes internal debt to reduce long-run individual utility by more than an equal-sized external debt.

Swapping external for internal debt (holding the total constant) unambiguously raises the equilibrium interest rate – since it adds government demand to the capital market without changing supply-side taxes – and Diamond shows this swap reduces individual utility in the dynamically efficient case, while its effect is ambiguous (depending on the relative sizes of specific debt and capital terms) in the inefficient case (Sec. 16, pp. 1145-1148, esp. Eqs. 35-38). Diamond situates his results relative to the literature: Bowen, Davis, and Kopf [1960] had emphasized only the tax effects of internal debt, while Modigliani [1961] and Vickrey [1961] had emphasized only the capital-stock displacement from the demand side; Diamond’s framework shows these are two additive, separately identifiable channels, and that Modigliani’s assumption of a one-for-one replacement of capital by debt “does not directly alter net output” is incomplete because the resulting fall in capital and output itself feeds back onto the equilibrium quantity of total wealth (p. 1149, n. 17).

Diamond concludes that “in the ’normal’ case external debt reduces the utility of an individual living in long-run equilibrium. Surprisingly, internal debt is seen to cause an even larger decline in this utility level” (Abstract, p. 1126; Conclusion, pp. 1147-1148), but he is explicit that these welfare rankings reverse in sign, or become ambiguous, whenever the underlying no-debt competitive equilibrium is itself dynamically inefficient (r below n) rather than efficient. He also flags the model’s specific assumptions as scope conditions on these results: no bequests (n. 2, p. 1127), one-period-maturity debt used to avoid capital-gains complications (Sec. 12, p. 1136), lump-sum taxes on the young rather than the old (n. 13, p. 1137, noting the old-generation-tax case only partially offsets the described effects), and an assumed unique, stable long-run equilibrium in the capital market (n. 8, p. 1132); an alternative capital-market slope assumption, worked out in Appendix A, reverses several of the paper’s directional results (pp. 1148-1149).