Running Primary Deficits Forever in a Dynamically Efficient Economy: Feasibility and Optimality

Running Primary Deficits Forever in a Dynamically Efficient Economy: Feasibility and Optimality

Research Question

The paper addresses two questions about government debt rollover. First, a positive question: what is the maximum ratio of government bonds to capital that can be sustained forever without any primary budget surpluses? Second, a normative question: among sustainable bond-capital ratios along a balanced growth path, which one maximizes the welfare (steady-state utility) of consumers? The analysis is motivated by Blanchard’s (2019) AEA presidential address and the fiscal responses to the COVID-19 pandemic.

Setting and Mechanism

The baseline environment is a standard two-generation (young and old) overlapping-generations model. Young consumers earn labor income and save; old consumers live off portfolio returns. The production function is Cobb-Douglas, Yt = (GtN)^(1−α) K^α, where G = 1+g is the gross growth rate of labor-augmenting productivity. Uncertainty enters exclusively through a stochastic i.i.d. durability shock ε_t to the depreciation rate of capital (δ − ε_t), so the rate of return on capital r = αk^(α−1) − δ + ε is stochastic even though the capital stock per unit of effective labor k is deterministic along a balanced growth path. Consumers have Epstein-Zin-Weil utility with an intertemporal elasticity of substitution equal to one. Because IES = 1 and labor income is earned only when young, aggregate saving of young consumers is a constant fraction β of their wage income, making total assets (capital plus bonds) non-stochastic.

This structure creates a key wedge: the expected rate of return on capital R can exceed the growth rate g (dynamic efficiency) while the riskfree interest rate rf — determined by the portfolio equilibrium between risky capital and riskless bonds — can remain below g. In deterministic economies these two rates coincide, so dynamic efficiency and the infeasibility of permanent debt rollover always go together. In this stochastic model they can be decoupled.

Main Findings

Positive finding. The maximum sustainable bond-capital ratio, Bmax, is attained precisely when rf = g (equivalently, when the adjusted gross riskfree rate Rf = 1). Starting from a bond-less economy with rf < g (which may itself be dynamically efficient), introducing government bonds crowds out capital, raises the marginal product of capital and the constellation of returns, and drives rf upward toward g. Once rf = g is reached, any further increase in bonds would require rf > g, making rollover infeasible without primary surpluses. The maximum sustainable ratio Bmax is characterized as the unique root of f(Bmax, 1) = 0, and it is finite.

Normative finding. The welfare-maximizing sustainable bond-capital ratio equals Bmax. Proposition 6 establishes that u′(B) ≥ 0 for all B ∈ [0, Bmax] whenever Rf ≤ 1, with strict inequality unless Rf = 1. Proposition 7 therefore concludes that the welfare-maximizing B is the corner solution Bmax. Intuitively, increasing B reduces capital and wages but raises the rate of return on capital. When rf ≤ g, the welfare gain from a higher return on capital in old age dominates the welfare loss from a lower wage when young (via the factor-price frontier and the intertemporal optimality condition E{uo′(co)} ≥ uy′(cy)). When rf = g (at Bmax), a marginal increase in bonds also provides no additional welfare improvement if all seignorage is transferred to young consumers (ζ = 1), but still raises welfare if some seignorage is wasted (ζ < 1). In either case, Bmax is the optimum. Critically, at the optimum the economy is dynamically efficient — even though the government is running permanent primary deficits.

Dual role of bonds. At the optimal bond-capital ratio, government bonds serve two purposes simultaneously: (1) they crowd out any dynamically inefficient overaccumulation of capital that might prevail without bonds, and (2) they supply riskfree assets to risk-averse consumers who would otherwise hold only risky capital, improving risk sharing.

Quantitative Illustration

The paper calibrates a 30-year-period OLG model with α = 0.33, β = 0.353 (annual discount rate 2%), annual productivity growth g = 1% (G = 1.35), and target mean return on unlevered equity m = 3% per year. Risk aversion γ ∈ {1, 3, 8, 10} and annualized standard deviation of capital returns s ∈ {0.02, …, 0.22}. Key results (ζ = 0): at γ = 10 and s = 0.22, Bmax = 0.478 and B∗ (the bond-capital ratio needed just to eliminate dynamic inefficiency) = 0.083, so there is a wide interval [0.083, 0.478] of dynamically efficient, permanently rollable bond-capital ratios. For a capital-output ratio of 2, the debt-GDP ratio corresponding to Bmax = 0.478 is approximately 0.956. Bmax is strictly increasing in both γ and s, and is invariant to ζ (the share of seignorage transferred rather than wasted).

Scope Conditions

• Results hold along balanced growth paths with constant g and constant rf; the sustainability characterization is more complex if either rate is stochastic.
• The key sufficient condition for Rf to be increasing in B (Proposition 1) is that risk aversion γ < Λ, a model-dependent upper bound that is always positive. All subsequent propositions assume R′f(B) > 0, which is satisfied for a potentially larger set of γ.
• The paper focuses on welfare along the balanced growth path; it does not study transition dynamics or welfare during convergence from an initial state.
• The No Ponzi Game (NPG) condition is violated by design in the feasible-rollover region (rf ≤ g); the value of government bonds is positive even though the present value of all future primary surpluses is non-positive.

Q&A

Q1: Why can an economy be both dynamically efficient and able to roll over government bonds forever, when this is impossible in deterministic models?

In a deterministic economy, the riskfree rate rf and the rate of return on capital r are equal, so the conditions rf < g (feasibility of rollover) and r < g (dynamic inefficiency) are identical. In a stochastic economy, aggregate uncertainty drives a wedge between rf and the expected return on capital. Risk-averse consumers require a premium to hold risky capital over riskless bonds, so rf < E{r}. It is therefore possible that E{ln R} > 0 (the Zilcha sufficient condition for dynamic efficiency holds) while Rf < 1, i.e., rf < g. This decoupling is the central theoretical contribution of the paper.

Q2: What is the formal criterion the paper uses for dynamic efficiency, and how does it relate to the AMSZ criterion?

Abel, Mankiw, Summers, and Zeckhauser (AMSZ, 1989) show that if the rate of return on capital exceeds g in all states (R > 1 always), the economy is dynamically efficient, and since rf < r, the economy has rf > g so rollover is infeasible; conversely if r < g always, the economy is dynamically inefficient. The AMSZ criteria are silent when R sometimes exceeds and sometimes falls short of one. Building on Zilcha (1991), the paper uses E{ln R} ≥ 0 as a sufficient condition for dynamic efficiency. In the five-region diagram (Figure 1), Region E satisfies E{ln R} > 0 (Zilcha-efficient) and Rf < 1 (rollover feasible simultaneously), which is the case of central interest.

Q3: How does the model achieve a deterministic capital stock despite stochastic capital returns?

The durability shock ε_t affects depreciation but is additively separable from the production function. Because (1) IES = 1 and (2) consumers earn income only when young, aggregate saving is the fixed fraction β of wage income, which depends only on capital k (itself non-stochastic). Total assets At+1 = Kt+1 + Bt+1 = St are thus non-stochastic. The stochastic shock to depreciation makes the rate of return on capital r = αkα−1 − δ + ε stochastic even though k is deterministic. Online Appendix B establishes that this model is isomorphic to a model with production function shocks, extending the scope of the results.

Q4: What is the financial market equilibrium condition that pins down the riskfree rate?

Young consumers optimally choose the portfolio share λ in riskfree bonds. The first-order condition for this portfolio problem along a balanced growth path is E{(λRf + (1−λ)R)^(−γ)(Rf − R)} = 0 (equation 20). In equilibrium, λ = B/(1+B) (the bond-capital ratio determines the portfolio share), so the equilibrium riskfree rate Rf satisfies the implicit equation f(B, Rf) = 0 (equation 21). Lemma 1 establishes that Rf = E{R^(1−γ)_a}/E{R^(−γ)_a}, a ratio-of-moments formula analogous to an Euler equation.

Q5: Why is the riskfree rate Rf an increasing function of the bond-capital ratio B, and what is the sufficient condition for this?

Lemma 2 shows ∂f/∂B > 0; intuitively, more bonds reduce capital, raise the marginal product of capital, and raise R, inducing consumers to demand more capital and less bonds, pushing Rf up to restore equilibrium. Lemma 3 provides a sufficient condition for ∂f/∂Rf < 0, namely γ < Λ (where Λ is a positive parameter-dependent bound). Under this condition, the implicit function theorem implies Rf′(B) > 0 (Proposition 1). The condition γ < Λ is sufficient but not necessary, so the results of all downstream propositions hold potentially for a wider parameter range.

Q6: What is the maximum sustainable bond-capital ratio Bmax, and how is it characterized?

By definition, a bond-capital ratio B is sustainable if and only if Rf(B) ≤ 1. If Rf(0) ≥ 1, then Bmax = 0 (no positive amount of bonds is sustainable). If Rf(0) < 1, Bmax is the unique positive root of Rf(B) = 1, i.e., f(Bmax, 1) = 0 (Proposition 4). At Bmax, the riskfree rate exactly equals the growth rate: rf = g. The paper also shows Bmax ≤ (1−α)β/α − 1, an upper bound that depends only on production and preference parameters. Notably, Bmax is invariant to the parameter ζ (the share of seignorage transferred to young consumers rather than wasted), because at Bmax transfers are always zero regardless of ζ.

Q7: Why does the welfare-maximizing sustainable bond-capital ratio equal Bmax rather than some interior value?

Proposition 6 shows that u′(B) ≥ 0 for all B ∈ [0, Bmax] whenever Rf ≤ 1, with strict inequality unless Rf = 1 and (1−ζ)B = 0. Since utility is weakly increasing throughout the feasible set, the optimum is the corner solution Bmax (Proposition 7). The mechanism: increasing B reduces k, lowering wages (bad for utility when young) but raising the marginal product of capital and hence the rates of return on capital and bonds (good for utility when old). The factor-price frontier ensures that the wage reduction equals the income gain accruing to initial capital, and the intertemporal optimality condition uy′(cy) = Rf E{uo′(co)} implies that when Rf ≤ 1 (so E{uo′(co)} ≥ uy′(cy)/Rf ≥ uy′(cy)), the welfare gain in old age dominates.

Q8: How does Proposition 5 square with the optimality of Bmax? Does reducing expected consumption not reduce welfare?

Proposition 5 shows that when ζ = 1, a marginal increase in B at Bmax reduces expected aggregate consumption (dE{c}/dB < 0). However, welfare is not simply expected aggregate consumption: it also depends on the distribution of consumption across states. At Bmax, even though expected consumption falls, the increased risk sharing from holding more riskfree bonds — which smooth consumption between the high-return and low-return states of capital depreciation — is large enough to leave welfare unchanged (u′(Bmax) = 0 when ζ = 1) or to increase it (u′(Bmax) > 0 when ζ < 1). This illustrates that in stochastic economies, the welfare criterion diverges from the aggregate consumption criterion that characterizes dynamic inefficiency in deterministic economies.

Q9: How does the paper’s welfare analysis relate to the No Ponzi Game (NPG) condition and the fiscal theory of the price level?

The standard NPG condition requires that the value of government debt equals the present value of future primary surpluses. In the paper’s feasible-rollover region (rf ≤ g), the NPG condition is violated by design: the present value of future primary surpluses is non-positive (all primary balances are deficits or zero), yet the market value of outstanding bonds is strictly positive. This is possible because, as Santos and Woodford (1997) show, when the present value of aggregate consumption is infinite, the NPG can fail. The market value of the capital stock remains finite (it is the value of profits on a depreciating capital stock approaching zero), but the bubble value of government bonds is positive.

Q10: What does the quantitative calibration reveal about the range of dynamically efficient, permanently rollable bond-capital ratios?

With α = 0.33, β = 0.353, g = 1% per year, G = 1.35, target mean equity return m = 3% per year, and risk aversion γ = 10 with annualized return standard deviation s = 0.22, the paper finds Bmax = 0.478 and B∗ = 0.083 (ζ = 0, Table 1). The interval [B∗, Bmax] = [0.083, 0.478] is the range of bond-capital ratios for which the economy is both dynamically efficient and able to roll over bonds permanently. For an economy with a capital-output ratio of 2, these bond-capital ratios correspond to debt-GDP ratios of up to 0.956. Both Bmax and B∗ are increasing in risk aversion γ and in the standard deviation of capital returns s; Bmax is independent of γ in any given column of the table for the ζ = 0 case (since R is independent of γ there), but rises substantially with γ in the ζ = 1 case.

Q11: What is the role of the parameter ζ (the share of seignorage transferred vs. wasted)?

The parameter ζ governs what the government does with seignorage revenue: transfer it to young consumers (ζ = 1) or waste it (ζ = 0), or some mix. Corollary 1 shows that Bmax is completely invariant to ζ, because at Bmax, rf = g so seignorage (g − rf)Bt = 0 in any case. The value ζ does affect u′(Bmax): if ζ < 1, u′(Bmax) > 0; if ζ = 1, u′(Bmax) = 0. Both configurations yield Bmax as the welfare-maximizing level. The parameter ζ matters for welfare levels and for B∗ (only in the ζ = 1 case, where transfers are positive and boost saving capacity), but not for the main positive or normative results.

Q12: In what sense is the model tractable, and what are its key limitations?

Tractability comes from three design choices: (i) the durability shock is additively separable from the production function, so labor income and aggregate saving are non-stochastic; (ii) IES = 1 with Epstein-Zin-Weil preferences, making saving a constant fraction of income; (iii) along balanced growth paths, g and rf are constant, so sustainability reduces to comparing two constants. Limitations acknowledged by the authors: the paper analyzes only balanced growth paths and does not characterize transition dynamics; the framework does not directly address economies where g or rf are stochastic; and the two-period OLG structure is stylized. The authors pose as an open question whether the result that optimal borrowing equals maximal borrowing generalizes to settings with random g.

Key Concepts

Bond-capital ratio (B): The ratio of outstanding government bonds to the capital stock, Bt/Kt. This is the paper’s central state variable and policy instrument. A value B is “sustainable” if the government can roll over its debt forever at the riskfree interest rate without any primary budget surpluses. The paper distinguishes B from the more commonly reported debt-GDP ratio (which equals B times the capital-output ratio).

Adjusted gross rate of return / riskfree rate (R, Rf): R ≡ (1+r)/G and Rf ≡ (1+rf)/G, where r is the net return on capital, rf is the riskfree interest rate on bonds, and G = 1+g is the gross growth rate. Expressing returns in these “adjusted” gross units scales out balanced growth and simplifies the sustainability condition to Rf ≤ 1 (equivalently, rf ≤ g).

Dynamic efficiency (Zilcha criterion): In the paper’s stochastic setting, the relevant criterion for dynamic efficiency is E{ln R} ≥ 0 (Zilcha 1991, as amended by Rangazas-Russell 2005 and Barbie-Kaul 2009), meaning the geometric mean of the adjusted gross return on capital is at least one. This differs from the deterministic condition r ≥ g. The paper’s Region E in Figure 1 is the key zone where E{ln R} > 0 (dynamically efficient) and Rf < 1 (rollover feasible) simultaneously.

Bmax (maximum sustainable bond-capital ratio): The largest value of B for which the bond-capital ratio is sustainable, defined as the unique root of Rf(B) = 1. At Bmax, the riskfree rate exactly equals the growth rate (rf = g). The paper proves Bmax is finite, invariant to ζ, and equals the welfare-maximizing sustainable bond-capital ratio.

B∗ (dynamic efficiency threshold): The bond-capital ratio at which the economy crosses from Zilcha-inefficiency into Zilcha-efficiency, defined by E{ln R} = 0. For B ∈ [B∗, Bmax], the economy is dynamically efficient and debt rollover is feasible. B∗ < Bmax when risk aversion γ or return volatility s is large enough, defining a non-trivial interval of dynamically efficient, permanently rollable bond levels.

Durability shock (ε): An i.i.d. random variable with mean zero that enters the capital depreciation rate as δ − ε_t. This shock makes the rate of return on capital r = αkα−1 − δ + ε stochastic while leaving the capital stock per unit of effective labor, aggregate wages, and aggregate saving non-stochastic. It is the only source of aggregate uncertainty in the model and is the mechanism that drives a wedge between rf and E{r}.

No Ponzi Game (NPG) condition: The condition that the present discounted value of government debt converges to zero (equivalently, debt equals the present value of future primary surpluses). Standard fiscal sustainability analyses assume this condition holds. The paper explicitly violates it: in the feasible-rollover region rf ≤ g, the present value of aggregate consumption is infinite and the NPG fails, yet government bond values are positive and debt rollover is sustainable.

Seignorage (ζ): The revenue the government obtains by issuing new bonds in excess of interest payments on existing bonds, equal to (g − rf)Bt when rf < g. The parameter ζ ∈ [0,1] governs the share transferred to young consumers (as lump-sum transfers τt) versus wasted (captured by the government but yielding no utility). A key finding is that Bmax is invariant to ζ, since seignorage is zero at rf = g regardless of ζ.