A Goldilocks Theory of Fiscal Deficits

Mon, 01 Jan 0001 00:00:00 +0000

This paper develops a tractable continuous-time model to study the fiscal sustainability of government deficits and the joint dynamics of public debt, with two main ingredients: an endogenous interest rate R that rises with the debt level through a convenience yield mechanism (savers value holding government bonds), and a potentially binding zero lower bound (ZLB) on the nominal interest rate. The paper’s central theoretical contribution is deriving the correct free-lunch condition: not the commonly cited $R < G$, but the stricter condition $R < G - \varphi$, where $\varphi$ captures the sensitivity of $R - G$ to debt. Even when $R < G$, accumulating more debt raises R through reduced convenience yields, and this endogenous feedback tightens fiscal sustainability. The paper maps the full deficit-debt space with a hump-shaped locus, analyzes ZLB dynamics where the deficit-debt relationship can invert, and studies the role of income inequality and tax policy. Calibrating to U.S. and Japan as of December 2019, the paper finds little room for free-lunch policies in the U.S. — a maximum permanent deficit of just over 2% of GDP at a stable debt-to-GDP ratio of 110% — while Japan is in the “inverted” ZLB regime where deficit increases can reduce debt through higher nominal growth.

Summary of a published paper based on the NBER working paper full text (w29707), AI-assisted, pending human review. See the linked original for the authoritative claims and full conditions.

Layer 1: Overview

Mian, Straub, and Sufi construct a tractable deterministic continuous-time model with savers who derive convenience utility from holding government bonds, hand-to-mouth spenders, and a monetary authority that targets inflation (except at the ZLB), to systematically analyze when deficits can be “free lunches.” The core insight is that the standard r < g analysis treats interest rates as exogenous to the debt level, but if R rises as debt accumulates — through the declining marginal convenience yield of bonds — then the condition for a free-lunch policy is not R < G but R < G − φ. This matters empirically: the paper estimates φ (the debt-to-interest-rate sensitivity) from empirical estimates of the convenience yield elasticity, and calibrates the model to U.S. and Japan December 2019 conditions. The U.S. calibration finds a maximum free-lunch deficit of just over 2% of GDP at a stable debt ratio of 110%, implying the U.S. was barely inside the free-lunch region pre-Covid. By contrast, the paper finds ample free-lunch space for Japan and an “inverted” ZLB regime in which higher deficits can reduce the debt-to-GDP ratio by stimulating nominal growth. The analysis is extended to incorporate aggregate risk, capital, debt maturity structure, and inequality — each with distinct implications for the size and location of fiscal space.

In depth

Q1. What is the deficit-debt diagram, and what is the free-lunch condition?

The deficit-debt diagram is the locus of steady-state combinations of the primary deficit z and the debt-to-GDP ratio b, derived from the government budget constraint $\dot{b} = -(G^ - R^(b))b + z$ at steady state; this locus is hump-shaped, with the maximum sustainable permanent deficit z occurring at the debt level b where $R^(b^) = G^* - \varphi(b^)$.** The hump shape arises because at low debt levels the convenience yield is high (R is low relative to G, allowing large deficits), while at high debt levels the convenience yield is saturated (R rises toward G, leaving little deficit room). The left branch of the locus — where debt levels are below b — is the free-lunch region: any permanent increase in the deficit to a value below z* raises the steady-state debt level but requires no future tax increases. The right branch — debt above b* — is the conventional region: any deficit increase must eventually be accompanied by higher taxes. The key departure from the standard r < g analysis is that R is endogenous; Proposition 1 and Corollary 1 formally establish that the correct free-lunch threshold is $R^(b_0) < G^ - \varphi(b_0)$, not simply R < G.

Q2. Why is R < G insufficient as a free-lunch condition, and what does φ capture?

The condition R < G fails as a free-lunch criterion because, when the government borrows an additional dollar and rolls it over forever, it faces two opposing budget effects: a positive cash flow of G − R from rolling over the existing debt, and a tightening of the budget constraint from the endogenous rise in R on all infra-marginal outstanding debt; the parameter φ measures the magnitude of this second effect as the semi-elasticity of R − G with respect to the log of debt. When φ is positive — as it is empirically because convenience yields are declining in debt supply — the net fiscal benefit of rolling over additional debt is G − R − φ, not G − R. An economy can exhibit R < G yet be in the conventional debt region if φ is sufficiently large that R > G − φ at the current debt level. The U.S. calibration illustrates this: the traditional R < G condition holds up to a debt ratio of 220% of GDP, but the stricter R < G − φ condition breaks down already at 110%, which is the actual boundary of the free-lunch region for the U.S.

Q3. How does the analysis change at the zero lower bound, and what is the “inverted” fiscal regime?

At the ZLB, the direction of causality reverses: instead of the debt level determining the interest rate, the debt level determines the nominal growth rate G (via aggregate demand and the Phillips curve), creating an “inverted” regime in which higher deficits can reduce rather than increase debt by stimulating nominal growth and inflating away the debt. The mechanism is: when the nominal rate is constrained at zero, fiscal expansion raises aggregate demand, which via the Phillips curve (slope κ) raises inflation, which raises nominal growth G, which accelerates the inflation of the debt ratio. If the fiscal multiplier times κ times the debt level exceeds one — a sufficient statistic condition — then higher deficits reduce the debt ratio. The paper finds this condition plausible for Japan (debt ratio ~225%, estimated κ = 0.1-0.3, multipliers of 1.5-2) but not for the U.S. in 2019. The deficit-debt locus in this regime is “backward-bending”: as the ZLB binds more tightly (lower debt), the locus can curve back and eventually allow the inverted relationship between deficits and debt.

Q4. How does income inequality affect fiscal space, and why does the ZLB reverse the sign?

Outside the ZLB, greater income inequality (a larger income share of savers relative to hand-to-mouth spenders) expands fiscal space, because savers have a higher propensity to save, which reduces the natural interest rate R and thus raises G − R and allows larger sustainable deficits; at the ZLB, greater inequality shrinks fiscal space, because it reduces aggregate demand and hence nominal growth G rather than R.* Formally, outside the ZLB: $z(b) = (v’(b)(1-x-\mu) - \rho)b$, which increases as the spender share μ falls (Corollary 3). At the ZLB, nominal growth G becomes demand-determined via equation (20), and lower μ reduces demand, lowering G and hence z(b). The policy implication is a potential conflict between redistributive policies and deficit finance: redistribution (raising μ) reduces fiscal space outside the ZLB but expands it at the ZLB. The paper notes that roughly 69% of U.S. government debt held by households is directly or indirectly held by the top 10% of the wealth distribution, making savers’ saving propensity the primary driver of the convenience yield.

Q5. What are the U.S. and Japan calibration results for fiscal space as of December 2019?

For the U.S. in December 2019, the model calibrates a maximum permanent primary deficit z of just over 2% of GDP at a stable debt-to-GDP ratio of b ≈ 110%, implying the U.S. was just inside the free-lunch region; for Japan, the model finds the economy in the inverted ZLB regime where higher deficits reduce debt by raising nominal growth.** The calibration uses empirical estimates of φ from the literature on convenience yield demand elasticities (Krishnamurthy and Vissing-Jorgensen 2012, Laubach 2009, Presbitero and Wiriadinata 2020). For the U.S., the standard r < g condition holds up to a debt ratio of 220% (the upper bound), but the binding free-lunch condition R < G − φ limits fiscal space to 110%. Deficits beyond the 2%-of-GDP limit must be financed by future tax increases or spending cuts, even though R < G throughout the range. The Japan calibration illustrates the ZLB regime: with a debt ratio already above 200%, the fiscal multiplier effect on inflation is large enough that the backward-bending locus applies, and Japan’s economy lies in the inverted region.

Q6. How does the analysis extend to aggregate risk, capital crowding-out, and debt maturity?

With aggregate risk, the free-lunch condition R < G − φ remains informative: when the condition holds on average, free-lunch policies can be designed with probability approaching one; when it fails on average, no free lunch is possible. The risk extension follows Mehrotra and Sergeyev (2020) and confirms numerically that the deterministic condition provides a valid signal for the stochastic case. Adding capital and crowding-out (Section 7.2) yields a counterintuitive finding: greater crowding-out of capital actually increases fiscal space by reducing the sensitivity of interest rates to debt (lower φ), because each additional unit of government debt displaces private capital rather than reducing convenience yields as sharply. Regarding debt maturity: issuing long-term debt reduces fiscal space at low debt levels (locking in higher interest costs), but increases it at high debt levels; this suggests that QE-style maturity shortening may constrain fiscal space as debt rises. These extensions confirm that the φ parameter — and the R < G − φ condition — is robust to a range of model ingredients, making it a practically useful criterion beyond the baseline model.

Key Concepts

free-lunch fiscal policy : a permanent increase in the primary deficit that raises steady-state debt to a new higher level without requiring any future tax increases or spending cuts; feasible only when $R^(b_0) < G^ - \varphi(b_0)$, which is strictly tighter than the standard r < g condition when φ > 0.

debt-rate sensitivity (φ) : the semi-elasticity of R − G with respect to the log of debt, capturing how much the endogenous convenience yield on government bonds falls (and hence interest rates rise) as the debt supply increases; the paper’s addition to the standard r < g framework that tightens the sustainability condition from R < G to R < G − φ; estimated empirically from convenience yield demand curves.

deficit-debt diagram : the hump-shaped locus of sustainable steady-state combinations of the primary deficit and the debt-to-GDP ratio; the left (increasing) branch is the free-lunch region where fiscal expansion is self-sustaining, and the right (decreasing) branch is the conventional region where fiscal expansion requires future tax increases.

inverted ZLB fiscal regime : the case where the nominal interest rate is zero and the deficit-debt locus bends backward, so that higher deficits reduce rather than increase the debt ratio; occurs when the fiscal multiplier is large enough that deficit-induced nominal growth more than offsets the direct debt accumulation effect; found to apply to Japan as of December 2019 but not the U.S.

convenience yield : the non-pecuniary benefit savers derive from holding government bonds (capturing liquidity, safety, and regulatory premia), modeled as the utility function v(b) for savers; the mechanism making R endogenous to debt: as debt supply rises, the marginal convenience yield v’(b) falls, pushing R toward G and shrinking fiscal space.

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