**Risk, Discounting, and Dynamic Efficiency
**

In the presence of risk, a comparison of the risk-free interest rate and the expected growth rate is insufficient to assess whether an economy is dynamically efficient or inefficient. Stochastic discount factors—not risk-free interest rates—enter the government’s budget constraint, even if debt is safe.

These points are made, for example, by Andrew **Abel**, N. Gregory **Mankiw**, Lawrence **Summers**, and Richard **Zeckhauser** (Assessing Dynamic Efficiency: Theory and Evidence, REStud 56(1), 1989),

the issue of dynamic efficiency can be resolved by comparing the level of investment with the cash flows generated by production after the payment of wages … dynamic efficiency cannot be assessed by comparing the safe rate of interest and the average growth rate of the capital stock, output, or any other accounting aggregate,

or Henning **Bohn** (The Sustainability of Budget Deficits in a Stochastic Economy, JMCB 27(1), 1995),

discounting at the safe interest rate is usually incorrect. … popular fiscal policy “indicators” like deficit levels or debt-GNP ratios may provide very little information about sustainability. … the intertemporal budget constraint imposes very few restrictions on the average primary balance.

Recent work in which these themes appear include papers by Zhengyang **Jiang**, Hanno **Lustig**, Stijn **Van Nieuwerburgh**, and Mindy **Xiaolan** (Manufacturing Risk-free Government Debt, NBER wp 27786, 2020), Robert **Barro** (r Minus g, NBER wp 28002, 2020), or Stan **Olijslagers**, Nander **de Vette**, and Sweder **van Wijnbergen** (Debt Sustainability when r−g<0: No Free Lunch after All, CEPR dp 15478, 2020).

**Intergenerational Risk Sharing**

With overlapping generations the way the government manages its debt has implications for intergenerational risk sharing, see for example Henning **Bohn** (Risk Sharing in a Stochastic Overlapping Generations Economy, mimeo, 1998), Robert **Shiller** (Social Security and Institutions for Intergenerational, Intragenerational, and International Risk Sharing, Carnegie-Rochester Conference on Public Policy 50, 1999), or Gabrielle **Demange** (On Optimality of Intergenerational Risk Sharing, Economic Theory 20(1), 2002).

**Long-Run Debt Dynamics and Fiscal Space**

Dmitriy **Sergeyev** and Neil **Mehrotra** (Debt Sustainability in a Low Interest World, CEPR dp 15282, 2020) offer an analysis of long-run debt dynamics under the assumption that the primary surplus systematically, and strongly responds to the debt-to-GDP ratio such that the government’s intertemporal budget constraint is necessarily satisfied:

Population growth and productivity growth have opposing effects on the debt-to-GDP ratio due to their opposing effects on the real interest rate. Lower population growth leaves the borrowing rate unchanged while directly lowering output growth, shifting the average debt-to-GDP ratio higher. By contrast, when the elasticity of intertemporal substitution is less than one, a decline in productivity growth has a more than a one-for-one effect on the real interest rate, lowering the cost of servicing the debt and thereby reducing the average debt-to-GDP ratio. To the extent that higher uncertainty accounts for low real interest rates, we find that

the variance of the log debt-to-GDP ratio unambiguously increases with higher output

uncertainty. However, uncertainty also has an effect on the mean debt-to-GDP ratio that

depends on the coefficient of relative risk aversion. Higher uncertainty lowers the real

interest rate but this effect may be outweighed by an Ito’s lemma term due to Jensen’s

inequality that works in the opposite direction.

Sergeyev and Mehrotra also consider the effects of rare disasters as well as of a maximum primary surplus which implies that debt becomes defaultable and the interest rate on debt features an endogenous risk premium, generating the possibility of a “tipping point” with a slow moving debt crises as in Guido Lorenzoni and Ivan Werning (Slow Moving Debt Crises, AER 109(9), 2019).

Ricardo **Reis** (The Constraint on Public Debt when r<g But g<m, mimeo, 2020) analyzes a non-stochastic framework under the assumption that the marginal product of capital, m, exceeds the growth rate, g, which in turn exceeds the risk-free interest rate, r. Reis considers the case where m is the relevant discount rate, for example because r features a liquidity premium:

there is still a meaningful government budget constraint once future surpluses and debt are discounted by the marginal product of capital.

He shows the following:

- The debt due to a one-time primary deficit can be rolled over indefinitely and disappears asymptotically as long as r<g.
- With permanent primary deficits that grow at the same rate as debt and output, the government’s intertemporal budget constraint features a bubble component due to r<m. This corresponds to the usual seignorage revenue measure (see p. 173 in Niepelt, Macroeconomic Analysis, 2019).
- Suppose that from tomorrow on, the primary deficit and debt quotas are given by d and b, respectively. Then, the present value of total net revenues in the government’s budget constraint equals [- d + (m – r)*b] / (m-g). Both m>g and g>r relax the constraint, as does a lower r.
- Along a balanced growth path, b = [- d + (m – r)*b] / (m-g) and thus, d = (g-r)*b where d is assumed to be positive. Reis argues that b cannot be larger than total assets relative to GDP. Accordingly, the deficit cannot exceed total assets times (g-r).

Reis concludes that most of the bubble component “has already been used.” In addition to developing a model that yields m>g>r in equilibrium he also discusses the role of inflation (stable inflation generates fiscal space because it renders debt safer and thus increases demand for debt) and inequality (more inequality increases fiscal space).

**Blanchard’s Presidential Address**

In his presidential address, Olivier **Blanchard** (Public Debt and Low Interest Rates, AER 109(4), 2019) argues that the risk-free interest rate has fallen short of average US growth rate (and similarly, in other countries). Importantly—and implicitly addressing Abel, Mankiw, Summers, Zeckhauser, and Bohn (see above)—he also argues that risk is not that much of an issue as far as the sustainability of public debt is concerned:

Jensen’s inequality is thus not an issue here. In short, if we assume that the future will be like the past (admittedly a big if), debt rollovers appear feasible. While the debt ratio may increase for some time due to adverse shocks to growth or positive shocks to the interest rate, it will eventually decrease over time. In other words, higher debt may not imply a higher fiscal cost.

Most of his formal analysis doesn’t focus on debt though. Instead he analyzes the effects of risk-free social security transfers from young to old in a stochastic OLG economy. (There are close parallels between debt and such transfers to the old that are financed by contemporaneous taxes on the young.) In a steady-state with very low interest rates higher transfers have two effects on welfare, by (i) providing an attractive substitute for savings and by (ii) reducing capital accumulation and thereby lowering wages and raising the interest rate. If the economy initially is dynamically inefficient both effects are welfare improving because (i) capital accumulation with a low return is replaced by higher yielding intergenerational transfers and (ii) lower wages and higher interest rates are attractive, starting from a situation with a low interest rate. In a stochastic economy the first channel yields welfare gains as long as the growth rate exceeds the risk-free rate, and the second channel yields welfare gains (approximately) when the growth rate exceeds the marginal product of capital. Blanchard argues

[b]e this as it may, the analysis suggests that the welfare effects of a transfer may not necessarily be adverse, or, if adverse, may not be very large.

In the corresponding case with debt there is another effect because the intergenerational transfer is not risk-free; the size of this additional effect depends on the path of the risk-free interest rates (Blanchard assumes that the debt level is stabilized which requires net tax payments by the young to reflect the contemporaneous risk-free rate). In the slightly different case where debt is increased once and then rolled over, without adjusting taxes in the future, the sustainability and welfare implications are ambiguous and critically depend on the production function:

In the linear case, debt rollovers

typically do not fail[my emphasis] and welfare is increased throughout. For the generation receiving the initial transfer associated with debt issuance, the effect is clearly positive and large. For later generations, while they are, at the margin, indifferent between holding safe debt or risky capital, the inframarginal gains (from a less risky portfolio) imply slightly larger utility. But the welfare gain is small … . In the Cobb-Douglas case however, this positive effect is more than offset by the price effect, and while welfare still goes up for the first generation (by 2 percent), it is typically negative thereafter. In the case of successful debt rollovers, the average adverse welfare cost decreases as debt decreases over time. In the case of unsuccessful rollovers, the adjustment implies a larger welfare loss when it happens. If we take the Cobb-Douglas example to be more representative, are these Ponzi gambles, as Ball, Elmendorf, and Mankiw (1998) have called them, worth it from a welfare viewpoint? This clearlydepends on the relative weight the policymaker puts on the utility of different generations[my emphasis].

Blanchard argues that the marginal product of capital may be smaller than commonly assumed, implying that it is more likely that the welfare effects working through (ii) are positive (those working through (i) are very likely positive). Finally, he also presents some additional potential arguments pro and con higher public debt.

Blanchard’s work has attracted substantial criticism, for instance at the January 2020 ASSA meetings (see this previous post). In a short paper presented at the meetings, Johannes **Brumm**, Laurence **Kotlikoff**, and Felix **Kubler** (Leveraging Posterity’s Prosperity?) point out that a negative difference between average interest and growth rates is not necessarily indicative of dynamic inefficiency (see the discussion above) and that Blanchard’s analysis disregards tax distortions as well as the welfare effects from intergenerational risk sharing (again, see above):

To see the distinction between risk-sharing and a Ponzi scheme, modify B’s two-period model to include agents working when old if they don’t randomly become disabled. Now workers face second-period asset income and labor earnings risk. The government has no safe asset in which to invest. If it borrows, invests in capital, and taxes bond holders its excess return, “safe” debt is identical to risky capital. But if the net taxes are only levied on the non-disabled, bonds become a valued risk-mitigating asset and their return can be driven far below zero. This scheme could be, and to some extend it is, implemented through progressive taxation. If, observing this gap between growth and safe rates, the government decides to institute an “efficient” Ponzi scheme with a fixed pension benefit financed on a pay-go basis by taxes on workers, net wages when young will be more variable, raising generation-specific risk and potentially producing an outcome in which no generation is better off and at least one is worse off.

Brumm, Kotlikoff, and Kubler also note that the effective interest rate at which US households are borrowing is much higher than the borrowing rate of the government; this undermines Blanchard’s approach to gauge the welfare implications. And they point out that the scheme suggested by Blanchard could harm other countries by reducing global investment.

Jasmina **Hasanhodzic** (Simulating the Blanchard Conjecture in a Multi-Period Life-Cycle Model) simulates a richer OLG model and rejects the Blanchard conjecture of Pareto gains due to higher transfers:

It shows that the safe rate on government debt can, on average, be far less than the economy’s growth rate without its implying that ongoing redistribution from the young to the old is Pareto improving. Indeed, in a 10-period, OLG, CGE model, whose average safe rate averages negative 2 percent on an annual basis, welfare losses to future generations resulting from the introduction of pay-go Social Security, financed with a 15 percent payroll tax, are enormous—roughly 20 percent measured as a compensating variation relative to no policy.

Relative to Blanchard’s simulations, her model implies more negative consequences of crowding out on wages, a higher tax burden from the transfer scheme, and more induced old-age consumption risk.

Michael **Boskin** (How, When and Why Deficits Are Dangerous) offers a broad discussion of potential weaknesses of Blanchard’s analysis. Richard **Evens** (Public Debt, Interest Rates, and Negative Shocks) questions Blanchard’s simulations on calibration grounds and notes that he couldn’t replicate some of Blanchard’s findings.

On his blog, **John Cochrane** argues along similar lines as Ricardo Reis: Even if r<g, expected primary deficits are so large that debt quotas will explode nevertheless.

**Note**: This post was updated several times.