C63 | Macro Paper Warehouse

An endogenous gridpoint method for distributional dynamics

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

This paper introduces the Distributional Endogenous Gridpoint Method (DEGM), a novel numerical technique for solving the distributional dynamics that arise in heterogeneous agent macroeconomic models. The core problem is how to efficiently update the distribution of agents over the state space as the economy evolves. The dominant existing approach — the “lottery method” of Young (2010) — discretizes the state space and represents policy functions as lotteries over nearby gridpoints, producing a transition matrix that is linear in optimal policies. This linearity renders the lottery method incapable of capturing nonlinear effects in distributional dynamics, a limitation that becomes quantitatively significant for higher-order perturbation solutions.

DEGM extends Carroll’s (2006) endogenous gridpoint method from individual optimization to the distributional level. Rather than discretizing the density and integrating forward, DEGM works directly on the cumulative distribution function (CDF). The key insight is that when the policy function is monotone — as savings functions typically are — the endogenous gridpoints generated by the policy function trace out exact points on the post-policy CDF without requiring integration. Specifically, if A*_{i,j} = a*(A_i, Y_j) are optimal asset choices from grid point A_i at income Y_j, then the CDF values at those endogenous points are known analytically as F_t(A_i | Y_j). An interpolant using shape-preserving splines constructed through these points allows evaluation of the updated CDF at any point without integration. The income transition step is handled separately via standard quadrature over the discretized income process.

The paper demonstrates DEGM’s performance with two applications. First, in the Aiyagari (1994) economy, DEGM converges to the stationary equilibrium an order of magnitude faster than the lottery method in terms of gridpoints. At nk=40 gridpoints, the lottery method deviates from the benchmark capital stock by 1.72% and the wealth Gini by 2.24% (for nh=5), while DEGM deviates by only 0.09% and 0.12% respectively. Both methods converge to the same solution as the number of gridpoints increases, but DEGM reaches this limit far faster.

Second, the authors introduce a Krusell-Smith style model with aggregate investment risk (capital depreciation shocks calibrated following Barro, 2006, as a 0.4% quarterly probability of 7.5% capital destruction causing a 10% annual GDP drop) as a new baseline for studying aggregate nonlinearities with household heterogeneity. This model overcomes the near-linearity of aggregate capital dynamics in the original Krusell-Smith specification. Using a third-order perturbation solution with DEGM, aggregate investment risk lowers the capital stock by 5 to 11 basis points and increases wealth inequality by up to 11 basis points relative to the non-stochastic steady state, depending on idiosyncratic income risk calibration. The lottery method systematically mispredicts these effects: it always predicts a decrease in wealth inequality in the presence of investment risk, while DEGM predicts an increase. At third order, the lottery method predicts wealth Gini changes of +2.0 bp (persistent calibration) and -149.7 bp (transitory calibration), while DEGM predicts +10.7 bp and +2.1 bp respectively.

The mechanism for increased inequality under investment risk is heterogeneous: for less wealthy households the substitution effect dominates (they reduce saving more in response to risky returns), while for wealthy households the income effect is stronger and precautionary saving motives dominate. The lottery method, by making the distributional transition matrix linear in policies, zeros out the second derivative of the transition matrix with respect to the policy function, missing the term capturing how the density at the pre-image of each asset level is affected nonlinearly. DEGM’s cubic spline interpolant captures all nonlinearities up to third order, enabling economically meaningful results that qualitatively differ from lottery-method predictions on wealth inequality.

Q: What is the fundamental numerical problem that DEGM solves? A: Evolving the distribution of agents forward over time in heterogeneous agent models requires evaluating a Kolmogorov forward equation, which naively demands numerical integration. The lottery method avoids integration by discretizing the state space and expressing transitions as a linear matrix operation, but this forces the distributional dynamics to be linear in optimal policies. DEGM avoids integration by exploiting policy function monotonicity: the endogenous policy gridpoints are the interpolation nodes, so the CDF update requires only interpolation, not integration. This preserves nonlinear effects up to the order of the splines used.

Q: How does DEGM handle the borrowing constraint and the resulting mass point? A: Savings policy functions are typically weakly monotone: constant at the borrowing constraint for sufficiently poor households, then strictly monotone above a threshold. DEGM accommodates this by starting the endogenous grid at the EGM solution corresponding to the borrowing constraint (the threshold a_j above which the policy is strictly monotone), restoring strict monotonicity on the relevant domain. The mass point at the borrowing constraint is captured by evaluating F_t(a_j, Y_j). Echoes of the borrowing constraint diminish as the number of income states increases, and in practice 10 income gridpoints are sufficient to smooth them.

Q: How much faster does DEGM converge relative to the lottery method for the stationary equilibrium? A: In the Aiyagari economy with nk=40 asset gridpoints, the lottery method’s capital stock deviates from the benchmark by 1.72% and the wealth Gini by 2.24% (nh=5), while DEGM deviates by only 0.09% and 0.12% respectively — roughly a 20-fold improvement in accuracy for the same gridpoints. At nk=80, the lottery method still shows 0.56%/0.78% deviations while DEGM shows 0.03%/0.00%. Although for a fixed number of gridpoints the lottery method is faster in wall-clock time (0.35s vs 0.82s at nk=40, nh=20), DEGM is faster for a given level of accuracy because it requires far fewer gridpoints.

Q: Why does the lottery method fail at higher-order perturbations? A: The lottery method constructs its transition matrix as a piecewise linear function of the optimal policy a*, so its second derivative with respect to a* is zero. As a result, it misses the second term in the second-order derivative of the end-of-period CDF: the term involving the derivative of the density at the pre-image of each asset level times the squared linear policy effect. This missing nonlinearity becomes quantitatively important at second and third order. DEGM’s cubic hermitian spline interpolant captures all nonlinearities up to third order, allowing it to correctly represent how the distribution responds nonlinearly to aggregate shocks.

Q: What does the paper find about the effect of aggregate investment risk on the capital stock and wealth inequality? A: Using a third-order perturbation solution with DEGM, aggregate investment risk lowers the capital stock by 5 to 11 basis points from the non-stochastic steady state, depending on whether income risk is persistent or transitory (DEGM third-order: -4.7 bp persistent, -11.4 bp transitory). Wealth inequality increases by up to 11 basis points (DEGM third-order: +10.7 bp persistent, +2.1 bp transitory). The lottery method diverges dramatically at third order, predicting Gini changes of +2.0 bp and -149.7 bp for the persistent and transitory calibrations respectively, compared to DEGM’s +10.7 bp and +2.1 bp.

Q: What is the mechanism through which aggregate investment risk increases wealth inequality? A: The mechanism operates through heterogeneous saving responses across the wealth distribution. For less wealthy households, capital income is a small share of total income, so the substitution effect of risky returns dominates: higher investment risk reduces their incentive to save. For wealthy households, capital income is central, so the income effect is stronger and precautionary saving motives intensify. A capital depreciation shock upon realization compresses the wealth distribution, but the risk of such a shock increases inequality on average because it disproportionately reduces saving among poorer households.

Q: How do the authors extend DEGM to handle aggregate risk and higher-order perturbations? A: The authors follow Reiter (2009) in including the distribution and value functions in the state space, defining a nonlinear difference equation over these objects. Higher-order perturbation of this system proceeds using the algorithms of Andreasen et al. (2018) and Levintal (2017), with second-order terms solved via a generalized Sylvester equation using Kim et al.’s (2008) doubling algorithm. The implementation handles up to 3,200 variables at second order and 220 variables at third order. For the second-order solution, the Bayer-Luetticke (2020) state-space reduction and its refinement in Bayer et al. (2024) yield results identical to the full unreduced system.

Q: What is the state-space reduction procedure and how much does it compress the system? A: The full system uses 402 states and 412 controls (persistent calibration). A copula representation of the distribution reduces this to 213 states and 412 controls; adding DCT compression of the value function gives 213 states and 98 controls; further adding a factor representation from the first-order solution yields 111 states and 98 controls — a 75% reduction. The R-squared-like IRF statistic remains 1.00 across all reductions, and ergodic moments are identical (capital: 25.54, Gini: 0.61 for the persistent calibration).

Q: Does DEGM produce different first-order impulse responses than the lottery method? A: For first-order perturbations, DEGM and the lottery method converge to the same solution as the number of gridpoints increases, but DEGM converges faster. For the first-order dynamics of the wealth distribution (wealth Gini IRFs), DEGM reaches convergence with nk=40 gridpoints while the lottery method requires nk=160. For aggregate capital stock IRFs, both methods converge quickly at first order. Quantitative differences become significant only at second and higher orders.

Q: What calibration is used for the investment risk model? A: Capital depreciation deviates from its steady-state value by a shock with second moment sigma_delta = 0.005 and third moment tau_delta = 0.012. This corresponds to a 0.4% quarterly probability that a disaster destroys 7.5% of the capital stock and causes a 10% drop in annual GDP, consistent with the evidence in Barro (2006). The model is solved under both a persistent income calibration (beta=0.98, rho=0.98, sigma_epsilon=0.14, implied Gini=0.66) and a transitory income calibration (beta=0.99, rho=0.88, sigma_epsilon=0.18, implied Gini=0.42).

Distributional Endogenous Gridpoint Method (DEGM): A numerical method for evolving the joint CDF of agents over the state space by constructing an interpolant at endogenous gridpoints A*_{i,j} = a*(A_i, Y_j) — the optimal policy values — at which CDF values are known analytically as F_t(A_i | Y_j), thus updating the distribution through interpolation rather than integration and preserving nonlinearities up to the order of the spline.

Lottery Method (LM): Young’s (2010) standard technique that replaces the continuous distribution with a discrete counterpart and represents optimal policy functions as probability weights over nearby gridpoints, yielding a single transition matrix A* such that f_{t+1} = f_t * A*. The transition matrix is linear in optimal policies, which zeroes out the second derivative of the distributional dynamics with respect to policies and causes systematic misprediction of distributional dynamics under higher-order perturbation.

Kolmogorov Forward Equation (Distributional Dynamics): The law of motion for the joint CDF F_t(a, y) describing how the distribution of households over assets and income evolves given optimal policies and the income transition process. In DEGM, this equation is split into a sub-period for asset choices (where endogenous gridpoints allow integration-free updating) and a sub-period for income transitions (handled by quadrature over the discretized income process).

Higher-Order Perturbation Solution: A Taylor expansion of the model’s nonlinear equilibrium conditions around the non-stochastic steady state beyond first order. Second-order solutions capture precautionary motives and mean deviations from the steady state; third-order solutions additionally capture asymmetric effects of shocks, requiring DEGM’s nonlinear distributional representation to produce accurate results.

Aggregate Investment Risk (Capital Depreciation Shocks): Shocks to the aggregate capital depreciation rate calibrated following Barro (2006) as a 0.4% quarterly probability of a disaster that destroys 7.5% of the capital stock and causes a 10% annual GDP drop. Proposed as a replacement for near-linear Krusell-Smith aggregate productivity shocks to generate genuine nonlinearities in aggregate capital dynamics while remaining equally parsimonious.

State-Space Reduction: A sequence of compression techniques — copula representation of the wealth distribution, discrete cosine transform (DCT) compression of the value function, and factor representation from the first-order solution — that reduce the Reiter (2009) system from 402 states and 412 controls to 111 states and 98 controls (a 75% reduction) with no measurable loss of accuracy in impulse responses or ergodic moments.

Shape-Preserving Interpolation: Interpolation methods (linear spline or piecewise cubic hermitian splines) that maintain the monotonicity of the CDF when constructing the interpolant from endogenous gridpoints. Cubic hermitian splines additionally preserve differentiability, making the distributional dynamics smooth enough for third-order perturbation and capturing all nonlinear effects that the lottery method misses.

From Interaction to Business Fluctuations: How Credit Network Explains Cycles

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

Layer 1: Overview

This paper investigates how the endogenous structure of credit, deposit, and interbank networks shapes business cycle fluctuations and large financial crises in the U.S. economy. Ciola and Tedeschi build and estimate a microfounded heterogeneous-agents macroeconomic model in which households, firms, and banks interact through decentralized matching in three markets — deposits, credit, and interbank lending — with agents choosing partners based on both posted interest rates and the size of the counterpart, generating a preferential-attachment mechanism that endogenously concentrates the financial sector. The structural parameters governing network formation are estimated on U.S. quarterly interest rate and GDP growth data from 1947 to 2019 via an Extended Method of Simulated Moments (EMSM) procedure combined with a Bayesian Adaptive Random Walk Metropolis–Hastings sampler; the calibrated model reproduces the empirical autocorrelation structure of these series. The model’s key finding is that preferential attachment endogenously concentrates roughly three-quarters of deposits, credit, and interbank transactions into a single hub bank, whose dominance raises markups, suppresses deposit rates, and depresses aggregate capital accumulation relative to the initial symmetric state. Bank runs against this hub — rare but endogenously generated when households reallocate deposits simultaneously — collapse the interbank market completely and produce deep recessions that last multiple quarters, with recovery requiring approximately five years.

Summary of a forthcoming paper, AI-assisted and human-reviewed. See the linked original for the authoritative claims and full conditions.

In depth

Q1. What is the model’s core structure and how do agents interact?

The model consists of a fixed number of households (N_H = 1,000), banks (N_I = 10), and firms (N_F = 1,000) who interact in deposit, credit, and interbank markets through a decentralized preferential-attachment matching mechanism in which agents assess both current interest rates and the size of potential counterparts. Households deposit savings in a single bank chosen based on a fitness index combining the bank’s promised deposit rate and its size (used as a proxy for long-run quality), and they search for a new partner each period with probability ζ_H. Firms borrow from one bank at a time, also choosing based on a fitness that weighs the promised profit share against bank size, and switch with probability ζ_F. Banks set interest rates in all three markets to maximize expected profits, exploiting their monopolistic power (higher when they are larger), subject to a balance sheet constraint that links deposits, credit extended to firms, and interbank borrowing. The interbank market exists specifically to cover unexpected deposit withdrawals: when a bank’s deposits fall below its outstanding credit, it borrows in the interbank market or closes credit lines.

Q2. How does the estimation methodology work and what parameters does it identify?

The paper employs the Extended Method of Simulated Moments (EMSM) of Smith (1993) and Gourieroux et al. (1993), which minimizes the weighted distance between the coefficients of a VAR auxiliary model estimated on observed U.S. data and on H simulated time series generated from a given structural parameter vector, with the optimal weighting matrix set to the inverse of the Newey–West covariance of the auxiliary parameter estimates. Because gradients of the criterion function are not analytically available for this nonlinear agent-based model, the authors use a two-step approach: first, a Particle Swarm Optimization (PSO) algorithm explores the parameter space to locate a neighborhood of the global minimum; second, a Bayesian Adaptive Random Walk Metropolis–Hastings (ARWMH) algorithm generates posterior draws from the structural parameter distribution using the chi-square distributional properties of the EMSM criterion function. The estimated structural parameters include the nine network formation parameters {ω_X, ζ_X, ψ_X} for each of the three markets — governing competition intensity, switching probability, and the weight agents assign to counterpart size — while the production coefficient (α = 0.37) and household discount factor (β = 0.997) are calibrated directly to U.S. labor share and real interest rate data. Estimation uses 1947:Q1–2019:Q4 U.S. real GDP growth and real interest rate data; with three VAR lags and d = 9 structural parameters, the overidentification chi-square test can be assessed.

Q3. What are the long-run dynamics and how does the financial network concentrate?

Starting from an equal distribution of agents across banks, the model converges to a pseudo-steady-state in which a single hub bank intermediates approximately three-quarters of deposits, credit lines, and interbank transactions, because the preferential-attachment mechanism is self-reinforcing: larger banks attract more depositors (providing more stable funding), more firms (generating more profit), and more interbank counterparts, which further enlarges their size and attractiveness. This concentration has clear aggregate consequences: as the hub’s monopolistic power grows, it widens the markup over the perfect competition interest rate in the credit market and the markdown below it in the deposit market, reducing the deposit rate paid to households and thereby depressing household capital accumulation. Simulations across 1,000 independent replicas show that the aggregate production level in the pseudo-steady-state is below the initial competitive equilibrium, credit and interbank interest rates rise, and approximately 10% of total capital circulates through the interbank market as periphery banks rely on the hub for liquidity provision.

Q4. How do cyclical fluctuations and crises emerge endogenously?

Business cycles arise from the continuous reallocation of household deposits across banks, which generates endogenous liquidity shocks that do not require an exogenous crisis trigger: when a critical mass of households simultaneously reallocates away from the hub — a rare but endogenous event driven by the stochastic matching process — the hub faces a severe liquidity shortage, must close credit lines and interbank lending, and produces a systemic economic contraction. In a representative 100-year simulation, aggregate production fluctuates around a stable trend with mild recessions most of the time, but the model occasionally generates a catastrophic bank run against the hub. When this occurs, the hub’s weighted degree in all three markets collapses to near zero within one or two quarters, the interbank market freezes completely, and firm production stops because firms cannot immediately reallocate their credit demand to alternative banks. The impulse response to a sudden reduction in hub deposit centralization shows that aggregate production falls sharply in the short run (as credit contracts) and only surpasses its pre-run level after approximately five years (20 quarters).

Q5. What does the VAR impulse response analysis reveal about recovery dynamics?

An estimated VAR on all simulations — with aggregate production and the volume, centralization, and interest rates of each of the three markets as endogenous variables — shows that a negative shock to deposit market centralization (i.e., a bank run against the hub) triggers an immediate spike in deposit interest rates (as competing banks compete for the displaced funds), a contraction in credit and interbank supply (as periphery banks lack sufficient liquidity to expand), and a rise in credit interest rates (as the pool of surviving credit lines is concentrated in the most profitable projects). In the medium run, higher deposit rates promote household capital accumulation, which ultimately expands the aggregate supply of productive capital; at the same time, the dissolution of the old hub reduces the sector’s average monopolistic markup, permanently lowering credit market interest rates. This self-correcting mechanism underlies the five-year recovery window and also illustrates why prompt policy intervention during hub-collapse crises is particularly effective — early stabilization prevents the reinforcing deposit-withdrawal spiral that deepens the contraction.

Q6. What is the paper’s contribution relative to existing macroeconomic network literature?

The paper makes three distinct contributions over prior agent-based macroeconomic network models: first, it treats households as active depositors whose reallocation choices generate endogenous liquidity shocks rather than simply passive shock absorbers; second, it models banks as profit-maximizing agents that optimally set interest rates exploiting market power rather than assuming perfect competition or regulatory constraints; and third, it produces a Bayesian estimator of all structural parameters rather than relying on calibration to observed moments. Prior work in this tradition (Delli Gatti et al. 2010; Riccetti et al. 2013; Lenzu and Tedeschi 2012) typically either omits households from the deposit market or assumes exogenous mechanisms of crisis formation. By endogenizing all three sources of network dynamics — deposit, credit, and interbank — and estimating the model on U.S. data, the paper provides a framework in which large financial crises emerge as intrinsic system properties rather than imposed scenarios, and quantifies the structural parameters driving them.

Key Concepts

preferential attachment : a matching mechanism in which agents preferentially form links with larger counterparts; in this model it causes households and firms to favor large banks, endogenously concentrating the financial sector into a hub-and-spoke structure with a dominant hub bank.

hub bank : the single largest financial intermediary that endogenously emerges in the model’s long-run equilibrium, intermediating approximately three-quarters of deposits, credit lines, and interbank transactions; its size confers monopolistic power but makes it the systemic node whose failure triggers economy-wide crises.

Extended Method of Simulated Moments (EMSM) : the estimation strategy used to identify the nine network formation structural parameters; it minimizes the weighted distance between VAR coefficients estimated on observed U.S. data and on model-simulated data, with a Bayesian ARWMH sampler used to generate the posterior distribution given the chi-square-distributed criterion function.

endogenous bank run : the crisis mechanism in this model — a simultaneous reallocation of household deposits away from the hub, triggered by the stochastic matching process rather than an external shock, that freezes the interbank market and produces a deep recession lasting approximately five years (20 quarters) in impulse response analysis.

Monetary Policy and the Drifting Natural Rate of Interest

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

This paper analyzes how monetary policy should respond to a long-run natural interest rate that can drift permanently — following a bounded random walk with upper bound 3 percent and lower bound 0 percent — when the zero lower bound (ZLB) on nominal interest rates is a binding constraint. The central result is that the long-run neutral rate (the real policy rate consistent with stable inflation in long-run equilibrium) should fall more than one-for-one with the long-run natural rate as the latter approaches zero, because the mere risk of future ZLB episodes — even when the economy is currently away from the ZLB — imparts a persistent downward bias on inflation expectations that can only be offset by maintaining a pre-emptive expansionary bias. Quantitatively, the model implies that the neutral rate should be zero as soon as the long-run natural rate falls to 75 basis points — well above the near-zero estimates prevailing in the late 2010s — and that the ZLB would bind one-third of the time under optimal policy when the natural rate fluctuates between 0 and 3 percent. Price level targeting with a 10-basis-point upward drift closely approximates optimal commitment policy and has the advantage of not requiring knowledge of the natural rate level.

Summary of a forthcoming paper, AI-assisted and human-reviewed. See the linked original for the authoritative claims and full conditions.

In depth

Q1. What empirical fact motivates the model?

Empirical analyses of the long-run natural rate — the real interest rate prevailing over a long-run equilibrium in which nominal rigidities are absent — consistently find that it is time-varying in a manner best described by a random walk, meaning it can drift without reverting to a constant long-run level. The paper cites Holston, Laubach, and Williams (2017), Fiorentini et al. (2018), and Hamilton et al. (2016) as the main empirical references. Holston et al. (2017) place the long-run natural rate at between 0 and 1 percent in the U.S. and possibly slightly negative in the euro area as of 2016. The paper draws one central lesson: because the natural rate is time-varying and its future level is uncertain, a model with constant natural rate will give unreliable guidance for monetary policy, especially at low natural rate levels near zero.

Q2. What is the model and what are the key equilibrium concepts?

The paper embeds a new Keynesian model in which the long-run natural rate follows a bounded random walk with upper bound 3 percent and lower bound 0 percent, calibrated to post-WWII U.S. TFP data, and studies optimal monetary policy under commitment while imposing the zero lower bound. A critical distinction separates two notions of the long-run equilibrium interest rate: the “long-run natural rate” (denoted ¯r) is the real rate that would prevail in flexible-price equilibrium, determined by fundamentals outside the central bank’s control; the “neutral rate” (r*) is the real policy rate consistent with stable inflation in the long run, which the central bank operationally targets. The two coincide in standard models with constant ¯r, but diverge in this paper because ZLB risk drives a wedge between them.

Q3. What is the main theoretical result?

Under optimal commitment, the neutral rate r should fall more than one-for-one with the long-run natural rate ¯r — that is, the central bank should maintain a negative gap (r < ¯r) that widens as ¯r falls toward zero — because permanent downward movements in ¯r make future ZLB binding episodes permanently more likely, creating a persistent downward bias on inflation expectations that requires pre-emptive accommodation even in periods when the ZLB is not currently binding.** This result contrasts with the existing literature on optimal commitment at the ZLB, which has emphasized forward guidance — the promise to maintain low rates even after the economy recovers from a ZLB episode — as the primary stabilization tool. The paper shows that forward guidance alone is not sufficient when ¯r can permanently drift lower, because each downward drift permanently raises the probability of future ZLB episodes, reducing the central bank’s scope for fulfilling future inflation promises.

Q4. What are the quantitative implications?

The model implies that the neutral rate r reaches zero when the long-run natural rate ¯r is at 75 basis points — a level that was well above the near-zero estimates of ¯r prevailing at the end of the 2010s — and that the ZLB binds one-third of the time under optimal policy when ¯r fluctuates between 0 and 3 percent.* The 75 basis-point threshold means that a central bank operating in an environment where ¯r has declined to its estimated late-2010s levels would already be constrained to a neutral rate of zero under optimal policy. The one-third ZLB frequency is higher than what would be predicted by models with constant ¯r at typical calibrations, reflecting the permanent nature of ¯r shocks and their cumulative effect on the neutral rate.

Q5. What do the adjustment dynamics look like after a negative ¯r shock?

Following a permanent reduction in ¯r, the real policy rate adjusts gradually rather than immediately — remaining temporarily above the new long-run neutral rate during the transition — implying that monetary policy is contractionary along the adjustment path and that a permanent decline in ¯r is followed by a temporary disinflation before the economy settles at the new r.* This history-dependence of optimal commitment policy means the central bank does not immediately jump to the new, lower r* after a ¯r shock; it moves gradually, making the short-run policy stance more contractionary than the long-run position. The temporary disinflation is consistent with the general principle of history-dependence of optimal policy under commitment.

Q6. What role does price level targeting play?

Price level targeting variants — particularly a rule with an optimally chosen upward drift of 10 basis points — closely approximate the economic outcomes achieved under optimal commitment policy in the model, with the practical advantage that such rules do not require the central bank to know or estimate the current level of the long-run natural rate ¯r. The Eggertsson-Woodford (2003) price level target works well in models with constant ¯r by generating positive inflation expectations in the wake of deflationary ZLB episodes. Adding a small upward drift of 10 basis points strengthens this property under a drifting ¯r, because it provides additional buffer against the downward expectations bias that permanent ¯r drift generates. Under price level targeting rules, the neutral rate reaches the ZLB as soon as ¯r falls below 1 percent.

Key concepts

long-run natural rate (¯r) : the real interest rate prevailing over a long-run equilibrium in which nominal rigidities are absent; in this paper modelled as a bounded random walk with upper bound 3 percent and lower bound 0 percent, calibrated to post-WWII TFP data.

neutral rate (r)* : the real policy rate consistent with stable inflation in the long run; distinct from ¯r in this paper because ZLB risk drives a negative gap (r* < ¯r) that widens as ¯r approaches zero.

zero lower bound (ZLB) : the constraint that nominal policy rates cannot fall below zero; in this model the reason that permanent reductions in ¯r create a persistent downward bias on inflation expectations even when the ZLB is not currently binding.

expansionary bias : the paper’s finding that optimal commitment policy should maintain r* < ¯r — a pre-emptive accommodation away from the ZLB — to offset the downward bias on inflation expectations created by the risk of future ZLB episodes.

price level targeting : a monetary policy rule in which the central bank targets the price level rather than the inflation rate; shown in this paper to approximate optimal commitment policy and to have the practical advantage of not requiring knowledge of ¯r.