A Model of Post-2008 Monetary Policy
What this paper finds — and why it matters
Layer 1: Overview
Research question and motivation: Since 2008 the US economy has gone through two zero-lower-bound (ZLB) episodes (Dec 2008–Dec 2015 and Mar 2020–Mar 2022). Standard New Keynesian (NK) and monetarist models struggle with three broad facts about US inflation during these episodes, emphasized by Cochrane (2018): (1) no significant deflation, (2) little inflation volatility, and (3) no significant inflation following large quantitative-easing (QE) balance-sheet expansions. A fourth challenge is that money-market rates (federal funds, T-bills) were often below the interest rate on reserves (IOR rate), which many read as evidence of full satiation of reserve demand — undercutting any model relying on a monetary friction. Diba and Loisel build a model that can qualitatively account for all four facts and then draw out implications for policy normalization and the operational framework (floor system).
Model setup: They add banks and bank reserves to the basic NK model. Monopolistically competitive firms must borrow a fraction phi in (0,1] of their nominal wage bill from banks before producing (a cost channel); calibration uses phi=1. Households contain production workers and bankers; bankers produce real loans using their own labor and real reserves via a production function homogeneous of degree d in (0,1], so holding reserves reduces banking (labor) costs — i.e., reserves carry a convenience yield. The central bank sets TWO instruments directly: the IOR rate (I^m) and the nominal stock of reserves (M). A ZLB on the net IOR rate arises because non-interest vault cash is a perfect substitute for reserves. Calvo price rigidity (theta) is assumed.
Key analytical results: Under a permanent IOR-rate peg with an exogenous (or QE-rule) money supply, the model delivers a UNIQUE steady state and local-equilibrium determinacy, provided 1 <= I^m < I = 1/beta. Setting the IOR rate pins down real reserve demand, and given the exogenous nominal stock this pins down the price level; steady-state inflation equals the money growth rate. This rules out the Benhabib-Schmitt-Grohe-Uribe deflationary equilibria. The log-linearized model yields an IS equation, a modified Phillips curve (output enters net of real reserves, with delta_m and slope kappa depending on banking-cost cross-derivatives), and a reserves-demand equation. The characteristic roots satisfy 0 < rho < 1 < omega_1 < omega_2, so anticipated shocks decay exponentially with horizon — the opposite of the basic NK model (where 0<omega_1<1<omega_2 makes effects grow exponentially with ZLB duration). Hence deflation converges to a finite value kappa·z*/[beta·sigma·(omega_1-1)(omega_2-1)] rather than exploding, explaining no severe deflation and low inflation volatility. (In the basic NK model under their calibration, deflation reaches about 21% per year for an expected ZLB duration of two years.)
QE simulations (calibrated to US data, November 2010, start of QE2): Calibration: sigma=1 (log utility), eta=1 (unit Frisch), alpha=0.67, epsilon=6, theta=0.67, phi=1, net IOR rate = 25 bps p.a., benchmark net shadow-rate-minus-IOR spread (I - I^m) = 10 bps p.a. (alternatives 5 and 20 bps), beta=0.999 quarterly, reserves/loans ratio m/ell = 1/9, loan rate I^ell-1 = 3.25% p.a.; derived ical=0.0039, V_b=0.019. Two conditions make QE nearly non-inflationary: demand close to satiation (I^m close to I, Gamma_m near 0) and the expansion perceived as temporary. Results (Figure 1, 5-year expected duration): a single QE2 expansion ($1T to $1.6T over 3 quarters) lowers the I_t - I^m_t spread from 10 to 6.2 bps and raises annualized inflation by only 18 bps on impact. Double/triple/quadruple QE2 lower the spread to 4.5/3.5/2.9 bps and raise inflation by only 27/32/35 bps — strongly decreasing returns to QE. With a 5-bps steady-state spread the single-QE2 impact falls to 9 bps; with 20 bps it rises to 37 bps (inflation impact moves roughly one-for-one with the spread). Inflation impact scales roughly one-for-one with expected duration: single QE2 raises inflation 18 bps (5 yrs), 40 bps (10 yrs), 84 bps (20 yrs); up to 32xQE2 reaches 48/104/212 bps for 5/10/20 yrs (Table 1). The calibration makes omega_1 = 1.0003 (very close to 1) and omega_2 = 1.42.
Implications: A permanent reserve expansion would be fully inflationary (proportional long-run price rise) unless accompanied by a rise in money demand (e.g., a higher IOR rate). The 2021-22 inflation surge may partly reflect expansions coming to be seen as permanent plus adverse supply shocks raising the shadow rate I via a Fisher effect. Forward guidance about expansion duration is a powerful inflation-control tool. An extension with liquid government bonds reconciles non-satiation with T-bill rates below the IOR rate without changing any inflation implications. Normalization (IOR hikes and balance-sheet contraction) is always deflationary — no Neo-Fisherian effect. Under a floor system, determinacy holds for any non-negative IOR response to inflation (Taylor principle not required) and for a wide range of output responses (threshold 15.7 on the output coefficient under their calibration).
Layer 2: Deep Dive
What is the core modeling innovation relative to the basic New Keynesian model?
They introduce banks and bank reserves with a convenience yield: holding reserves reduces banks’ labor cost of making loans (banker production function f^b homogeneous of degree d in (0,1] in banker labor and reserves), and firms must prepay a fraction phi of their wage bill via bank loans (a cost channel). Crucially the central bank sets BOTH the IOR rate and the nominal stock of reserves, two instruments the Fed controls directly. This gives the model a ‘monetarist element’ while keeping NK price rigidity (Calvo theta).
Why does the model deliver determinacy and avoid the NK ZLB pathologies?
Because the central bank sets the money supply (exogenously or via a QE rule), the model has a unique steady state provided 1 <= I^m < 1/beta: setting the IOR rate pins down real reserve demand, and the exogenous nominal stock then pins down the price level. The third-order price-level dynamic equation has roots 0<rho<1<omega_1<omega_2, satisfying Blanchard-Kahn for one predetermined variable, so there is a unique bounded solution. Anticipated future shocks decay exponentially (weights omega_1^{-k}, omega_2^{-k} both <1), so deflation stays bounded and inflation volatility stays low. In the basic NK model the analogous roots are 0<omega_1<1<omega_2, so weights grow exponentially with ZLB duration, producing explosive deflation and volatility.
What exactly are the three (four) facts the model targets, and which mechanism handles each?
(1) No significant deflation and (2) little inflation volatility at the ZLB — handled by determinacy under a money-supply-setting central bank, giving bounded, duration-insensitive deflation. (3) No significant inflation after QE — handled by near-satiation (Gamma_m near 0, small steady-state spread) plus the expansion being temporary, so a large nominal-reserve increase is absorbed by a tiny fall in the IOR-vs-shadow-rate spread rather than by higher prices. (4) Money-market/T-bill rates below the IOR rate — handled by an extension where government bonds provide liquidity services to non-bank entities, generating T-bill returns below the IOR rate without requiring full reserve satiation.
What are the two key conditions for QE to be nearly non-inflationary, and how sensitive are the results?
Condition 1: demand for reserves is close to satiation, meaning I^m close to I (Gamma_m near 0) so the semi-elasticity of reserve demand is large and a flat Gamma_m absorbs large supply changes through small spread movements. Condition 2: the expansion is perceived as temporary. Sensitivity: the inflation impact moves roughly one-for-one with the steady-state I - I^m spread (single QE2 impact = 9, 18, 37 bps for spreads of 5, 10, 20 bps) and roughly one-for-one with expected duration (18, 40, 84 bps for 5, 10, 20 years). A permanent expansion would be fully (proportionally) inflationary in the long run.
How is the central spread calibrated given the shadow rate is unobservable, and why is that a limitation?
The shadow bond rate I is a rate on hypothetical bonds with no non-pecuniary services in zero net supply, hence unobservable. Using Nagel (2016) and the repo-T-bill spread (8 bps in Nov 2010), assuming the convenience yield of borrowed Treasuries is half that of T-bills held outright, they back out a net shadow rate I-1 of about 30-35 bps and an I - I^m spread of about 5 bps; to be conservative they set the benchmark spread to 10 bps (alternatives 5 and 20). The authors flag the unobservability of the relevant spread as a genuine limitation of the model’s quantitative QE implications and call for future work with observable spreads.
How does the liquid-government-bond extension reconcile non-satiation with T-bill rates below the IOR rate?
Workers derive utility from holding government bonds (a proxy for pension/money-market funds that hold bonds and supply financial services). Banks could use bonds instead of reserves for liquidity but choose not to in equilibrium, so the extended model’s equilibrium coincides with the benchmark for all common endogenous variables except the lump-sum transfer T_t. This lets the bond/T-bill return fall below the IOR rate (driven by strong non-bank demand, e.g., collateral or international reserve use) while reserve demand remains unsatiated, leaving all inflation results from Sections 3-4 intact.
What does the model imply for monetary-policy normalization and Neo-Fisherian effects?
In the log-linearized model under exogenous instruments, current and expected future IOR-rate hikes and balance-sheet contractions ALWAYS exert deflationary pressure: in the inflation solution (Equation 25), the coefficient on i^m_{t+k} is negative and on reserve growth mu_{t+k} is positive, because the unstable eigenvalues omega_1, omega_2 are positive real numbers >1 and delta_m·chi_y < 1. So the model has no Neo-Fisherian region (unlike some NK equilibria in Schmitt-Grohe-Uribe 2017 and Bilbiie 2022). The authors stress this hinges on the eigenvalues being positive reals; with complex or negative eigenvalues (as in MIU models) the sign could flip by horizon.
What does the model say about the floor system and the Taylor principle?
Under a floor system (nominal reserves exogenous, IOR rate set by a Taylor rule I^m = R(Pi, y)), local-equilibrium determinacy holds for ANY non-negative IOR response to current inflation (r_pi >= 0) — the Taylor principle is not required; even an IOR-rate peg works. If the rule also responds to output, a sufficient condition is r_y < (1 - delta_m·chi_y)/(delta_m·chi_i), whose right-hand side equals 15.7 under their calibration — comfortably above typical output coefficients (about an order of magnitude smaller), so determinacy is likely to prevail.
What robustness checks support the determinacy result?
Appendix C replaces the exogenous nominal reserve stock with a QE rule (reserves react to output and the price level): determinacy no longer holds for all parameter values but holds for all reasonable calibrations. Appendix D adds household cash via a cash-in-advance constraint: determinacy still holds under an exogenous IOR rate and exogenous monetary base, except for implausible calibrations. The QE simulation results are also stated to be insensitive to most parameters (e.g., raising theta to 0.75 only makes inflation impacts smaller) and to plausible variations in the loan-rate and reserves/loans targets.
How does this paper relate to and differ from closely related prior work?
It builds on Diba and Loisel (2021), which showed a small monetary friction resolves NK puzzles/paradoxes under an IOR peg. Reserve/banking-cost modeling is close to Curdia and Woodford (2011) and Ireland (2014), but with new analytical results (determinacy proof, closed-form inflation/output solution) and three differences: banking costs tied to time spent on banking, borrowers are firms borrowing the wage bill, and reserve demand is not satiated. It complements asset-side QE models (Gertler-Karadi 2011, Sims et al. 2023) by focusing on the liability side. Versus Andolfatto (2015), which links low inflation to full satiation, this paper generates low inflation WITHOUT full satiation. The determinacy analysis overlaps most with Piazzesi, Rogers, Schneider (2022).
What are notable caveats the authors themselves raise?
They state the model cannot explain why QE1 (starting from about $45 billion of reserves in 2008) was non-inflationary, since Gamma_m was unlikely to be flat at such low reserve levels; they attribute QE1’s non-inflationary effect to a rise in reserve demand (interbank-market collapse, IOR introduction Oct 2008, later Basel III liquidity-coverage and stress-test requirements). The unobservable shadow rate limits quantitative precision. Results are qualitative for the inflation facts. The Discussion subsection explicitly notes some views ‘go beyond the formal results.’