MPCs, MPEs, and Multipliers: A Trilemma for New Keynesian Models

Fact 1: “Average MPCs are high in the data, around 0.25 quarterly and 0.5 annually” (Section 1, p. 5-6, citing Kaplan and Violante 2014’s synthesis of the 2001 US tax-rebate literature). Fact 2: “Average MPEs are small in the data, around 0 to 0.04 annually,” based principally on Cesarini, Lindqvist, Notowidigdo, and Ostling’s (2017) precisely estimated Swedish lottery MPE of 0.01 (95% CI 0.005-0.015), together with an adjusted reading of Imbens, Rubin, and Sacerdote’s (2001) annuity-based estimates (Section 1, pp. 6-7). Fact 3: “Fiscal multipliers lie in a moderate range in the data. Under accommodative monetary policy, both impact and cumulative multipliers are between 0.6 and 2” (Section 1, p. 7, citing Ramey 2019’s review). Fact 1 rules out representative-agent models; the authors’ contribution is to show Facts 2 and 3 jointly rule out every way of fixing that within a frictionless-labor New Keynesian framework.

CI is defined (equation 5) as the marginal propensity to consume out of a change in earned income induced by a wage change, holding the marginal utility of wealth fixed; Lemma 1 shows CI = 0 for separable preferences and CI = 1 for GHH preferences, and in general CI = Unc/Un divided by Ucc/Uc. Proposition 1 then derives an exact individual-level relationship: MPE/MPC = (wn/c) x (Frisch/EIS) x (1 - CI) (Section 2, pp. 8-9). With Frisch and EIS calibrated to similar magnitudes and wn/c close to 1 for most agents, this simplifies (equation 7) to MPE ≈ (1 - CI) x MPC: under separable preferences (CI = 0) MPC and MPE are approximately equal, so “if preferences are separable, Facts 1 and 2 cannot hold simultaneously” (Section 2, p. 9) – matching a 0.5 annual MPC and a 0.04 annual MPE requires CI of at least 0.92.

Proposition 2 (Section 3, p. 12): under a constant real interest rate, dY_t/dG_s = [1/(1 - (1-tau)CI)] x 1{s=t}, where tau is the steady-state labor wedge. With separable preferences (CI = 0) this collapses to a multiplier of exactly 1, “the celebrated result from Woodford (2011).” With GHH preferences (CI = 1) the multiplier becomes 1/tau, which the authors note is typically calibrated to be large – for example, “Nakamura and Steinsson (2014) report a constant-r multiplier of 7.00 under GHH preferences” (Section 3, pp. 12-13). The intuition is a demand feedback loop: an increase in output requires more labor, which under positive CI raises desired consumption by CI(1-tau) per unit of the shock, generating a geometric “second-round effect” that sums to 1/(1-(1-tau)CI).

The authors build a HANK model with “GHH-plus” preferences (equation 16) that nest separable (alpha=0) and GHH (alpha=1) preferences via a single parameter alpha, with CI a simple increasing function of alpha (equation 17); at every value of alpha the model is recalibrated (using two-point discount-factor heterogeneity, following the permanent-heterogeneity approach) to hit a quarterly MPC of exactly 0.25, matching Fact 1 by construction (Section 4.1-4.2, pp. 13-17). Calibration otherwise follows McKay, Nakamura, and Steinsson (2016): a Frisch elasticity of 0.5, average EIS of 0.5, AR(1) idiosyncratic productivity with persistence 0.966, liquid assets at 1.4 times annual GDP, and an elasticity of substitution of 7 implying a steady-state labor wedge tau = 0.43.

Sweeping alpha (and hence CI) from 0 to 1, the separable calibration (alpha=0) produces an annual MPE of 0.23 – far above Fact 2’s upper bound – while the GHH calibration (alpha=1) produces a cumulative multiplier above 5, far above Fact 3’s upper bound of 2, versus the RANK-implied 1/tau = 2.32 for the same tau (Section 4.3, pp. 17-18). The best intermediate compromise, at alpha = 0.6-0.7, achieves annual MPEs of 0.040 and 0.028 (just inside Fact 2) and impact multipliers of 1.70 and 1.91 (just inside Fact 3), “but the cumulative multipliers of 2.71 and 3.09 are still well outside the acceptable range, and these calibrations therefore do not solve the trilemma” (Section 4.3, p. 18).

No single alternative parameterization the authors test succeeds for the cumulative multiplier (Section 4.4, pp. 18-19, detailed in Appendix B): raising the EIS to 1.0 or lowering the Frisch elasticity to 0.25 lowers MPEs and impact multipliers but leaves cumulative multipliers outside the acceptable range; raising the steady-state markup (lowering the elasticity of substitution from 7 to 3, raising tau from 0.43 to 0.56) produces only mild multiplier reductions; and switching to progressive tax financing (raising tax rates only on the 10 highest of 25 income states) “drastically increases the cumulative multiplier… thereby worsening the trilemma.” The authors conclude that “a highly specific combination of these assumptions… might bring both MPEs and multipliers into the acceptable range,” but pursue “a more general and robust solution” instead (Section 4.4, p. 19).

Table 2 shows that moving from a constant real rate to an active Taylor rule (coefficient 1.25 on inflation) lowers multipliers across all calibrations, while a temporary nominal-rate peg (3 years, mimicking the effective lower bound) raises them even further – for separable preferences the multiplier under a 3-year peg is 118.26, “near an asymptote” (Section 4.5, p. 19-20). On fiscal policy, the authors find a surprising invariance: varying the persistence of debt financing (rho_B = 0, 0.9, 0.95) leaves the cumulative multiplier completely unchanged in their flexible-wage model, which they attribute to an unusual form of Ricardian equivalence arising from the interaction of the tax system with tradable firm equity (formally established in Appendix F.1) – a property they flag as another empirical strike against the sticky-price, flexible-wage model, since “there is some empirical evidence that deficit financing tends to raise the fiscal multiplier” (Section 4.5, p. 20).

Following Erceg, Henderson, and Levin (2000) and borrowing the wage-Phillips-curve microfoundation from Auclert, Rognlie, and Straub (2018), households take hours worked n as given each period rather than choosing it from a static first-order condition, so “their MPE equal[s] 0 by construction,” while a labor union sets wages via a wage Phillips curve (equation 25) subject to aggregate labor demand from final-goods firms (Section 5.1, pp. 20-21). Calibrated with separable preferences (CI = 0) and the same target MPC of 0.25, “the impact fiscal multiplier is 1.21 and the cumulative fiscal multiplier is 1.18, well within the range of Fact 3,” and Table 2 shows this result is robust across the alternative monetary and fiscal policy rules the paper tests (Section 5.2, pp. 21-22).

The authors note that in the sticky-wage model, equity prices are “mildly procyclical instead of strongly countercyclical” in response to a government spending shock, giving the model “a chance of fitting both the cyclicality of the price-cost margin (e.g. Nekarda and Ramey 2020) and the response of equity prices to government spending shocks” – and it avoids the large redistributional effects tied to concentrated firm ownership that Broer, Hansen, Krusell, and Öberg (2020) show to be problematic for the sticky-price, flexible-wage alternative, and that the authors note “drive peculiar heterogeneity in impulse responses” in their own flexible-wage model (Section 5.2, p. 22, footnote 25; Appendix C).

The authors position their result alongside two contemporaneous critiques of the sticky-price, flexible-wage HANK model: Broer, Hansen, Krusell, and Öberg (2020), who “focus on the implied countercyclicality of profits,” and Nekarda and Ramey (2020), who “argue that the implied cyclicality of marginal costs is rejected by the data” (Introduction, p. 4). The paper frames its own contribution as adding “the trilemma to these voices against the flexible-wage model,” and explicitly welcomes “the recent trend in the HANK literature of assuming sticky wages in addition to, or as a substitute for, sticky prices (e.g. Hagedorn, Manovskii and Mitman 2019, Alves, Kaplan, Moll and Violante 2020)” (Introduction, p. 5, footnote 6).