Managing a Liquidity Trap: Monetary and Fiscal Policy

The paper reworks the standard New Keynesian liquidity-trap model of Eggertsson and Woodford (2003) in deterministic continuous time specifically because doing so “is well suited to focus on the dynamic questions of policy, such as the optimal exit strategy… and delivers several new results” that a discrete-time Poisson formulation cannot cleanly deliver (Introduction, pp. 2-3). The author states plainly that the continuous-time approach yields “the first formal results explaining” patterns (zero rates persisting past when the natural rate turns positive) that prior work had only shown via numerical simulation (Introduction, p. 3), and that it specifically reveals the discrete, discontinuous jump in the nominal rate at exit – “a property that can only be appreciated in continuous time” (p. 3).

The log-linearized equilibrium is a system of three conditions (Section 2, equations 1a-1c, p. 7): a consumption/output Euler equation in which the growth rate of the output gap is proportional to the gap between the nominal rate and the sum of the natural real rate and inflation; a forward-looking New Keynesian Phillips curve in which the growth rate of inflation depends on discounted inflation minus the output gap; and the zero bound on the nominal rate – where the output gap is measured relative to the flexible-price efficient outcome, and the natural rate of interest is “the real interest rate that would prevail in an efficient, flexible price, outcome with [a zero output gap] throughout” (p. 7). A liquidity trap is a scenario in which the natural rate is negative up to some date T and non-negative thereafter.

Without commitment, the discretionary central bank sets the nominal rate to zero throughout the trap and the unique resulting equilibrium features strictly negative inflation and output for the whole duration of the trap, with both becoming unbounded as the trap’s duration grows without limit (Proposition 1, Section 3.1, pp. 10-12). The mechanism is that “the main distortion is that the real interest rate is set too high during the liquidity trap. This depresses consumption. Importantly, this effect accumulates over time” – with a numerical illustration that a natural rate of -4% over a two-year trap depresses output by “at least 8%” even before accounting for the deflationary feedback (p. 12), and “matters are just made worse by deflation, which raises the real interest rate even more, further depressing output, leading to even more deflation, in a vicious cycle” (p. 12).

Proposition 2 (Section 3.2, pp. 13-14) shows that under discretion, a higher degree of price flexibility produces strictly lower (more negative) inflation and output throughout the trap, with both diverging to negative infinity as prices become fully flexible: “sticky prices hold back deflation and mitigate depressions.” The logic: “for a given negative output gap, a higher [degree of price flexibility] creates more deflation. More deflation, in turn, increases the real interest rate… By the Euler equation this requires higher growth in the output gap; since [the output gap is zero] at [the end of the trap], this translates into a lower level of [the output gap] for earlier dates” – flexible prices accelerate deflation, which raises the real rate and depresses output further, reinforcing the deflationary spiral (p. 13). The paper notes this echoes, and provides the first formal limiting explanation for, a related simulation finding in Christiano, Eichenbaum, and Rebelo (2011) – itself in this same reading list (footnote 7, p. 13).

Proposition 3 (Section 4.1, pp. 18-19) establishes that the optimal nominal rate equals either zero or an unconstrained target rule that adds to the natural rate a term proportional to inflation, and that the interest rate must be held at zero strictly longer than the interval over which that unconstrained target rate would itself be negative – that is, “the nominal interest rate should be held down at zero longer than what current inflation warrants” (p. 19). One direct implication is that “optimal policy requires a discrete upward jump, from zero, in the nominal interest rate” at the moment the zero bound stops binding, “even when economic fundamentals vary smoothly” (p. 19).

Proposition 4 (Section 4.1, pp. 20-21) shows inflation must be strictly positive at some point along the optimal path, and in a knife-edge parameter case inflation is nonnegative throughout the entire episode – so the complete absence of deflation is not, by itself, evidence against having been in a liquidity trap. Output, however, cannot avoid an initial recession: “output is initially negative, but becomes strictly positive at some point,” so optimal policy trades an unavoidable initial slump for a subsequent boom, and this boom “is larger than that stimulated by the inflationary promise” alone, because holding the rate at zero for longer than inflation dynamics require adds further stimulus on top of the pure expected-inflation channel (Section 4.1, p. 21).

The paper defines opportunistic spending as the level a purely static, forward-blind cost-benefit calculation would choose given the current consumption gap (spend more when the shadow wage is depressed in a slump), and defines stimulus spending as the residual – actual spending minus opportunistic spending – which exists only because spending can also relax the dynamic Phillips-curve constraint (Section 5.2, pp. 34-35). Proposition 7 proves stimulus spending is always exactly zero at the very start of a trap, and Proposition 8 shows that under a knife-edge parameter restriction, stimulus spending is identically zero for the entire episode, so that “government spending could be determined by a naive agency, lacking commitment, that performs a static cost-benefit calculation, ignoring the dynamic effects this has on the private sector” and still be optimal (p. 35).

Yes on both counts: Proposition 6 (Section 5.1, pp. 33-34) shows that whenever the zero bound binds over an interval, optimal spending is positive at the start of that interval and turns negative by its end. The intuition is that “initially, higher spending helps compensate for the negative consumption gap at the start of a liquidity trap. However… optimal monetary policy eventually engineers a consumption boom. If government spending leans against the wind, we should expect lower spending” once that boom materializes (p. 33) – so front-loaded spending and the later consumption boom are jointly optimal, not competing signals of policy failure.

In this mixed case (Section 6, pp. 38-41), stimulus spending becomes unambiguously positive and strictly increasing over the trap, because fiscal policy alone must substitute for the missing monetary commitment: “positive stimulus spending emerges as a way to fight deflation… back-loading stimulus spending provides a bigger bang for the buck, both in terms of inflation and output,” since price setting is forward-looking so spending promised near the end of the trap raises inflation both then and earlier (p. 40, Introduction pp. 4-5). If fiscal policy can also commit past the end of the trap itself, optimal spending turns negative immediately after exit and converges back toward its natural level, mimicking – through lower post-trap government spending and the resulting consumption boom – the same kind of forward-looking stimulus that committed monetary policy would otherwise have provided (Section 6.2, p. 41).