Determinacy and Identification with Taylor Rules
📄 Summarized from the full manuscript · Human-reviewed for faithfulness before publication
In brief
How does raising interest rates when inflation rises actually pin down the price level? The standard "new-Keynesian" story says the Federal Reserve's aggressive response rules out runaway inflation. John Cochrane argues that story's math actually works in reverse -- it selects one path only by silently assuming the government would let the economy fly apart, infinite inflation or deflation, rather than let the model's other, equally legitimate paths occur, an assumption he calls implausible. He also shows the key policy-response number cannot be measured from real data. A leading textbook explanation for how central banks control inflation may not actually explain it.
What this paper finds — and why it matters
This paper attacks the standard new-Keynesian explanation for how an interest-rate rule of the Taylor type determines the price level. The textbook story holds that because the Federal Reserve raises nominal rates more than one-for-one with inflation, the resulting “active” policy rules out the indeterminacy that plagues a fixed interest-rate peg and supplies the economy’s missing “nominal anchor.” Cochrane shows this account is incomplete on its own terms: within the standard model, the Taylor rule leaves a continuum of valid equilibria, indexed by an arbitrary initial inflation rate, all but one of which involve inflation or deflation that “explodes” or eventually leaves a neighborhood of target; the literature selects the single “locally bounded” path by simply disallowing the others, but the only economic device available to disallow them – an appeal to the consumer’s transversality condition – rules out explosive real quantities, not explosive nominal ones, so nothing in the model’s actual economics justifies the selection. Working through the specific proposals in the literature that try to trim these alternative equilibria by describing policies the government would follow if inflation or deflation began to run away, Cochrane argues that every version he examines either fails to rule out the unwanted path after all, or succeeds only by describing a policy under which no equilibrium at all could form – jointly infeasible commitments (for instance, pegging a commodity price while also enforcing an incompatible money-supply limit) that amount to a government threat to “blow up the economy,” which he argues is not a plausible description of how governments actually behave or of what people currently believe they would do. He then shows that even granting the theory’s own equilibrium-selection logic, the Taylor rule’s policy-response coefficient cannot be identified from time-series regressions of interest rates on inflation, because in the unique bounded equilibrium the “right-hand” variable is, by construction, a deterministic function of the unobserved policy disturbance itself, so such a regression recovers only the disturbance’s own serial-correlation parameter rather than the structural policy coefficient – undermining the central empirical claim, associated especially with Clarida, Galí, and Gertler (2000), that a measured shift in this coefficient around 1980 explains the end of 1970s U.S. inflation. Cochrane closes by pointing to a specific alternative: a “non-Ricardian” or “active fiscal, passive money” regime, in which the government-debt valuation equation, not the Taylor rule, pins down the price level directly, a mechanism that requires no equilibrium-trimming threats, remains fully consistent with the Fed appearing to follow a Taylor rule empirically, and is not itself the target of the paper’s critique of new-Keynesian modeling more broadly, since it retains the same forward-looking IS and pricing equations.
Summary of a classic paper, AI-assisted and human-reviewed. See the linked original for the authoritative claims and full conditions.
Questions & answers
Q1. What is the paper’s basic claim, stated as directly as possible?
“The new-Keynesian, Taylor rule theory of inflation determination relies on explosive dynamics. By raising interest rates in response to inflation, the Fed induces ever-larger inflation, unless inflation jumps to one particular value on each date. However, economics does not rule out explosive inflation, so inflation remains indeterminate” (Abstract, p. 565). Cochrane frames the mechanism as the opposite of the intuitive “old-Keynesian” stabilizing story: “new-Keynesian models do not say that higher inflation causes the Fed to raise real interest rates, which in turn lowers ‘demand,’ which reduces future inflation… In new-Keynesian models, higher inflation leads the Fed to set interest rates in a way that produces even higher future inflation” (Section I, p. 566). A second, independent claim follows from the same logic: “the parameters of the Fed’s policy rule are not identified, so regression evidence does not say anything about determinacy in a new-Keynesian model” (Section I, p. 566).
Q2. In the simplest possible model, why does a Taylor rule leave inflation indeterminate rather than pinned down?
Combining a Fisher equation with a linear Taylor rule yields a single first-order difference equation, E_t(pi_t+1) = phi*pi_t + x_t, which has infinitely many solutions indexed by an arbitrary initial inflation rate and arbitrary mean-zero “sunspot” shocks (Section II.A, pp. 571-572). If phi > 1 (“the Taylor principle”), all but one of these solutions explode; disallowing explosive paths picks out a unique, locally bounded solution, pi_t = -sum_j (1/phi)^(j+1) E_t(x_t+j). But Cochrane argues “there is nothing wrong with” the excluded explosive paths: “transversality conditions can rule out real explosions, but not nominal explosions. Hyperinflations are historic realities” (p. 572). He also stresses this logic is entirely distinct from “old-Keynesian” stabilizing intuition about inflationary or deflationary spirals – as King (2000) had already noted, a coefficient phi < 1 that produces oscillating hyperinflation and deflation “works just as well” as phi > 1 to deliver a unique bounded solution, “hard to describe by ‘stabilizing’ intuition” (p. 573).
Q3. Does this problem go away once we replace the toy model with a fully specified, microfounded frictionless economy?
No – Cochrane builds an explicit consumer optimization model with complete markets and a government debt valuation equation and shows the linearized Fisher-equation-plus-Taylor-rule system represents its exact equilibrium conditions, and that the “nonlocal” equilibria are indeed globally valid, not merely an artifact of linearization (Section III.A-C, pp. 574-578). Following Benhabib, Schmitt-Grohé, and Uribe (2002) and Woodford (2003), he shows graphically that with a zero lower bound on nominal rates, the policy function F must have a second stationary point PL besides the “good” target P*, one that necessarily violates the Taylor principle locally, and that many perfect-foresight paths converge to PL – all of which satisfy the consumer’s first-order conditions and can be validated by an appropriate ex post fiscal policy, so “all of the ’explosive’ equilibria… are, in fact, valid,” including a Friedman-rule-style deflationary path at the zero bound (Section III.C, pp. 576-578).
Q4. What specifically is wrong, in Cochrane’s view, with proposals in the literature that claim to rule out the unwanted equilibria?
He argues that to actually rule out an equilibrium path, the government must specify a policy under which it is impossible for any equilibrium to form at some point along that path – and every version of this he reviews does so only by describing jointly infeasible policy commitments, which is “a threat to blow up the economy,” not a plausible description of behavior (Section I.A, pp. 567-568). His example: a proposed policy switches to a commodity standard once inflation gets too high, but simultaneously maintains a money-supply limit or nominal-rate target that is inconsistent with that standard – “it is these inconsistent or overdetermined policies, not the inflation or deflation stabilization, that trim equilibria.” He distinguishes sharply between a government stopping an inflation, which is economically sensible and historically observed, and a government making it impossible for any equilibrium to have formed along an unwanted path in the first place – “stopping an inflation or deflation is completely different from disallowing an equilibrium” (p. 567). He also directly engages Woodford’s (2003) argument that hyperinflationary paths are simply “unreasonable” expectations, countering that in a model where the Fed is “absolutely committed to raising interest rates more than one-for-one with inflation, forever, no matter what,” a rational agent who takes the model literally “would confidently expect hyperinflation” – so if we find that forecast unreasonable, “it means we do not believe the model describes the world in which we live” (Section IV.A, p. 583).
Q5. Why does Cochrane say the Taylor rule’s key coefficient cannot be identified from regression evidence, even granting the theory?
Because in the unique bounded equilibrium, inflation is, by the model’s own construction, a deterministic linear function of the unobserved policy disturbance – “a Taylor rule regression of i_t on pi_t will estimate the disturbance serial correlation parameter rho rather than the Taylor rule parameter phi” (Section II.B, p. 573). The regressor (inflation) and the policy error are “perfectly correlated… no accident or statistical assumption; it is central to how the model behaves,” since the whole point of the equilibrium-selection mechanism is that endogenous variables jump in response to shocks specifically to head off explosions (p. 573). He shows lags do not rescue identification either: “If the structural disturbances are serially correlated, lagged endogenous variables are correlated with the monetary policy error term. If the structural disturbances are not serially correlated, lagged endogenous variables are uncorrelated with the right-hand side of the monetary policy rule” (Section I.B, p. 569).
Q6. What does this mean for the well-known empirical finding, associated with Clarida, Galí, and Gertler (2000), that the Fed shifted from “passive” to “active” policy around 1980?
Cochrane argues this reading cannot be sustained: not only is the coefficient uninterpretable as the structural Taylor-rule parameter even in the post-1980 “determinate” period, but the theory has nothing coherent to say about the pre-1980 period at all, since it is supposedly indeterminate, so “Taylor rule regressions in the 1970s are doubly uninterpretable in the new-Keynesian context” (Section VIII.B, p. 607). He also flags the estimated coefficients themselves as implausibly large under a literal structural reading – Clarida et al.’s estimates range from 2.15 to 3.13, implying that a return to 1970s-style 12% inflation would require the Fed to raise the funds rate by 21.5 to 31.3 percentage points: “If these predictions seem implausibly large, digesting the estimates as something less than structural helps a great deal” (p. 607).
Q7. Is Cochrane’s critique an attack on new-Keynesian modeling in general?
No – he is explicit that the paper “is not a criticism of new-Keynesian economics in general” and does not challenge the model’s core ingredients, the forward-looking IS curve or the forward-looking, frictional model of price-setting (Section I.C, p. 570). His target is specifically the “Ricardian” fiscal-regime assumption paired with a Taylor rule as the mechanism for price-level determination; he stresses that “the passive-money, active-fiscal regime of such a model can determine inflation” using essentially the same New Keynesian apparatus otherwise.
Q8. What alternative does Cochrane point to for determining the price level, and what does it require?
A “non-Ricardian,” active-fiscal/passive-money regime, following Leeper (1991): if the fiscal authority sets the path of real primary surpluses independently of the price level (e.g., via a proportional income tax), the government-debt valuation equation alone determines a unique price level – “the same mechanism by which stock market prices are determined as the present value of dividends” – and the central bank remains free to set interest rates via an ordinary Taylor rule without needing to threaten any “blow up the economy” policy (Section III.D, pp. 578-580). Because price-level determinacy in this regime does not depend on the measured size of the Taylor-rule coefficient phi at all, Cochrane notes that “problems in measuring phi are to some extent welcome” under this alternative, since it removes the pressure to treat noisy regression estimates as evidence about the economy’s determinacy regime, and it is explicitly consistent with the empirical observation that interest rates do appear to follow something like a Taylor rule (Section III.D, p. 580).
Q9. What is the paper’s own summary of its contribution, and what does Cochrane say is still missing?
Cochrane calls the paper’s contribution “entirely negative”: it establishes that the leading theory does not, in fact, determine the price level or inflation rate, without itself fully working out or testing a replacement (Section I.C and VIII.C, pp. 569-570, 607-608). He is candid about the stakes of leaving this open: “If inflation is, in fact, stabilized in modern economies by interest rate targets interacted with backward-looking IS and Phillips curves, economists really have no idea why this is so” (Section VIII.C, p. 608) – because that popular “stabilizing” story, while intuitively appealing and common in FOMC statements and textbook treatments, “throws out the edifice of theoretical coherence – explicit underpinnings of optimizing agents, budget constraints, clearing markets” that is the specific achievement of the new-Keynesian research program the paper critiques from within.
Key terms in this paper
Definitions below follow the paper's own usage.
- The "locally bounded" equilibrium-selection criterion
- the standard new-Keynesian selection criterion for picking a single equilibrium out of the many that solve a Fisher equation plus a Taylor rule: among the continuum of solutions indexed by an arbitrary initial inflation rate and arbitrary "sunspot" shocks, all but one either explode (E_t(pi_t+j) grows without bound) or eventually leave a local neighborhood of the target; if the policy responds more than one-for-one to inflation, ruling out these "nonlocal" paths leaves a single, locally bounded solution. Cochrane's central objection is that "transversality conditions can rule out real explosions, but not nominal explosions," so there is no economic force that actually disallows the other paths.
- The "blow-up-the-world" critique of equilibrium-trimming
- Cochrane's term for proposals in the literature that try to rule out unwanted equilibria (explosive inflation or deflation) by describing policies the government would follow if such an equilibrium began to emerge -- for example, pegging a commodity price while also holding a money-supply limit or interest-rate target that is jointly infeasible with that peg. He argues these proposals do not merely describe implausible behavior; because they specify a policy under which "it is impossible for an equilibrium to form" at all, "a threat to blow up the economy," they violate the basic Ramsey/Walrasian requirement that "the government must operate in markets just like agents." Ending an inflation, he stresses, is not the same as making that inflation never have been a valid equilibrium path in the first place.
- The Taylor principle (active versus passive monetary policy)
- the requirement, following Taylor (1993) as formalized in the model's reduced form, that the nominal interest rate respond to inflation with a coefficient greater than one (phi > 1) in order for the unique locally bounded new-Keynesian equilibrium to exist; a coefficient less than one leaves inflation indeterminate under the standard criterion. Cochrane shows this coefficient is precisely the parameter that, according to his identification argument, cannot be recovered from a regression of interest rates on inflation, because in equilibrium the right-hand variable moves in lockstep with the unobserved policy disturbance by construction.
- The identification problem
- Cochrane's result that regressing the nominal rate on inflation in the unique bounded equilibrium recovers only the serial-correlation parameter of the policy disturbance, rho, not the structural policy-rule coefficient phi -- "a Taylor rule regression of i_t on pi_t will estimate the disturbance serial correlation parameter rho rather than the Taylor rule parameter phi" -- because equilibrium inflation is, by the model's own construction, an exact function of the disturbance term, so the regressor and the unobserved policy error are perfectly correlated and no available instrument (including lags) escapes the problem.
- Active fiscal / passive money (non-Ricardian) regime
- the alternative regime, following Leeper (1991), in which fiscal policy sets the path of real primary surpluses independently of the price level, so the government-debt valuation equation alone determines a unique price level (not just inflation), "the same mechanism by which stock market prices are determined as the present value of dividends." In this regime the central bank remains free to set nominal rates via a Taylor rule, is not forced to "blow up the economy" to rule out alternative paths, and Cochrane notes this regime "is not inconsistent with empirical Taylor rule regressions," offering a coherent alternative account of how the price level is actually pinned down.