Fiscal theory · monetary policy · debt maturity
Unpleasant Interest-Rate Arithmetic
What happens to output and inflation when the central bank moves rates and fiscal policy does not move at all — and why the maturity of the debt decides the answer.
Response to the policy path
Deviations from the pre-announcement steady state, in percent. Time in years.
Where the jump comes from
The price level jumps on announcement by exactly the amount the government's debt has to be revalued. Two forces set it, and they usually pull in opposite directions.
Maturity sets the jump
The same policy path, run at every debt maturity from overnight to twenty years. Everything else held at the current settings.
Reading this run
On announcement the price level jumps −1.66%, which opens an output gap of −2.76% through the supply curve. The bond-revaluation term dominates: the rate rise moves long bond prices, the outstanding stock is revalued by −4.33%, and the price level has to follow.
Inflation starts at −0.56%, reaches +0.95% in year 6.2, and converges on +0.60% — the permanent change in the policy rate, because the model has long-run neutrality and is stable under the peg. Short run and long run genuinely disagree here: the rate rise lowers inflation at first and raises it eventually. That gap is the whole difficulty — you have to sit through the first move to reach the second.
Slide the maturity of the debt toward zero and watch the announcement jump disappear; slide the fiscal share toward one and watch the interest-cost channel switch off. Those two controls, not the structural parameters, are what decide whether the short run and the long run agree.
How the model works
Three things are moving at once, and the whole argument turns on the fact that they move on different clocks.
The short run behaves conventionally
Raise the policy rate above inflation and the real interest rate rises. Households postpone spending, the output gap opens up, and firms raise prices more slowly. Inflation falls. This is the mechanism every central banker has in mind, and in this model it is real — it just is not the end of the story.
The long run is Fisherian
Over a long enough horizon the nominal interest rate and inflation have to move together: nobody lends at 2 percent in a world of permanent 50 percent inflation. If inflation eventually settles wherever the central bank puts the nominal rate, then a permanently higher rate must mean permanently higher inflation. Two assumptions deliver this — long-run neutrality and stability under a rate peg — and neither is easy to argue away. So monetary policy on its own rearranges inflation across time rather than removing it: less now is bought with more later.
Why the maturity of the debt matters so much
When the central bank raises rates and everyone expects the higher rates to stick, long-term bond prices fall. Bondholders take a capital loss, so the real value of what the government owes falls. With no change in taxes or spending, the price level has to fall to keep the government's books balanced — a deflationary jump on the day of the announcement. Overnight debt has no price to fall. Roll the entire stock into bills and that channel disappears: the announcement produces no jump at all, and only the slower forces are left.
Why cutting rates can be disinflationary
Interest on the debt is a real cost. Cut the rate and, unless the Treasury hands the saving back through lower taxes, the government's interest bill falls, its finances improve, and less inflation is needed to balance them. With short debt this is the only channel left, so a rate cut lowers inflation on impact as well as eventually — no surge to sit through first.
The policy that follows
Buy back the long-term debt, then cut rates persistently. The buyback removes the bond-revaluation channel that would otherwise deliver an inflation surge; the cut lowers both the interest bill and, through the Fisher effect, the eventual inflation rate. Set the maturity slider to zero and cut the rate to see it.
How much to believe
The long-run proposition is hard to escape. What is far less secure is the claim that shortening the maturity structure reliably removes the short-run cost. Move the fiscal slider toward the conventional assumption and watch how much of the result is really about who pays the interest bill rather than about monetary policy at all.
A continuous-time model with a generalised Lucas supply curve, an interest-rate peg, and long-term nominal debt. All variables are deviations from the pre-announcement steady state; the announcement occurs at t = 0.
Private sector
pt = p*t + κ xt — Lucas supply: output responds to the price-level surprise
ṗ*t = πet + θ ( pt − p*t ) — reference price level catches up
π̇et = ν ( πt − πet ) — trend inflation adjusts
Writing dt ≡ pt − p*t = κxt reduces this to two states, with a ≡ κσ ⁄ (1 + κσ):
Both eigenvalues of this system are stable, so the peg leaves the price level indeterminate — the usual result. The steady state is x = 0, π = πe = i∞: long-run neutrality and a vertical Phillips curve are imposed, the Fisherian result is not. It follows.
Government: the valuation equation selects the jump
The real value of the debt equals the present value of surpluses, q0 − p0 = PV(s). Differentiating at the announcement gives the initial price-level jump, which is the model's only free variable:
Δq₀ = − ∫0∞ e−t/D it dt — log price of a geometric portfolio of duration D
x₀ = Δp₀ ⁄ κ, πe₀ = 0 — surprise maps into output through the supply curve
Primary surpluses respond to the extra real interest cost as Δst = φ·b̄·(it−πt), so φ = 1 is the conventional assumption that fiscal policy pays the added interest bill and φ = 0 asks what the central bank can do entirely alone. The discount-rate term is what Cochrane calls unpleasant interest-rate arithmetic: higher real rates raise interest costs, lower the present value of surpluses, and push toward inflation. Because Δp0 feeds back into π and hence into that integral, the condition is a fixed point; it is linear in Δp0 and solved exactly, and the feedback is stabilising (∂R ⁄ ∂Δp0 < 0), so no parameter setting makes it explode.
Policy path
it = ( 1 − e−λt ) [ i∞ + ( i0 − i∞ ) e−ηt ] — announced at 0, phased in at rate λ
In the phased-in case the announcement still moves bond prices and the price level at t = 0 even though the rate itself has not yet moved — which is why the dashed and solid paths in Cochrane's figures differ so little.
Baseline calibration
| Parameter | Default | Reading |
|---|---|---|
| σ = 1.00 | IS sensitivity | Unit intertemporal elasticity. |
| κ = 0.60 | Phillips slope | A one percent price-level surprise opens a 1.7 percent output gap. |
| θ = 0.90 | Catch-up speed | Reference price level closes the gap with a nine-month half-life. |
| ν = 0.90 | Expectations | Trend inflation adjusts at a similar pace. |
| ρ = 0.05 | Discount rate | Real rate applied to the surplus stream. |
| b̄ = 1.00 | Debt / GDP | Scales the whole interest-cost channel. |
| D = 5.00 | Duration, years | Roughly the post-war US average; set to 0 for bills only. |
Eigenvalues at the default are −0.35 ± 0.45i, so the system is comfortably damped and has settled well inside the ten-year window.
On fidelity. This is a stylised implementation of the argument in Cochrane's post, not a reproduction of his calibration. It reproduces the qualitative shapes and, at the default settings, lands close to his figures — a one-point rate rise against five-year debt opens a 2.8 percent output gap, and a rate cut against overnight debt produces no announcement jump and an inflation trough around year three. The parameters, the reference-price-level formulation of the supply curve and the exact form of the fiscal-response rule are my own.