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Lecture 1A: The Taylor rule and the Taylor principle

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The Taylor rule is a rule of thumb: a short formula that tells you roughly what interest rate a central bank "should" set, given two numbers you can look up (inflation and the output gap). Nobody at the Fed types it into a calculator and reads off the decision; it is a way for economists to describe and judge decisions. You will use it in two ways: fill in the numbers to get a prescribed rate, and read the formula to understand why the coefficient on inflation has to be bigger than one.

Everything in the formula is in percent. Inflation of 3% is 3, an output gap of −2% is −2, and the answer is a rate in percent. "0.5 ×" simply means "half of". Try the worked examples below with a pen; the exam asks exactly this kind of calculation.

The Taylor rule: a formula that describes central bank decisions

Central banks do not follow a fixed formula; a committee decides. But their behaviour can be described surprisingly well by a simple idea: raise the rate when inflation is high or the economy runs hot, lower it when inflation is low or the economy is weak. John Taylor (1993) showed that a simple formula described Fed policy in 1987-1992 well.

The rule has two uses: descriptive (what did the central bank do? You estimate it from data) and normative (what should it do? You compare the actual rate with the rule's prescription).

Taylor rule (1993)

Read it term by term:

  • it: the nominal policy rate the rule prescribes in period t.
  • r* ("r-star"): the neutral real interest rate, the real rate that neither stimulates nor slows the economy. Taylor used 2%.
  • πt ("pi"): inflation over the past four quarters (one year).
  • π*: the inflation target, 2%.
  • (πt − π*): how far inflation is above (+) or below (−) target. The rate rises by half of that gap.
  • yt: the output gap in %. The rate rises by half of it.

Logic of the first two terms: r* + πt is the nominal rate that keeps the real rate at its neutral level (because real ≈ nominal − inflation). The two 0.5 terms then add a push: tighter when inflation or activity is too high, looser when too low.

Neutral benchmark: with inflation at 2% and no output gap, i = 2 + 2 + 0 + 0 = 4%, the neutral nominal rate r* + π*.

The Taylor principle: why the rate must rise more than inflation

Rearrange the rule by collecting the πt terms: it = (r* − 0.5π*) + 1.5πt + 0.5yt. So when inflation rises by 1 percentage point, the rule raises the nominal rate by 1.5 points. Why does that matter? Because the real rate is i − π:

Step by step: where the 1.5 comes from
  1. Start: it = r* + πt + 0.5(πt − π*) + 0.5yt.
  2. Multiply out the bracket: 0.5(πt − π*) = 0.5πt − 0.5π*.
  3. Collect the two terms with πt: πt + 0.5πt = 1.5πt.
  4. Collect the constants: r* − 0.5π*. With r* = 2 and π* = 2 that is 2 − 1 = 1.
  5. Result: it = 1 + 1.5πt + 0.5yt. Every extra point of inflation raises the prescribed rate by 1.5 points.

Check with the neutral case: πt = 2 and yt = 0 give 1 + 3 + 0 = 4%, the same 4% as before.

Inflation +1 pp, rate +1.5 ppInflation +1 pp, rate +0.5 pp
Change in the real rate (Δi − Δπ)1.5 − 1 = +0.5: policy really tightens0.5 − 1 = −0.5: policy actually loosens

With a coefficient below 1, the central bank would seem to tighten (the nominal rate rises) while borrowing actually gets cheaper in real terms, which fuels inflation further. Taylor principle: the coefficient on inflation must be greater than 1. In the New Keynesian model (next lecture) this is the condition for a unique, stable equilibrium.

Worked examples (r* = 2%, π* = 2%)

SituationInflationOutput gapr* + π + 0.5(π − 2) + 0.5y
Overheating3%+1%2 + 3 + 0.5 + 0.5 = 6.0%
On target2%0%2 + 2 + 0 + 0 = 4.0%
Recession1%−2%2 + 1 − 0.5 − 1 = 1.5%
Deep recession0%−6%2 + 0 − 1 − 3 = −2.0%

Check the recession row yourself: inflation is 1 point below target, so 0.5 × (1 − 2) = −0.5; the gap is −2, so 0.5 × (−2) = −1. Total 2 + 1 − 0.5 − 1 = 1.5%.

The last row prescribes a negative rate. But central banks can hardly push rates far below zero (people would just hold cash, which pays 0%). This limit is the zero lower bound (ZLB); at the ZLB the central bank needs other tools, such as QE.

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