Deriving a Taylor-Rule Reaction Function

Why a Reaction Function Uses Gaps

A central bank faces changing economic conditions while pursuing its objectives. A reaction function summarizes how a policy-rate recommendation responds to those conditions. In a Taylor-style rule, the central bank responds to inflation relative to its target and economic activity relative to its sustainable level. The rule is a structured benchmark, not a claim that a central bank must mechanically set its rate to one exact number.

The inflation gap measures how far inflation is above or below the target. The output gap measures how far actual output is above or below potential output. Positive gaps call for a higher recommended rate in the basic rule; negative gaps call for a lower one. The signs and units matter: a gap of 1.0 percentage point is not the same as a gap of 1 percent of output.

Definition: Let \(\pi\) denote the inflation rate, \(\pi^*\) the inflation target, and \(x\) the output gap. The inflation gap is \(\pi-\pi^*\). The output gap is \(x=100(Y-Y^*)/Y^*\), where \(Y\) is actual real output and \(Y^*\) is potential real output. Thus \(x\) is measured in percent of potential output: \(x=1\) means output is 1 percent above potential. In the calculations below, rates and gaps are entered as percentage-point numbers, such as 2.5 for 2.5%.

A simple stabilization loss may assign costs to deviations of inflation from target and output from potential. For example, a quadratic loss is \(L=\tfrac12[(\pi-\pi^*)^2+\lambda x^2]\), where \(\lambda>0\) describes the relative weight on output stabilization. This loss helps explain why both gaps matter. But the loss function by itself does not uniquely determine a policy-rate formula: that requires assumptions about how interest rates affect inflation and output, as well as the timing of those effects. The rule below states its response assumptions explicitly.

Deriving the Taylor-Style Rule

Start by describing the desired real policy rate. Let \(r^*\) be the equilibrium real rate associated, in this simplified benchmark, with inflation at target and output at potential. Suppose the desired real rate rises linearly when inflation is above target or output is above potential:

r^d=r^*+\phi_\pi(\pi-\pi^*)+\phi_y x.

Here \(r^d\) is the desired real rate, while \(\phi_\pi\) and \(\phi_y\) are the response coefficients for the inflation gap and output gap. For the usual stabilizing direction, both coefficients are nonnegative. This linear response is a modeling assumption: it says how strongly the desired real stance changes for a one-percentage-point change in each gap.

To convert this real-rate recommendation into a nominal policy-rate recommendation, use the approximate Fisher relation \(r\approx i-\pi\), where \(i\) is the nominal rate. In this stylized rule, current inflation is used as a proxy for expected inflation. The approximation is suitable for a simple linear reaction function; it is not an exact identity when inflation and rates are large or when expected inflation differs materially from current inflation. Rearranging gives \(i\approx r+\pi\). Substituting the desired real rate and collecting terms yields the Taylor-style rule.

\begin{aligned} i^d &\approx r^d+\pi\\ &=r^*+\phi_\pi(\pi-\pi^*)+\phi_yx+\pi\\ &=r^*+\pi^*+(1+\phi_\pi)(\pi-\pi^*)+\phi_yx. \end{aligned}

The last step follows because \(\pi=\pi^*+(\pi-\pi^*)\): current inflation contributes one-for-one to the nominal rate under the Fisher approximation, in addition to the response to the inflation gap. Equivalently, the same rule can be written in the common form below, where \(\alpha=\phi_\pi\) and \(\beta=\phi_y\).

i^d=r^*+\pi+\alpha(\pi-\pi^*)+\beta x

At both target inflation and potential output, the gaps are zero, so the recommended nominal rate is \(r^*+\pi^*\). When the inflation gap rises by one percentage point and the output gap is unchanged, the recommended nominal rate rises by \(1+\phi_\pi\) percentage points. The extra \(\phi_\pi\) is the deliberate response in the desired real rate; the other one percentage point reflects the approximate conversion from real to nominal rates. With \(\phi_\pi>0\), the rule raises the nominal rate by more than the increase in inflation, so the recommended real rate rises rather than remaining unchanged.

The coefficients are not universal constants. They summarize the assumed strength of policy responses in the chosen model or benchmark. A larger \(\phi_\pi\) means a stronger reaction to an inflation deviation; a larger \(\phi_y\) means a stronger reaction to an output gap. Neither coefficient, by itself, states exactly how much inflation or output will change after a rate decision.

Applying the Rule

The safest calculation is to keep the baseline rate and the gap adjustments separate. Use percentage-point inputs throughout, preserve the sign of each gap, and only convert to a decimal rate if the answer specifically requires it.

Worked Example: Inflation Above Target and Output Above Potential

A hypothetical central bank uses \(r^*=1.5\%\), \(\pi^*=2.0\%\), \(\phi_\pi=0.5\), and \(\phi_y=0.25\). Inflation is 3.2%, and actual output is 1.2% above potential. Calculate the recommended nominal rate. Method: apply the gap form of the Taylor-style rule. Common mistake avoided: do not omit the one-for-one inflation component when using the gap form.

Step 1: Identify the gaps. The inflation gap is \(3.2-2.0=1.2\) percentage points. The output gap is \(x=1.2\).

Step 2: Find the baseline nominal rate. At both gaps equal to zero, the baseline is \(r^*+\pi^*=1.5+2.0=3.5\%\).

Step 3: Calculate the adjustments. The inflation adjustment in the gap form is \((1+\phi_\pi)(\pi-\pi^*)=(1+0.5)(1.2)=1.5(1.2)=1.8\) percentage points. The output adjustment is \(\phi_yx=0.25(1.2)=0.30\) percentage points.

Step 4: Add the components. The recommended rate is \(3.5+1.8+0.30=5.60\%\).

Check. The desired real rate is \(1.5+0.5(1.2)+0.25(1.2)=1.5+0.60+0.30=2.40\%\). Adding current inflation gives \(2.40+3.2=5.60\%\), consistent with the gap-form calculation.

Worked Example: Inflation Below Target and a Negative Output Gap

For a hypothetical economy, let \(r^*=2.0\%\), \(\pi^*=2.5\%\), \(\phi_\pi=0.75\), and \(\phi_y=0.50\). Inflation is 1.5%, while output is 2.0% below potential. Find the rule’s nominal-rate recommendation. Method: calculate each signed gap before applying its coefficient. Common mistake avoided: a negative output gap must reduce, not increase, the recommendation.

Step 1: Calculate the gaps. The inflation gap is \(1.5-2.5=-1.0\) percentage point. Output is 2.0% below potential, so \(x=-2.0\).

Step 2: Calculate the baseline. \(r^*+\pi^*=2.0+2.5=4.5\%\).

Step 3: Calculate the gap adjustments. The inflation adjustment is \((1+0.75)(-1.0)=-1.75\) percentage points. The output adjustment is \(0.50(-2.0)=-1.00\) percentage point.

Step 4: Add the components. The recommendation is \(4.5-1.75-1.00=1.75\%\).

Check. The desired real rate is \(2.0+0.75(-1.0)+0.50(-2.0)=2.0-0.75-1.00=0.25\%\). Adding current inflation gives \(0.25+1.5=1.75\%\). Both approaches agree.

Worked Example: Comparing Coefficients and Rate Recommendations

A hypothetical central bank has \(r^*=1.25\%\) and an inflation target of 2.25%. Inflation is 2.75% and output is 1.5% above potential. First calculate the recommendation when \(\phi_\pi=0.40\) and \(\phi_y=0.60\). Then find the change in the recommendation if the output response coefficient instead equals 0.80, with all else unchanged. Method: calculate the baseline and each gap contribution separately. Common mistake avoided: a coefficient change affects only the term multiplied by that coefficient.

Step 1: Calculate the baseline and gaps. The baseline is \(1.25+2.25=3.50\%\). The inflation gap is \(2.75-2.25=0.50\) percentage point, and \(x=1.5\).

Step 2: Apply the original coefficients. The inflation adjustment is \((1+0.40)(0.50)=1.40(0.50)=0.70\) percentage point. The output adjustment is \(0.60(1.5)=0.90\) percentage point. The recommendation is \(3.50+0.70+0.90=5.10\%\).

Step 3: Change only the output coefficient. With \(\phi_y=0.80\), the output adjustment is \(0.80(1.5)=1.20\) percentage points. The new recommendation is \(3.50+0.70+1.20=5.40\%\).

Step 4: Compare. The recommendation rises by \(5.40-5.10=0.30\) percentage point. This equals the change in the output coefficient, \(0.80-0.60=0.20\), multiplied by the output gap, \(1.5\): \(0.20(1.5)=0.30\).

Check. The desired real rate under the original coefficients is \(1.25+0.40(0.50)+0.60(1.5)=1.25+0.20+0.90=2.35\%\); adding inflation of 2.75% gives 5.10%. Under the changed output coefficient it is \(1.25+0.20+1.20=2.65\%\); adding 2.75% gives 5.40%.

Interpretation, Limits, and Common Traps

A Taylor-style reaction function provides a transparent way to organize a policy discussion: it shows the assumed response to inflation and output conditions, and lets the reader compare recommendations across scenarios. Its numerical result depends on its inputs and assumptions. In particular, changing the assumed equilibrium real rate, inflation target, response coefficients, or measured gaps changes the recommendation.

The rule is a benchmark, not a complete account of policy decisions. The approximate Fisher relation uses current inflation as a proxy for expected inflation; a different expectation measure can change the nominal-rate calculation. The linear specification also treats the coefficients as fixed and does not include every consideration that may enter a decision. A calculated recommendation should therefore be interpreted as conditional: it is what this particular rule implies for the supplied values.

Omitting the Fisher component. In the gap form, the inflation-gap multiplier is \(1+\phi_\pi\), not just \(\phi_\pi\). For example, if \(\phi_\pi=0.5\) and the inflation gap is 1.0 percentage point, the adjustment is \(1.5(1.0)=1.5\) points. Using \(0.5(1.0)=0.5\) understates the recommendation by 1.0 point.
Mixing coefficient conventions. In \(r^*+\pi+\alpha(\pi-\pi^*)+\beta x\), the inflation-gap coefficient is \(\alpha\). In the baseline-plus-gaps form, the corresponding multiplier is \(1+\alpha\). These are equivalent forms, not competing definitions.
Reversing a gap. The inflation gap is actual inflation minus target, and the output gap is actual output relative to potential. Reversing either subtraction can change the sign of the policy adjustment.
Mixing units. If inflation and the target are entered as 3.0 and 2.0, their difference is 1.0 percentage point. Do not treat that difference as 0.01 while leaving the other terms in percentage-point units.

Key takeaway: Under the approximate Fisher relation, a desired real-rate response to the inflation and output gaps produces \(i^d=r^*+\pi^*+(1+\phi_\pi)(\pi-\pi^*)+\phi_yx\). Define the gaps, preserve their signs, and keep the coefficient convention consistent.

Check Your Understanding

For each problem, use the stated Taylor-style rule and keep all rates and gaps in percentage-point units.

A hypothetical bank uses \(r^*=1.0\%\), \(\pi^*=2.0\%\), \(\phi_\pi=0.50\), and \(\phi_y=0.25\). Inflation is 2.8%, and output is 0.8% above potential. What nominal rate does the rule recommend? (Answer: 4.40%)
Let \(r^*=2.0\%\), \(\pi^*=2.0\%\), \(\phi_\pi=0.25\), and \(\phi_y=0.50\). Inflation is 1.4%, and output is 1.2% below potential. Calculate the recommended nominal rate. (Answer: 2.65%)
A rule is written \(i^d=r^*+\pi^*+1.6(\pi-\pi^*)+0.4x\). What are \(\phi_\pi\) and \(\phi_y\)? If the baseline \(r^*+\pi^*\) is 4.0%, the inflation gap is 1.25 percentage points, and \(x=-0.5\), find the recommendation. (Answer: \(\phi_\pi=0.6\), \(\phi_y=0.4\), and \(i^d=5.8\%\))
In the rule \(i^d=r^*+\pi^*+(1+\phi_\pi)(\pi-\pi^*)+\phi_yx\), \(\phi_\pi=0.5\). With the output gap unchanged, inflation rises by 0.8 percentage point. By how much does the recommended nominal rate change? (Answer: It rises by 1.2 percentage points.)
A hypothetical rule has \(r^*=1.5\%\), \(\pi^*=2.0\%\), \(\phi_\pi=0.50\), and \(\phi_y=0.25\). Inflation is 3.0%, output is above potential, and the recommended rate is 5.25%. Infer the output gap. (Answer: \(x=1.0\), so output is 1.0% above potential.)

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