From Short to Medium Run (IS-LM-PC)

The Taylor Rule

coreExam · medium

The Taylor rule prescribes how central banks set the policy rate in response to deviations of inflation from target and output from potential: iᵀ = r* + π + φπ(π − π*) + φy(Y − Yn)/Yn. Typical calibration φπ = 0.5, φy = 0.5. The rule enforces the Taylor principle (φπ > 0) so real rates rise with inflation — necessary for stability.

Derivation

Step 1 / 7

1Press Space or click Reveal next
(hidden)
2Press Space or click Reveal next
(hidden)
3Press Space or click Reveal next
(hidden)
4Press Space or click Reveal next
(hidden)
5Press Space or click Reveal next
(hidden)
6Press Space or click Reveal next
(hidden)
7Press Space or click Reveal next
(hidden)

The Reaction Function

i^T = r^* + \pi + \phi_\pi(\pi - \pi^*) + \phi_y \cdot \frac{Y - Y_n}{Y_n}

Four components:

$r^*$ — natural real rate (long-run equilibrium)
$\pi$ — current inflation (Fisher adjustment)
$\phi_\pi(\pi - \pi^*)$ — response to inflation gap
$\phi_y \cdot \text{(output gap)}$ — response to real-side gap

The Taylor Principle

For stability: $\phi_\pi > 0$ (often strengthened to $> 1$ ). When inflation rises, the CB raises $i$ by more than 1-for-1, pushing the real rate up and cooling demand. Passive responses ( $\phi_\pi \leq 0$ ) lead to self-reinforcing inflation spirals.

Worked Numerical Example

With $\pi = 3\%$ (target $2\%$ ), output 1% above potential, $r^* = 2\%$ , $\phi_\pi = \phi_y = 0.5$ :

| Term | Value | |------|-------| | $r^* + \pi$ | 5.0% | | $\phi_\pi(\pi - \pi^*)$ | +0.5 pp | | $\phi_y \cdot$ gap | +0.5 pp | | Prescribed $i^T$ | 6.0% |

Real rate: $r = i^T - \pi = 3\%$ , one pp above $r^*$ — a contractionary stance.

At the ZLB

The rule can prescribe $i^T < 0$ in deep recessions. The CB cannot comply → policy is constrained. Unconventional tools (QE, forward guidance) attempt to deliver the stance the Taylor rule implicitly calls for.

Historical Relevance

Volcker disinflation: Fed tightened aggressively in line with a strict Taylor rule.
Greenspan era 2002–06: Many critics argue the Fed held rates below Taylor prescription — contributing to the credit and housing booms.
Post-2008: Rule prescribes $i^T < 0$ ; Fed uses QE and forward guidance to simulate.

Worked Example

Target π* = 2%. Natural real rate r* = 2%. Current π = 3%. Output gap (Y − Yn)/Yn = 1%. φπ = φy = 0.5.

Baseline: r* + π = 2% + 3% = 5%.
Inflation-gap contribution: φπ(π − π*) = 0.5 × (3% − 2%) = 0.5 pp.
Output-gap contribution: φy × 1% = 0.5 pp.
Prescribed iᵀ = 5% + 0.5% + 0.5% = 6%.

Taylor rule prescribes iᵀ = 6%. Implied real rate r = iᵀ − π = 6% − 3% = 3%, one pp above r* — contractionary, as appropriate given both gaps are positive.

Common Mistakes

—Forgetting to include +π in the baseline — this is the Fisher-equation adjustment that keeps the rule in nominal terms.
—Confusing the Taylor principle (φπ > 0 or > 1) with a level condition.
—Applying the rule at the ZLB: when prescribed iᵀ < 0, the CB is constrained — unconventional tools needed.
—Using output growth instead of the output gap.

Exam Cues

→Formula: iᵀ = r* + π + φπ(π − π*) + φy·(Y − Yn)/Yn.
→Typical calibration: r* = π* = 2%, φπ = φy = 0.5.
→Taylor principle: φπ > 0 ensures stability. φπ > 1 ensures a real-rate response.
→ZLB: if Taylor prescribes iᵀ < 0, policy becomes constrained.