ECON6002 · Macroeconomic Analysis
Monetary Policy in the NK Model
Monetary Policy in the NK Model is Topic 9 of ECON6002 Macroeconomic Analysis at the University of Sydney, and one of the two extended-response topics the 55% final is built on. It closes the New Keynesian model by adding a Taylor rule to the dynamic IS curve and the NKPC, giving a three-equation system in the output gap, inflation and the nominal rate. The two ideas the exam returns to are the Taylor principle (the rate must respond more than one-for-one to inflation, φ_π > 1, for a determinate equilibrium) and the divine coincidence — that stabilising inflation also stabilises the output gap for demand and technology shocks, but not for a cost-push shock, which forces a genuine inflation–output trade-off.
What this chapter covers
- 011. The three-equation NK model — dynamic IS (demand) + NKPC (supply) + Taylor rule (policy), solved for the output gap ỹ, inflation π̂ and the nominal rate r̂
- 022. The transmission channel — with sticky prices, moving the nominal rate moves the real rate rr = r̂ − Eπ̂′, which shifts aggregate demand; under flexible prices this vanishes (neutrality)
- 033. The Taylor principle — φ_π > 1 is needed for a unique, stable (determinate) equilibrium, because the nominal rate must out-run inflation so the real rate rises
- 044. Indeterminacy — if φ_π < 1 the real rate falls when inflation rises, validating self-fulfilling 'sunspot' beliefs and giving a continuum of equilibria
- 055. The i.i.d. closed form — with i.i.d. shocks (φ_r = φ_y = 0) every expectation is zero, the model turns static and ỹ, π̂, r̂ solve algebraically
- 066. The shock taxonomy — demand and technology shocks move the gap and inflation together (no trade-off); a cost-push shock moves them in opposite directions (a trade-off)
- 077. Divine coincidence and its breakdown — stabilising inflation stabilises the gap for demand/tech shocks; a cost-push shock breaks it, so a higher φ_π cuts inflation variance but raises output variance
- 088. Optimal monetary policy — minimise the loss L = Var(π̂) + ϑ·Var(ỹ) (or the Ramsey welfare problem); the limits of standard NK (ZLB, finance, heterogeneity) motivate Topic 10
Solve the three-equation NK model under i.i.d. shocks
- +1i.i.d. shocks ⇒ every expected future deviation is zero: E_t ỹ_{t+1} = E_t π̂_{t+1} = 0. The IS curve collapses to ỹ_t = −(1/σ)r̂_t = −r̂_t (since σ = 1).
- +1The NKPC collapses to π̂_t = κ·ỹ_t + ε_t^π. The dynamic model is now static (two equations, plus the rule).
- +1Insert the rule r̂_t = φ_π·π̂_t + ε_t^m into the IS curve: ỹ_t = −φ_π·π̂_t − ε_t^m.
- +1Substitute the NKPC π̂_t = κ·ỹ_t + ε_t^π: ỹ_t = −φ_π(κ·ỹ_t + ε_t^π) − ε_t^m, so ỹ_t(1 + κφ_π) = −φ_π·ε_t^π − ε_t^m.
- +1Solve the three: ỹ_t = (−φ_π·ε_t^π − ε_t^m)/(1 + κφ_π), then π̂_t = (ε_t^π − κ·ε_t^m)/(1 + κφ_π), and r̂_t = φ_π·π̂_t + ε_t^m.
- +1(b) Denominator 1 + κφ_π = 1 + (0.15)(2) = 1.30. Output gap ỹ_t = −ε^m/1.30 = −0.25/1.30 ≈ −0.19 — a contraction; inflation π̂_t = −κ·ε^m/1.30 = −(0.15)(0.25)/1.30 ≈ −0.029 — inflation falls.
- +1Nominal rate r̂_t = ε^m/1.30 = 0.25/1.30 ≈ +0.19: the net rate rises even though π̂ falls, because the shock dominates the φ_π·π̂ feedback. The ex-ante real rate rr = r̂ − Eπ̂′ ≈ +0.19 > 0, so the tightening is real.
- +1(c) A demand shock ε^d raises ỹ and π̂ together (both = ε^d and κε^d over 1 + κφ_π), so the bank offsets both with no trade-off — divine coincidence. A cost-push shock ε^π raises π̂ but lowers ỹ, an unavoidable inflation–output trade-off — divine coincidence breaks down.
Key terms
- Three-equation NK model
- The core New Keynesian system: the dynamic IS curve (demand), the NKPC (supply) and a Taylor rule (policy), jointly determining the output gap, inflation and the nominal interest rate.
- Taylor rule
- An interest-rate feedback rule r̂_t = φ_r·r̂_{t−1} + (1−φ_r)(φ_π·π̂_t + φ_y·ỹ_t) + ε_t^m, in which the central bank sets the nominal rate in response to inflation and the output gap rather than fixing the money stock.
- Taylor principle
- The determinacy condition φ_π > 1: the policy rate must respond more than one-for-one to inflation so that the real rate rises with inflation and stabilises it; φ_π < 1 is destabilising.
- Determinacy
- The existence of a unique, stable rational-expectations equilibrium. Indeterminacy (φ_π < 1) admits a continuum of self-fulfilling 'sunspot' equilibria driven by arbitrary beliefs about future inflation.
- Real interest rate (transmission channel)
- The ex-ante real rate rr_t = r̂_t − E_t π̂_{t+1}. Because sticky prices keep expected inflation from moving one-for-one, changing the nominal rate changes the real rate — the sole reason monetary policy has real effects in the NK model.
- Divine coincidence
- The property that stabilising inflation simultaneously stabilises the output gap. It holds for demand and technology shocks (which move inflation and the gap together) but breaks down for a cost-push shock.
- Cost-push (mark-up) shock
- A supply disturbance ε_t^π that enters the NKPC directly, raising inflation while lowering the output gap. It moves the two goals in opposite directions and so creates a genuine inflation–output trade-off.
- Loss function / Ramsey policy
- The normative criterion for optimal policy: minimise L = Var(π̂) + ϑ·Var(ỹ), or equivalently maximise households' welfare (the Ramsey problem). The optimal rule responds to whatever the loss function penalises.
Monetary Policy in the NK Model FAQ
Is monetary policy in the NK model on the ECON6002 exam?
Yes — heavily. Topic 9 is one of the two extended-response topics on the 55% final exam (the other is Topic 8, the NKPC), and Q2 and Q3 are built on them. Any Topic-9 idea can also appear in the Q1 True/False/Uncertain set. The final is closed-book but provides a standard log-linearised NK formula sheet, so marks come from deriving the closed-form solution, signing the shock responses and stating the Taylor principle — not recall. Practice comes from Tutorials 7–10.
What is the Taylor principle and why does it need φ_π > 1?
The Taylor principle is the determinacy condition: the policy rate must respond more than one-for-one to inflation (φ_π > 1) for a unique, stable equilibrium. The mechanism runs through the real rate rr = r̂ − Eπ̂. If φ_π > 1, a rise in inflation lifts the nominal rate by more, so the real rate rises and pulls demand and inflation back down. If φ_π < 1 the real rate falls when inflation rises, which validates self-fulfilling inflation and gives indeterminacy. State the inequality strictly and give the real-rate story — an answer with only the number is a half-answer.
What is the divine coincidence, and when does it break down?
The divine coincidence is the property that stabilising inflation also stabilises the output gap. It holds for demand and technology shocks, which push inflation and the gap in the same direction, so a bank that leans against inflation automatically leans against the gap — the dual mandate is 'free'. It breaks down for a cost-push (mark-up) shock, which raises inflation while lowering the gap: now cutting inflation means deepening the recession, an unavoidable inflation–output trade-off. This distinction is the crux of Q2/Q3.
Why does the i.i.d. assumption make the NK model easy to solve?
With i.i.d. shocks and no interest-rate smoothing or output-gap feedback (φ_r = φ_y = 0) there are no state variables, so every expected future deviation is zero: E_t ỹ_{t+1} = E_t π̂_{t+1} = 0. The dynamic three-equation model collapses to a static two-equations-two-unknowns problem you can solve algebraically for ỹ, π̂ and r̂. With AR(1) (persistent) shocks the signs are identical but the responses become persistent and hump-shaped — same story, just propagated over time.
Can AI help me with monetary policy in the NK model?
Yes — ask Sia to walk through any monetary policy in the NK model problem or concept step by step, the way University of Sydney tests it. Sia is an AI tutor that explains the derivations — collapsing the i.i.d. system, signing each shock response, and showing why φ_π > 1 is needed — so you learn the method and can reproduce it closed-book, rather than just seeing a final number.
What is the most common mistake on Topic 9 questions?
Four recur: quoting the closed-form solution instead of deriving it; stating the Taylor principle as φ_π ≥ 1 or without the real-rate mechanism; claiming divine coincidence holds for all shocks (it fails for cost-push); and getting the φ_π comparative static backwards — a higher φ_π dampens both gap and inflation for demand/tech shocks, but for a cost-push shock it lowers inflation variance while raising output variance. Naming that asymmetry is the High Distinction point.
Studying with AI? Sia — free AI economics tutor works through ECON6002 step by step.
Exam move
Treat Topic 9 as one repeatable routine rather than a list of results, because the final is closed-book with only a formula sheet. Drill this sequence until it is automatic: (1) write the three equations — dynamic IS, NKPC, Taylor rule — and name each as demand, supply and policy; (2) impose the simplifying assumptions out loud (i.i.d. shocks ⇒ every expectation is zero ⇒ the model is static) and derive the closed form by substituting the rule into the IS curve, then the NKPC, and collecting ỹ; (3) sign each shock response and name the taxonomy — demand and technology shocks give divine coincidence (no trade-off), a cost-push shock gives a genuine trade-off; (4) state the Taylor principle φ_π > 1 strictly and explain it through the real rate rr = r̂ − Eπ̂; and (5) give the φ_π comparative static, flagging the cost-push asymmetry. Work Tutorials 7–10 by hand under timed, closed-book conditions during STUVAC, write the True/False/Uncertain explanations in full (the marks are in the mechanism plus one boundary case), and ask Sia to check each step of your derivations and generate fresh practice in the ECON6002 style, explaining the working as it goes.
