New Keynesian Economics

The three equations

Almost every modern central-bank model — the Federal Reserve's FRB/US, the European Central Bank's NAWM, the Bank of England's COMPASS, the New York Fed's DSGE — sits on top of the same three-equation skeleton. The equations look modest. They have shaped the past three decades of monetary policy.

Dynamic IS:        ŷ_t = E_t ŷ_{t+1} − σ⁻¹·(i_t − E_t π_{t+1} − r*_t)
Phillips curve:    π_t = β·E_t π_{t+1} + κ·ŷ_t
Taylor rule:       i_t = r* + π* + φ_π·(π_t − π*) + φ_y·ŷ_t + ε_t

Three variables: the output gap ŷ, inflation π, and the nominal policy rate i. Three equations. The system is closed. Given a path of shocks and an initial condition, the model produces a path for everything.

The dynamic IS curve is the household Euler equation. Households choose consumption today versus consumption tomorrow by setting expected marginal utility growth equal to the real return on saving. Linearised around steady state, this becomes the equation above: today's output gap depends on the expected future gap and on the real-rate gap (i − E π) − r*. The parameter σ is the inverse intertemporal elasticity of substitution — how aggressively consumption responds to interest-rate changes. Calibrations typically sit between 1 and 2.

The Phillips curve is derived from Calvo (1983) staggered price-setting. The discount factor β is around 0.99 quarterly. The slope κ depends on three deep parameters: the Calvo stickiness θ (probability of not resetting each quarter), the labour-supply elasticity, and σ. In the Smets-Wouters (2007) estimation, κ ≈ 0.05 — a one-percentage-point output gap produces just five basis points of additional inflation per quarter. The Phillips curve is flat, by construction, because Calvo prices are sticky.

The Taylor rule, written here in its simplest form, sets the policy rate as a response to the inflation gap and the output gap. The Taylor principle requires φ_π > 1; the standard calibration is φ_π = 1.5, φ_y = 0.5. The rule closes the model: it gives the central bank's reaction function.

Calvo pricing — the microfoundation that makes everything tick

Why does the New Keynesian Phillips curve look the way it does? Because Calvo's pricing assumption is exactly aggregable. Suppose at every period each of a continuum of monopolistically competitive firms gets a free chance to reset its price with probability 1−θ. With probability θ the firm holds its existing price one more period. The reset is independent across firms and across time, which makes the math tractable.

A firm that gets to reset chooses a price that maximises expected discounted profits over the entire random duration its new price will remain. Because that duration is geometrically distributed with parameter θ, the optimal reset price equals a markup over a discounted weighted average of expected future marginal costs. Aggregating across firms (a fraction 1−θ at the new price, θ at the old price index) gives the New Keynesian Phillips curve.

π_t = β·E_t π_{t+1} + κ·ŷ_t
κ = (1 − θ)(1 − βθ)·(σ + φ) / θ

The parameter φ is the inverse Frisch elasticity of labour supply. If θ = 0.66 (the standard US calibration — prices reset every three quarters on average), β = 0.99, σ = 1, φ = 1, then κ = (0.34)(0.34)(2)/0.66 ≈ 0.35. The Smets-Wouters estimate of κ is smaller — around 0.05 — because they allow indexation of non-reset prices to lagged inflation, which damps the slope. Either way, the structural derivation pins κ to underlying parameters; it's not a regression coefficient.

A worked numerical example — an inflation shock and the central bank's response

Suppose the economy starts at steady state — inflation at the 2% target, output at potential, the nominal rate at r* + π* = 2 + 2 = 4%. Then a cost-push shock raises current inflation by 1 percentage point, to 3%. What happens?

Under the Taylor rule (φ_π = 1.5, φ_y = 0.5), the central bank raises the nominal rate by 1.5 points, to 5.5%. Expected inflation rises by less than 1 point because the rule's response is credible: agents expect the central bank to fight the shock. Suppose expected inflation rises to 2.4%. Then the real rate is 5.5 − 2.4 = 3.1%, which is 1.1 points above the natural rate of 2%. The dynamic IS curve translates that real-rate gap into a fall in the output gap: with σ = 1, the gap drops by 1.1 points, to roughly −1%.

Now the Phillips curve feeds back. With κ = 0.05 and ŷ = −1%, the next-period contribution to inflation is −0.05 points. Combined with the discount-factor-weighted expected future inflation, the system marches back to steady state. Within five to ten quarters, depending on persistence parameters, inflation has returned to 2%, the output gap has closed, and the nominal rate has returned to 4%. The whole transmission runs through expectations — the central bank doesn't actually have to deliver the disinflation through grinding output losses; the credible promise to do so anchors expectations and pulls inflation back.

Comparison — Old Keynesian vs New Keynesian

Feature	Old Keynesian (1950s-70s)	New Keynesian (1990s-now)
Expectations	Adaptive, backward-looking	Rational, forward-looking
Microfoundations	Ad-hoc aggregate equations	Optimising households & firms
Price stickiness	Assumed (fixed wages)	Derived (Calvo, Rotemberg, menu costs)
Phillips curve	π = E π + α·u-gap (Friedman)	π = β·E π' + κ·ŷ (forward-looking)
Policy lever	Aggregate demand (G, M)	Interest rate via Taylor rule
Lucas critique	Vulnerable — coefficients shift	Immune — structural parameters
Canonical models	IS-LM, AD-AS	Smets-Wouters, FRB/US, NY Fed DSGE

Smets-Wouters and the rise of DSGE

The model that crystallised the New Keynesian framework into operational central-bank use is Smets and Wouters (2003, 2007). Frank Smets, then at the ECB, and Raf Wouters at the National Bank of Belgium, estimated a medium-scale DSGE — the three-equation skeleton augmented with habit formation in consumption, investment adjustment costs, variable capital utilisation, wage stickiness, and seven structural shocks — on euro-area and US data using Bayesian methods. The fit rivalled or beat reduced-form vector autoregressions on most variables.

The result was foundational. Bayesian DSGE estimation went from a research exercise to a central-bank workhorse within five years. By 2010, the ECB's NAWM, the Bank of England's COMPASS, the Norges Bank's NEMO, the Bank of Canada's ToTEM, and the New York Fed's DSGE were all Smets-Wouters-style models being used for quarterly forecasting and counterfactual policy analysis. The framework provides what no purely empirical method can: structural interpretation. A forecast change can be attributed to a specific shock — a productivity shock, a demand shock, a markup shock, a policy shock — because every variable in the model has a microeconomic origin story.

Extensions — financial frictions, heterogeneity, the zero lower bound

• Financial frictions. The baseline NK model has no banks. Curdia-Woodford (2010) introduce a credit spread; Gertler-Karadi (2011) add a banking sector with leverage constraints; Christiano-Motto-Rostagno (2014) integrate a financial accelerator. After 2008, these extensions became essential.
• Heterogeneous agents (HANK). The representative-agent core misses inequality and marginal propensities to consume. Kaplan-Moll-Violante (2018) develop heterogeneous-agent New Keynesian models with realistic wealth distributions. The implications for fiscal and monetary policy multipliers can differ substantially from the representative-agent baseline.
• Zero lower bound. The linearised model assumes unconstrained interest rates. At the ZLB, monetary policy must substitute forward guidance, balance-sheet policy, or fiscal stimulus. Eggertsson-Woodford (2003) develop the analytical framework; the post-2008 literature is enormous.
• Behavioural deviations. Gabaix (2020) studies bounded-rational expectations. Mankiw-Reis (2002) propose sticky information. Sims (2003) and Maćkowiak-Wiederholt (2009) introduce rational inattention. Each generates plausible alternatives to fully rational expectations.
• Open economy. Galí-Monacelli (2005) extend to a small open economy with exchange rates. Multi-country versions are used at the IMF and OECD for spillover analysis.

The Taylor principle inside the model

The most consequential property of New Keynesian models is determinacy. With rational expectations, the system has many solutions in principle — any path that satisfies the equations and transversality conditions is admissible. To pin down a unique stable equilibrium, the Taylor rule's long-run inflation coefficient must exceed one. This is the Taylor principle, and inside the NK model it has a sharp algebraic justification.

Linearise the three equations around steady state and solve for the dynamics. With φ_π > 1, the system has the right number of explosive roots for the right number of forward-looking variables — there's a unique non-explosive solution. With φ_π < 1, there's a continuum of non-explosive paths, including sunspot equilibria where arbitrary belief shifts move inflation. Clarida-Galí-Gertler (2000) argue the great inflation of the 1970s was at least partly a violation of the Taylor principle — the pre-Volcker Fed had estimated φ_π around 0.83 — and the Great Moderation was its restoration with post-Volcker φ_π estimates around 2.15.

Limits and common pitfalls

• Linearisation. The model is linearised around the steady state. At the zero lower bound or during deep recessions, non-linearities matter — the model can give misleading answers exactly when policy needs them most.
• No default risk. The basic three-equation model has no leverage, no defaults, no asset-price booms. Adding them is possible but adds complexity and parameters.
• Rational expectations. Households and firms forecast as if they know the model. Real agents do not. The mismatch is not fatal — models with imperfect information often match the data better — but it is real.
• Calvo's specific form. The fraction θ resets free per period, with no menu cost and no state-dependent timing. Real pricing decisions look more like menu-cost models (Golosov-Lucas 2007, Nakamura-Steinsson 2008), but Calvo dominates because it aggregates cleanly. The mismatch matters for moments like the response to large shocks.
• Steady-state real rate. The natural real rate r* is a key parameter, but it's unobserved and time-varying. Estimates have fallen by 150 basis points since the early 2000s, shifting every model prescription.
• Identification. Bayesian DSGE estimation depends heavily on priors. With weak data identification, priors drive results. Sensitivity analysis to priors is essential and often skipped.