The RBC model (Chapter 14) showed that technology shocks in a frictionless economy can generate realistic business cycle statistics. But it has a critical blind spot: monetary policy does nothing. In the RBC world, money is neutral, and the Fed is irrelevant. This contradicts overwhelming evidence that monetary policy affects real output, at least in the short run.
New Keynesian (NK) economics solves this by adding nominal rigidities (sticky prices or wages) to the RBC chassis. The result is a model where monetary policy has real effects, the central bank faces meaningful tradeoffs, and the Taylor rule becomes the central equation of modern central banking.
By the end of this chapter, you will be able to:
Explain why monopolistic competition is necessary for price stickiness to matter
Derive the New Keynesian Phillips Curve from Calvo pricing
Derive the dynamic IS curve from the household Euler equation
Analyze the 3-equation NK model (NKPC, IS, Taylor rule)
Explain the Taylor principle and its role in macroeconomic stability
Analyze the zero lower bound and the liquidity trap
Walkthroughs in This Chapter
This chapter's New Keynesian framework is a hub in the book's Walkthrough network. Each juncture appears after the section where the relevant model is developed; in the linear read they stay collapsed.
Monopolistic competition (Dixit-Stiglitz).A market structure where many firms produce differentiated goods and each firm faces a downward-sloping demand curve with elasticity $\varepsilon$. Unlike perfect competition, firms set prices above marginal cost. This is the prerequisite for price stickiness to have macroeconomic consequences.
In perfect competition, firms are price takers: there is no price to "stick." For price rigidity to matter, firms must have price-setting power. The standard NK setup uses Dixit-Stiglitz monopolistic competition:
Each firm faces a downward-sloping demand curve: $y_j = (p_j / P)^{-\varepsilon} Y$.
15.2 Calvo Pricing
Price stickiness (nominal rigidity).The empirical observation that firms do not continuously adjust their prices in response to changing demand or cost conditions. In the NK model, price stickiness is modeled via Calvo pricing and is the critical friction that gives monetary policy real effects.
Calvo pricing.Each period, fraction $(1-\theta)$ of firms randomly reset their price. Fraction $\theta$ keep their price unchanged. Expected price duration: $1/(1-\theta)$ periods. With $\theta = 0.75$, the average firm resets its price once per year.
The optimal reset price is a weighted average of current and expected future marginal costs:
Each period only a random slice of firms gets to change its price, so the price level as a whole moves slowly. A firm that resets today is stuck with that price for a while, so it sets it with an eye on where costs are heading, not just where they are. That lag between what the economy needs and what prices actually do is the whole reason monetary policy bites: change the money supply and, for a stretch, real spending moves before prices catch up.
New Keynesian Phillips Curve (NKPC).The equation $\pi_t = \beta E_t\pi_{t+1} + \kappa x_t$ relating current inflation to expected future inflation and the current output gap. Unlike the traditional Phillips curve, the NKPC is purely forward-looking and derived from firms' optimal pricing under Calvo frictions.
Output gap.The difference between actual output and the natural (flexible-price) level of output: $x_t = y_t - y_t^n$. A positive output gap means the economy is producing above its frictionless potential, putting upward pressure on inflation via the NKPC.
where $\pi_t$ is inflation, $x_t$ is the output gap, and $\kappa = \frac{(1-\theta)(1-\beta\theta)}{\theta} \cdot \frac{\sigma + \varphi}{1 + \varphi\varepsilon}$. Current inflation depends on expected future inflation (forward-looking!) and current marginal cost (proportional to output gap). With cost-push shocks:
Firms set prices looking forward, so today's inflation tracks two things: what firms expect inflation to be tomorrow, and how hot the economy is running right now. The old Phillips curve treated inflation as a trade-off you could ride; this one says the trade-off only exists in the gap between expectations and reality. Anchor expectations and a hot economy still pushes prices up, but the leverage runs through what people believe is coming, not just where output sits today.
Step 3: The optimal reset price is a discounted sum of expected future marginal costs: $\hat{p}_t^* = (1-\beta\theta)\sum_{k=0}^\infty(\beta\theta)^k E_t[\widehat{mc}_{t+k} + \hat{p}_{t+k}]$.
Step 1: From NKPC (one-period shock, $E_t\pi_{t+1} = 0$): $\pi = \kappa x + u = 0.3x$.
Step 2: From IS (one-period, $E_tx_{t+1} = 0$): $x = -(1/\sigma)(i - r^n) = -(i - 2)$.
Step 3: Taylor rule: $i = 2 + 1.5\pi + 0.5x$.
Step 4: Substitute Taylor into IS: $x = -(2 + 1.5\pi + 0.5x - 2) = -1.5\pi - 0.5x$, so \$1.5x = -1.5\pi$, giving $x = -\pi$.
Step 5: Substitute into NKPC: $\pi = 0.3(-\pi) = -0.3\pi$, so \$1.3\pi = 0$ and $\pi = 0$, $x = 0$, $i = 2\%$.
Result: With no shocks, the equilibrium is $\pi = 0$, $x = 0$, $i = r^* = 2\%$. Divine coincidence holds.
Example 15.3 — Optimal Taylor Rule Coefficients
The central bank minimizes $L = E_0\sum\beta^t[x_t^2 + \alpha_\pi\pi_t^2]$ with $\alpha_\pi = 0.5$, $\kappa = 0.3$.
Step 1: Under discretion, the central bank minimizes the one-period loss taking expectations as given: $\min_{x_t}\{x_t^2 + \alpha_\pi(\kappa x_t + u_t)^2\}$.
Step 4: The implied Taylor rule achieves this by responding aggressively to inflation. Higher $\alpha_\pi$ (inflation-averse) implies a larger $\phi_\pi$, reducing inflation at the cost of greater output gap volatility.
Cost-push shock.An exogenous disturbance $u_t$ that shifts the NKPC: $\pi_t = \beta E_t\pi_{t+1} + \kappa x_t + u_t$. Cost-push shocks (e.g., oil price spikes) break divine coincidence by creating a tradeoff between stabilizing inflation and stabilizing the output gap.
15.4 The Dynamic IS Curve
Natural rate of interest.The real interest rate that would prevail in the flexible-price equilibrium ($r_t^n$). When the central bank sets the real rate below the natural rate, it stimulates demand (positive output gap); above it, it contracts demand. The natural rate is the benchmark for whether monetary policy is expansionary or contractionary.
The output gap depends on the expected future gap minus the difference between the real interest rate and the natural rate. When the central bank sets the real rate below the natural rate, it stimulates demand.
People smooth their spending over time, so what they demand today depends on the whole expected path of real interest rates, not just today's rate. A central bank that promises cheap money for years stimulates now, because households and firms pull future spending forward. The benchmark is the natural rate, the rate that would prevail if prices were free to adjust: set the real rate below it and demand heats up, above it and demand cools.
15.5 The Taylor Rule
Taylor rule.A monetary policy rule $i_t = r^* + \phi_\pi\pi_t + \phi_x x_t$ prescribing how the central bank should set the nominal interest rate in response to inflation and the output gap. John Taylor (1993) showed that this simple rule approximates actual Fed behavior remarkably well.
Taylor principle.The requirement that $\phi_\pi > 1$: the central bank must raise the nominal interest rate by more than one-for-one with inflation. This ensures the real interest rate rises with inflation, stabilizing the economy.
Determinacy / indeterminacy.When $\phi_\pi > 1$ (Taylor principle satisfied), the NK model has a unique bounded equilibrium (determinacy). When $\phi_\pi < 1$, multiple bounded equilibria exist (indeterminacy), allowing sunspot-driven fluctuations unrelated to fundamentals.
The Taylor rule did not arrive fully formed. It is the operational distillate of a datable lineage: the inflation-targeting and DSGE consensus that crystallized between 1980 and 2008, downstream of the natural-rate counter-revolution it absorbed.
History of economic thought timeline
The framework's apparent payoff was the Great Moderation: across the rich economies, output-growth volatility fell sharply from the early 1980s. The cross-country GDP record over the 1971–2008 window is where that decline is visible.
GDP per capita map
15.6 The 3-Equation NK Model
Divine coincidence.In the basic NK model without cost-push shocks, stabilizing inflation automatically stabilizes the output gap. There is no policy tradeoff: zero inflation and zero output gap are simultaneously achievable. Cost-push shocks break this coincidence.
Commitment vs discretion (monetary policy).Under commitment, the central bank binds itself to a future policy path, improving outcomes by anchoring expectations. Under discretion, the central bank re-optimizes each period, which can lead to time-inconsistency problems (the inflation bias of Chapter 16) and suboptimal responses to cost-push shocks.
Three equations, three unknowns ($\pi_t$, $x_t$, $i_t$):
The whole model comes down to three rules working at once: how prices drift (firms set them forward-looking), how spending responds to interest rates (cheaper money means more demand), and how the central bank reacts (raise rates when inflation climbs). Plug them together and they pin down inflation and the output gap jointly. Nothing is set in isolation; a cost shock, a demand swing, or a more aggressive central bank all ripple through the same three-way feedback loop. The interactive below lets you push one lever and watch the equilibrium settle.
Interactive: 3-Equation NK Model
Adjust shocks and the Taylor rule aggressiveness to see how the NK equilibrium shifts. The left panel shows the NKPC and the monetary policy reaction (combining IS + Taylor rule) in $(\pi, x)$ space. The right panel shows the implied interest rate.
Figure 15.2. The 3-equation NK model. Left panel: NKPC (blue, upward slope) and monetary policy reaction function (red, downward slope) in ($x$, $\pi$) space. Right panel: Taylor rule interest rate. Adjust sliders to see how shocks and policy aggressiveness shift the equilibrium. Hover for values.
The Taylor Principle
The Taylor principle is the single most important operational rule in modern central banking. The pre-Volcker Fed (1960s–70s) had $\phi_\pi \approx 0.83 < 1$, producing the Great Inflation. The post-Volcker Fed had $\phi_\pi \approx 2.15 > 1$, producing the Great Moderation.
Interactive: Taylor Principle Explorer
Slide $\phi_\pi$ across the critical threshold of 1. Below 1, the economy is indeterminate: a rise in inflation lowers the real rate, fueling more inflation. Above 1, the real rate rises with inflation, stabilizing the economy.
Passive (0.5)Threshold: 1.0Aggressive (3.0)
DETERMINATE ($\phi_\pi = 1.50 > 1$): Unique stable equilibrium. A rise in inflation triggers a larger rise in the nominal rate, increasing the real rate and dampening demand.
Figure 15.3. Taylor principle visualization. The blue line is the Taylor rule ($i$ vs $\pi$). The gray dashed line is $i = \pi$ (constant real rate). When the Taylor rule is steeper than the 45-degree line ($\phi_\pi > 1$), real rates rise with inflation (stable). When flatter ($\phi_\pi < 1$), real rates fall with inflation (unstable).
The 3-equation model is not a free-standing invention. It is the formalization of a school — Mankiw and the New Keynesians, Woodford's microfounded monetary theory, Galí's textbook synthesis — whose intellectual lineage the history-of-thought volume traces in full.
History of economic thought
The framework's apparent vindication is an episode, not a theorem: the 1984–2007 Great Moderation, when the rich economies ran low, stable inflation under exactly this kind of rule. The economic-history volume tells that episode in full.
Economic history
15.7 Optimal Monetary Policy
Section 15.6 established divine coincidence: without cost-push shocks ($u_t = 0$), the central bank can achieve $\pi_t = 0$ and $x_t = 0$ simultaneously. There is no tradeoff. But when $u_t \neq 0$ (an oil price spike, a supply disruption, a wage-push shock), divine coincidence breaks. Now the central bank faces a genuine policy tradeoff: it can reduce inflation only by accepting a larger output gap, or close the output gap only by tolerating higher inflation. How should it choose?
The answer depends on the central bank's loss function, its formal objective. The standard specification penalizes both output gap deviations and inflation deviations quadratically:
The parameter $\alpha_\pi > 0$ is the relative weight on inflation stabilization. A central bank with high $\alpha_\pi$ (inflation-averse, like the Bundesbank or ECB) prioritizes price stability; one with low $\alpha_\pi$ (employment-focused) tolerates more inflation to stabilize output. The Federal Reserve's "dual mandate" corresponds to moderate $\alpha_\pi$.
Under discretion, the central bank re-optimizes each period, taking private-sector expectations as given. It minimizes the one-period loss $x_t^2 + \alpha_\pi \pi_t^2$ subject to the NKPC constraint $\pi_t = \beta E_t\pi_{t+1} + \kappa x_t + u_t$, treating $E_t\pi_{t+1}$ as fixed. The first-order condition yields:
When a cost shock hits and the bank cannot escape the trade-off, it splits the pain: it lets inflation rise a little and lets output fall a little, rather than absorbing all of one. How it splits depends entirely on how much it hates inflation. A hard-money bank lets output take more of the hit to keep prices steady; an employment-focused one tolerates more inflation to protect jobs.
The central bank partially accommodates the cost-push shock. With higher $\alpha_\pi$, it tolerates a larger output gap to keep inflation closer to zero. With lower $\alpha_\pi$, it accepts more inflation to protect output. This is the policy frontier under discretion: the set of achievable combinations of output gap variance and inflation variance as $\alpha_\pi$ varies.
Under commitment, the central bank binds itself to a state-contingent plan at time zero. Because it can promise future deflation after a cost-push shock, it manipulates the $\beta E_t\pi_{t+1}$ term in the NKPC. A credible promise of lower future inflation reduces current inflation directly, because private agents expect deflation and moderate their price-setting today. The optimal targeting rule under commitment (Clarida, Gali, and Gertler, 1999; Woodford, 2003) is:
A bank that can credibly promise future restraint gets cheaper disinflation today. The reason is the expectations channel: if people believe prices will be lower down the road, they moderate their own price-setting now, so inflation falls before the central bank has to squeeze output hard. The catch is credibility, the promise only works if everyone trusts the bank will keep it even when keeping it hurts. This is the Volcker lesson in one line.
This is history-dependent: inflation depends on its own past, not just the current shock. Under discretion, each period is a fresh optimization: the central bank cannot credibly commit to future deflation, so the expectations channel is unavailable. Under commitment, it can, and the result is a strictly better outcome: for any $\alpha_\pi$, the commitment frontier lies inside (southwest of) the discretion frontier in (var($x$), var($\pi$)) space.
The gain from commitment depends on shock persistence. When cost-push shocks are iid ($\rho_u = 0$), the future is irrelevant and commitment offers little advantage. When shocks are persistent ($\rho_u \to 1$), the expectations channel is powerful: the central bank's ability to promise future deflation dramatically reduces the current cost of disinflation. This is the Volcker lesson formalized: credible commitment to fighting inflation reduces the sacrifice ratio.
Interactive: Policy Frontier — Commitment vs Discretion
Adjust the inflation weight $\alpha_\pi$ to trace the policy frontier, and the shock persistence $\rho_u$ to see how persistence amplifies the commitment advantage. The commitment frontier (blue) always lies southwest of the discretion frontier (red): commitment achieves lower variance of both inflation and the output gap.
Fixed: $\kappa = 0.3$, $\beta = 0.99$, $\sigma = 1.0$. | Gain from commitment: —%
Figure 15.6. Policy frontier under discretion (red dashed) vs commitment (blue solid) in output gap variance–inflation variance space. Each curve shows the achievable (var($x$), var($\pi$)) pairs as $\alpha_\pi$ varies. Dots mark the current operating point. The commitment frontier lies strictly inside: commitment achieves lower variance of both variables. Increase $\rho_u$ to see the commitment advantage grow.
Example 15.7 — Commitment vs Discretion with Persistent Shocks
Step 3 (Commitment): Under commitment, the central bank promises future deflation. The optimal plan reduces $\pi_0$ below \$1.957\%$ because $E_0 \pi_1 < 0$ feeds back through the NKPC to lower current inflation. The history-dependent rule produces $\pi_0 \approx 0.71\%$, $x_0 \approx -0.21\%$: more output sacrifice on impact, but less inflation and faster convergence.
Step 4 (Comparison): $\mathcal{L}_C \approx 0.92$. Gain from commitment: $(1.33 - 0.92)/1.33 = 31\%$. The commitment advantage is substantial with persistent shocks because the expectations channel has multiple future periods to work through.
The optimal policy analysis assumes the central bank can set any interest rate. In practice, the nominal interest rate cannot go below zero: $i_t \geq 0$. When the natural rate falls below zero, even optimal policy is powerless: the zero lower bound binds, and conventional monetary policy is exhausted. Section 15.8 analyzes this constraint.
15.8 The Zero Lower Bound
The nominal interest rate cannot go below zero: $i_t \geq 0$. When the natural rate $r_t^n$ falls below zero during a severe recession, the Taylor rule calls for a negative nominal rate, which is infeasible. Conventional monetary policy is powerless.
Zero lower bound (ZLB).The constraint $i_t \geq 0$ on the nominal interest rate. When the natural rate falls below zero during a severe recession, the Taylor rule prescribes a negative nominal rate, which is infeasible. Conventional monetary policy is powerless at the ZLB.
Liquidity trap.A situation where the nominal interest rate is at zero and further monetary expansion cannot lower the real rate because agents are indifferent between money and bonds at $i = 0$. Demand remains depressed despite ample liquidity.
Forward guidance.Central bank communication about the future path of interest rates, used as a tool when current rates are at the ZLB. By promising to keep rates low even after the recession ends, the central bank can lower long-term rates and stimulate current spending. The effectiveness depends on the credibility of the commitment.
Forward guidance puzzle.The theoretical prediction that forward guidance about rates far in the future has implausibly large effects on current output and inflation. In the standard NK model, promising low rates $k$ periods ahead has effects that grow with $k$, which is unrealistic. This puzzle suggests the model overestimates agents' responsiveness to distant policy commitments.
Interactive: Zero Lower Bound Trap
Slide the natural rate from positive to negative. When $r^n$ goes negative, the Taylor rule calls for a negative nominal rate, but the ZLB binds at zero. The gap between the required rate and zero represents monetary policy impotence.
Deep recession ($-4\%$)Normal ($+2\%$)Boom ($+3\%$)
Normal conditions: Taylor rule rate = 2.0%. No ZLB constraint. Output gap = 0%.
Figure 15.4. ZLB trap. Left panel: Taylor rule prescribed rate (blue) vs actual rate (red, floored at 0). The shaded red region is the "monetary policy gap": the amount of stimulus the central bank cannot deliver. Right panel: resulting output gap. Drag $r^n$ below zero to see the trap engage.
Take
"The Fed is printing money and destroying the dollar" — Ron Paul, Peter Schiff, and a generation of viral clips
Ron Paul spent decades grilling Fed chairs on C-SPAN, and the clips became YouTube gold for the "End the Fed" movement. Peter Schiff turned "the Fed is debasing the currency" into a media empire. During 2020–2023, when the Fed's balance sheet ballooned from \$4 trillion to \$9 trillion and inflation hit 9%, "they're printing money" went from fringe libertarian talking point to dinner-table consensus. The New Keynesian model you just learned says the Fed controls the economy through interest rates, expectations, and the Taylor rule. The "End the Fed" crowd says the Fed is the problem. Who's right?
Advanced
The popular version
Ron Paul's "End the Fed" and Schiff's "money printer go brrr" rhetoric share a common structure: the Fed is a shadowy institution creating money out of nothing, enriching Wall Street, and impoverishing savers. The viral version strips out every mechanism and replaces it with a morality tale: sound money good, fiat money bad, gold standard natural. This is a person-level weakness; it treats the Fed as a conspiracy rather than a policy institution operating under constraints. On the other side, the financial media version is equally hollow. "The Fed raised rates by 25 basis points; here's what it means for your portfolio" treats every rate decision as if it mechanically determines outcomes, because that story is simpler than "the Fed's influence depends on the state of the economy, fiscal policy, global capital flows, and whether expectations are anchored." Paul and Schiff overstate the Fed's malice; CNBC overstates its omnipotence. Both miss the real question: how much control does the Fed actually have, and under what conditions?
The strongest case for Fed control
If Paul and Schiff had to make their case with the best available evidence, here is what they would say: the Fed's balance sheet expansion from \$900 billion (2008) to \$9 trillion (2022) coincided with asset price inflation that disproportionately benefited the wealthy, and eventually consumer inflation that punished everyone else. The Fed kept rates at zero for years while housing and stock prices soared, which is a wealth transfer by another name. But the NK model offers a powerful counter-case for Fed control working well. The Great Moderation (1984–2007) is the evidence: the Taylor principle was satisfied ($\phi_\pi > 1$), inflation expectations were anchored, and the 3-equation system produced stable equilibria. Clarida, Gali, and Gertler (2000) estimated that monetary policy was "good" (Taylor principle satisfied) after 1979 and "bad" (violated) before, and they argued that this regime shift explains the move from stagflation to stability. Even at the ZLB, the Fed used forward guidance and QE to prevent deflation. The formal argument is clean: as long as the Taylor principle holds and the ZLB doesn't bind, the Fed has a unique mapping from its instrument ($i$) to its targets (inflation and output gap).
The strongest case against
Three episodes give the "End the Fed" crowd more ammunition than the NK model's defenders would like. The first is the ZLB (2008–2015), when the Fed's primary tool became inoperative. The "forward guidance puzzle" (Del Negro, Giannoni, Patterson 2015) shows NK models predict absurdly large effects from forward guidance, suggesting the expectations channel is mismodeled. The second is post-COVID inflation (2021–2023). The Fed kept rates at zero while inflation rose to 9%. When rates were raised aggressively, inflation fell, but whether the Fed caused the decline or supply normalization did the work is genuinely uncertain. Ron Paul would say the Fed created the inflation and then took credit for fixing it. The identification problem is severe enough that we cannot dismiss this. The third is the FTPL challenge. The Fiscal Theory of the Price Level (Ch 16) argues that if Congress sets fiscal policy independently, the price level adjusts to satisfy the government budget constraint regardless of what the Fed does. Cochrane (2023) argues the post-COVID inflation was fundamentally fiscal: \$5 trillion in deficit-financed transfers increased nominal demand, and no amount of rate hiking could have prevented inflation without fiscal consolidation. If Cochrane is right, the Fed is less relevant than either its critics or its defenders believe.
The judgment
So were Ron Paul and Peter Schiff right? Partially, and for some of the wrong reasons. They are correct that the Fed's massive balance sheet expansion created distributional consequences the institution never acknowledged, and that "printing money" is not a crazy description of what QE does. They are wrong that the gold standard is the answer, that the Fed is a deliberate wealth-destruction machine, or that abolishing it would produce stability. The Fed has significant but conditional control over inflation, and the conditions are more fragile than the NK model suggests. The Great Moderation was real and the model's explanation is the best available. But the Fed controls inflation well only when: (1) the economy is away from the ZLB, (2) fiscal policy is not in "active" mode, (3) supply shocks are small, and (4) inflation expectations are anchored. When any condition breaks, control weakens sharply. The post-COVID experience violated conditions (2), (3), and (4) at once. The honest assessment: "usually, approximately, under favorable conditions." The "End the Fed" movement diagnosed a real loss of control in 2021–2023 but prescribed a cure (abolishing discretionary monetary policy) that would make the next crisis worse, not better.
The zero lower bound was the canonical model's first real failure. The model has no financial sector, so the 2008 crisis — a credit panic that pushed the natural rate below zero and pinned policy at the floor — is exactly the event it could not generate. The economic-history volume tells that crisis as it happened.
Economic history
15.9 NK vs. RBC: Impulse Responses Compared
Shock
RBC Response
NK Response
Technology +
Output up, hours ambiguous
Output up more slowly, hours may fall
Monetary expansion
No effect (neutral)
Output up, inflation up, rate down
Cost-push
Maps to tech shock
Inflation up, output down (stagflation)
Interactive: NK vs RBC Impulse Responses
Compare impulse responses side by side. Toggle between a technology shock and a monetary policy shock to see what nominal rigidities add.
Technology shock: Both models show output rising. RBC: immediate full adjustment. NK: sluggish adjustment due to sticky prices. Hours response differs.
Figure 15.5. Side-by-side impulse responses. Left column: RBC (flexible prices). Right column: NK (sticky prices). Top row: output. Bottom row: inflation. Toggle between shock types. The monetary shock has no effect in RBC but real effects in NK. This is what price stickiness adds.
15.10 Calvo Pricing Visualization
Interactive: Calvo Pricing Animation
A grid of 100 firms. Each period, a random fraction $(1-\theta)$ gets to reset their price (green). The rest are stuck with their old price (red). Adjust $\theta$ and step through periods to see how price stickiness works.
Flexible (0.00)Baseline (0.75)Very sticky (0.95)
Period 0 |
Reset this period: 100 / 100 |
Stuck: 0 / 100 |
Avg. price age: 0.0 periods
Figure 15.1. Calvo pricing visualized. Green cells = firms that reset their price this period. Red cells = firms stuck with an old price. With $\theta = 0.75$, only 25% of firms adjust each quarter, so aggregate prices are sluggish. This is the micro-mechanism behind the NKPC. Click "Step Forward" or "Auto-Play" to advance.
Example 15.4 — Indeterminacy When the Taylor Principle Is Violated
Set $\phi_\pi = 0.8 < 1$. Show that sunspot equilibria are possible.
Step 1: Suppose agents suddenly believe inflation will be 2% next period (a sunspot). From the IS curve: $x = E_tx_{t+1} - (1/\sigma)(i - E_t\pi_{t+1} - r^n)$.
Step 2: Taylor rule: $i = r^* + 0.8\pi + 0.5x$. With $\phi_\pi = 0.8$, a 1% rise in inflation raises $i$ by only 0.8%. The real rate $r = i - E\pi$ falls by 0.2%.
Step 3: Lower real rate stimulates demand: $x$ rises. Higher output gap raises inflation via NKPC: $\pi = \kappa x > 0$. This validates the original belief.
Step 4: The sunspot is self-fulfilling: belief in higher inflation causes lower real rates, higher demand, and higher actual inflation. With $\phi_\pi > 1$, this loop is broken: the real rate rises with inflation, choking off demand.
Example 15.5 — ZLB Scenario
A severe recession drives the natural rate to $r^n = -3\%$. Parameters: $\phi_\pi = 1.5$, $\phi_x = 0.5$, $\sigma = 1$, $\kappa = 0.3$.
Step 1: Without ZLB, Taylor rule: $i = 2 + 1.5(0) + 0.5(0) - 3 = -1\%$ (assuming $r^n$ enters). Negative rate is infeasible.
Step 2: ZLB binds: $i = 0$. Real rate: $r = 0 - E\pi \approx 0\%$ (if inflation near zero). But natural rate is $-3\%$. Monetary policy gap: $r - r^n = 0 - (-3) = 3\%$ too tight.
Step 3: From the IS curve: $x \approx -(1/\sigma)(r - r^n) = -3\%$. The output gap is severely negative.
Step 4: From NKPC: $\pi = \kappa x = 0.3(-3) = -0.9\%$. Deflation sets in, raising the real rate further and deepening the recession: the deflationary spiral.
Policy options: Forward guidance (promise low rates after recovery), fiscal stimulus (government spending has multiplier $> 1$ at ZLB), or unconventional monetary policy (QE).
Example 15.6 — NK vs RBC Impulse Responses to a Monetary Shock
Compare responses to a surprise 1% interest rate cut.
RBC model: Money is neutral. The nominal rate drop has no effect on any real variable. Output, consumption, investment, and hours are all unchanged. $\Delta y = \Delta c = \Delta i = \Delta h = 0$.
NK model: With $\theta = 0.75$ (prices reset once per year on average):
Step 1: The real rate falls by approximately 1% (prices are sticky, so lower $i$ passes through to lower $r$).
Step 2: From the IS curve, the output gap rises: $\Delta x \approx (1/\sigma)\Delta r = 1\%$.
Step 3: From the NKPC, inflation rises: $\Delta\pi = \kappa\Delta x = 0.3\%$.
Step 4: Over time, prices adjust. As more firms reset at higher prices, the price level catches up, the real rate returns to normal, and the output effect dissipates. Half-life: roughly $1/(1-\theta) = 4$ quarters.
Key insight: Nominal rigidities convert a nominal shock into a real one. As $\theta \to 0$, the NK response converges to the RBC response (no real effects).
The Historical Lens
The Volcker disinflation (1979–82): Raising rates to 20% to break inflation.
When Paul Volcker became Fed Chair in August 1979, U.S. inflation was running at 13% and accelerating. Inflation expectations had become unanchored: workers demanded higher wages, firms raised prices, and the Phillips curve had shifted up repeatedly. The pre-Volcker Fed under Arthur Burns had responded to inflation with modest rate increases ($\phi_\pi \approx 0.83 < 1$), violating the Taylor principle and allowing inflation to become self-fulfilling.
Volcker's strategy was radical: he raised the federal funds rate to a peak of 20% in June 1981. The real interest rate exceeded 8%, the most restrictive monetary policy in modern U.S. history. The economy plunged into recession: unemployment peaked at 10.8% in November 1982, and GDP fell by 2.7%.
The result: Inflation fell from 13% to 3% by 1983. More importantly, inflation expectations were broken. The sacrifice ratio — the cumulative output loss per percentage point of disinflation — was approximately 2.3, within the range predicted by NK models with moderate price stickiness ($\theta \approx 0.75$).
NK interpretation: Volcker's policy implemented the Taylor principle with a vengeance ($\phi_\pi \gg 1$). By demonstrating that the Fed would tolerate severe recession to reduce inflation, he shifted from an indeterminate regime to a determinate one. Post-Volcker, the Fed maintained $\phi_\pi > 1$, producing the Great Moderation (1984–2007), the longest period of macroeconomic stability in U.S. history.
Thread Example: The Kaelani Republic
NK Analysis of Kaelani's Monetary Policy
Kaelani's central bank adopts an inflation-targeting regime with target $\pi^* = 3\%$ and Taylor rule: $i_t = 0.04 + 1.5(\pi_t - 0.03) + 0.5x_t$.
Scenario 1 (demand shock): Commodity price boom raises inflation to 5%. Taylor rule: $i = 0.04 + 1.5(0.02) + 0.5(0.02) = 8\%$. The real rate rises, cooling demand.
Scenario 2 (ZLB): Global recession drives $r^n = -2\%$. Taylor rule calls for $i = -1\%$, but the ZLB binds at 0%. The economy remains in recession. Options: fiscal stimulus, forward guidance, or unconventional monetary policy.
Calvo pricing: Each period, fraction $(1-\theta)$ of firms reset prices. This generates the NKPC: $\pi_t = \beta E_t\pi_{t+1} + \kappa x_t$.
The dynamic IS curve is derived from the household Euler equation, giving it the microfoundations IS-LM lacks.
The Taylor principle ($\phi_\pi > 1$) is necessary for a unique stable equilibrium. Violation produces indeterminacy.
Divine coincidence holds without cost-push shocks. Cost-push shocks create a policy tradeoff.
Optimal monetary policy: With cost-push shocks, the central bank minimizes a loss function over inflation and output gap variance. Commitment to a policy rule dominates discretion by leveraging the expectations channel.
The ZLB constrains monetary policy when the natural rate is negative. Unconventional policies are the alternatives.
The NK model nests RBC: as $\theta \to 0$, NK reduces to RBC. Price stickiness gives monetary policy real effects.
In the 3-equation NK model with $\beta = 0.99$, $\kappa = 0.1$, $\sigma = 1$, $\phi_\pi = 1.5$, $\phi_x = 0.5$, $r^* = 0.02$: Verify that $\pi_t = 0$, $x_t = 0$, $i_t = 0.02$ is an equilibrium when $r_t^n = 0.02$.
A cost-push shock $u_t = 0.01$ hits for one period. Solve for $\pi_t$, $x_t$, $i_t$. Has divine coincidence broken down?
Derive the NKPC slope $\kappa$ as a function of $\theta$. What happens as $\theta \to 0$?
Apply
Explain intuitively why $\phi_\pi < 1$ leads to indeterminacy. Construct a sunspot scenario.
Compare IS-LM (Chapters 8-9) with the 3-equation NK model on the IS curve, the role of the LM curve, and what is gained.
Using the ZLB framework, explain Japan's "lost decades" of near-zero rates and deflation.
Compare the Smets-Wouters (2007) model with the IS-LM models it replaced. Has the Lucas critique been addressed?
Challenge
Derive the NKPC from the Calvo pricing setup (Eq. 15.3 to Eq. 15.4).
Prove divine coincidence when $u_t = 0$, then derive optimal commitment policy with $u_t > 0$.
Show that the forward guidance puzzle grows with the horizon $k$. Discuss model modifications to resolve it.
Compare NK and RBC impulse responses to a monetary shock. Explain the mechanism by which sticky prices convert a nominal shock into a real effect.
