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 — 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.
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: $y_j = (p_j / P)^{-\varepsilon} Y$.
The optimal reset price is a weighted average of current and expected future marginal costs:
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:
Step 1: Under Calvo pricing with parameter $\theta$, fraction $(1-\theta)$ of firms reset prices each period. The aggregate price level evolves as: $P_t = [\theta P_{t-1}^{1-\varepsilon} + (1-\theta)(p_t^*)^{1-\varepsilon}]^{1/(1-\varepsilon)}$.
Step 2: Log-linearize: $\hat{p}_t = \theta\hat{p}_{t-1} + (1-\theta)\hat{p}_t^*$. Since $\pi_t = \hat{p}_t - \hat{p}_{t-1}$: $\pi_t = (1-\theta)(\hat{p}_t^* - \hat{p}_{t-1})$.
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 4: Recursive substitution yields: $\pi_t = \beta E_t\pi_{t+1} + \frac{(1-\theta)(1-\beta\theta)}{\theta}\widehat{mc}_t$.
Step 5: Real marginal cost is proportional to the output gap: $\widehat{mc}_t = \frac{\sigma+\varphi}{1+\varphi\varepsilon}x_t$. Defining $\kappa = \frac{(1-\theta)(1-\beta\theta)}{\theta}\cdot\frac{\sigma+\varphi}{1+\varphi\varepsilon}$ gives the NKPC: $\pi_t = \beta E_t\pi_{t+1} + \kappa x_t$.
Parameters: $\beta = 0.99$, $\kappa = 0.3$, $\sigma = 1$, $\phi_\pi = 1.5$, $\phi_x = 0.5$, $r^* = 2\%$, $r^n = 2\%$, $u = 0$.
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.
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 2: FOC: \$1x_t + 2\alpha_\pi\kappa(\kappa x_t + u_t) = 0$. Solving: $x_t = -\frac{\alpha_\pi\kappa}{1 + \alpha_\pi\kappa^2}u_t = -\frac{0.5 \times 0.3}{1 + 0.5 \times 0.09}u_t = -\frac{0.15}{1.045}u_t = -0.144u_t$.
Step 3: Inflation: $\pi_t = \kappa x_t + u_t = -0.3(0.144)u_t + u_t = 0.957u_t$.
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.
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.
Three equations, three unknowns ($\pi_t$, $x_t$, $i_t$):
| Equation | Name | Role |
|---|---|---|
| $\pi_t = \beta E_t\pi_{t+1} + \kappa x_t + u_t$ | NKPC | Inflation determination |
| $x_t = E_tx_{t+1} - \frac{1}{\sigma}(i_t - E_t\pi_{t+1} - r_t^n)$ | Dynamic IS | Demand |
| $i_t = r^* + \phi_\pi\pi_t + \phi_x x_t$ | Taylor rule | Monetary policy |
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 is not an abstract theoretical curiosity — it 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.
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.
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 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.
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.
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.
| 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) |
Compare impulse responses side by side. Toggle between a technology shock and a monetary policy shock to see what nominal rigidities add.
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.
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.
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.
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.
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).
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 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.
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.
| Label | Equation | Description |
|---|---|---|
| Eq. 15.1–15.2 | Dixit-Stiglitz aggregation | Monopolistic competition |
| Eq. 15.4 | $\pi_t = \beta E_t\pi_{t+1} + \kappa x_t$ | New Keynesian Phillips Curve |
| Eq. 15.5 | $x_t = E_tx_{t+1} - \frac{1}{\sigma}(i_t - E_t\pi_{t+1} - r_t^n)$ | Dynamic IS curve |
| Eq. 15.6 | $i_t = r^* + \phi_\pi\pi_t + \phi_x x_t$ | Taylor rule |
| Eq. 15.7 | $\phi_\pi > 1$ | Taylor principle |
| Eq. 15.8 | NKPC with cost-push shock $u_t$ | Breaks divine coincidence |
| Eq. 15.10 | $i_t \geq 0$ | Zero lower bound |