Chapter 12 Model Apps

Monetary Policy and the Phillips Curve

Explore the models of Chapter 12 without any code. Every chart below is live: move its slider to change one part of the model and watch the curves respond, and hover over a line to read its values.

← Back to Chapter 12: Monetary Policy and the Phillips Curve

See also: Chapter 12 Data Apps, the chapter’s data without code.

The short-run model at a glance

Fig. 1: The structure of the short-run model

Move the slider to set the nominal interest rate, with inflation stuck at 2 percent; watch each circle of the chain respond.

\[ R_t = i_t - \pi_t \]

\[ \tilde{Y}_t = \bar{a} - \bar{b}\,(R_t - \bar{r}) \]

\[ \Delta\pi_t = \bar{\nu}\tilde{Y}_t + \bar{o} \]

where \(i_t\) is the nominal rate, \(\pi_t\) sticky inflation (2 percent here), \(\bar{r}\) the marginal product of capital and \(\bar{\nu}\) the slope of the Phillips curve.

Takeaway. Because inflation is sticky, the nominal rate moves the real rate one for one, the real rate sets short-run output through the IS curve, and output sets the change in inflation. A rate above 7 percent puts the real rate above \(\bar{r}\), so output falls below potential and inflation starts to fall; a rate below 7 percent does the opposite, and at 7 percent output is at potential and inflation stays put.

Source. Adapted from Fig. 12.1 in Jones (2024), Macroeconomics, 6th ed.

The MP curve and the IS/MP diagram

Fig. 3: The MP curve in the IS/MP diagram

Move the slider to change the demand shock; the MP line stays at 5 percent, so watch where it crosses the IS curve.

\[ R_t = i_t - \pi_t \]

\[ \tilde{Y}_t = \bar{a} - \bar{b}\,(R_t - \bar{r}) \]

where the MP line is the real rate \(R_t\) the central bank sets, here \(\bar{r} = 5\) percent, \(\bar{a}\) is the demand shock and \(\bar{b}\) the sensitivity of investment to the rate.

Takeaway. The central bank chooses the height of the MP line; the IS curve then decides how much output that rate brings. With the rate held at \(\bar{r}\), a positive demand shock shifts the IS curve right and raises output one for one with the shock, a negative one shifts it left into a recession, and with no shock the economy sits at potential, point A.

Source. Adapted from Fig. 12.3 in Jones (2024), Macroeconomics, 6th ed.

Fig. 4: Raising the interest rate in the IS/MP diagram

Move the slider to set the new real interest rate; the dashed line is the old MP line, and the arrow shows the fall in output.

\[ \tilde{Y}_t = \bar{a} - \bar{b}\,(R_t - \bar{r}) \]

\[ \tilde{Y}' = -\bar{b}\,(R' - \bar{r}) \]

where \(R'\) is the new real rate, \(\bar{r} = 5\) percent the marginal product of capital and \(\bar{b} = 2\) the sensitivity of investment to the rate.

Takeaway. Only the MP line moves: a higher real rate is a movement along the IS curve, not a shift of it. Above \(\bar{r}\), borrowing costs more than new capital earns, so investment falls and output drops by \(\bar{b}\) points for every point on the rate; at 5 percent nothing changes.

Source. Adapted from Fig. 12.4 in Jones (2024), Macroeconomics, 6th ed.

Fig. 5: Stabilizing the economy after a housing bubble

Move the slider to change the size of the demand shock; panel (a) shows the recession at B, panel (b) the rate cut that reaches C.

\[ \tilde{Y}_t = \bar{a}' - \bar{b}\,(R_t - \bar{r}) \]

\[ R' = \bar{r} + \frac{\bar{a}'}{\bar{b}} \]

where \(\bar{a}' < 0\) is the demand shock after the bust and \(R'\) the real rate that brings output back to potential.

Takeaway. The bust shifts the IS curve left, so at an unchanged rate output falls one for one with the shock. Cutting the real rate to \(R'\) makes borrowing cheap enough to make up for the lost demand and takes the economy back to potential at C; the larger the shock, the larger the cut. In the model, monetary policy can fully offset any demand shock.

Source. Adapted from Fig. 12.5 in Jones (2024), Macroeconomics, 6th ed.

The Phillips curve

Fig. 7: The Phillips curve

Move the slider to change short-run output and watch the economy slide along the Phillips curve; the title reads the change in inflation.

\[ \pi_t = \pi_{t-1} + \bar{\nu}\tilde{Y}_t + \bar{o} \]

\[ \Delta\pi_t = \bar{\nu}\tilde{Y}_t + \bar{o} \]

where \(\bar{\nu} = 0.5\) is the slope of the Phillips curve and \(\bar{o}\) a price shock (zero here).

Takeaway. Short-run output sets the change in inflation, not its level. Output above potential raises inflation by \(\bar{\nu}\) points for each point of boom, output below potential lowers it, and at potential (point A) inflation stays where it was. Every move of the slider is a movement along the curve, which itself never shifts.

Source. Adapted from Fig. 12.7 in Jones (2024), Macroeconomics, 6th ed.

Fig. 8: An oil price increase

Move the slider to change the price shock; the dashed line is the old Phillips curve, and point A shows inflation at potential.

\[ \Delta\pi_t = \bar{\nu}\tilde{Y}_t + \bar{o} \]

\[ \Delta\pi_t = 0 \quad\Longrightarrow\quad \tilde{Y}_t = -\frac{\bar{o}}{\bar{\nu}} \]

where \(\bar{o}\) is the price shock and \(\bar{\nu} = 0.5\) the slope of the Phillips curve; output stays at potential.

Takeaway. A price shock shifts the whole Phillips curve by \(\bar{o}\), so inflation changes even with output at potential: cost-push inflation. An oil price rise shifts the curve up, and holding inflation steady would then take a recession; a negative shock shifts it down, and at zero the curve does not move.

Source. Adapted from Fig. 12.8 in Jones (2024), Macroeconomics, 6th ed.

Using the short-run model: disinflation and the Great Inflation

Fig. 10: Tightening monetary policy

Move the slider to set the new nominal rate, with inflation stuck at 10 percent; watch the MP line rise and output fall.

\[ R_t = i_t - \pi_t \]

\[ \tilde{Y}_t = \bar{a} - \bar{b}\,(R_t - \bar{r}) \]

where sticky inflation is \(\pi_t = 10\) percent, as in 1979, so a nominal rate of 15 percent gives a real rate equal to \(\bar{r} = 5\) percent.

Takeaway. Because inflation is sticky, every point the Fed adds to the nominal rate is a point on the real rate, so the MP line shifts up and the economy moves along the IS curve into a recession. At 15 percent the real rate equals \(\bar{r}\) and nothing changes; the higher the rate above it, the deeper the recession.

Source. Adapted from Fig. 12.10 in Jones (2024), Macroeconomics, 6th ed.

Fig. 11: A recession and falling inflation

Move the slider to set how deep the recession is, and watch point B slide down the Phillips curve; the title reads the fall in inflation.

\[ \pi_t = \pi_{t-1} + \bar{\nu}\tilde{Y}_t + \bar{o} \]

where \(\pi_{t-1} = 10\) percent is inflation before the recession, \(\bar{\nu} = 0.5\) the slope and \(\bar{o} = 0\).

Takeaway. Firms facing weak demand raise their prices by less, so a recession lowers inflation by \(\bar{\nu}\) points for each point of output lost: the deeper the recession, the faster inflation falls. With output at potential the economy stays at A and inflation does not move. Lower inflation is bought with lost output.

Source. Adapted from Fig. 12.11 in Jones (2024), Macroeconomics, 6th ed.

Fig. 12: The disinflation over time

Move the slider to set how many years the Fed holds the rate high; watch inflation step down in panel (c) and the output lost in the title.

\[ \pi_t = \pi_{t-1} + \bar{\nu}\tilde{Y}_t \]

\[ \text{lost output} = \frac{\pi_{-1} - \pi^T}{\bar{\nu}} \]

where \(\pi_{-1} = 10\) percent is inflation before the disinflation, \(\pi^T\) the inflation it ends at, and output is held 2 percent below potential while policy is tight.

Takeaway. With output 2 percent below potential, inflation falls one point a year for as long as the Fed holds the rate high; when the rate returns to \(\bar{r}\), output goes back to potential and inflation stays at its new, lower level. Every point of inflation removed costs the same two percent-years of output, so a longer squeeze buys lower inflation at a proportionally higher cost.

Source. Adapted from Fig. 12.12 in Jones (2024), Macroeconomics, 6th ed.

Fig. 13: Mistaking a slowdown in potential for a recession

Move the slider to change true potential growth after 1973; watch how far actual output sits from each dashed potential path.

\[ \text{potential}_t = 100\,e^{g\,(t - 1973)/100} \]

\[ \Delta\pi_t = \bar{\nu}\tilde{Y}_{\text{true}} \]

where \(g\) is potential growth: 3.5 percent on the believed path, the slider’s rate on the true path after 1973; \(\tilde{Y}_{\text{true}}\) is how far actual output is above true potential.

Takeaway. The slower true potential grows after 1973, the further actual output falls behind the believed path, and the bigger the recession policymakers think they see; at 3.5 percent there is no slowdown and the stimulus is plainly a boom. Actual output stays above true potential at every step, so the Phillips curve raises inflation every year after 1973, by the same amount whatever the slider says.

Source. Adapted from Fig. 12.13 in Jones (2024), Macroeconomics, 6th ed.

How central banks control the nominal interest rate

Fig. 18: How the central bank sets the nominal interest rate

Move the slider to shift money demand; the money supply stays at 5, so watch the interest rate move from i to i**.*

\[ M^d = m_0 - i \]

\[ i^* = m_0 - M^s \]

where \(m_0\) is money demand at a zero interest rate and \(M^s = 5\) the money supply.

Takeaway. The nominal interest rate is the price of holding money, and it settles where people want to hold exactly the money supplied. With the supply fixed, a rise in money demand raises the rate one for one and a fall lowers it; at \(m_0 = 10\) the rate stays at 5 percent.

Source. Adapted from Fig. 12.18 in Jones (2024), Macroeconomics, 6th ed.

Fig. 19: Raising the nominal interest rate

Move the slider to set the new money supply; the dashed line is the old one, and the arrow shows how the interest rate moves.

\[ M^d = m_0 - i \]

\[ i^{**} = m_0 - M^{s\prime} \]

where \(m_0 = 10\), the old money supply is 5 units and \(M^{s\prime}\) is the new one.

Takeaway. Cutting the money supply leaves people wanting more money than there is, so the interest rate rises until demand falls back to the new supply: tight money means a higher rate. A supply above 5 units does the opposite and lowers the rate, and at 5 units nothing changes; each unit moves the rate by one point.

Source. Adapted from Fig. 12.19 in Jones (2024), Macroeconomics, 6th ed.

Fig. 20: Targeting the nominal interest rate

Move the slider to shift money demand with the rate pegged at 5 percent; watch the quantity of money change while the rate stays put.

\[ M^d = m_0 - i \]

\[ M^s = m_0 - i^T \]

where \(i^T = 5\) percent is the target rate and \(m_0\) money demand at a zero interest rate.

Takeaway. To peg the rate, the central bank supplies whatever money is demanded at its target, so the money supply becomes a horizontal line. A rise in money demand is met with more money and a fall with less, while the interest rate, and with it the real rate and output, does not move at all; at \(m_0 = 10\) nothing changes.

Source. Adapted from Fig. 12.20 in Jones (2024), Macroeconomics, 6th ed.

Worked exercise: the Great Inflation

Fig. 21: The Great Inflation

Move the slider to change the demand shock policymakers believe in; compare their belief in the top panels with the reality below.

\[ R' = \bar{r} + \frac{\bar{a}'}{\bar{b}} \]

\[ \tilde{Y}_t = -\bar{b}\,(R' - \bar{r}) = -\bar{a}' \]

\[ \Delta\pi_t = \bar{\nu}\tilde{Y}_t \]

where \(\bar{a}' < 0\) is the demand shock policymakers believe in; the true IS curve never moved.

Takeaway. Policymakers cut the real rate to offset a fall in demand that never happened, so on the true IS curve output overshoots potential by exactly the shock they believed in. With output above potential, the Phillips curve raises inflation every year the mistake lasts; the bigger the believed shock, the bigger the boom and the faster inflation rises.

Source. Adapted from Fig. 12.21 in Jones (2024), Macroeconomics, 6th ed., the worked answer to Exercise 12.