An Extended IS-PC-MR Model with Taylor Rule

The IS-PC-MR model helps us understand how central banks conduct monetary policy in modern economies. While the older IS-LM model assumed fixed prices and money supply control, today’s central banks set interest rates and must deal with changing inflation. The IS-PC-MR framework captures these realities through three key features: forward-looking expectations (people think about the future when making decisions), sticky prices (prices don’t adjust instantly), and systematic monetary policy (central banks follow predictable patterns when setting rates). This model has become a standard tool for analyzing monetary policy at central banks and universities worldwide.

The Rise and Fall of IS-LM

John Hicks (1937) created the IS-LM model to interpret Keynes General Theory. The IS curve showes combinations of interest rates and output where investment equals savings, while the LM curve depicts money market equilibrium. For decades, this framework dominated macroeconomic thinking. Policymakers used it to analyze fiscal and monetary interventions, while professors taught generations of students how exogenous money supply changes could stimulate economic activity.

Three fundamental problems undermined the IS-LM framework. First, the stagflation of the 1970s shattered the assumption that prices remain fixed in the short run. When oil shocks hit, inflation and unemployment rose together—something IS-LM couldn’t explain. Second, Robert Lucas (1976) demonstrated that ignoring expectations doomed policy analysis. When the Federal Reserve changed its behavior, people adjusted their expectations, altering the very relationships policymakers relied upon. Third, central banks had abandoned money supply targeting. Paul Volcker’s Fed and other central banks discovered they could control interest rates directly but not monetary aggregates. The LM curve, built on money supply control, did not describe how monetary policy actually worked.

The New Keynesian Revolution

During the 1990s, economists rebuilt macroeconomics from microeconomic foundations. John Taylor (1993) observed that the Federal Reserve systematically adjusted interest rates in response to inflation and output gaps. His “Taylor Rule” captured actual central bank behavior far better than any money supply rule. Meanwhile, Guillermo Calvo (1983) and Julio Rotemberg (1982) explained why prices adjust slowly: firms face costs when changing prices or can only adjust at random intervals. These “menu costs” and “staggered contracts” provided rigorous foundations for nominal rigidity. Richard Clarida, Jordi Galí, and Mark Gertler (1999) synthesized all this into what became the New Keynesian framework. They replaced Hicks’s static LM curve with two dynamic relationships: a forward-looking Phillips Curve and a monetary policy rule. Michael Woodford (2003) then demonstrated how this framework could guide optimal monetary policy.

The Model

The extended IS-PC-MR model incorporates realistic time lags that reflect how monetary policy actually transmits through the economy. Investment projects require planning, so interest rate changes affect output with a delay. Wage contracts are renegotiated periodically, so output gaps influence inflation gradually. These lags fundamentally alter the policy problem and its solution.

Central Bank’s Loss Function: $$L_t = (Y_{t+1} - Y_n)^2 + \beta(\pi_{t+2} - \pi^*)^2 \tag{1}$$

The central bank minimizes deviations of future output and inflation from their targets. The timing structure reflects transmission lags: today’s interest rate affects tomorrow’s output (through the IS curve) and inflation two periods ahead (through the Phillips curve). The parameter $\beta$ measures the central bank’s inflation aversion—higher values mean the bank tolerates larger output losses to stabilize inflation. When $\beta = 1$, the bank weights output and inflation deviations equally. As $\beta \to \infty$, the bank becomes hawkish, caring only about price stability. This quadratic specification implies that the central bank dislikes both positive and negative deviations symmetrically, and that larger deviations impose disproportionately higher losses.

IS Curve: $$Y_t = A - ar_{t-1} + \varepsilon^Y_t \tag{2}$$

Current output depends on the real interest rate from the previous period, capturing the implementation lag in investment and consumption decisions. The parameter $A$ represents autonomous spending—components of demand independent of interest rates, including government purchases, exports, and confidence-driven consumption. The coefficient $a > 0$ measures the interest sensitivity of aggregate demand: higher values mean output responds more strongly to rate changes. When the central bank raises $r_{t-1}$, output falls by $a$ units as firms delay investments and households postpone purchases. The demand shock $\varepsilon^Y_t$ captures temporary fluctuations in consumer confidence, fiscal policy, or foreign demand. This one-period lag implies that monetary policy cannot affect today’s output—the central bank must anticipate future conditions when setting current rates.

Phillips Curve: $$\pi_t = \pi^e_t + \alpha(Y_{t-1} - Y_n) + \varepsilon^\pi_t \tag{3}$$

Inflation emerges from three sources. Expected inflation $\pi^e_t$ enters because firms setting prices today must anticipate competitors’ actions and future costs. The output gap $(Y_{t-1} - Y_n)$ from last period drives current inflation through labor market pressure: when output exceeded potential yesterday, firms faced higher wage demands and pass these costs to prices today. The parameter $\alpha > 0$ measures nominal flexibility—higher values mean prices adjust more quickly to economic slack. In economies with flexible labor markets or frequent price adjustment, $\alpha$ is large. The supply shock $\varepsilon^\pi_t$ represents cost-push factors like energy prices, import costs, or supply chain disruptions. The one-period lag between output and inflation means monetary policy affects prices with a two-period delay, forcing central banks to be forward-looking.

Taylor Rule (Monetary Policy): $$r_t = \gamma_\pi(\pi_t - \pi^*) + \gamma_y(Y_t - Y_n) + r_n \tag{4}$$

The central bank systematically adjusts the real interest rate in response to economic conditions. When inflation exceeds the target $\pi^*$ or output exceeds potential $Y_n$, the bank raises rates above the natural rate $r_n$ to cool the economy. The response coefficients $\gamma_\pi$ and $\gamma_y$ determine policy aggressiveness. From optimization, these equal $\gamma_\pi = \alpha\beta/[a(1 + \alpha^2\beta)]$ and $\gamma_y = \alpha\gamma_\pi$. Higher inflation aversion ($\beta$) increases both coefficients—the bank responds more aggressively to deviations. Greater price flexibility ($\alpha$) also strengthens responses since policy becomes more effective. The Taylor Principle requires $\gamma_\pi > 0$ such that nominal rates rise more than one-for-one with inflation (specifically, $\partial i_t/\partial \pi_t = 1 + \gamma_\pi > 1$), ensuring real rates increase to stabilize the economy.

Fisher Equation (Real Interest Rate): $$r_t = i_t - \pi^e_{t+1} \tag{5}$$

The real interest rate equals the nominal rate minus expected future inflation. This relationship determines how monetary policy transmits to the real economy. When the central bank raises nominal rates $i_t$, the real rate increases only if inflation expectations $\pi^e_{t+1}$ don’t rise proportionally. This highlights why anchoring expectations is crucial: if rate hikes trigger higher inflation expectations, real rates may not increase, undermining the policy’s effectiveness. The forward-looking nature ($\pi^e_{t+1}$ rather than current inflation) reflects that borrowing and investment decisions depend on expected real returns over the life of projects.

Extrapolative Expectations: $$\pi^e_t = \pi_{t-1} \tag{6}$$

Agents form inflation expectations by extrapolating from recent experience. This adaptive behavior creates inflation persistence: yesterday’s inflation becomes today’s expectation, which influences today’s actual inflation through the Phillips curve. While simpler than rational expectations, this specification captures the inertia observed in actual inflation dynamics. The backward-looking nature means that disinflation requires temporarily accepting output below potential—the central bank cannot simply announce lower inflation and achieve it immediately. This contrasts with rational expectations models where credible announcements can reduce inflation without output costs.

Model Variables and Parameters

Endogenous Variables:

$Y_t$: Output level
$\pi_t$: Inflation rate
$r_t$: Real interest rate
$i_t$: Nominal interest rate
$\pi^e_t$: Expected inflation

Structural Parameters:

$A$: Autonomous spending
$a$: Interest sensitivity of demand
$\alpha$: Phillips curve slope

Policy Parameters:

$Y_n$: Natural output level
$\pi^*$: Inflation target
$\beta$: Inflation aversion

Stochastic Shocks:

$\varepsilon^Y_t$: Demand shocks
$\varepsilon^\pi_t$: Supply shocks

Deriving the Monetary Rule

The Monetary Rule tells us precisely how much output the central bank sacrifices to reduce inflation. To derive this, we minimize the Central Bank’s Loss Function (Equation 1) under the constraint of the Phillips Curve — the structural relationship that links output to inflation through wage and price-setting behavior in the economy.

$$\arg\min_{Y} (Y - Y_n)^2 + \beta(\pi - \pi^*)^2$$

$$\text{s.t.} \quad \pi = \pi^e + \alpha(Y - Y_n)$$

The first-order condition yields:

$$\frac{\partial L}{\partial Y} = 2(Y - Y_n) + 2\alpha\beta[\pi^e + \alpha(Y - Y_n) - \pi^*] = 0$$

Since $\pi^e + \alpha(Y - Y_n) = \pi$ from our constraint, this simplifies to:

$$(Y - Y_n) + \alpha\beta(\pi - \pi^*) = 0$$

Solving for output, we obtain the Monetary Rule:

$$\boxed{Y = Y_n - \alpha\beta(\pi - \pi^*)} \tag{7}$$

This fundamental relationship reveals that the central bank optimally accepts output below potential whenever inflation exceeds target. The sacrifice ratio—how much output falls per unit of inflation reduction—equals $\alpha\beta$. Higher inflation aversion ($\beta$) or more flexible prices ($\alpha$) intensify this trade-off, as both make anti-inflation policy more aggressive and effective.

Deriving Optimal Taylor Coefficients

The central bank’s optimization yields the fundamental trade-off between output and inflation (Equation 7). With transmission lags, this relationship applies to future periods:

$$Y_{t+1} = Y_n - \alpha\beta(\pi_{t+2} - \pi^*)$$

The IS curve links today’s interest rate to tomorrow’s output:

$$Y_{t+1} - Y_n = -a(r_t - r_n)$$

Substituting the Monetary Rule:

$$-\alpha\beta(\pi_{t+2} - \pi^*) = -a(r_t - r_n)$$

Therefore:

$$r_t = \frac{\alpha\beta}{a}(\pi_{t+2} - \pi^*) + r_n$$

Substituting the Monetary Rule for $Y_{t+1}$ into the forward looking Phillips-Curve:

$$\pi_{t+2} = \pi_{t+1} + \alpha(-\alpha\beta(\pi_{t+2} - \pi^*))$$

Solving for $\pi_{t+2}$:

$$\pi_{t+2} = \frac{1}{1+\alpha^2\beta}[\pi_{t+1} + \alpha^2\beta\pi^*]$$

Since $\pi_{t+1} = \pi_t + \alpha(Y_t - Y_n)$:

$$r_t = \frac{\alpha\beta}{a(1+\alpha^2\beta)}(\pi_t - \pi^*) + \frac{\alpha^2\beta}{a(1+\alpha^2\beta)}(Y_t - Y_n) + r_n$$

$$\boxed{\gamma_\pi = \frac{\alpha\beta}{a(1+\alpha^2\beta)}, \quad \gamma_y = \alpha\gamma_\pi = \frac{\alpha^2\beta}{a(1+\alpha^2\beta)}}$$

Economic Interpretation

Inflation aversion ($\beta↑$): Both coefficients increase—stronger preference for price stability demands more aggressive responses.

Price flexibility ($\alpha↑$): Both coefficients increase—when prices adjust quickly to output gaps, policy becomes more effective.

Interest sensitivity ($a↑$): Both coefficients decrease—when output responds strongly to rates, smaller adjustments suffice.

The constraint $\gamma_y = \alpha\gamma_\pi$ ensures internal consistency: the bank’s output response exactly compensates for future inflation effects through the Phillips curve.

The Taylor Principle follows automatically: $\partial i_t/\partial \pi_t = 1 + \gamma_\pi > 1$, guaranteeing real rates rise with inflation.

Deriving the Natural Real Interest Rate

The natural rate maintains output at potential when no shocks disturb the economy. It represents the interest rate that would prevail if prices adjusted instantly and monetary policy remained neutral.

Starting with the IS curve: $$Y = A - ar$$

At equilibrium, output equals its natural level: $$Y = Y_n$$

Substituting and solving for the interest rate: $$Y_n = A - ar_n$$

Therefore: $$\boxed{r_n = \frac{A - Y_n}{a}}$$

The natural rate depends on structural factors: autonomous spending $A$, potential output $Y_n$, and interest sensitivity $a$.

Central banks cannot observe $r_n$ directly—they must infer it from economic outcomes. When policy rates equal the natural rate and inflation remains stable, the economy operates at potential. Deviations signal misalignment: if inflation accelerates despite stable policy rates, the natural rate has likely risen.

The Complete Taylor Rule

Combining our derivations yields the operational Taylor Rule: $$r_t = \underbrace{\frac{A - Y_n}{a}}_{\text{natural rate}} + \underbrace{\frac{\alpha\beta}{a(1 + \alpha^2\beta)}}_{\gamma_\pi}(\pi_t - \pi^*) + \underbrace{\frac{\alpha^2\beta}{a(1 + \alpha^2\beta)}}_{\gamma_y}(Y_t - Y_n)$$

This rule operationalizes three key principles:

Baseline Setting: Start at the natural rate when the economy is at target
Inflation Response: Raise rates by $\gamma_\pi$ for each percentage point of excess inflation
Output Response: Raise rates by $\gamma_y$ for each percentage point of excess output

The Taylor Principle emerges naturally: nominal rates must rise more than one-for-one with inflation. Since $\gamma_\pi > 0$, we have: $$\frac{\partial i_t}{\partial \pi_t} = 1 + \gamma_\pi > 1$$

This ensures real rates increase when inflation rises, cooling the economy rather than accommodating the inflation

import numpy as np
import matplotlib.pyplot as plt

# ============================================================================
# PARAMETERS
# ============================================================================

# Structural parameters
a = 0.5        # Interest sensitivity of demand
alpha = 0.4    # Phillips curve slope (price flexibility)
A = 2      # Autonomous spending

# Policy parameters  
beta = 1.0     # Inflation aversion (1 = equal weight on output and inflation)
Y_n = 1.0      # Potential output
pi_star = 2.0  # Inflation target (percent)

# Derived parameters (from Taylor rule optimization)
r_n = (A - Y_n) / a
gamma_pi = (alpha * beta) / (a * (1 + alpha**2 * beta))
gamma_y = alpha * gamma_pi

# Simulation settings
T = 50               # Number of periods to simulate
shock_period = 20    # When shock occurs
shock_size = 0.3     # Magnitude of demand shock

# ============================================================================
# INITIALIZE TIME SERIES
# ============================================================================

Y = np.zeros(T)       # Output
pi = np.zeros(T)      # Inflation  
r = np.zeros(T)       # Real interest rate
eps_Y = np.zeros(T)   # Demand shocks
eps_pi = np.zeros(T)  # Supply shocks

# Start at equilibrium
Y[0] = Y_n
pi[0] = pi_star
r[0] = r_n

# ============================================================================
# INJECT SHOCK
# ============================================================================

eps_Y[shock_period] = shock_size

# ============================================================================
# SIMULATION LOOP
# ============================================================================

for t in range(T - 1):
    
    # Taylor Rule: Central bank sets rate based on current inflation and output
    r[t] = r_n + gamma_pi * (pi[t] - pi_star) + gamma_y * (Y[t] - Y_n)
    
    # IS Curve: Tomorrow's output depends on today's interest rate
    Y[t+1] = A - a * r[t] + eps_Y[t+1]
    
    # Phillips Curve: Tomorrow's inflation depends on today's output gap
    # (Expected inflation = today's inflation, due to adaptive expectations)
    pi[t+1] = pi[t] + alpha * (Y[t] - Y_n) + eps_pi[t+1]

# Calculate final interest rate
r[T-1] = r_n + gamma_pi * (pi[T-1] - pi_star) + gamma_y * (Y[T-1] - Y_n)

# ============================================================================
# VISUALIZATION
# ============================================================================

fig, axes = plt.subplots(3, 1, figsize=(10, 8), sharex=True)

# Output
axes[0].plot(Y, linewidth=2, color='steelblue', label='Output')
axes[0].axhline(Y_n, color='black', linestyle='--', alpha=0.5, label='Potential')
axes[0].axvline(shock_period, color='red', linestyle=':', alpha=0.4)
axes[0].set_ylabel('Output (Y)', fontsize=11, fontweight='bold')
axes[0].legend(loc='best')
axes[0].grid(alpha=0.3)

# Inflation
axes[1].plot(pi, linewidth=2, color='crimson', label='Inflation')
axes[1].axhline(pi_star, color='black', linestyle='--', alpha=0.5, label='Target')
axes[1].axvline(shock_period, color='red', linestyle=':', alpha=0.4)
axes[1].set_ylabel('Inflation (π) %', fontsize=11, fontweight='bold')
axes[1].legend(loc='best')
axes[1].grid(alpha=0.3)

# Interest Rate
axes[2].plot(r, linewidth=2, color='darkgreen', label='Interest Rate')
axes[2].axhline(r_n, color='black', linestyle='--', alpha=0.5, label='Natural Rate')
axes[2].axvline(shock_period, color='red', linestyle=':', alpha=0.4, label='Demand Shock')
axes[2].set_ylabel('Interest Rate (r) %', fontsize=11, fontweight='bold')
axes[2].set_xlabel('Time Period', fontsize=11)
axes[2].legend(loc='best')
axes[2].grid(alpha=0.3)

plt.suptitle(f'IS-PC-MR Model: Response to Demand Shock (ε = +{shock_size})', 
             fontsize=13, fontweight='bold')
plt.tight_layout()
plt.show()

# ============================================================================
# SUMMARY OUTPUT
# ============================================================================

print("\n" + "="*60)
print("MODEL PARAMETERS")
print("="*60)
print(f"Interest sensitivity (a):          {a}")
print(f"Phillips curve slope (α):          {alpha}")
print(f"Inflation aversion (β):            {beta}")
print(f"Natural interest rate (r_n):       {r_n:.2f}%")
print(f"Taylor coefficient on π (γ_π):     {gamma_pi:.3f}")
print(f"Taylor coefficient on Y (γ_y):     {gamma_y:.3f}")
print(f"\nTaylor Principle satisfied:        {1 + gamma_pi:.3f} > 1 ✓")

png

============================================================
MODEL PARAMETERS
============================================================
Interest sensitivity (a):          0.5
Phillips curve slope (α):          0.4
Inflation aversion (β):            1.0
Natural interest rate (r_n):       2.00%
Taylor coefficient on π (γ_π):     0.690
Taylor coefficient on Y (γ_y):     0.276

Taylor Principle satisfied:        1.690 > 1 ✓

# ============================================================================
# SUPPLY SHOCK SCENARIO
# ============================================================================

# Reset time series
Y = np.zeros(T)
pi = np.zeros(T)
r = np.zeros(T)
eps_Y = np.zeros(T)
eps_pi = np.zeros(T)

# Start at equilibrium
Y[0] = Y_n
pi[0] = pi_star
r[0] = r_n

# Inject negative supply shock (increases inflation directly)
shock_period = 20
supply_shock_size = 1.5  # Percentage point increase in inflation
eps_pi[shock_period] = supply_shock_size

# Run simulation
for t in range(T - 1):
    r[t] = r_n + gamma_pi * (pi[t] - pi_star) + gamma_y * (Y[t] - Y_n)
    Y[t+1] = A - a * r[t] + eps_Y[t+1]
    pi[t+1] = pi[t] + alpha * (Y[t] - Y_n) + eps_pi[t+1]

r[T-1] = r_n + gamma_pi * (pi[T-1] - pi_star) + gamma_y * (Y[T-1] - Y_n)

# Visualize
fig, axes = plt.subplots(3, 1, figsize=(10, 8), sharex=True)

axes[0].plot(Y, linewidth=2, color='steelblue')
axes[0].axhline(Y_n, color='black', linestyle='--', alpha=0.5)
axes[0].axvline(shock_period, color='red', linestyle=':', alpha=0.4)
axes[0].set_ylabel('Output (Y)', fontsize=11, fontweight='bold')
axes[0].grid(alpha=0.3)

axes[1].plot(pi, linewidth=2, color='crimson')
axes[1].axhline(pi_star, color='black', linestyle='--', alpha=0.5)
axes[1].axvline(shock_period, color='red', linestyle=':', alpha=0.4)
axes[1].set_ylabel('Inflation (π) %', fontsize=11, fontweight='bold')
axes[1].grid(alpha=0.3)

axes[2].plot(r, linewidth=2, color='darkgreen')
axes[2].axhline(r_n, color='black', linestyle='--', alpha=0.5)
axes[2].axvline(shock_period, color='red', linestyle=':', alpha=0.4, label='Supply Shock')
axes[2].set_ylabel('Interest Rate (r) %', fontsize=11, fontweight='bold')
axes[2].set_xlabel('Time Period', fontsize=11)
axes[2].legend(loc='best')
axes[2].grid(alpha=0.3)

plt.suptitle(f'IS-PC-MR Model: Response to Supply Shock (ε = +{supply_shock_size}%)', 
             fontsize=13, fontweight='bold')
plt.tight_layout()
plt.show()

png