Real Interest Rates in Deflation: A Fisher Equation Analysis

Hello! Welcome to the fourth lesson in our course on deflation.

Introduction

In our last lesson, we delved into the Quantity Theory of Money and empirically demonstrated that the velocity of money ($V$) is not a stable constant, particularly during a crisis like the Great Depression. We saw that its collapse was a rational "flight to quality" in response to heightened risk.

Today, we shift our focus to another critical variable that links the monetary and real economies: the real interest rate. This lesson directly addresses the learning outcome: "Calculate and analyze the behavior of ex-post real interest rates during a deflationary period using the Fisher Equation and historical data."

The real interest rate is arguably one of the most important prices in an economy, influencing saving, investment, and the real burden of debt. During deflation, its behavior can become perverse and deeply damaging. We will explore the theory behind this, formally define the Fisher Equation, and then, using Python, we will calculate and visualize this dynamic during the Great Depression.

This lesson should take approximately 60 minutes to complete.

Recap from Last Lesson:
We discovered that the dramatic fall in the velocity of money during the 1930s was not random but could be explained by a model incorporating risk premia. This showed that simply increasing the money supply ($M$) might not stimulate the economy if fear causes people to hoard cash, causing $V$ to fall in lockstep.

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1. The Fisher Equation: Deconstructing Interest Rates

You'll be familiar with the distinction between nominal and real variables from your economics background. The Fisher Equation, named after the great American economist Irving Fisher, provides the formal link between the nominal interest rate, the real interest rate, and inflation.

A simple, and very common, approximation of the relationship is:

$r\approx i-\pi$

Where:

• $r$ is the real interest rate.
• $i$ is the nominal interest rate (the rate quoted by a bank).
• $\pi$ is the inflation rate.

However, given your mathematical background, you'll appreciate the more precise, formal derivation. Please read the following brief resource, which lays out both the approximation and the exact formula.

16.14 The Fisher Equation: Nominal and Real Interest Rates (The Fisher Equation: Nominal and Real Interest Rates; More Formally: Derivation of the Fisher Equation)
Reading time: ~5 minutes. Focus on the distinction between the simple approximation and the formal derivation: $(1+i)=(1+r)(1+\pi )$. Understand why the cross-product term $r\pi$ is often ignored.

Ex-Ante vs. Ex-Post

A crucial subtlety in this equation is the timing of inflation.

• Ex-Ante Real Interest Rate: This is the rate calculated using expected inflation (${\pi }^e$). It is the real return that lenders and borrowers expect to realize when they agree on a nominal rate. This is the rate that influences decisions.
  $r_e=i-{\pi }^e$
• Ex-Post Real Interest Rate: This is the rate calculated using actual inflation ($\pi$). It is the real return that is actually realized after the fact. This is the rate that determines outcomes.
  $r_p=i-\pi$

When actual inflation deviates from what was expected, it leads to an unplanned redistribution of wealth.

To quickly solidify this concept, please watch the following short video.

1. Highlight 1: 0:00 - 0:00
2. Highlight 2: 0:00 - 0:00
3. Highlight 3: 0:00 - 0:00

Viewing time: ~3.5 minutes. This video provides a clear, concise explanation of the difference between ex-ante and ex-post rates and the consequences for borrowers and lenders when inflation surprises them.

During periods of deflation, especially when it is unexpected, the gap between the ex-ante and ex-post real rate can become enormous and economically destructive.

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2. The Perverse Dynamics of Real Rates in a Deflationary Spiral

Now, let's analyze what happens to the real interest rate when the economy enters deflation.

1. Deflation means $\pi <0$. The Fisher equation $r=i-\pi$ becomes $r=i+|\pi |$. This immediately tells us that in a deflationary environment, the real interest rate is higher than the nominal interest rate. A borrower repaying a loan is paying back dollars that are worth more than the ones they borrowed, in addition to the nominal interest.

2. The Zero Lower Bound (ZLB). A central bank can cut the nominal interest rate to stimulate the economy, but it can't cut it (meaningfully) below zero. This creates a hard floor for conventional monetary policy.

When you combine these two facts, you get a toxic macroeconomic dynamic. As a deflationary crisis deepens, the central bank will cut the nominal rate $i$ towards zero. Once $i\approx 0$, the Fisher equation simplifies to:

$r\approx -\pi$

This is a critical and dangerous result. It means that the real interest rate is now determined by the rate of deflation. If deflation worsens from -1% to -3%, the real interest rate rises from approximately 1% to 3%.

Think about the implications: just when the economy is at its weakest and needs lower borrowing costs to encourage investment, the real cost of borrowing is actually increasing. Monetary policy becomes trapped. It has pushed the nominal rate to zero but is powerless to stop the real rate from rising as deflationary expectations take hold.

This is not just a theoretical curiosity. The BIS paper you have in your resources discusses this exact phenomenon in the context of the 1937-38 US recession.

Deflation in a historical perspective, November 2005 (Two episodes of bad deflation and the zero nominal bound)
Reading time: ~2 minutes. Focus on the paragraph describing the 1937-38 recession. Note the key observation that "Real interest rates were perversely related to the evolution of real output," which is exactly the dynamic we just described.

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3. Empirical Analysis: Real Interest Rates in the Great Depression

Let's make this tangible. We will now calculate and plot the ex-post real interest rate during the Great Depression to see this perverse dynamic in action.

Coding Exercise: Visualizing the Real Rate Spike

The Python script below will:

1. Download the monthly 4-6 Month Prime Commercial Paper Rate (CP4M) as our nominal interest rate ($i$).
2. Download the monthly Consumer Price Index (CPIAUCSL).
3. Calculate the year-over-year inflation rate ($\pi$) from the CPI.
4. Calculate the ex-post real interest rate ($r_p=i-\pi$).
5. Plot all three series from 1927 to 1939 to observe their interactions.

import pandas as pd
import matplotlib.pyplot as plt
from fredapi import Fred

# --- Your FRED API Key ---
# Replace with your actual key if you haven't set it as an environment variable
fred_key = 'YOUR_API_KEY_HERE' 
fred = Fred(api_key=fred_key)
plt.style.use('seaborn-v0_8-whitegrid')

# --- Download Data (1927-1939) ---
start_date = '1927-01-01'
end_date = '1939-12-01'

# 1. Nominal Interest Rate: 4-6 Month Prime Commercial Paper Rate
nominal_rate = fred.get_series('CP4M', observation_start=start_date, observation_end=end_date)

# 2. Price Level: Consumer Price Index for All Urban Consumers
cpi = fred.get_series('CPIAUCSL', observation_start=start_date, observation_end=end_date)

# --- Calculate Inflation and Real Rate ---
# 3. Inflation: Year-over-year percentage change in CPI
inflation_rate = cpi.pct_change(periods=12) * 100

# 4. Real Interest Rate: r = i - pi
# We need to align the series before subtracting
data = pd.concat([nominal_rate, inflation_rate], axis=1).dropna()
data.columns = ['Nominal Rate', 'Inflation Rate']
data['Real Rate'] = data['Nominal Rate'] - data['Inflation Rate']

# --- Create Plot ---
fig, ax = plt.subplots(figsize=(16, 9))

ax.plot(data.index, data['Nominal Rate'], label='Nominal Rate (i)', color='blue', linewidth=2)
ax.plot(data.index, data['Inflation Rate'], label='Inflation Rate (π)', color='green', linestyle='--')
ax.plot(data.index, data['Real Rate'], label='Ex-Post Real Rate (r)', color='red', linewidth=2.5)

# Add a horizontal line at zero
ax.axhline(0, color='black', linewidth=0.5, linestyle='-')

# Formatting
ax.set_title('Nominal, Inflation, and Real Interest Rates During the Great Depression', fontsize=18, pad=20)
ax.set_xlabel('Year', fontsize=12)
ax.set_ylabel('Percent (%)', fontsize=12)
ax.legend(fontsize=12)
ax.set_ylim(-15, 20) # Set y-axis limits for better visualization
ax.grid(True)

plt.show()

# Display the first few and last few values
print("Data from the early depression period:")
print(data.loc['1929-01-01':'1933-12-01'].head())

Analysis Questions:

1. Run the script and examine the plot. What happens to the nominal rate between 1929 and 1933?
2. What happens to the inflation rate over the same period?
3. Most importantly, how does the real interest rate behave between 1931 and 1933? Does the empirical data confirm the theoretical dynamic we discussed?

Click to reveal the analysis

1. The **nominal rate** (blue line) falls sharply, from over 5% in 1929 to under 1% by 1933, as the Federal Reserve attempted to ease monetary conditions. 2. The **inflation rate** (green line) plummets, entering deep deflationary territory and reaching below -10% in 1932. 3. The **real interest rate** (red line) spikes dramatically. Despite nominal rates falling to near-zero, the intense deflation pushed the ex-post real interest rate to over 10%. The data provides a stunning confirmation of the theory: as deflation worsened, the real cost of borrowing soared, choking off investment and massively increasing the real burden of existing debt.

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Conclusion

Today we have dissected the crucial relationship between nominal rates, inflation, and the real interest rate. You have seen how, under deflation, the normal mechanics of monetary policy can be inverted, leading to a tightening of financial conditions precisely when the economy needs stimulus.

Key Takeaways:

• The Fisher Equation ($r\approx i-\pi$) is the key to understanding the real cost of borrowing.
• During deflation ($\pi <0$), the real interest rate is higher than the nominal rate.
• At the Zero Lower Bound ($i\approx 0$), the real interest rate becomes the inverse of the inflation rate ($r\approx -\pi$). This means deeper deflation causes higher real interest rates.
• Our empirical analysis of the Great Depression vividly demonstrated this perverse dynamic: as the Fed cut nominal rates, deep deflation caused the ex-post real interest rate to spike, exacerbating the economic collapse.

Preview of the Next Lesson:

This lesson has laid the perfect foundation for understanding one of the most famous theories of economic depression. The sharp rise in the real interest rate you plotted directly increases the real value of outstanding nominal debts. In our next module, we will begin by analyzing Irving Fisher's powerful "debt-deflation" theory, which argues that this very process was the primary mechanism that made the Great Depression so "Great." We will also see how the ZLB and the inability to control real rates are central to the concept of a liquidity trap, the topic of our next lesson in this module.