Inflation, Unemployment, and Monetary Rules

Introduction

This chapter builds on earlier models to look at inflation and how central banks manage it by setting interest rates to keep the economy stable around an inflation target.
Phillips curves show that inflation stays steady when unemployment is at its normal level. If unemployment goes down, inflation goes up, and vice versa.
Short-term: You might lower unemployment by accepting higher inflation, but this won't last.
Long-term: There's no trade-off between unemployment and inflation. The Phillips curve becomes a vertical line.
Central banks adjust interest rates to keep inflation near the target and output at its normal level.
They step in when things like investment booms push inflation or output off course.
For example, if investment increases, it raises inflation. The central bank then raises interest rates to cool down demand and bring the economy back to normal.
This action is a 'reaction function' or a monetary policy rule.
The Phillips curve, IS curve, and monetary rule together form the 3-equation model (IS-PC-MR), used by central banks and in advanced economics.
Chapter 5 goes deeper into monetary policy, including issues with using interest rates.
Section 1 explains the Phillips curve and why inflation is slow to change.
Section 2 introduces the 3-equation model, explains the monetary rule, and looks at what happens when inflation or demand changes. It also compares quick ('cold turkey') and slow ('gradualist') ways to reduce inflation.
Section 3 looks at how the inflation rate is set in the long run when using interest rates or controlling the money supply.
Section 4 studies inflation using the IS/LM model.
The economy's supply side is simplified by assuming businesses can easily adjust production without affecting costs.
An appendix discusses Milton Friedman's idea of a 'natural rate of unemployment' with a competitive job market and controlling the money supply.

1 Inflation and Phillips Curves

Inflation is how fast prices change. The price level reflects past inflation.
Deflation is when prices fall (negative inflation).
Inflation over the past year is: $\frac{P - P{-1}}{P{-1}}$ , where $P$ is the current price level and $P _{-1}$ is the price level last year.
High inflation is unstable, creates uncertainty, and makes it hard to understand price signals.
Lowering high inflation is tough and can lead to higher unemployment.
These issues lead central banks to target a specific inflation rate. (Chapter 5 has more on inflation and its costs.)
This section explains where inflation comes from and what causes it to rise or fall.

1.1 Inflation Inertia

It's important to understand how people expect inflation to change and why it's slow to react.
Evidence shows that changes in production lead to changes in inflation.
Standard Model: Inflation depends on:
- Past inflation, $\pi _{t-1}$ .
- The difference between current unemployment and the normal rate.
Some models suggest inflation can predict future production increases, but there's not much proof.
Past inflation can be seen in two ways:
- Expectations.
- Inertia (the main idea in this book).
People expect inflation to continue as it has in the past: $\pi{t}^{e} = \pi{t-1}^{e} + a(\pi{t-1} - \pi{t-1}^{e})$ (adaptive expectations), where $a$ is a number between 0 and 1.
If people quickly correct their expectations ( $a = 1$ ), then $\pi{t}^{e} = \pi{t-1}$ (simple adaptive expectations).
This means lagged inflation, $\pi _{-1}$ , affects current inflation.
Critics say it's too simple to only look at the past when thinking about the future.
A better way to explain why past inflation matters is that wages and prices in the economy are slow to adjust.
Wage setters use past inflation to set current wages, protecting themselves from losing purchasing power.
They can't fully account for expected future inflation when setting wages.
Chapter 15 discusses the debate around inflation inertia and expectations, presenting a more complex model.
Here, we use a model where inflation inertia is defined as $\pi' = \pi_{-1}$ . We often refer to lagged inflation as $\pi$ to highlight the role of inertia.
Inertia-Augmented Phillips Curve: $\pi = \pi' + \alpha(y - ye) = \pi{-1} + \alpha(y - y_e)$
- $\pi$ = current inflation
- $\pi'$ = inflation inertia
- $\alpha(y - y_e)$ = output gap
Expectations-Augmented Phillips Curve: $\pi = \pi^{e} + \alpha(y - ye) = \pi{-1} + \alpha(y - y_e)$
- $\pi$ = current inflation
- $\pi^{e}$ = expected inflation
- $\alpha(y - y_e)$ = output gap

1.2 Deriving Phillips Curves

There's a specific unemployment rate where the job market is balanced.
At this point, wage and price setters are happy with the current real wage and don't want to change it.
Inflation stays constant when unemployment is at this equilibrium rate; otherwise, inflation changes.
Example: Inflation is 4% per year, unemployment is at the equilibrium rate, employment is $E1$ , and the real wage is $w1$ .
Wages are set based on the WS curve to maintain the real wage. To keep the real wage at $w_1$ , wages need to rise by 4% to match the past year's price increase. So, wages are set to rise by 4%.
Firms set prices based on their pricing rule, marking up unit labor costs ( $\frac{W}{\lambda}$ ).
$\frac{\Delta P}{P} = \frac{\Delta W}{W} - \frac{\Delta \lambda}{\lambda}$ .
If labor productivity ( $\lambda$ ) is constant, then price changes depend only on wage changes: $\frac{\Delta P}{P} = \frac{\Delta W}{W}$ .
Wages and labor costs rise by 4%, so firms raise prices by 4% to keep their profit margin the same. Inflation remains steady at 4% per year.
If employment is at $E2$ , higher than equilibrium, and past inflation is 4%, the real wage on the WS curve is 2% above $w1$ . Wages will rise by 4% to keep the real wage unchanged, plus another 2% to reach $w_2$ on the WS curve (totaling 6%). Firms then raise prices by 6% to maintain their profit margin. Thus, inflation rises to 6%.
Unemployment below the equilibrium rate leads to inflation rising from 4% to 6%.
High unemployment weakens workers' bargaining power, shown by the WS curve's slope. Below employment level $E_1$ , the WS curve is below the PS curve.
These results can be shown on a Phillips curve diagram, with inflation on the vertical axis and output on the horizontal one.
A 1% change in output growth above or below its trend leads to a smaller change in unemployment due to labor hoarding and changes in the labor force.
An inertia- or expectations-augmented Phillips curve shows possible combinations of inflation and output for a given rate of past inflation, $\pi'_{-1}$ .

1.3 Phillips's Original Curve

This scenario assumes average inflation is zero, and the economy faces random demand shocks.
Short-lived shocks mean wage setters see temporary price increases as unimportant and don't factor in past inflation.

1.4 Phillips's Original Curve May Exist but It Cannot Be Exploited

Can the government shift the economy to a higher level of activity?
If the government increases the money supply, interest rates fall and output increases due to higher investment.
Eventually, workers start to expect 2% annual inflation and include it in their wage demands. Expected inflation rises from zero to 2% per year, and the long-run trade-off between inflation and unemployment disappears because the Phillips curve shifts up.
This illustrates the Lucas critique: a stable Phillips curve only existed because governments didn't try to use it systematically!
The conclusion is that policymakers can only operate on a vertical line above the equilibrium output level, leading to the concept of a vertical long-run Phillips curve.

1.5 Disinflation is Costly

The Phillips curve, $\pi = \pi{-1} + \alpha(y-ye)$ , implies that lowering inflation requires a period of unemployment above the equilibrium rate.
From the equation, if (\pi - \pi_{-1}) < 0:
- \Rightarrow \alpha(y - y_e) < 0
- \Rightarrow y < y_e

1.6 Disinflation and Central Bank Preferences

The central bank aims to achieve equilibrium output with 2% inflation.
A more inflation-averse bank has flatter indifference curves, choosing a point with lower output but quicker disinflation.

1.7 Costless Disinflation and Rational Expectations

If past inflation didn't influence wage setters, the economy could move to a lower inflation rate without raising unemployment.
To eliminate the cost of disinflation, we need to assume:
- No inflation inertia: wages and prices adjust quickly without rigidities.
- Rational expectations: the central bank's new low inflation target, $\pi^{T}$ , is believed by everyone. This target must be consistent with $\pi = \pi^{T} + \epsilon$ , where $\epsilon$ is a random shock.
Rational expectations mean agents don't make systematic errors and their expectations equal the 'objective' expectation given all available information. Thus:
- $\pi^{e} = E\pi = E(\pi^{T} + \epsilon) = \pi^{T} + E\epsilon = \pi^{T}$ (rational expectations of inflation)
The Phillips curve then becomes:
- $\pi = \pi^{T} + \alpha(y - y_e) + \epsilon$ (Phillips curve; rational expectations)
Lucas surprise supply equation:
- $y = ye + \frac{1}{\alpha}(\pi - \pi^{e}) = ye + \xi$
Economies where changes in demand or policy are fully understood have different Phillips curves from those where they aren't.
Deviations from equilibrium due to imperfect information can be expressed as:

2 Monetary Rules and the 3-Equation IS-PC-MR Model

The three equations are:
- (1) the IS equation,
- (2) the Phillips curve equation, and
- (3) the monetary rule derived from the central bank's trade-off between output and inflation.
This chapter focuses on understanding the 3-equation model using diagrams. Chapter 5 provides a more detailed analysis of monetary policy using these equations, and Chapter 15 explains how to develop more advanced versions.

2.1 The 3-Equation Model: IS-PC-MR

The IS curve is $y = A - ar$ , where $A$ is total demand. The stabilizing interest rate is defined as $ye = A - ars$ .
The IS equation in output gap form is $y - ye = -a(r - rs)$ .
The central bank can't immediately change output by changing the interest rate. It takes time for changes to affect investment and output.
The Phillips curve equation remains: $\pi = \pi{-1} + \alpha(y - ye)$
The monetary rule, MR, is derived from the central bank's output-inflation trade-off: $y - y_e = -b(\pi - \pi^{T})$

2.2 An Inflation Shock

Starting with $y = y_e$ and inflation at the target of 2%, an inflationary shock pushes inflation to 4%. The economy moves to point B.
The central bank chooses the interest rate r' on the IS curve to reach point C on the Phillips curve.

2.3 A Temporary Demand Shock

The economy starts in equilibrium, then a temporary increase in demand shifts the IS curve, raising output and inflation.

2.4 A Permanent Demand Shock

With a permanent demand shock, the IS curve shifts and stays there. The analysis is the same as with a temporary shock, but the stabilizing interest rate rises to rs.

2.5 The MR Line and the Real Interest Rate

To show that an inflation-targeting central bank sets the nominal interest rate to achieve a specific real interest rate, consider the example in Fig. 3.10.
To bring inflation back to the target after a permanent demand shock, the central bank raises the interest rate, increasing the real interest rate.
If the central bank keeps the nominal rate unchanged, the economy moves along the new IS curve.
This shows that the central bank focuses on the real interest rate when setting the nominal rate, adjusting for expected inflation.

2.6 Sacrifice Ratios and Disinflation Strategies

A more inflation-averse central bank quickly reduces output through higher interest rates, leading to a sharper rise in unemployment but a quicker return to equilibrium.
'Cold turkey' and 'gradualist' approaches are compared. With linear Phillips curves, the sacrifice ratio is independent of the central bank's inflation aversion.
$y - y_e = -b(\pi - \pi^{T})$
With non-linear Phillips curves, the sacrifice ratio is higher for a more inflation-averse central bank. With linear curves, the total unemployment cost is the same, only the timing differs.

3 Inflation at the Medium-Run Equilibrium

3.1 Two Monetary Policies

3.1.1 Interest Rate Rule (MR Approach)

A stable equilibrium requires inflation to match the central bank's target, achieved by managing the economy around the equilibrium rate of unemployment.

3.1.2 Money Supply Rule (LM Approach)

With the IS/LM model, money supply growth shapes inflation in the medium run, relying on central bank control.
Output is determined by the intersection of the WS and PS curves at ye.
In the medium run, the equilibrium has a constant inflation rate equal to the constant growth rate of the money supply: $\pi = \overline{\gamma}_{M}$

3.2 How the MR Relates to the LM Curve

In an inflation-targeting regime, the LM curve is in the background and doesn't directly influence output, inflation, or the interest rate.

4 Inflation in the IS/LM Model

The IS/LM model presents a different view of policymaking compared to the IS-PC-MR model.
However, the adjustment path with a money supply rule will differ from the IS-PC-MR model.

5 Conclusions

This chapter provides a framework for systematically analyzing shocks and policies affecting the economy.
Adding the Phillips curve to the IS/LM model allows us to analyze shocks when the central bank targets the money supply growth rate.

Appendix

Inflation in the Competitive Model: Friedman's Model

Milton Friedman's model explains why unemployment deviates from its natural rate due to worker misperceptions.

Here are 10 potential exam questions based on the provided notes, along with their solutions:

Question: Explain the relationship between unemployment and inflation as described by the Phillips curve. What are the short-term and long-term implications?
Solution: The Phillips curve indicates an inverse relationship between unemployment and inflation. In the short term, lower unemployment may lead to higher inflation, and vice versa. However, in the long term, there is no trade-off, resulting in a vertical Phillips curve.
Question: Describe the 3-equation model (IS-PC-MR) and its components. How do central banks use this model to manage the economy?
Solution: The 3-equation model consists of the IS equation, the Phillips curve, and the monetary rule. Central banks use this model to adjust interest rates to keep inflation near the target and output at its normal level.
Question: What is inflation inertia, and how does it affect current inflation? Explain the concept of the inertia-augmented Phillips curve.
Solution: Inflation inertia refers to the tendency of inflation to persist at its current rate due to past inflation affecting wage and price settings. The inertia-augmented Phillips curve is \$\$\pi = \pi{-1} + \alpha(y - ye)\$\$, where \$\$\pi\$\$ is current inflation, \$\$\pi{-1}\$\$ is past inflation, and \$\$\alpha(y - ye)\$\$ is the output gap.
Question: Explain how inflation expectations are formed. Differentiate between adaptive expectations and rational expectations.
Solution: Adaptive expectations are formed based on past inflation rates, while rational expectations incorporate all available information, assuming agents do not make systematic errors. Adaptive expectations: \$\$\pi{t}^{e} = \pi{t-1}^{e} + a(\pi{t-1} - \pi{t-1}^{e})\$\$. Rational expectations: \$\$\pi^{e} = E\pi = E(\pi^{T} + \epsilon) = \pi^{T} + E\epsilon = \pi^{T}\$\$
Question: What is the significance of the equilibrium unemployment rate in the context of wage and price setting? How does deviation from this rate affect inflation?
Solution: The equilibrium unemployment rate is the rate at which wage and price setters are content with the current real wage, and inflation remains constant. If unemployment is below this rate, inflation rises, and if it is above, inflation falls.
Question: Describe the Lucas critique. How did it change the understanding of the Phillips curve and its use by policymakers?
Solution: The Lucas critique states that a stable Phillips curve exists only when governments do not systematically exploit it. Once governments try to use it, expectations change, and the Phillips curve shifts, invalidating the original relationship.
Question: Explain the costs of disinflation. What does the Phillips curve imply about lowering inflation?
Solution: Disinflation, according to the Phillips curve \$\$\pi = \pi{-1} + \alpha(y-ye)\$\$, requires a period of unemployment above the equilibrium rate (\$\$y < y*e\$\$). This leads to potential output losses and economic hardship.
Question: Differentiate between the 'cold turkey' and 'gradualist' approaches to disinflation. How does central bank aversion to inflation affect these strategies?
Solution: The 'cold turkey' approach involves quickly reducing output through high-interest rates for a sharp reduction in inflation. The 'gradualist' approach reduces output more slowly. A more inflation-averse central bank tends to prefer the 'cold turkey' approach.
Question: How does an inflation-targeting central bank use nominal interest rates to achieve a specific real interest rate? Why is the real interest rate important?
Solution: An inflation-targeting central bank adjusts the nominal interest rate to achieve a specific real interest rate by accounting for expected inflation. The real interest rate is important because it influences investment and aggregate demand, thereby affecting output and inflation.
Question: Explain how monetary policy rules differ between the interest rate rule (MR approach) and the money supply rule (LM approach) in achieving medium-run equilibrium.
Solution: Under the interest rate rule (MR approach), the central bank manages the economy around the equilibrium rate of unemployment to match its inflation target. Under the money supply rule (LM approach), the central bank relies on controlling the money supply growth rate (\$\$\pi = \overline{\gamma}_M\$\$) to shape inflation in the medium run.