Understand how the inflation can affect the unemployment in Phillip Curve new edition

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Let’s walk through this step-by-step so it’s easier to understand. The goal of this section is to explain how the **natural rate of unemployment (un)** is determined and how it relates to inflation.

### 1. **Natural Rate of Unemployment**:
The **natural rate of unemployment** (denoted as \( u_n \)) is the unemployment rate at which the **actual inflation rate** is equal to the **expected inflation rate**. This means that when the economy is at the natural rate of unemployment:
- There are **no surprises** in inflation; what people expect inflation to be and what inflation actually is are the same.

This condition helps us figure out the natural rate of unemployment by looking at the factors that influence wage and price-setting.

### 2. **Equation (8.3)**:
We start with **Equation (8.3)**, which links **inflation** with **unemployment**, **wage-setting factors**, and **price markup**. It can be written as:
\[
\pi_t - \pi_t^e = -\alpha (u_t - u_n)
\]
Where:
- \( \pi_t \) = **actual inflation rate** at time \( t \).
- \( \pi_t^e \) = **expected inflation rate** at time \( t \).
- \( u_t \) = **actual unemployment rate** at time \( t \).
- \( u_n \) = **natural rate of unemployment**.
- \( \alpha \) = a constant that reflects how sensitive inflation is to unemployment.
- \( m \) = **markup** (the difference between the price of a product and its production cost).
- \( z \) = **wage-setting factors** (such as labor laws, unemployment benefits, or union strength).

### 3. **Condition at the Natural Rate of Unemployment**:
At the **natural rate of unemployment (u_n)**, **actual inflation** equals **expected inflation**:
\[
\pi_t = \pi_t^e
\]
Substituting this condition into equation (8.3) gives:
\[
0 = m + z - \alpha u_n
\]
This equation tells us that at the **natural rate of unemployment**, the upward pressure on wages from wage-setting factors (z) and the price markup (m) exactly balances the impact of unemployment.

### 4. **Solving for the Natural Rate of Unemployment**:
To find the **natural rate of unemployment (u_n)**, we solve the equation:
\[
0 = m + z - \alpha u_n
\]
Rearranging this, we get:
\[
u_n = \frac{m + z}{\alpha}
\]
This equation shows that the **natural rate of unemployment** depends on:
- **\( m \)** = Markup. The higher the price markup set by firms, the higher the natural rate of unemployment.
- **\( z \)** = Factors that affect wage-setting. These include things like labor laws, unemployment benefits, or the power of unions. The higher these factors, the higher the natural rate of unemployment.
- **\( \alpha \)** = Sensitivity of inflation to unemployment. The higher \( \alpha \), the lower the natural rate of unemployment because inflation responds more strongly to changes in unemployment.

### 5. **Modified Phillips Curve**:
Now we rewrite **Equation (8.3)**, which was:
\[
\pi_t - \pi_t^e = -\alpha (u_t - u_n)
\]
This equation shows the relationship between **actual inflation** and **expected inflation**, based on the **difference between the actual unemployment rate (u_t)** and the **natural rate of unemployment (u_n)**.

If actual unemployment \( u_t \) is **above** the natural rate \( u_n \), inflation will be **lower than expected**. If unemployment is **below** the natural rate, inflation will be **higher than expected**.

### 6. **Inflation Dynamics**:
If we assume that **expected inflation** in the current period is based on **last year’s inflation rate** (a common assumption), we can simplify the equation further. If:
\[
\pi_t^e = \pi_{t-1}
\]
This means that people expect this year’s inflation to be the same as **last year’s inflation**. Substituting this into the equation gives:
\[
\pi_t - \pi_{t-1} = -\alpha (u_t - u_n)
\]
This is **Equation (8.10)**.

### 7. **Interpretation of Equation (8.10)**:
\[
\pi_t - \pi_{t-1} = -\alpha (u_t - u_n)
\]
This is the **expectations-augmented Phillips curve** (or modified Phillips curve), which shows that:
- The **change in inflation** (\( \pi_t - \pi_{t-1} \)) depends on the difference between the **actual unemployment rate (u_t)** and the **natural rate of unemployment (u_n)**.
- If unemployment is **below the natural rate** (\( u_t < u_n \)), inflation will **increase** (the left-hand side will be positive).
- If unemployment is **above the natural rate** (\( u_t > u_n \)), inflation will **decrease** (the left-hand side will be negative).

### Key Insights:
- **Natural Unemployment**: \( u_n \) is the rate of unemployment where inflation is stable (actual inflation equals expected inflation).
- **Modified Phillips Curve**: The equation \( \pi_t - \pi_{t-1} = -\alpha (u_t - u_n) \) shows that when unemployment is below \( u_n \), inflation will accelerate, and when unemployment is above \( u_n \), inflation will decelerate.
- **Inflation Expectations**: In this model, expectations of inflation (often based on past inflation) play a critical role in determining how inflation responds to changes in unemployment.

### Conclusion:
The **natural rate of unemployment (u_n)** is the unemployment rate where inflation remains stable (expected inflation equals actual inflation). The **modified Phillips curve** (Equation 8.10) explains how deviations from the natural rate of unemployment affect inflation: if unemployment is below the natural rate, inflation rises; if it’s above the natural rate, inflation falls.

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