Understanding the equation where unemployment is related with the national output

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Let's break down the steps of this calculation in a way that’s easier to understand, like explaining it step-by-step for someone who’s learning algebra or basic economics.

### What the Calculation is Trying to Do:
The goal is to **connect the price level (P)** with **output (Y)** by first using the **unemployment rate (u)** and the **nominal wage (W)**. The formula shows how prices are affected by expected prices, unemployment, and output.

### Step-by-Step Explanation:

#### **Step 1: Eliminate the Nominal Wage (W)**
You have two equations, and both involve the **nominal wage (W)**. To make things simpler, we want to get rid of the nominal wage by **substituting** it from one equation into the other.

Here’s what happens:
- You take the first equation, which gives an expression for the **nominal wage** \(W\).
- You then **replace** \(W\) in the second equation with this expression from the first equation.

After this substitution, you get the new equation:
\[ P = P_e (1 + m) F(u, z) \]

This equation tells you that the **price level (P)** depends on:
- \(P_e\) = the **expected price level**
- \(u\) = the **unemployment rate**
- \(m\) = the **markup** (which we're assuming stays constant)
- \(z\) = other factors (which we’re also assuming are constant)

#### **Step 2: Replace the Unemployment Rate (u) with Output (Y)**
Next, we need to relate **unemployment (u)** to **output (Y)**, because output (Y) is what really drives the economy.

Here’s how it works:

- The **unemployment rate (u)** is defined as:
\[
u = \frac{U}{L}
\]
Where:
- \(U\) = the number of unemployed people.
- \(L\) = the total labor force (both employed and unemployed people).

- Unemployment can also be written as the difference between total labor (\(L\)) and employment (\(N\))—because if someone is not employed, they are unemployed:
\[
u = \frac{L - N}{L}
\]

- Now, to simplify, we use the idea that **output (Y)** is related to employment (\(N\)), because every unit of output requires one worker (as stated in the production function). So we can say that:
\[
Y = N
\]

- Substituting this into the formula for unemployment, you get:
\[
u = 1 - \frac{Y}{L}
\]
This means the **unemployment rate (u)** depends on how much **output (Y)** is produced relative to the total labor force (\(L\)).

#### **Final Equation:**
So now you have an expression for **unemployment (u)** in terms of **output (Y)**:
\[
u = 1 - \frac{Y}{L}
\]

This connects the unemployment rate to the amount of goods and services (output) being produced. By combining this with the previous equation that linked **prices (P)** to **unemployment (u)**, we get the final expression for how **price levels** depend on **output**.

### What This Means:
In summary:
- First, we eliminated the nominal wage (\(W\)) to simplify the equation for the price level.
- Then, we replaced the unemployment rate (\(u\)) with an expression involving output (\(Y\)).
- Now, the **price level (P)** is shown to depend on **expected prices (P_e)** and how much **output (Y)** is produced. When more output is produced relative to the available labor force, it affects unemployment and, eventually, prices.

By linking prices to output, this shows how changes in the economy’s production level affect the overall price level.

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