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give the answer. explain al the mathimatical steps you did

To solve this problem, we need to derive the steady-state capital stock per worker (\( k^* \)) in the Solow growth model with a Cobb-Douglas production function. The central equation, as given, is:

\[
k_{t+1} = sA k_t^\alpha + (1 - \delta)k_t
\]

Where:
- \( k_t \) is capital stock per worker at time \( t \),
- \( s \) is the savings rate,
- \( A \) is the level of productivity,
- \( \alpha \) is the output elasticity of capital,
- \( \delta \) is the depreciation rate of capital.

We also have:
- Output per worker: \( y_t = A k_t^\alpha \),
- Consumption per worker: \( c_t = (1 - s) A k_t^\alpha \).

### Step 1: Steady State Condition
At the steady state, capital per worker does not change, i.e., \( k_{t+1} = k_t = k^* \). This implies:

\[
k^* = sA(k^*)^\alpha + (1 - \delta)k^*
\]

### Step 2: Isolate \( k^* \)
Move all terms involving \( k^* \) to one side of the equation:

\[
k^* - (1 - \delta)k^* = sA(k^*)^\alpha
\]

Factor out \( k^* \) on the left-hand side:

\[
k^*(1 - (1 - \delta)) = sA(k^*)^\alpha
\]

Simplify:

\[
k^* \delta = sA(k^*)^\alpha
\]

### Step 3: Solve for \( k^* \)
Divide both sides of the equation by \( k^* \) (note that \( k^* \neq 0 \)):

\[
\delta = sA(k^*)^{\alpha - 1}
\]

Now, solve for \( k^* \):

\[
k^* = \left( \frac{sA}{\delta} \right)^{\frac{1}{1 - \alpha}}
\]

### Final Result:
The steady-state capital stock per worker \( k^* \) is given by:

\[
k^* = \left( \frac{sA}{\delta} \right)^{\frac{1}{1 - \alpha}}
\]

### Explanation:
1. The steady-state occurs when capital per worker does not change, which leads us to set \( k_{t+1} = k_t = k^* \).
2. By simplifying the resulting equation, we were able to isolate \( k^* \) and express it in terms of parameters \( s \), \( A \), \( \delta \), and \( \alpha \).
3. The formula \( k^* = \left( \frac{sA}{\delta} \right)^{\frac{1}{1 - \alpha}} \) shows that the steady-state capital depends on the savings rate, productivity, depreciation rate, and output elasticity of capital.

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Would you like further clarification on any step, or do you have specific questions about the Solow model?

Here are some related questions for further exploration:

1. How does a change in the savings rate \( s \) affect the steady-state capital stock?
2. What happens to the steady-state capital stock if the depreciation rate \( \delta \) increases?
3. How does the parameter \( \alpha \) (output elasticity of capital) influence economic growth in the Solow model?
4. What is the economic interpretation of the steady-state in the Solow growth model?
5. How does technological progress affect the Solow model and steady-state capital stock?

**Tip:** In the Solow model, steady-state values provide important insights into long-term economic performance. Changes in parameters like savings rates or depreciation directly affect this equilibrium.

Solve for an expression for the steady state capital stock per worker k*. In doing so, assume that the level of productivity is fixed at some value A.

Explore how to solve for the steady-state capital stock per worker in the Solow Growth Model using a Cobb-Douglas production function. This solution provides a detailed step-by-step breakdown of the mathematical process, focusing on key variables like savings rate, depreciation, and productivity.

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