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In the Solow Growth Model, the equation provided represents the change in capital per worker (denoted as \(\hat{k}(t)\)) over time, given by:

\[
\hat{k}(t) = s f(\hat{k}) - (n + \delta + g) \hat{k}
\]

### Given Variables:
1. **\(f(\hat{k}) = \hat{k}^{0.5}\)**: This production function implies output per worker \(f(\hat{k})\) depends on \(\hat{k}\), capital per worker, and has diminishing returns.
2. **\(s = 0.20\)**: Savings rate.
3. **\(n = 0.02\)**: Population growth rate.
4. **\(\delta = 0.10\)**: Depreciation rate of capital.
5. **\(g = 0.08\)**: Technological growth rate.

### Step 1: Substitute \(f(\hat{k})\) and Other Variables

The equation becomes:
\[
\hat{k}(t) = s \hat{k}^{0.5} - (n + \delta + g) \hat{k}
\]

Plugging in the values of \(s\), \(n\), \(\delta\), and \(g\):
\[
\hat{k}(t) = 0.20 \hat{k}^{0.5} - (0.02 + 0.10 + 0.08) \hat{k}
\]
\[
\hat{k}(t) = 0.20 \hat{k}^{0.5} - 0.20 \hat{k}
\]

### Step 2: Find the Steady-State Capital per Worker \(\hat{k}^*\)

In the Solow Growth Model, the steady state occurs when \(\hat{k}(t) = 0\), meaning capital per worker is constant over time. Thus, we set the equation to zero and solve for \(\hat{k}\):

\[
0 = 0.20 \hat{k}^{0.5} - 0.20 \hat{k}
\]

Dividing through by 0.20:
\[
0 = \hat{k}^{0.5} - \hat{k}
\]

Rearrange terms:
\[
\hat{k}^{0.5} = \hat{k}
\]

Square both sides:
\[
\hat{k} = \hat{k}^2
\]

This simplifies to:
\[
\hat{k}(\hat{k} - 1) = 0
\]

Thus, \(\hat{k} = 0\) or \(\hat{k} = 1\).

Since \(\hat{k} = 0\) is trivial (it implies no capital), the non-trivial solution is:
\[
\hat{k}^* = 1
\]

### Conclusion
The steady-state level of capital per worker, \(\hat{k}^*\), is 1. At this level, capital per worker remains constant over time, balancing the effects of savings, depreciation, population growth, and technological progress.

Would you like a deeper explanation of any step or further applications of this result?

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### Related Questions:
1. How does a change in the savings rate \(s\) affect the steady-state capital per worker \(\hat{k}\)?
2. What happens to the steady-state if technological growth \(g\) increases?
3. Can we calculate steady-state output per worker given the production function \(f(\hat{k})\)?
4. How does population growth \(n\) influence the long-run growth of capital?
5. What is the impact of depreciation rate \(\delta\) on capital accumulation?

### Tip:
In the Solow Model, a higher savings rate increases the steady-state capital, while higher population or depreciation rates decrease it.

Explore the solution to finding the steady-state capital per worker using the Solow Growth Model. This involves differential equations and economic growth theory, with specific variables such as savings rate, depreciation, and technological growth.

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