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?

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.