To find the new steady state level of capital in the given scenario, we need to use the Solow Growth Model. The steady state level of capital (K) occurs where the investment in capital equals the depreciation of capital.
In the Solow Growth Model, the production function is often given as:
[tex]\[ Y = \sqrt{K} \][/tex]
where [tex]\( Y \)[/tex] is the output and [tex]\( K \)[/tex] is the capital.
Here are the steps for calculating the steady state level of capital:
1. Determine the savings and depreciation functions:
- Let [tex]\( s \)[/tex] represent the savings rate (the portion of output that is saved and invested).
- Let [tex]\( \delta \)[/tex] (delta) represent the depreciation rate of capital.
2. Set up the equation for savings and depreciation:
- Savings function: [tex]\( sY \)[/tex]
- Depreciation function: [tex]\( \delta K \)[/tex]
3. Plug the production function into the savings function:
Since [tex]\( Y = \sqrt{K} \)[/tex], we have:
[tex]\[ sY = s\sqrt{K} \][/tex]
4. Find the steady state condition:
In the steady state, savings equals depreciation:
[tex]\[ s\sqrt{K} = \delta K \][/tex]
5. Solve for the steady state level of capital (K):
Rearrange the equation:
[tex]\[ s\sqrt{K} = \delta K \][/tex]
Divide both sides by [tex]\( \sqrt{K} \)[/tex]:
[tex]\[ s = \delta \sqrt{K} \][/tex]
Square both sides to isolate [tex]\( K \)[/tex]:
[tex]\[ s^2 = \delta^2 K \][/tex]
Finally, solve for [tex]\( K \)[/tex]:
[tex]\[ K = \left( \frac{s^2}{\delta^2} \right) \][/tex]
Simplify the expression:
[tex]\[ K = \left( \frac{s}{\delta} \right)^2 \][/tex]
Hence, the new steady state level of capital [tex]\( K^* \)[/tex] is given by:
[tex]\[
K^* = \left( \frac{s}{\delta} \right)^2
\][/tex]
Therefore, the new steady state level of capital for the Earth Kingdom, one year after the Fire Nation attacks, is determined by the savings rate [tex]\( s \)[/tex] and the depreciation rate [tex]\( \delta \)[/tex] according to the above formula.
