<p><span class="math-container">$$
\begin{eqnarray}
\frac{\delta \cdot k_t}{s \cdot A \cdot k_t^\alpha} &amp; = &amp;  \left(\frac{P_e}{\gamma \cdot A \cdot k_t^\alpha}\right)^{\frac{\gamma}{\gamma-1}}
\\
\frac{\delta}{s \cdot A}  \cdot k_t^{1-\alpha} &amp; = &amp; \left(\frac{P_e}{\gamma \cdot A}\right)^{\frac{\gamma}{\gamma-1}} \cdot k_t^{-\frac{\gamma}{\gamma-1}\alpha}
\\
k_t^{1-\alpha} \cdot k_t^{\frac{\gamma}{\gamma-1}\alpha} &amp; = &amp; \frac{s \cdot A}{\delta} \cdot \left(\frac{P_e}{\gamma \cdot A}\right)^{\frac{\gamma}{\gamma-1}}
\\
k_t^{1-\frac{\alpha}{\gamma-1}} &amp; = &amp; \frac{s \cdot A}{\delta} \cdot \left(\frac{P_e}{\gamma \cdot A}\right)^{\frac{\gamma}{\gamma-1}}
\end{eqnarray}
$$</span>
And then you just substitute the parameters and take the right hand side to the proper power.</p>

CC BY-SA 4.0
2023-12-15T16:50:55.253Z
2023-12-15T16:50:55.253Z
Giskard
<p>I want to solve my modified solow equation for k_t to get the steady state. But since I also inclueded the marginal product for the price of Energy its pretty difficult for me to find a solution.</p>
<p>here is the equation</p>
<p><span class="math-container">$$
\frac{\delta \cdot k_t}{s \cdot A \cdot k_t^\alpha} =  \left(\frac{P_e}{\gamma \cdot A \cdot k_t^\alpha}\right)^{\frac{\gamma}{\gamma-1}}
$$</span></p>
<p>I want to solve for <span class="math-container">$k_t$</span></p>
<p>Here is the initial equation.</p>
<p><span class="math-container">$$
0 = s \cdot A \cdot k_t^\alpha \cdot \left(\frac{P_e}{\gamma \cdot A \cdot k_t^\alpha}\right)^{\frac{\gamma}{\gamma-1}} - \delta \cdot k_t
$$</span></p>
<p>I would be really thankful if someone could help me or give me a hint</p>

<macroeconomics><solow>
Solving an equation for k_t in a modified Solow model
Elias Chrostian
# Solving an equation for k_t in a modified Solow model

(Asked by: username_0 on 2023-12-15)

I want to solve my modified solow equation for k\_t to get the steady state. But since I also inclueded the marginal product for the price of Energy its pretty difficult for me to find a solution.

here is the equation

$$
\frac{\delta \cdot k\_t}{s \cdot A \cdot k\_t^\alpha} = \left(\frac{P\_e}{\gamma \cdot A \cdot k\_t^\alpha}\right)^{\frac{\gamma}{\gamma-1}}
$$

I want to solve for $k\_t$

Here is the initial equation.

$$
0 = s \cdot A \cdot k\_t^\alpha \cdot \left(\frac{P\_e}{\gamma \cdot A \cdot k\_t^\alpha}\right)^{\frac{\gamma}{\gamma-1}} - \delta \cdot k\_t
$$

I would be really thankful if someone could help me or give me a hint

**Tags:** macroeconomics, solow

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## 1 answers

### Answer by username_1 on 2023-12-15. Score: 1 | Accepted answer

$$
\begin{eqnarray}
\frac{\delta \cdot k\_t}{s \cdot A \cdot k\_t^\alpha} & = & \left(\frac{P\_e}{\gamma \cdot A \cdot k\_t^\alpha}\right)^{\frac{\gamma}{\gamma-1}}
\\
\frac{\delta}{s \cdot A} \cdot k\_t^{1-\alpha} & = & \left(\frac{P\_e}{\gamma \cdot A}\right)^{\frac{\gamma}{\gamma-1}} \cdot k\_t^{-\frac{\gamma}{\gamma-1}\alpha}
\\
k\_t^{1-\alpha} \cdot k\_t^{\frac{\gamma}{\gamma-1}\alpha} & = & \frac{s \cdot A}{\delta} \cdot \left(\frac{P\_e}{\gamma \cdot A}\right)^{\frac{\gamma}{\gamma-1}}
\\
k\_t^{1-\frac{\alpha}{\gamma-1}} & = & \frac{s \cdot A}{\delta} \cdot \left(\frac{P\_e}{\gamma \cdot A}\right)^{\frac{\gamma}{\gamma-1}}
\end{eqnarray}
$$
And then you just substitute the parameters and take the right hand side to the proper power.


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