<p>When we talk about steady state level of capital, talking in terms of complex numbers never makes sense. What this solution for steady state <span class="math-container">$k^*$</span> tells us is:</p>
<ol>
<li>Its possible that no unique steady state exists if <span class="math-container">$4\left(\frac{\delta^2}{s^2}\right)k_0&gt;1$</span></li>
<li>If <span class="math-container">$4\left(\frac{\delta^2}{s^2}\right)k_0&lt;1$</span> there are two unique steady states.</li>
</ol>
<p>stability of equilibria is a anther story but what this model tells us is that there is at least two steady states with complex solutions being meaningless.</p>

CC BY-SA 4.0
2022-05-04T00:11:37.980Z
2022-05-04T14:13:19.797Z
2022-05-04T14:13:19.797Z
EconJohn
<p>I have been looking at a simple solow model with stone geary production technology and law of motion of capital specified as follows:</p>
<p><span class="math-container">$$f(k_t)=(k_t-\bar{k})^{0.5}$$</span>
<span class="math-container">$$k_{t+1}=(1-\delta)k_t+sf(k_t)$$</span>
<span class="math-container">$$k_0&gt;\bar{k}$$</span></p>
<p>where <span class="math-container">$\bar{k}$</span> is some input requirement for our production technology. Visual inspection with some values give us the picture below. Mathematically however I noticed that there can be some issues. if we derive the solow equation from our law of motion by subtracting <span class="math-container">$k_t$</span> from both sides of our law of motion and and sub in our production function we get:
<span class="math-container">$$(1-\delta)k_t+s(k_t-\bar{k})^{0.5}-k_t=k_{t+1}-k_t$$</span>
simplifying and noting in steady state that <span class="math-container">$k_{t+1}-k_t=0$</span> we get:
<span class="math-container">$$-\delta k_t+s(k_t-\bar{k})^{0.5}=0 $$</span>
rearranging further we get a quadratic.
<span class="math-container">$$\left(\frac{\delta}{s}\right)^2k_t^2-k_t+\bar{k}=0$$</span>. Noting the quadratic formula we note our steady states are:
<span class="math-container">$$k_t^*=\frac{1\pm\sqrt{1-4\left(\frac{\delta}{s}\right)^2k_0}}{2\left(\frac{\delta}{s}\right)^2}$$</span></p>
<p>Immediately I see that for such a simple problem no solution may exist because our roots may be complex. Looking at this picture though there should be a solution to what is the simplest case of multiple equilibria in a non game theoretic model.</p>
<p>Do complex roots pose an issue for identifying multiple steady states and if they do what is the solution (if there is any)?</p>
<p><a href="https://i.sstatic.net/fee4z.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/fee4z.png" alt="enter image description here" /></a></p>

<production-function><solow>
stone geary production and Multiple equilibria in a simple solow model: Do complex roots mean anything?
EconJohn
# stone geary production and Multiple equilibria in a simple solow model: Do complex roots mean anything?

(Asked by: username_0 on 2022-05-01)

I have been looking at a simple solow model with stone geary production technology and law of motion of capital specified as follows:

$$f(k\_t)=(k\_t-\bar{k})^{0.5}$$
$$k\_{t+1}=(1-\delta)k\_t+sf(k\_t)$$
$$k\_0>\bar{k}$$

where $\bar{k}$ is some input requirement for our production technology. Visual inspection with some values give us the picture below. Mathematically however I noticed that there can be some issues. if we derive the solow equation from our law of motion by subtracting $k\_t$ from both sides of our law of motion and and sub in our production function we get:
$$(1-\delta)k\_t+s(k\_t-\bar{k})^{0.5}-k\_t=k\_{t+1}-k\_t$$
simplifying and noting in steady state that $k\_{t+1}-k\_t=0$ we get:
$$-\delta k\_t+s(k\_t-\bar{k})^{0.5}=0 $$
rearranging further we get a quadratic.
$$\left(\frac{\delta}{s}\right)^2k\_t^2-k\_t+\bar{k}=0$$. Noting the quadratic formula we note our steady states are:
$$k\_t^\*=\frac{1\pm\sqrt{1-4\left(\frac{\delta}{s}\right)^2k\_0}}{2\left(\frac{\delta}{s}\right)^2}$$

Immediately I see that for such a simple problem no solution may exist because our roots may be complex. Looking at this picture though there should be a solution to what is the simplest case of multiple equilibria in a non game theoretic model.

Do complex roots pose an issue for identifying multiple steady states and if they do what is the solution (if there is any)?

[![enter image description here](https://i.sstatic.net/fee4z.png)](https://i.sstatic.net/fee4z.png)

**Tags:** production-function, solow

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## 1 answers

### Answer by username_0 on 2022-05-04. Score: 2 | Accepted answer

When we talk about steady state level of capital, talking in terms of complex numbers never makes sense. What this solution for steady state $k^\*$ tells us is:

1. Its possible that no unique steady state exists if $4\left(\frac{\delta^2}{s^2}\right)k\_0>1$
2. If $4\left(\frac{\delta^2}{s^2}\right)k\_0<1$ there are two unique steady states.

stability of equilibria is a anther story but what this model tells us is that there is at least two steady states with complex solutions being meaningless.


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