stone geary production and Multiple equilibria in a simple solow model: Do complex roots mean anything?

Question

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)?

Answer 1

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.