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


1 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.


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