# Global Asymptotic Stability in the Solow Model - Rigorous proof

Consider the (simplified) Solow equation, which is a first-order non-linear difference equation: $$$$k_{t+1} = sf(k_t) + (1-\delta)k_t$$$$ where $$s$$ and $$\delta$$ are two constants and $$s,\delta \in (0,1)$$. I assume that $$f: R_+ \to R_+$$ is twice continuously differentiable and $$f'(k)>0$$ and $$f''(k)<0$$ for all $$k>0$$. So, $$f(k)$$ is assumed to be strictly concave.

At the steady state, $$k^{\ast}$$ (fixed point) satsifies the following equation: $$$$s f(k^{\ast}) = \delta k^{\ast}$$$$

Now, pls, tell me if it's correct how I prove global stability:

First, $$$$k_{t+1} = g(k_t)$$$$ and for $$k^{\ast}$$ being a fixed point it must be $$$$k^{\ast} = g(k^{\ast})$$$$

Then, for all $$k_t \in (0,k^{\ast})$$, $$$$k_{t+1} - k^{\ast} = g(k_t) - g(k^{\ast}) = - \int_{k_t}^{k^{\ast}} g'(k) dk <0$$$$ The inequality holds because $$g'(k)>0$$ for all $$k$$.

This result shall ensure that for any $$k_t \in (0,k^{\ast})$$, then $$k_{t+1}$$ is always lower than $$k^{\ast}$$, and the convergence of $$k \to k^{\ast}$$ is ensured from the fact that $$$$\gamma(k_t) = \frac{k_{t+1} -k_t}{k_t} = \frac{s f(k_t) + (1-\delta)k_t - k_t}{k_t} = \frac{sf(k_t)}{k_t} - \delta$$$$ and since $$h(k_t) \equiv k_t/f(k_t)$$ is strictly increasing in $$k$$ and $$h(k^{\ast})=s/\delta$$, then the growth rate of $$k$$ is positive but strictly decreasing in $$k$$, with $$\gamma(k)>0$$ iff $$k< k^{\ast}$$ and $$\gamma(k)=0$$ iff $$k= k^{\ast}$$.

A similar argument applies to $$k_t > k^{\ast}$$. Is that correct?

UPDATE: For my understanding, global stability means that $$k^{\ast}$$ is globally asymptotically stable if for all $$k_0 > 0$$, for any solution $$\{k_t\}_{t=0}^{\infty}$$, we have $$k_t \to k^{\ast}$$. In plain English, it does not matter what initial value the capital takes as long as its initial value belongs to the domain of $$g()$$. Then, $$k_t$$ converges to the steady state $$k^{\ast}$$.

• I mean we already went over this once. Could you please include what you think is the definition of global stability in your question? Because you seem to think it has something to do with convergence. Mar 19 at 21:27
• @Giskard you are right, but going through Acemoglu's book gave me this idea. Let me elaborate on what I mean by global stability in the question above Mar 19 at 21:35
• Note that in all the definitions the word "asymptotically" is included. See also this question. Mar 20 at 20:24
• Nope, my education in economic dynamics was also pretty poor. It is my impression that books focusing on economic aspects are rather handwavy about mathematics, which is unfortunate when proving general statements. Do you need a rigorous proof? And were the recommendations by Bakerstreet under the other question not to your satisfaction? Mar 22 at 11:25
• @Maximilan @ Giskard Thank you for mentioning my references. But the book of Braun is only about differential equations, and in the book of Gandolfo, as far as I remember, there is no proof of the stability of the Solow model in discrete time. As far as I remember (now I am not at home and I cant consult books) Acemoglou provides what needed, including also the relevant theorems. Mar 22 at 13:31