Show the growth rate of technology and capital are eventually continuously increasing

(Romer 3.6.b) Consider an Endogenous Growth rate model including capital and knowledge with the following equations of motion and $\beta + \theta > 1$ and $n > 0$

\begin{cases} Y(t) = [(1-a_k) K(t)]^{\alpha} [A(t)(1-a_L) L(t)]^{1-\alpha}, & 0 < \alpha < 1 \\ \dot{K}(t) = s Y(t) & 0 < \alpha < 1 \\ \dot{A}(t) = B[a_k K(t)]^{\beta} [a_L L(t)]^{\gamma} A(t)^{\theta}, & B > 0, \; \beta \geq 0, \; \gamma \geq 0 \end{cases}

Show that regardless of the economy's initial conditions, eventually the growth rate of $A(t)$ and $K(t)$ (and hence the growth rate of $Y(t)$) are continuously increasing

We have computed

$$\frac{\dot{g}_A}{g_A} = \beta g_K + \gamma n - (1 - \theta)g_A$$

$$\frac{\dot{g}_K}{g_K} = (1-\alpha)[g_A + n - g_K]$$

and drawn in $(g_A, g_K)$ space the paths of each when $\dot{g}_A = 0$ and $\dot{g}_K = 0$.

Is the question asking to show either:

• $\exists t_0, s.t. \; \frac{d}{dt} g_A (t) = \dot{g}_A (t) > 0$ and $\frac{d}{dt} g_K (t) = \dot{g}_K (t) > 0$ $\; \forall t \geq t_0$
• $\dot{g}_A (t) > 0$ and $\dot{g}_K (t) > 0$ occur infinately often

That is, I am not sure if we need $\dot{g}_X (t) > 0$ to hold for all $t \geq t_0$ or if it may fluctuate between positive and negative and hits positive infinite times, I feel the latter is more reasonable to see in the real world so it would suffice to show (but most certainly not the method they intend to solve this problem)

$$\displaystyle P( \dot{g}_X > 0 \; i.o.) = 1$$

Where $i.o = infinately \; often$. But here have the problem of defining a probability measure on $\dot{g}_X (t)$

Can we give this the following probability?

| growth rate | positive | negative | zero
| probability |   1/3    |    1/3   |  1/3


Then summing over all possible initial values (countably infinite)

$$\sum_{n=0}^{\infty} P(\dot{g}_X > 0) = \sum_{n=0}^{\infty} \frac{1}{3} = \infty$$

And by Borel-Cantelli Lemma 2, since the events $A_n := \dot{g}_X > 0$ are independent and the sum of the probabilities is infinity then:

$$P( \dot{g}_X > 0 \; i.o.) = 1$$

Which tells us that the event the growth rate is positive occurs infinately often. Would this answer the question?