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(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?

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