<p>By the notation used, labor quantity appears to be constant. Then the only source of growth of output is the growth in $A_N(t)$, and they will have the same growth <em>rate</em>. Take logarithms and then time derivatives on the output per capita equation to see this.
Then remember introductory differential equations on how to express $x(t)$ as a function of $x(0)$ and a constant growth rate. And don't forget to explicitly solve for the <strong>10</strong>-factor difference required, at the end. </p>

CC BY-SA 3.0
2016-03-17T15:13:03.280Z
2016-03-17T15:13:03.280Z
Alecos Papadopoulos
<blockquote>
<p>Assume the world consists of two regions, the North and South with the following production functions and technology lags in the south by <span class="math-container">$\tau$</span> years:</p>
<p><span class="math-container">$$ Y_N = A_N (t) (1-a_L) L_N; \; \; \dot{A}_N = a_L L_N A_N (t)$$</span></p>
<p><span class="math-container">$$ Y_S = A_S (t) L_S  ; \; \; A_S = A_N (t - \tau) $$</span></p>
<p>The growth rate of output per worker in the North is <span class="math-container">$3 \% \; year^{-1}$</span> and if <span class="math-container">$a_L \simeq 0$</span>, what <span class="math-container">$\tau$</span> must be for output per worker in the North to exceed that in the South by a factor of 10?</p>
</blockquote>
<p>For output per worker in the north to exceed that in the south by a factor of ten:</p>
<p><span class="math-container">$$ 10 = \frac{Y_N/L_N}{Y_S/L_S} = \frac{A_N(t)(1-a_L)}{A_S(t)} $$</span></p>
<p>with <span class="math-container">$a_L \simeq 0$</span> this goes to</p>
<p><span class="math-container">$$ \to \frac{A_N (t)}{A_N (t-\tau)}$$</span></p>
<p>My book says this is equal to <span class="math-container">$e^{0.03 \tau}$</span></p>
<p>Could someone explain where this last result comes from please?</p>

<macroeconomics>
Delay in the transmission of knowledge to poorer countries
Sunhwa
# Delay in the transmission of knowledge to poorer countries

(Asked by: username_0 on 2016-03-17)

> Assume the world consists of two regions, the North and South with the following production functions and technology lags in the south by $\tau$ years:
>
> $$ Y\_N = A\_N (t) (1-a\_L) L\_N; \; \; \dot{A}\_N = a\_L L\_N A\_N (t)$$
>
> $$ Y\_S = A\_S (t) L\_S ; \; \; A\_S = A\_N (t - \tau) $$
>
> The growth rate of output per worker in the North is $3 \% \; year^{-1}$ and if $a\_L \simeq 0$, what $\tau$ must be for output per worker in the North to exceed that in the South by a factor of 10?

For output per worker in the north to exceed that in the south by a factor of ten:

$$ 10 = \frac{Y\_N/L\_N}{Y\_S/L\_S} = \frac{A\_N(t)(1-a\_L)}{A\_S(t)} $$

with $a\_L \simeq 0$ this goes to

$$ \to \frac{A\_N (t)}{A\_N (t-\tau)}$$

My book says this is equal to $e^{0.03 \tau}$

Could someone explain where this last result comes from please?

**Tags:** macroeconomics

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## 1 answers

### Answer by username_1 on 2016-03-17. Score: 1 | Accepted answer

By the notation used, labor quantity appears to be constant. Then the only source of growth of output is the growth in $A\_N(t)$, and they will have the same growth *rate*. Take logarithms and then time derivatives on the output per capita equation to see this.
Then remember introductory differential equations on how to express $x(t)$ as a function of $x(0)$ and a constant growth rate. And don't forget to explicitly solve for the **10**-factor difference required, at the end.


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