Delay in the transmission of knowledge to poorer countries

Question

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?

Answer 1

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.