An economy produces a consumption good and also has a research sector which produces new ideas. Time is discrete and indexed by ๐ก = 0, 1, 2, โฆ
The production function for the consumption good is given by
$๐_๐ก = ๐ด_๐ก ๐ฟ_{๐ฆ๐ก}$
where, at time ๐ก, $๐ฟ_{๐ฆ๐ก}$ is the amount of consumption good produced, ๐ด๐ก is the stock of existing knowledge, and $๐ฟ_{๐ฆ๐ก}$ is the amount of labour devoted to production of consumption good. It is known that $๐ด_0 = 1$.
The production function for new ideas is given by
$ ๐ด_{๐ก+1} โ ๐ด_๐ก = \frac{1}{250} ๐ด_๐ก $๐ฟ_{๐๐ก}
where ๐ฟ๐๐ก is the amount of labour devoted to production of new ideas at time ๐ก. Suppose that for all ๐ก, $๐ฟ_{a๐ก}$ = 10 and $๐ฟ_{๐ฆ๐ก}$ = 90. Then, the growth rate of the consumption good ($Y_{๐ก}$) at ๐ก = 50 is _____ percent (in integer).