By request of some viewers, and for anyone who is interested, this is the more elaborated Fel’dman model (as used in the video). For background, watch the video.
Note that superscripts indicate the sector in question. These can either be p (producer goods) or c (consumer goods).
Output at time t for the producer goods sector is,
where a is productivity in the producer goods sector. Consumer goods output is similar:
We then have investment functions for both sectors,
where e is the proportion of investment going to the producer goods sector and 1-e is therefore the proportion of investment going to the consumer good sector. We can substitute in the output functions to get:
Capital stock in the current period is going to consist of whatever capital stock is leftover from the previous period after any depreciation. Current period capital stock will also include whatever investment was made into the respective sector.
After a little algebra, we can write period t capital stock in the producer goods sector as:
Knowing this, we can also find capital stock in the consumer goods sector in period t:
Now capital stock in the consumer good sector is a function of past capital in both sectors. As mentioned in the video, the second term in the equation is a kind of spillover effect from heavy industry into light industry.
There are further applications and expansions of this model. However, the two capital stock equations were the focus of the video and what Robert Allen uses his exposition of the model in Farm to Factory.
Allen, R.C. 2003. Farm to factory: A reinterpretation of the Soviet industrial revolution. Princeton University Press.