Optimizing production for hypothetical factories

Question

You have $1$ factory that produces cars, and $1$ "builder" factory that produces other factories (either car factories, or other builder factories; the latter take twice as long). You want to maximize the number of cars produced over $4$ years. What is the optimal number of builder factories to produce before producing other car factories? Edit: “Builder” factories take a month to make, while car factories take half a month. Car factories produce a car a day.

Answer 1

There is not enough data for a meaningful answer; in particular, missing data include how long it takes to produce a factory, and how long it takes to produce a car.

For example, suppose producing a car factory takes 1 year, and a car also 1 year. Then, clearly, you should stop producing builder factories no later than 2 years down the road, as all builder factories finished after this time will not yield a single car. This means that you should not have more than 2 builder factories before switching to producing car factories.

Suppose instead that building a car factory takes 1 day, and a car 1 hour. It is not difficult to see that switching to car factories at any time during the first $3$ years is going to slow reduce car production: spendng 1 more month producing builder factories will yield more than 1000 times as many cars. Then, you are going to have far, far, far more than 2 builder factories by the time you hit the 3-year mark.