good question! i don't think ive ever done a formal analysis on this specific situation. here's my first attempt, kinda crude but it shows some interesting stuff.

quicky calculation based on quick-buy button resource conversion rates and the rate of return of an expansion asteroid with 3 resource rocks that is directly adjacent to your homeworld.

premises:

1) the tax income at full population from the asteroid is approximately 0.6 creds/sec. it takes several minutes to get to this level though. it will ramp up from 0.1 at initial settlement to 0.6 at full pop. assume that it takes 5 minutes to reach 20 population (its full level).

2) the resource income of each rock is .47 resources/sec. the total income from this is 1.41 resources/sec. it takes 30 seconds and 250 credits to build each extractor.

3) the underdevelopment penalty will subtract 2.4 creds/sec for 45 seconds while the upgrade builds. this total cost is 108 credits.

4) the cost of building the infrastructure upgrade is 450 credits, 100 metal, 75 crystal.

5) i will use the BUY rate conversion for resources on the black market since presumably we are not dealing with surplus situations in the early game and you will directly spend your resources. the BUY rate is used because it signifies resources you did not have to buy to make up for a shortfall. the analysis would be rather different if the SELL rate were applied, just something to be aware of.

cost analysis:

your outlay in cost is 450+750+108= 1308 credits

in addition you pay 100 metal and 75 crystal. assume conversions rates of 1 resource = 5 credits (median black market price on quick buy button), this amounts to 875 credit equivalent.

the total cost, all things considered, is 2183 credits.

benefit analysis:

the population based income really needs to be handled with integral calculus. i don't wanna do anything that complicated and lack the proper function of growth(time) anyway. i'm going to make alot of simplifying assumptions about this. bear with me.

assume linear growth at a constant rate for 6 minutes. .1 cred/sec for minute 1, .2 cred/sec for minute 2, etc. this is close enough to the truth that i'm satisfied by it but really its a damn crude approximation. oh well.

population based income then will produce 126 credits in the first 6 minutes and an additional 36 credits per minute each minute thereafter.

extractors follow a similar model except it hits its plateau after only 90 seconds, not 6 minutes. you will produce 84.6 resources in the first 90 seconds followed by a steady stream of 84.6 resources per minute each minute thereafter. using the 5:1 conversion rate this is a credit equivalent of 423 credits in the first 90 seconds and then 493 credits per minute.

conclusions:

clearly the resources are what you really want from the roid, the population income sucks. ignoring the population income since it is so low and takes so long to develop compared to the resource income, the amount of time it takes to make up your initial outlay of 2183 credits is... 3.57 minutes (about 3 minutes 34 seconds).

not too bad actually, those resources are worth alot if you consider what you'd have to pay to buy them from the black market. the roid is clearly a good initial investment. that pays dividends pretty quickly, only have to wait 3 and half minutes to see positive net revenue on it.

bigger planets take longer to develop obviously, but have potentially greater rewards due to non negligible income from population. feel free to apply my analysis technique (crude as it is) to a more complicated situation, like developing a terran world for population income. you'll probably get a similar result but over a longer time frame. higher initial outlay and longer time to recoup investment but result will be a larger net gain once it goes positive. what i'd be most curious to find out is which is most efficient credit for credit in terms of building revenue.