[kreios]

That's actually something close to my own setting.

First of all, looking at the numbers: You don't really need reactionless drives for cheap orbital transport. An SM+8 water-based Fusion Torch craft can put 750t into orbit for about 70t of water, for a cost of less than $2 per ton into orbit. Using SS2:40's $50/t should pay for handling, shipping, etc.
FTL cost will obviously depend on your FTL method. Let's assume the average distance between "interesting" systems is 10 days. We can use SS2:40's $100/t per day as a proxy. Total cost to travel between two interesting worlds is therefore $1050/t, call it $1000. With colonies being further away, $2000.

As another approach, we can look at the Genesis Colonial Transporter (SS5:19). A TL10^ design, it transports 4,000 colonists plus everything they need to a new home. At $1.8B, it's fairly expensive, and would probably be used in a "shuttle" service between a colony and a parent colony. Doing so, we can call it thirty days per round trip (total of 20 days travel due to FTL-2, plus five to unload and load). Call it ten trips per year.
Replacing the hydrogen-torch with a water-torch reduces fuel cost to $30,000 per trip. Let's assume that reduces passengers to 3000. Additional cost is for the crew (60 people, at $600,000 per trip for comfortable jobs), and to pay off the spacecraft itself ($3.6M per trip over fifty years). Call it a comfortable $5M per trip, including moving stuff into and out of orbit. That's only $16,000 per person, and includes space for industrial equipment and livestock.


Now, who can afford that? At TL10, monthly pay for an average job is $5,600. At status 1, you gain a profit of over $4,000 per month; $3000 if you have to support one dependant. It appears clear that, by saving for two or three years, most people would be able to buy a trip to a colony world.

And how many would there be? Well, look into genetic engineering. A human being with Longevity engineered will live to more than 200 years. You might well assume that UT medical technology and longer lives will lead to more children per parent - I find ericthered's 2% growth rate to look more and more plausible.

And why would they leave Earth? Well, aside from the usual - population pressure, resources, wars, devastation, climate change - those would be 200 billion people. You have to put them somewhere. But can you transport 200 billion people to other planets?
Let's say your 4 billion people, gross-world product of $22.4T, invest 0.5% into colony spacecraft. That's $112B in the first year, buying 60 Genesis transports. Those are able to transport 1.8 million people per year, and that only increases with each further year. Assuming 2% of the transports have to be replaced every year, the peak population is reached 64 years in, when about 5000 Genesis transport 150 million people per year. If the colonies continue to pay to transport Earth's population, it's actually going to be empty by the year 110.