All we need is the political will and we can fund a Mars base as a purely government funded research program.
1. The indomitable human spirit and drive to explore and expand
2. It's cool
3. More room for humans
There are also reasons that don't apply to Antarctica:
1. Hedging our bets against planet-ending catastrophes (global warming, giant asteroid/comet strike, ultra-pandemic)
2. Another stepping stone to exploring further interesting/important space goals, like gathering resources from the asteroid belt/moons of jupiter/etc, discovering life on Europa, and so on
Money is just a medium for the exchange of goods or services. What would the colony export in order to generate the revenue required to produce said taxes?
EDIT: this is also made worse by the fact that the first few colonists should be farmers, mechanics and doctors (aka human mechanics), since all the intellectual work can be done on earth.
A human being able to lift 3x as much without machines already opens up possibilities for greater productivity.
Stuff that must happen in the cold is cheaper to do too...
Think of Vegas - no economic output at all, tourist destination alone. Mars could do the same.
It only takes one thing - there is no need for a mixed economy.
If it takes 10,000 people to make $thing, then even at Musk's target price of $100k/person, the cost to develop and ship the automation[0] only has to come in less than a billion dollars to win.
[0] I guess the TCO would be more complex to determine, as the human side includes not just paying the humans (and presumably shipping good from Earth), but also figuring out how to do low-gravity and zero-gravity healthcare and surgery (on this scale there will be emergencies requiring surgery during transit), and planning for the colonists' desire to start families and retire.
The amount of money that a sizable and well funded group of people spend to get away from literally every other human on the planet and away from the government would easily fill a few rockets.
2. Martian Water.
Imagine all the disenfranchised homeopathics now have another woo-woo cure to turn to and will pay out the wazoo for. Make up a claim like "the purest water, untouched by human industry or nuclear tests, powered by billions of years of energy from sun, unfiltered by ozone and untouched by magnetism."
3. Tourism.
Vegas is basically a Martian tourist destination with an entire city built to support it. There's no other reason for Vegas to exist. If the accommodations were nicer, people would go to Antarctica as well. Rich people want to take their selfies with Olympus Mons in the background.
4. Low-G sports
Earth sports probably won't work the same, so entire new sports and leagues will form and provide entirely new season pass resell opportunities for streaming video providers. There's no way to simulate the low-gravity on Earth.
Also rich people sports like golf might take on an entire new ultra elite form when your par-4 hole 8 is 4500m long and you need a satellite to spot your ball.
5. Low-G food products
For similar reasons as the Martian water. Insert any combination of differences in nutrition/taste/look and it'll find it's way onto the plates of a three star Michelin restaurant or as supplements sold at a health store or something.
That's off the top of my head and could easily be a multiple billions of dollars per year of sustained economic output from Mars, mostly built on simple vices, novelty, entertainment, and pure human gullibility.
I expect it will be profitable in the future as it scales up. But there is probably only about 2x more scaling at the current price point (launching new countries, selling to people who aren't yet aware of it).
Also it has space accountancy pirates.
Even with a high volume and relatively low cost launch vehicle, the actual offworld base will be hugely expensive, and no commercial enterprise can realistically expect to make a return for their investors.
A government needs to step up with the rationale that it will eventually form a tax-producing colony - but a huge investment will need to be put in till it gets there.