>NASA's budget is about 17 billion so lets guesstimate that going to Mars will cost 100 billion. It's probably a lot more to establish a colony, let alone a self sufficient colony.

I would estimate that the constellation/SLS Orion program to Mars or some derivative thereof can be accomplished at a total cost of 50-100 billion USD (inflation adjusted to 2015 equivalent), as NASA projects $18 billon in development costs, and 1.5-2 billion per launch.[1] They are spending 2 billion per year on SLS/Orion, and will continue to do so for 10 years until first flight, followed by ~ one flight per 18 months at a similar cost for 10-15 years. For reference, the space shuttle program cost approximately 200 billion dollars.[2]

SpaceX/Blue Origin/Orbital Sciences could probably accomplish a similar goal with 10-25 billion dollars depending on speed of implementation, success of various reusability schemes, use of higher efficiency drives, and launch system success rates. From memory, I think SpaceX recently said that they could develop their super-heavy launch system in 5 years for 1.5 billion, and I believe Orbital said their system would have a development cost of 2 billion.

Colonization will probably cost more, though Musk has said that he aims to reduce the cost of moving to Mars to $500k USD (inflation adjusted to 2015) with a total colony cost of about $40 billion by about 2050.[3] I am not sure how to evaluate whether this is a worthy objective, but it seems that securing the future of humanity might be worth 0.13% of annual gross world product (100 billion dollars), and SpaceX aims to achieve this goal for much less money.[4]

>We aren't lacking space on Earth; it would be a lot easier to colonize the Saharan desert than to colonize Mars

Colonizing the Sahara desert is intellectually interesting, but there doesn't seem to be much point to the exercise. It is well established that people can live in deserts. Total human population will soon begin to decrease anyway, so overpopulation will not be a problem in the forseeable future, and living in the Sahara does nothing in terms of securing humanity's future against cataclysms.[5]

>Instead that money could go to scientific research (e.g. renewable energy generation, medical research, AI research, robotics research), or to some other cause. Wouldn't that be of much greater benefit to humanity?

It is not clear to me that the government is managing current research funding very well, so I would be most likely to support returning NASA's budget to the taxpayers (in the form of budget and later tax cuts), which are likely to improve everyone's standard of living (all over the world), and allow people to have better day-to-day lives.[6] However, it is very difficult to compare an improvement of daily life on Earth to the establishment of a multi-planetary species. I would likely favor the latter, though I do believe it would be much better if humanity made this great leap without coercing taxpayers (though this is probably one of the less morally problematic instances of government coercion).[7]

>If you're looking purely at space exploration it also seems to me that there are better ways to spend that money than to put humans on Mars. For the same money we could probably put a whole bunch of telescopes in space and send several probes/rovers to asteroids/planets/moons.

I would question this, on the basis that I am not sure what the goal of the robotic and telescopic exploration really is. When you say that human missions to Mars are esoteric and pointless, it is hard to simultaneously argue that it is imperative to figure out what happened long ago in a galaxy far far away or on a planet you never plan on going to. If there is to be exploration and study of space, it seems to me that establishing a multi-planetary presence, and expanding our horizons should be the top priority. In addition to providing inspiration and knowledge to those of us who remain here on earth, those astronauts (and future Martians) will be taking the first steps to securing humanity against disaster, and spreading life throughout the galaxy.

[1] https://en.wikipedia.org/wiki/Space_Launch_System

[2] https://en.wikipedia.org/wiki/Space_Shuttle_program

[3] http://www.outerplaces.com/science/item/9914-elon-musk-s-pla...

[4] https://en.wikipedia.org/wiki/Gross_world_product

[5] https://en.wikipedia.org/wiki/Projections_of_population_grow...

[6] http://eml.berkeley.edu/~dromer/papers/RomerandRomerAERJune2...

[7] https://en.wikipedia.org/wiki/The_Problem_of_Political_Autho...