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In 1873, Johns Hopkins University was started with an inflation-adjusted $144.5 million. In 1884, Stanford University began with an inflation-adjusted $139 million. Meanwhile, in 2009, KAUST was started with $10 billion. There's a difference of two orders of magnitude.
Why did it become so much more expensive to start a university in modern times?
Besides all the factors that the other answer already lists, the elephant in the room is that KAUST has explicitly been designed to be a world-class university (rather than organically growing into one, as was the case with your other examples).
In short, it is not so much more expensive nowadays to found just any university - in fact new universities get founded all the time, virtually always spending considerable less money than $10 billion - but founding a university with the expectation that it will compete with the best in the world in a very short time frame is expensive. You need to compensate for all the natural growth in budget and prestige that established universities had over the last one or two centuries.
I would add two other factors to the good answers already available.
I suspect that you’d probably need to look at the budgets for the institutions in question to get a definitive answer, but I can think of a few potential causes:
Increased administrative and bureaucratic costs. In the days when those universities opened, HR departments weren’t a thing, and there was significantly less regulation on businesses in general.
Increased cost and quality of facilities. Technology marches on - and the cost of building a university building before indoor plumbing was a thing is radically different to building a modern university building with electricity, running water, IT infrastructure, projectors and cameras for the lecture theatres, and all the inspections and certifications being done to make sure that all the work is being done in accordance with code.
Computers, in general. That’s an entire field of costs that didn’t exist back then - not only do they need the physical infrastructure and desktop computers for the staff and students to use, but they also need to pay for licenses for all the software that the students would use for their classes, for the university website, for their online learning system, for their class and room allocation system, for their centralised marking and enrolment system, etc.
Education inflation is currently running about 7-8% (source). I don't have any data on what this rate was a century ago, so I'll just assume it's been constant. The general rate of inflation in the US for the last century has averaged about 3.2%. This means that education inflation is about 4% in real dollars. If you compute 1.04^100, you get about 50, i.e., about two orders of magnitude in a century. This is pretty much the size of the effect you describe in the cost of starting up a new university.
The deeper question would be why inflation in certain areas, such as healthcare and education, is so much higher than the general rate of inflation. I don't think economists have a universally agreed upon explanation for this. My understanding is that there are several explanations, all of which are probably true:
Goods and commodities can always go down in value. The cost of a book, for example, is many orders of magnitude less than it was in the middle ages. But services can't become cheaper without bound. Services are provided by humans, who need to live. Therefore there is always a tendency for the non-service part of the economy to shrink in proportion to the service economy.
In most countries, healthcare and education are government monopolies, or nearly so. In such a setup, people are using something that they don't pay for, so there is no tendency to restrain their use of the resource.
In healthcare and education, people don't or can't make normal decisions on cost and value. For example, if the price of Greek yoghurt gets too high, I can decide to stop eating Greek yoghurt. But if education gets super expensive, I don't just tell my kids not to go to college.
Instruments and reagent price is on drastic rise. I purchased students grade compound microscopes at Rupees 7000 (each) around 2010, whereas at 2019 the same instruments costs more than Rupees 20,000. (each) This is immense. Previously if a college or university lacked a facility to provide it to students, it was easier to provide it. Whereas in recent years it is much harder to purchase any instruments except plastic vials or centrifuge tubes or micropipette tips.
Unlike household objects like computer, phone, sewing machines etc; there has been no visible effort to drop the price of scientific instruments. The price is ever rising.
Often a lack of mentainance drastically increases the cost. Say a part of building is damaged. While it could be readily repaired, due to carelessness of authority it is not repaired until and unless some devastating thing happens. Same for electrical wire mess, garbages, household appliences that provides electric shock, etc. Although it may not be the global scenario, inaction or showoff to hide the problem, actually increases the hidden costs.
