How do I determine the optimal price to maximize profit from a contest?

We are trying to setup a contest at my school where contestants have to pay to enter and compete. We have enough money to cover a grand prize, as well as other minor prizes. According to my naive understanding of supply and demand, the higher the price of the ticket, the less likely it is that people will enter the contest. However, if we don't set the price high enough, we won't generate enough revenue to make up for the cost of the prizes.

This is the first time I've ever tried to do this, so I don't have any data on how to set the price for maximum profit and/or maximizing the likelyhood of breaking even. We want to develop a business model to allow us to continue running this contest in the future, with prizes getting larger and larger each year. But before we can do that, we have to be successful with this first contest. How can we determine the optimal price of the ticket? Are there any particular case studies which have tested hypotheses of the optimal price as a function of the demand and/or the value of the grand prize?

To put into some context, here are some of the contest details:

The idea is to have a so-called "sales pitch" contest, where an independent panel of professionals from business, science, and humanities judge the quality of their "million dollar idea" presentations. There are $500, $150, and $50 first, second, and third prizes for this contest. We hope to obtain a 30-50% profit from the sales. The target audience is college students, but especially graduate students.

Obviously, as we increase the prize money, more people will be interested. But we also don't want to get "too many" such that we cannot guarantee a spot for everyone to participate in the 1 day contest. Are there any good theoretical/mathematical tools we can apply to predict the demand based upon the price in this context?

We haven't conducted any market studies on our target demographic. Are there any published case studies on university students to work from?

You need to start with your limits and bottom line requirements. If you can only accommodate X entries, then that will determine the minimum price (costs + req. profit)/max entrants (you may need to include relevant taxes in costs too).

OK, now is this based on a fixed date? - in which you have to sell tickets before that date. The alternative means you can hold back until all required tickets are sold. If so, then you need to estimate the amount of tickets you can sell (not an easy task, but that is what market research is all about I'm afraid). Give your self a good sized contingency (say 20%) and this gives you the price of the tickets - more sales is a bonus.

There is also the potential to scale the prizes based on the entrance fees (like a lottery) - winner takes home 50% of profit, second takes home 20% etc.

I helped a friend do such a thing as this for a charity - she did not listen to, and purchased a large amount of prizes on the assumption that all tickets would be sold until all prizes were exhausted - she ended up with lots of tickets, lots of prizes (of course all the major ones had gone) and not much in the way of profit for the charity ( indeed, barely broke even - and that is with some of us footing the bill for some of the prizes too!).

The problem was what economists call margin price - the lowest price someone will pay for something - this equates with what they think they can receive in this respect - they will pay high if the reward is great and less if it is not. Margin price determines value, not cost or profit to the vendor (service and enforcement also play a part, but is irrelevant here)