For web and mobile development the rate will be min £25/hour or around £200/day this rate will apply to freelance and it’s really the end of the bracket for freelance in the UK. If the client compare with a freelance in India for example it’s impossible to match their price. Than the client have to consider the “after sale” value and the proximity (timezone) to the developer and extra change and further development.
For an App or a website I will says that he shouldn’t ask the client for his budget because unless like a garden who is a physical accomplishment that people can evaluate the cost from their point of reference. A web app or mobile app who to them is totally virtual and most of the time have no idea to the level of complexity involved to implement it. Also have to consider that asking for budget is a tricky approach on the web or app business because of that unknown related to evaluation the client might think you try to get a maximum amount out of him.
An app complexity translate into more hours writing code and subsequently an higher price tag. Than yes the more complex an app and the more it will cost.
But this is kind of an erroneous way of thinking because complexity can be found in a single function with a simple rendering like the following example. And the client might not agree for you to spend a week writing that code remember this is not a proof of concept and you should be able to run the code 100 times a day without crashing.
Want to challenge your programming skills build me an app that display number partition.
This is extremely complex despite being extremely small piece of work
In comparison the PHD student research tools might be complex too but much simpler in implementation as it will be providing the algorithm in which to compile the information in the tool.
When I was junior I was charging fixed price for work this allowed me to get more job and further my skills, then money was secondary and I was doing this for the skills later on when I was well settled I charged by the hours. People expect spot on work when you charge by the hours far less room for mistake.
IP is irrelevant because we have NDA and if the client is worried about IP he should have Lucas sign a standard contract including an NDA clause.
I hope this help but the most important is what Lucas want to get out this job: further skills or money