In EU4, I have a nation that has no allies surrounding its borders and they all have an institution that I haven't embraced yet. However, I have noticed that if I increase the production value and base tax of some of my more wealthy provinces and if I have enough ducats, my nation embraces the institution on its own. Can someone tell me how exactly the development of provinces can lead to better institutions and if it would be worth it to develop said institution on my own?
Developing a province gives that province an immediate increase in the oldest non-present institution equal to 16.66% (slightly under one sixth) of the new development of the province. The development required to bring an institution with 0% presence to 100% presence is of 601 cumulative development points, because for 600 points (reaching 36 dev starting from 11) you get 99.96%. The required development within a province of a given starting development is summarized in the table below.
Regardless of initial development, the total cost of going from 0 to 100% institution presence in a province is always about 2000 monarch points before other modifiers (such as terrain or the merchant guild estate bonus) to development cost. The amount of monarch points required to reach 100%, without any modifiers or initial presence, is summarized by the table below:
Because all of your neighbhor are not allies. It is likely that you are not benefiting from "friendly" neighbhor institution spread modifier. It is worthwhile trying to temporarily befriend someone with institution next to you to help spread it faster even if you plan to beat them later on.
Converse the opposite is also true. If your neighbor is rival then institution spreader will be extremely painful. This can be easily observable when British start colonialism institution and only border France and nobody else.
This is when forcing institution via development is handy. Instead of waiting for it. You can just start it yourself. However you have to keep in mind that 2200 MP is a significant investment and try to find decent province to do it in. Like a province that is grassland (low penalty) and has a decent trade good.
Edit: There is also a state edict that allow you to make institution spread slightly faster but I don't recommend that method because it cost ducat monthly upkeep. I often forget about edicts that are still active and costing me upkeep.