There are several problems with the idea:
My problem is that although logically both idea looks the same (i.e. good getting cheaper with a constant level of income = income growth outpaces prize increase ), I find the latter method mathematically more complicated.
Here you are confusing economics as a science with economic policy. Yes in a science we prefer more parsimonious explanation to one that is less parsimonious (assuming they both have equal explanatory power) but that does not mean we should prefer more parsimonious policy. That is equivalent of saying that simpler medical treatment should be always preferred. Just because a more parsimonious biological theory/explanation is more efficient and hence preferred does not mean we might as well prefer herbal tea compared to complex antibiotics.
Its not that there are two theories one which says that inflation must occur as some sort of natural process when technology expands and second theory that says that technology can expand freely. Here you are talking about policy choice about what we choose the money supply to be along some path of technological progress.
From my perspective, the amount of money in an economy should remain constant (apart from replacing the old torn bills with a brand new one) and things should get progressively cheaper until we reach a technological utopia where everything becomes free.
There are several problems with this:
- First, what ultimately matters is not what the price tag is but how much real resources you have to spend for goods and services. If due to inflation apple costs $\\\$10,000$ but you earn $\\\$1,000,000,000$ per hour then in real terms the apple in such situation is much cheaper then in situation where apple costs $\\\$0.10$ but your hourly wage is $\\\$0.01$.
- You are ignoring well documented nominal rigidity especially when it comes to wages (this is so notorious you can see this mentioned in any textbook - for example Blanchard et al. Macroeconomics an European Perspective). For the sake of brevity I wont go into details but in the presence of nominal rigidity in short run deflation creates excess unemployment and depresses economic activity and inflation has the opposite effect. This is because in short run when aggregates prices fall firm might be still locked to higher nominal wages as those might be slower to adjust and this forces firms to unnecessarily lay off some workers. Due to this most economists advocate that over business cycle there should be steady slow (about $2\%$) inflation in order to help 'grease the wheels' of the economy.
A possible explanation to my question: By not increasing the money flow, we are not able to reward the individuals who brought about the technological advance as the things get cheaper uniformly for everyone, whereas if we print the extra money, then that extra money goes to person responsible for the change).
This is also incorrect. Even if money supply would remain constant that does not mean that there is any lack of money to reward people for their additional work or effort. In long run prices & wages would just adjust downward so even if people would earn less in nominal terms they could actually buy much more as before (an opposite situation as the one shown with apple before). What matters ultimately is not how much money you have but the amount of goods and services you can consume.
Economy actually existed and functioned well before money was invented at all. If there would be no nominal rigidity and we would live in a world where money is even in short run completely neutral then deflation would not matter at all.