What would have been the effects if the Portuguese Monarchy stayed in Brazil?

But they did, in a certain way. Emperor Pedro I of Brazil was also King Pedro IV of Portugal. It's true, however, that most members of the royal house departed. For the reasons, look after Pedro's article at wikipedia.
 
an interesting WI

but ultimately hard to see how much really changes. The main players weren't dynamic enough to lead Brazil (and drag Portugal) into a progessive state. A best case scenario leads to portugal accepting a secondary role. A semi-realistic case leads Brazil and the colonies to separate from Portugal. A worst case leads John and his heirs to losing the crown . Let's accept for argument's sake the best case. John stays in Brazil. He keeps a significant military/navy. This allows for a victorious cisplatine campaign, but beyond that, what really goes well for them? Brazil is a backward, ill developed country. With inspired leadership, there's a possibility Brazil progresses far beyond what they actually did, but there's no indication any of the crown has what it takes to do anything more than what Pedro 2 did - keep stability enough for the planter elite to maintain their hegemony. Likely a successful state, but not a power.

Now, if the WI were a dynamic John and/or Pedro, you might see an interesting development, but the first was a reactionary, and the second didn't have the temperment to conciliate his way to a bright future.
 
an additional note

ultimately, with Pedro declaring independance, you have the crown remaining in Brazil. you lose the bank account (which John took back to Portugal) and the military, which wasn't all that great, but certainly was enough to rule over Argentina/Paraguay. Otherwise, things remain the same as if John gave legitimacy to Brazil as a dominant player in the portugese empire. What screwed Brazil was Pedro being a poor emperor and then leaving it in the lurch to go adventuring in Portugal, which he could have had from the get go.
 
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