If you have many smallish cities (massive expansion in AA, and/or wartime acquisitions), then switching to Monarchy is better than staying in Despotism until your pop gets higher, particularly if you're Religious. The extra unit support is massive and could get your economy substantively up. Plus, your food and industry improve so you can seriously get your Civ up and cranking.
I'd like to point out that unit support for towns is 4 in despotism and 2 in monarchy. If you switch to monarchy, you can easily have the same initial economic problems as if you switch to republic.
I'd say that 5-7 is about the breakpoint. If most of your cities are size 4-6, Monarchy is the better plan. Greater pop growth sooner means more worked tiles and more comerce rather than waiting for the people to grow. Also, you have less of a need for temples and the like since your units can serve as a stopgap measure to allow you to concentrate on Science and Money buildings.
I'd say, if your cities are to small, it is better to stay in despotism a little longer than to have 2 anarchy's.
War is also a big issue. A defensive war can be fought under Republic, but if you're looking for a protracted invasion in the near future (for important resources for instance), then Monarchy is the way to go.
You can fight offensive wars extensively under republic. Up the lux slider a little bit and still earn more cash in the end.
You'll get lots of free unit upkeep, you can use your units to quell new cities, and each new city grants you even more unit upkeep. You can really field in massive multiple stacks of invaders this way.
Like I explained in my previous post. It doesn't matter whether you pay your upkeep with the bonus upkeep per city from monarchy, or the bonus commerce per tile from republic, except that the bonus commerce from republic is way higher. You can usually pay for a larger army in republic than you can keep under monarchy for free. And
still have more commerce for science.
However, most of the time, players will postpone invasions and wait for the Republic, since it is that great a peacetime government. You can really surge ahead technologically and economically with Republic.
Republic is superb for peacetime, and
great for war. Monarchy is mediocre for peacetime, and
o.k. for war. (Exception being the
Always-war variant.)
On occasion, I have played games with such massive commerce bonus tiles that I deemed Republic a wash. The extra commerce that would be gained just wouldn't be that much. Those games I go Monarchy then straight to Communism or Democracy.
Ok, lets see, your empire was only one city and that one city had 7 bonus-gold hills and 2 cows, and 2 rivers that covered all 21 tiles in the city radius? And even if that was the case, how would you beat +12 commerce? If you have 10 cities, we are talking about +120 commerce, thats 30 bonus-gold hills. And even
if, by some fluke, you had 30 bonus gold hills in your 10 city empire, the advantages of monarchy are still not going to beat an other +120 commerce.
For Religious Civs, it's also very useful to get Monarchy as a stepping stone to Republic, since you don't pay as much in terms of Anarchy, and you could use the additional growth.
That may be true, but only if, by some fluke, you got the monarchy tech way before republic. Otherwise you'd be better of going to republic right away.