For lumber:

• 50/h you get on an empty haven ("zero" production),
  
• all lumber yards maxed: 5*900/h,
  

then you have two groups of bonuses:

• from buildings: level 6 of the Royal Cannon (+25%), the sawmill (+40%), and the pirate's den (20%),
  
• from virtual boosts: all discoveries finished (7%), prestige (10%), relics (two automatons or wood axes: 2*5%), lumber production bonus (25%), all res production bonus (25%); not sure if the last two combine, but let's assume that they do.
  

In total:

(50+5*900)*(1+0.25+0.40+0.20)*(1+0.07+0.10+2*0.05+0.25+0.25)=14899/h

unless I forgot something.

Price in etchings:

• lumber yards: 5*(120+200+300+450+620),
  
• sawmill: 550+1800,
  

In total: 10800 etchings (110k rubies at full price, or 75.6k at 30% discount).

I expect it would be the same for gold (but you cannot have two relics for both gold and lumber at the same time), and you can apply this approach for the rum. The data is all available in the game, except for relics which you can find here (I'm pretty sure that Montezuma's Medallion was removed, so I didn't go with it in the above calculation).

Crowbar said:

For lumber:

• 50/h you get on an empty haven ("zero" production),
  
• all lumber yards maxed: 5*900/h,
  

then you have two groups of bonuses:

• from buildings: level 6 of the Royal Cannon (+25%), the sawmill (+40%), and the pirate's den (20%),
  
• from virtual boosts: all discoveries finished (7%), prestige (10%), relics (two automatons or wood axes: 2*5%), lumber production bonus (25%), all res production bonus (25%); not sure if the last two combine, but let's assume that they do.
  

In total:

(50+5*900)*(1+0.25+0.40+0.20)*(1+0.07+0.10+2*0.05+0.25+0.25)=14899/h

unless I forgot something.

Price in etchings:

• lumber yards: 5*(120+200+300+450+620),
  
• sawmill: 550+1800,
  

In total: 10800 etchings (110k rubies at full price, or 75.6k at 30% discount).

I expect it would be the same for gold (but you cannot have two relics for both gold and lumber at the same time), and you can apply this approach for the rum. The data is all available in the game, except for relics which you can find here (I'm pretty sure that Montezuma's Medallion was removed, so I didn't go with it in the above calculation).

Answered it completely, thanks!

That sucks. 

Anyway, in that case, one of the 25% bonuses in the third parentheses is off, so we get

(50+5*900)*(1+0.25+0.40+0.20)*(1+0.07+0.10+2*0.05+0.25)=12795/h