The coliseum increases your citizens' happiness by 1%, and provides a venue for holding Events. Some suggestions for how to use the coliseum can be found in the comment section
There are three events that can be held at the Coliseum.
+ The Tax Day reduces your happiness by 20% and gives you a different amount of silver based on your population.
-Spread the Wealth
+ Spread the Wealth reduces your silver different amounts based on your population (the same amount you gain for tax day) and you gain 10% happiness
+ Chariot Races use 20,000 wood, stone, and iron but you receive a 5% increase to both happiness and your march speed.
See Tax Rate