Requirements
To obtain a chicken, you usually need at least four patches of grass in the pasture. Once you have this and around 2400g, you can buy a chicken at the farmers' union for about 1000g. You can choose between a male (rooster) or female (hen) chicken, but unfortunately you are not able to name them. Female chickens, for obvious reasons, lay eggs.
To make mayonnaise you require a mayonnaise maker. After you've shipped enough eggs, you can purchase it from Lucus for about 30,000g. Mayonnaise has higher profit than eggs.
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Stages
There are four primary stages for chickens: egg, chick, chicken and golden chicken. Once you put an egg in the incubator, it will take approximately 7 days to hatch, 14 days to reach adult, and then 10 hearts (plus the equivalent of another heart)- about another 8-10 days-to reach its golden stage. To make the chicken reproduce, you need male and female adults. Now you can place fertilized eggs in the incubator.
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Produce
Egg 50G
Golden Egg 9000g
Mayonnaise 300G