Colouring in poultry feed is to give the yolks more colour. In battery hens these would be pale and insipid; the naturally golden yellow colour comes from free-range hens eating grass and other vegetation.
The shell colour is purely to do with the breed of bird. The reason for the idea that brown eggs are free-range and white are battery produced comes from the early days of battery farming, when smaller Mediterranean breeds such as white leghorns were used, which laid white eggs naturally. By selective breeding and crossing, the modern small hybrid was produced, which lays brown eggs to satisfy consumer demand.
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