The basic model considers two types of workers that choose occupation in one of two sectors.
Roy's original paper deals with workers selecting into fishing and hunting professions, where there is no uncertainty about the amount of goods (fish or rabbits) that will be caught in a given period, but fishing is more costly as it requires more skill.
The central question that Roy tries to answer in the original paper is whether the best hunters will hunt, and the best fishermen will fish.
[1] George Borjas was the first to formalize the model of Roy in a mathematical sense and apply it to self-selection in immigration.
Additionally, assume there is a cost C associated with migrating from country 0 to country 1 and workers know all parameters and their own realization of e0 and e1.
The third, and final, element needed for this is the correlation between the wages in the two countries, ρ.
[2] This leads to the famous central result that the expected wage for immigrants depends on the selection mechanism, as shown in equation (1), where ϕ is the standard normal pdf and, like before, Φ is the standard normal cdf: While Borjas was the first to mathematically formalize the Roy model, it has guided thinking in other fields of research as well.
A famous example by James Heckman and Bo Honoré who study labor market participation using the Roy model, where the choice equation leads to the Heckman correction procedure.
[3] More generally, Heckman and Vytlacil propose the Roy model as an alternative to the LATE framework proposed by Joshua Angrist and Guido Imbens.