Similarity of endowments and the factor price equalization condition

We give a geometric interpretation of the lens condition, proposed by Deardorff as a shortcut for checking the factor price equalization (FPE) condition. We identify the conditions under which the lens condition implies the HE condition. If the FPE zone is not a neighborhood of the diagonal allocations, however, the lens condition is irrelevant despite the implication since the FPE condition (hence the lens condition) is unlikely to be satisfied in that case. We give precise conditions under which the lens condition is equivalent to the HE condition and simultaneously, the FPE zone is a neighborhood of the diagonal allocations.