Factor price equalization beyond a "cubic" world

The importance of factor price equalization (FPE) is widely recognized in economics. The FPE theorem states that, absent any factor intensity reversal, factor prices are equal across countries with identical technologies and product mixes. In a two-factor-two-good-two-country Heckscher-Ohlin model this is equivalent to countries factor endowments being contained in the diversification cone defined by goods factor intensities. This paper identifies a condition, stated in terms of the allocation of factor endowments across countries relative to the demand for and the factor intensities of goods, that is necessary and sufficient for FPE in a world with arbitrary number of countries, goods and factors.