The small sample performance of the bivariate two-sample Wilcoxon type rank sum testWis studied by Monte Carlo evaluation of power under shift alternatives in the bivariate normal distribution. Two types of alternatives are considered: (a) shift in only one coordinate and (b) equal amounts of shifts in both coordinates. The estimated power of theWtest is compared with the power of Hotelling'sT2test. Interestingly, it is found that the empirical power ofWsubstantially exceeds the power ofT2for some normal shift alternatives although the latter is the uniformly most powerful invariant test for the problem.