Should civil servants (employees in the public sector) be allowed to bargain collectively? To answer this question, we construct a model of unionized mixed duopoly and examine the regulatory framework of public institutions, especially focusing on a wage regulation imposed on the public firm. The wage regulation turns out to yield critical welfare implications as it gives rise to two opposing strategic effects: the wage regulation intensifies downstream-market competition while it loosens upstream-market competition. The overall welfare effect is ambiguous, depending crucially on the degree of product differentiation between the firms. We also show that, in contrast to the popular belief, granting the right to bargain collectively to civil servants would not necessarily help them because they tend to demand excessively high wages when they are allowed to bargain collectively. Finally, we briefly discuss a new perspective on the role of profit motives in public institutions when the wages are determined endogenously.