We analyse a Cournot model of oligopoly in an industry which is subject to indirect taxation. We consider value added as well as excise taxes. We allow for various types of demand and cost conditions, as well as cost differences across firms. As the tax is raised, two types of outcomes can prevail. On the one hand, the net price can increase, which is shown to lead to an increase in market share of high cost firms. On the other hand, the net price can fall, in which case efficiency is rewarded in that low cost firms gain market share. This latter outcome, which accords with intuition, will obtain in most instances. It is more likely to occur (i) with VAT rather than with an excise tax, (ii) with concave rather than convex demand and (iii) with decreasing rather than increasing marginal cost. Finally, we show that an increase in VAT can lead to higher profits for low cost firms, provided cost differences across firms are large enough.