We analyse whether oligopolistic firms which maximise a weighted sum of profits and a Corporate Social Responsibility (CSR) objective avoid more or less taxes than a profit-maximising company. CSR is modelled as a dependence of the firm's objective on output and its contribution to the provision of a public good. Our theoretical model shows that making one CSR element more important induces the firm to adhere less to the other and to reduce tax avoidance. Hence, simultaneously a substitutive and a complementary relationship between CSR and tax avoidance can be observed. Therefore, empirical approaches employing composite indicators of CSR will be unable to identify the link.