This paper investigates the effects of price discrimination on prices, profits and consumer surplus, when one or more competing firms can use consumers' private information to price discriminate and consumers can pay a privacy cost to avoid it. While a monopolist always benefits from higher privacy costs, this is not true in the competing duopoly case. In this last case, firms' individual profits are decreasing while consumer surplus is increasing in the privacy cost. Finally, under competition, we show that the optimal selling strategy for the owner of consumer data consists in dealing exclusively with one firm in order to create maximal competition between the winner and the loser of data. This brings ineficiencies, and we show that policy makers should concentrate their attention on exclusivity deals rather than making it easier for consumers to protect their privacy.