Price capped firms enjoy a large degree of pricing discretion, which may damage captive customers and have adverse effects on the development of competition when regulated firms also operate in competitive industries. We study two alternative regulatory approaches to limit such a discretion. The first one places a fixed upper limit to the prices charged in captive markets; the other constrains the captive prices relatively to the price asset in the more competitive markets. We refer to the former approach as the Absolute regime, and to the latter as the Relative regime. We analyse the effects on prices, competition, and welfare stemming from the two regimes in a simple model where the regulated firm faces competition by a competitive fringe in some of the markets it serves. We find that the Relative regime is not much more effective in protecting captive customers since captive prices may be identical under both regimes. Moreover, since it makes more costly to the incumbent regulated firm to reduce its competitive price, this is weakly higher than under the Absolute regime. However, this is also the reason why the Relative regime turns out to be more pro-competitive: the total output supplied by competitors and/or the number of firms entering the potentially competitive market might be enhanced under this rule. The effects on aggregate welfare are ambiguous. We provide some evidence that the Relative regime is more likely to positively affect consumers' surplus and social welfare the more efficient is the competitive fringe.