Output-based allocations (OBAs) are typically used in emission trading systems (ETS) with a fixed cap to mitigate leakage in sectors at risk. Recent work has shown they may also be welfare enhancing in markets subject to supply and demand shocks by introducing some flexibility in the total cap, resulting in a carbon price closer to marginal damage. We extend previous work to simultaneously include both leakage and volatility. We study how OBA permits can be implemented under an overall cap that may change with the level of production in contrast with a design that deducts OBA permits from the overall permit allocation as is the current practice in the EU-ETS and California. We show that introducing OBA permits while keeping the overall cap fixed would only increase price fluctuations and induce severe welfare losses to non-OBA sectors.