We consider two privacy measures, local differential privacy and binary maximal leakage, simultaneously. We study the subsequent utility-privacy trade-off for two utility problems: information preservation and f-divergence maximization. We present new achievability results for both problems over all mechanisms that satisfy the combined privacy measure and compare this with the optimal utility over all ϵ-LDP mechanisms. In particular, we study the binary mechanism with erasure and the randomized response mechanism with erasure. Finally, we show that singular mechanisms are dominating for binary maximal leakage.