The opposite would be true with overshoots. The Bank studied variants of price-level targeting (PLT) extensively leading up to the renewal in 2011.16 This type of history-dependent framework could, in theory, make monetary policy more effective, by reducing the frequency of encounters with the lower bound for interest rates. And it would also make it easier to get up from that level.
Moreover, since lower- and middle-income households hold more 13 People in low-income households, especially the elderly, are more vulnerable to higher inflation  since they tend to rely more on nominal liquid assets like cash for transactions, which lose purchasing power when inflation is high.