In particular, if the equilibrium real wage falls, adjustment will be more painful if nominal wages fall and less painful if the price level rises. Menu costs probably pretty minor. Prices do decline in many cases. Money illusion, in contrast, seems to be a major force. Nominal wages hardly ever decline.
People eventually catch on about inflation, but it often takes a while.
Keynesian AS Both menu costs and money illusion change the shape of the AS curve in the short-run. Instead of vertical, it becomes upward-sloping. With menu costs, this means that if demand increases, some firms will increase output, while others increase prices in spite of menu costs. When AD increases, then, both output and the price level rise. With money illusion, if the price level rises, reaching the equilibrium wage is less likely to require nominal wage cuts.