How to distinguish between a good inflation and a bad inflation

Inflation can be put into two categories based on its causes: demand-pull inflation, cost-push inflation. However, this does not necessarily help to distinguish between a good inflation and a bad inflation. Usually, we distinguish based on the degree of volatility, we see a moderate inflation as a good inflation and a volatile inflation (hyperinflation) as a bad inflation.

I think we should find a way to distinguish a bad inflation and a good inflation based on its causes, in order to prevent further damage from a bad inflation when we can see a sign of it.

Both demand-pull inflation and cost-push inflation can be a bad inflation; however, we usually see that cost-push inflation is rather a bad inflation, as we expect a healthy economy tends to lower the costs of production due to the improvement of efficiency of resource allocation.

I see bad inflations always happen when the supply side loses its ability to meet the demand side due to the lack of production factors. Therefore, some cost-push inflations can be eased automatically by improving efficiency and productivity, when production factors are available, so they are not long term problems that need us to worry too much. On the other hands, many catastrophic hyperinflations are caused by the lack of production factors. For example, Germany after the first World War suffered a period of hyperinflation because many factories were destroyed and many men were killed in the War and Germany lost sufficient production factors to meet the domestic demand during its post-war period. Such inflation problem often requires a structural reform and help from the outside world.

Overall, we do not need to worry too much about inflation due to higher demand, higher costs due to bad management or inefficiency, the inflation that we need to worry is the one caused by the lack of production factors.