Consider CPI and PPI, Consumer Price Index and Producer Price Index respectively. Note that many book correct inflation using the above measures. However, I doubt inflation corrected by these values tells the true or whole story. Consider the following case. According to a microecon textbook I am reading, the real price of copper has been stable over the past decades despite the fact that demand soared over these decades. Let us assume that the real price is computed using PPI. If the PPI has increased siginificantly more than CPI in terms of percentage increase comparing to their common base year, then although PPI-corrected real price is stable for copper, it's relative price to basic commodities has soared. In this case, we note that PPI is not useful in assesing whether the real price of copper had risen. Rather, GDP deflator seems to be a better option. Am I correct? Any suggestion is helpful.
You are right that if the inflation rate is calcualted using PPI, then PPI-adjusted price for copper didn't not change, i.e. the real prie for copper did not change. However, we would also consider the real income and real prices of the necessities in determinging wether copper has increased in value reative to income and daily necessities. Since this time when we compute the real income, we would be using the PPI calculated inflation rate as well. If nomial income rose by $5\%$ by the end of the fifth year while the price of copper along with other PPI goods rose by $15\%$ at that time, we would get a real income that is lower than the real income five years ago, which then would be apparent that the value of copper, or the relative value of copper, has increased.