I agree wholeheartedly that incomparability is a big problem for the CPI. I conclude that the CPI, and thus indexed investments and pensions and various "real" estimates of GDP and productivity are heavily error-prone and bias-prone. It is doubtful, however, that these problems "systematically make us overestimate inflation." Some other proxy measures of inflation suggest the opposite is the case.
If one may generalize, technological progress over the last five years has increased value and decreased prices faster than other factors (e.g. rising regulations, rising money supply, and other political problems) have increased them in arguably luxury or zero-sum areas like entertainment and office bureaucracy (e.g. the oft cited rapidly falling costs and increased capabilities computers and associated products). But political factors have (to various degrees depending on your unit of measure) increased prices faster than rising technological productivity in traditional necessity areas such as food, housing, and energy. Furthermore, the problems Vermillion points out with choosing units of measurement (gallons of gas or miles driven?) are multiplied or exponentiated when it comes to comparing the value of gadgets that play music or movies or allow us to spiel on the Internet.
What is a luxury and what is a necessity is itself quite debatable, and most categories are part luxury and part necessity. Prices of underlying real estate are mostly necessity (we need to be within commuting distance of our jobs), but the prices of houses themselves (especially after the big increases in square footage) are increasingly luxury. I take a long view and a global view, under which agriculture and its inputs (over the last century including oil) has been the dominantly important industry for about ten millenia and is still a crucial industry. Just because water is far cheaper than diamonds, or flour is cheaper than cable TV, to such an extent that we can spend more on the latter than the former, doesn't mean water should be (or in fact subjectively is) weighed far less than diamonds or bread less than a month's subscription to cable. Even though Americans probably spend more on cable TV than on food calories (or at least more than we could spend if we didn't insist on ingredients and recipes and modes of delivery until recently considered luxuries), there would be far more suffering if the price of food calories went up by a factor of ten than if the price of cable TV did so. People once found and will find other ways to entertain themselves without cable, but we can't do without calories. (Many of us Americans would be healthier with a few less calories, but that goes for cable TV too).
There is a vast amount of interconnection between the different factors. Fuel mileage and thus the productivity of a mile of commuting may be up, but urban and suburban real estate prices and the associated commute distances probably have more than offset this factor in the ten years (not to mention value lost from extra time spent commuting instead of more enjoyable or productive activities).
We could just leap over these variously biased and error-prone indexes if we could observe how people value a dollar tomorrow versus a dollar today -- time preferences. Although there are other influences on time preferences, I hypothesize that the dominant influence on time preferences overall for users of a currency is the expectation of inflation in that currency. Even if we can't accurately measure an overall rate of inflation, due to the problems Vermillion describes, we may be able to find a good proxy measure for it if we can find a good proxy measure for the overall time preference of currency users that doesn't involve first computing a "real" rate of interest.
People sense price rises in the things they value, and thus come to their own estimates of past inflation, i.e. rises in the costs of their own budget due to external causes, and expectations of future inflation in the prices of things they value. I argue that this subjective inflation, and the expectation of future subjective inflation, are what a money issuer really ought to target: the issuer should observe proxy measures that most closely approximate overall subjective inflation expectation, and expectations of same, and expand or contract the money supply, via discount rates or otherwise, accordingly.
For the purposes of estimating subjective inflation, I don't think interest rates work as a proxy measure. Here's why, giving as an example today's U.S. dollar interest rates. If dollar inflation is lower than the CPI indicates, as Vermillion argues, then real interest rates are positive, showing that folks prefer to spend a dollar on todays's technologically inferior and more pricier goods over a dollar on tommorrow's superior and less expensive goods. But if consumers have such "Luddite" or risk averse commodity preferences, then inflation may be much higher than Vermillion or the CPI suggest, meaning real interest rates aren't really positive. Using interest rates as an indicator of time preferences introduces a circularity when arguing for or against such preferences for the purposes of properly weighing a basket of goods.
Another problem of using interest rates as a proxy measure for inflation expectations is that it is biased by investor behavior and in particular foreign inflows or outflows of dollars. If there is a larger dollar supply in the world returning home to the U.S. and chasing fewer investment opportunities, that will drive down interest rates despite subjective inflation expectations. If the economy using dollars is shrinking (either because the economy itself is shrinking or because some other currency is gaining favor, supplanting the use of dollars) and the dollar supply fails to shrink along with it, the result is both inflation and low interest rates.
A better measure of consumer time preferences, and thus per the argument above consumer inflation expectations, may be to look at savings rates. Savings rates here in the U.S. strongly suggest that people aren't saving up to buy computers or audio or video players two years from now because these gadgets will be very important to us and more functional and cheaper than today. Rather, increasingly U.S. consumer budgets are going into goods and services that have been rising much faster than the CPI, especially housing and education. The negative savings rate strongly suggests, as do the price trajectories of the most objectively comparable goods (commodities and housing), that the CPI has for the last five years been understating inflation, and that interest rates in the U.S. are too low compared to this inflation to induce Americans at least to save here. (Foreigners are investing vast sums here, but that's another story). If Americans have been spending as if their dollars will be worth less tomorrow than today, perhaps we should respect that perception, technological progress in entertainment and office bureaucracy notwithstanding.
The correlation between time preferences observed in U.S. consumer savings rates and a "consumer price index" far more heavily weighted towards necessities than today's CPI suggests either that (a) goods on the luxury end where productivity has been increasing tend to be "zero sum" status goods, or entertainment goods that only temporarily make us happier, so that the productivity increases won't actually make us much happier in the future, and thus don't effect much our time preferences compared to "positive sum" and "we'd be far less happy without them" necessities, or (b) that time preferences are very risk averse, and thus strongly weigh future water over future diamonds, despite our spending more on the latter in the present, or (c) a combination of both these factors. In both cases there are rational arguments to make for such preferences and in any case we should respect them and choose the units and weights of our price indexes consistent with them.
Another reason to believe the U.S. CPI, at least, underestimates inflation is the intuitive argument that the organization that created the formula for the CPI (i.e. the federal government) is also the organization that stands to pay out vastly greater sums on i-bonds, Social Security, and so on should the inflation rates it reports go up. Even small biases in reporting inflation today lower the federal government's future liabilities by vast sums. It is also hard to recognize and correct any bias, since due to incomparability CPI units and weights could be debated ad infinitum. There is a wide lattitude of government choice among CPI weights and measures that cannot be proven to be right or wrong. Why would they not choose the values that best meet their budget preferences? However, to make this argument correctly one should invoke public choice or similiar theory rather than treating government as if it has the incentives of an individual; such an analysis is beyond the scope of this post.
Whether or not we can find other proxy measures for inflation, Vermillion's basic point about incomparability and the dubious nature of the units and weights used in price indexes is correct. Unless there is a good substitute proxy measure for subjective time preferences and thus subjective inflation, (perhaps the savings rate as suggested above), we are stuck with weighted indexes. But measures based on these weighted indexes, including measuring the "real" values of GDP or of particular sectors or goods, measuring "inflation," and measuring "productivity" -- all these suffer from the incomparability of the goods and services in the index, and thus are much more error-prone, much more bias-prone, and indeed, much more prone to badly incentivized manipulation and fraudulent accounting than the usual economic pundits and advisors have been assuming.