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The BLS is a branch of the Dept. of Labor. And the Dept. of Labor is headed by the Secretary of Labor who was appointed by Obama. So yes, the administration has everything to do with the CPI-U. I noticed that you used the term "inflation", which cannot be used interchangeably with "CPI".
As far as an administration having nothing to do with inflation... that's a bit off the topic at hand, but it really isn't correct either. There are many things that presidential administrations can do (or not do) that can have an impact on inflation. Bank bailouts is one example. Regulating (or failing to regulate) Wall-street is another. Stimulus packages is another. There are many others.
As far as the DOL being "politically neutral"... I worked in government for many years. Despite what anyone tells you, there is very little in government that is politically neutral. In the case of the CPI, there is a massive amount of money at stake. A lower CPI benefits government and corporations while reducing the standard-of-living of ordinary citizens. Furthermore, the CPI has a direct impact upon the federal budget. Example: A deflated CPI reduces Social Security and SSI payments, perhaps delaying that day of reckoning when the SS Trust Fund must redeem Treasury Securities in the fund to pay benefits. Another example: TIPS & I-Bond rates are tied to the CPI. An artificially deflated CPI is a way to reduce spending on the sly. There are massive conflicts-of-interest in play here involving massive sums of money. It would be quite naive to believe that when it comes to the determination of the CPI, there are (as you state) "no interests except accurate figures".
Your last question is more complex and too lengthy to address here, but I'll mention three changes implemented in the 90s after a lot of beating of drums by Alan Greenspan and his ilk: 1) calculating housing costs by "equivalent rent" 2) hedonic adjustments 3) substitution/geometric weighting And now we have the US understating the cost of living relative to the foreign CPI composite by a very significant amount. I'd suggest you look into this in more detail.
And, of course, there is also the anecdotal evidence of millions of people who buy things every day, which should not be so easily dismissed.
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