the lag is that the cola adjustment is based on the inflation experienced in the last year. so you paid more first, then you get more income to adjust later. so at best there's a temporary cash flow problem.
the underadjustment comes in the way they calculate it. there are a number of reasons, one of which is the substitution adjustment. as prices for different product rise at different rates, people substitute an alternative product that has been rising at a slower rate over a product that has been rising at a higher rate. so the modern way to calculate inflation adjusts for this, giving more weight to the product that has been rising at a slower rate, which obviously lowers the end resulted calculated inflation rate.
there is certainly some merit to to making an adjustment for this. if you start substituting margarine for butter because of this and are really indifferent between the two products, then this adjustment makes sense. however, maybe you do care and see a big difference between butter and margarine, and this adjustment is punishing you for coping with the price fluctuations by effectively not reflecting the cost of switching to an inferior product due to inflation.
so in the end there is a slight underadjustment because of this. there other imperfections in the official inflation adjustments as well, largely motivated by a political desire to control government expenses without being obvious about it.
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