I will answer only from the numeric side. Of course, if someone already knows they have a shorter life expectancy or they NEED the money to live off of, the question would not apply to them.My dad is taking Social Security at 62, why would anyone not?
The SS monthly benefit is calculated off of the worker's salary history. Once the worker "retires" the numbers "freeze" except for annual COLA, which I'll address in a minute. The calculated amount is for their Full Retirement Age (FRA), which appears to be 67 in this case. The monthly benefit will increase 8% (guaranteed!) for each year the person delays taking SS and decrease a little less than that for each year they start collecting early. The increase/decrease is due to collecting for fewer/more months, so it is actuarially the same. But actuarial tables apply to a group of people--the general population, in this case, not to individuals. For example, women live longer than men overall, but the calculations are the same for them. A healthy person or someone whose family tends to have longer longevity would be calculated the same as for an unhealthy person, although they will tend to live longer overall. A higher income person would tend to live longer than a poorer person because she has access to better health care and likely knows more about how to stay healthy. These factors should be considered since they show whether a person's chances of living (and collecting SS) longer show they are "average" or not.
Each year, SS calculates the COLA. (You may disagree with it, but let's just say there is a COLA. For 2016, the COLA was 0% because there was very low inflation. For 2017, it is 0.3%.) The COLA is applied to not only those who are receiving SS, but to everyone over 62 who is waiting to start collecting it. It will show up in a current statement of how much you can expect to receive starting at FRA and at the lowest and highest age. If you compare a current statement to one a few years old (both for someone over age 62), you will see the difference. You will not see the difference in a year where the COLA was 0%.
Since the cost of living will impact both your living expenses and the monthly SS benefit pretty much the same, it is basically a "wash". So I wouldn't even bother with calculations using it.
1. Where else can you get a guaranteed yearly 8% increase in your "investment" (up to age 70)?
2. You will be collecting this for the rest of your life, and probably want to maximize your income in your final years, especially if you think you will live longer than average. Your chance to increase your income streams at age 80 or 90 is very limited, unless you take lots of risk. SS is relatively risk-free. (People have been saying that SS won't be there when they retire for 50 years, but when the program needs financial adjustments, it has been adjusted.)
3. If the person has tax-deferred accounts and has calculated what their RMD will be at age 70 and estimated their taxes then, they may see that they have a few years of low/no income tax, then many years of an extremely high income tax after age 70 due to SS and RMDs. (When calculating the RMDs, take into account that the account will grow between now and when RMDs are started. The account value (and RMDs) will continue to grow even after that as long as the RMD is less than the growth of the account.) It is often beneficial, in this case, to delay SS to age 70 and do Roth conversions during the low income tax years, so that the tax on the Roth conversions is lower than the tax on the future RMDs. This will also bring down the size of the after-70 RMDs. If the income is still needed, some of the Roth can be withdrawn tax-free instead. (There are tons of threads on this.)
4. If a widowed person remarries, he/she will want to know that their new spouse has a higher monthly income for as long as the spouse lives. (Don't laugh, it happens. My dad was widowed when he was about 60. He remarried several years later but was widowed again after 10 years. Then he remarried a third time and they are both still going strong--as well as someone in his 90s can be.)