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A tax-exempt savings account is just like a savings account at a bank, except that any money put into it isn't taxed, and the money can be spent only for specific, eligible expenses.
Tax exemption is like an itemized deduction except better, because you get the tax savings immediately. Deductions, as noted in the linked article, generally can be taken only after the expenses exceed a specific amount. Tax-exempt savings take effect immediately.
Some negatives of tax-exempt savings are:
There are many different types of savings plans, so they'll just be listed, not described:
Retirement savings: IRA, Roth IRA, SIMPLE IRA, 401(k), 403(b), 457
When you go to the drug store, you get a receipt with a little extra info about "Flexible Spending Account (FSA)" expenses, for specific items. You save these receipts, and total the FSA-eligible amounts, and find that you spend $300 a year on this stuff.
Your employer offers an FSA plan, so you sign up and start putting $20 per month into the account. You want to keep the amount below expected expenses, because FSAs have a "use it or lose it" rule, where you lose the money if you don't spend it.
So, during the year, you put in $240, and spend the whole $240. (You spend it by using an ATM card to pay for expenses at the drug store.) How much do you save?
It depends on your income tax bracket. Suppose you pay income taxes at the 20% rate. The question to ask is, "what would you have to earn to pay for $240 of expenses?".
The answer is $300.
So, instead of having to earn $300, you only had to earn $240. That's a $60 savings.
The savings changes drastically if you're in a higher tax bracket (wealthier people benefit more), and if you have a chronic illness.
If you have $4,000 of annual expenses, and are in the 25% tax bracket, your annual savings is around $1,300.
By and large, tax-exempt savings work if you're a middle income person (earning over $30,000) in a job with a fairly large employer. You'll have money to save, skills to deal with paperwork or access to someone who can help, and access to all the plans.
These plans are a big benefit to wealthy people, as well, because they can act like long-term insurance plans. You can save money for years, and spend it when you're old.
They really don't work out for poor people, because poor people generally lack savings, and need their savings to be flexible to deal with emergencies. They also tend to not have access to the plans, if they work in small businesses.
As you may have guessed, these savings plans are favored by Republicans. So if you're poor and a Republican, and voted that way in the past, consider this when you're voting next time. These savings plans end up costing the government tax revenue that could be used to pay for social services that everyone could use.