I see absolutely NO reason for us to continue making or using pennies. I think the US Mint should stop making them immediately, and announce that in 3 years pennies would no longer have value as a currency (but could still as a collectable), and thus should be exchanged for larger denominations of US currency. I'm not sure we shouldn't do the same with nickels too, as a 2nd step, maybe stop making those the same day the pennies are no longer currency.
Since US dollars and coins were established, inflation has increased their value 25-fold, so $1 in 1800 is worth $25 now (maybe more). Meaning the lowest US currency denomination in 1800 (penny) is worth 25 cents now. That supports the concept of getting rid of the penny, nickel, and even dime. However, I think also getting rid of the dime that quick might be too big of a change (no pun intended) for some people.
It costs about 3 cents for the US Mint to produce a penny, about 10 cents to produce a nickel, and about 5 cents to produce a dime. More reasons to stop making pennies and nickels, it cost more to make them than they are worth.
Some would argue that getting rid of the penny would harm the poor. Well yes, prices would have to be rounded, often up as opposed to down. But rounded down would happen often. Why are prices often like $1.99, $2.99, $399? Because it gives the appearance that they are significantly less than $2, $3, $400. Many retailers would rather risk making a fraction less per purchase, to not scare away one person from making the purchase (due to the appearance of a higher price). So, I think you would see some items that were say $2.99 become $2.95 instead of $3.00. If only 20% were rounded down & the other 80% were rounded up, then the overall cost across all purchases would be approximately unchanged.
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