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The word comes in two flavors, and I'd like to focus on the second; but both derive from the Latin debere, meaning to owe--hence also a "debit" to an account. In most common use, the word has to do with an obligation, contractual or moral; it is alleged for example that by some mysterious means everyone has incurred a duty to serve one's country, as in JFK's infamous inaugural speech. The word "country" itself is hard to define, when one thinks about it, and the source of any such duty is even more mysterious; his speech would have been much more truthful, if less appealing, had he said "Ask not what my government can do for you, ask what you can do for my government."
Here, though let's take a look at the other meaning of the word, a "tax or fee in imports, exports, etc." which dictionary.com  says was first found in 1474, while "'duty-free' as a noun is attested from 1958." Wow, five centuries without freedom from duty. Allegedly, those moving goods across borders "owe" a duty to pay the governments policing them a sum of money.
Among those today perceptive enough to find the smell of rotting fish in federal tax law, there is a sizeable subset who nonetheless accept that such duties (also known as excise taxes) are legitimate and proper. They point to Article I, Section 8 of the US Constitution: "Congress shall have power to lay and collect Taxes, Duties, Imposts and Excises," though it insists that such duties "shall be uniform" throughout the United States , so contrasting them with direct taxes, which Section 2 says "shall be apportioned." Big difference.
Those folk have it half right; import duties are certainly constitutional, but certainly not proper; and of course we need to agree what "proper" means. What I mean by it is something consistent with the self-ownership axiom, the bedrock of a free society. The notion that anyone can delegate any power to a Congress or anyone else to impose a duty on what a free person chooses to bring across a border defined by that Congress is anathema to that standard.
I've crossed many national borders, many times, and a couple of times I was pulled aside for baggage examination. Both of those were when I re-entered the UK and looked a lot younger (and perhaps even uglier). On the first occasion I admitted to having purchased some movie film in Paris, and the customs creep began to foam at the mouth at the prospect of viewing pornography in my briefcase; in fact, it just demonstrated the wonders of interactive computer CRT terminals, to amaze and convince an IBM prospect of mine. The creep's disappointment was palpable. The second was when I tried to sneak in a box of cigars from Amsterdam for a friend, and I must have looked guilty (or else the tobacco could be sniffed). Otherwise, customs agents seem a bored and sometimes cheerful bunch, doing a useless, deadly boring job with dedication to fit. French ones are, or were, the most bored and therefore the least troublesome.
In the case of commercial quantities of saleable merchandise, it's a different story. You import a container load of Stuff, and the border police not only X-ray the Stuff, they charge you money for their unwanted service--even if the Stuff declared is not subject to duty! Right; ordinarily travelers at the airport or road border crossing may have their bags opened without charge, but at the dockside the importer has to pay $250 for the privilege! And if the agent should have the whim that day to do a full inspection of the contents, the fee is three times as much! It's rather like China, where the family of an executed criminal has to pay for the bullet that shot him to death--except that nobody imputes guilt to the importer. This extra deadweight $500 cost is imposed by the government parasite in charge, and presumably depends on whether he got out of bed that day on the right side or the left; and must ultimately be paid by you and me. Vendors have many contingencies and hazards to assess and to build in to the prices they charge, whether their goods are manufactured in Dallas or imported from Dar-es-Salaam, and need that sort of extra one like a hole in the head.
Our Constitutionalist friends hold that an import tax is the fairest way to meet the expenses of government. Such a duty means that the burden is borne only by those choosing to import foreign goods, which is a voluntary and therefore avoidable act; on the premise that America was or should soon become self-sufficient back in the 18th Century, one can understand that such sophomoric economics might be popular. More fully, the Founders held three premises, whether stated or not:
1. A (federal) government was needed
2. Its expenses had to be paid
3. Import duties were the least unpopular way to pay them.
If the first of those had been correct, the second and third might have been too; but it's not, so all three are bogus. In fact, even if (arguendo) there had been some need of a federal union, its expenses could have been met simply by a grant of money from the founding States, of which there were already 13 too many; since they were supposedly delegating some of their powers to the central one, they could have taken the money previously used to pay for the discharge of those powers and handed it over to the Feds. That would even have served well the interests of the State governments, for they could ever since have held over the Feds' head the threat of cutting off those funds, should extra powers be grabbed without approval. But we know how eager government people are to give up money they have their claws on, so that didn't happen.
The third premise was correct as to popularity, but no further. The notion that somehow a free market should end at a national border is complete nonsense. If--despite the obvious additional costs of shipping--it is to the (domestic) buyer's advantage to import something rather than to buy it from down the road, everybody wins: the importer's customers (who get cheaper goods), the importer (who gets business) and of course the foreign producer, who gets customers. Even the producer down the road benefits, for the market is sending him a clear signal that he is inefficient and needs, in order to survive, to get his house in order. Assuming real money is employed, if nobody exports anything except gold to pay his bill, there will soon be a domestic shortage of gold, which will deflate the price of domestic products, and so remove the foreigner's advantage and stimulate domestic production. If, conversely, some other products are exported, to the extent of zeroizing the net flow of gold, even more players win because domestic producers of that other stuff will find new customers, and foreign buyers will be grateful to low-cost American producers.
It's not surprising that one society should produce some things more efficiently than others, even on an ongoing basis. It's really difficult to grow avocados in New Hampshire or run a ski lodge in Saudi Arabia, and different skill mixes made it at least temporarily more efficient to manufacture trucks in Michigan than in Guatemala, and so on; everybody wins when everyone produces what he does best, and that's as true internationally just as it is within a society. If it were otherwise--if there were some true advantage from erecting artificial barriers to trade over large distances, one would wonder why there are not similar advantages to restricting trade over short ones, so that for example there are customs duties payable at every county or city line. I hope that doesn't give ideas to any knee-jerk, pro-government reader.
That fiction--that domestic trade needs protecting from foreign competition--must have played some part in Premise #3 above, and certainly plays a part today. I feel nauseous every time I see a captain of some industry solemnly assure his TV audience that he "believes in free trade, but"--not in this case, not at this time, because some naughty foreign government is allegedly subsidizing the X trade and "dumping" competing products "too cheaply" at his customers' doors. One wonders in what currency that subsidy is paid, and for how long it can continue, and why he hasn't run his domestic factory in such a prudent way as to withstand such short-term competition. "Protectionism" doesn't protect, and never has; it may shield the idle or incompetent for a while, but in the medium and long terms it destroys an economy.
A spectacular example is the US auto trade, in which for decades employees were paid far more than their labor was worth, compared with other countries, thanks to trade union pressure, government interference, and management weakness; and as a result, their products became ever harder to export and now the companies have become basket cases. American car workers are very fortunate that foreign makers, more resistant to union anti-efficiency pressure, often chose to take advantage of high demand here by building factories within our shores.
They were probably too dumb to understand that, back in 1783, but "protection" is what the founders set out to do, when they authorized Congress to impose duties on imports. They tied a ball and chain around the country's leg, just as surely as they allowed them to be fastened to the ankles of imported slaves. (What, I wonder, was the duty on a slave?)
Now, in a free society there will of course be one duty to fulfill; the duty to honor a contract, freely signed--and ultimately that will be a "duty" to oneself, because contract-breakers will have short careers. But of this other kind of "duty," there will be no trace; for there will be no border, over which useful goods might pass, nor any duty-collector with guns, to patrol it and impede their free passage. And there will of course be no government with need of financing, to provide an excuse to impose such duties. Goods will therefore flow exactly as their producers and purchasers desire, and so everybody will win.