What were the reasons that the Prohibition in the United States was passed as a constitutional amendment, rather than a normal federal law, or a set of state laws?
Prohibition required a constitutional amendment, because the Federal government does not have the power to regulate intra-state commerce.
The majority of states and many localities had already banned the sale of alcohol. The progressive and women's suffrage movements saw banning alcohol as a way to improve living conditions for women and children and reduce the power of saloon-centric machine politics.
I believe this was to make the local State laws, that were a hodgepodge of enactments on a State level, more uniform and to bring the enforcement into the Federal realm. This was especially easy since the Temperance Movements had become politically powerful in the early 20th Century and in the election of 1917 the pro-Prohibitionists finally had a majority in Congress and moved things along. The actual text is:
Without the Volstead Act though the Amendment was very vague, the Volstead Act was more important as it was the one that set down enforcement of the ban. Note the text says nothing about consumption.
At the time the Constitution WAS seen as the enshrinement of laws, getting an Amendment gave weight to your cause and Federal enforcement. I don't have official sources on this but there are arguments that prior to the use of the Commerce Clause to give Federal enforcement powers an Amendment was the way to get National enforcement. Although looking more at varous sources the most common rationale given for a national-law is the Temperance Movement already had many States enacting Prohibitive Laws, but this would make it Federal and not allow Liquor interests to interfere locally and influence local politicians.
Prohibition existed in many localities around the country prior to the amendment, but the answer to your question seems to be because the powers that wished to enact prohibition had overwhelming public support. So, the group that wanted prohibition did it in part because they could.
To further flesh out some points of MichaelF's answer, prohibition was already starting to take hold around the country. According to Irving Fisher (a professor of economics who testified during congressional hearings on prohibition in April of 1926) in his book Prohibition at its Worst:
Professor Fisher notes that prohibition was essentially forced upon the large cities of the East Coast by the rest of the country. The Anti-Saloon League is cited as a particularly effective organization that saw prohibition through to its ultimate success.
It is also interesting to note, as the Howard McBain (a law professor at Columbia University) did in 1928, the sheer amount of popular support for prohibition:
Interestingly, the Prohibition Amendment invalidated all state and local regulations that dealt with prohibition because the Constitution became the supreme law in this area, and granted police power to the feds to enforce the amendment.
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