Jargon,
I agree with everything you write, but I think it makes an assumption, mainly that the manufacturers have no option but to make cigarettes, and the only variables at their command are how many to produce and how much to charge.
But AE assumes, given a free market, that they have a third option, closing down and going into another industry. The idea being that if Industry A is less profitable than Industry B, then people will move out of A and into B until profits equalize for both industries.
Here’s page 82 of HA:
Every firm and every branch of business is in the short run interested in
increased sales of its products. In the long run, however, there prevails a
tendency toward an equalization of returns in the various branches of
production. If demand for the products of a branch increases and raises
profits, more capital flows into it and the competition of the new enterprises
cuts down the profits.
Which means, as I understand it, that if any kind of tax is imposed on the cigarette industry, whether per pack or on profits, since it reduces profits in Industry A [=cigarettes], then some people will move on to Industry B [=anything else]. This means reduced supply of cigarettes, which means higher prices, which means the tax has been passed on to the consumer, albeit indirectly.
I think it is also implicit in the assumptions of every supply and demand curve that this will happen. When we look at a demand curve, the two axes are Amount Bought and Price Charged. This is accurate, because the buyer cares about the price, since it is the hit he has to take for buying his cigarette. But in a supply curve, strictly speaking, the two axes should be Amount Supplied and Profits Made, i.e the second axis should not be Price Charged, but Profits Made. The seller doesn’t care about the price per se. He cares about Profits. It’s only because we have to draw the supply and demand curves on one graph that we label one axis Price. But strictly speaking, Price only matters to the supplier in that the higher the price, the higher [presumably] the rate of profit. If a tax is imposed, no matter where, that reduces the rate of profit, and supply will be diminished.
In other words, if a given supplier is resolved to stay in the business, come what may, then your analysis will be correct. But a supply curve assumes that when profits are too low, for any reason, including a profit tax, some suppliers will drop out.