In a market with perfectly inelastic demand, how does the tax burden change between consumers and producers?

In a market with perfectly inelastic demand, consumers will bear the entire tax burden because they will buy the same amount of the product regardless of its price. This means that even if the price increases due to the tax, consumers will still need to purchase the same quantity since they cannot do without the good. For example, if a tax is added to life-saving medicine, people will continue to buy it even if the price goes up, so they end up paying the full amount of the tax. Producers, on the other hand, do not lose any sales from the tax because the demand does not change; they can pass the entire tax cost onto consumers. Therefore, in such a market, the tax does not affect the quantity sold but instead increases the price consumers pay.