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I am interested to know how one would expect a change in the quality of a product to affect its price elasticity of demand. Two example will illustrate my confusion:

Example 1

Suppose a producer is selling tap water at price P with quantity demanded Q. Let us call this good 'Beverage v1'. The producer changes the recipe; adding some grape juice and carbon dioxide to make Champagne. The producer continues to sell at price P but now with demand Q2>Q (in the short-run, which is probably not an equilibrium). Let us call this good 'Beverage v2'.

We would, I think, expect the price elasticity of demand for Beverage v2 to be lower than that of Beverage v1. The common intuition is that 'essential' goods have lower PEoD than 'luxury' goods.

Example 2

Now suppose a producer is selling a medicine ('Medicine v1') which treats the symptoms of an illness pretty well. It is the only medicine for this illness so patients must choose between going untreated (feeling pretty awful) and buying the drug, which makes them feel 50% better than going untreated. The medicine is sold at price P with quantity demanded Q. The producer then makes some changes and now the medicine makes patients feel 100% better. It is not quite a cure, as patients have to keep taking the medicine, but it is pretty close. Let us call this 'Medicine v2'. The producer continues to sell at price P but now with quantity demanded Q2>Q (in the short-run, which is probably not an equilibrium). Bear in mind that Medicine v1 is no longer available and that there are no other alternatives available either (except going untreated).

This seems to be similar to the beverage example above, but I feel as if the intuition is less clear in this case that PEoD for Medicine v2 would be higher than for Medicine v1. Is feeling 100% better more of a luxury than feeling 50% better? It is certainly better, but I am not certain that is therefore less essential. What do you think?

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It seems that your confusion is more about classifying goods into necessities vs. luxuries. In economics, the criterion for such classification is to look at whether the income (not price) elasticity of demand is "high" or not. If a good has a high (and positive) income elasticity, then it's a luxury; if it has a low (but still positive) income elasticity, it's a necessity. What counts as "high" or "low" is an empirical matter. Investopedia suggests $1$ as a threshold (high if $>1$ and low if $<1$).


Coming back to your question about price elasticity and quality improvement. The answer is: it depends on how you model the demand curve of the good. Recall that the price elasticity of demand is given by \begin{equation} -\frac{\mathrm dQ}{\mathrm dP}\frac{P}{Q}=-\frac{1}{\text{slope}}\frac{P}{Q} \end{equation} where the slope is evaluated at the point $(Q,P)$. The quality improvement shifts demand curve to the right (since consumer is getting more value for each dollar spent, so they are willing to pay more for each unit or buy more units at the same price). If price remains unchanged (as you assumed), then quantity demanded must increase, say from the original $Q$ to a higher $Q'$.

If the slope of demand stays the same at $(Q',P)$, then price elasticity would fall according to the formula. If demand becomes much flatter at $(Q',P)$ (with a magnitude greater than the increase in quantity), then price elasticity would increase instead.

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