In these lectures, the professor shows that in a Cournot Duopoly, firms will supply a total market quantity of identical goods between that of a monopoly and that of a perfect competition, while in a Bertrand Duopoly, the firms will undercut each others' prices until the market price of the good is equal to its production cost.

The professor makes a big point out of the fact that this result should be irksome, as the Cournot and Bertrand cases are two presentations of the same duopoly. The resolution is left for homework, but it involves the introduction of the Linear City Model to make more realistic assumptions about product differentiation than the initial assumption that they are identical.

However, the reason this is irksome to me is not that it fails to match empirical observations, but that it seems to fail more fundamentally. We generally expect price and quantity to correlate inversely, and while this relation fails for certain types of goods, it isn't clear that this should depend on whether competitors' goods are differentiated or identical.

How is this paradox resolved, even for the case when goods are identical and without introducing new pieces to obfuscate the paradox?

• Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. Commented Aug 18, 2022 at 10:50

It makes complete sense that if price of close substitute falls that will negatively affect demand for the first good. For example, if we are in market for cars it makes sense that demand for cars depends on price of other alternatives such as public transportation. If a price of public transportation drops we would expect many people to switch from buying cars to traveling with public transport (the exact number of people would depend on what the cross-price elasticity of demand is $$\frac{\partial Q_c}{\partial p_p} \frac{p_p}{Q_c}$$ where subscript c stands for car and p for public transport).
Furthermore, it makes complete sense this effect will be greater for perfect substitutes since cross-price elasticity for perfect substitutes is infinite (whereas for imperfect substitute it will be finite number). To understand why cross-price elasticity is infinite consider an example of perfect substitute to a 1 liter bottle of milk. What is a perfect substitute to 1 liter bottle of milk? Another 1 litter bottle of milk! Now suppose we conduct an experiment, someone offers you two absolutely identical bottles of milk, one bottle of milk will be offered at price \$10 second at price \$5. Would you ever in such experiment buy \$10 milk? Nobody would just waste their money for no reason. This will force the producer of second milk to either lower prices or they will never get any customers. However, suppose that now we allow products to be differentiated and one bottle of milk has chocolate milk and another bottle regular milk. Now the milks are no longer perfect substitutes (but still they are substitutes, they both quench thirst and taste somewhat similar they are just no longer perfect substitutes), and now actually some people who really love chocolate milk might be willing to pay \$10 for the bottle of chocolate milk even if you offer them alternative of regular milk that sells just for \\$5.