Linear Demands and Third-Degree Price Discrimination with Some Caveats

By focusing on the two intercepts — the price and quantity intercepts — of inverse linear demands, this note shows that compared to uniform pricing, third-degree price discrimination can be neutral. When all price intercepts of sub-markets' inverse demands are the same, not only will all the sub-markets be served under uniform pricing, but all prices before and after price discrimination will also be the same. In other words, all sub-markets sharing the same price intercept serves as a sufficient condition for no sub-market to be excluded. What is surprising is that having the same price intercept in all sub-markets is also both the necessary and sufficient condition for the price elasticity of demand in all the sub-markets to be the same at any given price, which implies all prices, outputs and welfare in all the sub-markets before and after the price discrimination are the same. However, when all quantity intercepts of sub-markets' inverse demands are the same, even though the price elasticity of demand in all the sub-markets are the same at any given quantity, price discrimination now is no longer neutral