Market research shows that there are two types of consumers for iPhone Pro (Pro) and iPhone SE (SE). The iPhone SE has similar functionalities as the iPhone Pro but has lower specifications (e.g., a smaller screen size). Assume both types of iPhones cost 300 dollars to produce per unit. Both consumer types, h and l, get utility m if they have m dollars and no iPhones: uh(0, m) = ul(0, m) = m. High type h consumers with m dollars (after paying for an iPhone) get utility uh(Pro, m) = 1500 + m and uh(SE, m) = 800 + m, where Pro and SE denote have one iPhone Pro and SE, respectively. Low types l get utility ul(Pro, m) = 600 + m and ul(SE, m) = 500 + m. Apple knows that there are one million consumers of high type and 2 million consumers of low type. Once Apple announces a price, it must serve all consumers who demand its product at that price. The consumers start with enough money to buy an iPhone if it raises their utility. Each consumer demands at most one iPhone.

Suppose Apple offers iPhone Pro at price ph and iPhone SE at price pl . Show that no prices are such that high types buy iPhone SE and low types buy iPhone Pro.