Price Discrimination allows a firm with market power to set different prices for the same good to different consumers. Optimal pricing for a price taking firm For a firm facing a horizontal demand curve, a price taking firm, the elasticity is infinite. Note that the higher the elasticity, the lower the markup of price over marginal cost.
Elasticity tends to be higher when there are many competitors and substitutes.
The lower the elasticity, the higher the markup. No, because a firm has to cover its fixed costs. It would not be in business if it were forced forced to price with no margin. Note that the low marginal cost producer produces more. Examples of price discrimination Coupons in the newspaper offer lower prices to consumers with lower opportunity cost of time.