Recently, a home delivery shaving company, "Harry's Razors" claimed it can sell higher quality razors at a lower price because it owned the factory that makes the razors. What conditions must hold for this assertion to be true?

The reason is no Rental cost (No Fixed cost)

Explanation:

The company don't need to make additional sales to earn additional contributions to cover the fixed cost to remain above the no profit & loss position. So the benefit that it can generate is making higher sales by keeping its price lower than its competitors to win the market share. Of-course customers want same quality product at the lower price and whoever is selling at their desired requirements wins the customer. So this lowering of cost will add value to their business.