After gaining access to the Secret Auction on the S.S. Prime Treasure during Chapter 8, Tim and Pikachu will attend it with Emilia. Once in the auction room, you will need to scope out your competitors to see how high they are willing to bid on the items you are needing to go after. You will have up to 90 million dollars to spend, towards both the Detective's Cape and the Henry Brothers Recipe Book.
By talking with your fellow competitors, you will learn that you will have one primary opponent for each item you are intending to buy. When you're ready to participate in the Auction, simply talk to Emilia. For the Detective's Cape, your Main competitor will be willing to bid as high as 50 million for it; however, you should be able to buy it for around 48 million. For the Henry Brothers Recipe Book, your main competitor will be willing to bid as high as 40 million; however, you should be able to secure it for only about 38 million.