> That makes me think they have control over the price
Well yeah, registrar is a business, of course you have some say on the price you sell your domains... But there's a base price that is set by the registry behind the registrars
As an example, the .com is sold by Verisign (the registry) around 9 USD for the create/renew, but around 50 USD for the restore, depending on the negotiated price between the registrar and Verisign. So basically no registrar will allow you to restore a domain in redemption period for less than 50 USD.