Carbocations genereally won't be the final product of a reaction because they aren't stable at all and exist only as short-lived reaction intermediates. Yes, carbocations can rearrange to more stable carbocations, but those more stable carbocations will eventually react with a nucleophile or perform an elimination reaction in order to fill the carbocation's octet.
However, I do agree that an E1 reaction would occur to form an alkene. Now, the question is, where is the preferred position for the double bond?