One theory for what happens when the reward drops (i.e. no new bitcoins are rewareded anymore) is that the already high fees will further increase. If that is not the case, the competition will lower and Bitcoin will become unsafer because it is backed by less mining power.

For some blocks I checked a week ago the transaction fees amounted for approx. 1/3 of the reward. Like, 12.5btc of newly generated coins and 6btc of fees. Although, to compensate for the next reward halving the fees would need to go further up, and they already aren't acceptable for smaller transactions.