There would be 'two' difficulty levels, one exactly half of the other. The highest difficulty level would adjust until the block generation rate was every 10 minutes regardless of the mining reward.
If you accepted half of award, you would be able to submit a block at the 50% difficulty level. If everyone moved to half the block reward then you would still get 1 block every 10 minutes at what ever difficultly level exists based upon competition of miners for 12.5 vs 25.
There is no point in doing that, everything would be exactly the same - miners would make as much money and the same amount of hashing would be done.
The rate of block production would be the same, every 10 minutes, but some percent of the miners are working for 'half pay' and thus there would be less mined. These miners work for half pay because they make more money via reduced devaluation of their current holdings due to inflation. This is 'intangible' but still real. Everything else being 'equal' the total hash power would be the same and all you would do is change the distribution of wealth... except... the miners working for half-fees are undermining the profits of those working at higher fees because the lower-fee miners have an alternative means of making money indirectly (reduced inflation) that is not available to the higher fee miners. In fact, if you own enough Bitcoin and have a small enough hashing power, you would make more money by mining for 'free' than mining in a pool or solo.
Lets work this math out: Lets assume I own $1 Million in BTC and the money supply is $1 Billion in BTC growing at 10% per year. This means that in one year (all else being equal), I have lost $100,000 in inflation paid to miners. If I want to prevent this loss I would have to invest in 0.1% of the hashing power of the network so that my earnings from mining offset my losses due to inflation.
Now if I had the option of only investing 0.05% of the hashing power to earn enough BTC to offset inflation, then I am still maintaining my position and have profited in the process by having to purchase less hashing power! The network would also have a slightly lower hash rate even thought he BTC/difficulty yield would be unchanged.
Conclusion: those who own more BTC than hashing power can 'vote' on the level of hash security they want by specifying the mining reward they want relative to the difficulty. At some point, those who hold a large number of BTC would be self-motivated to mine for 0 reward and still secure the network. The alternative is to debase all of their holdings to pay for unnecessary security.
Mining doesn't work this way.