This is a good question. Plato's commitment to reason or philosophy is the main point of the Republic. We can see this in three ways. 

First, if you look at Plato's ideal state, the leaders are philosopher kings. In other words, they are people who are committed to the study of philosophy, which is rooted in reason. So, all the decisions of his ideal republic is based on reason. 

Second, one of the most famous parts of his work is the allegory of the cave. In this allegory he is basically stating that the average person only sees illusions of the truth. He writes:

"He will be unable to see the realities of which in his former state he had seen the shadows; and then conceive some one saying to him, that what he saw before was an illusion."

What enables a person to see the truth and not the shadows is philosophy. From this perspective, it is all about reason. 

Finally, Plato famously banishes the poets from his ideal republic. The reason for this is because poets do not focus on reason but on emotions. 

All of these points reinforce the point that the most important aspect of a republic is reason. 

What was Socrates's belief about living a life of reason in the Republic?

We can say that Socrates is totally committed to the pursuit of reason in the Republic. In fact, almost every dialogue in which Socrates speaks, he is pursuing reason. 

One of the most memorable passages of the Republicis Socrates' allegory of the cave. In this allegory, people are tied up and they are facing the wall. There is a fire behind them and objects are moving. Of course, the people cannot see the objects. They only see the shadows on the wall. In this sense, their perception is not true. 

Socrates uses this allegory to show that without reason, we only see the shadows. What we need to see is the real object. Socrates calls these the forms. It is only through reason or philosophy that we can actually come to understand the forms.