To get the car to respond quicker, tire manufacturers use softer rubber that improves grip but wears faster. Generally, the higher the mileage the manufacturer promises, the harder the rubber is. Hard rubber will make more noise when they wear down.
....... Mmmmmm ...... Not exactly!
Getting a tire to respond faster to steering input is different than the level of grip. Grip is about the tread compound, and speed of response is about sidewall stiffness.
Inflation pressure can greatly affect sidewall stiffness, but so can the size and stiffness of the filler (see the cross section of the tire posted earlier.) - not to mention the aspect ratio. That's one of the reasons racing cars use low profile tires. Quick response makes it easier to control in a corner. (The exception to this is drag racing where it is better at launch to get the tire to absorb the shock by winding up and high profile tires are better at that!)
But improvements in steering response come by sacrificing ride quality - specifically impact harshness. So soft riding tires don't respond quickly and vice versa.
When it comes to tread compound, there is a 3 way technological triangle between grip, wear, and rolling resistance (fuel economy). To get improvements in one area, one or both of the other areas have to be sacrificed. So tires with great grip don't wear well and consume more fuel. It's all about the compromise.
However, small changes in this triangular relationship can be made by some changes in the make up of the tread compound. Replacing carbon black with silica improves the grip - if I understand this correctly - and that means that it is possible to get a bit more fuel economy out without affecting the wear and grip - and the term LRR has been applied to those tires that have that kind of tread compound change.
Please note: LRR is a relative term, not an absolute one. It means better fuel economy compared to tires with similar grip and wear properties - so it is possible for a non-LRR tire to get better fuel economy than an LRR one!
As an aside, good wearing tread compounds tend to be hard, but both high grip and low rolling resistance compounds tend to be soft.
To sum this up: Grip level and quickness of response are not tied to the same things, so it is possible to get a tire to have fantastic grip (and poor wear and fuel economy), but slow response and a soft ride - and it is possible to have a good wearing tire (with a moderate to low level of grip and poor fuel economy), and quick response and poor ride - and all the other combinations!