Is it because the narrower tire at the same pressure is under inflated?
Given 2 tires of the same construction, it is about the shape of the contact patch and the amount of deflection of the tire carcass. A wider tire will have a wider, shorter part of the tire tread in contact with the road and the tire carcass will flex less resulting in lower rolling resistance. On rough roads, it is possible to run the tire with less air pressure allowing the tire to conform better to irregularities for both better comfort and lower rolling resistance That is why pro cyclists are using much wider tires at lower pressure for races like Paris Roubaix where they ride over stretches of rough cobblestones. Even on better roads, the trend in the peloton is towards riding tires wider than the 700 x 23 that used to be the standard for road racing.
The setting where that discussion is relevant is on high performance road bikes. And even there, narrow means around 23-25 mm and wide means maybe 28-30 mm. MTB wide isn't part of the discussion. Why depends on several things. Sometimes a wider tire will make a certain rim more aerodynamic, which can be important on a high performance road bike. If you run a narrower tire, you have to have higher pressure to avoid snakebite punctures. Higher pressure means that the wheel/bike will get bouncier. Bounce means losing ground contact. Losing ground contact means losing traction and scary handling, and lower speed. Last is rolling resistance, which comes from the rubber deforming around the contact patch. A narrower tire will have a longer contact patch, with more deformation and more rolling resistance.
34 psi front, 36 psi rear, you won't go too far wrong. there's an element of personal preference involved, so you could vary that +/- 2psi if you wish.
Tire rolling resistance on the road is caused by: 1. Internal friction and hysteresis within the tire’s materials, and 2. On rough roads, small bumps lifting the bike and rider slightly on each little impact. The wider tire will have a shorter contact patch and hence less tire deflection. if the tire pressure is the same, the area of the contact patch must be the same to support the same load. However, the shorter the contact patch, the lower the vertical depth of tire deflection and hence the less the internal friction and hysteresis within the tire’s materials.
