Is light particle Elastic?
Light is an emergent phenomenon from the quantum mechanical superposition of a large number of elementary particles called photons. Superposition is not interaction. It is an addition of probabilities for finding a photon at an (x,y,z,t) spot. This double slit , single photon answer of mine might help to understand this .
A building is made of bricks, but a brick is not a building. In the same way light is built up by photons, but photons are not light and cover a large spectrum of electromagnetic energy, a small part of which is visible light..
If by elastic you mean "can they scatter" off each other, the answer is yes, with very small probability.
Do they collide?
At high energies photons are called gammas, and the probability of scattering off each other rises with energy. They are even considering gamma gamma colliders. Due to the relativistic nature of the collisions pairs of particles may be generated.
Do they exchange energy
They may collide elastically , no change in energy and momentum, or exchange energy and momentum, depending on the probability of two photon interactions/scattering.
Light has very low energy photons , a small part of the electromagnetic spectrum, and as the other answer says,the light beams, even though composed out of zillions of photons, interact very very rarely. It is only at the invisible energies of gamma radiation that interactions are measurable.