How does elasticity impact market efficiency?
Good question

If we examine edge cases of full elasticity, then we get:
supply or demand with zero impact of quantity on price: no matter the quantity, price will remain the same

Looking at elastic demand, that mean that for suppliers the MC function will arrive to the particular demand price. No special change in efficiency

Looking at elastic supply, that means that the consumer surplus will derive from the fixed price

Based on theory, there will be no loss of efficiency (a certain quantity that will be lost) when adding taxes or subsidies

The common real case of full elasticity, is for imported or exported goods, where the world market's equilibrium for the product imported /exported is unaffected by local market
In this case, either consumers get cheaper prices for products, comparing to a similar product manufactured locally 
or get extra prices for local goods sold abroad