The thing about solids is that they can have awe inspiring amounts of thrust compared to liquid engines, as you can see in this collection of boosters and thrust levels:

* Shuttle SRB - 2.8 Mlbs
* Ariane 5 boosters - 1.5 Mlbs
* Titan IV booster - 3.2 Mlbs (wow!)

Compared to liquid engines:

* SSME - 660 Klbs
* F-1 - 1.5 to 1.8 Mlbs
* RD-170 - 1.6 Mlbs
* RD-180 - 800 Klbs
* Merlin 1D - 160Klbs

But their [ISP](https://www.grc.nasa.gov/WWW/k-12/airplane/specimp.html) (explanation of ISP at bottom of link) usually sucks.

So with ATK (maker of US solid boosters) pushing to replace Antares first stage with a solid, and maybe even Atlas V first stage with a solid, the question I have is, do the numbers work? 

ISP is less important than thrust (per se) during the first few seconds and minutes of flight. You have to get the darn stack off the ground before it matters about efficiency. Upper stages are ISP dependent, but less so the first stage.

The question I have is, how much so? I.e. Can a Solid first stage really make sense in competition with high-performance liquid engines?

Contrary examples would be: 

* Ares-1x flight (SRB first stage)
* Ariane 6 - proposed solid first and second stage
* Vega light launcher
* Minotaur
* Taurus
* Pegasus

Of course Ares-1 was cancelled. And Ariane 6 is unlikely to end up as a solid based booster. Minotaur, Taurus, and Pegasus barely launch these days. Vega is very overpriced.