There seem to be two issues here - case capacity to bore ratio, and then using same weight bullets in different calibre combinations. The latter makes little or no sense to me. If output comparisons are wanted, ME values from the different ratio cartridges in a family, using the 'standard' bullet weight for each variant of that cartridge would provide better comparisons.
Increase bore size for a more or less fixed case capacity and as
@Steve Blair says, you get a more efficient combination in terms of ft/lb ME per grain powder burned, sometimes fps MV too, but not always as bullet weights rise according to calibre. (Look at reloading manual tables and do the simple math for such cartridge 'families' - eg 338 Federal gets you far more ft/lb ME per gn weight powder than any 243 Win load using the same basic case.) You may or may not get a more efficient projectile in subsequent flight, usually 'not' once calibre passes a ballistically ideal point - so 308 Win produces higher 140gn bullet MVs than 260 Rem with same weight .264 cal bullets, but the latter is far better for 1,000 yard and beyond shooting because of the higher SD / lower drag of its bullets. (However, read Bryan Litz's stuff on 'scaling' bullets between calibres and optimum resulting weights. The 308 equivalent to a 142gn 0.264 is up at 220gn or thereabouts.)
The classic example of how to get your cake and eat out of this conundrum it is ultra high-vel anti-tank ammunition which first saw sabot-encased projectiles widely used in late WW2 and has been the norm in tank cannon A/T ammunition for many years now - have a 120mm cal gun, and suitable dia. lightweight sabot enclose a sub-calibre ultra dense dart, depleted uranium nowadays. Expel them from the barrel at 4,000 + fps, the sabot petals separate and drop away and you have a projectile with a BC that would make us gasp cover a 2-mile flight in a tiny timespan to accurately hit a moving target with staggering retained energy and penetrative powers against inches of armour.
(I wonder if we'll see this principle scaled-down to use in ELR competition as King of Two Miles trends to ever longer distances?)
The varying ratio values you get from changing calibres on a near-fixed capacity common case is often referred to a the 'expansion ratio'. I believe that's technically incorrect though as the ER also needs barrel length as part of the input. So, those who investigate this factor often call it the 'Effective expansion ratio', or EER In effect, it's how many times the original unfired combustion chamber volume expands as the bullet travels along the barrel; or looking at it from the other side, how many inches of bullet travel are needed to double, triple etc the original combustion chamber volume. The lower that figure / lower the EER value the more thermally efficient the cartridge (within limits) and also faster burning powders can/should be employed to maximise velocities while keeping pressures within limits.
ER/EER seems to hold little interest these days. Not surprising as it doesn't tell you that much of great usefulness. As ER values increase alongside barrel lengths, you could use it to theoretically predict where an extra inch of barrel gives a poor return in MV terms - but why bother, there are much easier ways based on actual tests and/or experience. Although not called ER/EER these days, interest remains, in predicting barrel life in so-called 'Over bore capacity' tables. This takes nominal case capacity in grains water and divides that by the bore area in sq inches. As in here:
http://bulletin.accurateshooter.com/?s=over+bore+cartridge+chart&submit=Search
In theory, the results should also be a good predictor for powder burning rate classes, but IME it's not as you can only compare fairly similar size cartridges. The larger the cartridge, the slower burning the powder needed for any given ratio.
