Vlad wrote:
What about secondary objectives? Yes it's cheaper to send some cruisers here than to send a battleship or two but the cost of this is not wasted. Think of it as insurance. If an objective is not important enough for either side to commit their carrier fleet (or that fleet is tired out and indisposed) then the enemy cruisers are not going to have a very good time if they find a battleship in their way. If you send an equal cruiser force you could lose the fight.
You are right that there were not enough carriers available. The US Navy and Japanese navy lost most of their carriers in the battles of 1942 (sunk or damaged) and therefore had to make for most of 1943 without any significant contribution of carriers.
That could have been the big time of battleship, if you are right - and there were many battleships available, especially Japanese. But how often were battleships actually used, where they were actually needed and could not have been defeated by other means?
The Japanese battleships were used several times to bombard Henderson field on Guadalcanal - for sure that could have been done by smaller ships also (and was done by smaller ships also!). They were stopped once by a mixed force of cruisers and destroyers with same help of aircraft to finish of the crippled Hiei. The next time they were stopped by a mixed force of battleships and destroyers - in a battle, which also could have easily resulted in the loss of at least one of the USN battleships.
Any other examples in the Pacific War? Kuritas fleet? Hmm, not really successful.
In the European theatre the aircraft carrier was anyway much less needed, because land-based aircraft were heavily used in most battles. Here there was no clear superiority of the carrier over the battleship.
Are there examples of successful use of battleships?
Warspite in Narvik? For sure successful, but very risky and any other ship stronger than a destroyer would have done as good (and would have been sunk, if German submarine torpedoes would have been working).
Rodney and King George V finishing off Bismarck? That could be such an example, if Bismarck would not have been already heavily damaged (including some damage caused by the battleship Prince of Wales, but the serious one was the torpedo hit by a Swordfish from Ark Royal).
The British battleships at Battle of Cape Matapan: a crushing British victory, but would have some heavy or even light cruiser caused the same, if they would have surprised the Italian cruisers the same way and had the same advantage in radar technology?
There should be more examples, if you were right, because there were battleships available - for some time even more battleships than carriers.
Some numbers for prices (Wikipedia):
Essex class 68-78 million $
Iowa class 100 million $ (not surprising that a battleship was more expensive than a carrier!)
Baltimore class 40 million $
Hopefully someone else finds more about that question. But should be obvious that in the interest of numbers for secondary objectives to build more cruisers and destroyers would have been more useful - and easier to afford.