Following links may clarify the posed question,
1. https://www.bellchem.com/news/caustic-soda-in-water-treatment#:~:text=Caustic%20soda%20(sodium%20hydroxide%2C%20or,absorbing%20water%20and%20carbon%20dioxide.
2. https://www.astm.org/DIGITAL_LIBRARY/STP/PAGES/STP47165S.htm
3. https://www.calcinor.com/en/news/2017-03-16/functions-lime-water-treatment/
I think the caustic soda is more expensive but convenience, the powdered lime is cheaper but trouble, because the sludge will be produced.The caustic soda is more efficient, the sludge is not easy to deal.
Lime is a lot less expensive than caustic up front but the gap narrows when you factor the cost of storage, handling and maintenance or the lime and effects of calcium carbonate scaling. Depending on the alkalinity of your water the lime will add more hardness, which can be good or bad, and it will provide some corrosion protection to your system. Lime will aid in the removal of iron, manganese, and TOC/NOM Thru absorption in addition to pH adjustment. You may even want to consider a dolomitic lime. Caustic is easier to store and dispense and the elevated freeze point can be handled either by dilution of the product or implementing freeze protection. It reacts rapidly as compared to lime and pH control is easier. From a safety in handling perspective the lime is a lot more forgiving.
For a somewhat related RG discussion, yet on brines, see: https://www.researchgate.net/post/Anyone-can-advise-me-on-how-can-we-reduce-the-High-concentration-of-Magnesium-Calcium-and-sodium-from-highly-saline-brine
About predicting the pH of Ca(OH)2 aq. solutions (saturated or not) ― cf. my posts (§ III., IV, V.) at:
https://www.researchgate.net/post/How-to-increase-pH-for-a-solution-without-making-a-chemical-reaction-with-CaCl2
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