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I have long been conflicted on this point: do Wizards need to pay to write spells they learn for leveling into their spellbook, or just when copying from a scroll or another spell book?

It seems that if you require paying with each level, you are nerfing your spellcaster until they can write down their new spells, but if you don't, then it's actually smarter of the players to avoid spells of their chosen arcane school, since the "savant" ability with each arcane school makes it faster and cheaper to add spells to your spellbook.

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No, you don't have to pay for spells you gain when leveling up.

From the PHB, p. 114 sidebar:

Your Spellbook

The spells that you add to your spellbook as you gain levels reflect the arcane research you conduct on your own, as well as intellectual breakthroughs you have had about the nature of the multiverse. You might find other spells during your adventures. You could discover a spell recorded on a scroll in an evil wizard's chest, for example, or in a dusty tome in an ancient library.

Copying a Spell into the Book

When you find a wizard spell of 1st level or higher, you can add it to your spellbook if it is of a level for which you have spell slots and if you can spare the time to decipher and copy it.

As you see, the rules establish a clear difference between spells that add to your spellbook and those that you find and copy onto it. If you take a look at the next part, you will now why.

Copying a spell into your spellbook involves reproducing the basic form of the spell, then deciphering the unique system of notation used by the wizard who wrote it. You must practice the spell until you understand the sounds or gestures required, then transcribe it into your spellbook using your own notation.

For each level of the spell, the process takes 2 hours and costs 50 gp. The cost represents material components you expend as you experiment with the spell to master it, as well as the fine inks you need to record it. Once you have spent this time and money, you can prepare the spell just like your other spells.

As you see, the process of copying involves a lot of time, effort and materials which are dependent on you not yet knowing the spell and reproducing it yourself. For spells which you gain as you level, you skip all of this. The final sentence in the section makes it clear that once the spell is copied through this process, you can prepare it just like your other spells. This implies you do not go through this process for your other spells and that this is a different method of adding a spell to your spellbook.

Lastly, on the same page you are given these simple instructions on what happens when you gain new spells and when you do it.

Learning Spells of 1st Level and Higher

Each time you gain a wizard level, you can add two wizard spells of your choice to your spellbook.

And there is nothing more to it (aside from level restrictions on the spells).

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    \$\begingroup\$ you beat me to it >.< Good game, good game. :P \$\endgroup\$ – Airatome Nov 21 '15 at 18:57
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No, there is not a cost incurred for spells learned during level up; but there IS a price incurred for OTHER Wizard spells you might come across in your adventures.

PHB, p. 114

"...the spells that you add to your Spellbook as you gain levels reflect the Arcane research you conduct on your own, as well as intellectual breakthroughs you have had.

This tells us that the spells you get to chose as you level up (2 spells each time), is from your natural research and breakthroughs in magical study, which doesn't require you to pay yourself to do such.

Same page

"..you might find other spells during your adventures.... when you find a wizard spell of 1st level or higher, you can add it to your spellbook if...and involves reproducing the basic form of the spell, then deciphering the notation used by who wrote it...you must practice the spell and then transcribe it..."

This tells us that the Wizard spells you find that are of a level you have spell slots for can be put into your spellbook for a price; and that price includes everything required to continue practicing the spell, deciphering the notations of the spell, and then putting it into your own spellbook; since these are not spells you learned through your own breakthroughs.

Here is a simpler way of looking at it. "You can add this pie you made with your own ingredients to your inventory every level up. On your adventure you might find places that have ingredients, but you must pay for them because they don't belong to you."

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