Change in needle=more yarn?

I really like this pattern but it calls for a size 13 needle and 3 balls of yarn. I’d like to use a size 9 needle, I like the look of stitches with a small needle, but my guess is that a smaller needle means more yarn needed. How do I figure out how much yarn I’ll need for a smaller needle with this pattern?

I should add that I’ll be using Worsted yarn and figured out it should need 4 balls instead of the 3 the pattern calls for. I added up the number of total yards needed (using the original yarn stats) and divided it by the yards in the balls I’ll be using.

You can use this yarn calculator to estimate yardage if you knit a swatch with your needles and the worsted yarn that you want to use.

You’ll also have to cast on more stitches to get it the same size as the original.