Eh...
It is true that it's cheaper in the gift shop, being 10 less per frag. (Which, 60*80=4800, whereas it's 800 cheaper in this, being 50*80=4000,)
But, the boxes do provide with additional stuff, and I'd say that if you were one of the lucky ones who got Ay, it'd be worth it.
Math time! 
1 Summer Badge Pack= 20 cp and gives 10 badges- it means 1 badge= 2cp.
For whole clothing you need 50 fragments of it, so it's 50frags x 30 badges= 50 frags x 60cp= 3k cp of total price.
Then! If you collect all free badges from Summer event, which is 210 (420cp or 7 frags)- price of clothing will be 2580cp.
+ additional items from boxes- with a chance for something nice.
In Gift Shop total price of clothing is 2,5k and no other items from boxes.
So, is it really such a big difference?
(don't know where you got that 80 from- each clothing needs 50 frags to collect)
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