To the original posters, and as others have mentioned, NO the US dollar is not \"more valuable\" and does not have \"more bargaining power\".
It is only as good as the rate of exchange on any given day. The vendor/shop will have to go to the bank to exchange US into peso just like you would. Plus you are basically changing money at the discretion of a vendor or store or restaurant, rather than a bank/cambio window/ATM which gives standard rate. You get a tiny bit better at a bank than an ATM but usually have to wait in long lines (not worth the pennies saved to me).
\"Better\"
restaurants may take charge cards but lots of smaller, great, local places, especially in El Centro or Old Town do not accept them.