Probably a negligible effect on circulation from week to week, but probably a sizeable effect in terms of people not canceling their subscription. Even if you only redeem a few dollars' worth of coupons, the newspaper subscription pays for itself!

I'm guessing that many coupons are for 5%-25% of the item [hmmm, this suggests another analysis!], so your wife probably spends 5-10 times what she saves. Some serious Coupon Queens claim they save 33%-50% by using coupons!