Budget rods these days are probably far better than they have ever been. Our business sells a lot of rods to the beginner/intermediate angler, many of which would be very pleased with a 10lb carp and many not really looking for more than a good low double. I'd be fairly confident going to 20lb with one, possibly beyond.
Pricing is definitely a matter of choice. Regardless of opinion you do 'not always' get what you pay for, although I do agree that paying more will generally buy better gear. Reels are a classic example, and many of the cheaper 10 and 12 bearing reels are so smooth Shimano would have struggled to beat them a few years ago. Still though, people will spend up to ten times the price on a piece of equipment that is maybe better quality but one has to look at whether the 'improvement' is truly worth the massively hiked price tag - especially to a beginner.
Last season I witnessed a lad take a carp of at least 15lb last year on a six or seven foot shakespeare bright red effort that probably cost less than a tenner (with a single bearing rear drag reel). Didn't even have a bite alarm (just threw a ledger in and watched his line). A fish he'll probably remember for life, and frankly the experience was probably worth many times what the rod cost.
I and most others watching blanked that day....it definitely is not all in your tackle.