In my experience, varying jurisdictions can and do differ as to the myriad ways these disputes are resolved. Contract law is one area where the judge has a lot of discretion. This is definitely true in state courts, even from judge to judge, and can even be true in the federal level-The 9th Circuit has some wildly different appellate decisions when compared to the 1st Circuit, and so on. I say this not to be argumentative, but to highlight the importance of careful and concise drafting that fully explicates the bargained for exchange, as there can be a vast amount of judicial subjectivity that goes into determining which rules pertain to certain situations.
"Conflicting or competing clause" cases are now some of the most commonly litigated contract disputes. This is largely because the last 20 years has seen a huge influx of people "drafting" (more like piecing together) contracts without benefit of qualified counsel. This is particularly true because lay people do not generally create a specific insturment like an attorney would - from scratch, with definitions and terms specific to the transaction. Rather, they go online and find "form" or model contracts that they feel are close enough (which are almost always missing key components), and then they type in their own terms, or even write them in. Because this is so common, most jurisdictions follow the rule that hand written terms supersede pre-printed terms; likewise, type written terms will take precedence over pre-printed terms.
Specific terms also carry more weight than general terms. Specific terms will usually be given precedence over general terms, as these are seen as creating a specific exception to the general terms. For example, if Clause A in your scenario said: Written notice must be provided at least five days in advance of (any) change... (leaving out "to price"), then clause B would prevail because it would be more specific than the more general term (A), which in my scene would pertain to any change whatsoever (this is assuming the whole of the agreement did not shed light on the issues more fully).
In your hypothetical, these are both specific terms. In that case, the court would first examine the entire contract and all addendum, specs, plans, etc. when interpreting competing or conflicting clauses applying the fundamental principal that a contract should always be interpreted as a whole - not clause-by-clause - and not section-by-section. Contracts will often have numerous parts with portions incorporated specifically by reference, or numerous documents that may be integral to the transaction, If the parties agree to what constitutes the various parts of the contract (even if not incorporated) the entirety of the transactional documents may be considered by the trier of fact (and law). Once examined, if a proposed interpretation makes other portions of the global agreement meaningless, illogical or unenforceable, and another party's interpretation is in keeping with the document as a whole, that is the interpretation that will typically be adopted.
Assuming this analysis doesn't work to resolve the issue, then the court would look to see if there is an order-of-precedence clause, which is a clause that lays out what parts of the contract / types of clauses take precedence over others (ex. written requirements take precendee over performance requirements, addendum hold less import than the signed agreement, schematics hold less import than addendum, and so on).
Assuming this there is no order of precedence, the court will look first to see if the contract was negotiated back and forth, with terms being modified with each draft. If Yes, then the court will except extrinsic evidence (parole evidence) that goes to the intent of the parties bargained for exchange. If not, the contract term(s) will be construed against the drafter and in favor of the one who signed the others' instrument.
So, as you can see, there is no clear answer to what seems to be a simple issue.
This just goes to show: Lawyers seem expensive when you decide to hire them - Lawyers are expensive when you have to hire them, because you decided not to in the first place!
