An authorized expenditure is simply one that have been approved. How the approval process works is subject to legal, corporate, and/or project governance. Some companies allow the PM the freedom to spend the budget as long as the budget is pre-approved. Others may approve the budget, but still want item by item approval. Still others allow spending according to an approval chart specifying $ amounts and the role that must approve an expenditure of that amount.
Having a good cost management plan and financial checks and balances is place is usually a criteria for giving more freedom.