The employer-employee relationship doesn't involve authority because it is an ongoing, voluntary trade of labor for goods (typically money) or occasionally services. There is often confusion on this issue because it might appear that when a "boss" tells his employee what to do, he is exercising authority over that employee. Consider this example to help clarify the situation:
Boss: "Hey, Employee, I need you to stop working on Project X and start working on Project Y."
Employee: "No. I want to finish Project X first."
Boss: "Do it, or you're fired."
Employee: "Okay, I'll do it."
It probably looks like the boss just threatened to punish the employee if he didn't yield to the boss's authority, correct? Let's consider what actually happened beyond the common language used.
Boss: "Hey, voluntary trading partner, up to now, I have been offering to trade you money to work on Project X, but I am now choosing to discontinue this arrangement. I would like to continue trading with you, however; but now I want you to work on Project Y."
Employee: "No, I'm not really interested in this new arrangement, I'd prefer to keep trading for my labor on Project X."
Boss: "Unfortunately, I no longer wish to engage in such a trade. You are free to decline all offers or you can accept my offer to trade for your labor on Project Y."
Employee: "Okay, I'm willing to accept that trade."
The language used is clearer in the second example, but the underlying offer, counter-offer, etc. are fundamentally the same. Both the employer and the employee are engaging in a voluntary trade. The employee can discontinue his part in the trade if the employer refuses to agree to certain terms (a raise, certain days off, etc.) just as the employer can discontinue the trade if the employee doesn't meet his needs. It's not a hierarchical relationship, it's a symbiotic one.
Authority is what most politicians, cops, and some parents (illegitimately) claim to possess—they believe they have the right to hurt people who don't obey them. When a client hires me, there's no authority involved, merely an agreement to exchange goods and services. The entire scope of the relationship is within the terms of the contract, and we are equals in that we both have the right to terminate the business relationship at any time.