Economics of when to upgrade an iPhone ?

What's the right time to upgrade an iPhone after you become eligible for an upgrade? If you think a new iPhone is worth $200 or more then the answer is simple. Since you value the iPhone more than your $200, the new iPhone increases your quality of life and therefore buy it immediately but this question can become harder for those who do not see any additional value in a newer version of an iPhone.

When should someone upgrade an iPhone if they don't see any value in the newer version ? Or in other words if the value they associate for the new iPhone is $0 ? If iPhone was not subsidized by telephone companies and you had to pay full price of the iPhone then again the answer would have been simple. Never upgrade, as you value your $200 more than the new iPhone. The carriers subsidize iPhone when you buy a new iPhone and you end up paying back the subsidy on a monthly basis in form of higher monthly fee for data plans, text messages etc. Since apple sells the iPhones to carriers for roughly $600 and carries sells it to customers at $200 there's approximately $400 subsidy for a new iPhone. Typically Apple refreshes it's iPhones after 18 months which means that the carriers will have to collect back $400 within 18 months. This shows that a normal user gets charged roughly 400 / 18 = $ 22 every month as a subsidy recouping mechanism.

A customer eligible for an upgrade can save $200 by not buying a new iPhone on the first month but they will still have to pay $22 every month to the carrier for the iPhone subsidy for an iPhone they don't even use! This means that they are subsidizing other iPhone users who elect to buy an iPhone. A customer is better off by paying $22 in first month as subsidy and not spending the $200 for the new phone as they will still be saving $178 but as months go by they will be paying more and more money as subsidy to avoid paying $200. How much should someone pay to carriers to save $200 ? The maximum one should pay to save $200 is $200. As you can see in the chart below, this means that after 10 months the customer will have paid $220 to save $200 which doesn't make much sense.

I believe that if you think that the iPhone is worth $200 then you should upgrade your iphone immediately after you become eligible for an upgrade and if you think that a new iPhone is worthless then you should upgrade your iPhone after 10 months after you become eligible for an upgrade.