Friendship Equation

Work is preventing me from spending enough time with friends lately.  Rather than deal with this problem head on, I got curious about defining the relationship between friendship and time and came up with the following formula for calculating Friendship Value:

This assumes that Friendship Value = [1. Discounted perceived value of past interactions] + [2. perceived value of current interactions] + [3. discounted perceived value of future interactions]. Working backwards:

3. "Discounted Perceived Value of Future Interactions" can be expressed as the summation of all future interactions (t) years from the present (t=0) where "i" = the discount rate at which the net present value of the opportunity costs of a friendship equals the net present value of the benefits of the friendship:

Or, for those that want to graph friendship as a continuous rate (where d=discount rate and λ = log(1+i) ), by the integration:

2. "Perceived Value of Current Friendship Interaction" may be expressed as:

1. "Discounted Perceived Value of all Past Interactions" may be expressed as:

And thus, total Friendship Value can be expressed as =

By this we see that friendship is in a constant state of entropy, buoyed only by the value of our current interactions and the perceived value of our future interactions.  Without the hope of future interactions, the value of a friendship will decline asymptotally, approaching but never reaching zero.

If the value of perceived future interaction declines, it affects the net present value of the friendship.  So if I am going to be busy for the next six months, this dramatically affects the current value of my friendship.

We can calculate the relationship between time and friendship using an inverse square law:

Where FV₁ = The Friendship value of a friend, FV₂ = The Friendship value of me, and t = the amount of time spent apart.

By this equation, as the net present perceived value of either or both friends decreases, the force of attraction between the friends drops proportionately. But when time is spent apart, the overall value of the friendship drops exponentially.

Thus, friendship is a function of time.  And if I value friends, logic compels me to leave work alone at some point to spend some time with them. I probably need to get out more.