Given the choice between money today and money tomorrow, one would prefer the money today. This is true because money today can be hoarded until tomorrow, or alternately used for other purposes before tomorrow - "money tomorrow" is just one out of many options available to someone with "money today" and hence "money today" must be of greater than or equal value to "money tomorrow". Similarly, a greater sum of money is always preferable to a lesser sum, because I have the option of discarding a part of the greater sum to get the smaller sum. It is then plausible that there be a greater sum of money tomorrow (s2,t2) which is of equal value to a smaller sum today (s1,t1).

What is the relationship between (s2,t2) and (s1,t1)? So far we have found that (s2-s1)(t2-t1) >= 0. Only by assuming that investment opportunities are of much shorter duration and of much smaller size than the quantities under consideration, and also available equally throughout time (i.e.  having multiple instantiations at (t1+dt,t2+dt) for all dt), do we get the conventional discounting rule, where (s2/s1) = r^(t2-t1) for some r > 0.

To go from the inequality to the equality requires work. This specific equality only emerges because of the additional assumptions made. Given access to a pool (defined by our assumptions of time-invariance and scale-invariance) of investments with discount rate r, we can now arbitrage any cash flow to its present value. The net present value (NPV) is this concept, and when positive it represents situations in which choosing the investment over choosing the pool results in a cash gain.

So far, I have only dealt with the opportunity cost of money, as represented by participation in the pool of investments. I will discuss the risk of the investment itself next.

http://en.wikipedia.org/wiki/Two_envelope_problem

You are given two envelopes to choose from, and are told that one has twice the amount of money as in the other. You pick one and open it, finding $20. The host asks you if you would like to switch. It is apparent from the random nature of the initial choice that switching would not change the payoff.

The only possible amounts in the other envelope are $10 and $40. I became tempted to assign 2/3 probability to $10 and 1/3 probability to $40, so that the expected value came out to $20. This is wrong, however, because the host could announce that he would pay a square of whatever amount of money I ended up with ($100 for $10, $400 for $20, $1600 for $40) and still switching would not matter, yet those probabilities (2/3, 1/3) would prescribe switching.

This is apparently a state of ignorance in which the possibilities are finite and known, and yet probabilities cannot be assigned to them. This disturbs me.

HT: Lee Hsien-Yang's New York talk last Friday

When I went from undergrad to grad school, the introduction of a monthly stipend into my life gave me a natural unit in which to measure my monthly consumption. I never did itemized budgeting, but net positive savings was definitely a goal. This goal was easily reached, mostly due to my realization that scrimping on rent got me very far. As of a month ago, I started working, and for the first time ever, have substantial savings.

This introduces a degree of freedom, and with that, one more thing to optimize. There were a few decisions necessitated by the living environment, like eating out and using the wash-n-fold service instead of the laundromat. Other than that, however, it seems like the other decisions are going to be made by my choice of company more than anything else. Tonkatsu curry is the most delicious dish I can think of, and I really don't think anything will change that. I do follow along when people go to fancy restaurants though.