Optimal Stopping

When do you know when to quit? When to buy? This is a hard question, so as first pass, we make some assumptions.

Money has value R that you discount at rate r. The cost of the good is C, and the cost falls at rate (r+w). This is a process innovation that makes producing the good cheaper over time, and, we assume that this savings is passed on to consumers. Clearly, there is a benefit to waiting because cost is falling to zero. A standard method for discounting is exponential discounting, which gives the result that cost is falling faster earlier in time, and slower later in time. Last, the good itself has value S, which the person discounts at rate r, but enjoys from time T to forever (we assume infinitely durable goods for convenience — it won’t affect the conclusions later).

The problem is to decide the value of T, i.e. when to exchange money for the good.

Note that it must be true that S>R, otherwise, the value of the good will forever be less than the value of money. You end up never buying, which means T is infinite.

What is the basic trade-off? At the best T, it must be that waiting longer isn’t worth it. In other words, the benefit of waiting (lower cost) is less than the cost of waiting (which is foregone benefits of owning the good, which is a function of S-R). It must also be true that buying sooner won’t help, coz the cost of buying early (high cost) is far greater than the benefit of owning the good.

So, suppose we decide the value of the best/optimal T according to the rules above. What then is is the effect of a faster rate of cost reduction (a higher w)? This should push you to wait; BUT only when the cost reduction benefits of waiting are huge, which as we’ve determined is when the optimal is small. The process improvement forces you to buy earlier if the optimal T is large.

What helps determine whether optimal T is large or small? The size of (S-R). The larger S-R is, the larger the benefit of buying sooner, coz the gain you get from buying, S, is larger than the value of money, R. The larger S-R is , the lower the optimal T — in this situation, a cost improvement leads to waiting longer.

Finally, lets say there is no secular price declines for this good, w is zero. What then? Pretend you are at time zero, deciding whether to buy. If you delay, you earn a net of R-S+rC, or R-(S-rC). If R>S-rC, you think, if i delay, i get R (money happiness), which is greater than S (good happiness) minus rC (one time cost of buying the good). The opposite holds when R<S-rC.