Retrenchment

Jane Austen’s Persuasion begins with retrenchment. Our heroine’s family needs to cut expenses to avoid bankruptcy. But how drastic a cut is really required? How drastic a cut can really be tolerated? Do we need to reduce fossil fuel consumption in order to avoid a climate catastrophe? These are instances of a fascinating class of problems. The power curve in airplane flight is another nice instance. I first learned about this from Ran Prieur’s blog.

I find simple mathematical models helpful in understanding these sorts of puzzles. Here is a first attempt to capture the core structure.

In this retrenchment model, the state of affairs is a pair of numbers that I call a long-term balance and a short-term balance. The long–term balance earns steady interest while the short-term balance earns no interest. Money can be moved back and forth between the accounts. Moving money from the short-term balance to the long-term balance is free, but a penalty is incurred when moving money from the long-term balance to the short-term balance.

The evolution of the system consists of a series of alternating moves. First the world makes a move: interest is earned on the long-term balance, but also a random transfer occurs on the short-term balance. This random transfer might be positive or negative. Then the account holder makes a move: funds can be moved between the balances. After this, both balances must be positive or the sequence ends.

The core problem is to devise a strategy to keep the sequence going. Keeping funds in the long-term balance is good because interest is earned there. But funds must also be kept in the short-term balance in order to cover random negative transfers. The penalty incurred in moving funds from long-term to short-term make it prohibitively expensive to move as frequently as a small short-term balance would require.

The effectiveness of any such strategy depends on the details of the random transfers, for example whether the transfer at one time is correlated with the transfers in the recent past. In a realistic scenario the nature of this random sequence will not be known. The best a strategy can do is to look at the past and infer that the future won’t look too much different.

I coded a simple little simulation. Here the transfers are drawn from a Gaussian distribution and are not correlated across time. The strategy was very simple: when the transfer pulls the short-term balance below zero, move enough funds from the long-term balance to bring the short-term balance up to a fixed threshold. When the short-term balance rises above this threshold, move the excess funds to the long-term balance. As long as the short-term balance is between zero and the threshold, no funds are moved to or from the long-term balance.

Here is one sequence that emerges from this interplay:

Here are two runs where the sequence of transfers is exactly the same, but the threshold differs by less than 1%. The slightly higher threshold maintains a slightly higher short-term balance, thereby incurring fewer penalties for the movement of funds from the long-term balance.