The prolonged struggle Svetlana Danilkina and I have had with the Becker-Murphy (BM) rational addiction model took a new turn today. How to account for rehab and ‘cold-turkey’ withdrawal? BM mumble a bit about this but don’t say much.
In the limiting case where expenditure on withdrawal costs zero people will never get addicted. They will consume addictive goods then instantaneously withdraw to never accumulate any stiock of addictive capital. Initially there will be less constraints on pigging out on the addictive good since this will not drive increased future consumption – adjacent complementarity (the effect of current on future consumption) will not arise.
One way of inducing the desirability of withdrawal would be introducing costs of withdrawal. If disintoxication costs increase with the degree of addiction then people might consume addictive goods for a while and, when the disutility gets bad enough, switch to maintaining an equilibrium addiction level. But this might be an ongoing program of maintenance such as use of an opiate substitute such as methadone.
Cold-turkey could be introduced by supposing withdrawal costs have a high fixed cost component. Then you will withdraw either only periodically or perhaps once-and-for-all. The latter might occur if, with a substantial disintoxication effort, one can drive one’s stock of addictive capital back to the level where it is ever optimal to consume.
Analytically the problem reminded me of forestry issues where you have fixed costs of tree-cutting and ongoing flow benefits that decline. Colin Clark in his book Mathematical Bioeconomics, refers to such problems as ‘orcharding problems’.
A lot more work can be done on treatment issues and on the effects of cheaper treatment on initiating current use.
The BM argument regarding ‘cold turkey’ is a bit strange. Basically they introduce a random downward utility shock and rely on non-concavities in consumption utility to establish a case for ‘cold turkey’. The ‘exogenous shock’ idea isn’t the intuition behind ‘cold turkey’ which one thinks of as a state of desparation reached via addiction that makes one want to give up. But if you do introduce such shocks then you can simply shift users from utilising an interior approach path of addiction to one that hits a lower boundary. You can do this without introducing non-concavities in consumption utility.
There is lots of detail in BM that seems iffy and insecure.