(A quick note: it’s almost job market time here in the world of economics. Today’s paper is a job market paper by Guy Arie, a theorist here in MEDS who I saw present a really interest model the other day. If your department is looking for a micro theorist this year, definitely give a skim through the Northwestern candidates’ papers; we have 8-10 students in theory, and the job market talks I’ve been to have been very nice overall. OK, on to Arie’s paper…)
All but the simplest dynamic mechanism design problems prove difficult, particularly because the intuition of the revelation principle fails. In a dynamic problem, I can not only lie about my type, but I can also lie about my type conditional on the history generated by the mechanism. A particular worry among agents is that if, in a truthful mechanism, the principle can elicit my type in period 1, he will then use that information to punish me in future periods (so-called “ratcheting”). The constraints in dynamic mechanism design, then, are much larger: I am going to need to worry about history-dependent deviations, and I am going to need to be sure that ratcheting is not a concern.
Arie studies what looks like a straightforward contracting problem. Imagine a risk-neutral salesman selling over a year (a technical note: the salesman have limited liability so I can’t just sell them the firm right away). They have N potential buyers which they can try to sell to. If they try to sell to one in a given period, a sale is made with probability p. Under any sensible contract, the salesmen will try to sell to the easiest clients first, so that the effort needed to make a sales attempt is increasing in the number of sales attempts I’ve made previously. The salesmen’s boss can only observe the number of sales made, but does not know how many sales have been attempted. A successful sale is worth V to the boss.
Now hopefully you see the problem: if I knew how may sales had been attempted, I would pay the salesmen just enough to cover his cost of effort at any given time, and I would ask him to keep making sales until the wage payment, adjusted for the probability of success, was higher than the value of the additional sale. The salesman will make exactly zero profits after accounting for his cost of effort. But when the boss does not know how many sales are attempted, the salesman has a nice deviation: just sit on his laurels in period 1, and only then start trying to make sales. The payment for each successful sale is now higher than the cost to the salesman of attempting the sale because the salesman has “delayed” making the easy sales until later in the year when the “bonus” is higher. Worse, if I try to condition payments on past sales, a salesmen who is unlucky with his first few “easy” sale attempts will just stop working altogether because he will be getting paid for “easy” sales but will in fact be trying to make relatively hard sales. Evidently these problems actually comes up empirically.
So what can the boss do? The standard dynamic moral hazard solution without increasing costs but with limited liability would be something like the following: have the salesman try to sell, giving him “credit” for each sale. Once he has enough credit, give him the firm (meaning give him the profits from each future sale). The promise of this big bonus in the future is incentive enough for the employee to work hard now. If he gets unlucky and does not make early sales, fire him, because no matter how lucky he gets in the future, he’ll never make back enough credit to get the firm.
With increasing costs, things are not so simple. Arie manages to write the problem as a linear program – binary effort and risk neutrality are quite important here – and then notes that it is not obvious that we can apply a one-stage deviation principle; indeed, a simple example shows that checking one-stage deviations is not equivalent to checking incentive compatibility here. But, the problem can be transformed, as all linear programs (as well as many other mathematical programs!) into a dual. The dual has nice properties. Essentially, the choice variables of the dual will be the shadow prices on the ex-ante probability I will ask the salesman to try to sell after a given public history. If that shadow price is strictly positive, then asking the salesman to try harder after a given history increases the principle’s profits, even accounting for the dynamic effects such a request has on incentives to shirk in the past. The dual formulation offers a number of other nice interpretations which make the potential solution easier to see – the usefulness of shadow prices should come as no surprise to economists, given that it has been the critical element in results going back at least to Ramsey’s taxation paper and Hotelling’s resource extraction model, both in the early part of the 20th century.
So what is the optimal contract? Don’t pay the agent anything except “credit”. When this credit gets too low, fire the agent. When credit is sufficiently high, pay the agent a fixed rate per success over the next N periods, where the pay each period is just high enough to incentivize the agent to exert effort N periods in the future. Essentially, the agent is paid a very high piece rate at the end of the contract. Until that point, successful sales are rewarded only by conditions making it easier for the agent to begin getting his bonuses; e.g., “if you make a sale today, then after two more consecutive sales, I’ll put you on the bonus schedule for a month, but if you don’t make a sale today, you’ll need four more sales, and will only get the bonuses for two weeks”. And why do I tell the employee only to work for N more periods rather than just give him the firm and let him work until the moment it is too costly to exert any more effort? Essentially, I am destroying the “value” of the future firm after every failed sale in order to keep the agent from shirking; by reducing the future value of the firm after a failure, I can still give the agent who was unlucky some reason to keep working, but in such a way that the destroyed future value of the firm makes shirking unprofitable.
http://www.kellogg.northwestern.edu/faculty/arie/GuyArie_DynamicCosts.pdf (Job market working paper)
A Fine Theorem 