“Contracting with Repeated Moral Hazard and Private Evaluations,” W. Fuchs (2007)

Firms often want to evaluate employees subjectively or using private information – feedback from an employee’s clients, for instance – not available to the agent. Solving repeated games with private monitoring and no verification is difficult. Using some clever mathematics, William Fuchs merges the results of MacLeod (2003), where in a one-shot game firms must burn money sometimes if they are to incentivize workers, and Levin (2003), where optimal infinite reputational contracts are considered. Fuchs is different from MacLeod in that he considers a finite repeated game, rather than a one-shot game, and different from Levin in that he shows that the “full review” property Levin uses to solve for a pseudo-optimal contract is actually restrictive: a firm can do better by bundling reviews and termination periods such that reviews are only held every T periods.

Under risk neutrality, the usual tradeoff will guide any solution: firms need to punish workers for bad realizations of output, but firms must not have an incentive to lie to the employee, since otherwise they will report output is low when it is actually high, and the proposed contract will not be an equilibrium.

In either the finite or the infinite period case, the intuition above suggests that after any realization, the agent’s continuation value must be higher if the output was higher (to incentivize him to work) and the principle’s continuation value must be the same no matter the output (to incentivize her not to lie). In the finite case, this requires money to be burned at some point if the agent is going to be incentivized to always put in effort, since sometimes that effort will result in low output, and the principle can’t earn surplus by reporting low surplus: rather, he just has to burn the money. A footnote in this paper notes that this type of money burning actually does sometimes occur: in professional baseball, when players are fined by their teams, the team gives the money to charity rather than keeping it.

It is also straightforward to show that for any finite-period relational contract in the Fuchs setting, there is a payoff-equivalent contract that just pays efficiency wages to the agent each period (i.e., pays the agent his expected production given full effort) until he is fired. No bonuses are necessary. Essentially, the principle’s full value from the original contract in period 0 is paid to her by the agent; the relational contract thereafter makes the principle indifferent between firing and not firing the agent in every period. That is, the principle has no incentive problem. Let the agent collect the remaining surplus in every period. The agent will not want to quit because he collects all of the surplus after period 0. Making the agent work with full effort until termination just requires setting the termination date such that the appropriate amount of money burning occurs.

The previous results lead immediately to the following link between unlimited money-burning in dynamic games and equilibrium results in static games: if I can burn as much value in the last period as I want, and I can also just pay the agent (accounting for discounting) his wage in the final period, then in this equilibrium there is no need to give the agent updates about how he is doing (since I expect full effort in every period anyway), and the whole problem just collapses to a static game.

In the infinite-period game, the finite results suggest that as the length of the game goes to infinity, an optimal contract burns an arbitrarily large amount of money arbitrarily far in the future. This isn’t satisfying; for one, we only release information to the agent “at the end” whch is infinitely far away. Fuchs instead endogenizes money burning by capping the amount of money burned at the total surplus of the game. He then extends Levin by considering T-period review contracts, where the principal reveals her evaluation of the agent every T periods, rather than every period as in Levin. The results above that termination contracts can be found which are payoff-equivalent to contracts involving complicated wage and bonus schemes still hold, so let a T-period review contract fire the agent with probability B if performance is “unsatisfactory” after T periods. If the employee passes evaluation, a new T-period review starts with a clean slate. Linking as many periods as possible together is optimal because the amount of money the principle needs to burn in each evaluation period is independent of the length of the period; the intuition here is that if get effort from the agent if the principle pledges to burn money far in the future, then that same pledge will be even stronger in future periods since the future where money is burned is not as far away.

There are two simple notes at the end of the paper on how to avoid money burning. If there are two agents, as in Lazaer and Rosen tournaments, the principle can credibly commit to just pay whichever agent produces higher output, so no burning is necessary. Alternatively,the principle can hire a manager, pay her a fixed wage, and have the manager report publicly whether output was good or not; since the manager’s wage is independent of the report, there are no longer any incentive problems.

Even with these nice results, the general problem of optimal relational contracting is still open; dynamic mechanisms with imperfect monitoring are hard.

ftp://ftp.cemfi.es/pdf/papers/wshop/paper%20W.Fuchs.pdf (Working paper – Final version published in AER 2007)

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