# Why does the incentive compatibility constraint bind (moral hazard)?

Consider a very basic model of moral hazard with two possible effort levels $$e_L and two possible levels of output $$y_L. If the agent doesn't participate, they get utility of $$\bar{u}$$. If they participate, they get $$E[u(w)|e]-c(e)$$ where $$u'(w) > 0$$, $$u''(w) < 0$$ and $$c(e_L). In other words, the agent is risk averse and dislikes effort. Putting in more effort raises the chance that output is high, i.e. $$P(y=y_H)=p$$ if $$e=e_H$$ and $$P(y=y_H)=q$$ if $$e=e_L$$ where $$1>p>q>0$$. The principal chooses a contract $${w_H, w_L}$$ which specifies a wage following $$y=y_H$$ and $$y=y_L$$ respectively. Since they are risk neutral, they choose $$w_H$$ and $$w_L$$ to maximise $$E[y - w]$$.

Let us suppose that the principal wishes to induce $$e=e_H$$.

The optimal contract must satisfy an 'incentive compatibility' constraint:

$$pu(w_H)+(1-p)u(w_L)-e_H\geq qu(w_H)+(1-q)u(w_L)-e_L$$

In the optimal contract, this must hold with equality ('bind'). If I recall correctly, this has something to do with optimal risk sharing (more precisely, that the principal doesn't want to expose the agent to `unnecessary' risk). However, I would be very grateful if anyone could provide a more precise (but ideally still intuitive) explanation for why this must be the case.

• I don't think whether IC binds has anything to do with optimal risk sharing. Optimal risk sharing occurs in the first best, wherein the risk-neutral principal bears all the risk and conditions wage to the risk-averse agent only on effort (which is observable in the first best scenario). IC is a condition relevant only in the second best scenario, not the first best. – Herr K. Jan 21 '19 at 17:09
• You may well be right about this... at any rate, do you know of any intuitive explanation as to why IC must bind? (Whether this involves notions of risk sharing or not.) – user17900 Jan 21 '19 at 17:35
• It would help if you could write down the model you have in mind, since ultimately the argument hinges on the mathematical details. – Herr K. Jan 21 '19 at 20:00
• Sure, which details of the model are missing? I could introduce some notation, but I'm not sure whether this will add anything to the description in the first paragraph. – user17900 Jan 22 '19 at 10:17
• @HerrK. Have introduced some symbols, hope this makes things clearer! – user17900 Jan 22 '19 at 14:10

The principal faces two constraints: The individual rationality (IR, or participation) constraint: $$$$pu(w_H)+(1-p)u(w_L)-e_H\ge \overline u, \tag{IR}$$$$ and the incentive compatibility (IC) constraint: $$$$pu(w_H)+(1-p)u(w_L)-e_H\geq qu(w_H)+(1-q)u(w_L)-e_L. \tag{IC}$$$$ Suppose IR holds with $$>$$, i.e. it doesn't bind. Then it means principal is paying too much for the bad outcome; the principal can increase profit by lowering $$w_L$$ without violating either IR (by supposition it's not binding) or IC (since $$p>q$$). It follows that IR must hold with $$=$$.

Now that IR binds, if IC doesn't, then it means the principal is "over-paying" the agent to induce high effort. In particular, it must be that the principal is over-paying for the good outcome. To increase profit, the principal could reduce $$w_H$$. At the same time, to ensure that IR is not violated, $$w_L$$ has to increase by an appropriate amount so that IR still binds.

Lowering $$w_H$$ and raising $$w_L$$ effectively reduces the risk in the agent's compensation, since the two outcomes are now closer. As a result, the efficiency cost incurred by letting the agent bear risk due to incomplete information is reduced. The saving would thus accrue to the principal as profit. Therefore, IC must bind at the optimum.

• Thanks for this answer, not sure why it was downvoted! Just to clarify, do we imagine the principal reducing $w_H$ and increasing $w_L$ in a ratio $p$:$(1-p)$? (And thus making the agent strictly better off?) Also, I think you mean 'bind' not 'hold' in your last sentence. – user17900 Jan 24 '19 at 18:52
• @afreelunch: The exact magnitude of the adjustment in the wages will depend on $p$ as well as the curvature of the utility function. In essence, the wage adjustment is done in such a way that maintains the certainty equivalent implied by the utility function under distribution $(p,1-p)$. Thus, it's unlikely to be exact the ratio of $p:(1-p)$. Thanks for noting the wording at the end; it's corrected. – Herr K. Jan 24 '19 at 20:32

The idea behind the incentive compatibility constraint is that the expected utility when doing the high effort level needs to be as least as large as the one the agent would obtain from doing the low effort level. There are many contracts that satisfy the IC. Given that the agent is risk averse, i.e., has strictly convex indiference curves, the further the asymmetric information contract deviates from the symmetric information contract, the higher will be the expected payoff the agent receives. Of course, the higher the expected payoff for the agent, the lower the expected payoff for the principal. Hence, the principal will choose that contract that a) provides the agent with just her reservation utility and b) the IC is satisfied as an equality.

Perhaps what you have in mind is that, since the expected utility for the agent is the same irrespective of the effort level she exerts, she may as well end up doing the low effort level anyway. But recall that the agent is indifferent between the two effort levels, and, therefore, we can assume that she chooses to do the high effort level.

• This seems more like a request for clarification than an answer. – Giskard Jan 21 '19 at 21:43
• @denesp, I think my first paragraph provides a full answer. In the second, I just wanted to suggest a possible source of confusion – Patricio Jan 21 '19 at 21:48
• @Patricio You write: 'the incentive compatibility constraint needs to be binding, otherwise, the agent would choose to do the low effort level'. But this is not true! If IC holds with strict inequality, the agent will still want to choose the high effort level. – user17900 Jan 22 '19 at 10:16
• @afreelunch, I've edited my answer to clarify – Patricio Jan 23 '19 at 17:58