Consider the following principal-agent problem. You are a manager in charge of two workers, indexed with $i=1,2$. Each worker is asked to perform a particular task, and each can either work hard at the task or loaf (i.e., each worker can choose his action $a_i\in\{a_h,a_l\}$).
The value to you of each task will be either $\\\$10$ or $\\\$0$. The value is determined in part by how hard the worker works, but there are also random factors not within the control of the worker:
$\bullet$ If the worker works hard (i.e., $a_i=a_h$), the probability that the value to you of his task is $\\\$10$ is $0.7$. If the worker loafs (i.e., $a_i=a_l$), this probability is $0.3$;
$\bullet$ Moreover, there is a correlation between the outcomes of the two tasks. If both workers work hard, then there is $0.6$ joint probability that each will produce a $\\\$10$ outcome;
$\bullet$ If both loaf, then there is $0.2$ joint probability that each will produce a $\\\$10$ outcome;
$\bullet$ If one loafs and the other works hard, then there is a $0.25$ joint probability that each will produce a $\\\$10$ outcome.
Each worker has a utility function $U(w,a)=\sqrt{w}-a$, where $w$ is the amount of money received, and $a=0$ for loafing and $a=0.8$ for working hard. The two workers have the same utility function, and each has a reservation utility level of $1$ (i.e., the worker's expected utility must be at least $1$ to get the worker to take the job). You are risk-neutral. The problem is as follows:
Suppose that you wish to offer contracts that will induce both to work hard: Each worker must be induced to take the contract; each worker, assuming that his fellow worker is going to work hard, must be induced to work hard. What is the optimal contract to offer?
I'm familiar with the standard principal-agent problem, but I cannot understand the situation of this problem and cannot set up the conditions for individual rationality (IR) and incentive compatibility (IC).
(For reference, this is actually Problem 16.3 in Kreps' textbook "A Course in Microeconomic Theory." Any suggestion or hint would be much appreciated!)