Principal-Agent Problem with Two Agents

Question

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 $a_iin{a_h,a_l}$).
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!)

Herr K. · Accepted Answer

While it's not explicitly mentioned in the question, it seems safe to assume that the manager gets to observe separately the outcome of each worker, i.e. the value of $v_i$ for $i=1,2$. If this is the case, the IR and IC conditions depend on the exact terms of the contract.
For example, suppose the manager conditions each worker $i$'s wage on both $v_i$ and $v_{-i}$. That is,
begin{equation}
w_i(v_i,v_{-i})=
begin{cases}
w^{11}&text{if $v_i=10,v_{-i}=10$}
w^{10}&text{if $v_i=10,v_{-i}=0$}
w^{01}&text{if $v_i=0,v_{-i}=10$}
w^{00}&text{if $v_i=0,v_{-i}=0$}
end{cases}
end{equation}
Further, let $p_h^{11}$ denote the probability that both workers produce the high-value outcome (indicated by the superscript $11$) when $i$ works hard (indicated by subscript $h$) and the fellow worker $-i$ also works hard (we suppress the second $h$ in the subscript as we always assume that $-i$ works hard). In more compact notations,
begin{equation}
p_h^{11}=Pr(v_i=10,v_{-i}=10 mid a_i=a_h, a_{-i}=a_h).
end{equation}
Probabilities $p_h^{10},p_h^{01},p_h^{00},p_l^{11},p_l^{10},p_l^{01},p_l^{00}$ are similarly defined.
Then, the IR constraint for worker $i$, assuming worker $-i$ is going to work hard, would be
begin{equation}
p_h^{11}sqrt{w^{11}}+p_h^{10}sqrt{w^{10}}+p_h^{01}sqrt{w^{01}}+p_h^{00}sqrt{w^{00}}-a_hge 1.
end{equation}
The IC constraint for worker $i$, again assuming worker $-i$ is going to work hard, would be
begin{multline}
p_h^{11}sqrt{w^{11}}+p_h^{10}sqrt{w^{10}}+p_h^{01}sqrt{w^{01}}+p_h^{00}sqrt{w^{00}}-a_h ge p_l^{11}sqrt{w^{11}}+p_l^{10}sqrt{w^{10}}+p_l^{01}sqrt{w^{01}}+p_l^{00}sqrt{w^{00}}-a_l.end{multline}
From the information given by the problem, you should be able to figure out the eight probabilities.
If the wage is conditioned simply on $v_i$, or
begin{equation}
w_i(v_i)=
begin{cases}
w^1&text{if $v_i=10$}
w^0&text{if $v_i=0$}
end{cases}
end{equation}
then the IR and IC for each worker would be
begin{align}
0.7sqrt{w^1}+0.3sqrt{w^0}-a_h&ge 1 tag{IR}
0.7sqrt{w^1}+0.3sqrt{w^0}-a_h&ge 0.3sqrt{w^1}+0.7sqrt{w^0}-a_l. tag{IC}
end{align}

Principal-Agent Problem with Two Agents

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