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

CC BY-SA 4.0
2022-08-24T11:24:13.593Z
Dear Herr K. I have a principal agent problem. And I confused too much. Please can you look at my question ? Thank you so much. https://economics.stackexchange.com/questions/52468/principal-agent-problem
studentp
CC BY-SA 4.0
2021-01-04T22:27:07.663Z
2021-01-04T22:27:07.663Z
Herr K.
<p>Consider the following principal-agent problem. You are a manager in charge of two workers, indexed with <span class="math-container">$i=1,2$</span>. 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 <span class="math-container">$a_i\in\{a_h,a_l\}$</span>).</p>
<p>The value to you of each task will be either <span class="math-container">$\\\$10$</span> or <span class="math-container">$\\\$0$</span>. 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:</p>
<p><span class="math-container">$\bullet$</span> If the worker works hard (i.e., <span class="math-container">$a_i=a_h$</span>), the probability that the value to you of his task is <span class="math-container">$\\\$10$</span> is <span class="math-container">$0.7$</span>. If the worker loafs (i.e., <span class="math-container">$a_i=a_l$</span>), this probability is <span class="math-container">$0.3$</span>;</p>
<p><span class="math-container">$\bullet$</span> Moreover, there is a correlation between the outcomes of the two tasks. If both workers work hard, then there is <span class="math-container">$0.6$</span> joint probability that each will produce a <span class="math-container">$\\\$10$</span> outcome;</p>
<p><span class="math-container">$\bullet$</span> If both loaf, then there is <span class="math-container">$0.2$</span> joint probability that each will produce a <span class="math-container">$\\\$10$</span> outcome;</p>
<p><span class="math-container">$\bullet$</span> If one loafs and the other works hard, then there is a <span class="math-container">$0.25$</span> joint probability that each will produce a <span class="math-container">$\\\$10$</span> outcome.</p>
<p>Each worker has a utility function <span class="math-container">$U(w,a)=\sqrt{w}-a$</span>, where <span class="math-container">$w$</span> is the amount of money received, and <span class="math-container">$a=0$</span> for loafing and <span class="math-container">$a=0.8$</span> for working hard. The two workers have the same utility function, and each has a reservation utility level of <span class="math-container">$1$</span> (i.e., the worker's expected utility must be at least <span class="math-container">$1$</span> to get the worker to take the job). You are risk-neutral. The problem is as follows:</p>
<blockquote>
<p>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?</p>
</blockquote>
<p>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).</p>
<p>(For reference, this is actually Problem 16.3 in Kreps' textbook &quot;A Course in Microeconomic Theory.&quot; Any suggestion or hint would be much appreciated!)</p>

CC BY-SA 4.0
2021-01-04T17:54:17.507Z
Call the two agents Ann and Bob. To find the appropriate contract for Ann, simply assume that Bob works hard. Afterward, switch the roles of Ann and Bob. In either case, you have to solve only a single agent principal-agent problem.
Michael Greinecker
<microeconomics><principal-agent><contract-theory>
Principal-Agent Problem with Two Agents
rudgns55
# Principal-Agent Problem with Two Agents

(Asked by: username_0 on 2021-01-04)

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!)

**Tags:** microeconomics, principal-agent, contract-theory

> **Comments:**
> **username_1**: Call the two agents Ann and Bob. To find the appropriate contract for Ann, simply assume that Bob works hard. Afterward, switch the roles of Ann and Bob. In either case, you have to solve only a single agent principal-agent problem.  

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## 1 answers

### Answer by username_2 on 2021-01-04. Score: 4 | 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\_h\ge 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}
0.7\sqrt{w^1}+0.3\sqrt{w^0}-a\_h&\ge 1 \tag{IR}\\
0.7\sqrt{w^1}+0.3\sqrt{w^0}-a\_h&\ge 0.3\sqrt{w^1}+0.7\sqrt{w^0}-a\_l. \tag{IC}
\end{align}

> **Comments:**
> **username_3**: Dear username_2. I have a principal agent problem. And I confused too much. Please can you look at my question ? Thank you so much. https://economics.stackexchange.com/questions/52468/principal-agent-problem  

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