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\noindent \textbf{Answers to Odd-Numbered Problems, 4th Edition of Games and
Information, Rasmusen }

\begin{center} {\bf  CHAPTER 8: Further Topics in Moral Hazard} \end{center}

\noindent     26 March 2005. 12 November 2005. 
Erasmuse@indiana.edu. Http://www.rasmusen.org.

\noindent This appendix contains answers to the odd-numbered problems in the
fourth edition of \textit{Games and Information} by Eric Rasmusen, which I am
working on now and perhaps will come out in 2006. The answers to the even-
numbered problems are available to instructors or self-studiers on request to me
at Erasmuse@indiana.edu.

Other books which contain exercises with answers include Bierman \& Fernandez
(1993), Binmore (1992), Fudenberg \& Tirole (1991a), J. Hirshleifer \& Riley
(1992), Moulin (1986), and Gintis (2000). I must ask pardon of any authors from
whom I have borrowed without attribution in the problems below; these are the
descendants of problems that I wrote for teaching without careful attention to
my sources.

\newpage

\begin{center} {\bf PROBLEMS FOR CHAPTER 8: Further Topics in Moral Hazard}
\end{center}

 \noindent
 {\bf 8.1. Monitoring with error} (easy) \\
  An agent has a utility function $U= \sqrt {w} - \alpha e$, where $ \alpha =
1$ and $e$ is either 0 or 5. His reservation utility level is $\overline{U} =
9$, and his output is 100 with low effort and 250 with high effort.  Principals
are risk neutral and scarce, and agents compete to work for them. The principal
cannot condition the wage on effort or output, but he can, if he wishes, spend
five minutes of his time, worth 10 dollars, to drop in and watch the agent. If
he does that, he observes the agent  $Daydreaming$ or $Working$, with
probabilities that differ depending on the agent's effort. He can condition the
wage on those two things, so the contract will be $\{\underline{w},\overline{w}
\}$. The probabilities are given by Table 1.
  \begin{center} {\bf Table 1:
Monitoring with Error}

 \begin{tabular}{ l|cc }
  &  \multicolumn{2}{c}{{\bf Probability of }}\\
  {\bf Effort}  & {\bf
$Daydreaming$} &{\bf $Working$}\\
  \hline $ Low $($e=0$) & 0.6 &0.4\\
  \hline $
High $($e=5$) &   0.1 & 0.9\\
 \end{tabular}
  \end{center}

\begin{enumerate}

    \item[(a)]   What are profits in the absence of monitoring, if the agent is
paid enough to make him willing to work for the principal?
\noindent \underline{\textit{Answer}.} Without monitoring, effort is low. The
participation constraint is $  \sqrt{w} - 0 \geq 9$, so $w= 81$. Output is 100, so
\textrm{profit is 19}.

\item[(b)] Show that high effort is efficient under full information.

\noindent \underline{\textit{Answer}}. High effort yields output of 250.
$\overline{U} \geq \sqrt{w} - \alpha e$ or $9= \sqrt{w} - 5$ is the
participation constraint, so $14 = \sqrt{w}$ and $w=196$. Profit is then 54.
This is superior to the profit of 19 from low effort (and the agent is no worse
off), so high effort is more efficient.

\item[(c)] If $\alpha = 1.2$, is high effort still efficient under full
information?

\noindent \underline{\textit{Answer}}. If $\alpha = 1.2$, then the wage must
rise to 225, for profits of 25, so \textrm{high effort is still efficient}. The
wage must rise to 225 because the participation constraint becomes $ 
\sqrt{w } - 1.2(5) \geq  9$.

\item[(d)] Under asymmetric information, with $\alpha = 1$, what are the
participation and incentive compatibility constraints?

\noindent \underline{\textit{Answer}}. \textrm{The incentive compatibility
constraint is} \[ {0.6 \sqrt{\underline{w}} + 0.4 \sqrt {\overline{w}} \leq 0.1
\sqrt{ \underline{w}} + 0.9 \sqrt{\overline{w}} - 5}. \]

\textrm{The participation constraint is} ${9 \leq 0.1\sqrt{\underline{w}} + 0.9
\sqrt{\overline{w}}-5}$.

\item[(e)] Under asymmetric information, with $\alpha = 1$, what is the optimal
contract?

\noindent \underline{\textit{Answer}.} From the participation constraint, $14 =
0.1 \sqrt{\underline{w}} + 0.9 \sqrt{\overline{w}} $, and $\sqrt{\overline{w}} =
\frac{14}{0.9} - (\frac{1}{9}) \sqrt{\underline{w}}$. The incentive
compatibility constraint tells us that $0.5 \sqrt{\overline{w}} = 5 + 0.5
\sqrt{\underline{w}}$, so $\sqrt{\overline{w}} = 10 + \sqrt{\underline{w}}$.
Thus, \begin{equation}  \label{e89} 10 + \sqrt{\underline{w}} = 15.6 - 0.11
\sqrt{\underline{w}} \end{equation} and $\sqrt{\underline{w}} = 5.6/1.11 =
5.05.$ Thus, ${\ \underline{w} = 25.5} $. It follows that $\sqrt{\overline{w}} =
10 + 5.05$, so ${\ \overline{w} = 226.5}$.

\end{enumerate}

 %-----------------------------------------------------------

\bigskip

\noindent 
 \textbf{8.3. Bankruptcy Constraints } \\ 
 A
risk- neutral principal hires an agent with utility function $U= w-e$ and
reservation utility $ \overline{U} = 5$. Effort is either 0 or 10. There is a
bankruptcy constraint: $w \geq 0$. Output is given by Table  4.

\begin{center} \textbf{Table  4: Bankruptcy}

\begin{tabular}{l|cc|c} & \multicolumn{2}{c}
 {\textbf{Probability of Output of}}
& \\ \textbf{Effort} & 0 & 400 & Total \\
 \hline $Low$ ($e=0$) & 0.5 & 0.5 & 1
\\ $High$ ($e=10$) & 0.2 & 0.8 & 1 \end{tabular} \end{center}

 
 

\begin{enumerate}

 \item[(a)] What would be the agent's effort choice and utility if he owned the
firm?

\noindent \underline{ {\it Answer}.}  $ e=10$, because expected output is then
360 instead of the 200 with low effort, and the agent's {  \rm utility is 350}
instead of 200.

\item[(b)] If agents are scarce and principals compete for them what will be the
agent's contract under full information? His utility?

\noindent \underline{ {\it Answer}.} Effort is high, as found in part (a).  {
\rm The wage is 360 for high effort and 0 for low} (though there are other
possibilities). {  \rm Agent utility is 350}.

 \item[(c)] If principals are scarce  and agents compete to work for them, what
will the contract be under full information? What will the agent's utility be?

  \noindent \underline{\it Answer}. Because principals are scarce, $     { U =
\overline{U}=5}$. Effort is high. {  \rm The wage is 15 if effort is high, and 0
if it is low}.

\item[(d)] If principals are scarce and agents compete to work for them, what
will the contract be when the principal cannot observe effort?  What will the
payoffs be for each player?\\

 \noindent \underline{ {\it Answer}.} An efficiency wage must be paid so that
the incentive compatibility constraint of part (d) is satisfied.  The
participation constraint is thus not binding. The {  \rm low wage will be 0},
since the principal wants to make the gap as big as possible between the low
wage and the high wage. The high wage must equal 25  to get incentive
compatibility. Hence,
  \begin{equation} \label{e88} 
 U =   0.1(0) + 0.9 (25) -10 =
{  12.5} \end{equation} $     {\pi(H) = 337.5} (= 0.1 (0-0) + 0.9 (400-25)$).
This exceeds $\pi(L) = 195  (= 0.5 (0-5) + 0.5 (400-5))$.

 \item[(e)] Suppose there is no bankruptcy constraint. If principals are the
scarce factor and agents compete to work for them, what will the contract be
when the principal cannot observe effort? What will the payoffs be for principal
and agent?

  \noindent \underline{\it Answer}. Since agents are risk neutral, selling the
store works well.  The expected wage must be 15 for the agent so that $U =
\overline{U}=5$, and an incentive compatibility constraint must be satisfied to
obtain high effort: \begin{equation} \label{e87} 0.5 w(0) + 0.5 w(400) \leq
0.1 w(0) + 0.9w(400)-10, \end{equation} which can be rewritten as $w(400) - w(0)
\geq 25.$ Many contracts can ensure this. One is to sell the store for 360 minus
10 for the high effort minus 5 for the opportunity cost, which is equivalent to
letting the agent keep all the output for a lump-sum payment of 345: $     {w(0)
= 0 + 15 -360 = -345}$ and $     {w(400) = 400+15-360= 55}$, which averages to
an expected wage of 15 and an {  \rm expected utility of 5}.   The principal's
payoff is 345.

\end{enumerate}



 




\bigskip

\noindent \textbf{8.5.  Efficiency Wages and Risk Aversion} (see Rasmusen
[1992c]) \\
 In each of two periods of work, a worker decides whether to steal
amount $v$, and is detected with probability $\alpha$ and suffers legal penalty
$p$ if he, in fact, did steal. A worker who is caught stealing can also be
fired, after which he earns the reservation wage $w_0$. If the worker does not
steal, his utility in the period is $U(w)$; if he steals, it is $U(w+v) - \alpha
p$, where $U(w_0 +v) - \alpha p > U(w_0)$. The worker's marginal utility of
income is diminishing: $U^{\prime}> 0$, $ U^{\prime\prime}<0$, and $lim_{x
\rightarrow \infty} U^{\prime}(x) = 0.$ There is no discounting. The firm
definitely wants to deter stealing in each period, if at all possible.

\begin{enumerate}

\item[(a)] Show that the firm can indeed deter theft, even in the second period,
and, in fact, do so with a second-period wage $w_2^*$ that is higher than the
reservation wage $w_0$.

\noindent \underline{\textit{Answer}.} It is easiest to deter theft in the first
period, since a high second-period wage increases the penalty of being fired. If
$w_2$ is increased enough, however, the marginal utility of income becomes so
low that $U(w_2+v)$ and $U(w_2)$ become almost identical, and the difference is
less than $\alpha P$, so theft is deterred even in the second period.

\item[(b)] Show that the equilibrium second-period wage $w_2^*$ is higher than
the first-period wage $w_1^*$.

\noindent \underline{\textit{Answer}.} We already determined that $w_2 > w_0$.
Hence, the worker looks hopefully towards being employed in period 2, and in
Period 1 he is reluctant to risk his job by stealing. This means that he can be
paid less in Period 1, even though he may still have to be paid more than the
reservation wage.

\end{enumerate}

% -----------------------------------------------------------

\bigskip

 \noindent {\bf 8.7. Machinery }  \\ Mr. Smith is thinking of buying  a custom-
designed machine from either Mr. Jones  or Mr. Brown. This machine costs 5000
dollars   to build, and it is useless to anyone but Smith.  It is common
knowledge that with  90 percent probability the machine  will be worth  10,000
dollars to Smith at the time of delivery, one year from today, and with 10
percent probability it will only be worth 2,000 dollars. Smith owns assets of
1,000 dollars.   At the time of contracting,  Jones and Brown  believe there is
there is  a 20 percent chance that Smith is actually acting as an ``undisclosed
agent'' for Anderson, who has assets of 50,000 dollars.

  Find  the price be under the following two legal regimes: (a)  An undisclosed
principal is not responsible for the debts of his agent; and (b) even an
undisclosed principal is responsible for the debts of his agent. Also, explain
(as part [c]) which rule a moral hazard model like this would tend to support.

\noindent \underline{Answer.}   (a) The zero profit condition, arising from
competition between Jones and Brown, is \begin{equation} \label{e1} -5000 + 0.9P
+ 0.1 (1000) =0, \end{equation} because Smith will only pay for the machine with
probability 0.9, and otherwise will default and only pay up to his wealth, which
is 1. This yields $P \approx 5,444$.

\noindent (b) If Anderson is responsible for Smith's debts, then Smith will pay
the 5,000 dollars. Hence, zero profits require \begin{equation} \label{e2} -5000
+ 0.9P + 0.1 (0.2) P  + 0.1(0.8)(1000) =0, \end{equation} which yields $P
\approx 5,348$.

\noindent (c)  Moral hazard tends to support rule (b). This is because it
reduces bankruptcy and the agent will be more reluctant to order the machine
when there is a high chance it is unprofitable. In the model as constructed,
this does not arise, because there is only one type of agent, but more generally
it would, because there would be  a continuum of types of agents, and some who
would buy the machine under rule (b) would find it too expensive under rule (a).

Even in the model as it stands, rule (a) leads to the inefficient outcome that a
machine worth 2,000 to Smith is not give to Smith. Rather, he pays his wealth
and lets the seller keep the machine, which is inefficient since the machine
really is worth 2000 to Smith.

This  is a question about zero-profit prices. Guessing would have been a good
idea here: it is very intuitive that the price would always be above \$5,000,
and that it would be higher if the principal never had to cover the agent's
debts.   You should be able to tell that $P>10,000$ is impossible, because Smith
would never pay it. Also, the sellers compete, so it is their profits that
provide a participation constraint, not the benefit to the buyer.


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