\documentstyle[12pt,epsf]{msart} \begin{document} \textheight 9in \begin{large} \noindent JUne 26, 1993 Feb. 12, 1996 \section*{ 7 MORAL HAZARD: HIDDEN ACTIONS OCERHEDS} \subsection{Categories of Asymmetric Information Models} \noindent We will divide games of asymmetric information into five categories, to be studied in four chapters. \noindent (1) {\bf Moral hazard with hidden actions} (Chapter 7)\\ Smith and Jones begin with symmetric information and agree to a contract, but then Smith takes an action unobserved by Jones. Information is complete. \noindent (2) {\bf Moral hazard with hidden knowledge } (or {\bf hidden information}) (Chapter 8) \\ Smith and Jones begin with symmetric information and agree to a contract. Nature then makes a move observed by Smith but not Jones, and Smith takes some action, which may be simply a report of Nature's move. Information is complete. \noindent (3) {\bf Adverse selection} (Chapter 9)\\ Nature begins the game by choosing Smith's type (his payoff and strategies), unobserved by Jones. Smith and Jones then agree to a contract. Information is incomplete. \noindent (4, 5) {\bf Signalling } and {\bf Screening} (Chapter 10)\\ Nature begins the game by choosing Smith's type, unobserved by Jones. To demonstrate his type, Smith takes actions that Jones can observe. If Smith takes the action before they agree to a contract, he is signalling; if he takes it afterwards, he is being screened. Information is incomplete. %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \newpage We will make heavy use of the ``principal-agent model'' to analyze asymmetric information. Usually this term is applied to moral hazard models, since the problems studied in the law of agency usually involve an employee who disobeys orders by choosing the wrong actions, but the paradigm will be useful in all of these contexts. The two players are the principal and the agent, who are usually representative individuals. The principal hires an agent to perform a task, and the agent acquires an informational advantage about his type, his actions, or the outside world at some point in the game. It is usually assumed that the players can make a binding {\bf contract} at some point in the game, which is to say that the principal can commit to paying the agent an agreed sum if he observes a certain outcome. In the implicit background of such models are courts which will punish any player who breaks a contract in a way that can be proven with public information. \noindent {\it The {\bf principal} (or {\bf uninformed player}) is the player who has the coarser information partition.} \noindent {\it The {\bf agent} (or {\bf informed player}) is the player who has the finer information partition.} %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \newpage \begin{tiny} \begin{center} {\bf Table 7.1 Applications of the principal-agent model} \begin{tabular}{|llll|} \hline & & & \\ & {\bf Principal} & {\bf Agent} & {\bf Effort or Type and Signal} \\ \hline & & & \\ {\bf Moral hazard with} &Insurance company & Policyholder & Care to avoid theft\\ {\bf hidden actions} & Insurance company & Policyholder & Drinking and Smoking \\ & Plantation owner & Sharecropper & Farming Effort\\ & Bondholders & Stockholders & Riskiness of corporate projects\\ & Tenant & Landlord & Upkeep of the building\\ & Landlord & Tenant & Upkeep of the building\\ & Society & Criminal & Number of robberies\\ & & & \\ \hline & & & \\ {\bf Moral hazard with} & Shareholders & Company president & Investment decision\\ {\bf hidden knowledge} & FDIC & Bank & Safey of Loans\\ & & & \\ \hline & & & \\ {\bf Adverse selection} & Insurance Company & Policyholder & Infection with the HIV virus\\ & Employer & Worker & Skill\\ & & & \\ \hline & & & \\ {\bf Signalling and } & Employer & Worker & Skill and Education\\ {\bf screening} & Buyer & Seller & Durability and Warranty\\ & Investor & Stock-issuer & Stock value and Percentage Retained\\ & & & \\ \hline \end{tabular} \end{center} \end{tiny} %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \newpage \begin{center} {\bf ``The Production Game'' } \end{center} {\bf Players}\\ The principal and the agent. \noindent {\bf Order of Play} \\ (1) The principal offers the agent a wage $w$. \\ (2) The agent decides whether to accept or reject the contract.\\ (3) If the agent accepts, he exerts effort $e$. \\ (4) Output equals $q(e)$, where $q'>0$. \noindent {\bf Payoffs}\\ If the agent rejects the contract, then $\pi_{agent} = \bar{U}$ and $ \pi_{principal} = 0$.\\ If the agent accepts the contract, then $\pi_{agent} = U(e,w)$ and $ \pi_{principal} = V(q - w).$ Denote the monetary value of output by $q(e)$, which is increasing in effort, $e$. The agent's utility function $U(e,w)$ is decreasing in effort and increasing in the wage $w$, while the principal's utility $V(q-w)$ is increasing in the difference between output and the wage. %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \newpage \noindent {\bf ``Production Game I'': Full Information.} In the first version of the game, every move is common knowledge and the contract is a function $w(e)$. The agent must be paid some amount $\tilde{w}(e)$ to exert effort $e$, where $\tilde{w}(e)$ is defined to be the $w$ that solves the participation constraint \begin{equation} \label{e1} U(e, w(e)) = \overline{U}. \end{equation} Thus, the principal's problem is \begin{equation} \label{e1} \begin{array}{cl} Maximize& V(q(e) - \tilde{w}(e ))\\ e & \\ \end{array} \end{equation} The first order condition for this problem is \begin{equation} \label{e7.1} V'(q(e) - \tilde{w}(e )) \left(\frac{\partial q }{\partial e} - \frac{\partial \tilde{w} }{\partial e} \right)=0, \end{equation} which implies that \begin{equation} \label{e7.1a} \frac{\partial q }{\partial e} = \frac{\partial \tilde{w} }{\partial e}. \end{equation} From the implicit function theorem (see Section 13.4) and the participation constraint, \begin{equation} \label{e7.2} \frac{\partial \tilde{w} }{\partial e} = - \left( \frac{\frac{\partial U }{\partial e} }{ \frac{\partial U }{\partial \tilde{w}}}\right). \end{equation} Combining equations (7.\ref{e7.1a}) and (7.\ref{e7.2}) yields \begin{equation} \label{e7.3} \frac{\partial U}{\partial \tilde{w}} \frac{\partial q }{\partial e} = - \frac{ \partial U }{\partial e}. \end{equation} Under perfect competition among the principals the profits are zero, so the reservation utility $\overline{U}$ is chosen so that at the profit-maximizing effort $e^*$, $\tilde{w}(e^*) = q(e^*)$, or \begin{equation} \label{e1} U(e^*, q(e^*)) = \overline{U}. \end{equation} The principal selects the point on the $U= \overline{U}$ indifference curve that maximizes his profits, which is at the effort $e^*$ and wage $w^*$. He must then design a contract that will induce the agent to choose this effort level. %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \newpage A number of contracts are equally effective under full information. (1) The {\bf forcing contract} sets $w(e^*) = w^*$ and $w(e \neq e^*) =0$. This is certainly a strong incentive for the agent to choose exactly $e=e^*$. (2) The {\bf threshold contract} sets $w(e \geq e^*) = w^*$ and $w(e 0$. In this game, the agent has all the bargaining power, not the principal. The participation constraint is now that the principal must earn zero profits, so $q(e) - w(e) \geq 0$. The agent will maximize his own payoff by driving the principal to exactly zero profits, so $w(e)= q(e)$. Substituting $q(e)$ for $w(e)$ to account for the participation constraint, the maximization problem for the agent in proposing an effort level $e$ at a wage $w(e)$ can therefore be written as \begin{equation} \label{e1} \begin{array}{cl} Maximize& U(e, q(e))\\ e & \\ \end{array} \end{equation} The first order condition is \begin{equation} \label{e7.100} \frac{\partial U }{\partial e} + \left(\frac{\partial U }{\partial q} \right) \left(\frac{\partial q }{\partial e} \right) =0. \end{equation} Since $\frac{\partial U }{\partial q}=\frac{\partial U }{\partial w}$ when the wages equals output, equation (7.\ref{e7.100}) implies that \begin{equation} \label{e1} \frac{\partial U}{\partial w } \frac{\partial q }{\partial e} = - \frac{ \partial U }{\partial e}. \end{equation} %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \newpage \bigskip \noindent {\bf ``Production Game III'': A Flat Wage under Certainty}. In this version of the game, the principal can condition the wage on neither effort nor output. This is modelled as a principal who observes neither effort nor output, so information is asymmetric. The outcome of ``Production Game III'' is simple and inefficient. If the wage is non-negative, the agent accepts the job and exerts zero effort, so the principal offers a wage of zero. \bigskip \noindent {\bf ``Production Game IV'': An Output-Based Wage under Certainty}. In this version, the principal cannot observe effort but can observe output and specify the contract to be $w(q)$. Now the principal picks not a number $w$ but a function $w(q)$. His problem is not quite so straightforward as in ``Production Game I'', where he picked the function $w(e)$, but here too it is possible to achieve the efficient effort level, $e^*$, despite the unobservability of effort. The principal starts by finding the optimal effort level $e^*$, as in ``Production Game I''. That effort yields the efficient output level $q^* = q(e^*)$. To give the agent the proper incentives, the contract must reward him when output is $q^*$. Again, a variety of contracts could be used. The forcing contract, for example, would be any wage function such that $U(e^*,w(q^*)) = \bar{U}$ and $U(e,w(q)) < \bar{U}$ for $e \neq e^*$. %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \newpage \bigskip \noindent {\bf ``Production Game V'': An Output-Based Wage under Uncertainty}. In this version, the principal cannot observe effort but can observe output and specify the contract to be $w(q)$. Output, however, is a function $q(e,\theta)$ both of effort and the state of the world $\theta \in {\bf R}$, which is chosen by Nature according to the probability density $f(\theta)$ as the new move (3.5) of the game. Move (3.5) comes just after the agent chooses effort, so the agent cannot choose a low effort knowing that nature will take up the slack. (If the agent can observe Nature's move before his own, the game becomes moral hazard with hidden knowledge as well as hidden actions). Because of the uncertainty of the state of the world, effort does not map cleanly onto the observed output in ``Production Game V''. A given output might have been produced by any of several different effort levels, so a forcing contract will not necessarily achieve the desired effort. Unlike in ``Production Game IV'' the principal cannot deduce that $e = e^*$ from the fact that $q=q^*$. Moreover, even if the contract does induce the agent to choose $e^*$, if it does so by penalizing him heavily when $ q \neq q^*$ it will be expensive for the principal. The agent's expected utility must be kept equal to $\bar{U}$ because of the participation constraint, and if the agent is sometimes paid a low wage because output happens not to equal $q^*$, he must be paid more when output does equal $q^*$ to make up for it. If the agent is risk averse, this variability in his wage requires that his expected wage be higher than the $w^*$ found earlier, because he must be compensated for the extra risk. There is a tradeoff between incentives and insurance. Moral hazard is a problem when the functions $q$ and $V$ are not invertible. Often if the principal could observe the state $\theta$, he could deduce effort from observed output $q(e, \theta)$, or from his own observed utility $V(q-w)$. That $q(e, \theta)$ is {\bf invertible} means that every point in the space in which $(e, \theta)$ lies is mapped onto exactly one point in the space in which $q$ lies, and every point in $q$-space is mapped onto by some point in $(e, \theta)$-space. Knowing $q$, you know $(e,\theta)$, and vice versa. If $q$ and $V$ are not invertible, the principal can neither observe the agent's effort level nor deduce it without assuming equilibrium behavior. The combination of unobservable effort and lack of invertibility in ``Production Game V'' means that no contract can induce the agent to put forth the efficient effort level without incurring extra costs, which usually take the form of imposing extra risk on the agent. We will still try to find a contract that is efficient in the sense of maximizing welfare given the informational constraints. The terms ``first-best'' and ``second-best'' are used to distinguish these two kinds of optimality. \noindent {\it A {\bf first-best contract} achieves the same allocation as the contract optimal when the principal and the agent have the same information set and all variables are contractible. } \noindent {\it A {\bf second-best contract} is Pareto-optimal given information asymmetry and constraints on writing contracts.} The difference in welfare between the first-best world and the second-best world is the cost of the agency problem. %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \newpage The first four production games were easier because the principal could find a first-best contract without searching very far. Even defining the strategy space in a game like ``Production Game V'' is tricky, because the principal is choosing a function $w(q)$ and the agent is choosing effort as a function of that function: $e(w(\cdot))$. Finding the optimal contract when a forcing contract cannot be used becomes a difficult problem, without general answers because of the tremendous variety of possible contracts. The principal's problem in ``Production Game V'' is to maximize his utility knowing that the agent is free to reject the contract entirely and that the contract must give him an incentive to choose the desired effort. These two constraints come up in every moral hazard problem, and they are named the {\bf participation constraint } and the {\bf incentive compatibility contraint}. Mathematically, the principal's problem is \begin{equation} \label{ebig} \begin{array}{cl} Maximize& EV(q(\tilde{e},\theta) - w(q(\tilde{e},\theta)))\\ w(\cdot) & \\ \end{array} \end{equation} subject to\\ \begin{tabular}{lll} & \\ & $ \tilde{ e} = \stackrel{argmax}{e}\; EU(e, w(q(e,\theta)))$& ({\bf incentive compatibility constraint}) \\ & \\ & $ EU(\tilde{e}, w(q(\tilde{e},\theta))) \geq \bar{U}$ & ({\bf participation constraint})\\ & \\ \end{tabular} %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \newpage %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \newpage \begin{center} {\bf ``Broadway Game I''} \end{center} {\bf Players}\\ The producer and the investors. \noindent {\bf Order of Play}\\ (1) The investors offer a wage contract $w(q)$ as a function of revenue $q$.\\ (2) The producer accepts or rejects the contract.\\ (3) The producer chooses to $Embezzle$ or {\it Do not embezzle}.\\ (4) Nature picks the state of the world to be $Success$ or $Failure$ with equal probability. Table 7.2 shows the resulting revenue, $q$. \noindent {\bf Payoffs.}\\ The producer is risk averse and the investors are risk neutral. The producer's payoff is $U(100)$ if he rejects the contract, where $U' >0$ and $U''<0$, and the investors' payoff is 0. Otherwise, $$ \begin{array}{l} \pi_{producer} = \left\{ \begin{array}{ll} U( w(q) + 50) & if \;he\; embezzles \\ U(w(q)) & if\;he\; is\;honest \\ \end{array} \right.\\ \pi_{investors} = q - w(q) \\ \end{array} $$ \bigskip \begin{center} {\bf Table 7.2 ``Broadway Game I'': Profits} \begin{tabular}{lllccc} & & &\multicolumn{3}{c}{\bf State of the World}\\ & & & {\it Failure} (0.5) & & {\it Success} (0.5) \\ & & $ Embezzle$ & $-100$ & & $+100$ \\ & {\bf Effort} && & & \\ & & {\it Do not embezzle } & $-100$ & & +500 \\ \end{tabular} \end{center} \newpage Another way to tabulate outputs is shown in Table 7.3. \begin{center} {\bf Table 7.3 ``Broadway Game I'': Probabilities of Profits } \begin{tabular}{lllcccc} & & &\multicolumn{4}{c}{\bf Profit}\\ & & & {\bf -100} & {\bf +100} & {\bf +500} & {\bf Total} \\ & & $Embezzle$ & 0.5 & 0.5 & 0 & 1\\ & {\bf Effort} && & & & \\ & & {\it Do not embezzle } & 0.5 & 0 & 0.5 & 1\\ \end{tabular} \end{center} The investors will observe $q$ to equal either $-100, +100$, or $+500$, so the producer's contract will specify at most three different wages: $w(-100), w(+100)$, and $w(+500)$. The producer's expected payoffs from his two possible actions are \begin{equation} \label{e7.2a} \pi(Do\; not\; embezzle) = 0.5 U ( w(-100)) + 0.5 U ( w(+500)) \end{equation} and \begin{equation} \label{e7.2b} \pi(Embezzle) = 0.5 U ( w(-100)+50) + 0.5 U ( w(+100)+50). \end{equation} The incentive compatibility constraint is $\pi(Do\; not\; embezzle) \geq \pi(Embezzle)$, so \begin{equation} \label{e7.2c} 0.5 U ( w(-100)) + 0.5 U ( w(+500)) \geq 0.5 U ( w(-100)+50) + 0.5 U ( w(+100)+50), \end{equation} and the participation constraint is that \begin{equation} \label{e7.2d} \pi(Do\; not\; embezzle) = 0.5 U ( w(-100)) + 0.5 U ( w(+500)) \geq U(100). \end{equation} The following {\bf boiling-in-oil contract} provides both riskless wages and effective incentives. \noindent $w(+500) = 100$.\\ $w(-100) = 100$.\\ $w(+100) = -\infty$. \newpage Milder contracts than this would also be effective. Two wages will be used in equilibrium, a low wage $\underline{w}$ for an output of $q=100$ and a high wage $\overline{w}$ for any other output. The participation and incentive compatibility constraints provide two equations to solve for these two unknowns. To find the mildest possible contract, the modeller must also specify a function for $U(w)$ which, interestingly enough, was unnecessary for finding the first boiling-in-oil contract. Let us specify that \begin{equation} \label{e7.2e} U(w) = 100 w - 0.1w^2. \end{equation} A quadratic utility function like this is only increasing if its argument is not too large, but since the wage will not exceed $w=1000$, it is a reasonable utility function for this model. Substituting (7.\ref{e7.2a}) into the participation constraint (7.\ref{e7.2d}) and solving for the high wage $\overline{w}= w(-100) = w(+500)$ yields $\overline{w}=100$ and a reservation utility of 9000. Substituting into the incentive compatibility constraint, (7.\ref{e7.2c}), yields \begin{equation} \label{e7.2f} 9000 \geq 0.5 U ( 100+50) + 0.5 U ( \underline{w} +50). \end{equation} When $(7.\ref{e7.2f})$ is solved using the quadratic equation, the answer is (with rounding error), $\underline{w} \leq 5.6$. A low wage of $-\infty$ is much more severe than is needed. \newpage The conditions favoring boiling-in-oil contracts are (1) The agent is not very risk averse. (2) There are outcomes with high probability under shirking that have low probability under optimal effort. (3) The agent can be severely punished. (4) It is credible that the principal will carry out the severe punishment. Another first-best contract that can sometimes be used is {\bf selling the store.} Under this arrangement, the agent buys the entire output for a flat fee paid to the principal, becoming the {\bf residual claimant}, since he keeps every additional dollar of output that his extra effort produces. This is equivalent to fully insuring the principal, since his payoff becomes independent of the agent's and Nature's moves. In ``Broadway Game I'', selling the store takes the form of the producer paying the investors 100 ($= 0.5[-100] + 0.5[+500]-100$) and keeping all the profits for himself. The drawbacks are that (1) the producer might not be able to afford to pay the investors the flat price of 100; and (2) the producer might be risk averse and incur a heavy utility cost in bearing the entire risk. These two drawbacks are why producers go to investors in the first place. \end{large} \end{document}