Moral Hazard In Teams - Atlas of Economic Models
 

Moral Hazard In Teams

Contents

  1. Moral Hazard In Teams
    1. Introduction
    2. Model
    3. Solving Under Certainty
      1. Balanced Budget Constraint
      2. Principal As Budget Breaker
    4. Solving Under Uncertainty
    5. Applications and Extensions

Introduction

This model presents a form of the Principal-Agent problem with multiple agents, taken from Holmstrom's paper 'Moral Hazard In Teams' (holmstrom_1981). As with the single agent model, the actions of agents cannot be directly observed and the problem is to incentivise agents to act in the best interests of the Principal. The main issue to overcome is that of free riding - only total output is observable so individual agents may not put in effort if they do not face the right incentives to work. The Principal's primary role in this model is to break the budget constraint.

Model

There are $$ n $$ agents, who each take action $$ a_i $$ with private cost $$ v_i(a_i) $$ such that $$ v_i $$ is strictly convex, differentiable and increasing in $$ a_i $$ . The joint monetary outcome $$ x(a) $$ (total output valued in monetary terms) depends on the vector of agents' efforts $$ a=(a_1,...a_n) $$ and is strictly concave and increasing in $$ a $$ . Each agent $$ i $$ gets a share of the outcome $$ s_i(x) $$ and has the utility function $$ u_i(m_i,a_i) = m_i - v_i(a_i) $$ where $$ m_i $$ is total money received (i.e. $$ m_i = s_i(x) $$ if initial endowments are assumed to be 0) .

The aim is to find a set of sharing rules $$ s_i(x) \ge 0 \forall i $$ such that a Pareto-optimal Nash equilibrium $$ a* $$ is reached in the resulting non-cooperative game (i.e. there is an optimal tradeoff between output and agents' disutility from taking action). The condition for Pareto optimality is that $$ a* = \arg \max [x(a) - \sum_{i=1}^{n}v_i(a_i)] $$ .

Solving Under Certainty

In this case, 'certainty' means that total output is perfectly observed. However, the inputs of individual team members are imperfectly observed (where team simply refers to a group of agents such that their productive inputs are related).

Balanced Budget Constraint

The initial question is whether the Nash equilibrium $$ a* $$ can be achieved subject to a 'balanced budget' constraint $$ \sum_{i=1}^{n}s_i(x) = x $$ . Agents maximise their utility with respect to $$ a $$, such that $$ \frac{\partials_i}{\partiala_i} \frac{\partialx}{\partiala_i} - \frac{\partialv_i}{\partiala_i} = 0 $$ . Pareto optimality additionally requires that $$ \frac{\partialx}{\partiala_i} - \frac{\partialv_i}{\partiala_i} = 0 $$ , and when taken together these imply that $$ \frac{\partials_i}{\partiala_i} = 1 $$ .

However, differentiating the 'balanced budget' constraint with respect to $$ a_i $$ gives $$ \sum_{i=1}^{n} \frac{\partials_i}{\partiala_i} = 1 $$ . Therefore, each agent has only $$ \frac{1}{n} $$ of the share of output under the 'balanced budget' constraint as they would at the Pareto-optimal Nash equilibria (assuming a monotonic transformation of $$ s_i $$), so there is an incentive to underprovide action and there will be an inefficient level of output.

In general, wherever there is a 'balanced budget' constraint and there are externalities present ($$ \frac{\partialx}{\partiala_i} \ne 0 $$) efficiency will not be achieved. If agents cannot be penalised when total output is less than the Pareto-optimal level (since all output is returned to agents) and individual action cannot be perfectly observed then there will always be an incentive to free-ride.

Principal As Budget Breaker

If the 'balanced budget' constraint is relaxed so that $$ \sum_{i=1}^{n}s_i(x) \le x $$ then there will exist an $$ a* $$ that is an efficient Nash equilibrium.

Now let the sharing rule become: $$ \[ s_i(x) := \left\{\begin{array}{l l} b_i & \textrm{if } x \ge x(a*), \\ 0 & \textrm{if } x<x(a*) . \end{array}\right. \] $$

where $$ \sum b_i = x(a*) $$ and $$ b_i>v_i(a*_i)>0 $$. Pareto-optimality implies that $$ x(a*) - \sum v_i(a*_i) > 0 $$, so using the sharing rule $$ s_i(x) $$ described above $$ a* $$ is an efficient Nash equlibrium.

By relaxing the 'balanced budget' constraint there is now scope for sufficient bonuses (or penalties) to incentivise all agents. However, there is a problem with self-imposed penalties as if $$ x<x(a*) $$ ex-post it is not optimal for agents to enforce this sharing rule, and if it is expected that this will occur then the problem reverts to the balanced budget free rider problem as described above. This gives a role for a Principal who does not provide any productive input and takes any residual output left after the sharing rule has been implemented, ensuring that there is no enforcement problem. Here the Principal acts primarily as a 'budget breaker' rather than directly monitoring the actions of agents.

Solving Under Uncertainty

Under uncertainty, total output $$ x(a, \theta) $$ depends on both agents' action $$ a $$ and the state of nature $$ \theta $$ . However, in this case $$ \theta $$ is suppressed, with $$ F(x,a) $$ the conditional distribution of $$ x $$ given $$ a $$ and $$ f(x,a) $$ the conditional density function. The following assumptions are made:

  1. $$ F(x,a) $$ is convex in $$ a $$
  2. $$ \frac{\frac{\partialF(x,a)}{\partiala_i}}{F(x,a)} \rightarrow -\infty $$ as $$ x \rightarrow -\infty $$
  3. $$ \frac{\frac{\partialF(x,a)}{\partiala_i}}{(1-F(x,a))} \rightarrow -\infty $$ as $$ x \rightarrow +\infty $$

Agents' payoffs are given as before under certainty, and the sharing rule $$ s_i(x) $$ used satisfies $$ \sum_{i=1}^{n}s_i(x) \le x $$ . Under assumptions (1) and (2), a first-best solution (i.e. that under certainty) can be approximated using group penalties. If $$ \bar{x} $$ is the critical output level then let the sharing rule become: $$ \[ s_i(x) := \left\{\begin{array}{l l} s_ix & \textrm{if } x \ge \bar{x}, \\ s_ix-k_i & \textrm{if } x< \bar{x} . \end{array}\right. \] $$

where $$ k_i>0 $$ and $$ \sum s_i = 1 $$ . All output is shared if the critical output is achieved, otherwise each agent gets a penalty $$ k_i $$ .

For $$ a* $$ to be a Nash equilibrium under this $$ s_i(x) $$, it is necessary and sufficient that $$ s_i \frac{\partialE[x(a*)]}{\partiala_i} - k_i \frac{\partialF(\bar{x},a*)}{\partiala_i} - \frac{\partialv_i(a*_i)}{\partiala_i} = 0 $$ . For a fixed $$ \bar{x} $$, a value of $$ k_i $$ is chosen such that this equality holds, with the expected residual given by $$ W = \sum k_iF(\bar{x},a*) = \sum \frac{A_iF(\bar{x},a*)}{\left[\frac{\partialF(\bar{x},a*)}{\partiala_i} \right] $$ . $$ W $$ can be made arbitrarily small (i.e. the solution can come arbitrarily close to the first-best) since $$ W $$ goes to zero with $$ \bar{x} $$ from assumption (2).

However, $$ k_i $$ generally tends to infinty as $$ \bar{x} $$ decreases, and as the number of agents increases $$ a* \rightarrow 0 $$ . Unlike under certainty, endowment constraints limit the size of the team that can be effectively incentivised with a penalty scheme.

Applications and Extensions

Bibliography