Typically, we study games under given rules, but we can also examine the optimal rules of the game, i.e., mechanism design. For example,

• optimal design of auction, e.g., first-price or second-price auction;
• negotiation versus auction;
• voting rules;
• allocation rules.

Typically, mechanism design is a broader concept than contract theory, which focus more on bilateral relationships (principal-agent problems).

There are two classes of contract design problems:

• Hidden Action (Moral Hazard), for example:
  • Manager and Worker
    • Manager employs worker
    • Worker exerts effort (not observed), which correlates to output with noises
    • Manager pays worker according to output
  • Car Insurance
    • Car insurance company offers insurance contract
    • Driver chooses quality of driving (not observed)
    • Insurance company pays for accidents
  • Shareholders and CEO
    • Shareholders choose compensation for CEO
    • CEO puts effort
    • CEO paid according to stock price
• Hidden Type (Adverse Selection), for example:
  • Manager and Worker: Worker can be hard-working or lazy
  • Car Insurance: Drivers ex ante can be careful or careless
  • Shareholders and CEO: CEO is high-quality or thief

There exists information asymmetry under both problems, however, for Moral Hazard, principal can convince agents to exert effort with the appropriate incentives, and for Adverse Selection, agents' behavior (outcome) is not affected by incentives, but by her type.

Hidden Action (Moral Hazard)

Principal-Agent Model (Linear Contract Case) [2]

<aside> Sequence of events

1. The principal offers a linear contract: $w(q) = \alpha + \beta q$.
   • $\alpha$ is the salary, $\beta$ is the bonus rate, $q$ is the output.
   • risk netral, i.e., profit is $\mathbb{E}[q-w(q)]$
2. The agent chooses whether to accept or reject the contract (with a outside option $U$).
3. If accept, the agent chooses effort $a \in A \equiv[0, \infty]$.
   • risk averse, i.e., utility is $\mathbb{E}\left[-e^{-r(w(q)-c(a))}\right]$, where $c(a)=c \frac{a^{2}}{2}$. This means that the agent has no incentive to exert effort.
4. Output $q=a+\varepsilon$ is realized, where $\varepsilon \sim \mathcal{N}\left(0, \sigma^{2}\right)$. </aside>

Consider the first-best case, where the principal chooses the action $a$ to solve

$$ \begin{array}{ll}{\max _{a, w(q)}} & {\mathbb{E}[a+\epsilon-w(q)]} \\ {\text { s.t. }} & {\mathbb{E}\left[-e^{-r(w(q)-c(a))}\right] \geq U. \quad \text { Individual Rationality (IR) }}\end{array} $$

<aside> Optimal Solution

$a^{*}=\frac{1}{c}$ and $w(q)=-\frac{\ln (-U)}{r}+\frac{1}{2 c}$.

• Since the agent coincides with the principal, there is no need for a bonus rate to provide motivation, i.e., $\beta=0$.
• (IR) must bind since the profit increases in $a$.
• The objective function is concave in $a$. </aside>

Moral hazard occurs when the principal cannot choose and observe the agent’s action. Because the principal cannot enforce a particular action, she must provide incentives to the agent. Note that there are two extreme cases of the contract:

• Full insurance (but no incentives): $w(q)=\alpha$.
• Full incentives (but no insurance): $w(q)=\beta q$.

<aside> Optimal Solution

$a(\beta) = \frac{\beta}{c}$, $\beta^{}=\frac{1}{1+r c \sigma^{2}}$, $\alpha^{}=\frac{\bar{u}}{r}-\frac{1-r c \sigma^{2}}{2 c^{2}\left(1+r c \sigma^{2}\right)^{2}}$, where $\bar{u}=\ln (-\bar{U})$.

• Unless $\beta\geq1$, the effort is less than the case of first-best.
• (IR) must bind. </aside>