Topic 11_ Principal agent problem

Symmetric information

General case

One principal, one agent.

Pricipal offers agent a take-it-or-leave-it contract, no bargaining

Pareto efficient contract:

Suppose contract has two conponents, efforts and payoffs.

We start by assuming some level of effort, let it be efficient $e^0$$e^0$ and corresponding payoff $w^0(x_i)$$w^0(x_i)$

FOC:

By rearranging,

We know that ${\lambda }^0$$\lambda^0$ has to be strictly positive.

If there are only two plausible results (contingencies), (1) becomes:

Or

The ratio of two marginal utilities for outcomes 1 and $2=$$2=$ MRS (slope of the indifference curve) has to be equal for $\mathrm{P}$$\mathrm{P}$ and $\mathrm{A}$$\mathrm{A}$.

Then we can analyze three cases:

• Principal is risk neutral, $B^{\prime }=const$$B^\prime = const$, and Agent is risk averse. Then,

  $$\begin{array}{}\text{(6)}& U^{\prime }\left(w^0\left(x_i\right)\right)=\frac{c}{{\lambda }^0}=\text{ const. }\mathrm{\forall }i\end{array}$$

  This means that in order for :

  $$\begin{array}{}\text{(7)}& U^{\prime }\left(w^0\left(x_i\right)\right)=U^{\prime }\left(w^0\left(x_j\right)\right)\end{array}$$

  it has to be that $w^0\left(x_i\right)=w^0\left(x_j\right)$$w^0\left(x_i\right)=w^0\left(x_j\right)$ (marginal utilities of two points are the same if the two points are the same).

  Agent receives a payment independent of the result (same in both outcomes; complete insurance). In this case, principal takes responsibility of all the risks.

   

• Principal is risk averse, $B^{\prime \prime }<0$$B^{\prime \prime }< 0$, $U^{\prime }=const$$U^\prime = const$

  Then we have

  $$\begin{array}{}\text{(8)}& B^{\prime }(x_i-w^0\left(x_i\right))=\text{ const. }\mathrm{\forall }i.\end{array}$$

  which means that principal's profit has to be independent of the result, or:

  $$\begin{array}{}\text{(9)}& x_1-w^0\left(x_1\right)=x_2-w^0\left(x_2\right)=\dots =x_n-w^0\left(x_n\right).\end{array}$$

  Optimal contract is a franchise contract $w^0\left(x_i\right)=(x_i-k)$$w^0\left(x_i\right)=\left(x_i-k\right)$ where agent keeps $X$$X$ and pays the principal a fixed amount $k$$k$ (sell the store to the agent at price $k$$k$ ).

  这个意味着Principal保留的钱在不同状态是一样的。这种情况下 agent承担了全部风险。

• Suppose both of them are risk averse

  Rewrite (FOCs) as:

  $$\begin{array}{}\text{(10)}& -B^{\prime }(x_i-w^0\left(x_i\right))+{\lambda }^0{U}^{\prime }\left(w^0\left(x_i\right)\right)=0;\mathrm{\forall }i.\end{array}$$

  Differentiate w.r.t. $x_i$$x_i$ (not entirely correct because our framework is discrete, but O.K.):

  $$\begin{array}{}\text{(11)}& -B^{\prime \prime }(1-\frac{d{w}^0}{d{x}_i})+{\lambda }^0{U}^{\prime \prime }\frac{d{w}^0}{d{x}_i}=0\end{array}$$

  Factoring out $\lambda$$\lambda$ and using (1) gives:

  $$\begin{array}{}\text{(12)}& \frac{-B^{\prime \prime }}{B^{\prime }}(1-\frac{d{w}^0}{d{x}_i})+\frac{U^{\prime \prime }}{U^{\prime }}\frac{d{w}^0}{d{x}_i}=0\end{array}$$

  which can be rewritten using the definition of the coefficient of absolute risk aversion:

  $$\begin{array}{}\text{(13)}& \frac{d{w}^0}{d{x}_i}=\frac{{\gamma }_P}{{\gamma }_P+{\gamma }_A}\in (0,1)\end{array}$$