Principal_Agent_Model - Atlas of Economic Models
 

Principal Agent Model

Introduction

Use to model interactions between an uniformed 'Principal' (P) and an informed 'Agent(s)' (A). Central is the fact that the Agent take some action for or on behalf of the Principal (hence the the term 'agent') which affects the Principal's payoff and which is imperfectly observed by the Principal. Classic examples:

  • House-seller (P) and Estate Agent (A).
  • Shareholder(s) (P) and Manger(s) (A) of a company.
  • Citizens (P) and Government (A).

Most of the interest in Economics focuses on determining the optimal contract for the Principal to offer the Agent given the particular assumptions about the constraints faced by these participants and their payoff functions.

Model

P has a payoff function $$U$$ which depends on an outcome variable $$x$$. This outcome is in turn a function of some action $$a$$ taken by A: $$x=f(a)$$ ($$f$$ may also be a function of other variables but this dependency is suppressed here for notational simplicity). This action while known to A is unobserved by P who only observes $$x$$. P may offer A a (monetary wage) contract whose payoff may depends only upon observables: $$w = w(x)$$ and then A has some utility function: $$V(a,w)$$ and P's utility function $$U = U(x,w)$$

The problem: determine $$w$$ so as to maximize P's expected utility.

Remark: For this to be interesting one must have that there is some uncertainty in the function $$f$$ linking the action and the outcome (otherwise P could deduce $$a$$ from $$x$$ and the problem is trivial -- P just chooses $$a$$ to maximize $$ U(x(a),w(a))$$.

Examples

The classic basic example involves:

  • $$x = a + \epsilon$$ where $$\epsilon$$ is some stochastic shock with mean 0 and s.d. $$\sigma$$.
  • P being risk-neutral and with payoff function: $$U(x,w) = k x-w$$
  • A being risk-averse and maximizing expected utility.
    • Taking $$\epsilon$$ to be $$N(0,\sigma^{2})$$ and the classic CRRA form for the utility function one has a reduced form for (expected) utility function: $$ V(a,w) = E(w) - \frac{rVar(w)}{2} - \frac{ca^2}{2} $$ where $$r$$ is a measure of relative risk aversion and $$c$$ is the cost of effort.

    • Outside option equal to 0.
  • Restricting to linear contracts: $$w = \alpha + \beta x$$

Solving for $$a$$ (FOC + SOC) gives:

$$ \[ a^{*} = \frac{\beta}{c} \] $$

Now substitute back in to $$V$$ to get:

$$ \[ V(a^{*},w(a^{*})) = \alpha + \frac{\beta^{2}}{2c} - \frac{r \sigma^{2} \beta^{2}}{2} \] $$

Finally the principal maximizes:

$$ E U(x,w) = (k-\beta) a - \alpha = \frac{(k-\beta) \beta}{c} - \alpha $$

subject to $$ V(a^{*},w(a^{*})) \geq 0 $$.

Using the fact that the constraint must bind (if not the principal may simply reduce $$\alpha$$) and hence that $$\alpha$$ is defined by $$V = 0$$ we have:

$$ \[ max_{\beta} E(U) = \frac{k \beta}{c} - \frac{\beta^{2}}{2c} - \frac{r\beta^{2}\sigma^{2}}{2} \] $$

Solving we have:

$$ \[ \beta = \frac{k}{1+rc\sigma^{2}} \] $$

Applications and Extensions

ToDo

  • Correlated Shocks
  • Dynamic Models
  • Imperfect information (multiple types)