Consider the setting where a principal hires an agent to do a project. Payoff from project is $\pi = \beta e$, where $\beta \in \{1,2\}$ is the degree of the agent's talent and $e \in [0, +\infty]$ is the agent's effort level. The Agent has a cost of effort equal to $e^2$, and agent's utility is given by $w - e^2$ where $w$ is wage. The principal's utility is given by $\pi - w$ where $\pi = \beta e$. Probability of agent being high type ($\beta = 2$) is $0.5$. The principal moves first and makes a take it or leave it offer. The Agent privately knows $\beta$ at the time of contracting.

Here is what I (think) I know:

When principal observe agent's type

In the first-best contracts the principal will not leave any rents to the agent (i.e. $w=e$), so substituting principal's objective function, taking the derivative w.r.t to $e$ and setting equal to zero gives the desired answer (when $\beta = 1$ the agent elicit $0.5$ units of effort, for $\beta = 2$ the agent elicit $1$ unit of effort). Principal will offer $w=1$ when agent is high type and $w=0.25$ when agent is low type.

When principal does not observe agent's type

If the principal still offers the first-best contracts, the high talent agent has incentives to choose the highest payment scheme (i.e. the first-best contracts with the highest wage) but minimize his effort (i.e. less than the 1 unit of effort the principal expect when she has full information.)

What I don't understand:

In the environment where the principal does not observe the type of agent, I'm struggling to derive the individual rationality condition (I think it's $w_1 - e_1^2 \ge 0$) and incentive compatibility condition (substituting with IRC yields $ w_2 - e_2^2 \ge w_1 - e_1^2 \to w_2 = e_2^2$).

Because the agent's utility is not dependent on the type parameter, I get that the optimal set of contracts simply is the same as the first-best contracts? I feel there is something I'm missing.

  • 1
    $\begingroup$ "the principal will not leave any rents to the agent (i.e. $w=e$)" - should be $w=e^2$ I think. $\endgroup$ – VARulle Apr 22 at 12:12
  • $\begingroup$ Is effort contractible? $\endgroup$ – Theoretical Economist Apr 22 at 17:07

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