Strange screening game were FS contracts equals optimal contracts

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.

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