<p>Consider the setting where a principal hires an agent to do a project. Payoff from project is <span class="math-container">$\pi = \beta e$</span>, where <span class="math-container">$\beta \in \{1,2\}$</span> is the degree of the agent's talent and <span class="math-container">$e \in [0, +\infty]$</span> is the agent's effort level. The Agent has a cost of effort equal to <span class="math-container">$e^2$</span>, and agent's utility is given by <span class="math-container">$w - e^2$</span> where <span class="math-container">$w$</span> is wage. The principal's utility is given by <span class="math-container">$\pi - w$</span> where <span class="math-container">$\pi = \beta e$</span>. Probability of agent being high type (<span class="math-container">$\beta = 2$</span>) is <span class="math-container">$0.5$</span>. The principal moves first and makes a take it or leave it offer. The Agent privately knows <span class="math-container">$\beta$</span> at the time of contracting.</p>
<p><strong>Here is what I (think) I know:</strong></p>
<p><em>When principal observe agent's type</em></p>
<p>In the first-best contracts the principal will not leave any rents to the agent (i.e. <span class="math-container">$w=e$</span>), so substituting principal's objective function, taking the derivative w.r.t to <span class="math-container">$e$</span> and setting equal to zero gives the desired answer (when <span class="math-container">$\beta = 1$</span> the agent elicit <span class="math-container">$0.5$</span> units of effort, for <span class="math-container">$\beta = 2$</span> the agent elicit <span class="math-container">$1$</span> unit of effort). Principal will offer <span class="math-container">$w=1$</span> when agent is high type and <span class="math-container">$w=0.25$</span> when agent is low type.</p>
<p><em>When principal does not observe agent's type</em></p>
<p>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. <em><strong>less than the 1 unit of effort</strong></em> the principal expect when she has full information.)</p>
<p><strong>What I don't understand</strong>:</p>
<p>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 <span class="math-container">$w_1 - e_1^2 \ge 0$</span>) and incentive compatibility condition (substituting with IRC yields <span class="math-container">$ w_2 - e_2^2 \ge w_1 - e_1^2 \to w_2 = e_2^2$</span>).</p>
<p>Because the agent's utility is not dependent on the type parameter, <em><strong>I get that the optimal set of contracts simply is the same as the first-best contracts?</strong></em> I feel there is something I'm missing.</p>

CC BY-SA 4.0
2021-04-22T12:12:29.833Z
"the principal will not leave any rents to the agent (i.e. $w=e$)" - should be $w=e^2$ I think.
VARulle
CC BY-SA 4.0
2021-04-22T17:07:11.737Z
Is effort contractible?
Theoretical Economist
<microeconomics><game-theory><dynamic-games><contract-theory>
Strange screening game were FS contracts equals optimal contracts
Screeningdude
# Strange screening game were FS contracts equals optimal contracts

(Asked by: username_0 on 2021-04-22)

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.

**Tags:** microeconomics, game-theory, dynamic-games, contract-theory
> **Comments:**
> **username_1**: "the principal will not leave any rents to the agent (i.e. $w=e$)" - should be $w=e^2$ I think.  
> **username_2**: Is effort contractible?  

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## 0 answers

