No Commitment with Formal Requirements

Chapter IV: Formal Requirements on Costly Information

4.5 No Commitment with Formal Requirements

he always requires a maximum effort level 𝑒^𝑟 = 1 and makes the policy decision solely based on the realization of the public signal.

This is trivial since in case of pooling, the only information he takes into account is the public signal and therefore the principal wants it to be as accurate as possible.

Separating- Different Effort Levels

In this chapter, we consider only full separation- when the principal requires two different amounts of effort and agent types do not mix.9 The objective function that the principal maximizes is:

max

𝑒₁,𝑒₂

−𝑃(𝑠 =0)

𝑃(𝑖=0|𝑠 =0)

𝑃(𝜔 =1|𝑖 =0, 𝑠 =0) (1−𝑥₀₀)²+𝑃(𝜔 =0|𝑖 =0, 𝑠 =0)𝑥²

00

+ 𝑃(𝑖 =1|𝑠=0)

𝑃(𝜔 =1|𝑖 =1, 𝑠=0) (1−𝑥₀₁)²+𝑃(𝜔=0|𝑖=1, 𝑠=0)𝑥²

01 − 𝑃(𝑠=1)

𝑃(𝑖 =0|𝑠=1)

𝑃(𝜔 =1|𝑖=0, 𝑠=1) (1−𝑥₁₀)²+𝑃(𝜔 =0|𝑖 =0, 𝑠 =1)𝑥²

10

+ 𝑃(𝑖 =1|𝑠=1)

𝑃(𝜔=1|𝑖=1, 𝑠=1) (1−𝑥₁₁)²+𝑃(𝜔 =0|𝑖 =1, 𝑠 =1)𝑥²

11 , subject to two intensive compatibility (IC) constraints for each realization of the private signal𝑠:

𝐼 𝐶₀: 𝐸 𝑈^𝐿

𝐿(𝑒₀) =−𝑃(𝑖 =0|𝑠=0) (1−𝑥₀₀) −𝑃(𝑖 =1|𝑠=0) (1−𝑥₀₁) −𝑐 𝑒²

0 ≥

−𝑃(𝑖=0|𝑠 =0) (1−𝑥₁₀) −𝑃(𝑖 =1|𝑠=0) (1−𝑥₁₁) −𝑐 𝑒²

1=𝐸 𝑈^𝐻

𝐿 (𝑒₁)

𝐼 𝐶₁: 𝐸 𝑈^𝐻

𝐻(𝑒₁) =−𝑃(𝑖 =0|𝑠=1) (1−𝑥₁₀) −𝑃(𝑖 =1|𝑠=1) (1−𝑥₁₁) −𝑐𝑒²

1 ≥

−𝑃(𝑖=0|𝑠 =1) (1−𝑥₀₀) −𝑃(𝑖 =1|𝑠=1) (1−𝑥₀₁) −𝑐 𝑒²

0=𝐸 𝑈^𝐿

𝐻(𝑒₀). Lemma 27 When the principal imposes formal requirement that induce separation, for the optimal effort levels, the IC constraint of the low agent is binding and the IC constraint of the high type is slack.

Intuitively, if 𝐼 𝐶₀does not bind and the low agent does not exert maximum effort, then the principal would profit by increasing 𝑒₀. The low-type agent cannot exert maximum effort since this would violate her IC constraint (for both agents, it is the worst-case scenario to be considered a low type and be required to exert the maximum effort). This lemma shows that in separating equilibria, the principal

9Pure strategies let us concentrate on truthful revelation of the agent’s type in separating equi- librium ˜𝑠=𝑠.

primarily takes care of the IC constraint of the low agent, since this agent has the most incentive to hide her private information.

Corollary 28 When the principal imposes formal requirements and induces sepa- ration, the optimal strategy requires a high-type agent to exert maximum effort, i.e., 𝑒₁=1.

This result follows directly from the previous lemma. Both incentive compatibility constraints cannot be binding for the given cutoffs and 𝑒₀ ≠ 𝑒₁. Thus, 𝐼 𝐶₁is not binding. If optimal 𝑒₁ < 1, increasing it with 𝜖 small enough does not violate any IC constraints and increases the objective function, which gives us a contradiction.

Substituting all these results in𝐼 𝐶₀and solving for𝑒₀provides the optimal solution for the case when the principal imposes different requirements based on the private report of the agent.

Proposition 29 If the principal imposes formal requirements on the effort level for the public signal, separation is only possible for𝑐 > 𝑐¯. Moreover, the optimal effort levels in this case are: maximum effort for the high type𝑒^𝑟

1 = 1and intermediate effort𝑒^𝑟

0=

q−3−3𝑐+6𝜃−4𝑐𝜃+4𝑐𝜃²

𝑐(−3−4𝜃+4𝜃²) < 1for the low type.

This proposition shows that, similar to the case when the agent decides the effort level, when the principal imposes formal requirements he can only induce separation for high enough cost. The effort levels have two functions for the principal: (1) more effort increases the precision of the public signal and therefore leads to better decision rules, (2) the principal uses effort requirements to satisfy the incentive compatibility constraints of the agent and extract their private information. The optimal strategy, when separation is possible, has an intuitive structure. When the agent is reporting a low private signal, she is going against her preferences and is rewarded by being required to exert less effort in the public signal. Meanwhile, the agent with a high private report has to provide maximum effort in the public signal to support her proposal. As the cost increases, formal requirements on effort become a more effective tool for inducing separation and, therefore, the low-type agent’s required effort level increases as well. So far, we have characterized optimal strategies for the required effort in both separating and pooling cases. In equilibrium, the principal will evaluate his expected utility from both of these strategies and make a decision accordingly.

Equilibrium

In order to calculate the equilibrium effort levels when the agent can impose for- mal requirements, we compare 𝐸 𝑈_𝑝(𝑒₀ = 𝑒₁ = 1) and 𝐸 𝑈_𝑝(𝑒₀ = 𝑒^𝑟

0, 𝑒^𝑟

1 = 1).

Substituting the optimal values gives us the following result:

Proposition 30 When the principal chooses the effort levels for the public signal, in equilibrium:

• For

𝜃 < 𝜃¯ and 𝑐 < 𝑐˜

or

𝜃 > 𝜃¯ and 𝑐 < 𝑐¯

: the principal imposes the same maximum effort level after both private reports and the policy decision is made solely based on the public signal.

• For

𝜃 ≤ 𝜃¯and𝑐 ≥ 𝑐˜

or

𝜃 ≥ 𝜃¯and𝑐 ≥ 𝑐¯

: the principal imposes maximum effort level after the high private report 𝑒^𝑟

1 =1 and intermediate effort level after the low private report𝑒^𝑟

0=

q−3−3𝑐+6𝜃−4𝑐𝜃+4𝑐𝜃² 𝑐(−3−4𝜃+4𝜃²) < 1.

Figure 4.2: This figure shows equilibrium with formal requirements. For low enough costs, both agents exert maximum effort and the principal follows only the public signal. For high costs, the high-type agent exerts maximum effort and the low-type agent exerts intermediate effort level.

Case 1: The equilibrium for parameters on the white region of the graph has the following form:

𝑒^𝑟 =1;

𝑥^∗(𝑒^𝑟 =1).

There are two different situations in this case. (1) The cost of acquiring additional information is not high enough to satisfy the incentive compatibility of the low agent even when the high type agent is required maximum effort and the low type is required no effort (𝑐 < 𝑐¯). Therefore, the principal disregards the reported private signal from the agent, always requiring the maximum amount of effort and making a decision based solely on this information. In the RUC example, this would mean that the cost of filling out surveys and of providing hard information is so low that it does not give the committee enough power to extract private information from the proposer. Consequently, the RUC requires the maximum number of surveys and makes a decision only based on this "hard information." (2) The cost is high enough (𝑐 ∈ (𝑐,¯ 𝑐˜)) to induce separation, but the separation is not worth it. When the principal induces separation, he has to incentivize the low-type agent by decreasing the required effort level for her. This means forgoing some precision in the public signal after a low private signal. For low enough 𝜃 and the cost, this "forgone"

public signal is significant and it is not worth learning the private signal of a relatively incompetent agent. Therefore, even though separation is possible, the principal always requires the same maximum level of effort from the agent, and the final decision is made solely based on the public signal.

Case 2: The equilibrium for parameters on the highlighted region of the graph has the following form:

𝑒^𝑟

0 < 1, 𝑒^𝑟

1 =1.

This corresponds to a situation in which the cost is high enough to give leverage to the principal for extracting private information. In our example, for these parameters, the proposer truthfully reports her professional opinion. If this opinion supports high prices, the principal requires the maximum amount of public information (surveys), while the low agent is asked to exert intermediate effort. In both cases, the principal makes a final decision based on his updated posterior, after observing all the information. Note that, 𝑒₀ is increasing in cost 𝑐. This result is intuitive,

since a higher cost gives more leverage to the principal and enables him to require more hard information.

Next, we do the welfare comparison between the two models to see the advantages of formal requirements for the principal.