Welfare Comparison

Chapter IV: Formal Requirements on Costly Information

4.6 Welfare Comparison

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.

The proof of this proposition is rather straightforward. Separating equilibia (where the agent truthfully reveals their soft information by choosing different effort levels) in both models only exists for a cost high enough 𝑐 > 𝑐¯. When the cost is lower than this threshold, in the main model, the principal knows that he cannot impose separation (the effort is cheap, and pretending to be a high type increases the policy choice of the agent for any realization of public signal). Therefore, in equilibrium, the principal requires both types to exert maximum effort and only makes a decision based on public signal 𝑒^𝑟 = 1. As for the case when the agent decides the effort level, in the best equilibrium for the principal, for low enough costs, both types exert maximum effort𝑒^𝑛 = 1. Since the effort is not very costly, they would rather pool on the same (maximum) effort level than be considered the low type. When the cost increases, however, the pooling effort starts to decrease𝑒^𝑛 < 1 in order to make up for a higher cost and not give an incentive to the low type to deviate to no public signal. Overall, we have:

for𝑐 < 𝑐,¯ 𝑒^𝑟 ≥ 𝑒^𝑛 =⇒ 𝐸 𝑈_𝑟 ≥ 𝐸 𝑈_𝑛.

Since the principal always prefers a higher effort level in pooling equilibria, when both models have pooling equilibria 𝑐 < 𝑐¯, the principal is weakly better off by imposing formal requirements. Observe that when the agent controls the effort level, we select the best-case scenario for the principal. Even with this selection criterion, the principal is strictly better off with formal requirements for high enough cost𝑐 > 0.375(2𝜃−1)(when𝑒^𝑛 < 1=𝑒^𝑟).

When the cost is high enough,𝑐 > 𝑐¯, separation can exist in both models. However, this does not mean that when the principal controls the effort level he would always prefer to separate. In fact, he might just benefit from ignoring the private signal and always requiring the maximum level of effort. In the equilibrium of the model with formal requirement, the principal is weakly better off than if he imposed the separation with 0 < 𝑒^𝑟

0 < 𝑒^𝑟

1 = 1. For the comparison between the two models, we can show that even with these effort levels (and separation), the principal would benefit from formal requirements on effort. When𝑐 = 𝑐¯, both models have equilibrium with pure separation i.e., 0 = 𝑒₀ < 𝑒₁ = 1. However, when the cost increases further, the effort levels change to:

𝑒^𝑟

0 >0, 𝑒^𝑟

1=1;

𝑒^𝑛

0=0, 𝑒^𝑛

1 < 1.

Overall, we have𝑒^𝑟

0 > 𝑒^𝑛

0, 𝑒^𝑟

1 > 𝑒^𝑛

1and since in a separating equilibrium the principal always prefers higher efforts, he is better off with formal requirement.10 The logic behind this result is straightforward. Both start out with the same pure separation:

no effort for the low agent and maximum effort for the high agent (red solid and gray dotted lines start from the same point). When the agent decides effort, the low-type agent exerts no effort, while the high-type agent exerts just enough effort to separate themselves from the low type. Consequently, when the cost increases, the high type can more easily separate themselves even with a lower effort level and𝑒^𝑛

1decreases.

This is the reason why the expected utility of the principal (red solid line on the graph) is decreasing in cost.

On the contrary, in the main model with formal requirements, the effort levels are used to extract soft information 𝑠 but also collect the maximum possible hard information (higher effort means a more precise public signal 𝑖). When the cost increases, the effort becomes a more effective mechanism for the principal to impose separation: he requires maximum effort from the high type, but can now increase the required effort for the low type as well, without violating her incentive compatibility constraint. Therefore, the difference between the effort levels shrinks with higher cost and𝑒^𝑟

0slowly converges to𝑒^𝑟

1 =1. This is the reason why the expected utility of the principal (gray dotted line and blue solid line) is increasing in cost.

The reason we chose this particular value of 𝜃 is to illustrate one more interesting feature that equilibria exhibit. Sometimes, when the agents separate themselves, revealing their soft information, the principal would rather ignore it altogether and make a decision just based on the maximum required hard information (the area with the gray dotted line on the graph). This is the case when𝜃is small enough so the soft information is not too valuable and the cost is intermediate, so the high-type agent would use it to separate themselves from the low-type agent if they were to choose their own effort levels. In fact, when the principal imposes formal requirements, he could also induce the separation by satisfying the incentive compatibility constraints of the agents, but with the intermediate cost, it would mean letting the low type off the hook by requiring relatively lower effort (𝑒^𝑟

0low). This, in combination with a weak private signal (not too high𝜃) does not make the separation worth forgoing hard information from the low type. Therefore, the principal would rather ignore

10When the principal chooses the effort level, his expected utility is always weakly more than expected utility from pooling at maximum effort level𝑒=1. For this reason, the principal is weakly better off with formal requirements comparing to any pooling equilibrium of the model where the agent chooses her own effort level.

the private reports and make a decision with the maximum possible effort in hard information.