Preliminary analysis

Before proceeding to the main analysis, we briefly describe the first-best bench- mark, and analyze the model’s solution when either Assumption 1 or Assumption 2 is relaxed.

Suppose that the principal is both able to observe the state of the world and to monitor search by the agent. Then, in the first best, search by the agent maximizes the (joint) surplus of search. Therefore first-best search policies are identical to the optimal search policies in the standard search model. We skip the derivation and simply state the solution in the following observation. For details, see for instance McCall (1970).

Observation: In the first best the agent searches as long as for all previously sampled solutions it holds that x ≤x¯FB(θ). Otherwise he stops search and adopts the last-sampled solution. The first-best stopping rule is given by a functionx¯FB∶ Θ→X, which is defined pointwise, such thatx¯FB(θ)for a given state θ is pinned down by the following condition:

c=

∫

In the first best, an agent who knows the world to be in stateθ, searches for better solutions until he finds one of at least a value of ¯xFB(θ). The optimal “stopping rule” ¯xFB(θ)equates the marginal expected benefits of finding a better

solution than ¯xFB(θ)with the marginal costs of searchingc.

To better understand the mechanics of our model consider now a situation where the principal is able to observe and verify the selection of solutions which the agent has sampled, but faces uncertainty from not knowing the true state of the world. This resembles a situation where assumption 1 holds, but 2 does not. In this case, the first-best outcome, in which the agent pursues the first-best optimal search policies in every state of the world, can be implemented by using a simple contractual arrangement. Essentially all we have to do is compensate the agent for his search costs independently from the search policy he pursues, and then he finds it (weakly) optimal to search according to the first best. We state the precise result in the following.

Proposition 1: Suppose that Assumption 1 holds, but that the principal is able to observe and verify the selection of solutions which the agent has sampled. Then the first-best search policies can be implemented by paying a transfer T(N)to the agent after he adopts a solution, where T(N) =N c, and N is the number of solutions in the final sample.

Proof Sketch. Since for the above described contract the agent breaks even in- dependently of his search behavior, there exists a first-best equilibrium, where the agent accepts the contract, pursues first-best search policies, and adopts the solution that yields the highest value. This is trivially true as none of these choices is payoff-relevant under the considered contract. Moreover, it is also trivially true that the principal has no incentive to deviate from the first-best equilibrium by offering another contract.

The point here is that by offering a contract that fully compensates the agent for his search costs, the principal can provide a contract, in which the agent’s private knowledge about the state of the world is not payoff-relevant to him. As a result, the agent is willing to reveal the state of the world without any explicit incentives. Critical to this contract is that the principal is able to verify the sampled selection of solutions, allowing him to assess theactualcosts of the agent. This is not possible anymore once we introduce Assumption 2, preventing the principal from differentiating bad luck while searching from a fundamentally bad distribution. In this sense Assumption 2catalyzesAssumption 1 by rendering the agent’s private information necessarily payoff-relevant for any non-constant contract.

Note, however, that also in the case where Assumption 2 holds but Assump- tion 1 does not, there exists again a simple contract which implements the first-best search policies. In this case it is sufficient that the agent is the residual claimant, as it then will be in his own interest to pursue first-best search policies.

Proposition 2: Assume 2. Suppose that the principal learns the state of the world prior to contracting the agent. Then first-best search policies can be implemented by paying a transfer T(x)to the agent after he adopts a solution, where T(x) = −x¯FB(θ) +x, andx¯FB(θ)is the first-best stopping rule in state θ.

Proof Sketch. Since the agent effectively becomes the residual claimant under the described contract, search obviously is efficient. The only question is, whether both the principal and the agent would agree to the price ¯xFB(θ)that the agent pays to become the residual claimant. To see that this is indeed the case, note that by condition (1), the first-best expected surplus, [F(¯ x¯FB(θ)∣θ)]−1 × ×(∫x¯BF B(θ)x

′

dF(x′∣θ) −c), is equal to ¯xFB(θ). Hence the principal reaps all the surplus, and therefore happily proposes this contract, which the agent accepts in equilibrium since he breaks even.

As before, this contract is not feasible anymore as soon as we introduce both informational asymmetries simultaneously. The reason is that withθ unknown

the price ¯xFB(θ)will be subject to adverse selection, as originally pointed out by Akerlof (1970). In this sense, adverse selection unleashes the moral hazard problem, much alike risk aversion and limited liability do in other moral hazard setups.

We conclude that whenever the principal is either able to perfectly monitor search by the agent, or is equally well informed about the state of nature, delegated search comes without any efficiency loss and is identical to search by a single agent. Only when the agent has some informational advantage about the state of world (e.g., because he is an expert)andsearch cannot be perfectly monitored, delegated search may differ from the standard search model. The remainder of the chapter analyzes how to optimally deal with such a situation.