Breakthroughs, Deadlines, and Self-Reported Progress: Contracting for Multistage Projects

Q: What is the principal’s payoff conditional on being in the first stage?

Given u, the principal’s payoff conditional on being in the first stage (s = 1) is G(u); i.e.,F unobs(u|s = 1) = ∫ u/φ0λ2te−λt(Π− ct)dt− e−λu/φ(1 + λu/φ)cu/φ− u= (1− e−λu/φ)(Π− 2c/λ)− λu φ e−λu/φ(Π− c/λ)− u.Conditional on being in the second stage, the principal’s value function is the benchmark payoff V̄ (u):F unobs(u|s = 2) = ∫ u/φ0λe−λt(Π− ct) + e−λu/φ(−cu/φ)− u = (1− e−λu/φ)(Π− c/λ)− u.

Q: What is the boundary condition for u2(t)?

Imposing the boundary condition u2(T ) = 0, the authors arrive at a unique solution for u2(t), which the authors can then substitute back into (S.10), to arrive atw(t) = φ ( T − t+ 1λ + e−λ(T−t) λ2T + eλt λ (q(−λT )− q(−λt))) ,where q(z) = − ∫∞ −z e−x/xdx.

Q: What is the case for the principal?

Suppose that the principal can implement a simple contract in which the incentive compatibility condition holds with equality for all t (this is the best possible case for the principal).

Q: What is the boundary condition for the HJB?

λρ/φ)( 2 + λus(ρ) φ ) e−λus(ρ)/φ ≡ 0.Step 4. The authors wish to show that for ρ sufficiently small and any u1 > 0, F1(u1; ρ) > F1(u1; 0) =F1(u1).

Question

1. What are the contributions in this paper?

2. What is the reason for requiring a costly report?

3. What is the way to show that the principal can benefit from the second stage?

4. What is the value function of the contract?

Accepted Answer

In Step 4 the authors prove that, for ρ sufficiently small, the principal indeed benefits by using the formal channel.. First observe that the only possible reason for requiring a costly report is to relax the no-false-progress constraint so as to add more time to the clock following the first reported breakthrough.. This implies that the formal communication channel should only be used to report progress ( not lack of progress ).. Next, observe that it can not be optimal for the principal to require the costly formal report at date t and take no action until date t′ > t, because this is dominated by waiting until t′ to require the report.. ( The project might be completed between t and t′. ). Let y be the highest value of the low type ’ s continuation utility at which the principal requires a formal report, and let F2 ( u1, u2 ; ρ ) denote the principal ’ s value function in the second stage.

Accepted Answer

First observe that the only possible reason for requiring a costly report is to relax the no-false-progress constraint so as to add more time to the clock following the first reported breakthrough.

Accepted Answer

Let u2(t) be the promised continuation value conditional on being in the second stage and u(t) be the unconditional continuation value at t.

Accepted Answer

There exists an incentive-compatible simple contract under which the principal’s ex-anteexpected payoff is maxuG(u).(iii) For all u > 0, G(u) < F1(u).

Accepted Answer

Given u, the principal’s payoff conditional on being in the first stage (s = 1) is G(u); i.e.,F unobs(u|s = 1) = ∫ u/φ0λ2te−λt(Π− ct)dt− e−λu/φ(1 + λu/φ)cu/φ− u= (1− e−λu/φ)(Π− 2c/λ)− λu φ e−λu/φ(Π− c/λ)− u.Conditional on being in the second stage, the principal’s value function is the benchmark payoff V̄ (u):F unobs(u|s = 2) = ∫ u/φ0λe−λt(Π− ct) + e−λu/φ(−cu/φ)− u = (1− e−λu/φ)(Π− c/λ)− u.

Accepted Answer

Imposing the boundary condition u2(T ) = 0, the authors arrive at a unique solution for u2(t), which the authors can then substitute back into (S.10), to arrive atw(t) = φ ( T − t+ 1λ + e−λ(T−t) λ2T + eλt λ (q(−λT )− q(−λt))) ,where q(z) = − ∫∞ −z e−x/xdx.

Accepted Answer

Suppose that the principal can implement a simple contract in which the incentive compatibility condition holds with equality for all t (this is the best possible case for the principal).

Accepted Answer

λρ/φ)( 2 + λus(ρ) φ ) e−λus(ρ)/φ ≡ 0.Step 4. The authors wish to show that for ρ sufficiently small and any u1 > 0, F1(u1; ρ) > F1(u1; 0) =F1(u1).

Accepted Answer

If the agent does not shirk,1 then the probability that the project succeeds at t ∈ [0, T ] is given by λ2te−λt and the probability that the project does not succeed prior to T is given by e−λT (1 + λT ).

Accepted Answer

The principal’s value function for u ≤ w∗ is therefore given byF unobs(u;w∗) = µ(u;w∗)F unobs(u|s = 2) + (1− µ(u;w∗))F unobs(u|s = 1).

Accepted Answer

Conditional on progress, the evolution of u2 is given byλu2(t) = λw(t) + u ′ 2(t). (S.10)The authors want to find w(t) such that u(t) = φ(T − t) for all t ∈ [0, T ].

Breakthroughs, Deadlines, and Self-Reported Progress: Contracting for Multistage Projects

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Most frequently asked questions

1. What are the contributions in this paper?

2. What is the reason for requiring a costly report?

3. What is the way to show that the principal can benefit from the second stage?

4. What is the value function of the contract?

5. What is the principal’s payoff conditional on being in the first stage?

6. What is the boundary condition for u2(t)?

7. What is the case for the principal?

8. What is the boundary condition for the HJB?

9. What is the probability that the project succeeds at t?

10. What is the principal’s value function for u w?

11. What is the condition for the evolution of u2?

Citations

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Making agency theory work for supply chain relationships: a systematic review across four disciplines

Experimentation in Organizations

Optimal Monitoring Schedule in Dynamic Contracts

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References

Multitask Principal–Agent Analyses: Incentive Contracts, Asset Ownership, and Job Design

Managing the development of large software systems

Application of a Technique for Research and Development Program Evaluation

Critical-Path Planning and Scheduling: Mathematical Basis

A Continuous-Time Version of the Principal-Agent Problem

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