Principal-Agent: Incentives, Monitoring, Multitasking

Theory of principal-agent studies how to structure contracts under asymmetric information about effort. Applications cover corporate governance (CEO–shareholders), insurance, regulation, government procurement, healthcare. Holmstrom and Milgrom (Nobel Prize 2016) laid the foundation of this theory.

Basic PA Model (LEN-framework)

Structure: The principal (risk-neutral, P-firm) hires an agent (risk-aversion r, Arrow-Pratt coefficient). Effort e ∈ [0,∞) is unobservable. Output: y = e + ε, ε ~ N(0, σ²). Linear contract: w(y) = α + β·y (fixed part α + variable β·y).

Optimum under LEN:

Agent's utility: U = E[w] − (r/2)Var[w] − c(e) = α + βe − (rβ²σ²)/2 − e²/2

Agent's optimality condition (IC): differentiating with respect to e: e*(β) = β (when c(e) = e²/2).

Agent's participation condition (IR): U ≥ 0 → α = (rβ²σ²)/2 − β²/2 + U₀.

Principal's problem: max_{α,β} E[y − w] = e*(β) − α − β·e*(β) = (1−β)·β − α. Substituting IR: max_β β − β²/2 − rβ²σ²/2 = β − (1/2 + rσ²/2)β².

Optimal β:*

β* = 1/(1 + rσ²)

Decoding: r = 0 (risk-neutral agent) or σ² = 0 (no risk): β* = 1 (first best — agent takes the entire outcome). As r or σ² increase: β* decreases → incentives weaken, agent is more protected from risk. The compromise: incentives vs insurance.

Loss of efficiency: e*(β*) = β* = 1/(1+rσ²) < 1 = e*_FB. Information asymmetry creates an irreducible loss.

Multitasking (Holmstrom-Milgrom, 1991)

Extended model: Agent solves K tasks (e₁,...,e_K). Principal measures only part of outputs. Contract w = α + β₁y₁ (only for task 1).

Effort distortion: Agent allocates effort as follows: e₁/e₂ = β₁/0 → with β₁ > 0: e₂ → 0. The higher β₁ — the greater the distortion away from task 2 (unobservable).

Multitasking theorem: Optimal β₁ is lower than the “one-sided” optimum, if tasks are interconnected (complementary) or if task 2 is important and unobservable.

Practical consequences:

• Teachers based on KPIs for tests → teaching to the test, ignoring critical thinking
• Managers on quarterly sales → neglect of R&D and reputation
• Doctors based on number treated → excessive procedures, ignoring chronic patients
• Workers based on one metric → all efforts towards it (Goodhart’s Law: “when a measure becomes a target, it ceases to be a good measure”)

Tournaments (Lazear-Rosen, 1981)

Idea: Instead of absolute incentive — relative ranking. Winner gets W₁, loser — W₂ < W₁. Each’s effort: e* = (W₁ − W₂) · f'(e* − e*) where f is the density of noise in difference. With symmetric noise: e* follows from 1 = (W₁ − W₂)·g(0) (g is the density of difference in noise).

Advantages: Eliminates common noise (if both agents experience the same shock — it is “subtracted” when comparing). Does not require knowledge of absolute production level.

Problems: Incentives for sabotage (lowering competitor’s result). “Collusion”: with few participants — it is beneficial that both choose low effort. Ineffective with heterogeneous abilities.

Optimal tournament structure: Sport: large gap W₁ − W₂ is needed to compensate for high effort and noise. Corporate “tournaments”: CEO salary is explained not by marginal productivity, but by the need to incentivize top managers below the level (Gabaix & Landier, 2008).

Numerical example

PA: r = 1, σ² = 2, c(e) = e²/2. β* = 1/(1+1·2) = 1/3. e* = 1/3. α* from IR: α + (1/3)² − (1·(1/3)²·2)/2 = 0 → α = −1/9 + 1/9 = 0 (for U₀ = 0). Principal: E[y − w] = e* − α − βe = 1/3 − 0 − (1/3)² = 1/3 − 1/9 = 2/9. First-best (e_FB = 1): principal gets 1 − 0 − 0 = 1. Loss: 1 − 2/9 = 7/9 ≈ 78% due to asymmetry with r=1, σ²=2.

With σ² = 0 (no noise): β* = 1, e* = 1, principal gets 1 − 0 − 0 = 1. First best! Information asymmetry without risk creates no losses.

Assignment: Agent with r=1, σ²=2, c(e)=e²/2, principal with U₀=0. (1) Find β*, e*, α*, principal's profit. (2) Calculate efficiency loss vs first-best. (3) How will β* change if σ²=4? Interpret. (4) Multitasking: add task 2 with e₂, β₂=0. Task 1 weighs w₁=0.6, task 2 — w₂=0.4 in the principal’s “true” utility. Agent allocates e₁=β₁·w₁/(β₁+0), e₂=0. Find the “cost” of measuring only task 1.