Appunti Advanced Microeconomics

Principal-Agent Model

1. P is risk neutral and A is risk averse

e_L R_L R₂↓ ↓1-p R₁e_H R_H R₄↓ ↓p R₃ R₂ > R₄R₂ > R₄p < R

A Symmetric Information

→ 1^st best contract

Low Effort

Principal's problem: min pU(w₂) + (1-p)U(w₁)s.t. pU(w₂) + (1-p)U(w₁) - e_L ≥ U̅ (PC)Write ℒ and take ∂ℒ/∂w₁ = 0 & ∂ℒ/∂w₂ = 0

⇒ we obtain λ > 0 (RC binding) andλ μ'(w₁) = 1 ⇒ w₁ = w₂ ⇒ fixed wage

to find w^* substitute Nᵣ = U₁ = w^*into PC and solve for w^*(e_L)

High Effort

Principal's problem: pU(w₂) + (1-p)U(w₁)s.t. pU(w₂) + (1-p)U(w₁) - e_H ≥ U̅ (PC)Write ℒ and take ∂ℒ/∂w₁ = 0 & ∂ℒ/∂w₂ = 0

⇒ obtain λ > 0 and λ μ'(w₂) = 1 ⇒ U₁ = U₂ = w^*λ μ'(w₁) = 1

to find w^* substitute Nᵣ = U₁ = w^*into PC and solve for w^*(e_H)

⇒ since e_H > e_L, we get w^*(e_H) > w^*(e_L)P will choose the contract based on his net expectedprofit = E[R̅ | e_H] - w^*(e_H)

1. If E[R̅ | e_H] - w^*(e_H) > E[R̅ | e_L] - w^*(e_L) then the1^st best contract is (high effort, w^*(e_H))
2. If E[R̅ | e_H] - w^*(e_H) < E[R̅ | e_L] - w^*(e_L) then the1^st best contract is (low effort, w^*(e_L))

Principal-Agent Model

• P is risk neutral and A is risk averse

Symmetric Information → 1^st best contract

Low Effort

Principal's problem: min pLw₂ + (1-p_L)w₁

s.t. pL u(w₂) + (1-p_L) u(w₁) - e_L ≥ ū (PC)

write L and take ∂L/∂w₁ = 0 & ∂L/∂w₂ = 0

⇒ we obtain λ > 0 (RC binding) and λ u'(w₁) = 1, λ u'(w₂) = 1 ⇒ w₁ = w₂ ⇒ fixed wage

to find w^* substitute NA = w₁ = w₂ into PC and solve for w^* (e_L) So, low effort entails no risk sharing since P bears the whole risk: R₂-R₁

High Effort

Principal's problem: min p_H w₂ + (1-p_H) w₁

s.t. p_H u(w₂) + (1-p_H) u(w₁) - e_H ≥ ū (PC)

write L and take ∂L/∂w₁ = 0 & ∂L/∂w₂ = 0

⇒ obtain λ > 0 and λ u'(w₁) = 1, λ u'(w₂) = 1 ⇒ w₁ = w₂ = w^* to find w^* substitute NA = w₁ = w₂ = w^* into PC and solve for w^*(e_H) ⇒ Since e_H > e_L we get w^*(e_H) > w^*(e_L) P will choose the contract based on his net expected profit = E [ R̅ - w*(e) ]

• If E [ R(e_H) ] - w*(e_H) < E [ R(e_L) ] - w*(e_L) then the 1^st best contract is [high effort, w*(e_H)]
• If E [ R(e_H) ] - w*(e_H) < E [ R(e_L) ] - w*(e_L) then the 1^st best contract is [low effort, w*(e_L)]

B) Asymmetric Information

2nd best contract

Low effort

• null contract as under symmetric info since there's no incentive to deviate
• EH s.t. U(W(_EH)) - e_L > U(W(_eL)) - e_H

High effort

moral hazard problem since U(W(_H(_eL))) - e_H < U(W(_eH)) - e_L

Principal's problem = min AW(_H) + (1-P_H)W_L

s.t. P_HU(W(₂)) + (1-P_H)U(W(₁)) = u (PC)

A: P_HU(W(₂)) + (1-P_H)U(W(₁)) - eh + E₂P_H(U(W(₁)) - eh + PLU(W(_H)) - P_HU(W(₁))

Write L and take ∂² = 0

1. and ∂ = 0

we obtain

• λ > 0
• μ > 0