Appunti Advanced Microeconomics

Principal-Agent Model

1. P is risk neutral and A is risk averse

e_L < e_H, R₁ < R₂

e_H - P_H R₂ , R₂ > R₄

1-P_L R₄ , P_H > P_L

A) Symmetric Information

⇒ 1st best contract

Low Effort

Principal's problem: min P_LU(w₂)+(1-P_L)U(w₁)

subject to: P_LU(w₂)+(1-P_L)U(w₁)-e_L ≥ U̅ (PC)

Write ℓ and take ∂ℓ/∂w₁=0 & ∂ℓ/∂w₂ = 0

⇒ we obtain λ > 0 (PC binding) and λU'(w₁)=1 ; λU'(w₂)=1 ⇒ w₁=w₂ ⇒ FIXED WAGE

⇒ λU'(w)=1; w=w̅=U̅

⇒ actual risk sharing since P bears the whole risk : R₂-R₄

High Effort

Principal's problem: min P_HU(w₂)+(1-P_H)U(w₁)

subject to: P_HU(w₂)+(1-P_H)U(w₁)-e_H ≥ U̅

Write ℓ and take ∂ℓ/∂w₁ = 0 & ∂ℓ/∂w₂ = 0

⇒ obtain λ > 0 and λU'(w₂) = 1 ⇒ w₁ = w₂ = w*

to find w* substitute N = λU'(w*)

into PC and solve for U*( = e_L)

⇒ < U̅ = w̅* = w̅_H ⇒ FIXED WAGE

⇒ since R_L > e_L we get U*(e_H) > U*(e_L)

P will choose the contract based on his net expected profit = E[R̅ - w̅*(e)]

1) If E[R̅|e_H] - w*(e_H) > E[R̅|e_L] - w*(e_L), then the

1st 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 the

1st best contract is [low effort, w*(e_L)]

(b) Asymmetric Information -> 2nd best contract

Low effort

• null contract or sign symmetric info
• since there's no incentive to deviate

E^h = μL(w(E^h)) - eL > μL(w(eL)) - eL

High Effort -> moral hazard problem

s.t.

μH(w(EH)) - eH < μL(w(eH)) - eL

Principal's problem: min Aw1u1 + (1-pH)w2

s.t. pHλ(u2) + (1-pH)μ(u2) = \bar{\pi}\ \bar{\pi} >

pHW(w2) + (1-pH)u(W + w2) - eH 2μ[u2]< -eL -pHμ(u(W))-eL

Write L and take λ

• we obtain \lambda > 0 μ > 0 (PC & IC binding)

FDC becomes:

μ^L(w2) = + μ{1 −

μ(W2) = λ + μ[1 − pL

Given that pH > PL we

FDC:

μ^L(W2) E[R(E^L]- W^*(E^L), the 2nd best contract is

Agency cost paid by p

AC = E[W~^ˉ - W^*(E^H)

deadweight loss

s is worse off while A is not better off

E[R̅-W̅|eH] > E[R̅-W̅|eL]

1/3 (X1-16) + 2/3 (X2-100) > X1-161/3 (X2-100) - 2/3 (X1-16) > 02X2 - 200 - 2X1 - 32 > 02(X2-X1) > 168   ⇒ X2-X1 > 84 ⇒ H = 84

c) If X2-X1 = 100 ⇒ eH is the 2nd best contract

AC = 1/3 .16 + 2/3 .100 - 64 = 8

CREDIT RATIONING - CONFLICT BTW LENDER & BORROWER

Project's value = _X^H X_H > X_L

EL(X) = PHX_H + (1-PH)X_L

• Bank's payoff = _RB
• 1-PH ¯̅ ¯̅ ¯̅ ¯̅
• Borrower's ¯̅ ¯̅ ¯̅ ¯̅ _RF
• (firm)

1- PH ¯̅ ¯̅ ¯̅ ¯̅ ¯̅

Bank is risk-averse ⇒ bank's _RB = min(D(u+t), X¯̅ )

Firm is risk lover ⇒ firm’s _RF = max(X¯̅ - D(u+t), 0)

credit rationing with MORAL HAZARD (B.I & F riskneutral) PROJECTS' VALUE

Firm is self funded (no debt)

⇒ It selects project X¯̅ since it has a higher expected value (less risky) ⇒ efficient choice

CREDIT MKT

Type I

• 0.9 X₁ = 10
• 0.1 X₂ = 7

π = type I / type I + type II = 0.4

Type II

• 0.8 Y₁ = 12
• 0.2 X₂ = 4

Firm borrows only if E[value asset - D(1 + r)] ≥ 1

a) D = 8 r = 10% ⇒ D(1 + r) = 8.8

E(X) = 0.9 (10 - 8.8) = 1.08 ⇒ type I borrows

E(Y) = 0.8 (12 - 8.8) = 2.56 ⇒ type II borrows

Expected value of loan for bank = 0.4[0.9·8.8 + 0.1·7] + 0.6[0.8·8.8 + 0.2·4] = 8.15

b) D = 8 r = 12% ⇒ D(1 + r) = 8.96

E(X) = (10 - 8.96)·0.9 = 0.936 < 1 ⇒ type I doesn't borrow anymore

E(Y) = (12 - 8.96)·0.8 = 2.432 > 1 ⇒ type II still borrows

Expected value of loan for bank = 0.8·8.96 + 0.2·4 = 7.968

⇒ the expected value for the bank decreases ⇒ it’s not convenient for the bank to increase r since it will end up lending only to the riskier individuals

⇒ despite the nominal value of the credit increase, the expected return decreases

Unfair premium

p > π L (when mkt is not competitive)

u'(w₂)

T(L-p) / (1-π) p < λ ⇒ u'(w₁) ≥ u'(w₂)

Partial insurance

• Solve for α & λ ⇒ y₂-αp > y₂ + α(L-p) and find that α* < 1

Partial insurance leads to an increase of expected utility but lower than under full insurance

E[u(w_j)] > E[u(q)]

E[u(w)] = utility without insurance

E[u(w_j)] = μ E(q)

Maximum price a risk-averse individual is willing to pay for full insurance ( α = 1 )

Indifference condition

u(y₁-p*) = E[u(q)] ⇒ u(y₁-p*) = (1-π)u(y₁) + π u(y₂)

p* = reservation price

Assume sells for p ≤ p*

Consumer buys for p ≤ p*

Consumer doesn't buy for p > p*

Is it fair?

u(y₁-p*) = E₀u(q)

y₁-p* ≤ E₂y ⇒ y₁-p* ≤ y₁-π L

p* > π L

E[u(w)] > E[q] = π u(y₂) + (1-π)u(y₁)

μ E[q] = μ [π y₂ + (1-π) y₄]

E[u(q)] < μ E[q]