Course Notes

ADVANCED MICROECONOMICS A.Y. 2015-2016

Asymmetric information → PRINCIPAL-AGENT problem

P-A problem: generally they have to conclude a deal, but the agent is in the condition that allows him to have more information.

• problem 1: HIDDEN ACTION → MORAL HAZARD
• problem 2: HIDDEN INFORMATION → ADVERSE SELECTION

The principal cannot observe what the agent does. (hidden actions). This fact implies that the actions purchased by the agent generate effects upon the principal wealth.

→ In this context, the 2 parties cannot conclude the contract

CONTRACT THEORY: When an action is not observable, it is also not contractable → The payment is no more a function of the action taken, but on the other hand, a function of the outcome

MORAL HAZARD: When an agent can do what is not in the interest of the principal, but is in the interest of the agent himself.

CONTRACT: Payment = f(outcome)

This kind of adjustment does not come for free

→ This implies a stochastic process → Also the agent becomes risk-averse hence the agent will bear a loss risk, reducing his utility → AGENCY COST

INSURANCE MARKET

Insurer PRINCIPAL → Exp.Π ⇩

Insured AGENT → behaviour → costly risky → safe

The insurer cannot observe the insured behaviors → Moral Hazard prob

The agent behaves in his interests damaging the insurer → Hidden info.

BONUS/MALUS covenant: is a mean through which the insurance can distinguish btw good and bad agents (drivers)

Price = f(outcome)

• ACTION
• 0 accident
• 1 accident
• 2 accidents

→ when N° of acc. ↑ also P ↑ If the insured want a low P, it should drive safely

RISK-AVERSE agents : want to put all the risk in the insurance company

vs

RISK-NEUTRAL principal → INCENTIVE COMPATIBLE contract

The agent bears some risk → agent exposibility → agency cost

⇒ The equilibrium contract under hidden info: 2^nd Best Contract

Other examples:

- CREDIT MARKET: Bank (P) lends money to the firm (A) without

observing the actions taken by the borrower

BANK (P)

• Firm₁
• Firm₂
• Firm₃

More or less risky firms

Firms know their behaviour

Bank cannot classify all firms

Hence, the bank offers only one price (interest rate)

LR firms will find r too high → do not accept!

HR firms will find r convenient → ACCEPT!

At the end the Bank lends money only to HR firms w/an higher

probability of default → Adverse selection problem

- Bond-Holder (P) vs Bank (A)

- Professor (P) vs Students (A) → High/low effort

↓

Excess demand situation: D > S ⇒ price ↑

In an ad. selection situation a bank will not ↑ r, but will

limit the quantity of loans (S) → CREDIT RATIONING of

customers (Agency cost)

min

• 0.1w₁ + 0.3w₂ + 0.6w₃

st. 0.1w₁ + 0.3w₂ + 0.6w₃ - 5 ≥ 9

0.1w₁ + 0.3w₂ + 0.6w₃ - 5 ≥ 0.6w₁ + 0.3w₂ + 0.1w₃ - 0

ℒ(x, λ, μ) = 0.1w₁ + 0.3w₂ + 0.6w₃ - λ(P.E.C.) - μ(I.E.C.)

ℒ'_w1 = 0

0.1 - λ( 0.1^- 1 / w₁) - μ( 0.5 · 1 / 2 1 / w₁) = 0

0.1 - λ( 1 / 2 1 / w₁) - μ( 0.5 / 2 1 / w₁) = 0

0.1 - 0.05λ + 0.25μ = 0

0.05λ / √w₁ + 0.25μ / √w₁ = 0.1

√w₁ = 0.05λ - 0.25μ / 0.1

ℒ'_w2 = 0

0.3 - λ(0.3 · 1 / w₂) = 0

0.3 - λ(0.3 · 1 / w₂) = 0

0.3 - 0.15λ / √w₂ = 0

0.15λ / √w₂ = 0.3

√w₂ = 0.15λ / 0.3

ℒ'_w3 = 0

0.6 - λ(0.6 · 1 / 2 1 / w₃) - μ(0.5 · 1 / 2 1 / w₃) = 0

0.6 - λ(0.6 / 2 1 / w₃) - μ(0.5 / 2 1 / w₃) = 0

0.6 - λ / √w₃ - μ / √w₃ = 0

0.30λ / √w₃ + 0.25μ / √w₃ = 0.6

√w₃ = 0.3λ + 0.25μ / 0.6

λ and μ sign:

- λ = μ = 0 gives Π₁₂ = 0 Not compatible

- λ > 0, μ > 0

λ = 1 - μ (1 - _Π₁₂/^Π₁₂) → λ > 1 → contradiction

λ = 1 - μ (1 - _Π₂₂/^Π₂₂) → λ < 1 Not compatible

- λ = 0, μ > 0

Π₁₂ = μ(Π₁₂ - Π₁₁) sign contradict Not compatible

- λ > 0, μ = 0 YES!

Setting I.C.C > is like we have μ = 0 and λ > 1 as in the Symmetric info scenario → Hence ∞ N° of solutions w₁*, w₂*

Can I choose the same couple of numbers?

I cannot choose 2 equal or different numbers, I should choose 2 numbers near to each other. → if equal I.C.C is violated

With reverse assumptions: P: Risk Averse

A: Risk Neutral

1. SYMMETRIC INFO

Max Π₁₂ . h(R₁ - W₁) + Π₂₂ . h(R₂ - W₂) where h = utility f(x) of P

s.t. Π₁₂W₁ + Π₂₂W₂ - a₂ ≥ M₂ Participation constr.

ℒ(w,λ) = Π₁₂ . h(R₁ - W₁) + Π₂₂ . h(R₂ - W₂) + λ (Π₁₂ W₁ + Π₂₂ W₂ - a₂ - M₂)

∂ℒ/∂W₁ = 0 {λ = ^{h'(R₁ - W₁)}/₀ + λΠ₁₂ = 0, λ = ^{h'(R₂ - W₂)}/₀

∂ℒ/∂W₂ = 0 {λ = ^{h'(R₂ - W₂)}/₀, λΠ₂₂ = 0

R₁ - W₁* = R₂ - W₂* ⇒ W₂* - W₁* = R₂ - R₁

P receives a constant return. All the risk is given to A

2. SYMMETRIC INFO → 2^nd Best contract

If P proposes 2 contracts (C_E; C_I) to the agents without knowing their actions, all A's will choose C_E because of its higher paym. even if their acting as A_ineff. → MORAL HAZARD

P could propose a pooling contract in the middle btw. the 2 contracts C_E, C_I. In this case A_eff. will leave the market and only A_ineff. will accept the pooling contract → ADVERSE SELECTION

→ Solution: SEPARATING CONTRACT where each agent will choose their contract C_A^E or C_A^I (where A stays asymmetric) also called SELF-SELECTIVE CONTRACT → self selection equilibrium

• MAX Exp. Π of P
• Exp. U_eff (C_eff1) ≥ μ₀^E
• Exp. U_ineff (C_ineff1) ≥ μ₀^I Part. consth.
• st. Exp. V_eff (C_eff1) > Exp. V_eff (C_ineft1) Incentive comp. consth.
• Exp. U_ineff (C C_ineff1) > Exp. U_ineff (C_ineft1) → Each A chooses his preferred contract

• MAX q [P_E (R₂−W₂^E) + (1−P_E) (R₁−W₁^E)] + (1−q) [P_I (R₂−W₂^I) + (1−P_I) (R₁−W₁^I)]
• st. P_E u (W₂^E) + (1−P_E) u (W₁^E) ≥ μ₀^E
• P_I u (W₂^I) + (1−P_I) u (W₁^I) ≥ μ₀^I Participation constn.
• st. P_E u (W₂^E) + (1−P_E) u (W₁^E) > P_E u (W₁^I) + (1−P_E) u (W₁^I)
• P_I u (W₂^I) + (1−P_I) u (W₁^I) > P_I u (W₂^E) + (1−P_I) u (W₁^E) Incent. comp. constn.
• Not needed to check the sign of λ and μ in the 2 cases
• P_E > P_I
• The A_ineff. bears an higher risk wrt A_eff. to get a lower payment