Graduate Microeconomics II Lecture 8: Insurance...

Graduate Microeconomics IILecture 8: Insurance Markets

Patrick Legros

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Outline

Introduction

Contingent Markets

Insurance firms

Moral Hazard

Adverse Selection

Akerlof; Outside Options

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Outline

Introduction

Contingent Markets

Insurance firms

Moral Hazard

Adverse Selection

Akerlof; Outside Options

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Outline

Introduction

Contingent Markets

Insurance firms

Moral Hazard

Adverse Selection

Akerlof; Outside Options

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Outline

Introduction

Contingent Markets

Insurance firms

Moral Hazard

Adverse Selection

Akerlof; Outside Options

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Outline

Introduction

Contingent Markets

Insurance firms

Moral Hazard

Adverse Selection

Akerlof; Outside Options

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Outline

Introduction

Contingent Markets

Insurance firms

Moral Hazard

Adverse Selection

Akerlof; Outside Options

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Introduction

I Insurance is a way for society to share risk

I Independent versus aggregate risks

I Parallel with contingent markets

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Introduction

I Insurance is a way for society to share risk

I Independent versus aggregate risks

I Parallel with contingent markets

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Contingent MarketsBasic Model

I I (identical) agents and a single good.

I Endowment of an agent is stochastic: w with probability1− π and w − L with probability π. Probability areindependent across agents

I Interpretations: loss of an asset, risk of being laid-off, risk ofbeing sick.

I vNM utility function: u increasing concave

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Q: What is a state of nature?

A: Who are the agents suffering a loss. There are 2I states of theworld, and from Arrow-Debreu, need to organize 2I markets.

x is is agent i ’s consumption of the good in state s, s = 1, 2, · · · ,S

with S = 2I .

The probability of state s is πs = πns (1− π)I−ns

We can write state-contingent endowments: w is = w if i did not

suffer a loss and w is = w − L if s/he suffered a loss. Note that only

the i-th component of s matters.

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Q: What is a state of nature?

A: Who are the agents suffering a loss. There are 2I states of theworld, and from Arrow-Debreu, need to organize 2I markets.

x is is agent i ’s consumption of the good in state s, s = 1, 2, · · · ,S

with S = 2I .

The probability of state s is πs = πns (1− π)I−ns

We can write state-contingent endowments: w is = w if i did not

suffer a loss and w is = w − L if s/he suffered a loss. Note that only

the i-th component of s matters.

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Let {ps} be a vector of contingent prices. An Arrow-Debreucompetitive equilibrium is a vector of prices p∗s and quantitiesconsumed x∗s , s = 1, 2, · · · ,S such that:

I Each agent finds x∗ to be the optimal contingentconsumption plan given prices p∗:

maxx is

S∑s=1

πsu(x is)

s.t.S∑

s=1

psxis =

S∑s=1

pswis

I The aggregate budget constraint is satisfied:

S∑s=1

x∗is =S∑

s=1

w is

An equilibrium exists and is Pareto Optimal. (Note necessarilysymmetric, but focus on symmetric allocations later on.)

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Contingent MarketsI large

When I is large, the law a large numbers implies that the numberof agents who suffer a loss is equal to πI almost surely.

Hence, almost surely, the per-capita endowment in the economy is

π(w − L) + (1− π)w = w − πL

Because agents are risk-averse, it is Pareto optimal to have eachagent consume the quantity in each state of nature; that is

x is = w − πL, for all agent i and for all state s

Hence when I goes to infinity the egalitarian optimum tendstoward the constant allocation w − πL. However, rather than usingincomplete markets, this allocation can be implemented ifinsurance is provided by firms on a competitive market.

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Insurance firms

Firms serve as intermediaries and provide insurance - smoothing ofconsumption - on an individual basis.

A typical contract specifies a premium P that the insuree pays -independently of the state of the world - and a reimbursement Rthat is paid to the insuree if there is a loss.

When contracts are individual (do not depend on what happens toother agents), the total premium paid is

PI

and the expected reimbursement is

πIR

.

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The expected utility from an individual contract is

πu(w − L + R − P) + (1− π)u(w − P)

Competition between insurers → zero profit condition

P = πR

To “recover” the Pareto optimal egalitarian allocation we need

w − L + R − P = w − P

= w − πL

Hence,P = πL & R = L

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Insurance firmsGraphical representation: state space diagram

state 1 (“no loss”)

state 2 (“loss”)

full insurance

ww − πL

w − L −1−ππ

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Insurance firmsThe insurance firm can allow agents to purchase partial insurance:if the agent insures a loss z then the price s/he pays is pz . In thiscase an agent will solve

maxx1,x2,z

(1− π)u(x1) + πu(x2)

x1 = w − pz

x2 = w − L + z − pz

(Note the change of variable: a contract gives an allocation xs

which is equivalent to paying a premium of pz and gettingreimbursed z . Hence

⇔ maxz

(1− π)u(w − pz) + πu(w − L + z − pz)

The FOC is

−(1− π)pu′(w − pz) + π(1− p)u′(w − L + z − pz) = 0

and when p = π (that is when there is zero profit) z = L, that is,the agent chooses to be fully covered.

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Moral hazard

Suppose that the probability π of a loss is a function of the care ataken by the agent:

I drive carefully to avoid a car accident

I close doors to avoid theft

I eliminate some foods to decrease the risk of heart failure

If the a has cost a, the income per capita is now w − π(a)L− a.The social optimum: full insurance and maximizeu(w − π(a)L− a), hence

(1 + π′(a)L)u′(w − π(a)L− a) = 0

⇒ 1 = −π′(a)L

The optimum cannot be incentive compatible since full insurancewould lead the agent to choose a = 0.

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Moral hazardSecond best

Cannot provide full insurance in a competitive market with moralhazard: the agent must bear some of the risk. How much?

Let z the amount of insurance and p the price. The problem forthe agent is now,

maxz,a

π(a)u(x2(z)) + (1− π(a))u(x1(z))

p = π(a)

x1(z) = w − a− pz

x2(z) = w − L− a + z − pz

π′(a)[u(x2(z))−u(x1(z))]−π(a)u′(x2(z))−(1−π(a))u′(x1(z)) = 0(1)

The incentive constraint (1) is compatible with a > 0 only ifu(x2(z))− u(x1(z)) > 0

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Adverse SelectionUnique price of coverage

Agents face different risks (Good and bad drivers). The populationis composed of good risks (πG ) and bad risks (πB > πG ). Then,

1− πG

πG>

1− πB

πB. (2)

The insurance market should discriminate between the two types.Suppose however that the price per unit of coverage is the same pfor type t. Each type chooses full insurance

u′(w − L− pzt + zt)

u′(w − pzt)=

1− πt

πt× p

1− p

By (2), we havezG < zB

Bad types choose to purchase more insurance than good types ifthe price of insurance is uniform.

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Adverse selectionState space

state 1 (“no loss”)

state 2 (“loss”)

full insurance

−1−πBπB

ww − πL

w − L

−1−πGπG

B

G

c

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Adverse selectionUniform price - continued

I Zero profit condition given price q per unit of coverage: linegoing through the initial endowment with slope between(1− πG )/πG and (1− πB)/πB .

I If maximum price: sell at zero profit to B but G types preferto bear the risk without insurance.

I If minimum price: zero profit with G but loses with B.

I Rather than uniform price: discrimination between types.I Firm offers two contracts at most. If different: separation of

types in equilibrium, in one only: pooling.I Still problem with existence of an equilibrium

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Adverse selectionExistence - Pooling

A special screening model, but must take into account thepossibility of free entry in the insurance market.

I If firms offer a pooling equilibrium: they must make zeroprofit, and the G must subsidize the B.

I why can’t a new firm offers contracts that are attractive tothe G only? Always can find a contract on the iso-profit linegoing thru the pooling equilibrium that attracts only the Gtypes: must then be consistent with positive profit!

Pooling contracts can never be part of an equilibrium.

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Adverse selectionState space

Candidate separating equilibrium: no profit from G and no profitfrom B

state 1 (“no loss”)

state 2 (“loss”)

O

ww − πL

w − LB G

pooling contract

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Adverse selectionState space

Candidate separating equilibrium: no profit from G and no profitfrom B

state 1 (“no loss”)

state 2 (“loss”)

O

ww − πL

w − LB G

pooling contract

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Therefore if the measure of type G is sufficiently high, a poolingcontract will induce both B and G to deviate from the separatingcontracts: there is no equilibrium!

I However we know that there does not exist a poolingequilibrium, hence the firm that enters and offers a poolingcontract will later face entry by a firm offering to cater only toG types.

I Refinements; Riley’s reactive equilibrium.

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AkerlofFirst best

N agents. Productivity θ and outside option r(θ). For instancer(θ) is what the agent could produce by home production.

θ ∼F Θ = [θ0, θ1]

Firms compete for workers by offering wages. Hence if the type ofthe agent is known the competitive wage is

w∗(θ) = θ

as long as the participation constraint

θ ≥ r(θ)

holds.

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AkerlofSecond best

Since there is only one instrument (the wage), it is not possible tooffer separating contracts: conditional on participating in the labormarket, all agents prefer a higher wage! remember that separationworks only if there are at least two instruments.

However, can “separate” the agents by inducing some toparticipate and the others not to participate in the labor market.

Let Θ∗ ⊆ Θ be the types who participate. Then the competitivewage must be

w∗ = µ =

∫Θ∗θdF (θ)

Now since Θ∗ = {θ : r(θ) ≤ µ}, we have

w∗ = E (θ|r(θ) ≤ µ).

This equilibrium is clearly not Pareto optimal

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AkerlofThe Lemon’s problem

[Exemple] r(θ) = r for all theta. Either w ≥ r and all workersparticipate or w < r and no worker participates. Now, sincew = µ = E (θ), if µ < r since no worker wants to participate.

[Exemple] r(θ) ≤ θ for all θ and r increasing. Then all workerswould work in firms. In a competitive equilibrium, only thoseworkers with r(θ) ≤ w work. This is the adverse selection problem:only worse types work; impossible to attract the good types.

I E.g., if θ uniform on [0, 1], and r(θ) = rθ, with r < 1. ThenE (θ|r(θ) ≤ w) = w

2r which is equal to w only if r = 1/2. Forr 6= 1/2 there is market failure.

I Possibility of multiple equilibria: in previous example, ifr = 1/2, an infinity of equilibria.

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AkerlofThe Lemon’s problem

[Exemple] r(θ) = r for all theta. Either w ≥ r and all workersparticipate or w < r and no worker participates. Now, sincew = µ = E (θ), if µ < r since no worker wants to participate.

[Exemple] r(θ) ≤ θ for all θ and r increasing. Then all workerswould work in firms. In a competitive equilibrium, only thoseworkers with r(θ) ≤ w work. This is the adverse selection problem:only worse types work; impossible to attract the good types.

I E.g., if θ uniform on [0, 1], and r(θ) = rθ, with r < 1. ThenE (θ|r(θ) ≤ w) = w

2r which is equal to w only if r = 1/2. Forr 6= 1/2 there is market failure.

I Possibility of multiple equilibria: in previous example, ifr = 1/2, an infinity of equilibria.

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