chapter thirty -three · cartel’s problem is now solution is not generally the same as for the...

Chapter Thirty-Three

Law and Economics

Effects of Laws

Property right assignments affect– asset, income and wealth

distributions;e.g. nationalized vs. privately

owned industry.

Effects of Laws

Property right assignments affect– asset, income and wealth

distributions;e.g. nationalized vs. privately

owned industry.– resource allocations; e.g. the tragedy of the commonse.g. patents encourage research.

Effects of Laws

Punishments affect– incentives for illegal behavior;e.g. high speeding fines can

reduce the amount of speeding.

Effects of Laws

Punishments affect– incentives for illegal behavior;e.g. high speeding fines can

reduce the amount of speeding.– asset, income and wealth

distributions;e.g. jail time results in lost

income.

Crime and Punishment

x is the quantity of an illegal activity produced by an individual.

C(x) is the production cost.B(x) is the benefit.Gain is B(x) - C(x).What is the rational choice of x?

Crime and Punishment

x≥0max B x C x( ) ( ).−

First-order condition is

′ = ′B x C x( ) ( ).

Notice that marginal costs matter more thando total costs.

Crime and Punishment

′ = ′B x C x( ) ( )

B(x)

C(x), low MC

xx *

Crime and Punishment

′ = ′B x C x( ) ( )

B(x)

C(x), low MC

xx *

C(x), higher, butsame MC

No change to illegal activity level.

Crime and Punishment

′ = ′B x C x( ) ( )

B(x)

C(x), low MC

xx *

Crime and Punishment

′ = ′B x C x( ) ( )

B(x)

C(x), low MC

C(x), high MC

xx *Higher marginal costs deter crime.

Crime and Punishment

Detection of a criminal is uncertain.e is police effort.π(e) is detection probability;

π(e) = 0 if e = 0π(e) ↑ as e ↑.

Crime and Punishment

Given e, the criminal’s problem is

x≥0max B x e C x( ) ( ) ( ).− π

Crime and Punishment

Given e, the criminal’s problem is

First-order condition isx≥0max B x e C x( ) ( ) ( ).− π

′ = ′B x e C x( ) ( ) ( ).π

Crime and Punishment

Given e, the criminal’s problem is

First-order condition is

Low e ⇒ low π(e) ⇒ low marg. cost.High e ⇒ high π(e) ⇒ high marg. cost.

x≥0max B x e C x( ) ( ) ( ).− π

′ = ′B x e C x( ) ( ) ( ).π

Crime and Punishment

′ = ′B x e C x( ) ( ) ( )π

B(x)

xx *Higher police effort deters crime.

MC = ′π ( ) ( )e C xh

MC = ′π ( ) ( )e C xl

e el h<

Crime and Punishment

Higher fines and larger police effort both raise marginal production costs of illegal activity.

Which is better for society -- higher fines, or more police effort?

Crime and Punishment

Higher fines and larger police effort both raise marginal production costs of illegal activity.

Which is better for society -- higher fines, or more police effort?

Police effort consumes resources; higher fines do not.

Better to fine heavily.

Liability Law

An injurer, IN, and a victim, V.x is effort by IN to avoid injuring V.cIN(x) is IN’s cost of effort x;

cIN(x) ↑ as x ↑.L(x) is V’s loss when IN’s effort is x;

L(x) ↓ as x ↑.

Liability Law

Society wishes to minimize total cost; i.e.

xmin c x L xIN( ) ( ).+

Liability Law

Society wishes to minimize total cost; i.e.

Social optimality requires

I.e. IN’s private marginal cost of effort equals marginal benefit of her extra effort.

xmin c x L xIN( ) ( ).+

′ = − ′c x L xIN( *) ( *).

Liability Law

Liability rules:– no liability rule– strict liability rule– negligence rule.

Which is best?

Liability Law

No Liability Rule: IN faces only private cost, cIN(x).Hence chooses effort levelNo liability results in suboptimal low

care level and excessive injury.

′ =x 0.

Liability Law

Full Liability Rule: IN faces private cost and V’s costs,

cIN(x) + L(x).Hence chooses the socially optimal

effort level where′ = − ′c x L xIN( *) ( *).x *

Liability Law

Negligence Rule: IN is liable for V’s loss if and only if care effort level

, a legally determined effort level.x x< ~

Liability Law

Negligence Rule: IN is liable for V’s loss if and only if care effort level

, a legally determined effort level.

What if the court sets , the socially optimal effort level?

x x< ~

~ *x x=

Liability Law

So ⇒ full liability for IN; hence she chooses

x x< *x x= *.

Liability Law

So ⇒ full liability for IN; hence she chooses

And ⇒ no liability for IN; hence she chooses

x x< *x x= *.

x x≥ *x x= *.

Liability Law

So ⇒ full liability for IN; hence she chooses

And ⇒ no liability for IN; hence she chooses

I.e. the negligence rule is socially optimal when

x x< *

x x≥ *

~ *.x x=

x x= *.

x x= *.

Liability LawBoth full liability and negligence

rules are socially optimal, but full liability fully insures V always,

and the negligence rule fully insures V

only if IN’s care effort level .x x< *

Liability LawBoth full liability and negligence

rules are socially optimal, but full liability fully insures V always,

and the negligence rule fully insures V

only if IN’s care effort level .Victims prefer full liability; injurers

prefer the negligence rule.

x x< *

Bilateral Accidents

V and IN can each exert effort to avoid a loss.

cV(xV) and cIN(xIN).Loss is L(xV,xIN).Society wishes to

x xV INmin,

c x c x L x xV V IN IN V IN( ) ( ) ( , ).+ +

Bilateral Accidents

Society wishes to

Social optimality requiresV’s MC of effort = MB of his effortIN’s MC of effort = MB of her effort.

I.e.′ = −c x L x x xV V

*V*

IN V( ) ( , ) /*∂ ∂

′ = −c x L x x xIN IN*

V*

IN IN( ) ( , ) /*∂ ∂

x xV INmin,

c x c x L x xV V IN IN V IN( ) ( ) ( , ).+ +

Bilateral Accidents

No Liability: Both V and IN face only their private effort costs, not the full social costs of their actions.

Bilateral Accidents

No Liability: Both V and IN face only their private effort costs, not the full social costs of their actions.

Hence V and IN both provide too little effort.

No liability is socially suboptimal.

Bilateral Accidents

Full Liability: V is fully compensated for all injury costs.

Bilateral Accidents

Full Liability: V is fully compensated for all injury costs.

Hence V chooses Full liability is socially suboptimal in

bilateral accidents.

xV = 0.

Bilateral Accidents

Strict Division of Losses: IN must pay a fixed fraction, f, of loss caused.

IN minimizes IN chooses effort satisfying

c x f L x xIN IN V IN( ) ( , ).+

c x f L x x xIN IN V IN IN( ) ( , ) / .′ = − ∂ ∂

′xIN

Bilateral Accidents

IN chooses effort satisfying

Optimality requires

Since f < 1, IN chooses less than the optimal effort level;

c x f L x x xIN IN V IN IN( ) ( , ) / .′ = − ∂ ∂

′ <x xIN IN* .

′xIN

′ = −c x L x x xIN IN*

V*

IN IN( ) ( , ) /*∂ ∂

Bilateral Accidents

IN chooses effort satisfying

Optimality requires

Since f < 1, IN chooses less than the optimal effort level;

Strict division of losses is a socially suboptimal liability rule.

c x f L x x xIN IN V IN IN( ) ( , ) / .′ = − ∂ ∂

′ <x xIN IN* .

′xIN

′ = −c x L x x xIN IN*

V*

IN IN( ) ( , ) /*∂ ∂

Bilateral Accidents

Negligence Rule: IN is fully liable for loss only if her effort level , a legally determined effort level.

Social optimality requires V and IN to choose effort levels

x x< ~

′ = −c x L x x xV V*

V*

IN V( ) ( , ) /*∂ ∂

′ = −c x L x x xIN IN*

V*

IN IN( ) ( , ) / .*∂ ∂

x xV V*= x xIN IN

*=and , where

and

Bilateral Accidents

Suppose V choosesThen IN is fully liable and wishes to

I.e. IN chooses

x xV V*= .

xINmin c x L x xIN IN V IN( ) ( , ).*+

x xIN IN*= .

Bilateral Accidents

Now suppose IN choosesThen V wishes to

I.e. V chooses

x xIN IN*= .

xVmin c x L x xV V V IN( ) ( , ).*+

x xV V*= .

Bilateral Accidents

Now suppose IN choosesThen V wishes to

I.e. V choosesThe Nash equilibrium of the

negligence rule game is the socially optimal outcome.

x xIN IN*= .

xVmin c x L x xV V V IN( ) ( , ).*+

x xV V*= .

Bilateral Accidents

Strict Liability with Defense of Contributory Negligence Rule: IN is fully liable unless V’s care level is less than a specified level ~.x

Bilateral Accidents

IN is fully liable unless V’s care level is less than a specified level

If society chooses and V chooses , then IN is fully liable, so her best reply is

~.x~x x= V

*

x xIN IN*= .

x xV V*=

Bilateral Accidents

IN is fully liable unless V’s care level is less than a specified level

If society chooses and V chooses , then IN is fully liable, so her best reply is

If IN chooses , then V’s best reply is

~.x

x xIN IN*=

x xV V*= .

~x x= V*

x xIN IN*= .

x xV V*=

Bilateral Accidents

IN is fully liable unless V’s care level is less than a specified level

If society chooses and V chooses , then IN is fully liable, so her best reply is

If IN chooses , then V’s best reply is

I.e. the rule causes a socially optimal Nash equilibrium.

~.x

x xIN IN*=

x xV V*= .

~x x= V*

x xIN IN*= .

x xV V*=

Bilateral Accidents

Notes:– socially optimal liability rules do

not generally fully compensate the victim.

– socially optimal accident deterrence is distinct from optimal accident compensation.

Treble Damages & Antitrust Law

The Sherman and Clayton Acts allow an agent damaged by price-fixing to sue and recover treble damages.

How does such a penalty affect the behavior of a price-fixing cartel?

Treble Damages & Antitrust Law

Assume firms collude to form a cartel with a constant marginal production cost,

Market demand isc.

x p( ).

Treble Damages & Antitrust Law

Assume firms collude to form a cartel with a constant marginal production cost,

Market demand isCartel’s goal is

c.x p( ).

pmax π ( ) ( ) ( ).p p c x p= −

Treble Damages & Antitrust Law

Assume firms collude to form a cartel with a constant marginal production cost,

Market demand isCartel’s goal is

Solution is

c.x p( ).

pmax ( ) ( ).p c x p−

p p x x pm m m= ⇒ = ( ).

Treble Damages & Antitrust Law

Suppose fixing price at results in damages to a victim V.

V’s probability of winning suit against the cartel is

If V wins, the cartel must pay

D p( )p

π .γ D p( ).

Treble Damages & Antitrust Law

Suppose fixing price at results in damages to a victim V.

V’s probability of winning suit against the cartel is

If V wins, the cartel must payCartel’s problem is now

D p( )p

π .γ D p( ).

pmax ( ) ( ) ( ).p c x p D p− − πγ

Treble Damages & Antitrust Law

Cartel’s problem is now

Solution is not generally the same as for the original problem

So generally cartel behavior is affected by the penalty.

pmax ( ) ( ) ( ).p c x p D p− − πγ

pmax ( ) ( ).p c x p−

Treble Damages & Antitrust Law

Special case -- suppose is the cartel’s profit. The cartel’s goal is

D p( )

pmax ( ) ( ) ( ) ( )( ) ( ).p c x p D p p c x p− − = − −πγ πγ1

Treble Damages & Antitrust Law

Special case -- suppose is the cartel’s profit. The cartel’s goal is

Maximizing after-penalty profit requires maximizing before-penalty profit.

D p( )

pmax ( ) ( ) ( ) ( )( ) ( ).p c x p D p p c x p− − = − −πγ πγ1

Treble Damages & Antitrust Law

Special case -- suppose is the cartel’s profit. The cartel’s goal is

Maximizing after-penalty profit requires maximizing before-penalty profit.

The cartel’s behavior is unaffected by the penalty.

D p( )

pmax ( ) ( ) ( ) ( )( ) ( ).p c x p D p p c x p− − = − −πγ πγ1

Treble Damages & Antitrust Law

What if consumers can seek to be damaged?

Treble Damages & Antitrust Law

What if consumers can seek to be damaged?

Suppose consumer utility is quasi-linear;

Consumer can win damages

So consumer’s goal is

u x m px( ) .+ −

D p c x= −πγ ( ) .

u x m px p c x( ) ( ) .+ − + −πγmaxx

Treble Damages & Antitrust Law

Consumer’s goal is

I.e.u x m px p c x( ) ( ) .+ − + −πγmax

x

u x m p p c x( ) [ ( ) ] .+ − − −πγmaxx

Treble Damages & Antitrust Law

Consumer’s goal is

I.e.u x m p p c x

p( ) [ ( ) ] .

+ − − −πγeffective pr ice, max

x

u x m px p c x( ) ( ) .+ − + −πγmaxx

Treble Damages & Antitrust Law

Consumer’s goal is

Since consumer’s action depends upon the effective price, rewrite the cartel’s problem as

u x m p p c xp

( ) [ ( ) ] .

+ − − −πγeffective pr ice,

maxx

max

p( ) ( ).p c x p−

Treble Damages & Antitrust Law

Consumer’s goal is

Since consumer’s action depends upon the effective price, rewrite the cartel’s problem as

Solution is the sameas the original problem;

u x m p p c xp

( ) [ ( ) ] .

+ − − −πγeffective pr ice,

maxx

max

p( ) ( ).p c x p−

.p pm=

Treble Damages & Antitrust Law

Solution is the same as the original problem;

is the price paid by buyers. Then .p pm=

p*p pm = −*p p p cm = − −* ( * ).πγ

Treble Damages & Antitrust Law

Solution is the same as the original problem;

is the price paid by buyers. Then

So

.p pm=p*

Nie można wyświetlić obrazu.

p p p cm = − −* ( * ).πγ

p p c p p c pm

mm

m* ( ) .=−−

= +−

−>

πγπγ

πγπγ1 1

Treble Damages & Antitrust Law

The cartel’s price , the price set in the absence of damage penalties.

But the effective price to both consumers and the cartel is the same as in the no damages case.

Nie można wyświetlić obrazu.

p pm* >

p p c p p c pm

mm

m* ( ) .=−−

= +−

−>

πγπγ

πγπγ1 1