Optimal Collusion with Limited Liability and Policy Implications

Optimal Collusion with Limited Liability and Policy Implications

Etienne Billette de Villemeur (Toulouse School of Economics, GREMAQ, IDEI) Laurent Flochel (Charles River Associates International)

Bruno Versaevel (EMLYON Business School, GATE)

June 2011

Objectives

Our objectives are:

1) to characterize the ability of oligopolistic …rms to implement a collusive strategy when their ability to punish deviations over one or several periods is limited;

2) to draw policy implications.

The limited liability constraint formalizes:

structural conditions (e.g., …nite demand);

a regulatory mechanism (e.g., prudential ratio);

…nancial market pressure (e.g., pro…tability target).

Objectives

Our objectives are:

1) to characterize the ability of oligopolistic …rms to implement a collusive strategy when their ability to punish deviations over one or several periods is limited;

2) to draw policy implications.

The limited liability constraint formalizes:

structural conditions (e.g., …nite demand);

a regulatory mechanism (e.g., prudential ratio);

…nancial market pressure (e.g., pro…tability target).

Objectives

Our objectives are:

1) to characterize the ability of oligopolistic …rms to implement a collusive strategy when their ability to punish deviations over one or several periods is limited;

2) to draw policy implications.

The limited liability constraint formalizes:

structural conditions (e.g., …nite demand);

a regulatory mechanism (e.g., prudential ratio);

…nancial market pressure (e.g., pro…tability target).

The Main Results

When the limited liability constraint binds:

there exists an in…nity of optimal multi-period punishment paths that permit …rms to implement a given collusive strategy;

the lowest discount factor for a collusive strategy to be implementable can always be reached with a …nite number of punishment periods;

this discount threshold is strictly lower with a multi-period punishment pro…le than with a single-period punishment scheme;

a longer punishment is only an imperfect substitute for more immediate severity.

As a policy implication, a well-adjusted limited liability constraint can restore competition

by iteration.

The Main Results

When the limited liability constraint binds:

there exists an in…nity of optimal multi-period punishment paths that permit …rms to implement a given collusive strategy;

the lowest discount factor for a collusive strategy to be implementable can always be reached with a …nite number of punishment periods;

this discount threshold is strictly lower with a multi-period punishment pro…le than with a single-period punishment scheme;

a longer punishment is only an imperfect substitute for more immediate severity.

As a policy implication, a well-adjusted limited liability constraint can restore competition

by iteration.

The Main Results

When the limited liability constraint binds:

there exists an in…nity of optimal multi-period punishment paths that permit …rms to implement a given collusive strategy;

the lowest discount factor for a collusive strategy to be implementable can always be reached with a …nite number of punishment periods;

this discount threshold is strictly lower with a multi-period punishment pro…le than with a single-period punishment scheme;

a longer punishment is only an imperfect substitute for more immediate severity.

As a policy implication, a well-adjusted limited liability constraint can restore competition

by iteration.

The Main Results

When the limited liability constraint binds:

there exists an in…nity of optimal multi-period punishment paths that permit …rms to implement a given collusive strategy;

the lowest discount factor for a collusive strategy to be implementable can always be reached with a …nite number of punishment periods;

this discount threshold is strictly lower with a multi-period punishment pro…le than with a single-period punishment scheme;

a longer punishment is only an imperfect substitute for more immediate severity.

As a policy implication, a well-adjusted limited liability constraint can restore competition

by iteration.

The Model

Symmetric …rms in N = f ^{1, . . . , n} g play a repeated stage-game over t = 1, 2, . . . , ∞.

Initially all …rms play a

2 ^{A R}

for an individual payo¤ π

π ( a

) in each period.

If a …rm deviates from a

in period t, all …rms switch to the punishment action a

, and earn π ( a

) π

, in period(s) t + 1, . . . , t + k, . . . , t + l , with l 1.

If a …rm deviates from a

in period t + k, with k = 1, . . . , l , the l-period punishment phase restarts.

Otherwise after l punishment periods all …rms switch back to a

.

The Model

The main assumptions:

(A1) Firms incur a …xed cost f 0, and a variable cost c ( q

) 0, to sell substitutable goods (possibly di¤erentiated), with either price (a = p) or quantity (a = q) as a strategic variable;

(A2) Each …rm i ’s inverse demand function p

: R

! ^R

is non-increasing and continuous;

(A3) p

( 0 ) > c and lim

p

( q

, q

) = 0, any q

in R

.

The Model

To compare, the main assumptions in Abreu (1986) are:

(e A1) Firms sell a homogeneous good at constant marginal cost c > 0, and their strategic variable is quantity;

(e A2) The market inverse demand function p ( q ) : R

! ^R

^{is strictly} decreasing and continuous in q = _∑

q

;

(e A3) p ( 0 ) > c and lim

p ( q ) = 0.

)

^lim

i!∞

( p ( q ) c ) q

= ∞

The Model

The notation:

a

: collusive action

a

: stage-game Nash equilibrium action a

: punishment action in period k = 1, . . . , l

π ( a ) : stage pro…t when all …rms play the same a (with π

π ( a

) )

π

( a ) : …rm i’s one-shot best reply bene…ts to a as played by all rivals in N nf ⁱ g

δ 2 ( 0, 1 ) : discount factor

The Benchmark

Incentive constraint (no deviation from collusion):

π

( a

) π

δ [ π

π ( a

)] (IC0 ) Incentive constraint (no deviation from punishment):

π

( a

) π ( a

) δ [ π

π ( a

)] (IC1) Participation constraint:

( 1 δ ) [ π

π ( a

)] π

(PC )

Limited Liability constraint:

π ( a

) π, (LLC)

with π π ( a

) .

The Benchmark

Given a

, the single-period punisment δ-minimization problem in a

is

a

min

δ

s.t. IC 0, IC 1, PC, LLC

The Benchmark

Proposition 1

The collusive action a

a

is implementable with a single-period punishment if and only if δ δ

, with

δ

= 8 >

> >

<

> >

> :

δ

d i(am) πm

πm π(a_P)

if a

a

, a

(regime 1);

δ

d i(am) πm

πm π

if a

a

, a

(regime 2);

δ

di(am) πm

πm π

if a

a

, a

(regime 3).

(1)

If π > π

π

( a

) π

then δ > 1 and a

cannot be implemented for any δ.

Remark 1

If a

a

, a

, so that regime 1 applies, δ δ, δ.

The Multi-Period Setup

If a …rm does not deviate from the punishment path, the continuation pro…ts it earns from period s + 1 onward is

V

( a

, δ ) =

∑

δ

π ( a

) +

∑

δ

π

.

The Multi-Period Setup

Multi-period Incentive constraints:

π

( a

) π

δ [ V

( a

, δ ) V

( a

, δ )] , (MIC 0) and

π

( a

) π ( a

) δ [ V

( a

, δ ) V

( a

, δ )] , . . .

π

( a

) π ( a

) δ [ V

( a

, δ ) V

( a

, δ )] , . . .

π

( a

) π ( a

) δ [ V

( a

, δ ) V

( a

, δ )] ,

(MIC 1, ..., l )

with 1 s l .

The Multi-Period Setup

Multi-period Participation constraint:

( 1 δ ) [ V

( a

, δ ) V

( a

, δ )] π

, (MPC) all s = 0, 1, . . . , l.

Multi-period Limited Liability constraint:

π ( a

) π, (MLLC )

with 1 k l , all l 2, and π π ( a

) .

The Multi-Period Setup

Given all constraints, the multi-period punishment problem is

(aP,1,...,a

min

δ

s.t. ( MIC 0 MIC l ) ; MPC; MLLC

The Multi-Period Setup

Lemma 2

Given a

, the lowest discount factor δ verifying (MIC 0) and (MIC 1) results from punishment actions a

, with k > 1, such that these two multi-period incentive constraints bind.

Proposition 2

In the multi-period punishment scheme, if a

a

, a

the collusive action a

a

is

implementable if and only if δ δ , and a

( a

, a

, . . . , a

) is optimal.

The Multi-Period Setup

Lemma 3

The lowest δ compatible with ( MIC 0 ) and ( MPC ) is δ

di(am) πm

π^d_i(am)

.

Proposition 3

In the multi-period punishment scheme, if a

a

, a

, the collusive action a

a

is

implementable if and only if δ δ, and a

( a

, a

, . . . , a

) is optimal.

The Multi-Period Setup

Lemma 4

The lowest δ compatible with ( MIC 0 ) and ( MLLC ) is δ

d i(am) πm

π^d_i(am) π^d_i(a_P)

.

Proposition 4

In the multi-period punishment scheme, if a

a

, a

collusion at a

is implementable

if and only if δ δ

sup f ^{δ, δ}

g ^{, with a}

( a

, a

, . . . , a

) of …nite length l .

The Multi-Period Setup

Remark 4

If ( MLLC ) is strictly binding, that is if a

a

, a

, there exits a continuum of optimal

punishments ( a

, a

, . . . , a

) of …nite length l 2 , s.t. a

is implementable for δ = δ

.

The Multi-Period Setup

Proposition 5

If a

a

, a

, and additional punishment periods are introduced, the lowest discount factor δ

that permits the implementation of a

a

cannot be as low as δ , and can attain δ only in particular circumstances. More formally, either a

a

so that

δ < δ

< δ, or a

a

and δ δ

< δ. In the latter case δ

= δ if and only if

˜

a

a

a

a

.

) ^l > 1 only an imperfect substitute to early severe punishment

A Linear Example

Firms in N = f ^{1, . . . , n} g incur a constant marginal cost c 0 to sell q to consumers with utility function

U ( q, I ) =

∑

q

1 2

∑

q

+ 2γ ∑

i6=j

q

q

! + I ,

all q

, q

0, j 2 ^N nf ⁱ g ^{, where γ} 2 ( 0, 1 ) measures product substitutability.

Limited Liability constraint:

p

( q

) 0, all i 2 ^N.

Proposition 6

The parameter space ( c, n, γ ) is partitioned in three subsets where either Regime 1, 2, or

3, as de…ned in ( 1 ) , applies.

A Linear Example

Regime 1 applies if and only if

(i) 2 n 3; 0 γ 1; 0 c < 1; or (ii) 4 n 5; 0 γ 1; c

c < 1; or (iii) 6 n; 0 γ γ; ˆ 0 c < 1; or (iv) 6 n; γ ˆ γ γ; ˇ c

c < 1.

Regime 2 applies if and only if 6 n; γ ˇ γ 1; c

c < 1.

Regime 3 applies if and only if

(i) n = 3; γ = γ ˆ = 1; c = c = 0; or

(ii) 4 n 5; γ ˆ γ 1; 0 c c

; or

(iii) 6 n; γ ˆ γ γ; ˇ 0 c c

; or

(iv) 6 n; γ ˇ γ 1; 0 c c

.

A Linear Example

Figure:

Collusion regimes in plane (

) for

6. The limited liability constraint binds in the

grey area (regime 3). In the benchmark single-period set-up, the collusive quantity is not

implementable below the frontier

.

Policy Implications

Regulatory constraint:

π ( a ) π

, (R)

with π

π ( a

) , and a

a

a

, implying that π π

π

.

Proposition 8

Suppose that …rms implement a

a

. By setting a price ‡oor slightly below the

observed transaction price, and by reducing the ‡oor incrementally by iteration, the

regulator drives the industry to the stage-game Nash equilibrium a

.

Policy Implications

Figure:

Cournot linear setup (n = 5,

= 1,

= 1/10,

= 3/5). Initially, all …rms implement

. A price ‡oor rules out large price reductions (i.e.,

R,1

, with

R,1

above

, but only limitedly so). A series of successive adjustments from

R,1

to

R,2

,

R,3

, ... drives the industry

toward the stage-game Nash equilibrium (point

) .

Policy Implications

Lemma 5

Consider any implementable collusive action a

. Then a

a

implies that a

a

.

***