EconS 503 - Microeconomic Theory IIHomework #5 - Answer key

1. Exercises from Tadelis:

� Exercise 9.3 from Chapter 9.

� Exercises 10.3, 10.7, 10.9, 10.12 from Chapter 10.

2. Collusion with probability of being caught - Harrington (2014).1 Consideran industry with N �rms. For generality, we do not assume whether they compete inquantities or prices yet, nor the inverse demand function or costs they face. Considerthat �rms are symmetric and in the Nash equilibrium of the unrepeated game, every�rm earns pro�ts �N , so we label the present value of the noncollusive stream as

V N � �N + ��N + ::: = 1

1� ��N .

When �rms collude, each of them earns pro�t �C , where �C > �N . When a �rmunilaterally deviates from the collusive outcome, it earns a deviating pro�t of �D, where�D > �C in that period. Consider a standard Grim-Trigger strategy (GTS) where every�rm chooses to collude in period t = 1, and continues to do so in subsequent periodst > 1 if all �rms colluded in previous periods. If one �rm did not cooperate in previousperiods, however, all �rms revert to the Nash equilibrium of the unrepeated game,earning �N thereafter (permanent punishment scheme). For simplicity, assume thatall �rms exhibit the same discount factor � 2 (0; 1).

(a) Find the minimal discount factor � that sustains this GTS as a subgame perfectequilibrium of the game.

� After a history of cooperation, every �rm i keeps cooperating as long as

1

1� ��C| {z }

Cooperation

� �D|{z}Deviation

+�

1� ��N| {z }

Permanent punishment

which, after solving for discount factor �, yields

� � b� � �D � �C�D � �N

The above number is a positive number since the pro�t from deviating, �D,satis�es �D > �C and �D > �N by de�nition.

� Comparative statics of cuto¤ b�:1Harrington, Joseph E. Jr. (2014) �Penalties and the Deterrence of Unlawful Collusion,� Economic

Letters, 124, pp. 33-36.

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�Cuto¤ b� is increasing in �N . Intuitively, as the pro�ts from reverting tothe Nash equilibrium of the unrepeated game �N increase, the punishmentfrom deviation become less severe, ultimately making the deviation moreattractive.

� In contrast, cuto¤ b� is decreasing in �C . In words, this indicates that,when the pro�ts from cooperation �C increase, deviation becomes lessattractive and can be sustained under larger values of discount factor �.

�Finally, cuto¤ b� is increasing in the deviating pro�t �D since @b�@�D

=�C��N(�D��N )2 > 0, since pro�ts from cooperation �C satisfy �C > �N . In-tuitively, when deviation becomes more attractive, the GTS can only besustained for more restrictive conditions on discount factor �.

(b) For the rest of the exercise, let us assume that the cartel faces a exogeneousprobability p of being discovered, prosecuted, and convicted, by a regulatoryagency such as the Federal Trade Commission. If caught and convicted in periodt, a �rm must pay a �ne F t, where F t = �F t�1 + f . Parameter 1 � � can beunderstood as the depreciation rate, which we assume to satisfy � 2 (0; 1) toguarantee that the penalty is bounded. In addition, assume that F 0 = 0, sothat F 1 = f , F 2 = �F 1 + f , and similarly for subsequent periods. Find thecollusive value V C(F ) given an accumulated penalty F . [Hint : Solve for V C(F )recursively.]

� The collusive value V C(F ) is de�ned recursively, for period t = 1 and t = 2,as follows

V C(F 1) = �C + p

��

1� ��N � F 2

�| {z }

Detected

+ (1� p)�V C(F 2)| {z }Not detected

where penalties are F 1 = �F 0 + f = f and F 2 = �F 1 + f = �F + f . Moregenerally for any two periods t and t+ 1, the collusive value is de�ned as

V C(F ) = �C + p

��

1� ��N � (�F + f)

�| {z }

Detected

+ (1� p)�V C(�F + f)| {z }Not detected

In words, the collusive value includes the collusive pro�t, �C , and:� If the cartel is detected, which happens with probability p, every �rmmust pay a penalty �F +f after being detected, and a generate stream ofNash equilibrium pro�ts �N thereafter since we assumed that after beingdetected �rms can never form a cartel in the future.

� If the cartel is undetected, which occurs with probability 1�p, the streamof collusive payo¤s starts in the next period, giving rise to continuationpayo¤ �V C(�F + f).

� Since the collusive value V C(F ) shows up in both the left- and right-handside of the above expression, we can solve for V C(F ) to obtain

V C(F ) = �C + p

��

1� ��N � (�F + f)

�+ (1� p)�V C(�F + f)

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which can be rearranged as follows

V C(F ) = �C + p

��

1� ��N � (�F + f)

�+ (1� p)��C + p(1� p)� �

1� ��N

� p(1� p)� (� (�F + f) + f) + (1� p)2�2V C(� (�F + f) + f)

=

��C + p�N

�

1� �

�[1 + � (1� p)]� pF

�� + (1� p)��2

�� pf [1 + � (1� p) + � (1� p) �] + (1� p)2�2V C(� (�F + f) + f)

=

��C + p�N

�

1� �

� TXt=0

[� (1� p)]t � p�FTXt=0

[(1� p) ��]t

� pfTXt=0

�t (1� p)t

tXs=0

�s

!+ (1� p)T+1�T+1V C

�T+1F + f

TXt=0

�t

!From the transversality condition that

limT!1

(1� p)T+1�T+1V C �T+1F + f

TXt=0

�t

!= 0

which means the present discounted value of collusion at in�nity is zero.Inserting this result in our above expression for V C(F ) we obtain

V C(F ) =

��C + p�N

�

1� �

�1� [� (1� p)]T+1

1� � (1� p) � p�F 1� [(1� p) ��]T+1

1� �� (1� p)

� pfTXt=0

�t (1� p)t 1� �t+1

1� �

=

��C + p�N

�

1� �

�1� [� (1� p)]T+1

1� � (1� p) � p�F 1� [(1� p) ��]T+1

1� �� (1� p)

� pf

1� �1� [� (1� p)]T+1

1� � (1� p) +p�f

1� �1� [�� (1� p)]T+1

1� �� (1� p)For an in�nite horizon game, we take T !1, such that

V C(F ) =

��C + p�N

�

1� �

�1

1� � (1� p) �p�F

1� �� (1� p)

� pf

1� �1

1� � (1� p) +p�f

1� �1

1� �� (1� p)

=�C + p�N �

1��1� � (1� p) �

p�F (1� �) [1� � (1� p)] + pf [1� �� (1� p)]� p�f [1� � (1� p)](1� �) [1� � (1� p)] [1� �� (1� p)]

=�C + p�N �

1��1� � (1� p)| {z }

Expected present valueof pro�ts from the product market

� p (� [1� � (1� p)]F + f)[1� � (1� p)] [1� �� (1� p)]| {z }

Expected discounted penalty

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Intuitively, the �rst term represents the expected present value of pro�ts fromthe product market, which includes the possibility of earning collusive pro�ts�C for the periods that the cartel is undetected, or Nash equilibrium pro�ts�N for all periods after the cartel is detected. The second term indicatesthe expected discounted penalty once the �rm is discovered, prosecuted, andconvicted.

(c) Write down the condition (inequality) expressing that every �rm has incentives tocollude, obtaining V C(F ) rather than deviating. For simplicity, you can assumethat if the cartel is convicted during the deviation period, it has no chances ofbeing caught during the permanent punishment phase.

� Every �rm cooperates if and only if

V C(F )| {z }Collusion

� �D � p(�F + f)| {z }Expected pro�t from deviation

+�

1� ��N| {z }

Permanent punishment

Intuitively, when the �rm deviates from the collusive price the cartel can stillbe detected by regulatory authorities, which occurs with probability p, andthus charged with a penalty �F + f .

(d) The steady-state penalty is F = f1�� , which is found by solving F = �F + f .

Evaluate the collusive value V C(F ) at this penalty, and insert your result in thecondition you found in part (c) of the exercise. Rearrange and interpret.

� Evaluating the collusive value V C(F ) at the steady-state penalty F = f1�� ,

we obtain

V C�

f

1� �

�=�C + p�N

��1���� pf

1��

1� � (1� p)� Substituting the above results into part (c) yields

�C + p��N

1�� �pf1��

1� � (1� p) � �D � pf

1� � +��N

1� �

Rearranging, we �nd

�C + � (1� p) pf

1� � + � (1� p)�D � �D + ��N

1� � [1� � (1� p)� p]

and solving for discount factor �, we obtain

� � b�(p) � �D � �C

(1� p)�pf1�� + �

D � �N�

Note that cuto¤b�(p) collapses to �D��C�D��N when the probability of cartel detec-

tion is zero (p = 0) as in part (a) of the exercise. In contrast, when detection isperfect, p = 1, cuto¤ b�(p) approaches in�nity, thus indicating that condition� � b�(p) cannot hold for any admissible discount factor � 2 [0; 1].

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� Next, we di¤erentiate the cuto¤ b�(p) with respect to p,@b�(p)@p

=

��D � �C

� h�D � �N � (1� 2p) f

1��

ih(1� p)

�pf1�� + �

D � �N�i2 � 0

Intuitively, as detection becomes more likely, the minimal discount factorsustaining collusion b�(p) increases, since �rms are less attracted to collude.

� Lastly, we di¤erentiate the cuto¤ b�(p) with respect to f,@b�(p)@f

= �p��D � �C

�(1� �) (1� p)

�pf1�� + �

D � �N�2 � 0

Intuitively, as the penalty becomes more severe, the minimal discount factorsustaining collusion b�(p) decreases, since �rms have to face a larger penaltyin expectation and have less to gain from collusion.

(e) Bertrand competition. Assume that �rms compete a la Bertrand, selling homo-geneous products with inverse demand function p(Q) = 1 � Q where Q denotesaggregate output. All �rms face a symmetric marginal cost c > 0. In this set-ting, every �rm obtains zero pro�ts in the Nash equilibrium of the unrepeatedgame, entailing �N = 0. If a �rm unilaterally deviates from the collusive price(charging a price in�nitely close, but below, the collusive price), it captures allindustry sales, earning a pro�t �D = N�C during the deviating period. Evaluateyour results from part (d) of the exercise in this context. Then discuss whethercollusion becomes easier to sustain when the penalty f increases; and when thenumber of �rms N increases.

� We �rst need to �nd the pro�ts that �rms obtain from cooperating by settinga collusive price. Since the inverse demand function is p(Q) = 1 � Q, thedirect demand function becomes Q = 1 � p, implying that the joint-pro�tmaximization problem in this setting is

maxp�0

pQ� cQ = p(1� p)� c(1� p)

Di¤erentiating with respect to price p yields 1 � 2p + c = 0, and solving forp we obtain a collusive price of pC = 1+c

2. Therefore, collusive pro�ts for the

industry are

N�C = pC(1� pC)� c(1� pC) =�1� 1 + c

2

��1 + c

2� c�=(1� c)24

which implies that every �rm�s collusive pro�ts, �C , is (1�c)24N

.

� Evaluating our above results from part (d) at pro�ts �N = 0, �C = (1�c)24N

,

and �D = N�C = (1�c)24, we �nd that every �rm cooperates after a history of

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cooperation if and only if

� � � =(1�c)24

� (1�c)24N

(1� p)hpf1�� +

(1�c)24

i=

1

1� p �(1�c)24

�1� 1

N

�4pf+(1��)(1�c)2

4(1��)

=N � 1N (1� p) �

(1� �) (1� c)2

4pf + (1� �) (1� c)2

� Comparative statics of the cuto¤ �:�Di¤erentiating the cuto¤ � with respect to N, we obtain

@�

@N=

1

N2 (1� p) �(1� �) (1� c)2

4pf + (1� �) (1� c)2� 0

so that as the number of �rm increases, a higher discount rate is neededto sustain collusion because the �rm can capture a larger pro�t fromdeviation, �D, relative to their collusion pro�t, �C .

�Di¤erentiating the cuto¤ � with respect to f, we obtain

@�

@f= � N � 1

N (1� p)4p (1� �) (1� c)2�

4pf + (1� �) (1� c)2�2 � 0

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