EconS 503 - Microeconomic Theory IIMidterm Exam #2 - Answer key

1. Temporary punishments in Bertrand competition. Consider an industry withtwo rms competing in prices a la Bertrand, facing a linear inverse demand functionp (Q) = 100 � Q, where Q denotes aggregate output. Firms face a common marginalcost c = 10. For simplicity, assume that both rms have the same discount factor� 2 (0; 1).

(a) Bertrand equilibrium. Find equilibrium prices in Nash equilibrium of the Bertrandgame when rms interact only once.

� When rms i and j, where i; j 2 f1; 2g, interact only once, each rm adoptsmarginal cost pricing, that is, p� = c = 10, yielding a total sales of Q =100 � 10 = 90 units, which are equally divided between the two rms, thatis, q�i = q

�j = 45. Equilibrium prots are zero in this setting.

(b) Innitely repeated game - Permanent reversion. Consider now a grim-triggerstrategy (GTS) where rms start setting a collusive price that maximizes theirjoint prots and continue to do so if both rms chose collusive prices in all previousperiods. Otherwise, every rm permanently reverts to the Bertrand equilibriumyou found in part (a). Under which conditions on � this GTS can be sustained asa SPNE of the innitely repeated game?

� Collusion. Rearrange the inverse demand function p (Q) = 100�Q to obtainthe demand function, Q (p) = 100 � p. The cartel comprising rms i and jchooses price p to solve the following joint prot maximization problem:

maxp�0

� (p) = p(Q)Q� cQ

= p (100� p)� 10 (100� p)= (p� 10) (100� p)

Di¤erentiating with respect to the price p, and assuming interior solutions,that is, p > 0, we obtain

110� 2p = 0,which yields a collusive price of pC = $55.Substituting the collusive price, pC = 55, into the demand function, we obtainthat aggregate output becomes QC = 100 � 55 = 45 units. Since rms aresymmetric, each rm would serve half of the market with qCi = q

Cj =

452= 22:5

units of output. As a result, every rm i in the cartel earns a collusive protof

�Ci =(p� 10) (100� p)

2=(55� 10) (100� 55)

2= $1; 012:5.

1

� Cooperation. If at any period t after a history of cooperation rm i chargesthe collusive price pC = 45 (as prescribed by the grim-trigger strategy), everyrm i obtains prots of 9621

2in every period, as follows

1; 012:5 + (� � 1; 012:5) +��2 � 1; 012:5

�+ : : :

=1; 012:5

1� �� Optimal deviation. Let us now analyze the payo¤ that rm i can obtain ifit deviates from cooperation. If at any period t after a history of coopa-ration rm i deviates from the collusive price of pC = 55, its optimal de-viation is to undercut rm js price by " > 0, such that it captures theentire market, selling 55 units of output, and earning a deviating prot of55(45) � 10(45) = $2; 025. Firms i and j detect this deviation immediately,and revert to marginal cost pricing thereafter, which entails zero prots forall subsequent periods. In this context, the payo¤ that rm i obtains fromdeviating at any period t is

2; 025| {z }�Devi

+ (� � 0) +��2 � 0

�+ : : :| {z }

Innite punishment

= 2; 025

� Comparison. Therefore, to sustain cooperation, it must be that the coopera-tion prot must be weakly higher than the deviation prot, that is,

1; 012:5

1� � � 2; 025

cross multiplying by 1 � �, and solving for discount factor �, we nd thatcooperation can be sustained as long as � � 1

2.

c. Repeated game, Temporary reversion. Consider again the grim-trigger strategyof part (c), but assume that, upon observing a deviation, rms revert to theBertrand equilibrium of part (a) during T periods, returning to the collusive priceif both rms chose the Bertrand equilibrium price during the last T periods (thatis, both rms return to cooperation if they observe that both implemented thepunishment during the prescribed T periods). Under which conditions on rmscommon discount factor this grim-trigger strategy can be sustained as a SPNE ofthe innitely repeated game? [Hint : Rather than solving for the minimal discountfactor sustaining cooperation, �, solve for the lenght of the temporary punishmentT .]

� Cooperation. If at any period t after a history of cooperation rm i chargesthe collusive price pC = 55 (as prescribed by the grim-trigger strategy), rmi obtains

1; 012:5 + (� � 1; 012:5) +��2 � 1; 012:5

�+ : : :+

��T � 1; 012:5

�+��T+1 � 1; 012:5

�+ : : :

=

�1; 012:5� 1� �

T+1

1� �

�+

�1; 012:5� �

T+1

1� �

�

2

� Optimal deviation. Let us now analyze the payo¤ that rm i can obtain if itdeviates from cooperation. By deviating, rm i earns all of the cartel prot inthis period, nothing in the following T periods, and half of the cartel protsbeginning the T + 1 period, yielding a deviation payo¤ of

2; 025| {z }�Devi

+ (� � 0) +��2 � 0

�+ : : :+

��T � 0

�| {z }Punishment for T periods

+��T+1 � 1; 012:5

�+ �T+2 � 1; 012:5 + : : :| {z }

Return to cooperation

= 2; 025 +

�1; 012:5� �

T+1

1� �

�� Comparison. Therefore, to sustain cooperation, it must be that the coopera-tion prot must be weakly higher than the deviation prot, that is,�1; 012:5� 1� �

T+1

1� �

�+

�1; 012:5� �

T+1

1� �

�� 2; 025 +

�1; 012:5� �

T+1

1� �

�which can be rearranged as follows

1� �T+1

1� � � 2

() 2� � 1 � �T+1

applying logs on both sides, we obtain

ln (2� � 1) � (T + 1) ln �

and solving for the lenght of the temporary punishment, T , we nd that

T � T̂ � ln (2� � 1)ln �

� 1

and the last inequality stems from the fact that ln � < 0 that reverses theinequality sign. The following gure depicts cuto¤ T̂ as a function of thediscount factor �. Intuitively, the punishment phase must be long enoughand rms must care enough about their future prots (as indicated by Tand � pairs on the northwest of the gure) for the GTS with temporarypunishment to be sustained as a SPNE of the innitely repeated game.

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When the punishment phase lasts two periods, as indicated by the dottedline at a height of T = 2, cooperation can be sustained for discount factorssatisfying � � 0:61, graphically represented by the range of � to the right-hand side of � = 0:61 in the gure. A similar argument applies when thepunishment phase lasts T = 3 periods, where we obtain that cooperation canbe supported as long as � � 0:54.

� After a history in which at least one rm deviated from cooperation, the GTSprescribes that every rm i implements the punishment during T rounds.This is rm is best response to rm j implementing the punishment, sothere are no further conditions on the discount factor, �, or the lenght of thepunishment phase, T , that we need to impose.

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2. Cournot competition when all rms are uninformed - Allowing for costcorrelation. Consider an industry with two rms competing a la Cournot and inversedemand function p(Q) = 1 � Q where Q = q1 + q2 denotes aggregate output. Everyrm i privately observes its marginal cost of production,MCi = 1=4 orMCi = 0, bothequally likely, but it does not observes its rivals marginal costs, MCj. The probabilitydistribution is, however, common knowledge among rms. Assume that rmscosts arepositive correlated, that is, when rm i observes that its costs are high (MCi = 1=4),it assigns a probability pH > 1=2 to its rivals costs being high and 1� pH < 1=2 to itsrivals costs being low. Similarly, after observing that its own costs are low, MCi = 0,rm i assigns a probability pL < 1=2 to its rivals costs being high and 1 � pL > 1=2to its rivals costs being low. Intuitively, observing that its own costs are high (low)increases the probability that its rivals costs are high (low) as well.

(a) Find the best response function for every rm i, qki (qHj ; q

Lj ), where k = fH;Lg

denotes rm is marginal cost (high or low).

� Low costs. When rm i has low costs, it chooses qLi � 0 that solves thefollowing expected prot maximization problem:

maxqLi �0

�Li�qLi�=

Prots if j is low costz }| {(1� pL)

�1� qLi � qLj

�qLi +

Prots if j is high costz }| {pL�1� qLi � qHj

�qLi

Assuming interior solutions, that is, qLi > 0, the rst order condition satises

@�Li�qLi�

@qLi= 1� 2qLi � qLj � pL(qHj � qLj ) = 0

such that the best response function of rm i when its costs are low becomes

qLi�qLj ; q

Hj

�=1

2�pLqHj + (1� pL)qLj

2

which originates at 1=2, and decreases in its rivals expected output, pLqHj +(1� pL)qLj , at the rate of 12 .

� High costs. When rm i has high costs, it chooses qHi � 0 that solves thefollowing expected prot maximization problem:

maxqHi �0

�Hi�qHi�=

Prot if j is low costz }| {(1� pH)

�1� qHi � qLj

�qHi +

Prot if j is high costz }| {pH�1� qHi � qHj

�qHi �

1

4qHi

=

�3

4� qHi � qLj � pH(qHj � qLj )

�qHi

Assuming interior solutions, that is, qHi > 0, the rst order condition satises

@�Hi (qHi )

@qHi=3

4� 2qHi � qLj � pH(qHj � qLj )

5

such that the best response function of rm i when its costs are high becomes

qHi�qLj ; q

Hj

�=3

8�pHqHj + (1� pH)qLj

2

which originates at 3=8, but decreases in its rivals expected output, pHqHj +(1� pH)qLj . Comparing it with rm is best response function when its costsare low, qLi

�qLj ; q

Hj

�, we can see that, for a given prole of rm js output,�

qLj ; qHj

�, rm i responds producing a larger output when its own costs are

low than when they are high.

(b) Use your results from part (a) to nd the Bayesian Nash Equilibrium (BNE) ofthe game.

� Since rms i and j are symmetric, we impose symmetry on the equilibriumoutput that

qL = qLi = qLj

qH = qHi = qHj

Substituting the above results into the best response functions we found inpart (a), yields

qL =1

2� p

LqH + (1� pL)qL2

qH =3

8� p

HqH + (1� pH)qL2

Solving for qL and qH in the simultaneous equations above, the equilibriumoutput satises

qL� =8 + 4pH � 3pL12(2 + pH � pL) and q

H� =5 + 4pH � 3pL12(2 + pH � pL) :

(c) Evaluate your results in the special cases of perfect positive (negative) cost cor-relation, where pH = 1 and pL = 0 (where pH = 0 and pL = 1, respectively).

� Under perfect positive cost correlation, where pH = 1 and pL = 0, equilibriumoutput levels we found in part (b) become

qL� =1

3and qH� =

1

4

In this case, rm js costs coincide with those of rm i. In contrast, underperfect negative cost correlation, pH = 0 and pL = 1, equilibrium outputlevels in part (b) simplify to

qL� =5

12and qH� =

1

6

6

Comparing the above output levels, note that, when rm i observes thatits own costs are low, its output level qL� is higher when it knows that itsrivals costs are high (under negatively correlated costs, where it produces5=12 units) than when it knows that its rivals costs are low (under positivelycorrelated costs, where it only produces 1=3 units). When rm i observesthat its costs are high, its output level qH� is higher when it knows that itsrivals costs are also high (under positive cost correlation, where it produces1=4 units) than when its rivals costs are low (under negatively correlatedcosts, where it only produces 1=6 units).

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3. First-price auction with entry fees. Consider a rst-price auction with N bidders.Every bidder is valuation, vi, is distributed according to a uniform distribution func-tion, that is, F (vi) = vi for all vi � U [0; v]. Consider the following two-stage game: inthe rst stage, the seller sets an entry fee E � 0 that every participating bidder mustpay, otherwise his bid is ignored; in the second stage, every bidder i independently andsimultaneously submit his bid for the object.

(a) Second stage. In this part of the exercise, let us focus on the second stage of thegame. For a given entry fee E, nd the optimal bidding function that bidder ichooses in the second stage, bi (vi; E). [Hint : Assume that there exists a criticalbidder whose valuation ve makes him indi¤erent between participation or not,given a positive entry fee E ].

� Every bidder is expected utility maximization problem is

maxbi�0

EUi (bi) = prob fwing (vi � bi)� E

where the entry fee, E, is a constant that bidder i must pay when he partici-pates in the auction, whether he wins the object or not.

� The probability of bidder i winning the object is analogous to the standardFirst Price Auction without entry fee, which is given by

prob fwing = [F (vi)]N�1

when his valuation exceeds other N � 1 bidders, vi � vj for j 6= i, j 2f1; : : : ; Ng. Note that for a given bidding strategy, b : [0; v] ! R+, thatis, bi (vi) = bi, we can dene its inverse, b�1i (bi) = vi, implying that thecumulative distribution function, which represents the probability mass ofvaluation below his, can be rewritten as

F (vi) = F�b�1i (bi)

�such that bidder is expected utility maximization problem becomes

maxbi�0

EUi (bi) =�F�b�1i (bi)

��N�1(vi � bi)� E

� We assume that the bidding function bi (vi) is monotonically increasing in vi;which we will demonstrate later. In addition, let us dene a critical bidderwith valuation ve and bid be (ve), where ve solves1

[F (ve)]N�1 (ve � be)� E = 0

which means that his expected utility from participating in the auction (giventhe entry fee E) is zero.

1We need this condition to ensure a one-to-one mapping between the entry fee and the critical biddersvaluation. The monotonically increasing bidding function bi(vi) ensures that bidders with valuations aboveve obtain a positive utility from participating in the auction (despite the entry fee E) and thus submit apositive bid for the object.

8

� Di¤erentiating bidder is expected utility with respect to his bid bi yields

(N � 1)�F�b�1i (bi)

��N�2f�b�1i (bi)

� @b�1i (bi)@bi

(vi � bi)��F�b�1i (bi)

��N�1= 0

Since the inverse of the bidding function yields the bidders valuation, b�1i (bi) =

vi, and the derivative of this inverse can be written asdb�1i (bi)dbi

= 1b0i(b

�1i (bi))

,

the above expression becomes

(N � 1) [F (vi)]N�2 f (vi) (vi � bi) = b0i (vi) [F (vi)]N�1

Further rearranging, we obtain

(N � 1) [F (vi)]N�2 f (vi) bi (vi)+[F (vi)]N�1 b0i (vi) = (N � 1) [F (vi)]N�2 f (vi) vi

The left-hand side isd[[F (vi)]N�1bi(vi)]

dvi. Hence,

@h[F (vi)]

N�1 bi (vi)i

@vi= (N � 1) [F (vi)]N�2 f (vi) vi

� Integrating the right-hand side of the above expression with respect to vi (weuse integration by parts), and taking the indi¤erent bidders valuation ve asthe lower bound of integration, yields

Z vive

dh[F (x)]N�1 bi (x)

idvi

dx =

Z vive

(N � 1) [F (x)]N�2 f (x)xdx

)h[F (x)]N�1 bi (x)

ivive=h[F (x)]N�1 x

ivive�Z vive

[F (x)]N�1 dx

We can then reorder the terms in the above expression as follows:

[F (vi)]N�1 bi (vi) = [F (vi)]

N�1 vi�[F (ve)]N�1 [ve � be (ve)]�Z vive

[F (x)]N�1 dx

Substituting the indi¤erent bidder condition, [F (ve)]N�1 (ve � be) � E = 0,

into the above expression, and rearranging, we obtain

bi (vi) = vi �E +

R vive[F (x)]N�1 dx

[F (vi)]N�1| {z }

bid shading

Note that when entry fees are absent, E = 0, the equilibrium bidding func-tion collapses to the standard expression found in previous exercises. Sincevaluations are uniformly distributed, this bidding function simplies to

bi (vi) = vi �E +

R vivexN�1dx

vN�1i| {z }bid shading

9

and, solving the integral, we obtain

bi (vi) = vi �NE +

�xN�vive

NvN�1i

= vi �NE + vNi � vNe

NvN�1i| {z }Bid shading

and the indi¤erent bidders bid, be (ve), solves

vN�1e (ve � be (ve)) = E

� Lastly, we show that the equilibrium bidding function, bi (vi), is monotonicallyincreasing in the bidders valuation, vi, since

dbi (vi)

dvi= 1�

[F (vi)]2N�2 � (N � 1) [F (vi)]N�2

�E +

R vive[F (x)]N�1 dx

�[F (vi)]

2N�2

=N � 1[F (vi)]

N

�E +

Z vive

[F (x)]N�1 dx

�> 0:

(b) How are equilibrium bids a¤ected by an increase in the entry fee E? Does a higherE limit participation in the auction?

� As in other games on rst-price auctions, the second term represents bidderis bid shading, which is increasing in the entry fee E. Intuitively, the entryfee a¤ects all bidders uniformly, inducing those with valuations below ve tonot participate in the auction and, in addition, reducing the bid of those whoparticipate. Intuitively, every participating bidder expects a lower expectedutility in equilibrium, both if he wins and if he loses the auction, leading himto decrease his bid (larger bid shading) to compensate for his lower expectedutility.

(c) First stage. Anticipating the optimal bidding function bi (vi) you found in part(a), what is the optimal entry fee E� that the seller sets in the rst stage tomaximize his expected revenue from the auction? For simplicity, assume thatthe critical bidder, who is indi¤erent between participation or not, submits a bid,be (ve) = 0.

� We rst nd the sellers expected revenue from the auction, and then di¤er-entiate it with respect to the entry fee E, to identify the revenue maximizingentry fee E�.

� Finding the sellers revenue from the auction. For compactness, let us deneG (x) = (F [x])N�1 to be the joint cumulative probability density function forN � 1 bidders, where valuation x satises x 2 [0; v]. Then the above optimal

10

bidding function can be rewritten as

bi (vi) = vi �E +

R viveG (x) dx

G (vi)

=1

G (vi)

�G (vi) vi � E +G (ve) ve �G (ve) ve �

Z vive

G (x) dx

�=

1

G (vi)

Z vive

xg (x) dx

by the fact that G (ve) ve = E and the opposite of integration by parts.From an ex-ante point of view (before observing his own valuation for theobject), bidder is expected payment to the seller is given by the probabilityof winning the auction times the bid he pays for the object upon winning,that is,

�i (vijvi � ve) = prob(win)� bi (vi) + E

= G (vi)�1

G (vi)

Z vive

xg (x) dx+ E

=

Z vive

xg (x) dx+ E

where the second line indicates that, as discussed in previous parts of theexercise, bidder i wins the auction if his valuation vi is above everyone elses,that is, vi � vj for every bidder j 6= i; and in addition, he participates inthe auction and pays a participation fee E. The probability of his valuationexceeding that of every other bidder is given by (F [vi])

N�1, and we repre-sent it more compactly as G (vi) = (F [vi])

N�1, and the last line inserts theequilibrium bidding function found above, bi (vi).

� Since the seller cannot observe biddersvalues, he nds the expected paymentfrom each bidder i, E [�i (vijvi � ve)], and then sums up for all N bidders,PN

i=1E [�i (vijvi � ve)], which gives us the sellers revenue from the auction(this is, of course, understood from an ex-ante perspective since the sellerdoes not observe biddersvaluations). We nd the sellers revenue as follows

E [� (ve)] =

NXi=1

E [�i (vijvi � ve)]

= N

Z vve

�E +

Z vive

xg (x) dx

�f (z) dz

Since the participation fee, E, is a constant, it is una¤ected by the integration,

11

helping us to rewrite the sellers revenue as

E [� (ve)] = NE

Z vve

f (z) dz +N

Z vve

�Z vvi

f (z) dz

�xg (x) dx

= NE [1� F (ve)] +NZ vve

[1� F (vi)]xg (x) dx

= NveG (ve) [1� F (ve)] +NZ vve

[1� F (vi)]xg (x) dx

where the rst line expands the integral into two parts; and the second lineintegrates the probability density function into the cumulative distributionfunction from the critical types valuation ve to the upper bound v, with thesecond part exchanges the order of integration that considers the density massof bidders whose valuation are above ve. Finally, the third line stems fromthe fact that E = [F (ve)]

N�1 (ve � be) = G (ve) ve when we assume that theindi¤erent bidder submits a bid of be (ve) = 0.

� Revenue-maximizing reservation price. We can now di¤erentiate the sellersrevenue with respect to the critical valuation ve,

dE [� (ve)]

dve= N [G (ve) (1� F (ve)� vef (ve)) + (1� F (ve)) veg (ve)� (1� F (ve)) veg (ve)]

= NG (ve) (1� F (ve))�1� ve

f (ve)

1� F (ve)

�Assuming interior solutions, we set the above rst order condition equal tozero.

ve =1� F (ve)f (ve)

The right-hand side of the above inequality is the inverse hazard rate, 1�F (ve)f(ve)

,which measures how sensitive is the distribution of the bidders valuationF (�) to a change in the critical valuation ve. That is, if the density massis concentrated in the region above the critical valuation ve, then 1 � F (ve)would be large relative to f (ve) so that the seller can further increase the entryfee E to raise his expected revenue by selling the object to those bidders withhigher valuations. Next, we will elaborate on the relationship between theentry fee E and the critical valuation ve.Evaluating ve =

1�F (ve)f(ve)

for a uniformly distributed valuation vi � U [0; 1], weobtain

ve = 1� ve, or v�e =1

2so that half of the bidders would participate in the auction in equilibrium.

� Substituting ve = 1�F (ve)f(ve) into the indi¤erent bidders valuation function, theoptimal entry fee E� solves

E� = [F (ve)]N�1 ve

= [F (ve)]N�1 1� F (ve)

f (ve)(4)

12

Substituting v�e =12into the expression of E�, we nd

E� = vN�1e (1� ve)

=1

2N

so that the entry fee E� decreases in the number of bidders N at a decreasingrate. Note that when N becomes innitely large, that is, N !1, the prot-maximizing entry fee that the seller sets approaches zero asymptotically; asdepicted in the next gure.

Figure 1. Optimal entry fee E� as a function of the number ofbidders, N .

For instance, when only N = 2 bidders compete for the object, the optimalentry fee becomes E� = 1

4, while when N = 3 bidders compete the entry fee

decreases to E� = 18.

� Therefore, the optimal bidding function, b�i (vi), becomes

b�i (vi) = vi �NE� + vNi � (v�e)

N

NvN�1i

= vi �N 12N+ vNi � 12NNvN�1i

=N � 1N

h1� (2vi)�N

ivi

For instance, when only N = 2 bidders compete for the object, this optimalbidding function simplies to

b�i (vi) =vi2� 18vi,

13

when N = 3 bidders compete, their optimal bidding function becomes

b�i (vi) =2vi3� 112v2i

,

while when N = 10 bidders compete in the auction, it becomes

b�i (vi) =9vi10� 910240v9i

.

The next gure depicts these bidding functions, where valuations are re-stricted in vi 2

�12; 1�since the bidder indi¤erent between participating and

not participating in the auction when valuations are uniformly distributed isv�e =

12, as shown above.

Figure 2. Optimal bidding function shifts up in N .

Every players bids are increasing in his valuation for the object, vi, and shiftupwards when he competes against a larger number of bidders, N . We nextconrm these two points more formally. First, let us di¤erentiate the optimalbidding function, b�i (vi), with respect to valuation vi,

db�i (vi)

dvi=N � 1N

h1� (2vi)�N +N (2vi)�N

i=N � 1N

h1 + (N � 1) (2vi)�N

i> 0

so that bidder is bid is increasing in his valuation for the object, vi. Second,let us now di¤erentiate the optimal bidding function, b�i (vi), with respect tothe number of bidders, N ,

@b�i (vi)

@N=

1

N2

h1� (2vi)�N

ivi +

N � 1N

[N log (2vi)] vi

= vi

"1� (2vi)�N

N2+ (N � 1) log (2vi)

#

14

and a su¢ cient condition for @b�i (vi)

@N� 0 is vi � 12 , which entails that log (2vi) �

0. However, we already showed that bidders who participate in this auctionhave a private value of vi � v�e = 12 , such that for those bidders who par-ticipate, their equilibrium bids increase when facing competition from morebidders.

15

4. Seltens horse. Consider the Seltens Horsegame depicted in Figure 1. Player 1is the rst mover in the game, choosing between C and D. If he chooses C, player 2 iscalled on to move between C 0 and D0. If player 2 selects C 0 the game is over. If player1 chooses D or player 2 chooses D0, then player 3 is called on to move without beinginformed whether player 1 chose D before him or whether it was player 2 who choseD0. Player 3 can choose between L and R, and then the game ends.

Figure 1. Seltens horse.

(a) Dene the strategy spaces for each player. Then nd all pure strategy Nash equi-libria (psNE) of the game. [Hint : This is a three-player game, so you can considerthat player 1 chooses rows, player 2 columns, and player 3 chooses matrices.]

� The strategy spaces of the players are as follows:

S1 = fC;DgS2 = fC 0; D0gS3 = fL;Rg

In Figure 2, we represent the strategies and payo¤s of the three players inthe following normal form representation of the game, where Player 1 choosesbetween the rows, Player 2 chooses between the columns, and Player 3 choosesbetween the matrixes.

16

Figure 2. Seltens horse - Matrix representation.

� We next underline the best responses of the three players in Figure 3, andidentify that (C;C 0; R) and (D;C 0; L) are the pure strategy Nash equilibriaof this game.

Figure 3. Seltens horse - Underlining best response payo¤s.

(b) Argue that one of the two psNEs you found in part (a) is not sequentially rational.A short verbal explanation su¢ ces.

� (D;C 0; L) is not sequentially rational. If Player 1 chooses D, then Player 3sbelief is � = 1, responding with L (see left-hand side at the bottom of thetree). Anticipating that Player 3 choosing L, Player 2 compares his payo¤from C 0, 1, against that from D0 (which is followed by Player 3 respondingwith L), 4, and thus chooses D0. Therefore, Player 2 choosing C 0 is notsequentially rational.

(c) Show that strategy prole fC;C 0; Rg can be sustained as a PBE of the game.(You dont need to show that this is actually the unique PBE we can sustain inthis game.) Discuss that this strategy prole is based on credible beliefs by player3.

� We check the pooling strategy prole, C;C 0, where Player 1 chooses C andPlayer 2 selects C.As depicted in Figure 4, since player 1 chooses C (as illustrated by the bluehorizontal arrow) and player 2 chooses C 0 (as illustrated by the green hori-zontal arrow), messages D and D0 are o¤-the-equilibrium path, leaving the

17

beliefs of Player 3 unrestricted, that is, � 2 [0; 1]. In other words, Player 3sinformation set should never be reached in this strategy prole.

Figure 4. Pooling Strategy Prole C;C 0

Therefore, if Player 3 is ever called out to move, he compares the expectedpayo¤ from responding with L and R, as follows:

EU3 (L) = 2� �+ 0� (1� �) = 2�EU3 (R) = 0� �+ 1� (1� �) = 1� �

Player 3 then responds with L if 2� > 1 � �, which simplies to � > 13.

Otherwise, he responds with R. This gives rise to two cases (one in which� > 1

3, and Player 3 responds with L; and another in which � � 1

3and Player

3 responds with R), which we separately analyze below.- Case 1, � > 1

3. As depicted in Figure 5a, Player 3 responds with L (as

illustrated by the red arrows) since � > 13. In this context, Player 2 can

improve his payo¤ by deviating from C 0, which yields a payo¤ of 1, to D0,which yields a payo¤of 4. Therefore, the pooling strategy prole C;C 0 cannotbe supported as a PBE of this game when Player 3s beliefs satisfy � > 1

3.

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Figure 5a. Pooling Strategy Prole C;C 0 when � > 13.

- Case 2, � � 13. As depicted in Figure 5b, Player 3 responds with R (as

illustrated by the red arrows) given that his beliefs are � � 13. In this context,

Player 2 does not deviate because his prescribed strategy, C 0, gives him apayo¤ of 1, while deviating to D0 would give him a payo¤ of 0. Similarly,Player 1 does not deviate because his prescribed strategy, C, gives him apayo¤ of 1, exceeds his payo¤ from deviating to D, zero. Therefore, strategyprole C;C 0 can be supported as a PBE of this game when Player 3s beliefssatisfy � � 1

3.

Figure 5b. Pooling Strategy Prole C;C 0 when � � 13.

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5. BONUS EXERCISE - A model of sales, based on Varian (1980).2 Firms o¤ersales at di¤erent times. In this exercise, we show that o¤ering sales (or, more generally,randomizing over prices) is a strategy that helps rms maximize their expected prots.This exercise belongs to the literature on price dispersionwhere rms face a shareof consumers who are uninformed about prices, and o¤er di¤erent prices, either at dif-ferent locations (spatial price dispersion) or at di¤erent points in time (temporal pricedispersion, as we analyze in this exercise). Price discrimination models, in contrast,assume that consumers can perfectly observe prices.

Consider an industry with N rms and free entry, so rms enter until the prots fromdoing so are zero. Consumers have a reservation price r for an homogeneous goodand purchase at most one unit. A share �I of consumers is informed about prices,buying from the cheapest rm, and a share 1��I are uninformed, who purchase fromany rm. Therefore, there are �U = 1��

I

Nuninformed consumers per rm. Firms face

a symmetric cost function C(q) = F + cq, where F > 0 denotes xed costs and crepresents its marginal cost. Every rm can only charge one price for its product.

As a reference, note that C(�I+�U) = F+c(�I+�U) denotes the cost from serving themaximum amount of customers (both informed and uninformed consumers). Therefore,the ratio

pL �F + c(�I + �U)

�I + �U

represents the average cost in this setting.

We next show that, in the above context, every rm has incentives to randomize itspricing over a certain interval. The following questions should help you nd the speciccumulative distribution function F (p) that every rm uses in the mixed-strategy Nashequilibrium of the game.

(a) Show that F (p) = 0 for all p < pL, and that F (p) = 1 for all p > r.

� This question essentially asks us to trimthe support of price randomizationin F (p) and characterize its lower and upper bounds.

� Lower bound. When charging prices below pL, a rm must be making losses,since its price lies below its cost in the most favorable scenario (when alltypes of consumers purchase the good). Therefore, the rm does not assigna probability weight on prices below pL.

� Upper bound. If a rm charges a price above the reservation price r, nocustomer buys from it, regardless of whether he is informed or uninformed.The rm then has no incentives to assign a probability weight on prices abover. Combining our above results, the price p in F (p) must lie in the interval[pL; r].

(b) Show that the cumulative distribution function F (p) is non-degenerated, that is,there is no pure strategy Nash equilibrium.

� If rm i uses a pure strategy, charging price pi = pL, it makes a loss, thushaving incentives to exit the industry. (Recall that, in equilibrium, rms

2Varian, H. (1980) A model of sales,American Economic Review, 70, pp. 65159.

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make zero prots.) If, instead, the rm sets a higher price pi that satisesr � pi > pL, other rms would have incentives to undercut rm is price bya small ". Therefore, rm i does not use a pure strategy.

(c) For simplicity, assume that F (p) is continuous.3 Find expected prots from thepricing strategy F (p).

� If a rm sets the lowest price, it attract all consumers, and its prot is

�s(p) = p(�I + �U)� F � c(�I + �U)

where the subscript s denotes that the rm is successful at attracting allconsumers.If, instead, the rm is unsuccessful, it only sells its product to uninformedconsumers, earning

�f (p) = p�U � F � c�U

where the subscript f denotes failure.The probability that rm i sets a price p higher than its rival j 6= i is

F (p) = Prob fp � pjg

so the probability that p < pj is the converse, 1 � F (p). As a result, theprobability that p is lower than the prices of all its N � 1 rivals is

[1� F (p)]N�1 ;

which represents the probability that rm i sells to informed consumers. Fi-nally, the probability that rm i does not sell to informed consumers is

1� [1� F (p)]N�1 :

� We are now ready to write rm is expected protZ rpL

264�s(p) [1� F (p)]N�1| {z }Success

+ �f (p)h1� [1� F (p)]N�1

i| {z }

Failure

375 f(p)dp(d) Using the no entry condition, nd the cumulative distribution function F (p) with

which every rm randomizes.

� Since rms make no prots in equilibrium (otherwise entry or exit would stillbe protable), the above expected prot must be equal to zero, which entails

�s(p) [1� F (p)]N�1 + �f (p)h1� [1� F (p)]N�1

i= 0

Rearranging,

F (p) = 1��

�f (p)

�f (p)� �s(p)

� 1N�1

The denominator is negative since �f (p) < �s(p) for any price p 2 [pL; r].Therefore, the numerator must also be negative, �f (p) < 0.

3That is, there is no mass point in the pricing strategy F (p) that every rm uses. Intuitively, therm chooses all prices in the [pL; r] interval with positive probability. More compactly, this means that thedensity function f(p) > 0 for all p 2 [pL; r].

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(e) Show that the cumulative distribution function F (p) has full support in p 2 [pL; r].That is, F (pL + ") > 0 and F (r � ") < 1 for any " > 0.� Prices slightly above pL. If, instead, F (pL + ") = 0, rm i is assigning noprobability weight to prices slightly higher than the lower bound pL. There-fore, rm i assigns probability weight to prices strictly above pL + ". In thatcase, another rm j could undercut rm is price and set for instance a pricepL +

"2to make positive prots. Hence, F (pL + ") > 0 for any " > 0.

� Prices slightly below r. If, instead, F (r� ") = 1, rm i assigns no probabilityto prices slightly below r. At ep < r, only uninformed consumers purchase thegood and the rm earns ep�U � F � c�U , yielding zero prots. However, adeviation to price p = r yields r�U �F � c�U which is positive, thus makingsuch deviation protable. Therefore, F (r � ") < 1 for any " > 0.

(f) Taking into account that �f (r) = 0, nd the equilibrium number of rms in theindustry, n�.

� Condition �f (r) = 0 entails

r�U � F � c�U = 0

Substituting �U = 1��I

Ninto the above expression, yields

F = (r � c) 1� �I

N| {z }�U

Solving for N , we obtain

N� =(r � c)

�1� �I

�F

.

Therefore, the higher the prot margin r � c, the larger share of the unin-formed consumers 1� �I , and the lower the entry cost F , the more rms inequilibrium.

(g) Taking into account that �s(pL) = 0, and the equilibrium number of rms N�,nd the lower bound of rmsrandomization strategy, pL.

� Condition �s(pL) = 0 entails

pL(�I + �U)� F � c(�I + �U) = 0

Substituting �U = 1��I

Ninto the above expression, yields

F = (pL � c)��I +

1� �IN

�Further inserting the equilibrium number of rms, N�, found in part (f), wehave

F = (pL � c) �I +

1� �I(r�c)(1��I)

F

!

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Rearranging, we obtain

F = (pL � c)�(r � c)�I + F

r � c

�Solving for pL, we nd the lower bound of rmsrandomization strategy

pL =c (r � c)�I + rF(r � c)�I + F

(h) Evaluate your above results in the special case in which all consumers are unin-formed.

� When all consumers are uninformed, �I = 0, the lower bound of rmsran-domization strategy, pL, becomes

pL =rF

F= F

which coincides with the upper bound of rmsrandomization strategy. Inother words, rms put full probability weight on one price, p = r, with everyrm extracting all surplus from a share 1

Nof consumers.

(i) Numerical example. Evaluate your results in parts (d), (f), and (g) at parametervalues r = 1, F = 2

9, c = 0, and �I = 1

3.

� In this setting, the equilibrium number of rms becomes

N� =(1� 0)

�1� 1

3

�29

= 3.

In addition, the lower bound is

pL =0 (1� 0) 1

3+ 2

9� 1

(1� 0) 13+ 2

9

=2

5.

In this context, the share of uninformed consumers for every rm becomes

�U =1� �IN�

=1� 1

3

3

=2

9:

� Finally, the cumulative distribution function is

F (p) = 1��

�f (p)

�f (p)� �s(p)

� 1N�1

23

where prots from successfully attracting all customers are

�s(p) = (p� c)��I + �U

�� F

= (p� 0)�1

3+2

9

�� 29

=5p� 29

and prots from only attracting uninformed consumers are

�f (p) = (p� c)�U � F

= (p� 0) � 29� 29

= �2 (1� p)9

Therefore, the above function F (p) becomes

F (p) = 1��

2 (1� p)5p� 2� 2 (1� p)

� 12

= 1�

s2 (1� p)7p� 4 .

which is distributed between the lower bound pL = 25 and the upper boundr = 1.Di¤erentiating F (p) with respect to p, we nd its probability density function

f (p) =3 (7p� 4)�

32p

2 (1� p)

which is positive so that rms randomize over the full support of the interval�29; 1�.

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