Game Theory with Applications to Finance and

Marketing

Lecture 5: Further Applications in Finance and Marketing, I

Chyi-Mei Chen, R1102x2964, cchen@ccms.ntu.edu.tw

1. We shall give a series of applications to finance and marketing.

2. Example 1. Consider the following order-driven batch trading modeldue to Kyle (1985), where there are three classes of traders: a monopo-listic insider, a bunch of noise traders, and a few Bertrand-competitivemarket makers. The insider and the noise traders submit market or-ders to the competitive market makers, and the latter are responsiblefor determining the asset (stock) price. The game goes as follows. First,nature moves by selecting a true value1 v from a prior normal distribu-tion N(µv, σ

2v) for the traded asset, and by selecting a demand quantity

x from a prior normal distribution N(0, σ2x) for the noise traders. The

latter may be irrational traders acting on noise rather than informa-tion (Black, 1986, Journal of Finance) or liquidity traders who do nothave time discretion (nondiscretionary liquidity traders; Admati andPfleiderer, 1988, Review of Financial Studies) and must complete tradeimmediately because of liquidity shocks. In any case, we assume thatx and v are independent, so that the noise traders’ trading behavior, x,contains no information regarding the true value v of the stock. Next,the insider gets to see the realization v but not x. The market mak-ers see neither v nor x. The insider is risk neutral, and given the vthat he saw and a pricing schedule p(x + y) that he expects to pre-vail in equilibrium, he selects his demand quantity y to maximize hisexpected profit from trade, knowing that v will be revealed after the

1In a simple model where the stock pays a one-time liquidation dividend, the truevalue corresponds to the (random) amount of the liquidation dividend. If the stock canbe traded more than once, then v may represent the resale value at the next trading date.Of course, in the latter case, a more robust approach is to analyze the entire course oftrading, especially when these traders are long-lived players. A long-lived player maygenerally want to adopt a dynamic trading strategy.

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trade is completed. Finally, the risk-neutral Bertrand-competitive mar-ket makers, given their correct expectation about the insider’s tradingstrategy y(v), compete in price upon seeing the total demand quan-tity submitted, i.e. x + y. We shall look for a linear perfect Bayesianequilibrium for this game, in which the insider’s quantity y is a linearfunction of his private information v, and the equilibrium asset price pchosen by the market makers is a linear function of x+ y.

3. Note that in the Bertrand equilibrium of the subgame where marketmakers engage in price competition to absorb the order imbalance,each market maker must earn a zero expected profit conditional on hisinformation x+ y, which implies that the asset price

p(x+ y) = E[v|x+ y(v)],

fulfilling the semi-strong form efficiency of the asset market since x+ yis the public information at the time trading takes place. On the otherhand, the insider’s demand quantity

y(v) ∈ argmaxy

E[y(v − p(x+ y))|v],

where the price schedule p(x+y) is correctly expected by the insider inequilibrium, and the expectation is taken over x. A pure strategy per-fect Bayesian equilibrium (PBE) of this game is a pair p(x+y), y(v),such that each type of each player correctly expects this strategy pro-file, and no one wants to deviate unilaterally.2 We shall look for a

2This is actually a signaling game with noise, where the insider is the signal sender,and the market makers are the receivers of the signal (insider’s market order). This isa signaling game with noise, because market makers do not observe the signal directly;rather, what they see is the signal plus a noise (the market order submitted by the noisetraders). In a linear perfect Bayesian equilibrium of this game, the assumption of theGaussian distributions ensures that no order imbalance can be detected as an outcomeof the insider’s off-the-equilibrium behavior. This greatly simplifies the determinationof the market makers’ supporting beliefs: all the posterior beliefs can be determined byBayes Law. Note that there is a unique linear PBE of this game under the Gaussianassumption, but there may be non-linear PBE’s for this game. When we replace theGaussian distributions by other probability distributions, the uniqueness result may ceaseto hold, as the multiplicity of PBE’s is the norm for signaling games. We shall demonstratethat this indeed is so via an example taken from Biais and Rochet (1996) in a subsequentsection.

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linear equilibrium in which these two schedules are linear. Hypothesizetherefore that in a perfect Bayesian equilibrium,

p(x+ y) = a+ b(x+ y), y(v) = α+ βv,

with some constants a, b, α, and β. Now, given p(·), y(v) must beoptimal for the insider of type v:

maxy∈ℜ

E[y(v − a− b(x+ y))|v] = y[v − a− b(E[x] + y)] = y(v − a− by)

⇒ y(v) =v − a

2b

⇒ α =−a

2b, β =

1

2b. (Θ)

Note that the insider will trade more agressively if b is small: if addingone more unit of trade will not move the price too much, then theinsider is encouraged to trade more. The situation is just like the insiderfacing an increasing supply curve (why does b have to be positive?). Inequilibrium the reciprocal of b, i.e. 1

b, will be referred to as the market

depth, which gives a direct measure of market liquidity: if 1bis large,

then the active traders (the insider and the noise traders) do not bearmuch liquidity loss when trading with the market makers.

On the other hand, expecting that y(v) = α+βv, the Bertrand marketmakers must set, using cov(x, v) = 0,3

p(x+ y) = E[v|x+ α+ βv] = E[v] +cov(v, x+ α+ βv)

var(x+ α + βv)(x+ y −E[y])

= µv +βvar(v)

var(x) + β2var(v)(x+ y − α− βµv)

3If X,Y are bivariate normal with their first two moments being (µx, µy, Vx, Vy, covxy),then

E[X|Y = y] = µx +covxyVy

(y − µy),

var(X|Y = y) = Vx − [covxy]2[Vy]

−1.

Note that E[X|Y = y] depends on y, but var(X|Y = y) does not. This follows fromTheorem M1 directly.

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⇒ a = µv, b =βσ2

v

σ2x + β2σ2

v

. (Γ)

Note that an increase in σv has two opposing effects on b: on the onehand, it implies that the order flow x + y is quite volatile so that theinformation contained in the order flow regarding the mean of v is quitenoisy, and hence E[v|x+ y] should not depend too much on x+ y (thiscorresponds to the fact that var(x+ y) appears in the denominator ofb); on the other hand, an increase in σv also implies that x+y becomesmore correlated with v, and hence the numerator of b, cov(v, x + y) isalso enlarged. Since in var(x+ y) =var(x) + β2var(v), the term var(x)does not grow when var(v) increases, the overall effect of an increase inσv is that b becomes higher. This discourages the insider from tradingaggressively (note that β is decreasing in the b perceived by the insider,and the perception is correct in a Bayes-Nash equilibrium), and henceβ is decreasing in σv!

Thus conditions (Θ) and (Γ) give 4 equations for the 4 unknownsa, b, α, β. Solving, we have

p(x+ y) = µv +σv

2σx

(x+ y),

y(v) =σx

σv

(v − µv).

Since all parties are risk neutral, µv can be regarded as the initial stockprice. Observe that the price rises if and only if x + y > 0; that is,there is a net buy order. The intuition is that when the market makerssee x+ y > 0, they partially attribute this event to a positive y, whichoccurs when v > µv. Thus the conditional expectation of v givenx + y > 0 rises above µv. (Of course x + y > 0 may also result fromx > −y > 0, but the latter event is less likely than y > 0, as E[x] = 0.)

Let us consider two extreme cases. If the market makers are sure thaty ≡ 0, then the price will not move (p = µv): noise trades x areuncorrelated with v, and hence if the order flow consists of x only,it is completely uninformative. On the other hand, if σx = 0, thenthere can be no trade in equilibrium. (This follows from the no-tradetheorem discussed in Lecture 3.) To see that this is so, imagine thattrade occurs at price p after the insider submits a buy order y > 0.

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Then, one feasible strategy (but not necessarily a best response) forthe insider is to submit y > 0 if and only if v > p. With this strategy,the insider has a strictly positive expected profit, implying that inequilibrium the market makers can never break even. A market makerwould be better off setting p = +∞ for any buy order and (for thesame reason) p = −∞ for any sell order.4 Note that the insider’s profitbecomes an arbitrage profit, since without σx, the insider faces no pricerisk so long as the market makers use pure strategies to determine p.Another way to interpret the no-trade pure strategy BE is to say thatarbitrage opportunities cannot exist in equilibrium, even for the insider.

Observe also that y > 0 if and only if the insider sees v > µv. Thisis quite natural. With a single opportunity of trading, we do not needto worry about the possibility that the insider sells in the good statesto manipulate the beliefs of the uninformed traders and then buy backsubsequently. Note also that in equilibrium, the magnitude of pricechange |p−µv| is increasing in the trading volume |x+y| with the marketmakers,5 since b is strictly positive. This result is consistent with a hugebody of empirical literature; see for example Karpoff (1987). Again,the idea is that |y| is proportional to |v − µv| while |x| has nothingto do with v (if x, v are independent, then so are |x|, v; see a lecturenote of mine in Stochastic Processes). Thus an order flow with large|x + y| indicates a more serious adverse selection problem, and henceself-protecting market makers will raise the ask price and lower the bidprice by a larger amount (here we call p an ask if x+ y > 0 and a bidif x + y < 0; there can be an infinite number of bid and ask prices inequilibrium, depending on the order flow).

Now we do comparative statics. As we argued above, the insider tradesless aggressively (meaning that β > 0 is lower) when σv is higher, sincea higher b will result. On the other hand, β is increasing in σx fora good reason. The presence of noise traders provides camouflage for

4This argument is robust even if market makers can use mixed strategies. For if a buyorder y > 0 may face a random p, then as long as p is in L1(Ω,F , P ) (i.e. it has a finitemean), a profitable feasible strategy (but not necessarily a best response) for the insiderwould be to submit y > 0 if and only if v > E[p]. Again, this implies that the marketmakers bear expected losses in equilibrium, a contradiction.

5Note that the total trading volume is 12 [|x|+ |y|+ |x+ y|].

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insider trading, and a higher noise trading level improves the insider’sprofit. At first, observe that with the Gaussian assumptions for x and v,the expected noise trading level, namely E[|x|], is proportional to σx.

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Thus σx does represent the level of noise trading. How does an increasein σx affect b, and hence β? A high σx implies that the order flow isvolatile, which then implies that rational market makers will not adjustthe price too much when the order flow changes. This corresponds tothe fact that σx appears in the denominator of b. Rationally expecting alower b when perceiving a high σx, the insider trades more aggressivelyin equilibrium, and hence a higher β results.

The insider’s profit, conditional on v, is

σx

2σv

[v − µv]2,

so that the insider’s expected profit, before seeing the realization of v,is

E[σx

2σv

[v − µv]2] =

σxσv

2,

which is consistent with our above discussions about the effects of in-creasing σx or σv.

One may get the impression that a high σx may lead to a more volatileprice. This is incorrect. With a higher σx, b is lower, which tends tostabilize the price. Does this mean that the price will be less volatile?No! With a high σx, the insider will correspondingly trade more ag-gressively in equilibrium, leading to more volatile order flow x+y. Theeffect of a more volatile order flow and the effect of a lower b cancel outeach other exactly inequilibrium, and it is easy to verify that var(p) isindependent of σx!

Define Σ0 ≡ σ2v and

Σ1 = var(v|p),

where Σt stands for the unresolved uncertainty in v at time t (taking

6Show by direct integration that E[|x|] =√

2πσx.

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the post-trade point in time as time 1). Using the formula7

var(v) = E[var(v|p)] + var(E[v|p]),

we now show that Σ1 =12Σ0; that is, exactly half amount of the insider’s

7Suppose that x, y ∈ L2(Ω,F , P ). Consider the following minimization program:

mina∈ℜ

f(a) ≡ E[(y − a)2|x].

Apparently, f(·) is convex, and hence the optimal a∗ solves the first-order condition

f ′(a∗) = 0 ⇒ a∗ = E[y|x].

Now, for any Borel function g(·), since for any x0 in the support of x,

E[(y − g(x))2|x = x0] = E[(y − g(x0))2|x = x0] ≥ E[(y − a∗)2|x = x0],

we conclude that, by the law of iterated expectations,

E[(y − g(x))2] ≥ E[(y − a∗)2] = E[(y − E[y|x])2].

In particular, consider the constant function g(·) = E[y]. We have

E[(y − E[y])2] ≥ E[(y − E[y|x])2].

In fact, observe that

E[(y − E[y])2] = E[(y − E[y|x])2] + E[(E[y|x]− E[y])2],

where E[(y − E[y])2] = var(y),

E[(y − E[y|x])2] = E[E[(y − E[y|x])2|x]] = E[var(y|x)],

and, by the law of iterated expectations, we have E[E[y|x]] = E[y] so that

E[(E[y|x]− E[y])2] = var(E[y|x]).

Thus we conclude that

var(y) = E[var(y|x)] + var(E[y|x]),

and moreover,var(E[y|x]) ≤ var(y).

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information is revealed through trade at time 1. Note that

Σ1 = var(v|p) = E[var(v|p)] = var(v)− var(E[v|p])

= Σ0 − var(p) = Σ0 − b2var(x+ y)

= Σ0 − b2(σ2x + β2Σ0)

= Σ0 − (σv

2σx

)2(σ2x + (

σx

σv

)2Σ0)

=1

2σ2v =

1

2Σ0.

Next, observe that8

var(v)− var(p) = var(v)− var(E[v|p]) = E[var(v|p)] ≥ 0.

This underlies a well-known puzzle uncovered by financial empiricists;see Shiller (1981, AER): the price volatility should not exceed the vari-ance of the stock’s intrinsic value, but this conclusion is violated inreality.

4. Example 2. (Competition among multiple insiders that agreeto disagree) Suppose that in a Kyle market, market makers are facedwith two insiders, called A and B, instead of just one as in Kyle’s 1985static model. The two insiders agree to disagree about the level of noisetrading. A believes that the standard deviation of x is σa, B believesthat it is σb, and market makers believe (correctly) that it is σx, whereσa > σx > σb. Assume that v is a standard normal random variable.The game is the same as in Kyle (1985), except that in the first stagethe two insiders submit market orders at the same time, as the noisetraders do. Suppose that σa = 3, σx = 2, σb = 1. Conjecture that inequilibrium, insider A’s market order is ya = βav, insider B’s marketorder is yb = βbv, and market makers’ pricing rule is p = λQ, whereQ = ya + yb + x. Find the equilibrium, βa, βb, and λ.

8We have used the fact that E[v|p] = E[v|p(x+ y)] = E[v|x+ y] = p. To see that theseequalities hold, recall from my notes in Stochastic Processes that E[v|p] is a short-handnotation for E[v|σ(p)], where σ(p) is the σ-algebra generated by the r.v. p. It is easy toshow that σ(p) = σ(x+ y), because p is a strictly monotonic transform of x+ y. Thus wehave, again in short-hand notation, E[v|p] = E[v|x+ y].

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Solution. In equilibrium, the two insiders only care about λ, whileλ reflects only market makers’ belief about the level of noise trading,which is summarized by the correct σx. Hence we conjecture that thereexists an equilibrium where the three parties’ equilibrium strategiescoincide with their strategies obtained in the symmetric equilibrium ina model where all traders agree that σx is the level of noise trading.

More precisely, market makers’ problem is, given βa and βb, to deter-mine the price after seeing Q,

p =cov(v, v(βa + βb) + x)

var(v(βa + βb) + x)[v(βa + βb) + x].

This gives rise to

λ =βa + βb

4 + (βa + βb)2.

For i, j ∈ a, b, insider i’s problem is, given βj and λ, to

maxyi

[v − λ(βjv + E[x] + yi)]yi,

and as long as λ > 0, the necessary and sufficient first-order conditionyields

yi(v) =(1− λβj)v

2λ.

Hence we obtain a system of three equations with three unknowns:

λ =βa + βb

4 + (βa + βb)2;

βa =(1− λβb)

2λ;

βb =(1− λβa)

2λ.

The last two equations give rise to

λβa = λβb =1

3,

and hence it is necessary that

βa = βb = β,

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which reduces the above system of equations to the following two equa-tions:

λ =2β

4 + 4β2, β =

1− λβ

2λ.

Solving, we obtain

βa = βb =√2, λ =

√2

6.

To fully understand the meaning of agreeing to disagree, consider thefollowing new assumption: insider A believes that market makers be-lieve that the level of noise trading is σa (and hence will choose λa),and insider B believes that market makers believe that the level ofnoise trading is σb (and hence will choose λb), whereas market mak-ers know the two insiders’ incorrect beliefs, and market makers believe(correctly) that the level of noise trading is σx (and hence will chooseλ). In this case agreeing to disagree result in the two insiders choosingdifferent β’s, because they believe in different λ’s.

This new assumption can be contrasted with our original problemabove, where insiders know that market makers believe that the level ofnoise trading is σx (and hence will choose λ). Given this belief, insidersfind their own beliefs regarding the level of noise trading totally irrel-evant, because they agree on the first moment of the noise trade, andbeing risk-neutral, the insiders do not care about the second momentof the noise trade.

5. Example 3. Kyle’s trading model is actually a signaling game withnoise, where the insider is the signal sender, and the market makersare signal receivers. It is well known that a signaling model may sufferfrom multiple equilibria problem, in the sense that there may existmultiple PBE’s for the same set of parametric values, which weakensthe predictive power of the theory.

To demonstrate the multiple equilibria problem, consider the followingdiscrete specifications. Suppose that instead of being Guassian, thenoise trade u is equally likely to be 1 and−1, and v is equally likely to be−2,−1, 1 and 2. The following proposition shows that the unique PBEobtained in Kyle’s model is a consequence of the assumed Gaussiandistribution for the random variables (v, x).

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Proposition 1 (Biais and Rochet 1996.) For each a ∈ (0, 23), define

the insider’s market order by

Xa(2) = −Xa(−2) = 1 + a, Xa(1) = −Xa(−1) = 1− a,

and market makers’ pricing rule by

∀z ∈ −2− a,−2 + a,−a, a, 2− a, 2 + a, Pa(z) = −Pa(−z),

and

Pa(a) =1

2, Pa(2 + a) = 2, Pa(2− a) = 1.

Then, Pa(·) and Xa(·) constitute a PBE.

Proof. Given Xa(·), define the order imbalance observed by the mar-ket makers by Z(u, v), with

Z(−1, 2) = Z(1,−1) = a, Z(1,−2) = Z(−1, 1) = −a.

Thus, for example, when seeing an order imbalance a, the marketmakers think that (u, v) = (−1, 2) and (u, v) = (1,−1) are equally

likely, and hence they set Pa(a) = 2+(−1)2

= 12. Similarly, they set

Pa(−a) = −2+12

= −12. On the other hand, the order imbalance 2 + a,

if it appears, is fully revealing: it must be that (u, v) = (1, 2), so thatPa(2 + a) = 2. The order imbalance 2 − a, likewise, can appear onlywhen (u, v) = (1, 1), and hence Pa(2 − a) = 1. Finally it is easy tocheck that

∀z ∈ −2− a,−2 + a,−a, a, 2− a, 2 + a, Pa(z) = −Pa(−z).

Note that discrete specifications differ from Kyle’s Gaussian formula-tion in one imporant aspect: there are off-the-equilibrium order imbal-ances, and posterior beliefs are needed for these order imbalances. Tothis end, we specify the following supporting beliefs:

∀z > 0, z = 2 + a, 2− a, a, prob.(v = 2|z) = 1,

and

∀z < 0, z = −2 + a,−2− a,−a, prob.(v = −2|z) = 1.

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It remains to check that Xa(·) is the insider’s best response given Pa(·).Given a, define

Ba(v, x) ≡ x(v − E[Pa(x+ u)]), ∀x ∈ ℜ, v = −2,−1, 1, 2.

Given the above supporting beliefs, x not contained in the set −2 −a,−2 + a,−a, a, 2− a, 2 + a can never be optimal. It is easy to showthat when a ∈ (0, 2

3),

Ba(2, 1 + a) ≥ Ba(2, 1− a), Ba(1, 1− a) ≥ Ba(1, 1 + a).

This finishes the proof. ∥

6. Example 4. Suppose that a common stock is traded at date 1 anddate 2 (t = 1, 2). The stock will pay a one-time liquidation dividendv1 + v2 at the end of t = 2, where v1 and v2 are independent and eachmay take values 0 or 1 with respectively probability 1−π and π, where0 < π < 1. Moreover, the realization of vt will be publicly announcedat the end of time t.

Unless otherwise mentioned, all market participants are risk-neutralwithout time preferences. At first, there are competitive specialists(market makers) who compete in setting bid and ask prices at each datet. In addition to market makers, there is a rational investor who candecide whether to spend k > 0 to acquire information at the beginningof t = 1, and if k is spent, then with probability 1

2this rational investor

may privately know v1 (but not v2) right before trading gets startedat date 1, and with probability 1

2she may know v2 (but not v1) right

before trading gets started at date 2.

At t = 1, 2, there is also a date-t liquidity trader who is equally likelyto buy or sell 1 share, where the sign of date-1 liquidity trade and thesign of date-2 liquidity trade are statistically independent, and theseliquidity trades are also independent of v1 and v2.

We shall assume that the specialists (market makers) know the rationalinvestor’s decision about information acquisition before trading getsstarted at time 1. For simplicity, let us make the unrealistic assumptionthat the specialists will also know whether the rational investor willlearn v1 or will learn v2, if they know that the rational investor haschosen to become informed.

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The timing of the game is as follows.

• At the beginning of t = 1, the rational investor first decides to ornot to spend k.

– If the rational investor does not spend k, then the marketmakers post bid and ask prices for t = 1, assuming the absenceof an informed trader at both t = 1 and t = 2.

– If the rational investor does spend k, she and the market mak-ers will then learn whether the rational investor will privatelysee the realization of v1 or the realization of v2. In either case,the market makers then post bid and ask prices for t = 1.

• Then the time-1 liquidity trader learns her trading need. If theinformed trader exists at t = 1, then she learns the realizationof v1. (An informed trader exists at t = 1, if and only if therational investor has chosen to become informed and her privateinformation is about the realization of v1.)

• Then, among the traders showing willingness to trade at t = 1,exactly one of them will be selected to trade with the marketmakers at t = 1. Also, exactly one market maker will be selectedto trade with the selected trader.

• Then v1 is realized, and the game moves on to t = 2.

• At the beginning of t = 2, the market makers know whether aninformed trader exists at t = 2 before posting bid and ask pricesfor t = 2.

• Then the time-2 liquidity trader learns her trading need. If theinformed trader exists at t = 2, then she learns the realizationof v2. (An informed trader exists at t = 2, if and only if therational investor has chosen to become informed and her privateinformation is about the realization of v2.)

• Then, among the traders showing willingness to trade at t = 2,exactly one of them will be selected to trade with the marketmakers at t = 2. Also, exactly one market maker will be selectedto trade with the selected trader.

• Then v2 is realized, and the dividend v1 + v2 is distributed to allshareholders.

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We shall assume that

(Γ)2π(1− π)

(2π + 1)(3− 2π)> k >

π(1− π)

(2π + 1)(3− 2π).

(i) Compute the date-1 ask and bid prices A1 and B1.(ii) Now, ignore part (i). Suppose instead that a circuit breaker existsso that the stock market will be closed at t = 2 if trade occurs at time1 at a price greater than or equal to 2π + ρ or lower than or equal to2π − ρ > 0. In particular, suppose that ρ = 1

4. How does the circuit

breaker affect the date-1 equilibrium?(iii) Suppose instead that a circuit breaker exists so that the stock mar-ket will be closed at t = 2 if trade occurs at time 1 at a price greaterthan or equal to 1+ρ or lower than or equal to 1−ρ > 0. In particular,suppose that ρ = 1

4and π = 3

4. How does the circuit breaker affect the

date-1 equilibrium?

Solution. If the rational investor chooses not to become informed,then A1 = B1 = E[v1] + E[v2] = 2π. By staying uninformed, the ra-tional investor gets a zero payoff. In computing the ask and bid prices,we must first conjecture (and verify later) the rational investor’s equi-librium decision about whether to become informed. Let us conjecturethat the rational investor does choose to become informed in part (i).

If the rational investor chooses to become informed, then there are twoequiprobable events. (Recall that market makers know which event didoccur before posting ask and bid prices.)

• Event 1: the rational investor will learn v1 but not v2.In this event, A2 = B2 = v1+π, and given that the market makersknow the rational investor has become informed before choosingbid and ask prices at date 1, we must have

A1 = E[v2] +12· π · 1 + 1

2· 12· π

12· π + 1

2· 12

= π +3π

2π + 1.

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Similarly, one can show that

B1 = E[v2] +12· (1− π) · 0 + 1

2· 12· π

12· (1− π) + 1

2· 12

= π +π

3− 2π.

• Event 2: the rational investor will learn v2 but not v1.In this event, we have A1 = B1 = 2π, and similarly, we have

A2 = v1 +12· π · 1 + 1

2· 12· π

12· π + 1

2· 12

= v1 +3π

2π + 1.

Similarly, one can show that

B2 = v1 +12· (1− π) · 0 + 1

2· 12· π

12· (1− π) + 1

2· 12

= v1 +π

3− 2π.

It follows that the rational investor’s payoff from becoming informedand engaging in stock trading is

−k + 2 · 12· 12[π · (1− 3π

2π + 1) + (1− π) · ( π

3− 2π− 0)]

=2π(1− π)

(2π + 1)(3− 2π)− k ≥ 0,

where the last inequality follows from condition (Γ). Thus our con-jecture that in equilibrium the rational investor is willing to becomeinformed turns out to be correct given that condition (Γ) holds. Itfollows that the bid and ask prices at date 1 are either (in event 2)

A1 = B1 = 2π,

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or (in event 1)

A1 = π +3π

2π + 1, B1 = π +

π

3− 2π.

This concludes part (i).

Consider part (ii). It is easy to see that in equilibrium the presenceof that particular circuit breaker alters nothing at date 1. Thus itseffect is leading to the likely unpleasant event that the liquidity tradefails to take place at date 2. Indeed, the circuit breaker will never betriggered in event 2, but in event 1, the informed trader does not alterher trading behavior because of the circuit breaker. The circuit breakerwill be triggered if either

A1 ≥ 2π +1

4⇔ 1

4≤ π ≤ 1

2,

or

B1 ≤ 2π − 1

4⇔ 1

2≤ π ≤ 3

4.

Thus for π lying between 14and 3

4, with a positive probability the date-2

liquidity traders get hurt by the presence of the circuit breaker.

Finally, consider part (iii). It follows from

A1 = B1 = 2π > 1 + ρ,

we know that the new circuit breaker will be triggered in event 2 forsure. This implies that the informed trader can at best make a tradingprofit in event 1. Condition (Γ) ensures that the rational trader, inanticipation of this, would rather stay uninformed in the first place(and would stay out of the stock market afterwards). It follows thatA1 = B1 = 2π in both events; the new circuit breaker minimizes thedate-1 price variability but it also maximizes the chance that the circuitbreaker may be triggerd at date 1 by forcing the informed trader toleave the market.

This exercise demonstrates the following important fact: the presenceof a circuit breaker may indeed reduce the date-1 price variability, buteven so it need not be welfare-improving for the liquidity traders. In

16

part (iii) above, among the public traders only the date-1 liquiditytraders get to trade at better terms; the date-2 liquidity traders fail totrade at all.

In general, the stock price moves for two reasons: either it results fromliquidity trades or it reflects new information. Here, the presence of thecircuit breaker may reduce the chance of informed trading in a laterperiod, which discourages the insider from acquiring information andbecoming informed in the first place, and with the adverse selectionproblem facing the market makers being alleviated, the market makersreduce the bid-ask spread, leading to a lower price variability.

7. Example 5. (Calling a convertible bond.)Firm A has a single shareholder, Mr. A, and the firm has issued aconvertible bond to Mr. B. The bond will mature at date 2 with a facevalue of 2. The firm generates cash flow only at date 2, which can beeither 4 or 1 with equal probability. The bond can be called by Mr.A at the price of y ≤ 4 at date 1, and in case Mr. A calls at date 1,Mr. B can still convert the bond into a fraction λ of the firm’s equity(that is, B can turn himself into the second shareholder) instead ofaccepting the y dollars and tendering the bond. For simplicity, assumethat if A does not call at date 1, then B cannot convert at date 1, butB can always convert the bond into a fraction λ of equity at date 2,one minute before the bond matures at date 2.

Assume that A learns privately the firm’s date-2 cash flow at date 1.The timing of the game is as follows. At date 1, after A privately seesthe firm’s date-2 cash flow, A must choose to or not to call the CB.(Signaling with two possible signals!) If A calls at dat 1, B can decidewhether to accept y dollars and tender the CB to A, or to turn the CBinto common stock that represents a fraction λ of the firm’s equity. IfA does not call at date 1, then the game moves on to date 2, and Blearns the firm’s date-2 cash flow before B decides whether to convertthe CB into equity before the CB matures in the next instant.

Suppose that all investors are risk neutral without time preferencesand in the security market there are no transaction costs and assetsare perfectly divisible.. Suppose that λ ≥ 1

2.

(i) Show that there does not exist a PBE where A calls the CB if and

17

only if his type is 4.9

(ii) Show that a separating PBE where the type-1 A chooses to call theCB but the type-4 A chooses not to exists if and only if y < λ ≤ 1.10

(iii) Show that a pooling PBE where A never calls the CB exists if andonly if y > 4λ, and the supporting beliefs are such that µ < 1

3[ yλ− 1],

where B believes that A is of type 4 with probability µ when seeing Acall the CB at date 1.(iv) Show that a pooling PBE where A always calls the CB exists ifand only if 5

2λ ≥ y.

(v) Conclude that the date-0 price of the CB is 2λ + 1 if y ≥ 4λ and5λ2if y ≤ 5λ

2.11

Solution.

• Consider a separating PBE where A calls if only if his type is 4.In such a PBE, seeing A’s calling B realizes that the date-2 cash

9Intuitively, the type-4 A knows that B will convert the CB at date 2 because 4λ ≥ 2,and so the type-4 A is willing to call the CB only if y ≤ 4λ. If such a separating PBEexists, then B learns that A is of type 4, and with y ≤ 4λ, B would rather convert theCB than accept y. If this were B’s best response after the CB is called, then the type-1 Awould want to call the CB also: the type-1 A knows that B will never convert the CB atdate 2 (and so the type-1 A will get nothing) if the CB is not called at date 1. But then,this is not a separating PBE.

10Intuitively, B would get max(y, λ) when the CB is called by the type-1 A, and so forthe latter to be willing to call the CB, it is necessary that y < 1. If B chooses to accept ywhen the CB is called, then y > λ ≥ 1

2 , so that by deviating and also calling the CB thetype-4 A would get 4− y > 3 > 2 ≥ 4(1−λ), where 4(1−λ) is the type-4 A’s equilibriumpayoff, which is a contradiction. Hence it is necessary that B chooses to convert the CBupon being called. Hence this separating PBE exists if and only if y < λ. In this case, thetype-4 A feels indifferent about calling and not calling the CB at date 1.

11Remark: Our separating PBE is consistent with empirical evidence which documentsa stock price fall when the firm announces that it will call a convertible bond issued earlier;see a more complicated model in Harris and Raviv (1985, JF). The idea is that a low-qualityfirm (here the type-1 firm) would bear a higher cost than its high-quality counterpart if itsignals that it is willing to abandon the opportunity of calling the CB before it matures.The low-quality firm is less confident that the CB will be converted into equity, and in caseconversion does not take place, then because equity is junior to the CB, the equityholdersmay get a very low payoff. This is why the security market believes that a low-qualityfirm is more likely to have made the call, and it adjusts the firm’s stock price to the lowerlevel that reflects the new information (i.e. using posterior rather than prior beliefs).

18

flow is 4, and B converts if and only if y ≤ 4λ. If y ≤ 4λ, thenA’s payoff following calling is (1−λ)4 in the event that the date-2cash flow is 4 and (1 − λ) in the event that the date-2 cash flowis 1. If y > 4λ, then A’s payoff following calling is 4 − y in theevent that the date-2 cash flow is 4 and 1−y in the event that thedate-2 cash flow is 1. If A does not call, then B converts at date 2if and only if the date-2 cash flow is 4. Thus A’s payoff followingno calling is (1 − λ)4 in the event that the date-2 cash flow is 4and zero in the event that the date-2 cash flow is 1.

Thus if y ≤ 4λ, given that B believes that only the type-4 Awould call, and hence calling always induces conversion, the type-1 A strictly prefers to call, a contradiction. We conclude that forthe separating PBE to exist, it is necessary that y > 4λ so thatcalling does not induce conversion. In this case, for the type-4 Ato call, we need

4− y ≥ (1− λ)4 ⇒ 4λ ≥ y,

which is another contradiction. Thus this kind of separating PBEdoes not exist.

• Consider a separating PBE where A calls if only if his type is 1.If the type-1 A does not call, then the game moves on to date2, and B will get min(2, 1) = 1, and A will get nothing. If thetype-1 A does call, then following A’s calling, B converts if andonly if λ ≥ y. Thus for the type-1 A to call, it is necessaryand sufficient that 1 − max(λ, y) ≥ 0, or equivalently, y ≤ 1.On the other hand, if y ≤ 1, and if the type-4 A deviates andcalls, then his payoff is 4 − y if y ≥ λ and (1 − λ)4 if y < λ, sothat A prefers his equilibrium payoff (1− λ)4 if and only if eithery < λ or y ≥ 4λ > λ. The latter implies that y ≥ 4λ ≥ 2 > 1,contradicting the other needed condition y ≤ 1. To sum up, thegame has a separating PBE where the type-1 A calls and the type-4A does not if and only if y < λ ≤ 1.

• Consider a pooling PBE where A never calls.In such a PBE, B converts at date 2 if and only if the date-2 cashflow is 4. A’s payoff is zero if he is of type 1, and if (1 − λ)4if he is of type 4. These payoffs should be higher than what A

19

would otherwise get by making a deviation. Let B believe that Ais of type 4 with probability µ when seeing A call at date 1. IfB converts following A’s calling, his expected payoff (based on hisposterior µ) is

λ[4µ+ (1− µ)] = (1 + 3µ)λ,

so that B converts following A’s calling if and only if y ≤ (1+3µ)λ.If (1 + 3µ)λ ≥ y, the type-4 A feels indifferent about calling andnot calling, but the type-1 A strictly prefers to call, which is acontradiction. Thus µ must be such that

y > (1 + 3µ)λ.

When this last condition holds, then following A’s calling, B willtake y, and in this case A gets 4− y and 1− y respectively whenhis type is 4 and 1. We need y > 4λ so that neither type ofA wants to deviate from the no-calling equilibrium strategy. Tosum up, such a pooling PBE exists if and only if y > 4λ, and thesupporting belief is such that µ < 1

3[ yλ− 1].

• Consider a pooling PBE where A always calls.In such a PBE, seeing A’s calling, B’s posterior is the same as hisprior: he believes that A is equally likely to be of type 4 and oftype 1. B converts following A’s calling if and only if 5

2λ ≥ y. If

52λ ≥ y, A’s equilibrium payoff is (1− λ)4 and 1− λ respectively

when he is of type 4 and of type 1. If the type-1 A deviates by notcalling, then he would get zero, and hence he is not gonna deviate.If the type-4 A deviates, then he would get the same payoff. Weconclude that the pooling PBE does exist when 5

2λ ≥ y. What if

52λ < y? In this case, A’s equilibrium payoff is 4 − y and 1 − y

respectively when he is of type 4 and of type 1. The type-4 Adoes not deviate if and only if y ≤ 4λ, and the type-1 A does notdeviate if and only if y ≤ 1. However, y ≤ 1 is inconsistent with52λ < y and λ ≥ 1

2. Thus we conclude that such a pooling PBE

does exist if 52λ ≥ y.

Based on the above discussions, we conclude that pure-strategy PBE’sexist when y ≥ 4λ or y ≤ 5

2λ, and the date-0 price of CB is 2λ+1 in the

20

former case and 52λ in the latter case. (The date-0 price of a security

is equal to its expected future payoff; one can verify that the date-0price of the CB is the same in the separating PBE and in the poolingPBE where A never calls whenever the two PBE’s are both possibleto arise–that is, when 0 < y ≤ λ.) This game also has semi-poolingPBE in the region where 5

2λ < y < 4λ, but I shall spare that part of

analysis.

8. Example 6. Firm YGO is all-equity financed, and it has three share-holders, the manager M and two outside investors O and Z, each hold-ing one share of the equity. All investors are risk neutral without timepreference.

It is common knowledge that the firm has 20-dollar cash at date 0,and the firm will generate cash x at date 1, which is either 40 dollars(with probability π) or 25 dollars. At date 0, M privately learns therealization of x.

At date 0, O and Z both know that M is considering buying back oneshare from O and Z. The game proceeds as follows. First, M decidesto or not to repurchase 1 share from either O or Z (signaling withtwo possible signals!). If M announces the share repurchase program,then O and Z compete in price to tender one share to M. Competitionbetween O and Z ensures that the stock price P equals one-third of theexpected value of the firm (expectation is taken using the informationrevealed by M’s announcement). Clearly, in any PBE, P ≤ 20 (why?).(i) Show that before this signaling game gets started at date 0, thestock price is P−1 = 15 + 5π.(ii) Show that this game has a unique separating PBE. Find the date-0stock price P0(j) in the separating equilibrium, where j = r (denotingthe event that M announces a share repurchase program) or j = nr(denoting the event that M does not announce the share repurchaseprogram). Show that P0(r) > P−1 > P0(nr).(iii) Show that this game has a unique pooling PBE where M does notrepurchase shares regardless of M’s type.(iv) Explain why P−1 > P0(nr), when apparently no public informationarrives at the stock market at date 0. (The finance literature has raisedthe question why no bad news was found to arrive at the U.S. stock

21

market before the 1987 crash.)Solution.Since the total cash at date 1 cannot exceed 20+40=60, we know thatP ≤ 20. Consider part (i). At date 0, when all three investors areequally uninformed, the stock price is

1

3[π(20 + 40) + (1− π)(20 + 25)] = 15 + 5π.

Consider part (ii). First consider the separating PBE where the type-40 firm announces share repurchase but the type-25 firm chooses notto. In such a PBE, announcing share repurchase is taken by O andZ as direct evidence that the firm is of type 40, and hence the shareprice is 20. Since the share is fairly priced for the type-40 M, he hasno incentive not to announce share repurchase: paying 20 to either Oand Z, and dividing the date-1 cash flow 40 between the remaining twoshares, M will get 20, which is also his payoff if he just sits back anddoes nothing. Will the type-25 M want to announce share repurchasealso? If he does not, then he waits to get his share of cash flow at date1, which is 1

3[20 + 25] = 15; but if he announces share repurchase and

gets accepted by either O or Z, then he gets

1

2[20 + 25− 20] = 12.5 < 15.

Thus such a separating PBE does exist.

Next consider the other type of separating PBE. If share repurchaseannouncement is taken as evidence that the firm is of type 25, then thetransaction price will be 1

3[20+ 25] = 15, following O and Z’s Bertrand

competition. But then the type-40 firm has an incentive to deviate:M’s payoff would become

1

2[20 + 40− 15] = 22.5 > 20,

where 20 is what the type-40 M would get if M follows his supposedequilibrium strategy of announcing nothing. We conclude that theseparating PBE of this game is unique.

Consider part (iii). Consider the pooling PBE where M never an-nounces share repurchase. If deviation occurs, let µ be the probability

22

that O and Z assign to the type-40 M. Then the transaction price fol-lowing O and Z’s Bertrand competition would be

P =1

3[60µ+ 45(1− µ)] = 15 + 5µ.

Apparently, the type-40 M would deviate unless µ = 1. This defines apooling PBE. Is there the other PBE where M always announces sharerepurchase? The answer is negative, for if otherwise the equilibriumtransaction price will again be 15+5π < 20, so that the type-40 M willdeviate.

Finally, consider part (iv). When “trapped” in the above separatingPBE, no news is taken as bad news by the stock market, and hence thedate-0 stock price P0(nr) drops below P−1. The investors (O and Z inour example) believe that the firm will repurchase shares if and only ifM receives good news. Game theory and information economics haveoffered one useful explanation to the documented phenomenon that nobad news was known to arrive in the U.S. stock market during the weekbefore the 1987 crash.

9. Example 7. (Stock price and corporate investment efficiency.)Consider a two-period model where at date 1, firm W is all equityfinanced and is managed by a large shareholder X who owns f ∈ (0, 1)shares of the equity (there is in total one share outstanding). X mustchoose an action d = 0, 1 at date 1, and the action is costless. Thereare two equally likely states G and B at date 1. If d is taken in states ∈ G,B, then the firm generates a date-2 cash flow C(d, s). Assumethat

C(1, G) > C(0, G) = C(0, B) > C(1, B) > 0.

The firm’s common stock is traded at date 1, before X makes the deci-sion about d.

There are 2 classes of participants in the date-1 stock market: an activetrader and many Bertrand competitive market makers. All people inthis world are risk neutral without time preferences. The active traderis equally likely to be a speculator or a liquidity trader. A speculatorcan spend k ≥ 0 to privately observe the true state s before stock

23

trading. A liquidity trader is equally likely to (be forced to) sell or buy1 share using market orders.

The game proceeds as follows. At date 1, the active trader finds outwhether he is a specular or a liquidity trader. If he is a speculator,he must decides whether to spend k and observe s. If he is a liquiditytrader, then he realizes whether he must buy or sell 1 share. Then theactive trader submits a market order to market makers, and the lattercompete in price to absorb the order. Then X can spend a tiny costg ≥ 0 to see the transaction price in the stock market, and chooses d.Then at date 2, s becomes public information, and C(d, s) is realized.(i) Suppose that C(1, G) = 10, C(1, B) = 4, C(0) = 6. Conjecture (andthen confirm) that there is a PBE where the speculator spends k andX spends g in equilibrium. Moreover, conjecture that there are twopossible equilibrium prices Pb and Ps, such that the equilibrium stockprice is Pb (respectively, Ps) after the active trader submits a buy order(respectively, a seller order), and such that X chooses d = 1 and d = 0after he sees respectively the stock prices Pb and Ps. Show that such aPBE exists if and only if

Pb =17

2, Ps = 6,

k ≤ 1

2[10− Pb] +

1

2[Ps − 6] =

3

4,

and

g ≤ f × 1

2× [6− 1

4C(1, G) +

3

4C(1, B)] =

f

4.

(ii) Suppose that C(1, G) = 7, C(1, B) = 4, C(0) = 6. Conjecture (andthen confirm) that there is a PBE where the speculator does not spendk and X does not spend g in equilibrium. Moreover, conjecture thatX chooses d = 0 in equilibrium. What may be the equilibrium stockprice?12

Solution. Consider part (i). Suppose that C(1, G) = 10, C(1, B) = 4,

12This problem intends to demonstrate the fact that the financial analysts in the financialmarkets (the speculator in our model) may possess valuable information which the firmmanager does not have, and that information may be partially revealed by the transactionprice of the firm’s securities. Hence by observing the prices of the firm’s securities, themanager can learn information, which improves the quality of the investment decisions

24

C(0) = 6. Without information revealed by the stock price, X wouldchoose d = 1 over d = 0. The information revealed by the stockprice may improve X’s investment decision because X can now makethe decision d contingent on the realized stock price. Indeed, we shalldemonstrate a PBE in which X chooses d = 1 upon seeing a stocktransaction that occurs at the ask price Pb, and X chooses d = 0 uponseeing a stock transaction that occurs at the bid price Ps.

Thus let us conjecture that in equilibrium the speculator always spendsk and then buys (respectively, sells) 1 share if the state is G (respec-tively, B). Seeing a 1-share buy order, the market makers then believethat with prob. 3

4the state is G, and if X is expected to choose d = 1

if and only if the stock price is A, then

Pb =3

4C(1, G) +

1

4C(1, B) =

17

2.

Similarly, seeing a 1-share sell order, the market makers then believethat with prob. 3

4the state is B, and if X is expected to choose d = 0

if and only if the stock price is Ps, then

Ps =3

4C(0, B) +

1

4C(0, G) = 6.

For this to be an equilibrium, the following IC constraints for X andfor the speculator must hold:

k ≤ 1

2[10− Pb] +

1

2[Ps − 6] =

3

4,

made by the manager. Since the investment decisions are made contingent on the theinformation revealed by the prices of securities, these prices affect the firm’s future cashflows, and the latter certainly determine the prices of the securities at the same time. (Fora story that a high stock price may attract better employees and hence raise the future cashflows of the firm, see Subrahmanyam, A. and S. Titman, 2001, Feedback from Stock Pricesto Cash Flows, Journal of Finance, 56, 2389-2413.) Some of you may be concerned aboutthe fact that the stock trading volume may exceed the number of shares outstanding. Oneinterpretation is that the equilibrium trade consists of short sales. Another interpretationis that the traded security is a European call option written on 1 unit of the stock andthe option has a zero exercise price and will expire at date 2. Note that this option hasa date-2 payoff which is identical to the firm’s date-2 earnings, and hence is equivalent tothe stock from traders’ perspective.

25

and

g ≤ f × 1

2× [6− 1

4C(1, G) +

3

4C(1, B)] =

f

4.

The above two IC’s say that for the speculator, the benefit of spendingk and becoming informed only occurs in state G, where the speculatorpays Pb and gets C(1, G) = 10, and the benefit must exceed the costk of information acquisition; and that for X, the benefit of spending gto look up the transaction price occurs only when the stock price is Ps

(which occurs with probability 12), where X benefits from altering the

investment decision from d = 1 into d = 0. Thus when the above twoconditions hold, and if C(1, G) = 10, C(1, B) = 4, C(0) = 6, then thisgame has a PBE where the stock price affects the firm’s future cashflow, and the stock price itself is also determined by the firm’s futurecash flow.

Next consider part (ii). Suppose that C(1, G) = 7, C(1, B) = 4, andC(0) = 6. With these new numerical values, for example, there is aPBE where X does not spend g and the speculator does not spendk. To see that this is a PBE, note that with these numerical values,X will choose d = 0 if X is uninformed. In equilibrium, X rationallyexpects that the speculator will not spend k, and hence the stock pricewill not contain useful information about the true state. In this casespending g is a waste of money, and hence X will choose not to doit, and being uninformed, X then chooses d = 0. On the other hand,rationally expecting X to not spend g, and that X will choose d = 0,the speculator knows that the state will not affect the future cash flow;in fact the future cash flow is expected to be 6, which will be the stockprice. In this case, spending k and learning the true state is a wasteof money for the speculator, and hence the speculator chooses not todo it. Rationally expecting the behavior of X and the speculator, themarket makers set the stock price at 6. This is indeed a PBE.

10. Example 8. (Optimal debt contract: the static case.)At date 0, Mr. A is endowed with an investment project that needs animmediate cash outflow of 3 dollars. At date 0, A has no money andmust raise the 3 dollars from competitive investors. Investors, like A,are risk neutral without time preferences. One investor, say B, will bepicked out to contract with A. If A gets the 3 dollars and makes the

26

investment, then at date 1 the project generates a cash inflow x, whoseoutcome is equally likely to be 1, 2, and 9. At date 1, however, onlyA can costlessly observe the outcome of x, but B can spend 1 dollarto hire an accountant to find it out (or equivalently, to audit A’s cashearnings, or to verify the true earnings state).

Any feasible financial contract13 will ask A to report his earnings x ∈1, 2, 9 at date 1 (A is free to lie about the true x; that is, x = x), andthe contract specifies two functions d(x) ∈ 0, 1 and R(d(x)x + (1 −d(x)x), where d = 1 and d = 0 indicate respectively that B should andB should not spend 1 dollar for state verification given each possiblereport x, and R(·) specifies the date-1 money transfer from A to B aseither a function of x or of x, depending on whether d(x) = 1 or d(x) =0, such that under contract (d,R), (i) at date 1 it is one of A’s optimalstrategies to report x = x for each true state x ∈ 1, 2, 9; (ii) at date0 B expects to break even; (iii) for all x ∈ 1, 2, 9, 0 ≤ R(x) ≤ x; and(iv) at date 0 A is better off offering this contract to raise the 3 dollarsthan giving up the investment project.

To paraphrase the definition, a feasible contract (d,R) must be (i) ex-post incentive compatible (IC)–A always tells the truth at date 1; (ii)consistent with B’s date-0 individual rationality (IR) constraint; (iii)consistent with the limited liability constraint; and (iv) consistent withA’s date-0 individual rationality constraint. A Pareto optimal contractis a feasible contract (d∗, R∗) that maximizes A’s expected payoff; thatis, no other feasible contract that gives A a strictly higher expectedpayoff. Our task is to find a Pareto optimal financial contract (d∗, R∗) =d∗(1), d∗(2), d∗(9), R∗(1), R∗(2), R∗(9) such that (d∗, R∗) is a feasiblecontract and it maximizes A’s expected payoff in the class of feasiblecontracts.

More specifically, A seeks to

maxd(·),R(·)

1

3[1−R(1)] + [2−R(2)] + [9−R(9)]

subject to

(B’s date-0 IR:)1

3[R(1)− d(1)] + [R(2)− d(2)] + [R(9)− d(9)] ≥ 3;

13This is a consequence of the revelation principle discussed in Lecture 4.

27

(A’s date-0 IR:)1

3[1−R(1)] + [2−R(2)] + [9−R(9)] ≥ 0;

(Limited Liability:) 1 ≥ R(1) ≥ 0, 2 ≥ R(2) ≥ 0, 9 ≥ R(9) ≥ 0;

and A’s 6 date-1 IC’s:

[1−R(1)] ≥ [1−R(d(2)+2(1−d(2)))], [1−R(1)] ≥ [1−R(d(9)+9(1−d(9)))];

[2−R(2)] ≥ [2−R(2d(1)+(1−d(1)))], [2−R(2)] ≥ [2−R(2d(9)+9(1−d(9)))];

[9−R(9)] ≥ [9−R(9d(1)+(1−d(1)))], [9−R(9)] ≥ [9−R(9d(2)+2(1−d(2)))].

(i) Show that at optimum d(1) = 1. (Hint: If instead d(1) = 0, thenit follows from A’s date-1 IC’s and the limited liability constraint thatB’s date-0 IR condition will be violated.)(ii) Show that d(2) = 1 at optimum. (Hint: If otherwise, then A’s date-1 IC’s together with part (i) will show that B’s date-0 IR condition willbe violated.)(iii) Show that a contract specifying d(1) = d(2) = d(9) = 1 satisfiesall A’s date-1 IC’s, B’s date-0 IR, and the limited liability constraint,but it yields a zero payoff for A.(iv) Show that there exists a feasible contract with d(9) = 0 that givesA a positive payoff, so that at optimum d(9) = 0. Show that withd(1) = d(2) = 1 and d(9) = 0, A’s maximization problem becomes

maxR(1),R(2),R(9)

1

3[1−R(1)] + [2−R(2)] + [9−R(9)]

subject to

(B’s date-0 IR:)1

3[R(1)− 1] + [R(2)− 1] + [R(9)− 0] ≥ 3;

(Limited Liability:) 1 ≥ R(1) ≥ 0, 2 ≥ R(2) ≥ 0, 9 ≥ R(9) ≥ 0;

and A’s 2 date-1 IC’s:

R(9) ≥ R(1), R(9) ≥ R(2).

Conclude that given that d(9) = 0 and d(1) = d(2) = 1, three contractsare equivalently optimal; they are (i) R(1) = 0, R(2) = 2 and R(9) = 9;(ii) R(1) = R(2) = 1 and R(9) = 9; and (iii) R(1) = 1, R(2) =

28

2, R(9) = 8. Show that A’s payoff is 13under each of these three

contracts.14

Solution.First I claim that at optimum d(1) = 1. To see this, suppose insteadthat d(1) = 0. It follows from A’s date-1 IC’s that

[2−R(2)] ≥ [2−R(2d(1) + (1− d(1)))] ⇒ R(2) ≤ R(1) ≤ 1,

[9−R(9)] ≥ [9−R(9d(1) + (1− d(1)))] ⇒ R(9) ≤ R(1) ≤ 1,

where we have used the limited liability constraint, so that

1

3[R(1)− d(1)] + [R(2)− d(2)] + [R(9)− d(9)] ≤ 1

3× 1 = 1 < 3,

violating B’s date-0 IR condition.

Similarly, d(2) = 1 at optimum. Suppose otherwise, but then A’s date-1 IC’s imply

[9−R(9)] ≥ [9−R(9d(2) + 2(1− d(2)))] ⇒ R(9) ≤ R(2) ≤ 2,

which implies that

1

3[R(1)−d(1)]+[R(2)−d(2)]+[R(9)−d(9)] ≤ 1

3[1−1]+[2−0]+[2−0] < 3,

violating B’s date-0 IR condition.

Now if d(9) = 1, then all A’s date-1 IC’s are satisfied automatically,and by B’s date-0 IR and the limited liability constraint, we have

3 =1

3[1−1]+[2−1]+[9−1] ≥ 1

3[R(1)−1]+[R(2)−1]+[R(9)−1] ≥ 3,

implying thatR(1) = 1, R(2) = 2, R(9) = 9,

14The problem is to minimize the verification cost and still make sure that B is willingto lend the money to A. We have shown that one among the 3 optimal contracts is astandard debt contract, with which A promises B a face value of debt equal to 8, and if Acannot honor the debt, then A’s firm goes bankrupt and B must spend 1 dollar to hire anaccountant to check how much money the firm exactly has.

29

yielding a zero payoff for A.

We shall show that there exists a feasible contract with d(9) = 0 thatgives A a positive payoff, so that at optimum d(9) = 0.

Note that given d(9) = 0 but d(1) = d(2) = 1, the type-9 A has noincentives of claiming that he is of type 2 or type 1, since such a claimwill lead to state verification, and A’s money transfer to B is still R(9).Similarly, a type-1 A has no incentive of claiming to be type 2, anda type-2 A has no incentive of claiming to be type 1. The remainingproblem is that a type-1 or type-2 A may have an incentive to claimthat he is of type 9. To rule out this possibility, R(9) must be suchthat

[1−R(1)] ≥ [1−R(d(9) + 9(1− d(9)))] ⇒ R(9) ≥ R(1);

[1−R(2)] ≥ [1−R(d(9) + 9(1− d(9)))] ⇒ R(9) ≥ R(2).

Now let us find the optimal R(1), R(2) and R(9) given that d(9) = 0but d(1) = d(2) = 1.

The maximization problem becomes

maxR(1),R(2),R(9)

1

3[1−R(1)] + [2−R(2)] + [9−R(9)]

subject to

(B’s date-0 IR:)1

3[R(1)− 1] + [R(2)− 1] + [R(9)− 0] ≥ 3;

(Limited Liability:) 1 ≥ R(1) ≥ 0, 2 ≥ R(2) ≥ 0, 9 ≥ R(9) ≥ 0;

and A’s 2 date-1 IC’s:

R(9) ≥ R(1), R(9) ≥ R(2).

We claim that R(2) > 0. To see this, note that if R(2) = 0, then wehave

1

3[R(1)− 1] + [R(2)− 1] + [R(9)− 0] ≥ 3

⇒ 1 + 9 ≥ R(1) +R(9) ≥ 3× 3 + 1 + 1 = 11,

30

which is a contradiction.

We conclude that given that d(9) = 0 and d(1) = d(2) = 1, it is optimalto set (i) R(1) = 0, R(2) = 2 and R(9) = 9; (ii) R(1) = R(2) = 1 andR(9) = 9; or (iii) R(1) = 1, R(2) = 2, R(9) = 8. A’s payoff is 1

3under

each of these three contracts. This establishes that d(9) = 0 is indeedoptimal.

To sum up, there are three possible optimal contracts, and all of themare equivalent in the sense that they generate the same payoffs forA and for B. Moreover, one of them is a standard debt contract: Apromises B a face value of debt equal to 8, and if A cannot honor thedebt, then B is required to spend 1 dollar to verify the state (the dollaris hence interpreted as a bankruptcy cost).

11. Example 9. (Optimal debt contract: the dynamic case.)Consider an entrepreneur E who can invest in the same project twice.The entrepreneur has no money at date 0 and must raise I = 8 dollarsfrom competitive investors. Investors and E are all risk neutral withouttime preferences. The project, if invested at date t, t = 0, 1, willgenerate a random cash inflow xt+1 at date t+ 1, which may equal

H = 14 or L = 7.5

with equal probability. At date t+1 only E can costlessly see the out-come xt+1, but an investor can spend K = 3.5 to audit the earnings. Ehas committed to consume only at date 2–all the money he makes atdates 0 and 1 will be re-invested. Assume that other than this invest-ment project, E can put money in a deposit account with zero interestrate.(i) First suppose that only short-term debt contracts can be signed.That is, at date 0, a debt contract with face value F1 must be signed,and at date 1, after all investors observe E’s earnings report to hislender, and how E makes the repayment to the lender, a new debt con-tract will be signed with face value F2(x1). Note that F2 depends on x1

because if in equilibrium E does truthfully report x1 to his old lender,then he will have cash max(x1 − F1, 0) at hand, and only need to raise8 − max(x1 − F1, 0) at date 1; different amounts borrowed of courselead to different face values of debt. Find F1 and F2(x1). Show that

31

E’s expected date-2 wealth is 238.

(ii) Ignore part (i). Suppose instead that E can sign a long-term con-tract with investors. Again, one of the competitive investors will bepicked out to contract with E. We focus on the long-term contractsthat (A) avoid costly verification at date 1; and (B) allow the lender tobreak even during the date-0-date-1 period.15 Show that there exists anincentive-compatible16 long-term contract that specifies (1) no auditingby the lender at date 1; (2) a date-1 repayment function R(x1) ∈ [0, x1],where x1 ∈ 14, 7.5 is E’s date-1 earnings report; (3) E must borrow8 − R(x1) at date 1 if he reports x1; (4) the lender may audit at date2 depending on E’s report x2. Show that E’s expected date-2 wealthunder this long-term contract is 37

8. Explain verbally what is going

on.17

(iii) Now we assume a new set of parametric values for the above prob-lem: K = 2, I = 6.5, H = 12, and L = 6. (Hint: If you can find afeasible contract that allows state verification to occur only at date 2in the event that x1 = x2 = L = 6, then this feasible contract mustalso be optimal. (Why?) Hence, you can confine attention to such acontract.)

15Again, here “break-even” means zero expected profits. Requirement B is made sothat we can compare directly the efficiency of this long-term contract with the sequenceof short-term debt contracts derived in part (i).

16This means that for t = 1, 2, xt = xt for all xt ∈ 14, 7.5, where xt is the trueearnings that E finds out, and xt is E’s earnings report at date t.

17The intuition to be learned here is that in the repeated game there is an opportunityof making the second-period borrowed amount contingent on how much E repays at thematurity of the first-period debt, and this will reduce E’s incentive of lying at date 1,which in turn reduces the likelihood of having to spend on state verification at date 1,raising E’s wealth at date 1 and reducing the need of borrowing a lot in the second period.The latter, in turn, reduces the state verification cost at date 2. The key is that with along-term contract E can be asked to repay more at date 1 if and only if the date-1 cashflow is 14 rather than 7.5. This will not be incentive compatible if the game ends at date1, but it becomes incentive compatible if the game ends at date 2: if E’s date-1 cash flowis 14 and in order to save repayments E lies, then E will have to borrow more at date 1than he would if he told the truth. Borrowing more than needed at date 1 incurs a costat date 2, since the date-2 state verification cost increases with the amount borrowed atdate 1! Thus comparing the gain from repaying less at date 1 by lying to the loss from anincrease in date-2 verification cost, E may choose to tell the truth even telling the truthmeans that he has to repay more at date 1. The next question is: Why does allowing thedate-1 repayment to vary with E’s date-1 cash flow improve efficiency?

32

Solution.Consider part (i). The investor must break even at date 0, and henceF1 must solve

1

2F1 +

1

2(7.5− 3.5) = 8 ⇒ F1 = 12,

where we have conjectured (correctly) that F1 is higher than 7.5 andlower than 14, so that state verification (or bankruptcy) cost 3.5 is spentonly when the earnings at date 1 is 7.5. It follows that at date 1, E’swealth before making the second-period investment is either 14−F1 = 2or 0, with equal probability, depending on whether the date-1 earningsx1 ∈ 14, 7.5.In case x1 = 14, then the second-period debt has a face value F2(14)that solves the investor’s second-period break-even condition:

1

2F2(14) +

1

2F2(14) = 8− (14− F1) = 6 ⇒ F2(14) = 6,

where we have conjectured (correctly) that the second-period debt inthis case is default-free and hence F2(14) = 6. (You can try the conjec-ture that F2(14) > 7.5 so that state verification will take place whenthe date-2 earnings x2 = 7.5, but you will get a contradiction to sucha conjecture!)

On the other hand, if x1 = 7.5, then E is penniless at date 1, justlike at date 0. Hence the first-period debt is again optimal, so thatF2(7.5) = 12, implying also that state verification will take place whenthe date-2 earnings x2 = 7.5.

Having solved the sequence of one-period debt contracts, we can nowcompute E’s payoff at date 0. At date 0, it is equally likely that x1 = 14and x1 = 7.5, and in the former case E’s expected date-2 wealth is

1

2(14− 6) +

1

2(7.5− 6) = 4.75;

and in the latter case E’s expected date-2 wealth is

1

2(14− 12) +

1

2× 0 = 1.

33

It follows that E’s payoff is

1

2[4.75 + 1] = 2.875.

Now consider part (ii). Unlike F1 in part (i), now the repayment Rdepends on x1. Consider R(14) = 8.5 and R(7.5) = 7.5. Note that ina static case these repayments would violate E’s IC: when x1 = 14, Ewould like to lie and repay only 7.5. In the current two-period case,however, these repayments satisfy E’s IC’s. Why? This is because if Elies and repays 7.5 when x1 = 14, then he has to borrow 8 at date 1and to incur a verification cost 3.5 when x2 = 7.5 at date 2; but if Econfesses that x1 = 14, then he can borrow 8− (14− 8.5) only at date1, and a default-free second-period debt is available which avoids thestate verification cost 3.5 when x2 = 7.5 at date 2! More specifically,when E tells the truth at date 1, then following x1 = 14, F2(14) = 2.5with no state verification, and following x1 = 7.5, F2(7.5) = 12, whichincurs a state verification cost 3.5 only when x2 = 7.5.

Computing E’s date-0 payoff under this long-term contract, we have

1

2[1

2(14 + 7.5)− 2.5] +

1

2[1

2(14− 12) +

1

2× 0] = 4.625 > 2.875.

Now consider part (iii). Note that if d2(6, 6) = d1(6) = 0 then it isfeasible for E to always report x1 = 6 at date 1 and following that, toalways report x2 = 6 at date 2. This implies by the limited liabilityconstraint that R1 ≤ 6 and R2 ≤ 6, so that the investor would bebetter off not lending to E.

Thus for the investor to be willing to lend, either d1(6) = 1 or d2(6, 6) =1 (or both). Note that a feasible contract that allows state verificationonly at date 1 when x1 = 6 is dominated by a feasible contract thatallows state verification only at date 2 when x1 = x2 = 6 (where recallthat feasibility means that the contract satisfies the investor’s date-0 IR and E’s IC’s at dates 1 and 2), since in the former case stateverification occurs with probability 1

2, while in the latter case state

verification occurs only with probability 14.

34

Hence if we can find a feasible contract that allows state verificationonly at date 2 when x1 = x2 = 6, then this contract must be optimal(I did not say that there is a unique optimal contract though).

So, let us confine attention to such a contract, with the additionalproperty that it allows the investor to break even period by period.18

Note that if such a contract exists, then it must satisfy E’s date-1 anddate-2 IC’s. Since state verification does not occur at date 2 if x1 = 12,it must be that

6.5− [12−R1(12)] ≤ 6,

where the left-hand side is the amount that the investor lends to Eat date 1, after E reports x1 = 12 and repays the investor R1(12). Itfollows that

R2(12, 6) = R2(12, 12) = 6.5− [12−R1(12)].

On the other hand, state verification does occur at date 2 if x1 = x2 = 6,and to minimize E’s incentive of lying at date 1 when x1 = 12, we wouldnaturally conjecture that R1(6) = 6; that is, R1(6) < 6 may encourageE to claim that x1 = 6 when actually x1 = 12. From here we deducethat, for the investor to break even in the first period, R1(12) = 7,19

which in turn implies that

R2(12, 6) = R2(12, 12) = 6.5− [12−R1(12)] = 1.5.

Given our conjecture that R1(6) = 6, we know that the amount thatthe investor lends to E following E’s report that x1 = 6 should be 6.5,and from here we deduce that, for the investor to break even in the

18Since E and the investor are committed to a long-term contract, there is no need toassume that the investor has to break even period by period; that is, it is all right thatthe investor has net losses in one period but he expects to break even over the entiretwo periods. However, our purpose is to find one feasible contract under which stateverification occurs only at date 2 in the state x1 = x2 = 6, and if we can find such acontract with the additional property, then we are done, and there is no need to find thosecontracts under which the investor can break even only over the entire two periods—evenif the latter contracts exist, they cannot beat the contract that we have already found.

19The first period refers to the date-0-date-1 period, and the second period refers to thedate-1-date-2 period.

35

second period,

R2(6, 6) = 6,1

2[(6− 2) +R2(6, 12)] = 6.5 ⇒ R2(6, 12) = 9.

Having obtained all the Rt’s, now let us verify that these Rt’s togetherwith the dt’s define a contract that satisfies the investor’s date-0 IRand E’s date-1 and date-2 IC’s. In fact, the former holds true surely,because we have actually used the investor’s break-even condition tofind these Rt’s.

It remains to check E’s IC’s. It is obvious that E’s date-2 IC’s are allsatisfied: when x1 = 12, R2 is independent of E’s report about x2, andhence there is no need to lie; when x1 = 6 on the other hand, E doeshave an incentive to lie, but the contract has specified a contingent stateverification to make sure that E will not claim x2 = 6 when actuallyx2 = 12 (we do not need to worry the other way around, for E neverwants to repay R2(6, 12) = 9 when he has only 6 at hand).

So, we are left with E’s date-1 IC’s. When x1 = 6, E cannot afford torepay R1(12) = 7, and hence we focus on the IC when x1 = 12. If Etells the truth, then E repays 7 to and borrows 1.5 from the investor.In this case, E’s date-1 payoff is zero, but his expected date-2 payoff ispretty good:

1

2× (12− 1.5) +

1

2× (6− 1.5) = 7.5.

What if E lies and reports x1 = 6? Then E repays 6 only, and he canput 12-6=6 in his pocket. It would be efficient to borrow 0.5 only forthe second period production, and that would avoid state verificationat date 2 because 0.5 < 6. Unfortunately, this would not be feasible:borrowing only 0.5 shows exactly that E has at least 6 at hand. There-fore, to pool with the type of E with x1 = 6, the E with x1 = 12 mustalso borrow 6.5 at date 1. This implies a lower expected date-2 payoff:

1

2× (12− 9) = 1.5.

Thus by lying, E’s payoff is the 6 at date 1 plus 1.5 that he expects toget at date 2, and the sum is again 7.5.

36

We conclude that E has no incentives to lie when x1 = 12 at date 1,and this completes the proof that the suggested contract

R1(12) = 7; R1(6) = 6 = R2(6, 6); R2(12, ·) = 1.5; R2(6, 12) = 9;

d1(·) = 0 = d2(12, ·) = d2(6, 12); d2(6, 6) = 1

is feasible, and hence is optimal.

The intuition to be learned from parts (ii) and (iii) is that in the re-peated game there is an opportunity of making the second-period bor-rowed amount contingent on how much E repays at the maturity of thefirst-period debt, and this will reduce E’s incentive of lying at date 1,which in turn reduces the likelihood of having to spend on state verifi-cation at date 1, raising E’s wealth at date 1 and reducing the need ofborrowing a lot in the second period. The latter, in turn, reduces thestate verification cost at date 2.

Like what has been emphasized in other areas of microeconomics, re-lationship (meaning that E and the investor must interact with eachother not just for once) can improve efficiency. This exercise is takendirectly from Webb, D., 1992, Two-period Financial Contracts withPrivate Information and Costly State Verification, Quarterly Journalof Economics, 1113-1123.

12. Example 10. (Trading volume measures illiquidity.)Recall the stock trading game with two market makers that we de-scribed in Lecture 3. Here we shall introduce some modifications to thegame. There are two risk-neutral Bertrand competitive market makersfacing one active public investor, who can either be an informed specu-lator (insider) or an uninformed liquidity trader. The true value of thestock may be one dollar with prob. π or zero with prob. 1 − π. Theprob. that the public investor is informed is µ. If the public investoris uninformed, then he may want to buy and sell with equal prob., butwith prob. q, he wants to trade one unit and with prob. 1− q he wantsto trade 2 units. For example, the probability that the public investoris uninformed and wants to buy 2 units is

(1− µ)× 1

2× (1− q).

37

The equilibrium bid and ask prices are denoted by B1, B2, A1, andA2, where subscripts denote the different quantities of trade the publicinvestor may demand.

This is a signalling game where the public investor can first send onesignal (there are 4 feasible signals: buy 1 unit, sell 1 unit, buy 2 units,and sell 2 units), and the market makers respond by giving the bid andask prices that ensure themselves with zero expected profits from thetrade. (In reality, market makers may need to post bid and ask pricesfirst receiving orders from the public investors, but this will not alterour results.) Since the uninformed liquidity trader must trade exactlythe quantity the liquidity shock asks him to trade, the PBE of thisgame critically depends on the informed trader’s behavior in equilib-rium.(i) Find a PBE where the insider randomizes between 1 unit and 2units in equilibrium. (He cannot randomize between buying and sell-ing, because for example, when he knows that the stock’s value is 1instead of zero, then buying dominates selling.) Find the equilibriumB1, B2, A1, and A2 and the conditions on parameters sustaining thisequilibrium.(ii) Find a PBE where the insider trades 2 units for sure in equilib-rium. Find the equilibrium B1, B2, A1, and A2 and the conditions onparameters sustaining this equilibrium.(iii) Verify that there are no other PBE’s of this game.Solution.Let d(Q) be the posterior prob. that the stock is worth one dollar aftermarket makers observe the order Q ∈ s1, s2, b1, b2 , where si andbi denote the sell order and buy order with the quantity of trade beingi units of the stock. Note that d(Q) will also be the equilibrium priceof the stock given Q. Thus it is only relevant to compute d(Q).

Take the case Q =s1 for example. Note that the prob. that the orderappears to be s1 is

µ · [π · 0 + (1− π) · a] + (1− µ) · 12· q,

where a is the probability that the insider sells 1 unit given that thestock’s true value is zero. Given that the order s1 has reached the

38

market makers, the conditional probabiltiy for the event that the orderis submitted by the insider that has learned that the stock is worth 1dollar is

µ · π · 0µ · [π · 0 + (1− π) · a] + (1− µ) · 1

2· q

= 0;

the conditional probabiltiy for the event that the order is submitted bythe insider that has learned that the stock is worth nothing is

µ · (1− π) · aµ · [π · 0 + (1− π) · a] + (1− µ) · 1

2· q

;

and the conditional probabiltiy for the event that the order is submittedby the liquidity trader is

(1− µ) · 12· q

µ · [π · 0 + (1− π) · a] + (1− µ) · 12· q

.

Hence we have

d(s1) =µ · π · 0 · 1 + µ · (1− π) · a · 0 + (1− µ) · 1

2· q · π

µ · [π · 0 + (1− π) · a] + (1− µ) · 12· q

=q2(1− µ)π

a(1− π)µ+ q2(1− µ)

.

Similarly, we have

d(s2) =1−q2(1− µ)π

(1− a)(1− π)µ+ 1−q2(1− µ)

,

d(b1) =[fµ+ q

2(1− µ)]π

fπµ+ q2(1− µ)

,

d(b2) =[(1− f)µ+ 1−q

2(1− µ)]π

(1− f)πµ+ 1−q2(1− µ)

,

where f is the prob. that the insider buys one unit. Now it remains todetermine a, f and the supporting IC’s. For the insider to not deviate,the expected profit of trading one unit must equal that of trading twounits, and hence we require, in case the stock is worth one dollar,

[1− d(b1)] · 1 = [1− d(b2)] · 2,

39

and in case the stock is worth zero,

[d(s1)− 0] · 1 = [d(s2)− 0] · 2.

These two equations yield solutions for a, f as functions of parameters.Since a, f ∈ (0, 1), we obtain a set of inequalities that must be satisfiedby the parameters in order to sustain this equilibrium. This concludespart (i).

For part (ii), plugging a = f = 0 into the above formulae for d(Q)’s,we then require

[1− d(b1)] · 1 ≤ [1− d(b2)] · 2,

and[d(s1)− 0] · 1 ≤ [d(s2)− 0] · 2.

This is a set of conditions that the parameters must satisfy to sustainthis PBE.

For part (iii), it is easy to see that there is no PBE where the insidertrades 1 unit for sure: trading two units will be treated as the liquiditytrader with the best terms of trade, and hence trading two units canmore than double the profit from trading one unit. Thus if we clas-sify the possible PBE’s according to the insider’s possible equilibriumstrategies, we conclude that all possibilities have been covered.

13. Example 11. (A borrower-lender reputation game.)At date 0, there is a borrowing firm that has three possible types, G,GB, and B. The firm needs to invest 1 dollar. A type-G firm has ariskless project that will return X > 1 dollars for sure at date 1. Atype B firm has a risky project that may return Y > X dollars withprobability π and nothing with probability 1 − π at date 1, whereπY < 1. A type-GB firm has both the riskless project and the riskyproject, and it must choose one of them to invest. The firm triesto borrow from a bank. The firm and the bank are both risk neutralwithout time preferences. At the beginning of date 0, the bank believesthat the firm is of type G and of type B both with probability p < 1

2

40

respectively. The bank must choose a face value F , and the firm canaccept or reject the bank’s offer.(i) First suppose that the game ends at date 1. Suppose that thefollowing condition holds:

π(Y − F ′) > X − F ′, π(Y − F ′′) < X − F ′′,

F ′ =1

p+ (1− p)π< X, F ′′ =

1

pπ + (1− p)< X.

Show that this game has two equilibria, and in one equilibrium the type-GB firm chooses the riskless project, and in the other equilibrium, thetype-GB firm chooses the risky project.(ii) Now, consider a repeated version of the above static game. Supposethat at date 1, after repaying its date-0 debt, the firm must borrow 1dollar from another bank (refer to the old bank as the date-1 bank andthe new bank as the date-2 bank; the date-t bank is the one that willbe repaid at date t). Assume that the date-2 bank does not know howmuch exactly the firm repaid the date-1 bank when default occurred atdate 1. The only information that the date-2 bank has about the firmis the latter’s default history.

Show that under the following conditions on the parameters,

X +π

1− π[1− pX

(1− p)π− Y ] <

p+ (1− p)π

p+ (1− 2p)π + pπ2<

X − πY

1− π,

1 < X < min(1

1−2p1−p

+ pπ1−p

,1

p+ π(1− p)),

and1− pX

(1− p)π< Y <

1

π,

there exists a PBE where the type-GB firm chooses the risky projectat date 0, but switches to the riskless project at date 1, whenever itobtains re-financing from the bank at date 1.Solution. Consider part (i). First we conjecture that there exists anequilibrium where the type-GB firm will adopt the riskless project in

41

equilibrium. Since the bank lends 1 dollar and is competitive, the bankmust choose a face value F ′′ of debt such that

F ′′(pπ + 1− p) = 1 ⇒ F ′′ =1

pπ + (1− p)< X < Y.

The condition0 < π(Y − F ′′) < X − F ′′

implies that the type-GB firm’s IC and IR are indeed satisfied, andhence the conjectured equilibrium does exist.

Next, we conjecture that there exists another equilibrium where thetype-GB firm adopts the risky project in equilibrium. If the bankholds this pessimistic belief, then the bank will choose a face value F ′

of debt such that

F ′(p+ (1− p)π) = 1 ⇒ F ′ =1

p+ (1− p)π< X < Y,

and the conditionπ(Y − F ′) > X − F ′ > 0

implies that the type-GB firm’s IC and IR are again satisfied, provingthat this conjectured equilibrium also exists.

Thus we have shown that two pure-strategy equilibria co-exist in thisstatic game, as asserted. This concludes our analysis for part (i).

Next, consider part (ii). We are asked to produce a set of conditionson parameters which support the equilibrium where the type-GB firmadopts the risky project at date 1, and at date 2 it switches to theriskless project if and only if one bank lends another dollar to the firmat date 2. Equivalently, either no bank is willing to lend to the firm atdate 2, or when a bank does lend to the firm at date 2, the type-GBfirm is induced to adopt the riskless project.

We now derive one set of conditions ensuring such an equilibrium exists.

We assume that the date-2 bank maintains the belief that the type-GBfirm has adopted the risky project at date 1, and the date-2 bank formsthe posterior belief based only on whether or not the firm defaulted itsdebt obligations to the date-1 bank.

42

• Suppose that default has occurred at date 1. Denote by dj thedate-2 bank’s posterior probability for the event that the firm isof type j ∈ G,B,GB after the date-2 bank observes the date-1default. We have

dG =p · 0

p · 0 + p · (1− π) + (1− 2p) · (1− π)= 0 < p,

dB =p · (1− π)

p · 0 + p · (1− π) + (1− 2p) · (1− π)=

p

1− p> p,

dGB =(1− 2p) · (1− π)

p · 0 + p · (1− π) + (1− 2p) · (1− π)=

1− 2p

1− p> 1− 2p.

If the date-2 bank decides to lend one dollar to the firm, whatshould be the face value of debt? Let µ denote the probabilitythat the type-GB firm may adopt the riskless project at date 2,after the type-GB firm obtains a loan with face value F (µ) fromthe date-2 bank. Then the face value F (µ) of debt must satisfy

F (µ)[dGB(µ+ (1− µ)π) + dBπ] = 1,

as long as F (µ) ≤ X (otherwise F (µ) would have to be higher).Note that F (µ) is decreasing in µ, and

∀µ ∈ [0, 1], F (µ) ≥ F (1) =1

1−2p1−p

+ pπ1−p

> X.

Thus at date 2, following its date-1 default, the type-GB firmrealizes that it would never get a face value of debt lower thanF (1), and if the date-2 bank does approve the loan to it, adoptingthe riskless project is always its dominated response at date 2.Thus after seeing the firm’s date-1 default, the date-2 bank knowsthat the firm will adopt the risky project with probability one atdate 2 if it lends to the firm: the firm can never be of type G,and the above reasoning tells the date-2 bank that the type-GBfirm will adopt the risky project, just like the type-B firm. Thisimplies that, following the firm’s date-1 default, there does notexist a face value of debt which is both profitable to the date-2bank and acceptable to the borrowing firm at date 2. We concludethat there will be no refinancing to the firm at date 2 followingthe date-1 default.

43

• Next, suppose that the firm did not default at date 1. Denote bynj the date-2 bank’s posterior probability for the event that thefirm is of type j ∈ G,B,GB after the firm has honored its date-1debt obligations. We have

nG =p · 1

p · 1 + p · π + (1− 2p) · π> p,

nB =p · π

p · 1 + p · π + (1− 2p) · π< p,

nGB =(1− 2p) · π

p · 1 + p · π + (1− 2p) · π< 1− 2p.

If the date-2 bank decides to lend one dollar to the firm, whatshould be the face value of debt? The face value of debt F ′′′

chosen by the date-2 bank should be

F ′′′[(1− nB) + nBπ] = 1,

given that the date-2 bank believes that the type-GB firm willadopt the riskless project. Note that we have assumed that

F ′′′ =1

1− nB + πnB

≤ X < Y,

which is actually implied by the assumed condition F ′′ < X, be-cause F ′′′ < F ′′, which in turn follows from the fact that nB < p.The following IC condition for the type-GB firm must be satisfied:

X − F ′′′ ≥ π(Y − F ′′′).

We have just shown that if

F ′′′ =1

1− nB + πnB

=p+ (1− p)π

p+ (1− 2p)π + pπ2< min(X,

X − πY

1− π),

then the type-GB firm’s date-2 strategy is to adopt the riskless projectwhenever the date-2 bank is willing to lend it another dollar.

Now, we consider the date-1 equilibrium. The type-GB firm is free todeviate and adopt the riskless project at date 1, and hence we need

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to make sure that such a deviation does not generate for the type-GBfirm a payoff higher than its equilibrium payoff.

The above analysis shows that the type-GB firm will get nothing atdate 2 if there is a date-1 default. Thus by deviating and adopting theriskless project, the type-GB firm’s payoff is

0 +X − F ′′′,

assuming that the face value of debt required by the date-1 bank, whichis 1−pX

(1−p)π, is greater than X.20 This latter assumption is exactly F ′ > X.

The above deviation payoff is required to be less than (or equal to) thetype-GB firm’s equilibrium payoff,

π(Y − 1− pX

(1− p)π+X − F ′′′),

where we have also assumed that the face value of debt required by thedate-1 bank, 1−pX

(1−p)π, is less than Y .

Thus we have obtained the following set of conditions,

F ′′′ =p+ (1− p)π

p+ (1− 2p)π + pπ2< min(X,

X − πY

1− π),

X <1− pX

(1− p)π< Y,

F ′′′ =p+ (1− p)π

p+ (1− 2p)π + pπ2> X +

π

1− π[1− pX

(1− p)π− Y ],

which altogether support the equilibrium where the type-GB firm adoptsthe risky project at date 1, and at date 2, either refinancing is notgranted, or refinancing is granted by a date-2 bank and the type-GBfirm always adopts the riskless project at date 2.

The above conditions can be further simplified as follows. Note that

min(X,X − πY

1− π) =

X − πY

1− π,

20If we assume that it is less than X, as in part (i), then we shall have a contradictionif we also assume F ′′′ = 1

1−nB+πnB< min(X, X−πY

1−π ).

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and hence the above first and third inequalities can be compactly writ-ten as

X +π

1− π[1− pX

(1− p)π− Y ] <

p+ (1− p)π

p+ (1− 2p)π + pπ2<

X − πY

1− π.

Thus this last inequality together with

1 < X < min(1

1−2p1−p

+ pπ1−p

,1

p+ π(1− p))

and1− pX

(1− p)π< Y <

1

π

gives the required set of conditions.Remark. This exercise intends to show two things. First, a borrow-ing firm’s incentive to take excessive risks increases with its financialleverage. That is, when the borrowing firm is faced with a higher in-terest payment, it is more likely to choose a risky project over a safeproject. Second, whether a borrowing firm is faced with a high interestpayment depends on its default history: in this model, the bank knowsthat only type GB has a moral hazard problem, but the bank is alsofaced with an adverse selection problem because it cannot distinguishone type of borrowing firm from another type. Having observed thefirm’s capability of repaying its past debts, the bank can update itsbeliefs about the borrowing firm’s type, and the bank becomes moreoptimistic after knowing that the firm honored its first debt. The firmis then granted with a lower face value for its second debt in the lattersituation, and this further encourages type GB to take the safe project.

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