TOF IProblems - Game Theory

1. The inverse-demand function in an industry is given by p = a− bq, where p is the marketprice, q is the aggregate supply in the market, and a and b are positive constants. Thereare n firms in this industry, and each firm produces the output at a marginal cost c, wherec < a.

(a) Assume that n = 2, and firms choose output levels to maximize individual profits.Compute the Nash equilibrium of this game.

(b) If the firms could collude by some means, could they increase their profits above thosein part (a)? If so, can such profits be sustained?

2. Consider the following two-player game. Player 1 chooses a row (Top, Middle, or Bottom)and, simultaneously, player 2 chooses a column (Left, Centre, or Right). Each cell in theoutcome matrix specifies the payoffs to players 1 and 2 respectively, for each combinationof choices made by them.

Player 2

Left Middle Right

Top 7,2 6,5 3,3

Player 1 Middle 3,5 3,3 5,6

Bottom 5,6 2,6 4,7

(a) What does the term “dominated strategy” mean in the context of games such as this?Are there any dominated strategies in this game?

(b) What is meant by a Nash equilibrium? Identify all the pure-strategy Nash equilibriain the above game.

(c) Are there any mixed-strategy Nash equilibria? If so, which strategies are not used inthe equilibrium randomization?

3. Consider the two-player game in extensive form below.

A

B

-1-1

1 0

-1-1

2 1

02

Payoff to APayoff to B( ) ( )

( )( )( ) ( )

a1a2 a3 a4

b1 b1b2 b2

B2B1

B

-1-1

( ) 12( )

(a) Explain the difference between an action and a strategy. For each player A and B,list the actions and strategies.

(b) Define the Nash equilibrium. What are the Nash equilibria in the preceding game?What are the Nash outcomes?

(c) Define a subgame perfect Nash equilibrium. What are the subgame perfect Nashequilibria in the preceding game? What are the subgame perfect outcomes?

(d) Do you consider the notion of subgame perfection appealing? Is it adequate in thecontext of the game above, and if not, why?

2

4. Consider the following extensive form game. The entrant moves first and decides whetherto enter. If he decides to enter, he then decides whether to produce bathtubs (B) orjacuzzis (J). The incumbent observes the decision about entry, but not the decision aboutproduct choice. The incumbent faces the same choice of products (b,j).

(a) Write down the actions and strategies available to each player.

(b) Define Nash equilibrium and subgame perfect equilibrium. For any game, what isthe relationship between the set of Nash equilibria and the set of subgame perfectequilibria?

(c) Find the pure strategy Nash equilibria of the game.

(d) Find the pure strategy subgame perfect Nash equilibria of the game. How does theset of Nash equilibria of this game differ from the set of subgame perfect equilibria?

Entrant

Entrant

Incumbent

O

I

B J

b j b j

-3-3

1-1

-1 1

-6-6

02

Payoff to EntrantPayoff to Incumbent( )

( )

( )( )( ) ( )

3

TOF IProblems - Adverse Selection

1. Consider a second-hand car market with two kinds of cars, type A which are completelyreliable, and type B which break down with probability 1/2. There are 20 car owners,10 with each kind of car, and 19 potential buyers. The car owners and buyers value acar at £1000 and £1500, respectively if it works and both owners and buyers value anon-working car at zero. Both buyers and owners are assumed to be price takers andrisk-neutral. Finally each seller is aware of the type that her car belongs to, but thisinformation is not available to buyers.

(a) What value do buyers and sellers place on type A and type B cars?

(b) How many cars, and of what quality will be supplied at each price? (Quality is definedas the proportion of cars that are of type A.)

(c) Given that buyers can observe the market price, and have rational expectations, whatwill be the demand for cars at each price?

(d) How many equilibria are there in the market?

2. There is a large number of sellers and even more buyers. Each seller has one used carto sell. Let Suppose the quality of a used car can be indexed by θ, which is uniformlydistributed on [0, 1]. If a seller of type θ sells his car for a price p, he receives utilityus(θ, p). A buyer who buys a car of quality θ at price p receives a utility of θ − p. If aseller does not sell his car, he receives a utility of zero. Similarly, a buyer who does notbuy a car receives a utility of zero. A seller knows the quality of his car, but buyers onlyknow the distribution of quality.

(a) Argue that in a competitive equilibrium under asymmetric information, we must haveE(θ|p) = p.

(b) Show that if us(θ, p) = p − θ/2, every p ∈ (0, 1/2] is an equilibrium price.

(c) Find the equilibrium price when us(θ, p) = p−√

θ. Which cars are traded in equilib-rium?

(d) Find an equilibrium price when us(θ, p) = p − θ3.

(e) Are any of the preceding outcomes Pareto efficient? Describe Pareto improvementswhenever possible.

3. A seller sells a unit of a good of quality q at a price t. The cost of producing at qualitylevel q is given by q2. There is a buyer who receives a utility of θq − t by consuming theunit of quality q at price t. If he decides not to buy, he gets a utility of zero. θ can taketwo values 1 and 2. Assume that the seller has all the bargaining power.

(a) Suppose the seller can observe θ. Derive the profit maximizing price-quality pairsoffered when θ = 1 and when θ = 2. Show that the quality offered when θ = 2 istwice the quality offered when θ = 1.

(b) Prove that the full information price-quality pairs are not incentive compatible if theseller cannot observe θ.

TOF IProblems - Moral Hazard

1. Consider the following hidden action model with three possible actions for the agent E ={e1, e2, e3}. There are two possible profit outcomes for the Principal : πH = 10 andπL = 0. The probabilities of πH conditional on the three effort levels are f(πH |e1) = 2/3,f(πH |e2) = 1/2, and f(πH |e3) = 1/3. The agent has a separable utility function u(w, ei) =v(w)− g(ei), where v is the von-Neumann Morgenstern utility of wage w, and g(ei) is thedisutility of effort. We are told that g(e1) = 5/3, g(e2) = 8/5, and g(e3) = 4/3. Finally,v(w) =

√w and that the manager’s reservation utility is u = 0.

(a) What is the optimal contract when effort is observable?

(b) Show that if effort is not observable, then e2 is not implementable. For what level ofg(e2) would e2 be implementable? [Hint: focus on the utility the manager will getfor the two outcomes].

(c) What is the optimal contract when effort is not observable?

2. A principal hires an agent to work on a project at wage w. The agent’s utility function is

U(w, ei) = 100 − 10

w− g(ei),

where g(ei) is the disutility associated with the effort level ei exerted on the project. Theagent can choose one of two possible effort levels, e1 or e2, with associated disutility levelsg(e1) = 50, and g(e2) = 0. The agent’s choice of effort affects the project’s output in aprobabilistic fashion: if the agent chooses effort level e1, an error occurs in the productionprocess with probability 1/4, and if he chooses e2, an error occurs with probability 3/4.The project yields a revenue of 20 if the production process is error free, and a revenue 0otherwise. The principal is risk neutral: she aims to maximize the expected value of theoutput, net of any wage payments to the agent. The agent has a reservation utility of 0.

(a) If the effort level chosen by the agent is observable by the principal, it is optimal forthe principal to choose a fixed-wage contract. Explain the intuition for this result.

(b) If effort is observable, what wage w∗ should the principal offer if she wants to imple-ment e1? What wage implements e2? Which of these will the principal implement?

(c) Suppose that the agent’s choice of effort level is not observable. What wage schedule

will implement e1 in this case? What expected net return does the principal getin this case? How does this compare with the value in part (b), where effort wasobservable?

(d) If effort is not observable, should the principal implement e1 or e2?

TOF IProblems - Market Microstructure

1. Consider a risky asset with final payoff V , which is a normally distributed random variablewith mean 0 and variance σ2

0 . There is a risk neutral informed trader who knows therealization v of the random variable V . The trader has no initial endowment of the riskyasset.

After observing v, the informed trader chooses his trading quantity X. This net order flowconsists of the informed trader’s order and the total order from liquidity traders (denotedby u) which is normally distributed with mean 0 and variance σ2

u. All random variablesare pairwise independent.

The market maker observes the net order flow X +u and sets a price P (X +u) = E(V |X +u).

Consider the linear equilibrium given by

X(v) =σu

σ0v,

andP (X + u) =

σ0

2σu(X + u).

(a) Show that if liquidity trade quantity doubles, P (X + u) remains unchanged in equi-librium. Explain the intuition behind this result.

(b) Derive the expected profit of the informed trader conditional on his information, andshow that it is decreasing in σ0. Explain the intuition behind this result.

(c) Derive the expected ex-ante profit of the informed trader, and show that it is increas-ing in σ0. Explain the intuition behind this result.

2. Consider the following sequential trading model. Suppose an asset can have two values 10and 20. The market maker attaches a prior probability of 1/2 that the value equals 10.Now a trader arrives in the market and submits either a buy or a sell order. Suppose 60%of the potential traders are informed and the rest are uninformed. An uninformed tradersubmits a buy or a sell order with equal probability. Suppose the market makers earns azero expected profit.

(a) Calculate the bid price and the ask price.

(b) What is the bid-ask spread if there are no informed traders?

(c) Trading in this market depends crucially on the presence of uninformed traders.Explain this statement.

(d) Briefly compare (in words only) the sequential trading model to the Kyle (1985)model.

Auction Theory - Proofs only.

TOF ISolutions - Game Theory Problems

Exercise 1: The inverse demand function in an industry is given by p = a− bq, where p is themarket price, q is the aggregate supply in the market, and a, b are positive constants. There aren firms in this industry, and each firm produces the output at a marginal cost c, with c < a.

(a) Assume n=2, and firms choose output levels to maximize individual profits. Compute theNash equilibrium of this game.

(b) If the firms could collude by some means, could they increase their profits above those inpart (a)?

The question only asks for an analysis of quantity (‘Cournot’) competition, but we take theopportunity to also discuss the related idea of ‘Bertrand’ competition (price competition).

(A) Cournot competition

(a) n = 2, p = a − bQ, Q = q1 + q2, MC = c < a.

Π1(q1, q2) = (p − c)q1 = (a − bQ − c)q1 = (a − bq1 − bq2 − c)q1

∂Π1

∂q1= 0 : q1 =

a − c − bq2

2b

After optimization, we assume that firms are identical: q1 = q2 = qDi .

qi =a − 2bqi − c

b⇒ qD∗

i =a − c

3b. (1)

pD∗ = a − 2ba − c

3b=

3a

3− 2a

3+

2c

3=

1

3a +

2

3c. (2)

ΠDi =

(

a − 2b

(

a − c

3b

)

− c

)

a − c

3b=

(a − c)

3

a − c

3b=

(a − c)2

9b. (3)

(b) n = 1, qM = q1 + q2.ΠM (qM ) = (a − bqM − c)qM

∂ΠM

∂qM= 0 : qM =

a − c

2bfor the whole market

Assuming equally split profits

qMi =

qM

2=

a − c

4b< qD

i (4)

pM = a − ba − c

2b=

a + c

2. (5)

ΠMi =

(

a + c

2− c

)

a − c

4b=

(a − c)2

8b> ΠD

i . (6)

7

(B) Bertrand competition

Suppose two firms play a one-shot game of setting prices p = (p1, p2) with no capacity constraintsand constant, equal marginal cost (c1 = c2 = c). Let demand be given by:

Di(p) =

Di(pi) if pi < pj

0 if pi > pj

Di(pi)/2 if pi = pj

(7)

It is easy to see that in equilibrium, p1 = p2 = c because in every other case, one firm couldappropriate all market profit by lowering the price a tiny bit under the competitor’s price. Thissolution leads to the so-called Bertrand paradox: (i) Prices are at marginal cost of one of thefirms, (ii) if marginal costs are different, one firm does not produce and the other makes profits,(iii) if marginal costs are equal, neither firm makes profits (i.e. we have a fully competitiveoutcome with only 2 firms).

8

Exercise 2: Consider the two-player game in extensive form from Figure 1.

a1(0,2)a4A

a3a2

b2

(1,0)

b1

(-1,-1)

b2

(2,1)

b1

(-1,-1)

B

B2 (1,2)

B1(-1,-1)

B

Figure 1: An extensive form game; payoffs are given as (payoff to A, payoff to B).

(a) Describe in words: The sequence of play, potential choices of agents and information ateach stage.

(b) Explain the difference between an action and a strategy. For each player A and B, providean example of an action, and one of a strategy.

(c) Define the Nash equilibrium. What are the Nash equilibria in the preceding game? Whatare the Nash outcomes?

(d) Define (informally but carefully) a subgame perfect Nash equilibrium. What are thesubgame perfect Nash equilibria in the preceding game? What are the subgame perfectoutcomes?

(e) Do you consider the notion of subgame perfection appealing? Is it adequate in the contextof the above game, and if not, why?

(a) (Omitted.)

(b) The strategy sets of players A,B specify ‘complete, contingent plans’ of how to playthe game. That means that a strategy must specify an action at each of the players’information sets (regardless of whether players will reach them ‘on the equilibrium path’or not). Therefore strategy sets are given as

SA = {a1, a2, a3, a4}, SB = {b1B1, b1B2, b2B1, b2B2}.

And the action sets of players A,B are given as

AA = {a1, a2, a3, a4}, AB = {b1, b2, B1, B2}.

(c) (Standard definition) We transform the game into the strategic form first and then searchfor equilibria.

9

b1B1 b1B2 b2B1 b2B2

a1 0,2 0,2 0,2 0,2a2 -1,-1 -1,-1 1,0 1,0a3 -1,-1 -1,-1 2,1 2,1a4 -1,-1 1,2 -1,-1 1,2

The set of Nash equilibria in pure strategies is

E = {(a1, b1B1), (a4, b1B2), (a3, b2B1), (a3, b2B2)}

leading to the outcomesO = {(0, 2), (1, 2), (2, 1), (2, 1)} .

(d) A subgame perfect Nash equilibrium is a Nash equilibrium of a game that also inducesNash equilibrium play in all subgames of the original game. Here we have:

1. the trivial subgame - i.e. the whole game, leading to the above set E of equilibria,and

2. the subgame starting at B’s information set after A’s action a4. Clearly, the equilib-rium of this subgame is just B2.

Therefore the set of subgame perfect Nash equilibria in pure strategies is given by:

ESGP = {(a4, b1B2), (a3, b2B2)}

leading to the outcomesOSGP = {(1, 2), (2, 1)} .

(e) Yes. No—not all ‘incredible threats’ are removed by the subgame perfection requirement.In particular we would like to remove equilibrium play of b1 which is strictly dominated byb2. Subgame perfection, however, is not strong enough to do this. To do this we would needmore restrictive concepts like ‘sequential equilibrium’ or ‘perfect Bayesian equilibrium’ (notcovered in this course).

10

Exercise 3: Consider the following two-person game. Player 1 chooses a row (Top, Middle, orBottom) and, simultaneously, player 2 chooses a column (Left, Centre, or Right). Each cell inthe outcome matrix specifies the payoffs to players 1 and 2 respectively, for each combination ofchoices made by them.

Player 2

Left Middle RightTop 7,2 6,5 3,3

Player 1 Middle 3,5 3,3 5,6Bottom 5,6 2,6 4,7

(a) What does the term ‘dominated strategy’ mean in the context of games such as this? Arethere any dominated strategies in this game?

(b) What is meant by a Nash equilibrium? Identify all pure-strategy Nash equilibria in theabove game.

(c) Are there any mixed strategy Nash equilibria? If so, which strategies are not used in theequilibrium randomization?

(a) A ‘strictly (weakly) dominated strategy’ has a strictly (weakly) lower payoff under everypossible strategy of the opponent than another strategy. Notice that

– ‘Right’ strictly dominates ‘Left’; hence ‘Left’ can be eliminated.

– Once this is done, ‘Middle’ strictly dominates ‘Bottom’ which we eliminate.

(The above procedure comes under the pretty name ‘iterated deletion of strictly dominatedstrategies.’) We end up in the reduced game

Middle RightTop 6,5 3,3

Middle 3,3 5,6

(b) (Standard definition) The set of pure-strategy Nash equilibria in the above game is

E = {(Top,Middle), (Middle,Right)}

We could also derive Nash equilibria by looking at the reduced game after eliminatingstrictly dominated strategies. This behaviour generalizes: We never reduce the set ofNash equilibria by eliminating strictly dominated strategies. If we also eliminate weaklydominated strategies we may eliminate Nash equilibria (also in this case the sequence ofelimination matters).

(c) First, note that strictly dominated strategies are never played with positive probability inequilibrium. Thus we can simply consider the reduced game derived above to compute themixed strategy Nash equilibrium.

To compute the mixed strategy, note that the randomization of player 2 (putting a weightof q on ‘Middle’ and the residual weight of 1− q on ‘Right ’) must be such that player 1 isindifferent between Top and Middle :

6(q) + 3(1 − q) = 3(q) + 5(1 − q)

5q = 2

q =2

5.

11

Similarly player 1 must choose a randomization (putting a weight of p on ‘Top’ and theresidual weight of 1 − p on ‘Middle’) such that player 2 is indifferent between Middle andRight:

5(p) + 3(1 − p) = 3(p) + 6(1 − p)

5p = 3

p =3

5.

Hence the mixed strategy Nash equilibrium is given by:

Em =

{

3

5Top +

2

5Middle,

2

5Middle +

3

5Right

}

leading to the outcome

Om =

{

21

5= 3

3

5+ 6

2

5,21

5= 3

2

5+ 5

3

5

}

.

Exercise 4: Consider the following 2-player simultaneous move game:

L RT 5,5 0,6B 6,0 1,1

(a) Solve the game without invoking Nash equilibrium.

(b) How would the equilibrium behaviour change if the game is played twice?

(a) Note that each player has a strictly dominant strategy. Therefore we can solve the gamesimply by allowing players to play their dominant strategies. The solution is (B,R).Games which can be solved in this way are called ‘dominance solvable.’

(b) There is no change. In the second period, players must play (B,R). Therefore in period1 they must play (B,R).

12

Exercise 5: Consider the extensive form game of Figure 2. The entrant E moves first anddecides whether to enter. If he decides to enter, he then decides whether to produce bathtubs(B) or jacuzzis (J). The incumbent I observes the decision about entry, but not the decisionabout product choice. The incumbent faces the same choice of products (b, j).

O(0,2)E

I

JB

E

j

(1,-1)

b

(-3,-3)

j

(-6,-6)

b

(-1,1)

I

Figure 2: Another extensive form game; payoffs are given as (payoff to E, payoff to I).

(a) Write down the actions and strategies available to each player.

(b) Define Nash equilibrium and subgame perfect equilibrium. For any game, what is therelationship between the set of Nash equilibria and the set of subgame perfect equilibria?

(c) Find the pure strategy Nash equilibria of the game.

(d) Find the pure strategy subgame perfect Nash equilibria of the game. How does the set ofNash equilibria of this game differ from the set of subgame perfect equilibria?

(a) The strategy sets of players E, I are given as

SE = {IB, IJ,OB,OJ}, SI = {b, j}.

The action sets of players E, I are given as

AE = {I,B,O, J}, AI = {b, j}.

(b) (Standard definitions) Since subgame perfection is a restriction on the set of Nash equi-libria, the set of subgame perfect Nash equilibria is always included in the set of Nashequilibria.

(c) We convert to the strategic form

13

b jIB -3,-3 1,-1IJ -1,1 -6,-6

OB 0,2 0,2OJ 0,2 0,2

the pure strategy Nash equilibria of which are

E = {(IB, j), (OB, b), (OJ, b)}

with outcomesO = {(1,−1), (0, 2), (0, 2)} .

(d) There are two subgames

1. the whole game (leading to the above equilibrium set E), and

2. the subgame consisting of the simultaneous move game after E’s choice of ‘I.’ Itsequilibria can be found by evaluating

b jB -3,-3 1,-1J -1,1 -6,-6

which has the two pure strategy Nash equilibria

E′ = {(J, b), (B, j)}

with outcomesO′ = {(−1, 1), (1,−1)} .

From E and E′, we find the Nash equilibria that give rise to Nash equilibria in all subgames.This is the set of subgame perfect equilibria

ESGP = {(IB, j), (OJ, b)}

with outcomesOSGP = {(1,−1), (0, 2)} .

14

TOF ISolutions - Adverse Selection Problems

Exercise 1: Consider a second-hand car market with two kinds of cars, type A which arecompletely reliable, and type B which break down with probability 1

2 . There are 20 car owners,10 with each kind of car, and 19 potential buyers. The car owners and buyers value a car at£1000 and £1500, respectively if it works and both owners and buyers value a non-working carat zero. Both buyers and owners are assumed to be price takes and risk-neutral. Finally, eachseller is aware of the type that her car belongs to, but this information is not available to buyers.

(a) What value do buyers and sellers place on a type A and type B car?

(b) How many cars—and of what quality—will be supplied at each price? (Quality is definedas the proportion of cars that are of type A.)

(c) Given that buyers can observe market price and have rational expectations, what will bethe demand for cars at each price?

(d) How many equilibria are there in the market?

Answer:

(a) The 20 (risk-neutral) owners’ valuations of a failsafe type A and the faulty type B are

voA = {1 : 1000, 0 : 0} = 1000, vo

B =

{

1

21000 +

1

20

}

= 500.

The 19 (risk-neutral) buyers’ valuations for the two types are

vbA = {1 : 1500, 0 : 0} = 1500, vb

B =

{

1

21500 +

1

20

}

= 750.

Under full information, 19 cars will be sold. Since there is competition among car-owners(20 > 19), a type-A car will sell for pA = 1000, and a type-B will sell for pB = 500. Underasymmetric information, the average value of buyers is 1

21500 + 12750 = 1125.

(b) Supply:

price supply quality

p < 500 0 0500 ≤ p < 1000 10 0

1000 ≤ p 20 1/2

(c) Demand.

price demand expected qualityp ≤ 750 19 0

750 < p < 1000 0 01000 ≤ p ≤ 1125 19 1/2

1125 < p 0 1/2

quantity

p

10 19 20

500

750

1000

1125

1500

supply demand

0

El

Eh

Figure 3: There is a low-price and a high-price equilibrium.

(d) We draw supply (solid) and demand (dashed) in figure 3 and count two situations wheresupply meets demand. Market transaction is possible at the intervals p ∈ [500, 750]and p ∈ [1000, 1125]. Assuming the short side determines the price, we get two equilibria:El = (q = 10, p = 750), and Eh = (q = 19, p = 1000).

Exercise 2: There is a large number of sellers and even more buyers. Each seller has oneused car to sell. Let Suppose the quality of a used car can be indexed by θ, which is uniformlydistributed on [0, 1]. If a seller of type θ sells his car for a price p, he receives utility us(θ, p). Abuyer who buys a car of quality θ at price p receives a utility of θ − p. If a seller does not sellhis car, he receives a utility of zero. Similarly, a buyer who does not buy a car receives a utilityof zero. A seller knows the quality of his car, but buyers only know the distribution of quality.

(a) Argue that in a competitive equilibrium under asymmetric information, we must haveE(θ|p) = p.

(b) Show that if us(θ, p) = p − θ/2, every p ∈ (0, 1/2] is an equilibrium price.

(c) Find the equilibrium price when us(θ, p) = p−√

θ. Which cars are traded in equilibrium?

(d) Find an equilibrium price when us(θ, p) = p − θ3.

(e) Are any of the preceding outcomes Pareto efficient? Describe Pareto improvements when-ever possible.

(a) In equilibrium, we must have E[θ|p] = p for the following reason. If E[θ|p] > p thenBertrand competition (more buyers than sellers!) drives prices down to p = E[θ|p]. Next,if E[θ|p] < p, buyers do not participate in trading.

16

(b) If at some price p cars of all qualities are supplied, E(θ|p) = 1/2. Thus the highest possibleprice is p = 1/2.

Here, a seller of type θ participates if p − θ/2 ≥ 0. Thus only θ ≤ 2p participate. For anyp ≤ 1/2, the market expectation of the buyers is E(θ|p) = E(θ|θ ≤ 2p) = 2p/2 = p. Thusthe equilibrium condition (E[θ|p] = p) is satisfied for any p ∈ (0, 1/2].

(c) us(θ, p) = p −√

θ. Now sellers θ ≤ p2 participate. Thus E(θ|p) = E(θ|θ ≤ p2) = p2/2.But p2/2 = p only at p = 0 for p ≤ 1/2. There is no trade.

(d) us(θ, p) = p − θ3. Thus sellers θ ≤ p1/3 participate. Therefore E(θ|p) = p1/3/2.

Equilibrium requires p1/3/2 = p, which implies

p =1

2√

2≈ 0.35.

Therefore, in equilibrium all sellers of type θ ≤(

1

2√

2

)(1/3)

≈ 0.7 participate.

(e) In all cases, the seller’s utility from any type θ is lower than that of the buyer. ThereforePareto efficiency implies that all types should exchange hands. This can be achieved withperfect information. Under asymmetric information this could happen in some cases - inpart (b), if p = 1/2, efficiency is achieved. In all other cases, some trades do not happen.In the case in (c) there is no trade at all. The outcome in (d) is not fully efficient becauseonly types up to approximately 0.7 trade.

Exercise 3: A seller sells a unit of a good of quality q at a price t. The cost of producing atquality level q is given by q2. There is a buyer who receives a utility of θq − t by consuming theunit of quality q at price t. If he decides not to buy, he gets a utility of zero. θ can take twovalues 1 and 2. Assume that the seller has all the bargaining power.

(a) Suppose the seller can observe θ. Derive the profit maximizing price-quality pairs offeredwhen θ = 1 and when θ = 2. Show that the quality offered when θ = 2 is twice the qualityoffered when θ = 1.

(b) Prove that the full information price-quality pairs are not incentive compatible if the sellercannot observe θ.

(a) The sellers problem for any i ∈ {1, 2} is to

maxqi,ti ti − q2i subject to θiqi − ti = 0.

This impliesmax

qi

θiqi − q2i .

Thus the optimal qi, denoted by q∗i is given by

q∗i =θi

2.

The optimal price t∗i is given by t∗i =θ2i

2.

17

Using θ1 = 1 and θ2 = 2, we have

q∗1 =1

2, t∗1 =

1

2

andq∗2 = 1, t∗2 = 2.

(b) The incentive compatibility (IC) constraints are

(ICθ1) θ1q1 − t1 ≥ θ1q2 − t2

which says that type θ1 should choose the contract (q1, t1) and not (q2, t2). Similarly, weneed incentive compatibility for type θ2:

(ICθ2) θ2q2 − t2 ≥ θ2q1 − t1

Now, since θ2 > θ1, we have

θ2q∗

1 − t∗1 > θ1q∗

1 − t∗1 = 0, (8)

where the last equality follows from part (a).

We also know from part (a) that θ2q∗

2 − t∗2 = 0. Using this, and the inequality from (8),we have

θ2q∗

2 − t∗2 < θ2q∗

1 − t∗1

which violates (ICθ2).

Thus the full information contracts are not incentive compatible. Type θ2 would find itprofitable to take contract (q1, t1).

(Alternatively, you can substitute the values of q∗i and t∗i from part (a) in (ICθ2). This

gives 0 ≥ 1/2, which is impossible.)

18

TOF ISolutions - Moral Hazard Problems

Exercise 1: Consider the following hidden action model with three possible actions for the agentE = {e1, e2, e3}. There are two possible profit outcomes for the Principal: πH = 10 and πL = 0.The probabilities of πH conditional on the three effort levels are f(πH |e1) = 2

3 , f(πH |e2) = 12 ,

and f(πH |e3) = 13 . The agent has a separable utility function u(w, ei) = v(w)− g(ei), where v is

the von-Neumann Morgenstern utility of wage w, and g(ei) is the disutility of effort. We are told

that g(e1) = 53 , g(e2) = 8

5 , and g(e3) = 43 . Finally, v(w) = w

1

2 and the manager’s reservationutility is u = 0.

(a) What is the optimal contract when effort is observable?

(b) Show that if effort is not observable, then e2 is not implementable. For what level of g(e2)would e2 be implementable? [Hint: focus on the utility the manager will get for the twooutcomes].

(c) What is the optimal contract when effort is not observable?

(a) The first best contracts are

e1: (IR1)√

w1 = 53 ⇒ w1 = 25

9 ; π(e1) = 2310 − 25

9 = 359 .

e2: (IR2)√

w2 = 85 ⇒ w2 = 64

25 ; π(e2) = 1210 − 64

25 = 6125 .

e3: (IR3)√

w3 = 43 ⇒ w3 = 16

9 ; π(e3) = 1310 − 16

9 = 149 .

Hence the principal will choose to implement e1.

(b) To implement the unobservable e2 we need to satisfy IR2

1

2

√wH +

1

2

√wL − 8

5≥ u0 = 0

and both(IC2,1)

12

√wH + 1

2

√wL − 8

5 ≥ 23

√wH + 1

3

√wL − 5

3 ,(IC2,3)

12

√wH + 1

2

√wL − 8

5 ≥ 13

√wH + 2

3

√wL − 4

3 .

But (IC2,1) implies that√

wH −√wL ≤ 2

5(9)

while (IC2,3) implies that√

wH −√wL ≥ 8

5. (10)

But (9) and (10) cannot be satisfied at the same time. Hence e2 cannot be implemented.

If we net g(e2) such that both

(IC2,1)16(√

wH −√wL) ≤ 5

3 − g(e2), and(IC2,3)

16(√

wH −√wL) ≥ g(e2) − 4

3

we get5

3− g(e2) ≥ g(e2) −

4

3or

g(e2) ≤3

2.

(c) The principal’s problem in implementing e1 is to minimize the expected wage cost subjectto

(IR1)23

√wH + 1

3

√wL − 5

3 ≥ u0 = 0,(IC1,2)

23

√wH + 1

3

√wL − 5

3 ≥ 12

√wH + 1

2

√wL − 8

5 , and(IC1,3)

23

√wH + 1

3

√wL − 5

3 ≥ 13

√wH + 2

3

√wL − 4

3 .

which implies(IC1,2)

√wH −√

wL ≥ 25 , and

(IC1,3)√

wH −√wL ≥ 1.

Hence (IC1,3) binds and√

wH = 1 +√

wL. We plug this into (IR1) and obtain

2

3(1 +

√wL) +

1

3

√wL =

5

3

which implies (wL, wH) = (1, 4). This gives a profit of

π(e1) =2

3(10 − 4) +

1

3(0 − 1) =

11

3.

To implement e3 the principal optimally uses the first best contract. As before, π(e3) = 149 .

Since 149 < 11

3 , it is optimal to implement e1.

20

Exercise 2: A principal hires an agent to work on a project at wage w. The agent’s utilityfunction is

U(w, ei) = 100 − 10

w− g(ei)

where g(ei) is the disutility associated with the effort level ei exerted on the project. The agentcan choose one of two possible effort levels, e1 or e2, with associated disutility levels g(e1) = 50,and g(e2) = 0. The agent’s choice of effort affects the project’s output in a probabilistic fashion:if the agent chooses effort level e1, an error occurs in the production process with probability14 , and if he chooses e2, an error occurs with probability 3

4 . The project yields a revenue of 20if the production process is error free, and a revenue 0 otherwise. The principal is risk neutral:she aims to maximize the expected value of the output, net of any wage payments to the agent.The agent has a reservation utility of 0.

(a) If the effort level chosen by the agent is observable by the principal, it is optimal for theprincipal to choose a fixed-wage contract. Explain the intuition for this result.

(b) If effort is observable, which wage w should the principal offer if she wants to implemente1? Which wage implements e2? Which of these will the principal implement?

(c) Suppose that the agent’s choice of effort level is not observable. Which wage schedule willimplement e1 in this case? Which expected net return does the principal get in this case?How does this compare with the value in part (b), where effort was observable?

(d) If effort is not observable, should the principal implement e1 or e2?

(a) Optimal risk-sharing as explained in the example in the lecture notes.

(b) The first best wages given e1 are

100 − 10

w1− 50 = 0 ⇒ w1 =

1

5,

and given e2

100 − 10

w2= 0 ⇒ w2 =

1

10.

The associated profits are

π(e1) =3

420 − 1

5⇒ π(e1) =

148

10,

and

π(e2) =1

420 − 1

10⇒ π(e2) =

49

10.

Hence the principal will optimally implement e1.

(c) The principal’s problem to implement the unobservable ei is to minimize the expectedwage cost subject to

(IR1) 50 − 34

10wH

− 14

10wL

≥ u0 = 0,

(IC1) 50 − 34

10wH

− 14

10wL

≥ 100 − 14

10wH

− 34

10wL

.

From (IR1) we get 1wL

≤ 20 − 3wH

. Since the principal wants to minimize the wage cost,the wage payments should be set so that this holds with equality:

1

wL= 20 − 3

wH.

21

Next, simplifying the (IC1), we get

1

wL− 1

wH≥ 10.

From the two equations above,

20 − 3

wH− 1

wH≥ 10 ⇒ wH ≥ 2

5.

Since it is optimal for the principal to pay the lowest possible wage, this is satisfied withequality. Therefore, the optimal contract under asymmetric information is given by:

(wH , wL) =

(

2

5,

2

25

)

.

(d) The principal’s payoff from implementing e1 is given by

π(e1) =3

4

(

20 − 2

5

)

+1

4

(

0 − 2

25

)

=734

50= 14.68.

To implement e2, the principal optimally uses the first best contract. Therefore π(e2) = 4.9as before.

Clearly, it is optimal to implement e1.

22

TOF ISolutions - Market Microstructure Problems

Exercise 1. Consider a risky asset with final payoff V , which is a normally distributed randomvariable with mean 0 and variance σ2

0. There is a risk neutral informed trader who knows therealization v of the random variable V . The trader has no initial endowment of the risky asset.

After observing v, the informed trader chooses his trading quantity X. This net order flowconsists of the informed trader’s order and the total order from liquidity traders (denoted by u)which is normally distributed with mean 0 and variance σ2

u. All random variables are pairwiseindependent.

The market maker observes the net order flow X + u and sets a price P (X + u) = E(V |X + u).

Consider the linear equilibrium given by

X(v) =σu

σ0v,

andP (X + u) =

σ0

2σu(X + u).

(a) Show that if liquidity trade quantity doubles, P (X +u) remains unchanged in equilibrium.Explain the intuition behind this result.

(b) Derive the expected profit of the informed trader conditional on his information, and showthat it is decreasing in σ0. Explain the intuition behind this result.

(c) Derive the expected ex-ante profit of the informed trader, and show that it is increasingin σ0. Explain the intuition behind this result.

(a)

P (X + u) =σ0

2σu(X + u) =

σ0

2σu

(

σu

σ0v + u

)

=v

2+

σ0

2

u

σu

If u doubles, so does σu - so P is unchanged.

Intuition: if liquidity trade quantity doubles, the order flow of the informed trader doubles aswell, as he can now better hide behind the uninformed order flow.

(b)

E(π|v) = E(X(v − P )|v) =σu

σ0v(v − v

2),

using Eu = 0.

Thus

E(π|v) =σu

σ0

v2

2,

which is decreasing in σ0.

Intuition: Rise in σ0 lowers ability of informed bidder to hide his trade behind noise trade.

(c)

E(π) =σu

σ0

Ev2

2=

σu

σ0

σ20

2=

σuσ0

2,

which is increasing in σ0.

Intuition: A rise in σ0 has the effect described in (b), but also has the additional effect ex-anteof making higher absolute values of v more likely, which increases profit. The calculation showsthat this second effect dominates.

Exercise 2. Consider the following sequential trading model. Suppose an asset can have twovalues 10 and 20. The market maker attaches a prior probability of 1/2 that the value equals10. Now a trader arrives in the market and submits either a buy or a sell order. Suppose 60% ofthe potential traders are informed and the rest are uninformed. An uninformed trader submitsa buy or a sell order with equal probability. Suppose the market makers earns a zero expectedprofit.

(a) Calculate the bid price and the ask price.

(b) Briefly compare (in words only) the sequential trading model to the Kyle (1985) model.

(a)

Prob(V = 10|S) =Prob(V = 10)Prob(S|V = 10)

Prob(V = 10)Prob(S|V = 10) + Prob(V = 20)Prob(S|V = 20)

Prob(S|V = 10) = (.6)(1) + (.4)(.5) = .8

Prob(S|V = 20) = (.6)(0) + (.4)(.5) = .2

Thus

Prob(V = 10|S) =(.5)(.8)

(.5)(.8) + (.5)(.2)= .8

and therefore Prob(V = 20|S) = .2. Now,

E(V |S) = 20 × Prob(V = 20|S) + 10 × Prob(V = 10|S) = 4 + 8 = 12.

Thus the bid price is 12.

Prob(V = 10|B) =Prob(V = 10)Prob(B|V = 10)

Prob(V = 10)Prob(B|V = 10) + Prob(V = 20)Prob(B|V = 20)

Prob(B|V = 10) = (.6)(0) + (.4)(.5) = .2

Prob(B|V = 20) = (.6)(1) + (.4)(.5) = .8

Thus

Prob(V = 10|B) =(.5)(.2)

(.5)(.8) + (.5)(.2)= .2

and therefore Prob(V = 20|B) = .8.

E(V |B) = 20 × Prob(V = 20|B) + 10 × Prob(V = 10|B) = 16 + 2 = 18.

Thus the ask price is 18.

The bid ask spread is 18 - 12 = 6.

(b) Sequential trading explains bid ask spread, and the effect of the proportion of informedtraders on the size of this spread. But trade size is fixed. The batch trading model of Kyle(1985) allows for trade size to be determined strategically by the informed trader. Thisexplains how the informed trader adjusts trade size - taking into account the uninformedtrading volatility - to control how much information is released through his order.

24