EC476 Contracts and Organizations, Part III: Lecture 3 Leonardo - - PowerPoint PPT Presentation

EC476 Contracts and Organizations, Part III: Lecture 3

Leonardo Felli

32L.G.06

26 January 2015

Failure of the Coase Theorem

◮ Recall that the Coase Theorem implies that two parties, when

faced with a potential externality, if they can costlessly trade and ownership rights are well defined, will be able to achieve efficiency.

◮ We are now going to consider a first environment in which the

Theorem fails.

◮ This is a situation where parties bargain under bilateral

asymmetric information (with no externalities).

◮ We will show that in this situation efficiency cannot be

achieved.

◮ Clearly, since efficiency cannot be achieved even in the

absence of externalities efficiency is not achievable when externalities are present.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 2 / 50

Bilateral Asymmetric Information

◮ Recall that when proving at the Coase Theorem we had to

focus on a specific extensive form of the parties negotiation.

◮ In general, this does not imply that we cannot find an

extensive form that will achieve efficiency.

◮ Fortunately an other fundamental principle of contract theory

will help in this case: Revelation Principle.

◮ Using the revelation principle we will be able to conclude that

efficiency cannot be achieved whatever bargaining protocol governs the negotiation between the two parties under bilateral asymmetric information.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 3 / 50

Bilateral Trade (Chatterjee and Samuelson, 1983):

◮ Consider the following simple model of bilateral trade (double

auction).

◮ Two players, a buyer and a seller: N = {b, s}. ◮ The seller names an asking price: ps. ◮ The buyer names an offer price: pb.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 4 / 50

Bilateral Trade (2)

◮ The action spaces:

As = {ps ≥ 0}, Ab = {pb ≥ 0}.

◮ The seller owns and attaches value vs to an indivisible unit of

a good.

◮ The buyer attaches value vb to the unit of the good and is

willing to pay up to vb for it.

◮ The valuations for the unit of the good of the seller and the

buyer are their private information of each player.

◮ Player i ∈ {b, s} believes that the valuation of the opponent

v−i takes values in the unit interval.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 5 / 50

Bilateral Trade (3)

◮ The type spaces:

Ts = {0 ≤ vs ≤ 1}, Tb = {0 ≤ vb ≤ 1}

◮ Player i ∈ {b, s} also believes that the valuation of the

• pponent is uniformly distributed on [0, 1]:

µs = 1, µb = 1.

◮ The extensive form of the game is such that:

◮ If pb ≥ ps then they trade at the average price:

p = (ps + pb) 2 .

◮ If pb < ps then no trade occurs. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 6 / 50

Bilateral Trade (4)

◮ The payoffs to both the seller and the buyer are then:

us(ps, pb; vs, vb) =    (ps + pb) 2 if pb ≥ ps vs if pb < ps and ub(ps, pb; vs, vb) =    vb − (ps + pb) 2 if pb ≥ ps if pb < ps

◮ Players’ strategies: ps(vs) and pb(vb). ◮ We consider strictly monotonic and differentiable strategies.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 7 / 50

Seller’s Best Reply

◮ Consider now the seller’s best reply. ◮ This is defined by the following maximization problem:

max

ps

Evb {us(ps, pb; vs, vb) | vs, pb(vb)}

◮ Consider now the seller’s payoff, substituting pb(vb) we have:

us =    (ps + pb(vb)) 2 if pb(vb) ≥ ps vs if pb(vb) < ps

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 8 / 50

Seller’s Best Reply (2)

◮ or

us =    (ps + pb(vb)) 2 if vb ≥ p−1

b (ps)

vs if vb < p−1

b (ps) ◮ The seller’s maximization problem is then:

max

ps

p−1

b

(ps) vb=0

vs dvb + 1

vb=p−1

b

(ps)

(ps + pb(vb) 2 dvb

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 9 / 50

Seller’s Best Reply (3)

◮ Recall that by Leibniz’s rule:

∂ ∂y β(y)

α(y)

G(x, y)dx

• =

= G(β(y), y) β′(y) − G(α(y), y)α′(y) + + β(y)

α(y)

∂G(x, y) ∂y dx

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 10 / 50

Seller’s Best Reply (4)

◮ Therefore the first order conditions are:

vs dp−1

b (ps)

dps − 1 2

• ps + pb(p−1

b (ps))

dp−1

b (ps)

dps + + 1

p−1

b

(ps)

1 2 dvb = 0

◮ or from ps = pb(p−1 b (ps)):

(vs − ps) dp−1

b (ps)

dps + 1 2

• vb

1

p−1

b

(ps) = 0 ◮ which gives:

(vs − ps)dp−1

b (ps)

dps + 1 2

• 1 − p−1

b (ps)

• = 0

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 11 / 50

Buyer’s Best Reply

◮ The buyer’s best reply is instead defined by:

max

pb

Evs {ub(ps, pb; vs, vb) | vb, ps(vs)}

◮ Consider now the buyer’s payoff obtained substituting ps(vs):

ub =    vb − (ps(vs) + pb) 2 if vs ≤ p−1

s (pb)

if vs > p−1

s (pb) ◮ we then get

max

pb

p−1

s

(pb) vs=0

• vb − (ps(vs) + pb)

2

• dvs

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 12 / 50

Buyer’s Best Reply (2)

◮ Therefore the first order conditions are:

• vb − (ps(p−1

s (pb)) + pb)

2 dp−1

s (pb)

dpb + − 1 2 p−1

s

(pb) vs=0

dvs = 0

◮ or

[vb − pb] dp−1

s (pb)

dpb − 1 2

• vs

p−1

s

(pb)

= 0

◮ which gives:

(vb − pb)dp−1

s (pb)

dpb − 1 2 p−1

s (pb) = 0

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 13 / 50

Equilibrium Characterization

◮ To simplify notation we re-write p−1 b (·) = qb(·) and

p−1

s (·) = qs(·). ◮ The two differential equations that define the best reply of the

seller and the buyer are then: [qs(ps) − ps] q′

b(ps) + 1

2 [1 − qb(ps)] = 0 [qb(pb) − pb] q′

s(pb) − 1

2 qs(pb) = 0

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 14 / 50

Equilibrium Characterization (2)

◮ Solving the second equation for qb(pb) and differentiating

yields: q′

b(pb) = 1

2

• 3 − qs(pb)q′′

s (pb)

[q′

s(pb)]2

• ◮ Substituting this expression into the first differential equation

we get: [qs(ps) − ps]

• 3 − qs(ps)q′′

s (ps)

[q′

s(ps)]2

• +
• 1 − ps − qs(ps)

2q′

s(ps)

• = 0

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 15 / 50

Equilibrium Characterization (3)

◮ This is a second-order differential equation in qs(·) that has a

two-parameter family of solutions.

◮ The simplest family of solution takes the form:

qs(ps) = α ps + β

◮ Then the values α = 3/2 and β = −3/8 solve the

second-order differential equation.

◮ The definition of qs(·) and qb(·) imply that:

ps = 2 3 vs + 1 4, pb = 2 3 vb + 1 12

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 16 / 50

Equilibrium Characterization (4)

◮ This is the (unique) Bayesian Nash equilibrium of this game. ◮ Notice now that it is efficient to trade whenever:

vb ≥ vs

◮ However in this double auction game trade occurs whenever:

pb ≥ ps

• r

2 3 vb + 1 12 ≥ 2 3 vs + 1 4

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 17 / 50

Inefficient Trade

In other words, in equilibrium trade occurs whenever: vb ≥ vs + 1 4 ............................................................. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

✻ ✲

• q

1 vb vs 1 (0, 0) vs = vb

• Trade

✻

vb = vs + 1

4

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 18 / 50

Bilateral Trade (Myerson and Satterthwaite 1983):

◮ The key question is whether the inefficiency we found in the

double auction can be eliminated by choosing a different trading mechanism.

◮ The revelation principle provides the right tool to get an

answer to the question above.

◮ Obviously, there is no principal, but the two parties at an

ex-ante stage, before they learn their private information act as the mechanism designer.

◮ Assume further that this is a pure bilateral contract transfers

cannot involve a third party.

◮ In jargon the mechanism has to be budget balancing.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 19 / 50

Unavoidable Inefficincy

Theorem (Myerson and Satterthwaite 1983)

When parties to a contract bargain in a situation of bilateral asymmetric information efficiency of trade cannot be achieved. There always exist configurations of the parameter values that yield no trade when trade is efficient. The reason is that the inefficiency is necessary to induce separation

• f the different types of buyer and seller.

In other words, the incentive compatibility constraints of the parties imply that efficiency is no longer guaranteed.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 20 / 50

Transaction Costs

Consider now a different cause for the failure of the Coase Theorem: the presence of transaction costs. Of course for this to be an interesting argument we need to abstract from other sources of failure of the Coase Theorem such as asymmetric information.

“Theorem” (Strong version of the Coase Theorem)

The Coase theorem guarantees efficiency:

• 1. regardless of the way in which property rights are assigned,

and

• 2. whenever the mutual gains from trade exceed the necessary

transaction costs.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 21 / 50

Transaction Costs (2)

◮ We are going to show that this is not necessarily the case.

The reason is the strategic role of transaction costs.

◮ Key factor: some transaction costs have to be paid ex-ante,

before the negotiation starts.

◮ These ex-ante transaction costs generate an inefficiency

usually known as a hold-up problem.

◮ The hold-up problem yield an outcome that is constrained

inefficient.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 22 / 50

Numerical Example

◮ Potential Surplus = 100, ◮ Ex-ante cost to each negotiating party = 20, ◮ Distribution of bargaining power = (10%, 90%), ◮ Ex-ante Payoff to party A = (10% 100 − 20) = −10, ◮ Ex-ante Payoff to party B = (90% 100 − 20) = 70, ◮ Social surplus = 60. ◮ Coasian negotiation opportunity is left unexploited.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 23 / 50

Coasian Solution

◮ Natural question: whether it is possible to find a Coasian

solution to this inefficiency.

◮ In other words we are asking whether the parties can agree

ex-ante on a transfer contingent on each party entering a future negotiation.

◮ We are going to show that under plausible conditions a

Coasian solution of this form may not be available.

◮ The reason is that any new negotiation may itself be

associated with (possibly small) ex-ante transaction costs.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 24 / 50

Coasian Solution (2)

◮ In the Numerical Example above Party B makes a transfer to

party A contingent on the cost of 20 being paid by A.

◮ Assume that B makes a take-it-or-leave-it offer to A, ◮ Ex-ante costs to each party associated with this ‘agreement

contingent on future negotiation’ = 1,

◮ A accepts this transfer if:

10% 100 − 20 + x ≥ 0,

◮ or x ≥ 10.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 25 / 50

Coasian Solution (3)

◮ There always exists an equilibrium in which x = 10. ◮ Ex-ante payoff to party A

πA = 10% 100 − 20 + 10 − 1 = −1

◮ No negotiation (contingent or not) will take place. ◮ A Coasian solution is not available.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 26 / 50

Examples of Transaction Costs

Ex-ante transaction costs:

◮ time to arrange and/or participate in a meeting, ◮ time and effort to conceive and agree upon a suitable

negotiation language,

◮ time and effort to collect information about the legal

environment in which the agreement is enforced,

◮ time to collect and analyze background information for the

negotiation,

◮ time and effort to think about the negotiation at hand.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 27 / 50

Examples of Transaction Costs (2)

◮ These costs can be monetized through the hiring of an outside

party: typically a lawyer.

◮ The problem does not disappear if the lawyer needs to be paid

independently of the success of the negotiation: no contingent fees.

◮ Indeed, monetizing the costs may increase the magnitude of

the inefficiency: moral hazard.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 28 / 50

Simple Coasian Negotiation

Consider the following simple coasian negotiation:

◮ two agents, i ∈ {A, B}; ◮ share a surplus, ◮ size of the surplus normalized to one, ◮ parties’ payoffs in case of disagreement to zero. ◮ Each party faces ex-ante costs: (cA, cB).

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 29 / 50

Complement Transaction Costs

Assume that (cA, cB) are:

◮ complements: each party i has to pay ci for the negotiation to

be feasible;

◮ affordable: party i’s endowment covers ci; ◮ efficient: cA + cB < 1.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 30 / 50

Complement Transaction Costs (2)

Timing:

✲ ✲ s s s

t = 0 t = 1 Contract Negotiated enforced

• Simult. Decisions
• n (cA, cB)

Contract t = 2

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 31 / 50

Complement Transaction Costs (3)

◮ We solve the model backward. ◮ We start with a simple bargaining rule:

◮ Let λ be the bargaining power of A. ◮ The division of surplus at t = 1 is then (λ, 1 − λ). Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 32 / 50

Complement Transaction Costs (4)

Result

For any given λ there exists a pair (cA, cB) of affordable and efficient ex-ante costs such that the unique SPE is (not pay cA, not pay cB)

Result

For any pair (cA, cB) of affordable and efficient ex-ante costs there exists a value of λ such that the unique SPE is (not pay cA, not pay cB)

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 33 / 50

Complement Transaction Costs (5)

Proof: The ‘reduced form’ of the two stage game is: pay cB not pay cB pay cA λ − cA, 1 − λ − cB −cA, 0 not pay cA 0, −cB 0, 0

◮ A pays cA iff λ ≥ cA and B pays cB, ◮ A does not pay cA if B does not pay cB, ◮ B pays cB iff 1 − λ ≥ cB and A pays cA, ◮ B does not pay cB if A does not pay cA.

Therefore the result holds when λ < cA or 1 − λ < cB.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 34 / 50

Substitute Transaction Costs

Assume now that (cA, cB) are:

◮ substitutes: either party has to pay ci; ◮ affordable: party i’s endowment covers ci; ◮ efficient:

min{cA, cB} < 1.

Result

Both results above hold. In the second result only one type of inefficiency may occur.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 35 / 50

Substitute Transaction Costs (2)

Proof: The reduced form game is now: pay cB not pay cB pay cA λ − cA, 1 − λ − cB λ − cA, 1 − λ not pay cA λ, 1 − λ − cB 0, 0

◮ A pays cA iff λ ≥ cA and B does not pay cB, ◮ A does not pay cA if B pays cB, ◮ B pays cB iff 1 − λ ≥ cB and A does not pay cA, ◮ B does not pay cB if A pays cA.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 36 / 50

Substitute Transaction Costs (3)

We then have two types of inefficiencies:

◮ an inefficiency that leads to a unique SPE with no agreement:

λ < cA, and (1 − λ) < cB

◮ an inefficiency that leads to an agreement obtained paying too

high a cost: if cA < cB, λ < cA, and (1 − λ) > cB Results 1 and 2 also generalize to the case in which (cA, cB) are substitutes and strategic complements.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 37 / 50

The Impossibility of a Coasian Solution

Is there a Coasian solution to this hold-up problem? Consider the following simple contingent agreement: A transfer σB ≥ 0 (σA ≥ 0) payable contingent on whether the other party decides to pay cA, (cB). Key assumption:

◮ this new negotiation is associated with a fresh set of ex-ante

costs (c1

A, c1 B); ◮ the two sets of ex-ante costs are assumed to be complements,

affordable and efficient.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 38 / 50

A Potential Coasian Solution

Assume: λ < cA

s s s s s s ✲ ✲ ✲ ✇

−2 −1 1 B makes offer to A A Accepts/Rejects

• n (c1

A, c1 B)

• n (cA, cB)

Tranfers Contract A/B does

✻ s

Negotiation

• Simult. Decision

not pay

• Simult. Decision

enforced 2

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 39 / 50

A Potential Coasian Solution (2)

Result

There always exists a SPE of this game in which both agents pay neither the second tier, (c1

A, c1 B), nor the first tier, (cA, cB), of

ex-ante costs. Proof: At each stage the two agents decide simultaneously and independently whether to pay their ex-ante costs. An agreement is achieved only if both agents pay (c1

A, c1 B) and

(cA, cB). Either agent will never pay if he expects the other not to pay his ex-ante cost.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 40 / 50

A Potential Coasian Solution (3)

Result

There always exists a SPE of this game in which both agents pay the second tier, (c1

A, c1 B), and the first tier, (cA, cB), of ex-ante

costs and an agreement is successfully negotiated . Proof: Assume that:

◮ both parties have paid the ex-ante costs (c1 A, c1 B) at t = −2

and

◮ party A has accepted the transfer σB ≥ 0.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 41 / 50

A Potential Coasian Solution (4)

The parties’ continuation game is then: pay cB not pay cB pay cA λ − cA + σB, 1 − λ − cB − σB −cA, 0 not pay cA 0, −cB 0, It follows:

◮ A pays cA if B pays cB and λ + σB > cA and ◮ B pays cB if A pays cA and 1 − λ − σB > cB.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 42 / 50

A Potential Coasian Solution (5)

Therefore if 1 − λ − cB > σB > cA − λ the subgame has two Pareto-ranked equilibria:

◮ one in which an agreement is successfully negotiated, ◮ an other one in which an agreement does not arise.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 43 / 50

A Potential Coasian Solution (6)

We can then construct a SPE of the model such that at t = 0

◮ if

λ + σB ≥ cA + c1

A

the (constrained) efficient equilibrium is played.

◮ if

λ + σB < cA + c1

A

the no-agreement equilibrium is played.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 44 / 50

A Potential Coasian Solution (7)

In this equilibrium necessarily σB = cA + c1

A − λ

Therefore the agreement is successfully negotiated. Observations:

◮ All equilibria of the model are constrained inefficient: costs

paid are inefficiently high.

◮ The equilibrium described in last result is not

renegotiation-proof.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 45 / 50

Continuous Costs

◮ Consider now the case with continuous ex-ante costs

(Holmstr¨

• m 1982): the more detailed the agreement is, the

higher the surplus.

◮ The surplus is a monotonic and concave function of costs:

x(cA, cB) the costs are complements: ∂2x(cA, cB) ∂cA∂cB > 0.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 46 / 50

Continuous Costs (2)

◮ Payoff to party A is:

λ x(cA, cB) − cA

◮ while the payoff to B is:

(1 − λ)x(cA, cB) − cB.

Result

Given λ, every equilibrium is such that c∗

A < cE A

c∗

B < cE B .

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 47 / 50

Continuous Costs (3)

Proof: The equilibrium costs (c∗

A, c∗ B) are such that:

max

cA

λ x(cA, cB) − cA max

cB

(1 − λ) x(cA, cB) − cB. The first order conditions of both these problems are: ∂x(c∗

A, c∗ B)

∂cA = 1 λ and ∂x(c∗

A, c∗ B)

∂cB = 1 1 − λ

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 48 / 50

Continuous Costs (4)

The (constrained) efficient level of costs (cE

A , cE B ) are such that:

max

cA,cB x(cA, cB) − cA − cB

The first order conditions of this problem are: ∂x(cE

A , cE B )

∂cA = 1 and ∂x(cE

A , cE B )

∂cB = 1 Concavity of x(·, ·) and the fact that: ∂2x(cA, cB) ∂cA∂cB > 0.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 49 / 50

More Pervasive Inefficiency

imply c∗

A < cE A

c∗

B < cE B . ◮ One key difference between this result and the one we found

for discrete costs is that this result holds for every λ ∈ (0, 1).

◮ In other words, when costs are continuous the inefficiency is

more pervasive.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 50 / 50