EC476 Contracts and Organizations, Part III: Lecture 5 Leonardo - - PowerPoint PPT Presentation

EC476 Contracts and Organizations, Part III: Lecture 5

Leonardo Felli

32L.G.06

9 February 2015

The Jungle

Through the history of mankind, it has been quite common [...] that economic agents, individually or collectively, use power to seize control of assets held by

• thers.

The Jungle: a simple world where power governs transactions. We want to understand where and how enforcement of trade arises.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 9 February 2015 2 / 44

Power and the Jungle

◮ Power: a stronger agent is able take resources from a weaker

agent.

◮ A jungle is a society that consists of a set of individuals, each

having exogenous preferences over bundles of desirable goods and exogenous strength.

◮ We focus on a model that mirrors a basic exchange economy. ◮ Define a jungle equilibrium as a feasible allocation of

commodities such that no agent would like and is able to take goods held by a weaker agent.

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Allocation of Houses

◮ Set of agents: I = {1, 2, 3, 4} ◮ Set of houses (indivisible commodities): H = {h1, h2, h3, h4}. ◮ Assume that the initial allocation of houses is:

h = (h1, h2, h3, h4), in other words agent i owns house hi, where i ∈ {1, 2, 3, 4}.

◮ Clearly an arbitrary allocation of houses is a permutation of h,

e.g. g = (h2, h1, h3, h4).

◮ The total number of allocations (permutations) is 4! = 24.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 9 February 2015 4 / 44

Preferences over Houses

◮ Assume that no agent can consume more than one house. ◮ Assume also that each agent has a strict preference ordering

• ver houses ≻i and strictly prefers having a house to no house.

◮ For example, we take the preferences to be:

1 2 3 4 h2 h4 h1 h2 h1 h3 h3 h4 h3 h2 h2 h1 h4 h1 h4 h3 ∅ ∅ ∅ ∅

◮ In other words, for agent 1:

u1(h2) > u1(h1) > u1(h3) > u1(h4) > 0.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 9 February 2015 5 / 44

Competitive Equilibrium

Result (Kaneko 1983)

There exists a competitive equilibrium that assigns one house to

• ne agent for any initial endowment of houses.

Proof: (intuition using example) We want to create a partition of the set of individuals I. Each group in the partition is such that agents have a reason to trade within their group in order of priority: first trade the individuals that care most, then the others.

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Competitive Equilibrium (2)

Consider only the first choice for each agent in the list of preferences: 1 2 3 4 h2 h4 h1 h2 Start from one individual, say 2, clearly he would like to trade with 4, while 4 would like to trade with 2. The group {2, 4} has then a viable trade that would allocate to all members their most preferred house. This is the first group in the partition.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 9 February 2015 7 / 44

Competitive Equilibrium (3)

Now, remove persons 2 and 4 and houses h2 and h4 from the preference list above: 1 3 h1 h1 h3 h3 ∅ ∅ Clearly in this case person 1 will not want to trade with person 3: he owns the most preferred house among the remaining ones. Therefore the next group is {1} and by the same procedure the remaining group is {3}. The partition is then: P =

• {2, 4}, {1}, {3}
• .

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 9 February 2015 8 / 44

Competitive Equilibrium (4)

Consider now the prices: ph2 = ph4 > ph1 > ph3 The competitive equilibrium price vector is: ¯ p = (ph1, ph2, ph3, ph4) . The associated competitive allocation is: ¯ w = (h1, h4, h3, h2) At these prices only one trade occurs between person 2 and 4.

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Observations

◮ In this construction, we can associate each group in the

partition with a round of trade.

◮ Some agents in each round exchange their houses and receive

their best house from the set of houses not allocated in earlier rounds.

◮ The group of agents who obtain a house in each round has

the property that each of its members can obtain his preferred house among the houses held within the group.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 9 February 2015 10 / 44

Power Relationship

◮ Let S be a power relationship among agents. ◮ Assume S is a binary relation that is

◮ irreflexive: (i S i) ◮ asymmetric: (i S j) → ( j S i) ◮ complete: (i S j) or ( j S i) ∀i, j ∈ I ◮ transitive: (i S j) & ( j S k) → (i S k)

◮ An example of allocation of power is:

2 S 4, 4 S 1, 1 S 3

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Equilibrium in the Jungle

We start by defining a jungle equilibrium.

Definition (Jungle Equilibrium)

A jungle equilibrium is an allocation such that no agent can assemble a more preferred bundle by combining his own bundle either with a bundle that is held by one of the agents weaker than him or with the bundle that is freely available. We can now construct a jungle equilibrium.

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Equilibrium in the Jungle (2)

◮ Houses are also allocated in rounds: in each round an agent

picks the best house among those that were not allocated earlier.

◮ Only one agent makes a choice in each round. This is the

strongest agents among the ones that have not made their choices earlier.

Result (Piccione and Rubinstein 2007)

In the house allocation problem every Pareto-efficient allocation is a jungle equilibrium for some strength relation S.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 9 February 2015 13 / 44

Equilibrium in the Jungle (3)

Proof: (intuition using example)

◮ In the jungle equilibrium houses are also allocated in rounds. ◮ Only one agent makes a choice in each round: he is the

strongest agent among those agents who have not made their choices earlier.

◮ Consider now the Walrasian allocation: ¯

w = (h1, h4, h3, h2).

◮ Recall that by First Welfare Theorem the Walrasian

equilibrium allocation ¯ w is Pareto efficient.

◮ Consider now the following allocation of power:

2 S 4, 4 S 1, 1 S 3

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Equilibrium in the Jungle (4)

◮ Consider now the following jungle equilibrium where

preferences are: 2 4 1 3 h4 h2 h2 h1 h3 h4 h1 h3 h2 h1 h3 h2 h1 h3 h4 h4 ∅ ∅ ∅ ∅

◮ In the first round all agents meet:

R1 = {1, 2, 3, 4}

◮ Given that 2 is the strongest agent then:

2 appropriates h4 clearly then 4 is left with h2.

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Equilibrium in the Jungle (5)

◮ In the second round agents 1, 3 and 4 meet:

R2 = {1, 3, 4}

◮ Given that 4 is the strongest among these agents then:

4 keeps h2 and the other agents keep their houses.

◮ Finally in the last round agents 1 and 3 meet:

R3 = {1, 3}

◮ Since 1 is the strongest agent then: 1 keeps h1 while

3 keeps h3

◮ In other words the jungle equilibrium allocation is:

¯ e = (h1, h4, h3, h2)

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Equilibrium in the Jungle (6)

◮ Consider now a model with K commodities and a set of

agents, I = {1, . . . , N}.

◮ The aggregate bundle ω = (ω1, . . . , ωK) is available for

distribution among the agents.

◮ Each agent i is characterised by a preference relation (strongly

monotonic and continuous) i on the set of bundles RK

+ and

by a consumption set X i.

◮ The set X i is interpreted as the bounds on agent i’s ability to

consume.

◮ Assume that X i is compact and convex, and satisfies free

disposal: xi ∈ X i, y ∈ RK

+, and y ≤ xi implies that y ∈ X i.

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Equilibrium in the Jungle (7)

◮ A feasible allocation is a vector of non-negative bundles

z = (z0, z1, . . . , zN) such that: z0 ∈ RK

+,

zi ∈ RK

+, ∀i ∈ {1, . . . , N}, N

• i=0

zi = ω

◮ The bundle z0 is the bundle of goods that are not allocated. ◮ A feasible allocation is efficient if there is no other feasible

allocation for which at least one agent is strictly better off and none of the other agents is worse off.

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Existence of Jungle Equilibrium

Definition

A jungle equilibrium is a feasible allocation z such that there are no agents i and j, iSj, and a bundle yi ∈ X i such that yi ≤ zi + z0 or yi ≤ zi + zj and yi ≻i zi.

Result (Piccione and Rubinstein 2007)

A jungle equilibrium exists. Proof: Construct an allocation ˆ z = (ˆ z0, ˆ z1, . . . , ˆ zN). Let ˆ z1 be

• ne of agent 1’s best bundles in {x1 ∈ X 1|x1 ≤ w}.

Define inductively ˆ zi to be one of the agent i’s best bundles in   xi ∈ X 1|xi ≤ w −

i−1

• j=1

ˆ zj    and ˆ z0 = w − N

j=1 ˆ

zj.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 9 February 2015 19 / 44

Existence of Jungle Equilibrium (2)

A jungle is smooth if, for each agent i:

• 1. the preferences are represented by a strictly quasiconcave, and

continuously differentiable utility function ui.

• 2. there exists a quasiconvex and differentiable function gi with

∇gi > 0 such that X i = {xi ∈ RK

+|gi(xi) ≤ 0}.

Result

If a jungle is smooth then ˆ z is the unique jungle equilibrium.

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Properties of the Jungle Equilibrium

Theorem (First Welfare Theorem for the Jungle)

If a jungle is smooth then the jungle equilibrium is efficient. Proof: (intuition using example) Consider the following jungle equilibrium. Let the power relationship be: 1 S 2, 2 S 3, 3 S 4

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 9 February 2015 21 / 44

Properties of the Jungle Equilibrium (2)

◮ Recall once again the individual preferences:

1 2 3 4 h2 h4 h1 h2 h1 h3 h3 h4 h3 h2 h2 h1 h4 h1 h4 h3 ∅ ∅ ∅ ∅

◮ In the first round all agents meet:

R1 = {1, 2, 3, 4}

◮ Given the power relationship then:

1 appropriates h2 clearly then 2 is left with h1

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Properties of the Jungle Equilibrium (3)

◮ In the second round all agents, except agent 1, meet:

R2 = {2, 3, 4}

◮ Given the power relationship then:

2 appropriates h4 clearly then 4 is left with h1

◮ In the third round all agents, except agent 1 and 2, meet:

R3 = {3, 4}

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Properties of the Jungle Equilibrium (4)

◮ Given the power relationship then:

3 appropriates h1 clearly then 4 is left with h3

◮ The jungle equilibrium allocation is:

ˆ e = (h2, h4, h1, h3)

◮ The allocation ˆ

e is Pareto efficient: agents 1, 2 and 3 obtain the best house, given their preferences, so clearly it is impossible to reallocate any house to agent 4, the only one whose utility can be improved.

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Properties of the Jungle Equilibrium (5)

Theorem (Second Welfare Theorem for the jungle)

Suppose that the jungle is smooth. In the exchange economy in which wi = ˆ zi, i ∈ {1, . . . , N} there exists a sequence of price vectors pn such that, for every agent i, the sequence of demands of agent i given pn converges to ˆ zi.

◮ When agents have the same preferences and consumption

sets, and prices supporting a jungle equilibrium as a competitive equilibrium exist, the relationship between power and wealth is unambiguous.

◮ The value of an agent’s jungle equilibrium bundle increases

with his strength pz1 ≥ p z2 ≥ . . . ≥ p zN

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Enforcement

◮ Two individuals A and B. We assume that A is more powerful

than B: A SB

◮ When A and B meet A, takes her own decisions as well as the

decisions of the less powerful one, B.

◮ This is a richer definition of power.

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Enforcement (2)

Timing:

◮ Each individual simultaneously and independently decides

whether to show at a predetermined meeting place or whether to hide: {S, H}

◮ If a meeting occurs both A and B decide whether to given the

endowment to the other individual or whether not to give it: {G, NG}

◮ Given the power ranking above these decisions are both taken

by A.

◮ If a meeting does not occur then each individual consumes

his/her endowment.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 9 February 2015 27 / 44

Enforcement (3)

◮ A meeting occurs if and only if both individuals show, S. ◮ Alternatively it also occurs if the more powerful individual, A,

shows, S, and the less powerful individual, B hides, H.

◮ In this case, however, a meeting occurs only with 1/10

probability.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 9 February 2015 28 / 44

Enforcement (4)

◮ Assume there exist gains from trade. ◮ If A consumes his/her endowment payoff is 10. ◮ If A consumes B’s endowment payoff is 100. ◮ Symmetrically for B.

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Equilibrium without Enforcement

◮ Using backward induction the date 1 game is (A is the row

player): A \B G NG G 100, 100 0, 110 NG 110, 0 10, 10

◮ Clearly, it is a dominant strategy for A to choose NG for

herself and to choose G for B.

◮ In other words the outcome is (NG, G) with payoffs (110, 0).

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 9 February 2015 30 / 44

Equilibrium without Enforcement (2)

◮ Moving now to date 0 the game faced by both player is:

A \B S H S 110, 0 20, 9 H 10, 10 10, 10

◮ It is a dominant strategy for A to choose S. ◮ It is a best reply for B to choose H if A chooses S and B is

indifferent between S and H if A chooses H.

◮ In other words, the outcome is then (S, H) with payoffs

(20, 9).

◮ This is an inefficient outcome: social surplus is 29 instead of

200.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 9 February 2015 31 / 44

The Enforcer

◮ Assume now that there exist a third player E, the enforcer,

such that E SA SB

◮ Assume that E is an individual whose preferences are close to

a bliss point corresponding to a consumption of α.

◮ Assume that at the start of the game both A and B have the

additional choice to show and invite E to the meeting: SI.

◮ In other words, the strategy space of A and B is now:

{SI, S, H}.

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Equilibrium with Enforcement

◮ Consider the last period. If both A and B choose S at date 0

then the game is the same as above, equivalently if the

• utcome at date 0 is (S, H), (H, S) and (H, H).

◮ If instead at least one individual between A and B chooses SI

then the game faced by A and B is: A \B G NG G 100 − min{α/2, 100}, 100 − min{α/2, 100} 0, 110 − min{α, 110} NG 110 − min{α, 110}, 10 − min{α/2, 10}, 10 − min{α/2, 10}

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Equilibrium with Enforcement (2)

◮ Notice that we are assuming that neither A nor B can be

made worse off than a payoff of 0.

◮ We are also assuming that if possible E expropriates both

individuals in equal share up to its bliss point.

◮ We also assume that when E is present the remaining players

can engage in a chain of expropriations.

◮ That is, E expropriates A and/or B and A expropriates B.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 9 February 2015 34 / 44

Equilibrium with Enforcement (3)

◮ If α < 20 then the unique outcome is the same as before

(NG, G) with payoffs (110 − α, 0): 110 − α > 100 − α/2

◮ If instead α > 20 then the unique outcome is (G, G) with

payoffs (100 − min{α/2, 100}, 100 − min{α/2, 100})

◮ Finally, if α = 20 then both outcomes above are possible.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 9 February 2015 35 / 44

Equilibrium with Enforcement (4)

◮ Consider now date 0. We can distinguish again two cases. ◮ If α > 20 then the game faced by A and B is:

SI S H SI 100 − min{α/2, 100} 100 − min{α/2, 100} 100 − min{α/2, 100} 100 − min{α/2, 100} 0, 9 S 100 − min{α/2, 100} 100 − min{α/2, 100} 110, 0 20, 9 H 9, 0 10, 10 10, 10

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 9 February 2015 36 / 44

Equilibrium with Enforcement (5)

◮ If 20 < α < 182 then the equilibrium outcomes are: (SI, SI)

and (S, SI) both with payoffs (100 − α/2, 100 − α/2)

◮ If S lexicographically dominates SI for A then (S, SI) is the

unique equilibrium outcome.

◮ The unique equilibrium outcome is for the less powerful

individual B to invite the most powerful individual E to the meeting so as to enforce trade.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 9 February 2015 37 / 44

Equilibrium with Enforcement (6)

◮ This is clearly a Pareto-superior outcome to the one achieved

in the absence of E.

◮ The equilibrium social surplus is 200. ◮ If instead α > 182 then the unique outcome is (S, H) as in

the absence of E.

◮ This is also the unique outcome if α < 20.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 9 February 2015 38 / 44

Externality

◮ Consider an environment with three agents A, B and C such

that A SB SC

◮ Assume that A’s ability to expropriate is unbounded. ◮ Assume that all three agents have an endowment equal to 10. ◮ Moreover, each agent has the possibility to use a protection

technology H (the ability to hide) that, comes at a cost κ.

◮ The cost κ is waisted, however, the technology guarantees full

protection from expropriation.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 9 February 2015 39 / 44

Externality (2)

◮ The game played by B (rows) and C (columns), we maintain

that the A player expropriates all players that choose strategy S, is: B \C S H S 0, 0 0, 10 − κ H 10 − κ, 0 10 − κ, 10 − κ

◮ If the protection technology is worth using κ < 10 then it is a

dominant strategy for both B and C to choose H.

◮ The payoffs are (10−κ, 10−κ, 10) (where the third element is

A’s payoff) and the social surplus in this case is W = 30 − 2κ.

◮ Clearly A exercises an externality of size κ on both B and C.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 9 February 2015 40 / 44

Externality (3)

◮ Assume that prior to the meeting and the choice of H and S

either B or C can invite to the meeting a fourth agent F that is more powerful than everybody else F SA

◮ As before we assume that F has a bliss point at α and has no

endowment.

◮ Assume further that if given the opportunity agent F

expropriates in decreasing order of power until he reaches his bliss point.

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 9 February 2015 41 / 44

Externality (4)

◮ Assume F is invited then the payoffs of all three remaining

players are (A matrix player), when α ≤ 10: B \C S H S 10, 10, 10 − α 10, 10 − κ, 10 − α H 10 − κ, 10, 10 − α 10 − κ, 10 − κ, 10 − α S B \C S H S 10 − α, 10, 10 − κ 10 − α, 10 − κ, 10 − κ H 10 − κ, 10 − α, 10 − κ 10 − κ, 10 − κ, 10 − κ H

Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 9 February 2015 42 / 44

Externality (5)

◮ Notice that if α < κ the unique equilibrium is (S, S, S) with

payoffs (10, 10, 10 − α, α) (the fourth element is S’s payoff) and total surplus W = 30.

◮ Notice that when α < κ both B and C find strictly convenient

to invite F.

◮ In other words, the availability of an agent F with a bliss

point α < κ eliminates the externality.

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Externality (6)

◮ If α > κ then the unique equilibrium is (H, H, H) with payoffs

(10 − κ, 10 − κ, 10 − κ, 0) and surplus W = 30 − 3κ.

◮ The availability of an agent F with a bliss point α > κ is

counterproductive: it extends the externality to the third agent A and further reduces the overall surplus.

◮ Finally notice that the range of agents F that would eliminate

the externality—the range of values of α such that the unique equilibrium is (S, S, S)—increases as the overall effect of the externality increases (κ increases).

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