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 others. 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. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 9 February 2015 3 / 44
Allocation of Houses ◮ Set of agents: I = { 1 , 2 , 3 , 4 } ◮ Set of houses (indivisible commodities): H = { h 1 , h 2 , h 3 , h 4 } . ◮ Assume that the initial allocation of houses is: h = ( h 1 , h 2 , h 3 , h 4 ), in other words agent i owns house h i , where i ∈ { 1 , 2 , 3 , 4 } . ◮ Clearly an arbitrary allocation of houses is a permutation of h , e.g. g = ( h 2 , h 1 , h 3 , h 4 ). ◮ 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 over houses ≻ i and strictly prefers having a house to no house. ◮ For example, we take the preferences to be: 1 2 3 4 h 2 h 4 h 1 h 2 h 1 h 3 h 3 h 4 h 3 h 2 h 2 h 1 h 4 h 1 h 4 h 3 ∅ ∅ ∅ ∅ ◮ In other words, for agent 1: u 1 ( h 2 ) > u 1 ( h 1 ) > u 1 ( h 3 ) > u 1 ( h 4 ) > 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 one 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. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 9 February 2015 6 / 44
Competitive Equilibrium (2) Consider only the first choice for each agent in the list of preferences: 1 2 3 4 h 2 h 4 h 1 h 2 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 h 2 and h 4 from the preference list above: 1 3 h 1 h 1 h 3 h 3 ∅ ∅ 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: p h 2 = p h 4 > p h 1 > p h 3 The competitive equilibrium price vector is: p = ( p h 1 , p h 2 , p h 3 , p h 4 ) ¯ . The associated competitive allocation is: w = ( h 1 , h 4 , h 3 , h 2 ) ¯ At these prices only one trade occurs between person 2 and 4. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 9 February 2015 9 / 44
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 Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 9 February 2015 11 / 44
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. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 9 February 2015 12 / 44
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 = ( h 1 , h 4 , h 3 , h 2 ). ◮ 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 Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 9 February 2015 14 / 44
Equilibrium in the Jungle (4) ◮ Consider now the following jungle equilibrium where preferences are: 2 4 1 3 h 4 h 2 h 2 h 1 h 3 h 4 h 1 h 3 h 2 h 1 h 3 h 2 h 1 h 3 h 4 h 4 ∅ ∅ ∅ ∅ ◮ In the first round all agents meet: R 1 = { 1 , 2 , 3 , 4 } ◮ Given that 2 is the strongest agent then: 2 appropriates h 4 clearly then 4 is left with h 2 . Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 9 February 2015 15 / 44
Equilibrium in the Jungle (5) ◮ In the second round agents 1, 3 and 4 meet: R 2 = { 1 , 3 , 4 } ◮ Given that 4 is the strongest among these agents then: 4 keeps h 2 and the other agents keep their houses. ◮ Finally in the last round agents 1 and 3 meet: R 3 = { 1 , 3 } ◮ Since 1 is the strongest agent then: 1 keeps h 1 while 3 keeps h 3 ◮ In other words the jungle equilibrium allocation is: ¯ e = ( h 1 , h 4 , h 3 , h 2 ) Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 9 February 2015 16 / 44
Equilibrium in the Jungle (6) ◮ Consider now a model with K commodities and a set of agents, I = { 1 , . . . , N } . ◮ The aggregate bundle ω = ( ω 1 , . . . , ω K ) 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 R K + 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: x i ∈ X i , y ∈ R K + , and y ≤ x i implies that y ∈ X i . Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 9 February 2015 17 / 44
Equilibrium in the Jungle (7) ◮ A feasible allocation is a vector of non-negative bundles z = ( z 0 , z 1 , . . . , z N ) such that: N z 0 ∈ R K z i ∈ R K z i = ω � + , + , ∀ i ∈ { 1 , . . . , N } , i =0 ◮ The bundle z 0 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. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 9 February 2015 18 / 44
Existence of Jungle Equilibrium Definition A jungle equilibrium is a feasible allocation z such that there are no agents i and j , i S j , and a bundle y i ∈ X i such that y i ≤ z i + z 0 or y i ≤ z i + z j and y i ≻ i z i . Result (Piccione and Rubinstein 2007) A jungle equilibrium exists. z 1 be z 0 , ˆ z 1 , . . . , ˆ z N ). Let ˆ Proof: Construct an allocation ˆ z = (ˆ one of agent 1’s best bundles in { x 1 ∈ X 1 | x 1 ≤ w } . z i to be one of the agent i ’s best bundles in Define inductively ˆ i − 1 x i ∈ X 1 | x i ≤ w − z 0 = w − � N � z j z j . ˆ and ˆ j =1 ˆ j =1 Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 9 February 2015 19 / 44
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