Approximability of Economic Equilibrium in Housing Markets with Duplicate Houses Katarína Cechlárová, PF UPJŠ Košice and Eva Jelínková, MFF UK Praha Economic equilibrium in housing markets K. Cechlárová & E. Jelínková
Basic notions Definition A housing market is a quadruple M = ( A, H, ω, P ) where A is a set of n agents, H is a set of m house types ω : A → H is the endowment function preference profile P is an n -tuple of agents’ preferences, i.e. linearly ordered lists P ( a ) of acceptable house types Example. A = { a 1 , a 2 , . . . , a 7 } ; H = { h 1 , h 2 , h 3 , h 4 } ω ( a 1 ) = h 1 ; P ( a 1 ) : h 4 , h 3 , h 2 , h 1 ω ( a 2 ) = h 4 ; P ( a 2 ) : ( h 1 , h 3 ) , h 4 ω ( a 3 ) = h 1 ; P ( a 3 ) : h 2 , h 4 , h 1 ω ( a 4 ) = h 2 ; P ( a 4 ) : ( h 1 , h 3 ) , h 4 , h 2 ω ( a 5 ) = h 2 ; P ( a 5 ) : h 4 , h 1 , h 2 ω ( a 6 ) = h 3 ; P ( a 6 ) : h 4 , h 3 ω ( a 7 ) = h 4 ; P ( a 7 ) : h 3 , h 1 , h 4 Economic equilibrium in housing markets K. Cechlárová & E. Jelínková
Basic notions Definition A housing market is a quadruple M = ( A, H, ω, P ) where A is a set of n agents, H is a set of m house types ω : A → H is the endowment function preference profile P is an n -tuple of agents’ preferences, i.e. linearly ordered lists P ( a ) of acceptable house types Example. A = { a 1 , a 2 , . . . , a 7 } H = { h 1 , h 2 , h 3 , h 4 } ω ( a 1 ) = h 1 ; P ( a 1 ) : h 4 , h 3 , h 2 , h 1 acceptable houses ω ( a 2 ) = h 4 ; P ( a 2 ) : ( h 1 , h 3 ) , h 4 ties ω ( a 3 ) = h 1 ; P ( a 3 ) : h 2 , h 4 , h 1 strict preferences ω ( a 4 ) = h 2 ; P ( a 4 ) : ( h 1 , h 3 ) , h 4 , h 2 trichotomous preferences ω ( a 5 ) = h 2 ; P ( a 5 ) : h 4 , h 1 , h 2 ω ( a 6 ) = h 3 ; P ( a 6 ) : h 4 , h 3 ω ( a 7 ) = h 4 ; P ( a 7 ) : h 3 , h 1 , h 4 Economic equilibrium in housing markets K. Cechlárová & E. Jelínková
Basic notions Definition A housing market is a quadruple M = ( A, H, ω, P ) where A is a set of n agents, H is a set of m house types ω : A → H is the endowment function preference profile P is an n -tuple of agents’ preferences, i.e. linearly ordered lists P ( a ) of acceptable house types Example. A = { a 1 , a 2 , . . . , a 7 } H = { h 1 , h 2 , h 3 , h 4 } ω ( a 1 ) = h 1 ; P ( a 1 ) : ( h 4 , h 3 , h 2 ) , h 1 trichotomous preferences ω ( a 2 ) = h 4 ; P ( a 2 ) : ( h 1 , h 3 ) , h 4 each agent has: ω ( a 3 ) = h 1 ; P ( a 3 ) : ( h 2 , h 4 ) , h 1 1. better house types ω ( a 4 ) = h 2 ; P ( a 4 ) : ( h 1 , h 3 , h 4 ) , h 2 2. type of his own house ω ( a 5 ) = h 2 ; P ( a 5 ) : ( h 4 , h 1 ) , h 2 3. unacceptable houses ω ( a 6 ) = h 3 ; P ( a 6 ) : h 4 , h 3 ω ( a 7 ) = h 4 ; P ( a 7 ) : ( h 3 , h 1 ) , h 4 Economic equilibrium in housing markets K. Cechlárová & E. Jelínková
Further notation Definition A function x : A → H is an allocation if there exists a bijection π on A such that x ( a ) = ω ( π ( a )) for each a ∈ A . Each allocation consists of trading cycles ω ( a 1 ) = h 1 ; P ( a 1 ) : h 4 , h 3 , h 2 , h 1 take trading cycles ω ( a 2 ) = h 4 ; P ( a 2 ) : ( h 1 , h 3 ) , h 4 ( a 1 , a 7 , a 6 , a 2 )( a 3 , a 4 , a 5 ) ω ( a 3 ) = h 1 ; P ( a 3 ) : h 2 , h 4 , h 1 this means ω ( a 4 ) = h 2 ; P ( a 4 ) : ( h 1 , h 3 ) , h 4 , h 2 x ( a 1 ) = h 4 ; ω ( a 5 ) = h 2 ; P ( a 5 ) : h 4 , h 1 , h 2 x ( a 7 ) = h 3 ; ω ( a 6 ) = h 3 ; P ( a 6 ) : h 4 , h 3 x ( a 6 ) = h 4 ; ω ( a 7 ) = h 4 ; P ( a 7 ) : h 3 , h 1 , h 4 x ( a 2 ) = h 1 etc. Economic equilibrium in housing markets K. Cechlárová & E. Jelínková
Economic equilibrium Definition A pair ( p, x ) , where p : H → R is a price function and x is an allocation on A is an economic equilibrium for market M if for each a ∈ A , house x ( a ) is of type that is among the most preferred house types in his budget set, i.e. S = B a ( p ) = { h ∈ H ; p ( h ) ≤ p ( ω ( a )) } . Lema If ( p, x ) is an economic equilibrium for market M then p ( x ( a )) = p ( ω ( a )) for each a ∈ A . Economic equilibrium in housing markets K. Cechlárová & E. Jelínková
Example: equilibrium ω ( a 1 ) = h 1 ; P ( a 1 ) : h 4 , h 3 , h 2 , h 1 ω ( a 2 ) = h 4 ; P ( a 2 ) : ( h 1 , h 3 ) , h 4 ω ( a 3 ) = h 1 ; P ( a 3 ) : h 2 , h 4 , h 1 ω ( a 4 ) = h 2 ; P ( a 4 ) : ( h 1 , h 3 ) , h 4 , h 2 ω ( a 5 ) = h 2 ; P ( a 5 ) : h 4 , h 1 , h 2 ω ( a 6 ) = h 3 ; P ( a 6 ) : h 4 , h 3 ω ( a 7 ) = h 4 ; P ( a 7 ) : h 3 , h 1 , h 4 Take p ( h j ) = p for all j and ( a 1 , a 7 , a 6 , a 2 )( a 3 , a 4 , a 5 ) Not equilibrium, since x ( a 5 ) = h 1 and this is the second-choice house Economic equilibrium in housing markets K. Cechlárová & E. Jelínková
Example: equilibrium ω ( a 1 ) = h 1 ; P ( a 1 ) : h 4 , h 3 , h 2 , h 1 ω ( a 2 ) = h 4 ; P ( a 2 ) : ( h 1 , h 3 ) , h 4 ω ( a 3 ) = h 1 ; P ( a 3 ) : h 2 , h 4 , h 1 ω ( a 4 ) = h 2 ; P ( a 4 ) : ( h 1 , h 3 ) , h 4 , h 2 ω ( a 5 ) = h 2 ; P ( a 5 ) : h 4 , h 1 , h 2 ω ( a 6 ) = h 3 ; P ( a 6 ) : h 4 , h 3 ω ( a 7 ) = h 4 ; P ( a 7 ) : h 3 , h 1 , h 4 Take p ( h j ) = p for all j and ( a 1 , a 7 , a 6 , a 2 )( a 3 , a 4 , a 5 ) Not equilibrium, since x ( a 5 ) = h 1 and this is the second-choice house Observation: In this example there is no equilibrium with equal prices, as demand for houses of type h 4 is 3, while the supply is only 2. Economic equilibrium in housing markets K. Cechlárová & E. Jelínková
Brief history Walras 1874: notion of equilibrium Arrow, Debreu 1954: notion of exchange economy equilibrium exist if commodites are infinitely divisible Deng, Papadimitriou, Safra 2002: if commodities are indivisible, decision about the equilibrium existence is NPC Shapley, Scarf 1974: housing market m = n ; each house different Gale 1974: proof of equilibrium existence by TTC algorithm Economic equilibrium in housing markets K. Cechlárová & E. Jelínková
Brief history Walras 1874: notion of equilibrium Arrow, Debreu 1954: notion of exchange economy equilibrium exist if commodites are infinitely divisible Deng, Papadimitriou, Safra 2002: if commodities are indivisible, decision about the equilibrium existence is NPC Shapley, Scarf 1974: housing market m = n ; each house different Gale 1974: proof of equilibrium existence by TTC algorithm Economic equilibrium in housing markets K. Cechlárová & E. Jelínková
Brief history Walras 1874: notion of equilibrium Arrow, Debreu 1954: notion of exchange economy equilibrium exist if commodites are infinitely divisible Deng, Papadimitriou, Safra 2002: if commodities are indivisible, decision about the equilibrium existence is NPC Shapley, Scarf 1974: housing market m = n ; each house different Gale 1974: proof of equilibrium existence by TTC algorithm Economic equilibrium in housing markets K. Cechlárová & E. Jelínková
Brief history Walras 1874: notion of equilibrium Arrow, Debreu 1954: notion of exchange economy equilibrium exist if commodites are infinitely divisible Deng, Papadimitriou, Safra 2002: if commodities are indivisible, decision about the equilibrium existence is NPC Shapley, Scarf 1974: housing market m = n ; each house different Gale 1974: proof of equilibrium existence by TTC algorithm Economic equilibrium in housing markets K. Cechlárová & E. Jelínková
Brief history Walras 1874: notion of equilibrium Arrow, Debreu 1954: notion of exchange economy equilibrium exist if commodites are infinitely divisible Deng, Papadimitriou, Safra 2002: if commodities are indivisible, decision about the equilibrium existence is NPC Shapley, Scarf 1974: housing market m = n ; each house different Gale 1974: proof of equilibrium existence by TTC algorithm Economic equilibrium in housing markets K. Cechlárová & E. Jelínková
Brief history Walras 1874: notion of equilibrium Arrow, Debreu 1954: notion of exchange economy equilibrium exist if commodites are infinitely divisible Deng, Papadimitriou, Safra 2002: if commodities are indivisible, decision about the equilibrium existence is NPC Shapley, Scarf 1974: housing market m = n ; each house different Gale 1974: proof of equilibrium existence by TTC algorithm Economic equilibrium in housing markets K. Cechlárová & E. Jelínková
Brief history Walras 1874: notion of equilibrium Arrow, Debreu 1954: notion of exchange economy equilibrium exist if commodites are infinitely divisible Deng, Papadimitriou, Safra 2002: if commodities are indivisible, decision about the equilibrium existence is NPC Shapley, Scarf 1974: housing market m = n ; each house different Gale 1974: proof of equilibrium existence by TTC algorithm Economic equilibrium in housing markets K. Cechlárová & E. Jelínková
Brief history Walras 1874: notion of equilibrium Arrow, Debreu 1954: notion of exchange economy equilibrium exist if commodites are infinitely divisible Deng, Papadimitriou, Safra 2002: if commodities are indivisible, decision about the equilibrium existence is NPC Shapley, Scarf 1974: housing market m = n ; each house different Gale 1974: proof of equilibrium existence by TTC algorithm Economic equilibrium in housing markets K. Cechlárová & E. Jelínková
Recommend
More recommend
