Two-sided investments and matching with multi-dimensional cost types - PowerPoint PPT Presentation

Match surplus, costs and ex-ante heterogeneity choosing an investment means choosing an attribute (deterministic investment technology) the sets of possible attribute choices are X (for buyers) and Y (for sellers) generic elements are denoted x and y gross match surplus v ( x , y ) agents are ex-ante heterogeneous characterized by cost types b ∈ B and s ∈ S cost functions c B ( x , b ) and c S ( y , s ) B , S , X and Y are compact metric spaces v : X × Y → R + , c B : X × B → R + and c S : Y × S → R + are continuous for simplicity: unmatched agents create zero surplus the heterogeneous ex-ante populations of buyers and sellers are described by probability measures µ B on B and µ S on S 10 / 33

Match surplus, costs and ex-ante heterogeneity choosing an investment means choosing an attribute (deterministic investment technology) the sets of possible attribute choices are X (for buyers) and Y (for sellers) generic elements are denoted x and y gross match surplus v ( x , y ) agents are ex-ante heterogeneous characterized by cost types b ∈ B and s ∈ S cost functions c B ( x , b ) and c S ( y , s ) B , S , X and Y are compact metric spaces v : X × Y → R + , c B : X × B → R + and c S : Y × S → R + are continuous for simplicity: unmatched agents create zero surplus the heterogeneous ex-ante populations of buyers and sellers are described by probability measures µ B on B and µ S on S µ B , µ S , v , c B and c S are common knowledge at stage 1 10 / 33

Stage 2: The matching market 11 / 33

Stage 2: The matching market Matching is frictionless, utility is transferable: the matching market is a continuum assignment game, described by 11 / 33

Stage 2: The matching market Matching is frictionless, utility is transferable: the matching market is a continuum assignment game, described by the match surplus function v the distributions of buyer and seller attributes that result from agents’ sunk investments: µ X on X and µ Y on Y 11 / 33

Stage 2: The matching market Matching is frictionless, utility is transferable: the matching market is a continuum assignment game, described by the match surplus function v the distributions of buyer and seller attributes that result from agents’ sunk investments: µ X on X and µ Y on Y The possible matchings of µ X and µ Y are the measures π 2 on X × Y with marginal measures µ X and µ Y : π 2 ∈ Π( µ X , µ Y ) 11 / 33

Stage 2: Stable outcomes 12 / 33

Stage 2: Stable outcomes Competition without frictions results in a stable outcome: a stable outcome of ( µ X , µ Y , v ) consists of 12 / 33

Stage 2: Stable outcomes Competition without frictions results in a stable outcome: a stable outcome of ( µ X , µ Y , v ) consists of an efficient, surplus-maximizing matching π ∗ 2 π ∗ � 2 ∈ Π( µ X , µ Y ) attains sup π 2 ∈ Π( µ X ,µ Y ) v d π 2 12 / 33

Stage 2: Stable outcomes Competition without frictions results in a stable outcome: a stable outcome of ( µ X , µ Y , v ) consists of an efficient, surplus-maximizing matching π ∗ 2 π ∗ � 2 ∈ Π( µ X , µ Y ) attains sup π 2 ∈ Π( µ X ,µ Y ) v d π 2 core payoff functions ψ ∗ X : Supp( µ X ) → R and ψ ∗ Y : Supp( µ Y ) → R ψ ∗ Y ( y ) + ψ ∗ X ( x ) = v ( x , y ) on Supp( π ∗ 2 ) for all ( x , y ) ∈ Supp( µ X ) × Supp( µ Y ): ψ ∗ Y ( y ) + ψ ∗ X ( x ) ≥ v ( x , y ) 12 / 33

Stage 2: Stable outcomes Competition without frictions results in a stable outcome: a stable outcome of ( µ X , µ Y , v ) consists of an efficient, surplus-maximizing matching π ∗ 2 π ∗ � 2 ∈ Π( µ X , µ Y ) attains sup π 2 ∈ Π( µ X ,µ Y ) v d π 2 core payoff functions ψ ∗ X : Supp( µ X ) → R and ψ ∗ Y : Supp( µ Y ) → R ψ ∗ Y ( y ) + ψ ∗ X ( x ) = v ( x , y ) on Supp( π ∗ 2 ) for all ( x , y ) ∈ Supp( µ X ) × Supp( µ Y ): ψ ∗ Y ( y ) + ψ ∗ X ( x ) ≥ v ( x , y ) Stable outcomes exist (Gretzky, Ostroy and Zame 1992; Villani 2009) 12 / 33

Stage 2: Stable outcomes Competition without frictions results in a stable outcome: a stable outcome of ( µ X , µ Y , v ) consists of an efficient, surplus-maximizing matching π ∗ 2 π ∗ � 2 ∈ Π( µ X , µ Y ) attains sup π 2 ∈ Π( µ X ,µ Y ) v d π 2 core payoff functions ψ ∗ X : Supp( µ X ) → R and ψ ∗ Y : Supp( µ Y ) → R ψ ∗ Y ( y ) + ψ ∗ X ( x ) = v ( x , y ) on Supp( π ∗ 2 ) for all ( x , y ) ∈ Supp( µ X ) × Supp( µ Y ): ψ ∗ Y ( y ) + ψ ∗ X ( x ) ≥ v ( x , y ) Stable outcomes exist (Gretzky, Ostroy and Zame 1992; Villani 2009) stable outcomes ( π ∗ 2 , ψ ∗ X , ψ ∗ Y ) are equivalent to competitive equilibria (Shapley and Shubik 1971; Gretzky, Ostroy and Zame 1992) 12 / 33

Stage 2: Stable outcomes Competition without frictions results in a stable outcome: a stable outcome of ( µ X , µ Y , v ) consists of an efficient, surplus-maximizing matching π ∗ 2 π ∗ � 2 ∈ Π( µ X , µ Y ) attains sup π 2 ∈ Π( µ X ,µ Y ) v d π 2 core payoff functions ψ ∗ X : Supp( µ X ) → R and ψ ∗ Y : Supp( µ Y ) → R ψ ∗ Y ( y ) + ψ ∗ X ( x ) = v ( x , y ) on Supp( π ∗ 2 ) for all ( x , y ) ∈ Supp( µ X ) × Supp( µ Y ): ψ ∗ Y ( y ) + ψ ∗ X ( x ) ≥ v ( x , y ) Stable outcomes exist (Gretzky, Ostroy and Zame 1992; Villani 2009) stable outcomes ( π ∗ 2 , ψ ∗ X , ψ ∗ Y ) are equivalent to competitive equilibria (Shapley and Shubik 1971; Gretzky, Ostroy and Zame 1992) ψ ∗ X and ψ ∗ Y are continuous 12 / 33

Stage 1: Best replies 13 / 33

Stage 1: Best replies In ex-post contracting equilibrium , agents’ attribute choices must “best-reply” to the correctly anticipated trading possibilities and the equilibrium outcome ( π ∗ 2 , ψ ∗ X , ψ ∗ Y ) of the endogenous market ( µ X , µ Y , v ) that results from others’ sunk investments. In particular, if x ∈ Supp( µ X ) is an equilibrium investment of type b , then x must satisfy ψ ∗ x ′ ∈ X , y ∈ Supp( µ Y ) ( v ( x ′ , y ) − ψ ∗ Y ( y ) − c B ( x ′ , b )) X ( x ) − c B ( x , b ) = max if y ∈ Supp( µ Y ) is an equilibrium investment of type s , then y must satisfy ψ ∗ Y ( y ) − c S ( y , s ) = y ′ ∈ Y , x ∈ Supp( µ X ) ( v ( x , y ′ ) − ψ ∗ max X ( x ) − c S ( y ′ , s )) 13 / 33

Ex-post contracting equilibrium Formal definition 14 / 33

Ex-post contracting equilibrium Formal definition Definition An ex-post contracting equilibrium is a tuple (( β, σ, π 1 ) , ( π ∗ 2 , ψ ∗ X , ψ ∗ Y )), in which ( β, σ, π 1 ) is a regular investment profile and ( π ∗ 2 , ψ ∗ X , ψ ∗ Y ) is a stable and feasible bargaining outcome for ( µ X , µ Y , v ), such that for all ( b , s ) ∈ Supp( π 1 ) it holds: ψ ∗ X ( β ( b , s )) − c B ( β ( b , s ) , b ) x ′ ∈ X , y ∈ Supp( µ Y ) ( v ( x ′ , y ) − ψ ∗ Y ( y ) − c B ( x ′ , b )) =: r B ( b ) , = max ψ ∗ Y ( σ ( b , s )) − c S ( σ ( b , s ) , s ) y ′ ∈ Y , x ∈ Supp( µ X ) ( v ( x , y ′ ) − ψ ∗ X ( x ) − c S ( y ′ , s )) =: r S ( s ) . = max 14 / 33

The efficiency benchmark 15 / 33

The efficiency benchmark The maximal net surplus that a pair ( b , s ) can generate is w ( b , s ) = x ∈ X , y ∈ Y v ( x , y ) − c B ( x , b ) − c S ( y , s ) max jointly optimal attributes ( x ∗ ( b , s ) , y ∗ ( b , s )) exist for all ( b , s ) w is continuous 15 / 33

The efficiency benchmark The maximal net surplus that a pair ( b , s ) can generate is w ( b , s ) = x ∈ X , y ∈ Y v ( x , y ) − c B ( x , b ) − c S ( y , s ) max jointly optimal attributes ( x ∗ ( b , s ) , y ∗ ( b , s )) exist for all ( b , s ) w is continuous The stable outcomes ( π ∗ 1 , ψ ∗ B , ψ ∗ S ) of the assignment game ( µ B , µ S , w ) provide the benchmark of ex-ante efficiency 15 / 33

The efficiency benchmark The maximal net surplus that a pair ( b , s ) can generate is w ( b , s ) = x ∈ X , y ∈ Y v ( x , y ) − c B ( x , b ) − c S ( y , s ) max jointly optimal attributes ( x ∗ ( b , s ) , y ∗ ( b , s )) exist for all ( b , s ) w is continuous The stable outcomes ( π ∗ 1 , ψ ∗ B , ψ ∗ S ) of the assignment game ( µ B , µ S , w ) provide the benchmark of ex-ante efficiency they describe how agents would match and divide net surplus if buyers and sellers could bargain in a frictionless market and write complete contracts before they invest, so that partners choose jointly optimal attributes 15 / 33

Efficient equilibria 16 / 33

Efficient equilibria Result Every stable outcome ( π ∗ 1 , ψ ∗ B , ψ ∗ S ) of ( µ B , µ S , w ) can be supported by an ex-post contracting equilibrium. In particular, an efficient equilibrium exists. 16 / 33

Two manifestations of inefficiency 17 / 33

Two manifestations of inefficiency Buyers and sellers may be mismatched from an ex-ante perspective the matching of cost types that is associated with the equilibrium investment behavior and the matching of attributes is not efficient for the benchmark assignment game ( µ B , µ S , w ) 17 / 33

Two manifestations of inefficiency Buyers and sellers may be mismatched from an ex-ante perspective the matching of cost types that is associated with the equilibrium investment behavior and the matching of attributes is not efficient for the benchmark assignment game ( µ B , µ S , w ) There may be inefficiency of joint investments agents’ attributes are not jointly optimal in a strictly positive mass of matches that arise in equilibrium 17 / 33

Full appropriation games 18 / 33

Full appropriation games Consider the following complete information, “full appropriation” (FA) game between a buyer of type b and a seller of type s strategy spaces are X and Y payoffs are v ( x , y ) − c B ( x , b ) and v ( x , y ) − c S ( y , s ) 18 / 33

Full appropriation games Consider the following complete information, “full appropriation” (FA) game between a buyer of type b and a seller of type s strategy spaces are X and Y payoffs are v ( x , y ) − c B ( x , b ) and v ( x , y ) − c S ( y , s ) Lemma The attributes of a buyer of type b and a seller of type s who are matched in equilibrium must be a Nash equilibrium (NE) of the FA game between them. 18 / 33

Technological multiplicity 19 / 33

Technological multiplicity Proposition Assume that for all b ∈ Supp( µ B ) and s ∈ Supp( µ S ), the FA game between b and s has a unique NE. Then ex-post contracting equilibria cannot feature inefficiency of joint investments. 19 / 33

Technological multiplicity Proposition Assume that for all b ∈ Supp( µ B ) and s ∈ Supp( µ S ), the FA game between b and s has a unique NE. Then ex-post contracting equilibria cannot feature inefficiency of joint investments. Note jointly optimal attributes x ∗ ( b , s ) and y ∗ ( b , s ) are always a NE of the FA game between b and s , as they maximize v ( x , y ) − c B ( x , b ) − c S ( y , s ) 19 / 33

Technological multiplicity Proposition Assume that for all b ∈ Supp( µ B ) and s ∈ Supp( µ S ), the FA game between b and s has a unique NE. Then ex-post contracting equilibria cannot feature inefficiency of joint investments. Note jointly optimal attributes x ∗ ( b , s ) and y ∗ ( b , s ) are always a NE of the FA game between b and s , as they maximize v ( x , y ) − c B ( x , b ) − c S ( y , s ) Definition An environment displays technological multiplicity if FA games have more than one pure strategy NE for some ( b , s ) ∈ Supp( µ B ) × Supp( µ S ). 19 / 33

The 1-d supermodular framework 20 / 33

The 1-d supermodular framework Condition (1dS) Let X \ { x ∅ } , Y \ { y ∅ } , B \ { b ∅ } , S \ { s ∅ } ⊂ R + . Assume that v is strictly supermodular in ( x , y ), c B is strictly submodular in ( x , b ), and c S is strictly submodular in ( y , s ). 20 / 33

The 1-d supermodular framework Condition (1dS) Let X \ { x ∅ } , Y \ { y ∅ } , B \ { b ∅ } , S \ { s ∅ } ⊂ R + . Assume that v is strictly supermodular in ( x , y ), c B is strictly submodular in ( x , b ), and c S is strictly submodular in ( y , s ). Lemma Let Condition 1dS hold. Then the induced matching of buyer and seller cost types is positively assortative in every ex-post contracting equilibrium. Mismatch is impossible. 20 / 33

The 1-d supermodular framework Condition (1dS) Let X \ { x ∅ } , Y \ { y ∅ } , B \ { b ∅ } , S \ { s ∅ } ⊂ R + . Assume that v is strictly supermodular in ( x , y ), c B is strictly submodular in ( x , b ), and c S is strictly submodular in ( y , s ). Lemma Let Condition 1dS hold. Then the induced matching of buyer and seller cost types is positively assortative in every ex-post contracting equilibrium. Mismatch is impossible. equilibrium attribute choices are increasing in type an equilibrium attribute x of type b must belong to � max y ∈ Supp( µ Y ) ( v ( x ′ , y ) − ψ ∗ Y ( y )) − c B ( x ′ , b ) � argmax x ′ ∈ X the matching of attributes is positively assortative 20 / 33

Mismatch without technological multiplicity 21 / 33

Mismatch without technological multiplicity Problem: beyond the 1-d supermodular framework, it is a priori unclear which matchings of buyers and sellers can occur in equilibrium 21 / 33

Mismatch without technological multiplicity Problem: beyond the 1-d supermodular framework, it is a priori unclear which matchings of buyers and sellers can occur in equilibrium An intuition for cases without technological multiplicity 21 / 33

Mismatch without technological multiplicity Problem: beyond the 1-d supermodular framework, it is a priori unclear which matchings of buyers and sellers can occur in equilibrium An intuition for cases without technological multiplicity equilibrium partners have jointly optimal attributes, and ( x ∗ ( b , s ) , y ∗ ( b , s )) is continuous on Supp( µ B ) × Supp( µ S ) 21 / 33

Mismatch without technological multiplicity Problem: beyond the 1-d supermodular framework, it is a priori unclear which matchings of buyers and sellers can occur in equilibrium An intuition for cases without technological multiplicity equilibrium partners have jointly optimal attributes, and ( x ∗ ( b , s ) , y ∗ ( b , s )) is continuous on Supp( µ B ) × Supp( µ S ) any attribute choice displays the preparation for the intended match, but it also strongly reflects the agent’s own type 21 / 33

Mismatch without technological multiplicity Problem: beyond the 1-d supermodular framework, it is a priori unclear which matchings of buyers and sellers can occur in equilibrium An intuition for cases without technological multiplicity equilibrium partners have jointly optimal attributes, and ( x ∗ ( b , s ) , y ∗ ( b , s )) is continuous on Supp( µ B ) × Supp( µ S ) any attribute choice displays the preparation for the intended match, but it also strongly reflects the agent’s own type marketed attributes x ∗ ( b , s ) are attractive targets for deviations by agents s ′ not too different from s , similarly for y ∗ ( b , s ) and buyers b ′ 21 / 33

Mismatch without technological multiplicity Problem: beyond the 1-d supermodular framework, it is a priori unclear which matchings of buyers and sellers can occur in equilibrium An intuition for cases without technological multiplicity equilibrium partners have jointly optimal attributes, and ( x ∗ ( b , s ) , y ∗ ( b , s )) is continuous on Supp( µ B ) × Supp( µ S ) any attribute choice displays the preparation for the intended match, but it also strongly reflects the agent’s own type marketed attributes x ∗ ( b , s ) are attractive targets for deviations by agents s ′ not too different from s , similarly for y ∗ ( b , s ) and buyers b ′ profitable deviations at stage 1 must be ruled out by sufficiently high net equilibrium payoffs 21 / 33

Mismatch without technological multiplicity Problem: beyond the 1-d supermodular framework, it is a priori unclear which matchings of buyers and sellers can occur in equilibrium An intuition for cases without technological multiplicity equilibrium partners have jointly optimal attributes, and ( x ∗ ( b , s ) , y ∗ ( b , s )) is continuous on Supp( µ B ) × Supp( µ S ) any attribute choice displays the preparation for the intended match, but it also strongly reflects the agent’s own type marketed attributes x ∗ ( b , s ) are attractive targets for deviations by agents s ′ not too different from s , similarly for y ∗ ( b , s ) and buyers b ′ profitable deviations at stage 1 must be ruled out by sufficiently high net equilibrium payoffs these requirements constrain mismatch if there is some differentiation of agents ex-ante 21 / 33

Mismatch and its constraints in a 2-d bilinear model (I) 22 / 33

Mismatch and its constraints in a 2-d bilinear model (I) The standard bilinear model Let Supp( µ B ) \ { b ∅ } ⊂ R 2 + \ { 0 } , Supp( µ S ) \ { s ∅ } ⊂ R 2 + \ { 0 } and X \ { x ∅ } = Y \ { y ∅ } = R 2 + . Surplus and costs are given by v ( x , y ) = x · y = x 1 y 1 + x 2 y 2 , c B ( x , b ) = x 4 1 + x 4 2 and c S ( y , s ) = y 4 1 + y 4 2 . 1 2 1 2 b 2 b 2 s 2 s 2 22 / 33

Mismatch and its constraints in a 2-d bilinear model (I) The standard bilinear model Let Supp( µ B ) \ { b ∅ } ⊂ R 2 + \ { 0 } , Supp( µ S ) \ { s ∅ } ⊂ R 2 + \ { 0 } and X \ { x ∅ } = Y \ { y ∅ } = R 2 + . Surplus and costs are given by v ( x , y ) = x · y = x 1 y 1 + x 2 y 2 , c B ( x , b ) = x 4 1 + x 4 2 and c S ( y , s ) = y 4 1 + y 4 2 . 1 2 1 2 b 2 b 2 s 2 s 2 FA games have unique non-trivial NE, given by ( x ∗ ( b , s ) , y ∗ ( b , s )) = 1 3 1 3 1 1 3 1 3 �� � � �� b 1 s 4 1 , b 4 2 s 4 4 , b 1 s 4 1 , b 4 2 s 4 4 2 2 2 22 / 33

Mismatch and its constraints in a 2-d bilinear model (I) The standard bilinear model Let Supp( µ B ) \ { b ∅ } ⊂ R 2 + \ { 0 } , Supp( µ S ) \ { s ∅ } ⊂ R 2 + \ { 0 } and X \ { x ∅ } = Y \ { y ∅ } = R 2 + . Surplus and costs are given by v ( x , y ) = x · y = x 1 y 1 + x 2 y 2 , c B ( x , b ) = x 4 1 + x 4 2 and c S ( y , s ) = y 4 1 + y 4 2 . 1 2 1 2 b 2 b 2 s 2 s 2 FA games have unique non-trivial NE, given by ( x ∗ ( b , s ) , y ∗ ( b , s )) = 1 3 1 3 1 1 3 1 3 �� � � �� b 1 s 4 1 , b 4 2 s 4 4 , b 1 s 4 1 , b 4 2 s 4 4 2 2 2 w ( b , s ) = 1 8 ( b 1 s 1 + b 2 s 2 ) 22 / 33

Mismatch and its constraints in a 2-d bilinear model (II) 23 / 33

b b Mismatch and its constraints in a 2-d bilinear model (II) Example 1 Let µ S = a H δ ( s H , s H ) + (1 − a H ) δ ( s L , s L ) , where 0 < s L < s H and 0 < a H < 1. Moreover, µ B = a 1 δ ( b ′ 1 , 0) + a 2 δ (0 , b ′ 2 ) + (1 − a 1 − a 2 ) δ b ∅ , where 0 < a 1 , a 2 , b ′ 1 , b ′ 2 and a 1 + a 2 < 1. Finally, let b ′ 1 > b ′ 2 and a H < a 1 + a 2 . b 2 , s 2 b ( s H , s H ) (0 , b ′ 2 ) b ( s L , s L ) b 1 , s 1 ( b ′ 1 , 0) 23 / 33

b b Mismatch and its constraints in a 2-d bilinear model (II) Example 1 Let µ S = a H δ ( s H , s H ) + (1 − a H ) δ ( s L , s L ) , where 0 < s L < s H and 0 < a H < 1. Moreover, µ B = a 1 δ ( b ′ 1 , 0) + a 2 δ (0 , b ′ 2 ) + (1 − a 1 − a 2 ) δ b ∅ , where 0 < a 1 , a 2 , b ′ 1 , b ′ 2 and a 1 + a 2 < 1. Finally, let b ′ 1 > b ′ 2 and a H < a 1 + a 2 . w (( b 1 , b 2 ) , ( s 1 , s 1 )) = 1 8 ( b 1 + b 2 ) s 1 b 2 , s 2 the ex-ante efficient matching is positively assortative in s 1 and b 1 + b 2 b ( s H , s H ) (0 , b ′ 2 ) b ( s L , s L ) b 1 , s 1 ( b ′ 1 , 0) 23 / 33

b b Mismatch and its constraints in a 2-d bilinear model (II) Example 1 Let µ S = a H δ ( s H , s H ) + (1 − a H ) δ ( s L , s L ) , where 0 < s L < s H and 0 < a H < 1. Moreover, µ B = a 1 δ ( b ′ 1 , 0) + a 2 δ (0 , b ′ 2 ) + (1 − a 1 − a 2 ) δ b ∅ , where 0 < a 1 , a 2 , b ′ 1 , b ′ 2 and a 1 + a 2 < 1. Finally, let b ′ 1 > b ′ 2 and a H < a 1 + a 2 . w (( b 1 , b 2 ) , ( s 1 , s 1 )) = 1 8 ( b 1 + b 2 ) s 1 b 2 , s 2 the ex-ante efficient matching is positively assortative in s 1 and b 1 + b 2 b ( s H , s H ) (0 , b ′ 2 ) b ( s L , s L ) b 1 , s 1 ( b ′ 1 , 0) 23 / 33

b b Mismatch and its constraints in a 2-d bilinear model (II) Example 1 Let µ S = a H δ ( s H , s H ) + (1 − a H ) δ ( s L , s L ) , where 0 < s L < s H and 0 < a H < 1. Moreover, µ B = a 1 δ ( b ′ 1 , 0) + a 2 δ (0 , b ′ 2 ) + (1 − a 1 − a 2 ) δ b ∅ , where 0 < a 1 , a 2 , b ′ 1 , b ′ 2 and a 1 + a 2 < 1. Finally, let b ′ 1 > b ′ 2 and a H < a 1 + a 2 . w (( b 1 , b 2 ) , ( s 1 , s 1 )) = 1 8 ( b 1 + b 2 ) s 1 b 2 , s 2 the ex-ante efficient matching is positively assortative in s 1 and b 1 + b 2 b ( s H , s H ) depicted case: a H < a 1 (0 , b ′ 2 ) b ( s L , s L ) b 1 , s 1 ( b ′ 1 , 0) 23 / 33

b b Mismatch and its constraints in a 2-d bilinear model (II) Example 1 Let µ S = a H δ ( s H , s H ) + (1 − a H ) δ ( s L , s L ) , where 0 < s L < s H and 0 < a H < 1. Moreover, µ B = a 1 δ ( b ′ 1 , 0) + a 2 δ (0 , b ′ 2 ) + (1 − a 1 − a 2 ) δ b ∅ , where 0 < a 1 , a 2 , b ′ 1 , b ′ 2 and a 1 + a 2 < 1. Finally, let b ′ 1 > b ′ 2 and a H < a 1 + a 2 . w (( b 1 , b 2 ) , ( s 1 , s 1 )) = 1 8 ( b 1 + b 2 ) s 1 b 2 , s 2 the ex-ante efficient matching is positively assortative in s 1 and b 1 + b 2 b ( s H , s H ) depicted case: a H < a 1 attributes in the endogenous market: x ∗ (( b ′ 1 , 0) , ( s H , s H )), x ∗ (( b ′ 1 , 0) , ( s L , s L )), (0 , b ′ 2 ) x ∗ ((0 , b ′ 2 ) , ( s L , s L )), y ∗ (( b ′ 1 , 0) , ( s H , s H )), b ( s L , s L ) y ∗ (( b ′ 1 , 0) , ( s L , s L )), y ∗ ((0 , b ′ 2 ) , ( s L , s L )) � � ′ 3 1 1 e.g. x ∗ (( b ′ 1 , 0) , ( s H , s H )) = 2 b 1 s 4 H , 0 4 , � � ′ 1 3 1 y ∗ (( b ′ 1 , 0) , ( s H , s H )) = 2 b 1 s 4 H , 0 4 b 1 , s 1 ( b ′ 1 , 0) 23 / 33

Mismatch and its constraints in a 2-d bilinear model (III) 24 / 33

b b Mismatch and its constraints in a 2-d bilinear model (III) Claim Consider the environment of Example 1. If a H < a 2 , then there is exactly one � sH � 2 3 − 1 b ′ additional, mismatch inefficient equilibrium if and only if 2 sL 2 1 ≥ . sH 3 b ′ sL − 1 Otherwise, only the ex-ante efficient equilibrium exists. b 2 , s 2 b ( s H , s H ) (0 , b ′ 2 ) b ( s L , s L ) b 1 , s 1 ( b ′ 1 , 0) 24 / 33

b b Mismatch and its constraints in a 2-d bilinear model (III) Claim Consider the environment of Example 1. If a H < a 2 , then there is exactly one � sH � 2 3 − 1 b ′ additional, mismatch inefficient equilibrium if and only if 2 sL 2 1 ≥ . sH 3 b ′ sL − 1 Otherwise, only the ex-ante efficient equilibrium exists. b 2 , s 2 b ( s H , s H ) (0 , b ′ 2 ) b ( s L , s L ) b 1 , s 1 ( b ′ 1 , 0) 24 / 33

b b Mismatch and its constraints in a 2-d bilinear model (III) Claim Consider the environment of Example 1. If a H < a 2 , then there is exactly one � sH � 2 3 − 1 b ′ additional, mismatch inefficient equilibrium if and only if 2 sL 2 1 ≥ . sH 3 b ′ sL − 1 Otherwise, only the ex-ante efficient equilibrium exists. b 2 , s 2 b ( s H , s H ) (0 , b ′ 2 ) b ( s L , s L ) b 1 , s 1 ( b ′ 1 , 0) 24 / 33

b b Mismatch and its constraints in a 2-d bilinear model (III) Claim Consider the environment of Example 1. If a H < a 2 , then there is exactly one � sH � 2 3 − 1 b ′ additional, mismatch inefficient equilibrium if and only if 2 sL 2 1 ≥ . sH 3 b ′ sL − 1 Otherwise, only the ex-ante efficient equilibrium exists. b 2 , s 2 attributes in the endogenous market: x ∗ (( b ′ 1 , 0) , ( s L , s L )), x ∗ ((0 , b ′ 2 ) , ( s H , s H )), b ( s H , s H ) x ∗ ((0 , b ′ 2 ) , ( s L , s L )), y ∗ (( b ′ 1 , 0) , ( s L , s L )), y ∗ ((0 , b ′ 2 ) , ( s H , s H )), y ∗ ((0 , b ′ 2 ) , ( s L , s L )) (0 , b ′ 2 ) b ( s L , s L ) b 1 , s 1 ( b ′ 1 , 0) 24 / 33

b b Mismatch and its constraints in a 2-d bilinear model (III) Claim Consider the environment of Example 1. If a H < a 2 , then there is exactly one � sH � 2 3 − 1 b ′ additional, mismatch inefficient equilibrium if and only if 2 sL 1 ≥ 2 . sH 3 b ′ sL − 1 Otherwise, only the ex-ante efficient equilibrium exists. b 2 , s 2 attributes in the endogenous market: x ∗ (( b ′ 1 , 0) , ( s L , s L )), x ∗ ((0 , b ′ 2 ) , ( s H , s H )), b ( s H , s H ) x ∗ ((0 , b ′ 2 ) , ( s L , s L )), y ∗ (( b ′ 1 , 0) , ( s L , s L )), y ∗ ((0 , b ′ 2 ) , ( s H , s H )), y ∗ ((0 , b ′ 2 ) , ( s L , s L )) (0 , b ′ 2 ) ( s H , s H )-sellers have no incentive to b ( s L , s L ) deviate by investing optimally for a match with x ∗ (( b ′ 1 , 0) , ( s L , s L )) if and only if the condition of the Claim holds b 1 , s 1 ( b ′ 1 , 0) 24 / 33

Mismatch and its constraints in a 2-d bilinear model (IV) 25 / 33

Mismatch and its constraints in a 2-d bilinear model (IV) Example 2 Supp( µ S ) = { ( s 1 , s 1 ) | s L ≤ s 1 ≤ s H } , for some s L < s H . µ B is compactly supported in the union of ( R + \ { 0 } ) × { 0 } , { 0 } × ( R + \ { 0 } ) and { b ∅ } . The restrictions of µ B to ( R + \ { 0 } ) × { 0 } and { 0 } × ( R + \ { 0 } ) have interval support. b 2 , s 2 Supp( µ S ) Supp( µ B ) b 1 , s 1 Supp( µ B ) 25 / 33

Mismatch and its constraints in a 2-d bilinear model (IV) Example 2 Supp( µ S ) = { ( s 1 , s 1 ) | s L ≤ s 1 ≤ s H } , for some s L < s H . µ B is compactly supported in the union of ( R + \ { 0 } ) × { 0 } , { 0 } × ( R + \ { 0 } ) and { b ∅ } . The restrictions of µ B to ( R + \ { 0 } ) × { 0 } and { 0 } × ( R + \ { 0 } ) have interval support. b 2 , s 2 result: the only ex-post contracting equilibrium is the ex-ante efficient one cost types are matched positively Supp( µ S ) assortatively in s 1 and b 1 + b 2 Supp( µ B ) b 1 , s 1 Supp( µ B ) 25 / 33

Mismatch and its constraints in a 2-d bilinear model (V) 26 / 33

Mismatch and its constraints in a 2-d bilinear model (V) Remarks 26 / 33

Mismatch and its constraints in a 2-d bilinear model (V) Remarks in Examples 1 and 2, results from the theory of assortative matching can be used to identify the efficient matching and to evaluate whether inefficient equilibria exist 26 / 33

Mismatch and its constraints in a 2-d bilinear model (V) Remarks in Examples 1 and 2, results from the theory of assortative matching can be used to identify the efficient matching and to evaluate whether inefficient equilibria exist this is not feasible in more complex environments 26 / 33

Mismatch and its constraints in a 2-d bilinear model (V) Remarks in Examples 1 and 2, results from the theory of assortative matching can be used to identify the efficient matching and to evaluate whether inefficient equilibria exist this is not feasible in more complex environments Characterization from optimal transport a matching π 1 ∈ Π( µ B , µ S ) is efficient if and only if it is concentrated on a w -cyclically monotone set Definition A set A ⊂ B × S is called w-cyclically monotone if for all K ∈ N , ( b 1 , s 1 ) , ..., ( b K , s K ) ∈ A and s K +1 = s 1 , the following inequality is satisfied. K K � � w ( b i , s i ) ≥ w ( b i , s i +1 ) . i =1 i =1 26 / 33