Hans Gersbach (ETH Zurich) Hans Haller (Virginia Tech) Hideo Konishi (Boston College) October 2012
Content Households and markets Literature Negative result (Gersbach and Haller 2011) An example This paper (assumptions, equilibrium concept) Framework The main result About the theorem Proof by Kakutani s theorem Concluding remarks
Households and Markets The textbook "household" is an individual who makes consumption decisions. A real household may be composed of several individuals, and the members need to decide on - how much each member spends, - members consumption vectors that may generate externalities to each other. Consider market equilibrium with households. Allow formation of households and exit from households.
Literature Chiappori (1988, 1992) - collective decisions (labor supply) within households Gersbach and Haller (2001): GE version - collective decisions within households - market equilibrium Gersbach and Haller (2010, 2011) - collective decisions within households - market equilibrium - formation of households
Negative Result (Gersbach and Haller, JET 2011) As long as the number of consumers is finite, equilibrium may fail to exist: - two commodities example - under each price vector, consumers decide whether or not to form a couple - equilibrium price system depends on the above decisions Even if economy is replicated, equilibrium may fail to exist.
Example A version of Example 3 in Gersbach and Haller (2011): - two commodities I and K & one type each for men and women - in order to make individuals indifferent between forming a couple or being single, need an irrational number of couples - atomless consumers - consumer characteristics endowments: e m = (0, 1), e w = (1, 1). preferences: utility representations u m (x I, x K; w, x K) = ln x K; u m (x I, x K; ) = ln x K; u w ( x I, x K; m, x K) = ρ ln x I + (1 ρ) ln (max{0, x K kx K}) + g; u w ( x I, x K; ) = ρ ln x I + (1 ρ) ln x K with parameters 0 < ρ < 1, 0 < k < 1, 0 < g and the convention ln 0 =.
Example (continued) Continuum Version: N m = 2 and N w = 1. For ρ = k = 1/2, g = ln 2, fraction ξ = 2( 5.5 2) of females is matched in equilibrium. Finite Economy: 3 individuals: ANITA and identical twins or clones, PETER and PAUL It is always ANITA who wants to change households.
Example (continued) PETER ANITA PAUL p p A N I T A P E T E R PAUL
This Paper (Assumptions) Atomless consumers with a finite number of types (observable types and taste types) Households: hetero-sexual marriage or being single (two-sidedness is not essential: roommate problem is OK) Externalities in households: - partner s observable types (appearance and hobbies) - partner s consumption vector - action choices discrete job choice (different wage rates and levels of commitment) child care commitment (or having no child) action choice affects feasibility of household consumption unobservable preferences
This Paper (Equilibrium Concept: Stable Matching Equilibrium) Stability There is no deviation of the following kind: - a member of household terminates the relationship unilaterally - a pair of single man and woman who form a couple by negotiating over pair s commodity consumption with externalities allowances for commodity consumption without externalities pair s actions (job choice, childcare commitment etc.) - a member of a couple and a single person who form a couple by negotiating over the above agendas - members of different couples who form a new couple Market equilibrium with households
Framework There are men and women Observable type sets of men and women: M and W (finite sets) may include task component representative elements m M and w W all possible preference types for M and W: Θ ( Θ m Θ w ) (finite set) representative element θ Θ a man s type (m, θ) M Θ, a woman s type (w, θ) W Θ population (measure) of type (m, θ): N (m,θ) similarly, for (w, θ): N (w, θ)
Externalities in a Household In a household composed of types (m, θ) and (w, θ), they choose where ( x I, x K }{{} consumption, a }{{} action } {{ } (m,θ) s choice ; x I, x }{{ K } consumption, ã }{{} action }{{} (w, θ) s choice K represents the set of commodities with externalities to the partner, K = K I represents the set of commodities without externalities, I = I a A is an unpriced action with externalities (job choice etc.) K, I, and A are all finite sets. )
Utility Function If (m, θ) and (w, θ) form a household: and that is, type (m, θ) cares about u (m,θ) = u (m,θ) (x I, x K, a; w, x K, ã) u (w, θ) = u (w, θ) ( x I, x K, ã; m, x K, a), - his own choice (x I, x K, a) - what observable type his partner has, w - his partner s part of choice ( x K, ã) - θ and x I do not affect (m, θ) s utility If (m, θ) is single, u (m,θ) = u (m,θ) (x I, x K, a; )
Consumption Sets and Endowments type (m, θ) s consumption set and endowment depend only on m M (not on θ Θ) his action (possibly job choice) affects his consumption set and endowment j J K is leisure associated with job j (wage rates can differ) j(a) J for all a A, with a A j(a) = J his endowment is: e m (a) = (e m 1 (a),..., em I (a); 0,..., 0, e m }{{} j(a) (a), 0,..., 0; e m I+J+1 }{{} (a),..., em I+K(a) ), }{{} i I j(a) J k K\J his consumption set is: X m (a) R I + {0}... [0, e m j(a) (a)]... {0} }{{}}{{} i I j(a) J R I+K J + }{{} k K\J.
Matching A continuum of atomless pairs and singles: sets of household types are (actions taken are included) - Γ C = M Θ A W Θ A, - Γ M = M Θ A, Γ W = W Θ A, and - Γ = Γ C Γ M Γ W. A matching is a mapping µ : Γ R + such that µ(γ) is the Lebesgue measure of type γ Γ. A matching µ is feasible if we have: (w, θ,a,ã) W Θ A A µ(m, θ, a; w, θ, ã) + a A µ(m, θ, a) = N(m,θ) for all (m, θ) M Θ. (m,θ,a,ã) M Θ A A µ(w, θ, ã; m, θ, a) + ã A µ(w, θ, ã) = N (w, θ) for all (w, θ) W Θ. measurement consistency (Kaneko and Wooders, 1986): measures add up nicely
Production Simple CRS convex and closed production technology Y R I+K with free disposal, Y + R I+K Y
Feasible Allocations A symmetric household consumption allocation x assigns each type of household γ Γ a household consumption vector - x(γ) = (x γ, x γ ) X m (a) X w (ã) for γ = (m, θ, a; w, θ, ã) - x(γ) = x γ X m (a) for γ = (m, θ, a) - x(γ) = x γ X w (ã) for γ = (w, θ, ã) A symmetric consumption allocation is a pair consisting of a symmetric household consumption allocation x and matching µ A feasible allocation is a pair of a symmetric consumption allocation (x, µ) and production vector y Y that satisfies the obvious feasibility constraint
Household Decisions Price simplex: = {p = (p I, p K ) R I+K + : i I p i + k K p k = 1} For p, a couple (m, w) has (for any θ, θ Θ) the budget set dependent on their actions (a, ã) A 2 : B (m,w;a,ã) (p) { (x, x) X m (a) X w (ã) p(x + x) p(e m (a) + ẽ w (ã)) }. - they choose (x K, a; x K, ã) and allowances B and B together - (m, θ) chooses x I using B, and (w, θ) chooses x I using B - negotiation can be done by reporting θ and θ to each other (potential misrepresentation) For single households, decisions are standard
Stable Matching Equilibrium A stable matching equilibrium is a list of price vector p, feasible allocation (µ, x, y) profit maximizing y stable matching - single household with µ(γ) > 0 makes optimal choice - no joint deviations from households γ, γ with µ(γ), µ(γ ) > 0 members of two matched couples cannot deviate jointly a member of a matched couple and a single cannot deviate jointly two singles cannot deviate jointly - neither member of a matched couple deviates to be single
The Main Result Theorem There exists a stable matching equilibrium when 1 consumption sets are closed and convex with positive endowment of commodity 1 in all circumstances 2 u (m,θ) (similarly, u (w, θ) ) is continuous; quasi-concave in (x I, x K, x K ); locally non-satiated in x I 3 essentiality of commodity 1 (Mas-Colell 1977) 4 there is a special type of men ( m, θ) with N ( m, θ) > 0 who 4.1 only cares about nonleisure consumption 4.2 has positive endowment in all goods relative to his job choice 4.3 strictly quasi-concave + strongly prefers commodity 1 (Mas-Colell 1977) 5 Y close, convex, free disposal, CRS, no free lunch
About the Theorem Kaneko and Wooders (1986) seminal paper - nonemptiness of f-core: a continuum of finite coalitions (bounded size) - cannot allow market interactions by small (finite) coalitions Recent club literature allows for market interactions (see below) Consumption externalities within households are key in the paper Gersbach and Haller (2010) stress that consumption externalities distinguish their model from the club literature in finitely populated economies.
About the Theorem (continued) This paper allows for - (i) consumption externalities - (ii) externalities from observable types - (iii) no discrimination by their preference type Ellickson, Grodal, Scotchmer, and Zame (1999) - allow for (ii) and (iii), but not (i) Allouch, Conley, and Wooders (2007) - allow for unbounded size coalitions, and (ii), but not (i) nor (iii) Konishi (2010 and 2011) - allow for (i) and (iii), but not (ii) - profit-maximizing club and land developer
Proof by Kakutani s Theorem Assign a consumption vector for each γ Γ C Γ M Γ W - for (m, θ, a; w, θ, ã) Γ C, assign a Pareto efficient allocation given p Shafer-Sonnenschein mapping (Greenberg 1979, Ray and Vohra 1997) here we use quasi-concave utility in (x I, x K, x K) - for (m, θ, a) Γ M and (w, θ, ã) Γ W, assign an optimal consumption plan population mapping (Konishi 1996) - assign N (m,θ) to the best γ for (m, θ) among the ones having m - assign N (w, θ) to the best γ for (w, θ) among the ones having w - consistency requirement others are standard mappings (though discrete consumption set is annoying)
Concluding Remarks Chiappori s (1988, 1992) collective household decision model with market interactions and stability of matching. Consumption externalities within households. Gersbach and Haller (2010, 2011) describe difficulties associated with consumption externalities in finite models. - Nonexistence of equilibrium in finite economy or its replica. This paper shows that with atomless consumers, existence of equilibrium is restored. As a byproduct, we can allow for job choice and non-priced action externalities (more desirable for analysis of households collective choices). A first welfare theorem holds. The same mapping can be applied to different (inefficient) household decisions.