Efficienc in a Search an Matching Moel with Enogenous Participation James Albrecht a Department of Economics, Georgetown Universit an IZA Lucas Navarro Department of Economics, ILADES-Universia Alberto Hurtao Susan Vroman Department of Economics, Georgetown Universit an IZA Abstract: We show that in a search/matching moel with enogenous participation in which workers are heterogeneous with respect to market prouctivit, satisfing the Hosios rule leas to excessive vacanc creation. JEL Coes: D8, J6 Kewors: Search, Matching, Efficienc, Participation, Hosios Rule a Corresponing author: Department of Economics, Georgetown Universit, Washington, DC 20057, USA, Telephone: 202 687 605, FAX: 202 687 602, albrechtgeorgetown.eu
In the stanar Pissaries (2000 search/matching moel, equilibrium is e cient when wages are etermine b Nash bargaining if the worker share of the net surplus of the match equals the elasticit of the matching function with respect to unemploment, i.e., if the Hosios (990 rule is satis e. As iscusse in Pissaries (2000, Chapter 8, this result hols more generall. In particular, the Hosios rule implements the e cient outcome when workers are heterogenous with respect to their outsie options, e.g., their leisure values, an labor force participation is enogenous. In this note, we also allow for enogenous labor force participation, but assume that workers are heterogeneous with respect to market prouctivit rather than with respect to leisure values. In this setting, the Hosios conition fails because the participation ecision a ects not onl labor market tightness but also the average prouctivit of matches. This average prouctivit e ect is not present in the Pissaries version of the moel with enogenous labor force participation. In our moel, an increase in participation causes average match prouctivit to fall, but the marginal participant oes not internalize this e ect. As a result, when wages are etermine b the Hosios rule, the labor force participation rate is too high. Equivalentl, there is excessive vacanc creation. To make our point as simpl as possible, we consier a one-perio version of the moel with a continuum of workers of measure one. Each worker chooses between searching for a job (participating an engaging in home prouction (not participating. A nonparticipant receives z with certaint, but a participant s expecte pao epens on his or her tpe an on labor market tightness. Prouctivit in market work is istribute across workers accoring to a continuous istribution function F (; 0 an F (0 = 0: All participants search an n a job with probabilit m(; where is market tightness. 2 A worker of tpe who ns a job gets a fraction of the output that he or she prouces; a worker who participates but fails to n a job gets a pao that is normalize to zero. The expecte pao of a participant of tpe is thus m(: A worker participates i m( z, A similar issue arises in the Albrecht, Navarro an Vroman (2009 moel of eveloping econom labor markets with an informal sector in which workers have i erent formalsector prouctivities. 2 As in Pissaries (2000, the matching function M(v; u is assume to have constant returns to scale so it can be written as m(u: We assume that m( is inepenent of ; i.e., all participants have an equal chance of ning a job. Our moel can be thought of as one in which emploers search sequentiall for caniates, e.g., an emploer hires the rst worker who applies for the job. Villena-Rolán (2008 consiers a moel of nonsequential emploer search in which a worker s chance of getting a job epens on his or her prouctivit.
so there is a cuto value of ; = z m( ( such that all workers with participate; the remaining workers are nonparticipants. Call this function = h(; an note that h 0 ( = m0 ( m( < 0: The participation rate is = F ( ; so labor market tightness is v F ( = v F (h( ; where v is the measure of vacanc creation. Di erentiating implicitl with respect to v, v = v + h0 ( v ; (2 where v = = F ( f( F ( : Note that =v > 0: Equilibrium requires that workers optimall choose whether to participate or not an that vacancies are create until the value of a vacanc equals zero. The value of a vacanc is V = c + m( Z ( F ( ; where c is the cost to open the vacanc, m(= is the probabilit that the vacanc hires a worker, an is etermine b worker choice. Setting this value to zero gives Z m( ( F ( = c: (3 Equations ( an (3 can be solve for the equilibrium values of an : 2
Next we consier the e cienc problem. The social planner chooses v to maximize = zf ( + ( F ( m( Z = zf ( + m( Z F ( cv taking into account that workers choose whether or not to participate. The rst term on the RHS is the value for nonparticipants, the secon term is the value of market output, an the thir term subtracts the costs of vacanc creation. The erivative of the social planner s objective with respect to v is v = (z m( f( h 0 ( Z v + m0 ( = (z m( f( h 0 ( v + m0 ( Substituting for v = z v cv; v Z an an rearranging gives Z +m 0 ( m( + m 0 ( Z c v + h0 ( v F ( f( h 0 ( v F ( c: (4 The Hosios rule sets the worker share of the net surplus equal to the elasticit of the matching function with respect to unemploment. 3 It can be written as = m0 ( m( : If this conition hols, then m 0 ( Z so from equation (3, F ( Z m( = ( Z m 0 ( F ( = c; F ( ; c: 3 As iscusse in footnote 2, the matching function is m(u: 3
an the nal two terms in equation (4 cancel. That is, v = z Z m( + m 0 ( F ( f( h 0 ( v : (5 Note that we are evaluating =v at the equilibrium level of vacanc creation when equals the Hosios value. If =v > 0 at the Hosios level of vacanc creation, there is not enough vacanc creation in equilibrium; if =v < 0; too man vacancies are being set up. 4 Since f( h 0 ( v < 0; the sign of =v at the Hosios level of v is the opposite of that of the term in parentheses in equation (5. The net output of the marginal participant is m( z: The marginal participant also reuces the average prouctivit of matches in the market sector. This is re ecte in the term m 0 ( R F ( : If workers were homogeneous with respect to market prouctivit, then this term woul be inepenent of ; i.e., participation woul have no e ect on average prouctivit. To sign =v; note that an, b the Hosios conition, Z m 0 ( z m( = ( m( Z F ( = ( m( We then have = ( m( + ( m( v Z ( f( = ( m( F ( Z F ( : F ( f( h 0 ( v f( h 0 ( v < 0: We have thus shown that when equals the Hosios value, too man vacancies are create in equilibrium. Equivalentl, is too low the rate of labor force participation is ine cientl high. 4 When z = 0; the equilibrium level of vacanc creation is e cient, i.e., =v = 0: The reason is that when z = 0; all workers participate, i.e., = 0; which in turn implies h 0 ( = 0: 4
References [] Albrecht, J., L. Navarro an S. Vroman, 2009, The E ects of Labour Market Policies in an Econom with an Informal Sector, The Economic Journal, 9, 05-29. [2] Hosios, A., 990, On the E cienc of Matching an Relate Moels of Search an Unemploment, Review of Economic Stuies, 57, 279-98. [3] Pissaries, C. 2000, Equilibrium Unemploment Theor, 2n eition, MIT Press, Cambrige, Mass. [4] Villena-Rolán, B., 2008, Aggregate Implications of Emploer Search an Recruiting Selection, mimeo, Universit of Rochester. 5