Matching with Multiple Applications: The Limiting Case

Transcription

1 Mtching with Multiple Applictions: The Limiting Cse Jmes W. Albrecht y Georgetown Uniersity Susn B. Vromn Georgetown Uniersity August 003 Pieter A. Gutier Ersmus Uniersity Tinbergen Institute Abstrct We gie n expression for the expected number of mtches between unemployed workers nd cncies when ech worker mkes = pplictions, correcting Albrecht, Gutier, nd Vromn (003). We lso show tht the limiting mtching probbility gien in our erlier note is correct for ny nite. Keywords: Serch, mtching JEL Codes: J4 J64 Introduction In Albrecht, Gutier, nd Vromn (003), we proposed generliztion of the urn-bll mtching function tht llowed for multiple pplictions. Speci clly, we considered sitution with u unemployed workers nd cncies. Ech unemployed worker submits pplictions, where f; ; :::; g is xed number. A worker s pplictions re rndomly distributed cross the cncies with the proiso tht ny prticulr worker We thnk Ken Burdett nd Serene Tn for lerting us to the mistke in the nite cse in our erlier note. We lso thnk Hrld Lng nd Misj Nuyens for helping us correct our mistke. Any remining errors re, of course, our own. y Corresponding uthor: Deprtment of Economics, Georgetown Uniersity, Wshington DC 0057; tel: ; FAX: ; e-mil: lbrecht@georgetown.edu

2 sends t most one ppliction to ny prticulr cncy. Once the pplictions re mde, ech cncy (of those tht receied t lest one ppliction) chooses one ppliction t rndom nd o ers tht pplicnt job. A worker my get more thn one o er. In tht cse, the worker ccepts one of the o ers t rndom. Let M(u; ; ) be the expected number of mtches, i.e., the expected number of ccepted o ers. We presented n expression for M(u; ; ) for nite u nd : We lso llowed u;! with =u = xed nd found n expression for the expected number of mtches per unemployed worker, i.e., the probbility tht n unemployed worker nds job. As pointed out by Tn (003), our mtching function for f; :::; g; u nd nite, ws incorrect. The mtching function for = nd = ws correct. In this note, we gie correct ersion of the mtching function for the cse of = : The formul for M(u; ; ) is complicted, but we use it to proe tht the expression we ge for the corresponding limiting probbility in our erlier note remins correct. We extend our limiting rgument from the cse of = to ny nite, xed number of pplictions per worker. The problem in the nite cse cn be understood when = : Our (incorrect) pproch ws to reson s follows. Consider ny cncy to which n unemployed worker pplies. The number of competitors the worker hs t this cncy is bin(; ): We cn use this fct to compute the probbility tht the worker fils to receie n o er t this cncy: Similrly, the number of competitors t the other cncy to which this worker pplies is bin(; ): Agin, we cn compute the probbility tht the worker fils to receie n o er from this cncy. The probbility tht worker receies t lest one o er equls minus the probbility he or she receies no o ers. Our mistke ws to ssume (implicitly) tht the probbility worker receies no o ers equls the probbility tht his rst ppliction doesn t generte n o er times the probbility tht his second ppliction doesn t generte n o er. The problem is tht the indictor rndom ribles, rst ppliction leds to n o er nd second ppliction leds to n o er re not independent. Equilently, the numbers of competitors tht worker hs t the cncies re not independent. This is obious (in retrospect). If, for exmple, u = = 3 nd = ; then the fct tht worker s rst ppliction fils to generte n o er implies tht t lest one of the other workers lso pplied to tht cncy, which in turn implies tht the chnce of the worker s second ppliction being successful increses. This description of where we went wrong suggests why we re correct in the limit. If u nd Tn (003) gies nother expression for the mtching function for nite u nd :

3 re lrge, then the fct tht the rst ppliction is unsuccessful implies next to nothing bout the probbility tht the second ppliction is successful. Equilently, if u nd re lrge, then the numbers of competitors tht worker hs t the cncies to which he or she pplies re pproximtely independent rndom ribles. In the next section, we derie the correct mtching function for the cse of = : The limiting cse for = is gien in the following section. The lst section proides the generl limiting result. The Mtching Function with = Let S be the number of competitors worker fces t the rst cncy to which he or she pplies, nd let T be the number of competitors t the second cncy. S nd T re ech bin(; ); but these rndom ribles re not independent. We wnt n expression for P [S = s; T = t] where 0 s; t : Once we he this, we cn compute the expected number of mtches s function of u nd when = s! X X s M(u; ; ) = ( s + )( t )P [S = s; T = t] ; t + s=0 t=0 tht is, s the number of unemployed times one minus the probbility tht n indiidul unemployed worker gets n o er t neither of the cncies to which he or she pplies. Let A(s; t) be the number of wys in which competitors cn send s pplictions to the rst nd t pplictions to the second cncy gien tht there re cncies ( 4). Then A(s; t) P [S = s; T = t] = : Let i be the number of competitors who pplied to both cncies, where i min[s; t]: This mens (s i) competitors pplied only to the rst cncy nd (t i) competitors pplied only to the second cncy. We then he A(s; t) = X i=mx[0;s+t ()] i i s i ( ) s i i (s i) t i ( ) t i i (s i) (t i): To understnd this expression, consider prticulr lue of i: There wys tht the i competitors who pplied to both cncies cn re i 3

4 be chosen from the unemployed. This lees i unemployed. There re i s i wys tht the s i competitors who pplied only to the rst cncy cn be chosen out of the remining i unemployed, nd there re ( ) s i wys tht these s i competitors cn spred their other ppliction cross the remining cncies. Now there remin i (s i) i (s i) unemployed. There re t i wys tht the t i competitors who pplied only to the second cncy cn be chosen from this group, nd these t i competitors cn spred their other ppliction cross the remining cncies in ( ) t i wys. Finlly, there re i (s i) (t i) workers who pplied to neither of the cncies. There re i (s i) (t i) wys tht their pplictions cn be spred cross the other cncies. To count ll the possible wys in which S = s nd T = t; we now need to sum oer ll possible i: The lower bound for the possible lues of i re ects the fct tht if u is smll reltie to s nd/or t, smll lues of i my not be possible. For exmple, if u = 4; s = 3 nd t = 3; only i = 3 is possible. Since i i (s i) ()! = i s i t i i!(s i)!(t i)!( + i s t)! we he A(s; t) = X i=mx[0;s+t ()] ( ) i ( 3) s t+i u++s+t i ()! i!(s i)!(t i)!( + i s t)! Substituting nd simplifying yields P [S = s; T = t] = s+t X nd the corresponding mtching function M(u; ; ) = i=mx[0;s+t ()] Our expression for A(s; t) does not pply when = 3: The reson is tht eery unemployed! must be competitor for t lest one cncy. This mens tht we need to set = nd to ccount for the fct tht for some lues of s; t nd i; P [S = s; T = t] = 0: This ltter is tken cre of i the indictor function in the following expression for A(s; t) : P i=mx[0;s+t ()] I[i = s + t ()] i i s i ( ) s i i (s i) t i i ( ) i ( 3) i ()! ( 3) s+t i!(s i)!(t i)!(+i s t)! ( ) t i : 4

5 u( u P P s=0 t=0 ( s s+ )( t t+ )s+t ( ) P i=mx[0;s+t ()] 3 The Mtching Function in the Limit ( = ) The formul we deried for M(u; ; ); lthough complicted; reduces to simple expression in the limit. The key is tht in the limit, S nd T re independent, so tht P [S = s; T = t] = P [S = s]p [T = t]: Since the mrginls for S nd T re ech bin(; ); we he (using the stndrd result on the Poisson s the limit of binomil) tht lim P [S = s] = u;!; u! s s e = h(s) for s = 0; ; ::: s! i ( ) i ( 3) i ()! ( 3) s+t i!(s i)!(t i)!(+i s t)! ) lim P [T = t] = u;!; u! t t e = h(t) for t = 0; ; ::: t! We thus need to show lim P [S = s; T = t] = s+t e 4= (s+t) u;!;=u= s!t! for s = 0; ; :: nd t = 0; ; :::: The rst step in doing this is to show tht in the limit, P [i > 0] = 0; tht is, the probbility tht ny competitor pplies to both of the cncies to which n indiidul hs pplied is zero. When this is true, only the i = 0 term suries in the summtion in our expression for P [S = s; T = t]: Note tht s u!, mx[0; s + t ()] = 0 for ech xed s nd t: To show tht in the limit P [i > 0] = 0, we rst note tht for lrge ; the number of pplictions tht competitor sends to the cncies to which the indiidul hs pplied is pproximtely bin(; ):3 Then, the probbility tht competitor pplies to neither or just one of these cncies, i.e., not to both, is pproximtely ( ) + 4 ( ); nd the probbility tht no competitor pplies to both cncies is pproximtely ( ) + 4 ( ) : Thus, P [i > 0] = ( ) + 4 ( ) : 3 The number of pplictions tht competitor sends to the cncies to which n indiidul hs pplied is hypergeometric rndom rible, so we re using the binomil distribution to pproximte the hypergeometric. Tht is, s! ; we re (legitimtely) ignoring the proiso tht the worker sends t most one ppliction to ny one cncy. 5

6 Setting t ; tking the limit s! ; nd pplying L Hôpitl s rule gies the result tht P [i > 0] = 0: lim u;!;=u= Gien tht we need only consider the i = 0 term in the summtion in the expression for P [S = s; T = t];, we he P [S = s; T = t] = s+t Finlly, note tht nd lim u;!; u! = lim u;!; u! ()! ( 3) s+t s!t!( s t)! : = e = ()! lim u;!; u! ( 3) s+t = (s+t) : s!t!( s t)! s!t! We thus he our result tht lim P [S = s; T = t] = s+t e 4= (s+t) u;!; u! s!t! = h(s)h(t): The limiting mtching function cn now be deried s follows: M(u; ; ) m(; ) lim = u;!; u! u = X ( x=0 X X s ( s + )( t t + )h(s)h(t)! = s=0 t=0 e = x + )(=)x x! ( e = ) This is precisely the expression tht we deried in our 003 note (pge 69) for the cse of = : 4 The Mtching Function in the Limit (generl ) To extend the limiting rgument from the cse of = to the generl cse of f; :::::; Ag; where A is n rbitrry (but xed) number of pplictions, we need to show tht in the limit the probbility tht ny competitor pplies to two or more of the cncies to which n indiidul hs pplied is zero. The rgument is the sme s the one used for = ; nmely, we need to show lim u;!;=u= ( ) + ( 6 ) = 0:

7 This is gin done using l Hôpitl s Rule rgument. Now, let S ; :::; S be the numbers of competitors tht n indiidul hs for the rst cncy to which he or she pplies, the second cncy to which he or she pplies,..., the lst cncy to which he or she pplies. Let A(s ; s ; :::; s ) be the number of wys tht potentil competitors cn mke s pplictions to cncy ; :::; s pplictions to cncy : Then P [S = s ; :::; S = s ] = A(s ; s ; :::; s ) : Gien tht no worker is competing with the indiidul for more thn one cncy (legitimte in the limit by the rgument gien boe), A(s ; s ; :::; s ) = s s s s s P j= s j s s P j= s j = Q ()! j= s j!( P j= s j)! = ()(u )(u P Q j= s j) j= s j! ( )! ( +)!( )! + P j= s P j ( )! j= s j ( )!! P j= s j ( )! ( )!! = () + (u ) + (u P j= s j) + P j= s j ( )( )( +)! Q j= s j! Since = ( ) ( + ) ; we he! P [S = s ; :::; S = s ] = P () (u ) (u j= s j ) + Q+ + j= s P j= s j j! + Finlly, lim P [S = s ; :::; S = s ] = u;!;=u= P j= s j Q j= s j! expf g ; which equls the product of independent Poissons, ech with prmeter =: The limiting mtching function for f; :::; Ag is then m(; ) M(u; ; ) lim = u;!;=u= u s ws gien in Albrecht, Gutier, nd Vromn (003). e ; 7

8 References: Albrecht, J.W., Gutier, P.A., nd Vromn, S.B., 003. multiple pplictions. Economics Letters 78, Mtching with Tn, S., 003, Mtching with multiple pplictions: A correction, mimeo. 8