Matching with Multiple Applications: A Correction Serene Tan1,2 12 July 2003 Abstract The matching function as derived in Albrecht et al. (2003a) is shown to be incorrect except in two special cases, and the correct expression for the matching function is proposed. Keywords: matching, search JEL codes: J41, J64

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Introduction

In a recent paper in this journal, Albrecht et al. (2003a) analyzed ”the implications of multiple applications by job seekers for the microfoundations of the matching function.” This note shows that their matching function is defined incorrectly through a simple counterexample in section two, and explains where the mistake was made in section three. The concluding section shows that their matching function is correct only in two special cases and proposes the right way to define the general matching function. This paper focuses on finite numbers of firms and workers whereas Albrecht et al. (2003b) focus on the limiting case and show that their original matching function is correct in the limit with each worker making a finite number of applications.

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A Simple Counter-Example

Albrecht et al. (2003a) derived the equilibrium matching function M (u, v; a) when u workers all exogenously make the same a number of applications to v firms. For finite u, v and a, 1

Graduate Group in Economics, University of Pennsylvania, 160 McNeil Building, 3718 Locust Walk,

Philadelphia PA 19104-6297, tel: +1 215 898-7701, fax: +1 215 573-2057, email: [email protected]. 2 I am extremely grateful to Ken Burdett, without whom this note could not possibly have been completed. Many thanks to Randy Wright, Rob Shimer, Nicolas Jacquet and Chun-Seng Yip for comments and feedback. Financial support from the National University of Singapore is gratefully acknowledged.

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a ∈ {1, ..., v},

³ ³ h v ³ a ´u ´´a i , M (u, v; a) = u (1 − (1 − q)a ) = u 1 − 1 − 1− 1− au v ³ ³ ´ ´ a u v 1− 1− where q = au v

(1) (2)

is the probability that any one of a worker’s applications leads to an offer.

The matching function in equation (1) is defined to be the number of workers multiplied by the probability that a worker gets at least one offer. Implicit in the way the matching function is defined is that the probability of success of each of a worker’s applications is independent. This is why (1 − q)a is the probability that none of a worker’s applications is successful, and 1−(1−q)a is the probability of at least one success on a worker’s applications. A simple counter-example that follows shows that equation (1) as defined is incorrect. Consider a case where there are three identical firms (firms 1, 2 and 3) and three identical workers (workers A, B and C). Each worker is allowed to sample two different firms without µ ¶ 3 = 3 ways of applying to replacement. As in Albrecht et al. (2003a), each worker has 2 firms, with each of the three ways carrying equal weight. Let worker A apply to firms 1 and 2 without any loss of generality. Hence, there are nine different outcomes as illustrated below. 1

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From the above diagram, it can easily be shown that the prob{worker A has no offers} = 41/162 ' 0.25, and the expected number of matches is 2.24. However, using equation (2),

1 − q = 14/27, and (1 − q)2 = (14/27)2 = 196/729 ' 0.27, with the expected number of matches being 2.19, which understates the actual number of expected matches formed. Hence, the matching function as defined in equation (1) is not correct in general.

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What Is Wrong?

It was implicitly assumed in equation (1) that each worker’s offer probabilities are independent across firms, and herein lies the problem. It is not true. Even though each firm independently makes its decision of who to give the job to (the inverse of its queue length, or the number of workers who turn up), each firm’s queue length is not independent of another firm’s queue length. Intuitively, each firm can make an inference about other firms’ queue lengths by looking at its own queue length. (Suppose there are 10 firms. If a firm has an extraordinarily large number of workers turning up, it knows that other firms will have fewer people than expected.) That is, queue lengths are not independent. What is of interest in defining the matching function is a worker’s probability of getting no offers when he makes a applications. To calculate this, we must consider the probability that all a firms he applied to jointly did not make him an offer. This in turn depends on the joint probability of the numbers of other people turning up at the a firms he had applied to. Since each firm’s queue length is not independent, the offer probabilities across the a firms will not be independent also. Hence, to express the matching function like in equation (1) where each worker’s offer probabilities are implicitly assumed to be independent yields a wrong and misleading matching function.

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The Correct Matching Function

In general, the matching function defined in equation (1) is incorrect. However, for any finite u and v, when a = 1 and when a = v, the matching function as derived by Albrecht et al. (2003a) is correct. When a = 1 and i + 1 workers turn up at any particular firm, u−1 µ X

¶ µ ¶i µ ¶ 1 u−1−i 1 u−1 1 1− prob {a worker gets exactly one offer} = i v v i+1 i=0 µ · ¶u ¸ 1 v , = 1− 1− u v µ ¶ ¸ · 1 u . This is the matching function as in and the number of matches is thus v 1 − 1 − v equation (1) with a = 1. When a = v, the probability of getting firm is ¶just 1/u. Hence, ¶v from ·any µ µ an offer v¸ 1 1 , and u 1 − 1 − is the number prob{a worker gets at least an offer} = 1− 1 − u u of matches, so the matching function in equation (1) is also correct when a = v. The reason why the matching function as defined in equation (1) is right under these two special cases is because when a = v, each firm’s queue length is known (it is just u), so it is a trivial case of queue lengths being independent across firms. Also, when a = 1, whether a worker gets an offer just depends on the queue length at that firm he applied to. The correct expression for any general number of applications is complicated. It will be constructed by induction. To begin, consider the case of a = 1 for any finite u and v. The probability that a worker applies to any particular firm is a/v. Hence, if there were i other people turning up at that u−1 X µ u − 1 ¶ ³ a ´i ³ a ´u−1−i i 1− ≡ firm, this worker does not get an offer with probability i v v i+1 i=0 P i , where ∆1 (i) is the probability that i other workers turned up at that firm. ∆1 (i) i + 1 i When a = 2 for any finite u and v, the probability that both applications turn out unsuccessful for a worker becomes:

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¶ u−1 µ X a ´u−1−i u − 1 ³ a ´i ³ 1− ∗ i v v i=0  à j µ ¶µ ¶µ ¶z µ ¶i−z µ ¶j−z µ ¶u−1−i−(j−z) ! u−1 X X i u − 1 − i a − 1 a − 1 a a  1− 1− ∗ z j−z v−1 v−1 v−1 v−1 j=0 z=0 ¸ j i 3 ∗ j + 1 i + " #1 P P j i (∆2 (i, j)) , ≡ ∆1 (i) j+1 i+1 i j

which is a convolution of two binomial terms.4 We have to examine whether the worker

gets offers from both firms jointly, and to do that, it is necessary to consider the queue P i ∆1 (i) lengths at both firms jointly, which is what the above expression takes care of. i+1 i is the probability of not getting an offer from the first firm a worker applies to when i other P j ∆2 (i, j) which is the probability people turned up there, and this is convoluted by j+1 j of not getting an offer from the second firm also when j other people turned up, given that i others had turned up at the first firm. (At this point, the keen reader may have intuited that the general expression for the matching function for any finite a is going to be horrendous. And it is.) Continuing this way, for any finite u, v and any finite a, prob{a worker did not get any offer from applying to a firms} ≡ Ψ = " · · · ¸ ¸ ¸ ¸ P P P P k j i l ∆1 (i) (∆2 (i, j)) (∆3 (i, j, k)) ... (∆a (i, j, k, ..., l)) ... l+1 k+1 j+1 i+1 i j k l (5) where ∆1 , and ∆2 , as before, are the non-independent queue lengths at the first and second firms this worker applied to respectively. This term is then convoluted by ∆3 , which represents the queue length at the third firm he applied to. We can do this successively across all the firms the worker had applied to. Hence, the queue length at the a-th firm he applied to is ∆a which is a function not just of the l other workers at this firm, but also of 3

Notice that when v = 3 and a = 2, (1 − a/(v − 1))u−1−i−(j−z) = (0)u−1−i−(j−z) ≡ ξ. If u−1−i−(j−z) =

0, let ξ = 1, and if u − 1 − i − (j − z) 6= 0, then let ξ = 0. In this paper, I let 0λ = 1 when λ = 0, and 0λ = 0 when λ 6= 0. 4

Throughout this paper,

³n´ r

where r > n or r < 0 is defined to be 0.

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the i others at the first firm, j others at the second firm, etc. More formally rather unfortunately), µ (and ¶ a ´u−1−i u − 1 ³ a ´i ³ 1− = , ∆1 (i) i v v ∆2 (i, j) =

¶ j µ ¶µ X i u−1−i

∗ j−z µ ¶ µ ¶ µ ¶j−z µ ¶u−1−i−(j−z) a − 1 i−z a a−1 z a 1− 1− , v−1 v−1 v−1 v−1 ³ z ´ µ i − z ¶ µ j − z ¶ µ u − 1 − i − (j − z) ¶ X ∆3 (i, j, k) = ∗ e f g h e+f +g+h=k µ µ ¶ µ ¶ ¶ µ ¶ a−2 e a − 2 z−e a − 1 f a − 1 i−z−f 1− 1− ∗ v−2 v−2 v−2 v−2 µ ¶g µ ¶j−z−g µ ¶h µ ¶u−1−i−(j−z)−h a−1 a a−1 a 1− 1− , v−2 v−2 v−2 v−2 and X µ s1 ¶ µ s2 ¶ µ s3 ¶ µ ss−1 ¶ µ ss ¶ ... ∆a (i, j, k, ..., l) = ∗ t1 t2 t3 ts−1 ts t1 +...+ts =l | {z } number of terms =2a−1 µ ¶ µ ¶ ¶ µ ¶ µ a − (a − 1) s1 −t1 a − (a − 2) t2 a − (a − 2) s2 −t2 a − (a − 1) t1 1− 1− ∗ ...∗ v − (a − 1) v − (a − 1) v − (a − 1) v − (a − 1) µ ¶ts−1 µ ¶ss−1 −ts−1 µ ¶ts µ ¶ss −ts a−1 a−1 a a 1− 1− v − (a − 1) v − (a − 1) v − (a − 1) v − (a − 1) z=0

z

Hence, the correct matching function is M (u, v; a) = u (1 − Ψ) ,

(6)

where Ψ is defined as in equation (5) above.5 Although it is not possible to simplify the convoluted binomial terms, it is possible (and very simple) to show that the expected number of matches as defined in equation (6) is always higher than that defined by Albrecht et al. (2003a) in equation (1)6 . 5

From the way Ψ is constructed, even though complicated, it can be shown that as v → ∞, for finite a,

the matching function as defined by Albrecht et al. in equation (1) is correct. The intuition is that since there are so many firms and relatively few applications made per worker, knowing one firm’s queue length adds absolutely nothing to another firm’s knowledge of its own queue length. 6 Proof upon request.

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References Albrecht, J., P. Gautier, and S. Vroman (2003a), Matching with Multiple Applications, Economics Letters, 78, 67-70. Albrecht, J., P. Gautier, and S. Vroman (2003b), Matching with Multiple Applications: The Limiting Case, Georgetown University and Erasmus University Rotterdam mimeo. Feller, W. (1968), An Introduction to Probability Theory and Its Applications, Volume 1, 3rd edition, John Wiley. Johnson, N.L., S. Kotz and X. Wu (1991), Inspection Errors for Attributes in Quality Control, Chapman and Hall.

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