Directed Search with Endogenous Search Effort.
The agents accepts a particular o¤er only if it is the highest one among all received o¤ers. This implies that F (W ) is equivalent to the cumulative distribution function of the highest o¤er received from other …rms. The …rm does not know the exact number of o¤ers that the agent has received, but it can construct F (W ) in order to compute the expected pro…t of a particular wage o¤er. Firms choose the wage o¤er in an, ex-ante, identical way. To be as general as possible I assume that the strategy space of …rms is B(W ), where B(W ) represents all possible cumulative distribution functions over W . Then F (W ) can be constructed as follows: F (w W ) = (1 O(S; )) S 1 + S P 1 if B(W ) is continuous in W , or i=1 S 1 i O(S; ) i (1 O(S; )) S 1 i B(w W ) i , (5) F (w W ) = F (w < W ) + S P 1 i=1 1 i S 1 i O(S; ) i (1 O(S; )) S 1 i B(w = W ) i , (6) if B(W ) has a discontinuity in W . This is identical to a sealed bid …rst-price auction with an unknown number of bidders, where all bidders value the good exactly the same. 4 Next I describe the behavior of …rms in a set of lemmas. Lemma 2.1 Any wage o¤er must be higher or equal than the reservation wage w and lower or equal than the production value Q. Proof. Any o¤er lower than the reservation wage yields negative pro…ts as the probability of acceptance is zero. Then it is dominated by the reservation wage. Any o¤er higher than the production value yields negative pro…t so it is also dominated. 4 The exact number of bidders is unknown, but it is bounded by S. 5
Lemma 2.2 The distribution B(W ) cannot have any discontinuity and, therefore, F (W ) has a unique discontinuity at the reservation wage, due to the probability that an agent has no other o¤er. Proof. If B(W ) has a discontinuity at some wage, this wage o¤er does not belong to W . An epsilon higher o¤er yields a higher expected pro…t since there is a discontinuity in F (W ) that increases drastically the probability of acceptance. The above results imply that F (W ) can be stated as XS 1 S 1 F (W w) = (1 O(S; )) S 1 + O(S; ) i (1 O(S; )) S 1 i B(W w) i , i i=1 (7) that is equivalent to F (W ) = ((1 O(S; )) + O(S; )B(W )) S 1 . (8) It also implies that the probability that a …rm makes a particular wage o¤er is zero, that is, B(w = W ) = 0. Lemma 2.3 The lowest o¤er is the reservation wage w and the probability of acceptance is equal to the probability of having no other o¤er. Proof. If the reservation wage were not the lowest o¤er, the probability of acceptance of the lowest o¤er and the reservation wage would be the same. Then, to o¤er the reservation wage w yields a higher pro…t. As B(w = w) = 0, the reservation wage is accepted only if there is no other o¤er. Lemma 2.3 implies that: F (w) = (1 O(S; )) S 1 . (9) Lemma 2.4 The highest o¤ered wage w must be equal to Q (Q w)(1 O(S; )) S 1 . 6