Shopping Externalities and Self Fulfilling Unemployment Fluctuations*

espect to u as (1 +S 0 (u) =u )(1 + e ) 2 u (1 u)b(u) 2 1 + u b(u)1(1 + u )(1 + e )b(u) 22 e (1 u)1 + e b(u) 2 e (1 u)y u1 + e b(u) (w y u )+2 1 + u (1 + u )u u(1 +yb(u) 2 u +e )(1 u) ewb(u) 1 + u b(u) 1 + e+ (1 + e)(1 u) 21 e (1 u)(r c)S 0 (u) :b(u)1 + e b(u)r(18)An increase in unemployment has four e¤ects on the …rm’s gains from trade in the BJmarket. First, an increase in unemployment increases the probability that— conditional onthe seller meeting a buyer— the buyer is unemployed. Since unemployed buyers are less likelyto purchase the good at the reservation price r, the increase in the conditional probability ofmeeting an unemployed buyer lowers the seller’s probability of making a sale. This negativee¤ect is represented by the …rst line on the right-hand side of (18) and we will refer to itas the market power e¤ect of unemployment. Second, since unemployed buyers have lessincome, the increase in the conditional probability of meeting an unemployed buyer alsolowers the seller’s average size of a sale. This negative e¤ect is represented by the second lineon the right-hand side of (18) and we will refer to it as the demand e¤ect of unemployment.Third, an increase in unemployment increases the probability that— conditional on the sellermeeting a buyer in a particular employment state (i.e. employed or unemployed)— the buyeris willing to purchase at the reservation price r. This positive e¤ect is represented by thethird line on the right-hand side of (18) and we will refer to it as the captivity e¤ect ofunemployment. Finally, an increase in unemployment has an e¤ect on the wage and, hence,on the quantity of the BJ good purchased by employed buyers. This e¤ect is measured inthe last line on the right-hand side of (18) and acts as a multiplier on the …rst three e¤ects.Thus, the sign of S 0 (u) depends on the relative strength of the market power, demand andcaptivity e¤ects of unemployment, which in turn depend on parameter values.The following theorem identi…es a set of necessary and su¢ cient condition for the existenceof multiple steady-state equilibria.Theorem 1 (Multiplicity of Steady States): (i) If and only if S 0 (u) < 0 for some u 2 [u; u],there exist a vacancy cost k and a labor market matching function M such that the modeladmits multiple steady-state equilibria. (ii) For any u 2 [u; u], there is a y e (u) y u such thatS 0 (u) < 0 if and only if y e > y e (u). (iii) There exists a ~u > 0 such that, for any u 2 [u; ~u],there is a e (u) u such that S 0 (u) < 0 if and only if 0 e < e (u).17