Shopping Externalities and Self Fulfilling Unemployment Fluctuations*

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of the AD good is set to clear the market and is normalized to 1. That is, the AD good isthe unit of account in our economy.Several remarks about the environment are in order. First, notice that the economydisplays constant return to scale. The production technology for goods has constant returns,the production technology for vacancies has constant returns, and the matching functions inboth the MP and the BJ markets have constant returns. Hence, the multiplicity of equilibriathat we obtain does not originate from increasing returns in production as in Kiyotaki(1988), Benhabib and Farmer (1994), Farmer and Guo (1994), Christiano and Harrison(1999) and Mortensen (1999), nor it does originate from increasing returns in matching as inDiamond (1982) and Diamond and Fudenberg (1989). Second, notice that we assume thatworkers cannot access credit markets. In the quantitative section of the paper, we will try toaddress this unrealistic feature of the model by making sure that the decline in expendituresexperienced by a worker who becomes unemployed is in line with what we observe in thedata. Third, we assume that …rms cannot store goods from one production period to thenext. While, it would be interesting to allow …rms to keep inventories, it would make theanalysis of the equilibrium much more complicated.2.2 Equilibrium ConditionsWe begin by deriving the equilibrium conditions for the Burdett-Judd product market. First,consider a buyer who enters the BJ market with z units of the AD good and who …nds alowest price of p. If p > r, the buyer does not purchase any of the BJ good. If p r, thebuyer purchases x units of the BJ good and y units of the AD good so as to maximize hisperiodical utility, x y 1solves the problemThe solution to the above problem is , subject to the budget constraint px + y = z. That is, the buyermaxx;y x y 1 ,s.t. px + y = z.(1)px = z, y = (1 )z. (2)The buyer …nds it optimal to spend a fraction of his income z on the BJ good and afraction 1 on the AD good.Next, consider a seller who posts the price p in the BJ market and denote as F t (p) thecumulative distribution of prices posted by the other sellers. If p > r, the seller’s expectedgains from trading in the BJ market are zero. If p r, the seller’s expected gains from trade10
are given byS t (p) = ((u t )) u t(1 + u )b(u t )1+((u t )) (1 u t)(1 + e )b(u t )2 u ((u t ))F t (p) yu (p c)1 + u p21 e ((u t ))F t (p) w(p c).1 + e p(3)The expression above can be understood as follows. The probability that a seller meets abuyer is ((u t )). Conditional on the seller meeting a buyer, the probability that the buyer isunemployed is u t (1 + u )=b(u t ). Conditional on the seller meeting an unemployed buyer, theprobability that the buyer is willing to purchase at the price p is 1 2 u ((u t ))F t (p)/ (1 +u), where 2 u ((u t ))F t (p)/ (1 + u ) is the probability that the buyer has contacted asecond seller and the second seller charges a price lower than p. As established in (2), thequantity of the BJ good purchased by an unemployed buyer is y u =p and the seller’s gainsfrom trade on each unit sold are p c. Hence, the …rst line on the right-hand side of (3)represents the seller’s expected gains from meeting an unemployed buyer.Similarly, thesecond line on the right-hand side of (3) represents the seller’s expected gains from meetingan employed buyer.The price distribution in the BJ market is consistent with the seller’s optimal pricingbehavior if and only if any price p on the support of F t maximizes the seller’s gains fromtrade. That is,S t (p) = S t maxp 0S t (p 0 ), all p 2 suppF t . (4)The following lemma characterizes the unique price distribution F t that satis…es (4). Theproof of this lemma follows arguments similar to those in Burdett and Judd (1983) and Head,Liu, Menzio and Wright (2012).Lemma 1 (Equilibrium Price Distribution): The unique price distribution consistent with(4) isF t (p) = u t (1 + u ) 1+(1 u t )(1 + e ) 12((u t )) fu t u y u + (1with support [p t; p t ], where c < p t c, aseller posting p 0 could increase its gains from trade by charging p 0. This deviation would11
Page 2: 1 IntroductionWe propose a novel th
Page 5 and 6: claims). However, they certainly in
Page 7 and 8: the evidence brought forward that
Page 9: function with boundary conditions (
Page 13 and 14: The …rm and the worker bargain ov
Page 15 and 16: the paper, which is mainly devoted
Page 17 and 18: espect to u as (1 +S 0 (u) =u )(1
Page 19 and 20: a seller meets a buyer is constant.
Page 21 and 22: The eigenvalues, the determinant an
Page 23 and 24: to denote the set of J’s such tha
Page 25: JJů = 0ů = 0E1E1J1sJ1sE2J3sE2J3sE
Page 28 and 29: understand that regime switches can
Page 30 and 31: De…nition 3. An i-j Regime Switch
Page 32 and 33: 5 Sentiments and the Great Recessio
Page 34 and 35: Table 1: Calibration targetsLabor M
Page 36 and 37: …ndings in Sorensen (2000) and Mo
Page 38 and 39: An increase in unemployment of 1 pe
Page 40 and 41: 11.51110.5109.5Mean posted priceMea
Page 42 and 43: with the pessimistic regime. The de
Page 44 and 45: etween employed and unemployed work
Page 46 and 47: Table 4: Robustness: steady-state u
Page 48 and 49: vacancy rate because they are assoc
Page 50 and 51: [22] Heller, W. 1986. “Coordinati
Page 52 and 53: AppendixA Proof of Lemma 1Consider
Page 54 and 55: Claim 5. The only F (p) that satis
Page 56 and 57: where the parameters 0 and 1 are
Page 58: After substituting (C5) into (C6),

Shopping Externalities and Self Fulfilling Unemployment Fluctuations*

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