Appendix Proof. Proposition 1. According to steady-state demand condition,

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1 Appendix roof. roposition. Accordin to steady-state demand condition, D =A f ss θ,a; D α α f ss,a; D α α θ. A,weref ss θ e,a; D is te steady-state measre of plants wit ae a and te expected idiosyncratic prodctivity θ e.morespecifically, f ss θ,a; D = f ss 0,D p a A2 f ss,a; D = f ss 0,D ϕ [ p a ] By definition, a steady state featres time-invariant distribtion of plants across a and θ e. Tisimpliestat A as to be time-invariant for A to old. In addition to A, f ss 0,D, and ave to satisfy te followin conditions. Te exit condition for a ood plant is: " α A γ ass # =0 A3 Te exit condition for an nsre plant is " 0 = α α α pϕ a= β a a Aθ γ ass ss # α Aθ γ a ; A4 Tefreeentryconditionis: c 0 c f ss 0,D = β a p a Aθ α γ a β a ϕ [ p a ] Aθ α γ a. A5

2 Frtermore, A3 sests: A = α " γ ass # A6 lin A6 and A2 into A ives D = α " γ ass # f ss 0,D p a α α ϕ [ p a ] α α θ θ. A7 lin A6 into A4 ives β ass ass pϕ β β γ β γ = µ θ γ γ ass ass A8 D β ass ass Notice tat D does not enter A8, so tat, as lon as A8 determines an niqe vale for, A7 and A5 wit A6 pled in wold jointly determine a ss tat θ <, wic olds by definition. Tis proves roposition.. It trns ot tat, for A8 to reveal a niqe soltion for roof. roposition 4. lin 3 into 2 ives D and f ss 0,D wit = D, itreqires c 0 c f ss 0,D = " β a p a γ ass θ " β a ϕ [ p a γ ass ] # θ γ a # A9 γ a, wic sests f ss 0,D= β a p a β a ϕ [ p a ] ss ss θ θ c 0 /c. A0 2

3 Combinin A7 and A0 ives D = α D " γ ass β a p a β a ϕ [ p a ] # ss ss p a α α ϕ [ p a ] α α θ θ θ /c. c 0 A were = D wit D determined by A8 independently. Apparently, te rit-and side of A increases monotonically in. Tis implies tat ier D leads to ier and ass. Moreover, te rit-and side of A0 also increases monotonically in D, wic sests tat, by casin ier, ierd will also ive ier f ss 0,D. Tis proves roposition Approximatin Vale Fnctions wit Krsell & Smit 998 Approac Te key comptational task is to map F, te plant distribtion across aes and idiosyncratic prodctivity, iven demand level D, into a set of vale fnctions V θ e,a; F, D. To make te state space tractable, we define a variable sc tat: F = a θ e f θ e,a qθ e,a A2 were f θ e,a, as a component of F, measres te mass of plants wit expected idiosyncratic prodctivity θ e and ae a. Apparently, F, D A = D F 0 = D H F, D. A3 F 0 is te pdated plant distribtion after entry and exit and F 0 = H F, D; F, D is te eqilibrim price in a period wit initial areate state F, D. lin A3 into 4 ives π a, θ; F, D = α D H F, D θ e γ a. A4 Ts, te areate state F, D and its law of motion elp plants to predict ftre profitability by sestin seqences of s from today onward nder different pats of demand realizations. Te qestion ten is: wat is te plant s critical level of knowlede of F tat allows it to predict te seqence of 0 s over time? Alto plants wold ideally ave fll information abot F, tis is not comptationally feasible. 3

4 {} H H x, D : lo 0 = lo H x, D l : lo 0 = lo R 2 for D : for D l : for D standard forecast error : % for D l : % for D maximm forecast error : % for D l : % Den Haan & Marcet test statistic χ 2 7 Table : Te Estimated Laws of Motion and Measres of Fit Terefore we need to find an information set tat delivers a ood approximation of plants eqilibrim beavior, yet is small eno to redce te comptational difficlty. We look for an tro te followin procedre. In step, we coose a candidate. In step 2, we postlate perceived laws of motion for all members of, denotedh, sc tat 0 = H,D. In step 3, iven H, we calclate plants vale fnctions on a rid of points in te state space of applyin vale fnction iteration, and obtain te correspondin indstry-level decision rles entry sizes and exit aes across areate states. In step 4, iven sc decision rles and an initial plant distribtion. We simlate te beavior of a continm of plants alon a random pat of demand realizations, and derive te implied areate beavior a time series of. Instep5,wesetestationaryreionoftesimlatedseriesto estimate te implied laws of motion and compare tem wit te perceived H ;ifdifferent, we pdate H, retrn to step 3 and contine ntil converence. In step 6, once H converes, we evalate te fit ofh in terms of trackin te areate beavior. If te fit is satisfactory, we stop; if not, we retrn to step, make plants more knowledeable by expandin, and repeat te procedre. We start wit = {} plantsobserve instead of F. We frter assme tat plants perceive te seqence of ftre comin 0 s as dependin on notin more tan te crrent observed and te state of demand. Te perceived law of motion for is denoted H x so tat 0 = H x, D. We ten apply te procedre described above and simlate te beavior of a continm of plants over 0000 periods. Te reslts are presented in Table 5. As sown in Table 5, te estimated H x is lo-linear. Te fit ofh x is qite ood, as sested by te i R 2, te low standard forecast error, and te low maximm forecast error. Te ood fit wen = {} implies tat plants perceivin tese simple laws of motion make only small mistakes in forecastin ftre prices. Toexploreteextenttowicteforecast error can be explained by variables oter tan, we implement te Den Haan and Marcet 994 test sin instrments [,,μ a,σ a,γ a,κ a,r ],wereμ a, σ a, γ a, κ a,r are te mean, standard deviation, skewness, and krtosis of te ae distribtion of plants, and 4

5 {, σ a } booms lo : lo 0 = lo 0.262σ a boomsσ a : σ H 0 a = lo σ a recessions lo : lo 0 = lo σ a recessionsσ a : σ 0 a = lo 0.975σ a booms lo : R 2 recessions lo : booms σ a : standard forecast error maximm forecast error Den Haan & Marcet test statistic χ 2 7 recessionsσ a : booms lo :. 0 8 % recessions lo : % booms σ a : % recessionsσ a : % booms lo : % recessions lo : % booms σ a : % recessionsσ a : % Table 2: Te Estimated Laws of Motion wit two moments and Measres of Fit te fraction of nsre plants, respectively. Te test statistic is , well below te critical vale at te % level. Tis sests tat iven te estimated laws of motion, we do not find mc additional forecastin power contained in oter variables. Neverteless, we expand frter to inclde σ a, te standard deviation of te ae distribtion of firms. Te reslts wen = {, σ a } aresownintable. 5