Online Appendix for Demand Fluctuations in the Ready-Mix Concrete Industry

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1 Onlne Appendx for Demand Fluctuatons n the Ready-Mx Concrete Industry Allan Collard-Wexler NYU Stern and NBER November 29, 2012 Contents A Stochastc Algorthm 2 A.1 Dscrete Acton Stochastc Algorthm: Termnaton Crtera B Modfed DASA to Compute the Gamma functon 5 C Market Fxed Effects 6 C.1 Condtonal Choce Probablty Estmaton C.2 Alternatve Market Categores from Market Fxed Effects D Identfcaton of Fxed Costs, Sunk Costs and Scrap Values 11 E Smulated Indrect Inference Estmaton 12 E.1 Consstency Proof F Seral Correlaton 15 G Prce Data 17 H Addtonal Tables and Fgures 18 1

2 A Stochastc Algorthm To compute the strateges assocated wth a Nash Equlbrum of the dynamc game, I adapt the stochastc algorthm of Pakes and McGure (2001) to the dscrete acton setup used n ths paper snce the state space has up to 1.4 mllon states. 1 I defne the ht counter, denoted h(a, x), as the number of tmes the locaton l = (a, x) has been vsted by my algorthm. The ht counter s mportant snce t allows me to keep track of the precson of the computaton of W (a, x) and Ψ[a x] usng the Dscrete Acton Stochastc Algorthm (henceforth DASA). Gven a reward and transton cost functon r( ) and τ( ), as well as a demand transton matrx D, the DASA computes a soluton to the dynamc game, characterzed by the choce specfc value functon W (a, x), and the condtonal choce probabltes Ψ. Algorthm Dscrete Acton Stochastc Algorthm (DASA) An teraton k of the DASA follows these steps: 1. Start n a locaton l k = {a k, xk }, wth values for W k, Ψ k, and h k n memory. 2. Draw an acton profle for other players a k r Ψk [a k r x k ]. Gven the acton profle a k = {a k, ak } draw a state n the next perod xk+1 : x k+1 a k D[M k+1 M k ] ι(x k+1 a k, x k ) (1) where ι(x k+1 a k, xk ) s the updatng functon, whch updates the frm s state based on a frm s acton and the frms largest sze n the past Increment the ht counter (how often you have vsted the state-acton par): h k+1 (a k, xk ) = h k (a k, xk ) Compute the value R of the acton as: where E(ε x k+1, Ψ k ) = R =r(a k+1, x k+1 ) τ(a k+1, x k ) + β j A W k (j, x k+1 )Ψ k [j x k+1 ] + βe(ε x k+1, Ψ k ) (2) ( γ ) j A ln(ψk [j x k+1 ])Ψ k [j x k+1 ] (where γ s Euler s Constant). 1 There are 10 frms, 7 possble states per frm and n the most complex model 50 demand states. I reduce the sze of the state from to 1.4 mllon by usng the assumpton of exchangeablty descrbed by Gowrsankaran (1999). 2 Later n the paper I make the frm s prevous state relevant to the transton cost. Specfcally, f the frm s state s x k = {x C,k a k, x k ) s: ι(x k+1, x P,k },.e. the current sze x C,k x k+1 = and the largest sze n the past x P,k, then the updatng functon { {a k, a k } f a k x P,k {a k, x P,k } f a k < x P,k 2

3 5. Update the W-functon: W k+1 (a k, x k ) = α[a k, x k ]R + ( ) 1 α[a k, x k ] W k (a k, x k ) (3) where α = 1 h k+1 (a k,xk ).3 6. Update the Polcy Functon Ψ for state x k : Ψ k+1 [a k x k ] = exp ( W k+1 (a k, xk ) ) j A exp (W k+1 (j, x k )) (4) for all actons a k A. 7. Draw a new acton a k+1 Ψ k+1 [ x k+1 ]. 8. Check the stoppng rule. 4 If t s not satsfed, update the current locaton to l k+1 = {a k+1, x k+1 }, ncrement k to k + 1 and return to step 1. The stoppng rule for ths algorthm s based on Fershtman and Pakes (2012) whch compares the W-functon to a smulated average based on rewards from steps 2 and 4 for states that are recurrent. If the W-functon s exact, then the squared dfference between between these two objects (weghted by the ergodc dstrbuton) can be accounted for by smulaton error. The stoppng rule s presented n the next secton. The ntal values of W n the stochastc algorthm are mportant, snce f I ntalze W (a, x) wth a hgh value, the algorthm mght get trapped at ths state. To fnd ntal values of W, I use Value Iteraton n whch I smulate the expectaton va Monte-Carlo. A.1 Dscrete Acton Stochastc Algorthm: Termnaton Crtera The stoppng rule s based on the fact that f I have the correct W functon, then t satsfes the Bellman equaton. However, t s computatonally expensve to calculate the W-functon exactly, nstead we can approxmate the value functon usng forward smulaton. Consder the locatons R A X defned as the state-acton pars vsted n the last 1 mllon teratons (keep a ht counter that tracks the last 1 mllon teratons denoted rh(l)). Algorthm Fershtman-Pakes Stoppng Rule (FPStop) 3 The man problem wth the stochastc algorthm s: 1- makng sure the entre state space s searched, 2- ensure fast learnng about the W functon at the start of the algorthm and 3- makng sure that the convergence propertes of the algorthm are satsfed. Frst, I ntalze the startng W usng farly hgh values so that the algorthm vsts all states before lowerng the estmate of W. Second, at the start of the algorthm I use α = 1/ h k+1 (a k, xk ) to ensure that ntally naccurate W s get updated quckly. As well, I reset the ht counter after 20 mllon teratons to ensure that the frst rounds of updates are down-weghted. Thrd, n the fnal stage of the algorthm I swtch to the α = 1/h k+1 (a k, x k ) update rule whch satsfes the convergence propertes of Stochastc Approxmaton Algorthms descrbed n Powell (2007) on page 216. However, n the context of a game ths condton on α may not be enough to guarantee convergence. 4 Snce the stoppng rule s computatonally ntensve relatve to a sngle teraton of the DASA, t s better to check the stoppng rule only every several mllon teratons. 3

4 For all locatons l = {a, x} A X whch have been vsted n the last 1 mllon teratons: 1. Compute an approxmaton to the the W-functon usng a one step forward smulaton (denoted W ). For q = 1,..., Q (I use Q = ): (a) Draw a state tomorrow x q gven locaton l = {a, x}. (b) Get rewards: R q =r(a q x q, θ) + τ(a q x, θ) + β W (j, x q )Ψ[j x q ] j A + β γ ln(ψ[j x q ])Ψ[j x q ] j A (5) (c) Compute the approxmaton to the W-functon: W (l) = 1 Q Q R q (6) 2. Compute the dfference n value functons weghted by the recent ht counter rh: q=1 1 τ = l rh(l) rh(l) ( W (l) W (l)) 2 (7) l If the test statstc τ s small enough, then we can argue that we have a good approxmaton. In practce I use the recent ht counter weghted R 2 between W (l) and W (l) beng greater than Ths usually happens after as lttle as 50 mllon teratons, and t s usually more effcent to run the DASA for 150 mllon teratons (.e. 15 mnutes) whch leads to a W functon whch satsfy the FPStop crtera. Furthermore, n ths applcaton there are only about state-acton pars (where the acton s not 0) that are vsted n the last 1 mllon teratons. Thus the ergodc class R s qute small compared to the sze of the entre state space. 4

5 B Modfed DASA to Compute the Gamma functon I use a modfed DASA to compute the Γ functon. The two dfferences are that () I shut down the polcy functon update n the DASA, and () I compute the net present value of the components of rewards rather than the rewards themselves (whch would requre me to have nformaton on the parameters θ). 5 Algorthm Γ-Compute Dscrete Acton Stochastc Algorthm (GC-DASA) An teraton k of the GC-DASA s gven by the followng steps: 1. Start n a locaton l k = {a k, xk } wth values for Γ k and h k n memory. 2. Draw an acton profle a k a k 1(a = a k ) ˆP [a k xk ] and a state n the next perod x k+1 gven acton profle a k : x k+1 a k ˆD[M k+1 M k ] ι(x k+1 a k, x k ) (9) where ι(x k+1 a k, xk ) s the updatng functon, whch updates the frm s state based on a frm s acton and the frms largest sze n the past. 3. Increment the ht counter (how often you have vsted the state-acton par): h k+1 (a k, xk ) = h k (a k, xk ) Compute th component of payoffs R of the acton a k as: R =r (a k, x k+1 ) + β j A Γ k, (j, x k+1 ) ˆP [j x k+1 ] (10) 5. Update the Γ-functon: where α = 1 h k+1 (a k,xk ). 6. Update current locaton to l k+1 = {a k, xk+1 }. The stoppng rule s Fershtman and Pakes (2012) s. 5 I could have computed the Γ P µ usng forward smulaton,.e.: Γ k+1, (a k, x k ) = αr + (1 α)γ k, (a k, x k ) (11) Γ P µ (a, s) 1 K K k=1 t=0 β t ρ(a tk, x tk ) (8) where the sequence of states x tk can be smulated usng demand transton process ˆD and the choce probabltes for frms ˆP. However, there are about states and 4 actons, thus I would need to do ths forward smulaton 1.4 mllon tmes the number of smulaton draws K. 5

6 C Market Fxed Effects C.1 Condtonal Choce Probablty Estmaton In the man model, I use a market categores model whch s meant to to mmc the ncluson of market fxed effects. These market fxed effects are crtcal to the estmaton of the model snce persstent market level dfferences n proftablty lead to upward bas on the effect of competton. Ths bas, especally when t nduces postve effects of competton, leads to very aberrant ndustry dynamcs such as havng a market flp between 0 and 10 plants due to a postve externalty due to competton. The goal of ths secton s to motvate the use of market category effects based on the average number of frms n a market over tme, and explan why other plausble correctons for market fxed effects usng average constructon employment or the number of plants n a pre-perod, do not gve the rght answer. I consder the followng dfferent specfcatons of the market category effects: a) No Market Effects. b) Average Number of Frms n Market (rounded to nearest nteger). In the man estmates of the model, I use the average number of frms n the market rounded to the nearest nteger. However, ths approach suffers from an endogenety problem. To put t most clearly, consder the followng dynamc, two frm model of the type: a t = αa t + βa t 1 + ɛ t (12) If I nclude a t+1 n the above regresson, then I am ncludng an endogenous regressor snce a t+1 s a functon of a t whch n turn depends on ɛ t, and more broadly on the entre hstory of ɛ t for t < t. c) Average Number of Frms n Market n years before ths one (rounded to nearest nteger). However, f I nclude lagged a t 2 then ths s not an endogenous regressor, snce there s no dependance on a t 2 except through a t 1 whch s already ncluded n the regresson. The only ssue wth usng the average number of frms n the market n prevous years s that a market can swtch categores over tme whch makes for a more dffcult state space to deal wth, whch s the reason that I do not use these market category controls n the man part of the paper. d) Average Number of Frms n the perod, wth data on the post 1983 perod. Notce that for ths model, I am usng the early perod to condton the number of frms n the market. Ths s a verson of model c), but but the pre-perod on whch I condton does not change wthn a market. e) Average Constructon Employment. Here I classfy markets by the average level of constructon employment from Ths s an exogenous classfcaton scheme snce t does not depend on what ready-mx concrete frms are dong. f) Market Fxed Effects (Condtonal Logt). Table 1 presents estmates from the bnary logt model of entry and ext for specfcatons (a)- (f). I have chosen the bnary logt model snce t allows me to use the condtonal logt wth market 6

7 fxed effects. 6 Column (a) shows estmate wthout market category controls (henceforth referred to as no market effects), whle column (f) shows estmates wth market fxed effects (henceforth referred to as market fxed effects), whle columns (b)-(e) show dfferent market category controls. Columns (b) and (c),.e. wth market controls based on the average number of plants and the average number of plants n the perods before ths one, are smlar to the market fxed effect estmates n column (f). Lkewse, columns (d) and (e) show estmates that are more smlar to the no market effects estmates n column (a). The effect of past plant sze on actvty are farly smlar n all of these estmates, wth smaller effects of plant sze n the market fxed effect specfcatons (f), (b) and (c) than the no market effect specfcaton (a), (d) and (e). Unobserved heterogenety between markets s loaded onto varable ndcatng state dependence, such as past plant sze. The effect of log constructon employment s hgher at to n the no market effect models (a), (d) and (e) than n the market fxed effect estmates, whch have estmates from to These hgher effects of demand are due to the fact that frms are far more lkely to react to cross-sectonal dfferences n demand (whch are more lkely to be persstent) than to year to year changes n demand. Lkewse, the effect of the second compettor (whch wll representatve of the effect of competton more broadly) vares from to n the no market effect columns (a), (d) and (e), but ranges from to n the market fxed effect columns (f), (b) and (c). Ths s ndcatve of the fact that unobserved dfferences n the proftablty of a market wll be correlated wth the number of plants n the market. There are two man conclusons from the table that are relevant for my choce of market categores. Frst, the market categores based on the ether the average number of frms (b) or the average number of frms n all perods before today (c) do a good job n mmckng true market fxed effects. However, usng categores based on the number of frms before 1983 (d), or usng nformaton about the average level of constructon demand (e) do not replcate market fxed effect estmates, and n fact mmc no havng any market controls whatsoever. Second, whle t s true that usng the average number of frms over tme condtons on an endogenous varable, I can equally easly use the lagged number of frms whch does not condton on an endogenous varable and obtan vrtually dentcal results. Thus the ssue of endogenety s of lmted practcal mportance n the use of the average number of frms over tme as a groupng. C.2 Alternatve Market Categores from Market Fxed Effects In ths secton, I present an alternatve procedure for constructng market categores based on values of the market fxed effect. Consder the bnary logt model: y mt = 1(αX mt + ξ m > ɛ mt ) I can construct market categores based on estmates of the market fxed effect varable ɛ mt, usng the followng procedure: 1. Step 1: Run a condtonal logt (wth market fxed effects) on the number of actve plants to recover parameters ˆα,.e. everythng except the market fxed effect ξ m. Note that we can get α wthout the problem of ncdental parameters usng a condtonal logt. 6 Techncally, I can also use a multnomal condtonal logt, but the number of categores I need to condton on becomes farly large. As well, I am not presentng margnal effects here snce the condtonal logt does not estmate the market fxed effects. 7

8 Dependant Varable: Actvty (a) (b) (c) (d) (e) (f) Condtonal Logt Log County 0.133*** ** *** 0.129*** 0.099*** Constructon Employment (0.011) (0.011) (0.010) (0.015) (0.019) (0.023) Frst Compettor *** *** *** *** *** *** (0.052) (0.051) (0.048) (0.066) (0.052) (0.043) Second Compettor *** *** *** (0.036) (0.037) (0.036) (0.047) (0.037) (0.030) Thrd Compettor *** *** *** (0.044) (0.044) (0.043) (0.058) (0.044) (0.036) Log Compettors above *** *** *** (0.029) (0.028) (0.028) (0.040) (0.029) (0.025) Small 5.889*** 5.703*** 5.720*** 5.977*** 5.887*** 5.585*** (0.037) (0.035) (0.035) (0.047) (0.037) (0.025) Small, 5.665*** 5.388*** 5.393*** 5.707*** 5.657*** 5.220*** Medum n Past (0.048) (0.045) (0.045) (0.057) (0.048) (0.033) Small, 4.866*** 4.636*** 4.643*** 4.944*** 4.865*** 4.450*** Large n Past (0.065) (0.063) (0.062) (0.075) (0.065) (0.041) Medum 7.503*** 7.292*** 7.315*** 7.696*** 7.495*** 7.234*** (0.057) (0.055) (0.055) (0.075) (0.057) (0.050) Medum, 7.511*** 7.237*** 7.251*** 7.585*** 7.503*** 7.122*** Large n Past (0.080) (0.079) (0.079) (0.094) (0.081) (0.074) Large 7.671*** 7.446*** 7.450*** 7.724*** 7.676*** 7.436*** (0.056) (0.054) (0.054) (0.068) (0.056) (0.050) Market Classfcaton Varable Average Number of Plants X Lagged Average Plants X Before 1983 Average Plants X Constructon Employment X Category *** 1.118*** 0.225*** 0.132** (0.036) (0.032) (0.062) (0.049) Category *** 1.836*** 0.348*** 0.199** (0.050) (0.047) (0.058) (0.061) Category *** 2.424*** 0.482*** 0.169* (0.063) (0.062) (0.061) (0.082) Constant *** *** *** *** *** (0.065) (0.066) (0.062) (0.089) (0.090) Observatons Markets Log-Lkelhood χ (Standard Errors are Clustered by Market). Table 1: Market Effects n the Bnomal Logt Regresson of Entry and Ext 8

9 2. Step 2: Create the varable Z ˆα = ˆαX, the part of the covarates that s not the market fxed effect. 3. Step 3: Run the logt on the model: y mt = 1(Z α mt + ξ m > ɛ mt ) whch can be done separately for each market n the data. Note that ths means that I am estmatng market fxed effects ˆξ m (for whch I need the number of tme perods to be large to do ths). 4. Step 4: Use estmated ˆξ m to form groups of markets. The estmated ˆξ m has the dstrbuton n Table 2. Percentle ˆξ m Table 2: Dstrbuton of Market Fxed Effect ˆξ m. Table 3 shows the bnary logt results on actvty, where columns IV, V, VI and VII shows fxed effects constructed by roundng ˆξ m nto 4, 10, 20 and 40 categores (where each category contans the same number of markets). Notce that usng these fxed effect categores yelds smlar results to Columns I (fxed-effects) and Column II (market categores). However, to match the market category effects n Column III, n partcular to get smlar effects of log compettors above 4, I need to have at least 10 market groups. In the estmates I wll show you, I use 20 market categores of ˆµ m. 9

10 Dependant Varable: Actvty I II (FE ) III (µ) IV V VI VII Log County Constructon Employment (0.01) (0.02) (0.01) (0.01) (0.01) (0.01) (0.01) Frst Compettor (0.05) (0.04) (0.05) (0.05) (0.05) (0.05) (0.05) Second Compettor (0.04) (0.03) (0.04) (0.04) (0.04) (0.04) (0.04) Thrd Compettor (0.04) (0.04) (0.04) (0.04) (0.04) (0.04) (0.04) Log Compettors above (0.03) (0.03) (0.03) (0.03) (0.03) (0.03) (0.03) Small (0.04) (0.03) (0.04) (0.03) (0.03) (0.03) (0.03) Small, Medum n Past (0.05) (0.03) (0.05) (0.05) (0.05) (0.05) (0.05) Small, Large n Past (0.07) (0.04) (0.06) (0.07) (0.07) (0.07) (0.07) Medum (0.06) (0.05) (0.06) (0.06) (0.06) (0.06) (0.06) Medum, Large n Past (0.08) (0.07) (0.08) (0.08) (0.08) (0.08) (0.08) Large (0.06) (0.05) (0.05) (0.06) (0.06) (0.06) (0.06) Market Classfcaton Varable 4 Fxed Effect Groups X 10 Fxed Effect Groups X 20 Fxed Effect Groups X 40 Fxed Effect Groups X Observatons Markets Log-Lkelhood χ Table 3: Bnary Logt Regressons of the decson to have an actve plant wth market fxed effects and market category effects. 10

11 D Identfcaton of Fxed Costs, Sunk Costs and Scrap Values To show the ntuton behnd the dentfcaton of fxed costs, sunk costs and scrap values, I wll use a smplfed model. Suppose that I have varable profts V (X) where X are tme nvarant proft shfters, fxed cost f, entry costs ψ and ext costs φ. Then the entry and ext rules n a statonary envronment are: Enter ff: Ext ff: t=0 t=0 β t (V (X) f) = V (X) f 1 β β t (V (X) f) = V (X) f 1 β ψ < φ In ths case t s clear that f, ψ and φ are lnearly ndependent. Addng future ext rates, δ (whch at ths pont are generated by shocks to the ext value φ + ɛ that I don t want to put n ths smple model). Ths wll adjust these equatons to Enter ff: Ext ff: V (X) f 1 β(1 δ) + φ 1 βδ ψ V (X) f 1 β(1 δ) + φ 1 βδ < φ Agan, we have the same collnearty problem as before. However, f future ext rates are dfferent n dfferent markets (say due to dfferences n future demand shocks such as a market at the top demand level, versus one at the bottom demand level), then we have a δ(x) whch depends on the state X. Ths allows us to separately dentfy f and φ n the ext equaton gven we know V (X) and δ(x). V (X) f 1 β(1 δ(x)) + φ 1 βδ(x) < φ Now gven we know ˆf and ˆφ, the entry equaton becomes V (X) ˆf 1 β(1 δ(x)) + ˆφ 1 βδ(x) ψ So formally I can separately dentfy f, φ and ψ. What makes ths dffcult s that I need enough varaton n δ(x) for ths to work, and ths varaton s not very mportant ether n Monte-Carlo smulatons, or n the data. 11

12 E Smulated Indrect Inference Estmaton The smulated ndrect nference estmator used n equaton (19) on page 14 uses the choce probabltes Ψ(a x, Γ, θ) as an outcome vector,.e. ỹ n = Ψ(a x, Γ, θ). Typcally, one would sample outcomes y n from the choce probabltes Ψ(a x, Γ, θ). I can show that usng the ỹ n s equvalent to samplng actons as the number of actons tends to nfnty. Denote the outcome vectors yn s as: 1(a s yn s n = small) = 1(a s n = medum) (13) 1(a s n = bg) where the acton a s n Ψ( x, Γ, θ) s drawn from the choce probabltes Ψ. The smulaton draws are ndexed from s = 1,, S. The β S (θ) coeffcent s estmated usng outcome vectors {yn} s {s=1,,s},n. The crteron functon usng S smulaton draws of actons s thus: ( ) ( ) Q S (θ) = ˆβ β(θ)) W ˆβ β(θ) (14) E.1 Consstency Proof In ths secton I wll show condtons under whch the procedure I use n ths paper s a consstent estmator of θ. Specfcally, I wll show the condtons that need to be satsfed for Proposton 1 on page S89 n Goureroux and Montfort (1993) dealng wth the consstency of ndrect nference estmators, to be satsfed. Defne the crteron functon used to compute S N,K (β, θ) = N K n=1 k=1 + [ 1 [ ( ) ] 2 1 a k n = small Z n β s ( a k n = medum β(θ) (for a gven value of θ) as: ) ] 2 [ Z n β m + 1 (15) ( ) ] 2 a k n = large Z n β l where N denotes the number of observatons and K denotes the number of smulaton draws to draw actons a k n from the polcy functon ψ(a n x n, θ, Γ( ˆP N, ˆD N )). Note that S N,K (β, θ) s the crteron used n OLS estmaton, just the sum of squared errors. The frst step s to show that I can replace draws of a k n wth the actual polcy functon ψ, or n other words S N,K (β, θ) a.s. S N, (β, θ) unformly as K. Theorem 1 As the number of smulaton draws K tends to nfnty, S N,K (β, θ) a.s. S N, (β, θ) unformly. Proof: I wll show the proof usng only the choce to be small to lghten the notaton, but the proof extends to as many actons as I want: S N,K (β, θ) = = N n=1 1 K K k=1 N (Z n β s ) 2 + n=1 [ ( ) ] 2 1 a k n = small Z n β s N K n=1 k=1 1 ( K 1 a k n = small ) 2 N 2 Z n β s n=1 K k=1 1 ( ) K 1 a k n = small (16) 12

13 As K, K k=1 1 K 1 ( a k n = small ) ψ(a n = small x n, θ, Γ( ˆP, ˆD)) snce ths s just an average, and K k=1 1 K 1 ( a k n = small ) 2 ψ(an = small x n, θ, Γ( ˆP, ˆD)) 2. Thus I can rewrte S N, (β, θ) as: Fx me S N, (β, θ) = = N (Z n β s ) 2 + n=1 2 N ψ(a n = small x n, θ, Γ( ˆP, ˆD) 2 n=1 N Z n β s ψ(a n = small x n, θ, Γ( ˆP, ˆD)) n=1 N [ψ(a n = small x n, θ, Γ( ˆP, ˆD)) ] 2 Z n β s n=1 Second, I need to show that S N, (β, θ) a.s. S 0, (β, θ) as N. The frst condton s that the lnear probablty estmator s consstent, whch s just an outcome of the OLS estmator beng a consstent estmator, whch s a standard proof. However, I am not usng the true Γ 0 (P 0, D 0 ) but an estmate of Γ( ˆP, ˆD) due to samplng error n the condtonal choce probabltes P and the demand transton process D, as well as approxmaton error n the computaton of Γ. The CCP s ˆP N P 0 whch happens snce I am usng a consstent estmator of the CCP s, just a parametrc multnomal logt, whch s consstent usng the usual proofs on the consstency of M- estmators. Lkewse ˆD N D 0 as N snce I am usng a consstent estmator of D, just a bn estmator where the number of bns s fxed as N. Now the next pont s to show that Γ L (P 0, D 0 ) Γ 0 (P 0, D 0 ) as the number of teratons L n the DASA goes to nfnty. It wll be dffcult to show convergence of the DASA, snce to my knowledge there s no proof of the convergence of algorthms that compute the solutons to games (n contrast to sngle agent problems). However, the Fershtman and Pakes (2012) convergence crteron can be used to check the convergence of the DASA, and I can send the tolerance of the Fershtman-Pakes crteron to 0 as N. 7 The convergence of Γ L (P 0, D 0 ) Γ(P 0, D 0 ) mples the convergence of S N,K (β, θ) S, (β, θ) as K and N. Ths satsfes assumpton (A2) n Indrect Inference. Assumpton (A3) of Indrect Inference requres that: (17) β(θ) = argmax β S, (β, θ) (18) be a contnuous functon and have a unque value. Contnuty s an outcome of the OLS structure of S, whle unqueness occurs f Z n s full rank and the dmenson of β s smaller than the dmenson of Z n. The fnal condton, (A4) requres that β(θ) be one to one and have full rank. I wll assume ths condton, but notce that the dmenson of β s larger than the dmenson of θ and I have checked that β(θ) s full rank n the estmaton of model. 7 Notce that snce there s a full support shock ɛ to the payoffs of any actons, Γ s computed correctly on the entre state space S, snce the set of recurrent ponts s the entre state space,.e. S = R. The DASA used to compute Γ s a verson of the Q-learnng algorthm, where consstency proof are provded for the sngle agent (non-game verson) n Propostons 5.5 and 5.6 on page n Bertsekas and Tstskls (1996) show condtons under whch the DASA s (whch s the game verson of a Q-learnng algorthm) computaton of Γ : converges wth probablty one to Γ 0. These condtons are (1) that polces are proper,.e. there s a postve probablty that a frm wll ext after t perod, whch s true n ths context due to the full support of the shock dstrbuton for each acton, ncludng the choce to ext; and (2) for mproper polces, there s a negatve nfnte value of W for at least one state. Unfortunately, there s to my knowledge no proof wth shows the convergence of the Q-learnng algorthm n the context of a game. 13

14 Snce condtons (A1), (A2), (A3) and (A4) are satsfed, then ˆθ defned as the mnmzer of: ( ) ( ) Q(θ) = ˆβ β(θ)) W ˆβ β(θ) (19) wll be a consstent estmator of θ as N. 14

15 F Seral Correlaton The assumpton that the unobserved state ɛ t a are..d. logts mples the followng assumpton: Assumpton 2 (Seral Independence) Unobserved states are serally ndependent,.e. Pr(ε t εk ) = Pr(ε t ) for k t. Seral ndependence of unobserved components of a frm s proftablty s volated by any form of persstent productvty dfference between frms, or long term reputatons of ready-mx concrete operators. Note that n the context of a dynamc game, unobserved states are a frst-order problem snce the sze of the frm-level state x t s severely restrcted by the dffculty of keepng track of the jont dstrbuton of the states of all frms. I smulate the age profle of ext usng the ext and sze changes n Table??, whch captures what the age profle of ext would look lke n the absence of selecton on an unobserved state. Wth a serally correlated unobserved state, as plants age ther ext rate falls due to the effect of selectng out plants wth a bad unobserved state. Fgure 1 shows the ext hazard wth age n the data and smulated data. Both the data and the smulaton have the same average ext rate of about 6%, but the data has a somewhat steeper declne n ext rates over tme, so a plant aged 20 years old has an ext rate of about 3.5% n the data, whle the smulated data yelds a ext rate of about 5.2%. Ths s consstent wth most models of ndustry dynamcs wth a serally correlated unobserved state, and the actve or passve learnng models of Pakes and Ercson (1998) and Jovanovc (1982), but s a small effect compared to other ndustres such as restaurants where we would worry more about unobserved states. I do not deal wth seral correlaton and both the estmates and counterfactuals wll be contamnated by ths problem. 15

16 Ext Hazard and Plant Age 7% Model Smulaton 6% Ext Rate 5% 4% Data 3% Plant Age Fgure 1: The data predcts a slghtly steeper declne of the ext hazard wth age. 16

17 G Prce Data The Census Bureau does not generally collect prce data. Ths job s left to the Bureau of Economc Analyss and the Bureau of Labor Statstcs. However, followng Syverson (2004a) we can generate prces usng the followng equaton: p t (c) = s t(c) q t (c) whch s just sales of the commodty dvded by quantty sold. Whle these prces may be good ndcators of prce dsperson (the applcaton Syverson consders), they are partcular poor measures of actual plant prces, wth an nterquartle range over 2 log ponts (the thrd quartle s 100 tmes bgger than the frst prce quartle). Ths s probably because of how measurement error n the numerator and especally the denomnator nteract. To reduce the mpact of mputed data and measurement error on the dsperson of prces, I apply a verson of Syverson (2004b) s procedures: 1. Hot Imputes n the data are dentfed as prces that satsfy the followng: (20) p t p t j < for some and j n the data (21) I drop all prces that are hot mputes. Notce that ths procedure wll also elmnate cold mputes, defned as prces whch equal the mode n the current year. 2. I trm the data by droppng observatons that are less than 1/5 or more than 5 tmes the medan prce for the current year. The deflated data s computed by p Dt = p t /cpt where I normalze the cp n 1977 to be equal to 1 (.e. cp t = raw cp t /raw cp 1977 ). Ths elmnate dfferences n prce level across tme, but does not ncorporate dfferences n prces between regons. 17

18 H Addtonal Tables and Fgures Year Brth Contnuer Death ,737 N.A , , , , , , , , , , , , , , , , , , , , , , , Table 4: The number of Brths, Deaths and Contnuers s farly stable over the last 25 years 18

19 References Bertsekas, D., and J. Tstskls (1996): Neuro-Dynamc Programmng. Athena Scentfc. Fershtman, C., and A. Pakes (2012): Dynamcs Games wth Asymmetrc Informaton: A Framework for Emprcal Work, Quarterly Journal of Economcs. Goureroux, C., and A. Montfort (1993): Indrect nference, Journal of Appled Econometrcs, 8, S85 S118. Gowrsankaran, G. (1999): Effcent Representaton of State Spaces for Some Dynamc Models, Journal of Economc Dynamcs and Control, 23(8), Jovanovc, B. (1982): Selecton and the Evoluton of Industry, Econometrca, 50(3), Pakes, A., and R. Ercson (1998): Emprcal Applcatons of Alternatve Models of Frm and Industry Dynamcs, Journal of Economc Theory, 79(1), Pakes, A., and P. McGure (2001): Stochastc Algorthms, Symmetrc Markov Perfect Equlbrum, and the Curse of Dmensonalty, Econometrca, 69(5), Powell, W. B. (2007): Approxmate Dynamc Programmng: Solvng the Curse of Dmensonalty, Wley Seres n Probablty and Statstcs. John Wley and Sons. Syverson, C. (2004a): Market Structure and Productvty: A Concrete Example, Journal of Poltcal Economy, 112(6), (2004b): Product substtutablty and productvty dsperson, Revew of Economcs and Statstcs, 86(2),