1 Lecture Notes - Production Functions - 1/5/2017 D.A.

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1 1 Lecture Notes - Producton Functons - 1/5/2017 D.A. 2 Introducton Producton functons are one of the most basc components of economcs They are mportant n themselves, e.g. What s the level of returns to scale? How do nput coeff cents on captal and labor change over tme? How does adopton of a new technology affect producton? How much heterogenety s there n measured productvty across frms, and what explans t? How does the allocaton of frm nputs relate to productvty Also can be mportant as nputs nto other nterestng questons, e.g. ndustry evoluton, evaluaton of frm conduct (e.g. colluson) dynamc models of For ths lecture note, we wll work wth a smple two nput Cobb-Douglas producton functon Y = e β 0K β 1 L β 2 e ε where ndexes frms, K s unts of captal, L s unts of labor, and Y s unts of output. (β 0, β 1, β 2 ) are parameters and ε captures unobservables that affects output (e.g. weather, sol qualty, management qualty) Take natural logs to get: y = β 0 + β 1 k + β 2 l + ε Ths can be extended to Addtonal nputs, e.g. R&D (knowledge captal), dummes representng dscrete technologes, dfferent types of labor/captal, ntermedate nputs. Later we wll see more flexble models 3 Endogenety Issues y = f (k, l ; β) + ε y = f (k, l, ε ; β) (wth scalar/monotonc ε ) y = β 0 + β 1 k + β 2 l + ε Problem s that nputs k, l are typcally choce varables of the frm. Typcally, these choces are made to maxmze profts, and hence wll often depend on unobservables ε. Of course, ths dependence depends on what the frm knows about ε when they make these nput choces. 1

2 Example: Suppose a frm operatng n perfectly compettve output and nput markets (wth respectve prces p, r, and w ) perfectly observes ε before optmally choosng nputs. Proft maxmzaton problem s: max K,L p e β 0K β 1 L β 2 e ε r K w L FOC s wll mply that optmal choces of K and L (k and l ) wll depend on ε. Intuton: ε postvely affects margnal product of nputs. Hence frms wth hgher ε s wll want to use more nputs. As a result, one cannot estmate y = β 0 + β 1 k + β 2 l + ε usng OLS because k and l are correlated wth ε. Generally one would expect coeff cents to be postvely based. Smlar problems would arse n more complcated models (e.g. non-perfectly compettve output or nput markets, ε only partally observed), except the specal case where the frm has no knowledge of ε when choosng nputs If k s a "less varable" nput than l, one mght expect the frm to have less knowledge about ε when choosng k (relatve to l ). Generally, ths wll mply k wll be less correlated wth ε than l s. So one mght expect more bas n the labor coeff cent. Note: we wll generally assume that the unobservables ε are generated or evolve exogenously,.e. they are not choce varables of the frm. Thngs get consderably harder when the unobservables are choce varables of the frm. WLOG, lets thnk about ε havng two components,.e. y = β 0 + β 1 k + β 2 l + ω + ɛ where ω s an unobservable that s predctable (or partally predctable) to the frm when t makes ts nput decsons, and ɛ s an unobservable that the frm has no nformaton about when makng nput decsons (e.g. ω represents average weather condtons on a partcular farm, ɛ represents devatons from that average n a gven year (after nputs are chosen)). ɛ could also represent measurement error n output. In ths formulaton, ω s causng the endogenety problem, not ɛ. Let s call ω the "productvty shock". 4 Tradtonal Solutons Two tradtonal solutons to endogenety problems can be used here: nstrumental varables and fxed effects model. I wll dscuss these before movng to more recent methodologcal approaches. 2

3 4.1 Instrumental Varables Want to fnd "nstruments" that are correlated wth the endogenous nputs, but do not drectly determne y and are not correlated wth ω (and ɛ ). Good news s that theory can provde us wth such nstruments. Specfcally, consder nput and output prces w, r, and p.theory tells us that these prces wll affect frms optmal choces of k and l. Theory also says that these prces are excluded from the producton functon as they do not drectly determne output y condtonal on the nputs. Last requrement s that w, r, and p are not correlated wth the productvty shock ω. When wll ths be the case (or not be the case)? One key ssue s the form of competton n nput and output markets. If output markets are mperfectly compettve (.e. frms face downward slopng demand curves), then a hgher ω wll ncrease a frm s output, drvng p down. In other words, p wll be postvely correlated wth ω, nvaldatng p as an nstrument. If nput markets are mperfectly compettve (.e. frms face upward slopng supply curves), then a hgher ω wll ncrease a frm s nput demand, drvng w and/or r up. So w and/or r are now correlated wth ω, nvaldatng them as nstruments. So for these nstruments want frms operatng n perfectly compettve nput or output markets. Typcally, ths s more belevable for nput markets than for output markets. Unfortunately, even f wllng to make these assumptons, IV solutons haven t been that broadly used n practce. Frst, one needs data on w and r. Second, there s often very lttle varaton n w and r across frms (often there s a real queston of whether frms actually operate n dfferent nput markets?). Thrd, one often wonders whether observed varaton n e.g. w, actually represents frms facng dfferent nput prces, or whether t represents thngs lke varaton n unobserved labor qualty (.e. the frm wth the hgher w s employng workers of hgher qualty). If the latter, then w s not a vald nstrument. Whle there mght be "true" varaton n nput prces across tme, ths s usually not helpful, because f one has data across tme, one often wants to allow the producton functon to change across tme, e.g. y t = β 0t + β 1 k t + β 2 l t + ω t + ɛ t (though there could be exceptons) That sad, I thnk f one can fnd a market where there s convncng exogenous nput prce varaton, IV approach s probably more convncng than the approaches I wll talk about n the rest of ths lecture note, as there seem to be less auxlary assumptons. Notes: Randomzed experments - ether drectly manpulatng nputs, or manpulatng nput prces. 3

4 As s typcally done n ths lterature, I have mplctly made a "homogeneous treatment effects" assumpton. A heterogeneous treatment effects model would be y = β 0 + β 1 k + β 2 l + ω + ɛ Ths affects the nterpretaton of IV estmators, e.g. Heckman and Robb (1985), Angrst and Imbens (1994, Ecta) If there are unobserved frm choce varables n ω, t becomes qute hard to fnd vald nstruments, even wth the above assumptons. 4.2 Fxed Effects Ths approach reles on havng panel data on frms across tme,.e. y t = β 0 + β 1 k t + β 2 l t + ω t + ɛ t Assume that ɛ t s ndependent across t (ths s consstent wth ɛ t not beng predctable by the frm when choosng k t and l t ) Suppose one s wllng to assume that the productvty shock s contant over tme (fxed effect assumpton),.e. ω t = ω Then one can ether mean dfference or frst dfference y t y = β 1 ( kt k ) + β2 ( lt l ) + (ɛt ɛ ) y t y t 1 = β 1 (k t k t 1 ) + β 2 (l t l t 1 ) + (ɛ t ɛ t 1 ) Snce the problematc unobservable ω t have been dfferenced out of these expressons (recall that we have assumed that the ɛ t s are uncorrelated wth nput choces) these equatons can be estmated wth OLS. Problems: 1) ω t = ω s a strong assumpton 2) These estmators often produce strange estmates. In partcular, they often generate very small (or even negatve) captal coeff cents. Perhaps ths s due to measurement error n captal (Grlches and Hausman (1986, JoE))? Other notes: The mean dfference approach requres all the nput choces to be uncorrelated wth all the ɛ t (strct exogenety). The frst dfference approach only requres current and lagged nputs to be uncorrelated wth current and lagged ɛ t. Usng k t 1 and l t 1 (or other lags) as nstruments for (k t k t 1 ) and (l t l t 1 ), one can allow current nputs to be arbtrarly correlated wth past ɛ t s (sequental exogenety) 4

5 Panel data approach can be extended to rcher error structures (Arellano and Bond (1991, ReStud), Arellano and Bover (1995. JoE), Blundell and Bond (1998, JoE, 2000, ER), Arellano and Honore (2001, Handbook)) e.g. or ω t = α + λ t I wll talk further about these these later. 4.3 Frst Order Condtons ω t = ρω t 1 + ξ t where λ t = ρλ t 1 + ξ t A thrd approach to estmatng producton functons s based on nformaton n frst order condtons of optmzng frms. For example, for a frm operatng n perfectly compettve nput and output markets, statc cost mnmzaton mples that Y L L Y Y K K Y = wl py = rk py.e. the output elastcty w.r.t. an nput must equal ts (cost) share n revenue. In a Cobb-Douglas context, these output elastctes are the producton functon coeff cents β 1 and β 2, so observatons on these revenue shares across frms could provde estmates of the coeff cents. Note that r can often be assumed known and often one drectly observes wl and py (rather than L and Y -.e. labor nput and output are measured n terms of dollar unts (that are mplctly assumed to be comparable across frms)) But: Ths assumes statc cost mnmzaton -.e. t assumes away dynamcs, adjustment costs, etc.. At the very least we often thnk about the captal nput beng subject to a dynamc accumulaton process, e.g. K t = δk t 1 + t 1 There are addtonal terms when frms are not operatng n perfectly compettve markets, e.g. when frms face downward slopng demand curve Y L L Y Y K K Y = µ wl py = µ rk py where µ = p mc,.e. percentage markup. Note that proft maxmzaton mples p mc = 1+ɛ, where ɛ s the elastcty of demand. So, for example, one could stll dentfy producton coeff cents usng ths method f the elastcty of demand was known (ths s done n Hseh and Klenow (2009, QJE)). Or, one mght be able to dentfy both wth addtonal restrctons, e.g. Constant Returns to Scale (related to Hall (1988, JPE)). ɛ 5

6 5 Olley and Pakes (1996, Ecta) Alternatve approach to estmatng producton functons. I wll argue that key assumptons are tmng/nformaton set assumptons, a scalar unobservable assumpton, and a monotoncty assumpton. Setup: y t = β 0 + β 1 k t + β 2 l t + ω t + ɛ t (1) Agan, the unobserved productvty shocks ω t are potentally correlated wth k t and l t.but the unobservables ɛ t are measurement errors or unforecastable shocks that are not correlated wth nputs k t and l t. Basc Idea: Endogenety problem s due to the fact that ω t s unobserved by the econometrcan. If some other equaton can tell us what ω t s (.e. makng t "observable"), then the endogenety problem would be elmnated. Olley and Pakes wll use observed frms nvestment decsons t to "tell us" about ω t. Assumptons: 1)The productvty shock ω t follows a frst order markov process,.e. p(ω t+1 I t ) = p(ω t+1 ω t ) where I t s frm s nformaton set at t (whch ncludes current and past ω t s). Note: Ths s both an assumpton on the stochastc process governng ω t and an assumpton on frms nformaton sets at varous ponts n tme. Essentally, frms are movng through tme, observng ω t at t, and formng expectatons about future ω t usng p(ω t+1 ω t ). The form of ths frst order markov process s completely general, e.g. t s more general than ω t = ω or ω t.followng an AR(1) process. Ths assumpton mples that and that we can wrte E [ω t+1 I t ] = g(ω t ) ω t+1 = g(ω t ) + ξ t+1 where by constructon E [ ξ t+1 I t ] = 0 g(ω t ) can be thought of as the "predctable" component of ω t+1, ξ t+1 can be thought of as the "nnovaton" component,.e. the part that the frm doesn t observe untl t + 1. Ths can be extended to hgher order Markov processes (see ABBP Handbook artcle and Ackerberg and Hahn (2015)) 2) Labor s a perfectly varable nput,.e. l t s chosen by the frm at tme t (after observng ω t ). 3) Labor has no dynamc mplcatons. In other words, my choce of l t at t only affects profts at perod t, not future profts. Ths rules out, e.g. labor adjustment costs lke frng or hrng costs. 6

7 4) On the other hand, k t s accumulated accordng to a dynamc nvestment process. Specfcally K t = δk t 1 + t 1 where t s the nvestment level chosen by the frm n perod t (after observng ω t ). Importantly, note that k t depends on last perod s nvestment, not current nvestment. The assumpton here s that t takes full tme perod for new captal to be ordered, delvered, and nstalled. Ths also mples that k t was actually decded by the frm at tme t 1. Ths s what I refer to as a "tmng assumpton". In summary: labor s a varable (decded at t), non-dynamc nput captal s a fxed (decded at t 1), dynamc nput We could also thnk about ncludng fxed, non-dynamc nputs, or varable, dynamc nputs. (see ABBP) Gven ths setup, lets thnk about a frm s optmal nvestment choce t. Gven t wll affect future captal levels, a proft maxmzng frm wll choose t to maxmze the PDV of ts future profts. Ths s a dynamc programmng problem, and wll result n an dynamc nvestment demand functon of the form: t = f t (k t, ω t ) (2) Note that: k t and ω t are part of the state space, but l t does not enter the state space. Why? f t s ndexed by t. Ths mplctly allows nvestment decsons to depend on other state varables (e.g. nput prces, demand condtons, ndustry structure) that are constant across frms. f t wll lkely be a complcated functon because t s the soluton to a dynamc programmng problem. Fortunately, we can estmate the producton functon parameters wthout actually solvng ths DP problem (ths s helpful not only computatonally, but also allows us to estmate the producton functon wthout havng to specfy large parts of the frms optmzaton problem (semparametrc)). Ths s a nce example of how semparametrcs can help n terms of computaton - lterature based on Hotz and Mller (1993, ReStud) s smlar n nature. One of the key deas behnd OP s that under some condtons, the nvestment demand equaton (2) can be nverted to obtan ω t = f 1 t (k t, t ) (3).e. we can wrte the productvty shock ω t as a functon of varables that are observed by the econometrcan (though the functon s unknown) What are these condtons/assumptons? 1) (strct monotoncty) f t s strctly monotonc n ω t. OP prove ths formally under a set of assumptons that nclude the assumpton that p(ω t+1 ω t ) s stochastcally ncreasng n ω t. Ths result s farly ntutve. 7

8 2) (scalar unobservable) ω t s the only econometrc unobservable n the nvestment equaton,.e. Essentally no unobserved nput prces that vary across frms (f there were observed nput prces that vared across frms, they could be ncluded as arguments of f t ). There s one excepton to ths - labor nput prce shocks across frms that are not correlated across tme. No other structural unobservables that affect frms optmal nvestment levels (e.g effcency at dong nvestment, heterogenety n adjustment costs, other heterogenety n the producton functon (e.g. random coeff cents)) No optmzaton or measurement error n 2) s a farly strong assumpton, but t s crucal to beng able to wrte ω t as an (unknown) functon of observables. Suppose these condtons hold. Substtute (3) nto (1) to get y t = β 0 + β 1 k t + β 2 l t + f 1 t (k t, t ) + ɛ t (4) Snce we don t know the form of the functon ft 1 (and t s a complcated soluton to a dynamc programmng problem), let s just treat t non-parametrcally, e.g. a hgh order polynomal n t and k t, e.g. y t = β 0 + β 1 k t + β 2 l t + γ 0t + γ 1t k t + γ 2t t + γ 3t k 2 t + γ 4t 2 t + γ 5t k t t + ɛ t (5) Man pont s that under the OP assumptons, we have elmnated the unobservable causng the endogenety problem In ths lterature, t s sometmes called a control varable and sometmes called a proxy varable. Nether s perfect termnology. So we can thnk about estmatng ths equaton wth a smple OLS regresson of y t on k t, l t, and a polynomal n k t and t. Problem: β 1 k t s collnear wth the lnear term n the polynomal, so we can t separately dentfy β 1 from γ 1t. Intutvely, there s no way to separate out the effect of k t on y t through the producton functon, from the effect of k t on y t through ft 1. But, there s no l t n the polynomal, so β 2 can n prncple be dentfed (though see dscusson of Ackerberg, Caves, and Frazer (ACF, 2015, Ecta) below) In summary, the "frst stage" of OP nvolves OLS estmaton of y t = β 2 l t + γ 0t + γ 1t k t + γ 2t t + γ 3t k 2 t + γ 4t 2 t + γ 5t k t t + ɛ t (6) where γ 0t = β 0 + γ 0t and γ 1t = β 1 + γ 1t. Ths produces an estmate of the labor coeff cent β 2 and an estmate of the "composte" term β 0 + β 1 k t + ω t Φ t = γ 0t + γ 1t k t + γ 2t t + γ 3t k 2 t + γ 4t 2 t + γ 5t k t t = β 0 + β 1 k t + ω t 8

9 To estmate the coeff cent on captal, β 1, we need a "second stage". Recall that we can wrte ω t = g(ω t 1 ) + ξ t where E [ξ t I t 1 ] = 0 Snce k t was decded at t 1, k t I t 1. Hence E [ξ t k t ] = 0 and therefore E [ξ t k t ] = 0 Ths moment condton can be used to estmate the captal coeff cent More specfcally, consder the followng procedure: 1) Guess a canddate β 1 2) Compute ω t (β 1 ) = Φ t β 1 k t for all and t. ω t (β 1 ) are the "mpled" ω t s gven the guess of β 1. If our guess s the true β 1, ω t (β 1 ) wll be the true ω t s (asymptotcally). If our guess s not the true β 1, the ω t (β 1 ) s wll not be the true ω t s asymptotcally. (Note: Actually, ω t (β 1 ) s really ω t + β 0, but the constant term ends up not matterng) 3) Gven the mpled ω t (β 1 ) s, we now want to compute the mpled nnovatons n ω t.e. mpled ξ t s. To do ths, consder the equaton ω t = g(ω t 1 ) + ξ t Thnk about estmatng ths equaton,.e. non-parametrcally regressng the mpled ω t (β 1 ) s (from step 2) on the mpled ω t 1 (β 1 ) s (also from step 2). Agan, we can thnk of representng g non-parametrcally usng a polynomal n ω t 1 (β 1 ). Call the resduals from ths regresson ξt (β 1 ) These are the mpled nnovatons n ω t. Agan, f our guess s the true β 1, ξ t (β 1 ) wll be the true ξ t s (asymptotcally). If our guess s not the true β 1, then the ξ t (β 1 ) s wll not be the true ξ t s. 4) Lastly, evaluate the sample analogue of the moment condton E [ξ t k t ] = 0,.e. 1 1 N T ξt (β 1 )k t = 0 t Snce E [ξ t k t ] = 0, ths sample analogue should be approxmately zero f we have guessed the true β 1. For other β 1, ths wll generally not equal zero (dentfcaton) 5) Use a computer to do a non-lnear search for the β 1 that sets 1 1 N T ξt ( β 1 )k t = 0 t 9

10 Notes Ths s a verson of the second stage of OP. It s essentally a non-lnear GMM estmator 1) Recap of key assumptons: Frst order markov assumpton on ω t (agan can be relaxed to hgher order (but Markov)) - note, for example, that the sum of two markov processes s not generally frst order markov (e.g. sum of two AR(1) processes wth dfferent AR coeff cents) Tmng assumptons on when nputs are chosen and nformaton set assumptons regardng when the frm observes ω t (ths can be strengthened or relaxed - see Ackerberg (2016)) Strct monotoncty of nvestment demand n ω t (can be relaxed to weak monotoncty - see below) Scalar unobservable n nvestment demand (tough to relax, though one can allow other observables to enter nvestment demand, e.g. nput prces) 2) Alternatve formulaton of the second stage (more lke OP paper) y t = β 0 + β 1 k t + β 2 l t + ω t + ɛ t (7) y t β 1 k t β 2 l t = β 0 + g(ω t 1 ) + ξ t + ɛ t (8) y t β 1 k t β 2 l t = β 0 + g( Φ t 1 β 0 β 1 k t 1 ) + ξ t + ɛ t (9) y t β 1 k t β 2 l t = g( Φ t 1 β 1 k t 1 ) + ξ t + ɛ t (10) So gven a guess of β 1, one can regress (y t β 1 k t β ) 2 l t on a polynomal n ( Φ t 1 β 1 k t 1 ) to recover mpled ξ t +ɛ t s,.e. and sample analogue to estmate β N T ξt + ɛ t (β 1 ), and then use the moment condton E [(ξ t + ɛ t ) k t ] = 0 t ( ) ξt + ɛ t (β 1 ) k t = 0 3) There are other formulatons as well. For example, Wooldrdge (2009, EcLet) suggests estmatng both frst stage and second stage smultaneously. Ths has two potental advantages: 1) eff cency (though ths s not always the case, see, e.g. Ackerberg, Chen, Hahn, and Lao (2014, ReStud), and 2) t makes t easer to compute standard errors (wth two-step procedure, t s typcally easest to bootstrap). On the other hand, a dsadvantage s that t requres a non-lnear search over a larger set of parameters (β 1 plus the parameters of g and ft 1 ), whereas the above two step formulatons only requre a non-lnear search for β 1 (or β 1 and g) 4) Note that there are addtonal moments generated by the model. The assumptons of the model mply that E [ξ t I t 1 ] = 0. Ths means that the mpled ξ t s should not only be uncorrelated wth k t, but everythng else n I t 1, e.g. k t 1, k t 2, l t 1,k 2 t... (though not l t ). These addtonal moments can potentally add eff cency, but also result n an overdentfed model, whch can lead to small sample bas. The extent to whch one utlzes these addtonal moments s typcally a matter of taste. 10

11 5) Intutve descrpton of dentfcaton Frst stage: Compare output of frms wth same t and k t (whch mply the same ω t ), but dfferent l t. Ths varaton n l t s uncorrelated wth the remanng unobservables determnng y t (ɛ t ), and so t dentfes the labor coeff cent. (But agan, see ACF secton below) Second stage: Compare output of frms wth same ω t 1, but dfferent k t s (note that frms can have the same ω t 1, but dfferent t 1 and k t 1 ). y t β 2 l t = β 0 + β 1 k t + g(ω t 1 ) + ξ t + ɛ t = β 0 + β 1 k t + g( Φ t 1 β 0 β 1 k t 1 ) + ξ t + ɛ t Ths varaton n k t s uncorrelated wth the remanng unobservables determnng y t (ξ t and ɛ t ), so t dentfes the captal coeff cent (However, note that the "comparson of frms wth same ω t 1 " depends on the parameters themselves, so ths s not completely transparent ntuton) 6) OP also deal wth a selecton problem due to the fact that unproductve frms may ext the market. The problem s that even f n the entre populaton of frms, E [ξ t k t ] = 0 E [ξ t k t, stll n sample at t] may not equal 0 and be a functon of k t Specfcally, f a frm s ext decson at t depends on ω t (and thus ξ t ), then ths second expectaton s lkely > 0 and depends negatvely on k t (snce frms wth hgher k t s may be more apt to stay n the market for a gven ω t or ξ t ). OP develop a selecton correcton to correct for ths, whch I dont thnk I wll go through (see ABBP for dscusson). On the other hand, f ext decsons at t are made at tme t 1 (a tmng assumpton lke that already beng made on captal), then there s no selecton problem, snce n ths case the ext decson s just a functon of I t 1. 6 Levnsohn and Petrn (2003, ReStud) Levnsohn and Petrn worry about the assumpton that nvestment s strctly monotonc n ω t. Intutvely, ths assumpton mples that any two frms wth the same k t and t must have the same ω t. But n many datasets, especally n developng countres, t s often 0 (e.g. n LP s Chlean dataset, approxmately 50% of observatons have 0 nvestment) It seems lke a strong assumpton that all these frms have the same ω t (gven k t ). It seems more lkely that there s some threshold ω t below whch frms nvest 0. One can extend OP to allow weak monotoncty for the observatons where t = 0, but ths requres dscardng these observatons from the analyss (Asde: n ths case, there s no selecton ssue as long as one uses the second stage moment E [(ξ t + ɛ t ) k t ] = 0 rather than E [ξ t k t ] = 0 (see Gandh, Navarro and Rvers (GNR, 2015)). Ths s because one cannot compute mpled ξ t s for observatons for whch t = 0 (but one can compute mpled (ξ t + ɛ t ) for these observatons)) 11

12 Anyway, gven these problems wth 0 nvestment and an unwllngness to throw away data, the basc dea of LP s to use a dfferent "control" varable to learn about ω t, one that s more lkely to be strctly monotonc n ω t. They use an ntermedate nput, e.g. nputs lke materals, fuel, or electrcty. These types of nputs rarely take the value 0. Producton Functon: y t = β 0 + β 1 k t + β 2 l t + β 3 m t + ω t + ɛ t (11) where m t s an ntermedate nput. m t s assumed to be a varable, non-dynamc nput, lke labor. Consder a frm s optmal choce of m t. Lke nvestment n OP, m t wll be chosen as a functon of the state varables k t and ω t,.e. m t = f t (k t, ω t ) (12) Assumng strct monotoncty, ths can be nverted and substtuted nto the producton functon y t = β 0 + β 1 k t + β 2 l t + β 3 m t + f 1 t (k t, m t ) + ɛ t (13) The rest follows exactly as n OP Estmate β 2 n frst stage (β 1 and β 3 cannot be dentfed because they are n f 1 t ) Estmate β 1 and β 3 n second stage (Need addtonal moment here to dentfy the second parameter. LP use. E [(ξ t + ɛ t ) m t 1 ] = 0 or E [ξ t m t 1 ] = 0, though see Bond and Soderbom (2005) and GNR ) 7 Ackerberg, Caves, and Frazer (2015) 7.1 Crtque Ths paper examnes the frst stage of LP and OP Our queston: Under what condtons s the labor coeff cent β 2 dentfed n the frst stage? The LP frst stage regresses y t on l t and a non-parametrc functon of k t and m t. (call ths non-parametrc functon np(k t, m t )) y t = β 2 l t + np(k t, m t ) + ɛ t (14) There s no endogenety problem here (snce ɛ t assumed uncorrelated wth everythng). But our queston s whether the labor nput l t moves around ndependently of np(k t, m t ). In other words, can two frms wth the same k t and m t have dfferent l t? To analyze ths, we need to thnk about a model of how frms chose l t Most natural model seems as follows. Snce m t and l t are both non-dynamc, varable nputs, and snce LP have already assumed that m t = f t (k t, ω t ) (15) 12

13 t seems logcal to treat l t symmetrcally and assume Of course, these wll be dfferent functons. If ths s the case, then note that l t = h t (k t, ω t ) (16) l t = h t (k t, ω t ) = h t (k t, ft 1 (k t, m t )) = h t (k t, m t ) The last lne mples that l t s a determnstc functon of k t and m t But ths s a problem for the frst stage estmatng equaton y t = β 2 l t + np(k t, m t ) + ɛ t (17) snce t mples that l t s functonally dependent on ("collnear" wth) np(k t, m t ),.e. l t doesn t move ndependently of np(k t, m t ). Another way of sayng ths s as follows: LP want to condton on k t and m t (.e. condton on ω t ) and look at remanng varaton n l t to dentfy β 2. But accordng to the above, l t s a determnstc functon of k t and m t. Hence, there s no remanng varaton n l t once we condton on k t and m t!!! Can also thnk about both h t (k t, m t ) and np(k t, m t ) beng polynomals. So f (16) s correct, then β 2 should not be dentfed n the frst stage. If OLS does n fact produce an estmate of β 2, then some assumpton of the model must be ncorrect. To get the frst stage of LP to work, we need to fnd somethng that moves around l t ndependently of np(k t, m t ). Unfortunately, ths s hard to do wthn the context of LP s other mantaned assumptons. For example, suppose one assumes that there s some frm-specfc unobserved shock to the prce of labor, v t. Ths wll clearly affect frms optmal labor choces,.e.. l t = h t (k t, ω t, v t ) The problem s that ths wll also generally affect the frms optmal choce of materals m t = f t (k t, ω t,, v t ) (18) whch then volates the scalar unobservable assumpton necessary to nvert f t and wrte ω t as a functon of observables. Our paper thnks about varous alternatve models of l t (.e. varous data-generatng processes (DGPs)) that mght "break" ths functonal dependence problem. We can only come up wth 2 such DGPs, and nether seems all that general. 13

14 1) Suppose there s "optmzaton" error n l t,.e. l t = h t (k t, ω t ) + u t where u t s ndependent of (k t, ω t ). In other words.e. for some exogenous reason frms do not get the optmal choce of labor correct. Ths breaks the functonal dependence problem (and does not seem completely unreasonable). On the other hand, one smultaneously needs to assume that there s 0 optmzaton error n m t (otherwse, the scalar unobservable assumpton s volated). It seems challengng to argue that there s a sgnfcant amount of optmzaton error n l t, but almost no optmzaton error n m t. (One example mght be f one s data measures "planned" or ordered materals, but actual labor (e.g. subject to sck days or unexpected quts)) 2) Suppose that m t s chosen at some pont n tme pror to l t, and that: a) The frm knows ω t when choosng m t b) Between these two ponts n tme, there s a shock to the prce of labor, v t, that vares across frms. c) v t s ndependent across tme (and other varables n the model) Ths second DGP also allows the labor coeff cent to be dentfed n the frst stage, because the shock v t moves l t around condtonal on m t and k t. Note that v t needs to be ndependent across tme, otherwse the choce of m t at t wll optmally depend on v t 1, volatng the scalar unobservable assumpton needed for nvertblty. Agan, ths DGP does not seem very general. Why s m t chosen before l t (f anythng, I would tend to thnk the reverse)? And t seems lke a stretch to assume that there are no unobserved frm specfc nput prce shocks except for ths very specal v t shock that must be realzed between these two ponts n tme. Notes: 1) Parametrc treatment of the ntermedate nput demand functon does not rescue the LP frst stage dentfcaton - see ACF for detals (though unlke wth t t s not hard to do ths, and t lkely adds eff cency) 2) The OP estmator s also affected by ths crtque. However, there s a 3rd DGP that breaks the functonal dependence problem. Ths nvolves l t beng chosen wth ncomplete knowledge of ω t, e.g. pror to the realzaton of ω t - see ACF for detals. 3)Bond and Soderbom (2005) make a related argument that crtczes the second stage dentfcaton of β 3 (the coeff cent on the ntermedate nput) n LP. The crux of the mplcatons of ther argument s that under the assumptons of the LP model the moment condton E [(ξ t + ɛ t ) m t 1 ] = 0 or E [ξ t m t 1 ] = 0 s not nformatve about the coeff cent β 3. The ntuton s that under the assumptons of the LP model, m t 1 s not correlated wth m t condtonal on k t and ω t 1 - hence t s unnformatve as an nstrument. A serally correlated, frm-specfc, unobserved prce shock to the ntermedate nput could generate such correlaton, but t volates the LP scalar unobservable assumpton. More generally, Bond and Soderbom show that wthout frm specfc nput prce varaton, coeff cents on perfectly 14

15 varable, non-dynamc nputs are not dentfed n Cobb-Douglas producton functons. GNR extend ths argument to more general producton functons, essentally showng that f a perfectly varable, non-dynamc nput s beng used as a proxy/control varable n the context of an OP/LP lke procedure, ts effect on output cannot generally be dentfed usng the above moments (and they argue that FOC approaches are needed for these nputs, see below). 7.2 Alternatve estmator suggested by ACF Gven that we feel these dentfcaton arguments rely on very partcular data generatng processes, we suggest a slghtly dfferent approach where we abandon tryng to dentfy the labor coeff cent n the frst stage. Instead, we try to dentfy β 2 along wth the captal coeffcent n the second stage. To do ths, assume that m t s chosen ether at the same tme or after l t s chosen. Ths mples we can wrte m t = f t (k t, ω t, l t ) (19) Ths can be thought of as a "condtonal" (on l t ) ntermedate nput demand functon, n contrast to LP s "uncondtonal" (on l t ) ntermedate nput demand functon: Uncondtonal nterm. nput demand (LP) : m t = f t (k t, ω t ) vs. Condtonal nterm. nput demand (ACF) : m t = f t (k t, ω t, l t ) Assumng strct monotoncty, ths condtonal ntermedate nput demand functon can be nverted and substtuted nto the producton functon to get y t = β 0 + β 1 k t + β 2 l t + f 1 t (k t, m t, l t ) + ɛ t (20) Treatng ft 1 non-parametrcally, t s obvous that now not even β 2 can be dentfed n the frst stage. However, we can stll dentfy the composte term: Φ t = β 0 + β 1 k t + β 2 l t + ω t (these are just the predcted values of y t from the regresson) Just lke n OP, gven guesses of β 1 and β 2, we can a) compute mpled ω t (β 1, β 2 ) s, then b) regress the ω t (β 1, β 2 ) s on the ω t 1 (β 1, β 2 ) s to obtan mpled ξ t (β 1, β 2 ) s (the resduals from the regresson). and c) compute the sample moment 1 1 N T ξt (β 1, β 2 )k t = 0 t But snce there s an addtonal parameter to estmate n the second stage, we need another moment condton. We suggest 1 1 N T ξt (β 1, β 2 )l t 1 = 0 t whch should be approxmately zero at the true β 1 and β 2 snce l t 1 I t 1 and hence E [ξ t l t 1 ] = 0. 15

16 So n summary, the second stage estmates of β 1 and β 2 are defned by ( 1 1 ξt ( β 1, β ) 2 )k t N T ξt ( β 1, β = 0 2 )l t 1 t Alternatvely, f we are wllng to assume that lke captal, l t s "fxed" and decded at t 1 (and thus l t I t 1 ), we could use ( 1 1 ξt ( β 1, β ) 2 )k t N T ξt ( β 1, β = 0 2 )l t t One can see how varous other tmng/nformaton assumptons would determne the dfferent moment condtons here (e.g. Ackerberg (2016)). Notes: 1) Even though frst stage does not dentfy any parameters, t s stll crucal n that t "separates" ω t from ɛ t. 2) The procedure does not rely on labor beng a non-dynamc nput,.e. labor choces could have dynamc mplcatons (e.g. hrng or frng costs). 3) The procedure allows frm specfc, serally correlated, unobserved shocks to the prce of labor (as well as to captal costs). We cannot allow such shocks to the prce of ntermedate nputs (t would volate the scalar unobservable assumpton necessary for the nverson), but n many cases ntermedate nputs are commodtes where we would expect very lttle prce varaton across frms. OP - rules out serally correlated, unobserved, frm specfc shocks to all nput prces ( t, l t,m t ) (note: can allow non-serally correlated shocks to prces of l t and m t ) LP - allows serally correlated, unobserved, frm specfc nput prce shocks to t, but not to (l t,m t ) ACF (wth ntermedate nput proxy) allows serally correlated, unobserved, frm specfc nput prce shocks to t and l t, but not to m t 4) Bond and Soderbom (2005) and GNR argument mples that we actually need some degree of 2) or 3) for dentfcaton of the labor coeff cent. 5) Can use t rather than m t as the proxy varable n ACF procedure, but lose ablty to allow serally correlated, unobserved, frm specfc nput prce shocks to t and l t. 6) Can also overdentfy the model by addng further lags of nputs as nstruments. 7) Also note that the producton functon (20) does not nclude m t. Ths s because the Bond- Soderbom (and GNR) arguments mply that we cannot use m t 1 as an nstrument to dentfy the coeff cent on m t. Not ncludng m t s mplctly usng a value-added producton functon (.e. y t s deflated revenues mnus deflated costs of ntermedate nputs). One structural nterpretaton of ths s that the producton functon s Leontef n the ntermedate nput (e.g. materals),.e. { } Y t = mn β 0 K β 1 t Lβ 2 t eω t, β 3 M t e ɛ t 16

17 whch mples that y t = β 0 + β 1 k t + β 2 l t + ω t + ɛ t (21) An alternatve would be to follow Appendx B of Levnsohn and Petrn, or Gandh, Navarro and Rvers (2015) and use a frst order condton to obtan the coeff cent on the ntermedate nput. 8) Less parametrc generalzatons Ackerberg and Hahn (2015) show that n ths "value added model" Y t = mn {F (K t, L t, ω t ), β 3 M t } e ɛ t y t = mn {f(k t, l t, ω t ), ln(β 3 ) + m t } + ɛ t f f s strctly monotonc n the scalar markov process ω t, t can be fully non-parametrcally dentfed. We formally consder the generc model y t = f (x t, ω t ) and show condtons on tmng and nformaton sets under whch f s non-parametrcally dentfed. We descrbe the result as showng how the tmng/nformaton set assumptons crucal to OP have power n a non-parametrc context. Note that these assumptons are startng to be used n other lteratures, e.g. demand wth endogenous product characterstcs. Gandh, Navarro and Rvers (2015) extend the frst order condton approach to estmatng the effect of the proxy varable (m t ) to a non-parametrc settng and show that n y t = f(k t, l t, m t ) + ω t + ɛ t f s non-parametrcally dentfed. I would descrbe ths approach as combnng the tmng/nformaton set assumpton dentfcaton approach (for k t and l t ) wth the frst order condton dentfcaton approach (for m t ) 9) More generally can mx-and-match the dfferent dentfcaton strateges (for dfferent nputs),.e. tmng/nformaton set assumptons (though as detaled above, ths does not work for a non-dynamc, varable nputs that s beng used to proxy for unobserved productvty). frst order condtons (at least for statc nputs) observed frm specfc nput prce shocks as nstruments 8 Dynamc Panel Approaches These are econometrc procedures (Arellano and Bond (1991, ReStud), Arellano and Bover (1995. JoE), Blundell and Bond (1998, JoE, 2000, ER), Arellano and Honore (2001, Handbook)) that generalze the fxed effects model to allow ω t to vary across tme. These have been used n many dfferent appled contexts, ncludng producton functons (e.g. Blundell Bond 2000, and emprcal work by John Van Reenan, Nck Bloom and coauthors). "Dynamc Panel" s somewhat of a msnomer n the context that I am usng these methodologes, as there s no lagged dependent r.h.s. varable. Many of these methods were developed n that context. 17

18 I wll focus on one very smple example, to try to hghlght the smlartes and dfferences between ths lterature and the lterature stemmng from OP. Producton functon where y t = β 0 + β 1 k t + β 2 l t + ω t + ɛ t (22) ω t = ρω t 1 + ξ t Suppose that ɛ t satsfes strct exogenety,.e. ɛ t s are uncorrelated wth all nput choces. Suppose that ω t not observed untl t, that k t s chosen at t 1, and that l t s chosen at t. These assumptons are analagous to the tmng/nformaton set assumptons made n OP, and mply the followng orthogonalty condtons E [ɛ t k s ] = E [ɛ t l s ] = 0 t, s E [ξ t k s ] = 0 for s t E [ξ t l s ] = 0 for s < t Consder "ρ-dfferencng" the producton functon,.e. or y t ρy t 1 = (1 ρ)β 0 + β 1 (k t ρk t 1 ) + β 2 (l t ρl t 1 ) + ξ t + (ɛ t ρɛ t 1 ) y t = ρy t 1 + (1 ρ)β 0 + β 1 (k t ρk t 1 ) + β 2 (l t ρl t 1 ) + ξ t + (ɛ t ρɛ t 1 ) Now, gven a guess of the parameters (ρ, β 0, β 1, β 2 ) one can compute the mpled values of the term ξ t + (ɛ t ρɛ t 1 ),.e. ξ t + (ɛ t ρɛ t 1 ) (ρ, β 0, β 1, β 2 ) = y t ρy t 1 (1 ρ)β 0 β 1 (k t ρk t 1 ) β 2 (l t ρl t 1 ) Then estmaton can proceed usng, e.g. by settng the sample moment N T ξ t + (ɛ t ρɛ t 1 ) (ρ, β 0, β 1, β 2 ) k t k t 1 = 0 t l t 1 Agan, there are actually many more potental moment condtons, snce all values of k, l, and y pror to (k t, l t 1, y t 2 ) are also uncorrelated wth ξ t + (ɛ t ρɛ t 1 ). Note: There are mplct assumptons here about what makes lags strong nstruments. Ths depends on dynamc ssues (e.g. adjustment costs) and ssues regardng seral correlaton n nput prces. See Blundell and Bond (1999) and Bond and Soderbom (2005) for dscusson of when they are strong nstruments, and when they are not so strong nstruments. 18

19 One can extend the model to allow for an addtonal unobservable that s fxed across tme,.e. y t = β 0 + β 1 k t + β 2 l t + α + ω t + ɛ t (23) where α s allowed to be correlated wth all nput choces. Ths model requres "double dfferencng",.e. (y t ρy t 1 ) (y t 1 ρy t 2 ) = β 1 [(k t ρk t 1 ) (k t 1 ρk t 2 )] + β 2 [(l t ρl t 1 ) (l t 1 ρ +ξ t ξ t 1 + (ɛ t ρɛ t 1 ) (ɛ t 1 ρɛ t 2 ) Agan, one can approprately lag k, l, and y to fnd vald moments. One problem s that double dfferencng can be demandng on the data, and estmates can be mprecse. Arellano and Bover (1995) and Blundell and Bond (1998, 2000) suggest some addtonal moment condtons based on statonarty assumptons that can help here, though these assumptons may be strong. So how do these dynamc panel approaches compare to the Olley-Pakes lterature? the man tradeoff s the followng I d say The dynamc panel approach does not requre the scalar unobservable and strct monotoncty assumptons that are requred for the OP/LP/ACF nversons. So n the dynamc panel lterature, for example, one does not need to worry about unobserved frm specfc nput prces, nor other sorts of unobservables lke optmzaton error. On the other hand, the dynamc panel approach requres that the seral correlaton n ω t s lnear, e.g. an AR or MA process. Ths s essental to beng able to construct usable moments. In contrast, the OP/LP/ACF lterature can allow the productvty shock to follow a completely general frst order markov process. Dynamc panel lterature can also allow addtonal fxed effects, although precse estmaton appears to be challengng. Intermedate assumpton of Arellano-Bover Blundell- Bover may be helpful. Gven that theory may provde lttle gudance between choosng between these two sets of assumptons (and because they are both strong), I would suggest tryng both approaches. 9 Other Issues 1) Often observe frm revenues as the output measure, not physcal quanttes. As ponted out by Klette and Grlches (1996, JAE), ths can be problematc when frms operate n dstnct mperfectly compettve output markets (and one does not observe output prce). Intuton: Suppose observe that frms that (exogenously) use double the nputs of others produce less than double revenue of other. There are two explanatons - 1) declnng returns to scale (but, e.g., perfect competton), 2) constant returns to scale wth a downward slopng demand curve. Even f one doesn t care about separatng the above two effects, there may now be two dstnct sources of unobservables n the (revenue) producton functon, whch can be problematc for the proxy based approaches. 19

20 See, e.g. Klette (1999, JIE), Foster, Haltwanger, and Syverson (2007, AER), DeLoecker (2011, Ecta), DeLoecker and Warzynsk (2012, AER). Note that just observng quanttes s not a complete panacea - need to be equvalent across frms for these quanttes to be meanngful. 2) Other types of nformaton structures (e.g. Greenstreet (2007)). 3) Addtonal nputs (e.g. Grlches Knowledge Captal model, Doraszelsk and Jaumandreu (2014, ReStud)) Grlches Knowledge captal where y t = β 0 + β 1 k t + β 2 l t + β 3 c t + ω t + ɛ t (24) c t = t τ=0 δ t τ c d τ where d τ s frm s chosen R&D expendtures n perod τ Doraszelsk and Jaumandreu (2014) y t = β 0 + β 1 k t + β 2 l t + ω t + ɛ t (25) where ω t = g(ω t 1, d t 1 ) Advantages DJ Doesn t have has ntal t = 0 R&D stock ssue that Grlches has (assumng use OP/LP/ACF related methods to estmate) Explctly has uncertanty n the contrbuton of R&D to productvty Dsadvantages DJ Unobserved component of "productvty" and R&D component of "productvty" affect future through scalar - ths s restrctve, e.g. ω t = ρω t 1 + d t 1 + ξ t d t 1 and ξ t forced to deprecate at same rate. Ths s not the case n Grlches -.e. δ c dfferent than ρ. Alternatve model of physcal captal stock where y t = β 0 + β 2 l t + ω t + ɛ t (26) ω t = g(ω t 1, d t 1, t 1 ) Note: both "endogenze productvty" f thnk about defnng productvty as (β 3 c t +ω t ) n the Grlches model 4) Emprcal questons - 20

21 Determnants of productvty - deregulaton (OP), trade openess, exportng - smlar to above generally best to nclude these factors as nputs n the producton functon Allocatve eff cency - recently Hseh and Klenow (2009, QJE), Asker, Collard-Wexler, and DeLoecker (2014, JPE) 21