Estimation of Cost and. Albert Banal-Estanol

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1 Esmaon of Cos and Producon Funcons ns

2 Movaon: Producon and Cos Funcons Objecve: Fnd shape of producon/cos funcons Evaluae effcency: Increasng reurns, economes of scale Complemenary/subsuably beween npus Synerges and mergers Producvy analyss: Measuremen of producvy across frms or over me Effecs of polcy (deregulaon, arffs, ) Reurns of R&D, reurns o adopon of new echnology Quanave Technques for Regulaon

3 Margnal/average g coss vs cos funcon In some nsances, Only need margnal/average cos funcons Compued drecly from company/ndusry y daa Bu, n ohers, Need o know f margnal cos vares wh quany Or, f here are economes of scale as sze changes Economc approach Make assumpons on shape Esmae model s parameers Engneerng approach: Inervew echncal exper abou effec of coss and scale effecs Quanave Technques for Regulaon

4 Ths chaper Accounng and economc coss Esmang producon funcons: Applcaons: reurns o scale and echncal change Problems and possble soluons E Esmang cos funcons: Applcaon: effcency measuremen Appendx: Index numbers and TFP Quanave Technques for Regulaon

5 Accounng and economc coss

6 Cos measures from company daa Need o allocae coss across mulple l operaons: Imporan for regulaed and non-regulaed busness Exsence of cos allocaon mehodologes Cos mgh no reflec value f vercally negraed: Transfer upsream/downsream for ax or regulaory reasons Coss and revenues mgh no occur a same me Buy plan now, revenues over 30 years. Cos per year? Subsanal dfferences beween accounng/economc coss Quanave Technques for Regulaon

7 Accounng versus economc cos of an npu "Accounng" " coss: Ou-of-pocke hsorcal coss, appropraely deprecaed "Economc" coss: Remuneraon receved n he nex bes alernave If marke avalable, equal o marke prce, e.g. prce of seel If no, need o fnd a value, e.g. economc cos of Invesng n new capacy: reurn of capal used f had been nvesed elsewhere (adjused for rsk) Company owner s me, hghes ncome n anoher occupaon How o compue economc coss? Quanave Technques for Regulaon

8 Example: opporuny cos of capal Cos of capal ofen repored as: Deprecaon charges May also nclude neres pad f frm needs o borrow Assumng consan deprecaon, accounng cos: Cos = δ Orgnal capal nvesmen Insead, economc cos: Cos = Opporuny cos of capal + Economc deprecaon, where Opporuny cos = r V where V : value of he good and r: approprae rae Economc deprecaon = V - V +1 Or n oher words: Cos = (r + Deprecaon rae ) V where Deprecaon rae = ( V V +1 ) / V Quanave Technques for Regulaon

9 Example: company car Buyng a Volkswagen Passa n Belgum, 2007: Prces: new: & smlar 1-year old: Ineres rae: 10% Accounng cos: Assumng 5-year lfe, deprecaed a consan rae (20%) Cos = / 5= 5610 Economc cos: Cos = ( ) = 9855 because Deprecaon rae = ( ) / = 25.1% Quanave Technques for Regulaon

10 Esmang producon funcons

11 Theorecal framework: npu subsuon effecs To bake a cake, need fxed proporon of ngredens: 1kg of flower, 6 eggs,. Gven I 1 kgs of flower, I 2 eggs, Q cakes can be baked: I 1 I2 I Q mn n,,..., 1 2 n where β 1, β 2, are he proporons Bu, f we also need capal and labour, one can use: () a small amoun of labour and a cake-mxer or () a large amoun of labour and a spoon Need general producon funcon, allowng for subsuon Q f I, I,...,,,,..., ) ( 1 2 I n 1 2 n Where β 1, β 2, are he parameers Quanave Technques for Regulaon

12 Example: Cobb-Douglas Two npus: labour and capal and K Q K 1L 2 Mn combnaons requred (soquan) O o produce q 0 uns L Quanave Technques for Regulaon

13 Esmaon Sraeges If frm mnmses coss, solves C ( Q,, u ) MnI 1, I 2,..., I p1 I1 p1 I 2... p I n n s.. Q f ( I, I,..., I,,,...,, u) 1 2 n 1 2 where 1,, n are observed npus and u an unobserved npu (e.g. producvy), whch wll be he economerc error erm Opon 1: Esmae producon funcon (cos ndrecly) Q need npu-oupu daa f ( I, I,..., I,,,...,, u) ) ( 1 2 n 1 2 n u Opon 2: Esmae cos funcon drecly C C ( 2 Q, p1, p,..., pn, u) need cos, oupus and npu prces n n Quanave Technques for Regulaon

14 AModel Assume producon follows a Cobb-Douglas funcon: Q e K k L l e u where Q s he oupu of plan (or frm) I ; L s labour npu (or more generally varable npu); K s capal npu (or more generally fxed npu); u s an error erm; α,β l, β k are parameers o be esmaed Remarks: We mgh also nclude: maeral, energy, dfferen ypes of labour Error erm ncludes: echnology or managemen dfferences, measuremen errors, varaon n exernal facors Takng logs (y =ln (Q ), l =ln (L ) and k =ln (K ) ), we oban y k k l l u Quanave Technques for Regulaon

15 Varables Producon: If mulproduc frm hen Oupu ypcally sales dvded by prce ndex (more on hs laer) Or usng a quany y ndex (more on hs laer) Labour: Usually hours or employees per year May adjus for ypes of labour Oher non-capal npus: E.g. fuel (lres) Oher (expendures dvded by prce ndex) Capal: Invesmen a dfferen mes compung deprecaon Quanave Technques for Regulaon

16 Applcaon 1: Reurns o Scale Esmang he prevous model by OLS one fnds esmaes of α, β l and β k β l and β k are he elasces of oupu wr npu Remember ha wh Cobb-Douglas model: Q AK k L l exhbs... CRS ff IRS ff DRS ff l l k k l We can perform F-ess o deermne he case k Quanave Technques for Regulaon

17 Applcaon 2: Techncal Change Measuremen Wh me seres (or panel) daa, we can esmae echncal change: y k k l l u where s a me rend (=1,2, T) Inerpreaon: β provdes an esmae of he annual percenage change n oupu resulng from echncal change (one year) Quanave Technques for Regulaon

18 Problem 1: Endogeney Wha f he rue model s bu we esmae y k y ~ k ~ k l l ~ l X x u Then our esmaes are based f (a) he omed varable has an effec on he dependen varable and (b) he omed varable s correlaed wh an ncluded varable k l u X may no be observable: Manageral effcency: Beer managers may need less labour o produce same oupu These frms produce more wh less labour and OLS wll underesmae β l Producvy shock: A hgher producvy shock may mean more labour OLS arbues all he ncrease n oupu o change n labour and overesmae β l Quanave Technques for Regulaon

19 Soluon 1: Insrumenal Varables (IV) Example: producvy shock unobserved by he economercan (bu he frm observes) when choosng s level of labour, he frm has observed he shock and herefore he npu s correlaed wh he shock OLS s nconssen. Vald nsrumen: prce of labour (a) uncorrelaed wh he error (f labour marke s compeve); (b) correlaed wh he level of labour Inuvely, usng a varable as an nsrumen means ha (a) regressng endogenous varable on he nsrumen (and oher varables) (b) use predced endogenous varable as a regressor n nal model Recen research challenges IV analyss: More recen echnques are dynamc panel daa Srucural esmaaon (see Olley and Pakes, 1996; Levnsohn and Pern, 2003). Quanave Technques for Regulaon

20 Soluon 2: Panel Daa Analyss Suppose ha we have panel daa : Several frms followed a dfferen pons n me Assume ha: y k l, k, l,, where s frm -specfc and, sasfes classcal assumpons Quanave Technques for Regulaon

21 Frm Fxed Effecs Assume ha γ s predeermned: No resul of random varaon bu fxed, long sandng characerscs For example: manageral ably (can be correlaed wh labour) If hs s rue hen panel daa. Allows us o oban unbased esmaes for he varables ncluded (especally for hose ha adap quckly as labour and maerals) Correc for unobservable heerogeney Acs lke a dummy shfng he nercep for each ndvdual (mpossble o do wh cross secon) Quanave Technques for Regulaon

22 Tme Fxed Effecs We may also nclude a me-varan varable: Common effec mpacng on all ndvduals Can accoun for ndusry-wde producvy shocks We can hen use: y k l, k, l,, where s frm -specfc and s me - specfc and, sasfes classcal assumpons Quanave Technques for Regulaon

23 Problem 2: Funconal form Funconal form needs o reflec Plausble npu subsuon possbles Plausble naure of reurns o scale If unsure, Use flexble economerc specfcaon Cobb-Douglas mposes same reurns o scale over whole oupu range Bu f overly flexble Implausble resuls mgh be obaned Daa mgh no be able o denfy As a resul, Impose wha you know Bu don mpose wha you do no! Quanave Technques for Regulaon

24 Esmang cos funcons

25 Cos Funcons Smlarly one can esmae a cos funcon: ( a) ln c o y ln y where c are he operang coss of plan (or frm) ε s an error erm (due for example o manageral effcency) β o and β y are parameers o be esmaed Bu, larger frms mgh oban raw maerals a lower prce: Ths does no reflec manageral effcency bu conaned n ε Correlaon beween error erm and ncluded varable OLS esmaor may be based. Need o conrol for! Quanave Technques for Regulaon

26 Cos Funcons If one has daa on npu prces: ( b ) ln c o w ln w r ln r y ln y where c are he operang coss of plan (or frm) w s he prce of he labour npu (wages) of plan (or frm) r s he prce of he capal npu of plan (or frm) I ε s an error erm (due agan for example e o manageral age a effcences) β o,β w, β r and β y are parameers o be esmaed Quanave Technques for Regulaon

27 Applcaon: effcency measuremen Assume, for smplcy, ha we esmae he model ( c ) c y where β o s he fxed cos and β 1 s he margnal cos of y Then c c ˆ o ˆ Where ĉ can be hough as an esmae of he coss of an average effcency company (benchmark) producng he same oupu And ˆ esmaes he effcency dfference beween company and hs average company The company s less effcen han average f ˆ 0 The company s more effcen han average f ˆ 0 y Quanave Technques for Regulaon

28 Effcency measuremen, graphcally Cos (c ) A A Oupu (y ) Quanave Technques for Regulaon

29 Exensons More han one cos drver: y L P c d ) ( M h ( l L b) y L p o y L P c d ) ( More han one year (panel, see Lab): y L P c e ) ( Non lnear funconal forms? (logs ) y L p o y L P c e,,,, ) ( Non-lnear funconal forms? (logs, ) Quanave Technques for Regulaon

30 References Davs, P. and E. Garces (2010), Quanave Technques for Compeon and Anrus Analyss, Prnceon and Oxford Unversy Press Greene, W., (2003) Economerc Analyss, Prence Hall Quanave Technques for Regulaon

31 Appendx: ndex numbers

32 Index Numbers Insrumen o measure change n levels of varous relaed economc varables (RPI, ) Here, some prce and quany ndex numbers relevan o producvy analyss Objecves: Measure changes n Toal Facor Producvy (TFP) (see chaper on benchmarkng) Aggregae daa no smaller ses of npus and oupus Quanave Technques for Regulaon

33 Prce Index Numbers In a mul-produc seng, how do we measure prce and quany changes, across me or ndvduals? (ndex number problem) Tornqvs prce ndex P T, e.g., from s o s defned as: N, s T, ln P, ln, ln, 1 2 s p p s where p and q are he prce and, s, s p q, s, s N p 1, sq, s, s s he value of he quany of produc n s - perod produc n perod s Ohers nclude Laspeyres, Paasche, Fsher Quanave Technques for Regulaon

34 Quany Index Numbers & TFP Smlar o before we have several possble ndces Tornqvs quany ndex Q T from s o, as before: ln N T, s, Q s, ln q, ln q, s 1 2 Wh Tornqvs quany ndces, e.g., compue TFP: ln Oupu T s, T T TFP s, ln ln Oupu T s, ln Inpus, Inpu s, Quanave Technques for Regulaon