Review of Network Economics

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1 Review of Nework Economics Volume 10, Issue Aricle 2 On he Relaionship Beween Hisoric Cos, Forward Looking Cos and Long Run Marginal Cos William P. Rogerson, Norhwesern Universiy Recommended Ciaion: Rogerson, William P. (2011) "On he Relaionship Beween Hisoric Cos, Forward Looking Cos and Long Run Marginal Cos," Review of Nework Economics: Vol. 10: Iss. 2, Aricle 2. DOI: /

2 On he Relaionship Beween Hisoric Cos, Forward Looking Cos and Long Run Marginal Cos William P. Rogerson Absrac This paper considers a simple model where a regulaed firm mus make sunk invesmens in long-lived asses in order o produce oupu, asses exhibi a known bu arbirary paern of depreciaion, here are consan reurns o scale wihin each period, and he replacemen cos of asses is weakly falling over ime due o echnological progress. I is shown ha a simple formula can be used o calculae he long run marginal cos of producion each period and ha he firm breaks even if prices are se equal o long run marginal cos. Furhermore, he formula for calculaing long run marginal cos can be inerpreed as a formula for calculaing forward looking cos (where he curren cos of using asses is based on he curren replacemen cos of asses). However, hrough appropriae choice of he accouning depreciaion rule, i can also be inerpreed as a formula for calculaing hisoric cos (where he curren cos of using asses is based on he hisoric purchase cos of asses). In paricular, he resuls derived in he simple benchmark model of his paper conradic he commonly expressed view ha measures of forward looking cos are superior o measures of hisoric cos in environmens wih declining asse prices. KEYWORDS: hisoric cos, forward looking cos, long run marginal cos, cos allocaion, depreciaion Auhor Noes: Research suppor from he Searle Fund for Policy Research is graefully acknowledged. I would like o hank Debra Aron, Kahleen Hagery, Michael Salinger, Sefan Reichelsein, Korok Ray, David Sappingon, William Sharkey, Nancy Sokey, T. N. Srinivasan, Jean Tirole, Julian Wrigh and wo anonymous referees for helpful discussions and commens.

3 Rogerson: Hisoric, Forward Looking and Long Run Marginal Cos 1. INTRODUCTION Under radiional rae of reurn regulaion, he per period cos of using long-lived asses o produce goods and services is calculaed by allocaing he original purchase cos of each asse across all of he periods ha he asse will be used. Since he cos of using long-lived asses in any given period will herefore depend on he original purchase coss of all of he asses being used in ha period, such coss are ofen referred o as hisoric coss. In he elecommunicaions indusry, he replacemen cos of many of he long-lived asses ha firms use o produce elecommunicaions services has been dropping dramaically over ime due o echnological progress. Moivaed by he inuiion ha hisorical cosing mehods may oversae he curren long run marginal cos of producion in his case, (because curren long run marginal cos should depend on he curren cos of replacing asses which is lower han he hisoric c o s o f p u r c h a s i n g a s s e s ), r e g u l a o r s i n m a n y c o u n r i e s, i n c l u d i n g h e U n i e d Saes and mos counries in Wesern Europe, have recenly begun o base prices on coss calculaed using he curren replacemen cos of asses insead of heir original purchase cos. Coss calculaed under such a mehodology are ofen referred o as 1 forward looking coss. Under a forward looking mehodology, he regulaor esimaes he oal cos of replacing he exising asses of he firm wih funcionally equivalen new asses and hen allocaes a share of he esimaed oal replacemen cos o he curren period. Noe ha eiher mehod of calculaing coss requires he regulaor o make a decision on how o allocae coss. In he case of hisoric cos, he regulaor mus choose wha share of he hisoric purchase price of an asse o allocae o each period of he asse s lifeime. This will be called he hisoric allocaion rule. In he case of forward looking cos, he regulaor mus choose wha share of he esimaed oal replacemen cos o allocae o he curren period. This will be called he forward looking allocaion share. There has been considerable conroversy and confusion over he relaed issues of how invesmen coss should be allocaed for purposes of calculaing eiher ype of cos, which cosing mehod is superior for purposes of seing cos-based prices, and how he answer o hese quesions depends on facors such as he rae of echnological progress and he depreciaion paern of he underlying asses. This paper provides a heory which addresses hese quesions. In paricular, wo major resuls are proven. Firs, i is shown ha a very simple formula can be used o calculae he long run marginal cos of producion in 1 In he Unied Saes his cos concep is ofen referred o as Toal Elemen Long Run Incremenal Cos (TELRIC) and in Wesern Europe, New Zealand, and Ausralia i is ofen referred o as Toal Service Long Run Incremenal Cos (TSLRIC). See Salinger (1998), Hausman (2000), Falch (2002), Rosson and Noll (2002), Tardiff (2002) and Federal Communicaions Commission (2003) as well as oher references cied below for furher discussion and insiuional background. 1

4 Review of Nework Economics, Vol. 10 [2011], Iss. 2, Ar. 2 each period, and ha he firm breaks even if price each period is se equal o long run marginal cos. Second, i is shown ha he formula for calculaing long run marginal cos can be inerpreed eiher as a formula for calculaing forward looking cos or as a formula for calculaing hisoric cos so long as he correc mehod for allocaing coss is used in eiher case. For he case of hisoric cos, he correc hisoric cos allocaion rule is a relaively simple and naural rule which is called he relaive replacemen cos (RRC) rule. For he case of forward looking cos, he correc forward looking allocaion share is he unique forward looking allocaion share ha allows he firm o break even. Noe ha he resuls of his paper show ha eiher m e h o d o f c a l c u l a i n g c o s s can be used o se opimal prices regardless of he rae of echnological progress. 2 In paricular, he inuiion ha basing prices on forward looking cos somehow becomes more appropriae as he rae of echnological progress increases is shown o be false in he simple benchmark model of his paper. This resul poenially has imporan public policy implicaions. I can be argued ha one advanage ha hisorically based cosing rules have over forward looking rules in real-world applicaions is ha hey are based on more objecive daa and hus reduce he cos of regulaory proceedings and also allow regulaors o make more binding commimens. If forward looking rules do no offer some compensaing advanage, hen i is no clear why hey should be used. In he formal model of his paper i is assumed ha asses have a known bu arbirary depreciaion paern and ha he purchase price of new asses decreases a a known consan rae over ime. The RRC allocaion rule is defined o be he unique allocaion rule ha saisfies he wo properies ha (i) he share of cos allocaed o each period is proporional o he cos of replacing he surviving amoun of he asse wih new asses and (ii) he presen discouned value of he cos allocaions calculaed using he firm s cos of capial is equal o he original purchase price of he asse. Propery (i) can be inerpreed as a mehod of maching he ime paern of cos allocaions for an asse o he ime paern of benefis creaed by he asse. Propery (ii) simply requires ha he firm be fully reimbursed for is invesmens. While he RRC allocaion rule is somewha differen han he sors of allocaion rules radiionally used in rae-of-reurn regulaion, i is inuiively reasonable, simple, and easy o calculae, and would herefore be exremely suiable for use in real regulaory proceedings. The welfare maximizaion problem of a social planner (choosing prices o maximize discouned oal surplus) is very similar in srucure o he profi 2 Of course, as will be seen, in eiher case he correc mehod of cos allocaion depends on he rae of echnological progress. DOI: /

5 Rogerson: Hisoric, Forward Looking and Long Run Marginal Cos maximizaion problem of an unregulaed firm (choosing prices o maximize discouned cash flows) sudied in he opimal invesmen lieraure. Therefore he resuls of his paper do no require he developmen of new analyic echniques, bu insead follow fairly direcly from exising resuls in he opimal invesmen lieraure ha show how he concep of user cos can be employed o dramaically simplify 3 he analysis of invesmen problems. For his paper s purposes, a significan limiaion of mos of his lieraure is ha i resrics iself o considering he case of exponenial depreciaion, where a consan fracion of he capial sock is assumed o depreciae each period. This assumpion dramaically simplifies he analysis because he age profile of he exising capial sock can be ignored. However, for he purposes of his paper s sudy of cos allocaion rules, i is imporan o allow for general paerns of depreciaion, because one of he mos ineresing quesions o invesigae regarding cos allocaion rules is how he naure of he appropriae cos allocaion rule should change as he depreciaion paern of he underlying asses changes. Obviously he paern of depreciaion mus be a facor which can be exogenously varied in order o invesigae his quesion. Furhermore, he case of exponenial depreciaion is no a paricularly naural case o consider for applicaions, since mos regulaors assume ha depreciaion occurs according o he so-called onehoss shay paern where asses have a fixed lifeime and remain equally useful over heir lifeime. In an early paper, Arrow (1964) showed ha he user cos approach could acually be generalized o apply o he case of general paerns of depreciaion. Recenly, Rogerson (2008) has exended hese resuls by deriving a simpler formula for user cos and showing ha he RRC allocaion rule can be used o calculae user cos. This paper s resuls follow from he resuls in Arrow (1964) and Rogerson (2008). In he conex of his paper s model, he Arrow/Rogerson resuls provide a simple formula for calculaing a vecor of user coss for any paern of depreciaion such ha he presen discouned cos of producing any vecor of oupus can be calculaed by assuming ha he firm has a consan marginal cos of producion each 4 period equal o ha period s user cos. Thus, he seemingly complex dynamic opimal invesmen problem acually collapses ino a series of addiively separable single period problems where he firm has a consan marginal cos of producion each period. Obviously, seing price each period equal o user cos generaes efficien consumpion and allows he firm o break even. I urns ou ha user cos in any period is equal o a consan muliplied by he oal cos of purchasing a uni 3 See Jorgensen (1963) for an early reamen and Abel (1990) for an exensive lieraure review. 4 Noe, however, ha his paper presens a differen and more direc derivaion of he vecor of user coss han in presened in Rogerson (2008). See foonoe 13 for deails. 3

6 Review of Nework Economics, Vol. 10 [2011], Iss. 2, Ar. 2 of he asse in ha period. By definiion, his formula can herefore be inerpreed as a formula for calculaing forward looking cos where he consan is he forward 5 looking allocaion share. Rogerson s (2008) resul, ha he RRC allocaion rule can be used o calculae user cos, direcly implies ha he RRC allocaion rule can also be used o calculae efficien prices. For purposes of describing he incremenal conribuions of his paper o he lieraure on dynamic regulaory pricing, his paper can be viewed as showing ha hree differen resuls hold rue for any exogenously specified asse depreciaion paern: (i) (ii) (iii) A simple formula exiss o calculae renal raes for capial which can be inerpreed as being hypoheical perfecly compeiive renal raes in he sense ha hey allow he firm o break even on an invesmen made in any period. The hypoheical perfecly compeiive renal rae in any period is acually equal o he long run marginal cos of using capial in ha period. Therefore he efficien price for oupu in any period is simply equal o he hypoheical perfecly compeiive cos of rening sufficien capial o produce one uni of oupu. If hisoric cos is calculaed using a simple and naural allocaion rule called he RRC allocaion rule, he uni accouning cos of producion in each period is equal o he long run marginal cos of producion in ha period. Thus cos-based pricing resuls in efficien prices when he RRC allocaion rule is used o calculae coss. The exising lieraure has esablished resul (i) for he case of any paern of depreciaion and resul (ii) for he case of exponenial depreciaion. Therefore he incremenal conribuion of his paper is o esablish ha resul (ii) holds rue for he case of depreciaion paerns oher han he exponenial paern and o esablish resul (iii). More specifically, Biglaiser and Riordan (2000), Laffon and Tirole (2000, Box 4.2, page 152), and Hausman (1997) consider he case of exponenial depreciaion and esablish resuls (i) and (ii) for his case. Salinger (1998) derives he formula for perfecly compeiive prices for he case of general paerns of depreciaion and Mandy and Sharkey (2003) derive he formula for he special case of one-hoss shay depreciaion bu neiher paper explicily invesigaes he welfare problem (or equivalenly, he problem of how o calculae he long run marginal cos of 5 This is rue when one uni of he asse is defined o be he amoun of he asse necessary o produce one uni of oupu. DOI: /

7 Rogerson: Hisoric, Forward Looking and Long Run Marginal Cos 6 producion in any given period). Regarding resul (iii), i is well undersood ha, as an accouning ideniy, any pricing rule ha allows he firm o break even can be hough of as being a cos-based rule for some mehod of depreciaing asses over 7 ime. Therefore resul (ii) immediaely implies ha here mus be some mehod of allocaing invesmen coss wih he propery ha prices will be efficien if hey are se equal o hisoric coss calculaed using his allocaion mehod. The conribuion of his paper is o show ha he allocaion rule akes a remarkably simple and naural form ha would be suiable for use in pracice. The main formal model in he economics lieraure ha analyzes he welfare effecs of iner-emporal cos allocaion rules in he conex of cos-based regulaion 8 is due o Baumol (1971). Baumol suppresses he issue of long run efficiency by simply assuming ha he firm has already made a single fixed exogenous expendiure on long-lived asses and no furher invesmen of any sor is possible. I is assumed ha he firm can vary is oupu from period o period only by varying he amoun of non-capial inpus i uses each period. In his analysis, i would be efficien o se price each period equal o shor run marginal cos and, in general, seing prices a his level would no allow he firm o recover is invesmen cos. Baumol solves for a second bes price pah ha maximizes oal surplus subjec o he consrain ha he firm mus be allowed o recover is invesmen cos. Therefore in Baumol s model, allocaing he cos of long-lived invesmens across ime is a sor of necessary evil ha has o be endured in order ha he firm be reimbursed for is invesmen expenses. This paper shows ha a dramaically differen sor of resul can occur in a model where i is assumed ha invesmen occurs every period and he issue of long run efficiency is considered. Namely, allocaing he cos of long-lived invesmens across ime can help play a role in making consumpion decisions more efficien by ensuring ha prices reflec long run marginal cos. 6 Also, see M andy (2002) for an esimaion of he exen o which prices se under exising regulaory pracices diverge from hypoheical perfecly compeiive prices. 7 The relevan depreciaion mehod is simply he Hoelling or economic depreciaion associaed wih he sream of revenues generaed by he pricing rule. Given any fixed sream of revenues, he Hoelling or economic depreciaion in any period is defined o be he change in he presen discouned value of he remaining revenue sream. See Hoelling (1925) for he original reamen of his concep of depreciaion. See Schmalensee (1989) for a clear reamen of he role of his concep in cos-based pricing. Salinger (1998) and Biglaiser and Riordan (2000) boh noe ha he pricing rules hey idenify could in principle be implemened by a cos based pricing rule. 8 Also see Crew and Kleindorfer (1992) who consider he issue ha i may be necessary o fron-load he reimbursemen of a firm s invesmen if fuure enry of compeiors is expeced. Rogerson (1992) analyzes he effec of various depreciaion rules on a regulaed firm s incenives o choose an efficien level of invesmen in he presence of regulaory lag. 5

8 Review of Nework Economics, Vol. 10 [2011], Iss. 2, Ar. 2 In a recen paper, Guhrie, Small and Wrigh (2006 )compare he performance of access pricing rules based on forward looking vs. hisoric cos measures and conclude ha hisoric access prices generally creae superior invesmen incenives o forward looking access prices. Guhrie, Small and Wrigh focus on he effecs of uncerainy and he opion value creaed by uncerainy and absrac away from issues relaing o ongoing invesmen by assuming ha here is a one-ime invesmen. This paper, in conras, focuses on issues creaed by ongoing invesmen and absracs away from issues relaed o uncerainy by considering a model wih no uncerainy. The wo papers hus provide complemenary analyses of differen economic facors ha affec he relaive performance of forward looking vs. hisoric cos measures. The paper is organized as follows. Secion 2 describes he basic model. Secion 3 describes he relevan user cos resul from he opimal invesmen lieraure and is implicaions for he efficien pricing rule in he model of his paper. Secion 4 briefly discusses he effec of changes in he rae of echnological progress on long run marginal cos. Secion 5 considers forward looking pricing rules. Secion 6 considers hisoric pricing rules. Secion 7 considers policy implicaions for he choice beween forward looking vs. hisoric pricing rules. Secion 8 draws a brief conclusion. More echnical proofs are conained in an appendix. 2. THE MODEL Le q [0, ) denoe he firm s oupu in period {1, 2,... } and le q = (q 1, q 2,... ) denoe he vecor of oupus. Similarly, le K [0, ) denoe he firm s capial sock in period {1, 2,... } and le K = (K 1, K 2,... ) denoe he vecor of capial socks. Assume ha a capial sock of K in period allows he firm o produce up o K unis of oupu in period. For simpliciy, assume ha no oher inpus are required o produce oupu and he firm has no asses a he beginning of period 0. Le I [0, ) denoe he number of unis of capial ha he firm purchases in period {0, 1,...} and le I = (I 0, I 1,... ) denoe he enire vecor of invesmens. Assume ha a uni of capial becomes available for use one period afer i is purchased and hen gradually wears ou or depreciaes over ime. I will be convenien o use noaion ha direcly defines he share of he asse ha survives, and is hus available for use in each period, raher han he share ha depreciaes. Le h s denoe he share of an asse ha survives unil a leas he period of he asse s lifeime and le s = (s 1, s 2,... ) denoe he enire vecor of survival shares. Then he vecor of capial socks generaed by any vecor of invesmens is given by (1) K = si i=1 i -i DOI: /

9 Rogerson: Hisoric, Forward Looking and Long Run Marginal Cos Assume ha s [0,1] for every, s 1 =1, and ha s is weakly decreasing in. Two naural and simple examples of depreciaion paerns are he cases of exponenial depreciaion given by (2) s = -1 for some (0,1) and one-hoss shay depreciaion given by (3) s = 1, {1, 2,..., T} 0, oherwise where T is a posiive ineger. Le (0,1) denoe he discoun facor. Le z [0, ) denoe he price of purchasing a new uni of he asse in period. Assume ha asse prices are weakly decreasing and change a a consan rae over ime. Formally, assume ha asse prices are given by (4) z = z0 9 for some (0, 1] and z0 (0, ). Thus, lower values of correspond o higher raes of echnological progress. Le E(I) denoe he presen discouned value of he expendiures required o creae he vecor of invesmens I, given by (5) E(I) = z 0 I () =0 A vecor of invesmens I will be said o be efficien for a vecor of oupus q if i minimizes he expeced discouned cos of invesmen subjec o he consrain ha sufficien capial is available every period o allow producion of q and subjec o he consrain ha invesmen every period mus be non-negaive. The 9 The assumpion ha asse prices change a a consan rae is necessary for some bu no all of he conclusions of his analysis. The basic user cos resul can sill be derived in he general case where i is only assumed ha asse purchase prices are weakly decreasing over ime. I is sill rue ha i is efficien o se price each period equal o ha period s user cos and ha he formula for calculaing efficien prices can be inerpreed as being eiher a formula for calculaing forward looking cos or hisoric cos. However he formulas are much more complicaed and have no simple inerpreaion. Thus, alhough he basic conclusion ha he efficien pricing rule can be inerpreed as being a formula for calculaing eiher forward-looking or hisoric cos remains correc, he formulas are no longer simple enough o be obviously suiable for applied use. See Rogerson (2008) for deails. 7

10 Review of Nework Economics, Vol. 10 [2011], Iss. 2, Ar. 2 assumpions ha 1 and < 1 imply ha i will never be efficien for he firm o sockpile asses ahead of ime. Therefore, for any vecor of oupus, he unique efficien vecor of invesmens can be calculaed sequenially beginning wih period 0. In each period he firm purchases he minimum number of asses required produce nex period s oupu. (If nex period s capial sock will already be greaer han he required level wihou any invesmen, hen no asses are purchased.) Le (q 1,..., q +1) denoe he efficien choice of I and le (q) = ( 0(q 1), 1(q 1, q 2),... ) denoe he enire vecor of efficien invesmen choices. Le C(q) denoe he minimum cos of producing he vecor of oupus q. This will be called he firm s cos funcion and is defined by (6) C(q) = E((q)) = z (q,..., q ) () A vecor of oupus will be said o saisfy he fully uilized invesmen (FUI) propery if here is never any excess capaciy when he vecor of oupus is produced efficienly. Formally, q saisfies he FUI propery if (7) q = si -i(q 1,..., q -i+1) i=1 For fuure reference, noe ha a sufficien condiion for a vecor of oupus o saisfy he FUI propery is ha oupu be weakly increasing over ime. 10 Le p [0, ) denoe he price of oupu in period and le P : (0, ) [0, ) denoe he period inverse demand funcion. Assume ha P (q ) is greaer han or equal o zero, sricly decreasing and differeniable where i is sricly posiive, ha P (q ) converges o 0 as q converges o, and ha P (q ) converges o as q 11 converges o 0. Le D (p ) denoe he period demand funcion. One assumpion wih real economic conen will need o be made abou demand. This is ha demand is weakly increasing over ime. Formally, i will be assumed ha =0 10 The proof is by inducion. The firm obviously operaes wih no excess capaciy in period 1. I is also easy o see ha if oupu is weakly increasing over ime and he firm operaes wih no excess capaciy in period, hen i will operae wih no excess capaciy in period The las assumpion is made simply o avoid he exra noaional burden of describing corner soluions a q = 0. DOI: /

11 Rogerson: Hisoric, Forward Looking and Long Run Marginal Cos (8) D +1(p) D (p) for every p [0, ) and {1, 2,... }. Le B (q ) denoe he consumer benefi of oupu q in period, given by (9) B (q ) = P (x)dx. x=0 q Le B( q) denoe he discouned social consumer benefi of he vecor of oupus q given by (10) B(q) = B(q) =1 Finally le W(q) denoe he discouned welfare of he vecor of oupus q given by (11) W(q) = B(q) - C(q) A vecor of quaniies will be said o be efficien if i maximizes W(q). A vecor of prices will be said o be efficien if i induces consumers o demand an efficien vecor of quaniies. I w i l l b e u s e f u l o i n r o d u c e o n e a d d i i o n a l p i e c e o f n o a i o n. L e (p) denoe he presen discouned value of he firm s cash flows if i charges prices according o p and invess efficienly o supply all demand a hese prices. I is given by (12) (p) = D(p)p - C(D(p)) =1 I w i l l b e s a i d h a h e f i r m e a r n s z e r o ( p o s i v e ; n e g a i v e ) p r o f i a p if (p) = (>; <) 0. Finally, i is worh drawing aenion o wo assumpions implicily inroduced in he descripion of he model which likely play an imporan role in generaing he specific resuls of his paper. The firs assumpion is ha he cos of acquiring capial in any given period is linear and ha capial can be acquired in infiniely divisible quaniies. If here were economies or diseconomies of scale in acquiring capial in any given period, or if capial was lumpy, i would be necessary o look muliple periods ahead o deermine he correc level of invesmen in any given period and 9

12 Review of Nework Economics, Vol. 10 [2011], Iss. 2, Ar. 2 he basic user cos resul from he opimal invesmen lieraure would no longer hold. The second assumpion is ha here is no uncerainy. The opimal invesmen problem changes dramaically in a number of ways once uncerainy is inroduced. Invesigaing he effecs of nonlinear capial acquisiion coss and uncerainy on efficien pricing rules and he exen o which hey can be achieved by forward looking vs. hisoric cos measures is an ineresing and imporan subjec for fuure research THE USER COST RESULT AND EFFICIENT PRICES Suppose for a momen ha, insead of having o purchase long-lived asses, he firm could ren asses on a period-by-period basis and he cos of rening one uni of he asse in period was equal o c. In his case, he welfare problem would collapse ino a series of simple addiively separable single period problems where he firm has a consan marginal cos of producion in period equal o c. Obviously he efficien soluion would be for he firm o charge a price of c in period and he firm would break even a his soluion. The essenial resul of he user cos approach is ha a very simple formula exiss o calculae a vecor of hypoheical perfecly compeiive renal prices or user coss and ha, over he relevan range of oupu, he firm s rue cos funcion, given ha i mus purchase asses, is acually equal o he hypoheical cos funcion i would have if i could ren asses a hese raes. In paricular, over he relevan range of oupu he firm has a consan marginal cos of producion in each period equal o he hypoheical perfecly compeiive renal rae of capial or user cos in ha period. The efficien soluion is herefore for he firm o charge a price for oupu in each period equal o ha period s hypoheical perfecly compeiive renal rae or user cos and he firm breaks even a his soluion. To derive he formula for he hypoheical perfecly compeiive renal raes, consider a hypoheical siuaion where here is a renal marke for asses and a supplier of renal services can ener he marke in any period by purchasing one uni of he asse and hen rening ou he available capial sock over he asse s life. Le c denoe he price of rening one uni of capial sock in period and le c = (c 1, c 2,... ) denoe he enire vecor of renal prices. Assume ha suppliers incur no exra coss besides he cos of purchasing he asse, ha hey can ren he full remaining amoun of he asse every period and ha heir discoun facor is equal o. Then he zero profi condiion ha mus be saisfied by a perfecly compeiive equilibrium is 12 As menioned in he inroducion, Guhrie, Small and Wrigh (2006)compare he performance of access pricing rules based on forward looking vs. hisoric cos measures in a model which focuses on he effecs of uncerainy. DOI: /

13 Rogerson: Hisoric, Forward Looking and Long Run Marginal Cos (13) z = c +i si i for every {0, 1, 2,...} i=1 Afer subsiuing equaion (4) ino (13), i is sraighforward o verify ha he vecor of renal raes c* = (c 1*, c 2*,... ) as defined below by equaions (14)-(15) saisfies equaion (13). (14) c * = k* z (15) k* = 1/ s() i i i=1 The vecor of renal raes c* will be called he vecor of hypoheical perfecly compeiive renal raes or he vecor of user coss. Noe ha he renal rae in period is equal o he posiive consan k* muliplied by he cos of purchasing a uni of capial in ha period. Therefore, all of he renal raes are sricly posiive and hey decline a he same rae ha he purchase price of asses declines a. Le H (q) denoe he cos funcion ha he firm would have if, insead of having o purchase asses, i was able o ren asses a he hypoheical perfecly compeiive renal raes. (16) H(q) = c *q =1 This will ofen be referred o simply as he hypoheical cos funcion. Recall ha C(q) denoes he firm s cos funcion given ha i is unable o ren asses bu insead mus purchase hem. This will someime be referred o as he firms rue cos funcion o disinguish i from he hypoheical cos funcion. Le p* and q* denoe he unique vecors of prices and quaniies ha would be efficien in he hypoheical case where he firm could ren asses a he hypoheical perfecly compeiive renal raes. These are obviously deermined by (17) p * = c * (18) q * = D (c *) Furhermore, i is also obvious ha he firm would break even a his soluion. 11

14 Review of Nework Economics, Vol. 10 [2011], Iss. 2, Ar. 2 If i c o u l d b e s h o w n h a h e f i r m s r u e c o s f u n c i o n w a s a l w a ys e q u a l o h e hypoheical cos funcion, i would herefore follow immediaely ha p* and q* were also he unique vecors of efficien prices and quaniies for he rue case of ineres and ha he firm breaks even a hese prices. I urns ou ha only a somewha weaker relaionship beween he wo cos funcions can be esablished. Lemma 1: (19) C(q) = H(q) for every q saisfying he FUI propery (20) C(q) H(q) for every q [0, ) Proof: See Appendix. QED Equaion (19) saes he firm s rue cos funcion is equal o he hypoheical cos funcion for every vecor of oupus saisfying he FUI propery. Equaion (20) saes ha he firm s rue cos funcion is always greaer han or equal o he hypoheical cos funcion. Boh resuls are very inuiive. The zero profi condiion (13) can be inerpreed as saing ha he cos of purchasing any single asse mus be equal o he hypoheical cos of rening asses o produce he vecor of oupus ha would be produced if he asse was fully uilized. Since he renal raes are all posiive his also means ha he cos of purchasing any single asse mus be greaer han or equal o he hypoheical cos of rening asses o produce any vecor of oupus ha his asse was able o produce. Obviously, he same condiions mus hold for he presen discouned value of any sequence of asses purchased over ime, which is wha is saed in equaions (19) and (20). (The proof of Proposiion 1 in he appendix is simply a formalizaion of his reasoning.) I u r n s o u h a Le m m a 1 i s s u f f i c i e n o e s a b l i s h h e r e s u l o f i n e r e s. T h e reason for his is ha he assumpion ha demand is weakly increasing implies ha he soluion o he hypoheical welfare problem, q*, is also weakly increasing and herefore saisfies he FUI propery. I follows immediaely from his and Lemma 1 ha q* mus herefore also be he soluion o he rue welfare problem. Proposiion 1: The unique vecors of efficien prices and quaniies are p* and q*. The firm breaks even a his soluion, i.e, (p*) = 0. Proof: See Appendix. QED DOI: /

15 Rogerson: Hisoric, Forward Looking and Long Run Marginal Cos In summary, he user cos approach essenially shows ha he firm s cos funcion is linear and addiively separable in each period s oupu over he relevan range of oupus. Therefore he seemingly complex muli-period problem acually collapses ino a series of addiively separable single period problems. Welfare is maximized by seing price each period equal o he marginal cos of producion and he firm breaks even a hese prices. Furhermore, here is a simple formula o calculae he marginal cos of producion in any period. 13 The resul ha he firm s cos funcion is linear and addiively separable over a broad range of oupus migh iniially seem somewha surprising in ligh of he fac ha each asse represens a join cos of producion across muliple periods. A widely acceped general principle in boh he economics and managerial accouning lieraures ha sudy cos allocaion is ha here is generally no economically 14 meaningful way assign a join cos o individual producs. Thus we migh expec ha he cos of producing a vecor of join producs would inherenly no be addiively separable in each produc. Ye his is precisely wha happens in he model of his paper. The resoluion o his apparen conflic lies in he fac ha here are muliple overlapping join coss in he model of his paper insead of a single join cos. When here is a single join cos for all producs, he only way o increase he oupu of a single produc is o increase invesmen in he join cos and his resuls in increased oupu of all producs. Thus, increasing he producion of one good necessarily resuls in increases in he producion of all goods. However, in he model 13 I is possible o direcly calculae he vecor of user coss wihou using he zero profi condiion (13) by invering he linear funcion in equaion (1) o direcly calculae he coefficiens of he linear funcion and subsiue hese ino (4) o direcly calculae C(q) as a linear funcion of q. This is he approach originally used by Arrow (1964) and yields a formula for user cos ha depends on he coefficiens of. While he coefficiens of have a naural inerpreaion (hey are he series of marginal changes o invesmen ha he firm would have o make in order o increase he sock of capial in one period while holding he sock of capial in all oher periods fixed) hey are difficul o calculae because hey are deermined by an infinie series of recursively defined equaions. Rogerson (2008) presens Arrow s derivaion and hen direcly shows ha he more simple formula for user cos in (14)-(15) is equivalen o Arrow s more complex formula. As seen above, his paper akes a differen approach which compleely avoids calculaing he coefficiens of and avoids deriving he more complex formula as an inermediae sage. I insead direcly observes ha he vecor of renal raes defined by (14)-(15) saisfies he zero profi condiion (13) and hen uses (13) o direcly show ha hese renal raes are equal o he marginal cos of producion over he relevan rage of oupu. While he approach presened in his paper is much simpler, Arrow s original approach provides some exra inuiion because i shows ha he marginal cos of producing one more uni of oupu in any period is equal o he presen discouned value of he series of marginal changes o invesmen ha would produce one more uni of capial in ha period while holding he level of capial fixed in all oher periods. See Rogerson (2008) for more deails. 14 See, for example, Demski (1981), Thomas (1978), and Young (1985). 13

16 Review of Nework Economics, Vol. 10 [2011], Iss. 2, Ar. 2 of his paper, where here are muliple overlapping join coss, his is no necessarily rue. An illuminaing way o see his poin is o direcly calculae marginal cos for a simple example by direcly deermining he adjusmens in asse purchases ha are necessary o produce an addiional uni of oupu in a given period while holding oupu in all oher periods fixed. The presen discouned value of hese adjusmens is, of course, by definiion he marginal cos of producion. Consider, for example, he case of he one-hoss shay paern of asse decay given by equaion (3), where an asse lass wih undiminished produciviy for T years. Suppose ha he firm has made invesmen plans o produce a paricular vecor of oupus over ime and is engaging in a leas one uni of invesmen in each period. Then he firm can increase oupu in period 1 by one uni while holding oupu in all oher periods consan by implemening he following series of adjusmens o is invesmen plans. The firm mus purchase an addiional uni of he asse in period 0 o increase producion by one uni in period 1. However, i will now be able o reduce is asse purchases by one uni in period 1. Now when period T arrives, he exra asse ha he firm purchased in period 0 will no longer be available he nex period, so he firm will have o purchase an exra uni of he asse in ha period o mainain is level of producion a he previously planned level in period T+1. However, as before, i will now be able o reduce is asse purchases by one uni in period T+1. This process coninues indefiniely. Tha is, he firm can produce exacly one more uni of oupu in he period 1 and hold oupu in all oher periods fixed by shifing he purchase of one uni of he asse forward in ime from period 1 o 0, T+1 o T, 2T+1 o 2T, ec. The presen discouned value calculaed in period 1 of he cos of hese adjusmens is, by definiion, he marginal cos of increasing oupu by one uni in period 1. I is sraighforward o direcly calculae his value and show ha i is equal o k*z 1. Thus, even hough each asse can be viewed as a join cos of producion over muliple periods, i is sill possible o increase producion in one period while holding oupu in all oher periods consan by adjusing he enire vecor of overlapping join coss. The resul is ha he cos funcion is linear and addiively separable even hough here are join coss of producion. 4. THE EFFECT OF THE RATE OF TECHNOLOGICAL PROGRESS ON MARGINAL COST The effec of changing he rae of echnological progress on he vecor of marginal coss and hus on he vecor of efficien prices can now be invesigaed. Recall ha lower values of correspond o higher raes of echnological progress in he sense ha asse prices fall more rapidly. Since a higher rae of echnological progress sricly reduces he purchase price of asses in all periods subsequen o period 0, i DOI: /

17 Rogerson: Hisoric, Forward Looking and Long Run Marginal Cos is clear ha an increase in he rae of echnological progress mus always reduce he presen discouned cos of producing any vecor of oupus whose producion requires any invesmen afer period 0. Based on his observaion, one migh suspec ha an increase in he rae of echnological progress would herefore reduce he marginal cos of producion in every period. This urns ou no o be rue. Insead, i is always he case ha an increase in he rae of echnological progress increases he marginal cos of producion in periods immediaely following he change and only reduces he marginal cos of producion in laer periods. To see his resul, rewrie equaion (14) so ha he user cos in any period is expressed as a funcion of he purchase price of asses in he previous period. This y i e l d s (21) c * = k**z-1 i-1 i (22) k** = k* = 1/ si i=1 I i s e a s y o s e e h a k * * i n c r e a s e s i n h e r a e o f e c h n o l o gi c a l p r o g r e s s ( i. e., k * * decreases in ). Obviously, he price of asses in he curren period, z 0, does no change wih he rae of echnological progress. However, he prices of asses in all subsequen periods decrease in he rae of echnological progress (i.e. hey increase in ). Therefore an increase in he rae of echnological progress will unambiguously increase he marginal cos of producion for period 1. However, is effec on marginal coss in subsequen periods will be ambiguous because he increase in k** will be couneraced by a decrease in asse prices. The decrease in asse prices will grow more significan over ime as he increased rae of echnological progress operaes over more periods. Therefore, we would expec he second effec o evenually dominae for periods far enough ino he fuure. Tha is, we would expec an increase in he rae of echnological progress o raise marginal cos in early periods bu o decrease marginal coss in laer periods. Proposiion 2 formally saes his resul. Proposiion 2: Le c *() denoe he user cos in period given. Then here exiss a value () (1, ) defined by equaion (49) in he appendix such ha > > (23) c *() = 0 = () < < 15

18 Review of Nework Economics, Vol. 10 [2011], Iss. 2, Ar. 2 Proof: See Appendix. QED Thus, even hough an increase in he rae of echnological progress will reduce he oal discouned cos of producing any given vecor of oupus, i will acually increase he marginal cos of producion in early periods. The explanaion for his resul is ha he firm produces more oupu in he curren period by shifing purchases of asses from he fuure o he presen. When here is a higher rae of echnological progress, asse prices decrease more rapidly over ime, and he opporuniy cos of shifing asse purchases ahead in ime is herefore higher. In he firs period of producion his is he only effec and marginal cos herefore rises. A second effec ha becomes more imporan over ime is ha echnological progress will reduce marginal cos by reducing he fuure purchase price of asses. This second effec evenually dominaes and causes he marginal cos of producion o fall in periods far enough in he fuure. 5. FORWARD LOOKING PRICES Recall ha forward looking cos in a given period is deermined by firs deermining he oal cos ha he firm would incur o purchase sufficien new asses o produce he desired level of oupu and hen allocaing a share of his cos o he given period. The share of he oal hypoheical cos allocaed o he curren period is called he forward looking allocaion share. In he simple model of his paper where one uni of he asse is required o produce one uni of oupu, a forward looking pricing mehod mus herefore be a rule of he form (24) p = k z where k is he forward looking allocaion share in period. Tha is, a forward looking pricing rule is simply a rule specifying a vecor of consans k = (k 1, k 2,...) where he regulaed price in period is se equal o he share k of he price of purchasing asses, and k is he forward looking allocaion share in period. Define a saionary forward looking rule o be a rule ha uses he same forward looking allocaion share in each period. A comparison of (24) and (14) shows ha he efficien vecor of prices is produced if and only if k is se equal o he consan k* in every period. Tha is, here is unique forward looking allocaion rule ha generaes efficien prices and i is he saionary rule where he forward looking allocaion share is se equal o k* in every period. Now consider any saionary forward looking rule ha uses he allocaion share k in every period. Obviously prices se equal o forward looking cos DOI: /

19 Rogerson: Hisoric, Forward Looking and Long Run Marginal Cos calculaed using he forward looking allocaion share k will be sricly greaer han (equal o, sricly less han) he efficien vecor of prices if and only if k is sricly greaer han (equal o, sricly less han) k*. I has already been observed ha he firm earns zero profi if k is se equal o k* and i is easy o see ha he firm earns sricly posiive (sricly negaive) profi if k is sricly greaer han (sricly less han) k*. 15 I herefore follows ha seing k=k * yields he unique saionary forward looking allocaion rule ha causes he firm o earn zero profi and i will earn higher (lower) profi if k is se higher (lower) han k*. Proposiion 3 summarizes hese conclusions. Proposiion 3: Suppose ha a regulaor ses prices equal o forward looking cos calculaed using he forward looking allocaion share k. Then (i) (ii) The resuling prices will be efficien if and only if k is se equal o k* defined by (15) Consider a saionary forward looking allocaion rule where he forward looking allocaion share is se equal o k in each period. Seing k=k* yields he unique saionary forward looking allocaion rule ha causes he firm o earn zero profi and i will earn higher (lower) profi if k is se higher (lower) han k*. Proof: As above. QED 6. Hisoric Cos 6.1. Allocaion and Depreciaion Rules And Hisoric Cos Define a depreciaion rule o be a vecor d = (d 1, d 2,... ) such ha d i 0 for every i and (25) d i = 1 i= 1 h where di is inerpreed as he share of depreciaion allocaed o he i period of he 15 Forward looking prices change a he same rae as asse prices for any value of k. This means ha he resuling vecor of quaniies saisfies he FUI propery. The conclusion follows immediaely from his. 17

20 Review of Nework Economics, Vol. 10 [2011], Iss. 2, Ar. 2 asse s life. Define an allocaion rule o be a vecor a = (a 1, a 2,...) ha saisfies ai 0 for every i and i (26) ai = 1 i= 1 for some discoun facor (0,1). Le (a) denoe he value of such ha (26) is saisfied. The allocaion rule a will be said o be complee wih respec o he discoun facor (a). Regulaors generally hink of hemselves as direcly choosing a depreciaion rule and a discoun facor insead of direcly choosing an allocaion rule. The cos allocaed o each period is hen calculaed as he sum of he depreciaion allocaed o ha period plus impued ineres on he remaining (non-depreciaed) book value of he asse. Formally, for any depreciaion rule d and discoun facor, he corresponding allocaion rule is given by (27) a i = d i + {(1- )/} d j. j=i I is sraighforward o verify ha he resuling allocaion rule deermined by (27 ) is complee wih respec o. I is also sraighforward o verify ha for any allocaion rule, a, here is a unique (d, ) such ha (27) maps (d, ) ino a. I is defined by = (a) and j-i j-i-1 (28) d i = a j - aj j=i+1 j=i+2 Therefore one can equivalenly hink of he regulaor as choosing eiher a depreciaion rule and discoun facor or as choosing an allocaion rule. For he purposes of his paper, i will be more convenien o view he firm as direcly choosing an allocaion rule. Le A (I 0,..., I -1, a) denoe he oal accouning cos assigned o period given he vecor of invesmens (I,..., I ) and he allocaion rule a. I is defined by 0-1 (29) A (I 0,..., I -1, a) = I-iz-iai i=1 DOI: /

21 Rogerson: Hisoric, Forward Looking and Long Run Marginal Cos 6.2. The RRC Allocaion Rule An allocaion rule a = (a 1, a 2,... ) can be said o allocae coss in proporion o he cos of replacing he surviving amoun of he asse wih new asses if i saisfies (30) a = ks i i i for some posiive real number k. I is easy o verify ha an allocaion rule of he form in (30) is complee wih respec o if and only if he consan k is equal o he value k* defined by (15). Le a* denoe he allocaion rule deermined by seing k equal o k*, i.e., (31) a * = k*s i i i This will be called he relaive replacemen cos (RRC) allocaion rule. I is he unique allocaion rule ha saisfies he wo properies ha: (i) i allocaes coss in proporion o replacing he surviving amoun of he asse wih new asses, and (ii) i is complee wih respec o. Propery (i) can be inerpreed as a version of he maching principle from accrual accouning ha suggess ha coss should be allocaed across he periods of an asse s lifeime in proporion o he benefis ha he asse creaes in each period where he benefi ha an asse creaes in a period is inerpreed o be he avoided cos of purchasing new asses. Propery (ii) is simply he requiremen ha he firm break even aking he ime value of money ino accoun. For applied purposes, noe ha he RRC allocaion rule akes he following simple form for he case where asses follow he one-hoss shay depreciaion paern defined by equaion (3). (32) a i* = i k*, i {1,..., T} 0, i {T+1,.... } Tha is, for he case of one-hoss shay depreciaion, he cos of purchasing an asse is allocaed across he periods of an asse s lifeime o saisfy he requiremens ha (i) he cos allocaions decrease a he same rae ha he purchase price of asses is decreasing a and (ii) he presen discouned value of he cos allocaions is equal o he iniial purchase price of he asse. While he RRC allocaion rule is simple and inuiive, i is somewha differen han he sors of allocaion rules acually used in pracice. As explained above, regulaors generally view hemselves as direcly choosing a depreciaion rule and hen calculaing he oal cos allocaed o any period as he sum of ha period s 19

22 Review of Nework Economics, Vol. 10 [2011], Iss. 2, Ar. 2 depreciaion plus ineres on he non-depreciaed book value. Perhaps for his reason, hey have ended o focus direcly on he ime paern of depreciaion shares insead of he ime paern of allocaion shares. In conras, he approach suggesed by his paper would require regulaors o focus direcly on he ime paern of allocaion shares ha a depreciaion rule induces. While he RRC rule is herefore somewha differen han he sors of rules radiionally used in pracice, i is very simple, inuiively reasonable, and easy o calculae, and would herefore be very suiable for use in real regulaory proceedings. Lem m a 2 no w d es cr ib es a k ey p ro per y o f h e R R C al l oca i on ru l e Lemma 2: (33) a i*z -i = c *si for every {1, 2,...} and i {1, 2,.., }. Proof: Subsiue equaion (4) ino equaion (31) and reorganize. QED To inerpre equaion (33), consider any period {1, 2,... } and suppose ha he firm purchases one uni of capial i periods earlier in period -i for any i {1,.., }. The LHS of (33) is he accouning cos allocaed o period if he firm uses he RRC allocaion rule. The RHS of equaion (33) is he user cos in period muliplied by he surviving share of he asse. Therefore equaion (33) saes ha he RRC allocaion rule has he propery ha he cos allocaed o any period of an asse s lifeime is equal o ha period s user cos muliplied by he surviving amoun of he asse. Tha is, under he RRC allocaion rule, he per uni accouning cos of capial in a given period is equal o ha period s user cos regardless of when he capial was purchased! I follows immediaely from his ha, under he RRC allocaion rule, he accouning cos in any period is simply equal o ha period s user cos muliplied by ha period s capial sock. This resul is saed as Lemma 3. Lemma 3: Le I denoe any vecor of invesmens and le K denoe he vecor of capial socks generaed by I according o equaion (1). Then (34) A (I 0,..., I -1, a) = c * K for every {1, 2,... } Proof: Subsiue equaion (31) ino equaion (29) and reorganize. QED DOI: /

23 Rogerson: Hisoric, Forward Looking and Long Run Marginal Cos 6.3. Regulaory Equilibrium An ordered pair of vecors of prices and oupus (p, q) will be defined o be a regulaory equilibrium given he allocaion rule a if i saisfies he following wo requiremens. (35) D (p) = q for every {1, 2,... } (36) pq - A(q 1,...,q, a) 0 for every {1, 2,.. } Equaion (35) simply requires ha he firm supply all demand a he prices i is charging. Equaion (36) requires ha he firm s revenue in any period is always less 16 han or equal o is accouning cos for ha period. Lemma 3 can be inerpreed as saing ha he RRC allocaion rule has he propery ha he hisoric accouning cos per uni of capial in any period is equal o ha period s user cos. I herefore immediae ha he RRC accouning rule also has he propery ha he hisoric accouning cos per uni of oupu in any period mus be equal o ha period s user cos so long as he vecor of oupus saisfies he FUI propery. Since he vecor of efficien oupus q* has already been shown o saisfy he FUI propery, i herefore follows ha (p*, q*) is a regulaory equilibrium under he RRC allocaion rule. I has already been observed ha he firm earns zero profi when a he price p*. I is sraighforward o show ha he consrains in (36) imply ha he firm s profi can never be greaer han zero. This means ha here can be no oher regulaory equilibrium ha he firm sricly prefers o (p*, q*). This resul is saed as Proposiion 4. Proposiion 4: The efficien vecor of prices and quaniies (p*, q*) is an regulaory equilibrium under he RRC allocaion rule. The firm earns zero profi in his equilibrium and here is no regulaory equilibrium where he firm earns sricly posiive profi. Proof: As above. QED 16 The LHS of equaion (36) is ofen referred o as he residual income of he firm in period. Thus equaion (36) simply requires ha he firm s residual income be less han or equal o zero in every period. 21