DNB Working Paper. No / June AndreaColciago. Imperfect Competition and Optimal Taxation DNB WORKING PAPER

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1 DNB Working Paper No / June 23 AndreaColciago DNB WORKING PAPER Imperfec Compeiion and Opimal Taxaion

2 Imperfec Compeiion and Opimal Taxaion Andrea Colciago* * Views expressed are hose of he auhor and do no necessarily reflec official posiions of De Nederlandsche Bank. Working Paper No. 383 June 23 De Nederlandsche Bank NV P.O. Box 98 AB AMSTERDAM The Neherlands

3 Imperfec Compeiion and Opimal Taxaion Andrea Colciago De Nederlandsche Bank and Universiy of Milano Bicocca Firs version May 22, his version June 23 Absrac This paper provides opimal labor and dividend income axaion in a general equilibrium model wih oligopolisic compeiion and endogenous rms enry. In he long run he opimal dividend income ax correcs for ine cien enry. The dividend income ax depends on he form of compeiion and naure of he sunk enry coss. In paricular, i is higher in marke srucures characerized by compeiion in quaniies wih respec o hose characerized by price compeiion. Oligopolisic compeiion leads o an endogenous counercyclical price markup. As a resul o seing he disorions over he business cycle requires deviaions from full ax smoohing. JEL Classi caion Numbers: E62, L3. Keywords: Firms Enry, Marke Sucures, Marke Disorions, Opimal Dividend Income Tax. I hank Russel Cooper, Marin Ellison, Chrisian Hellwig, Pierre Lafourcade, Andy Levin, Vivien Lewis, Tommaso Monacelli, Giorgio Moa, Marco Hoeberichs, Damjan Pfajfar, Raffaele Rossi, Andreas Shaber, Neelje van Horen and Jouko Vilmunen for useful commens. I would also like o hank seminar paricipans a he Bank of Finland, he Universiy of Leuven, Lancaser Universiy, Tilburg Universiy and he Duch Naional Bank. Correspondence: Andrea Colciago, De Nederlandsche Bank, Weseinde, 7 ZN Amserdam, Nederlands. a.colciago@dnb.nl.

4 Inroducion A recen macroeconomic lieraure emphasizes he imporance of he creaion of new rms for he propagaion of business cycle ucuaions. Bilbiie, Ghironi and Meliz (22, BGM henceforh), Jaimovich and Floeoo (2) and Colciago and Ero (2 a and b), among ohers, show ha accouning for rms dynamics helps improving he performance of dynamic general equilibrium models a replicaing he variabiliy of he main macroeconomic variables in response o exogenous disurbances. Mos of hese sudies are characerized by imperfec compeiion in he goods marke. As a resul an ine cien number of producers (and producs) may arise in equilibrium, leading o welfare losses for he sociey. For his reason, policy measures aimed a removing marke disorions could be desirable. This paper provides opimal Ramsey dividend and labor income axaion in a framework characerized by alernaive, imperfecly compeiive, marke srucures. The economy feaures disinc secors, each one characerized by many rms supplying goods ha can be imperfecly subsiuable o a di eren degree, aking sraegic ineracions ino accoun and compeing eiher in prices (Berrand compeiion) or in quaniies (Courno compeiion). As in BGM (22) he enry of a new rm in he marke amouns o he creaion of a new produc. Sunk enry coss allow o endogenize enry and he (sock marke) value of each rm in each secor. Preferences of agens are characerized by love for variey, such ha spreading a given nominal consumpion expendiure over a larger number of goods leads o an increase in uiliy. The degree of marke power, as measured by he price markup, depends endogenously on he form of compeiion, on he degree of subsiuabiliy beween goods and on he equilibrium number of rms. Imporanly, he price markup is counercyclical. During an economic boom pro s opporuniies arac rms ino he marke. This srenghens compeiion and, via sraegic ineracions, reduces price markups. An early references on he procyclicaliy of he number of rms in he U.S. is Chaerjee and Cooper (993), while a more recen one is Lewis and Poilly (22). The counercyclicaliy of he price markup is consisen wih he empirical ndings by Bils (987), Roemberg and Woodford (2) and Galì e al. (27). Neverheless noice ha aggregae pro s remain srongly procyclical, as in he daa. As emphasize in BGM (27), he marke equilibrium is characerized by wo disorions: a Labor Disorion and an Enry Disorion. As in oher models wih an imperfecly compeiive goods marke, he presence of a price markup leads o a wedge beween he marginal produc of labor and he marginal rae of subsiuion beween consumpion and hours. Oligopolisic compeiion renders his wedge ime varying. The Enry Disorion operaes hrough he ineremporal rms creaion margin and leads o an ine cien number of rms in equilibrium. A posiive dividend income ax is opimal in case of excessive Early conribuions o his lieraure are Chaejee and Cooper (993), Devereux e al. (996) and Devereux and Lee (2). More recen developmens are insead Bergin and Corsei (28) and Faia (22).

5 enry and viceversa. To minimize he welfare cos associaed o hese disorions, he Ramsey opimal scal policy, boh in he shor and in he long run, is provided. The Governmen levies axes on dividend income and labor income and issues sae coningen bonds o nance an exogenous sream of public spending. In he long run he dividend income ax removes he Enry Disorion, as in Chugh and Ghironi (2). The magniude, and sign, of he dividend income ax depends on he form of he enry cos and on he form of compeiion, and i is higher in marke srucures characerized by lower compeiion. In paricular, i is higher under Courno Compeiion wih respec o Berrand or monopolisic compeiion. The long run labor income ax is posiive under all marke srucures considered. As a resul he Labor Disorion canno be removed, and he e cien long run allocaion canno be achieved. As emphasized by Chugh and Ghironi (2), his sugges an analogy beween opimal axaion in he presen seing and ha in he sandard RBC model. As well know, he laer framework is characerized by a zero long run capial income ax, whereas he labor ax nances governmen spending and ineres paymen on deb. Such a policy delivers e ciency along he invesmen margin, bu disregards he social cos of labor disorion. Similarly, in he curren framework he dividend income ax, which is a form of capial axaion, o -ses he disorion along he ineremporal (enry) margin. Over he business cycle, Chugh and Ghironi (2) show ha opimal ax raes are consan under monopolisic compeiion. This is no he case in oligopolisic marke srucures. Due o he counercyclicaliy of he price markup, he disorions a ecing he economy are ime-varying. Couneracing hese disorion requires, hus, non consan ax raes. Finally, noice ha besides he form of compeiion, also he form of he sunk enry coss maers for opimal axaion purposes. For his reason, ogeher wih alernaive forms of compeiion, wo forms of he enry coss are considered. One feaures a consan enry cos measured in unis of oupu, he oher one feaures enry coss in erms of labor. As a resul he presen framework feaures as special cases wo models in he enry lieraure which also focus on opimal axaion problems: Coo Marinez e al. (27) and Chugh and Ghironi (2). Coo Marinez e al. (27) consider an environmen characerized by monopolisic compeiion under consan sunk enry coss. They nd ha he long run equilibrium is characerized by an ine cienly low number of rms. For his reason i is opimal o subsidize dividend income. Furher hey nd ha ax smoohing is opimal. Chugh and Ghironi (2) consider a framework wih monopolisic compeiion and sunk enry cos in erms of labor. In his case he opimal long run dividend income ax is zero and axes are consan over he business cycle. By neglecing sraegic ineracions and considering he appropriae form of he enry coss he provided framework reduces o eiher one of hese models. For his reason i can be regarded as a general framework where o sudy opimal axaion problems under various form of imperfec compeiion. Anoher paper closely relaed o he presen one is ha by Lewis (2). She sudies opimal scal policy under 2

6 oligopolisic compeiion and endogenous enry, bu in a saic environmen. The reminder of he paper is laid as follows. Secion 2 presens he model; secion 3 de nes he marke equilibrium; secion 4 characerizes he e cien allocaion; secion 5 discusses he disorion associaed o he marke equilibrium; secion 6 provided he Ramsey opimal scal; secion 7 concludes. Technical deails are lef in he Appendix. 2 The Model The economy feaures a coninuum of aomisics secors, or indusries, on he uni inerval. Each secor is characerized by di eren rms producing a good in di eren varieies, using labor as he only inpu. In urn, he secoral goods are imperfec subsiues for each oher and are aggregaed ino a nal good. Households use he nal good for consumpion and invesmen purposes. Oligopolisic compeiion and endogenous rms enry is modeled a he secoral level. A he beginning of each period Nj e new rms ener ino secor j 2 (; ), while a he end of he period a fracion 2 (; ) of marke paricipans exis from he marke for exogenous reasons. 2 As a resul, he number of rms in a secor N j, follows he equaion of moion: N j+ = ( )(N j + N e j) () where Nj e is he number of new enrans in secor j a ime. Following BGM (22) I assume ha new enrans a ime will only sar producing a ime + and ha he probabiliy of exi from he marke,, is independen of he period of enry and idenical across secors. The assumpion of an exogenous consan exi rae in adoped for racabiliy, bu i also has empirical suppor. Using U.S. annual daa on manufacuring, Lee and Mukoyama (27) nd ha, while he enry rae is procyclical, annual exi raes are similar across booms and recessions. Below alernaive forms of compeiion beween he rms wihin each secor are considered. In paricular, he focus is on he radiional monopolisic compeiion seing and he approach based on oligopolisic compeiion developed by Jaimovich and Floeoo (28) and Colciago and Ero (2 a and b). As in Ghironi and Meliz (25) and BGM(22) who gave new life o an ineresing lieraure on he role of enry in macroeconomic models, I inroduce sunk enry coss o endogenize he number of rms in each secor. The naure and he form of he enry cos will be speci ed below, where I will also consider alernaive speci caions. The household side is sandard. They supply labor o rms and choose how much o save in riskless bonds and in he creaion of new rms hrough he sock marke. 2 As discussed in BGM (22), if macroeconomic shocks are small enough N e j; is posiive in every period. New enrans nance enry on he sock marke. 3

7 2. Firms and Technology The nal good is produced aggregaing a coninuum of measure one of secoral goods according o he funcion Z =!!!! j dj where j denoes oupu of secor j and! is he elasiciy of subsiuion beween any wo di eren secoral goods. The nal good producer behave compeiively. In each secor j, here are N j > rms producing di ereniaed goods ha are aggregaed ino a secoral good by a CES aggregaing funcion de ned as 2 3 N X j j = 4 y j (i) " " 5 i= where y j (i) is he producion of good i in secor j, " > is he elasiciy of subsiuion beween secoral goods. As in Colciago and Ero (2 a), I assume a uni elasiciy of subsiuion beween goods belonging o di eren secors. This allows o realisically separae limied subsiuabiliy a he aggregaed level, and high subsiuabiliy a he disaggregaed level. Each rm i in secor j produces a di ereniaed good wih he following producion funcion " " (2) (3) y j (i) = A h c j(i) (4) where A represens echnology which is common across secors and evolves exogenously over ime, while h c j (i) is he labor inpu used by he individual rm for he producion of he nal good. The uni inersecoral elasiciy of subsiuion implies ha nominal expendiure, EXP, is idenical across secors. Thus, he nal producer s demand for each secoral good is P j j = P = EXP : (5) where P j is he price index of secor j and P is he price of he nal good a period. Denoing wih p j (i) he price of good i in secor j, he demand faced by he producer of each varian is where P j is de ned as y j (i) = pj P j 2 3 N X j P j = 4 (p j (i)) " 5 i= " j (6) " (7) 4

8 Using (6) and (5) he individual demand of good i can be wrien as a funcion of aggregae expendiure, y j (i) = p " j P " j EXP (8) As echnology, he enry cos and he exi probabiliy are idenical across secors, in wha follows he index j is disregarded o considered a represenaive secor. 2.2 Households Consider a represenaive agen wih uiliy: ( ) X U = E log C H+=' + =' = ; ' (9) where 2 (; ) is he discoun facor, H are hours worked and C is he consumpion of he nal good. The represenaive agen enjoys labor and dividend income. The household maximizes (9) by choosing hours of work and how much o inves in bonds and risky socks. The iming of invesmen in he sock marke is as in BGM (22) and Chugh and Ghrioni (2). A he beginning of period, he household owns x shares of a muual fund of he N rms ha produce in period, each of which pays a dividend d. Denoing wih V he value of a rm, i follows ha he value of he porfolio held by he household is x V N. During period, he household purchases x + shares in a fund of hese N rms as well as he N e new rms creaed during period, o be carried ino period +. Toal sock marke purchases are hus x + V (N + N e ). A he very end end of period, a fracion of hese rms disappears from he marke. 3 Following producion and sales of he N varieies in he imperfecly compeiive goods markes, rms disribue he dividend d o households. The household s oal dividend income is hus D = x d N, which is axed a he rae d. The variable w is he marke real wage, and h is he ax rae on labor income. The household s holdings of he sae-coningen one-period real governmen bond ha pays o in period are B ; and B j + are end-of-period holdings of governmen bonds ha pay o in sae j in period +, which has purchase price =R j in period. The Flow budge consrain of he household is X j R j B j + +C +x + V (N +N e ) = h w H +B +x V N + d x d N 3 Due o he Poisson naure of exi shocks, he household does no know which rms will disappear from he marke, so i nances coninued operaions of all incumben rms as well as hose of new enrans. 5

9 The FOCs (Firs Order Condiions) for he household problem are represened by a sandard Euler equaion for bonds holdings an asse pricing equaion = R j C C j + V = ( ) C C + E d + + (; N + ) + V + () and he condiion for opimal labor supply 2.3 Endogenous Enry C H ' = h w Upon enry rms faces a sunk cos, de ned as f. In each period enry is deermined endogenously o equae he value of rms o he enry coss. In wha follows I consider wo popular forms of he enry cos f, de ned as Form and Form 2, respecively. Form, adoped, iner alia, by Jaimovich and Floeoo (28) and Coo-Marinez e al. (27), feaures a consan enry cos measured in unis of oupu, f =. Form 2, adoped by Bilbiie, Ghironi and Meliz (27), feaures an enry cos equal o =A unis of labor, wih >. Noice ha, under his speci caion, echnology shocks a ec he produciviy of he workers ha produce goods and also of he workers ha creae new businesses. 2.4 Governmen The Governmen faces an exogenous expendiure sream fg g = in real erms. To nance his sream i issues real sae coningen bonds, B j, where he superscrip j refer o he sae of naure, and collecs axes on labor and dividend income. Also i provides enry-subsidies, s. Is period-by-period budge consrain is given by X B j + + = G + B () j R j where = h w H + d N d are oal ax revenues. The Governmen consumes he same index of goods faced by he household and opimizes he composiion of is expendiure across goods. 4 Public spending evolves exogenously over ime. 4 p (i) " Hence I follows ha g (i) = P G. 6

10 2.5 Sraegic Ineracions In each period, he same expendiure for each secor EXP is allocaed across he available goods according o he sandard direc demand funcion derived from he expendiure minimizaion problem of he household and he Governmen. I follows ha he direc individual demand faced by a rm, y (i), can be wrien as y (i) = p (i) P = p (i) P = p (i) EXP P P i = ; 2; :::; N (2) Invering he direc demand funcions, he sysem of inverse demand funcions can be derived: p (i) = y (i) EXP XN y (i) i= i = ; 2; :::; N (3) which will be useful in he remainder of he analysis. Firms canno credibly commi o a sequence of sraegies, herefore heir behavior is equivalen o maximize curren pro s in each period aking as given he sraegies of he oher rms. Each good is produced a he consan marginal cos common o all rms. A main ineres of his paper is in he evaluaion of he e ciency of equilibria characerized by popular forms of compeiion by rms such as compeiion in prices and quaniies. Firms ake as given heir marginal cos of producion and he aggregae nominal expendiure. 5 Under di eren forms of compeiion we obain equilibrium prices saisfying p (i) = (; N ) W A (4) where W A is he marginal cos and (; N ) > is he markup funcion. In he nex secions he mark up funcion under alernaive forms of marke compeiion is characerized Price Compeiion Consider compeiion in prices. In each period, he gross pro s of rm i can be expressed as: h i W p (i) A p (i) EXP [p (i)] = 2 3 (5) XN 4 p (j) 5 j= 5 Of course, boh of hem are endogenous in general equilibrium, bu i is reasonable o assume ha rms do no perceive marginal cos and aggregae expendiure as a eced by heir choices. 7

11 Firms compee by choosing heir prices. We consider wo alernaive approaches o his problem. The rs one is he radiional monopolisic compeiion approach, which neglecs sraegic ineracions beween rms. The second one is he Berrand approach, where sraegic ineracions are aken ino consideraion. The oucome of pro maximizaion under monopolisic compeiion is well known. Each rm i chooses he price p (i) o maximize pro s aking as given he price of he oher rms, neglecing he e ec of is price choice on he secoral price index. The symmeric equilibrium price is p = MC () W =A, which is ( ) associaed o he consan price markup MC () =. The laer does no depend on he exen of compeiion, bu jus on he elasiciy of subsiuion beween goods. Under Berrand compeiion, each rm i chooses he price p (i) o maximize pro s aking as given he price of he oher rms. The rs order condiion for any rm i is: p (i) p (i) W p (i) = A ( )p (i) p (i) XN i= p (i) W A p (i) Noice ha he erm on he righ hand side is he e ec of he price sraegy of a rm on he price index: higher prices reduce overall demand, herefore rms end o se higher mark ups compared o monopolisic compeiion. The symmeric equilibrium price p mus saisfy where he mark up reads as p = B (; N ) W A B (; N ) = + (N ) ( )(N ) (6) The mark up is decreasing in he degree of subsiuabiliy beween producs and in he number of rms. Imporanly, when N! he markup ends o MC (), he sandard one under monopolisic compeiion Quaniy Compeiion Consider now compeiion in quaniies in he form of Courno compeiion. Using he inverse demand funcion (3), he pro funcion of a rm i can be 6 Since oal expendiure EXP is equalized beween secors, we assume ha i is also perceived as given by he rms. Under he alernaive hypohesis ha he sum beween public and privae consumpion, C + G, is perceived as given, we would obain he higher mark up: ~ B (N ) (; N ) = ( )(N ) which leads o similar qualiaive resuls. This case would correspond o he equilibrium mark up proposed by ang and Heijdra (993). 8

12 expressed as a funcion of is oupu y (i) and he oupu of all he oher rms: W [y (i)] = p (i) y(i) = A = y (i) EXP XN y (j) j= W y (i) A (7) Assume now ha each rm chooses is producion y (i) aking as given he producion of he oher rms. The rs order condiions: y (i) EXP Pi y (i) y (i) 2 EXP h P i 2 = W i y (i) A for all rms i = ; 2; :::; N can be simpli ed imposing symmery of he Courno equilibrium. This generaes he individual oupu: y = ( )(N )A EXP N 2 W (8) Subsiuing ino he inverse price, one obains he equilibrium price p = C (; N ) W A, where C N (; N ) = (9) ( )(N ) is he markup under compeiion in quaniies. For a given number of rms, he mark up under compeiion in quaniies is always larger han he one obained before under compeiion in prices, as well known for models of produc di ereniaion (see for insance Vives, 999). Noice ha he mark up is decreasing in he degree of subsiuabiliy beween producs and in he number of compeiors. In he Courno equilibrium, he markup remains posiive for any degree of subsiuabiliy, since even in he case of homogenous goods, we have lim! Q (; N ) = N =(N ). 7 Finally, only when N! he markup ends o MC (), he markup under monopolisic compeiion. 3 Marke Equilibrium This secion conains he condiions characerizing he marke equilibrium (ME). Merging he household ow budge consrain wih he governmen budge leads o c + N e V = w H + N (2) 7 This allow o consider he e ec of sraegic ineracions in an oherwise sandard seup wih perfec subsiue goods wihin secors. See Colciago and Ero (2 b) 9

13 where c = C + G denoes he sum beween privae and public consumpion of he nal good. Noice ha he sum beween labor income and pro s income equals aggregae GDP. The Euler equaion for rms shares reads as V = ( C+ )E d V + C (2) while he se of Euler equaions for bond holdings provide he de niion of he sochasic discoun facor as E C+. C The real wage can be derived from he equilibrium pricing relaion as w = A p (; N )P = A. The rs order condi- where in he symmeric equilibrium = p P = N ion for labor supply is =( ) C H ' = l A (22) also we mus consider he equaion deermining he dynamics of he number of rms N + = ( ) (N + N e ) (23) I remains o impose he enry condiion and he clearing of he marke. To do so he analysis di ereniaes according o he form of he enry cos. Form. When he laer is measured in, consan, unis of oupu, he labor inpu is enirely employed for he producion of he nal good, hus he clearing of he labor marke requires H = N h c. The demand faced by rm i reads as y = N. In his case rm i s pro s are = y = Aggregaing pro s over rms and summing o labor income delivers GDP as A H. Since he enry condiion is simply V = he resource consrain (24) becomes c + N e = A H (24) Noice ha GDP here coincides wih he producion of he nal good. Euler equaion for rms share ranslaes ino C+ = ( )E C d + + N N + + The (25) Form 2. When he enry cos is measured in unis of labor, he economy amouns o one which feaures wo secors, one where N e A unis of labor are used o produce new rms, he oher one where N h c = H N e A unis of

14 labor are used o produce he nal good. This implies ha he se up of a new rm reduces he labor inpu available for he producion of he nal good. In his seing he individual demand faced by rms i is y = c N and rm level pro s can be wrien as 8 = y = c N Aggregaing pro s over rms and summing o labor income delivers GDP as GDP = c + A H. Also recall ha he enry condiion implies V = f = A w =. In his case, he resource consrain reads as c + N e = A H (26) The Euler equaion (2) for he value of he rm reduces, insead, o = ( )E C+ C d + + c N + + (27) De niion (Marke Equilibrium) Given he exogenous processes fa ; G g = and processes d ; l, he Marke Equilibrium (ME) consiss of an allocaion fc ; H ; N ; N e ; B + g = = which sais es equaions (22), (), (23), he de niion of he price markup and he de niion of he love for variey o which we mus add equaions (24) and (25) in he case of enry coss in Form, or equaions (27) and (26) in he case of enry coss in Form 2. 4 E cien Equilibrium This secion oulines a scenario where a benevolen Social Planner (SP) maximizes households lifeime uiliy by choosing quaniy direcly. In doing his, he SP is subjec o he same echnological consrains described in he previous secions. The SP maximizes (9) wih respec o fc ; N + ; N e ; H g =. The choice is subjec o wo consrains. The rs one is given by he dynamics of he number of rms, equaion (23), he second one is he resource consrain, which is represened by equaion (26) in he case of Form 2 of he enry coss and by (24) when he enry cos is measured in erms of oupu. The social 8 To see his noice ha P (C + G ) = in a symmeric equilibrium i follows ha Z N p i y i di P (C + G ) = N p y hus y = P (C + G ) (C + G) = : p N N

15 planner akes ino accoun he e ec of he number of varieies, N, on he relaive price,, which is a primiive of he problem. The FOC wih respec o H is independen of he form of he enry cos and reads as C H ' = A (28) his condiion simply saes ha he Planner equaes he marginal rae of subsiuion beween hours and consumpion o he marginal produc of labor, which depends on he number of varieies in he economy. The Planner s Euler equaion depends insead on he form of he enry cos. In he case of Form, i reads as = ( ) E C C + " # (N+ ) + N + + (29) while in he case of Form 2 i amouns o = ( ) E C C (N +) + c N + (3) where (N + ) = N;+ N+ is he bene of variey in elasiciy form. Under + CES preferences he laer is a consan equal o ", hence, in he remainder, we simply denoe i wih. These equaions sae ha a he social opimum he marginal rae of subsiuion beween presen and fuure consumpion mus be idenical o he ineremporal marginal rae of ransformaion (IMRT). In wha follows I discuss he naure of he IMRT and is dependence on he form of he enry cos. To gain inuiion, consider a non-sochasic version of he economy. In his case he marginal rae of subsiuion beween presen and fuure consumpion equals C + C. To analyze he IMRT and o provide conceps ha will be used in he nex secions we build on he analysis by Chugh and Ghironi (2) and BGM (27). The naure of he IMRT depends on he speci c form of he enry cos. However, under boh formulaions, he ransformaion of curren resources ino fuure ones can be undersood by considering rms as he capial sock of he economy. As such, he creaion of new rms conribues o omorrows consumpion hrough wo channels, a producion channel (PC) and an invesmen channel (IC). To each of his channel is associaed a parial ineremporal rae of ransformaion hrough which curren consumpion can be ransformed in fuure consumpion. The overall rae of ransformaion is he sum of he parial raes. Consider Form of he enry cos. One uni of consumpion oday can be ransformed ino new rms. A fracion ( ) of hese rms will produce in he nex period. Each of he addiional rms conribue o ime + consumpion hrough he producion process for + N + unis. Hence he parial 2

16 + ) N +, rae of ransformaion associaed o he producion channel reads as ( and we denoe i by P C IMRT SP, where he superscrip SP denoes ha i is he socially e cien rae, while he subscrip means ha i is he rae associaed o enry coss in form. Recall ha new rms represen invesmen and can be ransformed direcly ino ( ) unis of omorrows consumpion ( rms each wih a consumpion value ). Hence he ineremporal rae of ransformaion associaed o he invesmen channel, IC-IMRT SP, is ( ). The e cien ineremporal marginal rae of ransformaion, IMRT SP, is he ( ) + sum beween hese wo componens and reads as IMRT SP N = + + ( ). In he case of Form 2 of he enry cos he creaion of a new rms requires unis of labor. Foregoing one uni of consumpion leads o he creaion of new rms. This implies wo di erences wih respec o he previous case. The rs one is ha he IC IMRT2 SP is no consan over ime (and saes of naure), bu depends on he raio beween he bene of variey ino wo adjacen periods, +. In oher words, he cos of creaing a rm in erms of he nal good is no consan over ime. The reason is ha he love for variey a ecs he marginal produciviy of labor and hus he opporuniy cos of creaing a new rm in erms of he nal good. The second one is ha a ime + he creaion of N+ e new rms requires A N+ e unis of labor, which are divered from he producion of he nal good. As a resul he PC-IMRT SP depends on he sum beween privae and public consumpion and no on GDP. In paricular, he ne gain in erms of consumpion coming from he producion channel is A +(H A N +) e N;+ N + + c ( ) N + = N;+ + + (A +H N e + ) N + = c + N +. Hence, he PC- IMRT SP 2 is. Overall, he e cien ineremporal marginal rae of ransformaion associaed o his form of he enry cos reads as IMRT SP 2 = ( ) c + N Below we used he parial rae of ransformaion jus de ned o analyze he disorions characerizing he marke allocaion. To conclude his secion we provide a de niion of he socially e cien equilibrium. De niion 2 (E cien Equilibrium) The E cien Equilibrium consiss of an allocaion fc ; h ; N ; N e g = saisfying equaions (23) ; (28) ogeher wih equaions (26) and (3) in he case of enry cos in erms of labor, and equaions (24) and (29) in he case of enry coss in erms of oupu, for given N and fa ; G g =. 5 Marke Disorions The marke allocaion feaures wo disorions. To idenify hem i is convenien o se, for he momen being, scal insrumens o zero, d = h =. In he nex secions we reinroduce scal insrumens and design hem in order o 3

17 minimize he welfare losses associaed o he disorions ha we are abou o discuss. The rs disorion is re ered o as o he Labor Disorion. In he compeiive equilibrium labor is supplied up o he poin ha he following condiion is sais ed C H =' = A (3) A comparison beween equaion (28) and equaion (3) reveals ha in he decenralized equilibrium he marginal rae of subsiuion beween hours and consumpion, C h =', is lower han he marginal rae of ransformaion beween hours and oupu, A. As in oher models wih an imperfecly compeiive goods marke, his wedge is due o he presence of a price markup. Oligopolisic compeiion renders his wedge ime varying. Ruling ou he labor disorion requires =. However, since in his case rms would no recover he enry cos, he allocaion would be degenerae. 9 The second disorion involved in he decenralized allocaion is an Enry Disorion. This wedge operaes hrough he ineremporal rms creaion margin and could lead o an ine cien number of rms in equilibrium. To illusrae he enry disorion I will refer o he ineremporal marginal raes of ransformaion associaed o he producion and he invesmen channel inroduced above. By comparing he social raes of ransformaion o he marke counerpars we will be able o idenify he sources of he disorions involved in he ME. Denoe wih PC-IMRT ME he ineremporal rae of ransformaion associaed o he producion channel in he ME. The laer depends on he level of he markup, which provides he privae incenive o creae new rms, while he social rae depends on he love for variey e ec. As emphasized by Coo- Marinez e al. (27), here are wo exernaliies of opposie sign which could lead o a di erence beween PC-IMRT ME and PC-IMRT SP. A higher number of rms increases he welfare ha a household obains by spending a given nominal amoun. Given he individual rm neglecs his e ec, enry is oo low wih respec o he social opimum. The second e ec is a business sealing e ec. A higher number of rms in equilibrium reduces individual pro s. Since individual rms do no ake his e ec ino accoun when deciding abou enry, he resuling number of rms in equilibrium is oo high. Noice ha he business sealing e ec is sronger under oligopolisic compeiion wih respec o he monopolisic compeiion case, since enry erodes he markup. Nex consider he invesmen channel, and de ne IC-IMRT ME he relaive marginal rae of ransformaion. This channel is absen in saic enry model such as ha considered by Coo-Marinez e al. (27), bu i is speci c o he dynamic enry process which characerizes our seing, and oher models adoping he BGM enry framework. As menioned earlier, invesing ino new rms is a mean o save for he fuure, hence he rae a which rms can be 9 In a rs bes environmen, if lump sum axes were available, a labor income subsidy equal o h = would eliminae he Labor Disorion. 4

18 direcly ransformed ino consumpion ino wo adjacen period a ecs invesmen decisions. Suppose ha he unis of consumpion required o creae a new rm are lower in he privae equilibrium wih respec o hose required in he e cien equilibrium. In his case he ME could be characerized by over-enry and hus by a level of consumpion oo low wih respec o he opimal one. To provide he condiions o rule ou he enry disorion we disinguish according o he form of he enry cos. Form. The rae of ransformaion associaed o he producion channel is P C IMRT ME = + + N +. The laer is idenical o P C IMRT SP if = (32) + i.e., if he so called Lerner index equals he bene of variey. Since boh he planner and he marke can ransform rms ino consumpion a he same cos,, in erms of oupu he IC IMRT ME is no disored. As a resul condiion (32) is necessary and su cien o rule ou he enry disorion. Since = he condiion above implies ha he price markup mus be consan over ime, =. As a resul i canno be sais ed under oligopolisic compeiion. Noice also ha, alhough monopolisic compeiion leads o a consan markup =, he condiion for e ciency is no sais ed since <. Form 2. In his case boh he parial ineremporal raes of ransformaion described above could be disored, and hus wo condiions need o be imposed o reinsae e ciency. Recall ha P C while P C IMRT SP 2 = ( ) c + + IMRT ME 2 = N + : The laer are idenical if + + c + + N +, = (33) Also IC IMRT2 ME = ( ) + while IC IMRT SP + 2 = ( ) +. In his case he condiion for e ciency is + = (34) Combining condiions (33) and (34) we recover hose emphasized by BGM The Lerner Index is de ned as follows LI = p (i) MC (i) p (i) = = = p(i) P MC(i) P p(i) P = 5

19 (27), namely = + and + = which implies ha he price mark up should be consan over ime and ha he bene of variey in elasiciy form should equal he marke power as measured by he ne markup. Under oligopolisic compeiion boh condiions fail. Conrary o he previous case, hese condiions are sais ed under monopolisic compeiion, as emphasized by BGM (27) and Chugh and Ghironi (2). Thus, under monopolisic compeiion he enry margin is no disored. The analysis shows ha he condiions for e ciency depend on hree facors: i) preferences, which a ec he naure of he bene of variey; ii) he form of he enry coss, which a ec he marginal rae of ransformaion associaed o he invesmen channel, IC-IMRT and iii) he naure of compeiion, which deermines he form of he price markup and hrough his way disors he marginal rae of ransformaion associaed o he producion channel, PC-IMRT, and he supply of labor. Imporanly, under oligopolisic compeiion, he enry margin is always disored, no maer he form of he enry cos. This is no he case under monopolisic compeiion, which implies an e cien number of rms under enry coss in form 2, as emphasized by BGM (27) and Chugh and Ghironi (2). Noice ha he economic environmens in Chugh and Ghironi (2) and Coo Marinez a al. (27), can be considered as special cases of he one oulined here. In paricular, he presen seup collapses o ha considered by Coo Marinez e al. (27) when enry coss are in form and he sraegic ineracions beween rms are negleced. I coincides, insead, wih ha considered by Chugh and Ghironi (2) under enry coss in form 2 and sraegic ineracion are negleced. The nex oulines he scal policy aimed a minimizing welfare losses due o he marke disorions when he Governmen can raise revenues solely by imposing disorionary axes and issuing sae coningen bonds. 6 Ramsey Opimal Fiscal policy As emphasized by Chugh and Ghironi (2) a primal approach canno be applied o solve for he opimal policy. To see his consider he Euler equaion for rms shares, which in is general form reads as f = ( )E C+ C d f + Expeced fuure dividend income axes canno be removed from his equaion using oher equilibrium condiions. As a resul he se of allocaions ha he Planner can selec canno uniquely be characerized by means of he so called implemenabiliy consrain. Furher, compuing a rs order condiion wih respec o E d + would leave d + indeerminae from he poin of view of ime-. 6

20 To resolve hese indeerminacy issue I adop he soluion proposed by Chugh and Ghironi (2). In paricular, i is assumed ha he planner chooses a sae coningen schedule for he ime + dividend income ax rae d j;+ ;where j indexes he sae of he economy. This schedule is in he ime informaion se. Also we assume ha he Planner commis o he schedule, meaning ha he sae coningen ax rae is implemened wih cerainy a ime +. De niion 3 (Ramsey Equilibrium) Given he governmen expendiure fg g = and he iniial condiions fb ; N g he allocaions associaed o he opimal scal policy h ; d j;+ are derived by solving = ( ) X max E log C h+=' + =' = The choice variables are C ; N, H, N e, and d =+. The allocaion is resriced by four consrains, wo of hem depend on he form of he enry cos. The consrains which are independen of he form of he enry cos are equaion (23), deermining he dynamic of he number of rms, and he implemenabiliy consrain, which reads as X E = h ' = b C + C d d + V N s The allocaion is furher resriced by equaions (24) and (25) in he case of enry cos of form and by equaions (26) and (27) in he case of enry cos in form 2. The rs order condiions for he Ramsey problem are repored in he Appendix, for boh forms of he enry cos. Throughou he analysis, i is assumed ha he policy maker can credibly commi himself, bu he iniial period ( = ) is ignored. In deriving he Ramsey policy, he well known problem ha he policy maker s decision rules will be di eren for he rs period in which he policy is implemened is negleced. This is jusi ed by he fac ha he ineres is ha of making saemens abou he deerminisic seady sae as well as abou business cycle ucuaions around i, while he ransiion pah from he iniial values owards he seady sae is no analyzed. For he analysis, i is furher assumed ha he iniial values for he predeermined variables are equal o heir values in he deerminisic Ramsey seady sae. Since par of he following analysis is numerical, he calibraion of srucural parameers follows. The ime uni is mean o be a quarer. The discoun facor,, is se o he sandard value for quarerly daa.99, while he rae of business desrucion,, equals.25 o mach he U.S. empirical level of per cen business desrucion a year. Seady sae produciviy is equal o A =. The baseline value for he enry cos is se o = =. The baseline value for he inrasecoral elasiciy of subsiuion is = 6, which is in line wih he ypical calibraion for monopolisic compeiion and delivers markups levels belonging 7

21 o he empirically relevan range. In wha follows equilibrium allocaions under Courno, Berrand and Monopolisic Compeiion for each enry cos con guraion are compared. This is done holding parameers xed, in order o undersand he role of he di eren marke srucures. To his end, he following calibraion sraegy for he uiliy parameer is adoped. The value of is such ha seady sae labor supply is equal o one under monopolisic compeiion. In his case he Frish elasiciy of labor supply reduces o ', o which we assign a value of four as in King and Rebelo (2). Nex he values of and ' are held consan under boh Berrand and Courno compeiion. Turning o scal parameers, he raio of Governmen spending over GDP equals.22, as esimaed by Schmi-Grohè and Uribe (25), and he raio of Governmen deb o oupu equals.5 on an annual basis, in line wih he U.S. poswar average. There are wo exogenous processes in he economy, ha for Governmen spending and ha for echnology. They are boh assumed o be AR () processes in log deviaions from he seady sae: log G G = g log G G + "g log A A = a log A A + "a The auoregressive coe cien for he echnology process is a = :979 and he sandard deviaion of he disurbance, a, is :72, as in he RBC model by King and Rebelo (2). The parameerizaion of he Governmen spending process follows Chari and Kehoe (999) in seing g = :97 and g = : Ramsey Seady Sae By using he dividend income ax, he Ramsey Planner removes he ine ciency along he enry margin. This resul has rs been ideni ed by Chugh and Ghironi (2). In wha follows i is shown ha he level of he opimal dividend income ax rae and opimal enry subsidy di er according o he form of he enry cos and he form of compeiion. Proposiion provides he main resul of his secion. Proposiion 4 (Opimal long-run dividend income axes) Opimal longrun dividend income axes depend on he form of he enry coss (i) Form : d = (ii) Form 2: d = Oliveira Marins and Scarpea (999) provide esimaes of price mark ups for US manufacuring indusries over he period In broad erms mos of he secoral markups de ned over value added are in he range 3-6 per cen, while when de ned over gross oupu hey are in he range 5-25 per cen. In he laer case, high mark ups, over 4 per cen, are observed in few secors. 8

22 Proof. In he Appendix. The dividend income ax a ecs uniquely he PC-IMRT. The laer depends on he form of he enry cos and for his reason opimal ax raes di er accordingly. Consider condiion (i). Pro s are axed when he privae incenive o creae a new rm, as measured by he Lerner Index, ha is, is larger han he social incenive o inroduce a new variey,. In his case he number of rms in equilibrium is oo large wih respec o opimal and hus a posiive dividend income ax is required. A pro income subsidy is, insead, opimal in he opposie siuaion. Noice ha he number of rms is no he opimal one even if he ine ciency along he enry margin is removed. The reason is ha he Ramsey Planner canno remove he labor disorion. As a resul seady sae hours will be lower in he Ramsey seady sae wih respec o he hose observed in he long run allocaion reached by he Social Planner. In oher words he Ramsey Planner disregards he social cos of he labor disorion. However, as shown below, he Planner implemens he e cien level of unis of e ecive labor per rm, H N. Noice ha in he sandard neoclassical growh model he Ramsey Planner would arge he e cien raio beween capial and hours. This furher emphasizes he analogy beween he sock of capial in he neoclassical model and he sock of rms in he economy we have oulined. The opimal dividend income ax rae under Berrand compeiion is d Berrand = N while under Courno compeiion we obain d Courno = N ( ) 2 Noice ha he opimal ax rae in he case of monopolisic compeiion is d Monopolisic = ; Under oligopolisic compeiion he opimal dividend income ax could ake he form of a subsidy, depending on he parameerizaion of he model. On he conrary, removing he enry disorion under monopolisic compeiion requires a subsidy no maer he parameerizaion of he model. This means ha monopolisic compeiion always leads o a subopimal seady sae number of rms. Figure 3 displays he Ramsey seady sae number of rms, hours, oupu per rm and opimal ax raes as a funcion of he enry cos under Berrand, Courno and monopolisic compeiion. Allocaions are compared o he ef- cien ones. Firs, for any given value of he enry cos, he Ramsey Planner implemens he same allocaion for hours and he number of rms across marke srucures. These di er from he e cien ones, since here are less rms and hours worked are lower a he seady sae. As menioned above he Ramsey Planner 9

23 4 Number of firms 2 Hours Unis of effecive labor per firm.2.5 Tax rae on dividend income Tax rae on labor income Courno Berrand Social Planner Monopolisic.5.5 Figure : Ramsey Seady Sae under Enry Coss in Form. Enry cos ( ) on he horizonal axis. arges he level of unis of e ecive labor per rm, which are a heir e cien level. This, however, requires a di eren combinaion of opimal axes across marke srucures. The marke equilibrium under Courno compeiion would be characerized by excessive enry. Hence, resoring e ciency along he enry margin requires a posiive dividend income ax, no maer he level of he enry cos. As in Coo-Marinez e al. (27), he number of rms under monopolisic compeiion is oo low and dividend income should be subsidized o promoe enry. Berrand compeiion falls beween hese wo case, since we observe a dividend income subsidy in he case of a low enry cos and a ax in he case of a high enry cos. As saed by Vives (984), Berrand compeiion can be regarded as a more compeiive marke srucure wih respec o Courno and for his reason i is judged as more e cien. 2 This analysis suggess ha, for given enry coss, he dividend income ax should be lower in markes characerized by more compeiive marke srucures. Condiion (ii) is isomorphic o ha obained by BGM (27) and Chugh and Ghironi (2) under he case of monopolisic compeiion. In his case pro s should be axed whenever he ne markup exceeds he bene of variey in elasiciy form. However, under monopolisic compeiion condiion (ii) is auomaically sais ed and he marke equilibrium displays e ciency along he enry margin. This is no he case under oligopolisic compeiion. Under 2 Vives (984) provide he following inuiive explanaion o suppor his view. In Courno compeiion each rm expecs he ohers o cu prices in response o price cus, while in Berrand compeiion he rm expecs he ohers o mainain heir prices; herfore Courno penalizes price cuing more. One should expec Courno prices o be higher han Berrand prices. 2

24 Berrand compeiion he opimal ax rae is while under Courno compeiion is d Berrand = N d Courno = N + Hence enry coss in form 2 always imply a posiive dividend income ax rae under oligopolisic compeiion. Figure 4 displays he Ramsey Seady Sae number of rms and hours ogeher wih opimal ax rae as a funcion of he enry cos under enry cos in form 2. In his case allocaions slighly di er across marke srucures. Hours and he number of rms are lower wih respec o heir e cien counerpar, bu oupu per rm is equal o is e cien level. I is ineresing o noe ha under Courno compeiion he ax rae on labor income is always lower han ha on dividend income, hus in he long run he Governmen should rely more heavily on dividend income axaion, wih respec o labor income axaion, o nance public spending Ramsey Dynamics and Opimal Tax Volailiy This secion shows he business cycle implicaions of he Ramsey opimal policy. The dynamics under opimal policy are obained by solving a rs order approximaion o he Ramsey equilibrium condiions. As shown in various conribuions by Schmi-Grohè and Uribe, in models characerized by more fricions han he presen one, a rs order approximaion o he equilibrium condiions delivers dynamics ha are very close o he exac ones. Figures 3 depics percenage deviaions from he seady sae of key variables in response o a one sandard deviaion echnology shock under enry cos in form. For ax raes we repor deviaions from he seady sae level in percenage poins. Time on he horizonal axis is in quarers. Solid lines refer o he E cien (Social Planner) allocaion, dashed and dash-doed lines refer o Ramsey dynamics under Berrand and he Courno compeiion respecively, and doed lines o he case of Ramsey dynamics under monopolisic compeiion. The echnology shock creaes expecaions of fuure pro s which lead o he enry of new rms in he marke. This is so under boh he Ramsey and he 3 As shown by Chugh and Ghironi (2) he Ramsey Planner can also remove he enry disorion by using an enry subsidy insead of he dividend income ax. De ne s he enry subsidy such ha he ne enry cos is ( s ) f. I can be shown ha opimal subsidies depend on he form of he enry cos as follows (i) Form : s = (ii) Form 2: s = The Ramsey Planner will resor o an enry ax in he case of excessive enry or o a subsidy in he case of ine cienly low enry. 2

25 5 Number of firms.5 Hours Oupu per firm.2.5 Tax rae on dividend income Tax rae on labor income.5.5 Courno Berrand Social Planner Monopolisic Figure 2: Ramsey Seady Sae under Enry Coss in Form 2. Enry cos () on he horizonal axis..5 GDP.5 Hours Consumpion New enrans Aggregae profis Number of firms Labor income ax. 2 4 Markup Dividend income ax Efficien Dynamics Berrand Courno Monopolisic Compeiion Figure 3: Enry Coss in form. Response of he main macroeconomic variables o a one sandard deviaion shock o echnology. Solid lines refer o he social planner allocaion, dashed and dash-doed lines refer o Ramsey dynamics under Berrand and he Courno respecively and doed lines o he case of Ramsey dynamics under monopolisic compeiion. 22

26 .5 GDP.5 Hours Consumpion New enrans 2 4 Aggregae profis Number of firms Labor income ax. 2 4 Markup Dividend income ax Efficien Dynamics Berrand Courno Monopolisic Compeiion Figure 4: Enry Coss in form 2. Response of he main macroeconomic variables o a one sandard deviaion shock o echnology. Solid lines refer o he social planner allocaion, dashed and dash-doed lines refer o Ramsey dynamics under Berrand and he Courno respecively and doed lines o he case of Ramsey dynamics under monopolisic compeiion. e cien equilibrium. Under all marke srucures he dynamics of oupu, consumpion, hours and he number of rms are very close o he e cien ones. In he non sochasic economy we showed ha a necessary condiion for e ciency is he cosancy of he price markup. The following argumens suggess ha a similar resul applied o he sochasic case. Under monopolisic compeiion he markup is consan along he business cycle. As can be seen from Figure 3, his resuls in consan ax raes. This is no he case under Oligopolisic compeiion. Under Berrand and Courno compeiion he enry of new rms leads o higher compeiion which, in urn, leads o a counercyclical price markup. The price markup variabiliy enails a deviaion from opimal dynamics which is o se by he Ramsey Planner adjusing he ax raes. In paricular, we observe an increase in he labor income ax coupled wih a decrease in he dividend income ax. Changes in he ax raes are mild, bu sronger under Courno compeiion where he markup is characerize by a higher elasiciy o he number of rms wih respec o Berrand. The Ramsey policy under enry coss in Form is, hus, characerized by a counercyclical labor income ax rae and by a procyclical dividend income ax. Figure 4 displays he percenage changes in response o a echnology shock under enry coss in form 2. Lines have he same meaning as in he Figure 3. Previous consideraions exend o his case. A relevan di erence is he impac increase in he dividend income ax, which is revered afer few periods. Noice ha similar dynamics, alhough quaniaively less sizeable, can be observed in 23

27 C H N N e h d Monopolisic compeiion x (x) Cor( ; x ) Berrand Compeiion x (x) Cor( ; x ) Courno Compeiion x (x) Cor( ; x ) Table : Mean, Sandard deviaions and correlaions wih oupu of main-macro variables under alernaive marke srucures. Enry Coss in Form he case of a governmen spending shock. 4 Nex he variabiliy of he main macroeconomic variables in response o he echnology and governmen spending shocks is analyzed. Table displays he mean, he coe cien of variaion and he correlaion wih oupu of a number of variables of ineres under he Ramsey dividend and labor income axaion policy, under enry coss in form. For ax raes i is repored he sandard deviaion in percenage poins. Table 2 repors he same saisics for he case of enry coss in form 2. Under enry cos in form he variabiliy of he main macroeconomic variables under he opimal policy is idenical across marke srucures. However, his is reached by means of a di eren scal policy. As expeced from Figure 3, while under monopolisic compeiion axes are consan his is no he case under oligopolisic compeiion. Tax raes are more volaile under Courno compeiion wih respec o Berrand, wih he dividend income ax more variable han he labor income ax rae. Recall ha he elasiciy of he price markup o he number of rms is higher under Courno. As a resul minimizing he welfare cos of he disorions over he business cycle requires more variable axes when rms compee in quaniies. Under enry coss in form 2 allocaions and volailiies are no longer idenical across marke srucures. Ineresingly, while he overall variabiliy characerizing he economy in response o shocks, as measured by he sandard deviaion of oupu, is higher under enry cos in form, he variabiliy of ax raes is higher under enry coss in form 2. In paricular he sandard deviaion of he dividend income ax under Courno compeiion is sizeable. 4 Noice, however, ha aggregae consumpion drops in response o a Governem spending shock under all he marke srucures considered. 24

28 C H N N e h d Monopolisic compeiion x (x) =x Cor( ; x ) Berrand Compeiion x (x) =x Cor( ; x ) Courno Compeiion x (x) Cor( ; x ) Table 2: Mean, Sandard deviaions and correlaions wih oupu of main-macro variables under alernaive marke srucures. Enry coss in form 2 7 Conclusions This paper proposes an economy where he degree of marke power, as measured by he price markup, depends endogenously on he form of compeiion, on he degree of subsiuabiliy beween goods and on he number of rms. Imperfec compeiion leads o disorions in boh he goods and he labor marke and in boh he shor and he long run. The opimal long run dividend income correcs for ine cien enry, and i is higher in marke srucures characerized by lower compeiion. In paricular i is higher under Courno Compeiion wih respec o Berrand or monopolisic compeiion. Whereas opimal axes over he business cycle are consan under monopolisic compeiion, his is no he case in an oligopolisic marke srucure. Also he e ec of alernaive forms of sunk enry coss for he design of opimal axaion has been considered. The resuling framework feaures as special cases wo models in he enry lieraure which also focus on opimal axaion problems in he case of an endogenous dynamics of he number of rms. Coo Marinez e al (27) consider an environmen characerized by monopolisic compeiion under consan sunk enry coss. Chugh and Ghironi (2) consider a framework wih monopolisic compeiion and sunk enry cos in erms of labor. By neglecing sraegic ineracions and considering he appropriae form of he enry coss our model reduces o eiher one of hese models. For his reason i can be regarded as a general framework where o sudy opimal axaion problems under various form of imperfec compeiion. 25

29 References Bergin, Paul and Giancarlo Corsei, 28, The Exensive Margin and Moneary Policy, Journal of Moneary Economics, 55, Bilbiee, F., Ghironi, F. and Meliz, M. (27). Monopoly power and Endogenous Produc Variey: Disorions and Remedies. NBER WP Bilbiie, F., F. Ghironi and M. Meliz, (22). Endogenous Enry, Produc Variey, and Business Cycles, Journal of Poliical Economy, Vol. 2, Chari V. V., and Parick J. Kehoe, (999). Opimal Fiscal and Moneary Policy. In Handbook of Macroeconomics, edied by John B. Taylor and Michael Woodford, Vol. C, Elsevier. Colciago, A. and Ero, F. (2 a). Endogenous Marke srucures and he Business Cycle. Economic Journal, Vol. 2, Colciago, A. and Ero, F. (2 b). Real Business Cycle wih Courno Compeiion and Endogenous Enry. Journal of Macroeconomics, Vol. 32, -7. Chaerjee, S. and R. Cooper (993). Enry and Exi, Produc Variey and he Business Cycle, NBER WP Coo-Marinez, J., Garriga, C. and Sanchez-Losada, F. (27). Opimal axaion wih Imperfec Compeiion and Aggregae Reurns o Specializaion, Journal of he European Economic Associaion, 5(6), Chugh, S. and Ghironi, F. (2), Opimal Fiscal Policy wih Endogenous Produc Variey. Unpublished Manuscrip. Devereux, Michael, Allen Head and Beverly Lapham, 996, Aggregae Flucuaions wih Increasing Reurns o Specializaion and Scale, Journal of Economic Dynamics and Conrol, 2, Devereux, Michael and Khang Min Lee, 2, Dynamic Gains from Inernaional Trade wih Imperfec Compeiion and Marke Power, Review of Developmen Economics, 5, 2, Faia, E., 22, Oligopolisic Compeiion and Opimal Moneary Policy, Journal of Economic Dynamics and Conrol, 36 (), Ghironi, F. and M. Meliz (25). Inernaional Trade and Macroeconomic Dynamics wih Heerogenous Firms, Quarerly Journal of Economics, pp Jaimovich, N. and M. Floeoo (28). Firm Dynamics, mark up Variaions, and he Business Cycle. Journal of Moneary Economics, 55, 7, King, R. and S. Rebelo (2). Resusciaing Real Business Cycles, Ch. 4 in Handbook of Macroeconomics, J. B. Taylor & M. Woodford Ed., Elsevier, Vol., pp Lee., and Mukoyama, T., (27). Enry, Exi and Plan-Level Dynamics over he Business Cycle, Cleveland Fed Working Paper 7-8. Lewis, V. (29). Business Cycle Evidence on Firm Enry. Macroeconomic Dynamics 3(5), Lewis, V. (2). Produc Diversiy, Sraegic Ineracions and Opimal Taxaion. Unpublished manuscrip. 26

30 Lewis V., and Poilly, C., (22). Firm Enry and he Moneary Transmission Mechanism. Journal of Moneary Economics 59(7), Oliveira Marins, Joaquim and Sefano Scarpea, 999, The level and Cyclical Behavior of Mark-ups Across Counries and Marke Srucures, OECD Working Paper No. 23, OECD Publishing. Schmi-Grohè, S. and Uribe, M. (25). Opimal Fiscal and Moneary Policy in a Medium Scale Macroecomic Model. NBER Macroeconomics Annual. Vives, X. (999). Oligopoly Pricing. Old Ideas and New Tools, The MIT Press, Cambridge. ang, Xiaokai and Ben J. Heijdra, 993, Monopolisic Compeiion and Opimum Produc Diversiy: Commen, The American Economic Review, 83,,

31 Appendix Appendix A. Seady Sae of he Marke Equilibrium Enry Coss in Form The seady sae number of enrans is N e = The Euler equaion for shares implies Seady sae pro s are given by 5 = y wh = ( ) N ( ) d V = [ ( )] y = hence he share of pro s over oupu reads as N = and he value of rms over oupu is d NV ( ) = [ ( )] Invesmen over oupu is V N e = V ( ) N = V N The share of consumpion over oupu is C = N = d ( ) = [ ( )] Finally o ge he raio of labor income over oupu recall ha g y N e wh = N In order o x we assume ha H=. 6 = h w C In his case N ; 5 Noice ha his is he main di erence wr o cos 2 since in ha case pro s depend on c. 6 As menioned in he secion on calibraion I x H= under monopolisic compeiion and obain he corresponding value of. Under oligopolisic compeiion I consider he value of so obained and compue he corresponding value of H. 28

32 where boh raios are known. To compue he number of rms noice ha ( ) d V = [ ( )] Imposing he enry condiion and subsiuing for individual pro s ( ) d N = ( s ) [ ( )] AH he soluion o his equaion delivers he number of rms a he seady sae. This allows o compue all he oher variables. For a given H he number of rms a he seady sae is larger he higher he markup, hence is larger under oligopolisic compeiion. Enry Coss in Form 2 In his case he seady sae level of individual pro s is w (C + G) = y wh = y = y = = A N As a resul N c = To obain he share of invesmen over consumpion oupu noice ha ( ) d c V = [ ( )] N and V N e c = d ( ) [ ( )] = d ( ) (r + ) N e N = d [ ( )] To compue shares over aggregae oupu recall ha = c + N e V " c c = c + d [ ( )] hus c = + d [ ( )] # which implies ha he share of privae consumpion over oupu is C = c 29 g y c N

33 Since W H as + Given H, he laer leads o = we can compue he raio beween labor income and GDP W H = N c c = h wh CH + ' Labor marke equilibrium requires = h wh C H+ ' H = H C + H E = Nh + A N e = N y A + A N e = c A + A N e hus and or Nex consider N e = AH N = ( ) N + AH N = ( ) AH ( ) d V = [ ( )] c c ( ) c c N subsiuing for he enry condiion delivers or or ( s ) ( ) d w = A [ ( )] ( s ) c = ( s ) = ( ) d [ ( )] [ ( )] ( ) ( d ) c N c N Subsiuing he laer ino he equaion of moion for he number of rms delivers an equaion ha can be solved for N N N = ( ) AH ( s ) [ ( )] ( d ) ( ) N 3

34 or N = + ( s ) ( ) AH [ ( )] ( d )( ) Aa above a higher markups leads o a higher number of rms in equilibrium for any given H. Appendix B. E cien Equilibrium Enry Coss in Form The social Planner problem reads as and s.. max E fc ;N +;N e;hg = X = ( log C C + G + N e = A H N + = ( ) (N + N e ) ) H+=' + =' We aach he Lagrange Muliplier o he rs consrain and he muliplier o he second one. Firs order condiions are as follows C : C = N + : = E + N;+ A H + E + ( ) N e : = ( H : H =' ) = A Combining he rs and he hird condiion delivers C ( ) = Subsiuing he laer ino he hird condiion we obain = ( ) E C C + N;+ A H + which can be wrien as C = ( ) E + C + N nally he FOC wih respec o hours can be wrien as H =' C = A 3

35 To obain he seady sae we have he following equaions C + G + N e = AH Consider he resource consrain or C = From he hird equaion Combining we obain N = ( ) (N + N e ) = ( ) N + g y H =' C = A N e = g y C = g N y N = ( ( ) ) ( ) N C = Nex consider equaion g y ( ) ( ( ) ) hence we have H as H +=' = C AH = C H = Nex we wan o compue N. Noice ha given = N i follows +=' C N = ( ) ( ( ) ) AH or N 2 = ( ) ( ( ) ) AH N = ( ) ( ( ) ) AH 2 32

36 Enry Coss in form 2 The Social Planner problem can be wrien as follows ( ) X max E log C H+=' fc ;N +;N e + =' ;Hg = and s.. = C + G + N e = A H N + = ( ) (N + N e ) We aach he Lagrange Muliplier o he rs consrain and he muliplier o he second one. Firs order condiions are as follows C : C = N + : = E + N;+ (A H N e ) + E + ( ) N e : = ( H : H =' ) = A Subsiuing he rs condiion ino he hird delivers ( ) C = Subsiuing he laer and he de niion of ino he oher equaions we are lef wih and Since A H or = ( ) E C C + N;+ (A H H =' C = A N e ) + + N e = C+G, equaion (35) can be rewrien as = ( ) E C C + C + G N;+ + + C = ( ) E C + + G + + C + N + + To nd he seady sae we can consider he following equaions = ( ) c N + (35) 33

37 From he rs one H =' C = A c + N e = AH N = ( ) (N + N e ) c ( ( ) ) = N ( ) The aggregae resource consrain implies c = AH N Combining N = AH ( ( )) ( ) + we ge N as a funcion of H. Noice ha we have repeaedly used he seady sae version of he equaion of moion for he number of rms. Consider again he aggregae resource consrain or hen C AH = C + G + N e = AH g y N e AH = g y N AH H =' C AH = A AH delivers H implicily as a funcion of N H =' g y N = AH H The laer is equivalen o H +=' g y N = AH Nex subsiue for N as a funcion of H in he round bracke and ( " ( ( ) ) H = g y + #) ( ) ' (+') 34

38 Appendix C. The Implemenabiliy Consrain This Appendix follows closely Arsenau and Chugh (22) and Chugh and Ghironi (22). Consider he household ow budge consrain (in he symmeric equilibrium) X j R j B j + +V (N +N e )x + +C = l w H +B + d d + V N x Muliply boh sides by u c (c ) X u c (c ) R j B j + + u c (c ) V (N + N e )x + + u c (c ) C j = u c (c ) l w H + u c (c ) b + u c (c ) d d + V N x and sum over daes saring from =, where all erm are undersood as in expecaion as of ime = X X u c (c ) j R j B j + + X X u c (c ) V (N + N e )x + + u c (c ) C = = X u c (c ) h X X w H + u c (c ) b + u c (c ) d d + V N x = = = The euler equaion for bonds implies u c (C ) = R j u c C j +, using his in he rs erm on he LHS = X X + u c C j + B j + + X X u c (C ) V (N + N e )x + + u c (C ) C = = j = X u c (c ) h X X w H + u c (C ) b + u c (C ) d d + V N x = C j + = = = B j + can be undersood as he payo of Noice ha he erm P j u c a risk free bond. As such we can cancel ou he rs summaion on he LHS wih he respecive erms in he second summaion in he RHS, leaving jus ime erms = X X u c (c ) V (N + N e ) + u c (c ) C = = = X u c (c ) h X w H + u c (c ) b + u c (c ) d d + V N Noice ha he clearing of he asse marke implies x = a all. Considering = 35

39 ha u h(h ) u c(c ) = h w leads o X X X u c (c ) V (N + N e ) + u c (c ) C + u h (h ) H = = u c (c ) b + = X u c (c ) d d + V N = Nex consider u c (C ) V = E ( ) u c (C + ) d + d+ + V + and plug i ino he rs summaion in he LHS X + ( ) u c (C + ) d + d+ + V + (N + N e )x + + = X X + u c (c ) C + u h (h ) h = = u c (c ) b + considering ha i follows = X u c (c ) d d + V N x = (N + N e ) = N + = X + u c (C + ) d + d+ + V + N+ + = X X + u c (c ) C + u h (h ) h = = u c (c ) b + = X u c (c ) d d + V N = Simplifying he rs summaion on he LHS wih he second in he RHS delivers he implemenabiliy consrain X E = [u c (C ) C + u h (H ) H ] = u c (C ) B + u c (C ) d d + V N where we reinroduced he expecaion operaor. 36

40 Appendix D. The Ramsey Problem Enry Coss in form. Includes proof of resul (i) in Proposiion. The Ramsey problem reads as max E X = u (C ; H ) subjec o N + = ( ) (N + N e ) : C + G + N e = A H : 2 u c = ( )E u c+ += d C+ + G + + N + X E = E [u c C + u h H ] = u c B + u c : 3 d d + V N : Where i de ne he Lagrange mulipliers respecively aached o each consrain and is he (consan) lagrange muliplier aached o he implemenabiliy consrain. The choice variables are C ; N, H, N e, and eiher d =+ or s. Following Ljungqvis and Sargen (24) I de ne V (C ; H ; ) = u (C ; H ) + (u c C + u h H ) and = u c B + u c d d + V N As a resul he Lagrangian funcion can be wrien as 8 X >< V (C ; H ; ) + [( ) (N + N e ) N + ] L = E h + 2 ( A H C G N e ) + = >: 3 u c ( )E u c+ d + A += +H + + N + + i 9 >= >; The rs order condiions for periods are C : V c (C ; H ; ) u cc 3 ( )u cc d = A H N + = N + : + ( ) 3 E u c+ d += A + H + N + = ( ) E + + E 2+ N+ A + H N+ N + + N + + N

41 d += : ( )+ 3 E u c+ N e : ( + ) = 2 + A + H + N + = H : V h (C ; H ; ) + 2 A 3 ( )u c d = Since i follows and V (C ; H ; ) = u (C ; H ) + (u c C + u h H ) A N = V c (C ; H ; ) = u c (C; H) + u cc C + u c = C C 2 C + C = C V h (C ; H ; ) = H =' + ' ' + Consider now he seady sae. The FOC wih respec o d += reads as d += : ( )+ 3 C + G N u c = The laer implies ha a he seady sae 3 is equal o zero. In he Ramsey seady sae he rms enry condiion does no resric he allocaion. As a resul we can wrie he seady sae version of he FOCs as C : 2 = u c H : V h (C; H; ) + u c A = N e : = ( ) u c N + : N AH = [ ( )] ( ) The FOC wih respec o N can be wrien as = ( ) [ N AH + ] Since i follows N = N = N = ( ) AH N + The euler equaion for asse implies ha = ( ) d AH N + 38

42 For he laer wo equaions o be consisen wih each oher i has o be he case ha = d or d = which proves poin (i) in proposiion. Imporanly, he dividend income ax di ers from zero under monopolisic compeiion. Subsiuing he opimal dividend income ax ino he Euler equaion for shares we ge = ( ) N + Which implies N [ ( )] = ( ) Noice ha C = g N y Nex consider he implemenabiliy consrain, which can be wrien as B C + + V N A where B is exogenously given and C has been compued above. The Euler equaion wih respec o asses implies V N = ( ) [ ( )] and also we know = = B C + + V N ' ( ) +' H = Finally given H and recalling ha i follows N = [ ( )] ( ) ( ) ( ) N = [ ( )] AH 2 39

43 Enry Coss in Form 2. Includes proof of resul (ii) in Proposiion 2. In his case he Lagrangian is L = E X = 8 >< >: 3 " V (C ; H ; ) + [( ) (N + N e ) N + ] + 2 ( A H C G N e ) + u c + ( )u c+ d += C++G + + N # 9 >= >; As above, he choice variables are C ; N, H, N e, and eiher d =+ or s. The rs order condiions are C : V c (C ; H ; ) + 3 u cc ( )u cc d = +( )u c d = = 2 C +G N + + N 3 5 N + : + 2 ( ) 3 E u c+ 4+ d += + N ; N + N+ +( ) 3 E u + N+ + c+ 2 + = ( ) E + + E 2+ N+ A + H + N+ e + N+ +E 3+ u + N+ + c+ 2 + H : V h (C ; H ; ) + 2 A = N e : ( ) = 2 d += : ( C+ + G + )+ E u c+ 3 = N + Noice ha C = N = N + N ( )(N ) c + N + C N = ( )(N ) ( )N ( ) 2 (N ) 2 = (N ) N ( )(N ) 2 = ( )(N )

44 B = + (N ) ( )(N ) B N = ( )(N ) ( ) [ + (N )] ( ) 2 (N ) 2 = ( ) (N ) 2 Given 3 = we can wrie he seady sae sysem as above Since = ( ) V c (C; H; ) and V c and V h we ge C : V c (C; H; ) = 2 N + : = [( ) + 2 N (AH N e )] H : V h (C; H; ) + 2 A = N e : ( = ( ) V h (C;H;) V c(c;h;) ) = 2 + = A and given he de niions of c N Evaluaing he Euler equaion for asses a he seady sae implies! = ( ) d ( ) c N + For he wo o be consisen is has o be he case ha d ( ) = which proves poin (ii) in Proposiion. Noice ha di erenly from he monopolisic compeiion case his does no imply d =. The Euler equaion for asses evaluaed a he seady sae reads as hen = ( ) c + c N = ( ) ( ) N The aggregae resource consrain implies c = AH N e using he equaion for he dynamics of he number of rms c = AH N 4

45 Combing he laer wo equaions N = AH ( ( )) ( ) + = + ( ) AH ( ( )) we ge N as a funcion of H. This also implies ha we can compue he markup, under boh Courno and Berrand, as a funcion of H. Recall ha i has o be he case ha = wh + since i follows wh = AH; = = AH + c c using he aggregae resource consrain c = AH N e we obain = AH + (AH N e ) = AH A A N e Also noice To compue c and C which implies his allows o compue c = c + AH consider he euler equaion for asses V = ( ) c ( ) N V N e c = ( ) N e ( ) N = ( ) as follows c + N e V = hen and nally c = N e V c c = + V N e c c 42

46 From he laer we ge C as Knowing c AH we can deermine The FOC for hours AH H =' C = c = subsiuing he de niions of variables h or G c V h (C; H; ) V c (C; H; ) = A H =' = +' ' C h i + A C +' ' + = A Muliplying boh sides by H, he laer is equivalen o 2 H = 4 h AH C +' ' + Hence H is boh a funcion of H and. Nex consider he implemenabiliy consrain = B C + d N C + V N C As a resul = C B + 3 i i5 ' +' d + V N = B C + ( ) + V N where B is given and C is a funcion of H. Also from V = ( ) c ( ) N we ge and V N = ( ) c ( ) = c 43

47 Hence we can compue as a funcion of H. Nex using he seady sae version of he implemenabiliy consrain we ge or which implies h H +='i = H +=' = ( ) ( ) H = ' +' which is a funcion solely of H and can be solved numerically. Given he value H we can deermine he lagrange muliplier " # = ' AH C + ' Recall ha N can be compued as N = + H +' ' A ( ( )) which allows o compue he price markup a he ramsey seady sae. Also i implies a a value for. Since we ge In paricular noice ha H A = ( ) c ( ) N c = AN d = ( ) ( ) ( ) and h = CH ' w 44

48 Previous DNB Working Papers in 23 No. 367 No. 368 No. 369 No. 37 No. 37 No. 372 No. 373 No. 374 No. 375 No. 376 No. 377 No. 378 No. 379 No. 38 No. 38 No. 382 Andrea Colciago and Lorenza Rossi, Firm Enry, Endogenous Markups and he Dynamics of he Labor Share of Income Dirk Broeders, Paul Hilbers and David Rijsbergen, Wha drives pension indexaion in urbulen imes? An empirical examinaion of Duch pension funds Luca Arciero, Ronald Heijmans, Richard Heuver, Marco Massareni, Crisina Picillo and Francesco Vacirca, How o measure he unsecured money marke? The Eurosysem s implemenaion and validaion using TARGET2 daa Sijn Claessens and Neelje van Horen, Impac of Foreign Banks Gabriele Galai, Federica Teppa and Rob Alessie, Heerogeneiy in house price dynamics Jan Willem van den End, A macroprudenial approach o address liquidiy risk wih he Loan-o-Deposi raio Jon Fros and Ayako Saiki, Early warning for currency crises: wha is he role of financial openness? Ronald Heijmans, Lola Hernández and Richard Heuver, Deerminans of he rae of he Duch unsecured overnigh money marke Anneke Kosse and Rober Vermeulen, Migrans' Choice of Remiance Channel: Do General Paymen Habis Play a Role? Jacob Bikker, Is here an opimal pension fund size? A scale-economy analysis of adminisraive and invesmen coss Beaa Bieru, Global liquidiy as an early warning indicaor of asse price booms: G5 versus broader measures Xiao Qin and Chen Zhou, Sysemic Risk Allocaion for Sysems wih A Small Number of Banks Niels Gilber, Jeroen Hessel and Silvie Verkaar, Towards a Sable Moneary Union: Wha Role for Eurobonds? Mauro Masrogiacomo, Reform of he morgage ineres ax relief sysem, policy uncerainy and precauionary savings in he Neherlands Chrisiaan Paipeilohy, Jan Willem van den End, Mosafa Tabbae, Jon Fros and Jakob de Haan, Unconvenional moneary policy of he ECB during he financial crisis: An assessmen and new evidence Alexander Popov and Neelje Van Horen, The impac of sovereign deb exposure on bank lending: Evidence from he European deb crisis

49