Pigouvian Taxes as a Long-run Remedy for Externalities

Transcription

1 Pigouvian Taxe a a Long-run Remedy for Externalitie Henrik Vetter Abtract: It ha been uggeted that price taking firm can not be regulated efficiently uing Pigouvian taxe when damage are enitive to cale effect. Thi concluion follow becaue the Pigouvian tax cannot imultaneouly get right the number of firm and the output per firm. In thi paper, we argue that a imple nonlinear unit tax i efficient under tandard aumption about demand and damage. Jel No.: D61, D62. Keyword: Externalitie, Taxe. 1

2 1 Introduction A way to deal with externalitie i to impoe a per-unit tax on the externality producing action. On thi iue Carlton and Loury (1980, 1986) argue that a Pigouvian tax alone i inefficient in the long run when there are cale effect in the damage function. Thi follow becaue each firm, under a tax, produce until all economie of cale i exhauted, and thi production volume i only coincidently optimal when cale matter. Of coure, when the Pigouvian tax i combined with a lump-um tax or a lump-um ubidy, it i poible to realize an efficient outcome. But lump-um intrument are not practical becaue they are baed on the preumption that the regulator i eentially not contrained by informational limitation. Even if thee concern were omehow eaily dimied, Baumol and Bradford (1970), and later Baumol (1979) alone, argue that lump-um tranfer can be ruled out by the economy-wide budget. We rule out lump-um tranfer a a general rule and ak if it i the uphot of the reult reported by Carlton and Loury (1980, 1986) that one hould dimi unit taxe on ground of inefficiency. More preciely, a mentioned by Carlton and Loury (1980), a unit tax with ome exempting level can to a certain degree replace a unit tax without exempting and a ubidy. We ak in thi paper if uch a nonlinear unit tax the nonlinearity being the exemption level will correct pollution problem in common circumtance, i.e., when demand i negatively loped and when marginal pollution i poitive and ecalate at an increaing rate. In perfectly competitive market, entry and exit of firm regulate production per firm to the point where each firm fully utilize all economie of cale. Thi implie that average and marginal cot are eual and, in turn, that profit i zilch when price eual marginal cot. It i traightforward that lump-um tranfer are needed when concern about damage make it optimal to have price taking firm produce at a point of either increaing or decreaing cale. If firm ideally hould operate at a point of decreaing return a unit ubidy 2

3 and a lump um tax i called for. Oppoitely, if firm hould operate at a point of increaing return a unit tax and a lump um ubidy i called for. What we argue in thi paper i that the mot widepread aumption about damage function imply that firm hould ideally operate at a point of increaing cale. In turn, thi mean that firm mut be given a ubidy. We how that thi ubidy, a noted by Carlton and Loury (1980), for practical purpoe can take the form of a tax exemption threhold. That i, the proper Pigouvian tax will mot often provide the correct incentive in the hort run a well a in the long run. 1 It mut be treed that our reult can be found in paing in Carlton and Loury (1980), but the practical implication that a Pigouvian tax i a proper intrument in mot circumtance are not noticed. 2 Reult We conider a competitive indutry where the number of firm, called n, i regulated by the entry and exit of firm until the profit of each active firm i zero. Private production cot are given a a function of firm output according to C ( ), and average production cot per firm, ( ), C are aumed to be U-haped. It i uppoed that each firm utilize all economie of cale when it produce. Clearly, the potulation that the market i perfectly competitive reuire u to aume that actual production per firm i negligible relative to aggregate output. Since i what the untaxed firm produce, the aumption of perfectly competitive product market i euivalent to the aumption that i mall relative to aggregate production. With repect to regulation the aumption of perfectly competitive market extend traightforwardly to regulated market when output per firm goe down and the number of active firm goe up, while it i not traightforward that regulated market 1 When taxe are contructed thi way it i obviou that the problem lited by Baumol and Bradford (1970), and Baumol (1979) do not arie. 3

4 continue to be perfectly competitive if output per firm i regulated in the upward direction while there are fewer firm. Production ha external effect, and we aume that damage i a function of firm output and the number of firm according to D( n, ). Under tandard aumption damage increae at an ecalating rate when production increae, that i, ( n, ) > 0 ( n, ) > 0. D and D Likewie, damage increae at an increaing rate when the number of firm goe up, that i, D n ( n, ) > 0 and ( n, ) > 0. D nn Finally, it i uual to aume that the increae in damage that follow an increae in firm output depend poitively on the number of firm, ( n, ) > 0. D n Thee kind of aumption make ure that once an optimum level of damage exit, it will be uniue. Clearly, one can think of externalitie where it i proper to aume that damage increae with higher production when production i low, but after ome threhold level the marginal damage fall or goe to zero. For example, imagine a firm that emit chemical ubtance into a river and uppoe that damage i meaured by the number of dead fih. When the river i ufficiently affected, all of the fih are dead and further emiion are cotle. Such kind of aumption reult in nonconvexitie, and a coneuence of thi can be the exitence of multiple euilibria, or, for example, each firm hould either produce the profit-maximizing uantity or not produce at all (Baumol and Bradford, 1992). Our main concern i with the tandard cae of increaing marginal damage. If the invere demand function i P ( n), the long-run ocial optimum i given by (ee Carlton and Loury 1980 for detail): (1) P( n) = C ( ) + D ( n ) n and, (2) P( n) = C( ) + Dn ( n, ) 4

5 With the proper econd-order condition being atified, the uniue optimum allocation atifie (1) and (2), and we denote thi allocation by { *,n *}. Now, the ocial average cot, called, dsac d SAC i ( C ( ) D( n, ) ) 1 C + (3) = C ( ) + D ( n, ) +, and we have: ( ) D( n, ) Output per firm atifie ( ) C( ) C = when each firm fully utilize all economie of cale, and evaluating how ocial average cot change with production cale at we have dsac d (4) = D ( n, ) 1 D n (, ) When, for any number of firm, D ( n, ) > 0 and ( n, ) > 0 ( ) D( n ) D, it follow that D >,, and the ocial average cot i thu increaing in output per firm. In turn, thi how that the ocial optimum production cale atifie <. That i, each active * firm hould ideally produce at a point of increaing cale. Of coure, the concluion that optimum production cale atifie * < make it afe to aume that firm are price taker under regulation if it i alo the cae that the number of firm goe up. To ee that it actually doe o notice that C ( ) < C ( ). Alo, at the unregulated euilibrium, called { } *, we have P ( n ) = C ( ) while the optimum allocation i characterized by P( n * *) = C ( *). It follow that P( n ) P( n * *) > or n n * * n < when demand i downward loping. From n < n * * it i clear that n * > n ince * <. That i, when each firm i mall relative to aggregate production without regulation it will alo be mall under regulation that change the output-per-firm-number-of-firm combination in direction of the ocially optimal combination. Oppoitely, if output per firm were to be increaed by regulation the uppoition of market being competitive could be out of line with the aim of regulation 5

6 ince in thi cae regulation hould decreae the number of firm and let remaining firm increae production. Of coure, when each firm hould ideally produce at a cale where the average cot exceed the marginal cot, the competitive firm i unable to break-even unle the unit tax that regulate firm output to the ocial optimum i omehow modified. Once firm are ubject to a charge, called, P = + Let t on marginal output profit maximization call for ( n) C ( ) t. u uppoe that the tax ytem i nonlinear with a charge of t per unit of output that exceed S t. In thi cae, entry and exit i brought to a top when P ( n) C( ) t( S t) = 0. We can now how propoition 1. Propoition 1. The long-run competitive euilibrium i coincident with the full ocial optimum under a imple unit tax when firm output beyond S t i taxed at the rate of ( n*, *) n * t = D where = Proof. S ( t * D ( n*, *)) and i otherwie not taxed. n The reult follow traightforward from the analyi in Carlton and Loury (1980), but we hall give the proof for convenience. If we ue t D ( n*, *) n * ( n) = C ( ) t we have P( n) C ( ) D ( n*, *) n *. P + + = in euation = Plainly, when n = n * and = * euation (1) i atified. Uing the definition of S in euation P ( n) C( ) t( S t) = 0 we have ( n) C( ) t( ( * D ( n*, *) t) ) = 0 when n = n * and * =, * P( n * *) C( *) + D ( n*, *) = 0 P n, or, n howing that euation (2) i atified. Finally, we have implicitly aumed that each firm produce a uantity that i larger than S t, that i, S t < *. To verify that thi i actually atified, notice that S t = * D ( n*, *) D ( n*, *) when D ( n, ) > 0 and ( n, ) > 0 n D n a i aumed here. End of proof. 6

7 Propoition 1 demontrate that a nonlinear Pigouvian tax provide the right incentive in the hort a well a the long run when the damage function fulfill tandard aumption, when demand i downward loping, and when firm are price taker in the unregulated market. Thu, there are no eriou limitation on Pigouvian taxe a a cure againt externalitie a it eem to be the concluion et forth by Carlton and Loury (1980, 1986). With repect to the practical implication of propoition 1, notice that the exemption level i alway poitive; thi follow becaue each firm produce at a point of increaing cale. In turn, the tax ytem pecifie a zero unit tax for ome output and a poitive unit tax for the ret of each firm output o that revenue mut be poitive. In thi way, the problem of raiing revenue by lump-um taxation of other ector that are dicued by Baumol and Bradford (1970), and later Baumol (1979) doe not arie. Alo, ince the number of firm increae under regulation and each firm produce le the aumption of perfect competition continue to apply. A noted, there are certain kind of damage that are characterized by decreaing marginal damage, that i, D ( n, ) and ( n ) D nn, or both are negative. Thi implie that the olution to the problem of maximizing welfare doe not in general have a uniue olution. Thi follow immediately from the econd-order condition np' 1 2 ( n) C ( ) n D ( n, ) < 0 and P' ( n) Dnn ( n, ) < 0. Since ( n) P' i negative, thee condition are met for any uantity-number-of-firm pair when ( n, ) > 0 D ( n, ) > 0. Converely, if either D ( n, ), or ( n ) nn D and D nn,, or both are negative, the econdorder condition might not be atified. Auming however, that demand i ufficiently loped in a trong, negative manner a uniue ocial optimum till exit. Conider the cae where marginal damage of increaed output i poitive and increaing while the marginal damage of adding firm i poitive but at a decreaing rate. Going through the proof of 7

8 propoition 1, it can be een that the ign of D nn ( n, ) and ( n ) D n, play no role, o the reult till applie, and it i till the cae that the ocially optimal output per firm i le than the output that utilize all economie of cale and that the optimum number of firm i higher than the number urviving in the unregulated market, that i, * < and n * < n. We tate thi a propoition 2. Propoition 2. A long a D ( n, ) > 0, D n ( n, ) > 0, and ( n, ) > 0 D the nonlinear unit tax defined in propoition 1 i efficient when the problem of maximizing welfare ha a uniue olution. The explanation for propoition 2 i traightforward. Whenever ( n, ) > 0 ( n, ) > 0 D and D each firm production technology i convex and it i therefore poible to utain any deirable allocation a ha been hown for example by Baumol and Oate (1975, chapter 8). Alo it follow from ( n, ) > 0 D that regulation hould be aimed at reducing firm output to a point of increaing return and increae the number of active firm. In thi ene it i afe to potulate that regulated market are competitive when market are competitive without regulation. That i, all of the tandard aumption for price intrument to be efficient are met. The effect of auming that D nn ( n, ) i negative and ( n ) D, i poitive i that the ocial production-poibility et might be non-convex. Thi mean that it i uncertain whether or not a uniue olution exit, but propoition 2 make it clear that once uch a olution exit a tax i an efficient intrument. We are left with the cae where marginal damage of increaed output i poitive and decreaing. Again, thi produce a non-convexity of the ocial production poibility frontier but firm can be regulated by a tax a long a private cot atify the tandard condition. However, a noted, ocial average cot change according to dsac d ( D ( n, ) D( n, ) ) = when each firm produce at the point where all 8

9 economie of cale are exhauted. In turn, the lope of the ocial average cot curve at thi point i negatively loped when D ( n, ) > 0 and ( n, ) < 0. D Thi mean that output per firm hould be puhed into a region of decreaing return to cale and that the number of firm hould ideally decreae, that i, * > and n * < n. In relation to auming that market are perfectly competitive, thi i clearly a problem. In model of perfect competition, profit are nullified by the entry and exit of firm until each firm produce at the point where there are contant return to cale. Auming that firm are price taker i to aume that each firm produce a negligible amount of total production when the firm utilize all economie of cale. In the cae of ( n, ) > 0 D, firm production i adjuted downward, and we are therefore certain that firm production under regulation i mall relative to the aggregate output. Thi i not o when ( n, ) < 0; D here each firm hould grow and there are ideally to be fewer firm meaning that each firm produce at a level that might be non-negligible o that the aumption of price-taking behavior i uetioned. It i eay to ee thi point. When the marginal damage of increaing output from a firm i falling trongly, it i (relatively peaking) beneficial to have few large firm and, in uch cae, the cope of unit taxe a device againt externalitie can be limited becaue the number of firm make it implauible to aume that each firm i a price taker. 3 Concluion In thi paper, we have reviited the uetion of how efficient Pigouvian taxe are in the long run. We emphaize, once more, that the reult including the exitence of a unit tax with an exemption level are mentioned in paing by Carlton and Loury (1980, 564). However, the point that a imple nonlinear unit tax i efficient and doe not reuire the regulator to raie fund from other ector when we make tandard aumption about damage i miing. Thu, for many practical policy iue, the analyi ugget that Pigouvian taxe correct 9

10 externalitie efficiently. Clearly, the analyi doe not ugget that unit taxe are efficient in the preence of evere non-convexitie and non-uniuene problem. Alo, there i one further cae in which the working of Pigouvian taxe i ubtle. Thi i when the marginal damage of increae in production i poitive but falling. In thi cae, a unit tax with an exemption level would not work. 2 However, we ugget that one of the main problem in uch circumtance i whether one can think of a perfectly competitive indutry at all the reaon being that each firm hould ideally produce at a point of decreaing return. The natural or neceary aumption to make in model that are baed on competitive product market i that each firm i mall when the firm produce at the point of contant return to cale. But, once it i ideal to have fewer firm produce more, the idea of each firm producing a negligible amount of aggregate output i not naturally atified, and thi of coure uetion the general applicability of a unit tax which relie on firm being price taker. Reference Baumol, W.J Quai Optimality: The Price We Mut Pay for a Price Sytem, The Journal of Political Economy 87(3): Baumol, W.J. and D.F. Bradford Optimal Departure from Marginal Cot Pricing, American Economic Review 60(1): and. 1972, Detrimental Externalitie and Non-Convexity of the Production Set, Economica 39(154): Baumol, W.J. and W.E. Oate The Theory of Environmental Policy, Cambridge Univerity Pre. Carlton, D.W. and G.C. Loury The Limitation of Pigouvian Taxe a a Long-Run Remedy for Externalitie, The Quarterly Journal of Economic 95: and The Limitation of Pigouvian Taxe a a Long-Run Remedy for Externalitie: An Extenion of Reult, The Quarterly Journal of Economic 101: It i eay to demontrate, in thi cae, that each firm hould be given a ubidy according to ( ) where = D ( n*, *) n * and S ( D ( n )) S it i better for firm not to produce and collect the ubidy S. = * *, *. But, ince entry and exit nullifie profit, n 10