Monopoly vs. Competition in Light of Extraction Norms. Abstract

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1 Moopoly vs. Competitio i Light of Extractio Norms By Arkadi Koziashvili, Shmuel Nitza ad Yossef Tobol Abstract This ote demostrates that whether the market is competitive or moopolistic eed ot be the result of ideology, the relative political power of a certai producer or the ature (o-covexity) of the techology. The form of the market structure ca be determied by a govermet that imizes the extracted resources whe it cosiders the two alterative market structures. The extracted profits hige o the demad ad cost characteristics ad o the prevailig orms of extractio. Our claim is illustrated by assumig that both demad ad supply are liear fuctios ad that the govermet ca extract a fixed rate of every competitive producer's profit or the resources expeded by the producers whe they cotest o the moopoly profit. Keywords: moopoly, competitio, orms of extractio, JEL classificatio codes: D7; D78; O50 Departmet of Ecoomics, Bar Ila Uiversity, Ramat Ga, 5900 Israel, Correspodig author, Tel: , Fax: , itzas@mail.biu.ac.il. School of Idustrial Maagemet, Jerusalem College of Techology (JTC). toboly@gmail.com. 1

2 1. Itroductio Suppose that social orms allow the use of political office for privileged distributio such that society has a political culture of extractio, Cogleto, Hillma ad Korad (008), Epstei ad Nitza (007), Korad (008) or corruptio, Ades ad Di Tella (1999), Bliss ad Di Tella (1997), Treisma (000). Specifically, this culture is represeted by two optios available to the govermet. First, there exists a fixed profit share R the existig competitive producers are required to give up i order to secure the govermet approval of their productio activity. Secod, the govermet ca assig the moopoly status to oe of the existig producers extractig the moopoly profits from the producers that take part i the cotest o the award of the moopoly ret. The objective of this ote is to demostrate the determiatio of market structure by the govermet i the biary case of moopoly vs. competitio. Assumig that the cost ad demad fuctios are liear, we examie the effect of the market characteristics as well as of the prevailig orm of extractio or corruptio o the govermet's choice betwee moopoly ad competitio.. Moopoly vs. competitio Suppose that there are idetical producers i the idustry. Oe optio the govermet has is to award the exclusive right of productio to oe of these producers, the wier i the cotest o the moopoly profit 1. I such a case, we assume that the govermet is able to extract the moopoly profit A from the riskeutral cotestats. This assumptio is plausible uder certai coditios that are well kow i the cotest-theory literature. For a recet study of the class of cotests that satisfy this assumptio see Alcalde ad Dahm (009). The other possibility is to let these idetical producers get the profit of a competitive market, subject to the existig simple political culture that extracts a fixed rate R of the profit. I such a case the govermet receives RE, where E is the total profit of the competitive 1 The possible techological advatage of a moopoly due, for example, to his ability to ivest more i research ad developmet is disregarded. I other words, we assume that his cost fuctio is idetical to that of the competitive producers. Complete ret dissipatio i a cotest, where all moopoly profit is trasferred to the govermet, has already bee oted i the early studies of Krueger (1974) ad Poser (1975) ad i Tullock's (1980) ret-seekig cotest, where the govermet applies the simple lottery cotest success fuctio ad the umber of firms competig o the moopoly profit is sufficietly large I the later case, the 1 govermet actually extracts ( ) A. Although our complete dissipatio assumptio simplifies the illustratio, it is iessetial qualitatively.

3 producers. The market structure determied by the govermet higes o the relatioship betwee A ad RE. If R< A/E, the govermet prefers the moopoly market structure. If R > A/E, the -producer competitive idustry is the preferred market structure of the govermet. I the former case, moopoly is ot due to o covexity of the techology (icreasig returs to scale) or to the political power of some iterest group. I the latter case, competitio amog the existig producers is ot the result of ideology or the beevolece of the govermet. I both cases the preferred market structure reflects the imizatio of the extracted resources from the producers. Market structure is determied the by the prevailig orms of extractio (the ability to award the moopoly ret the fixed profit share R that ca be extracted from the competitive producers) ad the characteristics of the supply ad demad for the produced good. To illustrate how demad, supply ad the orms of extractio determie market structure, cosider the stylized case of liear demad ad supply fuctios. 3. Liear demad ad supply Suppose that the demad fuctio pxis ( ) liear ad the cost fuctio cx ( ) of every producer is quadratic ad there are o fixed costs. That is, the demad ad margial cost fuctio are liear, p( x) = δ κ x, κ > 0 ad c ( x) = µ x, µ > 0. Figure 1 presets the demad curve, margial reveue curve, margial cost curve of the moopoly ad the supply curve of the - producer idustry. The moopoly output x m is determied by the itersectio of the moopoly margial cost ad margial reveue curves. The competitive equilibrium output x e is determied by the itersectio betwee the market demad ad supply curves. The profit of a moopoly is represeted by the trapeze A. The total profit of the competitive producers E is give A µ by the triagle E. Let Κ= ad =. Κ is the relatio betwee the profits of the E κ moopoly ad the competitive idustry., which represets the market characteristics, is the relatio betwee the slopes i absolute terms of the margial cost ad the demad curves. 3

4 δ p m D c = µ x p( x) = δ κ x p e A E i= 1 c' = ( µ / ) x MR( x) 0 xm xe x Figure 1: Moopoly vs. competitio by producers The followig result clarifies how market structure is determied by the specific form of the demad ad supply fuctios, represeted by the parameter ad by the political culture, represeted by the parameter R. Notice that the parameter δ is icosequetial i determiig the selected form of market structure. Propositio 1: The govermet sets the moopoly market structure rather tha the competitive oe, if 1( + ) R< K =. The govermet prefers the competitive ( + ) market structure, if the iequality is reversed. 3 The proof of the propositio ad its corollaries is relegated to the Appedix. 3 Notice that R > K is a sufficiet coditio for the superiority of the competitive market structure, regardless of the proportio of the moopoly profit A extracted by the govermet (see footote ). 4

5 Oe implicatio of Propositio 1 is that K ca be larger tha or equal to 1. That is, the profit of the moopoly ca exceed the profit of the competitive idustry. I such a case the govermet certaily prefers the moopoly to the competitive market structure. Whether this situatio arises depeds o the parameters ad. Specifically, Corollary 1.1: The moopoly is always preferred to the competitive market structure if < 1. A secod implicatio of Propositio 1 relates to the effect of chages i ad o Κ. A icrease i the umber of the existig producers icreases the equilibrium quatity but reduces the equilibrium price. Hece, the effect of a chage i o the profit of the competitive idustry is ambiguous. I tur, the ecessary threshold of the extracted profit share that iduces the govermet to prefer the competitive system may icrease or decrease, depedig o the relatioship betwee ad. However, the effect of a chage i (a chage i the slope µ of the margial cost fuctio of a producer or a chage i the slope κ of the demad fuctio) is uequivocal. Specifically, Corollary 1.: Κ is decreasig with ad icreasig (decreasig) with, provided that is larger (smaller) tha. Corollary 1. implies that a less favorable techology (a icrease i µ - the margial cost becomes higher at ay quatity x) or a more favorable market demad (a decrease i κ - the demad becomes larger at ay price) result i a decrease i A Κ=. That is, the icetive to prefer the moopoly market structure becomes E weaker the rage of extractio rates that are sufficiet to iduce the govermet to prefer the moopoly is reduced. Corollary 1. also implies that if Κ is held fixed ad < ( > ), the a icrease i requires a decrease (a icrease) i. I other words, A typical iso- Κ curve is egatively or (positively) sloped whe < ( > ). Figure presets such 5

6 a iso- Κ curve for Κ = I this case, a govermet that ca extract 36% of the profit made by a competitive producer would prefer the competitive system rather tha the existece of a moopoly for ay combiatio of ad represeted by a poit above the iso-0.36 curve. K = 36% Figure : The iso- Κ curve, for Κ = 36%. Suppose that the govermet does ot take idustry size as give, but rather selects the umber of producers that yields the imal profit E. The secod part of Corollary 1. implies that the preferred idustry size is = * =. I this case, Propositio 1 takes the followig form: Corollary 1.3: If the govermet ca set the optimal competitive idustry size *, the it prefers the moopoly market structure rather tha the competitive oe, if 4 R< K =. The govermet prefers the competitive market structure, if the + iequality is reversed. 6

7 This corollary directly implies that a ecessary, but ot a sufficiet, coditio for the superiority of the competitive market structure is that >, that is, the slope of the margial cost must be larger tha two times of the absolute value of the slope of the demad fuctio. Notice that i this case the first part of Corollary 1. is still valid. Techologies ad cosumers may vary across differet regios or states as well as alog time i the same place. This variability certaily affects the threshold Κ that determies the govermet decisio. Oe might be iterested i computig the average threshold i differet geographical areas or alog time, takig ito accout the variability i both ad. Clearly if the average K i oe eviromet is smaller tha the average K i aother eviromet, the the probability that the govermet i the former eviromet prefers ad set the competitive market structure is higher. The followig corollary of Propositio 1 presets the average threshold K correspodig to, assumig that is uiformly distributed over [, ]. mi Corollary 1.4: If is uiformly distributed over [, ], mi ad mi the ( + ) = + ( + ). 1 ( ) K l l mi mi Corollary 1.3 implies that for =, K = 1/. I this case, o average, the miimum extractio rate required for the govermet to prefer the competitive market structure exceeds 1/. Assumig that mi = 0.1, we obtai the followig table that = = Table 1: K correspodig to =, 3, mi = 0.1 ad =3, 5, 7, 10, 15 ad. presets the values of the average threshold K correspodig to =,3 ad =3, 5, 7, 10, 15 ad. Notice that the miimum ecessary average threshold of the extracted profit share that iduces the govermet to prefer i higher probability the 7

8 two-producer or the three-producer competitive system requires that the imal relatio betwee µ ad κ is at least 10, 10. Otherwise, the profit of the competitive market structure is isufficiet to make it probabilistically advatageous relative to the moopoly market structure. 4. Summary We have demostrated how the techology ad market characteristics together with the prevailig extractio orms (the ability to extract the moopoly profit ad the fixed profit share that ca be extracted from the competitive producers) determie whether the market is moopolistic or competitive. The compariso betwee moopoly ad competitio with limited etry was based o the simplifyig assumptio of liear demad ad supply fuctios. We derived the criterio used by the govermet to decide which of the two possible market structures is preferred (Propositio 1). This criterio depeds o the slopes of the demad ad margial cost curves ad o the existig umber of producers. Propositio 1 was applied to compute the combiatios of liear demad ad margial cost fuctios ad umber of producers that esure the superiority of the moopoly (Corollary 1.1), to examie the effect of chages i demad, supply ad the existig umber of producers o the applied criterio (Corollary 1.), to derive the criterio used by the govermet to decide which of the two possible market structures is preferred, whe it ca set the optimal idustry size that yields the largest competitive profit (Corollary 1.3) ad to compute the average level of the criterio i a eviromet where the parameters of the demad ad supply fuctios are radom variables (Corollary 1.4). 8

9 Appedix: Proofs Propositio 1: I Figure 1, x ad x are determied, respectively, by the itersectio of the m e margial reveue ad margial cost curves ad the demad curve ad aggregate margial cost curves of the producers. Hece, (A1) δ κ x = µ x x m δ =, µ + κ (A) We ca ow use µ δ δ κ x = x xe = µ. + κ x ad x to compute A ad E, the profits of the moopoly ad m e the competitive idustry. Notice that i Figure 1, A= (A+D) D. + = 1 + ad Sice ( A D) x [ δ x ( κ µ )] m m x m D = κ, 1 (A3) A= (A+D) D = δ xm xm( κ + µ ). Sice 1 µ E = x, e (A4) A x m x m + = δ. E xe( µ / ) xe µ + κ Notice that (A5) µ δ κ δ x + m µ + κ = =. xe δ ( µ + κ) µ ( + κ ) Hece, by (A4), (A5), (A1) ad (A), we obtai that (A6) µ µ µ ( + κ)( + κ) ( ) A + κ + = E µ = ( µ + κ) ( µ + κ) / 9

10 ( + 1) 1( + ) = = K. ( + ) ( + ) = { 1} We have thus obtaied that the govermet sets the moopoly market structure rather 1( + ) tha the competitive oe, if ad oly if R< K = ( + ) competitive market structure if the iequality is reversed. ad it prefers the Corollary 1.1: By (A6), K 1 ( ) ( ) > + > +. That is, + + > + or + > < <. 1 1 Corollary 1.: The proof is directly obtaied by differetiatig ad. Specifically, it ca be readily verified that K 1( + ) = with respect to ( + ) [ ] K + (1 ) = = < 0, sice ad ( + ) ( + ) K 1 = > 0 > ( + ) < <. Corollary 1.4: 1 1 ( + ) 1 ( + ) K = d d. ( ) ( + ) ( + ) mi mi mi 10

11 Notice that sice = ( + ) ( + ), 1 1 K = + d ( ) + mi ( ). Now, sice ( + ) 1 d = + + l mi ad ( + ) 1 d ( ) ( )( ) ( ) = l( + ), ( + ) mi we get that 1 = = K ( 4 4) ( 4) l ( ) l( ) mi = l ( ) l( + ). That is, mi 1 K = { 6 4 l ( ) l( ) } = mi = 1 ( + ) ( mi ) l ( ) l + mi ( mi + ) = = 1 ( ) ( + ). + l l mi ( mi + ) 11

12 Refereces Ades, Alberto, ad Rafael Di Tella "Rets, competitio ad corruptio", America Ecoomic Review, 89(4): Alcalde, Jose, ad Matthias Dahm "Ret seekig ad ret dissipatio: A eutrality result", Joural of Public Ecoomics, forthcomig. Bliss, Christopher, ad Rafael Di Tella "Does competitio kill corruptio?", Joural of Political Ecoomy, 105(5): Cogleto, Roger D., Arye. L. Hillma, ad Kai A. Korad editors, The Theory of Ret Seekig: Forty Years of Research, Spriger, Heidelberg ad Berli. Epstei, Gil S., ad Shmuel Nitza Edogeous Public Policy ad Cotests, Spriger-Verlag, Heidelberg. Korad, Kai A Strategy i Cotests, Oxford Uiversity Press. Krueger, A The Political ecoomy of the ret seekig society, America Ecoomic Review, 64(3): Poser, R. 1975, The social costs of moopoly ad regulatio. Joural of Political Ecoomy, 83: Treisma, Daiel "The causes for corruptio: A cross atioal study", Joural of Public Ecoomics, 76(3):

EVEN NUMBERED EXERCISES IN CHAPTER 4

Joh Riley 7 July EVEN NUMBERED EXERCISES IN CHAPTER 4 SECTION 4 Exercise 4-: Cost Fuctio of a Cobb-Douglas firm What is the cost fuctio of a firm with a Cobb-Douglas productio fuctio? Rather tha miimie