BANDWIDTH ALLOCATION AND PRICING PROBLEM FOR A DUOPOLY MARKET

Transcription

1 Yugoslav Journal of Operatons Research (0), Nuber, DOI: 0.98/YJOR0065Y BANDWIDTH ALLOCATION AND PRICING PROBLEM FOR A DUOPOLY MARKET Peng-Sheng YOU Graduate Insttute of Marketng and Logstcs/Transportaton, Natonal ChaY Unversty, Tawan Chun-Cheh LEE Graduate Insttute of Marketng and Logstcs/Transportaton, Natonal ChaY Unversty, Tawan Y-Chh HSIEH Departent of Industral Manageent, Natonal Forosa Unversty,Tawan Receved: July 008 / Accepted: May 0 Abstract: Ths research dscusses the Internet servce provder (ISP) bandwdth allocaton and prcng probles for a duopoly bandwdth arket wth two copettve ISPs. Accordng to the contracts between Internet subscrbers and ISPs, Internet subscrbers can enjoy ther servces up to ther contracted bandwdth lts. However, n realty, any subscrbers ay experence the facts that ther on-lne requests are dened or ther connecton speeds are far below ther contracted speed lts. One of the reasons s that ISPs accept too any subscrbers as ther subscrbers. To avod ths proble, ISPs can set lts for ther subscrbers to enhance ther servce qualtes. Ths paper develops constraned nonlnear prograng to deal wth ths proble for two copettve ISPs. The condton for reachng the equlbru between the two copettve frs s derved. The arket equlbru prce and bandwdth resource allocatons are derved as closed for solutons. Keywords: Duopoly arket; equlbru prce; servce qualty; bandwdth allocaton. MSC: 9B4. INTRODUCTION Two of the ost popular technologes that offer speedy access to surf the Internet are the DSL (Dgtal Subscrber Lne) broadband and the cable Mode. DSL

2 66 P.S. You, C.C. Lee, Y.C. Hseh/ Bandwdth allocaton and prcng proble servce provdes Internet access over a sngle dedcated telephone lne, whle Cable broadband servce provdes Internet access over a cable televson lne. The advantages of DSL ncorporate cheaper servce than cable Internet, no dal up, no busy sgnals and a unque connecton. Several dfferent types of DSL exst. ADSL s one of the ost popular broadband eda aong the, for both coercal and resdental nternet users. ADSL s a od/deod technology provdng hgh-speed web servce through a tradtonal telephone lne. ADSL can also be used to provde hoe offce workers wth copany Internet servces or new style nteractve ulteda applcatons, ncludng web gaes and VOID servces. Regardng the Cable Mode, t uses the hgh bandwdth cable syste lad by the cable TV servce provder to enable users to watch TV progras, use telephone servces and surf the Internet sultaneously. The greatest advantage of Cable Mode s a huge bandwdth wth ts technologcal specfcatons, akng the data transsson speed ascend proptly, and consequently, the drea of transferrng ore data on a gven bandwdth s realzed. In addton, Cable offers a uch wder servce area than DSL. The bandwdth requreents are dstnctve accordng to the needs of the Internet users. Operatng n such an envronent, DSL and Cable provders usually offer a varety of bandwdth coodtes to serve ther custoers, wth each Internet coodty havng a dfferent sales prce and a unque bandwdth restrcton. To offer dfferent Internet coodtes to custoers, both DSL and Cable provders face the probles of allocatng a bandwdth resource to dstnct Internet coodtes. Accordng to the subscrpton contracts, Internet users are allowed to access Internet servce wthn the contracted speed. However, n realty, they are usually n a stuaton that ther on-lne requests are dened, or ther connecton speeds are far below ther contracted speed lts. The qualty of the Internet servces ay be nfluenced by the nuber of on-lne users. It s natural to assue that the qualty of the Internet servces wll becoe poorer f too any users access the Internet at the sae te. To prove ther corporaton ages, both DSL and Cable provders have to provde ther Internet subscrbers wth the agreed servce qualtes. One of the possble approaches for overcong ths proble s to control the nuber of subscrbers. To reach ths goal, both DSL and Cable provders can use the prcng strateges to control the nuber of subscrbers. It s possble to surf the Internet by ether DSL or Cable Mode. For copetton purposes, DSL and Cable provders ay offer slar Internet coodtes to coplete the sae custo pool. Accordng to ths and the prevously entoned fact, both DSL and Cable provders ay serously consder ther prcng strateges to coplete n the Internet arket. Ths paper consders a duopolstc arket whch conssts of a DSL provder and a Cable provder. The two Internet servce provders offer two slar Internet coodtes. To our knowledge, few works deal wth the ISP copetton proble wth servce qualty guarantees. Other research related to ths artcle ncludes the works on the equlbru prce n duopoly arket. Pala and Leruth (993) dealt wth a duopoly odel for two frs whch sell hoogeneous goods. They showed the exstence of a syetrc Nash equlbru prce. Harrngton (995) nvestgated the prce-settng behavor n a duopoly arket where the degree of product dfferentaton s uncertan. They showed that n arkets wth hghly substtutable products, prce dsperson can be enhanced, snce prce adjustent s used to obtan ore nforaton about the context of

3 P.S. You, C.C. Lee, Y.C. Hseh/ Bandwdth allocaton and prcng proble 67 product dfferentaton. Cho (996) dealt wth a prce copetton proble wth duopoly anufacturers and duopoly coon retalers. Ths paper proves that product dfferentaton s helpful to anufacturers, whle store dfferentaton s helpful to retalers. Peha and Tewar (998) exaned the result of copetton between two proftseekng telecouncatons carrers. In ther work, each fr has a lted capacty that can be used to offer two dfferent servces. They showed condtons for an equlbru where no carrer has any ncentve to change ts prces and outputs. Elberfeld and Wolfstetter (999) nvestgated a repeated gae wth sultaneous entry and prcng. They provded a syetrc equlbru soluton. Mason (000) dealt wth an Internet prcng proble. Ths paper showed that condtons for reachng equlbru for two copettve frs exst. Foros and Hansen (00) dealt wth a copetton proble between two ISPs provdng the qualty of nterconnecton. Mn et al. (00) dealt wth a prcng settng proble for two copettve frs that produce substtutable products. They derved equlbru solutons based on a two-stage gae fraework. Basar and Srkant (00) nvestgated a leader-follower gae proble for a network wth a sngle ISP and a larger nuber of users. In ths proble, the servce provder sets the sales prce and the custoers react to the prce, as well as to the network congeston. Bapna et al. (005) nvestgated a prcng settng proble for dgtal products. Ths study contans a servce provder that uses a prce settng echans to axze ts allocaton effcency. Syeonds (003) copared Bertrand and Cournot equlbru n a dfferentated duopoly wth substtute goods and product R&D. Ths paper provdes a condton that quantty copetton s ore benefcal than prce copetton for both consuers and frs. Da et al. (005) developed prcng strateges for a revenue anageent proble wth ultple frs provdng the sae servce for a coon pool of custoers. They showed that Nash equlbru exsts n a two-fr prcng gae when the deands of both frs are deternstc. The strategy n duopoly envronent has been wdely studed n any ndustres. The constrants on ost of the studes nclude the budget constrant, quantty constrant, etc. As entoned prevously, ISPs need to take ther servce qualty nto account. If ISPs are unable or have no plan to expand ther equpent over a future te nterval, endlessly acceptng custoers as ther subscrbers, they wll run the rsk of poor Internet access qualty. To avod such a stuaton, both DSL and Cable provders should serously set ther prcng strateges, n order to prove ther servce qualtes. Ths research dscusses the bandwdth allocaton and prcng probles by takng the servce qualtes nto account. The odel hypotheszes that there s only a DSL servce provder and a Cable servce provder n the arket; the two servce provders offer two slar Internet servce coodtes, and both of the rvalng partes confront lted total bandwdth and servce level. By establshng a atheatcal odel and Lagrange functon, reacton functons of the two servce provders can be obtaned. Through the derved reacton functons and soe condton, ths paper obtans closed for forulas for the arket equlbru prces and bandwdth resource allocatons for the two copettve ISPs.. MODEL ASSUMPTIONS, DESCRIPTION AND FORMULATION Ths secton of the paper wll develop a atheatcal odel for the proble. We assue that a DSL Internet servce provder and a Cable Internet servce provder

4 68 P.S. You, C.C. Lee, Y.C. Hseh/ Bandwdth allocaton and prcng proble exst. We use ndex = to denote the DSL servce provder and ndex = to denote the Cable servce provder. The total bandwdth unts avalable for provder are B. Both provders are assued to offer two slar Internet coodtes. The sales prce for coodty of provder s a decson varable and s denoted by p,. Custoers are able to purchase ther coodtes ether fro the DSL servce provder or the Cable servce provder. The deand curves for the two servce provders are gven as functons of prces. For convenence, we let = f =, = f =, = f = and = f =. Deand for coodty, {,} of provder s assued to follow the functon of q, = a, v, p, + h ( p,,,, p ) β ( p p ) () where the sybol h represents the substtuton coeffcent of slar coodtes offered by dfferent servce provders and s assued to be h, and β represents the substtuton coeffcent of dfferent coodtes offered by servce provder and s assued to be β. The deand functon llustrates that the deands for bandwdth coodtes are dependent not only on the sales prce of rvalng servce provder for the slar coodtes, but on the sales prce of ts dfferent coodty as well. For provder, we denote b, as the nuber of bandwdth unts allocated for coodty. Servce provder s confronted wth the ltaton of total bandwdth unts. That s, the total bandwdth unts that the servce provder allocates to all types of bandwdth coodtes shall never exceed the total bandwdth unt B. The fracton of the nuber of onlne users to the total nuber of subscrbers (on-lne users plus off-lne users) of coodty of servce provder at any te s assued to be an dentcal rando varable r,. The rando varable r, takes value between 0 and and s assued to follow a noral dstrbuton wth ean μ, and devaton σ,. The aount of bandwdth resource requred by an onlne subscrber of coodty of servce provder $$ s assued to be u, bandwdth unts. Suppose the total nuber of subscrbers for coodty of servce provder s q,, and the fracton of the nuber of on-lne subscrbers to the total nuber of subscrbers s r,. Consequently, the bandwdth unts requred for coodty s r, q, u, s the nuber of subscrbers of coodty of a servce provder that akes onlne requests. In order to keep the ISP s servce qualty, we assue that ISP wll let the probablty,.e., the rato of the nuber of bandwdth unts allocated for coodty exceeds the nuber of bandwdth resource needed by subscrbers of coodty, be no less than the prescrbed servce level 0 < α,. Ths paper as to set the equlbru coodty prce p, and deterne the bandwdth allocaton decsons b,. We have llustrated the odel descrptons and assuptons. We wll now forulate the odel. Frstly, we suarze our notaton and decson varables as follows.

5 P.S. You, C.C. Lee, Y.C. Hseh/ Bandwdth allocaton and prcng proble 69 Notaton: a, = the ntercept of deand functon for coodty of servce provder, h = the coeffcent of substtuton for coodty between servce provders, β = the coeffcent of substtuton between coodty and coodty ' of servce provder, u, = unt bandwdth consupton for coodty of servce provder, α, = guaranteed servce level of coodty of servce provder, r, = the fracton of the nuber of onlne users to the total nuber of subscrbers of coodty of provder at any te, μ, = ean of the rando varable r,, σ, = devaton of the rando varable r,, B = total avalable bandwdth unts for servce provder, p, = decson varables, representng the sales prce of coodty of servce provder, b, = decson varables, representng the nuber of bandwdth unts allocated for coodty of servce provder. Revenue functon Let R be the revenue for fr. Suppose servce provder sets the sales prce of coodty at p,. The total revenue for servce provder can then be expressed as follows. R = p q () =,, Resource and servce constrants Snce the su of the bandwdth unts allocated for all coodtes cannot exceed the avalable bandwdth unt B, we have the followng bandwdth resource constrant for servce provder., = b B 0. (3) Suppose b, s allocated for coodty. Snce the nuber of subscrbers of coodty of servce provder s expected to be q, the bandwdth unts requred by these subscrber s the nuber of q ur,. Because the probablty that the nuber of bandwdth unts allocated for coodty exceeds the nuber of bandwdth resource needed by subscrbers of coodty s requred to be no less than a prescrbed servce level 0 α,, we have <

6 70 P.S. You, C.C. Lee, Y.C. Hseh/ Bandwdth allocaton and prcng proble or, equvalently Pr ob( b, qur, ) α,,,. (4) b, Pr ob r, α,,,. (5) q, u, As r, s assued to be a noral rando varable wth ean μ, and devaton σ,, the above nequalty can be rewrtten as follows b, q, u, μ, / σ, Z,, (6) where Z, s called the z-statstc (Stone, 996). The value of Z, can be evaluated usng the Excel functon NORMSINV as Z, = NORMSINV ( α, ). The purpose of ths paper s to establsh the condtons for attanng the arket equlbru prce p, and bandwdth allocaton b,. Let p and b be -densonal vectors wth eleents p, and b, respectvely, at -th entry. Accordngly, the forulaton of servce provder can be establshed as follows. ax R p, b = p =, q, subject to nequalty (3) and b q u σ Z + ), (8),,, (,, μ, where nequalty (8) s a rewrtten for of nequalty (6). (7) 3. ANALYSIS For the purpose of shortenng our forulaton, we wll denote the followng functons. c = u + h + β, (9),, g = u σ Z + μ ). (0),, (,,, Afterwards, we can rewrte revenue functon R as follows.,,,,, ',, = = R = c p + β p p + ( α + h p ) p. () In addton, usng () and (0), we can rewrte nequalty (8) as follows.

7 P.S. You, C.C. Lee, Y.C. Hseh/ Bandwdth allocaton and prcng proble 7 g =, ( c, + β ) p, + g, ( α, + h p ', ) + β g, p b, 0. () Theore 3. R s concave functon of p Proof: Let H be the Hessan atrx of R. Consequently, the result follows f R R H s negatve defnte. Frst, we have =. Second, we have = β p p, pd and R R R H = = 4( c,, ) > 0 c β (3) p p p p Thus, H s negatve defnte and we have copleted the proof. The Lagrangan functon for servce provder can be forulated as follows. L = c, p, + β p, p, = ( α, + h p', ) p = = λ =, g + β ( c,, g, =, η b, B + x + β ) p + g ( α + h p, p b + y, +,, where λ,, and η Lagrangan ultplers, and y and x, are slack varables. Now, we are ready to develop the reacton functon for servce provder. Takng the partal dervatves of L wth respect to p,, b,, λ,, η, y and x,, we obtan the followng KKT condtons for servce provder. p λ,, g = c, c,, p λ,, ' + β p β g, ', ' = 0, + h p ',, + α, + ', ). (4) (5) b, = η + λ = 0. (6), η = b + B x = 0 =,, (7)

8 7 P.S. You, C.C. Lee, Y.C. Hseh/ Bandwdth allocaton and prcng proble λ, β g = g, = ( c + β ) p p + b + y g = 0, ( α + h p ' ) (8) x y, = η x = 0, (9) = λ y = 0. (0),,,, Let p, b, λ,, η, y and x, be a soluton pont to the above equaton syste of Eq. (5) to Eq. (0). In Theore 3., we have shown that R s concave functon of p zero, the functons of,,. Thus, f the values of p, b, λ,,, η, y and x, are no less than p and b, are servce provder 's reacton functons. For the purpose of splfyng our expresson, we wll denote the followng functon. A = B 0.5 g ( α + h p' ). () = Theore 3. Suppose A > 0. Then, the servce provder 's reacton functons are gven by p, and b, where p, β h' p' ' + hc' p' + α c' + β α ' =, () ( c c β ) b =.5g ( α + h p ). (3) 0 ' Proof: See Appendx A. Although the condton of A > 0 s restrctve to the proble, t exsts n any busness practces snce an Internet servce provder's avalable bandwdth unts are usually very large. Theore 3.3 Suppose A > 0. Then, the equlbru sales prce pˆ and the bandwdth allocaton decson bˆ are gven by the followng} pˆ 8( c = 4( βh' α h' ( h' c + ' ' ' ' c ' ' β )( α c' + βα ' ) E ' c' + hc' β ' α ' ' + hc' α E α α ' β ' h ) h' hα ' + E E c ' ' ' + β h β α ' ' ' ) (4)

9 P.S. You, C.C. Lee, Y.C. Hseh/ Bandwdth allocaton and prcng proble 73 where ( + h pˆ ) b ˆ = 0.5g α < (5) ' E = 6( c c β )( c c β ) + 8h h β β + 4( c,,,c,h + c,,c,h, ) h h. (6) Proof: Accordng to Theore 3., we see that the soluton to the equaton systes p, = p,, p, = p,, p, = p, and p, = p, are equlbru sales prce. By solvng the, we obtan p, = ˆp,, p, = ˆp,, p, = ˆp,, p, = ˆp,. Thus, the equlbru sales prce s gven by pˆ. Replacng p n () wth pˆ, we see that the bandwdth allocaton decson s gven by (5). Therefore, we have copleted the proof. 4. NUMERICAL EXAMPLES AND SENSITIVITY ANALYSIS Ths secton wll provde soe exaples to llustrate the functonng of the odel n the proposed fraework. Let us assue that the paraeters of a,, v, and u, are gven n Table, and the paraeters of α,, μ, and σ, are gven n Table where Z, s deterned by usng the Excel functon NORMSINV as Z, =NORMSINV( α, ). Table : paraeters B, α α, σ, σ, μ, μ, Z, Z, Table : paraeters ofα, μ, σ and,,, Z, α, α, σ, σ, μ, μ, Z, Z, Substtutng the above values nto (), we obtan A =43,93.9>0 and A =90,874.6>0. Thus, by Theore 3.3, pˆ n (4) and bˆ n (5) are the equlbru sales prce and bandwdth allocaton. The equlbru sales prces are ˆp, = , ˆp, =444.0, ˆp, = and ˆp, = The bandwdth allocatons are ˆb, =48,59, ˆb, =57,539, ˆb, =43,895 and ˆb, = 65,3.The sales revenues for servce provder = s 53,747,889.. The sales revenues for servce provder = s 45,847, The followng wll provde soe nsght nto the pact of dfferent nput characterstcs on the equlbru sales prces and bandwdth allocatons. We refer to the data set used n the above exaple as the basc paraeter set. We nvestgate the pact

10 74 P.S. You, C.C. Lee, Y.C. Hseh/ Bandwdth allocaton and prcng proble of the change of h, β and α on the equlbru values of pˆ and bˆ. In the followng cases, only one paraeter changes, whle others rean unchanged. The coputatonal results are descrbed n Table 3 to Table 8. Table 3 shows that the values of ˆp and ˆp slghtly decrease n the value of h, and the values of ˆb and ˆb slghtly ncrease n the value of h. Ths phenoenon could be explaned as follows. Note fro the deand functon n (0) that the deand for coodty of servce provder = ncreases n the value of h when the sales prce of ˆp s lower than ts rval s sales prce ˆp. In ths envronent, as the value of h ncreases, the pact of the dfference between ˆp and ˆp on deand becoes extensve. In case of ncreasng deand, servce provder = ay decrease ts sales prce to spur ore arket deand. On the other hand, the rval ay decrease ts sales prce to keep ts arket share. Thus, the values of ˆp and ˆp decrease n the value of h. In addton, snce both servce provders ay try to sell ore, they provde ore bandwdth unts for coodty. Thus, the values of ˆp and ˆp ncrease n the value of h. Table 3 The pact of h on equlbru prce and bandwdth allocaton h ˆp ˆp ˆp ˆp ˆb ˆb ˆb ˆb Table 4 shows that the values of ˆp and ˆp slghtly decrease n the value of h, and the values of ˆb and ˆb slghtly ncrease n the value of h. The explanaton of ths Table can be llustrated n a slar way to that of Table 3. Table 4 The pact of h on equlbru prce and bandwdth allocaton h ˆp ˆp ˆp ˆp ˆb ˆb ˆb ˆb Table 5 shows that the value of ˆp slghtly decreases n the value of β, and the value of ˆp slghtly ncreases n the value of β. Ths phenoenon could be explaned as follows. It s noted that the deand for coodty of servce provder =

11 P.S. You, C.C. Lee, Y.C. Hseh/ Bandwdth allocaton and prcng proble 75 decreases n the value of β when the sales prce of ˆp s hgher than ˆp. As the value of β ncreases, servce provder = ay try to lessen the dfference between ˆp and ˆp to avod the decrease n deand. Thus, servce provder = decreases the value of ˆp, and ncreases the value of ˆp as β ncreases. Table 6 shows that the value of ˆp slghtly decreases n the value of β, and the value of ˆp slghtly ncreases n the value of β. The explanaton of ths Table s slar to that of Table 5. Table 7 shows that the value of ˆb ncreases n the value of α. Ths phenoenon could be explaned as follows. It s noted that the bandwdth requred ncreases as the servce level ncreases. Thus, servce provder = ncreases the bandwdth allocaton to coodty when t expects to have a hgher servce level. Table 8 shows that the value of ˆb ncreases n the value of α. The explanaton of ths Table s slar to that of Table 7. Table 5 The pact of β on equlbru prce and bandwdth allocaton β ˆp ˆp ˆp ˆp ˆb ˆb ˆb ˆb Table 6 The pact of β on equlbru prce and bandwdth allocaton β ˆp ˆp ˆp ˆp ˆb ˆb ˆb ˆb Table 7 The pact of α on equlbru prce and bandwdth allocaton α ˆp ˆp ˆp ˆp ˆb ˆb ˆb ˆb

12 76 P.S. You, C.C. Lee, Y.C. Hseh/ Bandwdth allocaton and prcng proble Table 8 The pact of α on equlbru prce and bandwdth allocaton α ˆp ˆp ˆp ˆp ˆb ˆb ˆb ˆb CONCLUSIONS Ths research dscussed the proble of bandwdth allocaton and prcng n an Internet duopoly arket. The bandwdth coodtes servce level was taken nto consderaton. In order to fnd a soluton to the proble, ths research developed a Lagrange functon to develop the optal sales prce and bandwdth allocaton for both Internet servce provders. The results were used to establsh the reacton functons. Under certan condtons, ths paper proposed closed for forulas for the arket equlbru prce and bandwdth resource allocaton. A nuercal analyss s also provded to llustrate the pact of dfferent nput paraeters on the equlbru sales prces and bandwdth allocatons. The results are suarzed as follows:. For both servce provders, the equlbru sales prce of a coodty decreases, and the bandwdth allocaton of a coodty ncreases n the degree of substtutablty between coodtes of the two servce provders (Tables 3 and 4).. The equlbru sales prce of coodty- of servce provder- and the bandwdth allocaton of coodty- of servce provder- decrease, and the equlbru sales prce of coodty- of servce provder- and the bandwdth allocaton of coodty- of servce provder- ncrease n the degree of substtutablty between the coodtes offered by servce provder- (Table 5). 3. The equlbru sales prce of coodty- of servce provder- and the bandwdth allocaton of coodty- of servce provder- decrease, and the equlbru sales prce of coodty- of servce provder- and the bandwdth allocaton of coodty- of servce provder- ncrease n the degree of substtutablty between the coodtes offered by servce provder- (Table 6). 4. The bandwdth allocaton of a coodty- for a servce provder ncreases n the ntercept of deand functon for that coodty for that servce provder- (Tables 7 and 8). Appendx Settng the values of η, y and y n the equaton syste of (5) to (0) at η = 0, y 0 and y 0 and lettng the be equal to zero, we have = =

13 P.S. You, C.C. Lee, Y.C. Hseh/ Bandwdth allocaton and prcng proble 77 p p = c p + β p + a + h p + λ g c λ g β = 0 (7) = λ = 0 (8) η = b b+ B x = 0 (9) = g ( c + β ) p g h p βg ( p + p) λ (30) g a + b = 0 x y = η y = 0 (3) = λ y = 0 (3) Solvng the above equaton syste, we have λ = 0 for all and p, β h = p + h c ( c p c + a c β ) + β a (33) = g ( a + h p ) (34) b x = A. (35) Snce the values of p, b, λ, η, y b are optal solutons for servce provder. REFERENCES and x are no less than zero, p and [] Basar, T., and Srkant, R., Stackelberg gaes and ultple equlbru behavors on network, Journal of Optzaton Theory and Applcatons, 5 (3) (00) [] Bapna, R., Goes, P., and Gupta, A., Prcng and allocaton for qualty-dfferentated onlne servces, Manageent Scence, 5 (7) (005) [3] Cho, S.C., Prce copetton n a duopoly coon retaler channel, Journal of Retalng, 7 () (996) [4] Da, Y., Chao, X., Fang, S.C., and Henry, L.W., Prcng n revenue anageent for ultple frs copetng for custoers, Internatonal Journal of Producton Econocs, 98() (005) -6. [5] Elberfeld, W., and Wolfstetter, E., A dynac odel of Bertrand copetton wth entry, Internatonal Journal of Industral Organzaton, 7 (999) 53-5.

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