The Spatial Equilibrium Monopoly Models of the Steamcoal Market

Transcription

1 Advances n Management & Appled Economcs, vol.2, no.3, 2012, ISSN: (prnt verson), (onlne) Scenpress Ltd, 2012 The Spatal Equlbrum Monopoly Models of the Steamcoal Maret Hu Wen Cheng 1 and Chn We Yang 2 Abstract Ths paper replcates two spatal monopoly models by Taayama and Judge. The frst one s a smple spatal monopoly model. The second model allows the actvty of arbtrage between any pans of consumpton regons f the prce dfferental exceeds the correspondng unt transportaton cost. Wth the addtonal constrants, the proft level must decrease unless the constrants are all redundant. The smulaton of the Appalachan steamcoal maret ndcates that both models perform poorly ether n terms of flow varables or n the case of consumpton and producton levels. Ths mples that the steamcoal maret n our model s far from beng ether of the spatal monopoly models. The Appalachan steamcoal maret, characterzed by numerous coal mnes as well as utlty companes, smply cannot be modeled by the spatal monopoly models. JEL classfcaton numbers: L12, L71 Keywords: spatal model, monopoly, arbtrage, steamcoal maret 1 Department of Internatonal Busness, Mng Chuan Unversty, e-mal: hwcheng@mal.mcu.edu.tw 2 Department of Economcs, Claron Unversty of Pennsylvana, Department of Economcs, Natonal Chung Cheng Unversty, e-mal: yang@claron.edu Artcle Info: Receved : May 18, Revsed : June 11, 2012 Publshed onlne : August 31, 2012

2 126 The spatal monopoly of the steamcoal maret 1 Introducton The applcatons of the spatal equlbrum model have spanned over nearly three decades from Fox s free-lvestoc model [1] to the nternatonal alumnum model of Newcomb et al. [2]. However, they were n general formulated based on the assumpton of a compettve maret: a maret structure represents one extreme of a spectrum of varous marets. On the other hand, the formulaton of the spatal equlbrum monopoly model s due to Taayama and Judge ([3], Chapter 11) over twenty years ago; but the emprcal applcatons of ther model s at best scanty or none. The purpose of ths paper s to mplement the spatal equlbrum monopoly models to the Appalachan Steamcoal maret usng the parameters estmated by Yang [4], Labys and Yang [5], and Irwn and Yang [6]. The next secton dscusses the smple spatal equlbrum monopoly model; then a precautonary monopoly model wth the arbtrage actvtes. Last we evaluate the performance of the steamcoal maret based on the compettve and two spatal monopoly models. 2 A Smple Spatal Equlbrum Monopoly Model Followng the wor by Taayama and Judge [3], we mae four ey assumptons: (1) the product produced by the monopolst s homogeneous; (2) the producton and maret power s controlled by a sole monopolst or central authorty; (3) the nput marets (e.g., labor, raw materal) are compettve; and (4) the transportaton ndustry plays a passve role n the model,.e., unt transportaton costs are constant. Wthn ths framewor, the obectve functon of the spatal monopolst s to maxmze total proft from sales over n regons. Maxmze (1) ay by cx dx tx 2 2 Subect to x y 0, M ; (2) x x 0, N ; (3) y 0, x 0, x 0, ( ) M N. (4) where P a by s the demand prce of regon, 2 TC Cx dx s the total cost functon for regon, M, N = set of postve ntegers, M N = the Cartesan product of M and N, t = unt transportaton cost from regon to, x = commodty shpment from regon to, x = producton level of regon,

3 Hu Wen Cheng and Chn We Yang 127 y = consumpton level of regon. The formulaton s almost dentcal to that of the compettve equlbrum except that the coeffcent of 12 s mssng n both quadratc terms ndcatng the spatal monopolst can choose the best pont on each of the regonal demand functon gven hs cost structures n each supply regon n order to maxmze total proft. The frst-order condton of the lagrangan equatons can be expressed as: 2 2 L ay by cx dx tx ( x y ) ( x x ) ( L y) ay 2by 0, N ; (6) ( L x) C 2dx 0, M ; (7) ( L x ) t 0, ( ) M N ; (8) ( L ) x y 0, M ; (9) ( L ) x x 0, N. (10) where = mputed margnal revenue of regon, = mputed margnal cost of regon. Equaton (6) mples that optmum margnal revenue s less than or equal to the mputed margnal revenue of regon ; equaton (7) suggests that optmum margnal cost s greater than mputed margnal cost of regon ; and equaton (8) states that the dfference between mputed margnal prce and cost s no less than the correspondng transportaton cost. Note that all the decson varables are evaluated at optmum values and a strct equalty sgn holds for all postve values of decson varables. 3 A Precautonary Spatal Equlbrum Monopoly Model The smple spatal equlbrum monopoly model from the prevous secton gnores the possblty of resale between two regons,.e., the role of an arbtrager s not consdered. Ths assumpton s perhaps unrealstcally restrctve. To tae nto consderaton the possblty of arbtrage, we add the followng constrants : P P a by a b y t,. (11) Equaton (11) states that the optmum demand prces between demand regon and demand regon cannot exceed ts correspondng unt transportaton costs,.e., the arbtrage actvty between regon and are consdered n the feasblty regon. Addng equaton (11) to the smple spatal equlbrum monopoly model of equatons (1) through (4), we have the followng Lagrange functon:

4 128 The spatal monopoly of the steamcoal maret L ( t a b y a b y ), (12) where s the lagrange multpler of the arbtrage constrant between demand regon and. Note that f equaton (11) does not hold from the optmum soluton set of the smple spatal equlbrum monopoly model, such a constrant of the arbtrage actvty s not needed. We smply add numbers of the nc 2 constrants to equatons (1) through (4). The frst-order condtons are the same as that of the smple spatal equlbrum monopoly model except for the followng: ( y ) a 2b y b 0 t a by a by, ; (13) ( ) 0, ; (14) where varables are evaluated at optmum values; and the sgn of - s reversed for P P. Whle equaton (14) s the restatement of the arbtrage constrant, the nterpretaton of equaton (13) s not the same as equaton (6). It states that at optmalty, the margnal revenue of the th demand maret s equal to or less than the mputed margnal revenue plus the change n the dscrmnaton rent (see [3], p. 221). The equalty sgn holds for a postve consumpton level of regon. 4 An Evaluaton of the Spatal Equlbrum Monopoly Models Based on the estmated parameters by Labys and Yang [5], we assume that, perhaps to the contrary of the true maret structure of the Appalachan steamcoal, the producton process s controlled by one sngle suppler. The margnal cost functon n each regon reflects the smoothed step functons much le that of the world copper ndustry [7]. The margnal cost functons of seven supply regons by Yang [4] are lnearzed step functons. Under ths specfcaton, we smulated the smple spatal monopoly model wth the results reported n Table 1. Smlarly, the smulaton results of the precautonary monopolst model s agan shown n Table 1. The latter results are derved from the smple spatal monopoly assumpton wth 21 addtonal resalng possbltes (see Table 2) whch consders all the actvtes of arbtrage n the Appalachan maret. The unt transportaton costs between a par of demand regons are approxmated by the dfference of the observed demand prces. Of the 22 observed coal shpments, the smple spatal monopoly model predcts 12 flows whle the precautonary monopoly model predcts only 10 flows.

5 Hu Wen Cheng and Chn We Yang 129 Table 1: Optmum solutons under the compettve and monopoly maret structures 1 PA-MD 2 OH 3 Northern WV 4 Southern WV 5 VA 6 East. KY- TN 7 AL Total Demand Plant Prce New England (0.042) [0.023] {0.03} (0.042) [0.023] {0.03} (50.574) [59.157] {55.825} Md Atlantc (1.025) [0.597] {0.617} (0.195) [0.043] {0.37} (1.220) [0.639] {0.987} (46.074) [65.294] {53.8} IN-MI (0.304) [0.012] {0.172} (0.336) [0.354] {0.033} (0.640) [0.367] {0.205} (44.374) [49.976] {53.297} North Central (0.053) [0.05] (0.053) [0.05] {0} (47.574) [47.696] {49.7} Oho Valley (0.708) [0.547] {0.407} (0.035) (0.628) [0.214] {0.352} (1.37) [0.761] {0.759} (43.874) [51.627] {51.653} South Central (0.081) [0.057] (0.138) [0.107] {0.107} (0.219) [0.164] {0.107} (43.474) [44.482] {45.496} South Atlantc (0.138) [0.103] {0.156} (0.653) [0.336] {0.274} (0.791) [0.439] {0.43} (44.074) [51.885] {52.084} Total Supply (1.025) [0.597] {0.617} (1.012) [0.559] {0.579} (0.531) [0.397] {0.403} (0.035) [0] {0} (0.261) [0.183] {0.186} (1.334) [0.6] {0.621} (0.138) [0.107] {0.107} Mne 6 Prce cents 10 Btu or Margnal Cost (31.574) [29.663] {29.752} (30.574) [28.659] {28.744} (32.574) [30.661] {30.746} (31.274) [30.4] {30.4} (29.374) [27.456] {27.529} (30.474) [28.562] {28.629} (36.474) [39.579] {34.579} Sources: Mneral Year Boo (1974), Steam-Electrc Plant Factors (1974), Energy

6 130 The spatal monopoly of the steamcoal maret Economcs Vol.2 (Aprl 1980), Compettve Transportaton Costs of Supplyng Low- Sulfur Fuels to Md-western and Eastern Domestc Energy Maret (1972). Fgures n ( ) are optmal values of the compettve soluton; fgures n ( ) and { } are optmum values under the smple monopoly and precautonary monopoly respectvely. In order to evaluate the predctve power n terms of the mean percentage of absolute error, we calculate the followng ndex: I1 ( y y y), (15) I2 ( x x x), (16) I3 ( x x x ), (17) where barred varables denote optmum values; unbarred varables represent actual values. A calculaton reveals that the values of I 1 are 51.97% and 54.77% for the smple spatal monopoly and precautonary monopoly models respectvely. It mples that the former explans 48.03% and the latter explans 45.23% of the consumpton actvtes of the Appalachan steamcoal maret. Smlarly, the two spatal monopoly models explan only 43.88% (or I ) and 44.93% ( I ) of the coal producton actvtes. Fnally, of the 10 maor coal shpments,.e., x12, x23, x25, x32, x35, x57, x65, x66, x67 and x 76, the smple spatal monopoly model explans 33.45% ( I ) whle the precautonary monopoly model explans 42.8% ( I ) of the maor coal shpment actvtes. A perusal of Table 1 ndcates mmedately all the delvered peces under both spatal monopoly prces are hgher than that under the compettve spatal equlbrum prces. Ths comes as no surprse snce the obectve functon n both cases s to maxmze the monopoly proft whereas the obectve of the compettve spatal equlbrum model s consstent wth zero proft condton. The margnal costs of the mouth of mnes are all lower than that under the compettve spatal equlbrum level snce the producton levels at all regons are much lower than that of the compettve level. Hence, the spatal monopoly models would produce much hgher demand prces and lower output levels.

7 Hu Wen Cheng and Chn We Yang 131 Table 2: Ant-arbtrage constratnts optmum demand prce (hgher prce) optmum demand prce (lower prce) unt transportaton cost

8 132 The spatal monopoly of the steamcoal maret 5 Concludng Remars In ths paper, we replcate two spatal monopoly models by Taayama and Judge. The frst one s a smple spatal monopoly model, whch s no longer consstent wth the zero proft condton. The second model allows the actvty of arbtrage between any pans of consumpton regons f the prce dfferental exceeds the correspondng unt transportaton cost. Wth the addtonal constrants, the proft level must decrease unless the constrants are all redundant. The smulaton of the Appalachan steamcoal maret ndcates that both models perform poorly ether n terms of flow varables or n the case of consumpton and producton levels. Ths mples that the steamcoal maret n our model s far from beng ether of the spatal monopoly models. The Appalachan steamcoal maret, characterzed by numerous coal mnes as well as utlty companes, smply cannot be modeled by the spatal monopoly models. References [1] K.A. Fox, A spatal equlbrum model of the lvestoc feed economy, Econometrca, 21(4), (1953), [2] R.T. Newcomb, S.S. Reynolds and T.A. Musbruch, Changng Patterns of Investment Decson n World Alumnum, Resource and Energy, 11(3), (1990), [3] T. Taayama and G. Judge, Spatal and Temporal Prce and Allocaton Model, North-Holland Publshng Company press, [4] C.W. Yang, A Crtcal Analyss of Spatal Commodty Modelng: The Case for Coal, Ph. D. Dssertaton, Department of Economcs, West Vrgna Unversty, [5] W.C. Labys and C.W. Yang, A Quadratc Programmng Model of the Appalachan Steam Coal Maret, Energy Economcs, 2(2), (1980), [6] C.L. Irwn and C.W. Yang, Iteraton and Senstvty for a Spatal Equlbrum Problem wth Lnear Supply and Demand Functons, Operatons Research, 30(2), (1982), [7] R.S. Pndyc and P.L. Rubnfeld, Mcroeconomcs, Second edton, Macmllan Publshng Company press, 1992.