(*^ ::[ frontEndVersion = "Microsoft Windows Mathematica Notebook Front End Version 2.2"; microsoftWindowsStandardFontEncoding; fontset = title, "Arial", 24, L0, center, nohscroll, bold; fontset = subtitle, "Arial", 18, L0, center, nohscroll, bold; fontset = subsubtitle, "Arial", 14, L0, center, nohscroll, bold; fontset = section, "Arial", 14, L0, bold, grayBox; fontset = subsection, "Arial", 12, L0, bold, blackBox; fontset = subsubsection, "Arial", 10, L0, bold, whiteBox; fontset = text, "Arial", 12, L0; fontset = smalltext, "Arial", 10, L0; fontset = input, "Courier New", 12, L0, nowordwrap, bold; fontset = output, "Courier New", 12, L0, nowordwrap; fontset = message, "Courier New", 10, L0, nowordwrap, R65280; fontset = print, "Courier New", 10, L0, nowordwrap; fontset = info, "Courier New", 10, L0, nowordwrap; fontset = postscript, "Courier New", 8, L0, nowordwrap; fontset = name, "Arial", 10, L0, nohscroll, italic, B65280; fontset = header, "Times New Roman", 10, L0, right, nohscroll; fontset = footer, "Times New Roman", 10, L0, right, nohscroll; fontset = help, "Arial", 10, L0, nohscroll; fontset = clipboard, "Arial", 12, L0, nohscroll; fontset = completions, "Arial", 12, L0, nowordwrap, nohscroll; fontset = graphics, "Courier New", 10, L0, nowordwrap, nohscroll; fontset = special1, "Arial", 12, L0, nowordwrap, nohscroll; fontset = special2, "Arial", 12, L0, center, nowordwrap, nohscroll; fontset = special3, "Arial", 12, L0, right, nowordwrap, nohscroll; fontset = special4, "Arial", 12, L0, nowordwrap, nohscroll; fontset = special5, "Arial", 12, L0, nowordwrap, nohscroll; fontset = leftheader, "Arial", 12, L0, nowordwrap, nohscroll; fontset = leftfooter, "Arial", 12, L0, nowordwrap, nohscroll; fontset = reserved1, "Courier New", 10, L0, nowordwrap, nohscroll;] :[font = input; nowordwrap; ] (* The Differentiation Triangle by Marcel Canoy and Martin Peitz The programs below are written for Mathematica for Windows and should work under all versions. *) :[font = text; startGroup; ] (* The first part determines the equilibrium candidate in the case of one two-product and one single-product firm. In the text this corresponds to scenarios S1 and S2 *) i = .; x = .; y =.; z = .; c = .; xx = .;yy = .;zz = .; sh1 = .; sh2 = .; sh3 = .; pri = .; pre = .; lamL = .; lamH1 = .; lamH2 = . ; co = 0; (* Note that delta^m = (1 / 2 (y - x)) / ( 1 / 2 (x + y) - z) delta^b = (1 / 2) (y - x) *) (* p_H1 is denoted by x, p_H2 by y and p_L by z *) lamH1 = 1 / 2 + Log[ 2 / 3 (1 - (1 / 2 (y - x)) / ( 1 / 2 (x + y) - z))] (x - z) + (1 / 2) (y - x) (1 + Log[((1 / 2 (y - x)) / ( 1 / 2 (x + y) - z)) / ((1 / 2) (y - x)) ] ); lamH2 = 1 / 2 + Log[ 2 / 3 (1 + (1 / 2 (y - x)) / ( 1 / 2 (x + y) - z))] (y - z) - (1 / 2) (y - x) (1 + Log[((1 / 2 (y - x)) / ( 1 / 2 (x + y) - z)) / ((1 / 2) (y - x)) ] ); lamL = 1 - (1 / 2 + Log[ 2 / 3 (1 - (1 / 2 (y - x)) / ( 1 / 2 (x + y) - z))] (x - z) + (1 / 2) (y - x) (1 + Log[((1 / 2 (y - x)) / ( 1 / 2 (x + y) - z)) / ((1 / 2) (y - x)) ] )) - (1 / 2 + Log[ 2 / 3 (1 + (1 / 2 (y - x)) / ( 1 / 2 (x + y) - z))] (y - z) - (1 / 2) (y - x) (1 + Log[((1 / 2 (y - x)) / ( 1 / 2 (x + y) - z)) / ((1 / 2) (y - x)) ] )); For [i = 1, i < 21, i++, (* Here we determine the equilibrium candidate for marginal coosts c from 0 to 0.19 in steps of 0.01 *) c:= co + ( i - 1) * 0.01; (* step size *) k[x_, y_, z_] := lamL + z Log[ 4 / 9 (1 + (1 / 2 (y - x)) / ( 1 / 2 (x + y) - z)) (1 - (1 / 2 (y - x)) / ( 1 / 2 (x + y) - z))] + (x - c) Log[3 / (2 (1 - (1 / 2 (y - x)) / ( 1 / 2 (x + y) - z))) ]; g[x_, y_, z_] := lamH1 + (x -c) Log[(2 / 3) (1 - (1 / 2 (y - x)) / ( 1 / 2 (x + y) - z)) Sqrt[(x + y - 2 z) / 2] ] + z Log[3 / (2 (1 - (1 / 2 (y - x)) / ( 1 / 2 (x + y) - z))) ]; h[x_, y_, z_] := lamH2 + (y - c) Log[(2 / 3) (1 + (1 / 2 (y - x)) / ( 1 / 2 (x + y) - z)) Sqrt[(x + y - 2 z) / 2 ] ]; S = FindRoot[{k[x, y, z] == 0, g[x, y, z] == 0, h[x, y, z] == 0}, {x, 2.1 c + 0.2}, {y, 1.9 c + 0.1}, {z, 1.6 c}]; (* in this version we fixed the starting point of the algorithm (Newton's method). Varying the starting point we did not find any other admissible solution. For the secant method, one has to give each variable two starting points, e.g. {x, 0.1,0.4} *) xx = x /. S; yy = y /. S; zz = z /. S; sh1 = 1 / 2 + Log[ 2 / 3 (1 - (1 / 2 (yy - xx)) / ( 1 / 2 (xx + yy) - zz))] (xx - zz) + (1 / 2) (yy - xx) (1 + Log[((1 / 2 (yy - xx)) / ( 1 / 2 (xx + yy) - zz)) / ((1 / 2) (yy - xx)) ] ); sh2 = 1 / 2 + Log[ 2 / 3 (1 + (1 / 2 (yy - xx)) / ( 1 / 2 (xx + yy) - zz))] (yy - zz) - (1 / 2) (yy - xx) (1 + Log[((1 / 2 (yy - xx)) / ( 1 / 2 (xx + yy) - zz)) / ((1 / 2) (yy - xx)) ] ); sh3 = 1 - (1 / 2 + Log[ 2 / 3 (1 - (1 / 2 (yy - xx)) / ( 1 / 2 (xx + yy) - zz))] (xx - zz) + (1 / 2) (yy - xx) (1 + Log[((1 / 2 (yy - xx)) / ( 1 / 2 (xx + yy) - zz)) / ((1 / 2) (yy - xx)) ] )) - (1 / 2 + Log[ 2 / 3 (1 + (1 / 2 (yy - xx)) / ( 1 / 2 (xx + yy) - zz))] (yy - zz) - (1 / 2) (yy - xx) (1 + Log[((1 / 2 (yy - xx)) / ( 1 / 2 (xx + yy) - zz)) / ((1 / 2) (yy - xx)) ] )); (* sh stands for market share of the corresponding product *) pri = (xx - c) sh1 + zz sh3; pre = (yy - c ) sh2; cc[i] = c; xxx[i] = xx; yyy[i] = yy; zzz[i] = zz; fsha1[i] = sh1; fsha2[i] = sh2; fsha3[i] = sh3; proi[i] = pri; proe[i] = pre ] Table[{cc[i],xxx[i],yyy[i],zzz[i],fsha1[i],fsha2[i],fsha3[i],proi[i],proe[i]},{i,1,20}] (* marginal costs (1), prices (2-4), market shares (5-7), profits (8,9) *) (* For a longer table the number above (20) has to be increased accordingly *) :[font = output; inactive; formatted; output; endGroup; nowordwrap; ] {{0, 0.3429224049296811, 0.3024607020671747, 0.2068880043249172, 0.4029003115000465, 0.5067436171437511, 0.0903560713562025, 0.1568571310480382, 0.1532700302093585}, {0.01, 0.3587473918642208, 0.3168577145812109, 0.215808458620707, 0.3996829626856004, 0.5050021806604072, 0.0953148566539925, 0.1599581431073196, 0.1549638150159803}, {0.02, 0.3744695437344864, 0.3311038407942388, 0.2245723517879636, 0.3964222432288258, 0.503289431198965, 0.1002883255722091, 0.1630415968141515, 0.1565752750771458}, {0.03, 0.390090814328838, 0.3452062065301833, 0.233192318321021, 0.393128921531869, 0.5015999611725455, 0.1052711172955855, 0.1661105293850303, 0.1581074209568853}, {0.04, 0.4056143053896099, 0.3591719425993121, 0.241679637561724, 0.3898113466563605, 0.4999294860965503, 0.1102591672470892, 0.1691680003188877, 0.1595634652401118}, {0.05, 0.421043779057925, 0.3730079692028103, 0.2500443559777298, 0.3864760337113071, 0.4982745770431636, 0.1152493892455293, 0.1722169873742865, 0.1609466592361015}, {0.06, 0.4363833353026078, 0.3867208752799702, 0.2582954175754369, 0.3831280984900175, 0.4966324630525679, 0.1202394384574146, 0.1752603275232127, 0.1622601930209824}, {0.07000000000000001, 0.4516372007393046, 0.4003168580936451, 0.2664407904120139, 0.3797715816778318, 0.4950008840057442, 0.1252275343164241, 0.1783006865764815, 0.1635071367583543}, {0.08, 0.4668095924587951, 0.4138017003970282, 0.2744875834526144, 0.3764096921246571, 0.4933779801020518, 0.1302123277732912, 0.1813405467945095, 0.164690408696516}, {0.09, 0.4819046322735792, 0.4271807705151192, 0.2824421515342925, 0.3730449906046452, 0.4917622080771073, 0.1351928013182475, 0.1843822055406883, 0.1658127602296554}, {0.1, 0.4969262947765143, 0.4404590359074065, 0.2903101880916743, 0.369679529595039, 0.4901522771150589, 0.1401681932899021, 0.1874277804953452, 0.1668767717144128}, {0.11, 0.5118783779918084, 0.4536410841934579, 0.2980968062590393, 0.3663149603570788, 0.4885470993873128, 0.1451379402556084, 0.1904792185596485, 0.1678848549130252}, {0.12, 0.5267644890266979, 0.4667311478382945, 0.3058066093964211, 0.3629526155462063, 0.4869457515486891, 0.1501016329051046, 0.1935383066271324, 0.1688392593694579}, {0.13, 0.5415880395803213, 0.4797331301302751, 0.3134437522311991, 0.3595935733823623, 0.4853474445131233, 0.1550589821045143, 0.1966066830821183, 0.1697420809703046}, {0.14, 0.5563522478222862, 0.4926506310117669, 0.3210119937983134, 0.3562387078277722, 0.483751499538567, 0.1600097926336608, 0.1996858493259857, 0.1705952715651641}, {0.15, 0.5710601442771073, 0.5054869719198437, 0.328514743279574, 0.3528887280732811, 0.4821573291593186, 0.1649539427674003, 0.2027771809174866, 0.1714006489318055}, {0.16, 0.5857145801156671, 0.5182452191760472, 0.3359550997289255, 0.3495442097998977, 0.4805644218724163, 0.1698913683276861, 0.2058819380964375, 0.1721599066418941}, {0.17, 0.6003182367771215, 0.5309282057059955, 0.3433358865494469, 0.3462056200712454, 0.478972329753645, 0.1748220501751097, 0.2090012755766515, 0.1728746235608034}, {0.18, 0.6148736362020978, 0.5435385510202515, 0.3506596814713622, 0.3428733372623916, 0.4773806583766026, 0.1797460043610058, 0.2121362515670249, 0.1735462728313238}, {0.19, 0.6293831512029885, 0.55607867947933, 0.3579288426740173, 0.3395476670965971, 0.4757890585550125, 0.1846632743483905, 0.2152878360244401, 0.1741762302665326}} ;[o] {{0, 0.342922, 0.302461, 0.206888, 0.4029, 0.506744, 0.0903561, 0.156857, 0.15327}, {0.01, 0.358747, 0.316858, 0.215808, 0.399683, 0.505002, 0.0953149, 0.159958, 0.154964}, {0.02, 0.37447, 0.331104, 0.224572, 0.396422, 0.503289, 0.100288, 0.163042, 0.156575}, {0.03, 0.390091, 0.345206, 0.233192, 0.393129, 0.5016, 0.105271, 0.166111, 0.158107}, {0.04, 0.405614, 0.359172, 0.24168, 0.389811, 0.499929, 0.110259, 0.169168, 0.159563}, {0.05, 0.421044, 0.373008, 0.250044, 0.386476, 0.498275, 0.115249, 0.172217, 0.160947}, {0.06, 0.436383, 0.386721, 0.258295, 0.383128, 0.496632, 0.120239, 0.17526, 0.16226}, {0.07, 0.451637, 0.400317, 0.266441, 0.379772, 0.495001, 0.125228, 0.178301, 0.163507}, {0.08, 0.46681, 0.413802, 0.274488, 0.37641, 0.493378, 0.130212, 0.181341, 0.16469}, {0.09, 0.481905, 0.427181, 0.282442, 0.373045, 0.491762, 0.135193, 0.184382, 0.165813}, {0.1, 0.496926, 0.440459, 0.29031, 0.36968, 0.490152, 0.140168, 0.187428, 0.166877}, {0.11, 0.511878, 0.453641, 0.298097, 0.366315, 0.488547, 0.145138, 0.190479, 0.167885}, {0.12, 0.526764, 0.466731, 0.305807, 0.362953, 0.486946, 0.150102, 0.193538, 0.168839}, {0.13, 0.541588, 0.479733, 0.313444, 0.359594, 0.485347, 0.155059, 0.196607, 0.169742}, {0.14, 0.556352, 0.492651, 0.321012, 0.356239, 0.483751, 0.16001, 0.199686, 0.170595}, {0.15, 0.57106, 0.505487, 0.328515, 0.352889, 0.482157, 0.164954, 0.202777, 0.171401}, {0.16, 0.585715, 0.518245, 0.335955, 0.349544, 0.480564, 0.169891, 0.205882, 0.17216}, {0.17, 0.600318, 0.530928, 0.343336, 0.346206, 0.478972, 0.174822, 0.209001, 0.172875}, {0.18, 0.614874, 0.543539, 0.35066, 0.342873, 0.477381, 0.179746, 0.212136, 0.173546}, {0.19, 0.629383, 0.556079, 0.357929, 0.339548, 0.475789, 0.184663, 0.215288, 0.174176}} :[font = text; startGroup; backColorRed = 65280; backColorGreen = 65280; backColorBlue = 65280; fontColorRed = 0; fontColorGreen = 0; fontColorBlue = 0; plain; fontName = "Arial"; fontSize = 12; ] (* This part determines the equilibrium candidate for the scenarios with three single-product firms. This corresponds to scenarios S3 and S4. *) (* important: small cost asymmetry between H1 and H2 to be able to solve *) i = .; x = .; y =.; z = .; b = .; c = .; xx = .;yy = .;zz = .; sh1 = .; sh2 = .; sh3 = .; pri = .; pre = .; lamL = .; lamH1 = .; lamH2 = . ; co = 0.; b0 = 0.000000001; (* p_H1 is denoted by x, p_H2 by y and p_L by z *) lamH1 = 1 / 2 + Log[ 2 / 3 (1 - (1 / 2 (y - x)) / ( 1 / 2 (x + y) - z))] (x - z) + (1 / 2) (y - x) (1 + Log[((1 / 2 (y - x)) / ( 1 / 2 (x + y) - z)) / ((1 / 2) (y - x)) ] ); lamH2 = 1 / 2 + Log[ 2 / 3 (1 + (1 / 2 (y - x)) / ( 1 / 2 (x + y) - z))] (y - z) - (1 / 2) (y - x) (1 + Log[((1 / 2 (y - x)) / ( 1 / 2 (x + y) - z)) / ((1 / 2) (y - x)) ] ); lamL = 1 - (1 / 2 + Log[ 2 / 3 (1 - (1 / 2 (y - x)) / ( 1 / 2 (x + y) - z))] (x - z) + (1 / 2) (y - x) (1 + Log[((1 / 2 (y - x)) / ( 1 / 2 (x + y) - z)) / ((1 / 2) (y - x)) ] )) - (1 / 2 + Log[ 2 / 3 (1 + (1 / 2 (y - x)) / ( 1 / 2 (x + y) - z))] (y - z) - (1 / 2) (y - x) (1 + Log[((1 / 2 (y - x)) / ( 1 / 2 (x + y) - z)) / ((1 / 2) (y - x)) ] )); For [i = 1, i < 21, i++, c:= co + (i - 1) * 0.01; (* step size *) b:= b0 + (i - 1) * 0.01; k[x_, y_, z_] := lamL + z Log[ 4 / 9 (1 + (1 / 2 (y - x)) / ( 1 / 2 (x + y) - z)) (1 - (1 / 2 (y - x)) / ( 1 / 2 (x + y) - z))] ; g[x_, y_, z_] := lamH1 + (x - b) Log[(2 / 3) (1 - (1 / 2 (y - x)) / ( 1 / 2 (x + y) - z)) Sqrt[(x + y - 2 z) / 2] ]; h[x_, y_, z_] := lamH2 + (y - c) Log[(2 / 3) (1 + (1 / 2 (y - x)) / ( 1 / 2 (x + y) - z)) Sqrt[(x + y - 2 z) / 2 ] ]; S = FindRoot[{k[x, y, z] == 0, g[x, y, z] == 0, h[x, y, z] == 0}, {x, 1.5 c + 0.2}, {y, 1.4 c + 0.1}, {z, 1.0 c}]; (* see remark above on starting points *) xx = x /. S; yy = y /. S; zz = z /. S; sh1 = 1 / 2 + Log[ 2 / 3 (1 - (1 / 2 (yy - xx)) / ( 1 / 2 (xx + yy) - zz))] (xx - zz) + (1 / 2) (yy - xx) (1 + Log[((1 / 2 (yy - xx)) / ( 1 / 2 (xx + yy) - zz)) / ((1 / 2) (yy - xx)) ] ); sh2 = 1 / 2 + Log[ 2 / 3 (1 + (1 / 2 (yy - xx)) / ( 1 / 2 (xx + yy) - zz))] (yy - zz) - (1 / 2) (yy - xx) (1 + Log[((1 / 2 (yy - xx)) / ( 1 / 2 (xx + yy) - zz)) / ((1 / 2) (yy - xx)) ] ); sh3 = 1 - (1 / 2 + Log[ 2 / 3 (1 - (1 / 2 (yy - xx)) / ( 1 / 2 (xx + yy) - zz))] (xx - zz) + (1 / 2) (yy - xx) (1 + Log[((1 / 2 (yy - xx)) / ( 1 / 2 (xx + yy) - zz)) / ((1 / 2) (yy - xx)) ] )) - (1 / 2 + Log[ 2 / 3 (1 + (1 / 2 (yy - xx)) / ( 1 / 2 (xx + yy) - zz))] (yy - zz) - (1 / 2) (yy - xx) (1 + Log[((1 / 2 (yy - xx)) / ( 1 / 2 (xx + yy) - zz)) / ((1 / 2) (yy - xx)) ] )); pr1 = (xx - c) sh1; pr2 = (yy - c ) sh2; pr3 = zz sh3; the = 1 - 0.5 (xx + yy) + zz; cc[i] = c; bb[i] = b; xxx[i] = xx; yyy[i] = yy; zzz[i] = zz; fsha1[i] = sh1; fsha2[i] = sh2; fsha3[i] = sh3; pro1[i] = pr1; pro2[i] = pr2; pro3[i] = pr3; ] Table[{bb[i],cc[i],xxx[i],yyy[i],zzz[i],fsha1[i],fsha2[i],fsha3[i],pro1[i],pro2[i],pro3[i]},{i,1,20}] (* marginal costs (1,2), prices (3-5), market shares (6-8), profits (9-11) *) :[font = output; inactive; formatted; output; endGroup; nowordwrap; ] {{1.*10^-9, 0., 0.3319810890800416, 0.3319810882138184, 0.165990544323465, 0.4326966250475491, 0.4326966269543573, 0.1346067479980937, 0.1436470968245438, 0.1436470970827562, 0.02234344736981505}, {0.010000001, 0.01, 0.3448691784985117, 0.3448691776454338, 0.1724345890359864, 0.4300837897922469, 0.430083791637627, 0.1398324185701263, 0.1440218053732563, 0.1440218056243222, 0.02411194563004775}, {0.020000001, 0.02, 0.3576308394961616, 0.357630838654788, 0.1788154195377374, 0.4274965856909799, 0.4274965874804692, 0.1450068268285509, 0.1443360311085883, 0.1443360313530907, 0.02592945657518337}, {0.030000001, 0.03, 0.3702750472883801, 0.3702750464574889, 0.1851375234364673, 0.4249331931757923, 0.4249331949141178, 0.15013361191009, 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