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Bass Diffusion Question 1

Simulation Sales Cycle Q-Phone: 60 months Bass Diffusion Forecast Model for Q-Phone 1.0
Month Contagion (Network Effect) Q-Phone Prospects (Market Potential) Q-Phone Prabability to Buy Q-Phone Next Period Prospect Probability Fraction of Adopters Predicted Q-Phone Adopters Error Accumulative Sales
Given Input Data: t+1 q*Ft M(t+1) ft 1-ft Ft ft / 1- Ft Xt Nt
Market size M 1,000,000 0 - 0 1000000 0.01 0.99 0.00000000 0.01 1 0.000
Months after Q-Phone launch t 60 1 0.00199 990,058 0.0118692113 0.9881307887 0.00994200 0.0119884 9,942 9,942
Rate of adoption through innovation ip 0.01 2 0.00431 978,436 0.0140041588 0.9859958412 0.02156400 0.0143128 11,622 21,564
Rate of adoption through imitation (contagion) q 0.2 3 0.00706 964,701 0.0164576061 0.9835423939 0.03529900 0.0170598 13,735 35,299
Contagion (network effect) c 4 0.01023 948,854 0.0191945573 0.9808054427 0.05114600 0.0202292 15,847 51,146
Rate of adoption - Rate of innovation q-ip 0.19 5 0.01390 930,493 0.0222400854 0.9777599146 0.06950700 0.0239014 18,361 69,507
Month 0 sales Xt0 1 6 0.01803 909,861 0.0255014021 0.9744985979 0.09013900 0.0280278 20,632 90,139
Month 1 sales Xt1 9,942 7 0.02259 887,055 0.0289082354 0.9710917646 0.11294500 0.032589 22,806 112,945
Output Variables: 8 0.02774 861,298 0.032505731 0.967494269 0.13870200 0.0377404 25,757 138,702
Q-Phone single purchase adopters a 9 0.03332 833,397 0.0361032581 0.9638967419 0.16660300 0.0433206 27,901 166,603
Q-Phone single purchase prospects P 10 0.03931 803,426 0.0396207925 0.9603792075 0.19657400 0.0493148 29,971 196,574
In month t, the market potential with respect to number of prospects Mt 11 0.04568 771,617 0.0429610111 0.9570389889 0.22838300 0.0556766 31,809 228,383
In month t, the number of sales Xt 12 0.05229 738,558 0.0460035961 0.9539964039 0.26144200 0.0622884 33,059 261,442 Year 1 Accumulative Predictive Sales (Actual 260,000)
Net total accumulative sales Nt 13 0.05907 704,666 0.0486690257 0.9513309743 0.29533400 0.0690668 33,892 295,334
Single purchase adopter fraction in month t Ft 14 0.06583 670,840 0.0508711389 0.9491288611 0.32916000 0.075832 33,826 329,160 Month 14 Begin Sales Decline
Probability of prospect making a single purchase the following month ft 15 0.07259 637,033 0.0526147214 0.9473852786 0.36296700 0.0825934 33,807 362,967
16 0.07932 603,409 0.0538954057 0.9461045943 0.39659100 0.0893182 33,624 396,591
Given p<iq, AR peaks at T* 17 0.08579 571,029 0.0547012662 0.9452987338 0.42897100 0.0957942 32,380 428,971
0<=p,q<=1 18 0.09206 539,701 0.0550817761 0.9449182239 0.46029900 0.1020598 31,328 460,299
19 0.09801 509,946 0.0550796754 0.9449203246 0.49005400 0.1080108 29,755 490,054
20 0.10364 481,805 0.0547518384 0.9452481616 0.51819500 0.113639 28,141 518,195
1. We use the bass diffusion model theory because it is designed specifically for the type of forecasting that fits our Q-Phone business model. 21 0.10891 455,467 0.0541580324 0.9458419676 0.54453300 0.1189066 26,338 544,533
2. We use these parameters because the output data match real world observation within the acceptable error range. 22 0.11385 430,726 0.0533474826 0.9466525174 0.56927400 0.1238548 24,741 569,274
3. Given both the decision to use bass diffusion AND the specfiic parameters, we can forecast that... 23 0.11843 407,832 0.052379332 0.947620668 0.59216800 0.1284336 22,894 592,168
24 0.12278 386,124 0.0512676913 0.9487323087 0.61387600 0.1327752 21,708 613,876 Year 2 Accumulative Predictive Sales (Actual > 615,000)
Bass model published 1969 by Dr. Frank Bass, used for product sales forecasts. 25 0.12672 366,398 0.0500940811 0.9499059189 0.63360200 0.1367204 19,726 633,602
Produces product lifecylcle curve (introduction, growth, mature, decline Ft>+0.80) 26 0.13043 347,866 0.0488497092 0.9511502908 0.65213400 0.1404268 18,532 652,134
We use Bass Diffusion to find the Adoption Rate (AR), from both innovation and imitation, applied to the market potential for a given month. 27 0.13384 330,821 0.0475839032 0.9524160968 0.66917900 0.1438358 17,045 669,179
use error or confidence interval? 28 0.13697 315,174 0.0463196099 0.9536803901 0.68482600 0.1469652 15,647 684,826
29 0.13990 300,515 0.045046297 0.954953703 0.69948500 0.149897 14,659 699,485
30 0.14260 286,990 0.043795248 0.956204752 0.71301000 0.152602 13,525 713,010
31 0.14514 274,308 0.0425557042 0.9574442958 0.72569200 0.1551384 12,682 725,692
32 0.14745 262,749 0.0413698826 0.9586301174 0.73725100 0.1574502 11,559 737,251
33 0.14965 251,774 0.0401945106 0.9598054894 0.74822600 0.1596452 10,975 748,226
34 0.15166 241,692 0.0390723154 0.9609276846 0.75830800 0.1616616 10,082 758,308
35 0.15358 232,110 0.0379680896 0.9620319104 0.76789000 0.163578 9,582 767,890
36 0.15539 223,056 0.0368909642 0.9631090358 0.77694400 0.1653888 9,054 776,944
37 0.15703 214,850 0.0358863955 0.9641136045 0.78515000 0.16703 8,206 785,150
38 0.15856 207,196 0.0349251235 0.9650748765 0.79280400 0.1685608 7,654 792,804
39 0.15999 200,053 0.0340068894 0.9659931106 0.79994700 0.1699894 7,143 799,947
40 0.16134 193,298 0.0331197566 0.9668802434 0.80670200 0.1713404 6,755 806,702
41 0.16263 186,874 0.0322591616 0.9677408384 0.81312600 0.1726252 6,424 813,126
42 0.16382 180,901 0.0314441756 0.9685558244 0.81909900 0.1738198 5,973 819,099
43 0.16495 175,266 0.0306622258 0.9693377742 0.82473400 0.1749468 5,635 824,734
44 0.16603 169,831 0.0298959963 0.9701040037 0.83016900 0.1760338 5,435 830,169
45 0.16704 164,812 0.0291779209 0.9708220791 0.83518800 0.1770376 5,019 835,188
46 0.16800 160,005 0.02848073 0.97151927 0.83999500 0.177999 4,807 839,995
47 0.16891 155,454 0.0278121508 0.9721878492 0.84454600 0.1789092 4,551 844,546
48 0.16977 151,148 0.0271719364 0.9728280636 0.84885200 0.1797704 4,306 848,852
49 0.17058 147,093 0.0265622599 0.9734377401 0.85290700 0.1805814 4,055 852,907
50 0.17136 143,193 0.025969683 0.974030317 0.85680700 0.1813614 3,900 856,807
51 0.17211 139,438 0.0253933888 0.9746066112 0.86056200 0.1821124 3,755 860,562
52 0.17282 135,914 0.0248474169 0.9751525831 0.86408600 0.1828172 3,524 864,086
53 0.17351 132,450 0.0243058995 0.9756941005 0.86755000 0.18351 3,464 867,550
54 0.17416 129,181 0.0237904638 0.9762095362 0.87081900 0.1841638 3,269 870,819
55 0.17477 126,143 0.0233076187 0.9766923813 0.87385700 0.1847714 3,038 873,857
56 0.17535 123,227 0.0228406913 0.9771593087 0.87677300 0.1853546 2,916 876,773
57 0.17591 120,453 0.022393345 0.977606655 0.87954700 0.1859094 2,774 879,547
58 0.17646 117,689 0.0219445499 0.9780554501 0.88231100 0.1864622 2,764 882,311
59 0.17698 115,111 0.0215232015 0.9784767985 0.88488900 0.1869778 2,578 884,889
60 0.17747 112,641 0.021117011 0.978882989 0.88735900 0.1874718 2,470 887,359
by: J. Cervantes

Q-Phone 1.0 Sales (Xt)

Q-Phone Adopters Xt

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 1 9942 11622 13735 15847 18361 20632 22806 25757 27901 29971 31809 33059 33892 33826 33807 33624 32380 31328 29755 28141 26338 24741 22894 21708 19726 18532 17045 15647 14659 13525 12682 11559 10975 10082 9582 9054 8206 7654 7143 6755 6424 5973 5635 5435 5019 4807 4551 4306 4055 3900 3755 3524 3464 3269 3038 2916 2774 2764 2578 2470

Months since Launch Q-Phone 1.0

Quantity

Q-Phone 1.0 Accumulative Sales (Nt)

Accumulative Sales Nt

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 0 9942 21564 35299 51146 69507 90139 112945 138702 166603 196574 228383 261442 295 334 329160 362967 396591 428971 460299 490054 518195 544533 569274 592168 613876 633602 652134 669179 684826 699485 713010 725692 737251 748226 758308 767890 776944 785150 792804 799947 806702 813126 819099 824734 830169 835188 839995 844546 848852 852907 856807 860562 864086 867550 870819 873857 876773 879547 882311 884889 887359

Months since Launch Q-Phone 1.0

Quantity

Q-Phone 1.0 Prospects (Market Potential) M(t+1)

Q-Phone Prospects (Market Potential) M(t+1) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 1000000 990058 978436 964701 948854 930493 909861 887055 861298 833397 803426 771617 738558 704666 670840 637033 603409 571029 539701 509946 481805 455467 430726 407832 386124 366398 347866 330821 315174 300515 286990 274308 262749 251774 241692 232110 223056 214850 207196 200053 193298 186874 180901 175266 169831 164812 160005 155454 151148 147093 143193 139438 135914 132450 129181 126143 123227 120453 117689 115111 112641

Months since Launch Q-Phone 1.0

Quantity

Bass Diffusion Question 2

Simulation Sales Cycle Q-Phone: 60 months Bass Diffusion Forecast Model for Q-Phone 1.0
Month Contagion (Network Effect) Q-Phone Prospects (Market Potential) Q-Phone Prabability to Buy Q-Phone Next Period Prospect Probability Fraction of Adopters Predicted Q-Phone Adopters Error Accumulative Sales
Given Input Data: t+1 q*Ft M(t+1) ft 1-ft Ft ft / 1- Ft Xt Nt
Market size M 1,000,000 0 - 0 1000000 0.01 0.99 0.00000000 0.01 1 0.000
Months after Q-Phone launch t 60 1 0.00197 990,156 0.0118509791 0.9881490209 0.00984400 0.0119688 9,844 9,844
Rate of adoption through innovation ip 0.01 2 0.00431 978,430 0.014005247 0.985994753 0.02157000 0.014314 11,726 21,570
Rate of adoption through imitation (contagion) q 0.2 3 0.00705 964,745 0.016449867 0.983550133 0.03525500 0.017051 13,685 35,255
Contagion (network effect) c 4 0.01023 948,874 0.0191911664 0.9808088336 0.05112600 0.0202252 15,871 51,126
Rate of adoption - Rate of innovation q-ip 0.19 5 0.01387 930,655 0.0222138042 0.9777861958 0.06934500 0.023869 18,219 69,345
Month 0 sales Xt0 1 6 0.01797 910,130 0.0254599766 0.9745400234 0.08987000 0.027974 20,525 89,870
Month 1 sales Xt1 9,844 7 0.02259 887,027 0.0289122903 0.9710877097 0.11297300 0.0325946 23,103 112,973
Output Variables: 8 0.02774 861,294 0.0325062691 0.9674937309 0.13870600 0.0377412 25,733 138,706
Q-Phone single purchase adopters a 9 0.03336 833,209 0.0361264425 0.9638735575 0.16679100 0.0433582 28,085 166,791
Q-Phone single purchase prospects P 10 0.03937 803,170 0.0396492902 0.9603507098 0.19683000 0.049366 30,039 196,830
In month t, the market potential with respect to number of prospects Mt 11 0.04573 771,362 0.042986153 0.957013847 0.22863800 0.0557276 31,808 228,638
In month t, the number of sales Xt 12 0.05237 738,137 0.0460395238 0.9539604762 0.26186300 0.0623726 33,225 261,863 Year 1 Accumulative Predictive Sales (Actual 260,000)
Net total accumulative sales Nt 13 0.05926 703,690 0.0487389768 0.9512610232 0.29631000 0.069262 34,447 296,310
Single purchase adopter fraction in month t Ft 14 0.06611 669,449 0.0509518973 0.9490481027 0.33055100 0.0761102 34,241 330,551 Month 14 Begin Sales Decline
Probability of prospect making a single purchase the following month ft 15 0.07295 635,248 0.0526940757 0.9473059243 0.36475200 0.0829504 34,201 364,752
16 0.07965 601,749 0.0539469182 0.9460530818 0.39825100 0.0896502 33,499 398,251
Given p<iq, AR peaks at T* 17 0.08617 569,165 0.0547348906 0.9452651094 0.43083500 0.096167 32,584 430,835
0<=p,q<=1 18 0.09240 537,986 0.0550912728 0.9449087272 0.46201400 0.1024028 31,179 462,014
19 0.09830 508,502 0.0550705632 0.9449294368 0.49149800 0.1082996 29,484 491,498
20 0.10392 480,414 0.0547274177 0.9452725823 0.51958600 0.1139172 28,088 519,586
21 0.10921 453,933 0.0541148963 0.9458851037 0.54606700 0.1192134 26,481 546,067
22 0.11412 429,387 0.0532966308 0.9467033692 0.57061300 0.1241226 24,546 570,613
23 0.11869 406,568 0.0523197723 0.9476802277 0.59343200 0.1286864 22,819 593,432
24 0.12295 385,251 0.0512190434 0.9487809566 0.61474900 0.1329498 21,317 614,749 Year 2 Accumulative Predictive Sales (Actual > 615,000)
25 0.12694 365,291 0.0500236071 0.9499763929 0.63470900 0.1369418 19,960 634,709
26 0.13064 346,779 0.048772455 0.951227545 0.65322100 0.1406442 18,512 653,221
27 0.13407 329,654 0.0474929881 0.9525070119 0.67034600 0.1440692 17,125 670,346
28 0.13719 314,043 0.0462244288 0.9537755712 0.68595700 0.1471914 15,611 685,957
29 0.14009 299,526 0.0449572951 0.9550427049 0.70047400 0.1500948 14,517 700,474
30 0.14279 286,037 0.0437043369 0.9562956631 0.71396300 0.1527926 13,489 713,963
31 0.14525 273,737 0.042498381 0.957501619 0.72626300 0.1552526 12,300 726,263
32 0.14756 262,192 0.041311391 0.958688609 0.73780800 0.1575616 11,545 737,808
33 0.14974 251,309 0.0401436473 0.9598563527 0.74869100 0.1597382 10,883 748,691
34 0.15174 241,314 0.0390294507 0.9609705493 0.75868600 0.1617372 9,995 758,686
35 0.15362 231,914 0.0379451193 0.9620548807 0.76808600 0.1636172 9,400 768,086
36 0.15536 223,197 0.0369079898 0.9630920102 0.77680300 0.1653606 8,717 776,803
37 0.15702 214,914 0.0358943345 0.9641056655 0.78508600 0.1670172 8,283 785,086
38 0.15858 207,088 0.0349113921 0.9650886079 0.79291200 0.1685824 7,826 792,912
39 0.16002 199,894 0.0339862178 0.9660137822 0.80010600 0.1700212 7,194 800,106
40 0.16138 193,115 0.0330954694 0.9669045306 0.80688500 0.171377 6,779 806,885
41 0.16266 186,678 0.0322326449 0.9677673551 0.81332200 0.1726644 6,437 813,322
42 0.16386 180,693 0.031415538 0.968584462 0.81930700 0.1738614 5,985 819,307
43 0.16502 174,895 0.0306102978 0.9693897022 0.82510500 0.175021 5,798 825,105
44 0.16607 169,659 0.0298715547 0.9701284453 0.83034100 0.1760682 5,236 830,341
45 0.16708 164,584 0.0291450614 0.9708549386 0.83541600 0.1770832 5,075 835,416
46 0.16804 159,824 0.0284542978 0.9715457022 0.84017600 0.1780352 4,760 840,176
47 0.16893 155,345 0.0277960362 0.9722039638 0.84465500 0.178931 4,479 844,655
48 0.16981 150,968 0.0271450126 0.9728549874 0.84903200 0.1798064 4,377 849,032
49 0.17063 146,858 0.0265267256 0.9734732744 0.85314200 0.1806284 4,110 853,142
50 0.17141 142,945 0.0259317954 0.9740682046 0.85705500 0.181411 3,913 857,055
51 0.17214 139,304 0.0253727191 0.9746272809 0.86069600 0.1821392 3,641 860,696
52 0.17284 135,782 0.0248268697 0.9751731303 0.86421800 0.1828436 3,522 864,218
53 0.17352 132,402 0.0242983621 0.9757016379 0.86759800 0.1835196 3,380 867,598
54 0.17418 129,118 0.0237804884 0.9762195116 0.87088200 0.1841764 3,284 870,882
55 0.17478 126,077 0.023297088 0.976702912 0.87392300 0.1847846 3,041 873,923
56 0.17536 123,185 0.0228339412 0.9771660588 0.87681500 0.185363 2,892 876,815
57 0.17592 120,402 0.0223850917 0.9776149083 0.87959800 0.1859196 2,783 879,598
58 0.17645 117,744 0.0219535101 0.9780464899 0.88225600 0.1864512 2,658 882,256
59 0.17694 115,293 0.0215530348 0.9784469652 0.88470700 0.1869414 2,451 884,707
60 0.17744 112,798 0.0211429022 0.9788570978 0.88720200 0.1874404 2,495 887,202

Q-Phone 1.0 Sales (Xt)

Q-Phone Adopters Xt

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 1 9844 11726 13685 15871 18219 20525 23103 25733 28085 30039 31808 33225 34447 34241 34201 33499 32584 31179 29484 28088 26481 24546 22819 21317 19960 18512 17125 15611 14517 13489 12300 11545 10883 9995 9400 8717 8283 7826 7194 6779 6437 5985 5798 5236 5075 4760 4479 4377 4110 3913 3641 3522 3380 3284 3041 2892 2783 2658 2451 2495

Months since Launch Q-Phone 1.0

Quantity

Q-Phone 1.0 Accumulative Sales (Nt)

Accumulative Sales Nt

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 0 9844 21570 35255 51126 69345 89870 112973 138706 166791 196830 228638 261863 296310 330551 364752 398251 430835 462014 491498 519586 546067 570613 593432 614749 634709 653221 670346 685957 700474 713963 726263 737808 748691 758686 768086 776803 785086 792912 800106 806885 813322 819307 825105 830341 835416 840176 844655 849032 853142 857055 860696 864218 867598 870882 873923 876815 879598 882256 884707 887202

Months since Launch Q-Phone 1.0

Quantity

Q-Phone 1.0 Prospects (Market Potential) M(t+1)

Q-Phone Prospects (Market Potential) M(t+1) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 1000000 990156 978430 964745 948874 930655 910130 887027 861294 833209 803170 771362 738137 703690 669449 635248 601749 569165 537986 508502 480414 453933 429387 406568 385251 365291 346779 329654 314043 299526 286037 273737 262192 251309 241314 231914 223197 214914 207088 199894 193115 186678 180693 174895 169659 164584 159824 155345 150968 146858 142945 139304 135782 132402 129118 126077 123185 120402 117744 115293 112798

Months since Launch Q-Phone 1.0

Quantity

Sheet1

0

Bass Diffusion Question 3

Simulation Sales Cycle Q-Phone: 60 months Bass Diffusion Forecast Model for Q-Phone 1.0
Month Contagion (Network Effect) Q-Phone Prospects (Market Potential) Q-Phone Prabability to Buy Q-Phone Next Period Prospect Probability Fraction of Adopters Predicted Q-Phone Adopters Error Accumulative Sales Immitation (q)
Given Input Data: t+1 q*Ft M(t+1) ft 1-ft Ft ft / 1- Ft Xt Nt 668,120 0.2 0.4 0.6 0.8 1
Market size M 1,000,000 0 - 0 1000000 0.01 0.99 0.00000000 0.01 1 0.000 0.01 667842 407041 244863 161287 116688
Months after Q-Phone launch t 60 1 0.00198 990,083 0.0118645606 0.9881354394 0.00991700 0.0119834 9,917 9,917 0.02 533091 318159 199041 137152 102558
Rate of adoption through innovation ip 0.01 2 0.00438 978,076 0.0140694276 0.9859305724 0.02192400 0.0143848 12,007 21,924 Innovation (p) 0.03 454078 273661 176855 125646 94859
Rate of adoption through imitation (contagion) q 0.2 3 0.00714 964,314 0.0165256419 0.9834743581 0.03568600 0.0171372 13,762 35,686 0.04 401351 245097 162485 117059 89983
Contagion (network effect) c 4 0.01033 948,338 0.0192819876 0.9807180124 0.05166200 0.0203324 15,976 51,662 0.05 361722 224972 151407 110848 85948
Rate of adoption - Rate of innovation q-ip 0.19 5 0.01401 929,974 0.0223242119 0.9776757881 0.07002600 0.0240052 18,364 70,026
Month 0 sales Xt0 1 6 0.01819 909,032 0.0256288846 0.9743711154 0.09096800 0.0281936 20,942 90,968
Month 1 sales Xt1 9,917 7 0.02282 885,900 0.029075238 0.970924762 0.11410000 0.03282 23,132 114,100 Immitation (q)
Output Variables: 8 0.02795 860,234 0.032648633 0.967351367 0.13976600 0.0379532 25,666 139,766 34,294 0.2 0.4 0.6 0.8 1
Q-Phone single purchase adopters a 9 0.03356 832,195 0.0362512464 0.9637487536 0.16780500 0.043561 28,039 167,805 0.01 34235 47542 34445 22327 15117
Q-Phone single purchase prospects P 10 0.03959 802,073 0.0397711105 0.9602288895 0.19792700 0.0495854 30,122 197,927 Innovation (p) 0.02 34367 34534 23948 16444 11653
In month t, the market potential with respect to number of prospects Mt 11 0.04600 770,010 0.04311902 0.95688098 0.22999000 0.055998 32,063 229,990 0.03 31252 27873 19491 13752 10121
In month t, the number of sales Xt 12 0.05271 736,463 0.0461816799 0.9538183201 0.26353700 0.0627074 33,547 263,537 Year 1 Accumulative Predictive Sales (Actual 260,000) 0.04 28617 24076 17369 12486 9282
Net total accumulative sales Nt 13 0.05952 702,414 0.0488298545 0.9511701455 0.29758600 0.0695172 34,049 297,586 0.05 25776 21283 15648 11441 8465
Single purchase adopter fraction in month t Ft 14 0.06638 668,120 0.0510283331 0.9489716669 0.33188000 0.076376 34,294 331,880 Month 14 Begin Sales Decline
Probability of prospect making a single purchase the following month ft 15 0.07318 634,097 0.0527445689 0.9472554311 0.36590300 0.0831806 34,023 365,903 Immitation (q)
16 0.07985 600,764 0.0539769633 0.9460230367 0.39923600 0.0898472 33,333 399,236 331,880 0.2 0.4 0.6 0.8 1
Given p<iq, AR peaks at T* 17 0.08627 568,626 0.0547443544 0.9452556456 0.43137400 0.0962748 32,138 431,374 Innovation (p) 0.01 331552 591340 754937 838418 883933
0<=p,q<=1 18 0.09251 537,431 0.055094094 0.944905906 0.46256900 0.1025138 31,195 462,569 0.02 465941 681166 801279 862487 897584
19 0.09848 507,582 0.0550643227 0.9449356773 0.49241800 0.1084836 29,849 492,418 0.03 545497 726164 822584 875075 904574
20 0.10406 479,702 0.0547146182 0.9452853818 0.52029800 0.1140596 27,880 520,298 0.04 599065 755098 837611 883118 910091
21 0.10938 453,088 0.0540907329 0.9459092671 0.54691200 0.1193824 26,614 546,912 0.05 638305 774711 847609 889278 914263
22 0.11425 428,739 0.053271764 0.946728236 0.57126100 0.1242522 24,349 571,261
23 0.11883 405,867 0.0522864657 0.9477135343 0.59413300 0.1288266 22,872 594,133
24 0.12303 384,831 0.0511955303 0.9488044697 0.61516900 0.1330338 21,036 615,169 Year 2 Accumulative Predictive Sales (Actual > 615,000)
25 0.12695 365,273 0.0500224571 0.9499775429 0.63472700 0.1369454 19,558 634,727
26 0.13058 347,120 0.0487967411 0.9512032589 0.65288000 0.140576 18,153 652,880
27 0.13397 330,132 0.0475302925 0.9524697075 0.66986800 0.1439736 16,988 669,868
28 0.13712 314,415 0.0462557916 0.9537442084 0.68558500 0.147117 15,717 685,585
29 0.14005 299,729 0.0449755953 0.9550244047 0.70027100 0.1500542 14,686 700,271
30 0.14276 286,208 0.0437206761 0.9562793239 0.71379200 0.1527584 13,521 713,792
31 0.14526 273,676 0.0424922494 0.9575077506 0.72632400 0.1552648 12,532 726,324
32 0.14760 262,024 0.0412937247 0.9587062753 0.73797600 0.1575952 11,652 737,976
33 0.14975 251,264 0.0401387205 0.9598612795 0.74873600 0.1597472 10,760 748,736
34 0.15177 241,157 0.0390116303 0.9609883697 0.75884300 0.1617686 10,107 758,843
35 0.15366 231,687 0.0379184968 0.9620815032 0.76831300 0.1636626 9,470 768,313
36 0.15541 222,954 0.0368786428 0.9631213572 0.77704600 0.1654092 8,733 777,046
37 0.15703 214,832 0.0358841624 0.9641158376 0.78516800 0.1670336 8,122 785,168
38 0.15857 207,162 0.0349208012 0.9650791988 0.79283800 0.1685676 7,670 792,838
39 0.16003 199,873 0.0339834868 0.9660165132 0.80012700 0.1700254 7,289 800,127
40 0.16138 193,108 0.0330945401 0.9669054599 0.80689200 0.1713784 6,765 806,892
41 0.16266 186,698 0.0322353514 0.9677646486 0.81330200 0.1726604 6,410 813,302
42 0.16385 180,754 0.0314239383 0.9685760617 0.81924600 0.1738492 5,944 819,246
43 0.16498 175,094 0.0306381582 0.9693618418 0.82490600 0.1749812 5,660 824,906
44 0.16605 169,754 0.0298850559 0.9701149441 0.83024600 0.1760492 5,340 830,246
45 0.16708 164,595 0.0291466472 0.9708533528 0.83540500 0.177081 5,159 835,405
46 0.16805 159,726 0.028439981 0.971560019 0.84027400 0.1780548 4,869 840,274
47 0.16895 155,242 0.0277808043 0.9722191957 0.84475800 0.1789516 4,484 844,758
48 0.16983 150,849 0.0271272058 0.9728727942 0.84915100 0.1798302 4,393 849,151
49 0.17064 146,813 0.0265199186 0.9734800814 0.85318700 0.1806374 4,036 853,187
50 0.17141 142,938 0.0259307256 0.9740692744 0.85706200 0.1814124 3,875 857,062
51 0.17214 139,290 0.0253705592 0.9746294408 0.86071000 0.182142 3,648 860,710
52 0.17285 135,749 0.0248217318 0.9751782682 0.86425100 0.1828502 3,541 864,251
53 0.17354 132,310 0.0242839128 0.9757160872 0.86769000 0.183538 3,439 867,690
54 0.17414 129,278 0.0238058197 0.9761941803 0.87072200 0.1841444 3,032 870,722
55 0.17476 126,215 0.0233191048 0.9766808952 0.87378500 0.184757 3,063 873,785
56 0.17535 123,247 0.0228439054 0.9771560946 0.87675300 0.1853506 2,968 876,753
57 0.17592 120,388 0.0223828259 0.9776171741 0.87961200 0.1859224 2,859 879,612
58 0.17647 117,660 0.0219398249 0.9780601751 0.88234000 0.186468 2,728 882,340
59 0.17698 115,088 0.0215194305 0.9784805695 0.88491200 0.1869824 2,572 884,912
60 0.17748 112,581 0.0211071137 0.9788928863 0.88741900 0.1874838 2,507 887,419

Q-Phone 1.0 Sales (Xt)

Q-Phone Adopters Xt

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 1 9917 12007 13762 15976 18364 20942 23132 25666 28039 30122 32063 33547 34049 34294 34023 33333 32138 31195 29849 27880 26614 24349 22872 21036 19558 18153 16988 15717 14686 13521 12532 11652 10760 10107 9470 8733 8122 7670 7289 6765 6410 5944 5660 5340 5159 4869 4484 4393 4036 3875 3648 3541 3439 3032 3063 2968 2859 2728 2572 2507

Months since Launch Q-Phone 1.0

Quantity

Q-Phone 1.0 Accumulative Sales (Nt)

Accumulative Sales Nt

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 0 9917 21924 35686 51662 70026 90968 114100 139766 167805 197927 229990 263537 297586 331880 365903 399236 431374 462569 492418 520298 546912 571261 594133 615169 634727 652880 669868 685585 700271 713792 726324 737976 748736 758843 768313 777046 785168 792838 800127 806892 813302 819246 824906 830246 835405 840274 844758 849151 853187 857062 860710 864251 867690 870722 873785 876753 879612 882340 884912 887419

Months since Launch Q-Phone 1.0

Quantity

Q-Phone 1.0 Prospects (Market Potential) M(t+1)

Q-Phone Prospects (Market Potential) M(t+1) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 1000000 990083 978076 964314 948338 929 974 909032 885900 860234 832195 802073 770010 736463 702414 668120 634097 600764 568626 537431 507582 479702 453088 428739 405867 384831 365273 347120 330132 314415 299729 286208 273676 262024 251264 241157 231687 222954 214832 207162 199873 193108 186698 180754 175094 169754 164595 159726 155242 150849 146813 142938 139290 135749 132310 129278 126215 123247 120388 117660 115088 112581

Months since Launch Q-Phone 1.0

Quantity

Xt

Immitation (q) 34,294 0.01 0.02 0.03 0.04 0.05 Innovation (p) 0.2 34235 34367 31252 28617 25776 34,294 0.01 0.02 0.03 0.04 0.05 Innovation (p) 0.4 47542 34534 27873 24076 21283 34,294 0.01 0.02 0.03 0.04 0.05 Innovation (p) 0.6 34445 23948 19491 17369 15648 34,294 0.01 0.02 0.03 0.04 0.05 Innovation (p) 0.8 22327 16444 13752 12486 11441 34,294 0.01 0.02 0.03 0.04 0.05 Innovation (p) 1 15117 11653 10121 9282 8465

Nt

Immitation (q) 331,880 0.01 0.02 0.03 0.04 0.05 Innovation (p) 0.2 331552 465941 545497 599065 638305 331,880 0.01 0.02 0.03 0.04 0.05 Innovation (p) 0.4 591340 681166 726164 755098 774711 331,880 0.01 0.02 0.03 0.04 0.05 Innovation (p) 0.6 754937 801279 822584 837611 847609 331,880 0.01 0.02 0.03 0.04 0.05 Innovation (p) 0.8 838418 862487 875075 883118 889278 331,880 0.01 0.02 0.03 0.04 0.05 Innovation (p) 1 883933 897584 904574 910091 914263

Nt

668120 0.01 0.02 0.03 0.04 0.05 0.2 667842 533091 454078 401351 361722 668120 0.01 0.02 0.03 0.04 0.05 0.4 407041 318159 273661 245097 224972 668120 0.01 0.02 0.03 0.04 0.05 0.6 244863 199041 176855 162485 151407 668120 0.01 0.02 0.03 0.04 0.05 0.8 161287 137152 125646 117059 110848 668120 0.01 0.02 0.03 0.04 0.05 1 116688 102558 94859 89983 85948 Immitation (q) 331,880 0.01 0.02 0.03 0.04 0.05 Innovation (p) 0.2 331552 465941 545497 599065 638305 331,880 0.01 0.02 0.03 0.04 0.05 Innovation (p) 0.4 591340 681166 726164 755098 774711 331,880 0.01 0.02 0.03 0.04 0.05 Innovation (p) 0.6 754937 801279 822584 837611 847609 331,880 0.01 0.02 0.03 0.04 0.05 Innovation (p) 0.8 838418 862487 875075 883118 889278 331,880 0.01 0.02 0.03 0.04 0.05 Innovation (p) 1 883933 897584 904574 910091 914263

Bass Diffusion Question 4

Simulation Sales Cycle Q-Phone: 60 months Bass Diffusion Forecast Model for Q-Phone 1.0
Month Contagion (Network Effect) Q-Phone Prospects (Market Potential) Q-Phone Prabability to Buy Q-Phone Next Period Prospect Probability Fraction of Adopters Predicted Q-Phone Adopters Error Accumulative Sales
Given Input Data: t+1 q*Ft M(t+1) ft 1-ft Ft ft / 1- Ft Xt Nt
Market size M 1,000,000 0 - 0 1000000 0.01 0.99 0.00000000 0.01 1 0.000
Months after Q-Phone launch t 60 1 0.00201 989,932 0.0118926471 0.9881073529 0.01006800 0.0120136 10,068 10,068
Rate of adoption through innovation ip 0.01 2 0.00436 978,191 0.0140485835 0.9859514165 0.02180900 0.0143618 11,741 21,809
Rate of adoption through imitation (contagion) q 0.2 3 0.00711 964,442 0.0165031457 0.9834968543 0.03555800 0.0171116 13,749 35,558
Contagion (network effect) c 4 0.01029 948,548 0.0192464183 0.9807535817 0.05145200 0.0202904 15,894 51,452
Rate of adoption - Rate of innovation q-ip 0.19 5 0.01396 930,194 0.0222885645 0.9777114355 0.06980600 0.0239612 18,354 69,806
Month 0 sales Xt0 1 6 0.01809 909,552 0.0255489519 0.9744510481 0.09044800 0.0280896 20,642 90,448
Month 1 sales Xt1 10,068 7 0.02278 886,077 0.02904968 0.97095032 0.11392300 0.0327846 23,475 113,923
Output Variables: 8 0.02791 860,453 0.032619257 0.967380743 0.13954700 0.0379094 25,624 139,547
Q-Phone single purchase adopters a 9 0.03346 832,710 0.0361879112 0.9638120888 0.16729000 0.043458 27,743 167,290
Q-Phone single purchase prospects P 10 0.03948 802,576 0.0397153128 0.9602846872 0.19742400 0.0494848 30,134 197,424
In month t, the market potential with respect to number of prospects Mt 11 0.04586 770,698 0.0430514986 0.9569485014 0.22930200 0.0558604 31,878 229,302
In month t, the number of sales Xt 12 0.05248 737,596 0.0460855882 0.9539144118 0.26240400 0.0624808 33,102 262,404 Year 1 Accumulative Predictive Sales (Actual 260,000)
Net total accumulative sales Nt 13 0.05929 703,548 0.0487491223 0.9512508777 0.29645200 0.0692904 34,048 296,452
Single purchase adopter fraction in month t Ft 14 0.06613 669,370 0.0509564606 0.9490435394 0.33063000 0.076126 34,178 330,630 Month 14 Begin Sales Decline
Probability of prospect making a single purchase the following month ft 15 0.07301 634,961 0.0527067157 0.9472932843 0.36503900 0.0830078 34,409 365,039
16 0.07966 601,699 0.0539484527 0.9460515473 0.39830100 0.0896602 33,262 398,301
Given p<iq, AR peaks at T* 17 0.08619 569,045 0.0547370076 0.9452629924 0.43095500 0.096191 32,654 430,955
0<=p,q<=1 18 0.09241 537,950 0.0550914595 0.9449085405 0.46205000 0.10241 31,095 462,050
19 0.09838 508,107 0.0550679253 0.9449320747 0.49189300 0.1083786 29,843 491,893
20 0.10394 480,314 0.0547256323 0.9452743677 0.51968600 0.1139372 27,793 519,686
21 0.10918 454,121 0.0541202335 0.9458797665 0.54587900 0.1191758 26,193 545,879
22 0.11409 429,571 0.0533036612 0.9466963388 0.57042900 0.1240858 24,550 570,429
23 0.11868 406,620 0.0523222351 0.9476777649 0.59338000 0.128676 22,951 593,380
24 0.12294 385,322 0.0512230113 0.9487769887 0.61467800 0.1329356 21,298 614,678 Year 2 Accumulative Predictive Sales (Actual > 615,000)
25 0.12691 365,462 0.0500345253 0.9499654747 0.63453800 0.1369076 19,860 634,538
26 0.13057 347,157 0.0487993735 0.9512006265 0.65284300 0.1405686 18,305 652,843
27 0.13396 330,189 0.0475347349 0.9524652651 0.66981100 0.1439622 16,968 669,811
28 0.13711 314,467 0.0462601712 0.9537398288 0.68553300 0.1471066 15,722 685,533
29 0.14002 299,898 0.0449908179 0.9550091821 0.70010200 0.1500204 14,569 700,102
30 0.14269 286,531 0.0437515072 0.9562484928 0.71346900 0.1526938 13,367 713,469
31 0.14520 273,991 0.0425238964 0.9574761036 0.72600900 0.1552018 12,540 726,009
32 0.14753 262,344 0.0413273651 0.9586726349 0.73765600 0.1575312 11,647 737,656
33 0.14972 251,403 0.0401539363 0.9598460637 0.74859700 0.1597194 10,941 748,597
34 0.15173 241,359 0.0390345566 0.9609654434 0.75864100 0.1617282 10,044 758,641
35 0.15358 232,080 0.0379645747 0.9620354253 0.76792000 0.163584 9,279 767,920
36 0.15534 223,314 0.0369221115 0.9630778885 0.77668600 0.1653372 8,766 776,686
37 0.15699 215,075 0.0359142989 0.9640857011 0.78492500 0.166985 8,239 784,925
38 0.15856 207,205 0.0349262676 0.9650737324 0.79279500 0.168559 7,870 792,795
39 0.16001 199,971 0.0339962298 0.9660037702 0.80002900 0.1700058 7,234 800,029
40 0.16136 193,224 0.0331099372 0.9668900628 0.80677600 0.1713552 6,747 806,776
41 0.16264 186,825 0.0322525339 0.9677474661 0.81317500 0.172635 6,399 813,175
42 0.16382 180,884 0.0314418357 0.9685581643 0.81911600 0.1738232 5,941 819,116
43 0.16498 175,102 0.0306392779 0.9693607221 0.82489800 0.1749796 5,782 824,898
44 0.16604 169,818 0.0298941494 0.9701058506 0.83018200 0.1760364 5,284 830,182
45 0.16705 164,756 0.0291698521 0.9708301479 0.83524400 0.1770488 5,062 835,244
46 0.16804 159,805 0.0284515224 0.9715484776 0.84019500 0.178039 4,951 840,195
47 0.16893 155,365 0.0277989934 0.9722010066 0.84463500 0.178927 4,440 844,635
48 0.16979 151,071 0.0271604206 0.9728395794 0.84892900 0.1797858 4,294 848,929
49 0.17059 147,040 0.0265542477 0.9734457523 0.85296000 0.180592 4,031 852,960
50 0.17136 143,206 0.0259716683 0.9740283317 0.85679400 0.1813588 3,834 856,794
51 0.17210 139,525 0.0254068049 0.9745931951 0.86047500 0.182095 3,681 860,475
52 0.17280 135,985 0.024858466 0.975141534 0.86401500 0.182803 3,540 864,015
53 0.17348 132,616 0.0243319593 0.9756680407 0.86738400 0.1834768 3,369 867,384
54 0.17411 129,438 0.0238311408 0.9761688592 0.87056200 0.1841124 3,178 870,562
55 0.17473 126,348 0.0233403166 0.9766596834 0.87365200 0.1847304 3,090 873,652
56 0.17532 123,388 0.0228665603 0.9771334397 0.87661200 0.1853224 2,960 876,612
57 0.17587 120,628 0.0224216571 0.9775783429 0.87937200 0.1858744 2,760 879,372
58 0.17642 117,879 0.0219754983 0.9780245017 0.88212100 0.1864242 2,749 882,121
59 0.17691 115,460 0.0215803977 0.9784196023 0.88454000 0.186908 2,419 884,540
60 0.17739 113,028 0.0211808142 0.9788191858 0.88697200 0.1873944 2,432 886,972

Q-Phone 1.0 Sales (Xt)

Q-Phone Adopters Xt

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 1 10068 11741 13749 15894 18354 20642 23475 25624 27743 30134 31878 33102 34048 34178 34409 33262 32654 31095 29843 27793 26193 24550 22951 21298 19860 18305 16968 15722 14569 13367 12540 11647 10941 10044 9279 8766 8239 7870 7234 6747 6399 5941 5782 5284 5062 4951 4440 4294 4031 3834 3681 3540 3369 3178 3090 2960 2760 2749 2419 2432

Months since Launch Q-Phone 1.0

Quantity

Q-Phone 1.0 Accumulative Sales (Nt)

Accumulative Sales Nt

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 0 10068 21809 35558 51452 69806 90448 113923 139547 167290 197424 229302 262404 296452 330630 365039 398301 430955 462050 491893 519686 545879 570429 593380 614678 634538 652843 669811 685533 700102 713469 726009 737656 748597 758641 767920 776686 784925 792795 800029 806776 813175 819116 824898 830182 835244 840195 844635 848929 852960 856794 860475 864015 867384 870562 873652 876612 879372 882121 884540 886972

Months since Launch Q-Phone 1.0

Quantity

Q-Phone 1.0 Prospects (Market Potential) M(t+1)

Q-Phone Prospects (Market Potential) M(t+1) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 1000000 989932 978191 964442 948548 930194 909552 886077 860453 832710 802576 770698 737596 703548 669370 634961 601699 569045 537950 508107 480314 454121 429571 406620 385322 365462 347157 330189 314467 299898 286531 273991 262344 251403 241359 232080 223314 215075 207205 199971 193224 186825 180884 175102 169818 164756 159805 155365 151071 147040 143206 139525 135985 132616 129438 126348 123388 120628 117879 115460 113028

Months since Launch Q-Phone 1.0

Quantity