Optimizing the Revenue of Spotify with a new Pricing Scheme (MCM Problem 2)

Optimizing the Revenue of Spotify with a new Pricing Scheme (MCM Problem 2) November 4, 2018

Contents Non-technical Summary 1 1. Introduction 2 2. Assumption 3 3. Model 5 4. Data Presentation 13 5. Result and Analysis 14 6. Strengths, Limitations, and Improvement 19 7. Conclusion 22 Bibliography 23 Appendix 24

Non-technical Summary The invention of Internet fosters the growth of music streaming service. Though piracy exists, most of the users will pay for content as long as price is reasonable and content access is easy. Spotify is one of the companies that benefit from Internet. As a music streaming business, Spotify provides huge number of songs to users while earning a considerable revenue. Artists will also receive money for each stream played by users. But the complaints from artists about a low-income per stream force Spotify to change its 0 flat monthly fee charged from users. We have been hired by Spotify to propose a better pricing scheme. Our goal is to achieve higher revenue for both the company and the artists. An à la carte structure where the price would scale with number of listening hours is preferred by the company. With these in mind, we propose a new pricing scheme that charges user by hours with different price levels. Our mathematical model could also estimate the revenue under new pricing scheme. We evaluate the advantages of our pricing scheme by analyzing the change in company's revenue and artists' income. We also account for how our pricing scheme will affect piracy and different categories of artists. In summary, our central finding is that with our new pricing scheme, the company's revenue can boost 30% and artists will feel a substantial increase in their income. There will be a subtle increase in piracy. However, this can be ignored due to the rationality of pricing. We see a clear benefit from our pricing scheme compared to a fixed subscription for all customers. 1/

1. Introduction Online music streaming business is growing rapidly in recent years, satisfying people's daily demand for musics. Companies, like Spotify, earn a large amount of revenue from the market. In our scenario, Spotify is the only music streaming business. It currently employs a monthly pricing plan that charges users 0 per month to use all services provided by the company. However, Spotify is receiving complaints from artists due to the low-income per stream they can receive from their musics. We, as consultants of Spotify, come up with a new price scheme with an à la carte structure to benefit both the company and its artists. We propose a Step Pricing Scheme (SPS) for Spotify. The SPS takes into account the hours of music streamed per month by each user to satisfy Spotify's demand. Under SPS, we provide each user with 5 hours free usage of the music stream service each month. After that, we have 3 different levels to charge users. In the first level, we charge users per step with a maximum of. In the second level, we charge users.8 per step with a maximum of $4. In the third level, we charge users.6 per step with a maximum of. So the total monthly fee will not exceed 0. Sample Step Pricing Scheme is available in Appendix 1. With our SPS, we create a discrete piecewise model to estimate the revenue. We evaluate our pricing scheme by comparing the revenue of the company and income of artists before and after implementing our SPS. The remainder of the report is organized as follows: Section 2 states the assumptions of our problem and our model and Section 3 introduces our mathematical model for 3LPS. Section 4 is composed of the data mining and analysis and Section 5 presents the results of our model and the interpretations. Section 6 gives possible further improvements, and Section 7 concludes our report. 2/

2. Assumption The model contains several assumptions. We list the assumptions and arguments before getting into our model. 2.1 All users' time spent listening on Spotify is normally distributed with standard deviation 40. According to the data from Spotify, we have millions of users online. Using central limit theorem, we can approximate the distribution of users' time to normal distribution. We will use the assumption to help us calculate the profit of SPS. 2.2 Spotify is the only company in the music streaming business and the only other choice is piracy. This is the assumption given in Problem 2. This means we do not have to consider the effect of competition in our model. 2.3 The influence of piracy is trivial to our model. The Problem mentions that a 2% of people will always resort to piracy while that vast majority of users will pay for content when it is priced reasonably, can be accessed easily, and supports their favorite artists. We will discuss about the influence of our price scheme to piracy, but piracy will be considered having no effect on our model. 2.4 Users' time listening to music online will not be affected by price. With this assumption, we can use the normal distribution to calculate our revenue after implementing SPS. 2.5 Elasticity in our model is constant. Elasticity is the percentage of people who are willing to change from free user to premium user. Elasticity is fixed at 60% because according to the survey report by TechCrunch, only 42% will be willing to only use free music streaming service. 2.6 The operation cost of the company is trivial compared to the revenue. The cost of operating online music streaming service only takes a small percentage of the 3/

revenue of the company. So we only consider the revenue to artists and revenue to Spotify in the net revenue. 2.8 Spotify pays 50% of its revenue to artists. According to the data we obtain from Spotify, the reasonable estimated percentage of total revenue paid to artists is 50%. 2.9 One month contains 30 days in our model. 4/

3. Model 3.1 Parameter List Parameter Actual Meaning Table 1 - Parameter List e Δm ΔP S Price elasticity of free users with listening time more than T 0 per month Percentage of free users will switch to paid music plan Change in price to paid plan Total number of Spotify users in the current year k Average growth of user per year (from 2017) Y Current year b Total number of users in 2017 S free S paid O P 0 Ē g E 0 t P(t) T 0 T 1 p i SD N Φ i R c β ω Number of free users in total Number of paid users in total Original number of paid users An increment in the price when music listening time exceeds first step time. Step time: Time region to listen music without change in price. Expected average expense per user (on music) per month Expected growth for Spotify per month Current average expense per user (on music) per month Average music listening time per user per month Step price given a music listening time t Free threshold: Free listening time per month Step time: Time region to listen music without change in price. Cumulative price for users that fall into step time box i Standard deviation for normal distribution Normal distribution: number of users against music listening time per user Percentage of users that falls into step time i Total revenue of Spotify per month Percentage of Spotify new revenue shared with artists Average price paid to artist per stream Number of streams in total per month 5/

3.2 Helping Functions This formula is for calculating the price elasticity e of free users who listen to Spotify T 0 e = Δm ΔP music more than hours. The price elasticity defines percentage of free users will switch to paid music plans if there is no free music plan available. S = k * (Y 2017) + 159M This formula is for estimating total number of Spotify users S (including paid users and free users) given a year Y and annual growth rate k since 2017. Data for 2017 is collected to be 159 million. S free = S S paid This formula is for calculating total number of free users (before new paid plan) who could potentially switch to the new paid plan. This formula is for calculating total number of paid users (after new paid plan) consisting of all previously paid users and some free users switching to paid plan. Original paid user will not leave Spotify because their plan rate will very likely to decrease; Or in worst case that they listen to Spotify music all the day, their plan rate will be kept the same (0). See more details in the later discussion. P 0 p 0 S paid = O + e S free is the increment in the price when music listening time exceeds first step time. In the Step Pricing Scheme, after a user spends all his/her free music hours, P0 is the price the user will pay for his/her next step time. S paid 6/

Team 9 OSUMCM 2018 Ē = g E 0 S This formula is for calculating the target average expense Ē for Spotify users. The target average expense is derived from target company growth rate (in percentage defined by Spotify manager) and current average expense. 3.3 Modeling In the model, we will define a mapping between music listening time t and expense p i. The expense is calculated from cumulative step price until step i (as shown in figure 1 below). p i t E Figure 1 - Step Price and Cumulative Cost Step Price Cumulative Cost p i e.g. For Step i = 4, the cumulative cost p i will be $4 Step Price = Starting from step i=0 for the free music listening time each month $4 $3 $2 7/

Team 9 OSUMCM 2018 0 t T 0 P 0 T 0 t T 0 + 5T 1 P(t) = T 0 + 5T 1 t T 0 + 10T 1 0.8 * P 0 T 0 + 10T 1 t T 0 + 15T 1 P(t) is the step price function where T 0 is the free music time and T 1 is the step time. In order to encourage Spotify users to listen to more music (i.e. spend more time on streaming), two step price incentives are proposed: 1. When users reach every 5 time steps, Spotify will give out 5 free time steps. 2. When users reach every 10 time steps, Spotify will discount each step price by 20%. T 0 + 15T 1 t T 0 + 20T 1 0.6 * P 0 T 0 + 20T 1 t T 0 + 25T 1... 0 = P(t) dt When users listen to music all the day, Spotify will limit the upper bound of p i cumulative cost to 0, the previous premium monthly fee. So, as discussed above, original users (O) will not leave Spotify because of the decrease in price; or in worst case, same monthly fee ever. 0 T o +it 1 p i = P(t) dt 0 From the step price function, we are able to calculate the cumulative cost from step i, p i 8/

Figure 2 - Step and Cumulative Cost Cumulative Cost 5 Step Price + 1 Discounted Step Price 5 Step Price 2 Step Price Step Price 0 1 2 3 4 5 6 7 8 9 10 11 12 Step Ē = t 0 P(t) dt In order to calculate the step time, we first match the average listening time t with T 1 target average expense Ē. Then, we count number of steps and calculate the step time. For example, from data collected, the average listening time t is 137 hours and the target average expense Ē is.4. Then we get a total of 11 steps between t and Free Music Threshold, so each time step is calculated to be 12 hours as shown below. 9/

Figure 3 - Step Time Calculation Step Price Cost Per Month (cumulative) 137 hrs matches target average expense.4 11 Steps from 5 hrs to 137 hrs Free Music Hours 5 hrs 0 hrs.8.8.8.8.8 $9 $9 $8.2 $7.4 $6.6.8 $4 $3 $2 Incentive 2: Discounted step price Incentive 1: Free time steps 10/

N(μ, σ) = Normal( t, SD) Z = X μ σ Z = X μ σ n Φ(z) = 1 2π z e t2 /2 dt Assume distribution of Spotify users music time is normally distributed with mean t and standard deviation SD assumed to be 40. In the computation, we will use the classic formulas for normal distributions from above. Φ i (z) = 1 2π z i z i 1 e t2 /2 dt Φ i We use to estimate the percentage of population that lies in a single time step i. As shown in the figure below: Figure 4 - Normal Distribution for time step i 11/

R(t) = i=0 Φ i Sp i This formula will calculate number of users that fall into time step i by Φ i S and p i accumulative cost for step i by. By summing all the time steps, we will get the total revenue of Spotify company. β = R c ω This formula will calculate the average payment to artist per stream by total revenue R, percentage of artist sharing c and number of streams ω. And this will be the price we used to compare the original model. 12/

4. Data Presentation Data is required for our model for certain reasons. In our model, we assume the time spent listening of users is normally distributed. The mean and standard deviation are required for us to model the distribution. According to the article in Forbes, the average time American people listen music per week online is 32 hours. We generalize it to all users of Spotify and use it in our model to help calculating the revenue. We also want to take into account the change of users year over year in order to better predict the revenue after the implementation of SPS. From 2015 to 2017, the amount of users grew from 91 to 159 millions in a linear way with slope k = 34. Therefore, we create a linear function to model the growth of users' number and implement it into our revenue model. The number of streams online is also important factor in the calculation of the revenue. We acquire the information that 8.2 billion per 6 month online streams in Nielsen's 2018 music report. Assuming no change in the number of online streams, we use it as a constant in our calculation. As there exist free users and premium subscribers, the proportion between them and possible number of users transit from free member to premium member should be taken into account in our model. We find the proportion from Spotify's website, which is 54:46. However, there is no data directly related to the latter element. So, we make use of the survey in TechCrunch demonstrating 43% of users will prefer to only use free music stream service and interpret it as 57% will be willing to change to premium subscriber after we employ SPS. We will later measure SPS' effect on artists income, and the amount of Spotify's revenue paid to artists is essential to solve the problem. Through the article in The Verge, 50% of the net revenue will be used to pay artists. So we take it as given and calculate the income of artists under SPS. 13/

Team 9 OSUMCM 2018 5. Result and Analysis With our model, we put in real values from Spotify to see how the model works. We will pick 2017 as the example to analysis. 5.1 Parameters Value At first, we need to get some data from reality. As we explained before in data presentation, we can get the elasticity of people to transient from free users to paid users will be 0.6. e = 0.6 Also we can get the amount of the music stream users from the internet. S = 159M users Then,we can get the number of paid users with the percentage of member users in total users. O = 0.45 S = 71.55M users At here, for better computation and comparison, we assume the step price ( ) to be 1 dollar, the time people can listen to music freely ( ) be 5 hours. Also we can get the average time American people listen to music per year and the number of streams through Internet. T 0 P 0 P 0 = t = 137 hours T 0 = 5 hours ω = 8200M 6 = 44700M 14/

Team 9 OSUMCM 2018 Then consider the average time people listen to music per month as 137 hours then we need to shift the distribution to make it more reasonable so we take the standard deviation to become 40. SD = standard deviation = 40 Then assume Spotify s goal is to make about 20 percent more profits. 5.2 Function Evaluation At this section, we need to use help function to evaluate some parameters. We need to calculate the amount of people who switch to paid plan: S free = S O = 159M 71.55M = 87.45M S paid = O + e S free = 71.55M + (0.6 87.45M ) = 124.02M Then we can get the current average expense of each user in the Spotify. E 0 = CurrentSubscriptionFee O S 0 71.55M = = $4.5 159M Since our goal is to get about 20 percent, then we can derive expected average expense Ē as: Ē = g E 0 = 1.2 $4.5 =.4 From our model, we can get a table of the step price scheme. From out model, the music listening time decides the expense of users, so we can get an amount of steps. Also we know that t - T 0 = amount of steps times a step time, then we can get that T 1 is 12 hours. Here is the figure of step time and the money people will pay. 15/

Figure 5 - Step Price Scheme Time Listen to Music >= 9 hrs 257 hrs 245 hrs 233 hrs 221 hrs 209 hrs 197 hrs 185 hrs 173 hrs 161 hrs 149 hrs 137 hrs 125 hrs 113 hrs 101 hrs 89 hrs 77 hrs 137 hrs 53 hrs 41 hrs 29 hrs 17 hrs 5 hrs 0 minutes.4.6.8.8.8.8.8 Cost Per Month (cumulative) 0 $9.6 $9 $9 $9 $9 $9 $9 $8.2 $7.4 $6.6.8 $4 $3 $2 16/

Team 9 OSUMCM 2018 T 1 = 12 hours Then with the normal distribution, divide the people into different parts. The partition way is that people who listen the same step time music will be in the same partition. With the math model appling normal distribution's knowledge to get population apply Z = X μ σ Φ(z) = 1 z 2π e t2 /2 dt and Normal Distribution Cumulative Table (Appendix 2) We can derive the population of people who are in a same partition by multiplying the total number of users S. Because of the model we take is discrete, then with the model we take formula R(t) = i=0 Φ i Sp i to get the total revenue. Below is the table about revenue of different partitions. Table 2 - Population and Revenue i Price ($) Lower Bound Upper Bound Population (million) Φ Population Paid (million) 1 0 0 5 0.00017 0.02703 0 2 1 5 17 0.00086 0.13674 0.13674 3 2 17 29 0.0021 0.3339 0.6678 4 3 29 41 0.0047 0.7473 2.2419 5 4 41 53 0.0096 1.54 6.1056 6 5 53 65 0.018 2.862 14.31 7-11 5 65 125 0.3461 55.0299 275.1495 11 5.8 125 137 0.1179 18.7461 108.72738 17/

i Price ($) Lower Bound Upper Bound Population (million) Φ Population Paid (million) 12 6.6 137 149 0.1179 18.7461 123.724 13 7.4 149 161 0.1078 17.1402 1.83748 14 8.2 161 173 0.0901 14.3259 117.47238 15 9 173 185 0.0689 10.9551 98.5959 16-20 9 185 245 0.1116 17.7444 159.6996 21 9.6 245 257 0.0021 0.3339 3.20544 22 10 257 9 0.00086 0.13674 1.3674 Revenue: 1038.24138 According to the data we searched in the internet, we can find that Spotify share about 50 per cent of the interest to artists. So with this data we can compute the each song per stream now. ArtistShare = 50 % AvgPayArtistPerStream = TotalRevenue ArtistShare NumberOf Streams 038.24M 50 % = 44700M =.011613423 18/

6. Strengths, Limitations and Improvement 6.1 Strength 6.1.1 Customer To the customer of Spotify, this scheme will convert most of free users to paid users because the scheme starting at an affordable price. Moreover, this scheme will also attract new users because it offers 5 hours per month free music without advertisement. We assume users would like to pay as little as possible. So, the starting price is and typical price is less or equal to.8, which makes up 50% users. Price (less or equal than) Population 38%.8 50% $8.2 81% $9 99% 6.1.2 Artist Considering musical artist, the new model will resolve their complain as the new scheme will create more revenue for Spotify and thus more profit for musical artists. In comparison to the original pricing plans, artists incomes increases from.0072 to.0116 per stream, which is an 61% improvement. More incomes will also bring Spotify more sustainable music sources, which benefit the company and artists in the long run. In the model, we improve the income for all artist in general. So, we expect small-scale, mid-range and popular artist to benefit from SPS the same. 19/

6.1.3 Spotify From the Step Pricing Scheme, Spotify will earn a higher revenue. In comparison to its current revenue, $795M per month, the new pricing scheme will bring Spotify 038M per month, which is an increase of 30% net revenue. 6.1.4 Flexibility This model brings flexibility to the evaluation. Parameters including year, step price and step time are flexible to be changed. Although this report derive the scheme by step price.00, step time 12 hours and free threshold 5 hours, Spotify could also use this model to derive price scheme by other step prices (.01), step times (1 minute) or free threshold (1 hours per day). Because number of users, average music listening hours are changing, they are also parameters of this model, which could be changed by Spotify. The flexibility of this model brings Spotify more business values. 6.2 Limitation We use the normal distribution to compute and analyze, but in the real world the distribution about length of people's listening to the music time is not exactly fit to the normal distribution. Also, because of lack of data between listening time and population, we have to assume standard deviation to 40. However, normal distribution with standard deviation 40 is still a reasonable model. As for piracy, we expect it to increase because the new SPS plan limit the free plan to 5 hours per month. Unsatisfying users who would not like to pay for music are very likely to conduct piracy. 6.3 Improvement Besides, even to our model, there is something we can improve. We take the discrete model as a way to compute due to the time limitation, so to each separate time, we take the accumulated price to derive a fixed cumulative cost p i in a step time T 1, but 20/

time could be considered continuously and a continuous model could gain more accuracy. However, a discrete pricing scheme is reasonable for a business pricing and that is the reason we chose discrete model. 21/

7. Conclusion In this report, we derived a pricing scheme for Spotify that take music listening time into consideration. The main idea of the pricing scheme is to transfer free users into low cost paid users. Through discrete modeling, we obtained a Step Pricing Scheme that gives user 5 hours per month free music, 12 hours music for, 125 hours music for and unlimited music for 0. The estimated revenue for Spotify increases 30% and the price per stream for artist increases 61% to 0.012 per stream. In conclusion, this pricing scheme brings profit to all thresholds - customer, artists and Spotify. 22/

References Infographic: Spotify Reaches 83 Million Premium Subscribers. Retrieved November 4, 2018, from https://www.statista.com/chart/13098/spotify-subscribers-and-activeusers/ Monthly Active Spotify Users Worldwide 2017 Statistic. Retrieved November 4, 2018, from https://www.statista.com/statistics/367739/spotify-global-mau/ NIELSEN MUSIC MID-YEAR REPORT U.S. 2018. Retrieved November 4, 2018, from https://www.nielsen.com/content/dam/corporate/us/en/reports-downloads/2018- reports/us-midyear-music-report-2018.pdf Nielsen: Music Streams Doubled In 2015, Digital Sales Continue To Fall. Retrieved November 4, 2018, from https://techcrunch.com/2016/01/07/nielsen-music-streamsdoubled-in-2015-digital-sales-continue-to-fall/ Deahl, D. (2018, September 20). Spotify Will Now Let Artists Directly Upload Their Music To The Platform. Retrieved November 4, 2018, from https:// www.theverge.com/2018/9/20/17879840/spotify-artist-direct-upload-independentmusic McIntyre, H. (2017, November 9). Americans Are Spending More Time Listening To Music Than Ever Before. Retrieved November 4, 2018, from https:// www.forbes.com/sites/hughmcintyre/2017/11/09/americans-are-spending-more-timelistening-to-music-than-ever-before/#73cdc2c02f7f 23/

Appendix 1 - Sample Pricing Scheme Example of Step Pricing Scheme Time Listen to Music Cost Per Month (cumulative) 2300 minutes 2200 minutes 2100 minutes 2000 minutes 1900 minutes 1800 minutes 1700 minutes 1600 minutes 1500 minutes 1400 minutes 1300 minutes 1200 minutes 1100 minutes 1000 minutes 900 minutes 800 minutes 700 minutes 600 minutes 500 minutes 400 minutes 300 minutes 200 minutes 100 minutes 0 minutes.4.6.8.8.8.8.8 0 $9.6 $9 $9 $9 $9 $9 $9 $8.2 $7.4 $6.6.8 $4 $3 $2 24/

Appendix 2 - Normal Distribution Cumulative Table z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.0 0.00000 0.00399 0.00798 0.01197 0.01595 0.01994 0.02392 0.02790 0.03188 0.03586 0.1 0.03983 0.04380 0.04776 0.05172 0.05567 0.05962 0.06356 0.06749 0.07142 0.07535 0.2 0.079 0.08317 0.08706 0.09095 0.09483 0.09871 0.10257 0.10642 0.110 0.11409 0.3 0.11791 0.12172 0.12552 0.12930 0.13307 0.13683 0.14058 0.14431 0.14803 0.15173 0.4 0.15542 0.15910 0.16276 0.16640 0.17003 0.17364 0.17724 0.18082 0.18439 0.18793 0.5 0.19146 0.19497 0.19847 0.20194 0.20540 0.20884 0.212 0.21566 0.21904 0.22240 0.6 0.22575 0.22907 0.23237 0.23565 0.23891 0.24215 0.24537 0.24857 0.25175 0.25490 0.7 0.25804 0.115 0.424 0.730 0.27035 0.27337 0.27637 0.27935 0.28230 0.28524 0.8 0.28814 0.29103 0.29389 0.29673 0.29955 0.30234 0.30511 0.30785 0.31057 0.31327 0.9 0.31594 0.31859 0.32121 0.32381 0.339 0.32894 0.33147 0.33398 0.33646 0.33891 1.0 0.34134 0.34375 0.34614 0.34849 0.35083 0.35314 0.35543 0.35769 0.35993 0.36214 1.1 0.36433 0.36650 0.36864 0.37076 0.37286 0.37493 0.37698 0.37900 0.38100 0.38298 1.2 0.38493 0.38686 0.38877 0.39065 0.39251 0.39435 0.39617 0.39796 0.39973 0.40147 1.3 0.40320 0.40490 0.40658 0.40824 0.40988 0.41149 0.41308 0.41466 0.41621 0.41774 1.4 0.41924 0.42073 0.42220 0.42364 0.42507 0.447 0.42785 0.42922 0.43056 0.43189 1.5 0.43319 0.43448 0.43574 0.43699 0.43822 0.43943 0.44062 0.44179 0.44295 0.44408 1.6 0.44520 0.44630 0.44738 0.44845 0.44950 0.45053 0.45154 0.45254 0.45352 0.45449 1.7 0.45543 0.45637 0.45728 0.45818 0.45907 0.45994 0.46080 0.46164 0.46246 0.46327 1.8 0.46407 0.46485 0.46562 0.46638 0.46712 0.46784 0.46856 0.469 0.46995 0.47062 1.9 0.47128 0.47193 0.47257 0.47320 0.47381 0.47441 0.47500 0.47558 0.47615 0.47670 2.0 0.47725 0.47778 0.47831 0.47882 0.47932 0.47982 0.48030 0.48077 0.48124 0.48169 2.1 0.48214 0.48257 0.48300 0.48341 0.48382 0.48422 0.48461 0.48500 0.48537 0.48574 2.2 0.48610 0.48645 0.48679 0.48713 0.48745 0.48778 0.48809 0.48840 0.48870 0.48899 2.3 0.48928 0.48956 0.48983 0.49010 0.49036 0.49061 0.49086 0.49111 0.49134 0.49158 2.4 0.49180 0.49202 0.49224 0.49245 0.496 0.49286 0.49305 0.49324 0.49343 0.49361 2.5 0.49379 0.49396 0.49413 0.49430 0.49446 0.49461 0.49477 0.49492 0.49506 0.49520 2.6 0.49534 0.49547 0.49560 0.49573 0.49585 0.49598 0.49609 0.49621 0.49632 0.49643 2.7 0.49653 0.49664 0.49674 0.49683 0.49693 0.49702 0.49711 0.49720 0.49728 0.49736 25/

z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 2.8 0.49744 0.49752 0.49760 0.49767 0.49774 0.49781 0.49788 0.49795 0.49801 0.49807 2.9 0.49813 0.49819 0.49825 0.49831 0.49836 0.49841 0.49846 0.49851 0.49856 0.49861 3.0 0.49865 0.49869 0.49874 0.49878 0.49882 0.49886 0.49889 0.49893 0.49896 0.49900 3.1 0.49903 0.49906 0.49910 0.49913 0.49916 0.49918 0.49921 0.49924 0.499 0.49929 3.2 0.49931 0.49934 0.49936 0.49938 0.49940 0.49942 0.49944 0.49946 0.49948 0.49950 3.3 0.49952 0.49953 0.49955 0.49957 0.49958 0.49960 0.49961 0.49962 0.49964 0.49965 3.4 0.49966 0.49968 0.49969 0.49970 0.49971 0.49972 0.49973 0.49974 0.49975 0.49976 3.5 0.49977 0.49978 0.49978 0.49979 0.49980 0.49981 0.49981 0.49982 0.49983 0.49983 3.6 0.49984 0.49985 0.49985 0.49986 0.49986 0.49987 0.49987 0.49988 0.49988 0.49989 3.7 0.49989 0.49990 0.49990 0.49990 0.49991 0.49991 0.49992 0.49992 0.49992 0.49992 3.8 0.49993 0.49993 0.49993 0.49994 0.49994 0.49994 0.49994 0.49995 0.49995 0.49995 3.9 0.49995 0.49995 0.49996 0.49996 0.49996 0.49996 0.49996 0.49996 0.49997 0.49997 4.0 0.49997 0.49997 0.49997 0.49997 0.49997 0.49997 0.49998 0.49998 0.49998 0.49998 /