Adam and his friends Brigit , Cheryl , David , Emily , Frank, Gail , Henry, Ivan , and Juliet have two choices for weekend activities . They can...

urban economics question

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Adam and his friends Brigit , Cheryl , David , Emily , Frank, Gail , Henry, Ivan , and Juliet have two choices forweekend activities . They can either go to the local park or get together in Adam's hot tub . The local parkisn't much fun , which means that the benefits from being there are low on the friends' common utilityscale . In fact , each of the friends receives a benefit equal to 3 " utils" from being at the park . This benefitdoesn't depend on how many of the friends go to the park . Adam's hot tub , on the other hand , can be fun ,but the benefits of using it depend on how many of the friends are present . When the tub isn't toocrowded , it's quite enjoyable . When lots of people show up , however , the tub is decidedly less pleasant .The relationship between benefit per person ( measured in utils ) and the number of people in the hot tub( denoted T) isAB = 2 + & T - TZ, where AB denotes " average benefit "!a) Using the above formula , compute AB for T = 1, 2, 3. 8, 9, 10. Next compute total benefit from USE Ofthe hot tub for the above T values as well as T = O. Total benefit is just T times AB. Finally , compute*marginal benefit ( MB ) , which equals the change in total benefit from adding a person to the hot tub . To dothis , adopt the following convention :`define MB at T = T' to be the change in total benefit when T changes from T' - 1 to T' ( in other words , MBgives the change in total benefits from entry of the " last " person ) . Deviation from this convention will leadto inappropriate answers . For example , computation of MB using calculus will lead you astray given thatwe're dealing with a discrete rather than continuous problem .60) Recalling that the park yields 3 utils in benefits to each person , find the equilibrium size of the groupusing the hot tub . Show that ( aside from the owner Adam ) we can't be sure of the identities of the otherhot tub users . ( Hint : In contrast to the freeway case , the relevant benefit number won't exactly equal 3 atthe equilibrium , with a similar outcome occurring in the other Cases considered below! !\[ ) Find the optimal size of the hot tub group , and give an explanation of why it differs from the equilibriumsize . Next compute the grand total of benefits for all the friends , which is the sum of total benefits for thehot tub group and total benefits for those using the park . Perform this computation for both theequilibrium and the optimal group sizes . What do your results show ?