After graduating , Julia decided to pursue her dreams and bought a bakery in Lisbon. At the beginning of each day, she buys pasteis de nata from Tilman, a fellow graduate, at 1.00 EUR per unit and sells them at 3.00 EUR per unit. Tilman makes pasteis de nata at 0.10 EUR per unit. At the end of the day, Julia throws away any extra pasteis de nata she could not sell. On any given day, demand for pasteis de nata at Julia's bakery is normally distributed with mean 100 and standard deviation 30.

a. How many pasteis de nata should Julia purchase from Tilman on a given day? What are Julia and Tilman' expected profits from pasteis de nata sales?

Tilman now decides to start a charity where he gives away unsold pasteis de nata to homeless people. He knows the demand at Julia's bakery and thus offers Julia 0.30 EUR for each unsold pastel de nata in her bakery, at the end of the day.

b. Julia accepts Tilman' new offer. How many pasteis de nata should Julia now purchase from Tilman at the beginning of the day? Let us call this quantity Q. Under these settings, what are Tilman' and Julia's expected profits from pasteis de nata sales?

It is now summer time and students do not have regular classes. Therefore, demand for pasteis de nata at Julia's bakery is still normally distributed with mean 100, but the standard deviation is now 60. Tilman is aware of this change in demand, but is not very enthusiastic about making changes to his well-established manufacturing process just for the period of summer.

c. How much money should Tilman offer Julia or how much money should Tilman ask from Julia for each unsold pastel de nata so that Julia still orders Q from him in the morning? What is Tilman' expected revenue? Please disregard the extra information in part b and use the initial parameters given in the main question.