Mizuno Corporation produces and sells running shoes. The marketing division (the downstream division) of Mizuno faces the following the direct market demand equation as follows:

Q = 392 – 2P,

where Q is the number of pairs of shoes and P is the price of a pair of shoes.

Production of each pair of shoes requires 1 square yard of leather. The leather is shaped and cut by the Form division of Mizuno (the upstream division). The Form division’s only customer is the marketing division.

The total cost function for leather is

TC_U=0.5Q²_U+10Q_U+2,
where QU is the quantity of leather produced.

The total cost of assembling and selling shoes (excluding the leather) is
TC_D = 2Q² + 6Q + 4.

a) Write down the profit equations of downstream division (π_D), upstream division (π_U), and total profits (π).

b) Calculate the profit-maximizing quantity of shoes (Q*), quantity of leather (Q*_U ) and price of shoes (P*)

c) Determine the optimal transfer price (P*_U).

d) Using the profit equations of your answer in a), calculate the profits earned by the upstream division (π_U), the downstream division (π_D), and total profits (π).