The Rainbow Company manufactures and sells television sets. Its assembly division $($AD$)$ buys television screens from the screen division $($SD$)$ and assembles the TV sets. The SD$,$ which is operating at capacity, incurs an incremental manufacturing cost of $$90$ per screen. The SD can sell all its output to the outside market at a price of $$115$ per screen, after incurring a variable marketing and distribution cost of $$8$ per screen. If the AD purchases screens from outside suppliers at a price of $$115$

per screen, it will incur a variable purchasing cost of $$8$ per screen. Rainbow's division managers can act autonomously to maximize their own division's operating income.

$1.$ What is the minimum transfer price at which the SD manager would be willing to sell screens to the AD$.$

$2.$ What is the maximum transfer price at which the AD manager would be willing to purchase screens from the SD$?$

$3.$ Now suppose that the SD Can sell on $85\%$ of its output capacity of $20,000$ screens per month on the open market. Capacity cannot be reduced in the short run. The AD can assemble and sell more than $20,000$ TV sets per month.

a$.$ What is the minimum transfer price at which the SD manager would be willing tosell screens to the AD$?$

b$.$ From the point of view of Rainbows management, how much of the SD output should be transferred to the AD$?$

c$.$ If Rainbow mandates the SD and AD managers to $$split the difference$$ on the minimum and maximum transfer prices they would be willing to negotiate over, what would be the resulting price? Does this price achieve the outcome desired in requirement $3$b$?$