The Orsilo Corporation makes and sells $10,000$ multisystem music players each year. Its assembly division purchases components from other divisions of Orsilo or from external suppliers and assembles the multisystem music players. In particular, the assembly division can purchase the CD player from the compact disc division of Orsilo or from Johnson Corporation. Johnson agrees to meet all of Orsilo$$s quality requirements and is currently negotiating with the assembly division to supply $10,000$ CD players at a price between $$38$ and $$45$ per CD player.

A critical component of the CD player is the head machanism that reads the disc. To ensure the quality of its multisystem music players, Orsilo requires that if Johnson wins the contract to supply CD players, it must purchase the head mechanism from Orsilo$$s compact disc division for $$20$ each.

The compact disc division can manufacture at most $12,000$ CD players annually. It also manufactures as many additional head mechanisms as can be sold. The incremental cost of manufacturing the head mechanism is $$15$ per unit. The incremental cost of manufacturing a CD player $($including the cost of the head mechanism$)$ is $$25$ per unit, and any number of CD players can be sold for $$35$ each in the external market.

Required

$1.$ What are the incremental costs minus revenues from sales to external buyers for the company as a whole if the compact disc division transfers $10,000$ CD players to the assembly division and sells the remaining $2,000$ CD players on the external market?

$2.$ What are the incremental costs minus revenues from sales to external buyers for the company as a whole if the compact disc division sells $12,000$ CD players on the external market and the assembly division accepts Johnsons offer at $($a$)$ $$38$ per CD player or $($b$)445$ per CD player?

$3.$ What is the minimum transfer price per CD player at which the compact disc division would be willing to transfer $10,000$ CD players to the assembly division?

$4.$ Suppose that the transfer price is set to the minimum computed in requirement $3$ plus $$1,$ and the division managers at Orsilo are free to make their own profit$-$maximizing sourcing and selling decisions. Now, Johnson offers $10,000$ CD players for $$40\mathrm{.}50$ each.

a$)$ What decisions will the managers of the compact disc division and assembly division make?

b$)$ Are these decisions optimal for Orsilo as a whole?

c$)$ Based on this excersize, at what price would you recommend the transfer price be set?