A firm faces demand P$=100-$Q and constant marginal cost of $10.$ Due to asymmetric information, the manufacturing division of the firm determines the price charged to the sales division of the firm $($transfer price$)$ and the sales division has no information about the production cost of the manufacturing division.

i$.$ Determine the profit maximizing transfer price of the firm, that is$,$ the price that the manufacturing division charges the sales division for the product. Explain in economic terms.

ii$.$ Derive the transfer price, that is$,$ the price that the manufacturing division charges the sales division, when the manufacturing division exploits the asymmetric information. Show that the overall profit of the firm is lower than in $($i$).$ Compute the profit that the manufacturing division makes and the profit that the sales division makes. Explain in economic terms.