A Firm Has Two Semiautonomous Divisions: Production and Marketing $$\begin{array}{l}M C _ { P } = 2 Q\\M C _ { M } = Q\end{array}$$

A firm has two semiautonomous divisions: production and marketing. The production division manufactures a product that is purchased and then resold by the marketing division. The marginal cost functions for the production division and for the value added by the marketing division are defined below.
$$\begin{array}{l}M C _ { P } = 2 Q\\M C _ { M } = Q\end{array}$$ The demand function for the product is:
$$Q D = 100 - P$$ (i)Assume that there is no external market for the output of the production division. How many units should be produced and what transfer price should be paid to the production division by the marketing division?
(ii)Assume that the external market for the output of the production division is perfectly competitive and that the market price is $52. How many units should be produced by the production division, how many should be purchased by the marketing division, what transfer price should be paid to the production division by the marketing division, and what price should be charged for the product by the marketing division?