Solutiona. Let us consider the firm

Solution
a. Let us consider the firm as a whole to begin with. We can obtain the profit-maximizing position by finding the MC and MR for the firm as a whole and equating them. First we can obtain the total cost to the firm by adding the costs of the manufacturing (N) and marketing (K) divisions:
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Note that 100Q has been deducted from the costs of the marketing division, since this cost related to paying the transfer price to the manufacturing division, and was not a cost to the firm as a whole. Thus:
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Price can be derived by substituting the value of Q into the demand equation; price is the same as average revenue, thus the demand equation is given by:
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This is the external price charged by the firm’s marketing division. The total profit of the firm can be obtained from its profit function:
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The above prices, output and profit are optimal for the firm as a whole.

b. However, the above result will not be achieved with the current strategy of charging a transfer price of $100. Consider the manufacturing division: profit N is maximized where MRN =MCN
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Total profit of the firm is obtained by substituting Q=40 into the profit function in (10.21), giving a profit of $1,360.
Now considering the marketing division:
Profit is maximized where MR =MC
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Substituting this value of Q into the demand equation P= 500-8Q, the firm will charge the price of $340.
Total profit of the firm is obtained by substituting Q=20 into the profit function, giving a value of $4,560.
c. In order to obtain the transfer price we need to consider the transferring division’s marginal cost, that is MCN. With no external market this represents the firm’s optimal transfer price. The reason for this is that the firm wants to produce the prof it-maximizing output for the firm as a whole, and the way to motivate the manufacturing (transferring) division to produce this output is to ensure that the division maximizes its profit at this output. Setting the transfer price, which is the manufacturing division’s MR, equal to its MC at the overall profit-maximizing output ensures this. This condition can be seen more plainly by using the following analysis, which represents an alternative method of obtaining the optimal output for the firm, but which also shows how to compute the transfer price.