Question

4 . Individual Problems 22-1

Suppose that a paper mill “feeds” a downstream box mill. For the downstream mill, the marginal profitability of producing boxes declines with volume. For example, the first unit of boxes increases earnings by $30, the second by $27, the third by $24, and so on, until the tenth unit increases profit by just $3.

The cost the upstream mill incurs for producing enough paper (one “unit” of paper) to make one unit of boxes is $9.50.

Assume the two mills operate as separate profit centers, and the paper mill sets the price of paper. It follows that the marginal profitability of boxes represents the highest price that the box division would be willing to pay the paper division for boxes.. Furthermore, assume that fixed costs are $0 for the paper mill.

The following table summarizes the quantity, total revenue, and marginal costs from the perspective of the paper mill for selling paper to the box mill at various prices.

In the following table, fill in the marginal revenue, total cost, and total profit for the paper mill when selling paper to the box mill at each given price.

Price	Quantity	Total Revenue	Marginal revenue	Total Cost	Marginal Cost	Profit
(Marginal Profitability to the Box Mill)	(Units of Paper equivalent to One Box)	($)	($)	($)	($)	($)
($)
$30	1	$30		$9.50
					$9.50
$27	2	$54
					$9.50
$24	3	$72
					$9.50
$21	4	$84
					$9.50
$18	5	$90
					$9.50
$15	6	$90
					$9.50
$12	7	$84
					$9.50
$9	8	$72
					$9.50
$6	9	$54
					$9.50
$3	10	$30

If the paper mill sets the price of paper to sell to the box mill, it will set a price of (_____ ) and sell (______) units of paper to the box mill. Profits will be(_____)

for the paper mill. Companywide  profits will be(______). (Hint: Recall that the prices in the table represent the marginal profitability of each unit of paper, or box, to the box mill.)

Suppose the paper mill is forced to transfer paper to the box mill at marginal cost ($9.50).

In this case, the box mill will demand(_____) units of paper. This leads to companywide profits of (_____).

.

Answer 1

Price	Quantity	Total Revenue	Marginal revenue	Total Cost	Marginal Cost	Profit = Total Revenue - Total cost
(Marginal Profitability to the Box Mill)	(Units of Paper equivalent to One Box)	($)	($)	($)	($)	($)
($)
$30	1	$30	$30	$9.50	$9.50	$20.5
	$9.50
$27	2	$54	$27	$19	$9.50	$35
	$9.50
$24	3	$72	$24	$28.5	$9.50	$43.5
	$9.50
$21	4	$84	$21	$38	$9.50	$46
	$9.50
$18	5	$90	$18	$47.5	$9.50	$42.5
	$9.50
$15	6	$90	$15	$57	$9.50	$33
	$9.50
$12	7	$84	$12	$66.5	$9.50	$17.5
	$9.50
$9	8	$72	$9	$76	$9.50	$-4
	$9.50
$6	9	$54	$6	$85.5	$9.50	$-31.5
	$9.50
$3	10	$30	$3	$95	$9.50	$-65

If the paper mill sets the price of paper to sell to the box mill, it will set a price of (__$21___ ) and sell (___4__) units of paper to the box mill. Profits will be(__$46__)

for the paper mill. Companywide  profits will be(_$148____). (Hint: Recall that the prices in the table represent the marginal profitability of each unit of paper, or box, to the box mill.)

Suppose the paper mill is forced to transfer paper to the box mill at marginal cost ($9.50).

In this case, the box mill will demand(__7___) units of paper. This leads to companywide profits of (__$164.5___).

Explanation: The paper mill will set the price of paper at a level where its profit is highest in this case at 4 units of paper the profit of paper mill is highest so the price corresponding to 4 units is $21. The profit earned by paper mill at this level of output is $46 and the profitablity of paper box is $102 ($30+$27+$24+$21) so, total profit company wide is $148. At cost $9.50 the box mill would demand 7 units of paper because till this level their marginal profit is higher than the cost. The company wide profit in this case would be $147+$17.5 = $164.5 where $147 is the profit of box mill and $17.5 is the profit of paper mill.