demand conditions      Marginal Revenue      Marginal Cost
P1=$400-4Q              MR1=$400-8Q           MC1=$350+2Q
P2=$100-5Q              MR2=$100-10Q          MC2=$50

A. What else do we need to know to determine whether or not this firm
has any profitable choices?
B. How much of the primary product should the firm produce to maximize
its total profits?
C. If the market collapses on the by-product, how much of the primary
product should the firm produce to maximize its total profits?

A. While we know the price per product and can therefore calculate the
total revenues, we do not know the firm's fixed costs.  Therefore,  we
cannot calculate its total costs in order to determine if a net profit
is made from a given quantity of production and sales.

B.  "The maximum-profit price and quantity of a monopolist come where
its marginal revenue equals its marginal cost."  "Economics" 14th
Edition, by Paul A. Samuelson & William D. Nordhaus, McGraw-Hill Inc.,
1992, page 173

For Product 1, 350 + 2Q = 400 - 8Q results in Q equaling 5.  For
Product 2, 50 = 100 - 10Q also results in Q equaling 5.  Therefore,
the maximum quantity of the primary product that should be produced to
maximize the firm's profits is 5.

C.  Because Product 2 did not affect the equilibrium quantity of
Product 1 to be produced because its equilibrium quantity was also 5,
the firm should still produce only 5 units of Product 1 because it
will begin to decrease its profits if it produces more than 5.

Here is a spreadsheet detailing the marginal revenue, marginal cost,
and marginal profit for quantities of production varying from 1 to 6. 
You can see that marginal profit becomes negative when more than a
quantity of 5 is produced.

	Quantity	Price	MR	MC	MP
Product 1	1	396	392	352	40
Product 2		95	90	50	40
Product 1	2	392	384	354	30
Product 2		90	80	50	30
Product 1	3	388	376	356	20
Product 2		85	70	50	20
Product 1	4	384	368	358	10
Product 2		80	60	50	10
Product 1	5	380	360	360	0
Product 2		75	50	50	0
Product 1	6	376	352	362	-10
Product 2		70	40	50	-10

I hope the above information meets your needs.

Sincerely,

Wonko

I really appreciated the matrix for part C but it could have used an explanation.