A multi-product firm has total cost function C(q) = qtAq and faces inter-related but linear demand schedules for the n goods it produces: q = Bp + c. Both A and B are symmetric and B is invertible. Obtain an expression for total profit π(q) in the form π(q) = qtDq - etq where D and e are appropriate matrices.
We are given that Total Cost is, C(q) = qtAq.
We also are given a production function: q = Bp + c
Total profits is just Total Revenue, p*q, minus Total Costs.
π = p*q - C(q).
And we want to get it into a form like this:
π(q) = qtDq - etq
The Attempt at a Solution
So far I have done some manipulation of the production function:
q = Bp+c
and then substituted into the equation: π(q) = p*q-C(q) = (B-1(q-c))q - qtAq
The remaining dificulities I'm having is figuring out how to get this resulting equation to look something like the requested, π(q) = qtDq - etq. I'm also not sure if the algebra is fully legal.
While I know this isn't Physics, it's really just math. The only econ part of it is in the word problem and equation definitions. If you could help at all I would be most appreciative. If you need any more info please let me know.