Maximizing Multi-Product Firm Profits with Symmetric Matrices

[TheBestMilke]

Homework Statement

A multi-product firm has total cost function C(q) = q^tAq 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) = q^tDq - e^tq where D and e are appropriate matrices.

Homework Equations

We are given that Total Cost is, C(q) = q^tAq.
We also are given a production function: q = Bp + c

Total profits is just Total Revenue, p*q, minus Total Costs.
Hence:
π = p*q - C(q).
And we want to get it into a form like this:
π(q) = q^tDq - e^tq

The Attempt at a Solution

So far I have done some manipulation of the production function:
q = Bp+c
B^-1(q-c)=p
and then substituted into the equation: π(q) = p*q-C(q) = (B^-1(q-c))q - q^tAq

The remaining dificulities I'm having is figuring out how to get this resulting equation to look something like the requested, π(q) = q^tDq - e^tq. 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.

Thanks!

 

[foxjwill]

π = p*q - C(q).

Try thinking about this equation right here. Remember that both \mathbf{p} and \mathbf{q} are vectors, while n is a scalar, so the "*" is scalar multiplication. But scalar multiplication can be rewritten using matrix multiplication, right? And once you write it in that form, try looking at its transpose.

 

[TheBestMilke]

I'm not sure if I completely understand, but this is what I've managed to come up with.
If we substitute the solved p value into the equation as well as the total cost we come up with something like:

n=p*q - c(q) = [B^-1(q-c)]q - qAq^t

If I take the transpose of the substituted p, p^t, I get:

p^t = [B^-1(q-c)]^t = (B^-1)^t(q-c)^t = (B^-1)^t(q^t-c^t).

Am I then able to try and carry some terms through, such as:

qp^t = q*(B^-1)^t(q^t-c^t) = q(B^-1)^tq^t-B^-1c^tq

And then combine with the original, n = qp - c(q) into:

n = q(B^-1)^tq^t - B^-1c^tq - qAq^t.

In this case the B^-1c^t could be equal to the e^t but I would still have to find a way to combine the q(B^-1)tq^t - qAq^t.

Thank you for your help. It's been quite some time since I've done anything with matrices and my brain seems to be full of rust.

 

[TheBestMilke]

I've reworked this now, trying to pay a little more attention to the way the distributive property works with matrices as well as how transpose carries through. If I start from defining the profit, n.

n = TR - TC.
TR = pq
TC = C(q) = q^tAq

We know that q = Bp+c which means that B^-1(q-c) = p.
Now if I take the transpose of that, I get:
p^t=(B^-1(q-c))^t=(q-c)^t(B^-1)^t=(q^t-c^t)(B^-1)^t=(q^t-c^t)(B^t)^-1=(q^t-c^t)B^-1=q^tB^-1-c^tB^-1 since we knew B was symmetrical, and therefore B=B^t

Now when I substitute that back into the equation, I get:
p^tq=(q^tB^-1-c^tB^-1)q=q^tB^-1q-c^tB^-1q

If I do the same with taking C(q)^t, I get:
C(q)^t = (qAq^t)^t=q^tA^tq=q^tAq because A is symmetrical as well.

Now, putting that all into the equation we arrive at:
n = p^tq-C(q)^t
n = q^tB^-1q-c^tB^-1q-q^tAq=q^t(B^-1-A)q-c^tB^-1q
and that falls into the n = q^tDq - e^tq kind of format.

*I guess the issue is this: if I let n = pq - c = p^tq-c^t it works. Is this possible or do I have to start from the beginning and say n^t = (pq)^t-c^t?

 

[TheBestMilke]

I figured it out. I don't know what I was possibly thinking to begin with as it was actually a simple task. Thanks for bearing with my multiple posts and bumps.