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## Homework Statement

A multi-product firm has total cost function C(

**q**) =

**q**

^{t}A

**q**and faces inter-related but linear demand schedules for the n goods it produces:

**q**= B

**p**+

**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**

^{t}D

**q**- e

^{t}

**q**where D and

**e**are appropriate matrices.

## Homework Equations

We are given that Total Cost is, C(

**q**) =

**q**

^{t}A

**q**.

We also are given a production function:

**q**= B

**p**+

**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**

^{t}D

**q**- e

^{t}

**q**

## The Attempt at a Solution

So far I have done some manipulation of the production function:

**q**= B

**p**+

**c**

B

^{-1}(

**q**-

**c**)=

**p**

and then substituted into the equation: π(

**q**) =

**p***

**q**-C(

**q**) = (B

^{-1}(

**q**-

**c**))

**q**-

**q**

^{t}A

**q**

The remaining dificulities I'm having is figuring out how to get this resulting equation to look something like the requested, π(

**q**) =

**q**

^{t}D

**q**-

**e**

^{t}

**q**. 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!