I am desperate about some functions I would like to understand. I hope someone can help me to understand.
Lets suppose we have a demand-function like:
$$ D_i=\ \ 1/3+\ \ (2q_i-\ \sum_{j\neq i}\ q_j)/(2t) $$
And the corresponding profit function looks like:
$$ \pi_i=(p-c\ast q_i\ )\ast D_i. $$
In order to calculate quality and profit level in the Nash equilibrium, it is stated that the qualities are derived by the first order condition. The results are:
$$ q_i^*\ \ =\ \ p/c\ \ -\ \ t/3 $$
$$ \pi_i^*=ct/9 $$
I tried to derive the profit function by p and q, but I couldn't make it to the stated results.
May the solution be quite straightforward, I can't think about it and would appreciate any kind of help!
Thanks in advance KR
EDIT: For my understanding, the first order condition is the first derivative of the profit function that needs to set zero, what means the slope is zero. For identification, if its a minimum or maximum one need to evaluate the second derivative. I failed at the first derivative due to the summation sign. I know how to interpret it, but I don't know how to derive it. Following this, all I got is:
$$(d\pi)/dq = -c*(dD_i/dq)$$
EDIT: My recent solutions:
$$((d\pi)/(dq)) = (p-c*q)*(1/t)+(1/3+((2q-j)/(2t))*(-c)$$
$$ 0 \overset{!}{=} (p-c*q)*(1/t)+(1/3+((2q-j)/(2t))*(-c) $$
$$ q = (-3ct^2 + 6c-2t)/(6p-6ct^2) $$
Two things:
First, you should be differentiating $\pi_i$ with respect to $q_i$, i.e. find $\frac{\partial \pi_i}{\partial q_i}$, which means that all of the $q_j$ where $j \neq i$ will be treated as constant.
Second, your derivative is not quite right - remember that $(uv)' = uv' + u'v$, so in this case $\frac{\partial \pi_i}{\partial q_i} = (p - c q_i)\frac{\partial D_i}{\partial q_i} + D_i \frac{\partial}{\partial q_i}(p - c q_i)$.