The setting is simple, i.e. formula for demand of service/product is linear
$$
d = \alpha - \beta p
$$
where $ \alpha $ is maximum demand, $ \beta $ is some coefficient, and $ p $ is price. There are two firms, with different prices $p_1$ and $ p_2 $, and the prices are fixed, so there is no competition. The fraction of total demand for the first firm is captured by the logit rule
$$
f_1 = \frac{e^{-\gamma p_1}}{e^{-\gamma p_1} + e^{-\gamma p_2}}
$$
where $ \gamma $ is some sensitivity coefficient. Similar for the other price/firm.
Question: how can I compute total demand in this setting?
EDIT: I asked this question on QF forum, but without luck. I had an answer, but it turns out that it is wrong. Now I have another answer: $$ d(p_1, p_2) = W - \beta\cdot\min\{p_1, p_2\} $$ It seems OK, but I'm not sure.