Incorporate fixed effects in pricing decisions with linear models

Question

I have the following model the predict the log of volume:

fixed_effect + day_of_week_effect + elasticity * (Price - competitor_price).

I want to formulate an optimization problem that maximize my profit such that:

$max$ profit = $max$ log of volume + log of unitary margin.

With unitary margin being my price minus my costs.

My problem is that given the fact that my decision variable is the price. The optimal price is the same for whatever day of week or fixed effect, since their derivative is equal to zero.

How can I integrate the fact that I want to put higher prices when day effect leads to higher volumes and vice versa. I thought about having an interaction term between price and the fixed effects, or modifying elasticity by multiplying it by the day of week. But I would like to know if their is any other approach used for these kind of cases

Answer 1

say price $p$ is indexed by day of the week $d$ & so is the effect $E$. Then a constraint like
$E_d - E_{d+1} \le M(p_d - p_{d+1})$
$p_d - p_{d+1} \le M(E_d - E_{d+1})$