# Problem with Optimizing Profit in Log-Linear Demand Model

I've trained a linear model to predict log(volume) to capture the elasticity of demand with respect to price difference between my product and a competitor's:

$$log(volume)= constant+elasticityCoef×(myPrice−competitorPrice)$$

I want to optimize profit, and given the log-linear form of the demand model, I assume the objective function should be: $$Objective=log(volume)+log(myPrice−costs)$$ equivalent to $$Objective=constant+ elasticityCoef×(myPrice−competitorPrice)+log(myPrice−Costs)$$

But using this model, I've noticed that for any given competitor price, the optimal price for my product remains fixed (verified using simple differentiation). This seems counterintuitive. What could I be missing?

Your objective function is additive separable between $$myPrice$$ and $$competitorPrice$$. This means that you can write: $$Objective = f(myPrice) + g(competitorPrice).$$ For some functions $$f$$ and $$g$$. In such case, the optimal level of $$myPrice$$ will indeed be independent of the value of $$competitorPrice$$. Notice that maximizing $$Objective$$ will choose $$myPrice$$ to maximize $$f$$ (whatever the value of $$g$$).
If you want to have something non-separable, you need to change your first stage equation such that $$myPrice$$ and $$competitorPrice$$ is no longer additively separable.
• @MarcM If you want the optimal price to depend also on the price of the competitor, your regression should have a form such that $myPrice$ and $competitorPrice$ are non-separable. For example $\ln(Volume) = constant + \alpha \ln(ownPrice) + \beta \ln(competitorPrice) + \gamma \ln(ownPrice) \ln(competitorPrice)$.