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I am trying to reconcile the derivation of a demand equation with what I actually run in an OLS model. After solving $$max_{x_1,x_2}U(x_1,x_2)= \alpha ln(x_1) + \beta ln(x_2)$$ subject to $$p_1x_1+p_2x_2=m$$ I get the familiar derived demand equation $x_1^*=p_1m\frac{\alpha}{\alpha+\beta}$. So far, this lines up with intuition. I see a price, an income, and some parameters. However, for a lot of goods that are estimated in the literature via OLS, economists also include demographic information like race, age, and sex. How can this be justified by the demand equation? The only two options I see are that income or the parameters are functions of these variables. Typically, most data sets will have some measure of income observed so that rules income out, and leaves only the parameters. Does this mean when I write an OLS equation like this $$X_i=\beta_0 + \beta_1 M_i + \beta_2demographics_i$$ I am somehow implicitly imposing that my $\alpha$ and $\beta$ from above are functions of demographics?

Thanks for any input. I'd also really appreciate any papers/articles that talk about this topic specifically, so far I haven't had any luck finding them.

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