I have a question about the contents of this paper*, which links a building energy model and a utility-maximization component. In it, the author tests several electricity prices using a Cobb-Douglas utility function. As I understand, the C-D function ($U=X^\alpha*Y^\beta$) stipulates that $X=\frac{\alpha}{\alpha+\beta}*\frac{Income}{P_x}$ at optimality. If Income and share parameters are fixed, that would mean $(X)(P_x)$ is always constant; if $P_x$ doubles, $X$ is halved, etc..
That is not what the paper finds in the linkage of the two components. P_electricity*ElectricityUse is not always constant. The expenditure equation implied by the C-D function is violated, but the paper applies an expenditure equation from the building energy model instead of it. They just maximize the utility function as the objective, subject to the budget constraint, and given the electricity use and expenditures from the building model, but do not factor in the implied relations.
Is this a violation that would nullify such linkage?
*Sorry if it is blocked behind a paywall, there isn't an alternate free copy.
Citation: Matar, W. "Households' response to changes in electricity pricing schemes: Bridging microeconomic and engineering principles." Energy Economics 75.