Linking top-down and bottom-up models for analyzing electricity price-based demand response: Expenditure constraint is violated? 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.
Constant expenditure in monetary terms requires constant Income indeed. Does the paper you site adjusts its equations and the data to account for possible changes in income present in the data?  

What the CD utility function $U = X^aY^b$ does impose, is constant expenditure shares,

$$\frac{X^*p_x}{I} = \frac {a}{a+b}$$

So the question would be for the appropriateness of the utility function specification whether the data support $X^*p_x/I = const.$

On the methodological front the OP writes


  ...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,...


This begs the question: if consumer demand / consumer expenditure is postulated from outside the model, exactly what is the purpose of solving the utility maximization problem? The main purpose of utility maximization is to provide the demand functions, namely to show how utility considerations determine choice and consumer decisions.  

Are we sure about what the authors are doing exactly?