# 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.

• The main point is that a C-D representation alone would have an own-price elasticity of -1. The question then revolves around: By having the expenditures computed in the residential model, is the utility maximization still valid? Also, given that a CES function would still have the limitation of constant price elasticity, maybe it wouldn't be easily achievable? – Harry Silver Nov 12 '18 at 11:23
• Welcome here. Apart from applying mathjax to your mathematics, the least you could do related to the paper is to fully cite it. – Alecos Papadopoulos Nov 12 '18 at 11:48
• Citation: Matar, W. "Households' response to changes in electricity pricing schemes: Bridging microeconomic and engineering principles." Energy Economics 75. – Harry Silver Nov 12 '18 at 12:35

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?

• No. Income and the expenditure shares are all constant throughout. A colleague told me that the expenditure shares should change with changing P_xX, and the fact they're not in the paper, there is a mathematical issue. It is as if the author is using average values for all P_xX cases. – Harry Silver Nov 12 '18 at 12:39
• They find P_x and X in the residential model for every change of the thermostat, go to the utility component with them known, and derive a utility value corresponding to the change of thermostat. They then identify where the utility is maximum, and say that is where the thermostat should be set, and compute the corresponding price elasticity ex-post. – Harry Silver Nov 12 '18 at 12:42
• @HarrySilver This looks then as a normative model, that takes the CD utility function as given, as part of the maintained hypothesis - its appropriateness is not tested against the data. – Alecos Papadopoulos Nov 12 '18 at 13:14

Okay. I have a follow-up question about this integration. Given the author is discovering $$P_eX_e$$ outside of the utility component, he's not directly maximizing the utility in the problem. He's just computing what a utility would be given the budget equation, and then identifying the maximum value afterward.

Do the strict Lagrangian and subsequently the optimality conditions actually apply?