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This study employs duality theory to develop a theoretical model for small commercial and industrial (CIS) electricity usage. The CIS production function is posited such that output is a function of three variable inputs (electricity, natural gas, and labor) and one fixed input (capital). A profit function dual to this production function is specified using a normalized quadratic functional form. CIS profits are functionally dependent upon output price, an electricity input price, and natural gas and labor input prices for a fixed quantity of capital. The derived input-demand equation results from differentiating the profit function with respect to the price of electricity. The input-demand equation for electricity is dependent upon the own-price of electricity, the CIS output price, and input cross-prices. The model may be of use to utilities and regulators for the analysis of CIS electricity usage.

Because of the importance of electricity in modern economies, substantial research is devoted to the analysis of electricity usage. Historically, residential consumption patterns have been extensively researched with comparatively less attention devoted to small industrial and commercial (CIS) demand. CIS usage, defined here as usage that does not exceed 600 kilowatts in any two consecutive months, represents a large portion of total electricity consumption and deserves more scrutiny [

It is helpful for regional utilities and regulatory agencies to understand how changes in economic conditions affect CIS electricity consumption. Regional economic growth frequently mandates additional generation capacity investments [

This study proposes a theoretical approach for the analysis of CIS electricity consumption. Duality theory and derived demand are employed for specification of the input-demand equation for electricity. Sectorial output supply functions also result within this framework, but the focus of this effort is CIS usage of electricity as an input into production. Derived demand refers to demand that results for one good as a result of demand for another product. As shown below, partial differentiation of the profit function with respect to the price of electricity yields the input demand equation.

The rest of the study is organized as follows. Section 2 provides a review of related energy and microeconomic studies. Section 3 summarizes the model that is developed. Section 4 concludes with suggestions for future research.

Much of the prior research using duality theory and derived input demand has been for use in agricultural economics. In this effort, a normalized quadratic functional form for the underlying profit function dual to a production function is used to describe CIS usage for electricity as derived demand [

The quadratic functional form is widely used in empirical research of the dual approach [

A normalized quadratic restricted profit function is used to derive a model of CIS electricity demand using one output, three variable inputs, and one fixed input. A profit function, as opposed to a cost function, is utilized because it is simpler to estimate and no endogenous variables are needed as explanatory variables [

This effort attempts to model CIS electricity usage within a formal analytical context. The model developed in the next section is based upon duality theory [

The first step in developing a demand function using duality theory is to specify a production function. Electricity demand is derived demand, meaning it is used as an input in the commercial and industrial sector as a factor of production in the output of final goods and services [

Q = f ( X i ; Z K ) (1)

Q = f ( X E , X G , X L ; Z K ) (2)

CIS demand for electricity as an input is derived from CIS output using a restricted profit function [

CIS firms maximize profit by choosing the quantity of output supplied, Q, and the quantities of the three variable inputs, ( X i = X E , X G , X L ). Output price and the variable input prices ( P i = P Q , P E , P G , P L ) are exogenously determined [

Π = f ( P i ; Z K ) (3)

Π ( P Q , P E , P G , P L ; Z K ) = max X i , Q { P Q ∗ Q − P i T X i ; ( Q , X i ; Z K ) ϵ T } (4)

The normalized quadratic functional form is the most appropriate means for the specifying the restricted profit function as shown in Equation (5) [

Π ( P i ; Z K ) = ∑ i 4 β i P i + β K Z K + 1 2 ∑ i 4 ∑ j 4 β i j P i P j + 1 2 β K K Z K 2 + ∑ i 4 β i K P i Z K (5)

Π ( P Q , P E , P G , P L ; Z K ) = β Q P Q + β E P E + β G P G + β L P L + β K Z K + 1 2 [ β Q Q P Q 2 + β E E P E 2 + β G G P G 2 + β L L P L 2 + β Q E P Q P E + β Q G P Q P G + β Q L P Q P L + β E G P E P G + β E L P E P L + β G L P G P L + β K K Z K 2 ] + β Q K P Q Z K + β E K P E Z K + β G K P G Z K + β L K P L Z K (6)

Based on Hotelling’s lemma, the profit function is differentiable on output and input prices. Differentiation of Equation (6) with respect to P E using Hotelling’s lemma yields the negative input-demand function for electricity or

∂ Π ( P i ; Z K ) ∂ P E = − X E * . Because the focus of this paper is CIS electricity demand,

only the input-demand function for electricity is explicitly derived in Equation (7). The input-demand function is homogeneous of degree zero in prices, and symmetry constraints result for the coefficients of Equation (7) such that β i j = β j i [

− ∂ Π ( P i ; Z K ) ∂ P E = X E * = β E + β E E P E + β Q E P Q + β E G P G + β E L P L + β E K Z K (7)

Finally, the own-price, cross-price, and output-price elasticities of electricity demand are extracted from the derived input-demand function. Reciprocity constraints, also known as symmetry conditions, are imposed on the derivatives

of the input-demand function, so that ∂ X i * ∂ P j = ∂ X j * ∂ P i . One advantage of deriving

demand equations from flexible functional forms using duality theory is that the elasticities of demand are subject only to those restrictions implied by economic theory [

ε X E , P E = ∂ X E * ∂ P E ∗ P E X E * = − ( β E E ) P E ¯ X E ¯ (8)

ε X E , P j = ∂ X E * ∂ P j ∗ P j X E * = ( β E j ) P j ¯ X E ¯ (9)

ε X E , P Q = ∂ X E * ∂ P Q ∗ P Q X E * = ( β Q E ) P Q ¯ X E ¯ (10)

The own-price elasticity coefficient is hypothesized to be negative, meaning an increase in the price of electricity will reduce CIS electricity consumption. The signs of the cross-price elasticity estimates are ambiguous, depending on whether electricity and natural gas and labor are substitutes or complements. If electricity and the alternate inputs are substitutes, an increase in the prices of those inputs will increase the demand for electricity, resulting in positive elasticity coefficients. If electricity and the inputs are complements, an increase in the prices of those inputs will decrease the demand for electricity, resulting in negative elasticity coefficients.

The analytical framework developed herein provides a logical starting point for empirical analyses of CIS electricity usage. Doing so will require collecting a combination of electric utility data and broader economic measures. Data assembly will require some effort, but should prove manageable for many regions and/or nations. Whether econometric evidence is eventually compiled that confirms the usefulness of the approach, of course, remains to be seen.

Small commercial and industrial (CIS) electricity represents a large percentage of total loads for many electric utilities. In spite of that, CIS demand has historically received far less attention than residential usage. A natural step toward addressing that gap in the energy economics literature is development of a formal modelling construct. This study attempts to do that.

The duality theory framework employed here specifies a CIS production function where output is expressed as a function of three variable input quantities (electricity, natural gas, and labor) and one fixed input quantity (capital). The dual to this production function is a profit function. A normalized quadratic functional form characterizes the restricted profit function. The profit function is a function of an output price, an electricity price, and the prices of natural gas and labor, given a fixed quantity of capital. Using Hotelling’s lemma, the input-demand equation is derived by differentiating the profit function with respect to the price of electricity. From the input-demand equation for electricity, the own-price, output-price, and cross-price price elasticities are derived.

Estimation of the derived input-demand equation above and should yield reliable, comparable results for regional electric utility empirical research. One advantage of duality theory is the capability to derive an input-demand equation consistent with profit-maximizing behavior. Although endogeneity may come into play, a similar approach based on cost-minimizing behavior can likely be developed for cases involving publically owned utilities. The dual approach includes all the elements of prior studies that use simpler models [

Financial support for this research was provided by El Paso Water, City of El Paso Office of Management & Budget, National Science Foundation Grant DRL-1740695, the UTEP Center for the Study of Western Hemispheric Trade, and the Hunt Institute for Global Competitiveness at UTEP. Helpful comments were provided by Jim Holcomb, Richard Jarvis, and an anonymous referee. Econometric research assistance was provided by Omar Solis and Esmeralda Muñiz.

The authors declare no conflicts of interest regarding the publication of this paper.

Allen, K.R. and Fullerton Jr., T.M. (2018) Analyzing Small Industrial and Commercial User Demand for Electricity. Theoretical Economics Letters, 8, 3109-3115. https://doi.org/10.4236/tel.2018.814193