Sectoral Interfuel Substitution in Canada: An Application of NQ Flexible Functional Forms.

• Author: Jadidzadeh, Ali

1. INTRODUCTION

   Over the years, there have been two major approaches to the investigation of interfuel substitution (energy elasticities) and the demand for energy. One approach uses cointegration techniques and error-correction models to estimate long-run and short-run demand elasticities, respectively. Although this approach deals with econometric regularity issues, it lacks proper microeconomic foundations--see, for example, Bentzen and Engsted (1993) and Hunt and Manning (1989). The other approach allows the estimation in a systems context assuming a flexible functional form for the aggregator function, based on the dual approach to demand system generation developed by Diewert (1974). Using recent methodological advances in microeconometrics, this approach allows us to achieve theoretical regularity (in terms of curvature, positivity, and monotonicity of neoclassical microeconomic theory). It is difficult, however, to simultaneously achieve econometric regularity (in terms of stationary equation errors), because the combination of nonstationary data and nonlinear estimation in large demand systems is an extremely difficult issue and has not yet been addressed in the literature.

   The flexible functional forms approach was pioneered by Berndt and Wood (1975), Fuss (1977), and Pindyck (1979) in the context of interfactor and interfuel substitution. It involves the specification of a differentiable form for the cost function, the application of Shephard's (1953) lemma to derive the cost share equations, and the use of relevant data to estimate the parameters and compute the relevant elasticity measures like the income elasticities, the own- and cross-price elasticities, and the Allen and Morishima elasticities of substitution. However, the major contributions in this area are quite outdated by now, since their data incorporate observations before the 1970s. Moreover, most of these studies ignore the theoretical regularity conditions of neoclassical microeconomic theory or do not report the results of full regularity checks. An exception is a series of recent papers by Serletis et al. (2010, 2011) and Chang and Serletis (2014) which pay explicit attention to the theoretical regularity conditions and produce meaningful inference consistent with the theory.

   In this paper we take the flexible functional forms approach to examine interfuel substitution possibilities in energy demand within the residential, commercial, and industrial sectors in Canada as a whole and six of its provinces--Quebec, Ontario, Manitoba, Saskatchewan, Alberta, and British Columbia--as data limitations make it impossible to deal with all provinces. We focus on electricity, natural gas, and light fuel oil, ignoring energy forms with low shares in total energy expenditure like heavy fuel oil, kerosene, wood, and liquefied natural gas. Our objective is to provide updated empirical work using methodological improvements of the past twenty years, and improve our understanding of how changing energy prices and incomes will influence interfuel substitution and the demand for energy in the future.

   In order to achieve this, we use recent state-of-the art advances in microeconometrics. In particular, we use duality theory and the demand systems approach based on neoclassical consumer and firm theories which allows us to estimate the demand for electricity, natural gas, and light fuel oil in a systems framework. We also use a locally flexible demand system derived from the expenditure/cost function of the representative consumer/firm. Moreover, we are motivated by the widespread practice of ignoring the theoretical regularity conditions of neoclassical microeconomic theory and approximate the unknown underlying expenditure function using a flexible functional form that allows the imposition of global curvature without losing its flexibility property. In particular, in the case of the commercial and industrial sectors we use the normalized quadratic (NQ) cost function, introduced by Diewert and Wales (1987), and in the case of the residential sector energy demand we use the NQ expenditure function, introduced by Diewert and Wales (1988). The NQ cost function has also been used recently by Serletis et al. (2010, 2011), but the NQ expenditure function is (to our knowledge) used for the first time in the empirical energy demand literature. It is to be noted that McKitrick (1998) also used NQ functional forms to estimate consumer demand and producer input demand in the context of computable general equilibrium models in Canada.

   The rest of the paper is organized as follows. Section 2 sketches out related neoclassical theory and applied consumption and production analyses. Section 3 presents the NQ expenditure and cost functions and derives the associated systems of consumer demand and input demand functions. Section 4 discusses related econometric issues, paying explicit attention to the singularity problem and the imposition of global concavity. Section 5 discusses the data and Section 6 presents the empirical results for each of the residential, commercial, and industrial sectors. Section 7 compares the reported results to those obtained in analyses performed by others, and the final section concludes the paper.

2. THE STRUCTURE OF PRODUCTION AND PREFERENCES

   Our econometric approach requires certain assumptions about the structure of production and preferences. We assume that the group of n energy inputs in the production context (or goods in the consumer context) is homothetically weakly separable from the non-energy forms in the underlying aggregator function f (production function in producer theory or utility function in utility theory). Therefore, the aggregator function f has the form

   Q=f(E(x),M) (1)

   where Q is gross output (or utility, u), E(*) is a homothetic aggregator function over the n energy inputs (or goods), x = ([x.sub.1],... , [x.sub.n]), and M is a vector of non-energy inputs (or goods). The requirement of weak separability in x is that the marginal rate of substitution between any two components of x does not depend upon the value of M.

   Under these assumptions and duality theorems [see Diewert (1974)], the corresponding cost function (or expenditure function in utility theory) can be written as

   where p = ([p.sub.1],... , [p.sub.n]) is the corresponding price vector of the n forms of energy, [p.sub.M] that of non-energy forms, and [P.sub.E](.) is an energy price aggregator function which is a homothetic function and can represented by a unit cost or expenditure function.

   Our objective is to estimate a system of demand equations for the residential sector and a system of input demands for each of the commercial and industrial sectors and produce inference consistent with neoclassical microeconomic theory. In order to do so, we use flexible functional forms capable of approximating an arbitrary twice continuously differentiable function to the second order at an arbitrary point in the domain. Moreover, the flexible functional forms that we use allow for the imposition of global curvature without losing their flexibility property.

3. NORMALIZED QUADRATIC FUNCTIONAL FORMS

   We use the NQ expenditure function, developed by Diewert and Wales (1988), to investigate interfuel substitution possibilities in energy demand within the residential sector and the NQ cost function, developed by Diewert and Wales (1987), to investigate interfuel substitution possibilities in energy demand within the commercial and industrial sectors. In what follows, we briefly derive the demand system for the NQ expenditure function and the input demand equations for the NQ cost function.

   3.1 The NQ Expenditure Function

   For a given utility level and vector of pricesp, the NQ expenditure function is defined as

   [mathematical expression not reproducible] (2)

   where [theta] = [[[theta].sub.1],[[theta].sub.2],...,[[theta].sub.n]], b = [[b.sub.1],[b.sub.2],... ,[b.sub.n]], and the elements of the n x n matrix B = [[[beta].sub.ij]] are the unknown parameters to be estimated.

   The elements of the non-negative vector [alpha]= [[[alpha].sub.1],[[alpha].sub.2],...,[[alpha].sub.n]] are predetermined. In fact, according to Diewert and Fox (2009), the [alpha] vector can be either a vector of ones ([alpha]= [1.sub.n]) or the sample mean of the observed commodity vector, [mathematical expression not reproducible]. In this paper we use the former.

   To ensure the flexibility and Gorman polar form of the NQ form, we follow Diewert and Wales (1988) and impose the following restrictions

   [mathematical expression not reproducible] (3)

   [mathematical expression not reproducible] (4)

   and

   [mathematical expression not reproducible] (5)

   wherep* >> [0.sub.n] is a reference (or base-period) vector of normalized prices, determined in such a way thatp* = [1.sub.n].

   The NQ demand system in budget share form is

   [mathematical expression not reproducible] (6)

   where v = [[v.sub.1],[v.sub.2],... ,[v.sub.n]] is the vector of income normalized prices, with the jth element [v.sub.j] = [p.sub.j]/y, and [s.sub.i] = [v.sub.i][x.sub.i] is the share of the ith good in the total expenditure.

   We can use different elasticity measures, calculated from the Marshallian demand functions, [x.sub.i](v), i= 1,...,n, to conduct empirical demand analysis--for more details, see Barnett and Serletis (2008). In particular, the own- and cross-price elasticities,[[eta].sub.ij], can be calculated as

   [mathematical expression not reproducible] (7)

   We can also use the homogeneity of degree zero in (p,y) property of the Marshallian demand functions and calculate the expenditure (income) elasticities as

   [mathematical expression not reproducible] (8)

   In addition, we can use the Allen-Uzawa and Morishima elasticities of substitutions to investigate substitutability/complementarity relationships among goods. In particular, the Allen-Uzawa elasticity of substitution, [mathematical expression not reproducible], can be calculated as

   ...