Understanding and Analysing Energy Demand

Abstract

This chapter introduces the concept of energy demand using basic micro-economics and presents the three-stage decision making process of energy demand. It then provides a set of simple indicators (such as price and income elasticities and energy intensity) and discusses the decomposition method and econometric method that can be used to analyse energy demand.

Appendices

Annex 3.1: Consumer Demand for Energy—The Constrained Optimization Problem

Consider that the utility function of a consumer can be written as

$$ {\text{Utility}}\,u = U(X_1, X_2, X_3, \ldots, X_n) $$

(3.31)

The consumer has the budget constraint

$$ I = p_1 X_1 + p_2 X_2 + \cdots + p_n X_n $$

(3.32)

For maximization of the utility subject to the budget constraint, set the lagrange

$$ L = U(X_1, X_2, X_3, \ldots, X_n) - \lambda (I - (p_1 X_1 + p_2 X_2 + \cdots + p_n X_n )) $$

(3.33)

Setting partial derivatives of L with respect to X ₁, X ₂, X ₃,…X _n and λ equal to zero, n + 1 equations are obtained representing the necessary conditions for an interior maximum.

$$ \begin{gathered} \delta {{L}}/\delta {{X}}_{1} = \delta {{U}}/\delta {{X}}_{1} - \lambda {{p}}_{1} = 0; \hfill \\ \delta {{L}}/\delta {{X}}_{2} = \delta {{U}}/\delta {{X}}_{2} - \lambda {{p}}_{2} = 0; \\ \vdots \\ \delta {{L}}/\delta {{X}}_{{n}} = \delta {{U}}/\delta {{X}}_{{n}} - \lambda {{p}}_{{n}} = 0 \hfill \\ \delta {{L}}/\delta \lambda = {{I}} - {{p}}_{1} {{X}}_{1} + {{p}}_{2} {{X}}_{2} + \cdots + {{p}}_{{n}} {{X}}_{{n}} = 0 \hfill \\ \end{gathered} $$

(3.34)

From above,

$$ \left( {\delta {{U}}/\delta {{X}}_{1} } \right)/\left( {\delta {{U}}/\delta {{X}}_{2} } \right) = {{p}}_{1} /{{p}}_{2} \;{\text{or}}\;{\text{MRS}} = {{p}}_{1} /{{p}}_{2} $$

(3.35)

$$ \lambda = (\delta {{U}}/\delta {{X}}_{1} )/{{p}}_{1} = (\delta {{U}}/\delta {{X}}_{2} )/{{p}}_{2} = \cdots = (\delta {{U}}/\delta {{X}}_{{n}} )/{{p}}_{{n}} $$

(3.36)

Solving the necessary conditions yields demand functions in prices and income.

$$ \begin{gathered} {{X}}_{1}^* = {{d}}_{1} ({{p}}_{1} ,{{p}}_{2} ,{{p}}_{3} , \ldots {{p}}_{{n}} ,{{I}}) \hfill \\ {{X}}_{2} ^* = {{d}}_{2} ({{p}}_{1} ,{{p}}_{2} ,{{p}}_{3} , \ldots {{p}}_{{n}} ,{{I}}) \hfill \\ \vdots \hfill \\ {{X}}_{{n}} ^* = {{d}}_{{n}} \left( {{{p}}_{1} ,{{p}}_{2} ,{{p}}_{3} , \ldots {{p}}_{{n}} ,{{I}}} \right) \hfill \\ \end{gathered} $$

(3.37)

An individual demand curve shows the relationship between the price of a good and the quantity of that good purchased, assuming that all other determinants of demand are held constant.

Annex 3.2: Cost Minimization Problem of Producers

Consider a firm with single output, which is produced with two inputs X ₁ and X ₂. The cost of production is given by

$$ {\text{TC}} = {{c}}_{1} {{X}}_{1} + {{c}}_{2} {{X}}_{2} $$

(3.38)

This is subject to

$$ {\text{St}}\;{{q}}_{0} = {{f}}({{X}}_{1} ,{{X}}_{2} ) $$

(3.39)

Write the Lagrangian expression as follows:

$$ {{L}} = {{c}}_1{{X}}_1 + {{c}}_2{{X}}_2 + \lambda \left( {{{q}}_0 - {{f}}\left( {{{X}}_1,{{X}}_2} \right)} \right) $$

(3.40)

The first order conditions for a constrained minimum are:

$$ \begin{aligned} \delta {{L}}/\delta {{X}}_{1} & = {{c}}_{1} - \lambda \, \delta {{f}}/\delta {{X}}_{1} = 0 \\ \delta {{L}}/\delta {{X}}_{2} & = {{c}}_{2} - \lambda \delta {{f}}/\delta {{X}}_{2} = 0 \\ \end{aligned} $$

From above,

$$ {{c}}_1/{{c}}_2 = (\delta {{f}}/\delta {{X}}_{1} )/(\delta {{f}}/\delta {{X}}_{2} ) = {\text{RTS}}\,(X_1{\text{ for }} X_2 ) $$

(3.41)

In order to minimize the cost of any given level of input, the firm should produce at that point for which the rate of technical substitution is equal to the ratio of the inputs’ rental prices.

The solution of the conditions leads to factor demand functions.

Annex 3.3: Adaptive Price Expectation Model

Consider that Q _t is related to price expectation and not the actual price level in time t.

$$ {{Q}}_{{t}} = {{a}}^* + {{b}}^*{{P}}_{{t}} ^* + {{e}}_{{t}} ^* $$

(3.42)

where P ^* represents expected level of prices, not actual prices

A second relationship defines the expected level of P ^*. It is assumed that in each time period, the expectation changes based on an adjustment process between the current observed value of P and the previous expected value of P ^*. The relationship is

$$ {{P}}_{{t}}^* - {{P}}_{{{{t}} - 1}}^* = {{c(P}}_{{t}} - {{P}}_{{{{t}} - 1}}^* ) $$

(3.43)

$$ {\text{or}}\;{{P}}_{{t}}^* = {{cP}}_{{t}} + (1 - {{c}}){{P}}_{{{{t}} - 1}}^* $$

(3.44)

This implies that the expected price is a weighted average of present price and the previous expected level of price.

For econometric estimation, the above equation is rearranged as follows:

$$ \begin{gathered} (1 - {{c}}){{P}}_{{{{t}} - 1}}^* = {{c(}}1 - {{c)P}}_{{{{t}} - 1}} + (1 - {{c}})^{2} {{P}}_{{{{t}} - 2}}^* \hfill \\ (1 - {{c}})^{2} {{P}}_{{{{t}} - 2}}^* = {{c(}}1 - {{c)}}^{2} {{P}}_{{{{t}} - 2}} + (1 - {{c}})^{3} {{P}}_{{{{t}} - 3}}^* \hfill \\ \end{gathered} $$

(3.45)

Substituting and combing we obtain

$$ {{P}}_{{t}}^* = {{c[P}}_{{t}} + (1 - {{c}}){{P}}_{{{{t}} - 1}} + {{c(}}1 - {{c)}}^{2} {{P}}_{{{{t}} - 2}} + \cdots ]= {{c}}\sum (1 - {{c}})^{s} {{P}}_{{{{t}} - {{s}}}} $$

(3.46)

Substituting P _t ^* in Q _t , we get

$$ {{Q}}_{{t}} = {{a}}^* + {{b}}^*{{c}}\sum (1 - {{c}})^{{s}} {{P}}_{{t - s}} + {{e}}_{{t}}^* $$

(3.47)

$$ Q_{t} = a^* + b^*c\sum\limits_{s = 0}^{\infty } {(1 - c)^{s} P_{t - s} } + e_{t} ^* $$

(3.48)

Letting a = a ^*, b = b ^* c, w = (1 − c), and e _t  = e _t *

Q _t  = a + b ∑(w)^sP_t − s + e _t , the original geometric lag model

For econometric estimation, model is rewritten as:

$$Q_{t-1}=a^{\ast}+b^{\ast}c\sum(1-c)^{s}P_{t-s-1}+e_{t-1}^{\ast}$$

(3.49)

We calculate Q _t − (1 − c)Q _t−1 to obtain

$$ Q_t - (1 - c)Q_{t - 1} = a^*c + b^*c P_t + e_t^* - (1 - c)e_{t-1}^* $$

(3.50)

$$ {{Q}}_{{t}} = {{a}}^*{{c}} + {{b}}^*{{c}}\;{{P}}_{{t}} + \left( {1 - {{c}}} \right){{Q}}_{{{{t}} - 1}} + {{u}}_{{t}}^* $$

(3.51)