In production economics, a firm’s behavior can be framed as either a profit maximization or a cost minimization problem, which gives rise to duality of the solutions. That is, a firm either chooses the optimal level of output, at a given total cost, to maximize the profit, or chooses the optimal levels of inputs to minimize the total cost of production, at a given level of output. The derived demands for inputs in a four-input KLEM model are determined by the level of production, the production technology, and the relative prices of inputs.
A key question in estimating either a production function or the derived demand for inputs is the degree to which inputs can be either substitutes or complements of each other. For example, some studies suggest that energy use is a complement to capital (equipment, machinery) and a substitute for labor (Griffin and Gregory 1976), whereas others indicate that substitution does occur but to a limited extent (Berndt and Wood 1975). Establishing the property of the derived demand for inputs (either complement or substitute, and degree of complementarity and substitutability) is crucial to public policy and energy development strategy. If factors are highly substitutable, increasing the price of one input might serve to reduce its own demand while increasing the demand for the substitute. However, if inputs are either not close substitutes or are even complementary, then changing the price of one input might have unintended consequences. For example, if energy and capital are distant substitutes, as suggested by Berndt and Wood (1975), energy pricing policies that raise the cost of electricity might negatively affect production, as firms are not able to adjust by switching to other inputs.
The degree of either complementarity or substitutability is conveyed in the concept of elasticity. The elasticity, calculated as the percentage changes in quantity to a one percent change in price, indicates how easy it is for a firm to change an input demand in response to an external price shock. If the price of an input such as energy increases, it is expected that the firm will respond by reducing the use of energy. This response is governed by the substitution effect and the income effect. If energy is becoming relatively more expensive, firms will switch to other less expensive inputs, depending on either the substitutability or the technologies that allow for such an adjustment. The income effect is related to the cost share in the total cost. If an input is used in a small amount, a higher price would not affect the total cost by much, and, therefore, the income effect is expected to be low. However, if an input accounts for a large share of the total cost, the income effect would be high, and the firm is expected to respond more decisively to even a small price change. This is particularly relevant in energy pricing policy, because the cost share is often small. The average cost shares were approximately 5% for K, 27.5% for L, 4.5% for E, and 63% for M, on average, for the United States during 1947–1971 (Berndt and Wood 1975). Therefore, the responsiveness of the demand, and, thus, the effectiveness of energy pricing policy, depends largely on technological ability.
When there are multiple inputs in the production, the literature does not agree on either the pairwise complementarity or the substitutability between inputs. This was summarized in Chung (1987), where, for example, K and E were found to be either complementarities or substitutes in manufacturing in the United States and Canada, depending on the time duration (short-term or long-term), methodologies and assumptions, and data aggregation level (sector or country level). A short-term production function assumes that the capital input does not change, whereas a long-term model requires the capital stock to change to reflect the impact of capacity expansion or substitution between capital stocks and variable inputs. As a result, short-term functions are expected to show that variable inputs, such as L, E, and M, are substitutes, and K and E are complements. In the long term, K and E could be substitutes when firms are given sufficient time to invest in more energy-efficient equipment if the price of energy is expected to increase. The time duration is also expected to have a strong impact on the degree of substitutability, with a higher elasticity expected in the long term than in the short term (Uri 1979).
The demand for energy is a derived demand for end-use services rendered by combining capital equipment and energy inputs. Therefore, the long-term and short-term effects are influenced by the duration for which a firm can replace capital goods. Hartman (1979) divided the decision-making involving residential or industrial energy demand into three levels. The first level is the decision whether to either buy or replace fuel-burning capital goods. The second level is the decision whether to buy equipment of certain technical and economic characteristics. The third level is to decide the frequency and intensity of use. In the short term, the stock of capital goods is fixed; thus, firms are only allowed to change the level of variable inputs. In the long term, a firm can change both capital goods as well as input intensity. In Hartman’s framework, the short term corresponds to the third level, whereas the long term corresponds to the first and second levels. Typically, equipment used in heavy industries often last much longer than those in light industries or services sectors; therefore, complete adjustments in response to energy prices may take a lot of time. For example, the economic life of equipment in mining and construction business could be as long as 20 years (FAO 1992), while office equipment are replaced every 3–5 years.
Different modeling techniques produce different versions of the elasticity. Cross-sectional studies assume that the market is in equilibrium and that between-firm variations in the data reflect a long-term decision to utilize the observed patterns of inputs. In those models, the capital stock would adjust instantaneously to a change, or the expectation of changes, in the relative price of inputs. Accordingly, a cross-sectional study is assumed to produce the long-term elasticity. Meanwhile, models using repeated observations over a short period of time (such as time series or panel data) will identify the short-term elasticity because the capital stock normally does not change within the observed time. A firm may only change the level of variable inputs such as energy, materials, and labor, to some extent. Either modeling approach, whether cross-sectional or panel data, may encounter certain issues. A cross-sectional model is prone to omitted-variable bias, which may occur when a factor that influences the demand for energy is not included in the model (Greer 2012, Ch. 9). If the omitted factor has a direct impact on a firm’s performance, then the estimated elasticity may be biased. Consequently, cross-sectional studies may produce an overly sensitive estimate of the price elasticity (Hartman 1979). With panel data, a first-difference estimator could solve the omitted-variable bias arising from an unobserved factor that does not change over time. However, the disadvantage is that there are often little variations in energy prices in a short panel, leading to large standard errors and low accuracy. Estimation based on longer time series may also be problematic if there is a structural change over time (Greer 2012, Ch. 9).
Most of the aforementioned studies used either state-level or sectoral data, which aggregate inputs across firms and geographic locations. An example is that of Griffin and Gregory (1976), which relied on cross-country variations in energy price to estimate the long-term energy demand. Depending on the aggregation level of data, energy demand can be estimated for countries, regions, sectors, or firms. The estimated elasticity is affected by aggregation levels. The finer the aggregation level, the lower the ability to substitute one input for another—which is due to specific technology and the production process being employed. Sectoral studies, such as Dargay (1983), found that the elasticity of energy demand differs substantially between Swedish manufacturing sectors—which could be a result of the distinct production structure of each industry. Dargay shows that energy and capital are complementary in most industries, whereas not only energy and labor but also energy and intermediate goods are substitutes. This demand pattern indicates that raising the energy price could result in firms substituting away from energy, at the expense of investment in capital goods.
With the availability of micro-data at the firm level, a firm-level production function can be estimated (Kleijweg et al. 1990; Pitt 1985). The firm size is an important determinant of the energy demand. In Dutch manufacturing, small firms adjust more quickly to energy price changes than do large firms, reflecting the greater likelihood of large-scale capital-intensive production being fixed in the short term. It is also possible to decompose each input into different sub-categories, such as different types of energy used (electricity, coal, gasoline, diesel, and LPG). Then, within each input category, inter-fuel substitution can be observed to a greater extent (Pindyck 1979).
However, researchers have to balance between the aggregation level of the data and modeling complexity. Modeling the demand of specific fuel types is often complicated by the presence of corner solutions (Woodland 1993). Most firms do not use all types of fuel. Electricity and gasoline are most often used for operations and transportation. The use of fuels, such as coal, as a heat source is needed only in certain manufacturing process, such as steel and cement production. As a result, estimating the demand function for a specific fuel often runs into a corner solution; that is only some type of fuel is used in the production. The corner solutions arise from two possibilities: one is that some technology only requires certain inputs, and the other is that, given the relative prices of inputs, it is not economical to use all types of fuels. In econometric terms, the choice of fuel type is endogenous to prices and other factors. There are two remedies to this potential corner solution issue. One is to estimate a demand system for each sector separately, as in Woodland (1993). Another is to use the aggregate data-assuming implicit substitution between sub-types of fuel, as in most other studies.
Methodology
The most popular approach used in the literature is the transcendental (translog) model, which is an approximation of any twice-differentiable strictly quasi-concave homothetic production function with constant return to scale (Thompson 2006). It is assumed that any technical change affecting the aggregate inputs is Hicks-neutral (for example, changes in input quality may shift the balance of input ratios toward the use of better inputs). A KLEM function linking a firm’s output to the four aggregate inputs—assuming symmetry and constant return to scale, and a stochastic term, is specified as:
$$\begin{aligned} lnQ_{it} &= a_0 + a_K lnK_{it} + a_L lnL_{it} + a_E lnE_{it} + a_M lnM_{it} \nonumber \\&\quad+ \frac{1}{2}\big [b_{KK}lnK_{it}^2+ b_{\text{LL}}lnL_{it}^2 + b_{\text{EE}}lnE_{it}^2 + b_{\text{MM}}lnM_{it}^2 \big ] \nonumber \\&\quad+b_{\text{KL}}lnK_{it} lnL_{it} + b_{\text{KE}}lnK_{it} lnE_{it} + b_{\text{KM}}lnK lnM_{it} \nonumber \\&\quad+b_{\text{LE}}lnL_{it} lnE_{it} + b_{\text{LM}}lnL_{it} lnM_{it} + b_{\text{EM}}lnE_{it} lnM_{it} \nonumber \\&\quad+\sum _j \beta _j X_{it}^j + \gamma {\text{Year}}_t + \sigma _i + \nu _{it}, \end{aligned}$$
(1)
where it represents firm i at year t, X is a set of variables to control for firm’s characteristics, and \(\sigma _i\) is firm i’s fixed effects, assumed to be unchanged during the observation time.
Symmetries of cross derivatives imply that \(b_{\text{KL}} = b_{\text{LK}}\), \(b_{\text{KE}} = b_{\text{EK}}\), \(b_{\text{KM}} = b_{\text{MK}}\), \(b_{\text{LE}}=b_{\text{EL}}\), \(b_{\text{LM}}=b_{\text{ML}}\), and \(b_{\text{EM}}=b_{\text{ME}}\). Furthermore, the homothetic condition requires that:
$$\begin{aligned} a_K+a_L+a_E+a_M = 1 \end{aligned}$$
(2)
and
$$\begin{aligned} {\left\{ \begin{array}{ll} b_{\text{KK}} + b_{\text{KL}} + b_{\text{LE}} + b_{\text{KM}} = 0 \\ b_{\text{LK}} + b_{\text{LL}} + b_{\text{LE}} + b_{\text{LM}} = 0 \\ b_{\text{EK}} + b_{\text{EL}} + b_{\text{EE}} + b_{\text{EM}} = 0 \\ b_{\text{MK}} + b_{\text{ML}} + b_{\text{ME}} + b_{\text{MM}} = 0 \end{array}\right. } \end{aligned}$$
(3)
Assuming that energy is paid with its marginal product, \(Q_E = e\), with \(Q_E\) being the first-order derivative of the production function with respect to E, then the elasticity of output to energy input is also the cost share of energy, \(\theta _E\), and is derived as:
$$\begin{aligned} \theta _E = \frac{\partial lnQ}{\partial lnE} = a_E + b_{\text{EE}}lnE + b_{\text{KE}}lnK + b_{\text{LE}}lnL + b_{\text{ME}}lnM \end{aligned}$$
(4)
In production economics, Allen’s relative substitution elasticity (RSE) is an important concept. It measures the responsiveness of relative input uses to relative input prices. Firms change the input mix when relative prices change. The sign and speed of changes would indicate the type of inputs (complements or substitutes), and the degree of interdependence. Without explaining the intermediate steps, we have the general formula of the RSE for two factors Capital and Energy, under a constant return to scale assumption, as follows:
$$\begin{aligned} \alpha _{\text{EK}} = \frac{\% \Delta E/K}{\% \Delta r/e} = \frac{K*Q_K + L*Q_L + E*Q_E + M*Q_M}{K*E}.\frac{|H_{\text{EK}}|}{|H|} = \frac{Q}{E*K}.\frac{|H_{\text{EK}}|}{|H|}, \end{aligned}$$
(5)
where |H| is the determinant of the bordered Hessian matrix H of first- and second-order derivatives of the production function. \(H_{\text{EK}}\) is the EK cofactor of H.
$$\begin{aligned} H = \begin{bmatrix} 0&Q_K&Q_L&Q_E&Q_M \\ Q_K&Q_{\text{KK}}&Q_{\text{KL}}&Q_{\text{KE}}&Q_{\text{KM}} \\ Q_L&Q_{\text{LK}}&Q_{\text{LL}}&Q_{\text{LE}}&Q_{\text{LM}} \\ Q_E&Q_{\text{EK}}&Q_{\text{EL}}&Q_{\text{EE}}&Q_{\text{EM}} \\ Q_M&Q_{\text{MK}}&Q_{\text{ML}}&Q_{\text{ME}}&Q_{\text{MM}} \end{bmatrix} \end{aligned}$$
(6)
The elements in matrix H are derived as follows:
$$\begin{aligned} Q_{\text{EE}}&= Q\frac{b_{\text{EE}} - \theta _E + \theta _E^2}{E^2} \nonumber \\ Q_{\text{KE}}&= Q\frac{b_{\text{KE}} + \theta _K\theta _E}{KE} \nonumber \\ Q_E&= \theta _E * \frac{Q}{E} \end{aligned}$$
(7)
Once the substitution elasticity has been identified, the cross-price elasticity of energy demand to capital price is inferred directly from Allen’s RSE:
$$\begin{aligned} \varepsilon _{\text{EK}} = \frac{d(ln E)}{d(ln r)} = \theta _K * \alpha _{\text{EK}} \end{aligned}$$
(8)
These elasticities are calculated at given cost shares, typically at either the sample means or representative firms. The derivation of the mathematical formula and applications are detailed in Thompson (1997, 2006).
Estimation methods
The translog production function (1) can be estimated by either a single-equation or a system of equations approach. In a system of equations approach, the cost shares of all inputs, namely K, L, E, and M, will be estimated. Due to the singularity of covariance matrix of four equations (the cost shares of all inputs adding up to one), it is necessary to drop one equation prior to the estimation. In both approaches, the symmetries and homothetic constraints in (2–3) are imposed on the parameters of either the single Eq. (1) or the system of Eq. (9).
$$\begin{aligned} {\left\{ \begin{array}{ll} \theta _K = a_K + b_{\text{KK}}lnK + b_{\text{KL}}lnL + b_{\text{KE}}lnE + b_{\text{KM}}lnM \\ \theta _L = a_L + b_{\text{LK}}lnK + b_{\text{LL}}lnL + b_{\text{LE}}lnE + b_{\text{LM}}lnM \\ \theta _E = a_E + b_{\text{EK}}lnK + b_{\text{EL}}lnL + b_{\text{EE}}lnE + b_{\text{EM}}lnM \\ \theta _M = a_M + b_{\text{MK}}lnK + b_{\text{ML}}lnL + b_{\text{ME}}lnE + b_{\text{MM}}lnM \end{array}\right. } \end{aligned}$$
(9)
In this study, we have employed the single-equation approach. Due to the availability of panel data, observed in 2015 and 2016, the advantage of fixed-effect and random-effect estimators outweighs the efficiency gained by the system approach using three-stage regression. Panel data with fixed effects can address a potentially serious concern about unobserved firm-specific characteristics that affect both the use of inputs and firm’s performance. Theoretically, a 3SLS approach could potentially improve the estimation properties compared to the single equation (Thompson 2006); however, it does not take advantage of having repeated observations to control for omitted-variable bias.
We have presented five different methods to check the sensitivity of the results and shown the robustness of the choice of the single-equation approach. First, we estimated the translog production function in (1) with pooled data over the years without the homothetic constraints. Then, we estimated two models using panel data with fixed and random effects, in case there may be omitted factors that cause bias in the least-squares estimate. Third, we estimated the translog model with pooled data, with the homothetic constraints imposed on the parameters, which produces the long-term estimate of the elasticities. Finally, we applied a first-difference estimator to identify the short-term elasticities. To account for provincial difference, such as either the business environment or local policies, which may affect overall firm performance, we used standard errors clustered at the provincial level to adjust for correlation among firms in the same province.
To calculate the Allen’s elasticities of substitution and price elasticities, all elements of the bordered Hessian matrix H, and of the cofactors must be identified, and then their determinants, which must be semi-negative definite, must be calculated. These were examined thoroughly in the calculation of the elasticities.
Data
We used two recent Vietnam Enterprise Surveys (VES), 2015 and 2016. The annual survey collects information from all formally registered establishments, enterprises, and cooperatives, either operational or idling, during the preceding year of the survey. The totals in 2015 and 2016 were 415,656 and 455,296 firms, respectively. The survey covers information relating to the ownership, industrial sectors (up to five digits by Vietnam’s System of Economic Branches in 2007), locations, import/export activities, economic performance in the preceding year, the total number of employees and the total labor cost, assets and liabilities (including short- and long-term assets), tax and payables, R&D expenditures, and branches.
Most importantly, the surveys collected detailed information on which types of energy were used, the amount, and the total value. However, due to the survey design, only up to a quarter of the dataset contained information about energy consumption. To create a panel dataset from the two surveys, we matched tax identification numbers between the two individual datasets. There is an issue with sub-branches of the same companies that share the same tax code. It is difficult to separate individual branches’ performances and the characteristics of firms having the same tax code at different locations and sectors. As a consequence, we dropped firms belonging to the same branch and sharing a unique tax code from the analysis. We further restricted the sample to include only firms that were operational in the previous year. The final sample, after dropping firms that reported a negative revenue, a negative capital stock, and a negative input cost (labor, energy, or material), and were excessively large, includes 149,959 observations, with 72,499 observations in 2015 and 77,460 observations in 2016. The balanced panel dataset has 37,685 firms.
The most common energy category is electricity usage, which includes the amount consumed (in 1000kwh), self-production, sales, net production, and the value of purchase. Other types of energy, including gasoline, coal, diesel, mazut oil, liquefied petroleum gas (LPG), and liquefied natural gas (LNG), are grouped into four main categories: gasoline, coal, diesel, and LPG. As expected, gasoline is the second most common source of energy with almost 43,000 observations, followed by diesel, with 28,000 observations. Other primary energy sources are used either very sparsely or to a limited extent. In the model, we aggregated the value of all energy types into a single energy category (E), assuming perfect substitution between the source and purpose of uses (heating/cooling or electric equipment operation). Though inter-fuel substitution is possible, modeling that possibility requires specific industry information on energy use, which is not available. Details of energy types and consumption are presented in Additional file 1: Table S1.
Other variables in the model include the total revenue of production or services as the dependent variable (Q); the total value of long-term assets (K), which includes long-term receivables and values of fixed assets at the end of the reporting year, less depreciation; the total cost of labor (L); and the total value of short-term assets, intermediates, and inventories as the material cost (M). The use of the value (in monetary terms) instead of the quantity (in units) in the production function assumes that the value is directly proportional to the quantity, after controlling for either sectoral differences or types of firm ownership. This assumption is entirely justifiable because most firms in Vietnam are small; thus, prices are taken as exogenous. Furthermore, the market of inputs, including capital, labor, and energy, is well regulated in Vietnam. Energy prices, in particular, are strictly under government control. Thus, all firms are expected to face the same price schedule (although sectoral or ownership differences are allowable). Establishments are classified by 13 types of ownership (Additional file 1: Table S2), and exclude household business. We collapsed the five-digit standard industrial classifications into 21 major industrial sectors. However, there were no observations in two areas related to the communist party (\(\text {VSIC}=84\)) and international cooperation (\(\text {VSIC}=99\)). As a result, the sample contained 19 industrial sectors (Additional file 1: Table S3). The processed data are summarized in Additional file 1: Table S4.
