This paper uses the OO approach of the SFA to investigate the influence of various types of knowledge on TE. Aigner et al. (1977) and Meeusen and Van den Broeck (1977) introduced this parametric approach in the economic literature to examine firms’ performance via TE. Among others, Kumbhakar (1987) defines the stochastic production frontier as the maximum attainable level of output that can be produced using the current technology and a certain amount of inputs. Standard economic theory assumes that firms are fully efficient and deviations from the frontier are attributable to random shocks. However, the SFA relaxes such an assumption by allowing for inefficiencies in the production process. Considering technical inefficiencies, the OO method defines the production frontier as follows:

where Y is total output, x is a vector of inputs, f (.) stands for the functional form of the frontier, and u is non-negative and measures TI. For small values of u, exp(-u)≅1-u. Thus, TE=exp(-u)=1-u=1-TI .

Previous studies applying the same methodology typically use the Cobb-Douglas (CD) and/or the TransLog (TL) functions to identify the frontier in equation (1). After taking the natural logarithm of equation (1), the CD function is given by the following expression:

where y_i is the natural logarithm of total output of the i-th firm, β₀=ln(A), β_j are the parameters in the production function, x_ji are the corresponding inputs in their logarithm forms, β_t is the measure of technical change $\left(\frac{\partial \mathrm{l}\mathrm{n}\left(y\right)}{\partial t}\right)$ , t is an indicator of time and (v_i-u_i ) is the composite error term. On the other hand, the TL production function is defined as follows:

where β_jk=β_kj the measure of technical change is equal to $TC={\beta }_t+{\beta }_{tt}t+{{\sum }_{j=1}^J\mathrm{‍}}_{}{\beta }_{jt}{x}_{ji}$ and β_tt is the speed of technical change.⁵ To identify the vector of parameters, we apply the simultaneous maximum likelihood ( ML) estimation.⁶

For both equations, (2) and (3), v_i is the random component which is assumed to be independent and identically distributed (i.i.d.) with mean zero and constant variance, $N\left(0,{\sigma }_v^2\right)$. The random error term (v_i ) is independent of u_i and captures unobserved heterogeneity across firms and stochastic events. Here, u_i is the inefficiency term and is defined as the ratio of the gap between maximum attainable output and the actual output to the maximum attainable output. From (2) and (3), if u_i = 0 firms are fully efficient, and deviations from the frontier depend only on the distribution of v_i .

To disentangle the composite error term and empirically compute the values of u_i , we assume a parametric distribution for the inefficiency term. Empirical studies usually adopt either a half-normal $(u_i\sim i.i.d. N^{+}\left(0,{\sigma }_u^2\right))$, a truncated-normal $(u_i\sim i.i.d. N^{+}\left(\mu ,{\sigma }_u^2\right))$ or an exponential $(f\left(u_i\right)=\frac{1}{\eta }\mathrm{*}\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{u_{i }}{\eta } \right))$ distribution for u_i . The half-normal and exponential distributions cluster most of the observations close to full efficiency, that is, the expected value of u_i is zero or nearly zero. For the truncated-normal, the average of u_i is not necessarily zero. Since this distribution allows us to parameterise inefficiency and the half-normal is a special case of it, when μ=0, we assume a truncated-normal distribution in this analysis.

Assuming a truncated-normal distribution for u_i , the SFA model uses the Jondrow’s (JLMS) formula (Jondrow et al., 1982) to obtain $\widehat{u_i}$:

where $e_i=v_i-{u_i=y}_i-\widehat{\beta }\mathrm{\text{'}}{x}_i$, $\sigma =\sqrt{{\sigma }_v^2+{\sigma }_u^2}$, $\lambda =\frac{{\sigma }_u}{{\sigma }_v}$, ϕ and Φ are the standard normal and cumulative density functions, respectively. To identify the main determinants of firms’ inefficiencies, this strand of literature hypothesizes $\mu =\widehat{E}\left[u_i|{\epsilon }_i\right]$ to be a function of a vector of factors, z_i , that may lead to production inefficiencies. So that, the TI equation is as follows:

where γ_m are the parameter estimates of the (mean) TI equation. The marginal effect of the m-th inefficiency determinant on μ_i is given by the following expression (Wang, 2002):

where Λi = μ_i/ σ_u,i , it represents the ratio between the expected and the standard deviation of TI in firm i and is used as an argument of the standard normal and cumulative density functions, which produce the adjustment term in equation (6). The main advantage of Wang’s (2002) formulation is that these marginal effects are observation-specific.

The ENAPROCE collected firm-level data on 28 034 enterprises that hire between 11 and 250 workers in the manufacturing industry,⁷ and between 11 and 100 workers in the trade⁸ and service⁹ sectors.¹⁰ From the 28 034, 52% reported information about their economic activities in 2014 (INEGI, 2015), 48% in 2017 (INEGI, 2018), and 12 457 firms appeared in both surveys.¹¹ Following Taymaz and Saatci (1997), Baek and Pagan (2002), Admassie and Matambalya (2002), and Taymaz (2005), the dependent variable in the frontier function is the value of total sales plus the change in stocks (output stock in January 1st minus output stock in December 31st) from all commodities produced in the corresponding firm.¹² The total number of working hours hired in 2014 and 2017 measures the size of labour in the production function. To account for capital endowments, we use fixed-assets’ self-reported replacement value. The sum of annual expenses on raw materials, maquila, electricity, telecommunications, and other inputs represents the value of intermediate inputs. We also introduce a time indicator and 11- NAICS indicators in the set of estimations.

To guide the model specification of the TE equation, we use the previous empirical findings shown in table 1. Unfortunately, data availability prevents us from including all controls in the existing literature. We account for total firm’s expenses when it pays taxes because there is a large debate in Mexico about harmful effects of these procedures on firms’ performance, especially for micro¹³ and SMEs. To consider managerial practices, the set of inefficiency effects in this study includes the ratio of managers to total workers. We also include the share of exports to total sales in the TI equation, aiming to check whether SMEs selling abroad are more efficient than those selling their products in the domestic market.

To investigate the knowledge- TE relationship, we use different proxy variables to account for knowledge endowments (human capital), the creation of knowledge (innovation), and knowledge transfers (technology transfers) in the TI model. To identify the human capital- TI relationship, we use the ratio of employees with a bachelor’s degree in relation to the total number of workers. Regarding the innovation- TI association, we include a dummy variable indicating whether the firm applied for a patent in the last two years or not. The ENAPROCE also collects data on the frequency in which firms buy existing technologies and adapt them to the current production processes without any modification (licences). Moreover, the survey reports the frequency in which SMEs sell their new technologies to other firms (transfer). Together with the previous two variables, access to the Internet accounts for knowledge flows in the TI model.

Table 2 displays definitions, measurement units, descriptive statistics, and expected signs in the production and TI equations for each control variable. On average, we observe that total annual output in both years reaches $39.7 million Mexican pesos (MXN).¹⁴ To produce such an output, SMEs utilize, on average, 137 460 working hours,¹⁵ $7.5 million MXN of fixed assets and $29.7 million MXN of intermediate inputs. As indicated in the last column, we expect a non-negative contribution of the three inputs to total output (Kumbhakar et al., 2015). Looking at the 11 NAICS indicators, we observe that 17.0%, 13.2%, 12.7%, 11.7%, and 10.4% of SMEs (65% in total) belong to the retail trade, temporary accommodation and food and beverage preparation services, wholesale trade, professional, scientific and technical services, and the 33 manufacturing industries, respectively. For the relationship between output and such indicators, we expect either a negative or a positive association, which depends on the base category (excluded indicator).¹⁶

Table 2
Variables, definitions, descriptive statistics and priors (ENAPROCE-2015)

Source: Own elaboration based on INEGI (2015, 2018).

Table 2 shows that, on average, SMEs spend 5.7% of total revenue on all required procedures for paying taxes, and 4.6% of their revenues come from exports. We expect that firms investing so much time and money in the required tax payments procedures would likely be less efficient than other firms and that selling their products abroad would likely reduce inefficiencies given the competitiveness in external markets (Bigsten et al., 2000; Yang and Chen, 2009).

Regarding managerial practices, human capital, and innovation, table 2 indicates that, on average, 16.7% of total workers were employed as either manager or supervisor, 23.7% of total employees have a bachelor’s degree, and 4% of SMEs asked for a patent in the two previous years. We expect a positive association between TE and managers’ ratio when the number of managers per worker is low. However, once it reaches a certain threshold, additional managers would likely be redundant and cause a non-optimal use of inputs (Taymaz, 2005). The existing literature argues that increasing human capital and innovation typically reduces TI (Aguilar, 2011; Valderrama et al., 2015; Diaz et al., 2019).

Regarding knowledge flow, and based on the literature review, we expect a negative association between the frequency with which technologies are transferred from and to SMEs and TI. Very active firms buying and selling technologies are subject to more competitive markets and, therefore, would likely use inputs more efficiently (Taymaz and Saatci, 1997; Taymaz, 2005). We also expect that knowledge flows from and to the firm via the Internet relates with TE positively, as firms using the Internet tend to be more efficient (Becchetti et al., 2003). We test the abovementioned hypotheses in the following section.

For estimating the set of parameters in the production frontier, we group SMEs in ENAPROCE 2015 and 2018 (pooled sample). Although several firms appear in both surveys, the few periods for which we have data prevent us from estimating a panel data model as suggested in Kumbhakar et al. (2015). To identify the best functional form of the frontier, we use four different specifications: CD and TL models without and with inefficiency effects based on equations (2), (3), and (5). Most of the time, and using different specifications, the TL function suffers from convergence issues, which is not the case when using the CD function. Given the convergence issues and to show how parameter estimates differ among industries, we opt for the CD specification. Likelihood ratio tests fail to reject the alternative hypothesis that TI exists in the production process. Thus, we find empirical evidence that SMEs in Mexico do not use inputs efficiently in their production activities and therefore operate below the frontier.

Table 3 shows the parameter estimates of the frontier for the pooled sample of 2014 and 2017, and for the 11 NAICS industries using the CD function with inefficiency effects. We present the set of parameter estimates for the 11 NAICS to see whether there is heterogeneity among industries (or not) in terms of elasticities, returns to scale or technological change. We transform all variables into their logarithm forms, except binary and categorical variables. Results in table 3 indicate that a marginal increase of 1% in labour, capital, and intermediate inputs lead to output increments of 0.16%-0.52%, 0.02%-0.08%, and 0.48%-0.71%, respectively. In most cases, we observe decreasing returns to scale (from 0.81 to 0.97), except in the 33, 46 and 54 NAICS industries where we find constant returns to scale. Interestingly, we find that technical change is negative and statistically significant in all industries.

Table 3
Parameter estimates of the production frontier

Notes: Dependent variable: log(total sales+(output stock in Jan 1st minus output stock in Dec, 31st).) Robust standard errors in parentheses. *** p<0.01, ** p<0.05, * p<0.1. Source: Own elaboration based on INEGI (2015, 2018).

Using the set of parameter estimates in table 3, figure 1 displays ATE scores and ± one standard deviation in the corresponding samples. For the pooled sample, 2014 and 2017, and for the 11 NAICS industries, the ATEs are 54.6%, 56.4%, 54.3%, and 42.7%-64.2%, respectively. This finding indicates that SMEs in these industries can obtain on average 35.8%-57.3% additional output by using the same amount of inputs more efficiently. For the economic recovery, this finding has important implications since it may help SMEs to adapt to a new market configuration in the short-term by adjusting their existing production processes.

Figure 1
Technical efficiency scores
Source: Own elaboration based on parameter estimates in table 3.

Table 4 shows the parameter estimates $\left(\widehat{\gamma }\right)$ of the TI model defined in equation (5) for samples and subsamples in table 3. This allows us to check whether there is heterogeneity among industries regarding technical inefficiency effects. In most cases, the proportion of employees with a bachelor’s degree to the total number of workers has a negative and significant association with TI. This finding is in line with previous investigations arguing that better human capital endowments contribute to the efficient use of limited resources (Aguilar, 2011; Valderrama et al., 2015). In the following NAICS sectors, such an association is not statistically significant: 31 and 32 (manufacturing industries), 71 (cultural and sports entertainment services and other recreational services), 72 (temporary accommodation and food and beverage preparation services), and 81 (other services except government activities sectors).

Table 4
Parameter estimates of the technical inefficiency model

Notes: Dependent variable: Technical inefficiency. Robust standard errors in parentheses. *** p<0.01, ** p<0.05, * p<0.1. Source: Own elaboration based on INEGI (2015, 2018).

In most industries, applications to register patents are not statistically significant, which indicates that the creation of knowledge has no effect on TE scores of SMEs in Mexico. According to Holgersson (2013), who conduct a literature review on patent management in SMEs, the main constraint for patenting in SMEs is their lack of resources in the application stage and for monitoring and enforcing their patents. The no significance of these terms may indicate that liquidity constraints prevent SMEs from making large investments in creating new technologies. In this regard, Lanjouw and Schankerman (2004) and Kingston (2004) argue that the patent system does not work properly for SMEs because such enterprises are more exposed to litigation risks and threats than large firms, which can deal with such litigations and threats more easily. Given such obstacles, SMEs may be adapting existing technologies to their productions processes rather than allocating their production efforts to create new knowledge.

According to table 4, we do not find empirical evidence that validates the previous hypothesis. It seems that the acquisition of existing technologies does not influence TI scores. Buying licences and adapting existing technologies to the production process without any modification (more frequently) does not lead to higher efficiency. The cases in which such a hypothesis holds are the retail trade sector and other services. Regarding transfers of knowledge to another enterprise, SMEs that create and sell new technologies to another enterprise more often tend to be less efficient than other SMEs. This finding is not in line with our original expectations; however, it might be explained by the creation of comparative advantages of those firms creating new technologies but not selling them to other firms. Furthermore, since SMEs do not generally create new forms of knowledge, transfers to other enterprises might not take place frequently. Using the Internet in the production activities strongly improves SMEs performance in most industries, especially in the manufacturing, transport, mail and storage, and professional, scientific and technical services sectors, where Internet connection is used intensively.

Our results add further support to the argument that expenses related to the tax payment procedures push SMEs away from the frontier. This association is strongly significant in all industries and both years. For policymakers, such a finding represents an opportunity because simplifying the required procedures for paying taxes would likely benefit TE of SMEs and, at the same time, boosts the economic recovery. Managerial practices are also hypothesised to influence firms’ performance. In this paper, we discover that the number of managers per worker generally has no effect on TI. As exceptions, in the 33 manufacturing industries (wholesale trade and other services) increasing (reducing) the number of managers would likely reduce (increase) TI. Perhaps, the existence of non-monotonic effects explains such results. That is, there exists a turning point at which the number of managers per worker is optimal and the slope of this relationship is reversed. This is something that should be analysed using the non-monotonic effects suggested by Wang (2002).

Using parameter estimates in table 4 and equation (6), we compute firm-specific marginal effects of each control variable on TI. Figure 2 shows the average and ± 1.96 SD marginal effects for the 8 explanatory variables and for different samples. In line with parameter estimates in table 4, higher proportions of employees with a bachelor’s degree tend to reduce TI. On average, a 10% increase in such a proportion leads to a 3% reduction in TI. The only exceptions are the 31 and 71 NAICS industries, where most of the firm-specific marginal effects are close to zero (see figure 2a). Although SMEs that asked for a patent are on average 2.9% more efficient than other SMEs, most of the firm-specific marginal effects in different industries are close to zero. There are some exceptions. SMEs in the 43, 71, 72, and 81 NAICS industries that applied for a patent are on average 11.3%, 38.7%, 9.1%, and 18.7% more efficient than other SMEs in the corresponding industries (see figure 2b).

Figure 2
Distribution of firm-specific marginal effects

Figure 2 (continued)

Figure 2 (continued)

Figure 2 (continued)

Figure 2 (continued)

Note: We exclude the 71 NAICS industry in the 2g figure due to scale issues. The average marginal effect in the 71 NAICS industry is -0.23. Source: Own elaboration based on Wang (2002) and INEGI (2015, 2018).

We also observe heterogeneity of firm-specific marginal effects of licenses and transfers among industries. Regarding licences that allow SMEs to use existing technologies in their production processes, most of the marginal effects are close to zero. However, SMEs in the 46, 48-49, 56, and 81 NAICS industries that rarely buy these licences tend to be more inefficient than their counterparts. On average, reducing the frequency of technology acquisition by one unit increases TI scores by approximately 0.45% in the pooled sample (see figure 2c). In terms of transfers, we observe that SMEs selling their own technology to other firms less frequently are, on average, more efficient than those SMEs doing it regularly. Our results indicate that reducing the frequency of technology transfers by one unit shrinks TI scores by approximately 7.4% in the pooled sample (see figure 2d). Furthermore, if the firm uses the Internet in its production processes, it is, on average, 27.6% more efficient than those without access to the Internet (see figure 2e).

We confirm that costs related to tax payments generally boost TI. On average, a 1% increase in the ratio of such costs to total revenue rises 1.12% TI scores in the pooled sample. In all other samples, we observe that the average marginal effect is above the zero line, which suggests that the complexity of tax payments’ procedures may create technical inefficiencies in the production process thereby such financial resources should be used in a more efficient way (see figure 2f). Our findings for marginal effects of the share of exports on TI contradict the initial expectations in most industries. A 1% increase in exports share leads to 0.13% rise in TI scores in the pooled sample. The exceptions are the 48-49, 56, and 72 NAICS industries, for which the average marginal effects are -0.01%, -0.07%, and -22.97%, respectively. This finding deserves further exploration, especially in large firms that are more likely to be linked to the external market (see figure 2g). Finally, we find that the current ratio between managers and the total number of employees in SMEs may be beyond the optimal level, implying that additional managers would likely increase TI scores (see figure 2h).

The results indicate that SMEs in the manufacturing, trade, and service sectors are not fully efficient. This implies that such enterprises can either produce more output using the same inputs more efficiently or produce the same output using fewer inputs. We also encounter that human capital, measured by the share of employees with a bachelor’s degree to total workers and the use of the Internet in the production activities, reduce TIs. On the other hand, when SMEs spend large quantities of money on tax payments procedures, such firms tend to be less efficient than their counterparts. The results are not conclusive and sector-specific for the remaining inefficiency effects.

To contextualise our findings, we compare them with previous research in table 1. Figure 3 shows ATEs reported by studies where Mexico is the case of interest. The first 10 studies use the DEA and the rest the SFA models. Overall, we observe lower ATEs in SFA than in DEA studies, but our results coincide with those using the same approach. The comparisons among studies are not always possible because the corresponding samples and industries of interest are not the same. However, our results for ATEs are in line with those in Aguilar (2011), Chavez and Fonseca (2012), Valderrama et al. (2015) and Diaz et al. (2019), who analyse manufacturing and automotive industries.

Figure 3
Average Technical Efficiency: A comparison
Notes: 1. All (CRS); 2. All (VRS); 3. Manufacturing (micro); 4. Manufacturing (small); 5. Manufacturing (medium); 6. Manufacturing (large); 7. Automotive (2004); 8. Automotive (2009); 9. Automotive (2014); 10. Electric (BC); 11. All; 12. Manufacturing (footwear); 13. Manufacturing (garment); 14. Manufacturing (wood); 15. Manufacturing (textile); 16. Manufacturing (1988); 17. Manufacturing (1993); 18. Manufacturing (1988); 19. Manufacturing (2003); 20. Manufacturing (2008); 21. Manufacturing (1985); 22. Manufacturing (2009); 23. Automotive (1988); 24. Automotive (2008); 25. All (11 NAICS); 26. All (11 NAICS-2014); 27. All (11 NAICS-2017); 28. Manufacturing (31 NAICS); 29. Manufacturing (32 NAICS); 30. Manufacturing (33 NAICS); 31. Wholesale trade (43 NAICS); 32. Retail trade (46 ; NAICS); 33. Transport, mail and storage (48-49 NAICS); 34. Professional, scientific and technical services (54 NAICS); 35. Business support, waste management, and remediation services (56 NAICS); 36. Cultural and sports entertainment and other recreational services (71 NAICS); 37. Temporary accommodation and food and beverage preparation services (72 NAICS); 38. Other services except government activities (81 NAICS). Source: Table 1 and figure 1a.

Some of the results are in line with the existing empirical literature. For example, Taymaz (2005), Aguilar (2011), Charoenrat et al. (2013), Valderrama et al. (2015), and Diaz et al. (2019) also discover that better endowments of human capital significantly reduce TIs, which adds further empirical support to the argument that high-qualified workers help to use available resources more efficiently. Regarding the use of the Internet, or ICTs, Becchetti et al. (2003) also find that its use reduces TI in Italian manufacturing firms, arguing that such tools positively affect the creation of new processes or products. We also argue that the Internet facilitates the flow of knowledge and market information towards and out of SMEs. The positive association between the expenses incurred by the firm for tax payments and TI is in line with our initial expectations. There are many studies in the literature looking at the subsidies or taxes- TI link; however, this information is not available in ENAPROCE. Therefore, we could not account for it in the TI equation. Assuming that such expenses are highly correlated with tax payments, our results suggest that, on average, current taxes significantly increase TI in SMEs. On the other hand, if such costs are not correlated with tax payments, policymakers should simplify tax payments procedures to reduce TI in SMEs.

Taymaz and Saatci (1997) find that technology transfers significantly reduce TI. However, Becchetti et al. (2003) and Taymaz (2005) encounter that such a relationship is not statistically different from zero. Our results are in line with the latter findings, i.e., the way in which they use available inputs remain unchanged. Regarding knowledge creation, we find that requests for registering patents are not associated with TI. However, SMEs creating and selling their own technology to other firms less frequently are, on average, more efficient than those doing it more frequently. This result suggests that SMEs prefer not to sell their innovations and create comparative advantages instead.

Bigsten et al. (2000) and Tovar (2012) find that firms selling their output in the external market are, on average, more efficient than other firms; notwithstanding, Alvarez and Crespi (2003) encounter a null association, and Diaz et al. (2019) the opposite relationship. Our findings align with Diaz et al. (2019) and Alvarez and Crespi (2003). However, given our data’s cross-sectional nature, we think that omitted variables might cause a bias in the parameter estimates and can be explored carefully in future studies. For the share of managers over total employees, the results are in line with Taymaz (2005), who do not find a statistically significant association between administrative personnel and TI. We argue that such a relationship should be analysed at the firm level and see whether the current level of managers per worker is optimal or not.

Unfortunately, we could not account for all inefficiency effects in the TI equation. Among other things, the existing literature suggests the inclusion of investment in R&D activities, training for employees, debts, firm experience (years in the market), specialization indexes, etc. (see table 1). Although some of these variables appear in the ENAPROCE, there are many records with missing information, preventing us from including them as inefficiency effects in the TI equation. Therefore, the reader should consider this limitation, and we encourage future studies to analyse such effects once the data becomes available.