Demands for multidimensional information on the work environment: A methodological framework for regular studies

Abstract

BACKGROUND:

Development of methodologies for making economic decisions on designing work environment studies is a theoretical challenge for researchers in occupational health sciences. There are well-defined tools available in the relevant literature for analysis of cost-efficiency associated with the assessment of an occupational exposure of interest. However, these analytical tools are not appropriate for holistic studies of the work environment as a multidimensional reality.

OBJECTIVE:

This article introduces an appropriate methodology for designing cross-sectional comprehensive studies of the work environment, in order to optimize the production of information on the psychosocial, ergonomic, and physical dimensions of the work environment in regular studies.

METHODS:

The employment of a translog cost-utility function is suggested as a suitable way to provide cost-minimized designs for regular studies which are aimed at providing or developing multidimensional information systems of the work environment.

RESULTS:

The translog cost-utility function is not subject to predetermined restrictions, but has a flexibility property allowing it to be transformed to any specification that is adaptable to the specific work environmental characteristics and research requirements.

CONCLUSION:

The translog cost-utility function is an appropriate econometric model for optimizing the production of multidimensional information on occupational exposures in regular cross-sectional workplace studies.

Keywords

Occupational health risk factors ordinal utility quality criteria economic decision cost-utility function

1 Introduction

As the work environment is a multidimensional social reality, work-related disorders among workers can have multiple causes [1]. To provide effective intervention programs for preventing work-related disorders, we first need to produce information about all dimensions of the work environment [2]. There may be several risk factors behind these work-related disorders and their socio-economic consequences in terms of lost working hours and productivity, and these risk factors may stem from different dimensions of the work environment [1, 3]. The determinant factors have recently been classified into three groups of exposures concerning the psychosocial, ergonomic, and physical environment in the workplace [1 –4]. The psychosocial dimension of the work environment contains risk factors such as stress, anxiety, depression, and violent behaviour among workers; the ergonomic dimension is concerned with ways of working and performing tasks, and contains static and dynamic mechanical workloads on the body of workers; and the physical dimension includes factors such as temperature, noise, air quality, and lighting. These different groups have been shown to have different effects on occupational health, non-idle working time, and labour productivity [3]. Thus, multifactorial health interventions in the workplace and successive management of occupational health and safety (OHS) need access to information on all three groups of risk factors. The complete, multidimensional set of information is particularly useful for allocating human and economic resources to promote OHS. However, most studies in the occupational health literature focus only partially on the psychosocial [5 –7], ergonomic [8 –10], and physical work environment [11 –13]; and the usefulness of partial studies for decision makers is low. The ability of work environmental studies to satisfy decision makers increases with more complete information about the work environment. Thus, despite scarce resources, the usefulness of comprehensive studies of the work environment in improving OHS stimulates researchers to collect more and better information. Holistic work environment studies can lead to more effective health interventions with expected economic benefits higher than their costs [1, 2].

Researchers and decision makers on both the organizational and the social levels can benefit from environmental studies of workplaces. The information produced by these studies can be used to improve the work environment, occupational health, and labour productivity. The expected social benefits of improving the work environment are the grounds on which to assess the economic value of the information that would be produced during such studies [2]. In making an economic decision on proposed work environmental studies, the estimated economic value of information to be produced during the studies will be compared to the cost of producing the information [2]. However, in addition to single studies in specific workplaces, there are also regular studies of workplaces and/or occupational groups periodically performed by public authorities and company safety units. In regular studies of the work environment, the demands for information on different dimensions of the work environment should be determined optimally based on the ever-fluctuating costs of using resources; as well as the activities, exposure patterns, and work organization that influence the requirements in the study. Thus, for regular studies of workplaces which have already been decided to proceed, the decision problems refer to allocation of scarce resources between different groups of environmental risk factors in order to meet certain requirements. To resolve the allocation problem of such studies, the inputs to produce the multidimensional information system in the workplaces, the costs of using these inputs, and the determinants of the utility perceived by researchers from the statistical production should be distinguished and quantified. Information on different dimensions of the work environment can therefore be demanded by using a cost-utility approach with respect to the production costs and the ability of the study to meet the researchers’ requirements. The utility of the study can then be maximized in relation to the available resources, or the study’s cost may be minimized in relation to a given utility.

As previously pointed out, regular workplace studies are characterized by non-constant input costs (particularly labour costs) of producing information, non-constant research budgets, and also non-constant research requirements depending on changes in the work environment. Thus, different alternative designs can be chosen for regular studies of the work environment. A design to produce multidimensional information on the work environment contains a study plan, technical methods, measurement strategies, and a set of risk factors to be assessed in the psychosocial, ergonomic, and physical dimensions of the work environment. If the research budget increases, the design may expand to include advanced technical methods, more risk factors, and workers in several workplaces. Conversely, if there is an increase in input prices (such as salaries for investigators), the decision makers should decrease the scope of the study and be satisfied with assessments of fewer risk factors among a smaller number of workers and workplaces. A design should also be adapted to changes in the research requirements, depending on the reports received from the affected occupational groups and workplaces.

Besides economic constraints, there are other reasons why researchers in the field of work environment and occupational health might evaluate one design against another. These non-economic reasons are described as preferences. Researchers can also be indifferent between certain alternative designs (i.e. they may be satisfied with either design), regardless of the associated costs. The level of satisfaction delivered by the work environment studies, which is the basis of researcher preferences, can be defined and measured as a non-economic benefit, the so-called ordinal utility – a ranking of satisfaction that researchers get from the studies from the most to the least preferred. Despite different measurement units, there is a strong positive association between the expected economic benefits of using the information from the studies and the ordinal utility of the non-economic benefits. The work environment studies deliver economic benefits beyond the cost, and the utility of the studies as assessed by the researchers reflects the studies’ expected value [2].

All users of the studies’ output (i.e. stakeholders such as authorities, employers, safety units, employees, and researchers in the field) share the economic benefits and realize the cost savings when they manage to prevent work-related disorders [2]. As the economic benefits increase with the number of users [2], the utility increases with the usefulness of the studies’ output. Thus, the utility approach focuses not on how the exposure data is acquired, analysed, and combined, but rather on how the final output from the work environment studies can be useful and effective in improving the systemic health risk management in workplaces. Accordingly, the ordinal utility of these studies is determined by the quantity and quality level of the information produced about occupational risk factors, and also by some of the quantity and quality criteria such as the integration between collected data and their combination set. Here, the quantity refers to the number of risk factors and the number of events in each risk factor to be observed and registered in the workplace study; quality refers to data management and technical characteristics of the data collection procedure such as relevance, low-error and up-to-date micro-data (i.e. events at the workplace), maintenance, availability, and easy-to-use information systems as the output of the studies; integration refers to timeliness, comparability, and consistency; and the combination set refers to the property of the multidimensionality and full view in the study design.

A substantial part of the costs of producing information on different dimensions of the work environment consists of the labour and user capital costs of collecting, analysing, and transforming exposure data into useful information systems; that is, the costs of investigators, equipment, and buildings. Again, the utility of work environment studies refers to researchers’ satisfaction with these studies in terms of meeting certain requirements. Researchers utilize both more and better informational products on risk factors in the workplaces in order to provide a useful multidimensional information system as the final output used to change the work environment and improve occupational health and labour productivity. Thus, although the utility of work environment studies is not an objective measure like the costs, it is also not a purely subjective measure, due to its relation with the expected economic benefits of using the information system. On the other hand, the costs of such studies often increase with the provision of more and better informational products. Thus, both the cost and the utility of work environmental studies are dependent on the determinant factors of the quantity and quality of the information produced during the studies.

Current partial studies of occupational exposures are characterized by the collection of exposure data with the least statistical error [14 –21], while ignoring economic constraints. Further, the corresponding cost-efficiency studies of exposure assessments balance the statistical efficiency of the exposure assessments with the cost of achieving this efficiency [22, 23]. However, statistical efficiency is only one of the factors necessary for exposure assessment studies to be effective and useful [24]. Researchers in the field have other requirements that they expect to be met during the studies, such as timeliness, completeness, comparability, participation, collaboration, and accessibility [2, 24]. In summary, they utilize the effectiveness and usefulness of such studies in the improvement of the work environment and occupational health. The main challenge and difficulty in work environment studies is not to produce low-error data on an exposure, but rather to make the studies useful and effective. By considering statistical efficiency as the target, resources can be allocated between different stages of statistical production; for example, between direct technical measurement and indirect observational assessments [25], or between sampling units [26]. However, this target does not allow allocation between different psychosocial and physical dimensions of the work environment (i.e. between the informational products which make the studies useful and effective in terms of socio-economic consequences). While all occupational exposures in a mutual interaction with various effects can lead to work-related diseases, none of the existing cost-efficiency studies offer even an incomplete methodology for demanding information on different dimensions of the work environment.

Sampling methodology usually utilizes a similar variance formula based on an additive random effect model for assessing the precision of a mean exposure estimate as ‘utility’ or ‘output’ that certainly contains several determinants and criteria [27 –29]. The formula, however, exhibits a linear relationship between samples and precision, and also results in an unusual model structure for estimating the total variable cost of the statistical production. According to Rezagholi [24, 26], the employed variance model is characterized by homogeneity of degree one; however, the homogeneity leads to a linear cost-precision association. The linearity in the samples-variance relationship means that increasing the number of sampling units will indefinitely increase the precision of the mean estimate at the same rate. The linear property leads to an additional linearity, as previously mentioned, in the cost-precision association. A linear cost-precision means that the curves of average and marginal cost are identical, and the statistical production exhibits a constant return to scale at each level of precision. This linearity was caused by using a variance formula based on an additive effect model and defining precision as the inverse of variance. One attempt to remove the linearity involved defining precision as the inverse of the standard error [25, 26], the result of which was that the assessment study exhibited a decreasing return to scale. It can also be removed by defining multiplicative effect models for errors [30], particularly when the models convey different efficiency rates [24]. However, considering the precision of the mean estimate as utility or output does not remove the obscurities, and thus difficulties remain in deciding whether to expand the study in order to collect more and better information [24].

The purpose of this study was to provide an appropriate methodology to optimize the production of multidimensional information about groups of occupational exposures in terms of the psychosocial, ergonomic, and physical dimensions of the work environment. For this optimization, the cost-utility approach is employed for regular or already-accepted cross-sectional studies of the work environment. The information on the three dimensions of the work environment is referred to the related exposures during certain times at which signals of the respective exposures are perceived or experienced by the workers. Further, as the three groups of exposures may have different and ever-changing health and socio-economic impacts, their information is demanded relative to each other, to their usefulness, and thus to the utility of the studies. Important econometric issues, which are basic tools for making economic decisions regarding demands for the informational goods, are assessed using estimated regression coefficients and mathematical methods.

2 Materials and methods

2.1 Design of work environment studies, and important variables

The econometric model presented and developed in this article is intended for regular or already-accepted studies of workplaces aimed at producing multidimensional information on occupational health hazards. The work environment studies cover three dimensions: the psychosocial, ergonomic, and physical work environment. The total and average unit costs of producing the three informational goods and the ordinal utility of the studies will be estimated and used for economic decisions regarding inputs to utility of the work environmental studies (i.e. demands for information on different dimensions of the work environment).

2.2 Analytical tools and basic assumptions

For optimizing the demands for information on the three dimensions of the work environment, a cost-utility function is employed. The main assumption is that the investigators will minimize the costs of the studies while still fulfilling certain requirements valued in terms of utility. Mathematical methods are used for assessing important econometric measures, which are used as the basis for economic decisions on designing and conducting work environmental studies.

The complete multidimensional information on the work environment is assumed to be a result of combining partial information on exposures in the psychosocial, ergonomic, and physical work environment. The usefulness and effectiveness of the studies’ outcomes in changing the work environment is expressed as ordinal utility. As work environment studies are usually performed over a period of a few months rather than several years, the effect of technical progress in producing the information is assumed to be constant during the studies. Thus, the measurement instruments and other equipment, statistical methods, and competence embodied in the investigators remain unchanged during the study. The average variable costs of producing the informational goods are also assumed to be constant during the economic analysis, as are the requirements of the work environment studies which stem from the status of occupational health hazards and their socio-economic consequences.

2.3 Inputs to utility of work environment studies

Researchers in the field benefit from access to complete, high-quality multidimensional information on the work environment; that is, more and better information on risk factors for illnesses and impairments in the psychosocial, ergonomic, and physical dimensions of the work environment. The quantity and quality of the information produced influence the usefulness and effectiveness of the studies in improving the work environment, occupational health, and labour productivity. To produce more and better information, researchers in the field have two types of requirements regarding the performance of work environment studies: 1) those that are met by the workplaces to be studied, such as the workers’ participation and the companies’ collaboration, and 2) those that are met by the technical characteristics of the equipment, the statistical methods used, and the competence embodied in the investigators. These requirements consist of relevance, timeliness, completeness, accessibility, comparability, and statistical efficiency associated with the exposure data collected.

The quality criteria are addressed in order to produce ‘better information’, and have a constant linear effect on the utility of work environment studies, while the quantity criteria are addressed in order to produce ‘more information’, and have variable non-linear effects on the utility.

3 Results

3.1 Derivation of cost-utility function

The study design or the technology to produce a multidimensional information system for the work environment is represented by a concave monotonic continuous and twice-differentiable utility function: $U = A \cdot f (S, W, E),$ (1)

where:

S represents the amount of information on the psychosocial work environment (i.e. about organization, salaries and rewards, carrier opportunity, work demands, expectations, supervisory supports, and behaviours in the workplace).

W represents the amount of information on the ergonomic work conditions (i.e. static and dynamic corporal stressors or mechanical pressures on workers’ bodies).

E represents the amount of information on the physical characteristics of the workplace such as temperature, air quality, lighting, noise, and vibration.

Thus, S, W, and E are the composite informational products which will be combined and transformed to a complete multidimensional information system for the work environment as the final output. The three intermediate products refer to the amount of the respective exposures and risk factors with their signals of events at the workplace. It is assumed that the events in each risk factor area are captured and registered with the same adequacy.

U represents the ordinal utility of the work environment studies; that is, the ability of the studies to meet the requirement and thus satisfy researchers in the field. In order for researchers to be satisfied, the multidimensional information system processed by combining the three informational products S, W, and E must be useful and effective in practice. The complete multidimensional information system for each workplace is the final output of the work environment studies. The usefulness and effectiveness of this output is an important determinant of the economic value of the information [2]. The more complete the information on the three dimensions of the work environment, the greater the utility or ability of the study to meet researchers’ requirements, and thus the greater expected economic benefits of using the information output.

A is a coefficient which reflects an increase in the utility of the study when combining informational products S, W, and E of acceptable quality. It shows the power or the efficiency of the holistic study design – an overall effect on utility, which is also determined by technical characteristics of the study and other quality criteria. A remains unchanged while the work environment studies are in progress.

The utility function (1) gives the utility of a work environment study as a function of the composite informational products and the technical characteristics of the study design. For trading off utility against cost, a cost-utility function is derived from (1) by using the duality principle and by assuming that investigators/contractors minimize the total variable cost (TVC) of producing the multidimensional information for a given utility: $TVC = \frac{1}{A} \cdot g (p_{S}, p_{W}, p_{E}, U),$ (2) where p_S, p_W, and p_E are the average unit prices of the information produced on the psychosocial, ergonomic, and physical risk factors in the work environment, respectively; and the TVC is calculated as TVC = p_S · S + p_W · W + p_E · E, where p_S · S, p_W · W, and p_E · E are the costs imposed by the researchers for informational products S, W, and E, respectively. The costs differ not only because of the different number of risk factors in each dimension, but also because different instruments are used to measure these risk factors. Usually, psychosocial risk factors are assessed by interviews, ergonomic risk factors are estimated by observational methods, and physical exposures are measured directly by technical instruments.

All components of the TVC consist substantially of costs for capital (buildings and equipment) and labour (investigators for providing information on the work environment). The prices of labour and capital are assumed to be given as determined in the labour and capital markets, respectively. Further, it is assumed that the total cost of the study is non-decreasing in both the utility and the unit prices.

The next decision to be made is about the functional form (specification) of the cost-utility function (2). This is very important, because it can affect the empirical results. The following log-linear utility and cost-utility functions of Cobb-Douglas type are the simplest way to optimize the demands for the informational products: $U = A \cdot S^{a} \cdot W^{b} \cdot E^{c}$ (3) $TVC = k \cdot p_{S} β_{1} \cdot p_{W}^{β_{2}} \cdot p_{E}^{β_{3}} \cdot U^{γ},$ (4) where a, b, and c refer to the marginal utilities of the informational products S, W, and E, respectively (i.e. the partial effects of additional units of the informational products on the utility of the work environment study); β₁, β₂, and β₃ are cost elasticities of the average unit prices p_S, p_W, and p_E, respectively, as functions of the marginal utilities; γ is the cost elasticity of utility; and k denotes the autonomous cost of the study as a function of A. Note that these elasticities, although constants, may not be equal; they are estimated with a regression technique. These values are used as a basis for economic decisions on designing regular environmental studies of workplaces. The general properties of cost or expenditure functions such as Shepherd’s Lemma, increased cost in utility, and homogeneity of degree one in the average costs are valid for the cost-utility function (4). The utility function (3) shows the maximum utility that can be achieved by combining the three informational products. The derived study design is efficient if the utility cannot be increased by using other combinations of the available informational products (i.e. other alternative designs).

However, attempts to maximize the utility are costly, and can seldom be relevant for researchers in the field, considering their cost constraints. Further, the basic assumption in consumption theory that ‘more is better than less’ is not fully met regarding information on the work environment [2]. Thus, the assumption of cost minimization and the cost-utility function (4) are chosen for further analysis. It is however not possible to estimate the derived cost-utility function, due to its non-linearity, and so it must be transformed initially to a linear functional form of regression by employing a natural logarithm principle as follows:

$\begin{matrix} ln TVC & = α + β_{1} \cdot {\ln p}_{S} + β_{2} \cdot {\ln p}_{W} \\ + β_{3} \cdot {\ln p}_{E} + γ \cdot \ln U, \end{matrix}$ (5) where α = ln k denotes the autonomous cost of the work environment studies in logarithm. This is a cost that is independent of any combination set of the informational products, and reflects the effort to produce better information and to composite the informational products.

According to the cost-utility model introduced here, the optimal demand for informational products (i.e. information about the three dimensions of the work environment) is a function of the average unit prices of obtaining them, their marginal utilities, and the utility of the holistic work environmental study.

3.2 Transcendental logarithmic cost-utility function

To avoid restrictions as much as possible, the logarithmic cost-utility function (5) is developed into a transcendental logarithmic cost-utility function based on a second-order Taylor’s series approximation theorem in logarithms [31 –33]. The non-technological progress of the function’s general specification is expressed as:

$\begin{matrix} \ln TVC & = α + \sum_{i = 1}^{3} β_{i} \cdot {\ln p}_{i} + \frac{1}{2} \sum_{i = 1}^{3} \sum_{j = 1}^{3} β_{ij} \\ \cdot {\ln p}_{i} \cdot ln p_{j} + γ_{U} \cdot \ln U + \frac{1}{2} γ_{UU} \\ \cdot {(\ln U)}^{2} + \sum_{i = 1}^{3} δ_{iU} \cdot {\ln p}_{i} \cdot \ln U, \end{matrix}$ (6) where i and j refer to the informational products S, W, and E, respectively; β_i is the cost elasticity of average unit price p_i and β_ij its second-degree polynomial approximations; γ_U is the cost elasticity of utility and γ_UU its second-degree polynomial approximation; and δ_iU reflects the combined effect of utility and input prices on the TVC.

The cost share equations for informational products (i.e. the cost of producing information about each dimension of the work environment relative to the total variable cost of study) is derived from (6) by employing the envelope theorem and Shepherd’s Lemma as follows: $S_{i} = β_{i} + \sum_{1}^{3} β_{ij} \cdot {\ln p}_{j} + δ_{iU} \cdot \ln U$ (7)

The function of optimal quantity demands for the informational products $x_{i}^{*}$ (where i is S, W, and E), which shows the need to provide information about the three dimensions of the work environment, will then be obtained as: $x_{i}^{*} = \frac{TVC}{p_{i}} \cdot (β_{i} + \sum_{1}^{3} β_{ij} \cdot {\ln p}_{j} + δ_{iU} \cdot \ln U)$ (8)

3.3 Restricted specifications

It is worth mentioning that the transcendental logarithmic cost-utility function (6) has a flexible functional characteristic, so that as different restrictions (assumptions) are imposed, its general unrestricted specification transforms into increasingly limited specifications, such as those given below.

3.3.1 Homothetic specification

The homothetic specification prevails when the utility level of the entire work environment study has no effect on the partial demands for informational products on the psychosocial, ergonomic, and physical dimensions of the work environment. Thus, the partial demands for the informational products are determined by cost, unit prices, and marginal utilities of the informational products, and not by the utility. Mathematically, this means that δ_iU = 0.

Under the homothetic assumption, the translog cost-utility function (6) transforms to the following specification:

$\begin{matrix} - 12 pt TVC & = α + \sum_{i = 1}^{3} β_{i} \cdot {\ln p}_{i} + \frac{1}{2} \sum_{i = 1}^{3} \sum_{j = 1}^{3} β_{ij} \cdot {\ln p}_{i} \\ - 12 pt & \cdot l {n p}_{j} + γ_{U} \cdot \ln U + \frac{1}{2} γ_{UU} \cdot {(\ln U)}^{2} \end{matrix}$ (9) and the cost share equations and the demand functions for the informational products can be simplified to:

$\begin{matrix} S_{i} = β_{i} + \sum_{1}^{3} β_{ij} \cdot {\ln p}_{j}; \\ - 12 pt & x_{i}^{*} = \frac{TVC}{p_{i}} \cdot (β_{i} + \sum_{1}^{3} β_{ij} \cdot {\ln p}_{j}) \end{matrix}$ (10)

3.3.2 Homogeneous specification

The homogeneous specification of the cost-utility function prevails when the underlying design to provide the multidimensional information system has the property of a constant degree of utility. In this case, the marginal rate of substitutions between any two dimensions of the work environment is independent of the utility of the entire work environment study. Mathematically, this means that δ_iU =γ_UU = 0. The translog cost-utility function is then specified by:

$\begin{matrix} \ln TVC & = α + \sum_{i = 1}^{3} β_{i} \cdot {\ln p}_{i} + \frac{1}{2} \sum_{i = 1}^{3} \sum_{j = 1}^{3} β_{ij} \\ - 12 pt & \cdot {\ln p}_{i} \cdot ln p_{j} + γ_{U} \cdot \ln U \end{matrix}$ (11)

Under the homogeneous assumption, the cost share equations and the demand functions for the informational products are the same as in the case of the homothetic assumption; that is, they are given by Equations (10).

3.3.3 Constant power to change utility

If the homogeneity in utility also has the linear property, then γ_U = 1. Linear homogeneity can be described as constant power to change utility, which also shows the design’s efficiency. This property means that a given change in the amount of informational products (e.g. 30%) will result in the same change (i.e. 30%) in the utility of the entire study of the workplace. The case of constant power to change utility (CPU) means economically that there is no advantage or disadvantage to changing the design. The translog cost-utility function will transform to the following specification under this assumption:

$\begin{matrix} \ln C & = α + \sum_{i = 1}^{3} β_{i} \cdot {\ln p}_{i} + \frac{1}{2} \sum_{i = 1}^{3} \sum_{j = 1}^{3} β_{ij} \\ - 12 pt & \cdot {\ln p}_{i} \cdot ln p_{j} + \ln U \end{matrix}$ (12) and the cost share equations and the demand functions for the three informational products will be the same as with the homothetic and homogeneous assumptions; that is, they are again given by Equations (10).

Hence, for an efficient estimation, each specification of the translog cost-utility function (i.e. Equations 6, 9, 11, and 12) is estimated together with its own cost share equations in a system of equations.

3.3.4 Constraint elasticity of substitution (CES)

If in addition to the homogenous property (i.e. δ_iU =γ_UU = 0), the informational products can also substitute each other while keeping the utility of the workplace studies unchanged (i.e. the elasticity of substitution between any two informational products is equal to unity), then mathematically δ_iU =γ_UU =β_ij = 0. In this case, the translog cost-utility function can be simplified to the Cobb-Douglas type of cost-utility function; that is, the regression Equation (5). When these assumptions hold, the cost share equations are equal to the cost elasticities of average unit prices, meaning that S_i =β_i. The demand functions for the informational products are then calculated from the total cost, their own prices, and their own cost elasticities.

3.4 Statistical test of the validity of the model specification

As the measure of important econometric issues varies depending on which specification of the translog cost-utility function is applied, the validity of the abovementioned specifications should be assessed. There are some statistical tests available for assessing the validation of a specified cost function [31]. The likelihood ratio statistic test (LR) determines whether or not a restricted econometric model is well-specified and appropriate compared to the more general model according to λ = 2 (L_U - L_R), where L_R is the log-likelihood value of the restricted model, L_U is the log-likelihood value of the unrestricted specification, and λ is χ²-distributed with r degree of freedom equal to the number of imposed restrictions. If λ is greater than the fixed critical value, then the unrestricted model cannot be rejected.

3.5 Derivation of formulas to estimate important econometric issues

3.5.1 Formulae for use in economic decisions on informational products

Allen-Uzawa partial elasticities of substitution (σ_{x_i,x_j}) between any two informational products, which measure the effect of a change in the cost ratio p_i/p_j on the ratio of demands for the related informational products x_j/x_i, in percent, can be calculated as follows: $σ_{x_{i} {, x}_{j}} = 1 + (β_{ij} / - S_{i} S_{j}),$ (13) where S_i and S_j are the minimized cost share functions of informational products on the psychosocial, ergonomic, and physical work environment. Different cases can be considered regarding the partial elasticities of substitution:

If σ_{x_i,x_j}is positive, the informational products i and j are said to be substitutes, while a negative value of σ_{x_i,x_j}, which reinforces the need for holistic study of the work environment, claims that the informational products are complement.

The case σ_{x_i,x_j} > 0 . 5 indicates strict substitutability between the two informational products i and j, meaning that having more i makes the researcher desire less of j, and vice versa.

The case σ_{x_i,x_j} < -0 . 5 indicates strict complementarity between the two inputs, meaning that the demand for informational good i is linked to the demand for informational good j, and vice versa. If a higher quantity is demanded for i, a higher quantity will also be demanded for j; and if a lower quantity is demanded for one product, a lower quantity will also be demanded for the other.

Cases 2) and 3) are not expected to be realized in work environment studies when risk factors in different dimensions of the work environment have their own effects on certain work-related disorders and productivity, and thus unequal marginal utilities for research. Further, in addition to partial effects, the risk factors may have combined effects [3], which rather require them to be studied in composition with flexible and non-strict relationships.

The case -0.5 < σ_{x_i,x_j} < 0 . 5 exhibits unclear (flexible and mild) relationship between informational products i and j. This case is more likely to occur than the two previous cases.

The standard error of σ_{x_i,x_j} can also be calculated as S_e (σ_{x_i,x_j}) = S_e (β_ij)/S_iS_j, as a way to examine its statistical significance.

An important tool for decisions on the quantities of informational products is the concept of price elasticity of demand. According to common assumptions in microeconomics, the demand for a good or an input to production will usually decrease when its price increases; this is known as the own-price elasticity or the sensitivity of the good/input towards a change in its price. However, it is also possible for the demand for a good/input to change when the price of another composite good/input increases; this is the cross-price elasticity or the sensitivity of the good/input towards a change in the price of another composite good/input. The own-price elasticity ( $E_{D_{i}}^{p_{i}}$ ) and the cross-price elasticity ( $E_{D_{i}}^{p_{j}}$ ) of demands for the informational products on the three dimensions of the work environment (S, W, and E) can be calculated as: $E_{D_{i}}^{p_{i}} = \frac{\partial ln Q_{D_{i}}}{\partial ln p_{i}} = \frac{β_{ii}}{S_{i}} + S_{i} - 1$ (14) $E_{D_{i}}^{p_{j}} = \frac{\partial ln Q_{D_{i}}}{\partial ln p_{j}} = S_{j} \cdot σ_{x_{i} {, x}_{j}}$ (15)

An absolute value of own-price elasticity less than unity means that the informational product is inelastic, while a value exceeding unity shows that the informational product is elastic. Unlike the partial elasticities of substitution, the cross-price elasticities of demands are not symmetric (i.e. $E_{D_{i}}^{p_{j}} \neq E_{D_{j}}^{p_{i}}$ ), because that the cost shares of informational products are unlikely be equal (i.e. S_i ≠ S_j). Two informational products are substitutes if the cross-price elasticities are positive, and are complementary to each other if the values are negative.

3.5.2 Formulae for use in economic evaluations of work environment studies as a whole

Using the general functional form of the translog cost-utility function, the cost elasticity of utility (i.e. the sensitivity of the study’s total cost to changes in the utility as result of changes in the research requirements), which is also defined in economics literature as the ratio of marginal cost to average cost of utility, can be estimated as follows:

$\begin{matrix} E_{U}^{C} & = \frac{MC}{AC} = \frac{\partial C / \partial U}{C / U} = \frac{\partial ln C}{\partial ln U} = γ_{U} + γ_{UU} \\ \cdot ln U + \sum_{1}^{3} δ_{iU} \cdot ln p_{i} \end{matrix}$ (16)

Under the homothetic assumption, the cost elasticity of utility will be simplified to $E_{U}^{C} = γ_{U} + γ_{UU} \cdot ln U$ , and under the homogeneous assumption it will transform into its simplest form: $E_{U}^{C} = γ_{U}$ . Values of $E_{U}^{C} < 1$ reflect economies of scale, meaning that there is an advantage in expanding the work environmental studies and producing more and better information. Conversely, values of $E_{U}^{C} > 1$ reflect diseconomies, implying disadvantage in expanding the studies

The power to change utility (PTU), or the percentage change in the utility as a result of a percent change in all informational products, is obtained as the inverse of $E_{U}^{C}$ : $PTU = {(E_{U}^{C})}^{- 1}$ (17)

For values of PTU > 1, the underlying study design displays increasing power to change utility (IPU) and for values PTU < 1, a decreasing power to change utility (DPU) prevails. In the case of PTU = 1, the underlying design to produce information on work environment exhibits constant power to change utility (CPU). In this case, there is no advantage or disadvantage in expansion of the work environment studies.

The utility elasticity of demands for informational goods (i.e. the effect of changes in the demand of informational goods on the utility of study) can be calculated as: $E_{D_{i}}^{U} = E_{U}^{C} + \frac{δ_{iU}}{S_{i}}$ (18)

A value of $E_{D_{i}}^{U} > 1$ , indicates a decreasing demand for informational product i, while $E_{D_{i}}^{U} < 1$ suggests an increasing demand for the product and $E_{D_{i}}^{U} = 1$ reflects a constant demand. If $E_{D_{i}}^{U} < 0$ , i is an inferior informational product, meaning that the corresponding dimension of the work environment has not been problematic in the workplace. Under the homothetic assumption, the utility elasticity of demand is equal to the cost elasticity of utility; that is, $E_{D_{i}}^{U} = E_{U}^{C}$ .

4 Discussion

4.1 Estimation method and applications

Allocation problems in designing studies of the work environment can be resolved theoretically by introducing a translog cost-utility function. As mentioned above, for efficient estimations of the important regression coefficients, the translog function should be estimated in a system of equations together with the cost share equations. In order to assess econometric issues and make empirical comparisons of the restricted and unrestricted specifications of the cost-utility function, the researchers must have access to cross-sectional data on average unit prices and utilities for workplace studies. Although the model introduced in this article is concerned with regular studies of companies’ work environments, it can also be applied to emergency case studies of affected occupational groups in a region or over the whole of a country [5 , 34], as well as already-accepted proposals for work environmental studies [2].

4.2 Utility function versus cost-utility function

A utility function is basically a way of describing choice behaviour; the design (i.e. the combination set of inputs to studies and informational goods on different dimensions of the work environment) that produces a higher utility for the researchers is chosen regardless its cost [14–21 , 35 and 36]. However, as researchers in the field also utilize low-cost studies of workplaces [37, 38], the present article offers a way to determine the demands for composite informational goods while minimizing the cost of the study. That is, a cost-utility function is employed instead of the utility function. A cost-utility function, unlike a utility function, is a way of introducing the optimal choice (design) at minimum cost; the design in the preference curve that has the lowest cost [39, 40]. Thus, the cost-utility function conveys more objective determinants and fewer assumptions compared to the utility function [40, 41].

However, cost-utility functions have some limitations when it comes to estimation of econometric issues. The marginal utilities of informational products, which show the rate of change in utility associated with a small partial change in the amount of the informational products, cannot be directly accessed through estimation of a cost-utility function, nor can the power to change utility associated with the study design [cf. Equations (17) and (18)]. However, cost-utility functions offer the possibility to estimate cost elasticities of prices, which are functions of marginal utilities of the informational goods, and also the cost elasticity of utility, which is associated with the power to change utility or the design efficiency [cf. Equations (7), (16) and (17)].

An additional advantage of cost-utility functions is that the basic prices (i.e. salaries for investigators and user costs of equipment and buildings) would always be exogenous, given as determined in the labour and capital markets, and not in the model. The cost-utility functions applied in this article also give researchers the possibility to estimate more complex econometric issues such as the elasticity of substitution between any two informational goods, which is difficult to do using the primal utility function.

4.3 Translog cost-utility function

The cost-utility functions introduced in this study are different to each other, and give different optimal solutions. As the Cobb-Douglas cost-utility function is based on larger assumptions, it should be developed into a transcendental logarithmic cost-utility function, or translog cost-utility function for short, based on second-order Taylor’s series approximation of quantities [31, 32]. The simple cost-utility function (5) constrains the elasticity of substitution between any two informational products to be equal to unity, which is crucial. It is not certain that, for instance, one more unit of information about the physical work environment can substitute for one less unit of information about psychosocial work environment while keeping constant the utility of having the information output. The three dimensions of the work environment are very unlikely to have the same socio-economic impacts in terms of sickness absences, impairments at work, and labour productivity. Unlike the Cobb-Douglas functional form, the translog cost-utility function is not subject for predetermined restrictions, while it has the flexibility property allowing it to be transformed to any specification that is adaptable to the empirical data collected in the work environmental studies. The cost-utility function also offers the opportunity to estimate key econometric issues of complex designs for acquiring data and information on occupational health hazards in a simple way – as the econometric issues of complex production technologies in its original function [32]. All the key issues used in economic decisions are assessed, in principle, on the basis of the estimated regression coefficients.

4.4 Technical characteristics and quality criteria

In the analysis described here, the studies’ quality criteria were on the whole assumed to have a constant effect on the utility of having the final information output. In the cost-utility function, the effect substantially reflects the autonomous cost of the work environment study – a cost independent of the quantity of informational goods to be produced during the work environment study. Thus, the technical characteristics of measurement instruments and methods, as well as the level of competence and experience embodied in the investigators were assumed to be predetermined and known, respectively, in designing work environmental studies, and to remain unchanged during the studies and the economic analysis [26]. As a result, cost elasticities of input prices could not be decomposed into their contents regarding physical capital, investigators, and technical characteristics.

The utility value of the information output from the workplace studies was assumed to be a result of meeting different quantity and quality requirements in the studies. As already mentioned, the quality requirements are mostly fulfilled by the technical characteristics of the measurement methods and investigators. It is worth adding that the econometric model introduced here assumes that the technical characteristics are homogeneous for all occupational exposures in the three mentions of the work environment. For instance, investigator competence was assumed to be at the same level when assessing exposures in each dimension of the work environment; and all measurement methods, measurement instruments, and other equipment were assumed to be appropriate and to have the same technical performance in assessing exposures in the workplace. Thus, the competence embodied in the investigators, the appropriateness of measurement instruments, and the technical performance of equipment were necessarily assumed to be distributed fairly over the three dimensions of the work environment.

5 Conclusions

The general form of the transcendental logarithmic cost-utility function introduced in this article is not subject to predetermined restrictions, but has a flexibility property allowing it to be transformed to any specification that is adaptable to the specific work environmental characteristics and research requirements. The function is thus an appropriate econometric model for optimizing the production of multidimensional information on the work environmental health risk factors in regular cross-sectional studies.

Conflict of interest

None to report.

Footnotes

Acknowledgments

The Faculty of Health and Occupational Studies from University of Gävle-Sweden is gratefully acknowledged for their support in writing the article.

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