Comparative Analysis of Hedonic Wage and Discrete Choice Models in Valuing Job Safety

The discrete choice model is widely used in valuing environmental goods and recreation demand. By exploring how individuals make choices among discrete alternatives, such as different brands of commodities, transportation modes, fishing sites, occupation/industries, etc., the discrete choice model can be used to infer consumers’ WTP for an attribute and to predict the changes in demand for a certain alternative associated with the changes in attributes. The random utility model (RUM) is the most widely adopted discrete choice model, which is based on the work of economists including Manski and McFadden (Manski and Lerman, 1977; McFadden, 1974; McFadden, 1980; McFadden & Train, 2000).

Suppose a worker i’s utility from working in industry j consists of four parts: the consumption of the ordinary commodity Q, job risk R, other industry-specific job attributes (such as working hours and the unemployment rates) X, and Y, workers’ personal attributes (such as education, age, and gender). Including these personal attributes can help investigate the observed heterogeneity in preferences. A worker chooses an industry to maximise his/her utility under the budget constraint: (1) U=VQ,R,X,Y+ε,s.t.Q=W+I U = V\left( {Q,R,X,Y} \right) + \varepsilon ,s.t.Q = W + I where ε is the unobserved term that affects the utility, which has the IID extreme value distribution. V is the representative utility. W is the wage, and I is the non-wage income.

Following the random utility studies, this study also assumes a linear representative utility function. Worker i’s utility from choosing alternative j is: (2) U=αW+I+βR+θX+δY+ε U = \alpha \left( {W + I} \right) + \beta R + \theta X + \delta Y + \varepsilon Since job risk is bad, we expect β to be negative. j is set to zero in this study for simplicity. Usually, one income/cost variable is included to measure the marginal utility of money. Thus, a utility maximisation consumer will choose the amount of risk according to the first-order condition: (3) dV=αdW+βdR+θdX+δdY=0 dV = \alpha dW + \beta dR + \theta dX + \delta dY = 0 The role of these personal attributes can help to investigate the heterogeneity in preference. To be specific, we include workers’ gender, age, and education in the utility function. These worker attributes are case-specific. The other job attributes included here are working hours and the unemployment rates for each industry. These variables are alternative-specific. Under the specification, the utility of worker i working in industry j can be written as follows: (4) Uij=Vij+εij=αWij+βRj+θXj+δYi+εij {U_{ij}} = {V_{ij}} + {\varepsilon _{ij}} = \alpha {W_{ij}} + \beta {R_j} + \theta {X_j} + \delta {Y_i} + {\varepsilon _{ij}} where ε_ij is distributed IID extreme value. Thus, the probability that worker i will choose industry j is: (5) Pij∀l≠j=eαWij+βRj+θXj+δYi∑leαWil+βRl+θXj+δYi {P_{ij}}\left( {\forall l \ne j} \right) = {{{e^{\left( {\alpha {W_{ij}} + \beta {R_j} + \theta {X_j} + \delta {Y_i}} \right)}}} \over {\sum\nolimits_l {{e^{\left( {\alpha {W_{il}} + \beta {R_l} + \theta {X_j} + \delta {Y_i}} \right)}}} }} which can be estimated with the conditional logit model, as in most random utility models.

The fixed coefficient β in the conditional logit model indicates that the preferences over an attribute are the same across consumers, which is a major weakness of the conditional logit model. Another shortcoming of the conditional logit model is the IIA (independence from irrelevant alternatives) property, which assumes that the probability ratio between two alternatives is irrelevant to a third alternative. The IIA property is a strong restriction which is sometimes inappropriate in real life.

The mixed logit model is a very flexible model that can solve the limitations of the standard logit model by allowing for unobserved random taste variation and unrestricted substitution pattern. In addition, the panel data mixed logit model can also address the correlation of the unobserved tastes over time. McFadden and Train (2000) show that the mixed logit model can be used to approximate any random utility model.

In the mixed logit model, the coefficient (β) is a distribution rather than a fixed value, and the choice probability can be written as: (6) Pij=∫eαWij+βRj+θXj+δYi∑leαWil+βRl+θXj+δYifβdβ {P_{ij}} = \int {{{{e^{\left( {\alpha {W_{ij}} + \beta {R_j} + \theta {X_j} + \delta {Y_i}} \right)}}} \over {\sum\nolimits_l {{e^{\left( {\alpha {W_{il}} + \beta {R_l} + \theta {X_j} + \delta {Y_i}} \right)}}} }}f\left( \beta \right)d\beta } Further, considering workers are making repeated choices over time, let j=(j₁,…,j_T) denote the vector of alternatives a worker faces, then the probability of the worker choosing alternative j can be written as: (7) Pij=∫∏t=1TeαWijt+βRjt+θXjt+δYi∑leαWilt+βRlt+θXlt+δYifβdβ {P_{ij}} = \int {\prod\nolimits_{t = 1}^T {{{{e^{\left( {\alpha {W_{ijt}} + \beta {R_{jt}} + \theta {X_{jt}} + \delta {Y_i}} \right)}}} \over {\sum\nolimits_l {{e^{\left( {\alpha {W_{ilt}} + \beta {R_{lt}} + \theta {X_{lt}} + \delta {Y_i}} \right)}}} }}f\left( \beta \right)d\beta } } which can be estimated through the panel data mixed logit model. The coefficients α,β,θ,δ are parameters to be estimated.

As shown in the first order condition (equation 3), α measures the marginal utility of money, β,θ,δ measures the marginal utility associated with changes in job/worker attributes, dividing the β,θ,δ by the marginal utility of money (α) provides a monetary estimate of the MWTP of these job/worker attributes. Specifically, the marginal willingness to pay for risk reduction is given by: (8) MWTP=−∂U/∂R∂U/∂W=−βα MWTP = - {{\partial U/\partial R} \over {\partial U/\partial W}} = - {\beta \over \alpha } The change in consumer surplus is equal to: (9) ΔCS=1αV1−V0 \Delta CS = {1 \over \alpha }\left( {{V^1} - {V^0}} \right) where V¹ and V⁰ are the utility after and before the change (in job risk, for instance), respectively.

The hedonic wage model is built based on the hedonic price theory by Thaler and Rosen (1976). A job can be seen as a composite good with its attributes. From a worker’s perspective, a job can be considered as a differentiated good consisting of various job characteristics, such as flexibility in working hours, the number of paid vacation days, working conditions, prestige, training and enhancement of skills, the risk of accidental injury or exposure to toxic substances, etc. Different jobs have different bundles of characteristics and different wage rates. Wage differentials can be interpreted as the implicit prices of job characteristics. From the perspective of employers, they are the suppliers of the job attributes. At the same time, employers can be viewed as choosing from among a set of workers of different characteristics. Thus, the job market also functions as the implicit market of job attributes and worker attributes.

The hedonic wage equation is an equilibrium relationship that reflects the interaction of supply and demand for worker attributes. It is the double-envelope of both the firm’s iso-profit curves and workers’ indifferent curves. In the hedonic wage model, the wages in a job market reflect the equilibrium relationship between workers and firms. However, the discrete choice model considers the wage as a job attribute that is exogenous. The hedonic wage equation is a function of the worker attributes Y, job attributes X, and job risk R and is exogenous to each individual in this market. (10) W=WR,X,Y W = W\left( {R,X,Y} \right) The marginal implicit price of each characteristic can be calculated by taking the derivative of (10) with respect to each characteristic. For example, the implicit price of risk, P_R, is the partial derivative of W with respect to R: (11) PR≡∂W∂R {P_R} \equiv {{\partial W} \over {\partial R}} It is often assumed that the shape of (10) is exponential: (12) W=eπ0+π1R+π2X+π3Y W = {e^{{\pi _0} + {\pi _1}R + {\pi _2}X + {\pi _3}Y}} Econometrically, the hedonic wage equation is therefore estimated using a log-linear form: (13) lnW=π0+π1R+π2X+π3Y+ε1 lnW = {\pi _0} + {\pi _1}R + {\pi _2}X + {\pi _3}Y + {\varepsilon _1} where π₀, π₁, π₂, π₃ are parameters to be estimated, ε₁~N(0,σ²).

After the estimation, the implicit price of risk (P_R) can be calculated as follows: (14) PR=∂W^∂R=π1^eπ0^+π1^R+π2^X+π3^Y+0.5σ^2 {P_R} = {{\partial \widehat W} \over {\partial R}} = \widehat {{\pi _1}}{e^{\widehat {{\pi _0}} + \widehat {{\pi _1}}R + \widehat {{\pi _2}}X + \widehat {{\pi _3}}Y + 0.5{{\hat \sigma }^2}}} where σ^2 {\hat \sigma ^2} is the unbiased estimator of σ². The nonlinearity of the hedonic wage equation implies that the marginal price of risk is not a constant but a function of the quantity of risk. If π₁>0, there is a positive wage premium for risk. In addition, the partial derivative of P_R is also positive, suggesting that the price is increasing in risk.

The implicit price of risk (P_R) can only measure the value of a marginal change in risks. To measure the value of non-marginal changes in risks, the demand for safety needs to be estimated, which requires data from multiple labour markets to meet the identification criteria. Please see Zhang et al. (2023) for details about the identification and estimation of the second-stage hedonic wage model.

We do not observe the wage a worker can earn in every industry but the wage in the industry he/she has already chosen. The first step, which is the same as in the sorting literature, is to predict wages in alternative industries that are not chosen by a worker.

For each of the seven industries, a wage equation was estimated in both years to capture the wage structure in this specific industry and then used to predict the wage income one specific worker would get if he/she chose this industry. The control variables mainly include factors that affect one’s productivity, such as gender, age, education, etc. The results of the first stage wage regressions for 2016 and 2018 are shown in Tables A1 and A2 in the Appendix. For later analysis in the discrete choice model, wage_alt will be the reported real wage for industries chosen by workers, and the wages in alternative industries that workers do not choose will be the predicted wages based on these industry wage regressions.

This study estimates three different discrete choice models: the conditional logit model, which assumes all coefficients are fixed (that workers have homogenous tastes over industry attributes); the mixed logit model, which assumes the coefficient of job attributes to be random, thus allowing for the heterogeneity in tastes variety; and the panel-data mixed logit model, which allows the coefficients to be correlated over time.

First, the random utility model in equation (4) was estimated using the conditional logit model with pooled observations from 2016 and 2018. The results are shown in Columns (1)-(2), Table 2.

As can be seen from Columns (1) and (2) in Table 2, workers prefer industries that have higher wage income, longer working hours, lower risks, and lower unemployment rates. The preferences are heterogeneous across workers. Compared with Manufacturing, high-education workers are less likely to choose Construction and Catering but are more likely to work in the Finance and Service industries. Older workers are less likely to choose a job in the Catering and Science industries. Female workers are less likely to get a job in the Construction and Transportation industries and are more likely to work in Catering, Finance, Science, and Service, as expected. Including the worker attributes can help solve the observed heterogeneity in preferences. However, there are cases in which unobserved worker attributes could also affect workers’ industry choices and can be addressed with the mixed logit model.

Because of the advantages of the mixed logit model in the flexible substitution pattern and allowing for unobserved heterogeneity, this study estimates a mixed logit model with pooled observations of 2016 and 2018. Following the literature, such as Huang et al. (2018), the coefficients of industry attributes are assumed to follow a random distribution, while the coefficient of income/cost (in our case, wage) is assumed to be fixed. In Model 6, personal attributes such as gender, education, and age are included to explore the observable heterogeneity in tastes. The results are shown in Models 3–4 in Table 2. In Model 3, only industry attributes are considered, and in Model 4, worker attributes are also included.

We can see that workers still prefer industries with higher wages, longer working hours, and lower mortality rates, similar to what we find in the conditional logit model. However, the coefficients of unemployment become insignificant, suggesting that unemployment has no impact on workers’ industry choices. In Model 4, the coefficient of risk_alt has a mean of −0.1162 and a standard deviation of 0.3186, both the mean value and the standard deviation are significant, suggesting rich variations in the population.

Similarly, the mean of the coefficient of workhour is 0.104, and the mean of the standard deviation of workhour is 0.427. Both are significant. The relatively large standard deviations suggest that workers have various tastes over working hours. However, the mean and the standard deviation of the coefficients of unemployment rates are insignificant. The individual-specific attributes affect the industry choices in a similar way as those found in the conditional logit model.

The panel-data mixed logit model has the advantage of capturing the correlation of workers’ choices over time and is very useful for modelling consumer behaviour such as habit formation and/or variation-seeking. In our context, it is reasonable to assume that workers prefer to choose the same industry as they did in the last period; otherwise, their work experience and skills accumulated in the last job will be less appreciated. In this specification, we continue our assumption that all workers have the same preference regarding wages but heterogeneous preferences for risks, workhours, and unemployment. The regression results are shown in Models 5 and 6, Table 2.

As we can see, workers show heterogeneous preferences for job attributes: they prefer jobs with higher wages, longer working hours, and lower job risks. The coefficients of workhour are positive, with substation variations among the population, suggesting that average workers are more likely to be engaged in industries that have longer working hours. The coefficients of unemployment rates are negative but not significant, suggesting that overall, unemployment rates do not affect how workers choose industries.

Compared with Manufacture, high-education workers prefer the Finance and Science industries over Construction, Catering, and Transportation. Older workers are more likely to be working in Construction and less likely to work in Catering, Finance, and Service. Female workers prefer Service, Catering, Finance, and Science and are least likely to be engaged in Construction. By comparing the regression results of the three specifications using the AIC and log-likelihood, we find that the panel mixed logit models perform better than the other models.

In the discrete choice models, the marginal willingness to pay (MWTP) for one unit of risk reduction (1/1000,000) can be calculated as follows: MWTP=−β^α^ MWTP = - {{\hat \beta } \over {\hat \alpha }} where α^ \hat \alpha is the estimated coefficient of wage and β^ \hat \beta is the estimated coefficient of risk for the conditional logit model. In the mixed logit model and the panel data logit model, the coefficient of risk_alt β^ \left( {\hat \beta } \right) is assumed to have a normal distribution. The mean and standard deviation of MWTP can be calculated as follows: meanMWTP=−meanβ^α^ mean\left( {MWTP} \right) = - {{mean\left( {\hat \beta } \right)} \over {\hat \alpha }} sdMWTP=−sdβ^α^ sd\left( {MWTP} \right) = - {{sd\left( {\hat \beta } \right)} \over {\hat \alpha }} The mean, upper, and lower limits of MWTP are shown in Table 4.

First, a separate hedonic wage model (equation 14) was estimated using data from 2016 and 2018, respectively. Then, the hedonic wage model was re-estimated with pooled data from these two years. The hedonic wage model regression results are shown in Table 3.

From Table 3, we can see that, in both years, workers with higher job risks are compensated with higher wages, indicating that risk is a disamenity. Similarly, July temperature is a disamenity, and workers engaged with higher July temperature are paid more. At the same time, January’s temperature is an amenity, and workers are willing to sacrifice some wages for a warmer winter. Both industry unemployment and workhour have a negative and significant coefficient. The workers’ individual attributes have expected effects on wages. For example, senior married male workers with a higher education (or the respondent’s father has a higher education, higher feduyear) receive higher wages, and workers who work in firms with more employees have higher wages, other factors being equal. We can see that the regression results from the fixed effect model and pooled_2 are very close to each other and equally good.

Given the estimation results of the hedonic wage model, we can measure the marginal willingness to pay for one unit of risk reduction (1/1000,000) as follows: MWTP=π1^×wage¯ MWTP = \widehat {{\pi _1}} \times \overline {wage} where π1^ \widehat {{\pi _1}} is the estimated coefficient of risk.

The estimated MWTP from both discrete choice models and hedonic wage models are calculated and transferred to US dollars at the exchange rate of 30 TWD: 1 USD, as shown in Table 4.

The MWTP estimated from the two mixed logit models are comparable to each other but are significantly larger than the MWTP from the hedonic wage model. In fact, the MWTP estimated from the discrete choice model is unreasonably larger than any related studies in Taiwan or abroad. For example, Zhang et al. (2022) found a VSL of $14 million based on Taiwan labour market data during 1999 and 2014. Banzhaf (2022) conducted a meta-analysis using the results of five meta-analyses about the US and found a median VSL of $7 million, with a 90% confidence interval of $2.4 to $11.2 million. Based on the stated preference method, the OECD estimate of VSL is about $4 million (OECD, 2012). Giergiczny, M. (2008) estimated the VSL for Poland and found a value between $0.79 and $2.41 million. VSLs estimated in mid- or low-income countries are generally lower.

Given the linear utility function assumed in the discrete choice model, the nonmarginal value of risk reduction can be calculated by multiplying the changes in risks by the MWTP; thus, the unreasonably large MWTP from the discrete choice model also implies an unreasonably large nonmarginal value of risk reduction. Therefore, we believe the hedonic wage model outperforms the discrete choice model in evaluating safety with labour market data.

One possible explanation for the surprisingly large MWTP estimates by the discrete choice model stems from the well-acknowledged non-existence of moments of a ratio distribution resulting from dividing two normally distributed variables. As we know, the MWTP of risk is measured as the absolute value of the marginal utility of risk (β in equation (8)) divided by the marginal value of money (α in equation (8)). When the distribution of the denominator (α) spans zero, the WTP ratio distribution may be undefined. Carson and Czajkowski (2019) suggest that restricting the coefficient of the cost/income variable (in this case, wage) to be random and lognormally distributed could help solve the problem of non-existent moments in the WTP ratio. However, setting the coefficient of wage to be random could cause non-convergence problems. Unfortunately, the Carson and Czajkowski approach does not work in this study since the mixed logit models fail to converge and thus are not reported here.

Another possible explanation of the extremely large MWTP from the discrete choice model is that job choices are actually not very responsive to wages. In our discrete choice model results, the coefficients of wages are quite small. Though they are statistically significant, they suggest that wages do not have a large effect on workers’ industry choices. In our case, the wage varies across individuals rather than industries, which could cause the workers’ industry choices to be insensitive to wages.

The two approaches differ in their underlying assumptions and theoretical foundations. The hedonic wage method assumes a competitive market with numerous heterogeneous firms and consumers, while the discrete choice model typically assumes a known utility function that is linear in individual and job or housing attributes. The hedonic model results from consumers’ utility maximisation and firms’ profit maximisation, while the random utility model focuses solely on consumer behaviour (Jonker, 2001). Despite these differences, there is a connection between the two models. Wong (2010) demonstrates a duality between the hedonic price model and the discrete choice model: the gradient of the hedonic price function represents the ratio of a weighted average of individuals’ marginal utilities, where the weights are derived from choice probabilities in the discrete choice logit model.

Compared to the hedonic wage model, the discrete choice model has two widely recognised disadvantages, aside from the issue of non-existent moments in the WTP ratio distribution. First, the hedonic price equation reflects market equilibrium that balances consumers’ willingness to pay and firms’ marginal costs, while the discrete choice model only captures the demand side. This may explain why the discrete choice model often yields higher WTP estimates than the hedonic price model. For example, Palmquist and Israngkura (1999) analyse MWTP for air quality improvements using the same housing market data with both models. They find that the discrete choice model produces larger estimates. Similarly, Sinha et al. (2021) compare the two approaches in evaluating MWTP for climate attributes and find that the discrete choice model generates MWTP estimates for warmer winters that are twice as large as those from the hedonic price model.

A second disadvantage of the discrete choice model is that it relies heavily on assumptions about the utility function form and the distribution of the random term, which are, unfortunately, restrictive and hard to test (Freeman et al., 2014). Moreover, most studies adopting the discrete choice model in environmental goods valuation (Chay and Greenstone, 2005; Bayer et al., 2009; Freeman et al., 2019) have adopted a multinomial logit estimation procedure that is subjected to the IIA property. The IIA property imposes a strict substitution pattern so that introducing a third option does not change the ratio of the choice probabilities between two alternatives – a condition rarely met in real-world scenarios.

This is the first study to apply the discrete choice model to real labour market data to investigate the value of job safety and compare the results with those from the hedonic wage model. This study finds that the hedonic wage model performs better than the discrete choice model in modelling real labour market decisions and provides a more reliable marginal willingness to pay (MWTP) for job safety. Future research should further explore the comparison between these two approaches using real market data or simulation in both the housing and labour markets regarding both the marginal value and nonmarginal value.