Bayesian estimation of labor demand by age: theoretical consistency and an application to an input–output model

ABSTRACT

Extended input–output models require careful estimation of disaggregated consumption by households and comparable sources of labor income by sector. The latter components most often have to be estimated. The primary focus of this paper is to produce labor demand disaggregated by workers’ age. The results are evaluated through considerations of its consistency with a static labor demand model restricted with theoretical requirements. A Bayesian approach is used for more straightforward imposition of regularity conditions. The Bayesian model confirms elastic labor demand for youth workers, which is consistent with what past studies find. Additionally, to explore the effects of changes in age structure on a regional economy, the estimated age-group-specific labor demand model is integrated into a regional input–output model. The integrated model suggests that ceteris paribus ageing population contributes to lowering aggregate economic multipliers due to the rapidly growing number of elderly workers who earn less than younger workers.

KEYWORDS:

• Labor demand by age
• translog cost function
• Bayesian SUR
• regularity conditions
• Miyazawa’s input–output model

Disclosure statement

No potential conflict of interest was reported by the authors.

Acknowledgments

We are truly grateful to William E. Griffith for allowing the use of the SHAZAM code for the Bayesian SUR model and related technical notes. The helpful comments of Peter Huber, Kurt Kratena, Anil Bera, Kathy Baylis, Woong Yong Park, Andrew Crawley, Dongwoo Kim, and Sungyup Chung are greatly appreciated. Constructive comments from seminars in the Regional Economics Applications Laboratory and Department of Economics, University of Illinois, are acknowledged as well.

ORCID

Kijin Kim http://orcid.org/0000-0001-6046-5473

Geoffrey J. D. Hewings http://orcid.org/0000-0003-2560-3273

Notes

1 Under a translog cost function, monotonicity implies nonnegative factor cost shares. In the presence of very small or large input cost shares, estimated shares are likely to deviate from the 0–1 range unless the range of predicted values is imposed a priori.

2 Munnell (2011) calculates the recent average retirement age for men and women to be 64 and 62, respectively. She argues that the retirement age will continue to rise. The surveys in Hamermesh and Grant (1979) and Hamermesh (1996) cover studies on labor demand by age that had been published until the early 1990s. Among the papers in the surveys, Ferguson (1986) is the only study that includes workers aged 65 and over. We could not find any papers on labor demand for the elderly group henceforth. A most recent survey on demand for aggregate and heterogeneous (mostly by skill level) labor, including empirical studies released from 1980 to 2012, can be found in Lichter et al. (2014).

3 For example, Lee et al. (1990) find statistically significant aggregation bias when a disaggregate employment model with 41 industries is compared with an aggregate employment model for the UK.

4 In practice, many empirical studies on factor demand assume separability due to data availability (Atkinson and Manning, 1995). However, separability is essentially an empirical issue that requires statistical testing. If labor is not separable from other factors, the estimates of labor–labor substitution are biased when other factors are omitted in the model. Since this paper focuses on the regional level (i.e. the US states) where data on prices and quantities of other factors, especially capital among others, are usually not available, measurement errors due to constructing estimates for capital might be more problematic (Hamermesh and Grant, 1979). Furthermore, regional models are often developed upon a single-input (usually labor) assumption that inputs other than labor can be approximated by local employment (Glaeser et al. 1992; Bishop and Gripaios, 2010; Felipe and McCombie, 2012).

5 Although it is not explored here because of a relatively short time series data (13 years), a time varying trend, which can be estimated using the Kalman filter, might be a more sensible choice (Jorgenson et al., 2013).

6 Similar identification assumptions can be found, for example, in Slaughter (2001).

7 See Berndt and Wood (1979) for a geometric interpretation of differences between gross and net price elasticities.

8 One of the share equations is dropped due to the singularity of the covariance matrix. In addition, we use the maximum likelihood (ML) method to ensure that estimates are invariant to the choice of the omitted equation.

9 The objective of the Bayesian method is to obtain samples for statistical inference from a target (posterior) distribution. However, since a target density is often analytically intractable, the MH algorithm generates a sequence of samples from a proposal density instead. In the limit, these samples follow the target density. See Chib and Greenberg (1996) for more details on the MH algorithm.

10 When conventional non-informative prior distributions (e.g. the inverted-Wishart distribution for the variance–covariance matrix) are assumed, the target density f(λ|y) is proportionate to the determinant of the variance–covariance matrix for the errors in the SUR model (Griffiths, 2003).

11 Setting a proper value for the tuning parameter ensures a candidate parameter vector drawn from a proposal density to move around the parameter space more efficiently. For example, we first set c=0.05 that is the same value used in Griffiths et al. (2000). When we found the acceptance rate too low, say less than 10%, we then reset c=0.01 for candidate parameter vectors not to traverse too far from the target density.

12 The data can be downloaded from the IPUMS USA, the Minnesota Population Center, University of Minnesota (https://usa.ipums.org/usa/; Ruggles et al., 2015). According to the Employment Cost Trends (ECT) compiled by the BLS, wages and salaries make up around 70% of employee compensation costs and the remaining 30% is comprised of benefits such as health insurance, paid leave, legally required benefits, and retirement and savings. However, neither the ACS nor the ECT provides comprehensive benefits data by worker's age.

13 Self-employment in the ACS includes both the unincorporated (a dominant type) and incorporated self-employed while the CPS treats the incorporated self-employed as wage and salary workers.

14 In the CPS, median inflation-adjusted weekly earnings for wage and salary workers show similar trends over the 2000–2013 periods.

15 See Appendix A (supplementary material) for labor cost shares, employment and wages by sector for all age groups.

16 State fixed effects were initially explored, but majority of sectors showed a fair amount of insignificant state fixed effects. Thus region fixed effects were scaled down to the four Census regions, i.e. Northeast, Midwest, South and West. Time dummy variables accounting for the recent financial crisis in the US (2008 and 2009) did not significantly change the results, and thus they were not included in the final specification.

17 The same sets of results for other sectors are available upon request.

18 Barnett (2002) further explains that since monotonicity is less often violated, researchers commonly impose only curvature in practice.

19 An empirical example in Ryan and Wales (2000) shows that choosing one concavity-restricted point could make all points satisfy concavity. Our finding suggests that the choice of a restriction point affects the degree to which concavity holds at other points.

20 From this subsection on, the term “elastic” is used in the context of relative comparison among the age groups for easier illustration despite the convention referring to the elasticity of demand greater than unity.

21 These figures are based on the aggregate employment and wage at the US level in the CPS data.

22 We are measuring the effects of input price on quantity demanded: two inputs are p-substitutes if ${\eta }_{gh}=\frac{\mathrm{\partial }\log x_g}{\mathrm{\partial }\log w_h}>0$, p-compliments, otherwise. By contrast, q-substitute (${\in }_{gh}<0$) or q-compliment (${\in }_{gh}>0$) are based on the cross-demand elasticity of factor price ${\in }_{gh}=\frac{\mathrm{\partial }\log w_g}{\mathrm{\partial }\log x_g}$. In the case of three or more inputs, equal signs for η_gh and ${\in }_{gh}$ are not guaranteed (Hamermesh, 1996).

23 More details on the intermediate steps to solve the system can be found in Miyazawa (1968, pp. 43–45).

24 The Chicago region in this study includes seven counties in Illinois: Cook, Du Page, Kane, Kendall, Lake, McHenry, and Will.

25 More specifically, Kim et al. (2015) estimate the AIDS model for five nondurable goods and services using the data from the Consumer Expenditure Survey (CES). The five types of expenditures are then disaggregated into consumption in 45 sectors via a bridge matrix. Durable goods consumption is allocated across age groups, proportional to the number of households in each group.

26 Average propensity to consume by age group is calculated from the 2009 CES for the US. It is worth mentioning that average propensity to consume significantly varies by age group: 1.11 for the under 25 group, 0.77 for the 25–44 group, 0.73 for the 45–64 group, and 0.93 for the 65+ group.

27 According to the Census Bureau, the Illinois population aged 45–64 and 65+ is projected to increase 14% and 61%, respectively, between 2000 and 2030 while the total population is expected to grow only 8% over the same period.