Adult social care labour supply: pay increase impact assessment

Technical Appendix A2. Wage elasticity of labour supply to the sector

Based on a Dynamic Monopsony Model (Manning, 2003), and following Vadean et al.(2024), the wage elasticity of labour supply to the ASC sector () is determined from the overall wage elasticity of separation (), the wage elasticity of separation to other employment inside the sector (), and share of separations to employment inside the sector ():

(1)

given the steady-state assumptions that the overall flows of staff separation and recruitment are equal (), the overall recruitment elasticity equals the negative of the separation elasticity (), and the recruitment elasticity from employment inside the sector equals the negative of the separation elasticity to employment inside the sector (). The equality between the overall flows of staff separation and recruitment also implies that the shares of separations to and recruitment from employment inside the sector have to be equal as well ().

Econometric approach

Also following Vadean et al. (2024), the two wage elasticities of separation are estimated using discrete time proportional hazard model proposed by Jenkins (2005). The discrete hazard of the job spell to end during the tenure-year is:

(2)

where (D) is the baseline hazard, allowed to be piece-wise constant over the tenure periods (d). To account for time-invariant unobserved effects, the discrete time proportional hazard model is estimated by correlated random effects (CRE) probit. This is a quite flexible estimator for binary settings, including among covariates the average over time of the time-varying covariates () to remove the time-invariant unobserved heterogeneity associated with the explanatory variables (). The parameters are Mundlak-type ‘within’ estimates similar to those from a fixed-effects estimator but allowing the estimation of average partial effects (i.e., marginal effects) and elasticities (Wooldridge, 2010).^[6] Most unobservables () are time-invariant (or change very little over time) and, thus are captured by . Nonetheless, if they would change over time in a deterministic way, they would be captured by the included year dummies. We estimated CRE probit by pooled probit in Stata 17.0, with the Huber-White sandwich estimator used to obtain cluster-robust standard errors.

An important challenge in estimating wage elasticities of separation is related to the adequate control for other relevant factors in besides wages. Following related studies on determinants of job separation and wage elasticities of labour supply (Vadean et al., 2024; Vadean and Allan, 2023; Vadean and Saloniki, 2023;), the covariates included are: a) individual factors that can be associated with the likelihood of job separations (i.e., age, gender, ethnicity, and qualifications); b) a job and employer related characteristics, like job role, training incidence, employment without guaranteed working hours, sector (i.e., public, private, and voluntary), user type (i.e., younger adults, older people, and mixed), employer size, the vacancy rate in the previous year, and the turnover rate for the past 12 months to capture any potential ‘herd’ effect with respect to separations; and c) local market characteristics, like the local unemployment rate, the log of the 1^st quartile of the local wage distribution (as proxy for peer wages in alternative employment), the geometric mean of local house prices (as proxy for demand of self-funded care), the ASC tariffs paid by local councils (as proxy for demand of publicly funded care), and competition in the local ASC market. Additionally, we used regional and year fixed effects.

The above variables have been found to be significant factors in previous studies, with job separations and/or staff turnover in long-term care shown to be related to job characteristics (e.g., tenure, training provision, and job benefits and rewards) (Castle et al., 2007; Gaudenz et al., 2019; Karantzas et al., 2012; Morris, 2009; Park et al., 2017; Rosen et al., 2011), employer characteristics (e.g., employer’s size, lower staffing levels, guaranteed working hours, for-profit ownership, and home care provision) (Castle, 2008; Castle and Engberg, 2006; Kennedy et al., 2021, 2020), as well as local market factors (e.g., unemployment, wages in alternative jobs in the local area, and competition) (Castle, 2008; Donoghue, 2010; Morris, 2009).

Certain controls used in earlier studies (e.g., the ratio of staff per service users [as proxy for workload] and metrics for management quality) were not including in the analysis, as they were not available for certain care settings (e.g., community care). Moreover, we avoided including a control for full-time/part-time employment as this can be itself an outcome of wages and mediate some of the relationship between wages and extensive margin employment choice.

The reason behind using a panel fixed effects estimator (i.e., CRE probit) is that many factors related to job performance and separations (e.g., workers’ job commitment and motivation, workload, organisational culture, management style/quality, etc.) are often not observed in survey data. Not suitably controlling for unobserved factors has been shown to bias the separation elasticities towards zero, even if uncorrelated with wage (Manning, 2003).

We used the 3-degree polynomial of hourly wages in , to allow more flexibility with respect to the functional form.^[7] Wage elasticities of separation (i.e., proportional change in the probability of separation for a proportional change in wage) are obtained by expressing derivatives after CRE probit as:

(3)

where h is the job separation rate, and w is the hourly wage.