Part-time unemployment and optimal unemployment insurance

Abstract

A significant fraction of the labor force consists of employed workers who are part-time unemployed (underemployed) in the sense that they are unable to work as much as they prefer. This paper studies the design of optimal unemployment insurance in an economy with unemployment as well as part-time unemployment. Part-time work provides income insurance and serves as a stepping stone to full-time jobs. Unemployment benefits for part-timers reduce the outflow from part-time work to full-time employment. However, such benefits also increase the outflow from unemployment to part-time work, thereby reducing unemployment. We examine the optimal structure of benefits for unemployed and underemployed workers. The results indicate that there are welfare gains from part-time benefits. Moreover, there are welfare gains associated with time limits for unemployment benefits as well as for part-time benefits. The welfare gains from optimal insurance are larger when wages are fixed than when they are flexible.

Appendices

Appendix 1: The numerical model

The calibration is done with an eye on the Swedish labor market. The matching function is Cobb Douglas, $M=au^{\eta }v^{1-\eta }$, where $\eta =0.7$ is assumed. Productivity is normalized to unity for both part-time and full-time employees: $y=1$. The time period is taken to be a day. The rate of interest (equal to the rate of time preference) is set to zero. The annual separation rate is set to 15 % corresponding to daily rate of $0.15/365=0.000411$.

UI coverage is relatively low according to the labor force surveys; around one third of the unemployed received unemployment benefits over the period 2005–2009. We account for this by choosing a benchmark replacement rate of 30 % for both unemployed and part-timers, i.e., $\rho _{u}=\rho _{p}=0.3$.

Regarding work hours, we assume that the full-time employee works as much as she prefers. With log utility this implies $h_{e}=1/(1+\delta )$ when $T=1$ as is assumed. Working time among part-timers is given by $h_{p}=0.48h_{e}$, which is in line with Swedish data: involuntary part-timers in the labor force surveys report that they wish to work twice as much as they actually do.

Regarding wages, empirical work has documented wage penalties for part-timers. A recent Swedish study by Wahlberg (2008) suggests a wage penalty of 20 %, arguably implausibly large. We aim for a full-time wage premium of less than 10 %. When baseline wages are determined by Nash bargaining ($\beta =\eta$) we obtain $w_{e}=0.962$ and a wage premium for full-time workers of 3 %.

The matching parameter $a$, preferences for leisure $\delta ~$, and the vacancy cost $\kappa$ are chosen so as to obtain reasonable outcomes regarding unemployment rates and durations. We set $a=0.0091$, $\delta =0.12$, and $\kappa =3$ and obtain unemployment at 6 % and part-time unemployment at 4.4 %. Unemployment duration is 26 weeks, and the average vacancy lasts for 4 weeks.

The parameterized model produces outcomes as given by the first column in Table 3 in the main text.

Appendix 2: Time limits

With time limits of benefit receipt, the relevant utility functions are:

$$\begin{aligned} \upsilon _{u}^{I}&= \ln B^{I}+\ln \left( 1-t\right) +\delta \ln \left( T-s_{u}^{I}\right) , \\ \upsilon _{u}^{N}&= \ln B^{N}+\ln \left( 1-t\right) +\delta \ln \left( T-s_{u}^{N}\right) , \\ \upsilon _{p}^{I}&= \ln \left( w_{p}h_{p}+b^{I}\right) +\ln \left( 1-t\right) +\delta \ln \left( T-h_{p}-s_{p}^{I}\right) , \\ \upsilon _{p}^{N}&= \ln \left( w_{p}h_{p}+b^{N}\right) +\ln \left( 1-t\right) +\delta \ln \left( T-h_{p}-s_{p}^{N}\right) , \\ \upsilon _{E}&= \ln \left( w_{e}h_{e}\right) +\ln \left( 1-t\right) +\delta \ln \left( T-h_{e}\right) . \end{aligned}$$

Optimal search efforts are obtained as:

$$\begin{aligned} s_{p}^{I}&:\frac{\delta }{T-h_{p}-s_{p}^{I}}=\alpha \left( \theta _{E}\right) \left( E-P^{I}\right) , \\ s_{p}^{N}&:\frac{\delta }{T-h_{p}-s_{p}^{N}}=\alpha \left( \theta _{E}\right) \left( E-P^{N}\right) , \\ s_{u}^{I}&:\frac{\delta }{T-s_{u}^{I}}=\alpha \left( \theta _{P}\right) \left( P^{I}-U^{I}\right) +\alpha \left( \theta _{E}\right) \left( E-U^{I}\right) , \\ s_{u}^{N}&:\frac{\delta }{T {\acute{}}-s_{u}^{N}}=\alpha \left( \theta _{P}\right) \left( P^{I}-U^{N}\right) +\alpha \left( \theta _{E}\right) \left( E-U^{N}\right) . \end{aligned}$$

The flow equilibrium conditions for unemployment and part-time work are given as:

$$\begin{aligned} u^{I}&: \phi (1-u^{I}-u^{N})=\left( \lambda +s_{u}^{I}\left[ \alpha \left( \theta _{P}\right) +\alpha \left( \theta _{E}\right) \right] \right) u^{I}, \\ u^{N}&: \lambda u^{I}=s_{u}^{N}\left[ \alpha \left( \theta _{P}\right) +\alpha \left( \theta _{E}\right) \right] u^{N}, \\ p^{I}&:s_{u}^{I}\alpha \left( \theta _{P}\right) u^{I}+s_{u}^{N}\alpha \left( \theta _{P}\right) u^{N}=\left[ \phi +\mu +s_{p}^{I}\alpha (\theta _{E})\right] p^{I}, \\ p^{N}&:\mu p^{I}=\left[ \phi +s_{p}^{N}\alpha (\theta _{E})\right] p^{N}, \end{aligned}$$

and employment is obtained from the labor force identity: $1=e+u^{I}+u^{N}+p^{I}+p^{N}$.