# Quantitative Exercises

**Calibration Overview**

To calibrate the model, I first match the model to moments of the aggregate labor market with idiosyncratic productivity shocks, holding the realizations of aggregate shocks at their average value. Next, I set productivity, entry, and exit parameters to match the firm size and age distribution. The idiosyncratic shock process matches the average separation rate and the share of employment at firms with little to no employment growth. The aggregate shock process matches the quarterly autocorrelation and standard deviation of average labor productivity. Table 1.5 displays the values of parameters and the targets they match.

**Production**

I use a decreasing returns to scale function in labor:

*xzF* (L) = *x*_{0}x_{1}zE^{n}

where n = 0.7 is set to target a labor share of income of 0.66. Productivity of a firm is composed of an aggregate component z and idiosyncratic productivity x = x**0**x**1**.

Idiosyncratic productivity is a product of a fixed component x**0 **which is drawn upon firm entry and a transitory component x_{1} .

**Matching Function and Vacancies**

rate of a firm, VL:

Where Y = 2 as in Kaas and Kircher (2015), so that the recruitment cost is cubic.

The model period is set to weekly, so the matching function parameter k is set to target a weekly job-finding rate of 0.129 (corresponding to a monthly rate of 0.45 as in Shimer (2005)) at the steady-state queue length.

With multiple vacancies per firm and nonlinear posting costs, the rate of vacancy posting must be determined. The job-filling rate for vacancies is set to 0. 3 to match the observed monthly vacancy yield of 1.3 as reported in Davis et al. (2013).^{[1]} The parameter c in the vacancy cost function is set to match this weekly job-filling rate of 0.3. In steady state, the queue length is the ratio of the job-filling and job-finding rate, yielding a steady-state queue length of A = 2.326.

**Home Production and Part-time**

The benefit from home production b is set to match 70% of the average wage, corresponding with the calibrated value of non-market work in Hall and Milgrom (2008). To calculate the replacement ratio, I consider the special case of a constant wage per period for full-time and part-time workers. That is, the wage contract is a pair of constants *{w**f,w** _{p}}* for the duration of the match. The share of leisure gained by part-time workers, £, is used to target a steady-state fraction of employment working part-time of 5%.

**Permanent Firm Types and the Size/Age Distribution of Firms**

What remains to be determined is the set of productivity parameters xz and their shock process. To match the broad size distribution of firms and the fact that a small fraction of very large firms retains a large share of total employment, the permanent component of idiosyncratic productivity x**0 **is set to match the size distribution of firms. To match the firm share and employment distribution for 5 classes, x_{0} takes on 5 values at entry, with entry share *Ơ.* The exit probability of firms, *8, *is exogenous and also dependent on size classification. The vector *8* is chosen by matching the observed job destruction rate for closing firms in each size class from the Bureau of Labor Statistic’s Business Employment Dynamics data for 1992-2011. Using permanent firm types characterized by permanent productivity level x**0**, entry share a, and exit probability *Ỗ* given in Table 1.4, the model can closely match the firm size and age distribution in the data, as seen in Figure 1.11.^{[1]} The vector x**0 **matches the employment shares of firms in each size category, while *a* and *ố* determine the share of firms in each size category.

**Idiosyncratic Shocks**

The shock process x_{1} is evenly spaced between [1 — x, 1+x] and is redrawn with probability n each period. This shock process matches the fact that many firms in the data experience little to no net job growth in a given quarter. The idiosyncratic shock parameters match the average monthly separation rate and the share of employment at firms with monthly growth rates between —2% and 2%. The exogenous separation rate s**0 **is set to match a monthly quit rate of 2% per month from the Job Openings and Labor Turnover Survey (JOLTS) data.

**Aggregate Shocks**

The aggregate shock process affecting z is a mean-reverting Markov process as in Appendix C of Shimer (2005). The aggregate shock process is parameterized by a persistence parameter psi and range [z, 2 — z]. The parameters *(Ộ,* z) = 0.015, 0.95 are chosen to produce a quarterly standard deviation and autocorrelation of productivity shocks of *ơ _{z}* = 0.015 and p

_{z}= .76. The remaining parameter to be set is the entry cost K(z). The entrant firm’s value function is homogeneous in the vector {x

_{0}

*,b,c, ụ,K*} in steady-state, so the stationary value of parameter K(z) can be normalized arbitrarily. Tractability requires that there is positive entry of firms in every state, so that the value of

*ụ*for all workers is determined by the value at entrant firms. I allow K(z) to vary with the aggregate state so that job creation at entrant firms is stable over the business cycle to ensure positive entry in all aggregate states.

I numerically solve for the values of ụ, x_{0},*I, c,* x, and n that minimize the target moments in steady-state (holding z at its average level). These moments are the jobfinding rate, share of firms in each size category and share of employment by size categories, separation rate, ratio of b to average wages, and the employment share at firms with growth rate < ±2%.