The Minimum Wage in a Roy Model with Monopsony

Abstract

I propose and estimate a structural model of the labor market that features agents who are heterogeneous in both productivity and reservation wages, and a monopsony employer bound by the minimum wage. I examine the consequences of alternative minimum wage regimes. My results indicate that under a $15 federal minimum wage, at least 1.58 million lower-skilled women aged between 24 and 55 who worked for less than $15 per hour in 2017 would lose their jobs and be involuntarily excluded from labor market participation. The remaining 11.21 million would see their wage rise to the new minimum. The $15 minimum would also encourage an additional 2.74 million women to enter the labor force.

Appendices

Appendix A: Data Appendix

The data are from the merged outgoing rotation groups (ORG) of the twelve monthly Current Population Surveys conducted by the Bureau of Labor Statistics in 2017. The CPS data are made available by both IPUMS at the Minnesota Population Center Flood et al. (2015) and by NBER. Households selected into the CPS are surveyed eight times, over four consecutive months in two consecutive years. The outgoing rotation group in each month consists of all the households surveyed for either the fourth or eighth time. Certain questions about employment and earnings are only asked to members of an ORG. These are called the “earner study” questions. Individuals are eligible for the earner study if they are older than 15 and employed as a wage or salary worker (i.e., not self-employed).

For each employed, civilian adult in the survey, the data include the number of hours usually worked per week at the main job. For everyone in the earner study, the data also include weekly earnings. To determine weekly earnings, people are first asked their pay periodicity, either hourly, weekly, monthly, or on some other basis, and then for their earnings on that basis. Those who report that they are paid hourly are then asked the number of hours per week they work at that rate. Weekly earnings for those paid hourly are the hourly rate times the number of hours worked per week, plus any usual weekly overtime pay, tips, and commissions (OTC). For all others, weekly earnings are annualized pay including OTC divided by 52. Workers who report a non-hourly wage frequency are asked if they are paid at an hourly rate nonetheless, and if so, what the hourly rate is.

A large fraction of the people surveyed do not respond to questions about their earnings. The BLS replaces missing data with an “allocated” value from a randomly selected respondent with similar demographic and work characteristics.

My sample is drawn from women in the ORGs aged between 24 and 55 who have an associate degree or less. There are 49,689 such records. I consider a worker to be anyone eligible for the earner study who works at least five hours per week. Of these, I drop 73 observations of workers who either (1) usually work zero hours per week at the main job, (2) are paid on an hourly basis and either work zero hours at the specified hourly rate or are paid $0 per hour, or (3) earn $0 per week on a non-hourly basis.

I determine an hourly wage for each worker. For those who report an hourly wage and who do not receive OTC, I use the reported value. Since weekly earnings include OTC but the hourly wage does not, for all other workers I use the ratio of weekly earnings and the usual number of hours worked per week. Among respondents who report that their weekly hours vary, I set usual hours to 40 and 20 for those who work full and part-time, respectively.

To eliminate implausibly high or low hourly wages, following common practice as reported in (Schmitt 2003), I drop 4 observations with wages above $150.00 per hour, and 47 with wages less than or equal to $1 per hour. Weekly earnings in the data are topcoded at $2884.61, and hourly wages are topcoded at $99.99. The sample contains 99 observations with topcoded earnings. Following (Schmitt 2003), I multiply topcoded earnings by 1.4 times the topcode.

The final sample contains information about 49,565 women, of whom 30,128 are workers. For 11,538 of them, the hourly wage is drawn from allocated data. I retain these observations because I am concerned with both the distribution of earnings and labor force participation, and 38.3% of all workers have allocated wages.

Appendix B: Proof of Proposition 1

The proof relies on the following properties of the Gaussian Inverse Mills Ratio, $\lambda \left (x\right ) = \frac {\displaystyle \phi \left (x\right )}{\displaystyle {\Phi }\left (x\right )}$, which are established in Heckman and Honoré (1990).

$$ \begin{array}{@{}rcl@{}} -1 < \lambda^{\prime}\left( x\right) = -\lambda\left( x\right)\left[\lambda\left( x\right) + x\right] < 0 \end{array} $$

(11)

$$ \begin{array}{@{}rcl@{}} \lim_{x\to -\infty} \lambda\left( x\right) = \infty, \quad \lim_{x\to \infty} \lambda\left( x\right) = 0 \end{array} $$

(12)

$$ \begin{array}{@{}rcl@{}} \lim_{x\to -\infty} \lambda^{\prime}\left( x\right) = -1, \quad \lim_{x\to \infty} \lambda^{\prime}\left( x\right) = 0 \end{array} $$

(13)

The first order condition in Eq. 2 can be rewritten as follows:

$$ g\left( W\left( Z\right),Z\right) = 0 $$

(14)

where

$$ \begin{array}{@{}rcl@{}} g\left( W,Z\right) = W - Z + \log \left[1+\frac{\displaystyle \sigma_{y|z}}{\lambda\left( \widehat{W}\right)}\right] \end{array} $$

(15)

$$ \begin{array}{@{}rcl@{}} \widehat{W} = \frac{W - \mu_{y|z}}{\displaystyle \sigma_{y|z}} \end{array} $$

(16)

Since $\lambda \left (x\right )$ is a decreasing function with range $\left (0,\infty \right )$, given any Z, $g\left (W,Z\right )$ is an increasing function in W with range $\left (-\infty ,\infty \right )$. This implies that $W\left (Z\right )$ exists for all Z, and that $\frac {\partial W}{\partial Z} < 0$ if and only if $g_{Z}\left (W,Z\right ) > 0$, where g_Z is the partial derivative of $g\left (W,Z\right )$ with respect to Z. Differentiating Eq. 15, we get

$$ g_{Z}\left( W,Z\right) = -1 - \frac{\rho\sigma_{y}}{\sigma_{z}}h\left( \widehat{W}\right) $$

(17)

where the function $h\left (x\right )$ is defined as follows:

$$ h\left( x\right) = \frac{\displaystyle \lambda\left( x\right) + x}{\displaystyle \lambda\left( x\right) + \sigma_{y|z}} $$

Using Eqs. 11 through 13, we can show that

$$ \begin{array}{@{}rcl@{}} h^{\prime}\left( x\right) > 0 \end{array} $$

(18)

$$ \begin{array}{@{}rcl@{}} \lim_{h\to \infty} h\left( x\right) = \infty \end{array} $$

(19)

$$ \begin{array}{@{}rcl@{}} \lim_{x\to -\infty} h\left( x\right) = 0 \end{array} $$

(20)

This implies that if ρ ≥ 0, then $g_{Z}\left (W,Z\right ) < 0$ and so $\frac {\displaystyle \partial W}{\displaystyle \partial Z} > 0$.

Now suppose that ρ < 0. Then, Eqs. 18 to 20 imply that $g_{Z}\left (W,Z\right )$ is increasing in W, and that there exists a function $W_{0}\left (Z\right )$ defined as follows:

$$ g_{Z}\left( W_{0}\left( Z\right),Z\right) = 0 $$

Note that $\frac {\displaystyle \partial W}{\displaystyle \partial Z} < 0$ if and only if $g\left (W_{0}\left (Z\right ),Z\right ) < 0$. Using Eqs. 17 and 16, we find that

$$ W_{0}\left( Z\right) = \mu_{y} + \frac{\rho\sigma_{y}}{\sigma_{z}}\left( Z-\mu_{z}\right) + \sigma_{y|z}h^{-1}\left( \frac{\sigma_{z}}{-\rho\sigma_{y}}\right) $$

Substituting this into Eq. 15, we find that $g\left (W_{0}\left (Z\right ),Z\right )$ is a linear, decreasing function of Z. So define Z₀ as follows:

$$ g\left( W_{0}\left( Z_{0}\right),Z_{0}\right) = 0 $$

If Z > Z₀, then $g\left (W_{0}\left (Z\right ),Z\right ) < 0$, and so $\frac {\displaystyle \partial W}{\displaystyle \partial Z} < 0$. If Z < Z₀, then $g\left (W_{0}\left (Z\right ),Z\right ) > 0$, which implies that $\frac {\displaystyle \partial W}{\displaystyle \partial Z} > 0$.

Appendix C: Table 1

Table 7 Parameter estimates under alternate specifications