Thursday, November 2, 2023

Is pay at your law school keeping up with the cost of living? (Michael Simkovic)

On average, real pay for law professors has plummeted in the last 10 years by 24 percent.*  A large decline in real terms is also apparent in data from the Society of American Law Teachers.  However, pay has gone down by more at some law schools than others.  A few law schools, especially the most elite and best funded institutions, may have even seen real increases in pay. 

How can you tell if pay at your law school is keeping up with the cost of living? 

An individual high-skilled worker will typically see nominal increases in pay from year to year.  The pay increase is observed by the individual as a single pay increase or raise.  But there are really two components to this pay increase:

  • A cost of living adjustment, holding constant experience and productivity
  • A raise to reflect increased experience and productivity of the individual worker

We’re interested in (1), the cost of living adjustment. In particular, we want to know whether it is larger or smaller than a measure of inflation or growth in the cost of living.* 

However, to get at (1) the cost of living adjustment, we have to subtract out (2) the raise due to increased experience, from individual pay increases.

This means that we need an estimate of how much pay should go up or down with experience. 

I’ve created a spreadsheet, here which provides such an estimate. Inputs are highlighted in yellow. 

Download Law professor wage growth from experience and inflation

My spreadsheet enables users to enter an end year (for example, 2022) and a start year, input their age at the end year, and their actual nominal salaries in the start year and the end year.  It also lets users choose between cost of living adjustments based on either the average wage index or the Case Shiller Housing Index.

The spreadsheet then calculates how much of a pay increase the user would have to receive between the start year and the end year to stay even with cost of living, assuming that the user was also getting the expected increase in pay due to increased experience and productivity. 

This expected increase in pay is expressed both as a percentage pay increase and as an expected compensation in the end year. 

If compensation in the end year is lower than this amount, this indicates that pay at your school may not be keeping up with costs of living.  Or it could mean that you are receiving lower than average productivity increases.  If other members of your faculty also have experienced slower than expected pay growth, that provides stronger evidence that overall faculty compensation is growing slower than the cost of living.

What this means is, to get the most out of this spreadsheet, you’ll need to share it with your colleagues and have a conversation about the results that you’re getting.

Users might want to customize the spreadsheet to use an index of local housing prices rather than a national index.

More details on how I estimated the expected wage increase from experience is explained below.

Detailed Explanation of Estimates of Growth in Pay Due to Experience

Because data on experience is not available in the most common data sets such as Census data, it is customary to instead use age as a proxy for experience.**

Over the course of a lifetime, pay typically increases up to a point as workers gain experience and become more productive. But eventually, pay peaks and begins to decline as age reduces workers’ cognitive ability, health, and overall productivity and eventually leads to retirement.  For example, an aging professor might see lower compensation once the professor stops producing research and does not receive a summer stipend, steps down from an administrative role that is compensated, and when the professor enters a phase retirement by working part time.

The typical age of the peak and the slope of the curve at various ages varies by education level and occupation.  It might also vary by race, sex or other factors.  The percent increase or decrease—the slope of the curve—is different at different ages.

Because economists rarely have longitudinal data tracking each person every year throughout their career, researchers instead typically use data from many people at different ages in a single year (or a few years combined, with appropriate adjustments for inflation). If these people are similar enough to each other (same education, race, sex, etc.), and real wages have more or less been flat other than increases with experience, then we can assume that the cross-sectional age-adjusted averages across these people reflect a reasonable estimate of lifetime wage growth for a typical individual.  The result is called a synthetic lifetime wage profile.

The non-linear effects of age on wages are typically modeled using a regression analysis that includes a control variable both for age and for age-squared (that is for age multiplied by itself).   This generates a regression function in the form of

Regression 1


In the above equation,  β1 (“beta 1”) and β2 (“beta 2”) are regression coefficients that specify the relationship between age and wages. β0  ("beta 0") is a constant term.*** 

With sufficient data, separate equations can be estimated for different sub-populations, for example, men or women of a particular race or ethnicity with a particular level of education.  Or such differences can be included in the model through additional control variables and/or interaction terms, subject to constraints based on the amount of data available and concerns about overfitting or over-controlling.

When wages peak and then decline, they form an inverted u-shape. This typical pattern is reflected in the model as a positive  and a negative .****   

The change in wages expected from a change in age is given by the slope of the equation, or more precisely, the partial derivative with respect to age.  For example, if our regression equation is:

Equation 2

Recall that beta 1 is positive and beta 2 is negative.  Notice also that the change in wage for an additional year of age depends on the current age.  In other words, the increase or decrease in wages expected with a one more year of experience depends on the worker’s current age.

Once we have this equation which estimates (2) the raises due to increased experience, we can subtract this raise from the actual raises that individual professors at a particular institution received in a given year.  On average, the remaining increase or decrease in wages reflect the cost of living adjustment.  We can then compare this cost of living adjustment to the actual change in the cost of living to see if wages at a given institution are keeping up with changes in cost of living.

My analysis looked at post-secondary instructors age 30 to 74 working in colleges and universities or professional schools at least 35 hours per week who have professional degrees and who have undergraduate majors that are relatively typical of law graduates. The ACS data are from 2009 to 2021 and are inflation adjusted to 2022 dollars using AWI. 

I use professional degree holders and undergraduate majors to proxy for law professors because ACS does not precisely identify law professors or provide the field of graduate degrees.  Thus my sample is both over-and-under-inclusive.  BLS, which does identify law professors, does not have data on age or experience in the publicly accessible version of its data.

The results of my regression analysis are available here.

Download Law prof earnings by age simple regression

* Figures in this article are adjusted for inflation using the Social Security Administration’s Average Wage Index unless otherwise stated.  Changes in the cost of living can be estimated using the growth rate of the Average Wage Index, national or local housing costs, for example from the Case Shiller Indices, the Consumer Price Index, or another index tied to the cost of a basket of goods or services.

** When experience is available, as in administrative data or longitudinal data, it is preferable to use experience rather than age.

*** Changes in the cost of living can be estimated using the growth rate of the Average Wage Index, national or local housing costs, for example from the Case Shiller Indices, the Consumer Price Index, or another index tied to the cost of a basket of goods or services.

**** At low ages the absolute value of  is greater than the absolute value of   Thus, as age increase, predicted wages  initially go up. But at high ages, the absolute value of  is less than the absolute value of .  A since  is negative, at higher ages, wages fall as age increases.

The model can be made more flexible by including additional terms, for example age-cubed. Alternatively, average wages can be estimated at the center point of age ranges, for example ages 30 to 34, 35 to 39, 40 to 44, etc. These center points can then be used to estimate a slope.

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