I wrote about the possible effects of the December 2017 tax reform on capital accumulation in "The Real Test of the December 2017 Tax Reform Will Be Its Long-Run Effect." In this post, I am writing about the possible effects on labor hours.

This division of the discussion follows Robert Barro's. In "Tax Reform Will Pay Growth Dividends
The effects will be even larger thanks to last-minute cuts in marginal individual rates," he writes:

In a November letter to Treasury Secretary Steven Mnuchin, eight other economists and I argued that the bill’s corporate tax provisions would increase business investment and expand the economy. But what about the changes to the individual tax code? 

Here is the heart of Robert's claims:

The Tax Policy Center estimates that the weighted-average marginal tax rate from individual income and payroll taxes will fall in 2018 by 3.2 percentage points. By comparison, the cuts were 4.5 points from 1986-88 in the Reagan reform, 3.6 points from 1963-65 in the Kennedy-Johnson tax reduction, and 2.1 points from 2002-03 in the George W. Bush reform.

Moreover, the Tax Policy Center finds that marginal tax rates in 2018 will drop for taxpayers across the income distribution. Most cuts are near 3 percentage points, with the smallest being zero for incomes below $10,000 and 0.6 point for incomes between $500,000 and $1,000,000.

My research with Charles Redlick, published in 2011 by the Quarterly Journal of Economics, suggests that cutting the average marginal tax rate for individuals by 1 percentage point increases gross domestic product by 0.5% over the next two years. This means the tax bill’s average cut of 3.2 points should expand the economy by 1.6% through 2019, or extra growth of 0.8% a year. This growth effect is temporary, but what it adds to the level of GDP is permanent.

In line with standard economic theory, Robert is arguing that a reduced distortion in the labor market due to lower marginal tax rates is likely to increase the total amount that people work. The key mechanism is that with less of every extra dollar earned taken by taxes, people will be willing to work more even if they don't get a raise. With people willing to work more even if they don't get a raise, wage pressures will be muted and the Fed can afford to keep interest rates lower than it otherwise would, leading to higher GDP. 

How long does it take for the increase in employment and labor hours to happen? As I discussed in "The Real Test of the December 2017 Tax Reform Will Be Its Long-Run Effect," capital accumulation is painfully slow. By contrast, changes in employment and labor hours can happen relatively quickly. Robert suggests that adjustments in employment and labor hours would take about two years, which seems reasonable to me. 

How big is the effect? To get the total amount of extra work that people do, Robert is multiplying two numbers:

• An estimate of the semi-elasticity of GDP with respect to reductions in the average marginal tax rate: .5
• An estimate of the reduction in the average marginal tax rate: 3.2 percentage points

This leads to estimate of the overall increase in GDP of 1.6%. Spread out over a two-year adjustment period, that raises the growth rate of GDP by .8% over each of two years. (Of course, the extra growth wouldn't really be equal in both years. It might be tilted toward the second of the two years, since there is a lag in the effect of monetary policy.)

How reasonable is the estimate of how much a one percentage point reduction in the tax rate raises GDP? Here, I think making things as simple and transparent as possible is helpful. Let me use a toy model so simple that I can figure out a semi-elasticity of GDP with respect to reductions in the marginal tax rate in just a few lines. I will use "h" to represent the fraction an extra dollar of pay that a worker takes home. That is, 

h = 1 - marginal tax rate.

If there is monopolistic competition in the picture

h = (1- marginal tax rate)/(1 + markup)

This marginal tax rate doesn't have to equal the average tax rate, and so h doesn't have to have any particular relationship to government purchases G. 

Here are the key simplifications:

• There are no distortions other than the marginal tax rate and the markup. The economy is otherwise frictionless. The effect of the Fed getting the economy to its new natural level of employment, hours and output by appropriate monetary policy is approximated by using a model in which the economy is always at the natural level of employment, hours and output because prices adjust instantly.
• Workers get to choose the number of hours they work, given their wage.
• There is no capital and no investment. This means that all effects involving capital accumulation are ignored. It also means that the effect of a 1% increase in total amount people work is overestimated because the fact that they are more likely to be fighting to use the same tools at the same time and more crowded in the office is ignored. 
• Hours go up by an equal percentage for each worker rather than additional people being employed. Since I will use an estimate of the labor supply elasticity based only on increases in hours for a given worker, this tends to underestimate the total increase in amount worked. There is no need to assume that everyone is identical. The model still works fine if some people have, say, the productivity of two people and have each of their hours counted as two hours, and so earn twice the wage. Also, though not an exact result, if people's marginal tax rates differ, the concept of an average marginal tax rate is meant to give a good approximation to the right result when plugged into a model where people's marginal tax rates are all the same. 
• I will use a consumption-constant labor supply elasticity of 1. This is is close to the Frisch elasticity Matthew Shapiro and I found in “Labor Supply: Are the Income and Substitution Effects Both Large or Both Small?" but below the consumption-constant labor supply elasticity we found there. (This is a paper I mentioned in my first post on this blog "What is a Supply-Side Liberal?")
• The income and substitution effects for labor supply cancel.  What I mean by this is that—holding the number of hours per week fixed—a worker requires a wage 1% higher when the worker's consumption is 1% higher. Barring other unreasonable assumptions, something close to this assumption is needed in order to avoid the implication that the large increase in wages over most of the 20th century should have resulted in huge changes in weekly work hours that we don't see.

Note that the intertemporal aspects of the utility function don't matter, because there is no way for the economy to transfer resources intertemporally anyway. So the utility function can be represented simply as

C is consumption. N is total amount worked. N is measured in units so that if government purchases were zero and h = 1, the total amount worked would be equal to 1. Each unit of labor produces one unit of output. Since there is no investment, 

N = Y = C+G,

where Y is GDP and G is government purchases. From here on, I will just use N, but it is important to remember that Y = N in this simple model. 

If there is no markup, the pre-tax wage will be equal to the marginal product of 1. If there is a markup, the pre-tax wage will be 1/(1+markup). Whether there is a markup or not, the after-tax wage will be equal to h. With the worker choosing herhis hours, the after tax wage will equal the reservation wage for one extra hour. The reservation wage for an extra hour is equal to the marginal disutility of a little extra work—which for the utility function above is equal to N—divided by the marginal utility of a little extra consumption—which for the utility function above is equal to 1/C. That makes the reservation wage equal to N/(1/C) = CN. Setting the reservation wage for an extra hour equal to the after-tax wage, 

CN = h

Now, combine C = N - G from the equation  N =Y = C+G above with CN = h to get the quadratic equation

To get a starting point at which to calculate a semi-elasticity, take G = .2 and h = .8. Then, at this starting point, 

This has the solutions N = 1 and N = -.8. Only the solution N=1 makes sense. So this is the starting level of labor for this numerical example. Actually, you can fairly easily show that if G = 1-h, then the positive solution for N is equal to 1. (In particular, if G = 0 and h = 1, then N=1.)

To find the semi-elasticity of N (equal to GDP) with respect to h, take the total differential of 

The total differential is

(2N – G) dN – N dG - dh = 0. 

If government purchases G stay the same, dG = 0, and one can solve for the semi-elasticity in general and then plug in the starting values of N=1 and G=.2. 

(1/N) dN/dh = 1/[N(2N-G)] = 1/1.8 = .556

This is in the same ballpark as the estimate that Robert Barro uses. There are offsetting positive and negative biases relative to a more complex model. So the bottom line is that the semi-elasticity estimate Robert uses can be questioned, but is reasonable. 

How reasonable is the Tax Policy Center's estimate of the reduction in the average marginal tax rate from the December 2017 tax reform? I don't know the answer to this, but at this point I have no reason to doubt their estimate. I would be interested in any criticisms readers have of the Tax Policy Center's estimate of the effect of the tax reform on average marginal tax rates.