Calibration of the tax function

The crucial aspect in representing that fiscal reform is the calibration of the parameters 𝜌, 𝜁 and 𝜙, that represent in the model the empirical tax function (equation 1a), both before and after the implementation of the policy, while taking into account the standard deductions to household income, and the federal income tax brackets defined in the legislation. The tax code specifies that, after taking the standard deduction, the application of the marginal tax rate to each successive tranche of income, up to the tax bracket of the total income of the individual. For example, in the pre-reform scenario the average household in the middle quintile subtracts the standard deduction ($9,350), and pays a total tax of 10% on his first $13,350 plus 15% on the subsequent $37,450 plus 25%

on the balance. Table 3.1 details the tax schedules before and after the tax reform.

The estimation of the parameters before and after the TCJA follows a procedure analogous to the one described in Carneiro et al. (2022). First, use the tax schedules reported in Table 3.1 to calculate the amount of income tax paid by each household quintile as described in the previous paragraph. Thereafter, sets 𝜌 equal to the top marginal income tax rate and uses the equations (3.1a) and (3.1b) sequentially to estimate 𝜙_𝑖 and 𝜁_𝑖. Tables 3.2 and 3.3 report the pre- and post-reform scale and progressivity fiscal parameters. It is noteworthy that the estimated pre-reform income average tax rates paid by the head of household (column “Estimated Tax Rate” of Tables 3.2) are similar to the observed historical US federal tax rates.⁹⁸

98 According to CBO data reported by the quintiles, the average Federal Taxes from 2000 to 2017 were 3.15%, 9.59%, 13.97%, 17.60% and 25.37%, for Q1 to Q5 respectively. When analyzing the differences between these tax rates and those reported in Tables 3.2, one should keep in mind that the former considers several taxes – individual income taxes, payroll taxes, corporate income tax, and excise tax – while the latter considers only the income tax rates paid by the head of household.

Table 3.1 – Head of Household Taxable Income Tax Brackets and Rates

2017 2018

Standard Deduction: $ 9,350 Standard Deduction: $ 18,000 Tax Rate Taxable Income Bracket Tax Rate Taxable Income Bracket

10% < 13,350 10% <13,600

15% 13,351-50,800 12% 13,601-50,800

25% 50,801-131,200 22% 50,801-82,500

28% 131,201-212,500 24% 82,501-157,500

33% 212,501-416,700 32% 157,501-200,000

35% 416,701-444,550 35% 200,201-500,000

39.6% > 444,550 37% > 500,000

Table 3.2 – Pre-reform Tax Parameters by Quintiles (2017)

Quintile Average Income*

Marginal Tax Rate

Estimated Taxes

Estimated

Tax rate 𝜌 𝜙̂_𝑖 𝜁̂_𝑖

First (Q1) $ 13,258.00 10% $ 390.80 2.9% 0.396 0.192 0.256 Second (Q2) $ 35,401.00 15% $ 3,240.10 9.2% 0.396 0.192 0.257 Middle (Q3) $ 61,564.00 25% $ 7,306.00 11.9% 0.396 0.474 0.236 Fourth (Q4) $ 99,030.00 25% $ 16,672.50 16.8% 0.396 0.359 0.239 Highest (Q5) $ 221,646.00 33% $ 49,749.18 22.4% 0.396 0.615 0.307

*Mean income received in 2017 (CENSUS Data - available at https://www.census.gov/data/tables/ time- series/demo/income- poverty/historical- income- households.html).

Table 3.3 – Post-reform Tax Parameters by Quintiles (2018)

Quintile Average Income*

Marginal Tax Rate

Estimated Taxes

Estimated

Tax rate 𝜌 𝜙̂𝑖 𝜁̂𝑖

First (Q1) $ 13,258.00 10% - $ 474.20 -3.58% 0.370 0.335 0.217 Second (Q2) $ 35,401.00 12% $ 1,816.12 5.13% 0.370 0.216 0.263 Middle (Q3) $ 61,564.00 22% $ 4,232.08 6.87% 0.370 0.502 0.254 Fourth (Q4) $ 99,030.00 24% $ 12,475.20 12.60% 0.370 0.467 0.260 Highest (Q5) $ 221,646.00 32% $ 45,564.72 20.56% 0.370 0.696 0.317

*Mean income received in 2017 (CENSUS Data - available at https://www.census.gov/data/tables/ time- series/demo/income- poverty/historical- income- households.html).

The next step is to transform these quintile estimates into summary economy-wide average estimates 𝜙̂ and 𝜁̂ suitable for use as the fiscal parameters in equation (3.1a) of the model. This could be done in several different ways, but here, as in Carneiro et al.

(2022), this is done by minimizing the sum of squared residuals between the quintile rate calculated by the tax functions (Eqs. 3.1a and 3.1b) and the observed data, for both the

corresponding marginal tax rate (column “Marginal Tax Rate”) and the rate (column

“Estimated Tax Rate”) of Tables 3.2 and 3.3. The range of the parameters 𝜙̂_𝑖 and 𝜁̂_𝑖 constitute the upper and lower constraints to estimate these average fiscal parameters.

Table 3.4 reports the calibrated values of the parameters, both before and after the TCJA:

Table 3.4 – Intercept and Estimated Average Tax Parameters

𝜌 𝜙 ̂ 𝜁̂

Pre-Reform Economy 0.396 0.290 0.236 Post-Reform Economy 0.370 0.332 0.231

Figure 3.1 show calibrated tax functions adopting two sets of parameters: the one denoted 𝑓(𝑦_𝑖) uses the pre-reform parameters of Table 3.4, and the other, denoted 𝑔(𝑦_𝑖), uses the average parameters estimated by Chen and Guo (2013), which was the calibration used in chapter 2. It also displays the estimated pre-reform income tax rates paid by the head of household. The comparison of the two curves supports the progressive tax schedule adopted in this chapter, indicating that the calibration proposed here is vastly superior in its adherence to the estimated tax rates for all income brackets than the one adopted by Chen and Guo (2013).

Finally, the comparison of 𝑓(𝑦_𝑖) with the tax function calibrated with that in Carneiro et al. (2022), denoted 𝑧(𝑦_𝑖), in Figure 3.2 shows that both functions accurately reproduce the pre-reform tax rates for the quintiles.

Nevertheless, 𝑓(𝑦_𝑖) emulates more closely the tax rates paid by a super-rich household than 𝑧(𝑦_𝑖). Recall equations (3.1a) and consider, for example, the IRS data on the 400 Individual Income Tax Returns.⁹⁹ For 𝑧(𝑦_𝑖), using the fiscal parameters of Carneiro et al. (2022), the marginal income tax rate paid by these is approximately 62%, which is much larger than the observed the top marginal rate. In turn, if the tax function is 𝑓(𝑦_𝑖), the average marginal income tax rate paid by this group is equal to 37.7%, which is more plausible, considering the actual rate of 39.6%.

99 Available at https://www.irs.gov/pub/irs-soi/09intop400.pdf.

Figure 3.1 – Comparison of the tax function here with that in Chen and Guo (2013)

Figure 3.2 – Comparison of the tax function here with that in Carneiro et al. (2022)