Calculate employment_income from economic activity

[nikhilwoodruff]

With: c = cutoff e = economic activity r = marginal rate x = pay

x = (e + rc) / (1 + r)

[MaxGhenis]

As a bit more detail, the formula above follows algebraically from e = x + r (x - c), which is the total cost of the worker from the employer's perspective when x > c (employer NI = r (x - c)).

Since e = x when x<=c, the formula will be:

def pay_from_economic_activity(e, r, c):
    pay_if_subject_to_NI = (e + r * c) / (1 + r)
    return np.where(pay_if_subject_to_NI <= c, e, pay_if_subject_to_NI)

[MaxGhenis]

This seems equivalent to what the US Joint Committee on Taxation does (ht @jdebacker). Is there non-taxable compensation the employer could reallocate to? Here's page 15 of their report:

In estimating the effects of changes in payroll taxes, the Joint Committee staff generally assumes that the net effect on Federal receipts is less than the change in payroll tax receipts. Payroll taxes include taxes under the Federal Insurance Contributions Act (“FICA”) and the Federal Unemployment Tax Act (“FUTA”). Because these taxes are levied on virtually all employees,13 there are no shifting opportunities for avoiding this tax. Consequently, it is assumed that payroll taxes are part of the cost of compensation. In addition, it is assumed that employees’ compensation is equal to their marginal product of labor, and that their marginal product of labor is not affected by changes in payroll tax law. Therefore, any change in payroll taxes will be offset by changes in other forms of compensation. While it is possible that this reallocation of compensation in response to a change in payroll taxes could occur over several years, in the interest of consistency and simplicity of implementation, the offset is applied as if the reallocation occurs in the first year of the policy change.

Using these assumptions, the offset is derived by allocating the change in other compensation across taxable and non-taxable compensation proportionately and applying an average marginal tax rate on wages to the taxable portion of the reallocation. This computation results in an offset of 10 percent of the estimated changes in payroll taxes.14 As with the income and payroll tax offset for excise taxes, the payroll tax offset has not been recomputed in recent years.