The household optimization problem is subject to the constraint that leisure not exceed the endowment of time (equation (2)). For those h iseholds who would violate the constraint, the model calculates shadow wage rates at which they exactly consume their full-time endowment.
The household’s budget constraint is kinked due to the tax deductions applied against wage income. A household with wage income below the deduction level faces marginal and average tax rates equal to zero. A household with wage income above the deduction level faces positive marginal and average tax rates. Due to the discontinuity of the marginal tax rates, it may be optimal for some households to locate exactly at the kink.
Our algorithm deals with this problem as follows. We identify households that choose to locate at the kink by evaluating their leisure choice and corresponding wage income above and below the kink. We then calculate a shadow marginal tax rate from the first-order conditions that puts those households exactly at the kink. This procedure generates optimal forward-looking leisure and consumption choices for all periods of life.
The payroll tax ceiling introduces additional complexity by creating a non-convexity in the budget constraint. For those above the payroll tax ceiling, the marginal tax rate on labor falls to zero. We evaluate the utility on both sides of the non-convex section and put households on the side that generates highest utility.
The sequence of calculations is as follows. An initial guess is made for the time-paths of these aggregate variables as well as for the shadow wage rates, shadow tax rates, endogenous tax rates, the separate OASI / DI / HI payroll tax rates, and the Social Security and Medicare wealth levels. The corresponding factor prices are calculated along with the forward-looking consumption, asset and leisure choices for all income classes in each current and future cohort.
Shadow wages and shadow taxes are calculated to ensure that the time endowment and the tax constraints discussed above are satisfied. Households’ labor supply and assets are then aggregated by both age and lifetime income class at each period in time. This aggregation generates a new guess for the time-paths of the capital stock and labor supply. The tax rate which is endogenous for the particular simulation, is updated to meet the revenue-neutrality requirement. The payroll tax is also updated to preserve the pay-as-you-go financing of OASI and HI benefits.9 The tax rate for DI benefits is also updated. The algorithm iterates until the capital stock and labor supply time-paths converge.