Kramer Quist and Dr. Richard Evans, Economics
Optimal labor income taxation research has long relied on simplifying assumptions regarding the convexity of policy maker decisions in order to make the associated calculus of variations optimization problem analytically tractable. Furthermore, researchers have assumed that taxable agents have simplistic distributions of ability few degrees of heterogeneity. We developed a model that removes assumptions of convexity, while allowing the taxable agents to have more realistic distributions of wages as well as higher degrees of heterogeneity. In order to show the functionality of our model, we replicate the optimal taxation problem developed in Mirrlees 1971.
Mirrlees 1971 assumes a lognormal distribution of individual agents with only one degree of heterogeneity. Given a series of social welfare functions, the authors then demonstrate what the optimal taxation policy is, utilizing functional analysis and calculus of variations techniques. Instead of assuming that taxation is a continuous function, we assume that tax policies are bracketed (as they are in almost every developed country in the world). We use the same wage distribution, individual utility functions and societal welfare functions as Mirrlees 1971 employs. We then use equidistributed sequences to simulate 640 taxable agents and 2,000,000 potential tax policies. We evaluate each of these tax policies for both the revenue that it will produce and the social welfare that it achieves. We then throw out tax policies that are strictly dominated by some other policy with respect to revenue and social welfare. We take the remaining tax policies and use equidistributed sequences on a rectangular prism in order to simulate approximately 2,000 potential tax policies near each of the remaining policies. We then reiterate this process many times by throwing out inferior tax policies and simulating more policies near the potentially optimal policies to find frontier of optimal taxation policies.
To replicate the results of Mirrlees 1971, we assumed a model with 4 different tax brackets, and a wealth transfer. Mirrlees 1971 showed that with most social welfare functions, the optimal tax policy involved a positive welfare transfer with low tax rates at low incomes, higher tax rates for middle incomes, and low tax rates again for high incomes. In our replication of Mirrlees, the first tax bracket was low, the middle bracket was higher, and the third bracket was lower. Each of these 3 brackets were at similar tax rates to those found at corresponding incomes in Mirrlees 1971, though the match was not perfect. However, the top income bracket did not have any consistency along the social welfarerevenue frontier. This likely occurred because lognormal distributions do not have a sufficiently fat tail to accurately capture realistic wage or income distributions. Hence the highest income bracket affects an incredibly small portion of the population (and even for those it affects, it frequently affects a small portion of their income). Hence small variations in lower tax brackets affect social welfare and revenue more than even large shifts in the top bracket. Since we do not refine perfectly, there is thus wide variation in the top tax bracket between tax policies on the frontier.
Overall this project was a success. We developed a computational method that is capable of computing optimal taxation policy with large degrees of heterogeneity, exotic distributions, and a nonconvex policy makers problem. We replicated the results of Mirrlees 1971 and are now moving on to show the effects of heterogeneous disutility of labor on optimal taxation policy
Figure 1: The leftmost figure is an example of 10,000 tax policies simulated and graphed according to the social welfare and revenue generated. In the middle figure we “throw out” all the points that are not strictly dominated (make them gray), and highlight the points that we keep. The right most figure is a similar figure after several iterations. Notice that the frontier receives significant refinement
Figure 2: This is an example of one of the tax policies on the revenuesocial welfare frontier. Notice that the tax rates start low, then increase for the middle class, and then get lower for the upper class. The yaxis is the tax rate, and the xaxis is the income being taxed. The incomescale listed is as defined by Mirrlees and has no intuitive significance.
Figure 3: This is another tax policy that is close to the optimal tax policy in Figure 2 on the optimal taxation policy frontier. Observe that the tax policies are similar (different sizes as they have been rescaled), but the top tax bracket is significantly higher as discussed in the text.