Solving for Elasticity in a Notch

Setup

The agent faces the following budget line:

\[ c(z) = \begin{cases} z \cdot (1-t_1) & \quad \text{if} \quad z \leq z^* \\ z^* \cdot (1-t_1) + (z - z^*)\cdot (1-t_2) - T & \quad \text{if} \quad z > z^*\\ \end{cases} \]

Where \(c\) is consumption (net earnings after tax), \(z\) are pre-tax earnings, \(z^*\) is the notch point, \(t_1\) and \(t_2\) are the marginal tax rates before and after the notch point, and \(T\) is the Taxed “penalty” for crossing the notch.

An agent has an ability measure \(n_i\), and an elasticity of earnings w.r.t. net-of-tax rate \(e_i\). We assume a smooth ability distribution in the population. We also assume that elasticity is constant among all agents, or that its mean, conditional on ability, is constant (in the latter case, we are estimating the mean elasticity in the population). The agent has quasi-linear, iso-elastic utility:

\[u(c,z) = c - \frac{n}{1 + 1/e } \cdot \left( \frac{z}{n} \right)^{1+1/e}\]

Which has a first order condition of: \[ z = n \cdot (1-t)^e\]

Where \(z\) is the level of earnings. Note that when the marginal tax rate is zero, earnings equal ability. Thus we can interpret the ability parameter as the level of income this individual would earn in a world where marginal tax rate is zero.

Estimating \(e\)

There is an agent with ability \(n^*\), who optimally earns exactly the sum of money where the notch kicks in. This agent’s tangency condition for maximizing utility is \(z^* = n^* \cdot (1-t_1)^e\). There is another agent, the marginal buncher, with ability \(n^* + \Delta n^*\). This agent is indifferent between earning at the notch point \(z^*\) or earning at the point satisfying his first order condition, which we call \(z^* + \Delta z^*\). The marginal buncher is indifferent between two bundles, as the budget line is not convex. Agents with higher ability have only one optimal point, a tangency point to the right of the notch. They are unaffected by the notch.

When estimating elasticity for a notch, bunchr first tries to get an estimate of \(\Delta z^*\), using the amount of bunching and assuming that all that bunching comes from the right side of the distribution. After estimating this \(\Delta z^*\), or being provided one by the user with the force_after option,bunchr finds the elasticity that would equate utilities of this agent at both point: \(z^*\) and \(z^* + \Delta z^*\). To do so, it uses the convenient connection between ability and earnings given by the first order condition: \(z^* + \Delta z^* = (n^* + \Delta n^*)\cdot (1-t_2)^e\)