# Solving for Elasticity in a Notch

#### 2016-12-02

This vignette is meant to explain how bunchr estimates the earning elasticity from bunching induced by a notch. These calculations derive the formulas used for notch analysis. It closely follows the discussion by Kleven1 with one main difference: Kleven discusses a notch created by a change in average tax rates on income. bunchr’ is inspired by another setting: tax rates are marginal, and the notch is created by a “cash cliff” - where a fixed sum of money is taken from the agent for crossing a threshold. This “cash cliff” is observed in many settings of government transfers in developed countries (e.g. disability insurance in the US).

## 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: $$z^* + \Delta z^* = (n^* + \Delta n^*)\cdot (1-t_2)^e$$

### Utility of Marginal Buncher at Bunch Point $$z^*$$

\begin{align*} u(c,z^*) &= c - \frac{n^* + \Delta n^*}{1+1/e} \cdot \left( \frac{z^*}{n^* + \Delta n^*} \right)^{1+1/e}\\ &= c - \frac{1}{1+1/e} \cdot (n^* + \Delta n^*)^{-1/e} \cdot \left(z^* \right)^{1+1/e}\\ &= c - \frac{1}{1+1/e} \cdot \left(\frac{z^*+\Delta z^*}{(1-t_2)^e} \right)^{-1/e} \cdot \left(z^* \right)^{1+1/e}\\ &= c - \frac{1}{1+1/e} \cdot \frac{(1-t_2)}{(z^*+\Delta z^*)^{1/e}} \cdot \left(z^* \right)^{1+1/e}\\ &= z^* \cdot (1-t_1) - \frac{1}{1+1/e} \cdot \frac{(1-t_2)}{(z^*+\Delta z^*)^{1/e}} \cdot \left(z^* \right)^{1+1/e} \end{align*}

### Utility of Marginal Buncher at Tangency Point $$z^* + \Delta z^*$$

\begin{align*} u(c, z^* + \Delta z^*) &= c - T - \frac{n^* + \Delta n^*}{1+1/e} \cdot \left( \frac{z^*+\Delta z^*}{n^* + \Delta n^*} \right)^{1+1/e}\\ &= c - T - \frac{1}{1+1/e} \cdot (n^* + \Delta n^*)^{-1/e} \cdot \left(z^* + \Delta z^* \right)^{1+1/e}\\ &= c - T - \frac{1}{1+1/e} \cdot \left(\frac{z^*+\Delta z^*}{(1-t_2)^e} \right)^{-1/e} \cdot \left(z^* + \Delta z^*\right)^{1+1/e}\\ &= c - T - \frac{1}{1+1/e} \cdot \frac{(1-t_2)}{(z^*+\Delta z^*)^{1/e}} \cdot \left(z^* + \Delta z^* \right)^{1+1/e}\\ &= c - T - \frac{1}{1+1/e} \cdot (1-t_2) \cdot \left(z^* + \Delta z^* \right)\\ &= z^* \cdot (1-t_1) + \Delta z^* \cdot (1-t_2) - T - \frac{1}{1+1/e} \cdot (1-t_2) \cdot \left(z^* + \Delta z^* \right)\\ \end{align*}

### Equating these two, we can numerically solve for elasticity.

After calculating $$\Delta z^*$$, bunchr solves for elasticity, by minimizing the difference between these two utilities. The function elas_equalizer, included in bunchr, takes the marginal taxes, the Tax variable, and $$\Delta z^*$$, returning the squared difference between the utilities defined with some $$e$$. Using the optimize function in the stats package, bunchr finds the elasticity that minimizes the squared distance between these utilities. Note that, while by definition of the utility function, earnings cause disutility (from work). Hence elasticity should be positive. bunchr` bounds the elasticity search between 0 and 5, the latter being a very high elasticity in most settings, let alone labor supply.

1. Kleven, H.J., 2016. “Bunching”, Annual Review of Economics 8(1)