Package 'bunchr'

Bunching Analysis

Description

Given a kinked or notched budget set, this function gets a vector of earnings and analyzes bunching. The bunchr package has two main useful functions:

Usage

bunch(
  earnings,
  zstar,
  t1,
  t2,
  Tax = 0,
  cf_start = NA,
  cf_end = NA,
  exclude_before = NA,
  exclude_after = NA,
  force_after = FALSE,
  binw = 10,
  poly_size = 7,
  convergence = 0.01,
  max_iter = 100,
  correct = TRUE,
  select = TRUE,
  draw = TRUE,
  nboots = 0,
  seed = NA,
  progress = FALSE,
  title = "Bunching Visualization",
  varname = "Earnings"
)

Arguments

earnings	Vector of earnings, hopefully a very large one.
zstar	Place of kink (critical earning point).
t1	Marginal tax rate before kink.
t2	Marginal tax rate after kink.
Tax	"Penalty" tax for crossing zstar.
cf_start	Number of bins before the kink bin where counter-factual histogram should start.
cf_end	Number of bins after the kink bin where counter-factual histogram should start.
exclude_before	Number of excluded bins before the kink bin.
exclude_after	Number of excluded bins after the kink bin.
force_after	For notch analysis, should bunch be forced to use of the provided exclude_after for the end of the bunching, rather than trying to find the bin where the sum of the integral is zero? See details at notch_estimator documentation.
binw	Bin width.
poly_size	Order of polynomial used to calculate counter-factual histogram.
convergence	Minimal rate of change of bunching estimate to stop iterations.
max_iter	Maximum number of iterations for bunching estimates.
correct	Should the counter-factual histogram be corrected to compensate for shifting left because of the notch? See details.
select	Should model selection be used to find counter-factual histogram? See details.
draw	Should a graph be drawn?
nboots	how many bootstraps should be run?
seed	specify seed for bootstraps (earnings sampling).
progress	Should a progress bar be desplayed?
title	Title for Plot output
varname	Name for running variable, to be desplayed in the plot

Details

bunch checks if the specification has a kink, i.e. if the Tax parameter is greater than zero. If so, it applies notch_estimator. Otherwise, it applies kink_estimator. Additionally, bunch can bootstrap by sampling the earnings vector, returning a vector with the estimated elasticities.

Value

bunch returns a list comprising of the parameters returned by kink_estimator and notch_estimator. If bootstraps were asked for, bootstrapped values are added to the list. Drawing of histograms is suppressed when running the bootsraps.

See Also

kink_estimator, notch_estimator

Examples

# analyzing a kink
ability_vec <- 4000 * rbeta(100000, 2, 5)
earning_vec <- sapply(ability_vec, earning_fun, 0.2, 0, 0.2, 0, 1000)
# bunch_viewer(earning_vec, 1000, 20, 20, 1, 1, binw = 20)
estim <- bunch(earning_vec, 1000, 0, 0.2, Tax = 0, 20, 20, 1, 1,
binw = 20, draw=TRUE, nboots = 0, seed = 16)
estim$e

# analyzing a notch
earning_vec <- sapply(ability_vec, earning_fun, 0.2, 0.2, 0.2, 500, 1000)
bunch_viewer(earning_vec, 1000, 10, 40, 2, 22, binw = 50)
estim <- bunch(earning_vec, 1000, 0.2, 0.2, Tax = 500, 10, 40, 2, 22,
binw = 50, draw = FALSE, nboots = 0, seed = 16)
estim$e

---

Visualizing a histogram and potential excluded areas

Description

This function is meant to aid find excluded bins and analysis area for a bunching study. It displays a histogram with borders. Optionally, you can get the actual histogram back. This is convenient, as the kink/notch point is set as the center of a bin.

Usage

bunch_viewer(
  earnings,
  zstar = NA,
  cf_start = 10,
  cf_end = 50,
  exclude_before = 2,
  exclude_after = 20,
  binw = NA,
  trimy = TRUE,
  report = FALSE,
  title = "Count Histogram",
  varname = "Running Variable"
)

Arguments

earnings	Vector of earnings, hopefully a very large one
zstar	Place of notch/kink (critical earning point)
cf_start	Number of bins before the kink bin where counter-factual histogram should start.
cf_end	Number of bins after the kink bin where counter-factual histogram should start.
exclude_before	Number of excluded bins before the kink bin.
exclude_after	Number of excluded bins after the kink bin.
binw	Bin width.
trimy	Logical. Should the y-axis be trimmed to better show off-bunching histogram?
report	Should the function return the actual histogram?
title	Title for Plot output
varname	Name for running variable, to be desplayed in the plot

Value

A plot, the actual histogram if report is set to TRUE.

See Also

bunch

Examples

ability_vec <- 4000 * rbeta(100000, 2, 5)
earning_vec <- sapply(ability_vec, earning_fun, 0.2, 0.1, 0.2, 0, 1000)
bunch_viewer(earning_vec, 1000, 20, 40, 2, 2, 20, trimy = TRUE, report = FALSE)

---

Run bunchApp: an Interactive Bunching Simulation

Description

bunchApp is an interactive simulator for bunching analysis. It is meant to serve as a tool for understanding bunching analysis in general, and the use of bunchr for data analysis. This app is opened on a separate window.

Usage

bunchApp()

Details

This function merely runs the app. It accepts no parameters.

See Also

The machinery behind the simulation: bunch

This simulator is also offered online at https://trilnick.shinyapps.io/bunchapp/.

---

bunchr: A Package for Bunching Analysis

Description

The bunchr package is meant to help analyze bunching. Given a vector of earnings (or any other numeric vector), it creates a counter-factual count histogram and calculates the compensated elasticity of earnings w.r.t. the net-of-tax rate.

Main functions

bunchr has three main functions:

bunch

is the main function running the actual analysis.

bunch_viewer

serves as an aid to the second by visualizing some of the user-specified options without running an analysis. Use it to see what the histogram of your earnings vector looks like when setting specific bin size, where the counter-factual analysis should be done, and the bounds of the excluded area. You can also save the histogram bins and counts.

bunchApp

is an interactive simulator. Use it to explore bunching simulation and estimation of earning elasticity.

Author(s)

Maintainer: Itai Trilnick [email protected]

See Also

bunch, bunch_viewer

---

Finding optimal earning under kinked/notched budget set

Description

For an agent with quasi-linear iso-elastic utility, find the utility maximizing earning level.

Usage

earning_fun(n, elas, t1, t2, Tax, zstar)

Arguments

n	Ability of person (earnings with zero tax)
elas	elasticity of earnings w.r.t. net-of-tax rate
t1	Tax rate before notch/kink
t2	Tax rate after notch/kink
Tax	height of notch (zero for pure kink)
zstar	place of notch/kink (critical earning point)

Details

earn_funciton is intended to simulate earnings of agents under a kink or notch.

Value

Optimal earning level.

See Also

util_calc, bunch

Examples

earning_fun(1200,0.2,0.1,0.3,100,1000)

---

Using elasticity to calculating distance between utility at tangency and at notch point

Description

Given an elasticity, a budget set, and the earnings of the marginal buncher, calculate the utility at notch point and at marginal buncher's earning, and return the absolute difference. Equating these two utilities helps find the elasticity of the marginal buncher. See equations (3) and (4) at Kelven and Waseem (2013)

Usage

elas_equalizer(elas, t1, t2, Tax, zstar, delta_zed, binw)

Arguments

elas	elasticity of earnings w.r.t. net-of-tax rate
t1	Tax rate before notch/kink
t2	Tax rate after notch/kink
Tax	Height of notch (zero for pure kink)
zstar	Place of notch/kink (critical earning point)
delta_zed	The notch size in bin units
binw	Bin width

Value

Absolute value of utility at $zstar+delta_zed$ minus utility at kink/notch point.

References

Kleven, H. and Waseem, Mazhar (2013) Using notches to uncover optimization frictions and structural elasticities: Theory and evidence from Pakistan, The Quarterly Journal of Economics 128(2)

Examples

elas_equalizer(0.2, 0.1, 0.2, 100, 1000, 20, 10)
# The elasticity value to minimize this is ~0.0716:
elas_equalizer(0.0716, 0.1, 0.2, 100, 1000, 20, 10)

---

Analyzing Bunching at a Kink

Description

Given a kinked budget set, this function gets a vector of earnings and analyzes bunching. This function could be run independently, but best used through the bunch function.

Usage

kink_estimator(
  earnings,
  zstar,
  t1,
  t2,
  cf_start = NA,
  cf_end = NA,
  exclude_before = 2,
  exclude_after = 2,
  binw = 10,
  poly_size = 7,
  convergence = 0.01,
  max_iter = 100,
  correct = TRUE,
  select = TRUE,
  draw = TRUE,
  title = "Bunching Visualization",
  varname = "Earnings"
)

Arguments

earnings	Vector of earnings, hopefully a very large one.
zstar	Place of kink (critical earning point).
t1	Marginal tax rate before kink.
t2	Marginal tax rate after kink.
cf_start	Number of bins before the kink bin where counter-factual histogram should start.
cf_end	Number of bins after the kink bin where counter-factual histogram should start.
exclude_before	Number of excluded bins before the kink bin.
exclude_after	Number of excluded bins after the kink bin.
binw	Bin width.
poly_size	Order of polynomial used to calculate counter-factual histogram.
convergence	Minimal rate of change of bunching estimate to stop iterations.
max_iter	Maximum number of iterations for bunching estimates.
correct	Should the counter-factual histogram be corrected to compensate for shifting left because of the notch? See details.
select	Should model selection be used to find counter-factual histogram? See details.
draw	Should a graph be drawn?
title	Title for plot output
varname	Name for running variable, to be desplayed in the plot

Details

A histogram is created from the earnings vector, with the kink point zstar as the center of one of the bins.

Correction of the counter-factual is required, as the kink-induced bunching will shift the whole distribution on the right side of the kink to the left. This option follows Chetty et al (2009) in correcting for this.

Model selection works using the step function from the stats package. It runs backwards from the full polynomial model, trying to find the best explanatory model using the Akaike information criterion.

Value

kink_estimator returns a list of the following variables:

e

Estimated elasticity

Bn

The sum of total estimated extra bunching in the excluded bins

b

The rate of extra bunching in the excluded area, divided by the length of area in $

data

A data frame with bin mids, counts, counter-factual counts, and excluded dummy

References

Chetty, R., Friedman, J., Olsen, T., Pistaferri, L. (2011) Adjustment Costs, Firm Responses, and Micro vs. Macro Labor Supply Elasticities: Evidence from Danish Tax Records, Quarterly Journal of Economics, 126(2).

See Also

bunch, notch_estimator

Examples

ability_vec <- 4000 * rbeta(100000, 2, 5)
earning_vec <- sapply(ability_vec, earning_fun, 0.2, 0, 0.2, 0, 1000)
# bunch_viewer(earning_vec, 1000, 40, 40, 1, 1, binw = 10)
kink_estimator(earning_vec, 1000, 0, 0.2, 40, 40, 1, 1, binw = 10, draw = FALSE)$e

---

Analyzing Bunching at a Notch

Description

Given a notched budget set, this function gets a vector of earnings and analyzes bunching. This function could be run independently, but best used through the bunch function.

Usage

notch_estimator(
  earnings,
  zstar,
  t1,
  t2,
  Tax = 0,
  cf_start = NA,
  cf_end = NA,
  exclude_before = NA,
  exclude_after = NA,
  force_after = FALSE,
  binw = 10,
  poly_size = 7,
  convergence = 0.01,
  max_iter = 100,
  select = TRUE,
  draw = TRUE,
  title = "Bunching Visualization",
  varname = "Earnings"
)

Arguments

earnings	Vector of earnings, hopefully a very large one
zstar	Place of notch (critical earning point)
t1	Tax rate before notch
t2	Tax rate after notch
Tax	Lump sum penalty for crossing zstar.
cf_start	Number of bins before the notch bin where counter-factual histogram should start.
cf_end	Number of bins after the notch bin where counter-factual histogram should start.
exclude_before	Number of excluded bins before the notch bin.
exclude_after	Number of excluded bins after the notch bin.
force_after	Should bunch be forced to use of the provided exclude_after for the end of the bunching, rather than trying to find the bin where the sum of the integral is zero? See details.
binw	Bin width.
poly_size	Order of polynomial used to calculate counter-factual histogram.
convergence	Minimal rate of change of bunching estimate to stop iterations.
max_iter	Maximum number of iterations for bunching estimates.
select	Should model selection be used to find counter-factual histogram? See details.
draw	Should a graph be drawn?
title	Title for plot output
varname	Name for running variable, to be displayed in the plot

Details

By default, notch_estimator will try to find the end of the notch, i.e. a histogram bin where the incentive to bunch wears off. To do this, a counter factual distribution is interpolated using the bins inside the counter-factual area but outside of the excluded area. This is assumed to be the histogram we would have without the tax at the notch point. At the observed bunching area, the bins should be much greater than their corresponding counter-factual bins. Later on, the bunching should create the notch area where the observed bins are lower than the counter-factual. factual bin However, both observed and counter-factual histograms should have the same total mass. The end of the notch is determined as the bin where the running sum of differences between observed and counter-factual bins reaches zero. notch_estimator goes through an iterative process, setting the notch end bin and re-interpolating the counter-factual in the notch area using all the bins outside of the notch, trying to find a stable right-side boundary.

A user might want to force a visibly detectable end of notch, rather than let notch_estimator calculate one. Use force_after=TRUE with caution: this sets the notch size as exclude_after minus exclude_before in terms of bins. The notch size is used to calculate elasticity, and forcing the wrong notch size might bias the intensive margin elasticity estimate. In other settings, e.g. a labor market with extensive margins (entry and exit from labor force), forcing the notch size might be helpful.

For "impure" notches, where the marginal tax rate after the notch is different than the one before it, this function disregards the shifting of post-notch distribution to the right, as suggested by Kleven (2016). Assumption is that the notch effect is much stronger anyway.

The select option implements a step model selection. It runs backwards from the full polynomial model (specified by poly_size), trying to find the best explanatory polynomial model for counter-factual histogram.

Value

notch_estimator returns a list of the following variables:

e

Estimated elasticity

Bn

The sum of total estimated extra bunching in the area starting at cf_start and through the notch bin (zstar)

notch_size

Distance between notch bin and bin where the estimated influence of the notch ends, delta_zed

data

A data frame with bin mids, counts, counter-factual counts, and excluded dummy

References

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

See Also

bunch, kink_estimator

Examples

ability_vec <- 4000 * rbeta(100000, 2, 5)
earning_vec <- sapply(ability_vec, earning_fun, 0.2, 0.2, 0.2, 500, 1000)
bunch_viewer(earning_vec, 1000, 15, 30, 2, 21, binw = 50)
notch_estimator(earning_vec, 1000, 0.2, 0.2, 500, 15, 30, 2, 21, binw = 50,
draw = FALSE)$e

---

Calculating quasi-linear iso-elastic utility

Description

$u(z,n,elas,t1,t2,Tax,zstar) =$

$z * (1 - t1) + [z > zstar] * ((z - zstar) * (t2 - t1) - Tax) - n / (1 + (1 / elas)) * (z / n)^(1 + (1 / elas))$

Usage

util_calc(z, n, elas, t1, t2, Tax, zstar)

Arguments

z	Earnings
n	Ability of person (earnings with zero tax)
elas	elasticity of earnings w.r.t. net-of-tax rate
t1	Tax rate before notch/kink
t2	Tax rate after notch/kink
Tax	height of notch (zero for pure kink)
zstar	place of notch/kink (critical earning point)

Value

The utility of earning sum z given other parameters.

Examples

util_calc(900, 950, 0.2, 0.1, 0.2, 100, 1000)

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Calculating distance between utility at tangency and at notch/kink point

Description

Ability (n) and elasticity (e) determine an agent's earnings and utility. This function determines the tangency point of the agent's utility with the budget line and returns the distance between the utility of earning at that point and the utility of earning at the notch/kink point. This function is mostly used to find the marginal buncher.

Usage

util_equalizer(n, elas, t1, t2, Tax, zstar)

Arguments

n	Ability of person (earnings with zero tax)
elas	elasticity of earnings w.r.t. net-of-tax rate
t1	Tax rate before notch/kink
t2	Tax rate after notch/kink
Tax	height of notch (zero for pure kink)
zstar	place of notch/kink (critical earning point)

Value

Absolute value of utility at tangency minus utility at kink/notch point.

See Also

util_calc

Examples

util_equalizer(1200,0.2,0.1,0.3,100,1000)

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