feenberg
commented
8 years ago How do I read the equations?
dan
On Wed, 9 Nov 2016, Matt Jensen wrote:
Proposal for proxy welfare measures
This note outlines a methodology for computing proxy welfare measures for use in assessing the normative implications of changes in tax policy. It primarily relies on estimates of changes in tax liabilities from Tax-Calculator. In addition, users may assume an ad hoc revenue offset or use an estimate of the revenue offset from another economic model to estimate a balanced-budget welfare impact.[^1] See Table Shells.xlsx for sample output files.
User input: (i) welfare measures requested (step one only; steps one and two; or steps one, two, and three), (ii) income classifiers and ranges, (iii) choice of $\theta$ for use in estimating welfare impacts (default is 0, 2, and 4), (iv) consumption level at the assumed upper bound for marginal utility (default \$1,000), (v) assumed marginal utility of income for tax units with zero or negative income in the baseline (default zero), (vi) choice of revenue offset (default 15 percent), and (vii) income elasticity (default 0.4).
Step 1: Static change in after-tax income without financing, welfare impacts per assumed specification for utility of consumption
- Using the incidence assumptions otherwise reflected in the model, compute after-tax income under the baseline and under the policy alternative for each tax unit.
- For each income class compute the aggregate (for the class) percentage change in after-tax income.
For the specified set of values for $\theta,$ compute the change in welfare for each tax unit with positive after-tax income in the baseline as
$$\Delta u = \left( \frac{\overset{\overline{}}{c}}{10^{9}} \right)\frac{\left( \left( \frac{\max\left( c^{'},c{\min} \right)}{\overset{\overline{}}{c}} \right)^{1 - \theta} - \left( \frac{\max\left( c,c{\min} \right)}{\overset{\overline{}}{c}} \right)^{1 - \theta} \right)}{\left( 1 - \theta \right)}$$
$+ \left( \frac{\overset{\overline{}}{c}}{10^{9}} \right)\left( \frac{c{\min}}{\overset{\overline{}}{c}} \right)^{- \theta} \times \left( \min\left( 0,\frac{\left( c^{'} - c{\min} \right)}{\overset{\overline{}}{c}} \right) - \min\left( 0,\frac{\left( c - c_{\min} \right)}{\overset{\overline{}}{c}} \right) \right)$,
where $c$ is the baseline after-tax income, $c^{'}$ is the alternative after-tax income, $c_{\min}$ is the consumption level at the assumed upper bound for marginal utility, and $\overset{\overline{}}{c}$ is the average after-tax income in the baseline.[^2]^,^[^3] Compute the change in utility for each tax unit with zero or negative after-tax income in the baseline as
$\Delta u = u^{'} \times \left( c^{'} - c \right)$,
where $u^{'}$ is the assumed marginal utility of income for tax units with zero or negative income in the baseline.^4
Step 2: Add static change in after-tax income with financing (including behavioral impact in the determination of the amount of financing) and corresponding welfare impacts
- Compute aggregate static change in tax liability.
- Compute the required financing as the fraction of the tax change not offset by the revenue feedback.[^5]
- In principle, the revenue feedback should include all levels of government, including feedback to state and local government budgets.
- Allocate the required financing to tax units on a lump-sum basis (per person or per adult).[^6]
- For each income class compute the aggregate (for the class) percentage change in after-tax income including financing.
- For the specified set of values for $\theta,$ compute the change in utility for each tax unit as in step one using the change in after-tax income including financing.
Step 3: Add direct welfare impact of reoptimization in response to changes in tax rates (indirect welfare impact of the revenue offset is already included at step two)
- Compute an effective income change for the value of reoptimization for each tax unit as
$$\frac{1}{2}\text{εy}\frac{\left( \Delta\left( 1 - t \right) \right)^{2}}{1 - t_{0}},$$
where$\ y$ is pre-tax income, $\varepsilon$ is the assumed income elasticity, $\Delta\left( 1 - t \right)$ is the change in the keep rate, and $t_{0}$ is the baseline marginal tax rate.[^7]
- For each income class compute the aggregate (for the class) percentage change in after-tax income including both financing and the effective income change from reoptimization.
- For the specified set of values for $\theta,$ compute the change in utility for each tax unit as in step one using the change in after-tax income including financing and the effective income change from reoptimzation.
- Until this point, the distinction between curvature of utility at the family level and any potential additional curvature in the social welfare function could largely be ignored. At this point, the methodology is implicitly assuming linearity at the household level to derive the effective income change.[^8]
Relates to
403.
[^1]: Such an offset could reflect the microeconomic behavioral response in a conventional JCT score, macroeconomic feedback, or a combination of the two.
[^2]: The marginal utility of income increases without bound as income approaches zero in this specification of the utility of income. Imposing a bound on marginal utility prevents a small number of extremely low income tax units from driving the entirety of the results and could be appropriate if such tax units can finance consumption using transfers or loans from friends, family, or government programs not reflected in the model.
[^3]: In general, comparing changes in welfare across welfare measures is uninformative as the magnitudes of the differences do not necessarily have any meaning. However, within the class of CRRA specifications, normalizing by the baseline average after-tax income allows for rough comparisons consistent with the intuition that higher values of $\theta$ are consistent with a stronger redistributive motive. In addition, by further normalizing by 10^-9^ the $\theta = 0$ case is equal to the aggregate change after-tax income in billions of dollars.
[^4]: Assuming zero marginal utility of income for tax units with zero or negative incomes amounts to assuming they are high income families (or simply excluding them from the analysis).
[^5]: In general offsets can be greater than 100% and less than 0%. Offset must be specified by user or derived from another economic model.
[^6]: A typical (static) fully financed distribution table might include financing proportional to income or proportional to tax. Such options could be provided here, but the motivation for such options often is policies that would effectively increase marginal tax rates and thus allowing independence of the financing allocation and offset is potentially problematic. Additional allocation methods could be added as an option for user input. The superior approach would be to incorporate the implicit change in marginal rates reflected in a non-lump sum allocation of financing into the tax policy proposal contemplated thus allowing for consistent estimation of revenue effects and distributional impacts.
[^7]: This formula is an approximation based on a model in which there is a single consumption good and solely labor income and the elasticity is thus an elasticity of labor income with respect to the tax rate. Using a taxable income elasticity involves some conceptual slippage relative to the derivation, but would be broadly ok as an approximation for the additional margins of response. (The recommendation to use pre-tax income already reflects some of the same kinds of slippage.) Note that there is a relationship between the revenue offset of step two and the elasticity of step three, but allowing independent parameterizations is substantially simpler.
[^8]: In addition, note that the there is an important interaction between this approximation and the upper bound on marginal utility reflected in $\overset{\overline{}}{c}.$ If that bound is relaxed, the quality of this approximation becomes worse for tax changes that increase taxes for low-income families. However, in this case, even a more complex approximation for the welfare change due to reoptimzation such as $\frac{1}{2}\left( \left( \frac{c^{'}}{\overset{\overline{}}{c}} \right)^{- \theta}\left( 1 - t{1} \right) - \left( \frac{c}{\overset{\overline{}}{c}} \right)^{- \theta}\left( 1 - t{0} \right) \right)\text{εy}\frac{\Delta\left( 1 - t \right)}{1 - t_{0}}\ $ behaves poorly.
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MattHJensen
martinholmer
codykallen
gleiserson
@gleiserson (Greg Leiserson) emailed me a detailed proposal for adding proxy welfare measures to Tax-Calculator and potentially TaxBrain output tables.
Proposal for proxy welfare measures.docx
Table Shells.xlsx
I will be studying and thinking about this over the next week or so, and I will leave comments here. I'm hoping that others will find this interesting and do the same.
cc @martinholmer @feenberg @viard @andersonfrailey @Amy-Xu @GoFroggyRun @codykallen