4.8: A Tax-Rebate Proposal

Last updated
Save as PDF

Page ID: 107535

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $

$ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $

$ \newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$

( \newcommand{\kernel}{\mathrm{null}\,}\) $ \newcommand{\range}{\mathrm{range}\,}$

$ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$

$ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$

$ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$ \newcommand{\Span}{\mathrm{span}}$

$ \newcommand{\id}{\mathrm{id}}$

$ \newcommand{\Span}{\mathrm{span}}$

$ \newcommand{\kernel}{\mathrm{null}\,}$

$ \newcommand{\range}{\mathrm{range}\,}$

$ \newcommand{\RealPart}{\mathrm{Re}}$

$ \newcommand{\ImaginaryPart}{\mathrm{Im}}$

$ \newcommand{\Argument}{\mathrm{Arg}}$

$ \newcommand{\norm}[1]{\| #1 \|}$

$ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\AA}{\unicode[.8,0]{x212B}}$

$ \newcommand{\vectorA}[1]{\vec{#1}} % arrow$

$ \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$

$ \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $

$ \newcommand{\vectorC}[1]{\textbf{#1}} $

$ \newcommand{\vectorD}[1]{\overrightarrow{#1}} $

$ \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} $

$ \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} $

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $

$ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $

$\newcommand{\avec}{\mathbf a}$ $\newcommand{\bvec}{\mathbf b}$ $\newcommand{\cvec}{\mathbf c}$ $\newcommand{\dvec}{\mathbf d}$ $\newcommand{\dtil}{\widetilde{\mathbf d}}$ $\newcommand{\evec}{\mathbf e}$ $\newcommand{\fvec}{\mathbf f}$ $\newcommand{\nvec}{\mathbf n}$ $\newcommand{\pvec}{\mathbf p}$ $\newcommand{\qvec}{\mathbf q}$ $\newcommand{\svec}{\mathbf s}$ $\newcommand{\tvec}{\mathbf t}$ $\newcommand{\uvec}{\mathbf u}$ $\newcommand{\vvec}{\mathbf v}$ $\newcommand{\wvec}{\mathbf w}$ $\newcommand{\xvec}{\mathbf x}$ $\newcommand{\yvec}{\mathbf y}$ $\newcommand{\zvec}{\mathbf z}$ $\newcommand{\rvec}{\mathbf r}$ $\newcommand{\mvec}{\mathbf m}$ $\newcommand{\zerovec}{\mathbf 0}$ $\newcommand{\onevec}{\mathbf 1}$ $\newcommand{\real}{\mathbb R}$ $\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$ $\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$ $\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$ $\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$ $\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$ $\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$ $\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$ $\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$ $\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$ $\newcommand{\laspan}[1]{\text{Span}\{#1\}}$ $\newcommand{\bcal}{\cal B}$ $\newcommand{\ccal}{\cal C}$ $\newcommand{\scal}{\cal S}$ $\newcommand{\wcal}{\cal W}$ $\newcommand{\ecal}{\cal E}$ $\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$ $\newcommand{\gray}[1]{\color{gray}{#1}}$ $\newcommand{\lgray}[1]{\color{lightgray}{#1}}$ $\newcommand{\rank}{\operatorname{rank}}$ $\newcommand{\row}{\text{Row}}$ $\newcommand{\col}{\text{Col}}$ $\renewcommand{\row}{\text{Row}}$ $\newcommand{\nul}{\text{Nul}}$ $\newcommand{\var}{\text{Var}}$ $\newcommand{\corr}{\text{corr}}$ $\newcommand{\len}[1]{\left|#1\right|}$ $\newcommand{\bbar}{\overline{\bvec}}$ $\newcommand{\bhat}{\widehat{\bvec}}$ $\newcommand{\bperp}{\bvec^\perp}$ $\newcommand{\xhat}{\widehat{\xvec}}$ $\newcommand{\vhat}{\widehat{\vvec}}$ $\newcommand{\uhat}{\widehat{\uvec}}$ $\newcommand{\what}{\widehat{\wvec}}$ $\newcommand{\Sighat}{\widehat{\Sigma}}$ $\newcommand{\lt}{<}$ $\newcommand{\gt}{>}$ $\newcommand{\amp}{&}$ $\definecolor{fillinmathshade}{gray}{0.9}$

A Tax-Rebate Proposal

This section examines a tax-rebate plan that provides further practice with the logic of income and substitution effects. This application shows that they are more than an intellectual curiosity.

The heart of the idea is for the government to reduce consumption of a particular good, for example, gasoline, without hurting the consumer.

The idea is to tax a good and then turn around and rebate (give back) all of the tax revenue to the consumer. Can we alter the consumer’s choices without lowering satisfaction? We keep things simple by ignoring administrative costs of collecting the tax and rebating it so the tax and rebate leaves the consumer’s income unchanged. Proponents point out that the government is not making any money (all of the tax revenue raised is refunded back) so the consumer is not going to be hurt.

Opponents contend that this scheme will have no effect because the rebated tax will immediately be spent on the taxed good and we will end up right where we started.

Who is right? We use the Theory of Consumer Behavior to find out. Along the way, income and substitution effects will come into play.

A Concrete Example

STEP Open the Excel workbook TaxRebate.xls and read the Intro sheet, then go to the QuantityTax sheet.

We have a Cobb-Douglas utility function with an option to apply a per unit (quantity) tax on good 1. The workbook opens with no tax and the consumer maximizing satisfaction by buying the bundle 25,50, yielding $U^* = 1250$.

We begin by applying a quantity tax.

STEP Change cell B21 to 1. Notice that a new budget line appears. The consumer cannot afford the original bundle and must re-optimize. Run Solver to find the new optimal solution.

You should find that the consumer will now buy the bundle $16 \frac{2}{3}$,50 and maximum utility falls to 833.33. Cell B22 shows that the government collects $16.67 ($1/unit tax on the 16.67 units purchased).

The idea behind the tax-rebate proposal called for rebating the tax revenue so that the consumer would not be hurt by the tax. We need to implement the rebate part of the proposal.

STEP Change cell B18 to 116.67. This shifts the budget constraint out. Run Solver to find the optimal solution.

You should find that the consumer optimizes by purchasing 19.445 units of $x_1$ and 58.335 units of $x_2$.

This result presents us with a problem. This is not the tax-rebate scheme the government envisioned. After all, the government is collecting more tax revenue ($19.445) than the consumer is getting as a rebate ($16.67).

Instead of giving the consumer $16.67, let’s give her $19.445. What does the consumer do in this case?

STEP Change cell B18 to 119.445. This shifts the budget constraint out a little bit more. Run Solver to find the optimal solution.

Now the consumer buys a little more $x_1$, just over 19.9 units. But we still do not have a revenue neutral policy. We need to increase m again. This process of repeatedly doing the same thing is called iteration.

STEP Set the cell B18 value to $100 (initial m) plus the amount of tax revenue in cell B22. Run Solver.

You can see that we are converging because the increases to income keep getting smaller and smaller. There is a tax rebate that yields an optimal $x_1$ that generates a tax revenue that exactly equals the tax rebate. The value of this tax rebate is $20.

STEP Set cell B18 to $120. Run Solver.

You should see that the optimal solution is 20,60 and maximum utility is 1200. If Solver is off by a little bit (this is false precision), you can enter 20 and 60 in cells B11 and B12. Since they buy 20 units of $x_1$, the consumer is paying $20 in tax. Since they are getting a tax rebate of $20 (m is set is 120), the tax they pay is exactly canceled out. We are ready to evaluate this program.

Who’s Right?

Proponents argued that by taxing the good and then turning around and rebating (giving back) the tax revenues to the consumer, we can alter the consumer’s choices without lowering satisfaction. Since the government is not making any money (all of the tax revenue raised is refunded back), the consumer is not going to be hurt.

Clearly the supporters of the tax-rebate proposal are wrong. The consumer had an initial $U^* = 1250$ and now has a new $U^* = 1200$. While we cannot meaningfully say that utility has fallen by 50 (because utility is measured on an ordinal, not cardinal scale), we can say that utility has fallen. Thus, in fact, the consumer is hurt by the tax-rebate proposal.

Critics, on the other hand, believed that this scheme will have no effect since the rebated tax will immediately be spent on the taxed good and we will end up right where we started.

Because the consumer went from an initial bundle of 25,50 to 20,60 after the $20 tax-rebate, it is obvious that the critics are wrong also. This consumer has altered purchasing plans and is, in fact, buying less $x_1$.

So, wait, who’s rightthe critics or the supporters of the scheme? Neither. They are both wrong. Income and substitution effects will help us explain why.

We return to the original problem without a tax or rebate and the initial solution of 25,50. The $1/unit tax is just like a price increase. We can find point B and compute the substitution and income effects from such a price change.

We first use the Income Adjuster Equation. \[\Delta m = x_1 \mbox{*}\Delta p_1\] \[\Delta m = [25][+1]\] This result says that a $25 increase in income to $125 will allow us to buy the initial bundle.

STEP Set income in cell B18 to 125 (and confirm that there is a $1/unit tax in cell B21) and run Solver.

The optimal solution is $20 \frac{5}{6},62 \frac{1}{2}$. We have points A, B, and C so we can compute total, substitution, and income effects of the $1/unit price increase due to the tax without any rebate.

SE (A to B): $20 \frac{5}{6} - 25 = - 4 \frac{1}{6}$
IE (B to C): $16 \frac{2}{3} - 20 \frac{5}{6} = - 4 \frac{1}{6}$
TE (A to C): $16 \frac{2}{3} - 25 = - 8 \frac{1}{3}$

Figure 4.27 displays these results with each point signifying a tangency between the budget line and an indifference curve (not drawn in to make it easier to read the graph).

Fig27Effects copy.png — Figure 4.27: TE, SE, and IE for tax without rebate.

The tax-rebate proposal is closely related to Figure 4.27. The tax is like a price increase that moves the consumer from A to C and the rebate is like an income effect that moves the consumer from C to B.

However, if you look carefully, the changes in income are not the same. In the tax-rebate proposal, the revenue-neutral rebate is $20, whereas in our income and substitution effect work we gave the consumer $25 to be able to purchase the original bundle. A $25 rebate is not revenue neutral because the consumer buys only $20 \frac{5}{6}$ units of $x_1$ so the government ends up losing revenue. The rebate has to be $20 to be consistent with the break-even logic of the proposal.

In addition to the income and substitution effects, Figure 4.28 adds point D, which shows the optimal solution given the tax-rebate proposal. Point D (at coordinate 20,60) has utility of 1200, which is, of course, lower than point B (the combination $20 \frac{5}{6},62 \frac{1}{2}$ yields just over 1300 units of utility). More importantly for the purposes of evaluating the proposal, utility at point D is less than utility at point A (where 25,50 generates $U^* = 1250$).

Fig28PointD copy.png — Figure 4.28: Understanding the tax-rebate proposal.

The key to the analysis lies with point D in Figure 4.28. It has to be on the initial budget line to fulfill the revenue-neutral condition of the proposal. But we know point A was the initial optimal solution on that budget line, so we can deduce that the consumer prefers point A to point D (and any other point on the initial budget line) and will suffer a decrease in satisfaction if the tax-rebate proposal is implemented.

Tax-rebate Schemes

Taxes are often used to pay for government services and fund programs deemed worthy by society, but they can also be corrective. Taxes on specific products can discourage particular activities (think cigarettes and smoking).

Simultaneously taxing a good and rebating the tax revenue periodically appears as a policy proposal (often with regard to gasoline). Proponents claim the rebate cancels out the price increase from the tax. The scheme is related to income and substitution effects. The tax is like a price increase and the rebate is like an income effect.

Although similar to income and substitution effects, there is one important difference in tax-rebate proposals: a revenue-neutral rebate does not return enough income to allow the consumer to buy the pre-tax bundle or to reach the pre-tax level of satisfaction. Thus, the consumer cannot reach the initial level of satisfaction.

It is true, however, that a tax-rebate policy will alter consumption patterns. Whether the loss in utility is compensated by the changed consumption pattern is a different question.

Exercises

Analytically, we can show that the demand curves for goods 1 and 2 with a Cobb-Douglas utility function (where c = d) are $x_1 \mbox{*} = \frac{m}{2(p_1+Q_Tax)}$ and $x_2 \mbox{*} = \frac{m}{2p_2}$. Use these demand functions to compute the income, substitution, and total effects for $x_1$ for a $1/unit tax. Show your work.
We know that the tax-rebate scheme gives back too little income to return the consumer to the initial level of utility (1250 units). With a $1/unit tax, find that level of rebate where the consumer is made whole in the sense that $U^* = 1250$. Describe your procedure in answering this question.
At point D in Figure 4.28, is the MRS greater or smaller in absolute value than the price ratio before the tax-rebate scheme is implemented? How do you know this?

References

The epigraph is from page 87 of the fifth edition of Taxing Ourselves: A Citizen’s Guide to the Debate over Taxes published in 2017 by Joel Slemrod and Jon Bakija. The book does not discuss the tax-rebate proposal covered in this chapter, but it is an excellent, user-friendly guide to the ever-present debate over taxes.

Government spending, taxing, and budgeting is part of the subdiscipline of economics called Public Finance. If you are interested in government’s role in the economy or tax reform (including flat or consumption tax proposals), the history of the income tax in the United States, or how economists evaluate and judge taxes, this book is a great place to start.

Search

Text Color

Text Size

Margin Size

Font Type