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{\centerline{\LARGE{\bf
{Application Taxing petrol and compensating car users}
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\centerline{\bf {Lennart Stern}}
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Description: This capsule studies how tax reforms can be implemented
that ensure that no one is made worse off. We will assume that the policy makers have no
information on the consumers' preferences apart from what they reveal
about their preferences through their choices during the implementation
of the reform.
Resource:
Throughout this set of exercises we will be studying the following
model:
In a country there are two types of people, car users and cyclists. All
of the car users are identical as far as their petrol consumption is
concerned.
There are two negative external effects of petrol consumption (i.e. effects not born by the people consuming the petrol). Firstly, there is a health care cost of
\(\gamma\) per unit of petrol consumed. This cost is paid for by the
government who provides the health care. Secondly, some people do not recover
fully from the health problems caused by petrol consumption. We do not
model this effect explicitly.
We shall denote by \(x\left( z, t \right)\) the representative car
user's demand for petrol that would emerge if the tax rate on petrol was
set to be \(z\) and a transfer of \(t\) was given to the car user. We
intentionally use the formulation that `the demand for petrol would
emerge' because the car users do themselves not know how difficult it
would be for them to reduce their consumption of petrol. This they will
only learn by trying other modes of transportation etc., which they will
only do once they are exposed to the tax on petrol. In our model this
fact can be expressed as follows:
The car drivers initially only know that they prefer their current
consumption bundle to all others feasible in the absence of the tax on
petrol, but apart from that they do not know their own preferences.
The cyclists think about possible reforms that would introduce a tax on
petrol and pay transfers to the car users. The cyclists know that the
car drivers are very skeptical people. Indeed, the cyclists know that
the only way they could get the car users to accept a reform proposal
will be if they can show that whatever the car drivers' preferences turn
out to be (and assuming that the car drivers do not get health benefits
from any reduction in petrol consumption), they will be made no worse off by the reform.
This set of exercises aims to help the cyclists find the policy
proposals that will lead to the highest possible tax rate on petrol
under the constraint that car users are guaranteed to be made no worse
off by the reform.
\emph{Question 1}
Suppose the tax on petrol is introduced with a tax rate of \(z\) and car users are given the compensation of
\(\text{z\ x}\left( 0 \right)\). Is their initial consumption bundle
still feasible?
True: yes
False: no
Explanation:
If the car users stick to their initial consumption bundle, then their
expenditure increases by the amount of the tax, i.e. by \(z\ x(0)\).
Hence the transfer \(\text{z\ x}\left( 0 \right)\) is precisely what is
required to keep the initial consumption bundle affordable.
\emph{Question 2:}
In the diagram below are depicted the preferences of a car user with
preferences
\(u\left( x,y \right) = \min\left( x + 0.5\ y,y + 0.5\ x \right)\).
As always in this capsule, \(x\) denotes the amount of petrol consumed.
We shall denote by \(t(z)\) the transfer that if paid in the presence of
the petrol tax with rate \(z\) leads to the car user being exactly as
well off as without the tax and without the transfer.
\begin{figure}[h!]
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\includegraphics[width=0.70\columnwidth]{figures/image1/image1}
\caption{Couldn't find a caption, edit here to supply one.%
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What can we say about the transfer \(t(z)\)? (Hint: First identify the
consumer's optimal choices of petrol in the graph under the various
budget lines by looking at the indifference curves.)
True: The optimal quantities of petrol lie on the white line for all of
the three budget lines.
True: \(t(z)\) is by definition the compensating variation for the car
driver and the reform that consists of introducing the petrol tax at
rate \(z\)
True: in the example with preferences
\(u\left( x_{1},x_{2} \right) = \min\left( x_{1} + 0.5\ x_{2},x_{2} + 0.5\ x_{1} \right)\)
we have \(t\left( z \right) = z\ x(0,0)\)
True: in the example with preferences
\(u\left( x_{1},x_{2} \right) = \min\left( x_{1} + 0.5\ x_{2},x_{2} + 0.5\ x_{1} \right)\)
we have \(t\left( z \right) = z\ x(z,t(z))\)
True: Whenever
\(x\left( 0,0 \right) = x\left( z,t\left( z \right) \right)\) we have
\(t\left( z \right) = z\ x(0,0)\)
True: Whenever
\(x\left( 0,0 \right) = x\left( z,t\left( z \right) \right)\) we have
\(t\left( z \right) = z\ x(z,t(z))\)
Explanation:
\(t(z)\) is by definition the compensating variation for the car driver
and the reform that consists of introducing the petrol tax at rate
\(z\).
Suppose the tax reform accompanied by the compensating variation leads
to no change in the consumption of petrol, i.e. suppose that
\(x\left( 0,0 \right) = x\left( z,t\left( z \right) \right)\). Then the
pre-tax expenditure is the same as in the case with no reform. Therefore
the transfer \(t(z)\) must pay exactly for the tax payment, i.e.
\(t\left( z \right) = z x(z,t\left( z \right))\). We then also
have \(t\left( z \right) = z x(0,0)\).
\emph{Question 3}
In the diagram below are depicted the preferences of a car user with
preferences \(u\left( x,y \right) = x\ y\). As always in this set of
exercises, \(x\) denotes the amount of petrol consumed. As always in
this set of exercises, we denote by \(t(z)\) the transfer that if paid
in the presence of the petrol tax with rate \(z\) leads to the car user
being exactly as well off as without the tax and without the transfer.
\begin{figure}[h!]
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\includegraphics[width=0.70\columnwidth]{figures/image2/image2}
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The consumer chooses point A in the absence of the tax on petrol. If the
tax on petrol was introduced at rate \(z\) without any transfer being
paid to the consumer, he would choose point B.
What can we say about \(t(z)\) in this case?
True: \(t\left( z \right) < z\ x(0,0)\)
False: \(t\left( z \right) > z\ x(0,0)\)
False: \(t\left( z \right) = z\ x(0,0)\)
Explanation:
If the transfer \(t = z\ x(0,0)\) was paid to accompany the
introduction of the tax on petrol, then the budget line would be the
yellow line in the diagram below:
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image3/image3}
\caption{Couldn't find a caption, edit here to supply one.%
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\end{center}
\end{figure}
Here the consumer could choose consumption bundles strictly preferred to
\(x(0,0)\). Hence \(\tau = z\ x(0,0)\) is more than what is necessary to
compensate the consumer so as to make him as well off as in the case
with no tax on petrol and no transfer.
\emph{Question 4}
We have seen in the questions 1 and 2 that paying a transfer of
\(z\ x(0,0)\) to accompany the introduction of the tax on petrol at rate
\(r\) leads to the consumer being as well off as in the case with no tax
on petrol and no transfer (in question 1) or even better off (in
question 2). Does this always hold?
True: yes, always!
False: if we assume that preferences are convex, then yes, but otherwise
we can't be sure.
False: if we assume that preferences are strictly monotonic, then yes,
but otherwise we can't be sure.
False: if we assume that preferences are locally non-satiated, then yes,
but otherwise we can't be sure.
Explanation
If we pay a transfer of \(z\ x(0,0)\) then the original consumption
bundle is still feasible, so we can be sure that the consumer will be at
least as well off.
\emph{Question 5:}
The cyclists consider making the following reform proposal to the car
users:
Reform proposal of type 1: A tax on petrol is introduced with rate
\(z\). All the tax revenue raised and all the cost savings due to
reduced health care expenditure are used to pay a uniform transfer to
the car users.
We shall analyze for which values of \(z\) we could be sure that the
reform proposal of type 1 would make cyclists at least as well off as in
the situation with no tax on petrol. In preparation for answering this
question, you can connect the pairs for the reform proposal of type 1:
Categorizer:
tax revenue does the government gets \(x\left( z \right)z\)
the money that the government saves due to reduced health care costs
\(\gamma\left( x\left( 0 \right) - x\left( z \right) \right)\)
money required to pay the car users enough to be sure that they are not
made worse off \(z\ x(0)\)
Explanation:
From question 2 we know that it might be necessary to pay a transfer of
z \(x\left( 0 \right)\) car drivers in order to ensure that they are not
made worse off by the introduction of the tax on petrol with rate \(z\).
From question 4 we know that this is actually always enough to ensure
that car drivers are not made worse off by the introduction of the tax
on petrol with rate \(z\).
\emph{Question 6}
We shall make from now on the realistic assumption that if \(z > 0\)
then \(x\left( 0 \right) > x\left( z \right)\), i.e. that the
introduction of the petrol tax always reduces petrol consumption. What
conditions are necessary and sufficient in order to be sure that reform
proposal 1 will not make the car drivers worse off?
True:
\(x\left( 0 \right)z \leq x\left( z \right)z + \gamma\left( x\left( 0 \right) - x\left( z \right) \right)\)
True:
\(\left( x\left( 0 \right) - x\left( z \right) \right)\left( z - \gamma \right) \leq 0\)
True: \(z \leq \gamma\)
Explanation:
The transfer payment required to ensure that the car drivers are not
worse off has to be no more than the sum of the tax revenue and the
savings due to lower health care costs. This is the condition
\(x\left( 0 \right)z \leq x\left( z \right)z + \gamma\left( x\left( 0 \right) - x\left( z \right) \right)\).
The condition
\(\left( x\left( 0 \right) - x\left( z \right) \right)\left( z - \gamma \right) \leq 0\)
is obtained by rearrangement. Our assumption that
\(x\left( 0 \right) > x\left( z \right)\) allows us to divide by
\(\left( x\left( 0 \right) - x\left( z \right) \right)\) to obtain that
\(z \leq \gamma\) is an equivalent condition.
\emph{Question 7}
So far we have been very general in our analysis. To facilitate graphical illustration, we
shall for the rest of the capsule assume that the car drivers only consume one other good and that their utility is quasilinear in this other good. As illustrated, in the capsule 'Using a 2 good consumer choice model for understanding the n good consumer choice model', we can interpret the other good as 'the money spent on all the other goods (apart from petrol) combined'.
As shown in the capsule 'Equivalent Variation, Compensating Variation and Hicksian Demand', quasi-linearity implies that the demand for petrol only depends on the price of petrol relative to the price of good \(k\) and not on the consumer's available money. This allows us to write the demand for
petrol as \(x(z)\) without having to keep track of the transfer \(t\).
It also follows that \(x(z)\) is a decreasing function.
We now illustrate the reform proposals of type 1 in the diagram below. Such a
reform would introduce a tax on petrol with rate \(z\) and use all the
money saved due to the avoided health care costs to pay a uniform
transfer to the car users.
The red line and the dark green line are both possible demand functions
that would result in the same petrol demand upon the introduction of the
reform proposal of type 1. We note that under reform proposals of type 1 it would not be
known whether the true demand function corresponds to the dark green
line or the red line.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image4/image4}
\caption{Couldn't find a caption, edit here to supply one.%
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\end{center}
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What are the graphical and the algebraic way to find for a reform of type 1 the transfers
paid and the tax revenue collected?:
Category 1: transfers made to car drivers following a reform of type 1
introducing the petrol tax with rate \(z\)
Area of ABDE
z x(0)
category 2: tax revenue collected following reform of type 1
introducing the petrol tax with rate \(z\)
Area of FCDE
z x(z)
\emph{Question 8}
To facilitate the exposition, let us assume that all quantities are
expressed per month. Thus \(x(z)\) is the petrol that the cyclists would
consume in the presence of a tax on petrol with rate \(z\).
Having contemplated the limitations of reform proposal 1, the cyclists
consider making the following reform proposal instead:
Reform proposal of type 2: In a first step the tax on petrol is
introduced with a small rate, \(z_{1}\) for the first month. The car
users are given the transfer \(z_{1}x(0)\) for this month. For the
second month the tax rate on petrol is increased to \(z_{2}\). The
transfer paid to the car users for this second month is increased by
\((z_{2} - z_{1})x(z_{1})\) relative to the transfer paid in the first
month. Thus the overall transfer paid to the car users in the second
month is \(z_{1}x\left( 0 \right) + (z_{2} - z_{1})x(z_{1})\). For the
third month the tax rate on petrol is increased to \(z\). The transfer
paid to the car users for this third month is increased by
\((z - z_{2})x(z_{2})\) relative to the transfer paid in the first
month. Thus the overall transfer paid to the car users in the second
month is
\(z_{1}x\left( 0 \right) + \left( z_{2} - z_{1} \right)x\left( z_{1} \right) + (z - z_{2})x(z_{2})\).
We retake the diagram from the previous question:
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image5/image5}
\caption{Couldn't find a caption, edit here to supply one.%
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\end{center}
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What can we conclude?
True: Area of AMLE + area of GHKL+ area of IJDK= transfers made in month
3 to car drivers, if the true Marshallian demand function is the dark
green line
True: In each month the car drivers are at least as well off as in the
previous month.
False: Area of IJDK= transfers made in month 3 to car drivers, if the
true Marshallian demand function is the dark green line
Explanation:
Each month, the consumption choice made in the previous month is still
feasible, since the increase in the transfer relative to the preceding
month is exactly equal to the increase in tax payments that would arise
if the car user were to choose the same consumption bundle as in the
preceding period. Hence each month the car user can be no worse off than
in the preceding month.
\emph{Question 9:}
Consider the same diagram as before but now we look specifically at the
case where \(z = \gamma\). Our results from the previous questions about the
reform proposal of type 1 are
reflected in the diagram: If the tax on petrol is introduced immediately
(i.e. without any intermediate steps) with a rate
of \(z = \gamma\) the only way to make
sure that the car drivers are not made worse off is to pay them the
amount corresponding to the area ABDE. This is just about feasible,
since the tax revenue is FCDE and the savings from reduced health care
costs are ABCF.
In the diagram we have illustrated a reform proposal of type 2 that
increases the tax on petrol in four steps from 0 to \(z_{1}\), then from
\(z_{1}\) to \(z_{2}\), then from \(z_{2}\) to \(\gamma\) and then from
\(\gamma\) to \(z_{4}\). We shall say that a tax reform is `feasible' if
the payments made to the car users are less than the sum of the tax
revenue and the cost savings from reduced health care expenditure. What
can we say about this reform proposal, assuming that the dark green line
corresponds to the actual demand function?
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image6/image6}
\caption{Couldn't find a caption, edit here to supply one.%
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\end{center}
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True: Tax revenue = RPQE
False: tax revenue= area of FNQE
True: Cost savings from reduced health care expenditure = ABSR
True: The tax reform is feasible if and only if the area of CNPS is
smaller than the area of TBJIHG.
False: The tax reform is feasible if and only if the area of CNPS is
smaller than the area of TBMNCJIHG.
True: Adding an intermediate step in the tax reform can never result in
an increase in the transfers made to the car users.
True: If
\(z_{4} - \gamma \leq \frac{\text{Area\ of\ \ }\text{TBJIHG}}{z\left( 1 + \gamma \right)}\)
then the tax reform is feasible
True: If the tax reform is feasible then
\(z_{4} - \gamma \leq \frac{\text{Area\ of\ \ }\text{TBJIHG}}{z\left( 1 + \gamma \right)}\)
\emph{Question 10:}
Suppose the cyclist propose to the car drivers to introduce a reform of
type 2 where in the first three months the tax rate on petrol is
increased from \(0\) to \(z_{1}\) to \(z_{2}\) to \(\gamma\) and then in
the forth month to
\(\gamma + \frac{\text{Area\ of\ \ }\text{TBJIHG}}{z\left( 1 + \gamma \right)}\),
i.e. to
\(\gamma + \frac{\left( z_{2} - z_{1} \right)\left( x\left( 0 \right) - x\left( z_{1} \right) \right) + \left( \gamma - z_{2} \right)\left( x\left( 0 \right) - x\left( z_{2} \right) \right)}{z(1 + \gamma)}\).
In other words: Having introduced the tax on petrol and then increased
its rate in several steps to \(\gamma\), society sees how much excess
revenue the tax reform has generated thus far (taking into account the
cost savings due to lower health care expenditure) and then increases
the rate of the tax on petrol by an amount obtained by dividing this
excess revenue by the petrol consumption observed at that point.
What can we conclude?
True: This reform will always be feasible.
False: This reform might not be feasible.
True: This reform will make the car drivers at least as well of as they
are initially without any tax on petrol.
True: This reform might result in a tax on petrol higher than what is
achievable with a reform of type 1 under the constraint that car users
be guaranteed to be made no worse off.
True: This reform might result in a tax on petrol lower than what is
achievable with a reform of type 1 under the constraint that car users
be guaranteed to be made no worse off.
\emph{Recap:}
We have seen in our example how payments can be made that ensure that a
consumer is no worse off from a tax increase. Moreover, we have seen
that dividing tax reforms into several steps can reduce the amount of
payments that have to be made to ensure that a consumer is better off.
Comparing different tax reforms that ensure that no one is worse off, we
saw that a stepwise reform can reach higher tax rates than a reform that
introduces the tax all at once. Intuitively, this is because more
information on preferences is collected along the way, so that the
transfers are better tailored to be sufficient to make sure that no one
is worse off.
In the capsule 'Learning preferences over consumption bundles' we saw that under
convexity assumptions each consumer's preferences could in principle be learned
if we could observe the consumer's choices under all possible income levels and prices.
In this capsule we have illustrated a result that actually holds very generally:
If we want to introduce a reform and pay a consumer exactly the amount that makes
him indifferent to the reform being implemented or not, we can collect all the
required information on his preferences by dividing the reform up into sufficiently
many steps.
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