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\begin{document}
\title{Equivalent Variation, Compensating Variation and Hicksian demand}
\author[1]{Lennart Stern}%
\affil[1]{École normale supérieure}%
\vspace{-1em}
\date{\today}
\begingroup
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\sloppy
Resource~:
Consider a consumer who under the status quo faces price vector \(p\)
and has available money \(m\). Suppose there is a reform project that
results in the price vector instead being \(p'\) and in the available
money for the consumer instead being \(m'\). Such a reform project could
be a tax reform. In the example that we will analyze at the end of this
capsule the reform project will involve the introduction of a tax on
petrol.
Approximate duration: 30 minutes
The \textbf{equivalent variation} of a reform project transforming (p,m)
into (p',m') is a change in income that the consumer would be
indifferent about accepting in lieu of the price and income change
induced by the project. In other words, the equivalent variation is the
unique amount of money, denoted by \(\text{EV}\), such that not having
the reform project and instead receiving the transfer payment
\(\text{EV}\) results in the same utility for the consumer as not having
this transfer payment but having the project being implemented. Of
course if \(\text{EV}\) is negative this means that taking away the
amount \(|\text{EV}|\) from the consumer results in the same utility for
the consumer as not having this transfer payment but having the project
being implemented.
Let us denote by \(e(p,u)\) the expenditure function, which gives the
money that the consumer requires in order to be able to reach utility
\(u\). The definition of EV means that if we give the consumer the
amount EV and don't go ahead with the project then he will be able to
reach the same utility level as with the project, i.e. when facing
\((p^{'},m^{'})\). This is equivalent to saying that \(m + EV\) is equal
to the expenditure required to attain the utility \(v(p',m')\). Hence we
have:
\[EV(p,p^{'},m,m^{'}) = e\left( p,v(p',m') \right) - m\]
The \textbf{compensating variation} \(\text{CV}\) of a project
transforming (p,m) into (p',m') is defined to be the amount of money
that if this amount is taken away from the consumer and the project is
implemented then the consumer is indifferent between this situation and
the situation where the project is not implemented and no money is taken
away from the consumer. Of course if \(\text{CV}\) is negative this
means that giving \(|\text{CV}|\) to the consumer whilst having the
project being implemented leaves the consumer indifferent relative to
the situation without the project and without any transfers.
We have that the expenditure required und the new prices \(p'\) to reach
the original utility level \(v(p,m)\) is equal to \(m^{'} - CV\). Hence
we have:
\[e\left( p^{'},v\left( p,m \right) \right) = m^{'} - CV(p,p^{'},m,m^{'})\]
\[\text{CV}(p,p^{'},m,m^{'}) = m^{'} - e\left( p^{'},v\left( p,m \right) \right)\]
References: Varian, Microeconomic Analysis, Chapter 10
\emph{Question 1}
Throughout this capsule we will consider the following problem:
A consumer has quasilinear preferences over two goods~:
\[u\left( x_{1},x_{2} \right) = f\left( x_{1} \right) + x_{2}\]
The consumer maximizes his utility under the budget constraint
\[p_{1}x_{1} + p_{2}x_{2} \leq y\]
Suppose for \((p,y)\) we have an interior solution, as illustrated in
the following diagram:\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image1/image1}
\caption{{Couldn't find a caption, edit here to supply one.%
}}
\end{center}
\end{figure}
Suppose that the corresponding utility is \(u\). What can we infer?
True If \(y^{'} > y\) then \(x_{1}\left( p,y \right) = x_{1}(p,y^{'})\)
False If \(y' > y\) then \(x_{1}\left( p,y \right) > x_{1}(p,y^{'})\)
True If \(u' > u\) then \(h_{1}\left( p,u \right) = h_{1}(p,u^{'})\)
False If \(u' > u\) then \(h_{1}\left( p,u \right) > h_{1}(p,u^{'})\)
\emph{Question 3}
What do we know about \(h_{1}\) from capsule `The Hicksian demand
function'?
True: \(h_{1}\left( p,u \right)\) is decreasing in \(p_{1}\).
False: \(h_{1}\left( p,u \right)\) might not always be decreasing in
\(p_{1}\).
\emph{Question 4}
A useful result is that
\(\text{EV}\left( p_{1},p_{2},p_{1}^{'},\ p_{2},m,m \right) = - \int_{p_{1}}^{p_{1}'}{h_{1}\left( \tilde{p_{1}},p_{2},v(p_{1}^{'},p_{2},m) \right)}d\tilde{p_{1}}\).
It means that the EV for a reform project that only changes the price of good 1, whilst leaving the price of the other good and the consumer's available money unchanged can be calculated as the area under the Hicksian
demand curve with utility level equal to that attained at the new
prices, \(\left( p_{1}^{'},p_{2} \right)\).
In order to prove this result, which of the following two approaches is
fruitful?
False: Differentiating
\(EV\left( p_{1},p_{2},p_{1}^{'},\ p_{2},m,m \right) = e\left( p_{1},p_{2},v\left( p_{1}^{'},p_{2},m \right) \right) - m\)
with respect to \(p_{1}^{'}\)
True: Differentiating
\(EV(p_{1},p_{2},p_{1}^{'},\ p_{2},m,m) = e\left( p_{1},p_{2},v(p_{1}^{'},p_{2},m) \right) - m\)
with respect to \(p_{1}\)
Explanation:
Suppose \(p_{1}\) is changed to \(p_{1}'\), whilst \(m\) and \(p_{2}\)
are left unchanged. One way to write down the equivalent variation
\[V(p_{1},p_{2},p_{1}^{'},\ p_{2},m,m) = e\left( p_{1},p_{2},v(p_{1}^{'},p_{2},m) \right) - m\]
Now let us differentiate with respect to \(p_{1}\):
\[\frac{\partial}{\partial p_{1}}EV(p_{1},p_{2},p_{1}^{'},\ p_{2},m,m) = \frac{\partial}{\partial p_{1}}e\left( p_{1},p_{2},v(p_{1}^{'},p_{2},m) \right)\]
This is of course true for any value of \(p_{1}\), not just our
particular value \(p_{2}\) in the situation before the reform project.
To remind us of this fact let us write:
\[\frac{\partial}{\partial p_{1}}EV(\tilde{p_{1}},p_{2},p_{1}^{'},\ p_{2},m,m) = \frac{\partial}{\partial p_{1}}e\left( \tilde{p_{1}},p_{2},v(p_{1}^{'},p_{2},m) \right)\]
Now using Sheppard's theorem, we get:
\[\frac{\partial}{\partial p_{1}}EV(\tilde{p_{1}},p_{2},p_{1}^{'},\ p_{2},m,m) = h_{1}\left( \tilde{p_{1}},p_{2},v(p_{1}^{'},p_{2},m) \right)\]
Now integrating this with respect to \(\tilde{p_{1}}\) from \(p_{1}\) to
\(p_{1}^{'}\) gives:
\[\int_{p_{1}}^{p_{1}'}{\frac{\partial}{\partial p_{1}}EV(\tilde{p_{1}},p_{2},p_{1}^{'},\ p_{2},m,m)}d\tilde{p_{1}} = \int_{p_{1}}^{p_{1}'}{h_{1}\left( \tilde{p_{1}},p_{2},v(p_{1}^{'},p_{2},m) \right)}d\tilde{p_{1}}\]
By the fundamental theorem of calculus we have
\(\int_{p_{1}}^{p_{1}'}{\frac{\partial}{\partial p_{1}}EV(\tilde{p_{1}},p_{2},p_{1}^{'},\ p_{2},m,m)}d\tilde{p_{1}} = EV\left( p_{1}^{'},p_{2},p_{1}^{'},\ p_{2},m,m \right) - EV(p_{1},p_{2},p_{1}^{'},\ p_{2},m,m)\)
But
\(\text{EV}\left( p_{1}^{'},p_{2},p_{1}^{'},\ p_{2},m,m \right) = 0\),
so we obtain
\(\int_{p_{1}}^{p_{1}'}{\frac{\partial}{\partial p_{1}}EV(\tilde{p_{1}},p_{2},p_{1}^{'},\ p_{2},m,m)}d\tilde{p_{1}} = - EV(p_{1},p_{2},p_{1}^{'},\ p_{2},m,m)\),
hence we deduce:
\[\text{EV}\left( p_{1},p_{2},p_{1}^{'},\ p_{2},m,m \right) = - \int_{p_{1}}^{p_{1}'}{h_{1}\left( \tilde{p_{1}},p_{2},v(p_{1}^{'},p_{2},m) \right)}d\tilde{p_{1}}\]
\emph{Question 5}
It is a useful result that
\(\text{CV}\left( p_{1},p_{2},p_{1}^{'},\ p_{2},m,m \right) = - \int_{p_{1}}^{p_{1}'}{h_{1}\left( \tilde{p_{1}},p_{2},v(p_{1},p_{2},m) \right)}d\tilde{p_{1}}\).
It means that the CV can be calculated as the area under the Hicksian
demand curve with utility level equal to that attained at the initial
prices, \(\left( p_{1},p_{2} \right)\).
In order to prove this result, which of the following two approaches is
fruitful?
True: Differentiating
\(CV\left( p_{1},p_{2},p_{1}^{'},\ p_{2},m,m \right) = m - e\left( p_{1}',p_{2},v(p_{1},p_{2},m) \right)\)
with respect to \(p_{1}^{'}\)
False: Differentiating
\(CV\left( p_{1},p_{2},p_{1}^{'},\ p_{2},m,m \right) = m - e\left( p_{1}',p_{2},v(p_{1},p_{2},m) \right)\)
with respect to \(p_{1}\)
Explanation:
Suppose \(p_{1}\) is changed to \(p_{1}'\), whilst \(m\) and \(p_{2}\)
are left unchanged. One way to write down the compensating variation is
\[CV\left( p_{1},p_{2},p_{1}^{'},\ p_{2},m,m \right) = m - e\left( p_{1}',p_{2},v(p_{1},p_{2},m) \right)\]
Now let us differentiate with respect to \(p_{1}'\):
\[\frac{\partial}{\partial p_{1}'}CV(p_{1},p_{2},p_{1}^{'},\ p_{2},m,m) =- \frac{\partial}{\partial p_{1}'}e\left( p_{1}',p_{2},v(p_{1},p_{2},m) \right)\]
This is of course true for any value of \(p_{1}'\), not just our
particular value \(p_{2}'\) in the situation after the reform project.
To remind us of this fact let us write:
\[\frac{\partial}{\partial \tilde{p_{1}}}CV(p_{1},p_{2},\tilde{p_{1}},\ p_{2},m,m) = -\frac{\partial}{\partial \tilde{p_{1}}}e\left( \tilde{p_{1}},p_{2},v(p_{1},p_{2},m) \right)\]
Now using Sheppard's Lemma, we get:
\[\frac{\partial}{\partial p_{1}'}CV(p_{1},p_{2},\tilde{p_{1}},\ p_{2},m,m) = -h_{1}\left( \tilde{p_{1}},p_{2},v(p_{1},p_{2},m) \right)\]
Now integrating this with respect to \(\tilde{p_{1}}\) from \(p_{1}\) to
\(p_{1}^{'}\) gives:
\[\int_{p_{1}}^{p_{1}'}{\frac{\partial}{\partial \tilde{p_{1}}} CV(p_{1},p_{2},\tilde{p_{1}},\ p_{2},m,m)}d\tilde{p_{1}} = \int_{p_{1}}^{\tilde{p_{1}}}{h_{1}\left( \tilde{p_{1}},p_{2},v(p_{1},p_{2},m) \right)}d\tilde{p_{1}}\]
By the fundamental theorem of calculus we have
\(\int_{p_{1}}^{p_{1}'}{\frac{\partial}{\partial \tilde{p_{1}}}CV(p_{1},p_{2},\tilde{p_{1}},\ p_{2},m,m)}d\tilde{p_{1}} = CV\left( p_{1},p_{2},p_{1}^{'},\ p_{2},m,m \right) - CV(p_{1},p_{2},p_{1},\ p_{2},m,m)\)
But
\(CV\left( p_{1},p_{2},p_{1},\ p_{2},m,m \right) = 0\),
so we obtain
\(\int_{p_{1}}^{p_{1}'}{\frac{\partial}{\partial \tilde{p_{1}}}CV(p_{1},p_{2},\tilde{p_{1}},\ p_{2},m,m)}d\tilde{p_{1}} = CV(p_{1},p_{2},p_{1}',\ p_{2},m,m)\),
hence we deduce:
\[CV\left( p_{1},p_{2},p_{1}^{'},\ p_{2},m,m \right) = - \int_{p_{1}}^{p_{1}'}{h_{1}\left( \tilde{p_{1}},p_{2},v(p_{1},p_{2},m) \right)}d\tilde{p_{1}}\]
As a consistency check for this result, consider what we obtain if
\(p_{1}^{'} > p_{1}\). In this case the right hand side is negative.
Increasing a price has a negative effect on an agent's welfare. Going
back to the definition of the CV as the `the amount of money that if
this amount is taken away from the consumer and the project is
implemented then the consumer is indifferent between this situation and
the situation where the project is not implemented and no money is taken
away from the consumer', we note that the CV for a price increase must
be negative: the agent has to paid if he is to be made indifferent.
Hence the signs in the equation
\(CV\left( p_{1},p_{2},p_{1}^{'},\ p_{2},m,m \right) = - \int_{p_{1}}^{p_{1}'}{h_{1}\left( \tilde{p_{1}},p_{2},v(p_{1},p_{2},m) \right)}d\tilde{p_{1}}\)
make sense.
\emph{Question 6}
Consider again the case where the consumer has quasilinear preferences:
\[u\left( x_{1},x_{2} \right) = f\left( x_{1} \right) + x_{2}\]
We have seen that in this case we have
\(h_{1}\left( p,u \right) = h_{1}(p,u^{'})\).
What can we deduce from that?
True:
\(EV\left( p_{1},p_{2},p_{1}^{'},\ p_{2},m,m \right) = CV\left( p_{1}^{'},p_{2},p_{1}^{'},\ p_{2},m,m \right)\)
False:
\(EV\left( p_{1},p_{2},p_{1}^{'},\ p_{2},m,m \right) > CV\left( p_{1}^{'},p_{2},p_{1}^{'},\ p_{2},m,m \right)\)
False:
\(EV\left( p_{1},p_{2},p_{1}^{'},\ p_{2},m,m \right) < CV\left( p_{1}^{'},p_{2},p_{1}^{'},\ p_{2},m,m \right)\)
Explanation
In the formulas
\[EV\left( p_{1},p_{2},p_{1}^{'},\ p_{2},m,m \right) = - \int_{p_{1}}^{p_{1}'}{h_{1}\left( \tilde{p_{1}},p_{2},v(p_{1}^{'},p_{2},m) \right)}d\tilde{p_{1}}\]
and
\[CV\left( p_{1}^{'},p_{2},p_{1}^{'},\ p_{2},m,m \right) = - \int_{p_{1}}^{p_{1}'}{h_{1}\left( \tilde{p_{1}},p_{2},v(p_{1},p_{2},m) \right)}d\tilde{p_{1}}\]
The only difference on the right hand side is the utility level at which
the Hicksian demand is evaluated. But
\(h_{1}\left( p,u \right) = h_{1}(p,u^{'})\) says precisely that this
utility level does not matter. Hence the two expressions are equal.
\emph{Question 7}
Let us continue considering the case where the consumer has quasilinear
preferences:
\[u\left( x_{1},x_{2} \right) = f\left( x_{1} \right) + x_{2}\]
We have seen that (assuming that all the income and utility levels
\(y,y^{'},u,u'\) are sufficiently high that we have an interior
solution) the following two conditions hold:
\[h_{1}\left( p,u \right) = h_{1}(p,u^{'})\]
\[x_{1}\left( p,y \right) = x_{1}(p,y^{'})\]
These results allow us to write \(h_{1}(p)\) and call it `the demand',
without having to keep track of the consumer's available money (or even
of whether we are talking about the Hicksian or the Marshallian demand.
All this allows us to graphically represent the compensating
variation (which we know is also equal to the equivalent variation in this case). We will illustrate this for the case of a reform that
introduces a tax on petrol. We will now refer to our consumer as `the
car user'. We assume that the only effect of this reform is to increase
the price of petrol. In particular, we assume that the reform does not
affect the consumer's income or the prices of other goods. We assume
that there are two goods, good 1 being petrol, with utility given by
\(u\left( x_{1},x_{2} \right) = f\left( x_{1} \right) + x_{2}\).
The price of petrol prior to the introduction of the tax is normalized
to 1, the price after the introduction of the tax is \(1 + z\). In the
graph below two different demand functions are depicted, in dark green
and in red.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image2/image2}
\caption{{Couldn't find a caption, edit here to supply one.%
}}
\end{center}
\end{figure}
Suppose the actual demand function corresponds to the dark green line.
How can we find the equivalent variation and the compensating variation
in the graph?
True: the equivalent variation is given by the area ACDE, where A and C
are connected via the dark green line
True: the compensating variation is given by the area ACDE, where A and
C are connected via the dark green line
False: the equivalent variation is given by the area ACF, where A and C
are connected via the dark green line
False: the compensating variation is given by the area ACF, where A and
C are connected via the dark green line
\emph{Question 8}
Let us now interpret the graph:
To aid interpretation, you can drag and drop the items into the two
categories:
Category 1: the Marshallian demand function is given by the dark green
line
This corresponds to a case where substituting away from petrol turns out
to be relatively easy for car users.
The consumer must receive relatively small transfer payments in order to
be made as well off as without the tax.
Category 2: the Marshallian demand function is given by the red line
This corresponds to a case where substituting away from petrol turns out
to be relatively difficult for car users
The consumer must receive relatively large transfer payments in order to
be made as well off as without the tax.
Explanation:
When the Marshallian demand function is given by the red line, even a
substantial increase in the tax rate starting from 0 leads the car user
to hardly reduce his petrol consumption, as shown by the fact that the
red line is quite steep near the point \(A\). This reveals that the car
driver values continuing to consume petrol consumption. Substituting
away from using cars must be difficult, given that the car user is
willing to pay a high price to continue consuming petrol at similar
levels as before the tax reform. This is fact is picked up by the
compensating and equivalent variations, as the graph illustrates: The
area ACDE enclosed by the red line is relatively large.
When the Marshallian demand function is given by the dark green line,
even a small increase in the tax rate starting from 0 leads the car user
to substantially reduce his petrol consumption, as shown by the fact
that the green line is quite flat near the point \(A\). This reveals
that the car driver is not that bothered after all about reducing his
petrol consumption. This is fact is picked up by the compensating and
equivalent variations, as the graph illustrates: The area ACDE enclosed
by the dark green line is relatively small.
Recap:
For a consumer with quasilinear preferences over two goods~:
\[u\left( x_{1},x_{2} \right) = f\left( x_{1} \right) + x_{2}\]
The equivalent variation and the compensating variation for a reform
project consisting of an increase in the price \(p_{1}\) of good 1 are
equal. We have seen that such reform projects can be illustrated
graphically in an intuitive way.
\selectlanguage{english}
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\end{document}