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\begin{document}

\title{Equivalent and compensating variations for an example with
quasilinear utility}



\author[1]{Lennart Stern}%
\affil[1]{École normale supérieure}%


\vspace{-1em}



  
  \date{May 08, 2024}


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Description: In this capsule we will study a consumer with utility
\(u\left( x_{1},x_{2} \right) = \sqrt{x_{1}} + \ x_{2}\). We will
compute equivalent variations and compensating variations and see when
they are equal.

Approximate duration: 25 minutes

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
result 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 throughout this
capsule the reform project will be a trade deal.

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 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
\(EV\) is negative this means that taking away the amount
\(|EV|\) 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:

\[ V(p,p^{'},m,m^{'}) = e\left( p,v(p',m') \right) - m \]

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:

\[ \left( p^{'},v\left( p,m \right) \right) = m^{'} - CV(p,p^{'},m,m^{'}) \]

\[ CV(p,p^{'},m,m^{'}) = m^{'} - e\left( p^{'},v\left( p,m \right) \right) \]

Sheppard's Lemma states that:

\[ h_{i}\left( p,v \right) = \frac{\partial}{\partial p_{i}}e(p,v) \]

Where \(h_{i}\) is the Hicksian demand function for good \(i\).

References: Varian, Microeconomic Analysis, Chapter 10\subsubsection{}

\emph{Question 1}

Throughout this set of exercises we will consider a consumer with
preferences represented by the utility function
\(u\left( x_{1},x_{2} \right) = \sqrt{x_{1}} + \ x_{2}\).

Suppose we were to plot the quantity of good 1 on the x-axis and the
quantity of good 2 on the y-axis. What can we say about the indifference
curves?

True: For any \(z,z'\) the indifference curve for
\(u\left( x_{1},x_{2} \right) = z\) is a vertical translation of the
indifference curve for \(u\left( x_{1},x_{2} \right) = z'\).

false: For any \(z,z'\) the indifference curve for
\(u\left( x_{1},x_{2} \right) = z\) is a horizontal translation of the
indifference curve for \(u\left( x_{1},x_{2} \right) = z'\).

\emph{Question 2}

The consumer is faced with prices \(p_{1},p_{2}\) and has available
money \(m\). Which statements corresponds to which graphs?

Graph 1 Graph 2\selectlanguage{english}
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Graph 1:

Corner solution

\(x_{2} = 0\)

An additional dollar of disposable money is spent entirely on good \(1\)

At optimal choice:
\(\frac{\frac{\partial u}{\partial x_{1}}}{\frac{\partial u}{\partial x_{2}}} > \frac{p_{1}}{p_{2}}\)

\(\frac{p_{2}^{2}}{4p_{1}} \geq m\)

Graph 2

Interior solution

\(x_{2} > 0\)

An additional dollar of disposable money is spent entirely on good \(2\)

At optimal choice:
\(\frac{\frac{\partial u}{\partial x_{1}}}{\frac{\partial u}{\partial x_{2}}} = \frac{p_{1}}{p_{2}}\)

\(\frac{p_{2}^{2}}{4p_{1}} \leq m\)

Explanation:

\[\frac{\frac{\partial u}{\partial x_{1}}}{\frac{\partial u}{\partial x_{2}}} = \frac{p_{1}}{p_{2}}\]

Computing
\(\frac{\frac{\partial u}{\partial x_{1}}}{\frac{\partial u}{\partial x_{2}}} = \frac{1}{2\sqrt{x_{1}}}\),
we deduce that we would need:

\[\frac{1}{2\sqrt{x_{1}}} = \frac{p_{1}}{p_{2}}\]

\[\frac{1}{4\ x_{1}} = \frac{p_{1}^{2}}{p_{2}^{2}}\]

\[x_{1} = \frac{p_{2}^{2}}{4p_{1}^{2}}\]

We must need that the consumer has enough money to afford this, i.e. we
need that \(p_{1}x_{1} \leq m\), so we get:

\[ \frac{p_{2}^{2}}{4p_{1}} \leq m \]

If this condition is satisfied, we do indeed have an interior solution
with the remaining money being spent on good 2:

\[x _{2} = \frac{m - \frac{p_{2}^{2}}{4p_{1}}}{p_{2}} \]

\emph{Question 3}

From now on let us assume that we always have
\(m> \frac{p_{2}^{2}}{4p_{1}}\) , i.e. suppose we are always at an
interior solution.

What are the Marhsallian demands?

True: \(x_{1} = \frac{p_{2}^{2}}{4p_{1}^{2}}\)

True: \(x_{2} = \frac{m - \frac{p_{2}^{2}}{4p_{1}}}{p_{2}}\)

False: \(x_{1} = \frac{p_{1}^{2}}{4p_{2}^{2}}\)

False: \(x_{2} = \frac{m + \frac{p_{2}^{2}}{4p_{1}}}{p_{2}}\)

Explanation:

We have already carried out the required computations in the previous
question:

\[\frac{\frac{\partial u}{\partial x_{1}}}{\frac{\partial u}{\partial x_{2}}} = \frac{p_{1}}{p_{2}}\]

Computing
\(\frac{\frac{\partial u}{\partial x_{1}}}{\frac{\partial u}{\partial x_{2}}} = \frac{1}{2\sqrt{x_{1}}}\),
we deduce:

\[\frac{1}{2\sqrt{x_{1}}} = \frac{p_{1}}{p_{2}}\]

Squaring both sides:

\[\frac{1}{4\ x_{1}} = \frac{p_{1}^{2}}{p_{2}^{2}}\]

Solving for \(x_{1}\)

\[x_{1} = \frac{p_{2}^{2}}{4p_{1}^{2}}\]

\emph{Question 4}

What is the indirect utility function?

True
\(v\left( p_{1},p_{2},m \right) = \frac{p_{2}}{4p_{1}} + \ \frac{m}{p_{2}}\)

False
\(v\left( p_{1},p_{2},m \right) = \frac{p_{2}}{2p_{1}} + \frac{m}{p_{2}}\)

Explanation:

We can plug the Marshallian demand functions into the utility function
\(u\left( x_{1},x_{2} \right) = \sqrt{x_{1}} + \ x_{2}\)

\[\left( p_{1},p_{2},m \right) = u\left( x_{1}\left( p_{1},p_{2},m \right),x_{2}\left( p_{1},p_{2},m \right) \right) = \sqrt{x_{1}\left( p_{1},p_{2},m \right)} + \ x_{2}\left( p_{1},p_{2},m \right)\]

\[v\left( p_{1},p_{2},m \right) = \frac{p_{2}}{2p_{1}} + \frac{m}{p_{2}} - \frac{p_{2}}{4p_{1}}\]

\[v\left( p_{1},p_{2},m \right) = \frac{p_{2}}{4p_{1}} + \frac{m}{p_{2}}\]

\emph{Question 5}

What is the expenditure function?

True \(e(p_{1},p_{2},m) = p_{2}\ v - \frac{p_{2}^{2}}{4p_{1}}\)

False \(e(p_{1},p_{2},m) = p_{2}\ v - \frac{p_{2}}{4p_{1}}\)

Explanation:

From the indirect utility function

\[\left( p_{1},p_{2},m \right) = \frac{p_{2}}{4p_{1}} + \ \frac{m}{p_{2}}\]

We solve for \(m\):

\[ m= p_{2}\ v - \frac{p_{2}^{2}}{4p_{1}}\]

We have thus found the expenditure function:

\[e (p_{1},p_{2},m) = p_{2}\ v - \frac{p_{2}^{2}}{4p_{1}} \]

\emph{Question 6}

What is the Hicksian demand for good 1?

True: \(h_{1}(p_{1},p_{2},u) = \frac{p_{2}^{2}}{4p_{1}^{2}}\)

False: \(h_{1}(p_{1},p_{2},u) = \frac{p_{2}}{4p_{1}^{2}}\)


Explanation:

One way to obtain this result is to apply Sheppard's Lemma using our
expression \(e(p_{1},p_{2},m) = p_{2}\ v - \frac{p_{2}^{2}}{4p_{1}}\),
which gives:

\[h_{1}\left( p_{1},p_{2},u \right) = \frac{\partial e}{\partial p_{1}} = \frac{p_{2}^{2}}{4p_{1}^{2}}\]

In our particular case we could have gotten the result also by observing
that the Marshallian demand does not depend on \(m\):
\(x_{1} = \frac{p_{2}^{2}}{4p_{1}^{2}}\).

\emph{Question 7}

What is the Hicksian demand for good 2?

True: \(h\left( p_{1},p_{2},v \right) = v - \frac{p_{2}}{2p_{1}}\)
False: \(h\left( p_{1},p_{2},v \right) = v - \frac{p_{2}}{2(p_{1}+p_{2})}\)

Explanation:

We can apply Shephard's Lemma using our expression for the expenditure
function
\(e\left( p_{1},p_{2},u \right) = p_{2}v - \frac{p_{2}^{2}}{4p_{1}}\).

\[h_{2}\left( p_{1},p_{2},v \right) = \frac{\partial e}{\partial p_{2}} = v - \frac{p_{2}}{2p_{1}}\]

Alternatively, from our expression for the expenditure function

\[m = p_{2}v - \frac{p_{2}^{2}}{4p_{1}}\]

Now we can plug this into
\(x_{2} = \frac{m}{p_{2}} - \frac{p_{2}}{4p_{1}}\) and obtain:

\[x_{2} = \frac{p_{2}v - \frac{p_{2}^{2}}{4p_{1}}}{p_{2}} - \frac{p_{2}}{4p_{1}} = v - \frac{p_{2}}{4p_{1}} - \frac{p_{2}}{4p_{1}} = v - \frac{p_{2}}{2p_{1}}\]

We have thus found demand for good 2 as a function of utility and
prices, so that we know that
\(h\left( p_{1},p_{2},v \right) = v - \frac{p_{2}}{2p_{1}}\)

\emph{Question 8}

Now suppose prices change to \(p_{1}',p_{2}'\).

Which of the following expressions gives the compensating variation and
the equivalent variation, respectively?

Connect the pairs:

True:
\(CV\left( p,p^{'},m,m^{'} \right)\) =\( m^{'} - p_{2}\ \left( \frac{p_{2}}{4p_{1}} + \frac{m}{p_{2}} \right) - \frac{p_{2}^{2}}{4p_{1}}\)

True:
\(EV\left( p,p^{'},m,m^{'} \right)\) = \(p_{2}\ (\frac{p_{2}'}{4p_{1}'} + \ \frac{m^{'}}{p_{2}'}) - \frac{p_{2}^{2}}{4p_{1}} - m\)

Explanation:

We substitute our result
\(e(p_{1},p_{2},m) = p_{2}\ v - \frac{p_{2}^{2}}{4p_{1}}\) into the
formula

\[V(p,p^{'},m,m^{'}) = e\left( p,v(p',m') \right) - m\]

And obtain:

\[V(p,p^{'},m,m^{'}) = p_{2}\ v(p_{1}^{'},p_{2}^{'},m') - \frac{p_{2}^{2}}{4p_{1}} - m\]

Now substituting in
\(v\left( p_{1}',p_{2}',m' \right) = \frac{p_{2}'}{4p_{1}'} + \frac{m'}{p_{2}'}\)
yields:

\[EV\left( p,p^{'},m,m^{'} \right) = p_{2}\ (\frac{p_{2}'}{4p_{1}'} + \frac{m^{'}}{p_{2}'}) - \frac{p_{2}^{2}}{4p_{1}} - m\]

To compute the compensating variation we substitute our result
\(e(p_{1},p_{2},m) = p_{2}\ v - \frac{p_{2}^{2}}{4p_{1}}\) into

\[CV(p,p^{'},m,m^{'}) = m^{'} - e\left( p^{'},v\left( p,m \right) \right)\]

And obtain:

\[CV(p,p^{'},m,m^{'}) = m^{'} - p_{2}\ v\left( p,m \right) - \frac{p_{2}^{2}}{4p_{1}}\]

Now substituting in
\(v\left( p_{1},p_{2},m \right) = \frac{p_{2}}{4p_{1}} + \frac{m}{p_{2}}\)
yields:

\[CV\left( p,p^{'},m,m^{'} \right) = m^{'} - p_{2}\ \left( \frac{p_{2}}{4p_{1}} + \frac{m}{p_{2}} \right) - \frac{p_{2}^{2}}{4p_{1}}\]

Here is an alternative way of proceeding:

Using that \(v\left( p_{1},p_{2},m + EV \right) = v(p_{1}',p_{2}',m')\)

\[\frac{p_{2}}{4p_{1}} + \frac{m + EV}{p_{2}} = \frac{p_{2}'}{4p_{1}'} + \ \frac{m'}{p_{2}'}\]

\[V = \frac{p_{2}p_{2}'}{4p_{1}'} + \frac{p_{2}m'}{p_{2}'} - \frac{p_{2}^{2}}{4p_{1}} - m\]

Using that
\(v\left( p_{1},p_{2},m \right) = v(p_{1}^{'},p_{2}^{'},m' - CV)\)

\[\frac{p_{2}}{4p_{1}} + \ \frac{m}{p_{2}} = \frac{p_{2}'}{4p_{1}'} + \frac{m^{'} - CV}{p_{2}'}\]

\[V = - p_{2}^{'}\frac{p_{2}}{4p_{1}} - p_{2}^{'}\frac{m}{p_{2}} + \frac{p_{2}^{'2}}{4p_{1}'} + \ m^{'}\]

\emph{Question 9}

Of the following statements exactly 1 is correct:

True: If \(p_{2} = p_{2}'\) then \(EV = CV\)

False: If \(p_{1} = p_{1}'\) then \(EV = CV\)

Explanation:

Using our previously obtained expressions, we see that \(EV = CV\) is
equivalent to:

\[ p_{2}^{'}\frac{p_{2}}{4p_{1}} - p_{2}^{'}\frac{m}{p_{2}} + \frac{p_{2}^{'2}}{4p_{1}'} + \ m^{'} = \frac{p_{2}p_{2}'}{4p_{1}'} + \ \frac{p_{2}m^{'}}{p_{2}'} - \frac{p_{2}^{2}}{4p_{1}} - m\]

Suppose \(p_{2}^{'} = p_{2}\). Then the condition for \(EV = CV\)
becomes:

\[p_{2}\frac{p_{2}}{4p_{1}} - m + \frac{p_{2}^{2}}{4p_{1}'} + \ m^{'} = \frac{p_{2}^{2}}{4p_{1}'} + \ m^{'} - \frac{p_{2}^{2}}{4p_{1}} - m\]

This simplifies to

\[ 0= 0\]

Meaning that we always have \(EV = CV\) as long as we have
\(p_{2}^{'} = p_{2}\).

\[ p_{2}^{'}\frac{p_{2}}{4p_{1}} - p_{2}^{'}\frac{m}{p_{2}} + \frac{p_{2}^{'2}}{4p_{1}} + \ m^{'} = \frac{p_{2}p_{2}'}{4p_{1}} + \ \frac{p_{2}m^{'}}{p_{2}'} - \frac{p_{2}^{2}}{4p_{1}} - m\]

Now if we also assume that \(m = m^{'}\) we obtain:

\[ p_{2}^{'}\frac{m}{p_{2}} + \frac{p_{2}^{'2}}{2p_{1}} + \ m = \frac{p_{2}p_{2}'}{2p_{1}} + \ \frac{p_{2}m}{p_{2}'} - m\]

\[(2 - \frac{p_{2}^{'}}{p_{2}} - \frac{p_{2}}{p_{2}'}) = \frac{(p_{2} - p_{2}^{'})p_{2}'}{2p_{1}}\]

\[(p_{2} - p_{2}^{'})(\frac{1}{p_{2}} - \frac{1}{p_{2}'}) = \frac{(p_{2} - p_{2}^{'})p_{2}'}{2p_{1}}\]

Clearly, if \(p_{2} \neq p_{2}^{'}\) then this is only true for one
particular value of \(m\), so it is not true in general.

\emph{Recap:} We have seen in our example with
\(u\left( x_{1},x_{2} \right) = \sqrt{x_{1}} + \ x_{2}\) that for a
change in the price of good 1 the equivalent variation is equal to the
compensating variation. Actually, this is true for any utility function
that is quasilinear in the second good, as we will see in the capsule
`Equivalent Variation, Compensating Variation and Hicksian demand'.

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