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\begin{document}
\title{Indirect utility functions version 2.0 (taken in part from Varian,
exercise 7.4)}
\author[1]{Lennart Stern}%
\affil[1]{École normale supérieure}%
\vspace{-1em}
\date{\today}
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Description: This is a simple exercise to practice applying Roy's
identity and thinking about how much information is encoded in the
indirect utility function.
Approximate duration: 20 minutes
Resource:
Consider a consumer with utility function \(u(x)\) over consumption
bundles \(x\). Suppose he has available money \(m\) and is faced with
the price vector \(p\). We denote by \(x(p,m)\) the solution to the
problem of maximizing \(u(x)\) subject to the budget constraint
\(x\ p \leq m\), (where \(xp = x_{1}p_{1} + \ldots + x_{n}p_{n}\)
denotes the dot product of the consumption bundle \(x\) and the price
vector \(p\)). We call \(x(p,m)\) the Marshallian demand function.
The indirect utility function \(v(p,m)\) is defined by
\(v\left( p,m \right) = u\left( x\left( p,m \right) \right).\)
Suppose the consumer with strictly monotonic preferences is faced with
\((p,m)\) and chooses an optimal \(x\). From strict monotonicity we know
that the budget constraint is binding, i.e. \(px = m\). Now suppose the
price \(p_{i}\) of good \(i\) is slightly increased by \(dp_{i}\).
Suppose for a moment that the consumer stubbornly sticks to his choice
of \(x\) that was feasible and optimal before the price increase. Then
he will find himself short in money, namely by \(dp_{i}\ x_{i}\). Having
to respect his budget constraint, he will now have to cut back in
spending by \(dp_{i}\ x_{i}\). We can ask: How different is this in its
effect on the consumer's utility from the scenario where instead of the
price increase, the consumer's money is decreased by \(dp_{i}\ x_{i}\)?
Roy's identity tells us that for sufficiently small \(dp_{i}\) the two
changes have the same effect on the consumer's utility. Formally, we
have:
\[\frac{\partial v}{\partial m}x_{i}\left( p,m \right) = \frac{\partial v}{\partial p_{i}}\]
Rearranging this gives us the classical statement of Roy's identity,
which gives us a way to compute the Marshallian demand function from the
indirect utility function:
\[x_{i}\left( p,m \right) = \frac{\frac{\partial v}{\partial p_{i}}}{\frac{\partial v}{\partial m}}\]
Roy's identity can be proved using the envelope theorem.
Sheppard's Lemma states that:
\[x_{i}\left( p,v \right) = \frac{\partial}{\partial p_{i}}e(p,v)\]
Where \(h_{i}\) is the Hicksian demand function for good \(i\).
References: Varian, Microeconomic Analysis, Chapter 7
\emph{Question} 1:
Consider the indirect utility function given by
\(v\left( p_{1},p_{2},m \right) = \frac{m}{p_{1} + p_{2}}\)
What are the demand functions?
True:\(\ x_{1}\left( p,m \right) = \frac{m}{p_{1} + p_{2}}\)
True:\(\ x_{2}\left( p,m \right) = \frac{m}{p_{1} + p_{2}}\)
False:\(\ x_{1}\left( p,m \right) = \frac{m}{p_{1}}\)
False:\(\ x_{2}\left( p,m \right) = \frac{m}{p_{2}}\)
Explanation:
Applying Roy's identity,
\(x_{i}\left( p,m \right) = - \frac{\frac{\partial v}{\partial p_{i}}}{\frac{\partial v}{\partial m}}\),
we obtain:
\[x_{1}\left( p,m \right) = - \frac{\frac{- m}{\left( p_{1} + p_{2} \right)^{2}}}{\frac{1}{p_{1} + p_{2}}} = \frac{m}{p_{1} + p_{2}}\]
\[x_{2}\left( p,m \right) = \frac{m}{p_{1} + p_{2}}\]
\emph{Question 2: }
Continue considering the indirect utility function given by
\(v\left( p_{1},p_{2},m \right) = \frac{m}{p_{1} + p_{2}}\). What is
true about the expenditure function?
True:
\(e\left( p,v \right) = h_{1}\left( p,v \right)p_{1} + h_{2}\left( p,v \right)p_{2}\)
True: \(e\left( p,v \right) = v\left( p_{1} + p_{2} \right)\)
Explanation:
As a general method, we can solve \(v = \frac{m}{p_{1} + p_{2}}\) for
\(m\) and have thus the expenditure as a function of the utility and
prices. In this case it gives us
\(e\left( p,v \right) = v\left( p_{1} + p_{2} \right).\)
\(e\left( p,v \right) = h_{1}\left( p,v \right)p_{1} + h_{2}\left( p,v \right)p_{2}\)
is always true, since
\((h_{1}\left( p,v \right),h_{2}\left( p,v \right))\) is a solution to
the expenditure minimization problem.
\emph{Question 3: }
Continue considering the indirect utility function given by
\(v\left( p_{1},p_{2},m \right) = \frac{m}{p_{1} + p_{2}}\). What is the
Hicksian demand function?
True: \(h_{1}\left( p,v \right) = v\)
False: we cannot know
False: \(h_{1}\left( p,v \right) = p_{1} v \)
Explanation:
We can apply Shephard's Lemma to
\(e\left( p,v \right) = v\left( p_{1} + p_{2} \right)\) and obtain:
\[ h_{1} = \frac{\partial e}{\partial p_{1}} = v \]
\emph{Question 4: }
Continue considering the indirect utility function given by
\(v\left( p_{1},p_{2},m \right) = \frac{m}{p_{1} + p_{2}}\). What is the
set of points \((x_{1},x_{2})\) that are ever chosen for some
\((p_{1},p_{2},m)\)?
True: a straight line through the origin
False: the set of points \((x_{1},x_{2})\) with \(x_{1} \geq 0\) and
\(x_{2} \geq 0\).
Explanation:
From the Marshallian demand functions
\(x_{1}\left( p,m \right) = \frac{m}{p_{1} + p_{2}},\ x_{2}\left( p,m \right) = \frac{m}{p_{1} + p_{2}}\)
we see that we always have \(x_{1} = x_{2}\).
\emph{Question 5: }
Which of the following direct utility functions is consistent with the
indirect utility function
\(v\left( p_{1},p_{2},m \right) = \frac{m}{p_{1} + p_{2}}\)?
False: \(u\left( x_{1},x_{2} \right) = x_{1}\)
True: \(u\left( x_{1},x_{2} \right) = max(x_{1},x_{2})\)
False:
\(u\left( x_{1},x_{2} \right) = \left( \max\left( x_{1},x_{2} \right) \right)^{2}\)
True:
\(u\left( x_{1},x_{2} \right) = \max\left( x_{1},x_{2} \right) + \frac{1}{2}|x_{1} - x_{2}|\)
Explanation:
It might be tempting to take
\(v\left( p_{1},p_{2},m \right) = \frac{m}{p_{1} + p_{2}}\) and try to
eliminate \(m,\) \(p_{1}\) and \(p_{2}\) using that
\(x_{1}\left( p,m \right) = \frac{m}{p_{1} + p_{2}}\) and that
\(x_{2}\left( p,m \right) = \frac{m}{p_{1} + p_{2}}\). From this we will
obtain the relations \(u = x_{1}\) and \(u = x_{2}\). These equalities
must indeed hold on the set of points that are ever chosen. However, we
cannot conclude that these equalities hold on points that are never
chosen. Indeed, there are infinitely many preferences that are
consistent with the indirect utility function
\(v\left( p_{1},p_{2},m \right) = \frac{m}{p_{1} + p_{2}}\). For
example, the indifference curves corresponding to the utility function
\(u\left( x_{1},x_{2} \right) = \min\left( x_{1},x_{2} \right) - \frac{1}{2}|x_{1} - x_{2}|\)
are depicted in the diagram below, where the black lines draw out some
indifference curves and the bluer regions correspond to points ranking
lower in the preference ordering.\selectlanguage{english}
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Graphically, we see that that these preferences lead the consumer to
always choose \(x_{1} = x_{2}\). The same holds for the utility
functions
\(u\left( x_{1},x_{2} \right) = \min\left( x_{1},x_{2} \right)\) and
\(u\left( x_{1},x_{2} \right) = \left( \min\left( x_{1},x_{2} \right) \right)^{2}\).
Thus all of these three utility functions yield the right Marshallian
demand function. However, this is not necessarily enough for being
consistent with the indirect utility function
\(v\left( p_{1},p_{2},m \right) = \frac{m}{p_{1} + p_{2}}\). In
addition, we need that \(u\left( x_{1},x_{2} \right)\) evaluated at the
Marshallian demand
\(\left( x_{1}(p_{1},p_{2},m),x_{2}(p_{1},p_{2},m) \right)\) is actually
equal to \(\frac{m}{p_{1} + p_{2}}\), i.e. we need that
\(u\left( x_{1}(p_{1},p_{2},m),x_{2}(p_{1},p_{2},m) \right) = \frac{m}{p_{1} + p_{2}}\).
We thus compute
For \(u\left( x_{1},x_{2} \right) = \min\left( x_{1},x_{2} \right)\) we
have
\(u\left( x_{1}(p_{1},p_{2},m),x_{2}(p_{1},p_{2},m) \right) = \min\left( \frac{m}{p_{1} + p_{2}},\frac{m}{p_{1} + p_{2}} \right) = \frac{m}{p_{1} + p_{2}}\)
For
\(u\left( x_{1},x_{2} \right) = \min{\left( x_{1},x_{2} \right) - \frac{1}{2}|x_{1} - x_{2}|}\)
we have
\(u\left( x_{1}(p_{1},p_{2},m),x_{2}(p_{1},p_{2},m) \right) = \min\left( \frac{m}{p_{1} + p_{2}},\frac{m}{p_{1} + p_{2}} \right) - \frac{1}{2}|\frac{m}{p_{1} + p_{2}} - \frac{m}{p_{1} + p_{2}}| = \frac{m}{p_{1} + p_{2}}\)
For
\(u\left( x_{1},x_{2} \right) = \left( \min\left( x_{1},x_{2} \right) \right)^{2}\)
we have
\(u\left( x_{1}(p_{1},p_{2},m),x_{2}(p_{1},p_{2},m) \right) = \left( \min\left( \frac{m}{p_{1} + p_{2}},\frac{m}{p_{1} + p_{2}} \right) \right)^{2} = \left( \frac{m}{p_{1} + p_{2}} \right)^{2}\)
Hence both
\(u\left( x_{1},x_{2} \right) = \min\left( x_{1},x_{2} \right)\) and
\(u\left( x_{1},x_{2} \right) = \min\left( x_{1},x_{2} \right) + \frac{1}{2}|x_{1} - x_{2}|\)
are consistent with
\(v\left( p_{1},p_{2},m \right) = \frac{m}{p_{1} + p_{2}}\) but
\(u\left( x_{1},x_{2} \right) = \left( \min\left( x_{1},x_{2} \right) \right)^{2}\)
is not.
Recap:
We can use Roy's identity to deduce the Marshallian demand functions
from the indirect utility function. On the domain of consumption bundles
that are chosen for some \((p,m)\) we can recover the direct utility
function.
The indirect utility function may not allow us to infer the preferences
on the entire set of consumption bundles.
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