Proving the composite commodity theorem consists of showing that choices made
when we use a composite commodity are identical to those that would be made if
we specified the complete utility-maximization problem. This problem asks you
to show this using two different approaches. For both of these we assume there
are only three goods, \(x_{1}, x_{2},\) and \(x_{3}\) and that the prices of
\(x_{2}\) and \(x_{3}\) always move together-that is, \(p_{2}=t p_{2}^{0}\) and
\(p_{3}=t p_{3}^{0},\) where \(p_{2}^{0}\) and \(p_{3}^{0}\) are the initial prices
of these two goods. With this notation, the composite commodity, \(y,\) is
defined as $$y=p_{2}^{0} x_{2}+p_{3}^{0} x_{3}$$ a. Proof using duality. Let
the expenditure function for the original three-good problem be given by
\(E\left(p_{1}, p_{2}, p_{3}, \bar{U}\right)\) and consider the alternative
expenditure minimization problem; Minimize \(p_{1} x_{1}+\) tys.t.
\(U\left(x_{1}, x_{2}, x_{3}\right)=\bar{U}\). This problem will also yield an
expenditure function of the form \(E^{*}\left(p_{1}, t, \bar{U}\right)\)
i. Use the envelope theorem to show that $$\frac{\partial E}{\partial
t}=\frac{\partial E^{*}}{\partial t}=y$$ This shows that the demand for the
composite good \(y\) is the same under either approach.
ii. Explain why the demand for \(x_{1}\) is also the same under either approach?
(This proof is taken from Deaton and Muellbauer, \(1980 .)\) b. Proof using two-
stage maximization. Now consider this problem from the perspective of utility
maximization. The problem can be simplified by adopting the normalization that
\(p_{2}=1-\) that is, all purchasing power is measured in units of \(x_{2}\) and
the prices \(p_{1}\) and \(p_{3}\) are now treated as prices relative to \(p_{2}\).
Under the assumptions of the composite commodity theorem, \(p_{1}\) can vary,
but \(p_{3}\) is
a fixed value.
The original utility maximization problem is:
\\[
\text { Maximize } U\left(x_{1}, x_{2}, x_{3}\right) \text { s.t. } p_{1}
x_{1}+x_{2}+p_{3} x_{3}=M
\\]
(where \(M=I / p_{2}\) ) and the first-order conditions for a maximum are
\(U_{i}=\lambda p_{i}, i=1,3\) (where \(\lambda\) is the Lagrange multiplier). The
alternative, two-stage approach to the problem is
\\[
\text { Stage } 1: \text { Maximize } U\left(x_{1}, x_{2}, x_{3}\right) \text
{ s.t. } x_{2}+p_{3} x_{3}=m
\\]
(where \(m\) is the portion of \(M\) devoted to purchasing the composite good).
This maximization problem treats \(x_{1}\) as an exogenous parameter in the
maximization process, so it becomes an element of the value function in this
problem. The first-order conditions for this problem \(\operatorname{are}
U_{i}=\mu p_{i}\) for \(i=2,3,\) where \(\mu\) is the Lagrange multiplier for this
stage of the problem. Let the value (indirect utility) function for this stage
1 problem be given by \(V\left(x_{1}, m\right)\)The final part of this two-stage
problem is then:
\\[
\text { Stage } 2: \text { Maximize } V\left(x_{1}, m\right) \quad \text {
s.t. } p_{1} x_{1}+m=M
\\]
This will have first-order conditions of the form \(\partial V / \partial
x_{1}=\delta p_{1}\) and \(\partial V / \partial m=\delta,\) where \(\delta\) is
the Lagrange
multiplier for stage 2 Given this setup, answer the following questions:
i. Explain why the value function in stage 1 depends only on \(x_{1}\) and \(m .\)
(Hint: This is where the fact that \(p_{3}\) is constant plays a key role.)
ii. Show that the two approaches to this maximization problem yield the same
result by showing that \(\lambda=\mu=\delta .\) What do you have to assume to
ensure the results are equivalent? (This problem is adapted from Carter, \(1995
.)\)
