Download Problem set 9

Instituto Tecnológico Autónomo de México
Maestría en Teoría Económica / Licenciatura en Economía
Consumer and Producer Theory
Problem set 9, Fall 2013
Ricard Torres
1. Let X = RL
+ be the consumption set, and let u : X → R be a continuous utility function that is
continuously differentiable on the domain int(X) = RL
++ of consumption vectors with strictly positive
components. Suppose that ∂u/∂xi (x) > 0 for all 1 ≤ i ≤ L, and for all x ∈ int(X). For 1 ≤ i, j ≤ L,
define the marginal rate of substitution between i and j as:
MRSij (x) =
∂u/∂xi
(x),
∂u/∂xj
for x ∈ int(X).
(i) Show that the preferences are strictly monotonic.
(ii) Using the implicit function theorem, show that, for a given x̂ ∈ int(X) and 1 ≤ j ≤ L, the relation
u(x) = u(x̂) defines xj as a continuously differentiable function of the remaining components
(xi )i̸=j around the point x̂. Next show that the partial derivative of this function with respect
to xi evaluated at x̂ is MRSij (x̂).
(iii) Using discrete approximations, justify that MRSij (x) measures the value of the ith good in terms
of the jth good at the point x̂.
(iv) Let u(X) = A ⊂ R, and let f : A → R be a strictly increasing and continuously differentiable
function. Define v : X → R as v(x) = f [u(x)]. Show that, for any x ∈ int(X), MRSij (x) is the
same no matter whether we compute it from u or from v.
(v) Consider the utility maximization problem that corresponds to (p, w) ≫ 0. Suppose that, at the
optimum x∗ , x∗i > 0 and x∗j > 0. Show that MRSij (x∗ ) = pi /pj . Interpret it in the sense that
the subjective value of good i in terms of good j equals the market value of good i in terms of
good j.
(vi) Suppose that, for all 1 ≤ i ≤ L and for all x ∈ int(X),
lim
xi →0
∂u
(x) = +∞.
∂xi
Show that the solution to the utility maximization problem that corresponds to (p, w) ≫ 0 lies
always in int(X).
2.
(JR) Assume that v(p, w) is continuously differentiable, for (p, w) ≫ 0. Assume x0 solves the
utility maximization problem given (p0 , w0 ). For p ≫ 0, define g(p) = v(p, p · x0 ).
(i) Show that p0 minimizes g(p). Since this is an interior minimum, its partial derivatives must
equal zero.
(ii) Show that the above step results in Roy’s identity.
(iii) Interpret.
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3. (JR) Assume that the utility function is strictly monotonic and continuously differentiable, and
that e(p, u) is continuously differentiable, for p ≫ 0 and u > u(0). Assume x0 solves the expenditure
minimization problem given (p0 , u0 ). For p ≫ 0, define g(p) = p · x0 − e(p, u0 ).
(i) Show that p0 minimizes g(p). Since this is an interior minimum, its partial derivatives must
equal zero.
(ii) Show that the above step results in Shepard’s Lemma.
(iii) Interpret.
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