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ECON30020 Mathematical Economics

创建日期：2024-05-28 16:45:43

所属分类：经济Economics

标签： ECON30020

Mathematical Economics

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ECON30020 Mathematical Economics
Assignment 3. Constrained optimization and envelope theorems
Problem 1. Constrained optimization
For Q ∈ [0, 1], let
P (Q) =
1− 83Q, if Q ∈ [0, 1/4)4
9(1−Q) if Q ∈ [1/4, 1]
(1)
and
R(Q) = P (Q)Q.
Fix Q = 1/4 and solve the problem
max
Q1,Q2α
(1− α)R(Q1) + αR(Q2)
subject to Q1 ≤ Q, Q2 ≥ Q, α ∈ [0, 1] and
(1− α)Q1 + αQ2 = Q.
Which solution values vary with Q and which ones would be the same for, say, Q = 1/3?
Hints: i) The constraints Q1 ≤ Q and Q2 ≥ Q will be slack, so you may neglect them in your
approach; then you just need to verify that they are indeed slack at your solution values. ii) It
may be simpler and clearer to work with general expression R when setting up and solving the
Lagrangian, and to make use of the specific functional form only at the end when expressing
the optimal values.
Problem 2. Envelope theorems
Consider a monopoly facing the inverse demand function P (Q) = 1 − Q for Q ∈ [0, 1] that
has a constant marginal cost c ∈ (0, 1). Given Q and c, the monopoly’s profit is Π(Q, c) =
(P (Q)− c)Q. Denote the maximizer for given c by Q∗(c) and partial derivatives by subscripts.
Let pi(c) = maxQ Π(Q, c) = Π(Q
∗(c), c).
a) Show that the first-order condition Π1(Q
∗(c), c) = 0 determines Q∗(c).
b) Use the fact thatQ∗(c) is given by a first-order condition to establish that pi′(c) = Π2(Q∗(c), c).
Verify that this is the case by computing Π(Q∗(c), c) and differentiating it with respect to c..
2023 1 Page 1 of 2
ECON30020 Mathematical Economics Assignment 3.CO and ET
c) Argue now that differentiability in Q is not material for the fact that pi′(c) = Π2(Q∗(c), c)
by showing the payoff function Π(Q, c) satisfies all the relevant properties of the Milgrom-Segal
envelope theorem (Theorem 3 in L6).

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