ECON30020 Mathematical Economics Assignment 1
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
ECON30020 Mathematical Economics
Assignment 1. Proofs, Sets and Functions
Problem 1 (25 points). Cournot marginal cost effect.
Consider Proposition 1 from Lecture 1. In the proposition, we assumed that the inverse demand function P(Q) is concave: P__ (Q) ≤ 0. Assume instead that the function is convex: P__ (Q) > 0. Does the proof given in lectures still work? If yes, explain why. If no, explain why not.
Problem 2 (30 points). Marriage market
Consider four women (Oprah, Prim, Rita, Sophie) and four men (Austin, Boris, Conan, Dishi), who have the following preferences:
Oprah: Boris > Austin > Dishi > Conan
Prim: Dishi > Conan > Austin > Boris
Rita: Austin > Dishi > Conan > Boris
Sophie: Boris > Austin > Dishi > Conan
Austin: Prim > Sophie > Oprah > Rita
Boris: Rita > Oprah > Sophie > Prim
Conan: Prim > Rita > Oprah > Sophie
Dishi: Sophie > Oprah > Rita > Prim
Assume that all men and women are heterosexual and (once married) monogamous. For each of the following matchings, determine if it is stable. Explain.
(a) (Austin, Oprah), (Boris, Prim), (Conan, Rita), (Dishi, Sophie)
(b) (Austin, Sophie), (Boris, Oprah), (Conan, Prim), (Dishi, Rita)
(c) (Austin, Prim), (Boris, Rita), (Conan, Sophie), (Dishi, Oprah)
Problem 3 (20 points). A valuation function.
Letting Y be the set of available objects, we call v : p(Y) → R a valuation function, where p(Y) is the power set of Y . This function assigns a valuation to any subset of set Y . Valuation function v is said to have the properties of
❼ substitutes if v({y}) + v({z}) > v({y, z}) for any y, z e Y
❼ complements if v({y}) + v({z}) ≤ v({y, z}) for any y, z e Y
❼ additive payoffs if v({y}) + v({z}) = v({y, z}) for any y, z e Y
❼ homogeneity if for any two subsets X < Y and Z < Y , v(X) = v(Z) if and only if
lZl = lXl.
For each of the following four cases, say which of the above properties the valuation function satisfies. It is sufficient just to state your answer, you do not need to provide any explanations. Note: Any given valuation function may exhibit multiple properties.
(a) v({s}) = 3, v({t}) = 2, v({s, t}) = 4
(b) v({s}) = 5, v({t}) = 9, v({s, t}) = 15
(c) v({s}) = 7, v({t}) = 7, v({s, t}) = 12
(d) v({s}) = 2, v({t}) = 3, v({s, t}) = 5
Problem 4 (25 points). Convex sets, concave and convex functions.
(a) Which of the following sets are convex? It is sufficient just to state your answer, you do
not need to provide any explanations.
1) a ball
2) octahedron
3) letter S
4) exclamation mark
5) a glazed doughnut (with a hole inside)
6) a person
(b) Are the following functions concave, convex, both or neither? It is sufficient just to state
your answer, you do not need to provide any explanations.
1) C(q) = q2 /2
2) D(p) = 1 C p
3) R(Q) = Q C Q2
4) u(x1 , x2 ) = x1(0) · 5x2(0) · 5
5) u(x1 , x2 ) = x1(0) · 5x2(2)
6) u(x1 , x2 ) = x1(2)x2(2)
2022-03-22