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Foundations of Mathematical Economics 1st edition Michael Carter - Solutions

Using the natural order > on R, order the plane R2 by the lexicographic order. It is a total order?
Let X be the product of n sets Xi each of which is ordered by 7i. Show that the lexicographic order is complete if and only if the component orders are complete.
Show that set inclusion is a partial order on the power set of a set X.
Let A be a nonempty subset of X that is partially ordered by ≿. If A has a least upper bound, then it is unique. Similarly A has at most one greatest lower bound.
Characterize the equivalence classes of the relation ~ induced by a partial order ≿.
A chain has at most one maximal element.
Let T, ≻) be a game tree (arborescence). For every non initial node, call p(t) = sup ≺(t) the immediate predecessor of t. Show that 1. p(t) is unique for every t ∈ T\W. 2. There is a unique path between any node and an initial node in a game tree.
The operations ∨ and ∧ have the following consistency properties. For every x; y in a lattice (X; ≿), 1. x ∨ y ≿ x ≿ x ∧ y 2. x ≿ y =⇒ x ∨ y = x and x ∧ y = y 3. x ∨ x ∧ y = x = x ∧(x ∨ y)
Any chain is a lattice.
The product of two lattices is a lattice. The previous exercise implies that the product of n lattices is a lattice, with ∨ and ∨ defined componentwise, that is, x ∨ y = (x1 ∨ y1, x24y2; . . . ; xn ∨ yn) x ∧ y = x1 ∧ y1, x2 ∧ y2, . . . , xn ∧ yn)
Let X be a partially ordered set which has a best element x*. If every nonempty subset S of a X has a greatest lower bound, then X is a complete lattice.
Let a, b be elements in a lattice L with a ≾ b. Then the subsets ≿(b), ≾(a) and [a, b] are sublattices. The sublattices are complete if L is complete.
1. For any lattice X, the strong set order ≿
If S1, S2 are subsets of a complete lattice, S1 ≿
If S1, S2 are intervals of a chain, S1 ≿S S2, inf S1 ≿ inf S2 and sup S1 ≿ sup S2
Many economics texts list three assumptions-complete, transitive, and reflexive-in defining the consumer's preference relation. Show that reflexivity is implied by completeness, and so the third assumption is redundant.
Why would anti symmetry be an inappropriate assumption for the consumer's preference relation?
In a weakly ordered set, maximal and best elements coincide. That is, x is maximal ⇒ x is best.
A weakly ordered set has a most one best element. True or false? In a weakly ordered set, every element is related to every other element. Given any element y, any other element x A ∈ belongs to either the upper or lower contour set. Together with the indifference sets, the upper and lower
If ≿ is a weak order on X, then for every y ∈ X, 1. ≿(y) ∪ ≾(y) = X and ≿(y) ∩ ≾(y) = Iy 2. ≿(y) ∪ (y) = X and ≿(y) ∩ ≺(y) = ϕ 3. ≻(y), Iy and ≺(y) together partition X
If ≿ is a weak order on X, x ≿ y ⇒ ≿(x) ⊂ ≿(y) x ≻ y ⇒ ≻(x) ⊂ ≻(y) The principal task of optimization theory and practice is identify the best element(s) in a choice set X, which is usually weakly ordered by some criterion. To identify the best element, optimization theory
Investigate which definition of the Pareto order is used in some leading texts, such as Kreps (1990), Mas-Colell et al. (1995), Varian (1992), and Osborne and Rubinstein (1994). Even if all the constituent orders ≿i are complete, the Pareto order ≿P is only a partial order. Where x ≻i y for
An electricity grid connects n hydroelectric dams. Each dam i has a fixed capacity Qi. Assuming that the dams are operated independently, the production decision can be modeled as a game with n players. Specify the set of players and the action space of each player.
A group S of individuals is decisive over a pair of alternatives x, y ∈ X if x ≻i y for every i ∈ S ⇒ x ≻ y Assume that the social order is consistent with the Pareto order and satisfies the IIA condition. Show that, if a group is decisive over any pair of states, it is decisive over
Assume that the social order is consistent with the Pareto order and satisfies the IIA condition, and that |X| > 3. If any group S with |S| > 1 is decisive, then so is a proper subset of that group.
Using the previous exercises, prove Arrow's impossibility theorem.
Liberal values suggest that there are some choices that are purely personal and should be the perogative of the individual concerned. We say that a social order exhibits liberalism if for each individual i, there is a pair of alternatives x; y ∈ X over which she is decisive, that is, for which x
Find the core of the cost allocation game (example 1.44). Example 1.44 The Southern Electricity Region of India comprises four states: Andra Pradesh (AP), Kerala, Mysore, and Tamil Nadu (TN). In the past each state had tended to be self-sufficient in the generation of electricity. This led to
Formulate the cost allocation problem in example 1.44 as a TP-coalitional game. [Regard the potential cost savings from cooperation as the sum to be allocated.]
Show that the core of coalitional game with transferable payoff is
Specify the set of unanimity games for the player set N = {1, 2, 3}.
Show that the core of a simple game is nonempty if and only if it is a unanimity game.
List all the coalitions in a game played by players named 1, 2, and 3. How many coalitions are there in a ten player game?
In the cost allocation game, find d(x) for x1 = (180, 955, 395) and x2 = (200, 950, 380) Show that d(x1) ≺L d(x2) and therefore x1 ≻d x2.
Is the deficit order ≿d defined in example 1.49 • A partial order? • A weak order? on the set X.
x belongs to the core if and only if no coalition has a positive deficit, that is, core = {x ∈ X : d(S, x) < 0 for every S ⊂ g
Show that Nu ⊂ core assuming that core ≠ ϕ.
Show formally that the action profile (C, C) is a Nash equilibrium.
Let ‰¿Ê¹i denote the partial order induced on player i 's action space by her preferences over A. That is,Show that if there exists an action profile a* such that a*i is the unique maximal element in (Ai, ‰¿Ê¹i) for every player i, then a* is the unique Nash equilibrium of the game.
Show that p(x, y) = |x - y| is a metric for R.
Show that p∞(x, y) = maxni=1 |xi - yi| is a metric for Rn.
What is the boundary of the set S = {1/n : n = 1, 2, . . .}?
For any S ⊂ T, 1. int S ⊂ int T 2. ⊂
Show thatS ˆª Tc = Sc ˆ© Tc(S ˆ© T)c = Sc ˆª TcSet union and intersection have straightforward extensions to collections of sets. The union of a collection C of sets
A set is open if and only if its complement is closed.
In any metric space X, the empty set 0 and the full space X are both open and closed. A metric space is connected if it cannot be represented as the union of two disjoint open sets. In a connected space the only sets that are both open and closed are X and ∅. This is case for R, which is
A metric space is connected if and only it cannot be represented as the union of two disjoint closed sets.
A metric space X is connected if and only if X and ∅ are the only sets that are both open and closed.
A subset S of a metric space is both open and closed if and only if it has an empty boundary.
1. Any union of any collection of open sets is open. The intersection of a finite collection of open sets is open. 2. The union of a finite collection of closed sets is closed. The intersection of any collection of closed sets is closed.
For any set S in a metric space 1. int S is open. It is the largest open set in S. 2. is closed. It is the smallest closed set containing S.
The interior of a set S comprises the set minus its boundary, that is, int S = S\b(S)
A set is closed if and only if it contains its boundary.
A set is bounded if and only it is contained in some open ball.
Union and intersection are one way of generating new sets from old. Another way of generating new sets is by welding together sets of disparate objects into another set called their product. The product of two sets X and Y is the set of ordered pairsX Ã— Y = {(x, y): x ˆˆ X, y
Given an open ball Br(x0) in a metric space, let S be a subset of diameter less than r that intersects Br(x0). Then S ⊂ B2r(x0).
Show that free disposal (example 1.12) implies that the production possibility set has a nonempty interior.
A topological space is said to be normal if, for any pair of disjoint closed sets S1 and S2, there exist open sets (neighborhoods) T1 ⊃ S1 and T2 ⊃ S2 such that T1 ∩ T2 = ∅. Show that any metric space is normal.
Let S1 and S2 be disjoint closed sets in a metric space. Show that there exists an open set T such thatS1 ⊂ T and S2 ∩ = ∅
In the previous example, illustrate the open ball of radius 1/2 about (2, 0). Use the Euclidean metric p2. The following result links the metric and order structures of real numbers (see exercise 1.20).
A set S ⊂ R is connected if and only if it is an interval.
If a sequence converges, its limit is unique. Therefore we are justified in talking about the limit of a convergent sequence.
Every convergent sequence is bounded; that is, the set of elements of a convergent sequence is a bounded set. [If xn → x, show that there exists some r such that p(xn, x) < r for all n.]
At a birthday party the guests are invited to cut their own piece of cake. The first guest cuts the cake in half and takes one of the halves. Then, each guest in turn cuts the remainder of the cake in half and eats one portion. How many guests will get a share of the cake?
Let (xn) be a sequence that converges to x. Show that the points of (xn) become arbitrarily close to one another in the sense that for every ε > 0 there exists an N such that p(xm, xn) < e for all m, n > N
Let the two possible outcomes of coin toss be denoted H and T. What is the sample space for a random experiment in which a coin is tossed three times?
Any Cauchy sequence is bounded. Exercise 1.99 showed that every convergent sequence is a Cauchy sequence; that is, the terms of the sequence become arbitrarily close to one another. The converse is not always true. There are metric spaces in which a Cauchy sequence does not converge to an element
A monotone sequence in R converges if and only if it is bounded. [If xn is a bounded monotone sequence, show that xn → sup{xn}.]
For every Î² ˆˆ R+, the sequence Î², Î²2, Î²3,... converges if and only if Î²
Show that 1. 1/2 (x + 2/x) > √2 for every x ∈ R+. [Consider (x - √2)2 > 0]. 2. The sequence in example 1.64 converges to √/2.
Extend example 1.64 to develop an algorithm for approximating the square root of any positive number. The following exercises establish the links between convergence of sequences and geometry of sets. First, we establish that the boundary of a set corresponds to the limits of sequences of elements
Let S be a nonempty set in a metric space. x ∈ if and only if it is the limit of a sequence of points in S.
A closed subset of a complete metric space is complete. A sequence (Sn) of subsets of a metric space X is nested if S1 ⊃ S2 ⊃ S2 ⊃ ....
Let (Sn) be a nested sequence of nonempty closed subsets of a complete metric space with d(Sn) †’ 0. Their intersection contains exactly one point.
Let C be the set of all subsets of a metric space X with nonempty interior. Consider the following game with two players. Each player in turn selects a set Sn from C such thatS1 Šƒ S2 Šƒ S2 Šƒ ...
Assume that Y ⊂ Rn is a production possibility set as defined in the previous example. What is Y ∩ Rn?
A closed subset of a compact set is compact.
A Cauchy sequence is convergent ‡” it has a convergent subsequence.Actually compact spaces have a much stronger property than boundedness. A metric space X is totally bounded if, for every r > 0, it is contained in a finite number of open ball Br(xi) of radius r, that is,The open balls
A compact metric space is totally bounded.
A metric space if compact if and only if it is complete and totally bounded.
Step 1. d(T) > 0 for every T ˆˆ Î´.
Step 1. There exists a finite number of open balls Br(xn) such that
We will used this property in the following form (see exercise 1.108).
Let S1 Šƒ S2 Šƒ S3... be a nested sequence of nonempty compact subsets of a metric space X. Then
Let X = [0, 2]. Show that the correspondence
Let K be any subset of Y. The constant correspondence
A correspondence
A correspondence Discuss.
A compact-valued correspondence
A correspondence Discuss in detail.
Let
Suppose that Y is compact. The correspondence
If
Assume that the consumption set X is nonempty, compact, and convex.Letdenote the budget correspondence. Choose any (p, m) ˆŠ P such that m > minxˆŠX ˆ‘mi=1 Pixi, and let T be an open set such that X(p, m) ˆ© T ‰ Î˜. For n = 1, 2,,
1. || f || supx∊X | f (x) | is a norm on B (X). 2. B(X) is a normed linear space. 3. B(X) is a Banach space.
Let Discuss.
If X is a compact space and
Let Discuss in detail.
The dynamic programming problem (example 2.32)subject to xt+1 ˆŠ G(xt), t = 0, 1, 2, . . . , x0 ˆŠ X gives rise to an operator on the space B(X) of bounded functionals (exercise 2.16). Assuming that €¢ f is bounded and continuous on X Ã— X €¢ G(x)
To prove corollary 2.4.1, let f: X → X be an increasing function on a complete lattice (X, ≿), and let E be the set of fixed points of f. For any S ⊆ E define S* = {x ∊ X : x ≿ s for every s ∊ S} S* is the set of all upper bounds of S in X. Show that 1. S* is a complete sublattice. 2. f

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