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Foundations of Mathematical Economics 1st edition Michael Carter - Solutions

If S = {x1, x2, . . . , xn} is affinely independent, every x ∈ aff S has a unique representation as an affine combination of the elements of S; that is, there are unique scalars a1, a2, . . . , an such that x = a1x1 + a2x2 +........+ anxn
Consider the relations
Show that the open interval a, b and the closed interval a, b are both convex sets of R with the natural order (example 1.20). The hybrid intervals a, b and a, b are also convex. Show that intervals are the only convex sets in R.
The core of a TP-coalitional game is convex.
The intersection of any collection of convex sets is convex.
Devise a formal proof of Y convex ⇒ V(y) convex for every y Sums and products of convex sets are also convex, as detailed in the following exercises. Convexity of a sum is used in establishing the existence of a general equilibrium in an exchange economy, while convexity of the product is used in
If {S1, S2, . . . , Sn} is a collection of convex subsets of a linear space X, their sum S1 + S2 +...........+ Sn is also a convex set.
If S1, S2, . . . , Sn are convex subsets of the linear spaces X1, X2, . . . ,Xn, their product S1 × S2 ×............× Sn is a convex subset of the product space X1 × X2 ×.......× Xn.
S convex ⇒ aS convex for every a ∈ R.
If {S1, S2, . . . , Sn} is a collection of convex subsets of a linear space X, any linear combination a1S1 + a2S2 +............+ anSn, ai ∈ R is also a convex set. Exercise 1.168 is a useful characterization of convex sets.
A set S is convex if and only if S = aS + (1 - a) S for every 0 < a < 1.
The collection of all convex subsets of a linear space ordered by inclusion forms a complete lattice.
Show that any equivalence relation on a set X partitions X.
A set is convex if and only if it contains all convex combinations of its elements.
The convex hull of a set of vectors S is the smallest convex subset of X containing S.
For any finite collection of sets {S1, S2, . . . , Sn},[Establish the result for n = 2. The generalization to any finite n is immediate.]
Let S be a nonempty subset of a linear space, and let m = dim S = dim aff S. Suppose that x belongs to conv S so that there exist x1, x2, . . . , xn ˆˆ S and a1, a2, . . . , an ˆˆ R+ with a1,......, a2+...........+ an = 1 such thatx = a1x1 + a2x2 +..............+ anxn
If x is not an extreme point of the convex set S Š‚ X, then there exists y ˆˆ X such x + y ˆˆ S and x - y ˆˆ S.A convex subset F of a convex set S is called a face of S if no point of F is an interior point of a line segment whose end points are in S but not
1. Show that the cube C2 = {x ˆˆ R2: - c2. Suppose for any n = 2, 3, . . . , that the cube Cn-1 Š‚ conv{(±c, ±c,. . . ±)} Š‚ Rn-1. Show that n-dimensional cube Cn Š‚ conv{(±c, ±c;. . . ,±c)} Š‚ Rn. 3.
If F is a face of a convex set S, then S\F is convex.Exercise 1.179 will be used in chapter 3.
Let S be a convex set in a linear space: 1. S and ∅ are faces of S. 2. The union of a collection of faces of S is a face. 3. The intersection of any nested collection of faces of S is a face. 4. The collection of all faces of S (partially ordered by inclusion) is a complete lattice.
In a game played by members of the set N, is the set of proper coalitions (example 1.3) a partition of N?
Let E be the set of extreme points of a polytope S. Then S = conv E.
Let S be the simplex generated by the finite set of points E = (x1; x2; . . . ; xn). Show that each of the vertices xi is an extreme point of the simplex. Simplexes are the most elementary of convex sets and every convex set is the union of simplexes. For this reason results are often established
Every n-dimensional convex set contains an n-dimensional simplex. The following examples give some impression of the utility of simplexes in economics and game theory.
The set of imputations of an essential TP-coalitional game (N, w)is an(n - 1)-dimensional simplex in ℜ
Show that set ℜ
A subset S of a linear space is a convex cone if and only if αx + βy ∈ S for every x, y ∈ S and x, β ∈ ℜ+
A set S is a convex cone if and only if 1. αS ⊆ S for every α ≥ 0 2. S + S ⊆ S Convex cones arise naturally in economics, where quantities are required to be nonnegative. The set of nonnegative prices vectors is a convex cone Rn and the production possibility set is often assumed to be a
Another conventional (and trivial) assumption on technology cited by Debreu (1959, p. 41) is 0 ∈ Y, which he calls the possibility of inaction. Show that the three assumptions convexity Y is convex additivity Y + Y ⊆ Y possibility of inaction 0 ∈ Y together imply that the technology exhibits
A natural assumption for TP-coalitional games is super additivity, which requires that coalitions cannot lose through cooperation. Specifically, a TP-coalitional game is super additive iffor all distinct coalitions S, T, S ˆ© ˆ…. Show that the set of super additive games forms a
The following interactions between these orderings are often used in practice.
If {S1,S2,. . . , Sn} is a collection of cones in a linear space x, then€¢ Their intersection€¢ Their sum S1 + S2 + . . . + Sn are also cones in X.
Suppose that a firm's technology is based on the following eight basic activities:y1 = ( -3, -6,4,0)y2 = ( -7, -9,3,2)y3 = ( -1, -2,3,-1)y4 = ( -8, -13,3,1)y5 = ( -11, -19,12,0)y6 = ( -4, -3,-2,5)y7 = ( -8, -5,0,10)y8 = ( -2, -4,5,-2)which can be operated independently at any scal, The aggregate
The conic hull of a set of vectors S is the smallest convex cone in X containing S.
Let S be a nonempty subset of a linear space and let m ˆ dim cone S. For every x ˆˆ cone S, there exist x1,x2,...,xn ˆˆ S and Î±1, Î±2, . . ., Î±n ˆˆ „œ + Such thatx = Î±1x1 + Î±2x2 + . . .
Let S be a nonempty subset of a linear space, and let m = dim S =dim aff S. Consider the set1. Show that dim cone = dim S + 1.2. For every x ˆˆ conv S, there exists m +1 points x1, x2,...,xm+1 ˆˆ S such thatx ˆˆ conv {x1; x2; . . . ; xm+1}
Why can a sub simplex have no more than two distinguished faces?
Show that the metric p(x,y) = || x - y|| satisfies the properties of a metric, and hence that a normed linear space is a metric space.
Show that ∥y∥∞ satisfies the requirements of a norma on ℜ''.
Show that the average of the net outputsdoes not satisfy the requirements of a norm on the production possibility set.
Show that ≻ is asymmetric and transitive.
Prove the following useful corollary of the triangle inequality: for any x, y in a normed linear spaceThe preceding corollary of the triangle inequality implies that the norm converges along with a sequence, as detailed in the following exercise.
Let xn → x be a convergent sequence in a normed linear space. Then ||xn|| → ||x|| Furthermore the norm respects the linearity of the underlying space.
Let xn → x and yn → y be convergent sequences in a normed linear space X. The sequence ( xn + yn ) convergent to x + y, and αxn converges to αx.
If S and T are subsets of a normed linear space with • S closed and • T compact then their sum S + T is closed.
Show that the infinite geometric series x + Î²x + Î²2x + . . . converges provided that| Î²|
Show that
What is the present value of n periodic payments of x dollars discounted at b per period?A special feature of a normed linear space is that its structure or geometry is uniform throughout the space. This can be seen in the special form taken by the open balls in a normed linear space. Recall that
Let S1 and S2 be disjoint closed sets in a normed linear space with S1 compact. There exists a neighborhood U of 0 such that (S1 + U) ∩ ∅ Completeness is one of the most desirable properties of a metric space. A complete normed linear space is called a Banach space. Almost all the spaces
Let X, Y be Banach spaces. Their product X × Y with norm ||(x,y)|| = max {||x||, ||y||} Is also a Banach space. The natural space of economic models is Rn, the home space of consumption and production sets, which is a typical finite-dimensional normed linear space. In these spaces the interaction
Show that ~ is reflexive, transitive, and symmetric, that is, an equivalence relation.
To prove lemma 1.1, assume, to the contrary, that for every c > 0 there exists x ˆˆ lin{x1; x2; . . . , xn} such thatwhere x = a1x1 + a2x2 + €¢ €¢ €¢ +‡anxn. Show that this implies that 1. There exists a sequence (x''') with |x'''||†’0 2. There
Let (xm) be a Cauchy sequence in a normed linear space x of dimension n. Let {x1,x2,...,xn} be a basis for X. Each term x''' has a unique representation1. Using lemma 1.1, show that each sequence of scalars xmi is a Cauchy sequence in „œ and hence converges to some Î±i
Two norms ||x||a and ||x||b on a linear space are equivalent if there are positive numbers A and B such that for all x ˆˆ X,The following exercise shows that there essentially only one finite dimensional normed linear space.
In a finite-dimensional normed linear space, any two norms are equivalent. One implication of the equivalence of norms in a normed linear space is that if a sequence converges with respect to one norm, it will converge in every norm. Therefore convergence in a finite-dimensional normed linear space
A sequence (xn) in Rn converges if and only if each of its components xni converges in R.
Let SŠ†X be a closed and bounded subset of a finite-dimensional normed linear space X with basis {x1, x2, . . . ; xn}, and let xm be a sequence in S.Every term xm has a unique representation1.Using lemma 1.1, show that for every i the sequence of scalars (xmi) is bounded 2.Show that (xm)
In any normed linear space, the unit ball is convex.
Let S be a convex set in a normed linear space. Then int S and are convex. Similarly it can be shown that closure preserves subspaces, cones and linear varieties. For any convex set the line segment joining an interior point to a boundary point lies in the interior (except for the endpoints).
Let S be a convex set, with x1 ∈ S and y2 ∈ int S. Then ax1 + (1 - α )x2 ∈ int S for all 0 < α
Let Si, i ˆˆ I be a collection of open convex sets.We have encountered two distinct notions of the extremity of a set: boundary points and extreme points. Boundary points, which demark a set from its complement, are determined by the geometry of a space. Extreme points, on the other hand,
Show that the relation in example 1.18 is an order relation. That is, show that it is reflexive and transitive, but not symmetric.Figure 1.6 Integer multiples
If S is a convex set in a normed linear space, ext (S) Š† (S).The converse is not true in general; not all boundary points are extreme points. However, boundary points and extreme points coincide when a set is strictly convex. A set S in a normed linear space is called strictly convex if
If S is a strictly convex set in a normed linear space, every boundary point is an extreme point, that is, ext (S) = b(S).
If S is a convex set in a normed linear space, S open ⇒ S strictly convex
S open ⇒ conv S open.
Let S be a compact subset of a finite-dimensional linear space X of dimension n.1. Show that conv S is bounded.2. For every x ˆˆ conv S, there exists a sequence (xk) in conv S that converges to x (exercise 1.105). By CaratheÃ‚odory's theorem (exercise 1.175), each term xk is a convex
Let S be a closed bounded subset of ℜ
The unit simplex in ℜ
If S is a convex set in a finite-dimensional normed linear space
Assume that all prices and income are positive (p > 0,m > 0) and that the consumer can afford some feasible consumption bundle, that is,Then the consumer's budget set X (p,m) is nonempty and compact.
The budget set is convex. Remark 1.22 In establishing that the budget set is compact (exercise 1.231), we relied on the assumption that the choice was over n distinct commodities so that the consumption set is finite dimensional, X ⊂ Rn. In more general formulations involving intertemporal
1. Strong monotonicity) ⇒weak monotonicity. 2. Strong monotonicity) ⇒local non satiation. 3. Local non satiation ) ⇒non satiation. A useful implication of strong monotonicity or local non satiation in consumer choice is that the consumer will spend all her income, so every optimal choice lies
Assume that the consumer's preference relation is strongly monotonic. Then any optimal choice x*‰¿x for every x ˆˆ X (p,m) exhausts her income, that is,
Extend the previous exercise to encompass the weaker assumption of local non satiation.
1. Assume that the preference relation ≿ on a metric space X is continuous. Show that this implies that the sets > (y) = {x : x>y} and < (y) = {x : x (y) = {x : x>y} and < (y) = {x: xz0. a. Suppose there exists some y ∈ X such that x0>y>z0. Show that there exist neighborhoods S(x0) and S (z0)
Mas-Colell et al. (1995, p. 46) define continuity of preferences as follows: The preference relation ≿ on X is continuous if it is preserved under limits. That is, for any sequence of pairs (xn; yn) with xn≿yn for all n, with x limn→∞ yn, and limn→∞yn, we have x≿y. 1. Show that this
Assume that ≿ is continuous preference on a connected metric space X. For every pair x, z in X with x>z, there exists y such that x>y>z.
Let ≿ be a continuous preference relation on a compact set X. The set of best elements is nonempty and compact.
Assume that the set X = {a, b, x, y, z} is ordered as follows: x ≺ a ≺ y ≺ b ~ z Specify the upper and lower contour sets of y.
Assume that a consumer with lexicographic preferences over two commodities requires a positive amount of both commodities so that consumption set X = ℜ2++. Show that no optimal choice exists.
Why is the existence of an optimal choice essential for a well-defined formulation of the consumer's problem?
Let (N, w) be a TP-coalitional game with a compact set of outcomes X. For every outcome x ∈ X, let d (x) be a list of coalitional deficits arranged in decreasing order (example 1.49). Let di (x) denote the ith element of d (x) 1. Show that X1 = {x ∈ X : d1(x)Ud1y for every y A Xg is
The preference relation ≿ is convex if and only if the upper preference sets ≿(y) are convex for every y.
Let ≿ be a convex preference relation on a linear space X. The set of best elements X* ={x : x ≿ y for every y ∈ X} is convex. A slightly stronger notion of convexity is often convenient (example 1.116). A preference relation is strictly convex if averages are strictly preferred to extremes.
In a TP-coalitional game the deficit order ≾d (example 1.49) is strictly convex. Consequently the nucleolus contains a single outcome.
Assume that ‰¿ is a continuous order relation on a connected metric space with x0 > y0 for at least one pair x0; y0 ˆˆ X.1. Show that for any x0; y0 ˆˆ X such that x0>y0,2. Suppose that 7 is not complete. That is, there exists x; y A X such that neither
If the convex preference relation 7 is continuous, x ≻ y ⇒ αx + (1-α)y > y for every 0 < α < 1
If ≿ is strictly convex, non satiation is equivalent to local non satiation.
Remark 1.8 distinguished the strong and weak Pareto orders. Show that the distinction is innocuous in an exchange economy in which the agents preferences are monotone and continuous. Specifically, show that an allocation is weakly Pareto efficient if and only if it is strongly Pareto efficient.
Formulate analogous definitions for minimal and first or worst elements
Every competitive equilibrium allocation x* belongs to the core of the corresponding market game.
Every best element is a maximal element, and not vice versa.
Every finite ordered set has a least one maximal element. The following characterization of maximal and best elements in terms of upper contour sets is often useful. (See, for example, proposition 1.5.) Analogous results hold for minimal and first elements.
Let X be ordered by 7.
Give examples of finite and infinite sets.
Formulate analogous definitions for lower bound and greatest lower bound.
For the set of positive integers N ordered by m is a multiple of n (example 1.18), specify upper and lower bounds for the set A = {2, 3, 4, 5}. Find the least upper bound and greatest lower bound.
x is an upper bound of

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