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Foundations of Mathematical Economics 1st edition Michael Carter - Solutions
If f, g are continuous functionals on a metric space X, then their product fg defined by (fg)(x) = f(x)g(x) is continuous.
Show that the sequence of polynomialsconverges uniformly on any compact subset S Š† „œ.
Given two functionals f and g on X, define (f ∨ g)(x) = max{f (x), g(x)} (f ∧ g)(x) = min{f (x), g(x)} If f and g are continuous, then so are f ∨ g and f ∧ g.
Use proposition 2.3 to provide an alternative proof of theorem 2.2 Theorem 2.2 A continuous functional on a compact set achieves a maximum and a minimum. Proof Let M = supx∊X f (x). There exists a sequence xn in X with f (xn) → M. Since X is compact, there exists a convergent subsequence xm →
Given a metric space X, the C...X€ denote the set of all bounded, continuous functionals on X. Show that€¢ C(X) is a linear subspace of B(X)€¢ C(X) is closed (in B(X))€¢ C(X) is a Banach space with the sup normFor certain applications somewhat weaker or
f is upper semi continuous, ‡” - f is lower semi continuous.
For any f: X → ℜ the following conditions are equivalent: 1. f is upper semi continuous. 2. f (x) ≥ limn→∞ f (xn) for every sequence xn → x. 3. The hypograph of f is closed in X × ℜ. Semi continuity, as opposed to the more restrictive continuity, is often assumed in economic
An upper semi continuous functional on a compact set achieves a maximum.
Let X be any set. Let F(X) denote the set of all functions on X. Show that F(X) is a linear space.
Let f: X → Y be uniformly continuous. If (xn) is a Cauchy sequence in X,(f(xn) is a Cauchy sequence in Y.
A continuous function on a compact domain is uniformly continuous. In economic analysis, uniform continuity typically takes a slightly stronger form. A function f: X → Y is Lipschitz (continuous) if there is a constant b such that for every x, x0 ∊ X, p (f(x), f(x0)) ≤ βp(x, x0) β is called
A Lipschitz function is uniformly continuous. We frequently encounter a particularly strong form of Lipschitz continuity where the function maps a metric space into itself with modulus less than one. Such a function, which maps points closer together, is called a contraction. Specifically, an
Let B(X) be the space of bounded functionals on a metric space X (example 2.11). Let T: B(X) → B(X) be an increasing function with property that for every constant c ∊ ℜ T(f + c) = T(f) + βc for every f ∊ B(X) (21) for some 0 ≤ ft < 1. Show that T is a contraction with modulus β.
Show that operator T in example 2.83 satisfies the conditions of the previous exercise.
Let X be a compact metric space. A closed subspace of C(X) is compact if and only if it is bounded and equicontinuous.
If F ⊆ C(X) is equicontinuous, then so is .
Let Explain.
Regarding a correspondence
For the game in example 2.84, calculate
A function f : X → Y is linear if and only if f (α1x1 + α2x2) = α1f(x1) + α2f(x2) for all x1, x2 ∊ X, and α1, α2 ∊ ℜ.
A player i is called a null player in a game (N, w) if he contributes nothing to any coalition, that is, if w(S ∪ {i}) = w(S) for every S ⊆ N Verify that the Shapley value of a null player is zero, that is, i null ⇒ ϕiw = 0
The set of solutions to a nonhomogeneous system of linear equations Ax = c is an affine set. The converse is also true.
Every affine set in ℜn is the solution set of a system of a linear equations.
Prove that the linear equation system x1 + 3x2 = c1 x1 - x2 = c2
Let Ax = c be a linear equation system with A a nonsingular square n × n matrix (rank A = n). For every c ∈ ℜn there exists a unique solution x = (x1, x2,..., xn) given by where is the matrix obtained by replacing the jth column of A with c.
Show thatwhere Δ = det(A) = ad - bc.
A portfolio is called duplicable if there is a different portfolio y ‰ x which provides exactly the same returns in every state, that is, Rx = Ry orShow that every portfolio is duplicable if rank R
A state is called insurable if there exists a portfolio x which has a positive return if state occurs and zero return in any other state, that is,Show that every state is insurable if and only if rank R = S.
Draw analogous diagrams illustrating the possible cases for a system of three equations in two unknowns.
Every affine subset of ℜn is the intersection of a finite collection of hyperplanes.
The set of solutions to a system of linear inequalities Ax ≤ c is a convex set.
Recall that, for any coalition T Š† N, the T-unanimity game (example 1.48) uT ˆŠ GN isCompute the Shapley value of a T-unanimity game.
The set of solutions to a homogeneous system of linear inequalities Ax ≤ 0 is a convex cone.
Let x be a feasible solution to the linear system Ax = c, x ≥ 0 (23) where A is an m × n matrix and c ∈ ℜm. Then c = x1A1 + x2A2 + ... + xnAn where Ai ∈ ℜm are the columns of A and and xi ≥ 0 for every i. Without loss of generality, assume that the first k components are positive
Let {S1, S2,..., Sn} be a collection of nonempty (possibly non convex) subsets of an m-dimensional linear space, and letThen 1. 2. where 3. with bij ‰¥ 0 and bij > 0 for at most m + n components. 5. Define Then i = conv Si and Show that all but at most i actually belong to Si.
Assume that A is productive. Show that Az ≥ 0 implies z ≥ 0
The system A is productive if and only if A-1 exists and is nonnegative.
Augment the input-output model to include a primary commodity ("labor"). Let a0j denote the labor required to produce one unit of commodity j. Show that there exists a price system p = (p1, p2, . . . , pn) such that the profit of each activity (industry) is zero.
What steady state unemployment rates are implied by the transition probabilities in table 3.1?
Show that x2 is convex on ℜ.
For any TP-coalitional game (N; w) the potential function is defined to bewhere t =| T | and αT are the coefficients in the basic expansion of w (exercise 1.75). Show that
Show that the power functions f (x) = xn, n = 1, 2, . . . are convex on ℜ+.
Prove corollary 3.7.1 directly from the definition (24) without using proposition 3.9.
A function f: S †’ „œ is convex if and only iffor all
Show thatfor every x1, x2, . . . , xn ˆˆ „œ+. Deduce that the arithmetic mean of a set of positive numbers is always greater than or equal to the geometric mean, that is,
f is concave if and only if -f is convex.
A function f: S → ℜ is concave if and only if hypo f is convex.
Show that the cost function c(w, y) of a competitive firm (example 2.31) is concave in input prices w.
Suppose that a consumer retires with wealth w and wishes to choose remaining lifetime consumption stream c1, c2, . . . , cT to maximize total utilityAssuming that the consumer's utility function u is concave, show that it is optimal to consume a constant fraction = w/T of wealth in each period.
If f is convex on R F(x1 - x2 + x3) ≤ f(x1) - f(x2) + f(x3) for every x1 ≤ x2 ≤ x3 ∈ R. The inequality is strict if f is strictly convex and reversed if f is concave.
If f ∈ F(ℜ) is strictly concave, f(x - y) displays strictly increasing differences in (x, y). If a function is both convex and concave, it must be affine.
Every linear function f: X → Y maps the zero vector in X into the zero vector in Y. That is, f (0x) = 0Y.
If f, g ∈ F(X) are convex, then • f + g is convex • αf is convex for every α ≥ 0 Therefore the set of convex functions on a set X is a cone in F(X). Moreover, if f is strictly convex, then • f + g is strictly convex • αf is strictly convex for every α > 0
If f and g are convex functions defined on a convex set S, the function f ˆ¨ g defined byfor every x ˆˆ S is also convex on S.
If f ∈ F(X) and g ∈ F(ℜ) with g increasing, then f and g convex ⇒ g ο f convex f and g concave ⇒ g ο f concave
If f is a strictly positive definite concave function, then 1/f is convex. If f is a strictly negative definite convex function, then 1/f is concave.
Suppose that f ∈ F(X) is monotone and g ∈ F(ℜ) is increasing. Then f supermodular and g convex ⇒ g ο f supermodular f submodular and g concave ⇒ g ο f submodular
Let f be a convex function on an open set S that is bounded above by M in a neighborhood of x0; that is, there exists an open set f containing x0 such thatf(x) ‰¥ M for every x ˆˆ U1. Show that there exists a ball B(x0) containing x0 such that for every x ˆˆ B(x0),2.
Let f be a convex function on an open set S which is bounded above by M in a neighborhood of x0. That is, there exists an open ball Br(x0) containing x0 such that f is bounded on B(x0). Let x1 be an arbitrary point in S.1. Show that there exists a number t > 1 such thatz = x0 + t(x1 - x0)
Let f be a convex function on an open set S that is bounded at a single point. Show that f is locally bounded, that is for every x ∈ S there exists a constant M and neighborhood U containing x such that |f(x′)| ≤ M for every x′ ∈ U
If f: X → Y and g: Y → Z are linear functions, then so is their composition g ∘ f: X → Z.
Prove corollary 3.8.1 [Use Caratheodory's theorem (exercise 1.175) and Jensen's inequality (exercise 3.122).]
Let f be a functional on a convex open subset S of a Euclidean space X. f is convex if and only f is locally convex at every x ∈ S.
f is quasiconcave if and only if -f is quasiconvex.
Every concave function is quasiconcave.
Any monotone functional on ℜ is both quasiconvex and quasiconcave.
A functional f is quasiconvex if and only if every lower contour set is convex; that is,is convex for every a ˆˆ „œ.
Show that the indirect utility function is quasi convex.
If f is quasi concave and g is increasing, then g ο f is quasi concave.
The CES functionis convex on „œn+ if p ‰¥ 1.
Show that a linear function maps subspaces to subspaces, and vice versa. That is, if S is a subspace of X, then f (S) is a subspace of Y; if T is a subspace of Y, then f-1(T) is a subspace of X.
Let f and g be affine functionals on a linear space X, and let S Š† X be a convex set on which g(x) ‰ 0. The functionis both quasi concave and quasi convex on S.
Let f and g be strictly positive definite functions on a convex set S with f concave and g convex. Thenis quasi concave on S.
Let f1, f2, . . . , fn be nonnegative definite concave functions on a convex set S. The functionis quasi concave on S for any α1, α2, . . . , αn ˆˆ „œ+.
In the dynamic programming problem (example 2.32)subject to xt+1 ˆˆ G(xt) t = 0, 1, 2, . . . , x0 ˆˆ X given Assume that €¢ f is bounded, continuous and strictly concave on X × X. €¢ G(x) is nonempty, compact-valued, convex-valued, and continuous
Assuming that u is strictly concave, show that the optimal growth model satisfies the requirements of exercise 2.126. Hence conclude that the optimal policy converges monotonically to a steady state.
Suppose that f: X → Y is a linear function with rank f = rank Y ≤ rank X. Then f maps X onto Y.
Let X and Y be compact subsets of a finite-dimensional normed linear space, and let f be a continuous functional on X × Y. Thenif and only if there exists a point (x*, y*) ˆˆ X × Y such that for every x ˆˆ X and y ˆˆ Y
If f is a continuous functional on a compact domain X × Y,
A function f: R+ → R is homogeneous of degree a if and only if it is (a multiple of) a power function, that is, f(x) = Axa for some A ∈ ℜ
Show that the CES functionis homogeneous of degree one.
Show that the cost function c(w, y) of a competitive firm (example 2.31) is homogeneous of degree one in input prices w.
If the production function of a competitive firm is homogeneous of degree one, then the cost function c(w,y). is homogeneous of degree one in y, that is, c(w, y) = yc(w, 1) where c(w, 1). is the cost of producing one unit (unit cost).
Show that the indirect utility function v(p,m) (example 2.90) is homogeneous of degree zero in p and m. Analogous to convex functions (proposition 3.7), linearly homogeneous functions can be characterized by their epigraph.
A function f : S → ℜ is linearly homogeneous if and only if epi f is a cone. The following useful proposition show how quasiconcavity and homogeneity combine to produce full concavity. Quasiconcavity ensures convexity of the upper contour sets, while homogeneity of degree k ≤ 1 strengthens
If f ∈ F(S) is strictly positive definite, quasiconcave, and homogeneous of degree one, then f is superadditive, that is, f (x1 + x2) ≥ f (x1) + f(x2) for every x1, x2 ∈ S
If f ∈ F(S) is strictly positive definite, quasi concave, and homogeneous of degree one, then f is concave.
Generalize exercise 3.170 to complete the proof of proposition 3.12.
A preference relation ⋩ (section 1.6) on a cone S is said to be homothetic if x1 ~ x2 ) tx1~ tx2 for every x1, x2 ∈ S and t > 0 Show that a continuous preference relation is homothetic if and only if every utility representation (example 2.58) is homothetic. Every monotonic transformation of
Suppose that f is a monotonic transformation of a homogeneous function. Show that f is a monotonic transformation of a linearly homogeneous function.
If h is a homogeneous functional on S and g: ℜ → ℜ is strictly increasing, then f = g o h is hoamothetic.
Let f be a strictly increasing homothetic functional on a cone S in an linear space X. Then there exists a linearly homogeneous function h: S → ℜ and a strictly increasing function g: ℜ → ℜ such that f = g o h.
If the production function of a competitive firm is homothetic, then the cost function is separable, that is, C(w,y) = ∅(y)c(w,1)
A strictly positive definite, strictly increasing, homothetic, and quasi concave functional is concavifiable.
Assume that Hf c is a supporting hyperplane to a set S at x0. Show that either x0 maximizes f or x0 minimizes f on the set S. As the illustrations in figure 3.16 suggest, the fundamental requirement for separation is convexity. This is the content of following basic separation theorem, whose proof
In example 3.77 show that int A⋂B = ∅.
Suppose that f: X → Y is a linear function with kernel f = {0}. Then f is one-to-one, that is, f(x1) = f(x2) ⇒ x1 = x2 A linear function f: X → Y that has an inverse f-1: Y → X is said to be nonsingular. A function that does not have an inverse is called singular.
In example 3.77 Robinson the producer makes a profit of wh* + pq*. This is Robinson the consumer's income, so his budget set is X = {(h,q : wh + pq ≤ wh* + pq*} This is half space below the separating hyperplane in figure 3.17. Note that A⊆X. (h*, q*) is the optimal choice in the feasible set
Let f be a convex function defined on a convex set S in a normed linear space X. For every x0 ∈ int S there exists a linear functional g ∈ X* that bounds f in the sense that f (x) ≥ f (x0) + g(x - x0) for every x ∈ S Such a linear functional g is called a sub gradient of f at x0. Consider
Let S be a nonempty, closed, convex set in a Euclidean space X and y ∉ S. There exists a continuous linear functional f ∈ X* and a number c such that f (y) < c ≤ f (x) for every x ∈ S
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