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Foundations of Mathematical Economics 1st edition Michael Carter - Solutions

Let Discuss briefly.
To prove corollary 2.4.2, let S c E and s* = sup S. 1. For every x ∊ S there exists some zx ∊ p(s*) such that zx ≿ x. 2. Let z* = sup zx. Then a. z* ≾ s* b. z* ∊
Show that i is increasing for every i.
Show that the best response correspondence B(a) = B1(a-1) × B2(a-2) ×...× Bn (a-n) of a supermodular game satisfies the conditions of Zhou's theorem (corollary 2.4.2). Therefore the set of Nash equilibria of a supermodular game is a complete lattice.
Prove corollary 2.5.1.Let f: X †’ X be a contraction mapping on the complete metric space X. Let (xn) be the sequence constructed from an arbitrary starting point x0, and let x = lim xn be the unique fixed point. Then
The sample space for tossing a single die is {1; 2; 3; 4; 5; 6}. Assuming that the die is fair, so that all outcomes are equally likely, what is the probability of the event E that the result is even (See exercise 1.4)?
Example 1.64 outlined the following algorithm for computing the square root of 2:Verify that €¢ The function f(x) = 1/2 (x + 2/x) is contraction mapping on the set X = {x ˆŠ „œ: x ‰¥ 1} €¢ The fixed point of f is ˆš2 Estimate how many
Prove corollary 2.5.2. Let f : X → X be a contraction mapping on the complete metric space X with fixed point x. If S is a closed subset of X and f (S) ⊆ S, then x ∊ S.
Prove corollary 2.5.3. Let f: X → X be an operator on a complete metric space X. Suppose that for some integer N, the function fN: X → X is a contraction. Then f has a unique fixed point.
Let X and Θ be metric spaces, and let f: X × Θ → X where • X is complete • For every θ ∊ Θ, the function fθ(x) = f (x, θ) is contraction mapping on X with modulus β • f is continuous in θ, that is for every θ0 ∊ Θ, limθ→ θ0 fθ(x) = fθ0(x) for every x ∊ X Then fθ has
Suppose that the linear model (section 3.6.1) Ax = c has been scaled so that aii = 1 for every i. Show the following:
Let v be the value function for the dynamic programming problem (example 2.32)subject to xt+1 ˆŠ G(xt), t = 0, 1, 2,..., x0 ˆŠ X Assume that €¢ f is bounded and continuous on X Ã— X €¢ G(x) is nonempty, compact-valued, and continuous for every x
Consider a dynamic programming problem that satisfies all the assumptions of the previous exercise. In addition assume that the state space X is a lattice on which • f (x, y) is supermodular in y • f (x, y) displays strictly increasing differences in (x, y) • G(x) is increasing Show that 1.
Let f: [0; 1] → [0; 1] be continuous. Show that f has a fixed point.
Let f: S → S be an operator on an n simplex with vertices {x0, x1,..., xn}. Suppose that the elements of S are labeled using the rule x → min{i: βi ≤ αi, ≠ 0} where αi, and βi are the barycentric coordinates of x and f (x) respectively. Show that 1. The rule assigns a label in {0,
Generalize the proof of the Brouwer theorem to an arbitrary compact convex set as follows. Let f: S → S be a continuous operator on a nonempty, compact, convex subset of a finite-dimensional normed linear space. 1. Show that there exists a simplex T containing S. 2. By exercise 3.74, there exists
Where the firm produces just a single output, it is common to distinguish output from inputs. To do this, we reserve p for the price of the output, and let the vector or list w = (w1; w2; . . . ; wn) denote the prices of the inputs. Using this convention, define the profit function for a profit
To show that each of the hypotheses of Brouwer's theorem is necessary, find examples of functions f : S → S with S ⊆ R that do not have fixed points, where 1. f is continuous and S is convex but not compact 2. f is continuous and S is compact but not convex 3. S is compact and convex but f is
Let B denote the closed unit ball in a finite-dimensional normed linear space B = {x ∊ X: ||x|| ≤ 1} and let S denote its boundary, that is, S = {x ∊ X: ||x|| ≤ 1} There is no continuous function r: B → S such that r(x) = x for every x ∊ S.
Let f: B → B be a continuous operator on the closed unit ball B in a finite-dimensional normed linear space. Show that the no-retraction theorem implies that f has a fixed point.
Prove that the no-retraction theorem is equivalent to Brouwer's theorem. The following proposition, due to Knaster, Kuratowki, and Mazurkiewicz (K-K-M), is equivalent to the Brouwer theorem. It is often used as a step on the way to the Brouwer theorem. It is more useful than the Brouwer theorem in
Prove the K-K-M theorem directly, using Sperner's lemma. Let A0, A1,..., An be closed subsets of an n-dimensional simplex S with vertices x0, x1,..., xn. If for every I ⊆ {0, 1,..., n} the face conv{xi: i ∊ I} is contained in the corresponding union ⋃i∊I Ai, , then the intersection ∩ni=0
Prove that the K-K-M theorem is equivalent to Brouwer's theorem, that is, K-K-M theorem ⇔ Brouwer's theorem The classic application of the Brouwer theorem in economics is to prove the existence of competitive equilibrium. We extract the mathematical essence in the following corollary, and then
Let z: Î”n-1 †’ „œn be a continuous function satisfying pz(p) = 0 for every p ˆŠ Î”n-1 andShow that G(p*) = p* ‡’ z (p*) ‰¤ 0
Show that the aggregate excess demand function z(p) is continuous and homogeneous of degree zero.
Assuming that the consumers' preference relations ≿i are non satiated and strictly convex, show that the aggregate excess demand function z(p) satisfies Walras's law pT z(p) = 0 for every p
Consider a competitive firm with a constant returns to scale technology Y ⊆ ℜn (example 1.101). Let f (p, y) = ∑I pi yi denote the net revenue (profit) of adopting production plan t with prices p. 1. If production is profitable at prices p, that is, there exists some y ∊ Y such that f(y, p)
p* is a competitive equilibrium price if z(p*) ≤ 0.
Verify that x* = limkʹ→∞ xkʹ as defined in the preceding proof is a fixed point of the correspondence, that is x* ∊
Generalize the proof of the Kakutani theorem to an arbitrary convex, compact set S.
Show that the best response correspondence B: S ⇉ S is closed and convex valued.
Suppose, in addition to the hypotheses of example 2.96, that • The players' payoff functions ui: S → ℜ are strictly quasi concave • The best response mapping B: S → S is a contraction Then there exists a unique Nash equilibrium of the game.
Let K be a compact subset of a normed linear space X. For every ε > 0, there exists a finite-dimensional convex set S ⊆ X and a continuous function h: K → S such that S⊆ conv K and ||h(x) - x|| < ε for every x ∊ K
Let gk = hk ∘ f as defined in the preceding proof. Show that 1. gk: Sk → Sk 2. ||gk(x) - f(x)|| ≤ 1 / k for every x ∊ Sk
Verify that x* = limk →∞ xk as defined in the preceding proof is a fixed point of f, that is, f (x*) = x*. Schauder's theorem is frequently applied in cases where the underlying space is not compact. The following alternative version relaxes this condition to require that the image lie in a
Let F be a nonempty, closed and bounded, convex subset of C(X), the space of continuous functionals on a compact metric space X. Let T: F → F be a continuous operator on F. If the family T(F) is equicontinuous, then T has a fixed point.
For given θ = Θ, verify that x* ∊ G(θ) is a solution to (4) if and only if it satisfies (5).
Letbe the value function for the dynamic programming problem (example 2.32). Assume that €¢ f is bounded on X Ã— X €¢ G(x) is nonempty for every x ˆŠ X Show that v is a bounded functional on X (i.e., v ˆŠ B(X)) that satisfies the equation for
Letbe the value function for the dynamic programming problem (example 2.32). Assume that €¢ f is bounded on X Ã— X €¢ G(x) is nonempty for every x ˆŠ X Show that the plan x* = (x0, x*1, x*2, ...) ˆŠ Î“ (x0) is optimal if and only if it
In the dynamic programming problem (example 2.32), assume that€¢ f is bounded on X Ã— X€¢ G(x) is nonempty for every x ˆŠ XShow that the function T defined byis an operator on the space B(X) (example 2.11).
Suppose that the allocation of $1 among three persons is to be decided by majority vote. Specify the characteristic function.
Assume that the consumer's preferences are continuous and strictly convex. Show that the demand correspondence is single valued. That is, the demand correspondence is a function mapping P → X.
s* = (s*1, s*2,...,s*n) is a Nash equilibrium if and only if s*i ∊ B(s*) for every i ∊ N
In a strategic game, a strategy of player i is justifiable if it is a best response to some possible (mixed) strategy (example 1.98) of the other players, that is, si is justifiable ⇔ si ∊ Bi (∑-i) where ∑-i is the set of mixed strategies of the opposing players (example 1.110). Let B1i
Show that the value function (example 2.28) can be alternatively defined by
Let Y be the production possibility set for a single-output technology and V(y) denote the corresponding input requirements sets V(y) = {x ∊ ℜn+: (y, - x) ∊ Y} Then Y is convex if and only if V(y) is a convex correspondence.
Suppose that the constraint correspondence G(y) in the constrained optimization problem (example 2.30)is defined by a set of inequalities (example 2.40) g1(x, Î¸) ‰¤ 0, g2(x, Î¸) ‰¤ 0, ... , gm (x, Î¸) ‰¤ 0 If each functional
Show that the identity function IX (example 2.5) is strictly increasing.
If f: X → Y and g: Y → Z are increasing functions, so is their composition g ∘ f: X → Z. Moreover, if f and g are both strictly increasing, then so is g ∘ f.
If X and Y are totally ordered (chains) and f: X → Y is strictly increasing, then f has a strictly increasing inverse f -1: f-1 (X) → X.
The controls {f-1(y): y ∊ Y} of a function f: X → Y partition the domain X. For any particular y ∊ Y, its pre image f -1(y) may be • Empty • Consist of a single element • Consist of many elements Where f -1(y) consists of one and only one element for every y ∊ Y, the pre image defines
If f: X → ℜ is increasing, -f: X → ℜ is decreasing.
If f; g ∊ F(X) are increasing, then • f + g is increasing • αf is increasing for every α ≥ 0 Therefore the set of all increasing functionals on a set X is a cone in F(X). Moreover, if f is strictly increasing, then • f + g is strictly increasing • αf is strictly increasing for every
If f and g are strictly positive definite and strictly increasing functionals on X, then so is their product fg defined by (fg)(x) = f(x)g(x)
Show that ex is strictly increasing on ℜ+.
The general power function f (x) = xa is strictly increasing on ℜ+ for all a > 0 and strictly decreasing for a < 0.
Let u: X → ℜ be a strictly increasing function on the weakly ordered set (X, ≿). Show that x2 ≿ x1 ⇔ u(x2) ≥ u(x1)
Let u: X → ℜ be a utility function representing the preference relation ≿. Show that every monotonic transformation g ∘ u is a utility function representing the same preferences. We say that utility representation is invariant to monotonic transformations.
Let ≿ be a continuous preference relation on ℜn+. Assume that ≿ is strongly monotonic. Let Z denote the set of all bundles that have the same amount of all commodities (figure 2.15), that is, Z = {z = z1: z ∊ ℜ+} where 1 = (1, 1, ..., 1). 1. For any x ∊ ℜn+, show that a. The sets Z+x
Remark 2.9 implies that the lexicographic preference relation (example 1.114) cannot be represented by a utility function, since the lexicographicpreference ordering is not continuous. To verify this, assume, to the contrary, that u represents the lexicographic ordering ‰¿L on
The function f: X → Y has an inverse function f -1: Y → X if and only if f is one-to-one and onto.
Suppose that u1: A ∊ ℜ represents the preferences of the player 1 in a two-person strictly competitive game (example 1.50). Then the function u2 = - u1 represents the preferences of the player 2 and u1(a) + u2(a) = 0 for every a ∊ A Consequently a strictly competitive game is typically called
Show that super additivity implies monotonicity, that is, if v: P(N) → ℜ is super-additive, then v is monotonic.
Show that the operator T: B(X) †’ B(X) defined byis increasing.
For Θ ∊ ℜ, if g ∊ F (Θ) is increasing, then the correspondence G(θ) = {x: 0 ≤ x ≤ g(θ)} is increasing The significance of this definition of monotonicity for correspondences is that every monotone correspondence has a monotone selection. This and other useful properties of monotone
Let Briefly.
If Discuss.
If ????i: X ⇉ Yi, i = 1, 2, … , n, is a collection of increasing correspondences with common domain X, their product ????: X ⇉ Π Yi defined byis also increasing.
If Discuss briefly.
Prove that φ: X ⇉ Y is always increasing if and only if every selection f ∊ φ is increasing.
Every functional on a chain is super modular. The following properties are analogous to those for monotone functions (exercises 2.31 and 2.32).
If f; g ∊ F(X) are super modular, then • f g ∊ is super modular • αf is super modular for every α ≥ 0 Therefore the set of all super modular functions on a set X is a cone in F(X).
If f and g are nonnegative definite, increasing, and super modular functional on X, then so is their product f g defined by (fg)(x) = f(x)g(x)
If a firm produces many products, a straightforward generalization of the cost function c(w, y) measures the cost of producing the list or vector of outputs y when input prices are w. The production technology displays economies of joint production or economies of scope at y = y1, y2, . . . , ym)
Every convex game is super additive.
Show that a TP-coalitional game (N; w) is convex if and only if w(T ∪{i}) - w (T) ≥ w (S ∪ {i}) - w(S) for every i ∊ N and for every S ⊂ T ⊂ N / {i} The marginal contribution of every player increases with the size of the coalition to which the player is joined.
Is the cost allocation game (exercise 1.66) convex?
The definition of super modularity utilizes the linear structure of ℜ. Show that super modularity implies the following strictly ordinal property
Let f: X × Y → ℜ. Show that f displays increasing differences if and only if
Let f be a functional on X × Y where X and Y are chains. Show that f has increasing differences in (x, y) if and only if f is super modular on X × Y.
Using (2), show that for every x ∊ ℜ, • e-x = 1/ex • ex > 0 • ex → ∞ as x → ∞ and ex → 0 as x → - ∞ This implies that the exponential function maps ℜ onto ℜ+.
In the standard Bertrand model of oligopoly n firms each produce a differentiated product. The demand qi for the product of the ith firm depends on its own price and the price charged by all the other firms, that is,qi = f(pi, p-i)If each firm's production cost is measured by the cost function i,
Increasing differences implies the following ordinal condition, which is known as the single-crossing condition. For every x2 ≿ x1? and y2 ≿ y1?,
Prove corollary 2.1.1.If, in addition to the hypotheses of theorem 2.1, the objective function displays strictly increasing differences in (x, Î¸), the optimal correspondenceis always increasing. Every selection from
Prove corollary 2.1.1.If, in addition to the hypotheses of theorem 2.1, the objective function displays strictly increasing differences in (x, Î¸), the optimal correspondenceis always increasing. Every selection from
Show that the indirect utility functionis decreasing in p.
Consider the general constrained maximization problem where X is a lattice, Θ a poset and the feasible set G is independent of θ. The optimal solution correspondence is increasing in (θ, G) if and only if • f is quasi supermodular in X • and f satisfies the single crossing condition
f: X → Y is continuous if and only if the inverse image of any open subset of Y is an open subset of X.
f: X → Y is continuous if and only if the inverse image of any closed subset of Y is a closed subset of X.
f: X → Y is continuous if and only if f (x) limn→∞ f(xn) for every sequence xn → x. Care must be taken to distinguish between continuous and open mappings. A function f: X → Y is continuous if f-1(T) is open in X whenever T is open in Y. It is called an open mapping if f (S) is open in Y
Let f: X → Y be one-to-one and onto. Suppose that f is an open mapping. Then f has a continuous inverse f-1: Y → X.
The exponential function is ``bigger'' than the power function, that is,
If f is a continuous function from X to Y, the graph of f, graph(f) = {x, y) : y = f (x), x ∊ X} is a closed subset of X × Y. The converse of this result is not true in general. The following exercise details a partial converse in the special case in which the range Y is compact. Later we show
Suppose that Y is compact. f: X → Y is continuous if and only if graph(f) = {(x, y): y = f(x),x ∊ X} is a closed subset of X × Y.
If f: X → Y and g: Y → Z are continuous function, so is their composition g o f: X → Z.
Let ≿ be a continuous preference relation on ℜn+. Assume that ≿ is strongly monotonic. There exists a continuous function u: ℜn+ → ℜ which represents the preferences.
Prove proposition 2.3. Let f: X → Y be continuous. • f (X) is compact if X is compact • f (X) is connected if X is connected
Suppose that X is compact and f is a continuous one-to-one function from X onto Y. Then f is an open mapping, which implies that f -1 is continuous and f is a homeomorphism.
A functional f: X → ℜ is continuous if and only if its upper ≿f (a) = {x: f (x) > a} and lower contour sets ≿f (a) = {x: f (x) ≤ a} are both closed.
If f, g are continuous functionals on a metric space X, then • f + g is continuous • αf is continuous for every α ∊ R Therefore the set of all continuous functionals on X is a linear space.

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