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Foundations of Mathematical Economics 1st edition Michael Carter - Solutions

Let f: S †’ Y be a differentiable function on an open convex set S Š† X. Then for every x1, x2 ˆˆ S,where A = {y ˆˆ Y: y = Df[] (x1 - x2) for some ˆˆ [x1, x2]}. Exercise 4.45 is illustrated in figure 4.8. The affine functions f(x1) + Df[x1](x2 -
Assume that f and g continuous functionals on [a, b] Š† „œ that are differentiable on the open interval (a, b). Then there exists some x ˆˆ (a, b) such thatProvided g(a) ‰ g(b) and g€²(x) ‰ 0, this can be written as
Suppose that f and g are functionals on „œ such thatIf limx†’a f€²(x)/g€²(x) exists, then
Recall that the CES function is homogeneous of degree one (exercise 3.163), while the Cobb-Douglas function is homogeneous of degree a1 + a2 + ... + an (example 3.69). In the previous example we assumed that a1 + a2 + ... + an = 1. What is the limit of the CES function when a1 + a2 + ... + an ≠ 1?
Let c(y) be the total cost of output y. Assume that c is differentiable and that there are no fixed costs (c(y) = 0). Show that the average cost of the first unit is equal to marginal cost.
If f: X → Y is differentiable at x0, then f is continuous at x0.
Suppose that f and g are differentiable functionals on „œ such thatwhile Show that 1. For every
If f: X → Y and g: Y → Z are Cn, then the composite function g ∘ f: X → Z is also Cn.
Compute the second-order partial derivatives of the quadratic function
Compute the Hessian of the quadratic function
Let f be a twice differentiable functional on some open interval S Š† „œ containing x0. For every x1 ˆˆ S, there exists some between x0 and x1 such that
Let f be a (n + 1)-times differentiable functional on some open interval S Š† „œ containing x0. For every x ˆˆ S - x0, there exists some between x0 and x0 + x such that
Let f be C3 on some open interval S Š† „œ containing x0. For every x ˆˆ S - x0,where That is, the approximation error becomes very small as x †’ 0. In this sense, (18) is the best quadratic approximation to f in a neighborhood of x0.
Assume that f is a Cn+1 functional on a convex set S. For fixed x0 ˆˆ S and x ˆˆ S - x0, define g: „œ †’ S by g(t) = x0 + tx. Show that the composite function h = f ˆ˜ g: „œ †’ „œ is Cn+1 with derivatives
Compute the second-order Taylor series expansion of the quadratic Functionaround the point (0, 0).
Let f be C1 and suppose that Df[x0] is one-to-one. Then 1. f is locally one-to-one, that is, there exists a neighborhood S of x0 such that f is one-to-one on S 2. f has an inverse f-1: f(S) → S that is continuous 3. f is locally onto
Let f: S → ℜn be a differentiable function on a convex set S ⊆ ℜn. Suppose that the Jacobian Jf(x) is positive (or negative) definite for all x ∈ S. Then f is one-to-one.
Suppose that all the functions in the IS-LM are linear, for example, C(y, T) = C0 + Cy (y - T) I(r) = I0 + Irr L(Y, r) = L0 + Lrr + Lyy Solve for r and y in terms of the parameters G, T, M and C0, I0, L0, Cy, Ir, Lr, Ly.
Show that F is Cn with DF[x0, y0] is nonsingular.
Under what circumstances, if any, could the IS curve be horizontal?
Determine the slope of the LM curve. Under what conditions would the LM curve be vertical?
A differentiable functional f on an open, convex set S is convex if and only iffor every x, x0 ˆˆ S. f is strictly convex if and only if for every x ‰ x0 ˆˆ S.
Suppose that f and h are convex functionals on a convex set S in Euclidean space with f differentiable at x0 and f(x0) = h(x0) and f(x) ≥ h(x) for every x ∈ S (31) Then h is differentiable at x0, with Dh[x0] = Df[x0]. When X ⊆ ℜn, the derivative can be represented by the gradient, which
A differentiable function f on a convex, open set S in „œn is convex if and only iffor every x0; x ˆˆ S. f is strictly convex if and only if for every x ‰ x0 ˆˆ S.
A differentiable function f on an open interval S ⊆ ℜ is (strictly) convex if and only if f′ is (strictly) increasing. Combined with exercises 4.35 and 4.36, this means that convex and concave functions on R can be identified through their second derivatives.
What can we say about the concavity/convexity of the simple power functions f(x) = xn, n = 1, 2, . . . over ℜ.
A differentiable functional f on an open set S Š† „œn is quasiconcave if and only ifInequality (36) can be strengthened where f is regular.
Suppose that a differentiable functional f on an open set S Š† „œn is quasiconcave. At every regular point ˆ‡f(x0) ‰ 0,A restricted form of quasiconcavity is useful in optimization (see section 5.4.3). A function is quasiconcave if it satisfies (37) at
A differentiable function f: S → R is pseudoconvex if f(x) < f(x0) ⇒ ∇f(x0)T(x - x0) < 0 for every x, x0 in S
Show that 1. Every differentiable concave function is pseudoconcave. 2. Every pseudoconcave function is quasiconcave
Is the CES functionpseudoconcave?
If a differentiable functional f is homogeneous of degree k, its partial derivatives are homogeneous of degree of k - 1.
Evaluate the error in approximating the functionby the linear functionat the point (2, 16). Show that the linear functionis a better approximation at the point (2, 16).
If f satisfies (40) for all x, it is homogeneous of degree k.Euler's theorem can be thought of as a multidimensional extension of the rule for the derivative of a power function (example 4.15). Two special cases are worth considering. When n = 1, equation (40) becomeswhich is precisely the rule for
Two inputs are said to complementary if their cross-partial derivative D2xixjf(x) is positive, since this means that increasing the quantity of one input increases the marginal productivity of the other. Show that if a production function of two inputs is linearly homogeneous and quasi concave,
In a generalization of the notion of elasticity of univariate functions (example 4.41), the elasticity the elasticity of scale of a functional f is defined bywhere the symbol |t=1 means that the expression is evaluated at t = 1. In general, the elasticity of scale varies with x. Show that the
A differentiable functional f homogeneous of degree k ≠ 0 is regular wherever f(x) ≠ 0.
If f is C2 and homogeneous of degree k with Hessian H, then xTHf(x)x = k(k - 1) f(x) Note that this equation holds only at the point x at which the Hessian is evaluated.
If f: S †’ „œ, S Š† „œn is strictly increasing, differentiable and homothetic, then for every i, j,
A subtle example that requires some thought is provided by the rank order function, which sorts vectors into descending order. Let r: ℜ
A small Pacific island holds the entire world stock K of a natural fertilizer. The market price p of the fertilizer varies inversely with the rate at which it is sold, that is,p = p(x); pÊ¹(x) where x denotes the number of tons of fertilizer mined and sold per year. Giving equal regard
Show that (0, 0) is the only stationary point of the function F(x1, x2) = x21 + x22 Is it a maximum, a minimum or neither?
A popular product called pfillip, a non narcotic stimulant, is produced by a competitive industry. Each firm in this industry uses the same production technology, given by the production function y = k1/6l1/3 where y is the amount of pfillip produced, k is the amount of kapitose (a special
Suppose that h: R → R is a monotonic transformation (example 2.60) of f: X → R. Then h o f has the same stationary points as f.
Suppose that a random variable x is assumed to be normally distributed with (unknown) mean m and variance Ïƒ2 so that its probability density function isThe probability (likelihood) of a sequence of independent observations (x1, x2, . . . , xT) for given parameters m and Ïƒ2 is given by their
Solvesubject to g(x) = x21 + x22 = 1
Analyze the consumer's problem where u(x) x1 a log x2 ensuring that consumption is nonnegative. For simplicity, assume that p1 = 1. Sometimes non negativity constraints apply to a subset of the variables, for which we have the following variation of theorem 5.2 and corollary 5.2.1.
Suppose that (x*, y*) is a local optimum ofsubject to g(x, y) = 0 and y > 0and a regular point of g. Then there exist multipliers Î»1, Î»2, . . . , Î»m such thatWith
Characterize the optimal solution of the general two-variable constrained maximization problem Subject to g(x1, x2) = 0 using the implicit function theorem to solve the constraint.
The consumer maximization problem is one in which it is possible to solve the constraint explicitly, since the budget constraint is linear. Characterize the consumer's optimal choice using this method, and compare your derivation with that in example 5.15.
Prove corollary 5.2.3.We will not prove the second part of corollary 5.2.2, since it will not be used elsewhere in the book (see Luenberger 1984, p. 226; Simon 1986, p. 85). Instead we develop below (corollary 5.2.4, section 5.4.3) some global conditions that are useful in practice.Suppose there
In the constrained optimization problemsuppose that f is concave and G(Î¸) convex. Then every local optimum is a global optimum. Another distinction we need to note is that between strict and non strict optima. A point x* A ˆˆ y is a strict local optimum if it is strictly
Suppose that (x*, Î»)† is a stationary point of the Lagrangean
If x* maximizes f(x) on G = {x ∈ X : g(x) 0}, then x* maximizes the Lagrangean L = f(x) - ∑ λjgj x on G.
Show that the volume of the vat is maximized by devoting one-third of the material to the floor and the remaining two-thirds to the walls.
Design a rectangular vat (open at the top) of 32 cubic meters capacity so as to minimize the required materials.
Solve the problemsubject to 2x1 - 3x2 + 5x3 = 19
Generalize the preceding example to solvesubject to p1x1 p2x2 = m
Solve the general Cobb-Douglas utility maximization problem
A common functional form in production theory is the CES (constant elasticity of substitution) functionIn this case the competitive firm's cost minimization problem is subject to f(x) = (a1xp1 + a2xp2)1/p = y
In the vat design problem, suppose that 48 square meters of sheet metal is available. Show that if the shadow price of sheet metal is 1, designing a vat to maximize the net profit function produces a vat of the optimal shape and exactly exhausts the available metal. Example 5.27 gives the
Show how the shadow price can be used to decentralize the running of the power company, leaving the production level at each plant to be determined locally.
In the constrained optimization problemsuppose that f is strictly quasi concave in x and G(Î¸) convex. Then every optimum is a strict global optimum.
Solve the problem max x1x2 subject to x21 + 2x22 < 3 2x21 + x22 < 3
Develop the conclusion of the preceding example to show that the regulated firm does not produce at minimum cost (see example 5.16).Example 5.16Suppose that (x*, y*) is a local optimum ofsubject to g(x, y) = 0 and y > 0 and a regular point of g. Then there exist multipliers Î»1,
Inequality constraints gj(x) < 0 can be transformed into to equivalent equality constraints by the addition of slack variables sj > 0, gj(x) sj = 0 Use this transformation to provide an alternative derivation of theorem 5.3 from theorem 5.2. Disregard the regularity condition.
An equality quality constraint g(x) = 0 can be represented by a pair of inequality constraints g(x) < 0, g(x) > 0 Use this transformation to derive theorem 5.2 from theorem 5.3. Disregard the regularity condition.
Use the Kuhn-Tucker conditions to prove the Farkas alternative (proposition 3.19). [Consider the problem maximize cTx subject to Ax < 0.]
In the previous example, verify that the binding constraints are regular at x* = (4, 0).
Derive and interpret the Kuhn-Tucker conditions for the consumer's Problemsubject to pTx constraining consumption to be nonnegative, while allowing the consumer to spend less than her income. We will show later (example 5.41) that the regularity condition is satisfieed.
Suppose that x* is a local solution of maxx∈G f(x). Then H+(x*) ∩ D(x*) = ϕ. Unfortunately, the set of feasible directions does not exhaust the set of relevant perturbations, and we need to consider a broader class. Suppose that x ∈ D(x*) with x* + ax ∈ G for every 0 < a < . Let ak 1/k.
The cone T(x*) of tangents to a set G at a point x* is a nonempty closed cone. See figure 5.10.Figure 5.10 Examples of the cone of tangents
Show that D(x*) ⊂ T(x*). Clearly, D(x*) ⊂ T(x*), and for some x* A G it may be proper subset. The significance of this is revealed by the following proposition. To adequately determine local optima, it does not suffice to consider only perturbations in D(x*).
Prove proposition 5.1 formally.A small Pacific island holds the entire world stock K of a natural fertilizer. The market price p of the fertilizer varies inversely with the rate at which it is sold, that is,p = p(x); pÊ¹(x) where x denotes the number of tons of fertilizer mined and sold
Show that T(x*) ⊂ L(x*). The Kuhn-Tucker first-order conditions are necessary for a local optimum at x* provided that the linearizing cone L(x*) is equal to the cone of tangents T(x*), which is known as the Abadie constraint qualification condition. The Kuhn-Tucker conditions follow immediately
Show that
Show Quasi convex CQ ⇒ Cottle
Show that regularity ⇒ Cottle [Use Gordan's theorem (exercise 3.239).]
Show that gj concave ⇒ AHUCQ ⇒ Abadie For nonnegative variables, we have the following corollary.
Suppose thatG {x ˆˆ X : gj(x) with gj is quasi convex. Let x* ˆˆ G and Î» ˆˆ Rm+ satisfy the complementary slackness conditions Î»jgj (x*) = 0 for every j = 1, 2, . . . ,m. Thenfor every x ˆˆ G.
Solve the preceding problem starting from the hypothesis that xc > 0, xb = xd = 0. [If faced with a choice between xb > 0 and xd > 0, choose the latter.]
What happens if you ignore the hint in the previous exercise? Previous exercise Solve the preceding problem starting from the hypothesis that xc > 0, xb = xd = 0. [If faced with a choice between xb > 0 and xd > 0, choose the latter.]
Show that B is a convex set with a nonempty interior.
Show that A ∩ int B = ϕ.
Prove corollary 5.1.If x* is a local maximum of f in Rn+, then it is necessary that x* satisfywhich means that for every i,
Show that L(c, z) = az - λTc with a > 0 and λ > 0. [Use exercise 3.47 and apply (75) to the point c, z* + 1).]
The constraint g satisfies the Slater constraint qualification condition if there exist ^x A X with Show that this implies that a > 0.
Suppose that a public utility supplies a service, whose demand varies with the time of day. For simplicity, assume that demand in each period is independent of the price in other periods. The inverse demand function for each period is pi(yi). Assume that marginal production costs ci are constant,
Prove corollary 5.1.1. If 1. x* is a stationary point of f, that is, ∇f(x) = 0, and 2. f is locally strictly concave at x*, that is, Hf (x*) is negative definite then x* is a strict local maximum of f .
For x* to be an interior minimum of f(x), it is necessary that 1. x* be a stationary point of f, that is, ∇f(x*) = 0, and 2. f be locally convex at x*, that is, Hf (x*) is nonnegative definite If furthermore Hf(x*) is positive definite, then x* is a strict local minimum. The following result was
Suppose that f ∈ C[a, b] is differentiable on (a, b). If f (a) = f(b), then there exists x ∈ (a, b) where fʹ(x) 0.
Solve the problem
Show that the Jacobian J in (4) is nonsingular. Having determined that v is differentiable, let us now compute its derivative. To simplify the notation, we will suppress the arguments of the derivatives. Let fx denote Dx f [x0, θ0], the (partial) derivative of f with respect to x evaluated at (x0,
It is characteristic of microchip production technology that a proportion of output is defective. Consider a small producer for whom the price of good chips p is fixed. Suppose that proportion 1 - y of the firm's chips are defective and cannot be sold. Let c(y) denote the firm's total cost function
Suppose that there are only two inputs. They are complementary if D2fx1x2 > 0. Show that Dw1x2 < 0 if the factors are complementary and Dw1x2 > 0 otherwise. This is special case of example 6.10.With explicit constraints, it can be more straightforward to linearize the first-order
Prove proposition 5.2.
Suppose that f is bilinear and thatThen F(x1 - x2, Î¸1 - Î¸2) > 0
Suppose that the cost function of a monopolist changes from c1(y) to c2(y) with 0 < cʹ1(y) < cʹ2 (y) for every y > 0 Show that c2(y1) - c2 (y2) > c1(y1) - c1(y2) (15) where y*1 and y*2 are the profit maximizing output levels when costs are c1 and c2 respectively.
The preceding example is more familiar where the firm produces a single output and we distinguish inputs and outputs. Assume that a competitive firm produces a single output y from n inputs x = (x1, x2, . . . , xn) according to the production function y = f(x) so as to maximize profitAssume that
An input i is called normal its demand increases with output, that is, Dyx* (w, y) > 0. Otherwise (Dyx* (w, y) < 0, i is called an inferior input. Show that an input i is normal if and only if marginal cost Dyc [w, y] is increasing in wi.

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