| № |
Condition |
free/or 0.5$ |
| m82986 | Suppose that x* = (x*1, x*2,..., x*n) is a Pareto-efficient allocation in an exchange economy (example 1.117) in which each of the n consumers has an endowment wi ∈ ℜl+ of the l commodities. Assume that
• Individual preferences ≿i are convex, continuous and strongly monotonic
• x* is a feasible allocation, that is
Show that there exists
• A list of prices p* ∈ ℜl+ and
• A system of lump-sum taxes and transfers t ∈ ℜn with ∑i ti = 0
such that (p*, x*) is a competitive equilibrium in which each consumer s after-tax wealth is mi = (p*)Twi + ti. |
buy |
| m82987 | Suppose that (x*, y*) is a local optimum of
subject to g(x, y) = 0 and y > 0
and a regular point of g. Then there exist multipliers λ1, λ2, . . . , λm such that
With |
buy |
| m82989 | 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. |
buy |
| m82990 | Suppose that Y is compact. The correspondence |
buy |
| m83041 | 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 choice or uncertainty, it is not appropriate to assume that the consumption set is finite dimensional. Then, compactness of the budget set is more problematic. Note, however, that finite dimensionality is not required to establish that the budget set is convex (exercise 1.232). |
buy |
| m83045 | The CES function
is convex on ℜn+ if p ≥ 1. |
buy |
| m83051 | The collection of all convex subsets of a linear space ordered by inclusion forms a complete lattice. |
buy |
| m83054 | 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 |
buy |
| m83055 | The conic hull of a set of vectors S is the smallest convex cone in X containing S. |
buy |
| m83056 | The constraint g satisfies the Slater constraint qualification condition if there exist ^x A X with Show that this implies that a > 0. |
buy |
| m83057 | 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. |
buy |
| m83059 | 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 a function from Y → X whish is called the inverse function. It is denoted f -l. |
buy |
| m83061 | The convex hull of a set of vectors S is the smallest convex subset of X containing S. |
buy |
| m83062 | The core of a TP-coalitional game is convex. |
buy |
| m83066 | The definition of super modularity utilizes the linear structure of ℜ. Show that super modularity implies the following strictly ordinal property |
buy |
| m83068 | The derivative of a function is unique. |
buy |
| m83070 | The determinant of symmetric operator is equal to the product of its eigenvalues. |
buy |
| m83076 | The dynamic programming problem (example 2.32)
subject to xt+1 ∊ G(xt), t = 0, 1, 2, . . . , x0 ∊ 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 × X
• G(x) is nonempty, compact-valued, and continuous for every x ∊ X
show that T is an operator on the space C(X) of bounded continuous functionals on X (exercise 2.85), that is Tv ∊ C(X) for every v ∊ C(X). |
buy |
| m83078 | The elasticity of a function f: ℜ → ℜ is defined to be
In general, the elasticity varies with x. Show that the elasticity of a function is constant if and only if it is a power function, that is,
f (x) = Axa |
buy |
| m83085 | The exponential function is ``bigger than the power function, that is, |
buy |