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free/or 0.5$ |
| m83368 | Use the definition of a linear space to show that
1. x + y = x + z ^ y = z
2. ax = ay and a ≠ 0 ⇔ x = y
3. ax = βx and x ≠ 0 ⇔ a = fi
4. (α - β)x = αx - βx
5. α(x - y) = αx - αy
6. α0 = 0
for all x, y, z ∈ X and α, β ∈ R. |
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| m83384 | Use the Kuhn-Tucker conditions to prove the Farkas alternative (proposition 3.19). [Consider the problem maximize cTx subject to Ax < 0.] |
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| m83407 | 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+. |
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| m83414 | Using the natural order > on R, order the plane R2 by the lexicographic order. It is a total order? |
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| m83415 | Using the previous exercises, prove Arrow s impossibility theorem. |
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| m83435 | Verify that the Shapley value is a feasible allocation, that is,
This condition is sometimes called Pareto optimality in the literature of game theory. |
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| m83436 | 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 compact set. A function f: X → Y is called compact if f(X) is contained in a compact set of Y. |
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| m83437 | Verify that x* = limkʹ→∞ xkʹ as defined in the preceding proof is a fixed point of the correspondence, that is x* ∊ |
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| m83442 | Verify these assertions directly. |
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| m83450 | We will used this property in the following form (see exercise 1.108). |
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| m83467 | What are the coordinates of the vector (1, 1, 1) with respect to the basis {(1, 1, 1), (0, 1, 1,), (0, 0, 1)}? What are its coordinates with respect to the standard basis {(1, 0, 0, (0, 1, 0,), (0, 0, 1)}? |
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| m83482 | What can we say about the concavity/convexity of the simple power functions f(x) = xn, n = 1, 2, . . . over ℜ. |
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| m83497 | 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.] |
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| m83504 | What is the boundary of the set S = {1/n : n = 1, 2, . . .}? |
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| m83530 | What is the linear hull of the vectors {(1, 0), (0,2)} in R2? |
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| m83536 | 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 the open ball about x0 of radius r is the set
By linearity, this can be expressed as
The unit ball B is the open ball about 0 of radius 1
It is the set of all elements of norm less than 1. Any open ball can be expressed in terms of the unit ball as follows:
Br(x0) = x0 + rB
That is, any open ball in a normed linear space is simply a translation and scaling of the unit ball. Therefore many important properties of a normed linear space are related to the shape of its unit ball. Figure 1.13 illustrates the unit ball in the plane (R2) for some different norms. The uniform structure of a normed linear space enables the following refinement of exercise 1.93. |
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| m83584 | What steady state unemployment rates are implied by the transition probabilities in table 3.1? |
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| m83593 | When the roles are reversed in the general power function, we have the general exponential function defined as
f(x) = ax
where a ∈ ℜ+. Show that the general exponential function is differentiable with derivative
Dxf(x) = ax log a |
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| m83595 | 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 maximizing competitive firm producing a single output.
The following exercise shows the importance of allowing the value function to take infinite values. |
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| m83639 | Why can a sub simplex have no more than two distinguished faces? |
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