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About 3307 results. 1294 free access solutions
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Condition |
free/or 0.5$ |
| m82818 | Solve the problem
max x1x2
subject to x21 + 2x22 < 3
2x21 + x22 < 3 |
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| m82819 | Solve the problem
subject to 2x1 - 3x2 + 5x3 = 19 |
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| m82828 | Specify the set of unanimity games for the player set N = {1, 2, 3}. |
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| m82834 | Step 1. d(T) > 0 for every T ∈ δ.
Step 3. Construct a sequence (xn: xn ∈ Tn). This sequence has a convergent subsequence xm → x0.
Step 5. Consider the concentric ball Br/2(x). There exists some N such that xn ∈ Br/2(x) for every n > N.
Step 6. Choose some n > min{N, 2/r}. Show that Tn ⊂ Br(x) ⊂ S0.
This contradicts the assumption that Tn is a big set. Therefore we conclude that δ > 0. |
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| m82835 | Step 1. There exists a finite number of open balls Br(xn) such that
Step 3. {S1, S2,.......... Sn} is a finite cover for X.
Yet another useful characterization of compactness is given in the following exercise. A collection C of subsets of a set has the finite intersection property if every finite sub collection has a nonempty intersection. |
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| m82854 | 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. |
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| m82870 | Suppose that a consumer retires with wealth w and wishes to choose remaining lifetime consumption stream c1, c2, . . . , cT to maximize total utility
Assuming 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. |
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| m82872 | Suppose that a differentiable functional f on an open set S ⊆ ℜn is quasiconcave. At every regular point ∇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 regular points of f. It is pseudoconcave if it satisfies (37) at all points of its domain. That is, a differentiable functional on an open convex set S ∈ ℜn is pseudoconcave if
A function is pseudoconvex if -f is pseudoconcave. Pseudoconcave functions have two advantages over quasiconcave functions-every local optimum is a global optimum, and there is an easier second derivative test for pseudoconcave functions. Nearly all quasiconcave functions that we encounter are in fact pseudoconcave. |
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| m82874 | Suppose that a firm s technology is based on the following eight basic activities:
y1 = ( -3, -6,4,0)
y2 = ( -7, -9,3,2)
y3 = ( -1, -2,3,-1)
y4 = ( -8, -13,3,1)
y5 = ( -11, -19,12,0)
y6 = ( -4, -3,-2,5)
y7 = ( -8, -5,0,10)
y8 = ( -2, -4,5,-2)
which can be operated independently at any scal, The aggregate production possibility set is
Y = {y1; y2; y3; y4; y5; y6; y7; y8}
1. Show that it is impossible to produce output without using any inputs, that is,
This is called the no-free-lunch property.
2. Show that Y does not exhibit free disposal (exercise 1.12).
3. Show that activities y4, y5, y6, and y8 are ine½cient. (Compare with y2, y1, y7, and y3 respectively.)
4. Show that activities y1 and y2 are inefficient. (Compare with a combination of y3 and y7.)
5. Specify the set of efficient production plans.
The following results are analogous to those for convex hull. |
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| m82880 | 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, independent of capacity and independent across periods. Further assume that the marginal cost of capacity c0 is constant. With these assumptions, the total cost function is
The objective is to determine outputs yi(and hence prices pi) and production capacity Y to maximize social welfare as measured by total consumer and producer surplus. In any period i, total surplus is measured by the area between the demand and cost curves, that is,
So aggregate surplus is
The optimization problem is to choose nonnegative yi and Y so as to maximize (79) subject to the constraints
yi < Y, i = 1, 2, . . . , n
Show that it is optimal to price at marginal cost during o¨-peak periods, and extract a premium during peak periods, where the total premium is equal to the marginal cost of capacity c0. Furthermore, under this pricing rule, the enterprise will break even. Note that |
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| m82881 | Suppose that a random variable x is assumed to be normally distributed with (unknown) mean m and variance σ2 so that its probability density function is
The probability (likelihood) of a sequence of independent observations (x1, x2, . . . , xT) for given parameters m and σ2 is given by their joint Distribution |
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| m82887 | 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. |
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| m82903 | Suppose that f and g are differentiable functionals on ℜ such that
while
Show that
1. For every |
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| m82904 | Suppose that f and g are functionals on ℜ such that
If limx→a f′(x)/g′(x) exists, then |
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| m82905 | 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 provides the following useful alternative characterization of a convex function. |
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| m82906 | 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. |
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| m82907 | Suppose that f ∈ C[a, b] is differentiable on the open interval (a, b). Then there exists some x ∈ (a, b) such that
f(b) - f(a) = f′[x](b - a) |
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| m82908 | 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 |
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| m82910 | Suppose that f is a monotonic transformation of a homogeneous function. Show that f is a monotonic transformation of a linearly homogeneous function. |
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| m82911 | Suppose that f is bilinear and that
Then
F(x1 - x2, θ1 - θ2) > 0 |
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