| № |
Condition |
free/or 0.5$ |
| m82920 | Suppose that f: X → Y is a linear function with rank f = rank Y ≤ rank X. Then f maps X onto Y. |
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| m82921 | Suppose that f: X → Y is a linear function and B ⊆ X is a basis for X. Then f (B) spans f (X). |
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| m82922 | Suppose that f: X → Y is a linear function. If X is finite-dimensional, then rank f + nullity f = dim X |
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| m82923 | Suppose that f: X → Y is a linear function with kernel f = {0}. Then f is one-to-one, that is,
f(x1) = f(x2) ⇒ x1 = x2
A linear function f: X → Y that has an inverse f-1: Y → X is said to be nonsingular. A function that does not have an inverse is called singular. |
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| m82924 | Suppose that f: X → Y is differentiable at x and its derivative is nonsingular. Suppose further that f has an inverse f-1: Y → X that is continuous (i.e., f is a homeomorphism). Then f-1 is differentiable at x with |
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| m82925 | Suppose that f0 is a linear functional defined on a subspace Z of X such that
f0(x) ≤ g(x) for every x ∈ Z
where g ∈ X* is convex. Show that
1. The sets
and
are convex subsets of the linear space Y = X × ℜ (figure 3.21)
2. int A ≠ ∅ and int A∩B = ∅.
3. There exists a linear functional |
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| m82928 | Suppose that
G {x ∈ X : gj(x) < 0; j = 1, 2, . . . ,m}
with gj is quasi convex. Let x* ∈ G and λ ∈ Rm+ satisfy the complementary slackness conditions λjgj (x*) = 0 for every j = 1, 2, . . . ,m. Then
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 L(c, z) = az - λTc with a > 0 and λ > 0. [Use exercise 3.47 and apply (75) to the point c, z* + 1).] 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 ... As the previous example demonstrated, utility maximization places no restrictions on the slopes of individual demand functions. However, the consumer s demand must always satisfy the budget constraint, which places certain ... |
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| m82929 | 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. |
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| m82938 | Suppose that nominal GDP rose 10 percent in your country last year, while prices rose 5 percent. What was the growth rate of real GDP? |
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| m82945 | Suppose that the allocation of $1 among three persons is to be decided by majority vote. Specify the characteristic function. |
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| m82947 | 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, θ) ≤ 0, g2(x, θ) ≤ 0, ... , gm (x, θ) ≤ 0
If each functional gj (x; θ) ∊ F(X × Θ) is convex jointly in x and y, then the correspondence
G(θ) = {x ∊ X: gj(x, θ) ≤ 0, j = 1,2, ... , m}
is convex. |
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| m82948 | 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. |
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| m82953 | Suppose that the linear model (section 3.6.1)
Ax = c
has been scaled so that aii = 1 for every i. Show the following: |
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| m82959 | Suppose that the production possibility set Y is convex. Then every efficient production plan y ∈ Y is profit maximizing for some nonzero price system p ≥ 0. |
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| m82968 | 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 conditions before deducing the comparative statics, as we did in example 6.14. Again, we emphasize that the theoretical foundation for this analysis remains the implicit function theorem. The difference is merely a matter of computation. |
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| m82969 | 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 a zero-sum game. |
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| m82982 | 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. Then the sequence xk = x* + akx ∈ G converges to x*. Furthermore xk - x* = akx and therefore the sequence xk - x*) = ak converges trivially to x. We say that xk converges to x* from the direction x (figure 5.9). |
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| m82983 | Suppose that (x*, λ) is a stationary point of the Lagrangean |
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| m82984 | 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. |
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| m82985 | Suppose that x* = (x*1, x*2,..., x*n) is a Pareto efficient allocation in an exchange economy with l commodities and n consumers (example 1.117). Assume that
• Individual preferences are convex, continuous and strongly monotonic.
• x*
Show that
1. The set
is the set of all aggregate commodity bundles that can be distributed so as to make all the consumers at least as well off as at the allocation x*.
2. S = ≿(x*) - x* is nonempty, convex and contains no interior points of the nonpositive orthant ℜl-.
3. There exist prices p* ∈ ℜl+ such that (p*)Tx ≥ (p*)Tx* for every x ∈ ≿(x*).
4. For every consumer i, (p*)Txi ≥ (p*)Tx*i for every xi ∈ ≻i(x*i).
5. (p*, x*) is a competitive equilibrium with endowments wi = x*i |
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