| № |
Condition |
free/or 0.5$ |
| m82691 | Show that the cost function c(w, y) of a competitive firm (example 2.31) is concave in input prices w. |
buy |
| m82692 | Show that the cost function c(w, y) of a competitive firm (example 2.31) is homogeneous of degree one in input prices w. |
buy |
| m82694 | Show that the definition (2) can be equivalently expressed as |
buy |
| m82695 | Show that the derivative of a functional on ℜn can be expressed as the inner product |
buy |
| m82697 | Show that the eigenvectors corresponding to a particular eigen-value, together with the zero vector 0X, form a subspace of X |
buy |
| m82703 | Show that the function defined in the previous example is bilinear. There is an intimate relationship between bilinear functionals and matrices, paralleling the relationship between linear functions and matrices (theorem 3.1). The previous example shows that every matrix defines a bilinear functional. Conversely, every bilinear functional on finite dimensional spaces can be represented by a matrix. |
buy |
| m82704 | Show that the function f: ℜ3 → 3 ℜ2 defined by
f(x1, x2, x3) = (x1 = x2, 0)
is a linear function. Describe this mapping geometrically. |
buy |
| m82705 | Show that the function f defined in the preceding example is bilinear. |
buy |
| m82708 | Show that the function f(x) = 10x - x2 represents the total revenue function for a monopolist facing the market demand curve
x = 10 - p
where x is the quantity demanded and p is the market price. In this context, how should we interpret g(x) = 9 + 4x? |
buy |
| m82711 | Show that the gradient of a differentiable function f points in the direction of greatest increase. |
buy |
| m82712 | Show that the gradient of a differentiable functional on ℜn comprises the vector of its partial derivatives, that is, |
buy |
| m82715 | Show that the identity function IX (example 2.5) is strictly increasing. |
buy |
| m82716 | Show that the indirect utility function
is decreasing in p. |
buy |
| m82717 | Show that the indirect utility function is quasi convex. |
buy |
| m82718 | Show that the indirect utility function v(p,m) (example 2.90) is homogeneous of degree zero in p and m.
Analogous to convex functions (proposition 3.7), linearly homogeneous functions can be characterized by their epigraph. |
buy |
| m82719 | Show that the infinite geometric series x + βx + β2x + . . . converges provided that| β| < 1 with |
buy |
| m82721 | Show that the ith partial derivative of the function f: ℜn → ℜ at some point x0 corresponds to the directional derivative of f at x0 in the direction ei, where
ei = (0, 0, . . . , 1, . . . , 0)
is the i unit vector. That is, |
buy |
| m82722 | 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, θ0). |
buy |
| m82725 | Show that the metric p(x,y) = || x - y|| satisfies the properties of a metric, and hence that a normed linear space is a metric space. |
buy |
| m82727 | Show that the objective function in the preceding example displays increasing differences in (p, θ). |
buy |