By Claudio Irigoyen

ISBN-10: 067400888X

ISBN-13: 9780674008885

Ideas handbook to the vintage textbook Recursive equipment in financial Dynamics. The workouts within the e-book are critical to the textual content. This handbook offers exact solutions to the issues in Recursive tools. Softcover.

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**Example text**

At x. Given x, [0, f (x)] is compact and hence Γ(x) is compact-valued. Take arbitrary sequences xn → x and yn ∈ Γ(xn ). Define ² = supxn kxn − xk and let N (x, ²) denote the closed ²−neighborhood of x. Since the set £ ¤ ¡ ¢ {z : z ∈ 0, f , f = max f x0 }, x0 ∈N(x,²) is compact, there exists a convergent subsequence of yn call it ynk with lim ynk ≡ y. Since ynk ≤ f (xnk ) all k, we know that y ≤ f (x) by the continuity of f and standard properties of the limit. c. at x. Since x was chosen arbitrarily, Γ is a continuous correspondence.

We have to consider four cases: i) x = x0 ii) x = αx0 for any α ∈ R, and x 6= 0, x0 6= 0 iii) x 6= αx0 for any α ∈ R, and x 6= 0, x0 6= 0 iv) x 6= x0 and x = 0 or x0 = 0 i) and ii) are trivial. iii) Suppose f (x) ≥ f (x0 ), then f being homogeneous of degree one and non-negative implies that f (x)/f (x0 ) ≥ 1, so there exist a number γ ∈ (0, 1) such that γf (x) = f (γx) = f (x0 ). Hence for any λ ∈ (0, 1) we may write f (γx) = λf (γx) + (1 − λ)f (x0 ) = f (x0 ). By the assumed quasi-concavity of f , for any ω ∈ (0, 1), f [ωγx + (1 − ω)x0 ] ≥ ωf (γx) + (1 − ω)f (x0 ).

Then (1, 0) + (0, 1) = (1, 1) which is not an element of the unit circle. f. Choose α ∈ (0, 1). Then 1 ∈ I but α1 ∈ / I, which violates the definition of a real vector space. g. Let f : [a, b] → R+ , and α < 0, then αf ≤ 0, which does not belong to the set of nonnegative functions on [a, b] . 3 a. Clearly, the absolute value is real valued and well defined on S × S. Take three diﬀerent arbitrary integers x, y, z. The nonnegativity property holds trivially by the definition of absolute value. Also, ρ(x, y) = |x − y| = |y − x| = ρ(y, x) by the properties of the absolute value, so the commutative property holds.

### Solutions manual for recursive methods in economic dynamics by Claudio Irigoyen

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