The name is absent



(i) Monotonicity: If m(x) n(x)x R+ then

T(m)


max <^kl X2 + δ J


m(c(X, l) + v + u)f1(u)du g(v)dv+


δ

Xc


m(c(X, l) + r + v + u)f2(u)du g(v)dv


max <^kl X2 + δ J


n(c(X, l) + v + u)f1(u)du g(v)dv+


δ

Xc


n(c(X, l) + r + v + u)f2(u)du g(v)dv


as m(x) n(x) pointwise and the integral is a linear operator

T(n)


(ii) Discounting: For all a 0 there exits δ < 1 with


T(m + a)


max <^kl X2 + δ J


[m(c(X, l) + v + u) + a]f1(u)du g (v)dv+


δ                  [m(c(X, l) + r + v + u) + a]f2(u)du g(v)dv

Xcrit-c(X,l)  -∞


max
l


kl


X2 + δ


Xc

-∞


-c(X,l)


m(c(X, l)

-∞


+ v + u)f1 (u)du g(v)dv+


c(X,l)


m(c(X, l)

-∞


+ r + v + u)f2 (u)du g(v)dv


+ δa


T (m) + δa

The second line follows from the fact that the densities f1 (u), f2 (u) integrate to one.

Points (i) and (ii) are sufficient to show that T is a contraction mapping. This implies that we can
start with an arbitrary function
m() and repeated application ofT will converge to the unique fixed
point, the true value function.

We will next show that T maps concave functions into concave functions. Hence, if we start
with a concave function
m and repeatedly apply T , all resulting functions will be concave as well.
This implies that the unique attractor, the true value function, is concave as well.

Concavity: X1,X2 R+ define the optimal loading as l1 and l2, respectively. Note that
for the convex combination
X3 = θX1 + (1 θ)X2, where θ (0, 1), the convex combination

27



More intriguing information

1. How Offshoring Can Affect the Industries’ Skill Composition
2. The Determinants of Individual Trade Policy Preferences: International Survey Evidence
3. The name is absent
4. Commuting in multinodal urban systems: An empirical comparison of three alternative models
5. Innovation Policy and the Economy, Volume 11
6. PROJECTED COSTS FOR SELECTED LOUISIANA VEGETABLE CROPS - 1997 SEASON
7. Volunteering and the Strategic Value of Ignorance
8. The Interest Rate-Exchange Rate Link in the Mexican Float
9. Fiscal Rules, Fiscal Institutions, and Fiscal Performance
10. An Incentive System for Salmonella Control in the Pork Supply Chain