A variational problem under a monotonicity constraint

Question

Hi all,

We consider the set

$$ S = \left\lbrace (F,h)\;\;\middle\vert\;\genfrac{}{}{0pt}{}{F\text{ is a decreasing function from }R^{+}\text{ to }R^{+}, h\in R}{0=1- \dfrac{\theta + 1}{\theta} \dfrac {\int^{h}_{y=0} F(y) dy}{F(0)} \dfrac{F(0)-\frac{1}{2}F(h)}{F(0)-F(h)}} \right\rbrace $$

The function $L$ is defined on $S$ by

$$L(F,h) = \dfrac{\int^{h}_{x=0} \int^{h}_{y=x} F(y) dy dx}{\int^{h}_{x=0} \int^{h}_{y=0} F(y) dy dx} h$$

We denote by $L(\theta)$ the maximal value of L.

A special question: What is L(0.6) ?

A general question: What is $ max_{\theta \in (0.4 ,0.9)} \frac {L(\theta)}{\theta} $ ?

Context in which the problem arose: There is an increasing amount of congestion at sea straights through which ships pass. This increases the frequency of collisions and their damaging effects such as environmental pollution by oil tankers. It has been proposed that a tax be levied on each passage of a ship through a sea straight and that the revenue be used to fight AIDs, Malaria and TB. The tax would be charged for each ship that passes, irrespectively of the ship’s size. It has been raised as an objection to the tax that it will penalize transport via small ships more than transport via large ships. It is therefore useful to investigate whether in a market without such a tax ship sizes are below or above the social optimum (there is a trade-off: on the one hand, high-frequency transportation reduces the need for storage space in production sites, on the other hand larger ships are more fuel efficient.) In a model that I developed for that purpose the problem posted here arose. The model has only one parameter, namely θ, which is a positive constant that is estimated to be 0.6. If it turns out that the maximal value of the problem, $L(\theta)$, is smaller than θ for all realistic values of the parameter θ then this would suggest that the market without taxes leads to an inefficiently small ship size. This market failure would be corrected by the tax levied on each passage, so such a result could increase support for the proposal.

Answer 1

It seems that for the maximization problem, as it is, we have $\sup_{(F,h)\in S}=+\infty$, for all $\theta > 0$.

For $\lambda > 1$ consider the function $F_\lambda$ such that $F_\lambda(0)=\lambda$, and $F_\lambda(x)=1$ for all $x > 0$. Then there exists exactly one $h=h_\lambda > 0$ such that $(F_\lambda,h_\lambda)\in S\, ,$ namely, from the expression for the constraint,
$$h_\lambda:=\frac{\theta}{\theta+1}\frac{\lambda(\lambda-1)}{\lambda-1/2}\, .$$ For this choice we have

$$L(F_\lambda,h_\lambda)=\frac{h_\lambda}{2}\ ,$$ which implies that $L$ is unbounded over the class $S$. The same holds if we restrict to smooth decreasing functions, of course, approximating conveniently $F_\lambda$ and $h_\lambda$.