  Suppose that $c(x,y)$ is the cost of transporting a unit of mass from $x\in
X$ to $y\in Y$ and suppose that a mass distribution $\mu$ on $X$ is transported
optimally (so that the total cost of transportation is minimal) to the mass
distribution $\nu$ on $Y$. Then, roughly speaking, the Kantorovich duality
theorem asserts that there is a price $f(x)$ for a unit of mass sold (say by
the producer to the distributor) at $x$ and a price $g(y)$ for a unit of mass
sold (say by the distributor to the end consumer) at $y$ such that for any
$x\in X$ and $y\in Y$, the price difference $g(y)-f(x)$ is not greater than the
cost of transportation $c(x,y)$ and such that there is equality
$g(y)-f(x)=c(x,y)$ if indeed a nonzero mass was transported (via the optimal
transportation plan) from $x$ to $y$. We consider the following optimal pricing
problem: suppose that a new pricing policy is to be determined while keeping a
part of the optimal transportation plan fixed and, in addition, some prices at
the sources of this part are also kept fixed. From the producers' side, what
would then be the highest compatible pricing policy possible? From the
consumers' side, what would then be the lowest compatible pricing policy
possible? In the framework of $c$-convexity theory, we have recently introduced
and studied optimal $c$-convex $c$-antiderivatives and explicit constructions
of these optimizers were presented. In the present paper we employ optimal
$c$-convex $c$-antiderivatives and conclude that these are natural solutions to
the optimal pricing problems mentioned above. This type of problems drew
attention in the past and existence results were previously established in the
case where $X=Y=R^n$ under various specifications. We solve the above problem
for general spaces $X,Y$ and real-valued, lower semicontinuous cost functions
$c$.
