Digging into the paper "Dynamic Inventory Management with Cash Flow Constraints"

The paper “Dynamic Inventory Management with Cash Flow Constraints” published in Naval Research Logistics, 2008 gave a pioneering work on multi-period stochastic inventory problem with cash flow constraints.

The functional equation for the problem:

\[\begin{cases} V_n(x, w)=\max_{x\leq y\leq x+w/c}V_{n+1}((y-d)^+, p\min\{y, d\}-c(y-x)+w)\\ V_N(x,w)=w+rx \end{cases}\]

Now we need to prove the joint concavity of \(V_n(x,w)\). This is by induction.

\(V_n(x, w)\) is apparently jointly concave for \(n=N\).

Assume \(V_{n+1}(x, w)\) is jointly concave. We now prove the property for \(n\). This is by proving the joint concavity of \(V_{n+1}(x, y, w)\) in \(x\), \(y\) and \(w\).

For any \((x_1, y_1, w_1)\), \((x_2, y_2, w_2)$, and\)0\leq \lambda\leq 1\(,\)0\leq \overline{\lambda}\leq 1\(,\)\lambda+\overline{\lambda}=1$$, we need to prove:

\[\begin{align} &V_{n+1}((\lambda y_1+\overline{\lambda}y_2-d)^+,p\min\{\lambda y_1+\overline{\lambda}y_2,d\}-c(\lambda y_1+\overline{\lambda}y_2-\lambda x_1)+w)\\ \leq & \lambda V_{n+1}((y_1-d)^+, p\min\{y_1, d\}-c(y_1-\lambda x_1-\overline{\lambda} x_2)+w)\\ & + \overline{\lambda}V_{n+1}((y_2-d)^+, p\min\{y_2, d\}-c(y_2-\lambda x_1-\overline{\lambda} x_2)+w) \end{align}\]

Because of the convexity of \((y-d)^+\),

\[(\lambda y_1+\overline{\lambda}y_2-d)^+\leq \lambda(y_1-d)^++\overline{\lambda}(y_2-d)^+\]

Because of the concavity of \(\min\{y, d\}\),

\[p\min\{\lambda y_1+\overline{\lambda}y_2,d\}\geq \lambda p\min\{y_1, d\}+\overline{\lambda}p\min\{y_2, d\}\]

And

\[\begin{align} &V_{n+1}((\lambda y_1+\overline{\lambda}y_2-d)^+,p\min\{\lambda y_1+\overline{\lambda}y_2,d\}-c(\lambda y_1+\overline{\lambda}y_2-\lambda x_1-\overline{\lambda} x_2)+w)\}\\ &=V_{n+1}((\lambda y_1+\overline{\lambda}y_2-d)^+,p(\lambda y_1+\overline{\lambda}y_2)-p(\lambda y_1+\overline{\lambda}y_2-d)^+\\ &\quad -c(\lambda y_1+\overline{\lambda}y_2-\lambda x_1-\overline{\lambda} x_2)+w) \end{align}\]

Since \(V_n(z, A-pz)\) is decreasing in \(z\) (this is proved by that the domain of \(y\) narrows as the increasing of \(z\)), we can obtain:

\[\begin{align} &V_{n+1}((\lambda y_1+\overline{\lambda}y_2-d)^+,p\min\{\lambda y_1+\overline{\lambda}y_2,d\}-c(\lambda y_1+\overline{\lambda}y_2-\lambda x_1-\overline{\lambda} x_2)+w)\}\\ &\geq V_{n+1}(\lambda(y_1-d)^++\overline{\lambda}(y_2-d)^+,p(\lambda y_1+\overline{\lambda}y_2)-p(\lambda(y_1-d)^++\overline{\lambda}(y_2-d)^+\\ &\quad -c(\lambda y_1+\overline{\lambda}y_2-\lambda x_1-\overline{\lambda} x_2)+w)\\ &=V_{n+1}(\lambda(y_1-d)^++\overline{\lambda}(y_2-d)^+,p\lambda y_1-p\lambda(y_1-d)^--c\lambda (y_1-x_1)\\ &\quad +p\overline{\lambda} y_2-p\overline{\lambda}(y_2-d)^--c\overline{\lambda} (y_2-x_2)+w)\\ &\geq \lambda V_{n+1}((y_1-d)^+, p\min\{y_1, d\}-c(y_1-x_1)+w)\\ &\quad+\overline{\lambda}V_{n+1}((y_2-d)^+, p\min\{y_2, d\}-c(y_2-x_2)+w) \end{align}\]

The second inequality is justified by the concavity of \(V_{n+1}(x, w)\) from assumption. ( I am a little puzzled by this point. It may be a little too quick to reach this, but the conclusion is right. )

Another method of proving concavity is through deducting the Heissian maxtrix. This is more complex.