linear programming 'increasing profit' Consider, $$\max 1.000.000x_1 + 2.500.000x_2 $$ \begin{align} s.t. x_1 + 2x_2 \le 7 \\ x_1 + 3x_2 \le 10 \\ -3x_1 + x_2 \le 0 \\ x_1, x_2 \ge 0\end{align}
which is an LP-problem on a company's wishes to maximise profit given certain constraints on the production of product 1 ($x_1$) and product 2 ($x_2$). 
The company has 7 liters of oil, product 1 requires 1 liter per production, and product 2 requires 2. This is the first equation. 
Now, here are the questions I have a problem with: 


*

*Can the company increase its profits if more oil can be bought?. If yes, what should the price be for that to happen (i.e. increased profits)?

*Can the company increase its profits by selling oil? If yes, what should the selling price be for more profit?


I suspect by 'more profit', they are comparing to the current optimal solution, which I found to be $x_1 = 1, x_2 = 3$.
I haven't done an exercise like this before, (have only theoretical experience, so these types of exercises geared towards econ-students always cause me trouble) so would like some help. Any ideas?
 A: As we discussed in the comments this brings in the concept of the "shadow price" or the dual variable associated with the constraint $x_1+2x_2\leq 7$. But I think we can get the answers without appealing to the dual.
For each question we are asking what happens if we perturb the first constraint to $x_1+2x_2\leq 7+\epsilon$, where $\epsilon$ is small.
In the first case, $\epsilon>0$. The optimal solution satisfies $x_1+2x_2=7$, so initially we might think that a positive $\epsilon$ would relax this constraint and allow the objective to increase. But in fact, both the second and third inequalities are active as well, and there is no way to increase $x_1$ or $x_2$ without violating one or both of these constraints. Therefore, purchasing more oil will simply make the first inequality inactive: $x_1+2x_2<7+\epsilon$. It will not increase production, and so no price is worth paying.
In the second case, $\epsilon<0$. We must reduce $x_1$ and/or $x_2$ to compensate. Reducing either will make the second constraint inactive. But the third constraint will remain active, so the complete basis is given by the first and third constraints. This leads to
$$\begin{aligned} x_1 + 2 x_2 &= 7+\epsilon \\ -3 x_1 + x_2 &= 0\end{aligned}
\quad\Longrightarrow\quad \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} =
\begin{bmatrix} 1 \\ 3 \end{bmatrix} + \epsilon
\begin{bmatrix} 1 & 2 \\ -3 & 1 \end{bmatrix}^{-1} \begin{bmatrix} 1 \\ 0 \end{bmatrix}.$$
The objective value will be (in millions)
$$8.5 + \epsilon \begin{bmatrix} 1 & 2.5 \end{bmatrix}
\begin{bmatrix} 1 & 2 \\ -3 & 1 \end{bmatrix}^{-1} \begin{bmatrix} 1 \\ 0 \end{bmatrix} = 8.5 + (17/14)\epsilon \approx 8.5 + 1.214\epsilon.$$
Therefore, the reduction in sales is approximately $\$1.214M$ per liter of oil. If you can sell oil for more than that amount, it is profitable to do so.
This is an interesting problem because it shows the potentially asymmetric nature of shadow prices. If you had computed the shadow price associated with $x_1+2x_2\leq 7$, you would have arrived at the $\$1.214M$ figure. However, because there were more than 2 active constraints at the optimal solution, this shadow price applied only to a decrease and not an increase.
