Understanding and finding a marginal cost from a dual program

Question

A lumberjack has 60 square meter of forest which can be used as joinery wood of type A or B or let it as fuel wood. Joinery wood running cost are $60€.sqm^{-1}$ for type A, $70€.sqm^{-1}$ for type B and $40€.sqm^{-1}$ for fuel wood. Earnings are $90€.sqm^{-1}$, $90€.sqm^{-1}$ and $70€.sqm^{-1}$. The lumberjack has only $2600€$ to invest. As far as $x_1=10sqm$ and $x_3=50sqm$is optimal, what would be an acceptable price for him to rent some more fields? What would be an acceptable interest price to borrow money?

Therefore the Primal of this linear program is:

\begin{align*} (P): \begin{cases} \max &90 x_1 &+90x_2 &+60x_3\\ &60 x_1 &+70x_2 &+40x_3 &\le 2600\\ &x_1 &+x_2 &+x_3 &= 60\\ &\forall i, &x_i\ge 0 \end{cases} \end{align*}

And its dual would be

\begin{align*} (D): \begin{cases} \min &2600 y_1 &+y_2 &\\ &60 y_1 &+y_2 &\ge 90\\ &70y_1 &+y_2 &\ge 90\\ &40y_1&+y_2&\ge 60\\ &\forall i, &y_i\ge 0 \end{cases} \end{align*}

Unless we have something to do with the second contraint of the primal.

1. Can you help me understand what $y_i$ means in real life?
2. I understood the problem for leasing fields but it seems to me that it is the same dual program. Yet, I don't understand how do we now one price should be a minimal one and the other the maximal one. I understand that we have to maximize a leasing price and minimize a renting price but how do we show it mathematically? Economically? Do I have to create another Primal or the dual is sufficient to tell me so?
3. I don't know how to model the price of an acceptable interest price to borrow money at all...

Answer 1

Your dual is incorrect. You should have $y_2 \in \mathbb{R}$.

1. $y_1$ is by how much the primal objective value increases if the right hand side $2600$ is increased by $1$ (if no change of basis occurs). The same holds for $y_2$ and $60$.
2. The dual can be interpreted as minimizing the lease price. If you understand that, what would you like to show? Mathematically this is LP duality.
3. this should become clear from the answer to 1