# Shadow price of linear program constraints

$$Maximise \quad -c_1-c_2X_3-c_3X_4$$ \begin{align} s.t \quad X_1=c_3+c_4X_3-c_5X_4 \\ X_2=c_6-c_7X_3+c_8X_4\\X_1,X_2,X_3,X_4,c_1,c_2,c_3,c_4,c_5,c_6,c_7,c_8 \ge 0\end{align}

$c$ denotes constants and $X$ denotes variables.

Now the solutions which maximize the objective value are $c_3$ and $c_6$ since when$X_3$ and $X_4$ are $0$ this gives the maximum objective value of $-c_1$. Or have I totally misunderstood the definition of shadow price?

Wouldn't shadow price of $X_1$ be 0 because there is nothing I can do to improve the optimal obejctive value from $-c_1$?

• The objective value will be maximized when $X_3$ and $X_4$ are $0$, and this occurs when $X_1=c_3$ and $X_2=c_6$. – user430765 Apr 4 '17 at 6:26
• Ok, now I follow. Thanks. – quasi Apr 4 '17 at 6:37
• So am I correct in my conclusion? – user430765 Apr 4 '17 at 6:52
• What does it mean to relax the constraint on $X_1$ by $1$ unit? Does it mean change $c_3$ to $c_3-1$? If so, and if $c_3 < 1$, then you can't have $X_3 = 0$ and $X_4 = 0$. – quasi Apr 4 '17 at 6:55
• Question doesn't specify, it just asks what the shadow price of each of the constraints are. The $c's$ stay the same. – user430765 Apr 4 '17 at 6:58