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I have a (probably) stupid question but I can't find the answer. I have the following problem (my problem is much more complicated but as an example) :

\begin{equation} \begin{matrix} \displaystyle \min_{\substack{q_1,K}} & J= CK + \sum_{i=1}^{10} price(i)(q_1(i)+q_2(i)) \\ \textrm{s.t.} & q_1(i) + q_2(i) & \leq & K \\ \end{matrix} \label{Lagrangian_Multiplier} \end{equation}

So I have one fix storage cost, and then at each period I buy a quantity $q_1 + q_2 $ at price $price(i)$. I need to choose the capacity of my storage and how much $q_1$ do I buy.

The lagrangian is $CK + \sum_{i=1}^{10} price(i)(q_1(i)+q_2(i)) - \lambda(q_1 + q_2 - K)$

\begin{equation*} \begin{aligned} & \frac{\partial J}{\partial q_1}(t)& = & \sum_{i=1}^{10} p(i) - \lambda \\ & \frac{\partial J}{\partial \lambda}(t)& = & - q_1(i) - q_2(i) + K \end{aligned} \end{equation*}

and those two derivatives have to be zero. That would mean that $\lambda$ (my shadow cost) is equal to the sum of all what I bought during the period 1 to 10 ? That does not seems right to me intuitively. I must confuse different notion. Can you explain me where I am wrong. My though are that the shadow price should be greater than zero only when q_1 + q_2 = K

Best.

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I don't understand all of the notation or terminology but it seems you've misapplied the idea of Lagrange multipliers.

Look at your equation $\frac{\partial J}{\partial \lambda} = 0$. This means that $q_1 + q_2 = K$. This is a constraint which you enforce on the problem (not $q_1 + q_2 \le K$ which is what you seem to want).

Lagrange multipliers are used for optimisation of variables with strict constraints that reduce the dimensionality of the problem whereas your constraint does not seem strict enough to use Lagrange multipliers.

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