Lagrangian optimisation problem

Question

If labour $L > 0$ and capital $K > 0$ and they are used to produce 2 goods (1 and 2) so we have

$$ L = l_1 + l_2 , $$ $$ K = k_1 + k_2 , $$

and all $l_i$ and $k_i$ are non-negative. Production function of each goods is given by

$$ q_i = \sqrt{l_i * k_i} .$$

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The question is how can I set up a parametric constrained Lagrangian function to maximize $q_2$ with parameter $q_1$, $L$, $K$? More specifically, what should the constraint be?

Cheers!

Answer 1

The objective function is given by $\sqrt{l_2k_2}$. Let $q_1$ denote some fixed quantity of good $1$. The equality constraints are: $$\begin{align*}l_1+l_2&=L\\ k_1+k_2&=K\\ \sqrt{l_1k_1}&=q_1\end{align*}$$

The problem is (ignoring the non-negativity constraints): $$\max_{l_1,l_2,k_1,k_2} \sqrt{l_2k_2}\qquad \text{subject to $\qquad \begin{align*}l_1+l_2&=L\\k_1+k_2&=K\\\sqrt{l_1k_1}&=q_1\end{align*}$}$$

The Lagrangian is given by $$\sqrt{k_2l_2}+\lambda_L(L-l_1-l_2)+\lambda_K(K-k_1-k_2)+\lambda(q_1-\sqrt{l_1k_1}).$$

Alternatively, if we substitute for $l_1$ and $k_1$ using the first two constraints, then we can write the problem as $$\max_{l_2,k_2} \sqrt{l_2k_2}\qquad \text{subject to $\qquad \sqrt{(L-l_2)(K-k_2)}=q_1$}$$ for which the Lagrangian is

$$\sqrt{k_2l_2}+\lambda\left(q_1-\sqrt{(L-l_2)(K-k_2)}\right).$$

Answer 2

Solving the four equations as a function of $q_{1}$, $K$ and $L$, we obtain the relations:

$k_{1}=\frac{\sqrt{K}}{\sqrt{L}}q_{1}$,

$k_{2}=\frac{\sqrt{K}(\sqrt{K L}-q_{1})} {\sqrt{L}}$,

$l_{1}=\frac{\sqrt{L}}{\sqrt{K}}q_{1}$,

$l_{2}=\frac{\sqrt{L}(\sqrt{K L}-q_{1})}{\sqrt{K}}$.

The function to be maximized becomes:

$q_2 = \sqrt{l_2 * k_2} =\sqrt{K L}-q_1$.