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Suppose that a firm produces a good using capital, skilled labor, and unskilled labor. Let $K$ denote the amount of capital,$L_1$ unskilled labor, $L_2$ skilled labor. The production function is $f(L_1, L_2, K) = K^2 min\{L_1, L_2^{\frac{1}{3}}\}$. Furthermore, let capital rental rate, $r=200$, unskilled wage rate is $w_1=5$, and skilled wage rate is $w_2=6$.

Find long run cost function.

I am unsure as to how to approach this problem given the K^2 term in the function. Thus, I tried when $K=1$. However, I would like to know how to approach it for all $K$, where $K$ is a natural number.

My Attempt

Assume $K=1$. Then we have $$f(L_1, L_2, 1) = min\{L_1, L_2^{\frac{1}{3}}\}$$

From here we can see that in order to minimize cost at a particular level $q$, we have $L_1=L_2^{\frac{1}{3}}=q$ which leads to $$L_1=q$$ $$L_2=q^3$$. Thus, $$c(q)=200+5q+6q^3$$

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Your reasoning that $L_1=L_2^{1/3}$ is valid for any $K$. Indeed if this equality does not hold you can lower the cost by reducing the excess input of either skilled or unskilled labor. Thus you can rewrite the cost minimization problem in two dimensions instead of three

\begin{equation} \min_{K,L_1}{200 K + 5 L_1 + 6 L_1^3} \end{equation} subject to \begin{equation} K^2 L_1 = q \end{equation}

From the constraint you can replace $L_1$ by $q/K^2$ in the objective function, which becomes a minimization problem with one variable $K$ and a parameter $q$. Solving it will give you the cost function as the value function of the problem.

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