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Suppose $F(K,L)= 50L^{\frac{1}{2}}K^{\frac{1}{2}}$, the wage is $w = 5$ (euros) and rent is $r = 20$ (euros). What is the cost of producing $1000$ units? Derive the cost function $TC(Q)$.

I know how to find out the cost.

$L = 40$ and $K = 10$. So the total cost is $400$.

But now I am stuck. The solution for the cost function should be: $$TC(Q) = 5\cdot \dfrac{4Q}{100} + 20\cdot \dfrac{Q}{100} = \dfrac{2}{5}Q$$ How do I get to this function?

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You are trying to find the solution of $K$ and $L$ in terms of $Q$ to derive the total cost function with respect to quantity.

Page 13 of this lecture has the exact question you are interested in. (Google is your friend.)

  • From the Lagrangian you'll have $\frac{K}{L}=\frac{w}{r}$

  • From the production constraint you'll have $50K^{\frac{1}{2}}L^{\frac{1}{2}} = 1000$ and you have already derived the solution to this.

The first implies that $4K = L$ given your solution.

Then since, $50K^{\frac{1}{2}}L^{\frac{1}{2}} = Q$

$$\implies 50K^{\frac{1}{2}}(4K)^{\frac{1}{2}} = Q \\ \implies 100K = Q \\ \implies K(Q) = \frac{Q}{100}$$

A similar solution can be found for $L(Q)$, and then it should be smooth sailing for you from there.

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