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I am trying to find the long run total cost function, given the firm's production function $y=L^α K^β$ where $α,β>0$ and two inputs $L$ and $K$ where $ L,K∈R_+^2$, with factor prices $w$ and $r$ where $w,r∈R_{++}^2$

So far, I have tried to solve $min$ $⁡wL+rK s.t.y=L^α K^β$ which then I got $\mathcal{L}$$(L,K,λ)=wL+rK+λ(y-L^α K^β)$

Solving for FONC:

$\frac{∂L}{∂L}=0⟹w=αλL^{α-1} K^β$

$\frac{∂L}{∂K}=0⟹r=βλL^α K^{β-1}$

$\frac{∂L}{∂λ}=0⟹ y=L^α K^β$

Then dividing equation 1 by equation 2 I get:

$\frac{w}{r}= \frac{αλL^{α-1} K^β}{βλL^α K^{β-1} }⟹\frac{w}{r}= \frac{αK}{βL}⟹L=\frac{αr}{βw} K$

However, this is where I run into an issue...

When I substitute $L=\frac{αr}{βw} K$ back into equation 3 ($y=L^α K^β$), I get:

$y=(\frac{αr}{βw} K)^α K^β$

Which I don't know how to solve for $K$

I am wondering if I did something wrong, or how do I proceed to the next step, where I can solve for $K$ and $L$ and then proceed to use those values to find the long run total cost function.

Thanks

Pls. do like this $y=(\frac{αr}{βw} K)^α K^β = (\frac{αr}{βw})^α K^{(β+α)}$ and then

$$ y (\frac{βw}{αr})^α = K^{(β+α)}$$ and finally

$$ y^\frac{1}{{(β+α)}} (\frac{βw}{αr})^\frac{α}{(β+α)} = K := K^\star(w,r,y)$$

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    $\begingroup$ You have 3 unknowns $L, K$ and $\lambda$ and 3 first order conditions. This should allow you to compute $L, K$ and $\lambda$ as functions of the parameters. You have $L$ as a function of $K$ which is therefore not the final result. Next the solution should be substituted back into the budgets $w L + r K$ and not into the production function. $\endgroup$
    – tdm
    Commented May 4, 2021 at 15:00
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    $\begingroup$ Actually, this seems to be a duplicate of this older question: economics.stackexchange.com/questions/18736/… does answer there help you to find solution? $\endgroup$
    – 1muflon1 ♦
    Commented May 4, 2021 at 15:11
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    $\begingroup$ Remember that $y$ is treated as known. The cost function is a function of prices $w$ and $r$ and production level $y$. Similarly the conditional demand functions $K^\star$ and $L^\star$ are also functions of $y$ and prices. So if what you have done up until the last step is correct all you need to do is isolate $K$ then you have $K^\star(w,r,y)$. Do the same for labor and then $C(w,r,y) = wL^\star(w,r,y) + r K^\star(w,r,y)$. $\endgroup$
    – Jesper Hybel
    Commented May 4, 2021 at 15:52
  • $\begingroup$ @tdm Why do I sub it back into the budgets $wL+rK$? Since all my tutorial examples sub it back into the production constraint, so I am a little confused now $\endgroup$
    – DoubleRainbowZ
    Commented May 4, 2021 at 23:04
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    $\begingroup$ @DoubleRainbowZ I did it for you as an edit of your question ... now try to do it for labor on your own. $\endgroup$
    – Jesper Hybel
    Commented May 4, 2021 at 23:28

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