<p>I am trying to find the long run total cost function, given the firm's production function <span class="math-container">$y=L^α K^β$</span> where <span class="math-container">$α,β&gt;0$</span> and two inputs <span class="math-container">$L$</span> and <span class="math-container">$K$</span> where <span class="math-container">$ L,K∈R_+^2$</span>, with factor prices <span class="math-container">$w$</span> and <span class="math-container">$r$</span> where <span class="math-container">$w,r∈R_{++}^2$</span></p>
<p>So far, I have tried to solve <span class="math-container">$min$</span> <span class="math-container">$⁡wL+rK s.t.y=L^α K^β$</span> which then I got <span class="math-container">$\mathcal{L}$$(L,K,λ)=wL+rK+λ(y-L^α K^β)$</span></p>
<p>Solving for FONC:</p>
<p><span class="math-container">$\frac{∂L}{∂L}=0⟹w=αλL^{α-1} K^β$</span></p>
<p><span class="math-container">$\frac{∂L}{∂K}=0⟹r=βλL^α K^{β-1}$</span></p>
<p><span class="math-container">$\frac{∂L}{∂λ}=0⟹ y=L^α K^β$</span></p>
<p>Then dividing equation 1 by equation 2 I get:</p>
<p><span class="math-container">$\frac{w}{r}=  \frac{αλL^{α-1} K^β}{βλL^α K^{β-1} }⟹\frac{w}{r}=  \frac{αK}{βL}⟹L=\frac{αr}{βw} K$</span></p>
<p>However, this is where I run into an issue...</p>
<p>When I substitute <span class="math-container">$L=\frac{αr}{βw} K$</span> back into equation 3 (<span class="math-container">$y=L^α K^β$</span>), I get:</p>
<p><span class="math-container">$y=(\frac{αr}{βw} K)^α K^β$</span></p>
<p>Which I don't know how to solve for <span class="math-container">$K$</span></p>
<p>I am wondering if I did something wrong, or how do I proceed to the next step, where I can solve for <span class="math-container">$K$</span> and <span class="math-container">$L$</span> and then proceed to use those values to find the long run total cost function.</p>
<p>Thanks</p>
<p>Pls. do like this
<span class="math-container">$y=(\frac{αr}{βw} K)^α K^β = (\frac{αr}{βw})^α K^{(β+α)}$</span> and then</p>
<p><span class="math-container">$$ y (\frac{βw}{αr})^α = K^{(β+α)}$$</span> and finally</p>
<p><span class="math-container">$$ y^\frac{1}{{(β+α)}} (\frac{βw}{αr})^\frac{α}{(β+α)} = K := K^\star(w,r,y)$$</span></p>

CC BY-SA 4.0
2021-05-04T15:00:26.960Z
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.
tdm
CC BY-SA 4.0
2021-05-04T15:11:25.877Z
Actually, this seems to be a duplicate of this older question: https://economics.stackexchange.com/questions/18736/deriving-long-run-cost-function does answer there help you to find solution?
1muflon1
CC BY-SA 4.0
2021-05-04T15:52:30.120Z
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 $C(w,r,y) = wL^\star(w,r,y) + r K^\star(w,r,y)$.
Jesper Hybel
CC BY-SA 4.0
2021-05-04T23:04:15.303Z
@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
DoubleRainbowZ
CC BY-SA 4.0
2021-05-04T23:19:17.310Z
@1muflon1 I did take a look at the thread you posted, but I couldn't get the answer from that either (maybe I was making a mistake), but I did notice it was trying to find max $LaKb s.t. C=wL+rK$ which was not what my tutorial examples were doing, so I was confused. (They were doing min $ wL+rKs.t.y=LαKβ$ - not sure if they are equivalent)
DoubleRainbowZ
CC BY-SA 4.0
2021-05-04T23:22:18.083Z
@JesperHybel I can follow your process, however, as seen from my equation above, I can't seem to isolate $K$ to get $K⋆(w,r,y) $. I am not sure if I made an error earlier (I have triple checked and can't find anything wrong) or if I am missing something
DoubleRainbowZ
CC BY-SA 4.0
2021-05-04T23:28:15.320Z
@DoubleRainbowZ I did it for you as an edit of your question ... now try to do it for labor on your own.
Jesper Hybel
CC BY-SA 4.0
2021-05-04T23:33:02.963Z
Thanks @JesperHybel - so is $(\frac{αr}{βw}K)^αK^β=(\frac{αr}{βw})^αK^{β+α}$ because $α,β>0$ or is this equivalency always true?
DoubleRainbowZ
CC BY-SA 4.0
2021-05-04T23:42:23.523Z
$(ab)^c=a^cb^c$ always true, you can distributed the exponent across the factors.  $a^ba^c = a^{b+c}$ always true, you can add exponents when factors are the same. Hence $(ab)^c b^d = a^cb^cb^d = a^cb^{c+d}$ .... so it is always true.
Jesper Hybel
<microeconomics><production-function><cost-functions><long-run>
Finding long run total cost function
DoubleRainbowZ
# Finding long run total cost function

(Asked by: username_0 on 2021-05-04)

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

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

**Tags:** microeconomics, production-function, cost-functions, long-run
> **Comments:**
> **username_1**: 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.  
> **username_2**: Actually, this seems to be a duplicate of this older question: https://economics.stackexchange.com/questions/18736/deriving-long-run-cost-function does answer there help you to find solution?  
> **username_3**: 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 $C(w,r,y) = wL^\star(w,r,y) + r K^\star(w,r,y)$.  
> **username_0**: @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  
> **username_0**: @1muflon1 I did take a look at the thread you posted, but I couldn't get the answer from that either (maybe I was making a mistake), but I did notice it was trying to find max $LaKb s.t. C=wL+rK$ which was not what my tutorial examples were doing, so I was confused. (They were doing min $ wL+rKs.t.y=LαKβ$ - not sure if they are equivalent)  
> **username_0**: @JesperHybel I can follow your process, however, as seen from my equation above, I can't seem to isolate $K$ to get $K⋆(w,r,y) $. I am not sure if I made an error earlier (I have triple checked and can't find anything wrong) or if I am missing something  
> **username_3**: @DoubleRainbowZ I did it for you as an edit of your question ... now try to do it for labor on your own.  
> **username_0**: Thanks @JesperHybel - so is $(\frac{αr}{βw}K)^αK^β=(\frac{αr}{βw})^αK^{β+α}$ because $α,β>0$ or is this equivalency always true?  
> **username_3**: $(ab)^c=a^cb^c$ always true, you can distributed the exponent across the factors.  $a^ba^c = a^{b+c}$ always true, you can add exponents when factors are the same. Hence $(ab)^c b^d = a^cb^cb^d = a^cb^{c+d}$ .... so it is always true.  

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## 0 answers

