<p>To find the production function, you can solve for <span class="math-container">$\frac{v}{w}$</span> in <span class="math-container">$ {l}_{c}$</span> and <span class="math-container">${k}_{c}$</span> and set <span class="math-container">$\frac{v}{w}$</span> = <span class="math-container">$\frac{v}{w}$</span> then solve for <span class="math-container">$q$</span>.</p>
<p>This will yield</p>
<p><span class="math-container">$$ \frac{v}{w} = (\frac{4{l}_{c}}{3q})^{4} $$</span>
<span class="math-container">$$ \frac{v}{w} = (\frac{4{k}_{c}}{q})^\frac{-4}{3} $$</span></p>
<p>Set both equations equal</p>
<p><span class="math-container">$$ (\frac{4{l}_{c}}{3q})^{4} = (\frac{4{k}_{c}}{q})^\frac{-4}{3} $$</span>
Take both sides to the power of <span class="math-container">$1/4$</span> this eliminates the exponent in the left expression
<span class="math-container">$$(\frac{4{l}_{c}}{3q}) = (\frac{4{k}_{c}}{q})^{-1/3} 
$$</span></p>
<p>Then solve for <span class="math-container">$q$</span></p>
<p><span class="math-container">$$ (q^{-4/3}) \frac{4l_c}{3}= 4^{-1/3} k_c^{-1/3}$$</span>
take both sides to the power of <span class="math-container">$-3$</span>
<span class="math-container">$$ 
(q^4) \frac{3^3}{4^3 l_c^{3}}= 4 k_c
$$</span>
<span class="math-container">$$
q^4= \frac{256k_c l_c^{3}}{27}
$$</span>
This the production function
<span class="math-container">$$
q(k,l)= \frac{4k^{1/4} l^{3/4}}{27^{1/4}}
$$</span></p>

CC BY-SA 4.0
2021-11-13T21:52:21.263Z
2021-11-13T21:52:21.263Z
Macosso
<p>Given a total cost function, for example,
<span class="math-container">$$ C = q {w}^{3/4}{v}^{1/4}
$$</span>
and Shephard's Lemma, how do you find the underlying production function <span class="math-container">$q(k,l)$</span>?</p>
<p>For this example, Shephard's Lemma provides the constant output demand functions:
<span class="math-container">$$
{l}_{c} = \frac{3}{4}q({\frac{v}{w}})^{1/4}
$$</span>
<span class="math-container">$$
{k}_{c} = \frac{1}{4}q({\frac{w}{v}})^{3/4}
$$</span></p>
<p>How do we use this information to find <span class="math-container">$q(k,l)$</span>?</p>

<production-function><cost-functions>
Finding production given total cost (shephard's lemma)
kleinerde
# Finding production given total cost (shephard's lemma)

(Asked by: username_0 on 2021-11-13)

Given a total cost function, for example,
$$ C = q {w}^{3/4}{v}^{1/4}
$$
and Shephard's Lemma, how do you find the underlying production function $q(k,l)$?

For this example, Shephard's Lemma provides the constant output demand functions:
$$
{l}\_{c} = \frac{3}{4}q({\frac{v}{w}})^{1/4}
$$
$$
{k}\_{c} = \frac{1}{4}q({\frac{w}{v}})^{3/4}
$$

How do we use this information to find $q(k,l)$?

**Tags:** production-function, cost-functions

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## 1 answers

### Answer by username_1 on 2021-11-13. Score: 3 | Accepted answer

To find the production function, you can solve for $\frac{v}{w}$ in $ {l}\_{c}$ and ${k}\_{c}$ and set $\frac{v}{w}$ = $\frac{v}{w}$ then solve for $q$.

This will yield

$$ \frac{v}{w} = (\frac{4{l}\_{c}}{3q})^{4} $$
$$ \frac{v}{w} = (\frac{4{k}\_{c}}{q})^\frac{-4}{3} $$

Set both equations equal

$$ (\frac{4{l}\_{c}}{3q})^{4} = (\frac{4{k}\_{c}}{q})^\frac{-4}{3} $$
Take both sides to the power of $1/4$ this eliminates the exponent in the left expression
$$(\frac{4{l}\_{c}}{3q}) = (\frac{4{k}\_{c}}{q})^{-1/3}
$$

Then solve for $q$

$$ (q^{-4/3}) \frac{4l\_c}{3}= 4^{-1/3} k\_c^{-1/3}$$
take both sides to the power of $-3$
$$
(q^4) \frac{3^3}{4^3 l\_c^{3}}= 4 k\_c
$$
$$
q^4= \frac{256k\_c l\_c^{3}}{27}
$$
This the production function
$$
q(k,l)= \frac{4k^{1/4} l^{3/4}}{27^{1/4}}
$$


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